Perspectives in Theoretical Physics [1 ed.] 9781624176111, 9781611229608

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

PHYSICS RESEARCH AND TECHNOLOGY

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PERSPECTIVES IN THEORETICAL PHYSICS

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PHYSICS RESEARCH AND TECHNOLOGY

PERSPECTIVES IN THEORETICAL PHYSICS

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

THOMAS F. GEORGE RENAT R. LETFULLIN AND

GUOPING ZHANG EDITORS

Nova Science Publishers, Inc. New York

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Perspectives in theoretical physics / editors, Thomas F. George, Renat R. Letfullin and Guoping Zhang. p. cm. Includes index. ISBN:  (eBook) 1. Mathematical physics. 2. Thermodynamics. I. George, Thomas F., 1947II. Letfullin, Renat R. III. Zhang, Guoping, 1970QC20.5.P46 2010 530.1--dc22 2011003461

Published by Nova Science Publishers, Inc. + New York Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

CONTENTS vii 

Preface Chapter 1

Chapter 2

Chapter 3

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Chapter 4

Chapter 5

Chapter 6

Chapter 7

Matter-Density Distribution in Deuteron and Diffraction Deuteron-Nucleus Interaction Yu. A. Berezhnoy, V. Yu. Korda and A. G. Gakh  Study of Amplification Without Inversion in H2 Molecule: Effect of Homogeneous and Inhomogeneous Broadening in Three Level Lambda System Considering Bidirectional Pumping Sulagna Dutta and Krishna Rai Dastidar  Amplification Without Inversion and Absorption With Inversion in H2 Molecule: A Dressed-State Picture of Coherently Coupled Three Level Lambda System Sulagna Dutta and Krishna Rai Dastidar  Spectra and Bound States of the Energy Operator of a Two-Magnon System in a Non-Heisenberg Ferromagnet with Spin S=3/2 and Nearest-Neighbor Interactions S. M. Tashpulatov  Model-Based Optimization of the Growth Process Parameters for a Silicon Sheet Grown in a Vacuum by the Edge-Defined Film-Fed Growth Method L. Braescu, A. M. Balint, L. Nánai and St. Balint  Fibre Bundle Formulation of Relativistic Quantum Mechanics II: Covariant Approach Bozhidar Z. Iliev  On the Induction Period of Laser-Driven Thermochemical Processes László Nánai, Sándor Szatmári, Gregory J. Taft and Thomas F. George  

1 

15 

37 

59 

73 

85 

103 

Chapter 8

Dispersion Properties of Nano-Scale Systems V. Lozovski and A. Tsykhonya 

109 

Chapter 9

An Algorithm of Local Hidden Variables J. F. Geurdes  

131 

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Chapter 10

Contents Superconducting State Parameters of 3d-Transition Metals Binary Alloys Aditya M. Vora 

Chapter 11

Superconducting State Parameters of Be-Zr Glassy Alloys Aditya M. Vora 

Chapter 12

Engineering Solutions Using Acoustic Spectral Finite Element Methods Andrew T. Peplow 

Chapter 13

Fractional Oscillator: Phenomenological and Theoretical Issues A. Tofighi 

Chapter 14

Charge-Based Electronics is Under Control – Technology Welcomes Orbitronics for this Millenium Though Spintronics is Not Yet Established M. Idrish Miah 

137  149 

161  179 

191 

Chapter 15

Thermodynamics of Liquid Metals: A Variational Approach Aditya M. Vora 

195 

Chapter 16

Dominant Features of Interfaces in Nanometric Dielectrics O. P. Thakur and Anjani Kumar Singh

213

Chapter 17

On Matrix Representations of Deformed Lie Algebras L. A-M. Hanna and S. S. Hassan 

227 

Chapter 18

An Oscillating, Homogeneous and Isotropic Universe from Scalar-Tensor Gravity Christian Corda  

235 

Path Integral Approach to the Interaction of One Active Electron Atoms with Ultrashort Squeezed Pulses E. G. Thrapsaniotis 

249 

A Scheme for Remote Preparation of Multiparticle Entangled States Yu-Wu Wang and You-Bang Zhan 

273 

Chapter 19

Chapter 20

Index

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PREFACE Theoretical physics employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. This book presents and discusses current research in the study of theoretical physics, including interfaces in nanometric dielectrics; thermodynamics of liquid metals; fractional oscillators; superconducting state parameters of Be-Zr glassy alloys and 3d-transition metal binary alloys. Chapter 1 – A nonrelativistic deuteron wave function with account of the D-state and a correct asymptotic has been obtained on the basis of the experimentally measured charge and quadrupole deuteron form factors. The differential cross section of the elastic deuteronnucleus scattering has been calculated by using the wave function obtained. These predictions agree with the experimental data at the energy of 110 MeV. The integrated cross sections of various processes of the deuteron-nucleus interactions are also calculated. Chapter 2 – The authors have studied (ab-initio) the feasibility of amplification without inversion (AWI) with resonant and off-resonant driving and probe fields by using density matrix equations both numerically and analytically in H2 molecule. The authors have derived (ab-initio) the analytical expressions for coherences and populations in steady state limit in threelevel Λ scheme without any approximation i.e. keeping all orders of probe field Rabi frequency (G) and coherent field Rabi frequency (Ω). Previously approximate expressions for coherences and populations have been derived keeping only the first order terms in probe field Rabi frequency (G) and all orders of coherent field Rabi frequency (Ω). Hence AWI was studied under the condition that coherent field Rabi frequency (Ω) will be at least two orders ofmagnitude greater than the probe field Rabi frequency. Here the authors have explored the feasibility of AWI when coherent field Rabi frequency (Ω) is of the same order of probe field Rabi frequency and the authors have shown that AWI is more efficient than that in the previous case (when Ω >>G). From the time evolution of the coherences and populations, the authors discuss the conditions of transient light amplification mechanism with and without replenishment (i.e.bidirectional pumping) of the ground state. The authors found that when the replenishment of the ground state is considered AWI can be obtained at resonance of both the fields only when spontaneous decay rate on the coherent transition is greater than that on the probe transition. But when the replenishment of the ground state can be neglected, this condition between spontaneous decay widths need not be satisfied to get AWI at resonance of both fields. However under offresonant condition AWI is realized in both the cases (with and without replenishment) and there is no such restriction on the spontaneous decay widths. Dependence of AWI on the choice of vibrational levels as the upper lasing level has been explained. The authors have explored the effect of both the homogeneous and

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viii

Thomas F. George, Renat R. Letfullin and Guoping Zhang

inhomogeneous broadening of levels under the condition of bidirectional pumping in H2 molecule. It has been shown that in molecules AWI can be obtained on probe field of smaller wavelength than that of the coherent field which has not been observed in atoms so far. Chapter 3 – The physical basis for AWI in the bare state configuration for three level Λ system has been explored by analyzing the coherences and populations in the dressed state framework, obtained by solving density matrix equations derived from the master equation. The exact analytical expressions for populations and coherences in the bare state basis at the steady-state limit have been derived and the coherences and populations in the dressed state basis (at resonance) have been expressed in terms of these exact coherences and populations in the bare state basis which has not been done before. By the term ’exact’ the authors mean that to derive coherences and populations in the bare state basis (in the steady state) the authors included all the orders of both the probe field and coherent field Rabi frequencies, incoherent pumping rate, spontaneous emission rate and detunings. By solving the density matrix equations numerically the dynamics of coherences and populations in the dressed state basis have been studied. The numerical values of coherences and populations in large time limit (steady state) are in good agreement with the steady-state values obtained from the analytical expressions for coherences and populations in the dressed-state basis. From these analytical expressions the authors have explained the behavior of coherences and populations in the dressed-state basis leading to gain in the bare system. The authors have also explored the effect of both the homogeneous and inhomogeneous broadening of levels in the dressedstate basis. Amplification without inversion (AWI) and absorption with inversion (ADI) are feasible in the dressed-state basis. These complementary effects are the manifestation of quantum interference that occurs in these three level Λ systems. Chapter 4 – The authors consider the energy operator of two-magnon systems in a nonHeisenberg ferromagnet with a coupling between nearest neighbors and spin S = 3/2. The spectrum and the bound states of the systems in a ν− dimensional lattice are investigated. The change of the energy spectrum of the systems is described. Two-magnon systems have attracted the attention of many researchers. Chapter 5 – It is shown that the choice of the half-thickness w of the die determines a lower limit α c* of the contact angle α c which decreases when w increases. It is also shown that for a given w there exists a set of couples (v,T0), where v is the pulling rate and T0 is the melt temperature at the meniscus basis, for which the system of differential equations which governs the evolution of the sheet half-thickness and of the meniscus height has an asymptotically stable solution. This set of (v,T0) couples, representing the stable growth region, including the subset S(w,xf) for which the half-thickness of the sheet is equal to a prescribed value xf ,are found. Finally, those values of w,v,T0 are found for which small uncontrollable variations of v and T0 cause minimal variations of xf. Numerical results are given for a silicon sheet of half-thickness xf = 0.5 × 10-2 cm grown in a furnace in which the vertical temperature gradient is k = 3.59 × 102 K/cm. Chapter 6 – A fibre bundle formulation of the mathematical base of relativistic quantum mechanics is proposed. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of their investigation, the authors consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard)

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Preface

ix

fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered. Chapter 7 – Laser-induced chemical reactions on solid surfaces are investigated. The close relationship between the reaction rate and induction period is shown. Using a simple mathematical technique based on no thermal exchange during the reaction and applying a point-system description, it is demonstrated that the inverse dependence of the induction period on rate constants can be shortened by the use of external laser radiation. Chapter 8 – An approach allowing the determination of the dispersion properties for systems of nano-particles is proposed. The approach is based on a study of the absorption spectrum of the external field energy by the system. The Lippmann-Schwinger equation is used to calculate the local electromagnetic fields in the system and the dissipative function, which describes the absorption of external field energy by the system per unit of time. The main idea of the approach is using the resonant absorption of external radiation under conditions of excitation self-modes in the system. In the frame of the proposed approach the numerical calculation of the dispersion properties of a single (individual) particle having different forms (ellipsoids and parallelepipeds) as well as two spherical particles is provided. It is shown that spatial dispersion of the electromagnetic oscillation, which is localized on the system of nano-particles, appears in systems which consist of two or more particles. Chapter 9 – A simple Local Hidden Variables model is described that can violate Bell’s inequality in the form of CHSH with ‘raw’ correlations. Chapter 10 – The study of the superconducting state parameters (SSPs) viz. electron-

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phonon coupling strength isotope effect exponent

α

λ , Coulomb pseudopotential μ * , transition temperature TC , and effective interaction strength NOV of 3d-transition metals

binary alloys have been made extensively in the present work using a model potential formalism and employing the pseudo-alloy-atom (PAA) model for the first time. It is observed that the electron-phonon coupling strength λ and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential

μ * , isotope effect exponent α and effective interaction strength N OV

show weak dependences on the local field correction functions. The present results of the SSPs are found in qualitative agreement with the available experimental data wherever exist. Chapter 11 – The study of the superconducting state parameters (SSPs) viz. electronphonon coupling strength isotope effect exponent

α

λ , Coulomb pseudopotential μ * , transition temperature TC , and effective interaction strength NOV of 3d-transition metals

binary alloys have been made extensively in the present work using a model potential formalism and employing the pseudo-alloy-atom (PAA) model for the first time. It is observed that the electron-phonon coupling strength λ and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential

μ * , isotope effect exponent α and effective interaction strength N OV

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Thomas F. George, Renat R. Letfullin and Guoping Zhang

show weak dependences on the local field correction functions. The present results of the SSPs are found in qualitative agreement with the available experimental data wherever exist. Chapter 12 – The spectral finite element method is an advanced implementation of the finite element method in which the solution over each element is expressed in terms of a priori unknown values at carefully selected spectral nodes. These methods are naturally chosen to solve problems in regular rectangular, cylindrical or spherical regions. However, in a general irregular region it would be unwise to turn away from the finite element method since models defined in such regions are extremely difficult to implement and solve with a spectral method. Hence, for a complex waveguide, the method uses the efficiency and accuracy of the spectral method and is combined with the flexibility of finite elements to produce a high–performance engineering tool. Some contemporary examples from engineering including fluid-filled pipes, tyre acoustics, silencers and waveguides will be reviewed, presented and analysed. From simple examples to complex mixed materials configurations the study will highlight the strengths of the method with respect to standard methods. Chapter 13 – Fractional derivative is a convenient tool to model physical processes in the complex media. A general trend is to consider fractional generalization of an equation of physics and study its consequences. For the case of a fractional oscillator the authors provide justifications for this procedure. In a media with low-level of fractionality the order of the fractional derivative is close to a positive integer. And one can use perturbation theory to treat the deviation from the integer case. Using this method the authors study fractional oscillator in a media with low-level fractionality. The authors find expressions for the position, momenta, energy and the intrinsic damping force for this system. The authors show that the energy is a monotonously decreasing function of time. The authors propose a new model Hamiltonian for this system. And the authors make some remarks on non-casual fractional oscillator as well. Chapter 14 – Spintronics is a rapidly growing research field aimed at realizing practical devices that takes advantage of the electron spin as well as of its charge in conventional electronic devices. Much effort on the requirements for semiconductor spintronic devices reveals that the better understanding of spin currents in semiconductors is still a sufficiently challenging problem, though the first step in making the metal-based spin-technology amenable to industrial application came in 1988. However, in reality, an electron in strongly correlated-electron systems has three attributes: charge, spin and orbital. The orbital states of the electronic and magnetic phases could be controlled by external fields, such as electric, magnetic, stress, light or x-rays. Such control strategies for exploring the possibility of utilizing orbital degree of freedom may open the way to introduce the correlated-electron systems for a new type of electronics – orbitronics. However, future electronic devices with increased performance, capability and functionality would rely on a complete understanding of the three control parameters. Chapter 15 – On the bases of the Gibbs–Bogoliubov (GB) variational method and the Percus-Yevick (PY) hard sphere model as a reference system, a thermodynamic perturbation method has been applied with use of well known model potential. The thermodynamic properties of some liquid metals of the different groups of the periodic table are reported. The influence of local field correction function proposed by Hartree (HR), Taylor (TY), IchimaruUtsumi (IU), Farid et al. (FR) and Sarkar et al. (SR) is also investigated. The comparison with

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Preface

xi

available experimental or theoretical findings is found in qualitative agreement and establishes the use of the local field correction functions in such study. Chapter 16 – As the size of particle decreases to the nanometric scale, the number of molecules in contact with its surface increases sufficiently and as a result of this, the various short-range, known and unknown, interactions affect the electromechanical and the electrochemical properties of nanocomposite at the interface. The interfacial regions with non-Centro symmetry and molecular organization, between two phases in contact are a nanometric system with properties in variably different from the bulk phases on either side. The presence of a large number of point defects even in equilibrium and the influence of the electrode’s wall on the orientation distributions of interfacial molecules play a vital role in describing the electrochemical and electromechanical properties of the structurally modified interface as compared to the bulk in continuum. In the space charge regions, the DebyeHuckel length, which is based on the bulk concentration, has been severely modified due to differential ionic concentration and segregation of charged clusters at the interface. Hence the simulation of diffuse double layer with the nanometric interface is not valid up to a great extent. It has also been observed that the interface cannot be realized as electromechanical transducer up to a large extent due to its anisotropic and fluctuating dielectric behavior. Chapter 17 – Faithful matrix representations of least degree are discussed for the Lie algebra L: [X0, X±] = ± r X±, [X+, X−] = P(X0) , where, r 2 R, and P is a polynomial function. The operators X0,X+ and X− satisfy the physical properties: X0 is a real diagonal operator and X− = X†+ († is for hermitian conjugation). The physical condition X+ +X− must be real, is separately considered. Conditions are found for the polynomial function P to guarantee matrix representations of L. Chapter 18 – An oscillating, homogeneous and isotropic Universe which arises from Scalar- Tensor gravity is discussed in the linearized approach, showing that some observable evidences like the Hubble Law and the Cosmological Redshift are in agreement with the model. In this context Dark Energy appears like a pure curvature effect arising by the scalar field. Chapter 19 – The authors model in a fully quantum mechanical way the dynamics of an atom of one optically active electron interacting with a squeezed linearly polarized ultrashort photonic pulse. The authors use path integral methods. After integrating over the photonic field the authors consider the extracted propagator in its discrete form and derive the sign solved propagator to study the whole dynamics. Subsequently the authors develop a scattering theory. To avoid any additional complication the authors apply their methods to the ionization of atomic hydrogen from its ground state supposing that the photon energy is greater that the ionization threshold and give the ionization probability. Chapter 20 – In this paper a scheme for remote preparation of multiparticle entangled states is proposed. In this scheme, only two classical bits and one two-particle projective measurements are sufficient. Versions of these chapters were also published in International Journal of Theoretical Physics, Group Theory, and Nonlinear Optics, Volume 12 and Volume 13, Numbers 1-4, published by Nova Science Publishers, Inc. They were submitted for appropriate modifications in an effort to encourage wider dissemination of research.

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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In: Perspectives in Theoretical Physics ISBN: 978-1-61122-960-8 c 2011 Nova Science Publishers, Inc. Editors: T. F. George, R. R. Letfullin and G. Zhang

Chapter 1

M ATTER -D ENSITY D ISTRIBUTION IN D EUTERON AND D IFFRACTION D EUTERON -N UCLEUS I NTERACTION Yu.A. Berezhnoy1∗, V.Yu. Korda2 and A.G. Gakh1 1 Kharkov National University, Kharkov, Ukraine 2 Institute of Electrophysics and Radiational Technology, National Academy of Sciences of Ukraine, Kharkov, Ukraine

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Abstract A nonrelativistic deuteron wave function with account of the D-state and a correct asymptotic has been obtained on the basis of the experimentally measured charge and quadrupole deuteron form factors. The differential cross section of the elastic deuteron-nucleus scattering has been calculated by using the wave function obtained. These predictions agree with the experimental data at the energy of 110 MeV. The integrated cross sections of various processes of the deuteron-nucleus interactions are also calculated.

1.

Introduction

The two-body bound-state problem in quantum mechanics has been exactly solved for the Coulomb interaction (the hydrogen atom) which is well known. However, a similar problem in nuclear physics (the deuteron) is more complicated since the nuclear forces acting between two nucleons are not exactly known yet. Moreover, the nuclear forces depend on the particle spins, and the nucleon is a composite particle which consists of three quarks. It means that the nucleon-nucleon interaction is not elementary. In some sense it is similar to the van der Waals forces acting between atoms. Therefore, the various phenomenological potentials of the nucleon-nucleon interaction, reproducing its basic features [1]-[5], are usually used to solve the deuteron problem. ∗ E-mail address:

[email protected]

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2

Yu.A. Berezhnoy, V.Yu. Korda and A.G. Gakh

On the other hand, modern nuclear physics has at its disposal a wealth of knowledge about the nuclear structure obtained from the analysis of the experimental data on the scattering of various particles by nuclei. In particular, investigation of the scattering of electrons and protons by nuclei is an important source of information about the nuclear form factors, differential cross sections and distributions of charge and matter in the atomic nuclei. Therefore, such information is a reliable basis for constructing various models for the nuclei and phenomenological nuclear wave functions. The deuteron electromagnetic form factors measured experimentally with a sufficiently high accuracy [7]-[14] can be used for the determination of the deuteron wave functions. As it is known, the deuteron ground state is a superposition of the 3 S− and 3 D− states, and the corresponding wave function is a sum of the S- and D-waves. In order to determine these components of the deuteron wave function, one can use charge and quadrupole deuteron form factors. The asymptotic behavior of the S-state wave function must be determined by the value of the deuteron binding energy. In determining the deuteron wave functions, the internal nucleon structure is not usually included, i.e., the nucleons are considered as point particles. In reality, the nucleon structure essentially affects the values of the deuteron electromagnetic form factors, and it should be taken into account in corresponding calculations. The deuteron wave functions, obtained on the basis of such a procedure, allow us to construct the charge-density distribution in the deuteron. Assuming that the matter-density distribution is similar to the charge density, one can calculate the differential cross section of the elastic deuteron-nucleus scattering as well as determine the integrated cross sections of various processes of the deuteron-nucleus interaction. Thus, the problem of the determination of the deuteron wave functions and cross sections of the deuteron-nucleus interaction can be solved starting from the experimental data on the scattering of the charged particles by the deuterons. In this approach there is no necessity to use a model nucleon-nucleon potential since nuclear forces between two nucleons has not been sufficiently studied yet.

2.

Deuteron Wave Functions and Form Factors

The wave function of the deuteron ground state is u(r) w(r) YS , ΨD = YD , (1) r r where YS , YD are the spin-angular parts of the S- and D-states of the wave function. The radial functions u(r) and w(r) are given as Ψd (~r) = ΨS + ΨD , ΨS =

N nu u(r) = √ ∑ Ck exp(−αk r), 4π k=1   nw 3 3 N + w(r) = √ ρ ∑ Dk exp(−βk r) 1 + , βk r (βk r)2 4π k=1 !−1/2 nw nu 1 1 2 + ρ ∑ Dk D j , N = ∑ CkC j αk + α j βk + β j k, j=1 k, j=1

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(2) (3)

(4)

Matter-Density Distribution in Deuteron and Diffraction Deuteron-Nucleus...

3

where αi , βi , Ci , Di , ρ are the real adjustable model parameters: αi , βi , ρ are positive, and Ci and Di are arbitrary. Let us suppose that in the limit r → 0 the behavior of the wave functions is u(r) ∝ r2 and w(r) ∝ r3 . Then 1

∑ Ck = 0, ∑ Ck αk = 0, ∑ Dk = 0, ∑ Dk β2 = 0. k

k

k

k

(5)

k

In the limit r → ∞ the true deuteron wave functions must have the following asymptotic form [15]:   3 3 −αr −αr 1+ + , (6) u(r) ∝ e , w(r) ∝ e αr (αr)2 √ h = 0.2316 fm−1 , M is the deuteron reduced mass, and ε = 2.2245 MeV is where α = Mε/¯ the deuteron binding energy. Then the application of the condition (6) to the wave functions (2) and (3) leads to the following relation: α1 = β1 = α and αi > α1 , βi > β1 , i ≥ 2. The charge GC (q) and quadrupole GQ (q) deuteron form factors, as well as the structure function A(q), in neglecting of the contribution proportional to the magnetic form factor, are defined by the formulas (see, for example, [16]) Z ∞    qr  , (7) GC (q) = GSE (q) dr u2 (r) + w2 (q) j0 2 0    Z ∞  3 w2 (q) qr S GQ (q) = GE (q) √ dr 2u(r)w(r) − √ j2 , (8) 2 2 2η 0 2

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8 A(q) = GC2 (q) + η2 G2Q (q), η = q2 /4m2d , 9

(9)

where md is the deuteron mass, and GSE (q) is the electric isoscalar nucleon form factor  −2 which is conventionally chosen as [15]: GSE (q) = 1 + (0.278q)2) . Substituting the radial functions u(r) and w(r) in the form of (2) and (3) in the formulas (7) and (8) we obtain h i (S) (D) (10) GC (q) = GSE (q) FC (q) + FC (q) , N 2 nu q CkC j arctan , ∑ q k, j=1 αk + α j   N 2 ρ2 n w q 3  (D) Dk D j arctan FC (2q) = 1 + 2 q2 + (βk + β j )2 + ∑ q k, j=1 βk + β j 2βk )  3  + 2 2 q4 − (βk + β j )4 . 8βk β j ! nu nw 2 2 2 nw 3 N ρ N ρ (1) (2) GQ (2q) = GSE √ ∑ ∑ B − √2 ∑ Bk j , 2 2η 4q3 k=1 j=1 k j k, j=1    4  Ck D j q (1) 2 2 2 2 2 2 2 3qαk β j + 3q + q (6αk − 2β j ) + 3(αk − β j ) arctan , Bk j = 2 αk + β j βj (S)

FC (2q) =

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(11)

(12)

(13) (14)

4

Yu.A. Berezhnoy, V.Yu. Korda and A.G. Gakh ( 3 3  3 3β j − βk 1 (2) Bk j = Dk D j + −3q6 + q4 (−9β2k + 3β2j )+ 2 2 16 q2 βk β j 16q3 βk β j   q 2 4 2 2 4 2 2 2 2 2 . +q (−9βk + 2βk β j + 15β j ) − 3(βk − 3β j )(βk − β j ) arctan βk + β j

(15)

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Setting nu = 1 and ρ = 0 in formulae (2) and (3), we obtain the wave function of the Yukawa function type. For nu = 2 and ρ = 0 we have an expression for the Hulth´en wave function [17].

Figure 1. Dependence of the deuteron charge form factor GC (q) and structure function A (q) on momentum transfer. The solid lines are calculated by our model, while the dashed lines are calculated by the McGee model. The data sets are described in Table 1. McGee [18] suggest the wave function in the form (2) and (3) with nu = 5, nw = 6. This is an analytic approximation of the numerical wave function obtained as a result of the integration of the Schr¨odinger equation with the realistic model potential. The form factor GC (q) and the structure function A(q), calculated with the McGee wave function, are given in Fig. 1 by dashed lines. From this figure we can see that the charge form factor differs by an order of magnitude from the experimentally measured one in the region q > 5 fm−1 . Its minimum is shifted to the region of large values of q. The structure function A(q) is not in a good agreement with the experiment in the region q > 3.5 fm−1 The aim of the present investigation is to choose the radial wave functions (2) and (3) which will allow a better description of the experimental data on the electromagnetic form factors. In contrast to conventional potential approaches [1]-[5], we extract the wave function parameters (αi, βi ,Ci , Di , ρ) from the analysis of the experimentally measured form factors [7]-[14], i.e., by comparing them with the form factors calculated according to the formulas (7)-(13). We want to take into account that in reality nucleons are not point particles but have a complex internal structure. To do this, we assume that the functions u(r) and w(r) are related to deuteron consisting of nucleons having internal structure. This means that the multiplier GSE (q) in formulas (10) and (13) may be taken as one. Hence, the

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Table 1. Experimental data for A and Gc . The symbols are given for data shown in Figure 1. Symbol 4  5

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? • ⊕

/ .

Number of points Year and reference Measurements of Gc 3 1966 [6] 9 1994 [7] 4 2000 [8] Measurements of A 6 1962 [9] 5 1965 [10] 3 1966 [11] 10 1971 [12] 16 1981 [13] 11 1999 [14]

wave function parameters defined by this method contain the information about the nucleon structure as well. The search for the optimal values of the model parameters was made by using the genetic algorithm. The minimizing function includes the standard quantity χ2 for the experimental points GC (q) and A(q), as well as the following information about the experimentally measured deuteron static characteristics: the deuteron mean square radius with account for the proton dimension < r2 >1/2 = 2.094 fm and quadrupole moment Q = 0.2859 fm2 . The additional condition consists in the choice of the minimal possible number of terms in sums (2) and (3) for which we have agreement with the experiment in the region q < 7 fm−1 . This is explained by the fact that at present there exists complete information about the form factors just for this region. The results of the calculations are given in Figs. 1 and 2 as well as in Table 2. The wave function parameters with nu = 3 and nw = 3 as well as the obtained values of the static characteristics are given in Table 2. Our model contains 5 adjustable parameters. The form factor and the structure functions (7) and (9), calculated for the wave function (2), (3) by using the parameters from Table 2 are shown in Fig. 1 by a solid line. As seen from the figure, the calculated charge form factor provides a good description of the experimental data for all values of q for which such data exist. Its minimum lies in the region 4 .5 fm−1 that agrees with the model independent analysis (see, for example, [8]). The results of the calculation of the structure function A(q) by formula (9) are in good agreement with the experiment in the region q < 5 fm−1 . It should be noted that the calculated values of the structure function A(q) are not in agreement with the experiment (Fig. 1) in the region of large transfer momenta. It is a characteristic feature of all modern realistic nonrelativistic potentials. The difference between the experimental data and nonrelativistic theories may be only explained at additional account for the meson exchange currents, the relativistic effects and quark degrees of freedom [16]. The radial wave functions (2), (3) with the parameters from Table 2 are shown in Fig. 2 by a solid line.

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Yu.A. Berezhnoy, V.Yu. Korda and A.G. Gakh

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Figure 2. Dependences of the deuteron radial wave functions u (r) and w (r) on the relative distance between the nucleons. The solid lines are calculated by our model with finitesize nucleons, while the dashed ones are calculated by the McGee model with structureless nucleons. Table 2. Calculated values of parameters of the proposed deuteron wave function and values of the deuteron static characteristics calculated.

k αk fm−1 Ck βk fm−1 Dk 1 0.2316 1 0.2316 1 2 1.82007 -136.26749 1.07145 287.15485 3 1.83181 135.26749 1.03542 -288.15485 ρ = 0.033
1/2 r2 = 2.108 fm Q = 0.2859 fm2

3.

Differential Cross Section of Elastic Scattering

The general formula for the amplitude of the elastic diffraction deuteron-nucleus scattering in the model of strong absorption [19, 20] was obtained in Refs. [21, 22, 23]. It takes into account the Coulomb interaction, small value of the deuteron dimension in comparison with the target-nucleus radius R, finite values of the nuclear surface diffuseness d and surface refraction γ, but the nucleon spins and isospins are neglected. In the approximation Rd  R, d  R, γd  R, this amplitude is given by [22]       ∂ q f (q) FC + F(q) = 2 f c(q) + 1 + d ∂R 2   d Rd 2 exp [2iσc (KR)], (16) +iKR Φ(q) J0 (qR) + iγ [qRJ1 (qR) − J0 (qR)] R 4

nΓ(1 + in) exp [−2i ln (q/K)] , Γ(1 − in) (qR)2 Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, f c(q) = −KR2

(17)

Matter-Density Distribution in Deuteron and Diffraction Deuteron-Nucleus... f (q) = iK

Z ∞ 0

dbbJ0 (qb)ω(b) exp[2iσc (b)],

7 (18)

Γ(1 + Kb/2 + in) 2mZe2 , n= 2 , (19) Γ (1 + Kb/2 − in) h¯ K where m is the nucleon mass, Ze is the target-nucleus charge, q = 2K sin(θ/2) is the transfer momentum, θ is the scattering angle of the deuteron center of mass, K is the wave vector of the interacting deuteron, and σc is the Coulomb phase for the scattering of the proton, belonging to the deuteron, by the target nucleus. In the strong absorption model, the profile function is chosen in the form [24] exp[2iσc(b)] =

−1   b−R + iγ ω(b) = 1 + exp . d

(20)

The attenuation factor is determined by the expression [20] πqd . sh(πqd)

Φ(q) =

(S)

(21) (D)

We introduce the designation FC (q) = FC (q) + FC (q) in formula (16). The differential cross section of the elastic deuteron-nucleus scattering is dσ = |F(q)|2. dΩ

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4.

(22)

Integrated Cross Sections of Various Processes of Diffraction Deuteron-Nucleus Interaction

In the approximation Rd  R, d  R, γd  R, the general formulas for the integrated cross sections of various processes of the deuteron-nucleus interaction, in the strong absorption model [19, 20] with parameterization of the scattering matrix in the Ericson form [24], were found in Refs. [21, 22, 23]. The total cross section of all processes for the deuteron-nucleus, in the mentioned above approximation, is written in the form   1 (23) σt = 2πR2 + πRRd + 4πRd 1 + γ2 , 6 where the quantity Rd is Rd =

Z

d 3 r|Ψd (~r)|2 r.

(24)

Substituting the expression for the deuteron wave function in the form (1)-(3) into formula (24), we find (S)

(D)

Rd = Rd + Rd , (S) Rd

=N

2

nu

∑

k, j=1

CkC j

1 , (αk + α j )2

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(25) (26)

8

Yu.A. Berezhnoy, V.Yu. Korda and A.G. Gakh   nw 1 1 6 (D) 2 2 Rd = N ρ ∑ Dk D j + . βk + β j βk + β j β j k, j=1 The integrated cross section of the elastic scattering is defined by the expression   2 1 13 2 2 σe = πR + (1 − ln 2)πRR d + πRd 1 + γ . 3 3 5 The integrated cross section of all inelastic processes is given by   1 11 3 2 2 σin = πR + (1 + 2 ln 2)πRRd + πRd 1 − γ . 3 3 55 The integrated cross section of deuteron-nucleus scattering can be written as   1 1 13 σsc = πR2 + πRRd + πRd 1 + γ2 . 2 3 5

(27)

(28)

(29)

(30)

The integrated cross section of the deuteron diffraction disintegration is determined by the formula   2 1 ln 2 − πRRd . (31) σd = 3 4

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The integrated cross section of the reaction is given by the expression   3 2 1 11 2 σr = πR + πRRd + πRd 1 + γ . 2 3 55

(32)

The integrated cross section of the inclusive deuteron stripping reaction, is determined by the formula   1 5 2 2 (33) σsn = σsp = πRRd + πRd 1 + γ . 2 3 25 The formula for the integrated cross section of the absorption of the deuteron as a whole by the nucleus is   1 1 7 2 2 (34) σa = πR − πRRd + πRd 1 − γ . 2 3 5

5.

Discussion

The comparative analysis of the experimentally measured differential cross section for the elastic diffraction scattering of deuterons on 208 Pb nuclei has been carried out at the energy of 110 MeV [25]. In these calculations we use the Yukawa, Hulth´en and McGee wave functions as well as those proposed by us. The results of the calculations are given in Fig. 3 and Table 3. Figure 3 shows that the use of any of the above mentioned wave functions allows us to describe the differential cross section equally satisfactorily. And in this case, the parameter values for the target nucleus turn out to be close (Table 3). Hence, the level of inclusion of the details of the deuteron internal structure, with the help of

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Matter-Density Distribution in Deuteron and Diffraction Deuteron-Nucleus... Table 3.

208 Pb

9

target-nucleus parameters used in calculations of the elastic scattering differential cross section.

R, fm D, fm γ

Yukawa 7.90 0.55 0.70

Wave function Hulth´en McGee 7.80 7.80 0.55 0.55 0.50 0.50

This paper 7.80 0.55 0.50

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various realistic wave functions, has small impact on fitting the results of the experimentally measured differential cross section for the elastic diffraction scattering of the deuterons on 208 Pb nuclei at the energy of 110 MeV.

Figure 3. Deuteron-208Pb elastic scattering differential cross section at 110 MeV. The solid line is calculated with use of the wave functions (2), (3) (the parameters are from Table 2), the dashed line is for the Yukawa function, the dotted line is for the Hulthen function, and the dashed-dotted line is for the McGee function. The experimental data are taken from [25]. The values of the integrated parts of various processes of the diffraction deuteronnucleus interaction are given in Table 4. They are calculated according to the formulas (23), (28)-(34) using the target-nucleus parameters from Table 3. First of all, let us pay attention to the growth of the Rd quantity characterizing the deuteron dimension. As seen from the table, Rd increases in each subsequent model. This may be connected with more complete accounting of the deuteron internal structure. The difference of the Rd quantity, calculated with the Yukawa wave function and with Hulth´en, McGee and our wave functions, amounts to more than 50%. It is the reason for the significant change of the integrated cross sections related to the nucleon redistribution. The cross section of the deuteron-nucleus diffraction disintegration is most sensitive to the Rd value and increases by 50%. The cross section of

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Yu.A. Berezhnoy, V.Yu. Korda and A.G. Gakh Table 4. Values of the diffraction deuteron-nucleus interaction integrated cross sections (bn). The values in parentheses show the difference between the current cross section and the previous one (in percent).

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Rd , fm σt σe σin σsc σr σd σs,n , σs,p σa

Yukawa 2.161 5.048 2.174 2.874 2.332 2.716 0.1584 0.5046 1.707

Wave function Hulth´en McGee 3.274 (51.5) 3.428 (4.7) 5.187 (2.7) 5.224 (0.7) 2.150 (-1.1) 2.157 (0.3) 3.037 (5.7) 3.067 (1.0) 2.387 (2.4) 2.405 (0.7) 2.800 (3.1) 2.819 (0.7) 0.2370 (49.6) 0.2482 (4.7) 0.6303 (24.9) 0.6491 (3.0) 1.539 (-9.8) 1.521 (-1.2)

This paper 3.638 (6.1) 5.276 (1.0) 2.168 (0.5) 3.108 (1.3) 2.431 (1.1) 2.845 (0.9) 0.2634 (6.1) 0.6749 (4.0) 1.495 (-1.7)

the inclusive reaction of the deuteron stripping increases by 25%. The cross section of the deuteron-nucleus absorption decreases by 10%. The connection between the change of the integrated cross sections and Rd growth can be explained in the following way: the deuteron with the greater relative distance between nucleons has a greater probability to scatter (in the diffraction disintegration) and loses one nucleon (in the stripping) on the target nucleus. It is due to the increase of the range of the deuteron impact parameter where its interaction with the target nucleus takes place, and it leads to the increase of σd , σs,n , σs,p . The total absorption of the deuteron by the nucleus takes place when both nucleons from the deuteron interact with the target nucleus. The greater the distance between the nucleons in the deuteron, the more difficult it is for the target nucleus to simultaneously absorb both nucleons. This results in a decrease of the σa quantity. For the processes of the diffraction deuteron-nucleus interaction, connected with the scattering (for example, the integrated cross section of the elastic scattering, the integrated cross section of the deuteron-nucleus scattering), the corrections turn out to be less than 5.7%. Our calculations show that it is essential to take into account the finite radius of the nuclear force action when we determine the integrated cross section values. The Yukawa wave function has zero radius of the nuclear force action, and the finite value of the radius of the nuclear force action, between the neutron and proton in the deuteron, is taken into account in the rest three wave functions. The difference between the integrated cross sections, calculated by using the McGee, Hulth´en and our wave functions, is not so essential. The correction values for the integrated cross sections, calculated with the McGee wave function in comparison with the Hulth´en one, are: for the cross sections, related to the scattering, less than 1.0%; and for the cross sections, related to the particle redistribution, less than 4.7%. In turn, the use of our wave function changes the values of the integrated cross sections, in comparison with the McGee wave function, by not more than 1.3% and 6.1%, respectively. Thus, a more complete inclusion of the deuteron internal structure with

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the help of the McGee and the Hulth´en realistic wave functions and the function proposed by us does not turn out to be essential in the calculations of the integrated cross sections related to the scattering. However, it is essential to take into account the deuteron internal structure for the integrated cross sections related to the particle redistribution which is seen from the correction values. A detailed level of the deuteron internal structure may be also important in the calculation of the differential cross section for the diffraction disintegration of deuterons on nuclei and inclusive reaction of the deuteron stripping. If we know the analytic dependence of the deuteron radial wave functions u(r) and w(r) on the relative distance between the nucleons, one can obtain the central and tensor parts of the potential in accordance with the formulas √ h¯2 u00 (r) w(r) − ε − 8VT (r) , M u(r) u(r)  00  u00 (r) h¯2 w (r) 6 − − M w(r) u(r) r2  VT (r) = √  , u(r) 8 w(r) − w(r) − 2 u(r)

VC (r) =

(35)

(36)

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where u00 (r), w00 (r) are the second derivatives of the radial wave functions with respect to the relative distance between the nucleons in the deuteron. The result of the calculation of the potentials using our wave function is given in Fig. 4, where one can see that the central part of the potential has a standard form with the repulsion at small distances. It has its minimum near r = 1 fm, and its depth is about 50 MeV. In the limit r → 0, the central and tensor parts of the potential behave as VC (r) ∝ r12 and VT (r) ∝ − r12 , respectively. The behavior of the potentials obtained VC (r) and VT (r) qualitatively agree with the dependence of the realistic phenomenological potentials [1]-[5].

Figure 4. Dependence of the proton-neutron interaction potential on the relative distance between centers of the finite-size nucleons. The solid (dashed) line is the central (tensor) part of the potential. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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The radial wave functions (2), (3) in the McGee model and in our approach are shown in Fig. 2. While analyzing Fig. 2, it is necessary to keep in mind that the McGee wave function describes the deuteron consisting of the point nucleons while our wave function takes into account the nucleon finite size. The matter-density distribution in the deuteron obtained by the inversion of formula (7) is shown in Fig. 5 (dashed-dotted line). Here the solid line corresponds to the calculation in our approach, and the dashed line corresponds to the McGee model. The dashed and dashed-dotted lines both correspond to the structureless nucleons entering the deuteron. The analysis of Fig. 5 forces us to assume that the dependence of matter-density distribution in the deuteron on the distance between the point nucleons may have a form similar to the dashed-dotted line in Fig. 5. This assumption is necessary to correctly describe the experimental data on the electromagnetic form factors in the interval q < 7 fm−1 (Fig. 1) and on the deuteron static characteristics (Table 2), in the framework of the approach where the interaction potential between the nucleons, forming the deuteron, does not depend on their internal structure.

Figure 5. Dependence of the matter-density distribution in the deuteron ρ (r) =  2  2 2 u (r) + w (r) /r on the relative distance between the nucleons. The solid line is calculated in our model with accounting for the nucleon finite size. The dashed line is calculated in the McGee model with point nucleons, while the dashed-dotted one is calculated by the inversion of formula (7). Substituting the obtained expression for w(r) into the standard formula determining the D-wave admixture to the deuteron wave function, we find the value 1 .7%. Usually the magnitude of the D-wave admixture is determined by the deuteron quadrupole moment value. The magnitude of the quadrupole moment calculated with help of our wave function coincides with the experimentally measured value (see Table 2). Since the wave function suggested in this paper describes a deuteron as a bound state of nonpoint nucleons the obtained value cannot be considered as the deuteron D-wave admixture in a common sense.

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Acknowledgment The study partially supported by the State Fund of Fundamental Research of Ukraine (Grant N02.07/372).

References [1] M. Lacombe et al., Phys. Rev. C 21, 861 (1980). [2] R. B. Wiringa et al., Phys. Rev. C 51, 38 (1995). [3] V. G. J. Stoks et al., Phys. Rev. C 49, 2950 (1994). [4] R. Machleidt et al., Phys. Rev. C 53, R1483 (1996). [5] R. Machleidt, Phys. Rev. C 63, 024001 (2001). [6] D. Benaksas et al., Phys. Rev. 148, 1327 (1966). [7] Amroun, Breton et al., Nucl. Phys. A 579, 596 (1994). [8] D. Abbott et al., Eur. Phys. J. A 7, 421 (2000). [9] D. J. Drickey and L. N. Hand, Phys. Rev. Lett. 9, 521 (1962). [10] C. D. Buchanan and R. Yearian, Phys. Rev. Lett. 15, 303 (1965).

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[11] B. Grossette et al., Phys. Rev. 141, 1425 (1966). [12] S. Galster et al., Nucl. Phys. B 32, 221 (1971). [13] G. G. Simon et al., Nucl. Phys. A. 364, 285 (1981). [14] L. C. Alexa et al., Phys. Rev. Lett. 82, 1374 (1999). [15] T. Ericson and W. Weise, Pions and Nuclei (Clarendon Press, Oxford, 1988). [16] R. Gilman and F. Gross, J. Phys. G: Nucl. Part. Phys. 28, R37 (2002). [17] L. Hulth´en and S. Skalvem, Phys. Rev. 87, 297 (1952). [18] J. McGee Ian, Phys. Rev. 151, 772 (1966). [19] W. E. Frahn, R. H. Venter, Ann. Phys. 24, 243 (1963). [20] W. E. Frahn, Diffractive Processes in Nuclear Physics (Clarendon Press, Oxford, 1985). [21] Yu. A. Berezhnoy, V. Yu. Korda, Nucl. Phys. A 556, 453 (1993). [22] Yu. A. Berezhnoy, V. Yu. Korda, Int. J. Mod. Phys. E 3, 149 (1994). [23] Yu. A. Berezhnoy, V. Yu. Korda, Int. J. Mod. Phys. E 4 563 (1995). Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Yu.A. Berezhnoy, V.Yu. Korda and A.G. Gakh

[24] T. E. O. Ericson, in Preludes in Theoretical Physics, edited by A. de-Shalit, H. Feshbach and L. van Hove (North-Holland, Amsterdam, 1965) p. 321 f f .

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[25] A. C. Betker et al., Phys. Rev. C 48, 2058 (1993).

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In: Perspectives in Theoretical Physics ISBN: 978-1-61122-960-8 c 2011 Nova Science Publishers, Inc. Editors: T. F. George, R. R. Letfullin and G. Zhang

Chapter 2

S TUDY OF A MPLIFICATION WITHOUT I NVERSION IN H2 M OLECULE : E FFECT OF H OMOGENEOUS AND I NHOMOGENEOUS B ROADENING IN T HREE L EVEL Λ S YSTEM C ONSIDERING B IDIRECTIONAL P UMPING Sulagna Dutta ∗ and Krishna Rai Dastidar† Department of Spectroscopy, Indian Association for the Cultivation of Science, Kolkata, India

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Abstract We have studied (ab-initio) the feasibility of amplification without inversion (AWI) with resonant and off-resonant driving and probe fields by using density matrix equations both numerically and analytically in H2 molecule. We have derived (ab-initio) the analytical expressions for coherences and populations in steady state limit in threelevel Λ scheme without any approximation i.e. keeping all orders of probe field Rabi frequency (G) and coherent field Rabi frequency (Ω). Previously approximate expressions for coherences and populations have been derived keeping only the first order terms in probe field Rabi frequency (G) and all orders of coherent field Rabi frequency (Ω). Hence AWI was studied under the condition that coherent field Rabi frequency (Ω) will be at least two orders of magnitude greater than the probe field Rabi frequency. Here we have explored the feasibility of AWI when coherent field Rabi frequency ( Ω) is of the same order of probe field Rabi frequency and we have shown that AWI is more efficient than that in the previous case (when Ω >> G). From the time evolution of the coherences and populations, we discuss the conditions of transient light amplification mechanism with and without replenishment (i.e.bidirectional pumping) of the ground state. We found that when the replenishment of the ground state is considered AWI can be obtained at resonance of both the fields only when spontaneous decay rate on the coherent transition is greater than that on the probe transition. But when the replenishment of the ground state can be neglected, this condition between spontaneous decay ∗ E-mail address: † E-mail address:

sulagna [email protected] [email protected]

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Sulagna Dutta and Krishna Rai Dastidar widths need not be satisfied to get AWI at resonance of both fields. However under offresonant condition AWI is realized in both the cases (with and without replenishment) and there is no such restriction on the spontaneous decay widths. Dependence of AWI on the choice of vibrational levels as the upper lasing level has been explained. We have explored the effect of both the homogeneous and inhomogeneous broadening of levels under the condition of bidirectional pumping in H2 molecule. It has been shown that in molecules AWI can be obtained on probe field of smaller wavelength than that of the coherent field which has not been observed in atoms so far.

PACS numbers: 42.50.Gy, 42.50.Hz

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1.

Introduction

During the last decade there have been considerable interests in the study of amplification without inversion (AWI) and lasing without inversion (LWI), both theoretically [1-4] and experimentally [5]. Lasers based on inversionless systems show interesting statistical properties like narrower linewidths and amplitude squeezing [6]as compared to conventional lasers. Many models for LWI have been proposed, mostly three and four level schemes in Λ,V and Ladder configurations [1]. The key mechanism, which is common to most of the proposed schemes is that the absorption becomes minimum or equal to zero due to some destructive interference between different absorption channels keeping the emission intact. Thus a small population in the upper lasing level can give rise to light amplification. To minimize the absorption, atomic coherence is set either by (a) some external coherent field [1] or by (b) choosing a system where atomic coherence is inherent in the medium or by (c) the presence of two-photon coherence in diffused radiation field [2-4]. Here we will discuss AWI where molecular coherence is invoked by coherent field. Besides AWI/LWI there are also several effects like four-vector mixing,coherent population trapping, frequency up-conversion, electromagnetically induced transparency (EIT) which are developed as a consequence of quantum coherence and interference [7]. Theoretically, the feasibility of LWI in three-level schemes like ladder, V and Λ schemes has been studied parametrically [8] and also in real atomic systems [9]. The physical picture in these three-level systems can be found in various dressed-state analyzes [10,11]. Most of the theoretical works done on real systems have focussed on atoms and very few studies [3,12] have been conducted in molecules. The presence of rotational and vibrational states makes the study of LWI/AWI fascinating in molecules as these states influence the process of inversionless lasing. Choice of different ro-vibrational levels in the transition scheme results in interesting variations in the gain profile of a molecule. Also, gain can be obtained in a wide range of frequency covering these vibrational and rotational states. Thus in a molecular system, one has the flexibility to choose from a wide frequency range as well as various shapes of gain profiles. In atomic systems, the two upper levels and two lower levels respectively in the Λ and the V schemes are generally taken to be hyperfine levels. In some studies, for example in the Rb experiment by Zibrov et.al. in 1995 [13], fine structure components of an atomic level was considered. These levels being very closely spaced, very narrow bandwidth lasers are needed in order to single out the individual levels. In a molecular system, two widely spaced vibrational levels can be used and hence this

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Study of Amplification without Inversion in H2 Molecule

17

restriction is not required, as long as coherence can be maintained. However, the lasers used should have bandwidth short enough to avoid resonant excitation from other vibrational levels. Earlier our group has studied AWI in H2 molecule using resolvent operator technique [3], in which the dependence of gain on the choice of different vibrational levels was also shown considering the effect of two-photon near-resonant transitions to autoionizing states (the upper lasing level). In a recent calculation [14] on three-level ladder, V and Λ system in H2 and Li2 molecules we have shown that AWI can be obtained in VUV, red and far red region respectively from the first excited Rydberg states and AWI can be controlled by choosing different vibrational levels for coherent and probe transitions. Recently we have studied the feasibility of amplification without population inversion (AWI) in LiH molecule for three-level ladder, V and Λ schemes [15]. However in these studies the effect of bidirectional pumping on the AWI process has not been considered. In this paper we have studied AWI in H2 molecule considering Λ type transitions in the presence of coherent field. A probe field is also used to couple two lasing levels for the confirmation of the AWI. In the present work, we have set up the density matrix equation of the Λ system from the Master equation and have solved these equations numerically using ab-initio data for the potential energy curves and dipole transition moments of H2 [16]. We have also solved these equations analytically to give the general expressions for populations and coherences in steady state limit without any approximation i.e. keeping all the terms. From these expressions neglecting the terms containing second and higher order of probe Rabi frequency one can get the approximate expressions as obtained previously [17,20]. This approximation is valid only when strong field Rabi frequency is orders of magnitude greater than the probe field Rabi frequency. In this work we have used the exact expressions for coherences and populations to explore the feasibility of AWI without any approximation so that Ω can be chosen as Ω ∼ G. It is found that under the resonant conditions of coherent and probe fields, the condition for inversionless amplification is that the spontaneous decay width on the probe field transition (in this case γ13 ) should be less than the spontaneous decay width on the coherent field transition (in this case γ12 ) [17]. But this restriction is needed only when the replenishment of the lower lasing level (or bidirectional pumping) is taken into account (i.e the incoherent pumping rate (Λ) is added to γ13 ). It has been shown before that under the condition when the replenishment of the lower lasing level can be neglected, gain is feasible under resonant condition even when γ13 is greater than γ12 [14]. Here we have shown from the general solutions that when replenishment of the lower lasing level is considered, amplification can be obtained even if γ13 is greater than γ12 under the off-resonant condition. Here we have demonstrated the effect of replenishment of the ground state on AWI in H2 molecule by comparing gain profiles for both the cases (i.e. with and without replenishment). We have also studied the AWI for the same values of two detunings (i.e ∆1 = ∆2 = ∆). Earlier J. Mompart et al. have studied the Λ system in the frame work of both density-matrix and quantum-jump formalism [19] within the first order approximation. J. H. Wu et al. have given the analytical expressions of coherences and populations up to first order approximation in G to describe the spontaneously generated coherence (SGC) [20]. Here by using the exact solutions of the density matrix equations for the Λ-system in the steady state limit, we have shown that AWI can be more efficient when Ω ∼ G than for Ω >> G. We have chosen three level Λ-scheme in H2 molecule considering transitions between

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Sulagna Dutta and Krishna Rai Dastidar

vibrational levels of ground X 1 Σg state and the first excited B1 Σu electronic states. But in practice the excited molecule may spontaneously decay into different ro-vibrational levels of the allowed lower electronic states [14]. To include the effect of spontaneous decay to different ro-vibrational levels, we have considered the total spontaneous decay width in place of partial decay width. As the total spontaneous decay width is two orders of magnitude greater than the partial decay width in H2 , one will have to increase the intensity of coherent field so that the a.c stark splitting of the levels are greater than the total decay width, in order to realize AWI. The incoherent pumping rate should also be increased so that a small fraction of the molecule is excited in the upper lasing level to confirm AWI. We have also studied the effect of inhomogeneous broadening [14,21-23]on the gain profile and have explored the dependence of AWI on the choice of ro-vibrational levels for a doppler broadened system. In a recent study on Na2 molecule it has been shown that AWI is feasible although these molecules have large Doppler width [Popov et. al. in Ref. 12]. We have shown that although the Doppler width in H2 molecule is greater than that in Na2 molecule, AWI is feasible in presence of strong coherent coupling with and without considering the bidirectional pumping (replenishment) of the ground state. In general AWI is studied in atoms by considering three, four or multilevel schemes, where two closely spaced upper or lower levels are chosen to be hyperfine levels (the spacing between these levels is around 1 GHz depending on the specific atoms). In this work we have shown that for AWI in molecules one can also choose two rovibrational levels as two lower levels in three level Λ system. Although the spacing between vibrational levels are atleast one order of magnitude greater than that between rotational levels and several orders of magnitude greater than the hyperfine splitting (e.g. in the ground state of H2 molecule vibrational spacing, rotational spacing and hyperfine splitting are of the order of 65842 .5 GHz [16], 6584.25 GHz and of the order of 1 MHz [24] respectively), the coherence can be maintained between two vibrational levels, making the AWI feasible. Moreover the options for choosing different three level systems in molecules by considering transitions between different rovibrational levels are numerous. Hence within a set of two or three electronic states one can consider transitions between several rovibrational levels to get AWI in a wide range of frequencies which is not possible in atoms. In case of atoms since the hyperfine levels are very close to each other lasers used for probe and strong field should have bandwidth less than this spacing (less than 10 −3 cm−1 ) for singling out the individual states. In case of molecules since the vibrational levels are widely spaced this restriction is not required as long as coherence is maintained. It is to be noted that the Doppler width for the diatomic molecules e.g. that in H2 molecules is of the order of 25.26 GHz, which is orders of magnitude smaller than the rovibrational spacing but several orders of magnitude greater than the hyperfine splitting and is two orders of magnitude greater than the total spontaneous decay width of excited levels [16].

2.

Theory

We consider a closed Λ- type three level system with the ground state |3i and two excited states |2i and |1i, as shown in fig.1 . The transition |2i ↔ |1i of energy difference hω ¯ 12 is driven by a coherent coupling laser of frequency ωL with Rabi frequency Ω. The transition |3i ↔ |1i of energy difference hω ¯ 13 is pumped with a rate 2Λ by an incoherent field. 2γ13

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Figure 1. Schematic diagram for Λ transition scheme. The probe field of frequency ω p and the coherent field of frequency ωL has been applied between levels |1i → |3i and |1i → |2i respectively. The level |1i is pumped by the incoherent pumping field of rate 2 Λ from level |3i. 2γ13 (2γ12) is the spontaneous decay width from the state |1i to the state |3i (|2i). ∆1 and ∆2 are the detunings for coupling laser and probe laser respectively. Fig.1(a) and Fig.1(b) represent the case (1) and case (2) (see Section 3) respectively.

(2γ12 ) is the spontaneous decay width from the state |1i to the state |3i (|2i). There is no dipole allowed coupling between the states |2i and |3i. A probe laser of frequency ω p with Rabi frequency G is applied to the transition |1i ↔ |3i. Ω and G are chosen to be real. Here we have used the Master equation for our scheme described above [25]. The Master equation is chosen because, being an operator equation, it is independent of the representation and can be projected over any basis. The Master eqn. is given by 0 0 i σ˙ = − (Hσ − σH) − γ12 (S+S− σ + σS+ S− ) + 2γ12 S−σS+ − (γ13 + Λ)(S+ S− σ+ h¯ 0

0

0

0

0

0

0

0

0

0

σS+ S− ) + 2(γ13 + Λ)S− σS+ − Λ(S− S+ σ + σS− S+ ) + 2ΛS+ σS−

(1)

Here σ is the density matrix operator. S+ and S− (S+ and S− ) are the raising and lowering operators for the |2i ↔ |1i (|3i ↔ |1i) transition respectively. H is the fully quantum mechanical Hamiltonian of the total system. The Hamiltonian is given by 0

0

0 0 1 1 ¯ 13 |1ih1| +hω ¯ L (a† a + ) +hω ¯ p (a † a + ) H = hω ¯ 23 |2ih2| +hω 2 2 0

0

0

0

0

+g(S+a + S− a† ) + g (S+a + S− a † )

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20

Sulagna Dutta and Krishna Rai Dastidar 0

where g and g are coupling constants and are assumed to be real. The first two terms of the Hamiltonian represent the field free molecular system, the following two terms represent the coherent and probe fields respectively and the last two terms represent the interaction between the system and the field. The eigen states of the unperturbed part of the Hamiltonian form a 3D-manifold by the 0 molecular indexes, the strong field photon number N and by the probe photon no. N . The manifold is written as 0

0

0

0

ε(N, N ) = {|3, N, N + 1i, |2, N + 1, N i, |1, N, N i}

(3)

We represent the uncoupled eigen states of the atom/molecule and the two noninteracting field modes as     1 0 0 0 |3, N, N + 1i =  0  , |2, N + 1, N i =  1  , 0 0   0 0  |1, N, N i = 0  (4) 1 In this basis, the Hamiltonian can be written as   0 0 −G (∆2 − ∆1 ) −Ω  Hint =  0 −G −Ω ∆2

(5)

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where we define the Rabi frequencies and the detunings as ∆1 = ω12 − ωL ,

∆2 = ω13 − ω p p √ 0 hΩ ¯ = −g N + 1, hG ¯ = −g N 0 + 1

(6)

To obtain the semi-classical equations of motion for the elements of the density matrix, we project the Master eqn.(1) over the basis (4) and perform the following reduction operation by introducing the reduced populations and coherences as ρ11 =

∑ h1, N, N |σ|1, N, N i 0

N,N

ρ12 =

0

(7a)

0

∑ h1, N, N |σ|2, N + 1, N i 0

0

(7b)

N,N 0

and similar relations for other populations and coherences. Introducing the above mentioned reduced quantities, we obtain the density matrix equations as ρ˙ 11 = iΩ(ρ21 − ρ12 ) + iG(ρ31 − ρ13 ) − 2(γ12 + γ13 + Λ)ρ11 + 2Λρ33

(8a)

ρ˙ 22 = iΩ(ρ12 − ρ21 ) + 2γ12 ρ11

(8b)

ρ˙ 33 = iG(ρ13 − ρ31 ) + 2(γ13 + Λ)ρ11 − 2Λρ33

(8c)

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ρ˙ 12 = iΩ(ρ22 − ρ11 ) + iGρ32 − (γ13 + γ12 + Λ + i∆1 )ρ12

(8d)

ρ˙ 13 = iG(ρ33 − ρ11 ) + iΩρ23 − (γ12 + γ13 + 2Λ + i∆2 )ρ13

(8e)

ρ˙ 23 = iΩρ13 − iGρ21 + [i(∆1 − ∆2 ) − Λ]ρ23

(8 f )

The closure of the system requires ρ11 + ρ22 + ρ33 = 1. The gain-absorption co-efficient for the probe laser coupled to the transition |1i ↔ |3i (|1i ↔ |2i) is proportional to Im(ρ13 ) [Im(ρ12 )]. If Im(ρ13 ) < 0, the probe laser will be amplified. Similarly if Im(ρ12 ) < 0, the coupling laser will be amplified. We have analytically solved the above density matrix equations in the steady state limit without any approximation i.e keeping all the terms. The imaginary part of ρ13 is given by v13 =

d1 (a3 b2 + a2 b3 ) a1 (c2 b3 − b22 ) − a3 (c2 a3 + b2 a2 )

Where a1 = γ13 + γ12 + Λ +

(3Λ + γ13 )Ω2 ∆21 G2 + + Λγ12 γ13 + γ12 + Λ Λ d1 = Ω

a2 =

ΩG(γ13 − γ12 ) Λγ12

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c2 = γ13 + γ12 + 2Λ +

b3 =

Ω2 ∆22 G2 + + Λ γ13 + γ12 + 2Λ Λ

(9)

(10a) (10b) (10c) (10d)

b2 =

Ω(∆1 − ∆2 ) Ω∆2 − Λ γ13 + γ12 + 2Λ

(10e)

a3 =

G(∆1 − ∆2 ) G∆1 + Λ γ13 + γ12 + Λ

(10 f )

(∆1 − ∆2 )2 Ω2 G2 + + +Λ Λ γ13 + γ12 + 2Λ γ13 + γ12 + Λ

(10g)

When Ω ∼ G, it is clear from Eqns.10a, 10d, 10g that one can not neglect the terms containing G2 . Thus to give the exact analytical solution of Im(ρ13 ), we have considered all the terms i.e. all the orders of G and Ω and the analytical expression of Im(ρ13 ) is given as [A1 + Ω2 G3 cΛ(b + Λ)]Λb (11) v13 = R1 Here (12a) R1 = (D1 + G2 γ12 b)C1 + [G(∆1 − ∆2 )b + G∆1 Λ]P A1 = Ω4 GcbΛ + Ω2 GΛ2 b c(b + Λ) + Ω2 Gb(Λγ12 + bγ13 + Λγ13 )(∆1 − ∆2 )2 +Ω2 GΛ2 γ12 ∆21 − 2Ω2 GΛ2 γ12 ∆1 ∆2

(12b)

Where C1 = B1 Λb + G2 b(b + Λ)(∆1 − ∆2 )2 + Ω2 G2 Λb + G2 Λ(b + Λ) + G2 Λ2 b(b + Λ)

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22

Sulagna Dutta and Krishna Rai Dastidar P = Ω2 G∆2 cΛb − Ω2 Gc b(b + Λ)(∆1 − ∆2 ) − Qγ12 [G(∆1 − ∆2 )b + G∆1 Λ] D1 = Λγ12 b2 + (3Λ + γ13 )Ω2 b + ∆21 Λγ12

(13b) (13c)

B1 = [Ω2 + (∆1 − ∆2 )∆2 ]2 + (b + Λ)2 (∆1 − ∆2 )2 + 2Ω2 Λ(b + Λ)+ Λ2 (b + Λ)2 + Λ2 ∆22

(13d)

Q = Λ(b + Λ)2 + G2 (b + Λ) + Ω2 (b + Λ) + ∆22 Λ

(13e)

c = γ13 − γ12

(13 f )

b = γ13 + γ12 + Λ

(13g)

The imaginary part of ρ12 is calculated as v12 =

ΩbΛγ12C1 R1

(14)

ΩPΛb R1

(15)

The imaginary part of ρ23 is given by v23 =

With a strong coupling laser Ω  γi j , Λ, G ; we can approximate the above steady-state solutions considering all orders of Ω and only first order of G. The solutions are given by: A1 B1 D1

(16)

ΩΛγ12 b D1

(17)

ΩP B1 D1

(18)

Ω4 GcbΛ + Ω2 GΛ2 b c(b + Λ) − Ω2 GΛ2 γ12 ∆2 [(Ω2 + Λb + Λ2 )2 + Λ2 ∆2 ][Λγ12 b2 + (3Λ + γ13 )Ω2 b + ∆2 Λγ12 ]

(19)

(1)

v13 =

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(1)

v12 = (1)

v23 = When ∆1 = ∆2 = ∆, eqn.(16) reduces to (1)

v13 =

(1)

When c is −ve i.e. γ13 < γ12 ; v13 < 0 → the probe will be amplified.But when c is +ve i.e. γ13 > γ12 ; to achieve amplification we have to fulfill the condition Λγ12 ∆2 > Ω2 c b + Λb c(b + Λ) or ∆2 >

(γ12 + γ13 + Λ)(γ13 − γ12 )(Ω2 + Λ2 + bΛ) γ12 Λ

(20)

When ∆1 = ∆2 = 0 , eqn (16) reduces to (1)

v13 =

Ω4 GcbΛ + Ω2 GΛ2 b c(b + Λ) (Ω2 + Λb + Λ2 )2 [Λγ12 b2 + (3Λ + γ13 )Ω2 b]

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Study of Amplification without Inversion in H2 Molecule

23

Thus when ∆1 = ∆2 = 0 we can get amplification only if γ13 < γ12 . But when the replenishment of the lower lasing level is neglected ; v13 ∝ (γ13 − Λ − γ12 ) at resonance. Thus for (γ13 − Λ) < γ12 , we can get amplification. Λ can be greater than or less than γ13 . But for Λ > γ13 , the population will be inverted. Therefore Λ should be less than γ13 satisfying the condition (γ13 − Λ) < γ12 . Thus if we neglect the replenishment of the lower lasing level, we can get amplification for both the cases (a) γ12 > γ13 and (b) γ12 < γ13 at the resonant condition (∆1 = ∆2 = 0). The real parts of ρ12 , ρ13 , ρ23 are given below Re(ρ12) = u12 =

Gv23 + ∆1 v12 (γ12 + γ13 + Λ)

(22)

∆2 v13 − Ω v23 (23) (γ12 + γ13 + 2Λ) (∆1 − ∆2 )v23 + Ω v13 + G v12 (24) Re(ρ23 ) = u23 = − Λ With a resonant coupling laser and a resonant probe laser ( ∆1 = ∆2 = 0), we have found that ρ13 (t) = i Im[ρ13 (t)], ρ12 (t) = i Im[ρ12 (t)] and ρ23 (t) = Re[ρ23(t)]. Thus the dispersive response for the probe and coupling laser vanishes, and the two-photon coherence ρ23 is real. The general expression of the steady state populations are given Re(ρ13 ) = u13 =

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ρ11 =

Ω v12 γ12

(25)

ρ33 = (γ13 + Λ)ρ11 − G v13

(26)

ρ22 = 1 − ρ11 − ρ33

(27)

Gain Coefficient: The gain coefficient for a three-level system is given as −4πnω p |d p |2 Im(ρ p ) (28) h¯ c G where n is the number density of molecules, ω p is the frequency of the probe field, d p is the dipole transition moment for the probe transition and ρ p is the corresponding coherence term. Absorption Coefficient: The absorption coefficient for the coherent field is given by 0

G =

−4πnωc |dc|2 Im(ρc ) (29) hcΩ ¯ where ωc is the frequency of the coherent field, dc is the dipole transition moment for the coherent transition and ρc is the corresponding coherence term. A=

Doppler broadening: The Doppler width (FWHM) of a line of frequency ω is given r kT 2ω (30) 2ln2 ∆ω = c m where k is the Boltzmann constant, c is the velocity of light, T is the temperature and m is the mass of the molecule. as

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3.

Sulagna Dutta and Krishna Rai Dastidar

Schemes

Three cases are considered here : • Case (1):- Where γ13 < γ12 X 1 Σg (v = 0, j = 0) → B1 Σu (v = 1, j = 1), X 1 Σg (v = 1, j = 0) → B1 Σu (v = 1, j = 1), • Case (2):- Where γ13 > γ12 X 1 Σg (v = 0, j = 0) → B1 Σu (v = 5, j = 1), X 1 Σg (v = 1, j = 0) → B1 Σu (v = 5, j = 1), • Case (3):- Where γ13 > γ12 X 1 Σg (v = 0, j = 0) → B1 Σu (v = 7, j = 1), X 1 Σg (v = 1, j = 0) → B1 Σu (v = 7, j = 1),

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where v is vibrational quantum number and j is rotational quantum number for the molecular system. For all the transition schemes incoherent pump is on the probe transition. In the first transition scheme, the coherent laser couples the B1 Σu (v = 1, j = 1) state and v = 1, j = 0 level of ground state and the probe and pump field act between B1 Σu (v = 1, j = 1) state and the v = 0, j = 0 level of ground state. The frequencies of the two transitions are 87467.701 cm−1 and 91610.006 cm−1 respectively. Now let us consider the effect of further transitions from other rovibrational levels which will be populated due to spontaneous decay from the excited levels, on the amplification process. It is well known that under normal condition the population in v = 0 level of the ground state of a molecule is maximum and it decays exponentially with the increase in vibrational quantum number. The higher vibrational levels of the ground state will be populated due to spontaneous decay from these excited levels (v = 1 level of B1 Σu state ). But in the large time limit these molecules will decay to ground state due to collisional and vibrational relaxation. Within the life time of an excited vibrational level say v = 1 level of X 1 Σg state (which is closest to v = 0 level), the fraction of the molecules lying in this level may be further excited by the probe and coherent fields to higher vibrational levels of B1 Σu state e.g. coherent field will excite the molecule in between v = 1 and 2 levels and probe field will excite the molecule in between v = 4 and 5 levels and the amplification process will be repeated. But since this will be a second order process and since the frequencies are far away from the exact resonance from these levels, effect of this second order process will be orders of magnitude less than the first order process (i.e. the initial excitation from the v = 0 level of the ground state). In H2 molecules, the para molecules (with nuclear spin I = 0) will occupy the states with even total angular momentum quantum number ( j = 0, 2, 4 etc) and the ortho molecules (with I = 1) will occupy the states with odd angular momentum quantum numbers. By

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using molecular jet one can get most of the populations in j = 0 and 1 levels. The energy difference between v = 0, j = 0 of ground state of H2 molecule and v = 1, j = 1 of B1 Σu state is 91610.006 cm−1 . One can selectively excite v = 0, j = 0 level of the ground state by choosing a probe laser of this transition frequency. A simple calculation can show that if the v = 0, j = 1 of the ground state is excited by a photon of this energy, it will reach very close to the v = 1, j = 3 level of B1 Σu state (this is a forbidden transition) and it will be detuned from the j = 2 level by approximately 43.508 cm−1 . Again from the B1 Σu (v = 1, j = 1) level molecules may de-excite to v = 1, j = 0 and v = 1, j = 2 levels of the ground state. The energy difference between v = 1, j = 1 level of B1 Σu state of H2 molecule and v = 1, j = 0 level of ground state is 87467.701 cm−1 . Thus by choosing the coherent coupling laser of this frequency, one can set the X 1 Σg (v = 1, j = 0) → B1 Σu (v = 1, j = 1) transition to be a resonant transition. The X 1 Σg (v = 1, j = 2) → B1 Σu (v = 1, j = 1) transition will be detuned by approximately 337.618 cm−1 . The frequency of the coherent laser is much smaller than that for the probe laser. Therefore if the coherent coupling laser excite the molecules from the v = 0, j = 0 level of the ground state, it will be a dutuned transition from B1 Σu (v = 0, j = 1) level by approximately 2829.517 cm−1 . Again if the coherent coupling laser excite the molecules from the v = 0, j = 1 level of the ground state, it will be a dutuned transition from B1 Σu (v = 0, j = 0) level and B1 Σu (v = 0, j = 2) level by approximately 2663.45 cm−1 and 2780.31 cm−1 respectively. Since the probability for the detuned transition is much weaker than the resonant transition, this excitation channel can be ignored.

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4.

Results and Discussions

For the three-level Λ scheme, following aspects have been studied - a) the Gain and Absorption coefficient for the closed system of the H2 molecule under different transition schemes considering both unidirectional and bidirectional pumping, b) the behavior of gain profile when total spontaneous decay width is considered, c) the effect of Doppler broadenings. Due to space restrictions we have not presented all the features as figures, we have instead discussed these features in this section.

4.1.

Gain and Absorption for the Closed System Considering Unidirectional Pumping

We have presented here the results for gain in the three level Λ-system as a function of time and the detuning of the probe laser (∆2 ) in H2 molecule. Results are given for the steady state limit as well as at intermediate values of the evolution time. Since we have chosen a molecular system as a gain medium, it has been shown that the gain profile as well as its magnitude depend on the choice of different vibrational levels for the probe and strong field transitions. We have also shown that absorption on the coherent field can be controlled by choosing different vibrational levels as the upper lasing level. The energy difference between ground vibrational level ( v = 0, j = 0) of X 1 Σg state and the v = 1,5 and 7 ( j = 1) of B1 Σu state are 91610.006 cm−1 , 96535.739 cm−1 , and 98802.101 cm−1 respectively which correspond to the wavelength 109.09 nm , 103.53 nm and 101.15 nm respectively. Therefore in H2 molecule, AWI can be obtained in VUV spectral region.

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Sulagna Dutta and Krishna Rai Dastidar

Table 1. Gain of the probe field and absorption of the resonant coherent field in steady state for the closed system at the resonant condition of the coherent and probe lasers for the three schemes (see section 3). I1 and I2 refer to intensities of the probe field and the strong field respectively in Watt/cm2.

I1

I2

Λ(a.u.) (first

Gain(a.u.) Exact transition

Gain(a.u.) Approximate scheme)

Absorption(a.u.) Exact

0.001 0.01 0.1 1 1 1

2 2 2 2 20 200

0.03 × 10−10 0.11 × 10−10 0.39 × 10−10 1.9 × 10−10 0.98 × 10−10 0.89 × 10−10

0.402 × 10−12 0.137 × 10−11 0.386 × 10−11 0.783 × 10−11 0.894 × 10−12 0.884 × 10−13

0.402 × 10−12 0.138 × 10−11 0.402 × 10−11 0.985 × 10−11 0.928 × 10−12 0.887 × 10−13

−0.271 × 10−11 −0.928 × 10−11 −0.263 × 10−10 −0.577 × 10−10 −0.574 × 10−11 −0.563 × 10−12

(second

transition

scheme)

0.88 × 10−9 1.32 × 10−9

0.195 × 10−11 0.264 × 10−11

0.209 × 1011 0.446 × 10−11

(third

transition

scheme)

2.00 × 10−9

0.904 × 10−12

0.118 × 10−11

1 10

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7

20 20

70

−0.635 × 10−11 −0.717 × 10−11

−0.588 × 10−12

Generation of these wavelengths are possible in laboratory [26] and this can be used as probe field or coherent field [14]. We have found that while the probe field was amplified, the coherent field experienced absorption. The gain and absorption shown in Table.I for different vibrational levels (v=1,5,7 of B1 Σu state) as the upper lasing level, correspond to the resonant coherent field and probe field without considering bidirectional pumping. The intensities of the coherent field and the probe field are increased step by step and the value of the incoherent pumping (unidirectional) is chosen to get best value of gain satisfying the closure relation. Firstly we have considered the first transition scheme where the probe field couples v = 0, j = 0 level of X 1 Σg state and v = 1, j = 1 level of B1 Σu state whereas the coherent coupling is between v = 1, j = 1 of B1 Σu state and v = 1, j = 0 of X 1 Σg state (i.e. the case (1) of the transition schemes). In table I we have given gain and absorption coefficients (in a.u.) for different combination of intensities for probe and coherent fields. We found that for a fixed value of coherent field intensity both the coefficient increase with the increase in the probe field intensity. Maximum value of gain is obtained for I1 = 1 W /cm2 (G = 0.55 × 10−9a.u.) and I2 = 2 W /cm2 (Ω = 0.18 × 10−8 a.u.). One can obtain the intensity in the above mentioned range by focusing a pulse of energy 1 nanojoule and duration ∼ 1µsec into a focal area 0.1 mm2 . But when I2 is increased keeping I1 = 1 W /cm2 , value of gain decreases. Therefore we have shown here that when the probe field Rabi frequency is one third of the coherent field Rabi frequency one can get maximum gain. This shows that AWI becomes more prob-

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Figure 2. Gain profile for Λ-scheme in H2 molecule neglecting the replenishment of ground state : Solid line - Case (1) of the transition scheme. γ12 = 1.907 × 10−9 a.u., γ13 = 3.914 × 10−10 a.u. and Λ = 1.9× 10−10 a.u., the intensity for probe field = 1W /cm2 and the intensity for the coherent field = 2W /cm2 . Dashed line - Case (2) of the transition scheme, γ12 = 1.512 × 10−9 a.u., γ13 = 1.93 × 10−9 a.u. and Λ = 0.88 × 10−9 a.u., the intensity for probe field = 1W /cm2 and the intensity for the coherent field = 20W /cm2 . Dotted line - Case(3) of the transition scheme, γ12 = 0.31 × 10−9 a.u., γ13 = 2.1 × 10−9 a.u. and Λ = 2 × 10−9 a.u., the intensity for probe field = 7W /cm2 and the intensity for the coherent field = 70W /cm2 . In the above two calculations, the detuning of the strong field i.e. ∆1 = 0, τ = 1×1013 a.u.. Inset shows time evolution of gain (solid) and the population difference ( ρ11 − ρ33 ) (dashed) at the resonance condition of the two coupling lasers for the first transition scheme. able when the coherent field Rabi frequency is of the order of the probe field Rabi frequency. The analytical value of gain (exact) agrees well with the values obtained at large time by solving the density matrix equations numerically. We have tabulated the approximate value of gain considering only the terms containing first order of G (i.e. gain is calculated using Eqn. 14). This shows that when Ω is of the order of G, the approximate values of gain and absorption differ from the exact values appreciably. Previously all the calculation were done under the condition that Ω >> G. In this transition scheme the absorption of the coupling laser is approximately 7 times larger than the gain of the probe field. The reason for strong absorption is the high value of dipole transition moment for the coherent transition. In case (2) of the transition scheme, the probe field couples v = 0, j = 0 level of X 1 Σg state and v = 5, j = 1 level of B1 Σu state whereas the coherent coupling is between v = 5, j = 1 of B1 Σu state and v = 1, j = 0 of X 1 Σg state. Here the wavelength for probe

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Sulagna Dutta and Krishna Rai Dastidar

and coherent transitions are 103.09 nm and 108.303 nm respectively. In this case the dipole moment for the coherent transition is almost comparable with the dipole moment for the probe transition. Results in table I show that the absorption of the coherent field is approximately 3 times of the gain of the probe field. In case (3) of the transition scheme, the probe field couples v = 0, j = 0 level of X 1 Σg state and v = 7, j = 1 level of B1 Σu state whereas the coherent coupling is between v = 7, j = 1 of B1 Σu state and v = 1, j = 0 of X 1 Σg state. In this case the dipole moment for the the probe transition is almost twice the value of the dipole moment for the coherent transition. Therefore the gain of the probe field is approximately twice the value of absorption of the coupling laser. Thus the magnitude of the maximum gain and the absorption of the coupling laser can be controlled by choosing different ro-vibrational levels as the upper lasing level of the Λ-system. It is to be noticed that the value of Ω and hence the intensity of the coherent field should be chosen such that AC stark splitting is greater than the value of γ13 . Dependence of gain on the choice of different ro-vibrational levels as the upper lasing level is demonstrated in fig.2. We find that gain profile as well as the magnitude of maximum gain is different for different vibrational levels chosen as upper lasing levels. The solid line, dashed line and the dotted line represent the first, second and the third transition schemes respectively. The inset shows the time variation of gain (solid line) and the population difference i.e. (ρ11 − ρ33 ) (dashed line) at the resonance condition of the two coupling lasers (∆1 = ∆2 = 0) for the first transition scheme. This numerical value of gain at large time (shown in the inset) agrees with the analytical value of gain at the resonance condition of the two coupling lasers in the steady state (obtained from Eq. 11) shown in the main figure.

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4.2.

Gain and Absorption for the Closed System Considering Bidirectional Pumping

In Table II we have given the values of gain and absorption at the resonant condition of the two coupling lasers for the first transition scheme considering the bidirectional pumping. Comparing the results in table I and II, it is found that the value of gain and absorption changes in presence of bidirectional pumping. But the value of gain is maximum for I1 = 1 W /cm2 and I2 = 2 W /cm2 as obtained in absence of bidirectional pumping. Moreover, the dependence of gain and absorption on the probe field and the coherent field intensity is similar to that in case of unidirectional pumping. Fig.3 shows the gain profile for Λ transition scheme in H2 molecule (keeping ∆1 = 0) in the steady state limit, considering the replenishment of the ground state i.e. X 1 Σg (v = 0, j = 0) state for the first transition scheme. The inset shows the time variation of gain (solid line) and the population difference i.e. (ρ11 − ρ33 ) (dashed line) at the resonance condition of the two coupling lasers ( ∆1 = ∆2 = 0). These curves show that gain can be obtained without population inversion and the numerically obtained value of gain at large time agrees with the analytical value at the peak position of the gain profile under the steady state limit. For the second and the third transition schemes one can get gain in presence of bidirectional pumping of the ground state only at non-zero ∆1 and ∆2 . It is clear from Eqn. 11 and 12b. that when c is positive i.e. γ13 > γ12 , one may get positive gain if both ∆1 and ∆2 are non-zero and comparable to Ω. In fig.4 we have plotted gain profile for the second

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29

Figure 3. Gain profile for Λ-scheme in H2 molecule considering the replenishment of ground state for Case (1) of the transition scheme. The values of γ12 , γ13 , are given in the caption of case (1) fig. 2. In this calculation, the detuning of the strong field i.e. ∆1 = 0, τ = 1 × 1013 a.u., the intensity for probe field = 1W /cm2 and the intensity for the coherent field = 2W /cm2 and Λ = 2.6 × 10−10 a.u., .Inset shows time evolution of gain (solid) and the population difference (ρ11 − ρ33 ) (dashed) at the resonance condition (∆1 = ∆2 = 0) of the gain profile.

Table 2. Gain of the probe field and absorption of the coherent field in steady state for the closed system at the resonant condition of the coherent and probe lasers for the first scheme (see section 3) considering the replenishment of the ground state i.e. considering bidirectional pumping. I1 and I2 refer to intensities of the probe field and the strong field respectively in Watt/cm2.

I1

I2

Λ(a.u.)

Gain(a.u.) Exact

Gain(a.u.) Approximate

Absorption(a.u.) Exact

0.001 0.01 0.1 1 1 1

2 2 2 2 20 200

0.035 × 10−10 0.115 × 10−10 0.435 × 10−10 2.6 × 10−10 1.2 × 10−10 1.15 × 10−10

0.461 × 10−12 0.138 × 10−11 0.376 × 10−11 0.620 × 10−11 0.807 × 10−12 0.834 × 10−13

0.462 × 10−12 0.139 × 10−11 0.391 × 10−11 0.738 × 10−11 0.832 × 10−12 0.836 × 10−13

−0.312 × 10−11 −0.943 × 10−11 −0.264 × 10−10 −0.547 × 10−10 −0.555 × 10−11 −0.563 × 10−12

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Figure 4. Gain profile for Λ-scheme in H2 molecule considering the replenishment of ground state for Case (2) and case (3) of the transition scheme. Dotted line - case(2) of the transition scheme. The intensity for probe field = 1 W /cm2 and the intensity for the coherent field = 2 × 102W /cm2 and Λ = 0.2 × 10−9 a.u. and ∆1 = 0.55 × 10−7. The values of γ12 , γ13 , are given in the caption of case (2) of fig. 2. Solid line - case(3) of the transition scheme. The intensity for probe field = 7W /cm2 and the intensity for the coherent field = 7 × 102W /cm2 and Λ = 1.4 × 10−9 a.u. and ∆1 = 0.65 × 10−7. The values of γ12 , γ13 , are given in the caption of case (3) of fig. 2. The inset shows time evolution of gain (solid) and the population difference (ρ11 − ρ33 ) (dashed) at ∆2 = 0.516 × 10−7 a.u. for the second transition scheme. and the third transition schemes as a function of ∆2 considering the replenishment of the ground state. The dotted line represents the gain profile for the second transition scheme at ∆1 = 0.55 × 10−7a.u. and the solid line represents that for the third transition scheme at ∆1 = 0.65 × 10−7a.u.. In fig.3 gain peak is present around the zero dutuning ( ∆2 ) and the two minima are present around the AC stark splitted levels. But in this figure interference nature of gain profile is prominent. The inset gives the time variation of gain (solid curve) and the population difference i.e. (ρ11 − ρ33 ) (dashed line) at the peak position of the gain profile (∆2 = 5.61 × 10−8 a.u.) for the second transition scheme. The main utility of AWI is to generate lasing in shorter wavelength region and we have shown here that AWI in VUV region is possible in H2 molecule. Moreover, we have shown here that in this molecule amplification on probe field of shorter wavelength can be achieved using coupling laser of larger wavelength in Λ-transition scheme which has not been observed in atomic systems [27]. All the results given here are in a.u. To convert the gain and frequency in cm−1 , one has to multiply by 1.889 x 10 8 and 2.19475 x 10 5 respectively; and to convert the time in sec one has to multiply by 2.42 x 10 −17 .

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4.3.

31

Gain Profile Considering Total Spontaneous Decay Width

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In the calculations as described above, a closed system for the molecule is assumed i.e, it has been assumed that the molecule in the excited electronic state decays spontaneously only to the ground vibrational level, from where it is excited. In practice, molecules in the first excited electronic state also decay to the higher vibrational levels of the ground electronic state and this should influence the strength of the gain profile. For example, it was shown by Renzoni et.al. [28] that losses towards levels not excited by the laser field results in reducing the contrast and width of the coherent population trapping. Therefore, we have considered here an open system in the molecule by taking into account spontaneous decay to all the allowed vibrational levels. In fig.5 we have presented the gain profile as a function of ∆2 (keeping ∆1 = 1 × 10−7a.u.) in the steady state limit, considering the replenishment of the ground state for the first transition scheme. As both the γ’s (2.1×10−8 a.u.) are equal, no amplification of the probe occurs at the resonance condition (i.e. ∆1 = ∆2 = 0). This curve is similar to that in fig. 4. The inset in the fig.5 shows the time variation of gain (solid curve) and the population difference i.e. (ρ11 − ρ33 ) (dashed line) at the peak position of the gain profile (∆2 = 1 × 10−7 a.u.).

Figure 5. Gain profile for Λ-scheme in H2 molecule considering the replenishment of ground state and the total spontaneous decay width in place of partial decay width for Case (1) of the transition scheme; γ12 = γ13 = 2.105 × 10−8 a.u. and Λ = 0.5 × 10−8 a.u.; In this calculation, the detuning of the strong field ∆1 = 1 × 10−7, τ = 1 × 1011 a.u., the intensity for probe field = 1 × 10−1 W /cm2 and the intensity for the coherent field = 5 × 103 W /cm2 . Inset shows time evolution of gain (solid) and the population difference ( ρ11 − ρ33 ) (dashed) at ∆2 = 1 × 10−7 a.u.

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The calculation has been repeated for the first scheme, neglecting the replenishment of the ground state, where we get a non-zero gain at the resonant condition of the two coupling lasers. But due to the space restriction the figures are not shown here. The results for the second transition scheme (where γ’s ≈ 1.63 × 10−8 a.u.) show similar feature as in the case of first transition scheme. In all the three schemes, decay widths for an open system are larger than those for a closed system by at least two orders of magnitude. Hence to establish coherence in the system, intensity of the coherent field has been increased.

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4.4.

Gain Profile Considering Doppler Broadening

Figure 6. Gain profile for Λ-scheme in H2 molecule with Doppler broadening and considering the replenishment of ground state for case (1) of the transition scheme; γ12 = 3.479 × 10−6 a.u., γ13 = 3.643 × 10−6 a.u. and Λ = 0.9 × 10−6 a.u.; In this calculation, the detuning of the strong field i.e. ∆1 = 5 × 10−6, τ = 1 × 1010 a.u., the intensity for probe field = 10 W /cm2 and the intensity for the coherent field = 1 × 107 W /cm2 Inset shows time evolution of gain (solid) and the population difference ( ρ11 − ρ33 ) (dashed) at ∆2 = 0.51 × 10−5 a.u. At the room temperature, the Doppler broadening (see eqn.30) is much greater than the natural broadening. Therefore the total linewidth increases by many orders of magnitude (in the first case FWHM in the probe channel ≈ 3.64 x 10−6 a.u. ≈ 23.95 GHz and FWHM in the coherent coupling channel ≈ 3.47 x 10−6 a.u. ≈ 22.83 GHz). The states involved in the lasing schemes are broadened and hence high intensity of lasers is required to induce

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coherence. In this case, this is as high as ≈ 107 W /cm2 . To obtain such a high intensity, a pulse of energy 10 nanojoule and duration ∼ 10 nanosec should be focused in a focal plane 10µm2 i.e. a tight focusing is required. Fig.6 shows the gain profile (i.e gain as a function of detuning of probe ( ∆2 ), for ∆1 = 5 × 10−6 a.u.) for the Doppler broadened Λ-system (for the first transition scheme) where the replenishment of the ground state has been considered. Since in this case γ13 > γ12 , a non-zero value of ∆1 is chosen to amplify the probe laser. The nature of the profile is similar to that in fig.4. The time variation of gain (solid curve) and the population difference i.e. (ρ11 − ρ33 ) (dashed line) at the peak position of the gain profile ( ∆2 = 5.1 × 10−6 a.u.) are shown in the inset of this figure. The curves show that both the transient and the steady state gain can be obtained without population inversion in the Doppler broadened Λ-system.

Figure 7. Gain profile for Λ-scheme in H2 molecule with Doppler broadening and neglecting the replenishment of ground state for case (1) of the transition scheme; γ12 , γ13 are given in the caption of fig.6. In this calculation, the detuning of the strong field i.e. ∆1 = 0, τ = 1 × 1010 a.u., the intensity for probe field = 10 W /cm2 and the intensity for the coherent field = 1 × 107 W /cm2 Λ = 1.9 × 10−6 a.u.. Inset shows time evolution of gain (solid) and the population difference (ρ11 − ρ33 ) (dashed) at the resonance condition of the two coupling lasers. Fig.7 shows the same considering Doppler broadening of the above mentioned scheme, neglecting the replenishment of the ground state (keeping ∆1 = 0). In contrary to fig.6, we get here a considerable amount of gain at the resonant condition of the coherent field. The

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Sulagna Dutta and Krishna Rai Dastidar

nature of the curve is similar to that in fig. 2. The inset in fig. 7 shows the time variation of gain (solid curve) and the population difference i.e. ( ρ11 − ρ33 ) (dashed line) at the peak position of the gain profile. We have repeated the calculation considering the second transition scheme with Doppler broadened levels (FWHM in the probe channel ≈ 3.84 x 10−6 a.u. ≈ 25.26 GHz and FWHM in the coherent coupling channel ≈ 3.67 x 10−6 a.u. ≈ 24.19 GHz) and have found that similar conclusions can be drawn as mentioned above. Thus we can conclude that one can get AWI in both the transient and the steady state region even in presence of strong Doppler damping. We have shown that under Doppler free condition (closed/open) as well as in presence of Doppler broadening, AWI is feasible in H2 molecule and it can be tested in the laboratory. In this calculation, we have used molecular density of 10 9 /cm3 but by increasing the molecular density to the value as large as 10 11/cm3 (pressure is much less than 1 Torr), one can get amplification of the order of that in He-Ne laser in the Doppler free situation. But when we consider the Doppler broadened system where the bidirectional pumping of the ground state (replenishment) is taken into account, the molecular density should be increased to a value 1014 /cm3 (pressure is less than 1 Torr), one can get gain ( ∼ 10−3 cm−1 ) of the order of that in He-Ne lasers. At that high molecular density, the collisional linewidth ( ∼ 3 KHz at room temperature and pressure 3.1 millitorr [21]) is also high, but since the Doppler width (∼ 10GHz) considered here is much larger, inclusion of collisional broadening has no effect on the results.

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5.

Conclusion

We have derived the exact expressions of coherences and populations from the density matrix equations for three level Λ-system in the steady state limit keeping all the orders of probe field Rabi frequency and those analytical findings are well explained by the the gain profiles obtained in H2 molecule. It has been shown that AWI is more efficient when Ω ∼ G than that when Ω >> G. We have shown both analytically and numerically that in small diatomic molecules like H2 , AWI is feasible in VUV region both in case of inhomogeneous and homogeneous broadening of levels, with and without considering the replenishment of the ground state. The advantage of choosing molecular system is that the profile of AWI and the magnitude of the maximum gain can be controlled by choosing different ro-vibrational levels as the upper lasing level of the Λ-system. Moreover, we have shown here that in Λ-system one can generate high frequency lasing on probe channel using low frequency coherent coupling laser in this molecule.

Acknowledgement This work has been done under the BRNS project grant 2002/37/40/BRNS. One of the authors (S.D.) is thankful to BRNS.

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[19] J. Mompart et. al. Laser Physics, 9 No.4, 844 (1999) [20] J. H. Wu et. al. Phys. Rev. A, 65, 063807 (2002) [21] M. Fleischhaur, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich and Shi-Yao Zhu, Phys. Rev. A 46, 1468 (1992). [22] A. Kuhn, S. Stenerwald and K. Bergmann, European Phys. Journal D, 1, 57 (1998). [23] J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field and A. M. Lyyra, Phys. Rev. Lett., 88, 173003-I (2002). [24] F. Combes, D. Pfenniger, Astron. Astrophys. 327, 453 (1997) and references therein. [25] Clude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Atom-Photon Interactions (Wiley, New york. 1992), p. 428 [26] H.Rottke and K.H.Welge, Chem. Phys. Lett. 99, 456(1983). [27] H. X. Chen, A. V. Durrant, J. P. Marangos and J. A. Vaccaro Phys. Rev. A, 58, 1545 (1998); S. R. de Echaniz, Andrew D. Greentree, A. V. Durrant, D. M. Segal, J. P. Marangos and J. A. Vaccaro Phys. Rev. A, 64, 055801 (2001). [28] F. Renzoni, W. Maichen, L. Windholtz and E. Arimondo, Phys.Rev.A 55, 3710 (1997). Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

In: Perspectives in Theoretical Physics ISBN: 978-1-61122-960-8 c 2011 Nova Science Publishers, Inc. Editors: T. F. George, R. R. Letfullin and G. Zhang

Chapter 3

A MPLIFICATION WITHOUT I NVERSION AND A BSORPTION WITH I NVERSION IN H2 M OLECULES : A D RESSED -S TATE P ICTURE OF A C OHERENTLY C OUPLED T HREE L EVEL Λ S YSTEM

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Sulagna Dutta ∗ and Krishna Rai Dastidar† Department of Spectroscopy, Indian Association for the Cultivation of Science Kolkata, India

Abstract The physical basis for AWI in the bare state configuration for three level Λ system has been explored by analyzing the coherences and populations in the dressed state framework, obtained by solving density matrix equations derived from the master equation. The exact analytical expressions for populations and coherences in the bare state basis at the steady-state limit have been derived and the coherences and populations in the dressed state basis (at resonance) have been expressed in terms of these exact coherences and populations in the bare state basis which has not been done before. By the term ’exact’ we mean that to derive coherences and populations in the bare state basis (in the steady state) we included all the orders of both the probe field and coherent field Rabi frequencies, incoherent pumping rate, spontaneous emission rate and detunings. By solving the density matrix equations numerically the dynamics of coherences and populations in the dressed state basis have been studied. The numerical values of coherences and populations in large time limit (steady state) are in good agreement with the steady-state values obtained from the analytical expressions for coherences and populations in the dressed-state basis. From these analytical expressions we have explained the behavior of coherences and populations in the dressed-state basis leading to gain in the bare system. We have also explored the effect of both the homogeneous and inhomogeneous broadening of levels in the dressed-state basis. Amplification without inversion (AWI) and absorption with inversion (ADI) are feasible in the dressed-state basis. These complementary effects are the manifestation of quantum interference that occurs in this three level Λ systems. ∗ E-mail address: † E-mail address:

sulagna [email protected] krishna [email protected]

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Sulagna Dutta and Krishna Rai Dastidar

PACS numbers: 42.50.Gy, 42.50.Hz Keywords: Amplification without inversion; dressed-state

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1.

Introduction

During the last decade there have been tremendous interests in the study of amplification without inversion (AWI) and lasing without inversion (LWI), both theoretically [1-4] and experimentally [5]. Lasers based on inversionless systems show interesting statistical properties like narrower linewidths and amplitude squeezing [6] as compared to conventional lasers. Many models for LWI have been proposed, mostly three and four level schemes in Λ,V and Ladder configurations [1]. The key mechanism,which is common to most of the proposed schemes is that the utilization of external coherent fields, which induce quantum coherence and interference in multilevel system. If the atomic coherence between certain atomic states is established, different absorption processes may interfere destructively, leading to the cancelation or the reduction of absorption. Thus a small population in the upper lasing level can give rise to light amplification. To minimize the absorption, atomic coherence is set either by (a) some external coherent field [1] or by (b)choosing a system where atomic coherence is inherent in the medium or by (c) the presence of two-photon coherence in diffused radiation field [2-4]. Besides AWI/LWI there are also several effects like fourwave mixing,coherent population trapping, frequency up-conversion, electromagnetically induced transparency (EIT) which are developed as a consequence of quantum coherence and interference [7]. Theoretically, the feasibility of LWI in three-level schemes like ladder, V and Λ schemes has been studied parametrically [8,12] and also in real atomic systems [9]. Numerous studies have aimed at finding the physical origin of gain [10-15]. Agarwal [10] emphasized that the origin of gain without population inversion arises from the finite coherence between the appropriate dressed-states which is induced by coherent pumping. The physical interpretation of gain in the V -system was given by Grynberg et. al [11] in the density matrix formalism and also using S-matrix in the Hilbert-space of the dressed-state vectors. In the V -system, they found that AWI can occur both in the bare-state and in the dressedstate basis and the gain occurs due to quantum coherence. Y. Zhu analyzed the transient and steady-state proper ties of AWI in a closed three-level V -scheme and Λ-scheme in the bare state basis [12]. Agarwal [13] has shown the contribution of both dressed-state inversion and quantum coherence exists in the amplification process in the Ladder-scheme. The disappearance of dark state in Λ-system with quantum interference has been studied very recently by Hu et. al from the analogy of dressed-state approach in V system [14]. In most of the calculations only the dressing of two levels coupled by the driving field has been considered i.e. in the limit that coherent field Rabi frequency is much greater than the probe filed Rabi frequency (Ω >> G) [15]. Later on Braunstein and Shuker analyzed dressed state coherences and population considering dressing of three levels in V -configuration by coherent field and the probe field (at the resonances of both the fields) and explained the origin of two phenomena e.g. amplification without inversion (AWI) and absorption despite inversion (ADI) [16]. They have derived the approximate analytic time-dependent solutions for dressed state coherences neglecting dressed state populations and vice versa, and

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Amplification without Inversion and Absorption with Inversion in H2 Molecules

39

compared with full numerical solutions at resonances. In this paper we have studied a Λ-type three-level model using a dressed-state approach, and have derived the equations of motion for the elements of the density matrix from the master equation considering the dressing of all the three levels and at the resonances of both the probe and coherent fields. We have derived the elements of the density matrix in dressed states in terms of the density matrix elements in bare-system and vice-versa without any approximation. Previously we have derived the exact steady state expressions for coherences and populations in the bare state basis by considering all the orders of probe field Rabi frequency (G) and coherent field Rabi frequency (Ω) both in the resonance and off-resonance conditions [17]. Therefore one can get exact values of coherences and populations in the dressed state basis by using exact values of coherences and populations in bare state basis. In a previous work the analytical expressions of coherences and populations in the bare state basis have been derived in Λ system considering all orders of Ω and G and keeping γ12 = γ13 [18]. But this approximation is not valid in practice for real molecular and atomic systems. In the widely used first order approximation [12,19] only the terms containing the first order of G and all orders of Ω are considered which is valid only when Ω >> G. However it can be shown that when G and Ω are within the same order, AWI becomes more efficient than in the the case when Ω >> G in the bare state basis [17]. When Ω ∼ G, the calculations considering first order approximation fails to give correct values of coherences and populations in the bare state [17]. This paper presents the detail calculations of AWI in the dressed state basis which is valid to any lasing system (atom/molecule) dressed both by the coherent and probe lasers for the resonant condition of both the fields (i.e. ∆1 = ∆2 = 0). To our knowledge this type of study in three level Λ system using exact values of coherences and populations in dressed state basis has not been done before. The results show the feasibility of gain with and without inversion in the dressed-state basis, as well as gain without population inversion in the bare-state representation. Our study shows explicitly the existence of absorption despite inversion (ADI) in the dressed-state basis [8,13]. Although this effect is contrary to a simple physical explanation of absorption, it can be explained as the complementary process to amplification without inversion (AWI). This phenomenon can be interpreted as the constructive interference of the absorption channel, just as the reverse of AWI, which is obtained from the destructive interference of absorption. Moreover the origin of gain in the bare state has been explained in the light of dressed state approach. Most of the theoretical works done on real systems have focussed on atoms. Some studies [3,20,24] were conducted in molecules. Molecules possess more degrees of freedom than atoms and the presence of rotational and vibrational states makes the study of LWI/AWI fascinating in molecules. Choice of different ro-vibrational levels in the lasing system results in interesting variations in the gain profile of a molecule. Gain can be obtained in a wide range of frequency covering these vibrational and rotational states. Thus in a molecular system, one has the flexibility to get gain in wide frequency range as well as various shapes of gain profiles. Moreover, in a molecular system, in particular for Λ and V transition schemes, the two lower (for Λ) or two upper (for V ) levels need not be very closely spaced (hyperfine/fine levels) as in the case of atomic systems. In this paper we have studied AWI in H2 molecule considering Λ-type transitions which is dressed by both the probe field and the coherent field. Moreover in Λ-system, one can generate high frequency lasing on probe channel using low frequency coupling laser in this molecule unlike

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40

Sulagna Dutta and Krishna Rai Dastidar

atoms [21]. We have solved the dressed state equations numerically using ab-initio data for the potential energy curves and dipole transition moments of H2 [22].

2. 2.1.

Theory Hamiltonian And Master Equation hR -hR

ε(N, N’+1) ωp+R

hωL ε(N-1,N’+1)

1 2γ12

|α> h(ωp-2ωL)

ωL+2R

ωp-R

|γ> |β>

ε(N+1,N’)

|γ>

ωL-2R |β> |α> |γ> |β>

ωL+R

ε(N,N’) hωL hR

ε(N-1,N’)

-hR

(a)

2γ13

Probe Laser G

Coupling Laser Ω

|α> hωL

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|β> |α> |γ> |β>

∆1

2 Incoherent Pump 2Λ

∆2

3

|α> |γ>

Fig. 1

(b)

Figure 1. (a) Manifolds ε(N, N 0 + 1), ε(N − 1, N 0 + 1), ε(N + 1, N 0 ) etc. of uncoupled states of the molecule + lasers photons are shown in the right-hand side. The dressed levels are shown at the left-hand side. The circles represent steady-state populations and its size represents the relative population. The consecutive dressed levels are separated by an energy gap hR. ¯ (b) Schematic diagram for Λ transition scheme. The probe field of frequency ω p and the coherent field of frequency ωL has been applied between levels |1i → |3i and |1i → |2i respectively. The level |1i is pumped by the incoherent pumping field of rate 2 Λ from level |3i. 2γ13 (2γ12 ) is the spontaneous decay width from the state |1i to the state |3i (|2i). ∆1 and ∆2 are the detunings for strong coupling laser and probe laser respectively. We consider a closed Λ- type three level system with the ground state |3i and two excited states |2i and |1i, as shown in fig.1b . The transition |2i ↔ |1i of energy difference hω ¯ 12 is driven by a coherent coupling laser of frequency ωL with Rabi frequency Ω. The transition |3i ↔ |1i of energy difference hω ¯ 13 is pumped with a rate 2Λ by an incoherent field. 2γ13 (2γ12 ) is the spontaneous decay width from the state |1i to the state |3i (|2i). There is no dipole allowed coupling between the states |2i and |3i. A probe laser of frequency ω p with Rabi frequency G is applied to the transition |1i ↔ |3i. Ω and G are chosen to be real.

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Amplification without Inversion and Absorption with Inversion in H2 Molecules

41

Here we have used the Master equation for our scheme described above [23]. The Master equation is chosen because, being an operator equation, it is independent of the representation and can be projected over any basis. The Master eqn. is given by i 0 0 σ˙ = − (Hσ − σH) − γ12 (S+S− σ + σS+ S− ) + 2γ12 S−σS+ − (γ13 + Λ)(S+ S− σ+ h¯ 0

0

0

0

0

0

0

0

0

0

σS+ S− ) + 2(γ13 + Λ)S− σS+ − Λ(S− S+ σ + σS− S+ ) + 2ΛS+ σS− 0

(1)

0

Here σ is the density matrix operator. S+ and S− (S+ and S− ) are the raising and lowering operators for the |2i ↔ |1i (|3i ↔ |1i) transition respectively. H is the fully quantum mechanical Hamiltonian of the total system. The Hamiltonian is given by 1 0 0 1 H = hω ¯ 23 |2ih2| +hω ¯ 13 |1ih1| +hω ¯ L (a† a + ) +hω ¯ p (a † a + ) 2 2 0

0

0

0

0

+g(S+a + S− a† ) + g (S+a + S− a † )

(2)

0

where g and g are coupling constants and are assumed to be real. The first two terms of the Hamiltonian represent the field free molecular system, the following two terms represent the coupling and probe fields respectively and the last two terms represent the interaction between the system and the field. The eigenstates of the unperturbed part of the Hamiltonian form a 3D-manifold by the molecular indexes, the coherent field photon number N and by the probe photon number 0 N . The manifold is written as 0

0

0

0

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ε(N, N ) = {|3, N, N + 1i, |2, N + 1, N i, |1, N, N i}

(3)

We represent the uncoupled eigenstates of the atom/molecule and the two noninteracting field modes as     1 0 0 0    |3, N, N + 1i = 0 , |2, N + 1, N i = 1  , 0 0   0 0 (4) |1, N, N i =  0  1 In this basis, the Hamiltonian can be written as   0 0 −G (∆2 − ∆1 ) −Ω  Hint =  0 −G −Ω ∆2

(5)

where we define the Rabi frequencies and the detunings as ∆1 = ω12 − ωL ,

∆2 = ω13 − ω p

√ 0 hΩ ¯ = −g N + 1, hG ¯ = −g

N0 + 1

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(6)

42

Sulagna Dutta and Krishna Rai Dastidar

To obtain the semi-classical equations of motion for the elements of the density matrix, we project the Master eqn.(1) over the basis (4) and perform the following reduction operation by introducing the reduced populations and coherences as ρ11 =

∑ h1, N, N |σ|1, N, N i 0

N,N

ρ12 =

0

(7a)

0

∑ h1, N, N |σ|2, N + 1, N i 0

N,N

0

(7b)

0

and similar relations for other populations and coherences. Introducing the above mentioned reduced quantities, we obtain the density matrix equations as ρ˙ 11 = iΩ(ρ21 − ρ12 ) + iG(ρ31 − ρ13 ) − 2(γ12 + γ13 + Λ)ρ11 + 2Λρ33

(8a)

ρ˙ 22 = iΩ(ρ12 − ρ21 ) + 2γ12 ρ11

(8b)

ρ˙ 33 = iG(ρ13 − ρ31 ) + 2(γ13 + Λ)ρ11 − 2Λρ33

(8c)

ρ˙ 12 = iΩ(ρ22 − ρ11 ) + iGρ32 − (γ13 + γ12 + Λ + i∆1 )ρ12

(8d)

ρ˙ 13 = iG(ρ33 − ρ11 ) + iΩρ23 − (γ12 + γ13 + 2Λ + i∆2 )ρ13

(8e)

ρ˙ 23 = iΩρ13 − iGρ21 + [i(∆1 − ∆2 ) − Λ]ρ23

(8 f )

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Previously we have analytically solved the above density matrix equations in the steady state limit without any approximation i.e. keeping all the orders of both the probe field and coherent field Rabi frequencies [17]. The imaginary part of ρ13 is given by v13 =

[A1 + Ω2 G3 cΛ(b + Λ)]Λb R1

(9)

Here R1 = (D1 + G2 γ12 b)C1 + [G(∆1 − ∆2 )b + G∆1 Λ]P

(10)

A1 = Ω4 GcbΛ + Ω2 GΛ2 b c(b + Λ) + Ω2 Gb(Λγ12 + bγ13 + Λγ13 )(∆1 − ∆2 )2 +Ω2 GΛ2 γ12 ∆21 − 2Ω2 GΛ2 γ12 ∆1 ∆2

(11)

Where C1 = B1 Λb + G2 b(b + Λ)(∆1 − ∆2 )2 + Ω2 G2 Λb + G2 Λ(b + Λ) + G2 Λ2 b(b + Λ)

(12a)

P = Ω2 G∆2 cΛb − Ω2 Gc b(b + Λ)(∆1 − ∆2 ) − Qγ12 [G(∆1 − ∆2 )b + G∆1 Λ]

(12b)

D1 = Λγ12 b2 + (3Λ + γ13 )Ω2 b + ∆21 Λγ12

(12c)

B1 = [Ω2 + (∆1 − ∆2 )∆2 ]2 + (b + Λ)2 (∆1 − ∆2 )2 + 2Ω2 Λ(b + Λ)+ Λ2 (b + Λ)2 + Λ2 ∆22

(12d)

Q = Λ(b + Λ)2 + G2 (b + Λ) + Ω2 (b + Λ) + ∆22 Λ

(12e)

c = γ13 − γ12

(12 f )

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Amplification without Inversion and Absorption with Inversion in H2 Molecules b = γ13 + γ12 + Λ

43

(12g)

Gain coefficient for a three-level system (bare-state) is given as: 0

G =

−4πnω p |d p |2 Im(ρ p ) h¯ c G

(13)

where n is the number density of molecules, ω p is the frequency of the probe field, d p is the dipole transition moment for the probe transition and ρ p is the corresponding coherence term. Doppler broadening: The Doppler width (FWHM) of a line of frequency ω is given as r 2ω kT ∆ω = 2ln2 (13a) c m where k is the Boltzmann constant, c is the velocity of light, T is the temperature and m is the mass of the molecule.

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2.2.

Dressed States

By finding the eigenvectors of the interaction Hamiltonian (Eq.5), we have obtained the dressed states. To simplify the calculations, we have considered both the driving laser and the probe laser to be in exact resonance with the corresponding bare-state transitions, i.e, ∆1 = ∆2 = 0. Thus in perfect resonant condition, we have obtained the following dressed states: 0 G 0 Ω 0 (14a) |α(N, N )i = − |2, N + 1, N i + |3, N, N + 1i R R 0 1 0 Ω 0 G 0 |β(N, N )i = − √ |1, N, N i + √ |2, N + 1, N i + √ |3, N, N + 1i (14b) 2 2R 2R 0 1 0 Ω 0 G 0 (14c) |γ(N, N )i = √ |1, N, N i + √ |2, N + 1, N i + √ |3, N, N + 1i 2 2R 2R and the corresponding energies hR , E|βi = hR ¯ , E|αi = 0 E|γi = −¯

(15)

√ where R = Ω2 + G2 , the Rabi frequency at the resonance of the two fields. The dressed-state manifolds are shown in fig.1a. If we consider G = 0, Eqns.(14b) 0 and (14c) become the usual coupling and the non-coupling dressed-states, while |α(N, N )i 0 state, which is identical to the ground state |3, N, N + 1i, is not involved in the coupling. If 0 we consider strong coupling field i.e. Ω  G, it can be seen from Eqn.(14a) that |α(N, N )i 0 behaves like the ground state |3, N, N + 1i, and hence, is expected to be more populated 0 0 than the other two states. In contrast, the |β(N, N )i and |γ(N, N )i have the character of the excited bare states. From the eqns.(14b) and (14c), we can see that both the states |βi and |γi contain the same amount of the excited bare level coupled by the strong field, and thus, they are expected to possess the same population.

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2.3.

Sulagna Dutta and Krishna Rai Dastidar

Derivation of the Dressed-State Density Matrix Elements in Terms of the Bare-State Density Matrix Elements and Vice-versa

The eigenstates of Eqns.(14a)-(14c) define a transformation matrix  Ω  − GR 0 R 1  Ω  G T =  √2R √2R − √2  √G 2R

√Ω 2R

(16)

√1 2

that diagonalizes the Hamiltonian of Eqn.(5) via matrix multiplication T HT −1 . Thus, the density operator in the dressed-state basis, ρD , can be expressed as the matrix product ρD = T ρB T −1

(17)

where ρB is the bare-state density operator. The elements of the density matrix operator in dressed-state basis are given by ραα =

G2 Ω2 ΩG ρ + ρ33 − 2 (ρ23 + ρ32 ) 22 2 2 R R R

(18.a)

ΩG G Ω G2 Ω2 (ρ33 − ρ22 ) + √ ρ21 − √ ρ31 − √ ρ23 + √ ρ32 ραβ = √ 2R2 2R 2R 2R2 2R2

(18.b)

ΩG G Ω G2 Ω2 ραγ = √ (ρ33 − ρ22 ) − √ ρ21 + √ ρ31 − √ ρ23 + √ ρ32 2R2 2R 2R 2R2 2R2

(18.c)

ρ11 Ω2 G2 Ω G ΩG + 2 ρ22 + 2 ρ33 − (ρ12 +ρ21 )− (ρ13 +ρ31 )+ 2 (ρ23 +ρ32 ) (18.d) 2 2R 2R 2R 2R 2R ρ11 Ω2 G2 Ω G ΩG + 2 ρ22 + 2 ρ33 + (ρ21 − ρ12 ) + (ρ31 − ρ13 ) + 2 (ρ23 + ρ32 ) ρβγ = − 2 2R 2R 2R 2R 2R (18.e) ρ11 Ω2 G2 Ω G ΩG + 2 ρ22 + 2 ρ33 + (ρ12 + ρ21 ) + (ρ13 + ρ31 ) + 2 (ρ23 + ρ32 ) (18. f ) ργγ = 2 2R 2R 2R 2R 2R where ρi i and ρi j (i, j = 1, 2, 3) are the elements of the density matrix operator in the bare-state basis. In our previous study [17] we have derived (ab-initio) the exact analytical expressions for bare-state density matrix elements in the steady state limit. Resolving the above Eqns.(18a)-(18f) into real and imaginary parts and by using those exact analytical steady-state expressions of the bare-state density matrix elements, we have derived the exact analytical expressions for the populations and the coherences in the dressed-state basis under the steady-state condition. These analytical steady-state values agree well with the steady-state solutions obtained by solving numerically the dressed-state density matrix equations. Results are given in the next section. Similarly the density operator in the bare-state basis, ρB , can be expressed as the matrix product (19) ρB = T −1 ρD T

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ρββ =

Thus, the expressions of the probe field coherence and coupling field coherence are given by G G Ω (ργγ − ρββ ) − (ρβγ − ργβ ) + √ (ργα − ρβα ) (20a) ρ13 = 2R 2R Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated,2R 2011. ProQuest Ebook Central,

Amplification without Inversion and Absorption with Inversion in H2 Molecules ρ12 =

Ω Ω G (ργγ − ρββ ) − (ρβγ − ργβ ) + √ (ρβα − ργα ) 2R 2R 2R

45

(20b)

The imaginary part of the above eqns. are given by G Ω Im(ρ13 ) = − Im(ρβγ ) + √ [Im(ραβ ) − Im(ραγ )] R 2R

(21a)

Ω G Im(ρ12 ) = − Im(ρβγ ) + √ [Im(ραγ ) − Im(ραβ )] R 2R

(21b)

The gain (absorption) co-efficient for the probe (coherent)laser coupled to the transition |1i ↔ |3i (|1i ↔ |2i) is proportional to Im(ρ13 ) [Im(ρ12 )]. If Im(ρ13 ) < 0, the probe laser will be amplified. Similarly if Im(ρ12 ) < 0, the coupling laser will be amplified. Thus the dressed-state coherences are responsible for the inversionless amplification in the bare-state basis.

2.4.

Equation of Motion for the Density-Matrix Elements in the Dressed-State Basis

We have projected the master Eqn.(1) over the dressed-state basis which gives the following equation of motion for the elements of the density matrix operator in the dressed-state representation. ρ˙ αα = −2Λ0 ραα + (Γβ1 + Λ0 )(ρββ + ργγ ) − (Γβ1 + Λ0 )(ρβγ + ργβ ) − Γ0 (ραγ + ργα )

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−Γ0 (ραβ + ρβα )

(22a)

ρ˙ αβ = −(Γαβ − iR)ραβ + (Γαβ − Λ)ραγ + Γρβγ + (Γ − Γ0 )ργβ − Γ0 ραα − (Γ + Γ0 )ρββ −Γργγ

(22b)

ρ˙ αγ = −(Γαβ + iR)ραγ + (Γαβ − Λ)ραβ + Γργβ + (Γ − Γ0 )ρβγ − Γ0 ραα − (Γ + Γ0 )ργγ −Γρββ ρ˙ ββ = Λ0 ραα − (Γβ2 + Γβ1 + Λ)ρββ + (Γβ2 + Λ − Λ0 )ργγ + +Γ0 (ραγ + ργα )

(22c) (Γβ1 + Λ0 ) (ρβγ + ργβ ) 2 (22d)

ρ˙ βγ = −(Γβγ + 2iR)ρβγ − (Γβ2 + Λ − Λ0 )ργβ − Γ0 (ραβ + ργα ) − 2Γ0 (ρβα + ραγ ) − Λ0 ραα +(2Γβ2 +

Γβ1 Λ0 + )(ρββ + ργγ ) 2 2

ρ˙ γγ = Λ0 ραα − (Γβ2 + Γβ1 + Λ)ργγ + (Γβ2 + Λ − Λ0 )ρββ + +Γ0 (ραβ + ρβα ) where the constants are defined as Λ0 =

(22e) (Γβ1 + Λ0 ) (ρβγ + ργβ ) 2 (22 f )

ΛΩ2 R2

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(23a)

46

Sulagna Dutta and Krishna Rai Dastidar Γβ1 =

γ12 G2 + γ13 Ω2 R2

γ12 Ω2 + γ13 G2 2R2 ΛΩG Γ0 = √ 2R2 ΩG Γ= √ (γ12 − γ13 − Λ) 2R2 Γβ1 Λ0 Γαβ = + Γβ2 + Λ + 2 2 Γβγ = Γβ1 + 3Γβ2 + 3Λ − 2Λ0 Γβ2 =

(23b) (23c) (23d) (23e) (23 f ) (23g)

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2Λ0 can be interpreted as the effective incoherent pumping in the dressed-state picture which depopulates the |αi state. This population is distributed equally among the states |βi and |γi, as the co-efficient of ραα in Eqns.(22d) and (22f) is only the half width ( Λ0 ) of the total pumping rate 2Λ0 . (Γβ1 + Γβ2 ) is the total spontaneous decay rate of the |βi and |γi states. |βi (|γi) state decays by spontaneous emission with a rate Γβ1 to the level |αi and Γβ2 to the level |γi (|βi). The coherences ραβ and ραγ (ρβγ ) decay with a rate Γαβ (Γβγ ). Γ and Γ0 can be interpreted as the interference term which involve the product of Ω and G. From eqn.(22) it is clear that ραβ and ραγ oscillate with same frequency R during their evolution, however the oscillations are completely out of phase. The frequency of oscillation during the evolution of ρβγ is twice as large i.e 2R. Inspection of Eqns.(22) reveals that (ρ˙ αα + ρ˙ ββ + ρ˙ γγ ) = 0. Thus the closure of the system is satisfied in the dressedstate basis.

3.

Schemes

Two cases are considered here : • Case (1):- Where γ13 < γ12 X 1 Σg (v = 0, j = 0) → B1 Σu (v = 1, j = 1), X 1 Σg (v = 1, j = 0) → B1 Σu (v = 1, j = 1), is applied. • Case (2):- Where γ13 > γ12 X 1 Σg (v = 0, j = 0) → B1 Σu (v = 5, j = 1), X 1 Σg (v = 1, j = 0) → B1 Σu (v = 5, j = 1), is applied; where v is vibrational quantum number and j is rotational quantum number for the molecular system. For all the transition schemes incoherent pump is on the probe transition.

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In the first transition scheme, the coherent laser couples the B1 Σu (v = 1, j = 1) state and v = 1, j = 0 level of ground state and the probe and pump field act between B1 Σu (v = 1, j = 1) state and the v = 0, j = 0 level of ground state. The frequencies of the two transitions are 87467.701 cm−1 and 91610.006 cm−1 respectively. Now let us consider the effect of further transitions from other rovibrational levels which will be populated due to spontaneous decay from the excited levels, on the amplification process. It is well known that under normal condition the population in v = 0 level of the ground state of a molecule is maximum and it decays exponentially with the increase in vibrational quantum number. The higher vibrational levels of the ground state will be populated due to spontaneous decay from these excited levels (v = 1 level of B1 Σu state ). But in the large time limit these molecules will decay to ground state due to collisional and vibrational relaxation. Within the life time of an excited vibrational level say v = 1 level of X 1 Σg state (which is closest to v = 0 level), the fraction of the molecules lying in this level may be further excited by the probe and coherent fields to higher vibrational levels of B1 Σu state e.g. coherent field will excite the molecule in between v = 1 and 2 levels and probe field will excite the molecule in between v = 4 and 5 levels and the amplification process will be repeated. But since this will be a second order process and since the frequencies are far away from the exact resonance from these levels, effect of this second order process will be orders of magnitude less than the first order process (i.e. the initial excitation from the v = 0 level of the ground state). In H2 molecules, the para molecules (with nuclear spin I = 0) will occupy the states with even total angular momentum quantum number ( j = 0, 2, 4 etc) and the ortho molecules (with I = 1) will occupy the states with odd angular momentum quantum numbers. By using molecular jet one can get most of the populations in j = 0 and 1 levels. The energy difference between v = 0, j = 0 of ground state of H2 molecule and v = 1, j = 1 of B1 Σu state is 91610.006 cm−1 . One can selectively excite v = 0, j = 0 level of the ground state by choosing a probe laser of this transition frequency. A simple calculation can show that if the v = 0, j = 1 of the ground state is excited by a photon of this energy, it will reach very close to the v = 1, j = 3 level of B1 Σu state (this is a forbidden transition) and it will be detuned from the j = 2 level by approximately 43.508 cm−1 . Again from the B1 Σu (v = 1, j = 1) level molecules may de-excite to v = 1, j = 0 and v = 1, j = 2 levels of the ground state. The energy difference between v = 1, j = 1 level of B1 Σu state of H2 molecule and v = 1, j = 0 level of ground state is 87467.701 cm−1 . Thus by choosing the coherent coupling laser of this frequency, one can set the X 1 Σg (v = 1, j = 0) → B1 Σu (v = 1, j = 1) transition to be a resonant transition. The X 1 Σg (v = 1, j = 2) → B1 Σu (v = 1, j = 1) transition will be detuned by approximately 337.618 cm−1 . The frequency of the coherent laser is much smaller than that for the probe laser. Therefore if the coherent coupling laser excite the molecules from the v = 0, j = 0 level of the ground state, it will be a detuned transition from B1 Σu (v = 0, j = 1) level by approximately 2829.517 cm−1 . Again if the coherent coupling laser excite the molecules from the v = 0, j = 1 level of the ground state, it will be a detuned transition from B1 Σu (v = 0, j = 0) level and B1 Σu (v = 0, j = 2) level by approximately 2663.45 cm−1 and 2780.31 cm−1 respectively. Since the probability for the detuned transition is much weaker than the resonant transition, this excitation channel can be ignored. The energy difference between ground vibrational level ( v = 0, j = 0) of X 1 Σg state and the v = 1 and 5 ( j = 1) of B1 Σu state are 91610.006 cm−1 96449.623 cm−1 respectively which correspond to the wavelength 109.1 nm and 103.6 nm respectively. Therefore in H2

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Sulagna Dutta and Krishna Rai Dastidar 1.0

Second Transition Scheme

Population

0.8

ρα α

0.6

0.4

First Transition Scheme

ρβ β= ργ γ

0.2

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Figure 2. Time evolution of the dressed-state populations ραα and ρββ for both the schemes in H2 molecule. The solid line and the dashed line represent the time variation of ραα and ρββ respectively. For case (1) of the transition scheme, γ12 = 1.907 × 10−9 a.u., γ13 = 3.914 × 10−10 a.u. and Λ = 3.9 × 10−10 a.u.; the intensity for probe field = 1 × 10−7 W /cm2 and the intensity for the coherent field = 5 × 104 W /cm2 . For case (2) of the transition scheme, γ12 = 1.512 × 10−9 a.u., γ13 = 1.93 × 10−9 a.u. and Λ = 1.9 × 10−9 a.u. the intensity for probe field = 1 × 10−7 W /cm2 and the intensity for the coherent field = 5 × 105 W /cm2 . In the above two calculations the two coupling lasers are perfectly resonant i.e. ∆1 = ∆2 = 0. To convert the a.u. of time to second one has to multiply by 2.42 x 10 −17 . molecule, AWI can be obtained in VUV spectral region. Generation of these wavelengths are possible in laboratory [24] and this can be used as probe field or strong field [25].

4. 4.1.

Results and Discussions Temporal Dependence of the Populations and Coherences in the Dressed-State Basis

We have solved the Eqns.(22a)-(22f) numerically for both the schemes and the steadystate values of populations and coherences have been checked against the analytical results derived from Eqns. (18a)-(18f). These two results are found to match very well. In fig.2 we have plotted the temporal evolutions of ραα and ρββ for both the schemes (see section 3). The solid line and the dashed line indicate the evolution of ραα and ρββ respectively. ργγ coincides with ρββ curve. We can see that the population ραα is a monotonically decreasing function of time that reaches a steady-state value ρstαα ≈ 0.5 a.u.. The behavior of ρββ (ργγ ) is opposite i.e. it is a monotonically increasing function and it reaches the steadystate value ρββ = ργγ ≈ 0.25 a.u.. The steady-state values of the populations are almost equal for both the schemes. Inspection of fig.2 reveals that ρstαα > ρstββ = ρstγγ , and hence, population inversion is not possible in dressed-state basis for Λ-transition scheme. For the

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-10

2.0x10

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Fig. 3

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Figure 3. Time evolution of Re(ραβ ) for both the schemes in H2 molecule. The solid line represents the first transition scheme and the dashed line represents the second transition scheme. The chosen parameters are kept same as those given in the caption of fig. 2. transition |αi → |βi and |αi → |γi, the population difference is positive i.e non-inversion. For the transition |βi → |γi, the population difference is zero. Therefore it is to be seen that whether amplification is feasible on these transitions in absence of population inversion. In fig.3 we have plotted the temporal dependence of Re(ραβ ) for both the schemes. The dashed line indicates the time variation of Re(ραβ ) for the first transition scheme and the solid line indicates the same for the second transition scheme. The time variation of Re(ραβ) attains its steady-state value more quickly for second transition scheme than that for the first transition scheme. Again with the resonant coupling and probe lasers (∆1 = ∆2 = 0), we found that in the bare-state basis, ρ13 (t) = i Im[ρ13 (t)], ρ12 (t) = i Im[ρ12 (t)], and ρ23 (t) = Re[ρ23 (t)] [12,17]. Thus from the analytical expressions (18b) and (18c), it is obvious that Re(ραβ ) = Re(ραγ ) and the numerical solutions also confirm it. In fig.4 the temporal dependence of Re(ρβγ ) is plotted for both the schemes. The dashed line indicates the time variation of Re(ρβγ ) for the second transition scheme and the solid line indicates the same for the first transition scheme. The real part of ρβγ attains its steadystate value more quickly for the second transition scheme than that for the first transition scheme. In fig.5 we have plotted the time variation of Im(ραβ ) (solid line) and Im(ραγ ) (dotted line) respectively for the first transition scheme. Im(ραβ ) is negative throughout the time evolution, hence, the transition |βi → |αi is amplified without population inversion in the dressed state picture at the frequencies (ω p + R) and (ωL + R). Thus the amplification is due to the external field induced quantum coherences. The opposite transition |αi → |βi exhibits absorption with population inversion (ADI) at frequencies (ω p − R) and (ωL − R) [see fig.1a]. This is the reverse process of amplification without population inversion and it can be explained as a constructive quantum interference for the absorption process. The transition |γi → |αi is attenuated without population inversion at frequencies (ω p − R) and

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Sulagna Dutta and Krishna Rai Dastidar -4

Re(ρβ γ ) (a.u.)

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Figure 4. Time evolution of Re(ρβγ ) for both the schemes in H2 molecule. The solid line in lower panel represents the second transition scheme and the dashed line in the upper panel represents the first transition scheme. The chosen parameters are kept same as those given in fig. 2

-9

Im(ρα γ ) (a.u.)

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Fig. 5

Figure 5. Time evolution of Im(ραβ ) (solid line) and Im(ραγ ) (dotted line) for the first scheme in H2 molecule. The chosen parameters are kept same as those given for the case(1) in fig. 2

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(ωL − R). The opposite transition |αi → |γi is amplified with population inversion at the frequencies (ω p + R) and (ωL + R) (as ργγ < ραα ) since Im(ργα ) < 0 [see Fig.1a]. Thus both amplification and absorption are possible in the dressed-state basis under the condition of inversion and non-inversion. The solid and dotted arrows represent the amplification and absorption process respectively in fig.1a. Resolving the Eqns.(18b) and (13c) in real and imaginary parts, it can be shown that in the resonant condition of the coupling and probe lasers, Im(ρstαβ ) = −Im(ρstαγ ) = Im(ρstγα ). Inspection of Fig.5 also reveals the same fact. In these figures the nature of the evolution of coherences is very much different from that in the bare-state basis, where the coherences oscillate back and forth across zero, thus experiencing periodic transient amplification and absorption [12,17]. However in the second transition scheme we have got Im(ρstαβ ) ≈ 0. Let us now examine under which condition Im(ρstαβ ) ≤ 0. Resolving the Eqn.(18b) in real and imaginary parts, the dressed-state coherences can be expressed in terms of the bare-state coherences. The imaginary part of Eqn.(18b) is given by

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G Ω Im(ρ23 ) Im(ραβ ) = − √ Im(ρ12 ) + √ Im(ρ13 ) − √ 2R 2R 2

(21)

In the first case (as γ13 < γ12 ), Im(ρst13 ) < 0 and Im(ρst12 ) > 0 and Im(ρst23 ) = 0 [12,17]. Therefore if in the bare state, the probe laser is amplified, the coherent laser is attenuated and the two photon coherence is zero, then Im(ρstαβ ) is always negative. Hence the amplification exists in the |βi → |αi channel in the steady state when AWI exists in the bare state at the resonance condition. But if Im(ρst13 ) and Im(ρst12 ) both are positive, (Im(ρst23 ) = 0) the first two terms compete with each other to determine the sign of Im(ραβ ). In cases where these two terms are equal in magnitude Im(ραβ ) becomes zero or negligibly small. In the second scheme we have got Im(ρstαβ ) ≈ 0, i.e if in the bare state, the probe and the coupling lasers both are attenuated and the two photon coherence is zero, no amplification in the |βi → |αi channel in the steady state can be obtained. In fig.6 the temporal dependence of Im(ρβγ ) is plotted for both the schemes. The dashed line indicates the time variation of Im(ρβγ ) for the second transition scheme and the solid line indicates the same for the first transition scheme. Im(ρβγ ) < 0 for both the schemes, resulting in an amplification at frequencies (ω p − 2R) and (ωL − 2R) in |γi → |βi transition. Another feature seen in fig.6 is the strength of coherence ρβγ , which is six order of magnitude stronger than the imaginary part of other two coherences.

4.2.

Temporal Dependence of the Populations and Coherences in the Dressed-State Basis Considering Total Spontaneous Decay Width and Doppler Decay Width

In the calculations as described above, a closed system for the molecule is assumed i.e, it has been assumed that the molecule in the excited electronic state decays spontaneously only to the ground vibrational level, from where it is excited. In practice, molecules in the first excited electronic state also decay to the higher vibrational levels of the ground electronic state and this should influence the strength of the gain profile. For example, it was shown by Renzoni et.al. [26] that losses towards levels not excited by the laser field results in reducing the contrast and width of the coherent population trapping. Therefore, we have

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Sulagna Dutta and Krishna Rai Dastidar -3

6.0x10

-3

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Fig. 6

Figure 6. Time evolution of Im(ρβγ ) for both the schemes in H2 molecule. The solid line represents the first transition scheme and the dashed line represents the second transition scheme. The chosen parameters are kept same as those given in fig. 2

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(a.u.)

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Fig. 7

Figure 7. Time evolution of Im(ραβ ) (lower panel) and Im(ρβγ) (upper panel) for the first scheme in H2 molecule considering total spontaneous decay width. Here γ12 = γ13 = 2.105 × 10−8 a.u. and Λ = 2 × 10−8 a.u.; the intensity for probe field = 1W /cm2 and the intensity for the coherent field = 5 × 107 W /cm2 .

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-3

Im(ρβ γ) (a.u.)

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Fig. 8

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Figure 8. Time evolution of Im(ραβ ) (lower panel) and Im(ρβγ ) (upper panel) for the first scheme in H2 molecule considering Doppler width. Here γ12 = 3.479 × 10−6 a.u., γ13 = 3.643 × 10−6 a.u. and Λ = 3.64 × 10−6 a.u.; the intensity for probe field = 1W /cm2 and the intensity for the coherent field = 1 × 108 W /cm2 . considered here an open system in the molecule by taking into account spontaneous decay to all the allowed vibrational levels. In fig.7 the temporal dependence of Im(ρβγ ) (upper panel) and Im(ραβ ) (lower panel) for the first transition scheme are plotted. At the room temperature, the Doppler broadening (see eqn.13a) is much greater than the natural broadening. Therefore the total linewidth increases by many orders of magnitude (in the first case FWHM in the probe channel ≈ 3.64 x 10−6 a.u. ≈ 23.95 GHz and FWHM in the coherent coupling channel ≈ 3.47 x 10−6 a.u. ≈ 22.83 GHz). The states involved in the lasing schemes are broadened and hence high intensity of lasers is required to induce coherence. In this case, this is as high as ≈ 108 W /cm2 . To obtain such a high intensity, a pulse of energy 100 nanojoule and duration ∼ 10 nanosec should be focused in a focal plane 10µm 2 i.e. a tight focusing is required. In fig.8 the temporal dependence of Im(ρβγ) (upper panel) and Im(ραβ ) (lower panel) for the first transition scheme are plotted. Due to space restrictions we have not presented all the coherences and populations as figures. Therefore by analyzing the dressed state populations and coherences, one can provide the physical basis of AWI in bare state configuration considering all the decoherence effects.

4.3.

Dependence of Dressed State Populations and Coherences on the Relative Magnitude of Probe Field and Coherent Field Rabi Frequencies

In this paper we have considered the dressing of all the three levels which is much more relevant when the probe field Rabi frequency (G) is comparable to the coherent field Rabi frequency (Ω). In our previous work we have explored the feasibility of AWI in the bare Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Sulagna Dutta and Krishna Rai Dastidar 1.0

ρα α

Population

0.8

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ρβ β= ργ γ

0.4

0.2

0.0 0.0

10

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10

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Time (a.u.)

10

6.0x10

Fig. 9

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Figure 9. Time evolution of the dressed-state populations ραα (solid line) and ρββ (dashed line) for the first transition scheme in H2 molecule. Here Λ = 3.3 × 10−11 a.u. the intensity for probe field = 0.11W /cm2 and the intensity for the coherent field = 5 .5W /cm2 . The values of γ12 and γ13 are same as those given for the case(1) in fig.2 . state basis when coherent field Rabi frequency (Ω) is of the same order of probe field Rabi frequency and we have shown that AWI is more efficient in this case than that when Ω >> G [17]. Here we have studied the dependence of the dressed state populations and coherences on the relative values of Ω and G. In fig.9 we have plotted the temporal evolutions of ραα (solid line) and ρββ (dashed line) with the intensity for probe field = 0 .11 W /cm2 and the intensity for the coherent field = 5.5 W /cm2 (i.e.Ω ∼ 14G) for the first transition scheme. In fig.10 we have plotted the temporal dependence of Re(ραβ ) and the inset of the graph gives the temporal dependence of Re(ρβγ ). In fig.11 we have shown the time variation of the imaginary part of the coherences. It is to be noticed that the strengths of the real and imaginary part of the coherences increase by orders of magnitude when Ω ∼ 14G than that for Ω >> G, thus leading to amplification much larger than that for Ω >> G (see figs.3-6).

5.

Conclusion

The dynamics of dressed state coherences and populations have been studied (ab-initio) by solving the density matrix equations numerically (obtained from the master equation) for three level Λ system in H2 molecule. The exact analytical steady state expressions of coherences and populations in the dressed state basis have been derived using the exact analytical expressions of coherences and populations in the bare state basis at the resonances of both the coherent and probe fields. Comparison with these steady state solutions with the

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-3

0.0

Re(ρβ γ ) (a.u.)

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Figure 10. Time evolution of Re(ραβ ) and Re(ρβγ ) (inset) for the first transition scheme in H2 molecule. The chosen parameters are kept same as those given in fig. 9.

Im(ρα γ ) (a.u.)

Imaginary Part of Coherences (a.u.)

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-2

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Figure 11. Time evolution of the imaginary part of the dressed state coherences for the first transition scheme in H2 molecule. The solid line represents the temporal variation of Im(ρβγ ) and the dashed line represents the time variation of ten times of Im(ραβ ). The inset shows the time variation of Im(ραγ ). The chosen parameters are kept same as those given in fig. 9.

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Sulagna Dutta and Krishna Rai Dastidar

numerically calculated quantities shows excellent agreement. Analyzing the dressed state populations and coherences, one can provide the physical basis of AWI in bare state configuration. Absorption despite inversion (ADI) and amplification without inversion (AWI) in the three level Λ system are found in the dressed state framework. These complementary processes arise due to the destructive and constructive interferences respectively.

Acknowledgment This work has been done under the BRNS project grant 2002/37/40/BRNS. One of the authors (S.D.) is thankful to BRNS for providing Junior Research Fellowship.

References [1] H. Lee, Y. Rostovtsev and M. O. Scully, Phys. Rev. A 62, 063804 (2000) and references therein; Y. Zhu and J. Lin, Phys. Rev. A 53, 1767 (1996); O. Kocharovskaya, Phys. Rep. 219 175 (1992); G. S. Agarwal, Phys. Rev. A 55, 2467 (1997); L. M. Narducci, M. O. Scully, C.H. Keitel and S. Y. Zhu and H. M. Doss, Opt. Comm. 86, 324 (1991); L. M. Narducci, H. M. Doss, P. Ru, M. O. Scully, S. Y. Zhu and C. Keittel, Opts. Comm. 81, 379 (1991); see Special Issue on ”Lasing Without Population Inversion”, Laser Physics. 9, No. 4 (1999), Springer Verlag.

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[2] S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989); A. Immamoglu, J. E. Field and S. E. Harris, Phys. Rev. Lett. 66, 1154 (1991); V. G. Arkhipkin and Yu. I. Heller, Phys. Lett. A 98, 12 (1983); G. S. Agarwal, S. Ravi and J. Cooper, Phys. Rev. A 41, 4721 (1990); A. Apalategui, B. S. Mecking and P. Lambropoulos, Laser Physics, 9, 773 (1999), Springer Verlag. [3] S.Sanyal, L.Adhya and K.Rai Dastidar, Phys. Rev. A 49, 5135 (1994); L. Adhya, S. Sanyal and K. Rai Dastidar, Phys. Rev. A 52, 4078 (1995); ibid. Nuovo Cimento, 20D, 1283 (1998); K. Rai Dastidar, L. Adhya and R. K. Das, Pramana 52, 281 (1999); K. Rai Dastidar, Nature News India, July 1999. A. Bhattacharjee, S. Sanyal and K. Rai Dastidar, J. Mol. Spect., 232, 264 (2005). [4] G. Vemuri, K. V. Vasavada and G. S. Agarwal, Phys. Rev. A 52, 3228 (1995). [5] A. Nottleman, C. Peters and W. Lange, Phys. Rev. Lett. 70, 1783 (1993); W. E. Van der Veer et al. Phys. Rev. Lett.70, 3243 (1993); J. Kitching and L. Hollberg Phys. Rev. A 59, 4685 (1999) and references therein. A. S. Zibrov,M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky and H. G. Robinson, Phys. Rev. Letts, 75, 1499 (1995); E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandhu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson and Shi-Yao Zhu, Phys. Rev. Letts, 70, 3235 (1993) [6] Y. Zhu, A.I. Rubiera and Min Xiao, Phys. Rev. A 53, 1065 (1996); G. S. Agarwal, Phys. Rev. Lett. 67, 980 (1991) Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Amplification without Inversion and Absorption with Inversion in H2 Molecules

57

[7] A.S.Zibrov, M.D.Lukin, L.Hollberg and M.O.Scully, Phys. Rev. A 65, 051801 (2002); Y.Rostovtsev et.al, Phys.Rev.Lett.90, 214802(2003); J.P.Marangos, J.Mod.Opt. 45(3), 471-503 (1998); M.D.Lukin, P.R.Hemmer and M.O.Scully, Adv.At.Mol.Opt.Phy. 42, 347(2000); V.Kozlov, O.Kocharovskaya, Y.Rostovtsev and M.O.Scully, Phys.Rev. A 60(2), 1598(1999); S.E.Harris, Phys.Tod. 50(7), 36(1997). [8] M.O.Scully, S.Y.Zhu and Gavrielides, Phys.Rev.Lett62, 2813(1989); [9] S. Basile and P. Lambropoulos, Opt.Commun. 78, 163(1990); J. Zhang , P. Lambropoulos and X.Tang, Phys. Rev. A, 50, 1935(1994). [10] G. S. Agarwal, Phys. Rev. A 44, R28 (1991). [11] G. Grynberg, M. Pinard and P. Mandel, Phys.Rev A 54(1), 776(1996) [12] Y. Zhu, Phys. Rev A 53, 2742(1996); Y. Zhu, Phys. Rev. A, 55, 4568 (1997). Y.Zhu, Phys.Rev A 45(9), R6149(1992). [13] G. Bhanu prasad and G. S. Agarwal, Opt. Comm.86,409 (1991) [14] X. Hu and J. Peng, J.Phys.B, 33, 921(2000) [15] M. Fleischhauer, C.H. Keitel, L.M. narducci, M.O. Scully, S.-Y. Zhu and M.S. Zubairy, Optics Communications, 94, 599 (1992).

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[16] D. Braunstein and R. Shuker, Phys. Rev. A 64, 053812 (2001) and references therein. D. Braunstein and R. Shuker, Phys. Rev. A 68, 013812 (2003) and references therein. [17] S. Dutta and K. Rai Dasidar, International Journal of Theoretical Physics, Group Theory and Nonlinear Optics (in press) [18] S. Menon and G.S. Agarwal, Phys. Rev. A 57 4014 (1998) [19] J. H. Wu et. al. Phys. Rev. A, 65, 063807 (2002) [20] A. K. Popov, S. A. Myslivets and T. F. George, Opt.Exp.7, 148(2000); D.McGloin and M.H.Dunn, J.Mod.Opt.,47, 1887(2000). [21] A. S. Zibroz et.al., Phys.rev.Lett. 75, 1499(1995), H. X. Chen, A. V. Durrant, J. P. Marangos and J. A. Vaccaro Phys. Rev. A, 58, 1545 (1998); S. R. de Echaniz, Andrew D. Greentree, A. V. Durrant, D. M. Segal, J. P. Marangos and J. A. Vaccaro Phys. Rev. A, 64, 055801 (2001). [22] T. E. Sharp, Atomic Data 2, 119 (1971); A. C. Allison and A. Dalgarno, Atomic Data 1, 289 (1970); M. Glass-Maujean, P. Quadrelli and K. Dressler, Atomic Data and Nuclear Data Tables 30, 273 (1984). [23] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Atom-Photon Interactions (Wiley, New york. 1992), p. 428 [24] R. Das, S. Sanyal, K. Rai Dastidar, European Phys. Journal D, 32, 95 (2005). Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

58

Sulagna Dutta and Krishna Rai Dastidar

[25] H.Rottke and K.H.Welge, Chem. Phys. Lett. 99, 456 (1983).

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[26] F. Renzoni, W. Maichen, L. Windholtz and E. Arimondo, Phys.Rev.A 55, 3710 (1997)

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In: Perspectives in Theoretical Physics ISBN: 978-1-61122-960-8 c 2011 Nova Science Publishers, Inc. Editors: T. F. George, R. R. Letfullin and G. Zhang

Chapter 4

S PECTRA AND B OUND S TATES OF THE E NERGY O PERATOR OF A T WO -M AGNON S YSTEM IN A N ON -H EISENBERG F ERROMAGNET WITH S PIN S=3/2 AND N EAREST-N EIGHBOR I NTERACTIONS S.M.Tashpulatov∗ Institute of Nuclear Physics,Uzbek Academy of Sciences, Tashkent, Uzbekistan

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Abstract We consider the energy operator of two-magnon systems in a non-Heisenberg ferromagnet with a coupling between nearest neighbors and spin S = 3/2. The spectrum and the bound states of the systems in a ν− dimensional lattice are investigated. The change of the energy spectrum of the systems is described.

Two-magnon systems have attracted the attention of many researchers. Probably, such systems were first discussed by Bethe [1] in the context of one-dimensional integer-valued lattices. Bethe proved that no more than one bound state (BS) of the system can exist in the case of a one dimensional isotropic ferromagnet. Worts [2] examined the two-magnon system in a d-dimensional integer-valued lattice for arbitrary d and proved that in this case the system has 0,1,2,...,d BSs. Majumdar [3] investigated the two-magnon system in a one-dimensional Heisenberg ferromagnet with a coupling between nearest and second nearest neighbors for the full quasi momentum Λ = π. He found the spectrum and the BSs of the system numerically. In [4], such a system was examined for the case of a one- dimensional Heisenberg isotropic ferromagnet with a nearest- and second nearest-neighbor interactions for Λ = π and Λ = π2 . The spectrum and the BSs of the system for these values of Λ were studied with numerical methods. Gochev [5] considered the two-magnon system in a one-dimensional Heisenberg longitudinal ferromagnet with a coupling between nearest and second nearest neighbors for an arbitrary full quasi momentum. He investigated the spectrum and the BSs of the system analytically. ∗ E-mail address:

[email protected],[email protected]

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60

S. M. Tashpulatov

The two-magnon systems in the anisotropic Heisenberg model with a nearest-neighbor interaction were addressed in [6]. The focus in [7] was on two-magnon systems in a onedimensional anisotropic Heisenberg ferromagnet with a interaction between nearest and second nearest neighbors. The spectrum and the BSs of such systems were investigated for all values of the full quasi momentum. The usual starting point for theoretical studies of magnetically organized matter is the Heisenberg exchange Hamiltonian (with an arbitrary spin s) H = J ∑(~Sm~Sm+τ ),

(1)

m,τ

where J is the bilinear exchange interaction parameter for nearest-neighbor atoms, ~Sm = (Sxm; Sym ; Szm) is the atomic spin operator of the m th node of the ν− dimensional integervalued lattice Z ν , and τ denotes summation over the nearest neighbors. However, the actual isotropic spin exchange Hamiltonian with an arbitrary spin s has the form [8] 2s

H = ∑ ∑ Jn (~Sm~Sm+τ )n ,

(2)

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m,τ n=1

where Jn are the multipole exchange interaction parameters for nearest-neighbors atoms. Hamiltonian (2) coincides with Hamiltonian (1) only for s = 1/2, while there exist terms with higher powers of ~Sm~Sm+τ up to (~Sm~Sm+τ )2s inclucive for s > 1/2. These terms be taken into account. Hamiltonian (2) is called the non-Heisenberg Hamiltonian. The spectrum and the BSs of a two-magnon system in the non-Heisenberg ferromagnet with bilinear and biquadratic exchange interactions were studied in [9-16]. The spectrum and the BSs of two-magnon systems in a non-Heisenberg ferromagnet with coupling between nearest neighbors by bilinear and biquadratic interactions were investigated in [9-13]. Different methods, such as the Green’s function method, the molecular field approximation method, the random phase approximation method, numerical methods, and the use of the creation and annihilation operators through the Holsten-Primakoff transformation, Dyson transformation, Dyson-Maleev transformation, Golghirch transformation, and others, were applied in these works. In [14-15] , the spectrum and the BSs of this system were investigated for the case of a one-dimensional non-Heisenberg ferromagnet with s = 1 and with a coupling between second nearest and third nearest neighbors respectively. The values of the Hamiltonian parameters for which the BSs exist were found, and the energies of these BSs were calculated. In [16] , the spectrum and the BSs of two-magnon systems were investigated in a ν- dimensional non-Heisenberg ferromagnet with s = 1 and with a coupling between nearest neighbors. In the present work we consider a two-magnon system in a ν− dimensional integer-valued lattice Z ν with nearest neighbor interaction with the bilinear and biquadratic and quadrupole exchange couplings, i.e. in a ν− dimensional isotropic non-Heisenberg ferromagnet with spin value s = 3/2. The methods of this study differ by their simplicity and generality . This investigation is based on finding zeros of the Fredholm determinant of the Hamiltonian. The system Hamiltonian H = −J ∑(~Sm~Sm+τ ) − J1 ∑(~Sm~Sm+τ )2 − J2 ∑(~Sm~Sm+τ )3 , m,τ

m,τ

m,τ

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(3)

Spectra and Bound States of the Energy Operator of a Two-Magnon System ...

61

acts in the symmetrical Fock’s space H . Here ~Sm is the atomic spin s = 3/2 operator in the node m, J > 0 and J1 > 0 and J2 > 0 are the bilinear and biquadratic and quadrupole interaction parameters for nearest- neighbor atoms of the lattice, and τ denotes summation y x over the nearest neighbors. We set S± m = Sm ± iSm . Let ϕ0 be the so-called vacuum vector, z which is fully determined by the conditions S+ m ϕ0 = 0 and Sm ϕ0 = (3/2)ϕ0, ||ϕ0 || = 1. The − − vectors Sm Sn ϕ0 describe the state of the system of two magnons located at the nodes m and n. Let H2 be the closure of the space formed by all linear combinations of these two vectors. This space is called the two-magnon space of the operator H. Proposition 1. The space H2 is invariant with respect to the operator H. The operator H2 = H/H 2 is a bounded self-adjoint operator. It generates the bounded self-adjoint operators H¯2 , acting in the space l2 ((Z ν )2 ) according to the formula (H¯2 ) f )(p; q) = −J

∑ {(δp,q+τ + δp+τ,q − 6) f (p; q) + 1/2(3 − δp,q+τ) f (p − τ; q)+

(4)

p;q;τ

+1/2(3 − δ p+τ,q) f (p;q − τ) + 1/2(3 − δ p+τ,q) f (p + τ, q) + 1/2(3 − δ p,q+τ) f (p;q + τ)}− −J1

∑ {(−2δp,q+τ − 2δp+τ,q + 18 + 6δp,q) f (p;q) − 1/2(δp,q+τ + 6δp,q + 9) f (p − τ, q)−

p;q;τ

−1/2(δ p+τ,q +6δ p,q +9) f (p+τ, q)−1/2(δ p,q+τ +6δ p,q +9) f (p;q+τ)−1/2(δ p+τ,q +6δ p,q +9) f (p;q−τ)+ +3δ p,q f (p − τ;q − τ) + 3δ p,q+τ f (p − τ;q + τ) + 3δ p+τ,q f (p + τ;q − τ) + 3δ p,q f (p + τ;q + τ)}− −J2

∑ {(−2δp,q+τ −2δp+τ,q −54−48δp,q) f (p;q)+1/2(23δp,q+τ +48δp,q +27) f (p;q+τ)+1/2(23δp+τ,q+

p;q;τ

+48δ p,q + 27) f (p + τ;q) + 1/2(23δ p,q+τ + 48δ p,q + 27) f (p − τ, q) + 1/2(23δ p+τ,q + 48δ p,q + 27) f (p;q − τ)− −24δ p,q f (p − τ;q − τ) − 24δ p,q f (p + τ;q + τ) − 21δ p,q+τ f (p − τ;q + τ) − 21δ p+τ,q f (p + τ;q − τ)},

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whereδk, j is the kronecker symbol. The operator H2 acts on the vector Ψ ∈ H2 according to the formula − H2 Ψ = ∑(H¯ 2 f )(p; q)S− p Sq ϕ0 .

(5)

p,q

Proof. The proof is by direct calculation in which we use the well-known commutation − z relations between the operators S+ m , S p , and Sm . Lemma 1. The spectra of the operators H2 and H¯2 coincide. Lemma 1 can be proved using the Weyl criterion ( see [17]). The objective of this work is to investigate the spectrum and the BSs of the operator H¯2 , for which the momentum representation is convenient. Let F be the Fourier transformation

F : l2((Z ν )2) ⇒ L2 ((T ν )2 ) ≡ H˜ 2, where T ν is a ν− dimensional cube, T = [0; 2π] is a interval with the normalized Lebesgue measure dλ, λ(T ν ) = 1. Let H˜ 2 = F H¯ 2 F −1 . Proposition 2. Operator H˜ 2 is a bounded self-adjoint operator and it acts in the space ˜ H2 according to the formula (H˜ 2 f )(x; y) = h(x; y) f (x; y) +

Z Tν

h1 (x; y;t) f (t; x + y − t)dt,

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(6)

62

S. M. Tashpulatov

xi −yi i where h(x; y) = 12(J − 3J1 + 9J2 ) ∑νi=1 (1 − cos xi +y 2 cos 2 ), ν

h1 (x; y;t) = −12(J1 − 8J2 ) ∑ [1 − 2cos i=1

ν

−4(J + J1 − 23J2 ) ∑ [cos i=1

xi + yi xi − yi cos + cos(xi + yi )]− 2 2

xi − yi xi + yi xi + yi − cos ]cos( − ti ), x, y,t ∈ T ν . 2 2 2

Proof. The proof is by direct calculation in which we use the Fourier transformation in formula ( 4 ). It follows from Lemma 1 and from this fact that to investigate the spectrum of the operator H2 in the space H2 , it suffices to investigate the spectrum of the operator H˜ 2 acting in the space L2 ((T ν )2 ) according to formula ( 6 ). The following fact is important for further investigating the spectrum of the operator H˜ 2 . Let the full quasi momentum of the system x + y = Λ be fixed. let L2 (ΓΛ ) be the space of functions that are quadratically integrable over the manifold ΓΛ = {(x; y) : x + y = Λ}. It is known [17] that the operators H˜ 2 and the space H˜2 can be expanded into the direct integrals Z M Z M H˜ 2ΛdΛ, H˜ 2 = H˜ 2Λ dΛ H˜ 2 = Tν

Tν

of the operators H˜ 2Λ and the space H˜ 2Λ such that the spaces H˜ 2Λ are invariant with respect to the operators H˜ 2Λ and the operator H˜ 2Λ acts in the space H˜ 2Λ according to the formula

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(H˜ 2Λ fΛ )(x) = hΛ(x) f Λ(x) +

Z Tν

h1Λ(x;t) fΛ(t)dt,

where hΛ (x) = h(x; Λ − x), h1Λ(x;t) = h1 (x; Λ − x;t) and fΛ (x) = f (x; Λ − x).It is known that the continuous spuctrum of the operator H˜ 2Λ consists of the intervals GΛ = [mΛ; MΛ ], where mΛ = minxhΛ(x), MΛ = maxxhΛ (x). The eigenfunction ϕΛ ∈ L2 (T ν ) of the operator H˜ 2Λ corresponding to the eigenvalue zΛ ∈ / GΛ is called the BS the operator H˜ 2 , and the quantity zΛ is called the energy of this BS. We consider the operator KΛ , (KΛ (z) fΛ)(x) =

Z Tν

h1Λ(x;t) fΛ (t)dt. hΛ (t) − z

This operator is totally continuous in the space H2Λ for values of z not belonging to the set GΛ = ImhΛ (x) = [mΛ; MΛ ]. Let ∆νΛ (z) = detD, where d

1,1

 d2,1 D=  ... dν,1

d1,2 d2,2 .. . dν,2

Here d1,1 = 1 − 12B

Z Tν

d1,3 ..... d1,ν  d2,3 ..... d2,ν  . ..  .. .  ...... dν,3 ..... dν,ν gΛ (s)ds1ds2 ...dsν , hΛ(s) − z

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Spectra and Bound States of the Energy Operator of a Two-Magnon System ... dk+1,1 = −12B

Z

dk+1,k+1 = 1 − 4C dk+1,i+1 = −4C

Z Tν

fΛk (sk )gΛ(s)ds1 ds2 ...dsν , k = 1, 2, ..., ν, hΛ (s) − z

Tν

d1,k+1 = −4C

63

Z Tν

Z Tν

ϕΛk (sk )ds1 ds2 ...dsν , k = 1, 2, ..., ν, hΛ(s) − z

fΛk (sk )ϕΛk (sk )ds1 ds2 ...dsν , k = 1, 2, ..., ν, hΛ (s) − z

fΛk (sk )ϕΛi (si ) ds1 ds2 ...dsν, k = 1, 2, ..., ν, i = 1, 2, ..., ν,i 6= k. hΛ(s) − z

In these formulas, ν

gΛ (s) = ∑ [1 + cos Λi − 2 cos i=1

ϕΛk (sk ) = cos(

Λi Λi cos( − Si )], 2 2

Λk Λk − sk ) − cos , 2 2

Λk − sk ), k = 1, 2, ..., ν,Λ ∈ T ν , s ∈ T ν . 2 / GΛ is an eigenvalue of the operator H˜ 2Λ if and only if it Lemma 2. A number z = z0 ∈ ν is a zero of the function ∆Λ(z), i. e., ∆νΛ(z0 ) = 0. Proof. In the case under consideration, the equation for the eigenvalues is an integral equation with a degenerate kernel. It is therefore equivalent to a system of linear homogeneous algebraic equations. It is known that such a system has a nontrivial solution if and only if its determinant is equal to zero. In this case, the determinant of this linear homogeneous algebraic system is equal to function ∆νΛ(z). We set A = J − 3J1 + 9J2 , B = J1 − 8J2 ,C = J + J1 − 23J2 . Theorem 1. Let A = 0 and ν be arbitrary. Then the operator H˜ 2 has two BSs ϕ1 and ϕ2 ( not taking the order of the energy degeneration into account) with the energy values z1 = −8B and z2 = −8B[ν + 1 + ∑νi=1 cosΛi ] and z1 is degenerate ν − 1 times, while z2 is not degenerate, zi ∈ / GΛ , i = 1; 2, for all Λ ∈ T ν , i.e., the energy values of these BSs lie outside the continuous spectrum domain of the operator H˜ 2Λ. When B = 0, this BSs vanishes because it is incorporated into the continuous spectrum. Proof. If A = 0, then hΛ(s) ≡ 0, and

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f Λk (sk) = cos(

∆νΛ (z) = (1 +

2C ν−1 2C 8B ν 32BC ν Λi ) × {(1 + )[1 + (1 + cosΛi )] − 2 ∑ cos2 }. ∑ z z z i=1 z i=1 2

and C = 4B. Solving the equation ∆νΛ (z) = 0, we prove the theorem. Note: In the theorem, the zeroth-order degeneration corresponds to the case where there is no BS. Let π = (π; π; ...; π) ∈ T ν . Theorem 2. let Λ = π,C 6= 0. Then the operator H˜ 2 has only one BS ϕ with the energy value z = 12νA − 2C and this energy level is degenerate ν times. In addition, if C < 0, then z < mΛ , and if C > 0, then z > MΛ . When C = 0, thes BS vanishes because it is incorporated into the continuous spectrum.

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64

S. M. Tashpulatov

The proof is based on the equality hΛ (x) = 12νA with Λ = π and also on the corresponding form of the function ∆νΛ(z). Denote the triple (J; J1 ; J2 ) by P and introduce the following ranges of the triple P for ν = 1; F1 = {P : A < 0, B < 0,C < 0}, F2 = {P : A > 0, B > 0,C > 0}, F3 = {P : A > 0, B > 0,C < 0}, F4 = {P : A < 0, B < 0,C > 0}, F5 = {P : A < 0, B > 0,C < 0}, F6 = {P : A > 0, B < 0,C > 0}, F7 = {P : B = 0, A = C > 0}, F8 = {P : B = 0, A = C < 0}. In the case where ν = 1, the change of the energy spectrum is described by the following theorems. Theorem 3. 1. Let P ∈ F1 and Λ ∈]0; π[ (Λ ∈]π; 2π[) a) If C 6= 6B then the operator H˜ 2 has two BSs ϕ1 and ϕ2 with the corresponding energy levels z1 < mΛ and z2 > MΛ . b)If C = 6B then the operator H˜ 2 has only one BS ϕ with the energy level z < mΛ . 2. Let P ∈ F2 and Λ ∈]0; π[ (Λ ∈]π; 2π[) C , (C > 6B, A < 2B) theh the operator H˜ 2 has three BSs a) If 3A < C < 6B, cos Λ2 > 6B ϕi , i = 1, 2, 3; with the corresponding energy values zi < mΛ, i = 1, 2; z3 > MΛ. C , (C > 6B, A = 2B), then the operator H˜ 2 has two BSs b)If C < 3A < 6B, cos Λ2 > 6B ϕi , i = 1, 2; with the corresponding energy values z1 < mΛ , z2 > MΛ . In this case threed BS vanishes because it is incorporated into the continuous spectrum. C c)If C < 6B < 3A, cos Λ2 > 6B , (C > 6B, A > 2B) then the operator H˜ 2 has only one BS ϕ witn energy value z > MΛ . d)If C = 6B then the operator H˜ 2 has only one BS ϕ with energy value z < mΛ. e)If C > 6B (C < 6B) then the operator H˜ 2 has two BSs ϕ1 , ϕ2 with corresponding energy values z1 < mΛ , z2 > MΛ. 3. Let P ∈ F3 and Λ ∈]0; π[ (Λ ∈]π; 2π[). a) If C ≥ −6B then the operator H˜ 2 has a two BSs ϕ1 and ϕ2 with the corresponding energy values z1 < mΛ and z2 > MΛ . b) If C < −6B then the operator H˜ 2 has only one BS ϕ with energy value z < mΛ . 4. Let P ∈ F4 and Λ ∈]0; π[ (Λ ∈]π; 2π[) C C a)If 3A − 6B −C > 0, cos Λ2 > 3A−6B−C (cos Λ2 6= 6B ), then the operator H˜ 2 has three( two) BSs ϕi , i = 1, 2, 3; (ϕ j, j = 1, 2; ) with the corresponding energy values zk < mΛ, k = 1, 2; z3 > MΛ.(z1 < mΛ , z2 > MΛ.) C C C b) If 3A − 6B −C > 0, − 6B < cos Λ2 < 3A−6B−C or 3A − 6B −C < 0 (cos Λ2 = 6B ), then ˜ the operator H2 has only one BS ϕ with the energy value z > MΛ . 5. Let P ∈ F5 and Λ ∈]0; π[ (Λ ∈]π; 2π[). C C ,C ≥ 3A (cos Λ2 < 6B ,C ≥ 3A), then the operator H˜ 2 has three BSs a) If cos Λ2 > − 6B ϕ1 , ϕ2 , ϕ3 with corresponding energy levels zi < mΛ, i = 1, 2; z3 > MΛ . C (C < 3A, 3A − 6B − C < 0, cos Λ2 < b) If C < 3A, 3A − 6B − C < 0, cos Λ2 > 3A−6B−C C − 3A−6B−C ), then the operator H˜ 2 has three BSs ϕ1 , ϕ2 , ϕ3 with the corresponding energy values zi < mΛ , i = 1, 2; z3 > MΛ . C C c) If C < 3A, 3A − 6B − C < 0, − 6B < cos Λ2 ≤ 6B (C < 3A, 3A − 6B − C < 0, C Λ C − 3A−6B−C ≤ cos 2 < 6B ), or C < 3A, 3A − 6B −C ≥ 0, (C > 3A, 3A − 6B −C ≥ 0), then the operator H˜ 2 has only one BS ϕ with energy value z > MΛ.

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Spectra and Bound States of the Energy Operator of a Two-Magnon System ...

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C C d) If cos Λ2 = − 6B ,C ≥ 3A (cos Λ2 = 6B ,C ≥ 3A), then the operator H˜ 2 has two BSs ϕi , i = 1, 2; with the energy values z1 < mΛ , z2 > MΛ . C C e) If cos Λ2 = − 6B ,C < 3A (cos Λ2 > 6B ,C < 3A), then operator H˜ 2 has only one BS ϕ with the energy value z > MΛ . C C f) If cos Λ2 < − 6B (cos Λ2 > 6B ), then the operator H˜ 2 has two BSs ϕi , i = 1, 2; with the corresponding energy values z1 < mΛ, z2 > MΛ . 6. Let P ∈ F6 and Λ ∈]0; π[ (Λ ∈]π; 2π[). C C a) If cos Λ2 < − 6B (cos Λ2 > 6B ), then the operator H˜ 2 has two BSs ϕ1 , ϕ2 with the corresponding energy values z1 < mΛ , z2 > MΛ . C C b) If cos Λ2 ≥ − 6B (cos Λ2 ≤ 6B ), then the operator H˜ 2 has only one BS ϕ with the energy value z < mΛ . 7. Let P ∈ F7 and Λ ∈]0; π[ (Λ ∈]π; 2π[). Then the operator H˜ 2 has two BSs ϕ1 , ϕ2 with the corresponding energy values z1 < m Λ , z 2 > MΛ . 8. Let P ∈ F8 and Λ 6= 0. Then the operator H˜ 2 has two BSs ϕ1 , ϕ2 with the corresponding energy values z1 < m Λ , z 2 > MΛ . Theorem 4. Let Λ = 0. 1. a) If P ∈ F1 ,C > 6B then the operator H˜ 2 has two BSs ϕ1 , ϕ2 with the corresponding energy values z1 < mΛ , z2 > MΛ. b) If P ∈ F1 ,C ≤ 6B then the operator H˜ 2 has only one BS ϕ with the energy value z < mΛ . 2. a)If P ∈ F2 , 3A < C < 6B, then the operator H˜ 2 has three BSs ϕi , i = 1, 2, 3; with the corresponding energy values z j < mΛ , j = 1, 2; z3 > MΛ . b) If P ∈ F2 ,C ≤ 3A,C < 6B, or P ∈ F2 , 3A < 6B < C, then the operator H˜ 2 has two BSs ϕi , i = 1, 2; with the corresponding energy values z1 < mΛ, z2 > MΛ. c) If P ∈ F2 ,C = 6B,C > 3A, then the operator H˜ 2 has only one BS ϕ with the energy value z < mΛ . d) If P ∈ F2 ,C = 3A ≥ 6B, or P ∈ F2 , 6B < 3A < C, then the operator H˜ 2 has only one BS ϕ with the energy value z > MΛ . e) If P ∈ F2 ,C = 6B < 3A, or P ∈ F2 , 6B < 3A < C, then the operator H˜ 2 has no BS. 3. a) If P ∈ F3 ,C < −6B, A ≥ 2B, then the operator H˜ 2 has two BSs ϕi , i = 1, 2; with the corresponding energy values z1 < mΛ , z2 > MΛ . b) If P ∈ F3 , A < 2B, then the operator H˜ 2 has only one BS ϕ with the energy value z > MΛ . c) If P ∈ F3 ,C ≥ −6B, A ≥ 2B, then the operator H˜ 2 has only one BS ϕ with the energy value z < mΛ . 4. a) If P ∈ F4 ,C > −6B, then the operator H˜ 2 has two BSs ϕi , i = 1, 2; with the corresponding energy values zi < mΛ , i = 1, 2; b) If P ∈ F4 ,C = −6B, then the operator H˜ 2 has no BS. c) If P ∈ F4 ,C < −6B, then the operator H˜ 2 has only one BS ϕ with the energy value z < mΛ . 5. a) If P ∈ F5 , −6B < C < 3A,C > 3/2A − 3B, then the operator H˜ 2 has two BSs ϕi , i = 1, 2; with the corresponding energy values zi < mΛ , i = 1, 2;

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S. M. Tashpulatov

b) If P ∈ F5 , −6B < C < 3A,C ≤ 3/2A − 3B, or P ∈ F5 ,C = −6B < 3A, then the operator ˜ H2 has no BS. c) If P ∈ F5 ,C = −6B ≥ 3A, or P ∈ F5 ,C < −6B, then the operator H˜ 2 has only one BS with the energy value z < mΛ . 6.a) If P ∈ F6 , 3A ≤ C < −6B, then the operator H˜ 2 has two BSs ϕi , i = 1, 2; with the corresponding energy values zi > MΛ, i = 1, 2. b) If P ∈ F6 ,C = 3A > −6B, or P ∈ F6 ,C = −6B ≥ 3A, or P ∈ F6 ,C < −6B,C < 3A, then the operator H˜ 2 has no BS. c) If P ∈ F6 ,C = −6B < 3A, or P ∈ F6 ,C > −6B,C 6= 3A, then the operator H˜ 2 has only one BS ϕ with the energy value z > MΛ . 7. If P ∈ F7 (P ∈ F8 ), then the operator H˜ 2 has only one BS ϕ with the energy value z > MΛ (z < mΛ). A sketch proof of Theorems 3-4 is given below. In the case under consideration, the equation for the eigenvalues is an integral equation with a degenerate kernel. It is therefore equivalent to a system of linear homogeneous algebraic equations. It is known that such a system has a nontrivial solution if and only if its determinant is equal to zero. In this case, the equation ∆νΛ (z) = 0 is therefore equivalent to the equation stating that the determinant of the system is zero. Expressing all integrals in the equation ∆νΛ(z) = 0 through the integral J(z) =

Z T

dt , hΛ (t) − z

we find that the equation ∆νΛ(z) = 0 is equivalent to the equation

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Λ J(z) = {−C(z − 12A) + 12A(3A − 6B −C)cos2 }× 2 ×{C(z − 12A)2 + 12A(6B +C)(z − 12A)cos2 Because

1 hΛ (t)−z

(7)

Λ Λ + 864A2 Bcos4 }−1 . 2 2

is a continuous function for z ∈ / GΛ and 0

[J(z)] =

Z T

dt > 0, [hΛ(t) − z]2

the function J(z) is an increasing function of z for z ∈ / GΛ. Moreover, J(z) → 0 as z → −∞, J(z) → +∞ as z → mΛ − 0, J(z) → −∞ as z → MΛ + 0, and J(z) → 0 as z → +∞. Analysis of Eq.(7) outside the set GΛ = [mΛ; MΛ ] leads to the proof of Theorems 3-4. The energy spectrum in the case where ν = 2 for the full quasi momenta of the form Λ = (Λ1 ; Λ2 ) = (Λ0 ; Λ0 ) is described below. It is easy to see that if the parameters J, J1 , J2 and Λ0 satisfy the conditions of Theorems 3-4, the statements of the theorems are true. Only one additional BS ψ appears, whose energy value is z˜ < mΛ or z˜Λ > MΛ if C > 0 or C < 0. If C = 0, the operator H˜ 2 has no additional BS. The proof of this statement is based on the fact that if ν = 2 and Λ = (Λ0 ; Λ0 ), then the function ∆νΛ (z) has the form Z

[cos( Λ20 − s1 ) − cos( Λ20 − s2 )]2 ds] × Ψ(z), hΛ (s) − zIncorporated, 2011. ProQuest Ebook Central, T2 Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, ∆νΛ (z) = [1 − 2C

(8)

Spectra and Bound States of the Energy Operator of a Two-Magnon System ...

67

where

Ψ(z) = {1 − 24B

×{1 − 4C

Z T2

Z T2

1 + cosΛ0 − cos Λ20 [cos( Λ20 − s1 ) + cos( Λ20 − s2 )] ds1 ds2 }× hΛ(s1 ; s2 ) − z

cos( Λ20 − s1 )[cos( Λ20 − s1 ) + cos( Λ20 − s2 ) − 2cos Λ20 ] ds1 ds2 }− hΛ(s1 ; s2 ) − z −192BC

Z T2

×

Z

(9)

cos( Λ20 − s1 ) − cos Λ20 ds1 ds2 × hΛ (s1 ; s2 ) − z

cos( Λ20 − s1 ) × {1 + cosΛ0 − cos Λ20 [cos( Λ20 − s1 ) + cos( Λ20 − s2 )]} ds1 ds2 . hΛ(s1 ; s2 ) − z T2

The equation ∆νΛ(z) = 0 is therefore eqivalent to the equation Z

[cos( Λ20 − s1 ) − cos( Λ20 − s2 )]2 ds1 ds2 = 0 1 − 2C hΛ (s1 ; s2 ) − z T2

(10)

ψλ (z) = 0.

(11)

and

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It is easy to see Eq.(10) has a unique solution ˜z < mΛ if C > 0;if C < 0, this solution satisfies the condition z˜ > MΛ . If C = 0, Eq.(10) has no solution. Expressing the integrals in Eq.(11) throug the integral J(z) =

Z T2

ds1 ds2 , hΛ (s1 ; s2 ) − z

we obtain an equation of the form ηΛ (z)J(z) = ξΛ (z), where ηΛ (z) = C(z − 24A)2 + 24A(C + 6B)cos2 and

Λ0 Λ0 (z − 24A) + 3456A2Bcos4 2 2

Λ0 . 2 In turn, for ηΛ (z) 6= 0, the latter equation is equivalent to the equation of the form ξΛ (z) = −C(z − 24A) + 24A(3A − 6B −C)cos2

J(z) =

ξΛ (z) . ηΛ (z)

(12)

Analyzing Eq.(12) outside the set GΛ and taking into account that the function J(z) is monotonic for z ∈ / [mΛ; MΛ ], we obtain statements similar to the statements in Theorems 3-4. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

68

S. M. Tashpulatov

For all other quasi momenta Λ = (Λ1 ; Λ2 ), Λ1 6= Λ2 , these exists sets G j , j = 0,¯ 5, of the parameters J, J1 , J2 , and Λ such that in every set G j the operator H˜ 2 has exactly j BSs (taking the energy degeneration order into account) with the corresponding energy values zk ∈ / GΛ, k = 1,¯ 5. Indeed, in this case and for ν = 2, the function ∆νΛ (z) has the form ∆νΛ(z) = detD, where   d11 d12 d12 D =  d21 d22 d23  , d31 d32 d33 here d11 = 1 − 12B

Z T2

2 − 2 ∑2i=1 cos Λ2i cos( Λ2i − si ) + ∑2i=1 cosΛi ds1 ds2 , hΛ (s) − z

d1k+1 = −4C

dk+1,1 = −12B

Z T2

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d23 = −4C

d32 = −4C

T2

T2

cos( Λ2k − sk ) − cos Λ2k ds1 ds2 , k = 1, 2; hΛ(s) − z

cos( Λ2k − sk )[2 − 2 ∑2i=1 cos Λ2i cos( Λ2i − si ) + ∑2i=1 cosΛi ds1 ds2 , k = 1, 2; hΛ (s) − z

dk+1,k+1 = 1 − 4C

Z

Z

Z T2

Z T2

cos2 ( Λ2k − sk ) − cos Λ2k cos( Λ2k − sk ) ds1 ds2 , k = 1, 2; hΛ(s) − z

cos( Λ21 − s1 )cos( Λ22 − s2 ) − cos Λ22 cos( Λ21 − s1 ) ds1 ds2 , hΛ(s) − z

cos( Λ21 − s1 )cos( Λ22 − s2 ) − cos Λ21 cos( Λ22 − s2 ) ds1 ds2 , Λ ∈ T 2 , s ∈ T 2 . hΛ(s) − z

Expressing all integrals in the equation ∆νΛ (z) = 0 through J(z) and rearranging algebraically, we reduce the latter equation to the form ΘΛ(z)J(z) = χΛ (z),

(13)

where ΘΛ (z) is the fifth-order polynomial in z and χΛ(z) is a lower-order polynomial in z. Analyzing Eq.(13) outside the set GΛ and taking into account that the function J(z) with z∈ / [mΛ; MΛ ] is monotonic, we can easily verify that the equation has no more that five solutions outside the set GΛ . We now consider the case of ν = 3. Let the full quasi momentum have the form Λ = (Λ1 , Λ2 , Λ3 ) = (Λ0 , Λ0 , Λ0 ). If the parameters Λ0 , J, J1 , and J2 satisfy the conditions in Theorems3-4, then statements similar to those in the theorems are true. Only one additional BS η appears, whose energy value is z˜. This energy level is twice degenerate and z˜ < mΛ or z˜ > MΛ if C > 0 or C < 0. This additional BS vanishes when C = 0 because it is incorporated into the continuous spectrum.

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To prove this, we note that in this case, the function ∆νΛ(z) has the form ∆νΛ (z) =

[1 − 2C

Z T3

[cos( Λ20 − s1 ) − cos( Λ20 − s2 )]2 ˜ Λ (z), s ∈ T 3 , ds1 ds2 ds3 ]2 ψ hΛ (s) − z

where ˜ λ (z) = [1 − 12B ψ

×{1 − 4C −144BC

Z T3

Z

Z

[3 + 3cosΛ0 − 2cos Λ20 (∑3i=1 cos( Λ20 − si ))] ds1 ds2 ds3 ]× hΛ (s) − z T3

cos( Λ20 − s1 )[∑3i=1 cos( Λ20 − si ) − 3cos Λ20 ] ds1 ds2 ds3 }− hΛ (s) − z T3

cos( Λ20 − s1 )[3 + 3cosΛ0 − 2cos Λ20 ∑3i=1 cos( Λ20 − si )] ds1 ds2 ds3 × hΛ(s) − z ×

Z T3

cos( Λ20 − s1 ) − cos Λ20 ds1 ds2 ds3 . hΛ (s) − z

Therefore the equation ∆νΛ (z) = 0 is equivalent to the equations [1 − 2C

Z T3

[cos( Λ20 − s1 ) − cos( Λ20 − s2 )]2 ds1 ds2 ds3 2 ] =0 hΛ(s) − z

(14)

and ˜ Λ (z) = 0. ψ

(15)

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0

0

It is easy to see that Eq.(14) has a unique double solution z if C 6= 0 andR z < mΛ or 0 2 ds3 z > MΛ if C > 0 or C < 0. Expressing all integrals in Eq.(15) through J(z) = T 3 dshΛ1 ds , (s)−z we obtain the equation ˜ Λ (z), η˜ Λ (z)J(z) = Θ

(16)

where Λ0 Λ0 ηΛ˜(z) = C(z − 36A)2 + 36A(C + 6B)cos2 (z − 36A) + 7776A2Bcos 2 2 and

Λ0 ΘΛ˜(z) = −C(z − 36A) + 36A(3A −C − 6B)cos2 . 2 If ηΛ˜(z) 6= 0, Eq.(16) is, in turn, equivalent to the equation J(z) =

ΘΛ˜(z) . ηΛ˜(z)

Analyzing Eq.(16) outside the set GΛ and taking into account that the function J(z) for z∈ / GΛ is monotonic, we prove the statements made above . Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

70

S. M. Tashpulatov

If Λ 6= (Λ0 , Λ0 , Λ0 ), the system has at most seven BSs (taking the energy degeneration order into account), and there exist sets Gk , k = 0,¯ 7, of the parameters Λ, J, J1 , and J2 such that in every set Gk , the system has exactly k BSs. The energy values of these BSs lie outside the set GΛ. When passing from one of these sets to another, either some additional BSs of the operator H˜ 2 appear or some existing BSs vanish. In this case, function ∆νΛ (z) has the form ∆νΛ(z) = detD, where   d1,1 d1,2 d1,3 d1,4  d2,1 d2,2 d2,3 d2,4   D=  d3,1 d3,2 d3,3 d3,4  . d4,1 d4,2 d4,3 d4,4 Here d1,1 = 1 − 12B d1,k+1 = −4C dk+1,1 = −12B

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dk+2,2 = −4C

T3

Z

dk+1,k+1 = 1 − 4C d2,2+k = −4C

Z

T3

gΛ (s) ds1 ds2 ds3 , hΛ (s) − z

fΛk (sk ) ds1 ds2 ds3 , k = 1, 2, 3; hΛ (s) − z

ϕΛk (sk)gΛ (s) ds1 ds2 ds3 , k = 1, 2, 3; hΛ (s) − z

T3

Z T3

Z

Z

fΛk (sk )ϕΛk (sk ) ds1 ds2 ds3 , k = 1, 2, 3; hΛ(s) − z

T3

ϕΛ1 (s1 ) fΛk (sk ) ds1 ds2 ds3 , k = 1, 2; hΛ(s) − z

T3

ϕΛk (sk) fΛ1 (s1 ) ds1 ds2 ds3 , k = 1, 2; hΛ(s) − z

Z

d3,4 = −4C d4,3 = −4C

Z T3

ϕΛ2 (s2 ) fΛ3 (s3 ) ds1 ds2 ds3 , hΛ(s) − z

T3

ϕΛ3 (s3 ) fΛ2 (s2 ) ds1 ds2 ds3 . hΛ(s) − z

Z

In these formulas, 3

gΛ(s) = ∑ [1 + cosΛi − 2cos i=1

Λi Λi cos( − si )], 2 2

Λk Λk Λk − sk ) − cos , ϕΛk (sk ) = cos( − sk ), k = 1, 2, 3. 2 2 2 ν Expressing all integrals in the equation ∆Λ (z) = 0 through J(z) and rearranging algeNΛ (z) braically, we reduce this equation to the form J(z) = M , Λ (z) where MΛ (z) is a seventh-order polynomial in z, and NΛ (z) is a lower-order polynomial in z. Therefore, this equation has no more than seven solutions outside the set GΛ. f Λk (sk ) = cos(

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For an arbitrary ν > 3 and Λ = (Λ1 ; Λ2 ; ...; Λν) = (Λ0 ; Λ0 ; ...; Λ0) the change of energy operator spectrum is similar to that observed in the case of ν = 1. In this case the operator H˜ 2 with C 6= 0 has only one additional BS. The energy z of this additional BS is degenerated ν − 1 times. For all other values of the full quasi momentum Λ of the system, the operator H˜ 2 has no more that 2ν + 1 BSs (taking the energy degeneracy order into account) with energy values lying outside the set GΛ . The proof of these statements is based on finding zeros of the function ∆νΛ (z). Expressing all integrals in ∆νΛ (z) through J(z), we can bring the equation ∆νΛ (z) = 0 to the form J(z) =

CΛ (z) , DΛ (z)

(17)

where DΛ (z) is a (2ν + 1) th-order polynomial in z and CΛ (z) is also a polynomial in z whose order (with respect to DΛ (z)) is lower. Analysis of Eq.(18) outside the set GΛ leads to the proof of the statements made above. Theorem 5. Let C = 0 and ν be an arbitrary number. Then the operator H˜ 2 has at most one BS, and the corresponding energy level is not degenerated. Proof. If C = 0, the relations A = −4B, and h1Λ (x;t) = −12B ∑νi=1 [1 − 2cos Λ2i cos( Λ2i − xi ) + cosΛi ], hΛ(x) = 12A ∑νi=1 [1 − cos Λ2i cos( Λ2i − xi )] hold. Using the form determinant ∆νΛ (z) and solving the corresponding equation, we obtain the statement in Theorem 5.

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References [1] H. A. Bethe, Eigenverte und eigenfunction der linearen Atom kette. Z. Phys. v.71. p. 205 (1931). [2] M . Wortis, Bound states of two spin waves in the Heisenberg ferromagnet. Phys. Rev.B, v. 132.No 1.p. 85 (1963). [3] I. Majumdar, Bound states of two spin waves in the Heisenberg ferromagnet with nearest and next nearest neighbours interactions. J. Math. Phys. v. 132. No 10. p. 85 (1969). [4] I. Ono, S. Mikado, T. Oguchi, Two-magnon bound states in a linear Heisenberg chain with nearest and next nearest neighbours interactions. J.Phys. Soc. Japan, V. 30. No 2. p.358 (1971). [5] I. G. Gochev, Two-magnon states in a one-dimensional Heisenberg models whit second nearest neighbours interactions. Theor. Math. Phys.,15,402 (1973). [6] I. G. Gochev, Bound states of magnon systems in a linear anisotropic chains. JETP. 34, 892 (1972). [7] S. M. Tashpulatov, Investigation of the energy operator spectrum for a two-magnon system in a one-dimensional anisotropic Heisenberg ferromagnet with second neighbor interactions.Theor. Math. Phys., 107,544(1996). Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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S. M. Tashpulatov

[8] E. Shr¨odinger, On Non-Heisenberg Hamiltonians, Proc.Roy. Irich. Acad. A. 48,39 (1941). [9] R. Micnas, Heisenberg ferromagnet with biquadratic exchange in the Random phase approximation. Phys. stat. Sol. B,66, No. 2, 75 (1974). [10] A. A. Brown, Heisenberg ferromagnet with biquadratic exchange. Phys. Rev. B,4, No.1, 115 (1971). [11] H. H. Chen and P. Levy, Quadrupole phase transitions in magnetic Solids. Phys. Rev. Lett.,27,1383 (1971). [12] D. A. Pink and R. Ballard, Effect of biquadratic exchange.Can. J. Phys.,52, 33(1974). [13] D. A. Pink and P. Tremblay, Effect of biquadratic exchange upon ferromagnetic twomagnon bound states.Can. J. Phys., 50, 1728 (1972). [14] S. M. Tashpulatov, Investigation of the energy operator spectrum of the two-magnon system in a one-dimensional spin s=1 Non-Heisenberg ferromagnet with nearest and second-nearest neighbor interactions. Theor.Math.Phys., 1996. v. 107. No 2. pp. 620-628.

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[15] S. M. Tashpulatov, Two-magnon states in a one-dimensional Non-Heisenberg ferromagnet with spin one and nearest-neighbor coupling. Theor.Math.Phys., 1996. v. 107. No 2. pp. 629-634. [16] S. M. Tashpulatov, Spectra and bound states of the energy operator of two-magnon systems in a Non-Heisenberg ferromagnet with spin one and nearest-neighbor coupling. Theor.Math.Phys., 2000.v.125. No 2. pp. 1539-1551. [17] M. Reed and B. Simon, Methods of Modern mathematical Physics , Vol.1, Functional Analysis, Acad. Press, New York (1972). [18] M. A. Neimark, Normed Rings [in Russian], Nauka, Moskow (1968); English transl., Wolters-Noordhoff, Groningen (1970).

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In: Perspectives in Theoretical Physics Editors: T. F. George, R. R. Letfullin and G. Zhang

ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 5

MODEL-BASED OPTIMIZATION OF THE GROWTH PROCESS PARAMETERS FOR A SILICON SHEET GROWN IN A VACUUM BY THE EDGE-DEFINED FILM-FED GROWTH METHOD L. Braescu¹, A.M. Balint2,*, L. Nánai3 and St. Balint1

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¹ Department of Applied Mathematics, West University of Timisoara, Blv: V. Parvan 4, 300223-Timisoara, Romania 2 Department of Physics, West University of Timisoara, Blv: V. Parvan 4, 300223-Timisoara, Romania 3 Department of Physics, JGYTF, University of Szeged, H-6720 Szeged, Hungary

ABSTRACT It is shown that the choice of the half-thickness w of the die determines a lower limit

α c* of the contact angle α c which decreases when w increases. It is also shown that for a

given w there exists a set of couples (v,T0), where v is the pulling rate and T0 is the melt temperature at the meniscus basis, for which the system of differential equations which governs the evolution of the sheet half-thickness and of the meniscus height has an asymptotically stable solution. This set of (v,T0) couples, representing the stable growth region, including the subset S(w,xf) for which the half-thickness of the sheet is equal to a prescribed value xf ,are found. Finally, those values of w,v,T0 are found for which small uncontrollable variations of v and T0 cause minimal variations of xf. Numerical results are given fo ar silicon sheet of half-thickness xf = 0.5 × 10-2 cm grown in a furnace in which the vertical temperature gradient is k = 3.59 × 102 K/cm.

Keywords: single crystal growth; growth from the melt; edge-defined film-fed growth.

*

E-mail address: [email protected], Corresponding author: A. M. Balint, Tel/Fax: +40 256 494 002,

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1. INTRODUCTION The shape and quality of a sheet grown by the edge-defined film-fed growth (EFG) method are determined by the shape of the meniscus and its behavior during the growth. In the last twenty years, many experimental and theoretical studies have been reported regarding the EFG method [1-42]. This method is performed to achieve sheets with constant half-thickness in a vacuum when v and T0 are constant, and the bottom line of the melt/gas meniscus on the die is fixed to the inner or outer edge of the die. In reality, v and T0 can have small uncontrollable variations around an average value, and the bottom line of the melt/gas meniscus on the die can move. The effect of this kind of variation has been described in the literature [19-20,34,36,40]. In this paper, the mathematical model [32,33] for the EFG method is considered. It is assumed that the bottom line of the melt/gas meniscus on the die is fixed to the outer edge of

[

−3

the die top [38]. For a set of values of w in the range w ∈ 5.1 × 10 ; 6.0 × 10

−3

]cm, the

stable growth regions are determined, and those couples (v,T0) are found for which the grown sheet half-thickness is equal to a desired half-thickness xf. The couple (v,T0) is found for which the amplitude A of the variation of xf due to small uncontrollable variations of v,T0 is a minimum. Finally, those values of w,v,T0 are found for which the amplitude A is a minimum.

2. MATHEMATICAL MODEL

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The system of differential equations which governs the evolution of the half-thickness x = x(t) and the meniscus height h = h(t) is:

dx = − v ⋅ tg [α (x, h; w ) − α1 ] dt 1 dh =v− ⋅ [λ1 ⋅ G1 (x, h; v, T0 ) − λ2 ⋅ G2 (x, h; v, T0 )] dt Λ ⋅ ρ2

(1)

Details are provided in [19,20,32,33,39-40] and Fig. 1. The function α(x,h;w) is obtained from the Young-Laplace equation of the capillary surface in equilibrium in the absence of exterior pressure, i.e., p = 0: 3

2 ⎡ ⎛ ∂z ⎞ 2 ⎤ ∂ 2 z ⎡ ⎛ ∂z ⎞ 2 ⎛ ∂z ⎞ 2 ⎤ 2 ∂z ∂z ∂ 2 z ⎡ ⎛ ∂z ⎞ ⎤ ∂ 2 z ρ1 ⋅ g + ⎢1 + ⎜ ⎟ ⎥ ⋅ 2 = ⋅ z ⋅ ⎢1 + ⎜ ⎟ + ⎜⎜ ⎟⎟ ⎥ ⎢1 + ⎜⎜ ⎟⎟ ⎥ ⋅ 2 − 2 ⋅ ⋅ ⋅ ∂x ∂y ∂x∂y ⎢⎣ ⎝ ∂x ⎠ ⎥⎦ ∂y γ ⎢⎣ ⎝ ∂y ⎠ ⎥⎦ ∂x ⎢⎣ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥⎦

(2) For sheets, the solutions z = z(x) depending only on the coordinate x are searched, and for these functions Eq. (2) becomes

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3

2 d z ρ 1 ⋅ g ⋅ z ⎡ ⎛ dz ⎞ ⎤ 2 1 = ⋅ + ⎢ ⎜ ⎟ ⎥ . γ dx 2 ⎣⎢ ⎝ dx ⎠ ⎦⎥ 2

(3)

The significance of these quantities and their values for Si are given in Table 1 [15, 43]. Table 1. Material parameters for silicon. v

ρ1 ρ2 λ1 λ2

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Λ αe=π/2-α1 γ c1 c2 Bi Tm z Ten0 k Ten(z)=Ten0 - k·z T0 L g w χ1=λ1/(ρ1·c1) χ2=λ2/(ρ2·c2) μ1=(Bi·λ2)/x μ2=(Bi·λ2)/x α(x,h;w) G1(x,h;v,T0) G2(x,h;v,T0)

Nomenclature pulling rate [(10-2 cm)/s] density of the melt [g/(10-2 cm)3] density of the crystal [g/(10-2 cm)3] thermal conductivity coefficient in the melt [W/(10-2 cm)·K] thermal conductivity coefficient in the crystal [W/(10-2 cm)·K] latent heat [J/g] growth angle [rad.] superficial tension of the liquid [N/(10-2 cm)] heat capacity of the melt [J/(g·K)] heat capacity of the crystal [J/(g·K)] Biot number melting point temperature [K] coordinate in the pulling direction environmental temperature at quota z = 0 [K] vertical temperature gradient [K/(10-2 cm)] environmental temperature at quota z [K] melt temperature at the meniscus basis [K] length of the seed [10-2 cm] gravitational acceleration [10-2 cm/s2] half-thickness of the die thermal diffusitivity in the melt thermal diffusitivity in the crystal heat radiation coefficient in the melt heat radiation coefficient in the crystal angle between the Ox axis and the tangent to the meniscus in the point (x,h) temperature gradient in the melt temperature gradient in the crystal

Value 2.5 x 10-8 2.3 x 10-7 0.006 0.002 1,810 0.191 7.20 × 10-5 0.95 0.95 0.567 1,683 1,383 3.59

1 98,100

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L. Braescu, A. M. Balint, L. Nánai et al.

Figure 1. Two-dimensional model for a sheet grown by the EFG method.

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dz = −tgα dx , dα ρ1 ⋅ g ⋅ z 1 =− ⋅ dx γ cos α

(4)

Equation (3) is transformed into the system for which the following initial values are considered:

z ( w) = 0,

α ( w) = α c ;

⎛ π⎞ α c ∈ ⎜ 0, ⎟ . ⎝

2⎠

(5)

The solution of the initial values problem (4)-(5) is denoted by z = z(x;αc, w) and α = α(x;αc,w). For x < w, these functions are positive, and they decrease if x increases and are convex, i.e., z” > 0 and α” > 0 for x < w [39]. For physical reasons, they have to be considered only for x > 0. Therefore, αc has to satisfy the inequality

α c ≥ α c* ,

(6)

where α c* represents the solution of the equation 2 w2 ⋅ ρ1 ⋅ g ⎛ π ⎞ cos α c = ⎜ − αe − αc ⎟ ⋅ . 2 ⋅γ ⎝2 ⎠ sin α c

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(7)

Model-Based Optimization of the Growth Process Parameters…

77

The condition (6) expresses the lower limitation of the contact angle due to the choice of the half-thickness w of the die. In the above conditions, the angle αc is expressed from z = z(x;αc,w) as αc = αc(x,z;w) and is introduced into α = α(x,αc;w) to obtain the function α = α(x, z; w). The temperature gradients G1(x,h;v,T0) and G2(x,h;v,T0) at the interface in the melt and in the crystal side, respectively, are given by [19,20,32,33,39,40]:

G1 (r, h; v,T0 ) =

(

⎡⎛ v ⋅ k ⋅ λ1 ⋅ r ⎞ ⎟⎟ ⋅ − β1 ⋅ eδ1h ⎢⎜⎜ T0 − Ten0 − sh(β1 ⋅ h) ⎣⎝ 2 ⋅ μ1 ⋅ χ1 ⎠ 1

)

⎤ ⎛ v ⋅ k ⋅ λ1 ⋅ r ⎞ ⎟⎟ ⋅ (δ1 ⋅ sh(β1 ⋅ h ) + β1 ⋅ ch (β1 ⋅ h ))⎥ − k + ⎜⎜ Tm − Ten 0 + k ⋅ h − 2 ⋅ μ1 ⋅ χ1 ⎠ ⎝ ⎦ G2 (r, h; v, T0 ) =

(8)

⎡⎛ v ⋅ k ⋅ λ2 ⋅ r ⎞ ⎟ ⋅ ⎢⎜⎜ Tm − Ten 0 + k ⋅ h − 2 ⋅ μ2 ⋅ χ 2 ⎟⎠ sh(β 2 ⋅ L ) ⎣⎝ 1

× (δ 2 ⋅ sh (β 2 ⋅ L ) − β 2 ⋅ ch (β 2 ⋅ L )) −

⎤ v ⋅ k ⋅ λ2 ⋅ r ⋅ β 2 ⋅ e −δ 2 ⋅L ⎥ − k , 2 ⋅ μ2 ⋅ χ 2 ⎦

(9)

where δi and βI are defined for i = 1,2 by

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δi =

v 2χ i

,

2μ i v2 βi = + 2 λi ⋅ x 4χi

.

(10)

The significance and the values of the above quantities are given in the Table 1. In order to grow a sheet with constant half-thickness, the following system has to be satisfied:

α (x, h; w ) − α1 = 0 v−

. 1 ⋅ [λ1 ⋅ G1 ( x, h; v, T0 ) − λ2 ⋅ G2 ( x, h; v, T0 )] = 0 Λ ⋅ ρ2

(11)

For some w,v,T0, the system (11) has a unique solution (x*,h*), which is a steady-state for the system (1) and has to be asymptotically stable. The steady-state (x*,h*) depends on v, T0 and w too. For a given w, the set of the couples (v,T0) for which the steady-state (x*,h*) exists is asymptotically stable and represents the stable growth region. For obtaining the dependences x*=x*(v,T0,w), h*=h*(v,T0,w), several values of w have to be considered over a certain range. For each of these values of w, several couples of (v,T0) belonging to the stable growth region have to be determined, and the corresponding steady-states (x*,h*) have to be computed by solving Eq. (11). After that, by interpolation the dependences

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L. Braescu, A. M. Balint, L. Nánai et al. x*=x*(v,T0,w)

(12)

h*=h*(v,T0,w)

(13)

are found. The dependences (12)-(13) have to be considered only for those values of v,T0,w for which the following inequalities hold:

0 < x * (v , T0 , w) < w

0 < h * (v , T0 , w) < hmax (w)

.

(14)

In the last inequality, hmax (w ) represents the height of the highest meniscus, which can

be obtained using a die of half-thickness w. Mathematically, hmax (w) is the maximum with respect to x of the function z = z (x;α1,w). The function z = z(x;α1,w) is obtained by equating α(x,z;w) with α1 and expressing z from the equation α1 = α(x,z;w). The inequalities (14) define the stable growth region for a given w. For a given half-thickness w of the die, the equation:

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x*(v,T0,w) = xf

(15)

defines those couples (v,T0) which have to be used in order to grow a sheet with constant halfthickness equal to xf . The set of these couples depends on w and xf , and will be denoted by S(w,xf). For a couple (v,T0) which belongs to S(w,xf), using Eq. (15), we compute the halfthickness x*(v+Δv,T0+ΔT,w) corresponding to the small oscillations Δv and ΔT of v and T0, respectively. The amplitude A(v,T0,w) of the variation of xf (grown with the given parameters w, v and T0) due to the oscillations Δv and ΔT is given by A(v,T0,w) = max|x*(v+Δv,T0+ΔT,w) – xf | .

(16)

The system (v,T0,w) for which A(v,T0,w) is minimum defines the optimal growth parameters for the sheet.

3. NUMERICAL RESULTS Computations were performed for silicon sheets of half-thickness xf = 0.5×10-2 cm grown in a furnace having a vertical temperature gradient k = 3.59 × 102 K/cm. For the die half-

[

−3

thickness w, ten different values were considered in the range 5.1 × 10 ;6.0 × 10

−3

]cm.

For each of these values, 594 couples (v,T0) were considered in the ranges

[

]

v ∈ 1.0 × 10 −6 , 0.1 cm/s and T0 ∈[1684, 3000] K, respectively. For these cases, the steady-

states (x*,h*) were obtained. By interpolation the following dependences were found to a precision of 0.9999:

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a1 (w ) + a2 (w ) ⋅ v + a3 (w ) ⋅ T0 + a4 (w ) ⋅ T02 + a5 (w ) ⋅ T03 1 + a6 (w ) ⋅ v + a7 (w ) ⋅ v 2 + a8 (r0 ) ⋅ T0 . b1 (w ) + b2 (w ) ⋅ v + b3 (w) ⋅ v 2 + b4 (w) ⋅ v 3 + b5 (w ) ⋅ T0 h * (v , T0 , w ) = 1 + b6 (w ) ⋅ v + b7 (w ) ⋅ T0 + b8 (w ) ⋅ T02 + b9 (w ) ⋅ T03

x * (v , T0 , w ) =

(17)

The computed coefficients ai and bj (i = 1, 8 , j = 1, 9 ) are given in the Tables 2 and 3, for different values of w. Table 2. Computed coefficients ai for different values of the die half-thickness w. w (cm × 10-2) ai A1 A2 A3 A4 A5 A6 A7 A8

0.51

0.54

0.56

0.58

0.60

0.61339986 4.869732210-5 -0.000555745 1.2271265×10-7 -1.47517×10-11 0.00010067422 -9.10947×10-7 -0.0006463546

0.83589654 0.00014221295 -0.00085699897 2.6453806×10-7 -3.788855×10-11 0.00028689798 -1.7551599×10-6 -0.00063582066

1.075811 0.00014331942 -0.001201109 4.3507163×10-7 -6.669812×10-11 0.00027270035 -8.189981×10-7 -0.0006274255

0.69906464 7.2760456×10-5 -0.00063351106 1.4021627×10-7 -1.686759×10-11 0.0001296212 -7.6627595×10-7 -0.00064593142

0.72328084 0.00019461524 -0.0006544144 1.4446643×10-7 -1.733869×10-11 0.00033816061 -7.606957×10-7 -0.0006453956

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Table 3. The computed coefficients bj for different values of the die half-thickness w. W (cm × 10-2) bj B1 B2 B3 B4 B5 B6 B7 B8 B9

0.51

0.54

0.56

0.58

0.60

0.24220432 -5.9517×10-6 -1.6265×10-6 8.07487×10-8 -0.00014408 -1.5048×10-5 -0.00112878 3.89821×10-7 -5.139×10-11

0.13593779 -6.189823×10-6 5.0475117·10-7 0 -8.08295×10-5 2.3986809×10-5 -0.0013170321 5.7212789×10-7 -8.90281×10-11

0.09054272 -2.07021×10-6 4.465799·10-7 -3.18638×10-8 -5.38253×10-5 2.38732×10-5 -0.001406372 6.59760×10-7 -1.0791×10-10

0.70458797 -9.815285×10-5 3.2758465·10-6 0 -0.0004196687 7.9413325×10-6 -0.0008001163 7.9930041×10-8 0

0.29539406 -1.43797×10-5 1.4219016·10-6 0 -0.000175706 0.0001165453 -0.0011216821 3.830999×10-7 -5.03148×10-11

In order to obtain the stable growth regions, the dependences (17) are introduced into the inequalities (14) in which the maximum values for meniscus height hmax (w ) are given in Table 4. Table 4. The computed values for the maximal meniscus height hmax (w) . w (cm × 10-2)

hmax (w ) (cm × 10 ) -2

0.51 2.574

0.54 2.737

0.56 2.840

0.58 2.939

0.60 3.036

The set of couples (v,T0) for which it is possible to obtain a silicon sheet with constant half-thickness xf = 5.0×10-3 cm is defined by Eq. (15) and is denoted by S(w,xf). The equations of these curves, for different values of w, are: Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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L. Braescu, A. M. Balint, L. Nánai et al.

S (0.51; 0.5) : T0 = 1696.4775 + 0.024654599⋅ v + 0.007934057⋅ v 2 − 0.022384692⋅ v 3 + 0.015831994⋅ v 4 − 0.0063056197⋅ v 5 + 0.0015009136⋅ v 6 − 0.00021810102 ⋅ v 7 + 1.8969827⋅ 10−5 ⋅ v8

(18)

− 9.0748337⋅ 10−7 ⋅ v 9 + 1.8366725⋅ 10−8 ⋅ v10

S(0.54; 0.5) : T0 = 1760.6554+ 0.029350181⋅ v − 0.018497987⋅ v2 − 0.0035538758⋅ v3 + 0.0020321652⋅ v4 − 0.00069422145 ⋅ v5 + 0.0001477775⋅ v6 − 1.9774155×10−5 ⋅ v7 + 1.616405×10−6 ⋅ v8

(19)

− 7.3731575×10−8 ⋅ v9 + 1.4378665×10−9 ⋅ v10

S(0.56;05) : T0 = 1826.2838− 0.20285784⋅ v − 0.012020676⋅ v 2 + 1.777806×10−5 ⋅ v3 −1.9391653×10−5 ⋅ v 4 + 7.1841604×10−6 ⋅ v5 −1.5999059×10−6 ⋅ v6 + 2.1549041×10−7 ⋅ v7 −1.7152722⋅10−8 ⋅ v8

(20)

+ 7.3800754×10−10 ⋅ v 9 −1.312922×10−11 ⋅ v10 S(0.58; 0.5) : T0 = 1912.1 − 0.19550688⋅ v − 0.0095123761⋅ v 2 + 5.1211776×10−5 ⋅ v 3 − 3.1955391×10−5 ⋅ v 4

+ 1.0627968×10−5 ⋅ v5 − 2.1678645×10−6 ⋅ v 6 + 2.7429937×10−7 ⋅ v 7 − 2.0995336×10−8 ⋅ v8 (21) + 8.9020812×10−10 ⋅ v 9 −1.6052608×10−11 ⋅ v10

S(0.60;0.5) : T0 = 2025.4793− 0.63659335⋅ v − 0.0098757415⋅ v2 − 3.3962887×10−5 ⋅ v3 + 2.24288×10−5 ⋅ v4 −1.0652136×10−5 ⋅ v5 + 2.7931814×10−6 ⋅ v6 − 4.3302239×10−7 ⋅ v7 + 3.9537377×10−8 ⋅ v8

(22)

−1.9676542×10−9 ⋅ v9 + 4.1202738×10−11 ⋅ v10

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The stable growth regions, defined by the inequalities (14) and the corresponding S(w, xf) curves are presented in Fig. 2.

Figure 2. Stable growth regions and the sets S(0.51;0.5), S(0.54;0.5), S(0.56;0.5), S(0.58;0.5) and S(0.60;0.5). Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Model-Based M Optimization of the Growthh Process Paraameters…

81

Let us asssume that duuring the groowth process the followinng four kindss of small unncontrollable oscillations off v and T0 can occur: O1: Δv = ± 1.0×10-5 cm, ΔT Δ =±1K O2: Δv = ± 1.0×10-4 cm, ΔT Δ = ± 10 K O3: Δv = ± 2.0×10-4 cm, ΔT Δ = ± 20 K Δ = ± 30 K O4: Δv = ± 1.0×10-5 cm, ΔT o thhe variations of o the amplituddes Ai(v,T0,w),, i = 1, 4 , aloong a curve For these oscillations,

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S((w,xf) for five values of w arre presented inn Fig. 3.

Fiigure 3. Amplittudes Ai(v,T0,w) versus v along S(w,xf) for Oi, i =

1, 4 .

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This figuree shows that, if w = 5.1×100-3 cm, then thhe oscillationss O2, O3, O4 around a any pooint of S(0.51;0.5) lead to the t abandonment of the regiion of stable growth, g as it can c be seen -3 onn Fig. 2. The minimum m valuue of A1(v,T0,w w) is obtainedd for w = 6.0×10-3 cm, v = 6.4103×10 6 -3 -6 cm m/s, T0 = 2.0 02506×103 K: min A1(v,T0,6.0×10 , ) = 1.3×10 1 cm. The minimum m value of -3 -3 A2(v,T0,,w) is obtained o for w = 6.0×10 cm m, v = 6.49455×10 cm/s, T0 = 2.02506×103 K: min A2(v,T0,6.0×10-3 ) = 1.32×100-5 cm. The minimum m valuue of A3(v,T0,w) , is obtaineed for w = -3 6..0×10 cm, v = 6.6879×100-3 cm/s, T0 = 2.02506×103 K: min A3(v,T0,6.0×10-3) = 2.67×10-5 cm m. The minim mum value of A4(v,T0,w) is obtained o for w = 6.0×10-3 cm m, v = 6.7103× ×10-3 cm/s, 3 -3 -5 T0 = 2.02506×1 10 K: min A4(v,T ( 0,6.0×10 ) = 4.05×10 cm. c

Fiigure 4. Amplittudes Ai(v,T0,0.551), Ai (v,T0,0.554), Ai(v,T0,0.566), Ai(v,T0,0.58)), Ai (v,T0,0.60)), i = 1,2,3,4 vss. v along S(0.51;0.5), S(0.54;00.5), S(0.56;0.5)), S(0.58;0.5) annd S(0.60;0.5), respectively. r

The connecctions betweeen the amplituudes of the cryystal half-thickness variatioons and the osscillation leveel are represented in Fig. 4. This figgure shows thhat if the level of the teemperature oscillation increeases, then thee amplitude of o the crystal half-thickness h s variations inncreases too.

CON NCLUSIONS S Inn the case of a silicon sheett of half-thickkness, 5.0×10-33 cm, grown in a vacuum inn a furnace haaving a vertical temperaturee gradient k = 3.59 × 102 K/cm K for the oscillations o O1, 1 O2, O3, O4 iddentified in thee above sectioon, we have obbtained: 1. If the half-thickness of the die increases, theen the amplituude of the crrystal halfthickneess variations due to the aboove oscillationns decreases.

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2. If the level of the temperature oscillation increases, then the amplitude of the crystal half-thick-ness variation due to the above oscillations increases too. 3. Those values w and T0 for which the amplitude of the crystal half-thickness variation due to the above oscillations is minimum are independent on the oscillations: w = 6.0×10-3 cm and T0=2.025×103 K. 4. In the given model [32,33] it is possible to predict those values of w,v,T0 for which small uncontrollable oscillations of v and T0 cause minimal variations of the desired crystal half-thickness xf.

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REFERENCES [1] C. W Lan, M. C. Liang, C. R. Song and C. K. Fu, Journal of Crystal Growth 166 (1996) 485. [2] V. A. Borodin, V. V. Sidorov, S. N. Rossolenko, T. A. Steriopolo and T. N. Yalovets, Journal of Crystal Growth 198/199 (1999) 201. [3] A. Roy, H. Zhang, V. Prasad, B. Mackintosh, M. Ouellette and J. P. Kalejs, Journal of Crystal Growth 230 (2001) 224. [4] T. Surek, S. R. Coriell and B. Chalmers, Journal of Crystal Growth 50 (1980) 21. [5] V. A. Tatarchenko and E. A. Brener, Journal of Crystal Growth 50 (1980) 33. [6] G. I. Babkin, E. A. Brener and V. A. Tatarchenko, Journal of Crystal Growth 50 (1980) 45. [7] S. N. Rossolenko and A. V. Zhdanov, Journal of Crystal Growth 104 (1990) 8. [8] I. S. Pet’kov and B. S. Red’kin, Journal of Crystal Growth 104 (1990) 20. [9] V. A. Tatarchenko, J. Ph. Nabot, T. Duffar, E. V. Tatarchenko and B. Roux, Journal of Crystal Growth 148 (1995) 415. [10] V. A. Tatarchenko, V. S. Uspenski, E. V. Tatarchenko, J. Ph. Nabot, T. Duffar, B. Roux, Journal of Crystal Growth, 180 (1997) 615. [11] S. Despreaux, P. Witomski and T. Duffar, Journal of Crystal Growth 209 (2000) 983. [12] V. A. Tatarchenko, V. S. Uspenski, E. V. Tatarchenko and B. Roux, Journal of Crystal Growth 220 (2000) 301. [13] L. L. Kuanhykov and P. I. Antonov, Journal of Crystal Growth 222 (2001) 852. [14] J. P. Kalejs, Journal of Crystal Growth 61 (1983) 473. [15] H. M. Ettouney, R. A. Brown and J. P. Kalejs, Journal of Crystal Growth, 62 (1983) 230. [16] H. M. Ettouney, R. A. Brown and J. P. Kalejs, Journal of Applied Physics. 55 (1984) 4384. [17] I. Nicoara, D. Nicoara and D. Vizman, Journal of Crystal Growth 128 (1993) 152. [18] I. Nicoara, D. Vizman and J. Friedrich, Journal of Crystal Growth 218 (2000) 74. [19] L. Braescu, A. M. Balint, I. Jadaneantu and St. Balint, Journal of Crystal Growth, 240 (2002) 305. [20] L. Braescu, A. M. Balint, Z. Schlett and St. Balint, Journal of Crystal Growth 241 (2002) 374. [21] H. Machida, K. Hoshikawa and T. Fukuda, Journal of Crystal Growth 128 (1993) 829.

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L. Braescu, A. M. Balint, L. Nánai et al.

[22] P. I. Antonov, S. I. Bakholdin, E. A. Tropp and V. S. Yuferev, Journal of Crystal Growth 50 (1980) 62. [23] V. S. Yuferev and M. G. Vasil’ev, Journal of Crystal Growth 82 (1987) 31. [24] V. S. Yuferev, M. G. Vasil’ev, L. A. Stefanova and E. N. Kolesnikova, Journal of Crystal Growth 128 (1993) 144. [25] M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993). [26] J. L. Duranceanu and R. A. Brown, Journal of Crystal Growth 75 (1986) 367. [27] T. Tsukada, H. Hozawa and N. Imaishi, J. Chem. Eng. Japan, 27 (1994) 25. [28] M. Kobayashi, T. Hagino, T. Tsukada and M. Hozawa, Journal of Crystal Growth 235 (2002) 258. [29] C. M. Spuckler and R. Siegel, J. Thermophysics Heat Transfer 6 (1992) 596. [30] C. M. Spuckler and R. Siegel, Journal of Thermophysics Heat Transfer 8 (1994) 193. [31] H. Machida, K. Hoshikawa and T. Fukuda, Japanese Journal of Applied Physics 31 (1992) L974 Part 2. [32] V. A. Tatarchenko, Shaped Crystal Growth (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993), pp. 19- 145. [33] V. A. Tatarchenko, in Handbook of Crystal Growth, edited by D. T. J. Hurle (NorthHolland, Amsterdam, 1994), pp. 1017-1048. [34] M. Kobayashi and T. Tsukada, M. Hozawa, Journal of Crystal Growth 249 (2003) 230. [35] Q. Xiao and J. Derby, Journal of Crystal Growth 139 (1994) 147. [36] L. Braescu, A. M. Balint and St. Balint, Journal of Crystal Growth 259 (2003) 121. [37] A. M. Balint, St. Balint and S. Birauas, Bulletin for Applied Mathematics (Technical University of Budapest) 1103 (1995) 355. [38] J. P. Kalejs, Journal of Crystal Growth 230 (2001) 10. [39] S. Balint, Z. Schlett, A. M. Balint and I. Jadaneantu, Crystal Properties and Preparation 36-38, (1991), 586. [40] L. Braescu, A.M. Balint and S. Balint, Journal of Crystal Growth 268 (2004) 284. [41] L. Braescu, A.M. Balint and S. Balint, Journal of Crystal Growth 269 (2004) 617. [42] L. Braescu, PhD. Thesis (West University of Timisoara, Romania, 2001). [43] W. A. Tiller, The Science of Crystallization: Macroscopic Phenomena and Defect Generation (Cambridge University Press, Cambridge, 1991), pp. 19-93.

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Chapter 6

F IBRE B UNDLE F ORMULATION OF R ELATIVISTIC Q UANTUM M ECHANICS II: C OVARIANT A PPROACH Bozhidar Z. Iliev∗ Laboratory of Mathematical Modeling in Physics, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chauss´ee 72, 1784 Sofia, Bulgaria

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Abstract A fibre bundle formulation of the mathematical base of relativistic quantum mechanics is proposed. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.

PACS 02.40.Ma, 02.40.Yy 02.90.+p, 03.65.Pm. Keywords: Relativistic quantum mechanics, Fibre bundles, Geometrization of relativistic quantum mechanics, Relativistic wave equations, Dirac equation, Klein-Gordon equation.

∗

E-mail address: [email protected]; URL: http://theo.inrne.bas.bg/∼bozho/

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1.

Bozhidar Z. Iliev

Introduction

This paper is a second part of our investigation devoted to the fibre bundle description of relativistic quantum mechanics. It is a straightforward continuation of [1]. The developed in [1] bundle formalism for relativistic quantum wave equations has the deficiency that it is not explicitly covariant; so it is not in harmony with the relativistic theory it represents. This is a consequence of the direct applications of the bundle methods developed for the nonrelativistic region, where they work well enough, to the relativistic one. The present paper is intended to mend this ‘defect’. Here we develop an appropriate covariant bundle description of relativistic quantum mechanics which corresponds to the character of this theory. The organization of the material is the following. Sect. 2 contains a covariant application of the ideas of the bundle description of nonrelativistic quantum mechanics (see, e.g. [1, sect. 2] or [2, 3]) to Dirac equation. The bundle, where Dirac particles ‘live’, is a vector bundle over space-time with the space of 4-spinors as a fibre; so here we work again with the 4-spinor bundle of [1, sect. 4], but now the evolution of a Dirac particle is described via a geometric transport which is a linear transport along the identity map of space-time . The state of a Dirac particle is represented by a section (not along paths!) of the 4-spinor bundle and is (linearly) transported by means of the transport mentioned. The Dirac equation itself is transformed into a covariant Schr¨odinger-like equation. In Sect. 3, we apply the covariant bundle approach to Klein-Gordon equation. For this purpose we present a 5-dimensional representation of this equation as a first-order Diraclike equation to which the theory of Sect. 2 can be transferred mutatis mutandis. The goal of Sect. 4 is to be revealed some connections between the retarded Green functions (≡propagators) of the relativistic wave equations and the corresponding to them evolution operators and transports. Generally speaking, the evolution operators (resp. transports) admit representation as integral operators, the kernel of which is connected in a simple manner with the retarded Green function (resp. Green morphism of a bundle). Subsect. 4.1 contains a brief general consideration of the Green functions and their connection with the evolution transports, if any. In Subsect. 4.2, 4.3, and 4.4 we derive the relations mentions for Schr¨odinger, Dirac, and Klein-Gordon equations, respectively. Sect. 5 closes the paper with a brief summary of the main ideas underlying the bundle description of relativistic quantum mechanics. Appendix A contains some mathematical results concerning the theory of (linear) transports along maps required for the present investigation. In Appendix B are given certain formulae concerning matrix operators, i.e. matrices with operator entries, which arise naturally in relativistic quantum mechanics. The notation of the present work is the the same as the one in [1] and we are not going to recall it here. The references to sections, equations, footnotes etc. from [1] are obtained from their sequential numbers in [1] by adding in front of them the Roman one (I) and a dot as a separator. For instance, Sect. I.4 and (I.5.2) mean respectively section 4 and equation (5.2) (equation 2 in Sect. 5) of [1]. Below, for reference purposes, we present a list of some essential equations of [1] which

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are used in this paper. Following the just given convention, we retain their original reference numbers.

Ψγ : t → Ψγ (t) = Uγ (t, s) =

−1 lγ(t) −1 lγ(t)

ψ(t) = U (t, t0)ψ(t0),
−1 ψ(t) , Hγ : t → Hγ (t) = lγ(t) ◦ H ◦ lγ(t),

(I.2.3)

◦ U (t, s) ◦ lγ(s) : Fγ(s) → Fγ(t),

(I.2.4)

s, t ∈ J,

−1 Aγ (t) = lγ(t) ◦ A(t) ◦ lγ(t) : Fγ(t) → Fγ(t).

2.

(I.2.2)

(I.2.11)

Dirac equation

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The covariant Dirac equation [4, sect. 2.1.2], [5, chapter XX, § 8] for a (spin 12 ) particle with mass m and electric charge e in an external electromagnetic field with 4-potential Aµ is e D / := γ µDµ , Dµ := ∂µ − Aµ . (2.1) (i~D / − mc114)ψ = 0, i~c Here i ∈ C is the imaginary unit, ~ is the Planck constant (divided by 2π), 114 = diag(1, 1, 1, 1) is the 4 × 4 unit matrix, ψ := (ψ 0, ψ 1, ψ 2, ψ 3) is (the matrix of the components of) a 4-spinor, γ µ, µ = 0, 1, 2, 3, are the well known Dirac γ-matrices [6, 5, 4], and c is the velocity of light in vacuum. Since (2.1) is a first order partial differential equation on the Minkowski 4-dimensional spacetime, it does not admit an evolution operator with respect to the spacetime. More precisely, if x1 , x2 ∈ M , M being the spacetime, i.e. the Minkowski space M 4, then there does not exist a 4 × 4 matrix operator (see Appendix B) U (x2, x1) such that ψ(x2) = U (x2, x1)ψ(x1). (i~D /x − mc114 )U (x, x0) = 0,

U (x0 , x0) = idF ,

x, x0 ∈ M

e A / with ∂/ := γ µ∂/∂xµ and A / := γ µAµ , F is the space of 4-spinors, and where D / = ∂/ − i~c idX is the identity map of a set X. For this reason, the methods of [1] cannot be applied directly to the general 4dimensional spacetime descriptuion of the Dirac equation. An altternative approach to the problem is presented below. Suppose (F, π, M ) is a vector bundle with (total) bundle space F , projection π : F → M , fibre F , and isomorphic fibres Fx := π −1 (x), x ∈ M . There exist linear isomorphisms lx : Fx → F which we assume to be diffeomorphisms; so Fx = lx−1 (F ) are 4-dimensional vector spaces. Recall, here and below we identify the fibre F with the vector space of 4-spinors. To a state vector (spinor) ψ(x0) at a fixed point x0 , we assign a C 1 section1 Ψx0 of (F, π, M ), i.e. Ψx0 ∈ Sec1 (F, π, M ), by (cf. (I.2.3))
(2.2) Ψx0 (x) := lx−1 ψ(x0) ∈ Fx := π −1 (x), x ∈ M. 1

In contrast to the time-dependent approach [1] and nonrelativistic case [7] now Ψx0 is simply a section, not section along paths [7]. Physically this corresponds to the fact that quantum objects do not have world lines (trajectories) in a classical sense [8]. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Generally Ψx0 (x) depends on the choice of the point x0 ∈ M . Since in (F, π, M ) the state of a Dirac particle is described by Ψx0 , we call it state section; resp. (F, π, M ) is the 4-spinor bundle.2 The description of Dirac particle via Ψx0 will be called bundle description. If it is known, the conventional spinor description is achieved by the spinor
ψ(x0) := lx Ψx0 (x) ∈ F . (2.3) Evidently, we have Ψx0 (x2 ) = L(x2, x1)Ψx0 (x1),

x2 , x1 ∈ M.

(2.4)

x, y ∈ M.

(2.5)

with (cf. (I.2.4)) L(y, x) := ly−1 ◦ lx : Fx → Fy ,

Obviously, L is a linear 4 × 4 matrix operator satisfying the equations L(x3, x1) = L(x3, x2) ◦ L(x2, x1), L(x, x) = idFx ,

x1 , x2, x3 ∈ M,

x ∈ M.

(2.6) (2.7)

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idM is a linear Consequently, by definition A.1, the map L : (y, x) → L(y, x) = Kx→y transport along the identity map idM of M in the bundle (F, π, M ). Alternatively, as it is γ mentioned in Appendix A, this means that Lγ : (t, s) → Ls→t := L(γ(t), γ(s)), s, t ∈ J is a flat linear transport along γ : J → M in (F, π, M ). (Besides, L(y, x) is a Hermitian and unitary transport — see [7].) Equation (2.4) simply means that Ψx0 is L-transported (along idM ) section of (F, π, M ) (cf. [10, definition 5.1]). Writing (A.11) for the transport L and applying the result to a state section given by (2.2), one can prove that (2.4) is equivalent to

Dµ Ψ = 0,

µ = 0, 1, 2, 3

(2.8)

where, for brevity, we have put Dµ := DxidµM which is the µ-th partial (section-)derivation along the identity map (of the spacetime) assigned to the transport L. Now we shall introduce local bases and take a local view of the above-described material. Let {fµ (x)} be a basis in F and {eµ (x)} be a basis in Fx , x ∈ M . The matrices corresponding to vectors and/or linear maps (operators) in these fields of bases will be denoted
> by the same symbol but in boldface, for instance: ψ := ψ 0, ψ 1, ψ 2, ψ 3
and  (kernel)  µ µ lx (y) :=
µ (lx ) ν (y) are defined, respectively, by ψ(x) =: ψ (x)fµ(x) and lx eν (x) =: lx(y) ν fµ (y). We put lx := lx (x); in fact this will be the only case when the matrix of lx will be required as we want the ‘physics in Fx ’ to correspond to that of F at x ∈ M .
−1 A very convenient choice is to put eµ (x) = lx fµ (x) ; so then lx = 114 := δνµ = diag(1, 1, 1, 1) is the 4 × 4 unit matrix. The matrix
elements of the mapping L(y, x) are defined via the equation L(y, x) eµ (x) =: Lλµ (y, x)eλ(y) and, due to (2.5), we have L(y, x) = l−1 y · lx 2

The 4-spinor bundle is a known object — see, e.g., [9]

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(2.9)

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.

which is generically a matrix operator (see Appendix B). According to (A.13) the coefficients of the transport L form four matrix operators  λ 3 ∂L(x, y) ∂l(x) = l−1 (x) (2.10) µΓ(x) := µ Γ ν (x) λ,ν=0 := ∂y µ y=x ∂xµ where (2.9) was applied (see also theorem A.1). Applying (A.5) and (2.10), we find ∂L(y, x) = − µΓ(y) L(y, x), ∂y µ

∂L(y, x) = L(y, x) µΓ(x), ∂xµ

(2.11)

where denotes the introduced by (B.2) multiplication of matrix operators. Therefore /(y)L(y, x), ∂/y L(y, x) = −Γ

Γ /(x) := γ µ · µΓ(x).

(2.12)

Similarly to the nonrelativistic case [11], to any operator A : F → F we assign a bundle morphism A : F → F by (2.13) Ax := A|Fx := lx−1 ◦ A ◦ lx . Defining G µ (x) := lx−1 ◦ γ µ ◦ lx,

dµ := dµ |x := lx−1 ◦ ∂µ ◦ lx,

∂µ :=

∂ ∂xµ

(2.14)

and using the matrices G µ (x) and Eµ (x) given via (B.12), we get
dµ := 114 ∂µ + l−1 x ∂µ lx + Eµ (x)lx .

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µ G µ (x) = l−1 x G (x)lx ,

(2.15)

(Here and below, for the sake of shortness, we sometimes omit the argument x.) The anticommutation relations G µ G ν + G ν G µ = 2η µν idF , G µ G ν + G ν G µ = G µ G ν + G ν G µ = 2η µν 114


= γ µγ ν + γ ν γ µ ,

(2.16)

where [η µν ] = diag(1, −1, −1, −1) = [ηµν ] is the Minkowski metric tensor, can be verified by means of (2.14), (2.15), (B.10), and the well know analogous relation for the γ-matrices (see, e.g. [6, chapter 2, equation (2.5)]). For brevity, if aµ : F → F are morphisms, sums like G µ ◦ aµ will be denoted by / := γ µaµ ). ‘backslashing’ the kernel letter, a \ := G µ ◦ aµ (cf. the ‘slashed’ notation a µ Similarly, we put a \ := G (x)aµ for aµ ∈ GL(4, C). It is almost evident (see (2.14)) that \ = G µ (x) ◦ dµ : the morphism corresponding to ∂/ := γ µ ∂µ is d \ = lx−1 ◦ ∂/ ◦ lx . d

(2.17)

Now we can write the Dirac equation (2.1) in a bundle form. First of all, we rewrite (2.1) as e Dx := mc114 + A /(x), c Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, i~∂/x ψ(x) = Dx ψ(x),

(2.18)

90

Bozhidar Z. Iliev

the index x meaning that the corresponding operators act with respect to the variable x ∈ M . This is the covariant Schr¨odinger-like form of Dirac equation; ∂/x is the analogue of the time derivation d/dt and Dx corresponds to the Hamiltonian H. We call D the Dirac function, or simply, Diracian of a particle described by Dirac equation. Substituting (2.3) into (2.18), acting on the result from the left by ly−1, and using (2.14), we find the bundle form of (2.18) as i~d \x|y Ψx (y) = Dx |y Ψx (y)

(2.19)

with y ∈ M , d \x |y := G(y)dµ |x|y , dµ |x|y := ly−1 ◦ ∂x∂µ ◦ ly , and Dx ∈ Mor(F, π, M ) being the (Dirac) bundle morphism assigned to the Diracian. We call it the bundle Diracian. According to (2.13) it is defined by e Dx|y := Dx |Fy = ly−1 ◦ Dx ◦ ly = mc idFy + A \x|y , c

(2.20)

where Aµ |x = Aµ |x idF ∈ Mor(F, π, M ), Aµ |x |y = ly−1 ◦ (Aµ |x idF ) ◦ ly = Aµ |x idFy are the components of the bundle electromagnetic potential and A \x|y = G µ (y) ◦ Aµ |x|y = G µ (y) ◦ (ly−1 ◦ Aµ idF ◦ ly ) = ly−1 ◦ (A /|x idF ) ◦ ly = A \x |y . (2.21)

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3.

Other relativistic wave equations

The relativistic-covariant Klein-Gordon equation for a (spinless) particle of mass m and electric charge e in a presence of (external) electromagnetic field with 4-potential Aµ is [5, chapter XX, § 5, equation (30 0 )] 

Dµ Dµ +

m2c2  φ = 0, ~2

Dµ = ηµν Dν := ∂µ −

e Aµ . i~c

(3.1)

Since this is a second-order partial differential equation, it does not directly admit an evolution operator and adequate bundle formulation and interpretation. To obtain such a formulation, we have to rewrite (3.1) as a first-order (system of) partial differential equation(s) (cf. Sect. I.5). Perhaps the best way to do this is to replace φ with a 5×1 matrix ϕ = (ϕ0, . . . , ϕ4)> and
i to introduce 5 × 5 Γ-matrices Γµ , µ = 0, 1, 2, 3 with components Γµ j , i, j = 0, 1, 2, 3, 4 such that (cf. [12, chapter I, equations (4.38) and (4.37)]):   i~D0φ   i~D1φ for (i, j) = (µ, 4) 1  
i µ  φ = (3.2) i~D , Γ ϕ = a ηµµ for (i, j) = (4, µ) 2   j   i~D3φ 0 otherwise mcφ where the complex constant a 6= 0 is insignificant for us and can be (partially) fixed by an appropriate normalization of ϕ.

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Then a simple checking shows that (3.1) is equivalent to (cf. (2.1)) (i~ΓµDµ − mc115)ϕ = 0

(3.3)

with 115 = diag(1, 1, 1, 1, 1) being the 5 × 5 unit matrix, or (cf. (2.18)) i~Γµ ∂µ ϕ = Kϕ,

K := mc115 +

e µ Γ Aµ . c

(3.4)

Now it is evident that mutatis mutandis, taking Γµ for γ µ , ϕ for ψ, etc., the (bundle) machinery developed in Sect. 2 for the Dirac equation can be applied to the Klein-Gordon equation in the form (3.3). Since the transferring of the results obtained in Sect. 2 for Dirac equation to Klein-Gordon one is absolutely trivial, 3 we are not going to present here the bundle description of the latter equation. Since the relativistic wave equations for particles with spin greater than 1/2 are versions or combinations of Dirac and Klein-Gordon equations [13, 6, 5, 12], for them is mutatis mutandis applicable the bundle approach developed in Sect. 2 for Dirac equation or/and its version for Klein-Gordon one pointed above.

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4.

Propagators and evolution transports or operators

The propagators, called also propagator functions or Green functions, are solutions of the wave equations with point-like unit source and satisfy appropriate (homogeneous) boundary conditions corresponding to a concrete problem under exploration [6, 4]. Undoubtedly these functions play an important rˆole in the mathematical apparatus of (relativistic) quantum mechanics and its physical interpretation [6, 4]. For that reason it is essential to be investigated the connection between propagators and evolution operators or/and transports. As we shall see below, the latter can be represented as integral operators whose kernel is connected in a simple way with the corresponding propagator.

4.1.

Green functions (review)

Generally [14, article “Green function”] the Green function g(x0, x) of a linear differential operator L (or of the equation Lu(x) = f (x)) is the kernel of the integral operator inverse to L. As the kernel of the unit operator is the Dirac delta-function δ 4(x0 − x), the Green function is a fundamental solution of the non-homogeneous equation Lu(x) = f (x),

(4.1)

i.e. treated as a generalized function g(x0, x) is a solution of Lx0 g(x0, x) = δ 4 (x0 − x). Given a Geen function g(x0, x), the solution of (4.1) is Z 0 u(x ) = g(x0, x)f (x) d4x. 3

Both coincide up to notation or a meaning of the corresponding symbols.

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(4.2)

(4.3)

92

Bozhidar Z. Iliev

A concrete Green function g(x0, x) for L (or (4.1)) satisfies, besides (4.2), certain (homogeneous) boundary conditions on x0 with fixed x, i.e. it is the solution of a fixed boundary-value problem for equation (4.2). Hence, if gf (x0 , x) is some fundamental solution, then g(x0, x) = gf (x0, x) + g0(x0, x) (4.4) where g0(x0 , x) is a solution of the homogeneous equation Lx0 g0(x0, x) = 0 chosen such that g(x0, x) satisfies the required boundary conditions. Suppose g(x0, x) is a Green function of L for some boundary-value (or initial-value) problem. Then, using (4.2), one can verify that Z  Lx0 g(x0, x)u(x) d3x = δ(ct0 − ct)u(x0) (4.5) where x = (ct, x) and x0 = (ct0 , x0). Therefore the solution of the problem Lx u(x) = 0,

u(ct0 , x) = u0 (x)

(4.6)

is u(x) =

Z


g x, (ct0, x0) u0 (x0 ) d3x0 ,

x0 = (ct0, x0 )

for t 6= t0 .

(4.7)

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From here we can make the conclusion that, if (4.6) admits an evolution operator U such that (cf. (I.2.2)) (4.8) u(x) ≡ u(ct, x) = U (t, t0)u(ct0, x), then the r.h.s. of (4.7) realizes U as an integral operator with a kernel equal to the Green function g. Since all (relativistic or not) wave equations are versions of (4.6), the corresponding evolution operators, if any, and Green functions (propagators) are connected as just described. Moreover, if some wave equation does not admit (directly) evolution operator, e.g. if it is of order greater than one, then we can define it as the corresponding version of the integral operator in the r.h.s. of (4.7). In this way is established a one-to-one onto correspondence between the evolution operators and Green functions for any particular problem like (4.6). And a last general remark. The so-called S-matrix finds a lot of applications in quantum theory [6, 4, 12]. By definition S is an operator transforming the system’s state vector ψ(−∞, x) before scattering (reaction) into the one ψ(+∞, x) after it: lim ψ(ct, x) =: S lim ψ(ct, x).

t→+∞

t→−∞

So, e.g. when (4.8) takes place, we have S=

lim U (t+ , t− ) =: U (+∞, −∞).

t± →±∞

(4.9)

Thus the above-mentioned connection between evolution operators and Green functions can be used for expressing the S-matrix in terms of propagators. Such kind of formulae are often used in relativistic quantum mechanics [6]. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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93

Nonrelativistic case (Schr¨odinger equation)4

The (retarded) Green function g(x0, x), x0, x ∈ M , for the Schr¨odinger equation (I.2.1) is defined as the solution of the boundary-value problem [6, § 22]  ∂  i~ 0 − H(x0 ) g(x0, x) = δ 4(x0 − x), ∂t for t0 < t g(x0, x) = 0

(4.10) (4.11)

where x0 = (ct0 , x0 ) x = (ct, x), H(x) is system’s Hamiltonian, and δ 4 (x0 − x) is the 4-dimensional (Dirac) δ-function. Given g, the solution ψ(x0) (for t0 > t) of (I.2.1) is5 Z 0 0 θ(t − t)ψ(x ) = i~ d3 xg(x0, x)ψ(x) (4.12)

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where the θ-function θ(s), s ∈ R, is defined by θ(s) = 1 for s > 0 and θ(s) = 0 for s < 0. Combining (I.2.2) and (4.12), we find the basic connection between the evolution operator U and Green function of Schr¨odinger equation: Z 

(4.13) θ(t0 − t) U (t0 , t) ψ(ct, x0 ) = i~ d3 xg (ct0, x0 ), (ct, x) ψ(ct, x). Actually this formula, if g is known, determines U (t0 , t) for all t0 and t, not only for t0 > t, as U (t0 , t) = U −1 (t, t0) and U (t, t) = idF with F being the system’s Hilbert space (see (I.2.2) or [7, sect. 2]). Consequently the evolution operator for the Schr¨odinger equation can be represented as an integral operator whose kernel, up to the constant i~, is exactly the (retarded) Green function for it . To write the bundle version of (4.13), we introduce the Green operator which is simply a multiplication with the Green function: G(x0, x) := g(x0, x) idF : F → F .

(4.14)

The corresponding to it Green morphism G is given via (see (I.2.11) and cf. (I.2.4)) −1 −1 0 0 Gγ (x0, x) := lγ(t 0 ) ◦ G(x , x) ◦ lγ(t) = g(x , x)lγ(t0) ◦ lγ(t) : Fγ(t) → Fγ(t0 ) .

(4.15)

−1 Applying lγ(t 0 ) on (4.13) from the left and using (I.2.3) and (I.2.4), we obtain


 θ(t0 − t) Uγ (t0 , t) Ψγ (ct, x0) = i~

Z


d3 xGγ (ct0 , x0 ), (ct, x) Ψγ (ct, x).

4

(4.16)

The discussion of the Green’s functions (retarded Green’s equations) presented below is a Wick-rotated variant of heat kernels (equations). Recall, the change t 7→ +it or t 7→ −it, with t being the time coordinate, is called Wick rotation and it transforms the Schro¨ dinger equation (e.g. for a free particle) into the heat equation. 5 One should not confuse the notation ψ(x) = ψ(ct, x), x = (ct, x) of this section and ψ(t) from Sect. I.2. The latter is the wavefunction at a moment t and the
former is its value at the spacetime point x = (ct, x). −1 Analogously, Ψγ (x) ≡ Ψγ (ct, x) := lγ(t) ψ(ct, x) should not be confused with Ψγ (t) from Sect. I.2. A notation like ψ(t) and Ψγ (t) will be used if the spacial parts of the arguments are inessential, as in Sect. I.2, and there is no risk of ambiguities.

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Therefore for the Schr¨odinger equation the evolution transport U can be represented as an integral operator with kernel equal to i~ times the Green morphism G. Taking as a starting point (4.13) and (4.16), we can obtain different representations for the evolution operator and transport by applying concrete formulae for the Green function. For example, if a complete set {ψa(x)} of orthonormal solutions of Schr¨odinger equation satisfying the completeness condition 6 X ψa (ct, x0 )ψa∗(ct, x) = δ 3 (x0 − x) a

is know, then [6, § 22] g(x0, x) =

X 1 0 θ(t − t) ψa(x0)ψa∗(x) i~ a

which, when substituted into (4.13), implies 0

0

U (t , t)ψ(ct, x ) =

X

0

ψa(x )

Z

d3 xψa∗(ct, x)ψ(ct, x).

(4.17)

a

Note, the integral in this equation is equal to the a-th coefficient of the expansion of ψ over {ψa}.

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4.3.

Dirac equation

Since the Dirac equation (2.1) is a first-order linear partial differential equation, it admits both evolution operator and Green function(s) (propagator(s)). From a generic view-point, the only difference from the Schr¨odinger equation is that (2.1) is a matrix equation; so the corresponding Green functions are actually Green matrices, i.e. Green matrix-valued functions. Otherwise the results of Subsect. 4.2 are mutatis mutandis applicable to the theory of Dirac equation. The (retarded) Green matrix (function) or propagator for Dirac equation (2.1) is a 4 × 4 matrix-valued function g(x0, x) depending on two arguments x0 , x ∈ M and such that [4, sect. 2.5.1 and 2.5.2] (i~D /x0 − mc114 )g(x0, x) = δ(x0 − x) 0

g(x , x) = 0

(4.18) 0

for t < t .

(4.19)

For a free Dirac particle, i.e. for D / = ∂/ or Aµ = 0, the explicit expression g0 (x0, x) 0 for g(x , x) is derived in [4, sect. 2.5.1], where the notation K instead of g0 is used. In an external electromagnetic field Aµ the Green matrix g is a solution of the integral equation 7 Z e 0 0 g(x , x) = g0(x , x) + d4 y g0 (x0, y) A /(y)g(y, x) (4.20) c P Here and below the symbol a denotes a sum and/or integral over the discrete and/or continuous spectrum. The asterisk ∗ means complex conjugation. 7 The derivation of (4.20) is the same as for the Feynman propagators SF and SA given in [4, sect. 2.5.2]. The propagators SF and SA correspond to g0 and g respectively, but satisfy other boundary conditions [4, 6]. 6

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which includes the corresponding boundary condition. 8 The iteration of this equation results in the perturbation series for g (cf. [4, sect. 2.5.2]). If the (retarded) Green matrix g(x0, x) is known, the solution ψ(x0) of Dirac equation (for t0 > t) is Z θ(t0 − t)ψ(x0) = i~

d3 x g(x0, x)γ 0ψ(x).

(4.21)

Hence, denoting by U the non-relativistic (see Sect. I.4) Dirac evolution operator, from the equations (I.2.2), and (4.21), we find: Z 

(4.22) θ(t0 − t) U (t0, t) ψ(ct, x0) = i~ d3 x g (ct0, x0 ), (ct, x) γ 0ψ(ct, x). So, the evolution operator admits an integral representation whose kernel, up to a right multiplication with i~γ 0, is equal to the (retarded) Green function for Dirac equation. Similarly to (4.16), now the bundle version of (4.22) is Z 
 0 0 0 θ(t − t) U (t , t) Ψγ (ct, x ) = i~ d3 x Gγ (x0 , x)G0(γ(t))Ψγ (ct, x), (4.23) where G0 (x) is defined by (2.14) with µ = 0 and (cf. (4.15)) −1 0 Gγ (x0, x) := lγ(t 0 ) ◦ G(x , x) ◦ lγ(t) ,

0 G(x0, x) := lx−1 0 ◦ G(x , x) ◦ lx

(4.24)

with (cf. . (4.14))

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G(x0, x) = g(x0, x) idF .

(4.25)

(Here F is the space of 4-spinors.) Analogously to the above results, one can obtained similar ones for other propagators, e.g. for the Feynman one [6, 4], but we are not going to do this here as it is a trivial variant of the procedure described.

4.4.

Klein-Gordon equation

The (retarded) Green function g(x0, x) for the Klein-Gordon equation (3.1) is a solution to the boundary-value problem [4, sect. 1.3.1] 

Dµ Dµ +

m2c2  0 g(x0, x) = δ 4 (x0 − x), ~2 x g(x0, x) = 0 for t0 < t.

(4.26) (4.27)

For a free particle its explicit form can be found in [4, sect. 1.3.1]. A simple verification proves that, if g(x0, x) is known, the solution φ of (3.1) (for t0 > t) is given by (x0 = ct) Z h ∂g(x0, x)  ∂φ(x) i e 0 0 0 0 2 . (4.28) φ(x) + g(x , x) − A (x)φ(x) θ(t − t)φ(x ) = d3 x ∂x0 ∂x0 i~c 8

Due to (4.4), the integral equation (4.20) is valid for any Green function (matrix) of the Dirac equation (2.1).

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Bozhidar Z. Iliev Introducing the matrices   φ(x) , ψ(x) := ∂0 |xφ(x)

0

g(x , x) :=



 D0 |xg(x0, x) , 2g(x0, x)

(4.29)

we can rewrite (4.28) as 0

0

θ(t − t)φ(x ) =

Z

d3 x g> (x0 , x) · ψ(x)

(4.30)

where the dot (·) denotes matrix multiplication. An important observation is that for ψ the Klein-Gordon equation transforms into first-order Schr¨odinger-type equation (see Sect. I.5) with Hamiltonian K−GcH given by (I.5.3) in which id... is replaced by c id... . Denoting the (retarded) Green function, which is in fact 2 × 2 matrix, and the evolution    2 e 0 , x) = Gea (x0, x) 2 operator for this equation by G(x and Ue(t0 , t) = Ueab (t0, t) a,b=1, b a,b=1 respectively, we see that (cf. Subsect. 4.2, equation (4.12)) Z  0  0 0 0 0 e e 0, x)ψ(x). (4.31) θ(t − t) U (t , t)ψ(ct, x ) = θ(t − t)ψ(x ) = d3 x G(x

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Comparing this equations with (4.29) and (4.30), we find
Ge11 (x0, x), Ge12 (x0, x) = g> (x0, x). The other matrix elements of Ge can also be connected with g(x0, x) and its derivatives, but this is inessential for the following. In this way we have connected, via (4.31), the evolution operator and the (retarded) Green function for a concrete first-order realization of Klein-Gordon equation. It is almost evident that this procedure works mutatis mutandis for any such realization; in any case the corresponding Green function (resp. matrix) being a (resp. matrix-valued) function of the Green function g(x0, x) introduce via (4.26) and (4.27). For instance, the treatment of the 5-dimensional realization given by (3.2) and (3.3) is practically identical to the one of Dirac equation in Subsect. 4.3, only the γ-matrices γ µ have to be replace with the 5 × 5 matrices Γµ (defined by (3.2)). This results in a 5 × 5 matrix evolution operator U (x0, x), etc. Since the bundle version of (4.31) or an analogous result for U (x0, x) is absolutely trivial (cf. Subsect. 4.2 and 4.3 resp.), we are not going to write it here; up to the meaning of notation it coincides with (4.16) or (4.23) respectively.

5.

Conclusion

In this investigation we have reformulated the relativistic wave equations in terms of fibre bundles. In the bundle formulation the wavefunctions are represented as (state) liftings of paths or sections along paths (time-dependent approach) or simply sections (covariant approach) of a suitable vector bundle over the spacetime. The covariant approach, developed in the present work, has an advantage of being explicitly covariant while in the time-dependent one the time plays a privilege rˆole. In both cases the evolution (in time or in spacetime

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resp.) is described via a linear transport in the bundle mentioned. The state liftings or sections are linearly transported by means of the corresponding (evolution) transports. We have also explored some links between evolution operators or transports and the retarded Green functions (or matrices) for the corresponding wave equations: the former turn to have realization as integral operators whose kernel is equal to the latter ones up to a multiplication with a constant complex number or matrix. These connections suggest the idea for introducing ‘retarded’, or, in a sense, ‘causal’ evolution operators or transports as a product of the evolution operators or transports with θ-function of the difference of the times corresponding to the first and second arguments of the transport or operator. Most of the possible generalizations of the bundle non-relativistic quantum mechanics, pointed in [2], are mutatis mutandis valid with respect to the bundle version of relativistic quantum mechanics, developed in the present work. The only essential change is that, in the relativistic region, the spacetime model is fixed as the Minkowski spacetime. A further development of the ideas presented in this investigation leads to their application to (quantum) field theory which will be realized elsewhere.

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Appendix A. Linear transports along maps in fibre bundles In this appendix we recall a few simple facts concerning (linear) transports along maps, in particular along paths, required for the present investigation. The below-presented material is abstracted from [15, ?, 16] where further details can be found (see also [7, sect. 3]). Let (E, π, B) be a topological bundle with base B, bundle (total, fibre) space E, projection π : E → B, and homeomorphic fibres π −1 (x), x ∈ B. Let the set N be not empty, N 6= ∅, and there be given a map κ : N → B. By idX is denoted the identity map of a set X. Definition A.1. A transport along maps in the bundle (E, π, B) is a map K assigning to κ , where any map κ : N → B a map K κ , transport along κ, such that K κ : (l, m) 7→ Kl→m for every l, m ∈ N the map κ : π −1(κ(l)) → π −1 (κ(m)), Kl→m

(A.1)

called transport along κ from l to m, satisfies the equalities: κ κ κ ◦ Kl→m = Kl→n , Km→n κ Kl→l

= idπ−1 (κ(l)),

l, m, n ∈ N,

(A.2)

l ∈ N.

(A.3)

If (E, π, B) is a complex (or real) vector bundle and the maps (A.1) are linear, i.e. κ κ κ (λu + µv) = λKl→m u + µKl→m v, λ, µ ∈ C (or R), u, v ∈ π −1(κ(l)), Kl→m

(A.4)

the transport K is called linear. If κ belongs to the set of paths in B, κ ∈ {γ : J → B, J being R-interval}, we said that the transport K is along paths. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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For the present work is important that the class of linear transports along the identity map idB of B coincides with the class of flat linear transports along paths 9 (see the comments after equation (2.3) of [15]). The general form of a transport along maps is described by the following result. κ , l, m ∈ N Theorem A.1. Let κ : N → B. The map K : κ 7→ K κ : (l, m) 7→ Kl→m is a transport along κ if and only if there exist a set Q and a family of bijective maps {Fnκ : π −1 (κ(n)) → Q, n ∈ N } such that κ κ −1 Kl→m = (Fm ) ◦ (Flκ ) ,

l, m ∈ N.

(A.5)

The maps Fnκ are defined up to a left composition with bijective map depending only on κ, i.e. (A.5) holds for given families of maps {Fnκ : π −1 (κ(n)) → Q, n ∈ N } and {0Fnκ : π −1(κ(n)) → 0Q, n ∈ N } for some sets Q and 0Q iff there is a bijective map Dκ : Q → 0Q such that 0 κ Fn = Dκ ◦ Fnκ , n ∈ N. (A.6)

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For the purposes of this investigation, we need a slight generalization of [15, definition 4.1], viz. we want to replace in it N ⊂ Rk with an arbitrary differentiable manifold. Let N be a differentiable manifold and {xa : a = 1, . . ., dim N } be coordinate system in a neighborhood of l ∈ N . For ε ∈ (−δ, δ) ⊂ R, δ ∈ R+ and l ∈ N with coordinates la = xa (l), we define lb (ε) ∈ N , b = 1, . . . , dim N by lba(ε) := xa(lb (ε)) := la + εδba where the Kronecker δ-symbol is given by δba = 0 for a 6= b and δba = 1 for a = b. Let ξ = (E, π, B) be a vector bundle, κ : N → B be injective (i.e. 1:1 mapping), and Secp (ξ) 1 (resp. Sec(ξ)) be the set of C p (resp. all) sections over ξ. Let Lκ l→m be a C (on l) linear transport along κ. Now the modified definition reads: 10 Definition A.2. The a-th, 1 ≤ a ≤ dim N , partial (section-)derivation along maps generated by L is a map a D : κ 7→ a Dκ where the a-th (partial) derivation a Dκ along κ (generated by L) is a map

κ 1 0 (A.7) a D : Sec ξ|κ(N ) → Sec ξ|κ(N )
such that, if σ ∈ Sec1 ξ|κ(N ) , then
κ κ (A.8) a D (σ) (κ(l)) = a Dl (σ) where the a-th (partial) derivative a Dlκ along κ at l is a map
κ 1 −1 a Dl : Sec ξ|κ(N ) → π (κ(l)) defined by κ a Dl (σ) := lim

ε→0




1 κ Lla(ε)→l σ(κ(la(ε))) − σ(κ(l)) . ε

9

(A.9)

(A.10)

The flat linear transports along paths are defined as ones with vanishing curvature operator [17, sect. 2]. By [17, theorem. 6.1] we can equivalently define them by the property that they depend only on their initial and γ final points, i.e. if Ks→t , γ : J → B, s, t ∈ J ⊂ R, depends only on γ(s) and γ(t) but not on the path γ itself. 10 We present below, in definition A.2, directly the definition of a section-derivation along injective mapping κ as only it will be employed in the present paper (for κ = idN ). If κ is not injective, the mapping (A.7) could be multiple-valued at the points of self-intersection of κ, if any. For some details when κ is an arbitrary path in N , see [7, subsect. 3.3].

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Accordingly can be modified the other definitions of [15, sect. 4], all the results of it being valid mutatis mutandis. In particular, we have: Proposition A.1. The operators Dlκa are (C-)linear and κ a Dl (L)

≡ 0,

(A.11)

where L : κ(l) 7→ Ll0 →l (u0 ) for some fixed l0 ∈ N and u0 ∈ π −1(κ(l0)).
Proposition A.2. If σ ∈ Sec1 ξ|κ(N ) , then κ a Dl σ =

Xh ∂σ i(κ(l)) i

∂la

+

X

i i j Γ (l; κ)σ (κ(l)) ei (l), a j

(A.12)

j

where {ei (l)} is a basis in π −1 (κ(l)), σ(κ(l)) =: of L are defined by

P

i

σ i (κ(l))ei(l), and the coefficients

∂Lij (l, m; κ) ∂Lij (m, l; κ) = − . ∂ma ∂ma m=l m=l P j Here Lji (· · · ) are the components of L, Lκ j L i (m, l; κ)ej (m). l→m ei (l) =: i a Γ j (l; κ) :=

(A.13)

The above general definitions and results will be used in this work in the special case of linear transports along the identity map of the bundle’s base.

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Appendix B. Matrix operators In this appendix we point to some peculiarities of linear (matrix) operators acting on n × 1, n ∈ N, matrix fields over the space-time M . Such operators appear naturally in the theory of Dirac equation where one often meets 4 × 4 matrices whose elements are operators; e.g.  3 an operator of this kind is ∂/ := γ µ ∂µ = (γ µ)αβ ∂µ α,β=0 where γ µ are the well known Dirac γ-matrices [4, 6]. We call an n × n, n ∈ N matrix B = [bαβ ]nα,β=1 a (linear) matrix operator 11 if bαβ are (linear) operators acting on the space K 1 of C 1 functions f : M → C. If {fν0 } is a basis in µ the set M (n, 1) of n × 1 matrices with the µ-th element of fν0 being (fν0 )µ := δν ,12 then by definition Bψ := B(ψ) := B · (ψ) :=

n X


bαβ (ψ0β ) fα0 ,

ψ = ψ0β fβ0 ∈ M (n, 1).

(B.1)

α,β=1

P For instance, we have ∂/ψ = 3α,β,µ=0 (γ µ)αβ (∂µ ψ0β )fα0 . To any constant matrix C = [cαβ ], cαβ ∈ C, corresponds a matrix operator C := [cαβ idK 1 ]. Since Cψ ≡ Cψ for any ψ, we identify C and C and will make no difference between them. 11

Also a good term for such an object is (linear) matrixor. = 0 for µ 6= ν and δνµ = 1 for µ = ν. From here to equation (B.10) in this appendix the Greek indices run from 1 to n ≥ 1. 12 µ δν

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The multiplication of matrix operators, denoted by , is a combination of matrix multiplication (denoted by ·) and maps (operators) composition (denoted by ◦). If A = [aαβ ] and B = [bαβ ] are matrix operators, the product of A and B is a matrix operator such that AB := A B :=

X

 aαµ ◦ bµβ .

(B.2)

µ

One can show that this is an associative operation. It is linear in the first argument and, if its first argument is linear matrix operator, then it is linear in its second argument too. For constant matrices (see above), the multiplication (B.2) coincides with the usual matrix one. Let {fα (x)} be a basis in M (n, 1) depending on x ∈ M and f (x) := [fαβ (x)] be β defined by the expansion fα (x) = fα (x)fβ0. The matrix of a matrix operator B = [bαβ ] with respect to {fα }, i.e. the matrix of the matrix elements of B considered as an operator, is a matrix operator B := [B αβ ] such that Bψ|x =:

X α,β


B αβ (ψ β ) x fα (x),

ψ(x) = ψ β (x)fβ (x).

(B.3)

Therefore ϕ = Bψ ⇐⇒ ϕ = Bψ,

(B.4)

where ψ ∈ M (n, 1) is the matrix of the components of ψ in the basis given. Comparing (B.1) and (B.3), we get X
α
f −1 (x) µ bµν ◦ fβν (x) idK 1 . (B.5) B αβ = Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

µ,ν

For any matrix C = [cαβ ] ∈ GL(n, C), considered as a matrix operator (see above), we have 
α  (B.6) C = C αβ (x) , C αβ (x) = f −1 (x) µ cµν fβν , C(fβ ) = C αβ fα .  n Therefore, as one can expect, the matrix of the unit matrix 11n = δαβ α,β=1 is exactly the unit matrix, 11 n = 11n . (B.7) If {fµ (x)} does not depend on x, i.e. if fαβ are complex constants, and bµν are linear, than (B.5) implies
α (B.8) B αβ = f −1 (x) µ fβν (x)bµν for ∂fαβ (x)/∂xµ = 0. In particular, for a linear matrix operator B, we have B = B for fβα = δβα (i.e. for fα (x) = fα0 ).

(B.9)

Combining (B.2) and (B.5), we deduce that the matrix of a product of matrix operators is the product of the corresponding matrices: C = A B

⇐⇒

C = AB = A B.

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(B.10)

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101

After a simple calculation, we find the matrices of 114 ∂µ and ∂/ = γ µ∂µ :13
∂ µ = 114 ∂µ + Eµ (x), ∂ / = G µ (x) 114 ∂µ + Eµ (x) = G µ (x)∂ µ

(B.11)

 3  λ 3 where G µ (x) = G λµ α (x) α,λ=0 and Eµ (x) = Eµα (x) α,λ=0 are defined via the expansions γ µ fν (x) =: G λµ ν (x)fλ (x),

λ ∂µ fν (x) =: Eµν (x)fλ(x).

(B.12)

So, G µ is the matrix of γ µ considered as a matrix operator (see (B.6)).14 Evidently ∂ / = ∂/ for fα (x) = fα0 .

(B.13)

Combining (B.6),(B.7), and (B.11), we get that the matrix of the matrix operator i~D /− mc114 , entering into the Dirac equation (2.1), is
(B.14) i~D / − mc114 = i~G µ (x) 114Dµ + Eµ (x) − mc114 , so that / − mc114 for fα (x) = fα0 . i~D / − mc114 = i~D

(B.15)

References

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[1] Bozhidar Z. Iliev. Fibre bundle formulation of relativistic quantum mechanics. I. Time-dependent approach. Physica Scripta, 68(1):22–31, 2003. http://www.arXiv.org e-Print archive, E-print No. quant-ph/0105056, May 2001. [2] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantum mechanics. V. Interpretation, summary, and discussion. International Journal of Modern Physics A , 17(2):245–258, 2002. http://www.arXiv.org e-Print archive, E-print No. quant-ph/9902068, February 1999. [3] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantum mechanics (full version). http://www.arXiv.org e-Print archive, E-print No. quant-ph/0004041, April 2000. [4] C. Itzykson and J.-B. Zuber. Quantum field theory. McGraw-Hill Book Company, New York, 1980. Russian translation (in two volumes): Mir, Moscow, 1984. [5] A. M. L. Messiah. Quantum mechanics, volume II. North Holland, Amsterdam, 1962. Russian translation: Nauka, Moscow, 1979. [6] J. D. Bjorken and S. D. Drell. Relativistic quantum mechanics , volume 1. McGrawHill Book Company, New York, 1964. Russian translation: Nauka, Moscow, 1978. 13

Here and below the Greek indices run from 0 to 3. We denote the matrix of γ µ by G µ instead of γ µ (x) as usually [6, 4] by γ is denoted the matrix 3-vector 1 (γ , γ 2 , γ 3 ). 14

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Bozhidar Z. Iliev

[7] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantum mechanics. I. Introduction. The evolution transport. Journal of Physics A: Mathematical and General, 34(23):4887–4918, 2001. http://www.arXiv.org e-Print archive, E-print No. quant-ph/9803084, March 1998. [8] A. M. L. Messiah. Quantum mechanics, volume I. North Holland, Amsterdam, 1961. Russian translation: Nauka, Moscow, 1978. [9] T. Eguchi, P. B. Gilkey, and A. J. Hanson. Gravitation, gauge theories and differential geometry. Physics reports, 66(6):213–393, December 1980. [10] Bozhidar Z. Iliev. Linear transports along paths in vector bundles. II. Some applications. JINR Communication E5-93-260, Dubna, 1993. 24 p. http://www.arXiv.org e-Print archive, E-print No. math.DG/0412010, December 1, 2004. [11] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantum mechanics. II. Equations of motion and observables. Journal of Physics A: Mathematical and General, 34(23):4919–4934, 2001. http://www.arXiv.org e-Print archive, E-print No. quant-ph/9804062, April 1998. [12] N. N. Bogolyubov and D. V. Shirkov. Introduction to the theory of quantized fields . Nauka, Moscow, third edition, 1976. In Russian. English translation: Wiley, New York, 1980.

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[13] N. F. Nelipa. Physics of elementary particles . Vyshaya shkola, Moscow, 1977. In Russian. [14] Physical encyclopedia, chief editor Prohorov A. M., volume 1, Moscow, 1988. Sovetskaya Entsiklopediya (Soviet encyclopedia). In Russian. [15] Bozhidar Z. Iliev. Transports along maps in fibre bundles. JINR Communication E597-2, Dubna, 1997. 19 p. http://www.arXiv.org e-Print archive, E-print No. dg-ga/9709016, September 1997. [16] Bozhidar Z. Iliev. Linear transports along paths in vector bundles. I. General theory. JINR Communication E5-93-239, Dubna, 1993. 22 p. http://www.arXiv.org e-Print archive, E-print No. math.DG/0411023, November 1, 2004. [17] Bozhidar Z. Iliev. Linear transports along paths in vector bundles. V. Properties of curvature and torsion. JINR Communication E5-97-1, Dubna, 1997. 11 p. http://www.arXiv.org e-Print archive, E-print No. dg-ga/9709017, September 1997.

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Chapter 7

ON THE INDUCTION PERIOD OF LASER-DRIVEN THERMOCHEMICAL PROCESSES László Nánai Department of Physics, University of Szeged, Hungary

Sándor Szatmári Department of Experimental Physics,University of Szeged, Hungary

Gregory J. Taft Department of Physics and Astronomy, University of Wisconsin – Stevens Point, Stevens Point, WI, USA

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Thomas F. George Office of the Chancellor and Center for Molecular Electronics, Departments of Chemistry & Biochemistry and Physics & Astronomy, University of Missouri – St. Louis, St. Louis, MO, USA Dedicated to Prof. I. Hevesi on his 75th birthday

ABSTRACT Laser-induced chemical reactions on solid surfaces are investigated. The close relationship between the reaction rate and induction period is shown. Using a simple mathematical technique based on no thermal exchange during the reaction and applying a point-system description, it is demonstrated that the inverse dependence of the induction period on rate constants can be shortened by the use of external laser radiation.

INTRODUCTION Laser-driven processes, such as laser-material treatments and laser-induced chemical reactions, are very important in modern technology [1-2]. Essential to these processes are the

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heterogeneous chemical reactions that occur on solid surfaces in a gaseous (air and/or other gases) ambient atmosphere. The laser is a special “tool” since it provides a controllable and localized heat source for chemical microreactors. The chemical reaction can be divided into three periods: induction, increasing and decreasing. At the onset of the chemical reaction, the mixing of active reactant particles occurs without significant creation of product seeds. The time necessary for this is called the induction period. Applying laser radiation, one can affect the speed of both particle-mixing and the chemical reaction itself. The theoretical description of these processes normally involves the use of differential equations with distributed parameters, but, to get a preliminary qualitative picture, pointsystem methods are also applied [3-4]. If one supposes that the time scale for the chemical reactions is small compared to the duration of the laser irradiation, the resulting process time development follows the so-called parabolic law (diffusion determined). If the laser treatment time is comparable or only a little more than the reaction time itself, we can introduce the thin-film approximation, with the dominant role of built-in electric fields leading to different kinds of reaction kinetics, e. g., linear, logarithmic or inverse logarithmic (reaction-limited) laws. In our work, we suppose that the chemical reaction rate constant does not depend on time. The physical meaning of this is that the time period required for adsorption, diffusion and solution of gaseous particles in the subsurface layer prior to the seed’s appearance is considered as the induction period. More analysis is needed for taking into account the influence of seeding on the rate of chemical reactions. Although temperature will influence the reaction rate, we assume a constant temperature during the reaction. The following partial differential equations describe the details of these reactions.

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KEY EQUATIONS AND RELATED EXPERIMENTAL RESULTS Let us suppose a chemical reaction of the type

∂ ⎛ ∂n A ⎞ ∂n A = ⎟+S ⎜D ∂x ⎝ ∂x ⎠ ∂t ,

(1)

with the concentrations of participating particles (particles/area) as nA (gas), nB (chemically active particles of solid) and nC (chemically active particles of products). We assume a onedimensional model for the time dependent gas particle concentration, nA(χ,t), where χ is the direction perpendicular to the solid surface, S is the flux of gas particles arriving at the surface with units of area/time, and D is the coefficient of diffusion. The boundary condition at the surface and the initial condition are

nA (0, t ) = N 0 , nA ( x,0 ) = 0 ,

(2)

where N0 is the effective concentration of particles in the subsurface layer. For the formation of products, we have

∂nC = Kn Aμ nνB ∂t , Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

(3)

On the Induction Period of Laser-Driven Thermochemical Processes

105

where μ and ν are the stoichiometric coefficients, and K is the reaction rate constant with units of (area time)-1. For the units of each side of (3) to be equal, we must have the following relationship between the stoichiometric coefficients:

μ +ν = 0 .

(4)

We assume no heat generation and absorption during the process. We also assume that neither K nor D depend on time. Then,

∂n ⎛ ∂n ⎞ S = ⎜ A ⎟ = −μ C ∂t , ⎝ ∂t ⎠ R

(5)

⎛ ∂n A ⎞ ⎜ ⎟ ∂t ⎠ R is the reduction rate of the gas particle concentration n in some small where ⎝ A volume due to the reaction. At the same time (for any small volume),

nB + νnC = n0 ,

(6)

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where n0 is the original active solid particle concentration. During this process, particle conservation should hold just like energy or momentum conservation in closed systems. Inserting (6) into (3), we have

∂nC ν = K (n0 − νnC ) n Aμ ∂t ,

(7)

With

nC ( x,0) = 0 .

(8)

The solution has two forms: 1 ⎧ ⎫ t 1−ν ⎡ 1−ν ⎤ 1⎪ ⎪ μ nC ( x, t ) = ⎨n0 − ⎢n0 + ν (ν − 1)K ∫ nA ( x, t ′)dt ′⎥ ⎬ for ν ≠ 1 ν⎪ 0 ⎣ ⎦ ⎪ ⎩ ⎭

(9)

t ⎧⎪ ⎡ ⎤ ⎪⎫ 1 nC ( x, t ) = n0 ⎨1 − exp ⎢− K ∫ dt ′⎥ ⎬ n A ( x, t ′) ⎦ ⎪⎭ ⎪⎩ 0 ⎣

(10)

for ν = 1 .

In order to obtain nA(χ,t), we have to substitute these solutions into (5) and then into (1), which leads to a nonlinear equation whose solution is not an easy task. Even without a Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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László Nánai, Sándor Szatmári, Gregory J. Taft et al. n0

rigorous solution, we can see that Eqs. (9) and (10) show that nC(χ,t) reaches its limit ν for a limited time at ν < 1 and for an infinitely long time at ν > 1. The characteristic time for distance χ from the surface plane is determined by the equation t

νnν0 −1K ∫ nAμ ( x, t ′)dt ′ ≈ 1

. (11) For μ=1, Eq. (11) corresponds to setting the argument of the exponential function in (10) equal to –1. Let us call the induction period τ the time during which on the surface we find 0

n0

products in an amount compared to ν , so that

τ ≈ (νKN 0μ nν0 −1 ) . −1

(12)

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It is important to note the inverse relationship between the induction period τ and the reaction rate K. In the limit of an infinitely high reaction speed, K→∞, there would be no induction period, τ →0. Because the reaction speed increases with temperature, it can be increased with an external heat source like a laser beam. This laser-assisted increase in the reaction speed leads to a shorter induction period.

Fig. 1. Thickness ( d ) variation of vanadium films versus oxidation time, with oxidation at room temperature.

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On n the Inductionn Period of Laaser-Driven Thhermochemicaal Processes

107

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One experiimental exam mple of using a laser to conntrol the reaction rate is lasser-assisted oxxidation of vaanadium. In Fig. F 1 we dem monstrate the classical low-temperature (non-laserasssisted) oxidattion kinetics of o vanadium with w a relatively high inducttion period [5]. In Fig. 2 w see how thee V2O5 growtth rate is increeased by laserr irradiation. The we T growth of o the V2O5 laayer thickness is faster for higher h laser powers p [6]. In Fig. 2 the indduction periodd occurs in leess than a few w hundred millliseconds for all laser poweers, just at thee onset of the substantial thhickness grow wth. Note that the induction period is shoorter for higher laser powerss. In Fig. 3 w see the resu we ults of CW CO O2 laser-enhannced oxidationn of vanadium m. The bump reesults from raapid growth off the oxide layyer in the direcction opposite of the incidennt laser beam.

Fiig. 2. V2O5 layeer thickness verssus time for diffferent laser pow wers.

Fiig. 3. V2O5 layeers grown on a vanadium v surfaace under the influence of CW CO2 laser radiaation (15 W) unnder atmospherric conditions. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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CONCLUSION We have shown that the period of induction of heterogeneous reactions (without seed formation) depends inversely on the reaction rate (speed). Using a laser beam as an external heat source, the process can be accelerated while the induction period is dramatically decreased.

ACKNOWLEDGMENTS The authors express their thanks for financial support from the National Science Foundation (INT-0239211), the MTA Hungarian Academy of Sciences, and CBC. This work was supported by the U.S. Army Research Development and Engineering Command Acquisition Center. The content of the information does not necessarily reflect the position or the policy of the federal government, and no official endorsement should be inferred.

REFERENCES L. Nánai, D. A. Jelski, I. Hevesi and T. F. George, J. Mat. Res. 8, 945 (1993). U. Gratzke and G. Simon, J. Phys. D: Appl. Phys. 24, 827 (1991). A. T. Fremhold, Jr., Theory of Metal Oxidation (North-Holland, Amsterdam, 1976). L. Nánai, R. Vajtai and T. F. George, Thin Solid Films 298, 160 (1997). Lijun Jiang, Ph.D. dissertation, New Jersey Institute of Technology, 2004). D. A. Jelski, L. Nánai, R. Vajtai, I. Hevesi and T. F. George, Mat. Sci. Eng. A 173, 193 (1993).

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[1] [2] [3] [4] [5] [6]

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In: Perspectives in Theoretical Physics Editors: T. F. George, R. R. Letfullin and G. Zhang

ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 8

DISPERSION PROPERTIES OF NANO-SCALE SYSTEMS V. Lozovski* and A. Tsykhonya RadioPhysics Faculty of Kyiv T.Shevchenko National University, Kyiv, Ukraine

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ABSTRACT An approach allowing the determination of the dispersion properties for systems of nano-particles is proposed. The approach is based on a study of the absorption spectrum of the external field energy by the system. The Lippmann-Schwinger equation is used to calculate the local electromagnetic fields in the system and the dissipative function, which describes the absorption of external field energy by the system per unit of time. The main idea of the approach is using the resonant absorption of external radiation under conditions of excitation self-modes in the system. In the frame of the proposed approach the numerical calculation of the dispersion properties of a single (individual) particle having different forms (ellipsoids and parallelepipeds) as well as two spherical particles is provided. It is shown that spatial dispersion of the electromagnetic oscillation, which is localized on the system of nano-particles, appears in systems which consist of two or more particles.

PACS: 61.46.Df; 68.65.Hb; 78.67.Bf; 78.67.Hc Keywords: near-field, nano-particle, electromagnetic modes

dissipative

function,

dispersion

relations,

1. INTRODUCTION The miniaturization processes of electronic devices steadily lead to the necessity of making and using objects with nano-scale sizes: nano-composites, quantum wells, quantum *

E-mail address: [email protected]; Phone/Fax: +380(44)525-5530; Corresponding author: Valeri Lozovski, RadioPhysics Faculty, Kyiv T.Shevchenko National University, Glushkov ave. 2, build 5, 03022, Kyiv, Ukraine

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wires, quantum dots etc. [1, 2]. As a result of wide application of nano-particles in electronics, optics, optoelectronics, photonics and other branches of sciences and technology appear requirement of theoretical and experimental investigation of such objects [3-6]. The theoretical and practical aspects of micro- and nano-electronic devices very often need to know the dispersion properties of systems and conditions of characteristic oscillation excitation. Therefore, determining of the dispersion properties of nano-systems is an actual problem. The production of nano-objects and modern devices needs significant expenses. In this connection the interest in theoretical and numerical modeling of electrodynamical properties of nano-systems increases. In this work one discuss the dispersion properties of meso-particles. The term mesoparticle, in this context, means that linear dimensions of such object, on the one hand, are much smaller than characteristic wavelengths of probing fields and, on the other hand, the linear dimensions are in order of surface transition layer. The experimental research of the properties of nano-particles has begun yet in the middle of the last century and today is very relevant. It is well known experimental fact that small particles can efficiently absorb the light at the frequencies nearly the fundamental frequency of material of particles [7-9]. Moreover, for some metallic particles maximum absorption is situated in the visible or even in the ultraviolet spectrum region, while absorption band of material is situated in entirely another region of spectrum. Furthermore, absorption spectrum essentially depends on the shape of the particles, permittivity of the material and external medium, and on the location of particles in the medium. The first theoretical investigations of optical properties of small particles were carried out for spherical particles. One can note the important contribution in this field by Me [10] and Ruppin [11], where the surface modes of spherical microcrystals were obtained analytically. It should be noted that there are numerous works in which the problem of interaction between the small spherical particles or cube particles is considered [12-15]. It was shown a strong dependence of absorption spectra on the electrodynamical interaction between the particles. Furthermore, the substrate effects on the optical properties of small particles were investigated [16]. The calculations performed for a sphere near a dielectric or metallic surface demonstrate two main effects caused with the surface presence: the redshift of the main absorption peak and the appearance of the subsidiary peaks, which were observed experimentally. Nevertheless, in optical experiments on small particles, parallelepiped shaped crystallites often occur. The distinctive property of the parallelepiped particles is that the internal field is not homogenous even when the particle is placed in a long-wavelength external field. One can note a great contribution in this case by Fuchs [17], where the theory of the optical properties of small particles of arbitrary shape is developed. The absorption spectra of the small cube-like particles were calculated in Refs. [4,17]. Undoubtedly, in absorption processes the surface of the particle plays a vital part. In the modern scientific society is widely believed that absorption of the light is realized by some polarized oscillation modes. These electrodynamical oscillations which are localized at the particles usually are named as surface modes. In despite of the origin of these modes is closely connected with surface of the particle, in our opinion, such exploration is incorrect and doesn’t make sense at all. Let one consider this problem more detail. It is well known, that macroscopic Maxwell’s equations sufficiently describe optical properties of the system in that case when typical wave length of the radiation λ in the system is much more than characteristic length of

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Dispersion Properties of Nano-Scale Systems

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inhomogeneities of the microfields l (λ >> l ) , which is in general comparable with interatomic distances. For such systems one can introduce the conception of macrofield [18] which is the microfield averaged over the distances much less than λ but much more than l, and as well as using of boundary conditions which give the connections between fields on each side of interface and macroscopic permittivity of the system. As before, sizes of mesoparticles are less than the wavelength of the probing external electromagnetic field and are comparable with the thickness of transition surface layers where microscopic fields quickly change in the space. That’s why averaging of Maxwell’s equations becomes problem and leads to strong dependence of macroscopic parameters on the coordinate at characteristic distances about linear dimensions of the particles. Consequently, introduction of surface oscillations in small particles is, in principle, incorrect problem, because it is not known how to describe such system with parameters that can be controlled directly in experiment, but it is also incorrect to say that the resonances are absent in the system at all.

E ( 0 ) ( R , ω) E ( 0 ) ( R , ω)

R

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E ( R ) ( R , ω)

Figure 1. Forming of a local field near the particle.

The most useful parameters with the help of which one can correctly describe the system as a whole are characteristics of the external probing electromagnetic radiation: intensity of incident and absorbed light, wave vector and frequency of radiation. Then, dispersive laws must be determining by characteristics of external field when the energy of the field is the most effective absorbed by the system. Therefore, one needs to calculate the dissipative function as a function of the frequency and incident angle of the probe radiation and, next, to build a three-dimensional graph of this function. Obviously projection of the chains of the surface on the (k , ω) - plane gives us the dispersion relations of the system. Obtained in that way dispersion characteristics, maybe, will be not always dispersion relations of natural oscillations, it is depends on methods of excitation of oscillations, but always will determine conditions when we can excite them. In this connection proposed way gives us the requisite conditions of the excitation of self-modes in the system, but not sufficient conditions. To find sufficient conditions one may calculate the life-time of the oscillations which is connected

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with the half-width of the absorption profile. Since the half-width is in inverse proportion to the mode life-time, then the narrower absorption profile the more probability for establishment of oscillations in the system. However, the proposed approach in the case of distributed systems (as a thin film, for example) gives the possibility to obtain the true dispersion relations of wave-like excitations – surface waves or waveguide modes.

2. THE SELF-CONSISTENT EQUATION AND THE EFFECTIVE SUSCEPTIBILITY OF A NANO-PARTICLE To find the energy of external field, which is absorbed by the small particle we need to know the distribution of fields and currents inside the particle. For solving this problem the method of local field is widely used [19]. Let external electromagnetic field acts on a small, non-magnetic ( μ = 1 ) particle, which is located in the known medium. The intensity of probing electrical field is E

( 0)

(R, ω)

(Fig.1). Then, in every point of the medium will be two components of total field: external

(R, ω ) and field E ( R ) (R, ω) caused by current induced by the local field inside the particle. It is supposed that the electrodynamic Green function Gij (R, R′, ω) (photon

field E

(0)

propagator) of the medium, where the particle is located, is known. Then, the local field in any point of the system can be found from the equation [19-21]

Ei (R, ω) = Ei( 0 ) (R, ω) − iωμ 0 ∫ dR ′Gik (R, R ′, ω) J l (R ′, ω) ,

(1)

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V

where V is the volume of the particle, J l ( R ′, ω) is density of local current inside the particle, which is connected with the local field by constitutive equation

J k ( R ′, ω) = −iωχ kl (ω) E l ( R ′, ω) .

(2)

Here χ kl (ω) is the susceptibility of the material of the particle (linear response on local field). Substituting eq. (2) in (1) one obtains the Lippmann-Schwinger equation for the selfconsistent field [19]

Ei (R, ω) = Ei( 0) (R, ω) − ω2 μ 0 ∫ dR ′Gik (R, R ′, ω)χ kl (ω) El (R ′, ω) .

(3)

V

To write the solution of this equation one can introduce the linear response on the external field (so-called effective susceptibility) Χ kl (R, ω) which connects the local current and external field [22-24]

Ei (R, ω) = Ei( 0 ) (R, ω) − ω2 μ 0 ∫ dR ′Gik (R, R ′, ω) Χ kl (R ′, ω) El( 0) (R ′, ω) . V

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(4)

Dispersion Properties of Nano-Scale Systems The external field E

(0)

113

(R, ω) is long-range meaning that it changes in space at distances

much larger than the linear dimensions of the particle. That’s why, one can simplify expression (4) taking outside integral the external field. The effective susceptibility can be written in the form of standard function of linear response (see, Appendix):

[

Χ kl (R, ω) = χ kj (ω) δ lj + Slj (R, ω)

]

−1

= χ kj (ω) Pjl (R, ω) .

(5)

The second term in the brackets in Eq. (5) describes the processes of electrodynamic interaction and is a self-energy part:

∫

Slj (R, ω) = μ 0 ω2 dR′Glk (R, R′, ω)χ kj (ω) .

(6)

V

The self-energy part shows that changes of inoculating susceptibility are the result of interactions between electromagnetic field and current excited in the particle. Then, tensor Pij (R, ω) , in the right part of expression (5), describes the local field effects in formation of the response of the system on the external field. For isolated particle one can use the Green function of homogeneous, isotropic medium. Very often the particle can be located at the surface of the substrate or under surface of a solid. Then the Green function can be written as a sum of indirect and direct part [19]

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Gik ( R , R ′, ω) = Dik ( R , R ′, ω) + I ik ( R , R ′, ω) ,

(7)

where the direct part Dik ( R , R ′, ω) is the photon propagator of the external medium and has a singularity at the source region R = R ′ . Indirect part I ik ( R , R ′, ω) is responsible for the interaction electromagnetic field with interface (substrate surface) and secures completion of boundary conditions. Consequently, this approach allows to calculate the self-consistent field in arbitrary point of the system “external medium – particle” and, then, to define most electrodynamic properties of the system.

3. DISSIPATIVE AND DISPERSIVE PROPERTIES OF NANO-PARTICLES In the previous chapter one has shown how to calculate the self-consistent field in an arbitrary point of the system. The following step of the study will be finding of the dissipative function of the system. Let us determine dissipative function as the Joulean heat that absorb per unit of time by volume unit [18]. *

*

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with J i ( R , ω) local current excited inside the particles by the field Ei ( R, ω) . A sign ... means averaging over volume of the current localization, and (...) means the time averaging. It is necessary to remember, that time of averaging is much greater of oscillation period of external field. Because of this after time averaging one can obtain

Q(ω) = J i (R, ω) Ei* (R , ω) + Ei (R , ω) J i* (R, ω) .

(9)

Local current is connected with external field via effective susceptibility

J ( R , ω) = − i ω Χ ( R , ω) E ( 0 ) ( R , ω) .

(10)

Comparing Eqs.(2) and (10), one can see, that the local field is connected with external field via equation

Ei (R, ω) = χ ik−1 (ω) Χ kl (R, ω) El0 (R, ω) .

(11)

Substituting Eqs. (10), (11) in Eq.(9) one obtains the expression for the dissipative function

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Q(ω) = −iω(χ in (ω) − χ*ni (ω)) Pnj (R, ω) Pil* (R, ω) E 0j (R, ω)( El0 (R, ω))* .

(12)

Then, dissipative function is determined by the linear susceptibility of the material of the particle and by the factor which describes the local field effects. Notice that this factor is strongly depended on the particle shape and size. I(k,ω)

ω

k

Figure 2. Surface of dissipative function plane

Q (ϑ, ω) and projection of the chains of the surface onto the

(ϑ, ω) .

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It is well known, in order to find dispersion relations of infinite systems one has to, first, calculate the susceptibility (response on the local field) in the reciprocal space – χij (k , ω) . Second, equate the pole part of the susceptibility with zero [25]. In the case of small particle (mesoscopic system) one must use the effective susceptibility which is the response on the external field. Moreover, in this case the system is not homogeneous which means that one can not perform the Fourier transformation and work in reciprocal space. What does one must do when the system is mesoscopic? In this case one can not say that in each point exists own resonance, but we can not say also that resonances in the system are absent at all. Obviously in such case one need to work with parameters which one can control experimentally: reflection and transmission coefficient, dissipated energy, etc. Let one consider an experiment in which the dispersion law of surface waves is determined. The experiment consists in: the surface is illuminated by radiation at prescribed frequency with the prism of total internal reflection. The intensity of reflected radiation is detected at different incident angles ϑ. The curve of dependence of reflected radiation intensity on ϑ has a minimum when surface wave is excited. The exciting of surface wave means in this case the exciting of eigen mode. The curve ω = ω(ϑ) which is building on the base of these measurements is the dispersive relation of surface wave. In such a way one can propose the approach of theoretical calculation of dispersion relations for nano-particles. This approach consists in that one should: – calculate the dissipative function Q (ϑ, ω) as a function of the frequency ω and incident angle ϑ; – build the surface Q (ϑ, ω) in three dimensional space (fig. 2);

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– project chains of the surface onto the plane (ϑ, ω) . The projection curve is the curve which determines the dispersion relations of electromagnetic modes. Indeed, the curve defines optimal conditions for transfer of the external field energy to the system under mode excitation. To find life-time of oscillations one has to calculate the half-width of the absorption profile. Since the half-width is in inverse proportion to the mode life-time, then the narrower absorption profile the more probability for establishment of oscillations in the system.

4. NUMERICAL CALCULATIONS AND DISCUSSION As it was noted above, to investigate the dispersion properties of the system one have to built three-dimensional graph of the dissipative function Q (ϑ, ω) and, then, project the chains of the surface Q (ϑ, ω) onto the plane (ϑ, ω) . Let one consider a small particle is made of isotropic material which characterized the susceptibility

χ(ω) =

χ 0 ω02 , − ω2 + ω02 − iωγ

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116

V. Lozovskki and A. Tsykkhonya

w where χ 0 is thhe static susceeptibility; ω0 is the resonaance frequencyy of oscillatorr;

γ

is the

daamping factorr. Such single--oscillator model is used heere for simpliccity. For exam mple, alkaliIII

V

haaloid crystals,, semiconducttor crystals A B with structure s of ZnS Z in frequency region (nnear resonancee frequency ω0 ) belong to such materials. Nevertheleess one can pooint several m materials in wh hich the linear response has a form Eq.(133).

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dal particles a) ellipsoid To investig gate dispersionn properties of ellipsoids, leet us consider the dielectric ellipsoidal m meso-particles of three typess of the shape: the sphere, thhe oblate and prolate ellipsoids which arre located in free space (Fiig. 3). Geomeetry of calculaations is chossen in the nexxt way: the w wave vector off external fieldd is directed att the angle ϑ to OZ axis. The T OZ axis iss symmetry axxis of ellipsoid ds. Let the XO OZ plane of Cartesian C coorrdinate system m coincides wiith incident pllane. One con nsiders the situuation when the t electric fieeld vector of incident light E(0) lies in X XOZ plane (p-p polarized wavve).

a

b

c

Fiigure 3. Ellipsoidal particles: a – sphere; b – oblate o ellipsoid;; c – prolate elliipsoid.

To calculaate the dissiipative functtion one shoould define, at first, thee effective suusceptibilities of the partticles. Because the dielecctric ellipsoidds are situatted in the hoomogeneous field f (sizes off particles aree much smalleer than wave--length of the light), the efffective suscep ptibilities can be presented in i the form [18].

Χ ij (ω) = χ(ω)

1 δ ij , 1 + 4πχ χ(ω) Li

(14)

w where

Li =

axa y az 2

∞

∫ (s + a 2 ) 0

i

ds (s +

a x2 )(( s

+

a 2y )(s

+

a z2 )

iss the depolarizzing factors, a x , a y , a z – seemi-axis of elllipsoid. Usingg the designation

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Dispersion Properties of Nano-Scale Systems

Pij( El ) (ω) =

117

1 δ ij , 1 + 4πχ(ω) Li

(16)

one can write the effective susceptibility of the ellipsoidal particles in the standard form

Χ ij (ω) = χ(ω) Plj( el ) (ω) .

(17)

One should note that in this case the effective susceptibility doesn’t depend on coordinate. Substituting Eq.(17) in Eq.(12) one can calculate the dissipative function Q (ϑ, ω) . This calculation was carried out numerically. The results are shown in Figs. 4-6. The dissipative function and its projection onto the (ϑ, ω) -plane for spherical particle are shown in Fig. 4. First of all, one should note that form of the surface is awaiting because spherical symmetry of the particle. That’s why absorption spectrum doesn’t depend on the incident angle of the light. Secondly, the resonance frequency is shifted relatively the fundamental frequency of the material of the particle ω0 . It is easy to check up this frequency correspond to Fröhlich’s mode when permittivity of the material

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ε(ω) = 1 + 4πχ(ω)

(18)

is equal to -2. From the physical point of view it means that all excitations in the particle are inphase. One can see in the Fig. 4 (b) that the resonance frequency is not a function on the angle ϑ . That’s why the projection curve has a form of straight line parallel to ϑ-axes. In this sense the obtained mode is nondispersive. This is a result that dimensions of the particles are smaller than a wavelength of the probing field, and, also, one does not take into account a spatial dispersion of the material. Q(ω) , arb. units

3

16

12 2

ω ω0

8

1

4

0,0

3 0,5

ϑ , rad

1,0

1 1,5

0

ω ω0

2

0 0,0

0,5

1,0

1,5

ϑ , rad

(a)

(b)

Figure 4. Absorption spectrum of the spherical particle: (a) surface of dissipative function (b) projection of the chain of the surface onto the plane

Q (ϑ, ω) ;

(ϑ, ω) . Here and below the white dash line

designates the resonance frequency of particle material ω0 . Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

118

V. Lozovski and A. Tsykhonya Absorption spectrums of prolate ( a z = 10 a x ) and oblate ( a z = 0.1 a x ) ellipsoids are

shown in Figs. 5 and 6. One can see that in these cases absorption peaks splitting into two peaks which correspond to longitudinal and transversal oscillations. This means in the particle can be existed two modes. Since the resonance frequencies do not depend on the incident angle ϑ one can say that the modes are nondispersive. The existence of two modes in the system is a result of the uniaxial-symmetry of the system.

Q(ω) , arb. units 3

16

12

ω ω0

8

2

1

4 3

0,0

1 ω ω0

0,5

ϑ , rad 1,0

1,5

0

2

0 0,0

(a)

0,5

ϑ , rad

1,0

(b)

Figure 5. Absorption spectrum of the prolate ellipsoid: (a) surface of dissipative function projections of the chains of the surface onto the plane Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

1,5

Q (ϑ, ω) ; (b)

(ϑ, ω) .

Q(ω) , arb. units 16

4

12

3

ω ω0

8

4

0,0

2

1

0,5

ϑ , rad

1,0

1,5

0

1

2

ω ω0

3

4

0 0,0

(a)

0,5

ϑ , rad

1,0

(b)

Figure 6. Absorption spectrum of the oblate ellipsoid: (a) surface of dissipative function projections of the chains of the surface onto the plane

1,5

Q (ϑ, ω) ; (b)

(ϑ, ω) .

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b) parallelepiped particles The case of light absorption by parallelepiped particles is more complicated. In this case one should calculate the effective susceptibilities of the particles in the frame of near-field approximation using Eqs. (5) and (6). Then, the use of linear response method allows us to calculate the local field correction factor analytically for the simple shape of the particles in the frame of near-field approximation. It leads to essential simplification of numerical calculations. Namely, one only needed to calculate numerically the final equation [Eq.(12)]. To calculate the local field correction factor in the near-field approximation one used the photon propagator in the form

⎤ 1 ⎡ c2 3c 2 U e e , − ⎢ 2 3 2 3 R R⎥ 4π ⎣ ω ⋅ Rs ω ⋅ Rs ⎦

G (R , R′, ω) =

(19)

with U unit dyadic, R s = R − R′ , Rs = R − R ′ , e R =

Rs . Because of the Green Rs

function Gij (R , R ′, ω) has a pole at R ′ = R , evaluation of the integral in Eq. (6) becomes nontrivial problem. This problem is well known and related to the so-called radiation reaction field. The problem can be solved in the frame of scheme proposed by van Bladel [26] and Yaghjian [27]. The main idea of this approach is to introduce the exclusion volume Vδ whose depolarizing properties are accounted for by a special source dyadic [27]. Then, when calculating the self-consistent field the following relation should be applied

E(R ) = −iωμ0 lim Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

δ→0

∫

dR′G (R, R′)J (R′) −

V −Vδ

1 L ⋅ J (R ) , iωμ0

(20)

where L is the source dyadic, which depends solely on the geometry of the exclusion volume Vδ. The exclusion volume removes the singularity of G and becomes infinitesimally small when δ → 0 [28]. E(0)

z

x

ϑ

ϕ

E(0)

y

z

y x

z

z

y x

a

b

y x

c

Figure 7. Parallelepipedal particles: a – cube-like; b – plate-like; c – stick-like.

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120

V. Lozovski and A. Tsykhonya

To compare the absorption spectra of ellipsoidal and parallelepipedal particles one chose as having a cube-like, plate-like and stick-like shapes (Fig. 7). Geometry of calculations is chosen in the next way: coordinate system center is located into center of parallelepiped; coordinate axes are parallel to parallelepiped sides. The angle between the wave vector k and OZ axis is denoted as ϑ , the angle between the wave vector projection on the plane XOY and OX axis is denoted as ϕ (Fig. 7). As in the previous case, the calculations are carried out for p-polarized light when polarization vector lies in the incident plane. The results of these calculations are shown in Figs. 8–10. The results were obtained for fixed azimuth angle ϕ = 0 ( k y = 0 ). One can see that obtained spectra are qualitatively similar to absorption spectra of ellipsoidal particles shown earlier. Particularly, absorption peaks are shifted relatively to the fundamental frequency of the material of the particles. As one can see from Figs. 9-10, the two electromagnetic modes which correspond to longitudinal and transversal oscillations are existed in the case of parallelepiped particles. Nevertheless, in the case of the stick-like and plate-like particles the energy of electromagnetic oscillations pass from one mode to another. One mode is exited at small angles ϑ ( ϑ ≈ 0 ), another mode is exited at greater angles ϑ ( ϑ ≈

π ). Also, one can say that obtained modes are nondispersive because 2

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the resonance frequencies do not depend on the angle ϑ (Figs. 8(b)–9(b)). This result corresponds to small particles approximation. It needs to note, that in the case of parallelepipeds there are some peculiarities. Namely, as one can see from Fig. 8 there is an additional mode in the cubic particle, which is characterized by the smaller peak at higher (relatively the main mode) frequency. The half-width of the additional peak is broader than the main peak. This means the life-time of the excitation is shorter than the life-time of the main oscillations. It probably means that these addition oscillations could not be supposed as resonant. One should note that in contrast to prolate ellipsoidal particle when the peaks have

Q(ω) , arb. units 70 60

3

50

ω ω0

40 30

2

20 10 0,0

1 0,5

ϑ , rad

1,0 1,5

1

ω ω0

2

3

0,0

(a)

0,5

ϑ , rad

1,0

(b)

Figure 8. Absorption spectrum of the cube: (a) surface of dissipative function projections of the chains of the surface onto the plane

1,5

Q (ϑ, ω) ; (b)

(ϑ, ω) .

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similar maximal height, the high frequency peak of absorption spectrum of plate-like particle is much smaller than low-frequency peak (the main peak). Moreover, the half-width of this additional peak is rather broad (cf. Figs. 9 and 5).

Q(ω) , arb. units 180 150

3,00

120

ω ω0

90 60

2,25

1,50

30 0,0 0,5

ϑ , rad

1,0 1,5

0,75

ω ω0

1,50

2,25

3,00

0,75 0,0

0,5

(a)

ϑ , rad

1,0

1,5

(b)

Figure 9. Absorption spectrum of the stick-like particle: (a) surface of dissipative function (b) projections of the chains of the surface onto the plane

Q (ϑ, ω) ;

(ϑ, ω) .

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Q(ω) , arb. units 120 3

90

ω ω0

60 30 0,0

ϑ , rad

2

1

0,5 1,0 1,5

1

ω ω0

2

3

0,0

(a)

0,5

ϑ , rad

1,0

(b)

Figure 10. Absorption spectrum of the plate-like particle: (a) surface of dissipative function (b) projections of the chains of the surface onto the plane

1,5

(ϑ, ω) .

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Q (ϑ, ω) ;

122

V. Lozovski and A. Tsykhonya 2

E (0)

d

Z 1

Y X Figure 11. Two spheres in the external field.

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Consequently, optical properties of parallelepiped and ellipsoidal particles are similar, but there are some differences. Nevertheless, one can see that all modes that can exist in the systems, which consist of one nano-particle or non-interacting nano-particles (dilute system), are nondispersive. This means that the resonance frequencies do not depend on the incident angle ϑ and dispersion curves have a form of straight lines. c) two spherical particles As was shown above, in the case of single small particle the resonance frequencies do not depend on geometry of external excitation (on incident angle of radiation). It means that dispersion relations have form of straight lines, parallel to ϑ-axes. This behavior is result of particle dimensions being much less than wavelength of external radiation. It means that excitations in the particle are in-phase. One can suppose spatial dispersion must appear in the systems which consist of two or more particles. Let one consider two spherical particles with radii r1 , r2 and linear responses on the total field of the particles (the material characteristic) are χ (ω) and (1)

χ (2) (ω) , respectively. Let one choose the geometry of the system in the

follow way (Fig.11): as the particles have spherical symmetry the OZ axis crosses through centers of the particles and the distance between centers is equal to d. Let incident plane (0)

is parallel to OX coincides with YOZ plane of chosen coordinate system and vector E axes (s-polarized light). In this case the particles can be considered as two coupled oscillators under external force. As it is well know, the resonances in similar system depend on nature of the coupling between oscillators and on phase relations between oscillations. In this way in the system can appear a spatial dispersion which becomes apparent via dependence of resonant frequencies on the incident angle. To demonstrate this conception, one should calculate the absorption spectrum of two spherical particles. The calculations of the local field in the system of two spherical particles can be based on the pseudo-vacuum Green function approach [19, 23]. The main idea of the approach is that the pseudo-vacuum Green function is the Green function of the system without particle under consideration. In the case of two particle system the pseudo-vacuum Green function approach consists in two steps: Let us add

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Dispersion Properties of Nano-Scale Systems

123

a second particle to the system consisting of a single particle (number 1). Considering the currents generated inside the 2nd particle one can see that external (relative to this particle) field consists of two parts – the external field E

(0)

(R , ω) and the field reradiated by particle

number 1. To calculate the self-consisted field in the system one can introduce the generalized Green dyadic [23]

G ij( 2 ) ( R , R ′, ω) = Gij ( R , R ′, ω) + iωμ 0 ∫ dR ′′Gik ( R , R ′′, ω) Χ (kl1) ( R ′′, ω)Glj ( R , R ′′, ω) ,

(21)

V1

where Χ kl (R, ω) is the effective susceptibility of the 1st particle. The Green dyadic (1)

incorporates the indirect scattering channel, i.e., field scattering by the particle number 1. Substituting eq. (21) to Lippmann-Shwinger equation (3) and after some manipulations one can find expression for self-consisted field in the system ⎡ ⎤ (1+ 2 ) ( 2) E i ( R , ω) = ⎢ I ij( 2 ) ( R , ω) + iωμ 0 ∫ dR ′G il( 2 ) ( R , R ′, ω) Χ lm ( R ′, ω) I mj ( R ′, ω) e ik ( R ′− R ) ⎥ E (j 0 ) ( R , ω) , ⎢⎣ ⎥⎦ V2

(22)

where

I ij( 2 ) ( R , ω) = δ ij + iωμ 0 ∫ dR ′′Gik (R , R ′′, ω) Χ (kl1) ( R ′′, ω) e ik ( R ′′− R ) ,

(23)

V1

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−1

⎤ ⎡ Χ (jl1+ 2) (R, ω) = χ (jk2 ) (ω) ⎢δ lk − iωμ 0 ∫ dR ′Glm( 2) (R, R ′, ω)χ (mk2) (ω)⎥ . ⎥⎦ ⎢⎣ V2

(24)

One considers sufficiently small spheres and assumes that the distance between the particles is so large that one can suppose those to be situated in the homogeneous fields

(d >>1 ,r 2 r ). Using Eqs.(21) –(24) and taking into account that

e

ik ( R ′ − R )

⎧1, R - R ′ ≤ r1 , r2 ; ⎪ =⎨ d ⎪e 2 πi cos ϑ λ , R - R ′ > r , r , 1 2 ⎩

(25)

one can obtain

Ei (R1 , ω) = Lij (ω, d , ϑ) E (0) j ( R1 , ω) , with

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124

V. Lozovski and A. Tsykhonya

⎧ 1 + χ(2) (ω) 3 − F (2) (d , ω) ⎪ Lij (ω, d , ϑ) = ⎨ d ⎪1 + χ(2) (ω) 3 − F (2) (d , ω) F (1) (d , ω) e2 πi cos ϑ λ ⎩

⎫ d ⎪ 2 πi cos ϑ λ δ e ⎬ ij ⎪ ⎭

(27)

local field-factor [12] connecting local field with external field. Here the next designations

F (d , ω) = (1)

F (2) (d , ω) =

Χ (1) (0) (ω) 3

r13Gii( d ) (| R 2 − R1 |) ,

(28)

χ(2) (ω) 3 ( d ) r2 Gii (| R1 − R 2 |) , 3

where R1 and R 2 are coordinates of the centres of the 1st and the 2nd spherical particles, respectively, Χ ( 0 ) (ω) is the effective susceptibility of the 1st particle that determines by (1)

(14). And the following notation is used

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⎧ 1 ⎪ − d 3 , i = x, y , ⎪ Gii( d ) (| R 2 − R1 |) = Gii( d ) (| R1 − R 2 |) = ⎨ ⎪ 2 ⎪ 3 , i = z. ⎩ d

(29)

Then, Eq.(26) define the self-consistent field at the site of “1st” sphere as function on distance between spheres and incident angle. One should pay attention to phase factors exp(2 πi cos ϑ ⋅ d / λ ) . Moreover, one needs to note that Eq.(12) in this case transforms to 2

Q(ω) = −iω2 Im χ nn (ω) Lnj (R, ω) E 0j (R, ω)

2

(30)

It means that only phase factor in the denominator of Eq.(27) defines the dispersion properties of the system under consideration. Then, because the denominator of Eq.(27) depends on the incident angle ϑ of radiation, the resonant frequencies of the system will depend on ϑ . Consequently, the local field depends on the wave vector of the external field, the spatial dispersion of electromagnetic excitations, as expected, appears in the system of two spherical particles thus and in the systems which consist of more than two particles. Then, in the system consisting of at least two particles the spatial dispersion have to be observed. The results of calculations of absorption spectra of the system for different distances d between the particles are represented in Figs.12–14. The radii of the particles are chosen the same r1 = r2 = r . The absorption spectrum for d = 3r is shown in Fig.12. One can see that dispersion properties are rather

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125

Q(ω) , arb. units 2,4

10000

ω ω0

1000

100

2,2

2,0

10 0,0

0,5

ϑ , rad

1,0

2,20 2,15 1,8 2,10 ω 0,0 2,05 1,5 2,00

0,5

ω0

ϑ , rad

(a)

1,5

(b)

Figure 12. Absorption spectrum of two spherical particles for function

1,0

d = 3r :

(a) surface of dissipative

Q (ϑ, ω) ; (b) projections of the chains of the surface onto the plane (ϑ, ω) . Q(ω) , arb. units 2,4

ω ω0

100

2,2

2,0 10

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0,0

0,5

ϑ , rad

2,25

1,0

2,00 1,5

ω ω0

2,50 1,8 0,0

0,5

ϑ , rad

(a) Figure 13. Absorption spectrum of two spherical particles for function

1,0

1,5

(b)

d = 3.5r :

(a) surface of dissipative

Q (ϑ, ω) ; (b) projections of the chains of the surface onto the plane (ϑ, ω) .

specific in this case. The two high maximums appear against a background of main absorption peak. The appearance of these maximums can be explained by configuration resonances [29] because the local field is strongly depends on the distance between particles. Indeed, when the distance between the spheres increases, the absorption spectrum changes, maximums draw together and at some distance the maximums join to one (see Figs. 13 and 14). Let one more detail examines the absorption spectrum, which corresponds to d = 3.5r shown in Fig. 13. On one hand, one can see that maximums already have joined to one, and on the other hand, the projection of the chain onto the (ϑ,ω)-plane has a view some curve. If the distance increases this maximum is disappear and the spectrum tends to the spectrum of single particle (Fig. 14). One can see that this spectrum is similar to the spectrum of single sphere particle, but the dispersion is still rather visible. This means that the spatial dispersion in the system of two particles appears due to the interaction between particles.

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126

V. Lozovski and A. Tsykhonya

Q(ω) , arb. units 2,2

100

ω ω0

10

2,0

0,0

2,5

0,5

ϑ , rad

2,0

1,0 1,5 1,5

ω ω0

1,8 0,0

0,5

ϑ , rad

(a)

1,5

(b)

Figure 14. Absorption spectrum of two spherical particles for function

1,0

d = 5r :

(a) surface of dissipative

Q (ϑ, ω) ; (b) projections of the chains of the surface onto the plane (ϑ, ω) .

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CONCLUSION The determination of resonant properties of nano-particles has to be based on the simple and clear concepts connected with the experimental measurements – this is the main idea of this work. Most acceptable experiments for these purposes are measurements of absorption of external electromagnetic field energy. This energy absorbed by unit volume of the system per unit of time can be evaluated as dissipative function. The dissipative function is calculated via effective susceptibility of the system. The effective susceptibility depends as on material, dimension and shape of the particles as on configuration of the system. Then one can contend that resonant properties of the system will be depended on the system characteristics and parameters of external field which is absorbed. The frequency ω and incident angle ϑ (determined relatively any chosen direction) are the characteristics of external electromagnetic field, absorbed by the system. It is clear that in the case of single nanoparticle the absorbed energy should not be depended on the incident angle because all points inside the particle are equivalent for the external field since in this case the wavelength of the field is much longer than dimension of the particle. This assertion, of course, does not mean that the local field inside the particle is homogeneous. The strong dependence of the local field inside the particle on coordinates leads to strong dependence of absorption spectra on the frequency. The dependence of resonant conditions on the incident angle appears in the case of the many-particle systems when the phases of the fields shift from one particle to another. Because short-range part of local field (near-field) decreases very quickly (~ R-3) when distance increases, the dependence of the resonance frequency on the incident angle ϑ is stronger for the small (relatively) distances d between nano-particles, but d should be order of wavelength of the external field. It means that for small d the derivative dω is evident. One dϑ

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Dispersion Properties of Nano-Scale Systems should note that in spite of the derivative

127

dω defines the group velocity in the case of wavedϑ

like excitation, in the case under consideration the excitations inside the system are local ones. It means that dispersion relations, which are defined in this work as chain projection of dissipative function Q (ϑ, ω) onto (ϑ, ω) -plane, mean only the conditions under which localized electrodynamical excitations can be excited in the system. Of course, the proposed approach for description of dispersion relations in the case of distributed systems (as a thin film, for example) gives possibility to obtain the dispersion relations of wave-like excitations – surface waves or waveguide modes. Thus, the useful and reliable approach for determining the dispersion properties for systems of nano-particles is proposed in the work.

APPENDIX Let us represent the solution of the self-consistent Lippmann-Schwinger equation

∫

Ei (R, ω) = Ei( I ) (R , ω) − iωμ 0 dR ′Gij (R , R ′, ω)χ jl (ω) El (R ′, ω) ,

(1A)

V

where Gil ( R , R ′, ω) is the photon propagator of the medium in which the object is situated, in the form

∫

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Ei (R, ω) = Ei( I ) (R , ω) − iωμ 0 dR ′Gij (R , R ′, ω) Χ jl (R ′, ω) El( I ) (R ′, ω) ,

(2A)

V

with Χ ij ( R , ω) the tensor of the effective susceptibility. The term “effective susceptibility” means that in opposite to the standard constitutive equation, connecting the local field Ei (R , ω) and local current J i (R, ω)

J i (R, ω) = −iωχil (ω) El (R, ω) ,

(3A)

the effective susceptibility connects the local current and external (concerning the object) field Ei ( R , ω) . Namely (I )

J i (R, ω) = −iωΧ il (R, ω) El( I ) (R, ω) .

(4A)

To find the effective susceptibility, let us notes that it follows from Eqs. (3A) and (4A) that

El (R, ω) = ( χli (ω) ) Χ ij (R, ω) E (j I ) (R, ω) . −1

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(5A)

128

V. Lozovski and A. Tsykhonya

Substituting Eq.(5A) in the left hand part of Eq.(2A) and integrating both left and right parts of the equation over volume of the object, one obtains

∫ dR ( χ (ω) )

−1

il

Χ lj (R , ω) E (j I ) (R , ω)

V

∫

∫ ∫

= dREi( I ) (R , ω) − iωμ 0 dR dR ′Gij (R , R ′, ω) Χ jl (R ′, ω) El( I ) (R ′, ω) V

V

V

Taking into account that the external field can be represented in the form

E ( R , ω) = (I ) j

∑

E (j I ) (ω)eiK ⋅R , the equation can be rewritten as

K

⎧

∑ ∫ dR ⎪⎨⎪( χ (ω) ) ⎩ il

K

V

−1

⎫⎪ Χ lj ( R , ω) + iωμ 0 dR ′Gil ( R ′, R , ω) Χ lj (R , ω) − δij ⎬ eiK ⋅R E (j I ) (ω) = 0 . (6A) ⎪⎭ V

∫

Because the exponents make up the full set of orthonormalized functions, the expression in the brackets must be equal to zero. Namely

( χil (ω) )

−1

∫

Χlj (R, ω) + iωμ0 dR′Gil (R′, R, ω) Χ lj (R, ω) − δij = 0 .

(7A)

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V

This equation has a solution

[

]

[

]

−1 ~ −1 −1 Χ ij (R , ω ) = (χ ij (ω ) ) + S ji (R, ω ) = χ il (ω ) δ jl + S ji (R, ω ) ,

(8A)

with the self-energy part S ji ( R , ω ) represented in the form

∫

S ji (R, ω) = −iωμ0 dR′G jl (R′, R, ω)χli (ω) .

(9A)

V

We note that using the reciprocal theorem, one can write

∫

∫

S ji (R, ω) = −iωμ0 dR′G jl (R′, R, ω)χli (ω) = −iωμ 0 dR′G jl (R, R′, ω)χli (ω) . V

V

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REFERENCES [1] Science and Engineering of One- and Zero-Dimensional Semiconductors, (Ed. by S.P. Beaumont, New York, 1989). [2] Nanocomposite Science and Technology. (Ed. by P.M. Ajayan, L.S. Schadler, P.V. Braun [3] WILEY-VCH Verlag GmbH Co., Weinheim, 2003) [4] K. Ploog, Phys. in Tech. 19 (1988) 196-205. [5] R. Fuchs, Phys. Rev. B 35 (1987) 7700–7703. [6] E. R. Brown, A. Bacher, D. Driscol at al., Phys. Rev. Lett. 90 (2003) 077403-1-4. [7] J. P. Kottmann, O.J. Martine, D.R. Smith, and S. Schultz, Phys. Rev. B 64 (2001) 235402-1-10. [8] C.B. Murray, D.J. Norris, M.G. Bawendi, J. Am. Chem. Soc. 115 (1993) 8706-8715. [9] T. Sugimoto, G.E. Dirige, A. Muramatsu, J. Coll. Interf. Sci. 182 (1996) 444-456. [10] X. Peng, L. Manna, W. Yang, J. Wickham at. al., Nature 404 (2000) 59-61. [11] E.F. Venger, A.V. Goncharenko, M.L. Dmitruk, Optics of Small Particles and Disperse Media, Naukova Dumka, Kyiv 1999. [12] R Ruppin, J. Phys.: Condens. Matter 10 (1998) 7869–7878. [13] R. Fuchs and F. Claro, Phys. Rev. B 35 (1987) 3722. [14] R. Ruppin, Opt. Commun. 136 (1997) 395. [15] R. Ruppin, J. Phys.: Condens. Matter 10 (1998) 7869. [16] J.C. Weeber, Ch. Girard, J.R. Krenn, A. Dereux and J.P. Goudonnet, J. Appl. Phys. 86 (1999) 2576. [17] R. Ruppin, Surf. Sc. 127 (1983) 108-118. [18] R. Fuchs, Phys. Rev. B 11 (1975) 1732-1740. [19] L. Landau, E. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960). [20] O. Keller, Phys. Rep. 268 (1996) 85-262. [21] C. Girard, C. Joachim, S. Gauthier, Rep. Prog. Phys. 63 (2000) 893-938. [22] S. Patane, P.G. Gucciardi, M. Labardi, M. Allegrini, Riv. Nouvo Cimento 27 (2004)1-46. [23] S. Bozhevolnyi, V. Lozovski, Phys. Rev. B. 61 (2000) 11139-11150. [24] V. Lozovski, Physica E 19 (2003) 263 – 277. [25] S.I. Bozhevolnyi, V. Lozovski, Yu. Nazarok, Physica E 11 (2001) 323-331. [26] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics (Part 2), Course of Theoretical Physics, Vol. 9 (Pergamon, Oxford,1980). [27] J. van Bladel, IRE Trans. Antennas Progr. AP-9 (1961) 563. [28] A.D. Yaghjian, Proc. IEEE 68 (1980) 248. [29] Lakhtakia, Opt. Commun. 79 (1990) 1-5. [30] O. Keller, M. Xiao, and S. Bozhevolnyi, Surf.Sci. 280 (1993) 217.

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In: Perspectives in Theoretical Physics Editors: T. F. George, R. R. Letfullin and G. Zhang

ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 9

AN ALGORITHM OF LOCAL HIDDEN VARIABLES J. F. Geurdes* C.v.d. Lijnstraat 164, 2593 NN Den Haag, The Netherlands

ABSTRACT A simple Local Hidden Variables model is described that can violate Bell’s inequality in the form of CHSH with ‘raw’ correlations.

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1. INTRODUCTION The present paper will be based upon Gill’s (R. Gill, 2003 & R. Gill, 2007) representation of an EPR type of experiment. Like Gill, we employ a chain of ‘classical’ computers to simulate a measurement in an EPR Bell type experiment. However, we intent to model a completely perfect experiment as performed by Weihs (G. Weihs, T. Jennewein, C. Simon, H. Weinfurter & A. Zeilinger, 1998). We note however, that the computer model of physics is a too classical view on what needs to be added to quantum theory to make it local and causal. It is not at all by necessity that an addition to quantum theory should imply a return to classical physics. We propose to employ the chain of computers A-X-O-Y-B. Computers X-O-Y simulate ‘Nature’ while computers A and B simulate the experimental set-up. The protocol to be followed is as follows. i. ii. iii.

Computer X receives a message from computer O and is subsequently completely cut off from communication with O. Computer X receives a message from computer A. Computer X produces either a, +1, or a, -1.

An identical protocol is employed for computer Y and computer B. *

E-mail address: [email protected]

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In this representation of the experiment, one can stipulate that computers A and B generate randomly selected numbers, a, and, b, from the set, {1,2}. In addition, the computers X and Y generate measurements, x, and, y, from the set {-1,+1}. Based on the random selected parameters, a, and, b, we can define two total outcomes of an experiment of n=1,2,3,... .,N trials in total. We have

N

N

= a, b

= card{n : x ( n ) = y ( n ) , ( x ( n ) , y ( n ) , a ( n ) , b ( n ) ) = ( x, y, a, b)}

(1)

≠ = card{n : x ( n ) ≠ y ( n ) , ( x ( n ) , y ( n ) , a ( n ) , b ( n ) ) = ( x, y, a, b)} a, b

If, subsequently, the correlation is defined as a statistical raw ‘uncentered’ moment we may write

= ≠ −N a, b a, b . = ρ = ≠ a, b +N N a, b a, b N

(2)

For this correlation we may write a Bell type inequality like

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S=ρ

1,2

−ρ

1,1

−ρ

2,1

−ρ

2,2

≤2 2 .

(3)

Generally it is believed that the inequality in Eq. (3) cannot be violated with the use of local hidden variable models. However, the author would like to note his paper (J.F. Geurdes, 2007) in which the BI is violated with local hidden variables.

2. PROPOSED ALGORITHM In the experiment of Weihs et al., use is made of a timing mechanism between A and B to synchronize their clocks in such a way that equally occurring events can be marked as such. Of course, in a perfect situation, one would only need to sequentially match the occurrences after the experiment, but then still it would be necessary to have a perfect synchronization between A and B to mark the beginning of the measuring sequence. This means the use of a pair of perfectly synchronized clocks. If the experimenters are granted the right of perfectly synchronized clocks, then, in Gill’s model, nature’s computers should also be granted this same right. This means that the communications between X-O-Y are such that, each time, a pair of perfectly synchronized running ‘clocks’ should be at the disposal of the information arising from O. Let us call this information ‘simulated particles’. We propose to write, p X for the

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133

particle going into the X direction and, pY for the particle going to the Y. Again, if both, A and B are allowed to synchronize their time perfectly outside the light cone, then X and Y should be granted the same opportunity. Hence, we now propose that both, p X and, pY , somehow, each carry a clock with them that ‘ticks’ in four steps (1,i,-1,-i), i= − 1 . Moreover, this ticking of both clocks is related to the singlet state that is used in the EPR Bell type experiments. Let us define a clock for each simulated particle. We have, a clock for, p X , defined by, Clock X (1, i,−1,−i ) and similarly for pY . The clocks tick complementary, meaning that the sum of each clock position sums to zero. We have

Clock X (1, i,−1,−i ) + Clock Y (1, i,−1,−i ) = 0

(4)

One must read the previous spin-clock ‘equation’ such that each clock position of the ‘particle-simulation’ in computer X is ‘compensated’ by a clock position of the ‘particle simulation’ in the Y computer. E.g. when the clock of, p X , stands at, 1, then, the one of, pY , is, -1, when the clock of, p X , stands at, i, then, the one of, pY , is, -i, etc. In addition, let us propose the following relation between spin and clock. We propose the following relation.

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↑ X = Clock X (+1,+i ) and ↓ X = Clock X (−1,−i )

(5)

This up and down sign is arbitrary and can be changed but for the sake of the discussion it makes sense to hold on to the previous definition. Hence, simulated particle, p X , carries an up spin when its clock is either in the position +1 or in the position +I, etc. The same relation as in Eq. (5) is valid for, pY . Hence, one can see that the up-down or down-up pairing of the ‘singlet condition’ is realized in the proposed clock variant. The relation with the measurement settings can now be inserted. For, pY , we have, b=1 and b=2, are, in the following definition, associated to clock positions.

b = 1 ↔ Clock Y (1,−1) and b = 2 ↔ Clock Y (i,−i )

(6)

Similarly we have for p X

a = 1 ↔ Clock X (i,−1) and a = 2 ↔ Clock X (1,−i )

(7)

Hence, the clock positions, 1, and, -1, are, in case of, pY , associated to parameter value, 1, etc. Suppose that p X enters X. Its spin-clock position is then undisclosed. Suppose that a=2. According to Eq. (7) we might have, Clock-position-of-pX is either, 1, or, -i. The position

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1 corresponds to ↑, while -i corresponds to ↓. If, on the other hand, the parameter is equal to b=1, this is represented by Clock-position-of-pY is -1, which is a ↓ spin, or by Clock-positionof-pY is +1, which is an ↑ spin. Because of the singlet state that is observed all times through the use of the two spin-position-clocks, we have only the possibility, Clock-position-of-pX is 1, which is an ↑ spin, while Clock-position-of-pY is -1, which is a ↓ spin. Hence, when there is a simultaneous measurement of spins at X and Y, both spin-clocks stop at the position related to the setting, as described in Eqs. (6) and (7), but at all times observe the ‘singlet condition’ relation of their clocks, given in Eq. (4). According to Eq. (5) this observation of the ‘singlet condition’ is related to the up or down position of the spin. Hence X receives a spin ↑, with a +1 on its spin clock, together with a=2, and ‘deduces’ that b=1. This implies that X produces x=scos(aπ), when y=scos(bπ), s is a random number, set by O at either +1 or -1, prior to sending the simulated particles to X and Y. This number can be used to determine a spin-up or spin-down relation in, for instance, Eq. (5). If s=-1, up becomes down and down becomes up.

Figure 1. Spin-clocks used by the X-side particle (left) and the Y-side particle (right). For completeness, the ‘out of the paper’ projection is provided also. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Note that in the present case we have assigned the decision power solely to X, which is sufficient for the present model. Note also that it is useless to argue against the use of the spin-clock when on the side of the experimentalists, the clock is needed to synchronize the events in the experiment. This synchronisation is, at least one time, necessary to make sense out of the experimental results. In the figure in the appendix, the clocks and their projections are presented. It can be checked that all situations are covered. This means that X receives a ↓ with a=2 and the clock on -i, this implies that Y receives a ↑ with b=2 and the clock on i. X decides to send out x=-scos(aπ), when, y=scos(bπ). Similarly, X receives a ↑ with a=1 and the clock on +i, while Y receives a ↓ with b=2 and the clock on -i. X decides to respond with x=-scos(aπ), when, y=scos(bπ). Finally, X receives a ↓ with a=1 and the clock on -1 while Y receives a ↑ with b=1 and the clock on 1. X decides to responds with x=-scos(aπ), when, y=scos(bπ). The decisions are tabulated in Table I and Table II. Table I. Response function of X in relation to the setting of the parameter a, the spin of pX and the value on its spin clock. First column, a, second column, spin state, third column, clock state fourth column, response function x. a spin 2 up 2 down

clock +1 -i

response x +s.cos(a.π) -s.cos(a.π)

1 up 1 down

+i -1

-s.cos(a.π) -s.cos(a.π)

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Table II. Response function of Y in relation to the setting of the parameter b, the spin of pY and the value on its spin clock. First column b, second column spin state third column clock state, fourth column response function y. b spin 2 up 2 down

clock +i -i

response y +s.cos(b.π) +s.cos(b.π)

1 up 1 down

+1 -1

+s.cos(b.π) +s.cos(b.π)

In the following table the response functions x and y will be tabulated in relation to the signs of the correlation function defined by Eqs. (1) and (2). Note that the randomly defined s determines the sign. Hence s=+1 we take from +1/-1 the +1, when s=-1 we take -1, etc. This leads to the following table. Table III. Response functions x and y in relation to the setting of the parameters a and b. a 2 1 2 1

b 2 1 1 2

x -1/+1 +1/-1 +1/-1 +1/-1

y +1/-1 -1/+1 -1/+1 +1/-1

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J. F. Geurdes

3. EVALUATION From Table III it follows that

N a≠,b = 0

∀ ( a ,b )∈{(1, 2 )}

N a=,b = 0

∀ ( a ,b )∈{( 2, 2),(1,1),( 2,1)}

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This implies that ρ1,2=1, ρ2,2=-1, ρ1,1=-1 and ρ2,1=-1. Hence, ideally, we would find for the S in the CHSH inequality (Eq. (3)), S=4, which violates the limit of 2√2. The correctness of this limit for hidden variables was, I repeat, already questioned previously. However, should one accept this limit, then we may conclude that the BI can be violated. Note that this is no positive proof of local hidden variables, however, it is shown that BI’s like the one proposed in the cited literature (see also R.D. Gill, G. Weihs, A. Zeilinger and M. Zukowski, 2002) are no valid way of excluding locality and causality from quantum theory.

REFERENCES [1] R. D. Gill, Time, Finite Statistics, and Bell’s Fifth Position, arXiv: quant-ph/0301059v1 14 Jan 2003. [2] R. D. Gill, Better Bell inequalities, arXiv:math/0610115v2 [math.ST] 6 Sep 2007. [3] R. D. Gill, G. Weihs, A. Zeilinger and M. Zukowski, No time loophole in Bell’s theorem: The Hess-Philipp model is nonlocal, www.pnas.org/cgi/doi/10.1073/pnas.182536499, 2002. [4] J. F. Geurdes, Bell’s theorem refuted with a Kolmogorovian counterexample, to be published in Int. J. Theor. Phys, Group Theor. & Nonl. Optics 12(3). [5] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter & A. Zeilinger, Violoation of Bell’s inequality under strict Einstein locality conditions, arXiv:quant-ph/9810080v1 26 Oct 1998.

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In: Perspectives in Theoretical Physics Editors: T. F. George, R. R. Letfullin and G. Zhang

ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 10

SUPERCONDUCTING STATE PARAMETERS OF 3D-TRANSITION METALS BINARY ALLOYS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj – Kutch, 370 001, Gujarat, India

ABSTRACT

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The study of the superconducting state parameters (SSPs) viz. electron-phonon coupling strength

λ,

exponent

α

Coulomb pseudopotential

μ* ,

transition temperature

and effective interaction strength

TC , isotope effect

N OV of 3d-transition metals binary

alloys have been made extensively in the present work using a model potential formalism and employing the pseudo-alloy-atom (PAA) model for the first time. It is observed that the electron-phonon coupling strength λ and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential

μ* ,

isotope effect exponent

α

and effective interaction strength

N OV show weak dependences on the local field correction functions. The present results of the SSPs are found in qualitative agreement with the available experimental data wherever exist.

PACS: 61.43.Dq; 71.15.Dx; 74.20.-z; 74.70.Ad Keywords: Pseudopotential; superconducting state parameters (SSPs); 3d-transition metals binary alloys.

∗

Tel.: +91-2832-256424, E-mail address: [email protected].

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Aditya M. Vora

1. INTRODUCTION During last several years, the superconductivity remains a dynamic area of research in condensed matter physics with continual discoveries of novel materials and with an increasing demand for novel devices for sophisticated technological applications. A large number of metals and amorphous alloys are superconductors, with critical temperature TC ranging from 1-18K. The pseudopotential theory has been used successfully in explaining the superconducting state parameters (SSPs) for metallic complexes by many workers [1-13]. Many of them have used well known model pseudopotential in the calculation of the SSPs for the metallic complexes. Recently, we have studied the SSPs of some metallic superconductors using single parametric model potential formalism [3-11]. The study of the SSPs of the binary alloy based superconductors may be of great help in deciding their applications; the study of the dependence of the transition temperature TC on the composition of metallic elements is helpful in finding new superconductors with high TC . The application of pseudopotential to binary alloys involves the assumption of pseudoions with average properties, which are assumed to replace three types of ions in the binary systems, and a gas of free electrons is assumed to permeate through them. The electron-pseudoion is accounted for by the pseudopotential and the electron-electron interaction is involved through a dielectric screening function. For successful prediction of the superconducting properties of the alloying systems, the proper selection of the pseudopotential and screening function is very essential [3-11]. A well known empty core (EMC) model potential of Ashcroft [14] is applied here in the study of the SSPs viz. electron-phonon coupling strength

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transition temperature TC , isotope effect exponent

α

λ , Coulomb pseudopotential μ * , and effective interaction strength

N OV of 3d-transition metals binary alloys. To see the impact of various exchange and correlation functions on the aforesaid properties, we have used five different types of local field correction functions proposed by Hartree (H) [15], Taylor (T) [16], Ichimaru-Utsumi (IU) [17], Farid et al. (F) [18] and Sarkar et al. (S) [19]. We have incorporated for the first time more advanced local field correction functions due to IU [17], F [18] and S [19] in the investigation of the SSPs of the 3d-transition metals binary alloys. In the present work, the pseudo-alloy-atom (PAA) model was used to explain electronion interaction for binaries [3-11]. It is well known that the pseudo-alloy-atom (PAA) model is a more meaningful approach to explain such kind of interactions in binary systems [3-11]. In the PAA approach a hypothetical monoatomic crystal is supposed to be composed of pseudo-alloy-atoms, which occupy the lattice sites and form a perfect lattice in the same way as pure metals. In this model the hypothetical crystal made up of PAA is supposed to have the same properties as the actual disordered alloy material and the pseudopotential theory is then applied to studying various properties of an alloy and metallic glass [3-11]. The complete miscibility in the alloy systems is considered as a rare case. Therefore, in such binary systems the atomic matrix elements in the pure states are affected by the characteristics of alloys such as lattice distortion effects and charging effects. In the PAA model, such effects are involved implicitly. In addition to this it also takes into account the self-consistent treatment implicitly

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Superconducting State Parameters of 3d-Transition Metals Binary Alloys

139

[3-11]. Looking to the advantage of the PAA model, we propose a use of PAA model to investigate the SSPs of 3d-transition metals based binaries. To describe electron-ion interactions in the binary systems, the Ashcroft’s empty core (EMC) single parametric local model potential [14] is employed in the present investigation. The form factor W (q ) of the EMC model potential in wave number space is (in au) [14]

W (q ) =

− 8πZ cos(qrC ) . ΩO q 2ε (q )

(1)

here, Z , ΩO , ε (q ) and rC are the valence, atomic volume, Hartree dielectric function [15] and parameter of the model potential of 3d-transition metals binary alloys, respectively.

2. METHOD OF COMPUTATION In the present investigation for binary mixtures, the electron-phonon coupling strength λ is computed using the relation [3-11]

λ

m Ω = 2 b 0 2 4π k F M 〈 ω 〉

2k F

∫q

3

W (q ) dq . 2

(2)

0

Here mb is the band mass, M the ionic mass, ΩO the atomic volume and k F the Fermi Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

wave vector. The effective averaged square phonon frequency relation given by Butler [20],

ω2

12

= 0.69 θ D , where θ D is the Debye temperature of the

3d-transition metals binary alloys. 2 Using X = q 2k F and Ω O = 3 π Z

λ

=

12 m b Z M 〈ω 〉 2

1

∫X

3

ω 2 is calculated using the

W (X )

2

(k F )3 , we get Eq. (2) in the following form,

dX ,

(3)

0

where W(X ) is the EMC form factor of the 3d-transition metals binary alloys,

respectively. The expression of EMC form factor W(X ) in real space (in au) is given by

W (X ) =

− 2πZ cos (2k F XrC ) . Ω O X 2 k F2 ε ( X )

The Coulomb pseudopotential

μ * is given by [3-11]

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(4)

140

Aditya M. Vora

μ* =

mb π kF

1

dX

∫ ε (X ) 0

⎛ EF mb 1+ ln ⎜⎜ π k F ⎝ 10 θ D

⎞ ⎟⎟ ⎠

1

dX ∫0 ε (X )

.

(5)

Where EF is the Fermi energy and ε(X ) the modified Hartree dielectric function, which is written as [15]

ε(X ) = 1 + (ε H (X ) − 1) (1 − f (X)) .

(6)

Here ε H (X ) is the static Hartree dielectric function and the expression of it is given by [15], ε H (X ) = 1 +

⎛ 1 − η2 ⎞ 1+ η m e2 ⎜ ⎟ ;η = q . ln + 1 ⎜ 2η ⎟ 2 2 2k F 1 − η 2 π kF η ⎝

(7)

⎠

While f (X ) is the local field correction function. In the present investigation, the local

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field correction functions due to Hartree (H) [15], Taylor (T) [16], Ichimaru-Utsumi (IU) [17], Farid et al. (F) [18] and Sarkar et al. (S) [19] are incorporated to see the impact of exchange and correlation effects. The details of all the local field corrections are below. The H-screening function [15] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(8)

Taylor (T) [16] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [16],

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ ⎢1 + ⎥. π k F2 ⎦ ⎣

(9)

The Ichimaru-Utsumi (IU) local field correction function [17] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is

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Superconducting State Parameters of 3d-Transition Metals Binary Alloys

141

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎛ ⎤⎪ k ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ ⎝ kF ⎠ ⎪ . (10) f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ 3 ⎠ ⎝ kF ⎠ ⎛q⎞ ⎢⎣ ⎝ kF ⎠ ⎝ ⎥⎦ ⎪ ⎛ q ⎞ ⎝ kF ⎠ ⎝ kF ⎠ 4⎜⎜ ⎟⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩

On the basis of Ichimaru-Utsumi (IU) local field correction function [17] local field correction function, Farid et al. (F) [18] have given a local field correction function of the form ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎤⎪ ⎛q⎞ ⎛q⎞ ⎛q⎞ k ⎝ kF ⎠ ⎪ . (11) f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛q⎞ ⎢⎣ ⎝ kF ⎠ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 2 − ⎜⎜ ⎟⎟ ⎪ 4⎜ ⎟ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩ Based on Eqs. (10-11), Sarkar et al. (S) [19] have proposed a simple form of local field correction function, which is of the form

⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎛ ⎞ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎜ ⎟ ⎝ kF ⎝ ⎠

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(12)

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The parameters AIU , B IU , C IU , AF , B F , C F , DF , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [17-19]. After evaluating

α

λ and μ * , the transition temperature TC and isotope effect exponent

are investigated from the McMillan’s formula [3-11, 21]

TC =

⎡ − 1.04(1 + λ ) ⎤ θD exp⎢ ⎥, * 1.45 ⎣ λ − μ (1 + 0.62λ ) ⎦

(13)

α=

θD 1⎡ ⎛ * ⎢1 − ⎜⎜ μ ln 1.45TC 2⎢ ⎝ ⎣

(14)

2 ⎞ 1 + 0.62λ ⎤ ⎟⎟ ⎥. ⎠ 1.04(1 + λ )⎥⎦

The expression for the effective interaction strength N OV is studied using [4-9]

N OV =

λ − μ* . 10 1+ λ 11

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(15)

142

Aditya M. Vora

3. RESULTS AND DISCUSSION The constants and parameters used in the present investigation are tabulated in Table 1. To determine the input parameters and various constants for PAA model [3-11], the following definitions for 3d-transition metals binary alloys A1− x Bx are adopted,

Z = ( 1 − x )(Z A ) + x (Z B ) ,

(16)

M = ( 1 − x )(M A ) + x (M B ) ,

(17)

ΩO = ( 1 − x )(ΩO A ) + x (ΩO B ) ,

(18)

rC = ( 1 − x )(rCA ) + x (rCB ) ,

(19)

θ D = ( 1 − x )(θ DA ) + x (θ DB ) .

(20)

Where x is the concentration factor of the second metallic component of the alloy. The input parameters for Ti1-xVx and V1-xCrx (i.e. Cr>60%) binary alloys are computed from the metallic data of the pure components using PAA model. Table 1. Input parameters and other constants.

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Alloys Ti0.80V0.20 Ti0.70V0.30 Ti0.50V0.50 Ti0.25V0.75 Ti0.15V0.85 V0.90Cr0.10 V0.80Cr0.20 V0.75Cr0.25 V0.60Cr0.40 V0.50Cr0.50 V0.40Cr0.60 V0.20Cr0.80 V0.10Cr0.90 V0.055Cr0.945

Z 4.80 4.70 4.50 4.25 4.15 5.90 5.80 5.75 5.60 5.50 5.40 5.20 5.10 5.06

rC

ΩO 3

(au)

(au)

0.9148 0.7739 0.7135 0.6622 0.6407 0.9200 0.9612 0.9943 1.0111 1.0240 0.9945 0.8252 0.8481 0.8849

98.55 100.98 105.83 111.90 114.32 82.27 83.54 84.18 86.08 87.35 88.62 91.16 92.43 93.00

kF

M

θD

(au)

(amu)

(K)

1.1298 1.1128 1.0798 1.0399 1.0243 1.2853 1.2715 1.2646 1.2442 1.2308 1.2174 1.1909 1.1778 1.1719

50.33 50.03 49.42 48.66 48.36 51.89 51.78 51.73 51.57 51.47 51.36 51.15 51.05 51.00

354.40 362.60 379.00 399.50 407.70 370.00 400.00 425.00 450.00 470.00 454.80 396.40 367.20 354.06

ω2

2

x 10-6

(au)2 2.41508 2.52813 2.76199 3.06886 3.19614 2.63237 3.07655 3.47314 3.89376 4.24756 3.97727 3.02142 2.59268 2.41045

Table 2 shows the presently calculated values of the SSPs viz. electron-phonon coupling strength

λ , Coulomb pseudopotential μ * , transition temperature TC , isotope effect

exponent

α

and effective interaction strength N OV at various concentrations for 3d-

transition metals binary alloys with available experimental findings [21]. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Superconducting State Parameters of 3d-Transition Metals Binary Alloys

143

The calculated values of the electron-phonon coupling strength λ for 3d-transition metals binary alloys, using five different types of the local field correction functions with EMC model potential, are shown in Table 2 with the experimental data [21]. It is also observed from the Table 2 that, for Ti1-xVx binary alloys λ goes on increasing from the values of 0.5339→0.9094 as the concentration ‘ x ’ of ‘V’ is increased from 0.20→0.85, while for V1-xCrx binary alloys, the concentration ‘ x ’ of ‘Cr’ increases from 0.10→0.945, λ goes on decreasing from the values of 0.6906→0.20. The increase or decrease in λ with concentration ‘ x ’ of ‘V’ or ‘Cr’ shows a gradual transition from weak coupling behaviour to intermediate coupling behaviour of electrons and phonons, which may be attributed to an increase of the hybridization of sp-d electrons of ‘V’ and ‘Cr’ with increasing concentration ( x ). Table 2. Superconducting state parameters of the 3d transition metals alloys. H 0.5339 0.1180

T 0.6755 0.1262

Present results IU 0.6912 0.1273

F 0.6951 0.1275

S 0.6078 0.1219

Expt. [21] 0.54 −

3.5470

7.3062

7.7329

7.8454

5.4672

3.5

0.3960 0.2800 0.6193 0.1192

0.4201 0.3403 0.7954 0.1277

0.4215 0.3463 0.8228 0.1288

0.4220 0.3478 0.8239 0.1290

0.4117 0.3130 0.7204 0.1233

− − 0.62 −

6.1403

11.4071

12.2390

12.2578

9.2141

6.14

0.4197 0.3200 0.6508 0.1217

0.4379 0.3875 0.8557 0.1306

0.4399 0.3970 0.8901 0.1318

0.4397 0.3973 0.8923 0.1320

0.4330 0.3608 0.7630 0.1262

− − 0.65 −

7.3055

13.7760

14.8592

14.9126

10.9021

7.30

0.4225 0.3324 0.6429 0.1250

0.4414 0.4078 0.8649 0.1344

0.4436 0.4191 0.9033 0.1357

0.4436 0.4198 0.9070 0.1359

0.4355 0.3760 0.7601 0.1299

− − 0.65 −

7.1608

14.4797

15.7434

15.8533

11.0423

7.16

0.4149 0.3269 0.6378 0.1263

0.4379 0.4089 0.8656 0.1360

0.4405 0.4215 0.9051 0.1373

0.4407 0.4226 0.9094 0.1375

0.4298 0.3727 0.7568 0.1314

− − 0.65 −

7.0219

14.6426

15.9681

16.1030

11.0003

7.02

α

0.4111

0.4360

0.4389

0.4391

0.4271

−

N0V

0.3238

0.4083

0.4212

0.4226

0.3705

−

Alloys

SSPs λ μ*

Ti0.80V0.20

TC

(K)

α N0V λ μ*

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Ti0.70V0.30

TC

(K)

α N0V λ μ* Ti0.50V0.50

TC

(K)

α N0V λ μ* Ti0.25V0.75

TC

(K)

α N0V λ μ* Ti0.15V0.85

V0.90Cr0.10

TC

(K)

λ

0.5028

0.6357

0.6481

0.6557

0.5403

0.53

μ*

0.1110

0.1181

0.1190

0.1191

0.1136

−

3.2103

6.8969

7.2486

7.4943

4.1368

3.21

α

0.4010

0.4255

0.4266

0.4278

0.4086

−

N0V

0.2689

0.3281

0.3330

0.3362

0.2861

−

TC

(K)

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144

Aditya M. Vora Table 2. Continued Alloys

SSPs λ μ*

V0.80Cr0.20

TC

(K)

α N0V λ μ* V0.75Cr0.25

TC

(K)

α N0V λ μ* V0.60Cr0.40

TC

(K)

α N0V λ μ* V0.50Cr0.50

TC

(K)

α N0V λ μ*

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V0.40Cr0.60

TC

(K)

α N0V λ μ* V0.20Cr0.80

TC

(K)

α N0V λ μ* V0.10Cr0.90

TC

(K)

α N0V λ μ* V0.055Cr0.945

TC α N0V

(K)

H 0.4458 0.1126

T 0.5696 0.1199

Present results IU 0.5813 0.1208

F 0.5890 0.1210

S 0.4774 0.1154

Expt. [21] 0.48 −

1.9006

5.0153

5.3430

5.5890

2.5453

1.90

0.3666 0.2371 0.4176 0.1137

0.4043 0.2963 0.5379 0.1211

0.4061 0.3013 0.5497 0.1221

0.4081 0.3048 0.5576 0.1223

0.3768 0.2525 0.4460 0.1166

− − 0.45 −

1.3622

4.1828

4.5061

4.7540

1.8598

1.36

0.3407 0.2203 0.3488 0.1154

0.3895 0.2799 0.4511 0.1231

0.3919 0.2851 0.4613 0.1241

0.3945 0.2889 0.4679 0.1243

0.3524 0.2344 0.3743 0.1185

− − 0.38 −

0.3704

1.8167

2.0133

2.1688

0.5934

0.37

0.2383 0.1772 0.3071 0.1167

0.3302 0.2326 0.3985 0.1246

0.3346 0.2375 0.4076 0.1256

0.3392 0.2411 0.4136 0.1258

0.2629 0.1909 0.3306 0.1199

− − 0.33 −

0.1003

0.8187

0.9345

1.0290

0.1917

0.10

0.1108 0.1489 0.2800 0.1169

0.2620 0.2011 0.3606 0.1248

0.2692 0.2058 0.3684 0.1259

0.2763 0.2092 0.3733 0.1261

0.1543 0.1620 0.3041 0.1202

− − 0.28 −

0.0281

0.3470

0.4030

0.4490

0.0698

< 0.025

-0.0232 0.1300 0.2000 0.1164

0.1878 0.1776 0.2503 0.1243

0.1970 0.1817 0.2565 0.1253

0.2062 0.1846 0.2573 0.1255

0.0523 0.1441 0.2270 0.1198

− − 0.20 −

4 x 10-6

0.0014

0.0022

0.0023

0.0002

−

-1.4818 0.0708 0.2000 0.1160

-0.5177 0.1027 0.2508 0.1239

-0.4604 0.1064 0.2568 0.1249

-0.4556 0.1068 0.2578 0.1251

-0.7793 0.0889 0.2270 0.1195

− − 0.20 −

4 x 10-6

0.0014

0.0022

0.0023

0.0002

−

-1.4440 0.0711 0.2078 0.1158

-0.4949 0.1033 0.2613 0.1236

-0.4417 0.1069 0.2671 0.1246

-0.4346 0.1075 0.2686 0.1248

-0.7615 0.0892 0.2346 0.1193

− − 0.20 −

2 x 10-5

0.0035

0.0049

0.0053

0.0005

−

-1.0965 0.0774

-0.3421 0.1112

-0.3046 0.1146

-0.2935 0.1155

-0.5893 0.0951

− −

This may also be attributed to the increase role of ionic vibrations in the 3d-transition metals rich region. Also, the H-screening yields lowest values of λ , whereas the values obtained from the F-function are the highest. The present results are found in qualitative agreement with the available experimental data [21].

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Superconducting State Parameters of 3d-Transition Metals Binary Alloys

145

It is noticed from the present study that, the percentile influence of the various local field correction functions with respect to the static H-screening function on the electron-phonon coupling strength λ is 13.84%-30.19%, 16.32%-33.04%, 17.24%-37.11%, 18.23%-41.08%, 18.66%-42.58%, 7.46%-30.41%, 7.09%-32.12%, 6.80%-33.52%, 7.31%-34.15%, 7.65%34.68%, 8.61%-33.32%, 13.50%-28.65%, 13.50%-28.90% and 12.90%-29.26% for Ti0.80V0.20, Ti0.70V0.30, Ti0.50V0.50, Ti0.25V0.75, Ti0.15V0.85, V0.90Cr0.10, V0.80Cr0.20, V0.75Cr0.25, V0.60Cr0.40, V0.50Cr0.50, V0.40Cr0.60, V0.20Cr0.80, V0.10Cr0.90 and V0.055Cr0.945 3d-transition metals binary alloys, respectively. The calculated results of the electron-phonon coupling strength λ for Ti0.80V0.20, Ti0.70V0.30, Ti0.50V0.50, Ti0.25V0.75, Ti0.15V0.85, V0.90Cr0.10, V0.80Cr0.20, V0.75Cr0.25, V0.60Cr0.40, V0.50Cr0.50, V0.40Cr0.60, V0.20Cr0.80, V0.10Cr0.90 and V0.055Cr0.945 deviate in the range of 1.13%-28.72%, 0.11%-32.89%, 0.12%-37.28%, 1.09%-39.54%, 1.88%-39.91%, 1.94%23.72%, 0.54%-22.71%, 0.89%-23.91%, 1.50%-23.13%, 0.18%-25.33%, 0%-33.32%, 0%28.65%, 0%-28.90% and 3.90%-34.30% from the experimental findings [21], respectively. The computed values of the Coulomb pseudopotential

μ * , which accounts for the

Coulomb interaction between the conduction electrons, obtained from the various forms of the local field correction functions are tabulated in Table 2. It is observed from the Table 2 that for 3d-transition metals binary alloys, the

μ * lies between 0.11 and 0.14, which is in

accordance with McMillan [21], who suggested

μ * ≈ 0.13 for transition metals. The weak

screening influence shows on the computed values of the

μ * . The percentile influence of the

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various local field correction functions with respect to the static H-screening function on

μ*

for the 3d-transition metals binary alloys is observed in the range of 3.31%-8.05%, 3.44%8.22%, 3.70%-8.46%, 3.92%-8.72%, 4.04%-8.87%, 2.34%-7.30%, 2.49%-7.46%, 2.55%7.56%, 2.69%-7.71%, 2.74%-7.80%, 2.82%-7.87%, 2.92%-7.82%, 3.02%-7.84% and 3.02%7.77% for Ti0.80V0.20, Ti0.70V0.30, Ti0.50V0.50, Ti0.25V0.75, Ti0.15V0.85, V0.90Cr0.10, V0.80Cr0.20, V0.75Cr0.25, V0.60Cr0.40, V0.50Cr0.50, V0.40Cr0.60, V0.20Cr0.80, V0.10Cr0.90 and V0.055Cr0.945, respectively. Again the H-screening function yields lowest values of the

μ * , while the values

obtained from the F-function are the highest. The theoretical or experimental data of the Coulomb pseudopotential

μ * is not available for the further comparisons.

According to the PAA model, the input parameters are used in the computations of electron-phonon coupling strength λ and the Coulomb pseudopotential

μ * , which are used

in the computation of the transition temperature TC . Table 2 contains calculated values of the transition temperature TC for 3d-transition metals binary alloys computed from the various forms of the local field correction functions along with the experimental findings [21]. From the Table 2 it can be noted that, the static H-screening function yields lowest TC whereas the F-function yields highest values of the TC . The present results obtained from the H-local field correction functions are found to be in good agreement with available experimental data [21]. The calculated results of the transition temperature TC for Ti0.80V0.20, Ti0.70V0.30, Ti0.50V0.50, Ti0.25V0.75, Ti0.15V0.85, V0.90Cr0.10, V0.80Cr0.20, V0.75Cr0.25, V0.60Cr0.40 and V0.50Cr0.50 deviate in the range of 1.34%-124.15%, 0%-99.64%, 0.08%-104.28%, 0.01%-121.41%,

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146

Aditya M. Vora

0.03%-129.39%, 0.01%-133.47%, 0.03%-194.16%, 0.16%-249.56%,.0.11%-486.16% and 0.30%-929% from the experimental findings [21], respectively. The values of the isotope effect exponent α for 3d-transition metals binary alloys are tabulated in Table 2. The computed values of the α show a weak dependence on the dielectric screening, its value is being lowest for the H-screening function and highest for the F-function. Since the experimental value of α has not been reported in the literature so far, the present data of α may be used for the study of ionic vibrations in the superconductivity of alloying substances. Since H-local field correction function yields the best results for λ and TC , it may be observed that

α

values obtained from this screening provide the best

account for the role of the ionic vibrations in superconducting behaviour of this system. The negative value of the α is observed for V0.40Cr0.60, V0.20Cr0.80, V0.10Cr0.90 and V0.055Cr0.945 3dtransition metals binary alloys, which indicates that the electron-phonon coupling in these metallic complexes do not fully explain all the features regarding their superconducting behaviour. The theoretical or experimental data of the isotope effect exponent α is not available for the further comparisons. The values of the effective interaction strength N OV are listed in Table 2 for different local field correction functions. It is observed that the magnitude of N OV shows that the 3dtransition metals binary alloys under investigation lie in the range of weak coupling superconductors. The values of the N OV also show a feeble dependence on dielectric screening, its value being lowest for the H-screening function and highest for the F-screening function. The variation of present values of the N OV show that, the 3d-transition metals

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binary alloys under consideration falls in the range of weak coupling superconductors. The theoretical or experimental data of the effective interaction strength N OV is not available for the further comparisons. In most of the SSPs for 3d-transition metals binary alloys, the present results are computed from Hartree (H) local field correction function [17] gives the best agreement with the experiment [21] using EMC model potential and found suitable in the present case in comparison with other local field correction functions. There are various forms of the local field correction functions are available in the literature. But, we have adopted here only five local field correction functions in the present computation for showing which functions are generated consistent values of the SSPs of the alloys. From all the local field correction functions, H- and T-local field correction functions are older functions while IU-, F- and Slocal field correction functions are more recent one. From such kind of comparative study, one can choose the best suitable function among them, which is our main aim of the present study. Also, the main differences of the local field correction functions are played in important role in the production of the SSPs of 3d-transition metals binary alloys. The Hartree (H) dielectric function [15] is purely static and it does not include the exchange and correlation effects. Taylor (T) [16] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. The Ichimaru-Utsumi (IU) local field correction function [17] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also satisfies the self consistency condition in the

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Superconducting State Parameters of 3d-Transition Metals Binary Alloys

147

compressibility sum rule and short range correlations. Therefore, Hartree (H) local field correction function [17] gives the best agreement with the experiment with EMC model potential and found suitable in the present case. On the basis of Ichimaru-Utsumi (IU) local field correction function [17], Farid et al. (F) [18] and Sarkar et al. [19] have given a local field correction function. Hence, IU- and F-functions represent same characteristic nature of the SSPs. Also, the SSPs computed from Sarkar et al. [19] local field correction are found in qualitative agreement with the available experimental data [21]. From the study of the Table 2, one can see that among the five screening functions the screening function due to H (only static–without exchange and correlation) [15] gives the minimum value of the SSPs while the screening function due to F [18] gives the maximum value. The present findings due to T [16], IU [17] and S [19] local field correction functions are lying between these two screening functions. Therefore, most recent local field correction functions due to IU [17], F [18] and S [19] are able to generate consistent and comparable results regarding the SSPs of the 3d-transition metals binary alloys as those obtained form more commonly employed H [15] and T [16] functions. The effect of local field correction

λ and μ * in which the dielectric function is directly involved, makes drastic and significant variation on the values of the TC ,

functions plays an important role in the computation of

α

and N OV . Thus, the use of these more promising local field correction functions is

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established successfully. The computed results of

α

and N OV are not showing any

abnormal values for 3d-transition metals binary alloys. According to Matthias rules [22, 23] the alloys having Z>2 exhibit superconducting nature. Hence, the presently computed 3d-transition metals binary alloys are the superconductors. Also, for Ti1-xVx binary alloys when we go from Z=4.15 to Z=4.80 and for V1-xCrx binary alloys when we go from Z=5.06 to Z=5.90, the electron-phonon coupling strength λ changes with lattice spacing “a”. Similar trends are also observed in the values of TC for most of the 3d-transition metals binary alloys. Hence, a strong dependency of the SSPs of the 3d-transition metals binary alloys on the valence Z is found. Also from the presently computed results of the SSPs of 3d-transition metals binary alloys, we are observed that, for Ti1-xVx binary alloys except Ti0.25V0.75 and Ti0.15V0.85, as the atomic volume ΩO of the alloys increases, the SSPs increases while those for V1-xCrx binary alloys as the atomic volume ΩO of the alloys increases, the SSPs decreases. Lastly, we would like to emphasize the importance of involving a precise form for the pseudopotential. It must be confessed that although the effect of pseudopotential in strong coupling superconductor is large, yet it plays a decisive role in weak coupling superconductors i.e. those substances which are at the boundary dividing the superconducting and nonsuperconducting region. In other words, a small variation in the value of electron-ion interaction may lead to an abrupt change in the superconducting properties of the material under consideration. In this connection we may realize the importance of an accurate form for the pseudopotential.

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148

Aditya M. Vora

CONCLUSIONS The comparison of presently computed results with available experimental findings is highly encouraging in the case of 3d-transition metals binary alloys, which confirms the applicability of the model potential. The theoretically observed values of SSPs are not available for most of the 3d-transition metals binary alloys therefore it is difficult to draw any special remarks. However, the comparison with other such theoretical data supports the present computations of the SSPs. Such study on SSPs of other binary alloys and metallic glasses is in progress.

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REFERENCES [1] A. V. Narlikar, S. N. Ekbote: Superconductivity and Superconducting Materials (South Asian Publishers New Delhi Madras, 1983). [2] P. B. Allen: Handbook of Superconductivity, Ed. C. P. Poole, Jr. (Academic Press, New York, 1999) p. 478. [3] A. M. Vora, M. H. Patel, S. R. Mishra, P. N. Gajjar, A. R. Jani, Solid State Phys. 44, 345 (2001). [4] P. N. Gajjar, A. M. Vora, A. R. Jani, Mod. Phys. Lett. B18, 573 (2004). [5] Aditya M. Vora, Physica C450, 135 (2006); Physica C458, 21 (2007). [6] Aditya M. Vora, Phys. Scr. 76, 204 (2007). [7] Aditya M. Vora, J. Supercond. Novel Mag. 20, 355 (2007); J. Supercond. Novel Mag. 20, 373 (2007); J. Supercond. Novel Mag. 20, 387 (2007). [8] A. M. Vora, M. H. Patel, P. N. Gajjar, A. R. Jani, Pramana-J. Phys. 58, 849 (2002). [9] P. N. Gajjar, A. M. Vora, M. H. Patel, A. R. Jani, Int. J. Mod. Phys. B17, 6001 (2003). [10] P. N. Gajjar, A. M. Vora, A. R. Jani, Indian J. Phys. 78, 775 (2004). [11] Aditya M. Vora, Comp. Mater. Sci. 40, 492 (2007). [12] V. Singh, H. Khan, K. S. Sharma, Indian J. Pure & Appl. Phys. 32, 915 (1994). [13] R. C. Dynes, Phys. Rev. B2, 644 (1970). [14] N. W. Ashcroft, Phys. Lett. 23, 48 (1966). [15] W. A. Harrison: Elementary Electronic Structure (World Scientific, Singapore, 1999). [16] R. Taylor, J. Phys. F: Met. Phys. 8, 1699 (1978). [17] S. Ichimaru, K. Utsumi, Phys.Rev.B24, 7386 (1981). [18] B. Farid, V. Heine, G. Engel, I. J. Robertson, Phys. Rev. B48, 11602 (1993). [19] A. Sarkar, D. Sen, H. Haldar, D. Roy, Mod. Phys. Lett. B12, 639 (1998). [20] W. H. Butler, Phys. Rev. B15, 5267 (1977). [21] W. L. McMillan, Phys. Rev. 167, 331 (1968). [22] B. T. Matthias: Progress in Low Temperature Physics, Ed. C. J. Gorter (North Holland, Amsterdam, 1957), Vol. 2. [23] B. T. Matthias, Physica 69, 54 (1973).

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ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 11

SUPERCONDUCTING STATE PARAMETERS OF BE-ZR GLASSY ALLOYS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj – Kutch, 370 001, Gujarat, India

ABSTRACT The study of the superconducting state parameters (SSPs) viz. electron-phonon

λ , Coulomb pseudopotential μ * , transition temperature TC , isotope α and effective interaction strength NOV of 3d-transition metals

coupling strength Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

effect exponent

binary alloys have been made extensively in the present work using a model potential formalism and employing the pseudo-alloy-atom (PAA) model for the first time. It is observed that the electron-phonon coupling strength λ and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential strength

μ* ,

isotope effect exponent

α

and effective interaction

N OV show weak dependences on the local field correction functions. The

present results of the SSPs are found in qualitative agreement with the available experimental data wherever exist.

Keywords: Pseudopotential; superconducting state parameters; Be-Zr metallic glasses. PACS Number(s): 61.43.Dq; 71.15.Dx; 74.20.-z; 74.70.Ad

∗

E-mail address: [email protected]. Tel.: +91-2832-256424

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1. INTRODUCTION The field of electron correlation in condensed matter, especially superconductivity, is one of the dynamic areas in condensed matter physics which involves discoveries of new and existing phenomena, novel materials and devices for sophisticated technological applications. During the last few years, superconducting metallic glasses based on various simple as well as transition metals have been obtained and studied by various researchers. The study of the SSP of the metallic glasses may be of great help in deciding their applications; the study of the dependence of the transition temperature TC on the composition of metallic glass is helpful in finding new superconductors with high TC . Experiments also show that the

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superconducting transition temperature TC is grater for amorphous metals than for crystals, which also depends on the composition of the metallic elements in the crystalline as well as amorphous phases [1-8]. Though the pseudopotential theory is found very successful in studying the various properties of the metallic glasses, there are very few scattered attempts to study the superconducting state parameters (SSP) of metallic glasses based on model potential [7-19]. The application of pseudopotential to a metallic glass involves the assumption of pseudoions with average properties, which are assumed to replace two types of ions in the binary systems, and a gas of free electrons is assumed to permeate through them. The electron-pseudoion is accounted for by the pseudopotential and the electron-electron interaction is involved through a dielectric screening function. For successful prediction of the superconducting properties of the metallic glasses, the proper selection of the pseudopotential and screening function is very essential [4-19]. Out of very large numbers of the metallic glasses, the SSP of only few metallic glasses are reported based on the pseudopotential, so far. Recently, Vora et al. [4-9] have studied the SSP of some metallic elements using single parametric model potential formalism. The SSP of Ca70Mg30 metallic glass has been reported by Gupta et al. [15] and Sharma et al. [16]. The study on SSP of Mg70Zn30 glass was made by Agarwal et al. [17] and Gupta et al. [18]. They have used Ashcroft’s empty core (EMC) model potential [20] in the computation of the SSP. The screening dependence of the SSP of Ca70Mg30 metallic glass has been studied by Sharma and Sharma [19] using Ashcroft’s empty core (EMC) model potential [20], SharmaKachhava’s linear potential [21] and Veljkovic and Slavic [22] model potential. The theoretical investigation of the SSP of BeC Al100−C metallic glasses i.e. Be90Al10 and Be70Al30 has been reported by Sharma et al. [14]. Experimental studies on the superconducting properties of BeC Zr100−C ( C = 30, 35, 40 and 45 at %) metallic glasses have been reported by Hasegawa and Tanner [23]. In most of these studied, Ashcroft’s empty core (EMC) model potential [20] is adopted in the calculation. But, nobody has used Hartree (H) [24], Taylor (T) [25], Ichimaru-Utsumi (IU) [26], Farid et al. (F) [27] and Sarkar et al. (S) [28] local field correction functions in their computation of the SSP. Also, ‘Be’ and ‘Zr’ being a good conductor and exhibits superconducting nature, this class of glasses may be quite suitable for industrial applications. Hence, in the present article, we decided to study the SSP viz. electron-phonon coupling strength

λ , Coulomb pseudopotential μ * , transition temperature TC , isotope effect

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Superconducting State Parameters of Be-Zr Glassy Alloys

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exponent α and effective interaction strength N OV of BeC Zr100−C ( C = 30, 35, 40 and 45 at %) metallic glasses on the basis of Ashcroft’s empty core (EMC) potential [20]. In the present work, the pseudo-alloy-atom (PAA) model was used to explain electronion interaction for binaries [4-10]. It is well known that the pseudo-alloy-atom (PAA) model is a more meaningful approach to explain such kind of interactions in binary systems [4-10]. In the PAA approach a hypothetical monoatomic crystal is supposed to be composed of pseudo-alloy-atoms, which occupy the lattice sites and form a perfect lattice in the same way as pure metals. In this model the hypothetical crystal made up of PAA is supposed to have the same properties as the actual disordered alloy material and the pseudopotential theory is then applied to studying various properties of an alloy and metallic glass [4-10]. The complete miscibility in the alloy systems is considered as a rare case. Therefore, in such binary systems the atomic matrix elements in the pure states are affected by the characteristics of alloys such as lattice distortion effects and charging effects. In the PAA model, such effects are involved implicitly. In addition to this it also takes into account the self-consistent treatment implicitly [4-10]. Looking to the advantage of the PAA model, we propose a use of PAA model to investigate the SSP of binary metallic glasses.

2. COMPUTATIONAL METHODOLOGY In the present investigation for binary mixtures, the electron-phonon coupling strength λ is computed using the relation [4-19]

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λ

m Ω = 2 b 0 2 4π k F M 〈 ω 〉

2k F

∫q

3

W (q ) dq . 2

(1)

0

Here mb is the band mass, M the ionic mass, ΩO the atomic volume and k F the Fermi wave vector. The effective averaged square phonon frequency relation given by Butler [20],

ω2

12

= 0.69 θ D , where θ D is the Debye temperature of the

binary metallic glasses. 2 Using X = q 2k F and Ω O = 3 π Z

λ

=

12 m b Z M 〈ω 〉 2

1

∫X

3

ω 2 is calculated using the

W (X )

2

(k F )3 , we get Eq. (2) in the following form,

dX ,

(2)

0

where W(X ) is the EMC form factor of the binary metallic glasses, respectively. To describe electron-ion interactions in the binary systems, the Ashcroft’s empty core (EMC) single parametric local model potential [14] is employed in the present investigation. The form factor W(X ) of the EMC model potential in wave number space is (in au) [20]

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Aditya M. Vora

W (X ) =

− 2πZ cos(2k F XrC ) , Ω O X 2 k F2 ε ( X )

(3)

here rC is the parameter of the model potential of metallic glasses. The Ashcroft’s empty core (EMC) model potential is a simple one-parameter model potential [20], which has been successfully found for various metallic complexes [9-19]. When used with a suitable form of dielectric screening functions, this potential has also been found to yield good results in computing the SSP of metallic glasses. Therefore, in the present work we use Ashcroft’s empty core (EMC) model potential with more advanced IU, F and S-local field correction functions for the first time. The model potential parameter rC may be obtained by fitting either to some experimental data or to realistic form factors or other data relevant to the properties to be investigated. In the present work, rC is fitted in such a way that, the presently computed values of the transition temperature TC of the metallic glasses obtained from all local field correction functions are found as close as possible with the experimental data [23] of TC for metallic glasses whichever are available in the literature. After fitting the model potential parameter rC , same rC is then used in the computation of the SSP of binary metallic glasses. The Coulomb pseudopotential

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μ* =

mb π kF

1

μ * is given by [4-19]

dX

∫ ε (X ) 0

⎛ EF mb ln ⎜⎜ 1+ π k F ⎝ 10 θ D

⎞ ⎟⎟ ⎠

1

dX ∫0 ε (X )

.

(4)

Where EF is the Fermi energy and ε(X ) the modified Hartree dielectric function, which is written as [24]

ε(X ) = 1 + (ε H (X ) − 1) (1 − f (X)) .

(5)

Here ε H (X ) is the static Hartree dielectric function and the expression of it is given by [24], ε H (X ) = 1 +

m e2 2 π k 2 η2 F

⎛ 1 − η2 ⎞ 1+ η ⎜ ⎟ ;η = q . ln + 1 ⎜ 2η ⎟ 2k F 1− η ⎝ ⎠

(6)

While f (X ) is the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [24], Taylor (T) [25], Ichimaru-Utsumi (IU) Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[26], Farid et al. (F) [27] and Sarkar et al. (S) [28] are incorporated to see the impact of exchange and correlation effects. The details of all the local field corrections are below. The Hartree (H) screening function [24] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(7)

Taylor (T) [25] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [25],

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ . ⎢1 + 2 ⎥ π k F ⎦ ⎣

(8)

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The Ichimaru-Utsumi (IU) local field correction function [26] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎛ ⎤⎪ k ⎪ (9) k ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎝ F ⎠ ⎬ ⎛q⎞ 3 ⎠⎝ kF ⎠ ⎢⎣ ⎝ kF ⎠ ⎝ ⎥⎦ ⎪ ⎛ q ⎞ ⎝ kF ⎠ ⎝ kF ⎠ 2 − ⎜⎜ ⎟⎟ ⎪ 4⎜ ⎟ ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩

On the basis of Ichimaru-Utsumi (IU) local field correction function [26] local field correction function, Farid et al. (F) [27] have given a local field correction function of the form

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎤⎪ ⎛q⎞ ⎛q⎞ ⎛q⎞ k ⎝ kF ⎠ ⎪ . (10) f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛ q ⎞⎪ ⎢⎣ ⎝ kF ⎠ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 2 − ⎜⎜ ⎟⎟ 4 ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩ Based on Eqs. (9-10), Sarkar et al. (S) [28] have proposed a simple form of local field correction function, which is of the form

⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

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(11)

154

Aditya M. Vora The parameters AIU , B IU , C IU , AF , B F , C F , DF , AS , BS and C S are the atomic

volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [26-28]. After evaluating

α

λ and μ * , the transition temperature TC and isotope effect exponent

are investigated from the McMillan’s formula [4-19, 30]

TC =

⎡ − 1.04(1 + λ ) ⎤ θD exp⎢ ⎥, * 1.45 ⎣ λ − μ (1 + 0.62λ ) ⎦

θD 1⎡ ⎛ α = ⎢1 − ⎜⎜ μ * ln 1.45TC 2⎢ ⎝ ⎣

2 ⎞ 1 + 0.62λ ⎤ ⎟⎟ ⎥. ( ) 1.04 1 λ + ⎥⎦ ⎠

(12)

(13)

The expression for the effective interaction strength N OV is studied using [4-9]

N OV =

λ − μ* . 10 1+ λ 11

(14)

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3. RESULTS AND DISCUSSION The constants and parameters used in the present investigation are tabulated in Table 1. To determine the input parameters and various constants for PAA model [4-10], the following definitions for binary metallic glasses A1− x Bx are adopted,

Z = ( 1 − x )(Z A ) + x (Z B ) ,

(15)

M = ( 1 − x )(M A ) + x (M B ) ,

(16)

ΩO = ( 1 − x )(ΩO A ) + x (ΩO B ) ,

(17)

rC = ( 1 − x )(rCA ) + x (rCB ) ,

(18)

θ D = ( 1 − x )(θ DA ) + x (θ DB ) .

(20)

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Superconducting State Parameters of Be-Zr Glassy Alloys

155

Where x is the concentration factor of the second metallic component of BeC Zr100−C glassy alloy. The presently calculated results of the SSP are tabulated in Table 2 with the experimental [23] findings. Table 1. Input parameters and other constants.

Z

Metallic glass Be45Zr55 Be40Zr60 Be35Zr65 Be30Zr70

3.10 3.20 3.30 3.40

rC (au) 0.8786 0.8781 0.9040 0.9457

ΩO 3

(au) 113.54 117.75 123.81 129.67

M

θD

(amu)

(K) 785.55 730.60 675.65 620.70

54.23 58.34 62.45 66.56

ω2

2

(au)2 x 10-6

1.8412 2.1816 2.5509 2.9490

Table 2. Superconducting state parameters of the Be-Zr metallic glasses. Glass

Be45Zr55

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Be40Zr60

Be35Zr65

Be30Zr70

λ μ*

H 0.4173 0.1454

Present results T IU F 0.5601 0.5848 0.5859 0.1588 0.1607 0.1610

S 0.5062 0.1530

TC (K)

1.0020

4.9874

5.9899

6.0097

3.1961

≤ 1.0

α N0V λ μ*

0.1423 0.1971 0.4631 0.1441

0.2698 0.2659 0.6221 0.1572

0.2834 0.2769 0.6496 0.1590

0.2829 0.2772 0.6508 0.1593

0.2413 0.2419 0.5621 0.1515

− − 1.1 0.13

TC (K)

2.1018

7.7616

9.0308

9.0582

5.3736

2.10

α N0V λ μ*

0.2365 0.2245 0.4831 0.1430

0.3233 0.2969 0.6468 0.1559

0.3329 0.3085 0.6748 0.1577

0.3325 0.3088 0.6759 0.1580

0.3036 0.2717 0.5865 0.1504

− − 1.2 0.13

TC (K)

2.6012

8.5481

9.8081

9.8286

6.1643

2.60

α N0V λ μ*

0.2682 0.2363 0.4941 0.1417

0.3410 0.3091 0.6570 0.1544

0.3490 0.3205 0.6839 0.1562

0.3486 0.3208 0.6849 0.1565

0.3252 0.2845 0.5990 0.1490

− − 1.5 0.13

TC (K)

2.8027

8.4972

9.6286

9.6428

6.3322

2.80

α N0V

0.2866 0.2432

0.3504 0.3146

0.3572 0.3254

0.3568 0.3257

0.3375 0.2913

− −

SSP

Expt. [23] − 0.13

The calculated values of the electron-phonon coupling strength λ for BeC Zr100−C metallic glasses, using five different types of the local field correction functions with EMC model potential, are shown in Table 2 with other experimental data [23]. It is noticed from the Table 1 that, λ vales are quite sensitive to the local field correction functions. It is also

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156

Aditya M. Vora

observed from the present study that, the percentile influence of the various local field correction functions with respect to the static H-screening function on the electron-phonon coupling strength λ is 21.30%-40.40%, 21.38%-40.53%, 21.40%-39.91% and 21.22%38.62% for Be45Zr55, Be40Zr60, Be35Zr65 and Be30Zr70 metallic glasses, respectively. Also, the H-screening yields lowest values of λ , whereas the values obtained from the F-function are the highest. It is also observed from the Table 2 that, λ goes on increasing from the values of 0.4173→0.6849 as the concentration ‘ C ’ of ‘Zr’ is increased from 0.10-0.30. The increase in λ with concentration ‘ C ’ of ‘Zr’ shows a gradual transition from weak coupling behaviour to intermediate coupling behaviour of electrons and phonons, which may be attributed to an increase of the hybridization of sp-d electrons of ‘Zr’ with increasing concentration ( C ), as was also observed by Minnigerode and Samwer [31]. This may also be attributed to the increase role of ionic vibrations in the Zr-rich region [14]. The calculated results of the electron-phonon coupling strength λ for Be40Zr60, Be35Zr65 and Be30Zr70 metallic glasses deviate in the range of 40.84%-48.90%, 43.67%-59.74% and 54.34%-67.06% from the experimental findings [23], respectively. The presently computed values of the electronphonon coupling strength λ are found in the qualitative agreement with the available experimental data [23]. The computed values of the Coulomb pseudopotential

μ * , which accounts for the

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Coulomb interaction between the conduction electrons, obtained from the various forms of the local field correction functions are tabulated in Table 2 with other experimental data [23]. It is observed from the Table 2 that for all metallic glasses, the

μ * lies between 0.15 and

0.18, which is in accordance with McMillan [30], who suggested

μ * ≈ 0.13 for simple and

transition metals. Hasegawa and Tanner [23] have also been taken

μ * = 0.13 in their study

of Be-Zr metallic glasses. The weak screening influence shows on the computed values of the

μ * . The percentile influence of the various local field correction functions with respect to the static H-screening function on

μ * for the metallic glasses is observed in the range of 5.23%-

10.73%, 5.14%-10.55%, 5.14%-10.50% and 5.14%-10.43% for Be45Zr55, Be40Zr60, Be35Zr65 and Be30Zr70 metallic glasses, respectively. Again the H-screening function yields lowest values of the μ , while the values obtained from the F-function are the highest. The present *

results are found in good agreement with the available experimental data [23]. Here also, as the concentration ( C ) of ‘Zr’ (in at %) increases the present results of

μ * decreases.

Table 2 contains calculated values of the transition temperature TC for BeC Zr100−C metallic glasses computed from the various forms of the local field correction functions along with the experimental [23]. From the Table 2 it can be noted that, the static H-screening function yields lowest TC whereas the F-function yields highest values of TC . The present results obtained from the H-local field correction functions are found in good agreement with available experimental data [23]. It is seen that TC is quite sensitive to the local field correction functions, and the results of TC by using H-screening are in best agreement with the experimental data for the BeC Zr100−C metallic glasses under investigation. The percentile

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Superconducting State Parameters of Be-Zr Glassy Alloys

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influence of the various local field correction functions with respect to the static H-screening function on TC for the metallic glasses is observed in the range of 218.79%-499.77%, 155.67%-330.97% , 136.98%-277.85% and 125.93%-244.05% for Be45Zr55, Be40Zr60, Be35Zr65 and Be30Zr70 metallic glasses, respectively. The calculated results of the transition temperature TC for Be45Zr55, Be40Zr60, Be35Zr65 and Be30Zr70 metallic glasses deviate in the range of 0.2%-500.97%, 0.09%-331.34%, 0.05%-278.02% and 0.10%-244.39% from the experimental findings [23], respectively. Also, the above observations indicate that simple metallic glasses having high valence (more than two) tend to have higher TC . Perhaps only exception is divalent Be-Zr metallic glasses where high TC is likely to be due to unusually high Debye temperature. The higher values of TC may be due to the electron transfer between the transition metal and other metallic element. The increase in TC has also been

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attributed to the excitonic mechanism resulting from the granular structure separated by semiconducting or insulating materials [2]. The subtle difference of the shift among the glassy Be-Zr alloys may be reflected in the values of the electron-phonon coupling strength λ . Since both the density and the values of the electron-phonon coupling strength λ are larger for Be30Zr70 alloys than those for the Be40Zr60 alloy, a larger shift of the centre of gravity of the phonon spectrum toward lower phonon energy for the former than the latter alloy may be expected. If such a trend toward a softer phonon spectrum is related to the degree of the disorder increases as ‘Be’ content decreases in the Be-Zr glassy system, which was reported by Hasegawa and Tanner [23] from the glass transition temperature Tg . From this, we may thus conclude that the increase of TC or λ with decreasing ‘Be’ in the Be-Zr glassy system is due to the increase of the degree of structural disorder as was noted by Hasegawa and Tanner [23]. The presently computed values of the TC are found in the range, which is suitable for further exploring the applications of the metallic glasses for usage like lossless transmission line for cryogenic applications. While metallic glasses show good elasticity and could be drawn in the form of wires as such they have good chances of being used as superconducting transmission lines at low temperature of the order of 7K. The values of the isotope effect exponent α for BeC Zr100−C metallic glasses are

tabulated in Table 2. The computed values of the α show a weak dependence on the dielectric screening, its value is being lowest for the H-screening function and highest for the F-function. The negative value of the α is observed in the case of metallic glasses, which indicates that the electron-phonon coupling in these metallic complexes do not fully explain all the features regarding their superconducting behaviour. Since the experimental or theoretical values of α has not been reported in the literature so far, the present data of α may be used for the study of ionic vibrations in the superconductivity of amorphous substances. Since H-local field correction function yields the best results for λ and TC , it

may be observed that α values obtained from this screening provide the best account for the role of the ionic vibrations in superconducting behaviour of this system. The most important feature noted here is that as the concentration ( C ) of ‘Zr’ (in at %) increases the present results of α increases sharply.

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158

Aditya M. Vora The values of the effective interaction strength N OV are listed in Table 2 for different

local field correction functions. It is observed that the magnitude of N OV shows that the metallic glasses under investigation lie in the range of weak coupling superconductors. The values of the N OV also show a feeble dependence on dielectric screening, its value being lowest for the H-screening function and highest for the F-screening function. The variation of present values of the N OV show that, the metallic glasses under consideration fall in the range of weak coupling superconductors. Here also, as the concentration ( C ) of ‘Zr’ (in at %) increases the present results of N OV increases. In most of the SSP for BeC Zr100−C metallic glasses, the present results are computed from Hartree (H) local field correction function [24] gives the best agreement with the experiment [23] using EMC model potential and found suitable in the present case in comparison with other local field correction functions. There are various forms of the local field correction functions are available in the literature. But, we have adopted here only five local field correction functions in the present computation for showing which functions are generated consistent values of the SSP of BeC Zr100−C metallic glasses. From all the local

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field correction functions, H- and T-local field correction functions are older functions while IU-, F- and S-local field correction functions are more recent one. From such kind of comparative study, one can choose the best suitable function among them, which is our main aim of the present study. Also, the main differences of the local field correction functions are played in important role in the production of the SSPs of BeC Zr100−C metallic glasses. The Hartree (H) dielectric function [24] is purely static and it does not include the exchange and correlation effects. Taylor (T) [25] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. The Ichimaru-Utsumi (IU) local field correction function [26] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also satisfies the self consistency condition in the compressibility sum rule and short range correlations. Therefore, Hartree (H) local field correction function [24] gives the best agreement with the experiment with EMC model potential and found suitable in the present case. On the basis of Ichimaru-Utsumi (IU) local field correction function [26], Farid et al. (F) [27] and Sarkar et al. [28] have given a local field correction function. Hence, IU- and F-functions represent same characteristic nature of the SSP. Also, the SSP computed from Sarkar et al. [28] local field correction are found in qualitative agreement with the available experimental data [23]. From the study of the Table 2, one can see that among the five screening functions the screening function due to H (only static–without exchange and correlation) [24] gives the minimum value of the SSPs while the screening function due to F [27] gives the maximum value. The present findings due to T [25], IU [26] and S [28] local field correction functions are lying between these two screening functions. Therefore, most recent local field correction functions due to IU [26], F [27] and S [28] are able to generate consistent and comparable results regarding the SSPs of the 3d-transition metals binary alloys as those obtained form more commonly employed H [24] and T [25] functions. The effect of local field correction

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Superconducting State Parameters of Be-Zr Glassy Alloys functions plays an important role in the computation of variation on TC ,

α

159

λ and μ * , which makes drastic

and N OV . The local field correction functions due to IU, F and S are

able to generate consistent results regarding the SSP of the metallic glasses as those obtained from more commonly employed H and T-functions. Thus, the use of these more promising local field correction functions is established successfully. The computed results of α and

N OV are not showing any abnormal values for BeC Zr100−C metallic glasses. Lastly, we would like to emphasize the importance of involving a precise form for the pseudopotential. It must be confessed that although the effect of pseudopotential in strong coupling superconductor is large, yet it plays a decisive role in weak coupling superconductors i.e. those substances which are at the boundary dividing the superconducting and nonsuperconducting region. In other words, a small variation in the value of electron-ion interaction may lead to an abrupt change in the superconducting properties of the material under consideration. In this connection we may realize the importance of an accurate form for the pseudopotential.

CONCLUSIONS Lastly we concluded that, the H, T, IU, F and S-local field corrections when used with EMC model potential provide the best explanation for superconductivity in the BeC Zr100−C system. The values of the electron-phonon coupling strength

λ and the transition temperature TC

show an appreciable dependence on the local field correction function, whereas for the

μ * , isotope effect exponent α and effective interaction strength N OV a weak dependence is observed. The magnitude of the λ , α and N OV values shows

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Coulomb pseudopotential

that BeC Zr100−C metallic glasses are weak to intermediate superconductors. In the absence of experimental data for

α

and N OV , the presently computed values of these parameters may

be considered to form reliable data for BeC Zr100−C metallic glasses, as they lie within the theoretical limits of the Eliashberg-McMillan formulation. It is also concluded that, the BeC Zr100−C metallic glasses are favoured superconductors. The comparisons of presently computed results of the SSP of BeC Zr100−C metallic glasses with available theoretical and experimental findings are highly encouraging, which confirms the applicability of the EMC model potential and different forms of the local field correction functions. Such study on SSP of other multi component metallic glasses is in progress.

REFERENCES [1] P. B. Allen, in Handbook of Superconductivity, eds. C. P. Poole Jr., pp. 478, Academic Press, New York, 1999.

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Aditya M. Vora

[2] A. V. Narlikar and S. N. Ekbote, Superconductivity and Superconducting Materials, South Asian Publishers, New Delhi, 1983. [3] S. V. Vonsovsky, Yu. A. Izyumov and E. Z. Kurmaev, Superconductivity of Transition Metals, their Alloys and Compounds, Springer-Verlag, Berlin, 1982. [4] A. M. Vora, M. H. Patel, P. N. Gajjar and A. R. Jani, Pramana -J. Phys. 58, 849 (2002). [5] P. N. Gajjar, A. M. Vora, M. H. Patel and A. R. Jani, Int. J. Mod. Phys. B17, 6001 (2003). [6] P. N. Gajjar, A. M. Vora and A. R. Jani, Indian J. Phys. 78, 775 (2004). [7] A. M. Vora, M. H. Patel, S. R. Mishra, P. N. Gajjar and A. R. Jani, Solid State Phys. 44, 345 (2001). [8] P. N. Gajjar, A. M. Vora and A. R. Jani, Mod. Phys. Lett. B18, 573 (2004). [9] Aditya M. Vora, Physica C450, 135 (2006); Physica C458, 21 (2007); Physica C458, 43 (2007); Phys. Scr. 76, 204 (2007); Comp. Mater. Sci. 40, 492 (2007); J. Optoelec. Adv. Mater. 9 (2007) 2498; Frontiers Phys. 2 (2007) 430. [10] Aditya M. Vora, J. Supercond. Novel Mag. 20, 355 (2007); J. Supercond. Novel Mag. 20, 373 (2007); J. Supercond. Novel Mag. 20, 387 (2007); Chinese Phys. Lett. 24 (2007) 2624. [11] S. Sharma, K. S. Sharma and H. Khan, Czech J. Phys. 55, 1005 (2005); Supercond. Sci. Technol. 17, 474 (2004). [12] S. Sharma, H. Khan and K. S. Sharma, Phys. Stat. Sol. (b) 241, 2562 (2004). [13] S. Sharma and H. Khan, Solid State Physics 46, 635 (2003). [14] S. Sharma, H. Khan and K. S. Sharma, Ind. J. Pure & Appl. Phys. 41, 301 (2003). [15] M. Gupta, P. C. Agarwal and K. S. Sharma, L. Dass, Phys. Stat. Sol. (b) 211, 731 (1999). [16] K. S. Sharma, R. Sharma, M. Gupta, P. C. Agarwal and L. Dass, in The Physics of Disordered Materials, pp. 95, eds M. P. Saksena et al, NISCOM, New Delhi, 1997. [17] P. C. Agarwal, M. Gupta, K. S. Sharma and L. Dass, in The Physics of Disordered Materials, pp. 102, eds M. P. Saksena et al, NISCOM, New Delhi, 1997. [18] M. Gupta, K. S. Sharma and L. Dass, Pramana-J. Phys. 48, 923 (1997). [19] R. Sharma and K. S. Sharma, Supercond. Sci. Technol. 10, 557 (1997). [20] N. W. Ashcroft, Phys. Lett. 23, 48 (1966). [21] K. S. Sharma and C. M. Kachhava, Solid State Commun. 30, 749 (1979). [22] V. Veljkovic and I. Slavic, Phys. Rev. Lett. 29, 105 (1972). [23] R. Hasegawa and L. E. Tanner, Phys. Rev. B16, 3925 (1977). [24] W. A. Harrison, Pseudopotentials in the Theory of Metals, W. A. Benjamin, New York, 1966. [25] R. Taylor, J. Phys. F: Met. Phys. 8, 1699 (1978). [26] S. Ichimaru and K. Utsumi, Phys. Rev. B24, 3220 (1981). [27] B. Farid, V. Heine, G. Engel and I. J. Robertson, Phys. Rev. B48, 11602 (1993). [28] A. Sarkar, D. Sen, H. Haldar and D. Roy, Mod. Phys. Lett. B12, 639 (1998). [29] W. H. Butler, Phys. Rev. B15, 5267 (1977). [30] W. L. McMillan, Phys. Rev. 167, 331 (1968). [31] G. von Minnigerode and K. Samwer, Physica B108, 1217 (1981).

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In: Perspectives in Theoretical Physics ISBN: 978-1-61122-960-8 c 2011 Nova Science Publishers, Inc. Editors: T. F. George, R. R. Letfullin and G. Zhang

Chapter 12

E NGINEERING S OLUTIONS U SING ACOUSTIC S PECTRAL F INITE E LEMENT M ETHODS Andrew T. Peplow∗ Hoare Lea Acoustics, Aztec West Business Park, Bristol, UK

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Abstract The spectral finite element method is an advanced implementation of the finite element method in which the solution over each element is expressed in terms of a priori unknown values at carefully selected spectral nodes. These methods are naturally chosen to solve problems in regular rectangular, cylindrical or spherical regions. However in a general irregular region it would be unwise to turn away from the finite element method since models defined in such regions are extremely difficult to implement and solve with a spectral method. Hence for a complex waveguide the method uses the efficiency and accuracy of the spectral method and is combined with the flexibility of finite elements to produce a high–performance engineering tool. Contemporary examples from engineering including fluid-filled pipes, tyre acoustics, silencers and waveguides. Some of these will be reviewed, presented and analysed. From simple examples to complex mixed materials configurations the study will highlight the strengths of the method with respect to standard methods.

1.

Introduction

The advantage of the spectral finite element method is that stable solution algorithms and high accuracy can be achieved with a low number of elements under a broad range of conditions. Spectral element techniques are high order methods which allow for either obtaining very accurate results or reducing the number of degrees of freedom for fixed standard precision. The work described in this chapter is concerned with the development of the waveguide spectral finite element method (SFEM). Its beginnings in structures as a combination of the dynamic stiffness method and the finite element method based on a variational ∗ E-mail

address: [email protected]

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Andrew T. Peplow

formulation for a non-conservative motion. A dynamic stiffness approach for frame structures has been developed originally by Richard and Leung [25]. However, a major advance, was made by Gavric [17], where the cross-sectional motion of a given waveguide was approximated by standard finite element polynomials. Wave propagation along the waveguide could then be studied by finding eigenvalues, corresponding to propagating wavenumbers, from a system of differential equations. This innovation inspired the study for beam and plate structures where Finnveden used the spectral finite element method in [10] and [12]. Orrenius and Finnveden [21] and more recently Nilsson and Finnveden [20] used this approach for rib–stiffened plate structures used in train wagons. For two-dimensional modelling, one-dimensional finite element shape polynomial functions describe the motion’s z–dependence where, without loss of generality, it is assumed that the waveguide is aligned with the x–axis. It follows that nodal displacements, vertical and longitudinal, are functions of the x variable and may be found by the elastic waveguide boundary spectral finite element method. The underlying is through the solution of a matrix polynomial eigenvalue problem. This novel approach has been used to describe the dynamic motion of sandwich composite structures Bonfiglio et al. [6] and ground vibration in layered geomechanic media Peplow and Finnveden [22] modelled masses lying within bedrock layers of infinite extent. Handling complex geometries by spectral finite element methods are now an established alternative to finite difference and finite element methods to solve elliptic Partial Differential Equations (PDE). Spectral methods are naturally chosen to solve problems in regular rectangular, cylindrical or spherical regions. However in a general irregular region it would be unwise to turn away from the finite element method since models defined in such regions are extremely difficult to implement and solve with a spectral method. The examples here show how the spectral method uses boundary data to devise trial functions so that, in essence, no discretization of the interior domain is necessary. Thus the method maybe viewed as a boundary element method as the method is ”meshless” as possible but unlike most boundary element methods it is not derived from Greens theorem. Hence for a complex waveguide the method uses the boundary data to devise an efficient and accurate spectral method and is combined with the flexibility of finite elements to produce a high–performance engineering tool. In complex acoustic systems the underlying polynomial eigenvalue problem for acoustic problems are straightforward, however, see section 2. and 5.1. for large–scale three–dimensional problems. Waveguides with uniform cross sectional properties will be described here using the spectral finite element method. For such an analysis, a natural starting point is the variational formulation. In this context, this type of finite element analysis first appeared in Finnveden [13] and [11] for fluid–filled flexible pipes , and was used to compare with experimental techniques by Finnveden and Pinnington [15]. The effect of including distributed loading by turbulent boundary layers on pipes can be found in [3]. Peplow and Finnveden developed the method to study sound transmission in various acoustic waveguide configurations in [23]. Recently the SFEM has been used for the study of and comparison with measurements, by Birgersson et al. to predict the response of a structure by turbulence excitation [14]. [5], and [4]. Applications of the method can be found in the theses by Birgersson [2] and Nilsson [19] on applications in fluid–structure interaction and Fraggstedt [16] on modelling wave propagation in car tyres. For three-dimensional problems, two-dimensional polynomial shape functions describe the motion’s y– and z– dependence. Kirby used collocation to determine transmission loss for

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163

Γ2 Ω3

Ω2

Ω1

H

z x

L2 Figure 1. Two dimensional acousti ducts with expansion chamber. elliptic dissipative silencers with and without mean flow [18] and [8]. Although a specific three–dimensional finite element was not constructed the efficiency of the method is clearly described. In general the SFEM technique is an important development in engineering analysis. The flexibility of the waveguide boundary spectral finite element and the versatility of the modelling applications give new solutions to engineering dynamics and acoustic problems.

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2.

Weak Formulation for the Spectral Finite Element

A weak formulation is based on introducing the weight function χ(~x) and testing it with the Helmholtz operator such that integrating by parts gives Z Z h i ~∇χ(~x) · ~∇p(~x) − k2 χ(~x)p(~x) dΩ(~x) χ(~x) a v f (~x)dΓ(~x) − LΩ = =

k

+

Z

ΓZ

2

Γ

Ω

χ(~x)p(~x)dΩ(~x) −

χ(~x) a v f (~x)dΓ(~x) −

ZΩ

Z

Ω Ω

χz (~x)pz (~x)dΩ(~x)

χx(~x)px (~x)dΩ(~x) = 0.

(1)

Often, equation (1) represents the starting point for conventional finite element discretizations, e.g. Galerkin method. The second part (lower row) consists of a domain integral and a boundary integral. The variational statement Eq. (1) is used to obtain wave influence basis functions [W(x)]. An approximate solution to the original problem is also found by selecting a solution p j (x, z) from a discrete set of trial functions determined by a finite element discretization of a region with a uniform cross–section Ω j .

2.1.

Approximation Functions

We approximate the sound pressure p(~x) as N

p(~x) =

∑ φi(~x) pi

= φT (~x)p

i=1

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(2)

164

Andrew T. Peplow (b) 1

0.5

0.5 ne = 3

ne = 2

(a) 1

0

−0.5

−1 −1

0

−0.5

−0.5

0 z

0.5

−1 −1

1

−0.5

1

0.5

0.5

0

−0.5

−1 −1

0.5

1

0.5

1

(d)

1

ne = 5

ne = 4

(c)

0 z

0

−0.5

−0.5

0 z

0.5

−1 −1

1

−0.5

0 z

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Figure 2. Linear to quartic functions, φ(s), N = ne. Polynomials in (4) is illustrated in (b). where pi represents the discrete sound pressure at point ~xi and φi is the i−th basis function for our approximation. First we start the analysis with a two–dimensional formulation and then followed by an three–dimensional example. Figure 1 shows a typical two–dimensional waveguide section where Ω = Ω1 ∪ . . .∪ Ω5 . In the super–spectral method this section will define one waveguide finite element. Within one super–spectral finite element, Ω2 := D × C, D := [0, L2 ], C := [0, H] for example, the approximate solution p(.) may be represented by an expression of the form N

∑ piφi(z)Wi(x)

p(~x) =

(3)

i=1

using piecewise quadratic polynomial shape functions. Shape functions, φ(s), are defined sub–locally over the cross–sections. Elemental shape functions W (x) are defined later in this section. The complete set of quadratic polynomials defined over [−1, 1] is φ(s) = [φ1 (s) φ2 (s) φ3 (s) ] where φ1 (s) =




1 1 −1 1 − 2s + s2 , φ2 (s) = 1 + 2s + s2 , φ3 (s) = √ 1 − s2 . 4 4 2

(4)

Consider the functional LΩ , (1), for a single arbitrary region Ω2 say. Substitution of expression (3) into the resulting form, where subscript x denotes x–derivative, yields the approximation LΩ j

= k

2

+ ik

Z Z

T

Ωi

W K 1W dx − T

Z

W K 3W } dx −

Ωi

Z

W T K 2W dx W Tx K 4W x dx,

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(5)

Engineering Solutions Using Acoustic Spectral Finite Element Methods

165

where the matrices are defined by K1

=

φT (z)φ(z) dz,

(6)

C

K2

=

K3

=

K4

Z

=

Z

dφ T dφ dz, C dz dz  H 1 , ζ 0 Z

(7) (8)

φT (z)φ(z) dz.

(9)

C

Explicitly the mass and stiffness matrices in (6) and (7), defined for the interval C = [0, H], are given by √   6 1 −6√2 Z +1 H K1 = H (10) φT (s)φ(s) ds, = 1√ 6√ −6 2  , 15 −1 8 −6 2 −6 2

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K2 = H

√  2 −1 √2 H ds = 2 . −1 2 ds ds 3 √ √ 2 2 4

Z +1 dφ T dφ −1



(11)

For each geometrical sector Ω in Fig.1 the finite element cross–sectional matrices K 1 , . . ., K 4 are frequency independent, real valued and are fairly small in size (for a two– dimensional problem at least) and hence may be stored. In the case here these are clearly 3 × 3 real–valued symmetric matrices. For a full problem a dynamic stiffness matrix requires assemblage. To do this wave influence functions W for each sector Ω are required.

3.

Wave Influence Functions & the Dynamic Stiffness Matrix

Construction of the wave influence functions follows by consideration of the ordinary differential equations which correspond to Eq. (5) found by taking an appropriate first variation and ignoring any boundary conditions: K4

d2 W (x) + k2 K 1W (x) − K 2W (x) + i kK 3W (x) = 0. dx2

(12)

Crucial to the fundamental principle of determining waveguide boundary spectral finite elements is that the system of equations Eq. (12) are not defined over a specific region. It could be argued that Eq. (12) is defined over a region of infinite length. The differential equations have constant coefficients in the form of symmetric positive and semi–definite (N × N) real–valued matrices K 1 , . . ., K 4 . Hence, the solutions of the linear homogeneous system may be written as : W m (x) =

Φm eiλm−1 x , m = 1, . . ., N

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(13)

166

Andrew T. Peplow

where Φm is a vector representing the cross–sectional mode shapes. Under this assumption Eq. (13) reduces to a linear eigenvalue problem, K:  K(λ)Φ = k2 K 1 − K 2 + i kK 3 − λ2 K 4 Φ = 0 (14) of order N for the parameters λ2 . The solutions of Eq. (14) yield values for λ that occur in pairs, λ± = ±λ, indicating that pairs of eigenmodes result with the same phase speed propagating in the positive and negative axial directions. The dimension of the eigenvalue problem is (N × N) and a finite 2N number of propagating wavenumbers are obtained. The resolution of the matrix eigenvalue problem Eq. (14) itself may be achieved by a number of standard computational routines. In the present analysis a QZ algorithm was used as implemented in Matlab 7.0.2 and requires 46 N 3 operations to determine all eigenvalues and right eigenvectors. For large problems this numerical analysis procedure can dominate the total computation time for a single problem. This produces a complete set of cross– sectional mode shapes and corresponding eigenvalues λ2m , m = 0, . . ., N − 1. Now the finite element trial functions have been constructed the dynamic stiffness for a general spectral waveguide element Ω is described. The local dynamic stiffness matrix defined over a region Ω shown in Fig. 1 will now be described. A simple translation to T := {x : −D ≤ x ≤ +D}, where 2D is the length of the sector, is a key element in defining the wave influence trial functions. By consideration of the eigensolutions in Eq. (13) it is clear that each wave influence function may be written as 2N

W jk (x) =

∑ Φ jl Ell (x)Alk pk ,

j = 1, . . ., N, k = 1, . . ., 2N,

(15)

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l=1

where entries Φ jl and Ell , a diagonal matrix, take the values of the eigenmodes and wavefunctions respectively. Coefficients Alk are determined by appropriate scaling of the set of wave influence functions, and pk are the unknown coefficients. The local dynamic stiffness matrix for a certain element may be written in matrix form as:   T 2 (16) L = B k K 1 − K 2 B − dΛcBT K 4 BdΛc + ikB T K 3 B, where the matrix B, of order 2N × 2N, combines wave influence function matrices B

=

Φ dEc A.

(17)

The ansatz dynamic stiffness matrix above may seem a little contrived compared to direct FEM but it can be constructed very simply from Eq. (16) using matrix algebra operations. Within the computation of local dynamic stiffness matrix entries use is made of an important matrix generating function for the combination of matrix diagonal exponential terms which has an analytic form. The global dynamic stiffness matrix is generated by calculating local dynamic stiffness matrices for each region and enforcing continuity of pressure (and velocity) across neighbouring interfaces. For example, consider a uniform waveguide geometry consisting of three spectral elements, as in Fig. 1, with N degrees of freedom across each element cross–section. A N × N matrix eigenvalue problem is solved and a 2N × 2N local dynamic

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Engineering Solutions Using Acoustic Spectral Finite Element Methods

167

stiffness matrix generated for each element. Enforcing continuity across neighbours results in a 4N × 4N global dynamic stiffness matrix. Acoustic sources may be modelled as volume point sources but in the following examples normal accelerations are applied on the left– hand boundary. The total number of operations for solving this problem for an arbitrary length waveguide are around 57N 3 for all the Gaussian elimination operations and 138 N 3 operations for all eigenvalue and eigenfunction computations. Note that any code written can be made efficient by re–using elements and finite element matrices. The estimates above represent over–estimates of real computations. Construction of dynamic stiffness matrices for layered media follows exactly as with construction of stiffness matrices for a single layer waveguide as above. Hence it is possible to solve multi–layered waveguide problems using super–spectral elements bearing in mind the large generalised matrix eigenvalue problems to be solved. This is performed in Example 2 for multi–layered dissipative elements.

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4.

Examples for Two–Dimensional Spectral Finite Elements

One area in acoustical engineering that is well–suited to spectral waveguide finite element methods is the design of silencer systems for noise control. There is much work that has been done for smaller systems such as those used in automobiles and small engines, however, the design of much larger systems (such as the parallel baffle type used for gas turbines and other large industrial machines) is still largely guesswork and empirical extensions of previous results. Due to the large size, difficulties in testing and high costs of these silencer systems, the ability to accurately predict the performance before construction and commissioning would be very beneficial. To properly predict the performance of a silencer system, many factors need to be involved in the calculation. Geometrical concerns, absorptive material characteristics, flow effects (turbulence), break out noise, self-generated noise, and source impedance all need to be included in the design calculations of insertion loss (IL). It is very important to note that the method derivations, and their use with the numerical methods are based on plane wave propagation sound sources (i.e. the entire face of the inlet section moving in unison) and an anechoic termination, i.e. ζ = ρc at the left and right hand end sections of regions Ω1 and Ω4 respectively. Anechoic termination at the outlet and the inlet pipes are assumed. In both two–dimensional examples cubic polynomials corresponding to N = 4 for SFEM and cubic polynomials defined over 9–noded triangular elements for FEMLAB 3.1 were used. The total number of degrees of freedom (DOF) for the spectral waveguide finite element totalled 68 and using FEMLAB 3.1 [9] 2210 DOF over 472 triangles were required. The length of the inlet and outlet pipes for the SFEM and FEMLAB geomtries were 1.0 m. Note that the pipe lengths for SFEM could have been considerable longer but for the purposes of sensible computations it was necessary to keep the DOF for FEMLAB 3.1 to a minimum. The characteristic impedance ζ = ρc was applied to both inlet and outlet pipes outer boundaries. The height of the outer pipes was fixed at H = 0.05 m and chamber extended by an extra H1 = 0.2 m depth. In keeping with Belawchuk [1] the length of the chamber was fixed at L3 = 1.2 m. To compare efforts of the two methods the CPU timing for 55 frequencies was 3.9 s for SFEM and 26.3 s CPU expenditure for FEMLAB [9]. It seems that the profit speedup for SFEM is rather conservative. However, if the chamber was

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Andrew T. Peplow

L3 = 2.4 m long the CPU time would remain 3.9 s for the waveguide finite element method, but FEMLAB 3.1 would now require 849 elements with 3928 DOF and using UMFPACK CPU expenditure would increase to 46.5 s. Both a substantial increase in both storage and computation time compared to the spectral element method.

Γ2 Ω1

Ω2

Ω4

Ω1

H

Ω4

Ω2

z

z x

Ω3

H

Ω3 x H3

H1

x1 x2

L2

x3

H2

H1

Γ3 L3

(a)

L3

(b) (c)

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Figure 3. (a) Geometry used for expansion chamber examples. Inlet and outlet pipes are regions Ω1,2 and Ω4 respectively and Ω3 is the expansion chamber. Absorbing material may be placed along the boundary Γ3 ; (b) Geometry used for expansion chamber example including dissipative lining in chamber. Lining material material may be included in the regions of height H1 = 0.05 m and H2 = 0.1 m. Region of height H3 = 0.05m is air ; (c) typical standard FE mesh for silencer problem with 586 triangles and 354 nodes. Note that (a) and (b) are meshes for SFEM.

4.1.

Theory for Transmission Loss

The definition of transmission loss is the ratio of the incident sound power to the transmitted sound power. As long as the inlet and outlet regions of the silencer are of the same cross section, and the properties of the fluid (density, temperature) do not change, then the T L can be expressed as: Pi (18) T L = 20 log10 Pt where Pi is the rms pressure of the incident wave without silencer in place, Pt is the rms pressure of the transmitted wave with silencer in place. This can be simplified to the following equation: T L = SPLi − SPLt where it is understood that SPLi is obtained without the silencer in place, and SPLt is obtained with the silencer in place, on the exhaust side of the silencer. Fig. 3(a) illustrates the geometry used to calculate SPLi in region Ω2 and SPLt in Ω4 . The SPLi is calculated with the straight pipe (no expansion chamber) and the SPLt is calculated with the expansion chamber (no straight pipe). The inlet and outlet sections have the characteristic impedance (ζ = ρc) boundary condition applied. This models a completely anechoic source and termination. Also the inlet section is given a unit acceleration amplitude to model a sound source.

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169

45 40

Spectral finite element method FEMLAB

Transmission Loss (dB).

35 30 25 20 15 10 5 0 0

100

200

300

400 500 Frequency (Hz)

600

700

800

(−0.5,+0.25)

(1.7,+0.25)

(−0.5,+0.2)

(1.7,+0.2) (0.0,0.0)

(1.2,0.0)

Figure 4. Transmission loss results for expansion chamber shown in Fig. 3(a), height and length of chamber H3 = 0.2, L3 = 1.2m respectively and width of pipes H = 0.05m. Mesh for FEMLAB results, # of triangles = 402, # nodes = 260.

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4.2.

Example 1. Absorbing Boundary Material Lining Silencer Chamber

Starting from the one-dimensional wave equation, the so called 3-point method can be derived as Bilawchuk [1]: p1 − p2 eikx12 (19) pi = 1 − e2ikx12 where referring to Fig. 3: pi is the incoming contribution of rms sound pressure wave; p1 is the rms sound pressure at location x1 ; p2 =rms sound pressure at location x2 ; ; x12 = x2 − x1 (microphone spacing). Now that the incoming rms pressure values are known, the exiting rms pressure can be obtained and the TL can be calculated simply as follows: pi T L = 20 log10 (20) p3 where p3 is the rms sound pressure at point x3 . The rms pressure this point can obtained directly since the termination at the exit is given the characteristic impedance ( Z = ρc). The FEM and/or BEM calculations can then be started. In the post-processing stage, the pressures at points x1 , x2 , and x3 can be calculated and, knowing the distances x1 = −0.12 m and x2 = −0.1, x3 = 1.3 m and the wave number, k, the transmission loss can be determined. Fig. 4 shows results from SFEM and FEMLAB. Accuracy of the waveguide finite element method is clearly evident. Fig. 5 shows the change in transmission loss where the chamber lining, Γ3 , is lined with absorbing material. Flow resistivity values taken were from Delany and Bazley formula [7].

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170

Andrew T. Peplow 25 Material1 Material2

Transmission Loss (dB).

20

Material3 Rigid 15

10

5

0

100

200

300 400 Frequency (Hz)

500

600

700

Figure 5. Transmission loss for rigid expansion chamber and chamber with three types of absorbing lining along Γ3 . Flow resistivity values, from [7], for materials are σ1,2,3 = 400, 50, &1 × 103 Nsm−4 .

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4.3.

Example 2. Dissipative Fibrous Material Chambers

The acoustic performance of a dissipative expansion chamber lined with two layers of fibrous material with different resistances is investigated as a two–dimensional version of Selamet et al. [26]. A two–dimensional numerical approach is used to determine the transmission loss of this dissipative silencer. The flow resistivity of the fibre in the dissipative chamber greatly influences the acoustic performance. The model used describes complex valued characteristic impedance and wavenumber leading to complex values of wavespeed and density to input into SFEM scheme:
(21) Zˆ = ρc 1 + 0.0855( f /R)−0.754 − 0.0765( f /R)−0.732i,
−0.577 −0.595 kˆ = ω 1 + 0.1472( f /R) i (22) − 0.1734( f /R) Generally, the increasing resistance of fibre in the dissipative chamber improves the sound attenuation in the mid to high frequency range, while deteriorating to a degree at low frequencies. Thus, to improve the sound attenuation performance at all frequencies, it is a paradox to design a dissipative expansion chamber filled completely with a unique fibre. The present study considers a layered dissipative silencer to investigate the potential trade-offs. Thus a single–pass expansion chamber lined with two fibre layers of different fibre resistance is examined primarily by the SFEM approach, in Fig.6. Dissipative material be included in the regions of height H1 = 0.05 m and H2 = 0.1 m, see Fig.3(b). Respectively Dissipative material1 comprised fibrous materials R1 and R1 , Material2 comprised fibrous materials R2 and R1 , Material3 comprised materials R3 and R1 , and Material4 comprised materials R4 and R1 .

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171

50 Dissipative material1

Transmission Loss (dB).

45

Dissipative material2

40

Dissipative material3

35

No material

Dissipative material4

30 25 20 15 10 5 0 0

100

200

300 400 Frequency (Hz)

500

600

700

Figure 6. Transmission loss for rigid expansion chamber and chamber with four types of dissipative lining material configurations in Fig. 3(b). Resistivity values used R1 = 5000 rayls/m, R2 = 10, 000 rayls/m, R3 = 17, 000 rayls/m, and R4 = 25, 000 rayls/m. See text for nomenclature.

Ω3 Ω2

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Ω1

z x

R

y

Figure 7. Geometry for the three–dimensional waveguide problem. The system is finite in length with circular inlet and outlet pipes with radius R. The geometry of the silencer is arbitrary, but has an axially uniform cross–section.

5.

Results from a Three-Dimensional Analysis

A numerical technique has been developed for the analysis of rigid and absorbent lined silencers of arbitrary, but axially uniform, cross–section. The analysis begins by employing the spectral finite element method to extract the eigenvalues and associated eigenvectors for a silencer chamber. It is demonstrated also that the technique presented offers a considerable reduction in the computational expenditure when compared to a three-dimensional finite element analysis. The method for determining transmission loss predictions from section 4.2. may be used to compare with experimental measurements taken for automotive dissipative silencers with elliptical cross sections.

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172

Andrew T. Peplow 0.25 0.2

S1

0.15 0.1

Γ1

0.05 0 −0.05 −0.1

S2

−0.15

Γ2 z

−0.2 −0.25 −0.2

−0.1

0

0.1

0.2

(a)

y (b)

Figure 8. Typical mesh for expansion chambers and cross–sections Γ2 , Γ1 clearly shown in the right–hand figure. The dissipative silencer consists of a concentric tube of arbitrary cross section is surrounded by absorbent lined material, on S2 see Fig. 8(b). The silencer chamber, which has a length L, is assumed to be uniform along its length, the outer walls of which are assumed to be absorbent except for the final example. The inlet and outlet pipes regions Ω1 and Ω3 are identical, each having a circular cross section radius R with rigid walls.

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5.1.

Dispersion Relations for Three-Dimensional Examples

If the outcome of an analysis is a dispersion relation between frequency and propagating wavenumber it is prudent to use a sparse eigensolver, such as eigs in MATLAB 7.0.2. The finite element mesh for the silencer chamber cross–section consisted of three-noded triangular elements and was generated using distmesh by Persson and Strang [24], see Fig.8(a). The elliptical cross–section had major-axis radius 0 .25 m and minor–axis radius 0.15 m. For the silencer, 470 elements equating to 310 nodes (DOFs) were used to mesh the chamber. Finite element meshes, not shown here, for a square cross–section of width 0 .5 m and circular cross–section, radius 0.25 m, were also constructed. The finite element mesh (including inlet and outlet circular section) with 470 triangles and 310 nodes is shown in Fig. 8. The acoustic pressure in the three–dimensional problem is approximated by piecewise linear triangular elements in the cross–section and wave influence functions in the axial direction, similar to the two–dimensional case, Eq. (3) N

p(~x) =

∑ pJ φJ (y, z)WJ(x).

(23)

J=1

To arrive at the eigenvalue problem, in order to derive the wave trial functions, the following matrix entries are assembled across the elliptic and cylindrical cross–sections, see Eq. (6) Z

φTI (y, z)φJ (y, z) dy dz, Γ 1,2,3 Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, K1(IJ)

=

(24)

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173

14

12

Wavenumber

10

8

6

4

2

0 0

100

200

300 400 Frequency, (Hz)

500

600

700

Figure 9. Dispersion curves for various silencer chambers. Square cross–section of width 0.5 m (dashed line), circular cross–section radius 0 .25 m (solid line), and elliptical cross– section major-axis radius 0.25 m minor–axis radius 0.15 m. Mesh (including inlet and outlet circular section) used with 470 triangles and 310 nodes shown in Fig. 8. K2(IJ) K3(IJ)

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K4(IJ)

=

Z

=

Z

=

Z

Γ1,2,3

S1,2,3 Γ1,2,3

∇φTI (y, z) · ∇φJ (y, z) dy dz,

(25)

1 φT (y, z)φJ (y, z) dy dz, ζ(y, z) I

(26)

φTI (y, z)φJ (y, z) dy dz

(27)

where ζ(y, z) is the function defining specific surface impedance on the walls of the pipes and the chamber, that is the boundaries of S1,2,3 . The corresponding eigenvalue problem for the problem becomes  K(λ)Φ = k2 K 1 − K 2 + i kK 3 − λ2 K 4 Φ = 0 (28) Energy transmission through a waveguide system is possible when the propagating wavenumber λ, in Eq. (29) has zero imaginary component. For a given frequency or wavenumber k it is possible to solve the eigenvalue problem below seeking real–valued wavenumbers and their corresponding mode shapes W m (x) =

Φm eiλm−1 x , m = 1, . . ., N

(29)

giving a dispersion relation between frequency ω = 2π f and wavenumber λ. The eigenvalue problem corresponding to the elliptic expansion chamber equated to solving a 310 × 310 sparse eigenvalue problem. For 55 frequencies the sparse eigensolver, for the first seven eigenvalues λ2 with smallest imaginary part, CPU time costing around 15 s for the ellipse. For the circular and square cross–sections the CPU time increased to around 20 s due to the increased number of DOFs. Figure 9 shows dispersion relations for the three square, circular and elliptical configurations. The curves for the square and circular cross–sections

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could be easily compared with known solutions. Note that four propagating modes, Φ, exist for the elliptic geometry at 740 Hz. Omitting the plane–wave mode, corresponding to λ0 = 0, these are shown in Fig. 10 from lowest wavenumber to highest wavenumber respectively.

4

4

3 2

2

1

0

0 −1

−2

−2 −3

0.2

−4

0.1 0.2

0.1

0

−0.1

−4 −0.2

0 −0.1 −0.2

−0.2

x(m)

y(m)

0.2 0.1 −0.1

(a)

0

0.1

0.2

0 −0.1 −0.2 y(m)

(b)

x(m)

5 4 3

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2 1 0 −1 −2

0.2 0.1

−3 0.2

0.1

0 y(m)

−0.1

−0.2

0 −0.1 −0.2 x(m)

(c)

Figure 10. Propagating modes for elliptic cross–section expansion chamber, at excitation frequency f = 740 Hz for geometry in Fig. 8.

5.2.

Solutions for Elliptic Cylinder Silencer Problem

This section illustrates computations for a finite length combination of pipes and chamber assuming unit normal acceleration at the end of the inlet pipe. The mesh used, Fig. 9, shows the finite elements for the chamber and the cross–section mesh for the rigid inlet and outlet pipes, located just above the centre. The computation of acoustic pressure for the silencer problem is dominated by the assemblage of the wave influence functions Eq. (28)

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−3

x 10

−3

x 10

6

0.5

5

0

4

−0.5

3

−1

2

−1.5

1

−2

0 −0.2

175

−2.5 −0.2

−0.2 −0.1 −0.1

0 0

0.1 y(m)

0.1 0.2

0.2

−0.2 −0.1 −0.1

0 0

0.1

x(m)

(a)

0.1 0.2

0.2

y(m)

x(m)

(b)

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Figure 11. Acoustic pressure at two cross–sections of lined expansion chamber (length L = 0.4 m) due to unit normal acceleration at inlet pipe (length L = 1.0 m) at excitation frequency 740 Hz. Left hand plot shows acoustic pressure at entry of chamber and right hand plot acoustic pressure at exit of chamber. Outlet pipe length of 4 .0 m with rigid termination, geometry of pipes shown in Fig. 7 and Fig. 8. and the corresponding dynamic stiffness matrix for the silencer chamber Eq. (16). All the frequency independent matrices K 1 , . . ., K 4 are stored as sparse matrices. However, all the eigenvectors and eigenvalues are required for the elliptic chamber problem of size N2 = 310, requiring 46N23 operations. The dynamic stiffness matrix for the chamber costs a little over 5N23 operations, due to re–use of LU decomposition, from Eq. (??). For the inlet and outlet pipes the numbers of degrees of freedom are somewhat lower, N1,3 = 20. Computing the wave influence functions and the dynamic stiffness matrices for the inlet and outlet pipes (of lengths 1.0 m and 4.0 m) are negligible in comparison to the chamber. If one assumes the full dynamic stiffness matrix for this problem the total numbers of DOFs amount to Ntot = 660. For a unit normal acceleration at the left–hand end of Ω1 and a given excitation frequency the CPU expenditure time for solving the system of equation is 8.1 s, the total time for finding wave functions for chamber takes 30 .3 s, and the total CPU time for solving the problem in MATLAB 7.0.2 on Pentium M machine took 54 .8 seconds. Although the number of operations are proportional to 50 N23 as discussed in the section on two–dimensional analysis the CPU expenditure is generally higher than for an equivalent analytic matching procedure. However, CPU expenditure compares favourably with alternative fully three-dimensional treatments. Figures 11–12 show solutions at cross–sections of the silencer assembly. Specifically, solutions are shown at the beginning of the chamber (a) and at the midway point (b) for elliptic chambers of lengths L = 0.8, 1.0 and 1.2 m respectively with absorbent liner material (flow resistivity σ = 400 × 103 Nsm−4 ) covering the entire outer surface. Note that the size of the computational problem is not changed for each expansion chamber length.

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Andrew T. Peplow −3

x 10 3 2.5 2 1.5 1 0.5 0 −0.2

−0.2 −0.1

0

0

0.1

0.2

0.2 x(m)

(a)

y(m)

−3

x 10 0 −0.5 −1 −1.5

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−2 −2.5 −0.2

−0.2 −0.1 −0.1

0 0

0.1

y(m)

0.1 0.2

0.2 x(m)

(b)

Figure 12. Acoustic pressure at two cross–sections of lined expansion chamber (length L = 0.8 m) due to unit normal acceleration at inlet pipe (length 1 .0 m) at excitation frequency 740 Hz. Left hand plot shows acoustic pressure at entry of chamber and right hand plot acoustic pressure at exit of chamber. Outlet pipe length of 4 .0 m with rigid termination, geometry of pipes shown in Fig. 7 and Fig. 8.

6.

Conclusions

A new spectral method in the form of a finite element scheme has been used to treat the problem of sound transmission in non–uniform waveguides or ducts. A unique feature of the super–spectral finite element approach is the use of basis functions generated from linear eigenvalue calculations. The basis functions, themselves solutions to the homogeneous reduced wave equations, may be defined over regions of arbitrary length with sound absorbing sides.

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The waveguide geometries that have been given the most attention in the present study are that of a rectangular duct partially lined on one side or a geometrical non–uniform rectangular duct. These configurations are not easily amenable to analytical treatment and the use of a numerical method is appropriate here. The finite element formulation in this investigation is, however, sufficiently general that it can be extended to any non–uniform rectangular waveguide with varying linings and fluid densities, and may be coupled with a standard finite element code. The conclusions and remarks may be listed as: 1. Acoustic problems defined such that a propagating in one direction is amenable to SFEM in two or three dimensions. 2. Muffler problems with varying density or complex characteristic impedances may be solved extremely with SFEM. 3. Where a mesh generator is available three–dimensional duct problems are easily within reach using SFEM. Computational expenditure is reduced such that problems may be solved on Desktop PCs. This new finite element technique extends the family of computational methods for predicting acoustic wave transmission and may possess wider applications in general wave propagation analysis for solids, fluid–structure interaction and fluid flow analysis.

References [1] Bilawchuk S, Fyfe KR, Appl. Acoust 64, 903 (2003). Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

[2] Birgersson F, Trita-AVE, ISSN 1651-7660 PhD thesis, KTH, Stockholm (2004). [3] Birgersson F, Ferguson NS, Finnveden S, J Sound Vib. 259, 873 (2003). [4] Birgersson F, Finnveden S, J Sound Vib. 287, 315 (2005). [5] Birgersson F, Finnveden S, Nilsson C–M, J Sound Vib. 287, 297 (2005). [6] Bonfiglio P, Pompoli F, Peplow AT, Nilsson AC, J Sound Vib. 303, 780 (2007). [7] Delany ME, Bazley EN, Appl. Acoust 3, 105 (1976). [8] Denia FD, Selamet A, Fuenmayor FJ, Kirby R, J Sound Vib 302, 1000 (2007). [9] FEMLAB 3.1 (2004) Users manual. [10] Finnveden S, Acta Acust 2, 461 (1994). [11] Finnveden S, J. Sound Vib 208, 685 (1997). [12] Finnveden S, Acust. Acta Acust 82, 479 (1996). [13] Finnveden S, J. Sound Vib 199, 125 (1997). [14] Finnveden S, Birgersson F, Ross U, Kremer T, J. Fluid Struct 20, 1127 (2005). Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

178

Andrew T. Peplow

[15] Finnveden S, Pinnington RJ, J Sound Vib 229, 147 (2000). [16] Fraggstedt M, Trita-AVE, ISSN 1651-7660 Lic. thesis, KTH, Stockholm (2006). [17] Gavric L, J. Sound Vib 173, 113 (1994). [18] Kirby R, J. Acoust. Soc. Am 114, 200 (2003). [19] Nilsson C–M, Trita-AVE, ISSN 1651-7660 PhD thesis, KTH, Stockholm (2004). [20] Nilsson C–M, Finnveden S, J Sound Vib 305, 641 (2007). [21] Orrenius U, Finnveden S, J Sound Vib 198, 203 (1996). [22] Peplow AT, Finnveden S, Int. J. Num. Anal. Meths Geom DOI: 10.1002/nag.643, (2008). [23] Peplow AT, Finnveden S, J. Acoust. Soc. Am 116, 1389 (2004). [24] Persson PO, Strang G, SIAM Review 46, 329 (2004). [25] Richard TH, Leung AYT, J Sound Vib 55, 363 (1979).

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[26] Selamet A, Xu MB, Lee IJ, Huff NT, Int. Veh. Noise Vib 1, 341 (2005).

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In: Perspectives in Theoretical Physics ISBN: 978-1-61122-960-8 c 2011 Nova Science Publishers, Inc. Editors: T. F. George, R. R. Letfullin and G. Zhang

Chapter 13

F RACTIONAL O SCILLATOR: P HENOMENOLOGICAL AND T HEORETICAL I SSUES A. Tofighi∗ Department of Physics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran

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Abstract Fractional derivative is a convenient tool to model physical processes in the complex media. A general trend is to consider fractional generalization of an equation of physics and study it’s consequences. For the case of a fractional oscillator we provide justifications for this procedure. In a media with low-level of fractionality the order of the fractional derivative is close to a positive integer. And one can use perturbation theory to treat the deviation from the integer case. Using this method we study fractional oscillator in a media with low-level fractionality. We find expressions for the position, momenta , energy and the intrinsic damping force for this system. We show that the energy is a monotonously decreasing function of time. We propose a new model Hamiltonian for this system. And we make some remarks on non-casual fractional oscillator as well.

Keywords: Fractional derivative, Fractional oscillator.

1.

Introduction

Fractional calculus is a generalization of classical calculus [1-5]and has many applications in various fields of science and engineering [6-20]. As far as dynamical processes is concerned the pioneering work of Ref.[6] is focused on fractional generalization of oscillation, relaxation, diffusion and wave phenomena. Since then a lot of papers has been published on these processes. A popular and simple approach was to write down a fractional differential equation for a specific phenomena, such that it’s solution could explain empirical data. There was no discussion of the underlying dynamics at this stage. In this work we address the oscillation ∗ E-mail address:

[email protected]

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phenomena. For the process of oscillation there has been some phenomenological works [6,21]. The emphasis in these type of works were to explain a dynamical quantity like position or momenta by proposing some new definitions. However we try to provide a theoretical foundations for these works and to examine the validity and consistency of their proposals. We also study these processes in the case where the level of fractionality of the complex media is low. In this case the fractionality can be considered as a perturbation to the ordinary(integer) case. It is possible to obtain compact expressions for the variables of interest in this limit. And to unravel the intricate structure of dynamics of a complex media like the damping of a fractional oscillator or the slow decay of fractional relaxation phenomena [15]. The organization of this paper is as follows. In section two we describe the expansion method suitable for the media with low-level fractionality. In section three we find the physical basis of some proposed models of fractional oscillator within Newtonian mechanics. And in section four we discuss the damping of a fractional oscillator. We see that in the case of low-level fractionality the perturbation term undergoes a forced oscillation. A compact expression for the damping force is obtained in this section as well. To unravel the underlying theoretical structure of some models of fractional oscillator we propose a new Hamiltonian for this system in section five. And we show that the equation of motion obtained from this Hamiltonian identical to fractional generalization of equations of motions which has been proposed a decade ago. In section six we briefly comment on the case of non-causal oscillator. And finally in section seven we present our conclusions.

2.

Fractional Derivative

There exists a multitude of definitions [1-5]like Riemann-Lioville, Weyl, Riesz and Caputo for the fractional derivative. In this paper we use the Caputo fractional derivative. The left Caputo Fractional deivative is defined by C α 0 Dt f (t) =

1 Γ(n − α)

Z t 0

f (n) (τ) dτ, (t − τ)α+1−n

n − 1 < α < n,

(1)

where n is an integer number and f (n)(τ) denotes the nth derivative of the function f (τ). We notice that this derivative depends not only on the value of t (the upper limit of the integral)but also depends on the lower limit of the integral (namely zero), so it is not local. It does not have a simple form for the Leibniz’s rule for the derivative of product of two functions either.

2.1.

Non-uniform and Uniform-Expansion

In many situations the level of fractionality of the complex media is low, so the order of the fractional derivative is very close to an integer number n. One then defines a parameter Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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ε = n − α. The next thing is to consider a expansion in terms of this small parameter. This method has been developed in [14]. For the left Caputo derivative and in the limit of εt  1 they found D1−ε f (t) = f (1)(t) + εD11 f (t), (2) where D11 f (t) =

f

(1)

(0) ln(t) + γ f

(1)

(t) +

Z t 0

f (2) (t) ln(t − τ)dτ,

and γ = 0.5772156... is a constant. Similarly D2−ε f (t) = f (2)(t) + εD21 f (t), where D21 f (t) = f (2)(0) ln(t) + γ f (2)(t) +

Z t 0

(3)

(4)

f (3) (t) ln(t − τ)dτ.

(5)

One also perform an expansion for the dynamical variable related to the fractional differential equation. If we denote this variable by x(t) then the expansion is x(t) = x0 (t) + εx1 (t) + ......,

(6)

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where x0 (t) is the solution of integer (ordinary) differential equation and the second term is a correction term. However this perturbation technique is valid for εt  1. The authors of [14] have found that this expansion is not uniform with respect to t  1, an the two cases εt ,ε  1, are very different, they show that when t → ∞and in the first order in ε one has to utilize the following expansion −1 ˙ ), D2−ε x(t) ' ε(x(0)t −2 − x(0)t

(εt  1).

(7)

However, we have proved [15] that whenever the solution of the integer case vanishes exponentially, then perturbation theory alone is sufficient to explain the behavior of the dynamical system.

3.

Fractional Oscillation

In this section first we briefly review the case of a fractional oscillator and then we consider this system within Newtonian framework. For simplicity we assume m = k = 1 through out this paper.

3.1.

Fractional Oscillator a Mathematical Approach

In this approach one considers [6]the fractional generalization of equation of motion a simple harmonic oscillator is d α x(t) + x(t) = 0, dt α

1 < α ≤ 2.

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(8)

182

A. Tofighi

The solution of this equation is expressed in terms of Mittag-Leffler functions, namely α ˙ x(t) = x(0)Eα,1 (−t α ) + x(0)tE α,2(−t ),

(9)

where Mittag-leffler function of a complex variable z is defined by ∞

Eα,β (z) =

zk

∑ Γ(αk + β)

α ≥ 0,

(10)

k=0

where the parameter β is in general a complex number. Now if we choose x(0) = 1 and x(0) ˙ = 0 as the initial condition then x(t) = Eα,1 (−t α).

(11)

This solution has a finite number of zeros. The motion of this oscillator is composed of two parts. First the oscillator makes a finite number of damped oscillation. Next, it has a monotonic algebraic decay in the asymptotic region.

3.2.

Fractional Generalization of Newton’s Law

To discuss the dynamics of a fractional oscillator one has to define a momenta for this system which is defined by

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p(t) =

d η x(t) , dt η

0 < η ≤ 1.

The case η = 1 corresponds to ordinary definition of the momenta. The case η = been proposed in Ref.[21]. We also consider the following generalization of Newton’s law of motion F=

d ξ p(t) , dt ξ

ξ = α − η.

(12) α 2

has

(13)

As noted in Ref.[14] the composition law of integer derivative does not apply in the fractional case. we obtain the following identity 1−α ˙ p(0)t −ξ d ξ p(t) d α x(t) x(0)t = + − . dt α Γ(2 − α) Γ(1 − ξ) dt ξ

(14)

Therefore the equation of motion of a fractional oscillator is 1−α x(0)t ˙ p(0)t −ξ d α x(t) + x(t) = − + . dt α Γ(2 − α) Γ(1 − ξ)

(15)

Let us consider two special cases: Case A If we choose η = 1 the two terms in the right hand side of Eq. (15) cancel each other, and we will recover Eq. (8). This in turn provides a physical foundation for the generalization

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of Ref. [6]. Case B The choice η = α2 , ξ =

α 2

is the symmetric case. α

1−α d α x(t) x(0)t ˙ p(0)t − 2 + x(t) = − + . dt α Γ(2 − α) Γ(1 − α2 )

(16)

By using Caputo derivative it is easy to show that when ˙x(0) = 0 then p(0) = 0. The important contributions of Ref. [21] is to introduce the momenta and investigates other dynamical variable such as energy. However we find that the correct equation of motion in this case is Eq. (16). And their equation of motion is only valid when ˙x(0) = 0.

4.

Damping Charactristics of the Fractional Oscillator

In the symmetric case and with x(0) = 1 and x(0) ˙ = 0 one finds α

p(t) = −t 2 Eα,1+ α2 (−t α ).

(17)

It is useful to study the total energy of this oscillator which is defined by

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EF =

p2 x2 + . 2 2

(18)

The numerical study of Ref.[21] shows that EF is a monotonically decreasing function of time. However there is no damping term in the equation of motion. This led the present author to introduce the notion of the intrinsic damping force which is given by[22] α

α

d 2 x(t) d 2 EF = Fid . α α dt 2 dt 2

(19)

Damping of the fractional oscillator has also been considered in Ref.[23-24].

4.1.

Limiting Case of α = 2 − ε

If the level of a fractionality of a medium is low then the order of fractional derivative is close to an integer number. For the oscillation phenomena α = 2 − ε with small positive ε. The expansion method has been used to study fractional oscillator [14,25]. The correct equation of motion in this case can be obtained from Eq.(15) the result is ε

Therefore we have

−1+ε x(0)t ˙ p(0)t −1+ 2 d 2−ε x(t) + x(t) = − + . dt 2−ε Γ(ε) Γ( 2ε )

(20)

d 2 x0 (t) + x0 (t) = 0, dt 2

(21)

x(0) ˙ d 2 x1 (t) − D21 x0 (t) + x1 (t) = − x1 (0) = 0. x˙1 (0) = 0. 2 dt by Thomas F. George, et al.,2tNova Science Perspectives in Theoretical Physics, edited Publishers, Incorporated, 2011. ProQuest Ebook Central,

(22)

184

A. Tofighi

We see that while the unperturbed solution x0 (t) undergoes a free oscillation, but the motion of the perturbed part x1 (t)is a forced oscillation. Again if we consider x(0) = 1 and x(0) ˙ =0 then the dynamical variables are given by [25] ε x(t) = cos(t) + [cos(t) − 1 − tSi(t) cos(t) + tCi(t) sin(t)], (23) 2 and ε p(t) = − sin(t) + [− sin(t) + tSi(t) sin(t) + tCi(t) cos(t)], (24) 2 where Ci(t) and Si(t) are sine and cosine integral functions respectively; Si(t) =

Z t sin(x)

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0

x

dx,

Ci(t) = −

Z ∞ cos(x) 0

x

dx.

(25)

Figure 1. x(t) versus t for ε = 0.01. The figure shows the attenuation of the amplitude of a fractional oscillator in the region εt  1. In Figure 1 the variation of x(t) versus time is shown. It shows the attenuation of the amplitude of the oscillation. The total energy of this system is EF =

1 ε + [1 − cos(t) − tSi(t)]. 2 2

(26)

The total energy of ordinary harmonic oscillator is EH = 12 . In Figure 2 we depict the variation of the parameter y = EEHF versus time. we see that the total energy of the fractional oscillator is a decreasing function of time. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 2. y versus t for ε = 0.01. This shows that the ratio of the energy of a fractional oscillator and integer oscillator is a monotonously decreasing function of time.

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To discuss the intrinsic damping force in this limit we note that up to first order in ε. dEF d β EF = , β dt dt

ε β = 1− . 2

(27)

From Equations (19,26) we obtain ε

Fid = − 2

Si(t) εSi(t) = . pa 2 sin(t)

(28)

We notice that the direction of Fid is opposite to that of the velocity. Furthermore at t = nπ, where sin(t) = 0 and n = 1, 2, 3, ....... ε

Fid = − 2

5.

Si(nπ) (−1)n Si(nπ) = . p1 a nπCi(nπ)

(29)

Hamiltonian of Fractional Oscillator

In this section we discuss fractional oscillator within the framework of fractional generalization of Hamiltonian mechanics. Fractional generalization of Hamiltonian systems has been discussed in [26]. For a Hamiltonian of H(x, p,t) the equations of motion are

dσH dσH α , B p = − , σ= , t σ σ dp dx 2 Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, At x =

(30)

186

A. Tofighi

where At and Bt are linear(or nonlinear)operators that have derivative (integer or fractional order)with respect to time. For a Hamiltonian defined by p2 x2 + , 2 2

(31)

1 dσx [ σ ]2−σ Γ(3 − σ) dt

(32)

H= and assuming At x = we find

dσx (33) dt σ This provides the theoretical basis for the proposal of Ref. [21] for the generalized momenta in the framework of Hamiltonian dynamics. With similar consideration for the operator Bt we can obtain the symmetric case of section 3. Moreover this Hamiltonian corresponds to the total energy EF of the fractional oscillator as well. Here we propose another model Hamiltonian for the fractional oscillator, which is given by p=

H=

(x2 )µ (p2 )µ + , Γ(2 + ζ)Γ(2 − ζ) Γ(2 + ζ)

µ=

1+ζ 2

and

0 < ζ ≤ 1.

(34)

The equations of motions in the Hamilton’s approach [27]are

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(

∂ζ H dx ζ ) = Γ(2 − ζ)pζ−1 ζ , dt dp

(35)

∂ζ H dζ p = − . dt ζ dqζ

(36)

and

Using left Caputo derivative and from this model Hamiltonian we find dx = p, dt

and

dζ p = −x, dt ζ

(37)

but this corresponds to case A of section five. This provides the Hamiltonian structure of fractional equation of motion of Ref.[6].

6.

Non-casual Fractional Oscillator

The right Caputo fractional derivative is defined by C α t Db f (t) =

(−1)n Γ(α − n)

Z b t

f (n) (τ) dτ, (τ − t)α+1−n

n − 1 < α < n,

(38)

Again this is a non-local operator. While the left Caputo fractional derivative of a function at a time t is depending on the lower limit of the integral (i.e some time in the past). The right Caputo derivative is depending on the upper limit of the integral (namely some time Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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in the future). The right fractional oscillator is defined by [28] C α t D0 x(t) + x(t) =

1 < α ≤ 2.

0,

(39)

And they find an anti-causal solution in this case. In Ref.[29]the Lagrangian of a fractional oscillator is assumed to be 1 ρ L = [(C0 Dt x)2 − x2 ] = 0, 2

0 < ρ < 1.

(40)

The Euler-Lagrange equation within the framework of fractional variation calculus for this Lagrangian is [30-31] ∂L C ρ ∂L + D = 0, ∂x t b ∂(C0 Dtρ x)

0 ≤ t ≤ b.

(41)

In this framework the equation of motion of a fractional oscillator is C ρ C ρ t Db (0 Dt x(t)) =

x(t).

(42)

The canonical momenta [31]of this system is p=

∂L

ρ . ∂(C0 Dt x)

(43)

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Therefore the equation of motion in terms of canonical momenta is C ρ t Db p

= x.

(44)

Therefore it appears that the generalization of the Newton’s second law of motion in this case will be ρ (45) F(x) = −Ct Db p. However this is a non-causal force law. Fractional derivative breaks locality as well as time reversal invariance [3]. Should we add causality to the list?

7.

Conclusions

In our study of fractional oscillator we introduced the notion of the intrinsic damping force. In the limit of α = 2 − ε We showed that the total energy is a monotonously decreasing function of time, and we found expression for the intrinsic damping force. This in turn implies the dissipation interpretation of fractional derivative of this order. For a probabilistic interpretation see Ref. [34]. Considering left Caputo fractional derivative, our generalized Newton’s law suggests that the force at a time t is related to the value of the momenta at an earlier time. This is a non-local force law. This is also like a memory effect. Temporal memory has been discussed within the frame work of action functional in Ref. [35]. While using a right Caputo

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fractional the generalized Newton’s law has a non-causal character. We also pointed out the limitations of the mathematical and phenomenological approaches and we found the underlying force law and Hamiltonian for fractional oscillator. Here we summarize the main features of the expansion method For a complete analysis of the fractional oscillator [14]a separate asymptotic analysis for t → ∞ is needed so we use the term non-uniform expansion for this case. However for the relaxation phenomena the perturbation theory alone is sufficient. And the region εt  1 and εt  1 are treated in the same footing (uniform expansion)[15]. The expansion method is an efficient tool while considering the dynamical processes in the complex media with low-level fractionality. It opens the way for analytical calculations in this domain. However it has limitations when applied non-linear problem. For instance it is not possible to find a close analytical form for non-linear fractional oscillator [14]. This expansion method has been used in the context of classical gravity [36] and it is found that fractional dynamics in Minkowski space-time is equivalent to a field theory in curved space-time. This will have some implications on the meaning of fractional derivative. This method has been utilized by the present author in the context of non-relativistic quantum mechanics [37] as well. We plan to study other classical or quantum dynamical system within this framework.

References

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[1] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New-York, (1974). [2] I. Podlubny, Fractional Differential Equations , Academic Press, San Diego CA, (1999). [3] R. Hilfer, Applications of Fractional Calculus in Physics , World Scientific, Singapore, (2000). [4] B. J. West, M. Bologna, P. Grigolini, Physics of Fractals Operators , Springer-Verlag, New-York, (2003). [5] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics , Oxford University Press, Oxford, (2005). [6] F. Mainardi, Chaos, Solitons and Fractals 7(9) (1996)1461-1477. [7] K. M. Kolwankar, A. D. Gangal, Phys. Rev. Lett. 48 (1997) 49-52. [8] F. Mainardi, A. Mura, R. Gorenflo, M. Stojanovic, E-Print arxiv:cond-mat/070113v1. [9] T. M. Atanackovic, M. Budincevic, S. Pilipovic, J. Phys. A:Math. Gen. 38 (2005) 6703-6713. [10] A. Tofighi, Invited Commentary, to be published in Statistical Mechanics Research Focus, Nova Science Publishers, New-York. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[11] S. Saha Ray, Phys. Scr. 75 (2007)53-61. [12] S. Momani, Z. Odibat, Phys. Lett. A, 355 (2006)271-279. [13] A. M. A. El-Sayed , M. Gaber, Phys. Lett. A, 359 (2006) 175-182. [14] V. E. Tarasov, G. M. Zaslavsky, Physica A 368 (2006)399-415. [15] A. Tofighi, Submitted to Physica A . [16] M. Naber, J. Math. Phys. 45(8) (2004) 3339-3352. [17] R. Hilfer, Chem. Phys. 284 (2002) 399-408. [18] N. Laskin, Phys. Lett. A 268 (2000) 298-305. [19] N. Laskin, Phys. Rev. E 62 (2000) 3135-2145. [20] V. E. Tarasov, Chaos, 16 (2006) 033108. [21] B. N. Narahari Achar, J. W. Hanneken, T. Enck, T. Clarke, Physica A 297 (2001) 361-367. [22] A. Tofighi, Physica A 329 (2003) 29-34. [23] B. N. Narahari Achar, J. W. Hanneken, T. Clarke, Physica A 339 (2004) 311-319. [24] Y. E. Ryabov, A. Puzenko, Phys. Rev. B 66 (2002) 184201. [25] A. Tofighi, H. Nasrolah Pour, Physica A 474 (2007) 41-45. Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

[26] V. E. Tarasov, J. Phys. A :Math. Gen. 38 (2005)5929-5943. [27] V. E. Tarasov, J. Phys. A :Math. Gen. 39 (2006)8409-8425. [28] G. M. Zaslavsky, A. A. Stanislavsky, M. A. Edelman, Chaos 16 (2006) 013102. [29] O. P. Agrawal, J. Phys. A: Math. Theor. 40 (2007) 5469-5477. [30] O. P. Agrawal, J. Math. Anal. Appl. 272 (2002) 368-379. [31] S. I. Muslih, D. Baleanu, Czech. J. Phys. 55(6) (2005) 633-642. [32] H. Bateman, A. Erdelyi, Tables of Integral Transform Vol.1 , McGraw-Hill, New-York, (1954). [33] M. A. Abramowitz, I. A. Stegun, (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables , 9th printing. Dover, New York, (1972). [34] A. A. Stanislavsky, Physica A 354 (2005) 101-110. [35] V. E. Tarasov, G. M. Zaslavsky, Physica A, In Press. [36] E. Goldfain, Comm. Non. Sci. Num. Siml, In Press. [37] A. Tofighi, UMZ-Preprint. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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In: Perspectives in Theoretical Physics Editors: T. F. George, R. R. Letfullin and G. Zhang

ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 14

CHARGE-BASED ELECTRONICS IS UNDER CONTROL – TECHNOLOGY WELCOMES ORBITRONICS FOR THIS MILLENIUM THOUGH SPINTRONICS IS NOT YET ESTABLISHED M. Idrish Miah1,2,* 1

Nanoscale Science and Technology Centre, Griffith University, Nathan, Brisbane, Australia 2 Department of Physics, University of Chittagong, Bangladesh.

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ABSTRACT Spintronics is a rapidly growing research field aimed at realizing practical devices that takes advantage of the electron spin as well as of its charge in conventional electronic devices. Much effort on the requirements for semiconductor spintronic devices reveals that the better understanding of spin currents in semiconductors is still a sufficiently challenging problem, though the first step in making the metal-based spintechnology amenable to industrial application came in 1988. However, in reality, an electron in strongly correlated-electron systems has three attributes: charge, spin and orbital. The orbital states of the electronic and magnetic phases could be controlled by external fields, such as electric, magnetic, stress, light or x-rays. Such control strategies for exploring the possibility of utilizing orbital degree of freedom may open the way to introduce the correlated-electron systems for a new type of electronics – orbitronics. However, future electronic devices with increased performance, capability and functionality would rely on a complete understanding of the three control parameters.

*

E-mail address: [email protected]

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Spin

e-

Charge

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The control and manipulation of electronic charge by electric fields is well established for electronics and the progress in conventional electronics has proven that the spin of the electron was ignored in past in the mainstream of electronics, because the electron has spin as well as charge. Electron spin (self-rotation - rotation of the electron on its axis) is a quantum mechanical property associated with its intrinsic angular momentum, and as an intrinsic magnetic moment is associated with spin, spin is closely related to magnetic phenomena. Researchers [1-7] recently realized that spin of electrons can create a current, called spin current, as the movement of electrons creates a charge current, and adding the spin degree of freedom, i.e. the spin property of electrons to electronics could revolutionize today’s electronics, where the spin degree of freedom of the electron could also be utilized to sense, store, process and transfer information (which could extend the functionality of conventional devices), emerging a field called “spintronics” (“Spintronics” was coined by S. A. Wolf in 1996 and appeared in scientific journals from 1999) based on the control and manipulation of electron spin instead of, or in addition to, its charge, leading to novel electronic devices (spintronic devices) such as spin field effect transistors (SFETs), spin storage/ memory devices or spin quantum computers, which can hold promise of, e.g. reduced power consumption, faster operation and smaller size. Most essential requirements for such devices are the efficient injection of spin-polarized carriers into a semiconductor (such that they can be transported reliably over reasonable distances without spin-flipping or spin relaxation: if the spin relaxes too fast, the distance traveled by an electron without losing its spin will be too short to perform any practical operation) and then detecting them (since information is carried by the electron spin). Much effort on the requirements for semiconductor spintronic devices reveals that the better understanding of spin currents in semiconductors is still a sufficiently challenging problem, though the first step in making the metal-based spin-technology amenable to industrial application came in 1988 with the invention of the very high performance hard disk drives and magnetic RAM utilizing giant magnetoresistive [8] magnetic multilayer (sandwiched structures consisting of alternating ferromagnetic and nonmagnetic metal layers) – composed of transition metals. These first metallic spintronic components were passive devices whose electric resistance depends strongly on the external magnetic field. Depending on the relative orientation of the magnetizations in the magnetic layers, the device resistance changes from small (parallel magnetizations) to large (antiparallel magnetizations). The proposal of utilizing electron spin for quantum information processing using spin as the binary system (quantum bits, in short qbits), say, ‘1’ for spin-up and ‘0’ for spin-down in

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quantum computing is a natural extension of spintronics since the electron provides an ideal condition for a qubit. In comparison with metal-based spintronics, utilization of semiconductors promises more versatile design due to the ability to adjust potential variation and spin polarization in the device by, e.g. external voltage and device structure. However, the first active semiconductor spintronic device was suggested by Datta and Das in 1990 [9], where they proposed an electronic analogue of an electro-optical modulator, that was later termed SFET, in a twodimensional electron gas (2-DEG) contacted with two ferromagnetic electrodes: one as a source for the injection of spin-polarized electrons and the other as an analyzer for electronspin polarization. The Datta-Das SFET device relies on the basic concept of modulating the transistor’s source-to-drain current by varying the Rashba interaction in the channel with a gate voltage. The injection and detection of spin-polarized electrons or spin currents in semiconductors have been achieved either optically or by electrical methods with varying degrees of success [5-7]. More sophisticated experiments toward all-electrical spintronic devices (similar to the SPINFET) using controllable parameters, such as device structure or gate voltage, able to combine spin injection and detection are required. Such device and control strategies for realizing new high-performance devices taking advantage of the electron spin as well as of its charge might open the way to establish the semiconductor systems for spintronics. Given the intense work on injection and detection of spin currents in semiconductors and the recent breakthroughs in this field, there is no doubt that implementing semiconductor-based spintronic devices is only a matter of time. In reality, an electron in strongly correlated electron systems such as the transition-metal oxides, in which electrons interact between neighboring and often next-neighboring, has three attributes: charge (-e), spin (±1/2) and orbital which determine its behavior. Transition-metal oxides are often called correlated-electron systems because of many interacting electrons compared to doped-semiconductors. The orbital represents the electron’s probability-density distribution, as the shape of the electron cloud in a solid. A transition-metal ion in a crystal having perovskite structure, for example, is surrounded by six oxygen ions, which give rise to Coulomb potential of electrons from each lattice site [10]. While hybridization of d-electron of transition-metal ion with the oxygen p-electron states tends to delocalize the electrons, Coulomb repulsion tends to localize individual electrons at atomic lattice sites. The subtle balance results in a crystal-field potential, which partly lifts the degeneracy of the d-electron levels, where wave functions pointing toward oxygen ions have higher energy than those pointing in between them. For example, in the Mott-insulating state (conduction electrons are tied to the atomic sites) of a crystal, the d-electrons are almost entirely localized on the atomic sites, which make both the spin and orbital degeneracy. The degeneracy orbital states can be considered as a new degree of freedom that behaves like the spin degree of freedom, and the technology based on the orbital degree of freedom in the correlated-electron systems may be termed as “orbitronics” (The term “orbitronics” was first given by the Takura group). When an excitation (or a field) excites an orbital in a crystal at a particular lattice site, the orbital wave (and its quantized equivalent, “orbiton”) represents in principle the dynamical response of the orbital under excitation, and it propagates in the crystal through the interactions between orbitals at different transition-metal ions – this means

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modulations in the relative shape of the electron clouds in an orbitally ordered state could give rise to orbital waves. Typically, the orbital degree of freedom in the lattice and electronic response can be found in manganese oxide compounds with perovskite structure, where colossal magnetoresistance, i.e. a very large decrease of resistance, is observed upon the application of an external magnetic field, where conduction is known to be occurred through a process known as the double-exchange interaction [10,11]. In this type of compound, d-electrons on each transition-metal ion act like strongly coupled ferromagnets, where the angle between local spin moments on adjacent sites can determine the electron hopping from site to site, and by controlling the magnetic field intensity one might be able to control the orbital correlation which results in a control of electrical conduction due to varied electron hopping. Much intensive work is to be dedicated for searching the related possibility of controlling electrical conduction (or electric current) by changing the d-electron orbital shapes or states on a lattice site. Very recently, Tokura and co-workers [12,13] at the University of Tokyo in Japan have proposed that the orbital states of the electronic and magnetic phases of correlated-electron materials could be controlled by external fields such as electric, magnetic, stress, light or xrays fields. Such control strategies for exploring the possibility of utilizing orbital degree of freedom might open the way to introduce the correlated-electron systems for a new type of electronics – orbitronics. We don’t know for sure where this will go, but we do know that the potential is very exciting. However, future electronic devices with increased performance, capability and functionality would rely on a complete understanding of the three control parameters, namely carriers’ charge, spin and orbital.

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REFERENCES [1] S. D. Sarma, Am. Sci. 89 (2001), 516. [2] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova and D. M. Treger, Science 294 (2001), 1488. [3] H. – A. Engel, P. Recher and D. Loss, Solid State Commun. 119 (2001), 229. [4] D. D. Awschalom, D. Loss and N. Samarth, Eds., Semiconductor Spintronics and Quantum Computation (Springer, Berlin, 2002). [5] Žutić, J. Fabian and S. D. Sarma, Rev. Mod. Phys. 76 (2004), 323. [6] Y. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Science 306 (2004), 1910. [7] M. I. Miah, Mater. Lett. 60 (2006), 2863; J. Phys. D: Appl. Phys. 40 (2007), 1659. [8] G. Prinz, Phys. Today 48 (1995), 58. [9] S. Datta and B. Das, Appl. Phys. Lett. 56 (1990) 665. [10] C. Zener. Phys. Rev. 81 (1951), 440. [11] J. L. Cohn. J. Superconduc. Incorp. Novel Magnet. 13 (2000), 291. [12] Y. Tokura and N. Nagaosa. Science 288 (2000), 462. [13] Y. S. Lee, Y. Tokunaga, T. Arima and Y. Tokura, Phys. Rev. B 75 (2007), 174406.

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ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 15

THERMODYNAMICS OF LIQUID METALS: A VARIATIONAL APPROACH Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj – Kutch, 370 001, Gujarat, India

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ABSTRACT On the bases of the Gibbs–Bogoliubov (GB) variational method and the PercusYevick (PY) hard sphere model as a reference system, a thermodynamic perturbation method has been applied with use of well known model potential. The thermodynamic properties of some liquid metals of the different groups of the periodic table are reported. The influence of local field correction function proposed by Hartree (HR), Taylor (TY), Ichimaru-Utsumi (IU), Farid et al. (FR) and Sarkar et al. (SR) is also investigated. The comparison with available experimental or theoretical findings is found in qualitative agreement and establishes the use of the local field correction functions in such study.

Keywords: Thermodynamic properties, pseudopotential theory, PY hard sphere, model, liquid metals PACS Nos.: 61.25.Mv, 65.40.Gr, 71.15.Dx

1. INTRODUCTION The theoretical basis for an understanding of the thermodynamics of simple liquid metals has been forged in recent years to a point where it can be used to calculate the thermodynamics properties with some success [1-38]. This advance has been made possible due to the combination of the pseudopotential with thermodynamics perturbation theories. The pseudopotential theory enables one to formulate the energy in terms of the pseudopotential ∗

E-mail address: [email protected]. Tel.: +91-2832-256424

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196

Aditya M. Vora

and of the structure factor. On the other hand it is possible to write down closed form expressions for the thermodynamic quantities of hard sphere systems, including the structure factor, in the Percus-Yevick (PY) approximation [39]. A link between both theories is provided by a variational technique based on the Gibbs–Bogoliubov (GB) inequality [1-38]. This Gibbs–Bogoliubov (GB) inequality states that when the Hamiltonian of a given system is regarded as the Hamiltonian of a reference system plus a perturbation, the free energy of the system will always be smaller than that of the reference system plus the expectation values of the perturbation (calculated with the structure factors of the reference system). In our case, the reference system is one of the hard spheres, and the hard sphere diameters will be chosen to minimize the free energy. Hafner [4] has reported the internal energy of liquid alkali metals at their melting temperature using optimized pseudopotential. The results reported were deviating in the order of 3%-4% from the experimental findings. In the computation of the entropy and the internal energy of Na, K, Rb and Cs, Singh and Singh [5] have employed Ashcroft’s empty core (EMC) [40], Heine-Abarenkov’s [5] and Harmonic model potentials [5]. They concluded that, the thermodynamic properties of the alkali metals are very sensitive to the details of the potential inside the core region. In their study the local field correction due to Hubbard and Sham [5] was adopted. The temperature dependent thermodynamic properties of liquid Na, K, Rb and Cs were investigated by Singh and Singh [7] on the bases of Heine-Abarenkov model potential [7]. They have concluded that the internal energy depends very little on the temperature above the melting point. The application of generalized non-local model potential (GNMP) [8] and energy independent non-local model potential (EINMP) [9] have produced quite consistent results regarding the internal energy of liquid alkali metals. In all above studies, the PY theory [39] was adopted as a reference system for structural description. While, Ono et al. [10] have reported the use of one component plasma (OCP) system in the study of Helmholtz free energy of liquid alkali metals. The soft sphere (SS) reference system was used by Akinlade [12]. Also, Akinlade et al. [13, 14] have reported their results using modified generalized non-local model potential (MGNMP) with charged hard sphere (CHS) reference system [13, 14]. Very recently, Thakor et al. [15-18] have reported temperature dependent thermodynamic properties of some liquid metals using model potential formalism. Also, in the previous two decades, the considerable efforts have been made to the understanding of structure and thermodynamic properties of several liquid noble, transition and rare-earth metals [19-31] A large number of experimental measurements on various properties of these metals are now available in the literature [32, 33]. The interatomic potentials of the simple liquid metals [34-36] have been fully investigated and their thermodynamic properties could be derived with sufficient accuracy, but in the case of transition metals the hybridization of d-electron with s-electron makes the things complex. Despite the success of the theory in the solid state, results for the structure factor of liquid 3dtransition metals using molecular dynamics and other complicated liquid state theories have not been so reliable [11, 37]. Wills and Harrison [41] have derived the interatomic potentials for transition metals and obtained thermodynamic properties reasonably good. The WHpotentials [41] have also been used for the determination of the surface properties of noble metals [21] and thermodynamic properties of 3d-transition metals in the liquid state [11, 29]. The variational technique with a hard sphere fluid as a reference system have been used by Bretonnet and Derouiche [29] for the calculation of Helmholtz free energy of a series of liquid transition metals. The potentials of Wills-Harrison [41] and Bretonnet-Derouiche [29]

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197

are also successfully exploited for the computation of several equilibrium properties. A number of theoretical calculations based on the atomistic model and the use of interatomic potentials have been used for the study of the liquid metals [42-44]. Computer experiments are always intended to propose a plausible interpretation of experimental results in some cases to give the solution to an experimentally inaccessible problem. The reliability of the predicted values, however, is entirely depends on the validity of a given interatomic potential and the model used. Very recently, Baria [20] has reported thermodynamic properties of some d- and f-shell liquid metals using a variational approach. From the literature survey, it reveals that though the local pseudopotentials are proved very good for explaining various electronic, transport, static and vibrational properties of liquid metals, the local pseudopotentials are not rigorously applied to study the thermodynamics of large number of the liquid metals so far. The influence of various exchange and correlations on the thermodynamic is also not investigated from the aforesaid study. Hence we thought it worthwhile to apply well known Ashcroft’s empty core (EMC) model potential [40] in the detail investigation of Helmholtz free energy of some liquid metals of the different groups of the periodic table. The local field correction function is a very important factor for establishing a well local pseudopotential and for the proper reproduction of the property under investigation. To investigate influence of exchange and correlation effects, we have used five different types of the local field correction functions due to Hartree (HR) [45], Taylor (TY) [46], IchimaruUtsumi (IU) [47], Farid et al. (FR) [48] and Sarkar et al. (SR) [49]. The structural contribution to the Helmholtz free energy is accounted by adopting PY-hard sphere [39] reference system.

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2. COMPUTATIONAL METHODOLOGY Well known empty core (EMC) model potential of Ashcroft’s [40] used to explain electronion interaction in the liquid metals is of the following form (in Ryd. units)

WB (q) =

− 4π Z e 2 cos(qrC ) , Ωo q 2

(1)

Here Z , e , ΩO , q and rC are the valence, charge of the electron, atomic volume, wave vector, parameter of the model potential, respectively. A successful method for the theoretical calculation of thermodynamic properties in metallic systems has been discussed by Gibbs-Bogoliubov (GB) equation [32]. In this formulation, the Helmholtz free energy ( F ) of the system is written in the form [3-5 15-18, 20]

F = U − TS , with U is the internal energy and S the entropy of the system at a temperature T . The internal energy U can be expressed as [3-5 15-18, 20]

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(2)

198

Aditya M. Vora

U = U elec + U ion + U elec −ion ,

(3)

where,

U elec

2 ⎧⎪⎛ 3 ⎞ 2 ⎛ 3 ⎞ ⎛ 1 ⎞ ⎛ π k B ⎞ 2 ⎫⎪ ⎟⎟ T ⎬, = NZ ⎨⎜ ⎟ k F − ⎜ ⎟ k F − 0.0474 − 0.0155 ln k F − ⎜ ⎟ ⎜⎜ ⎝ 4π ⎠ ⎝ 2 ⎠ ⎝ kF ⎠ ⎪⎩⎝ 10 ⎠ ⎪⎭

U ion

⎛3⎞ = ⎜ ⎟ k BT ⎝2⎠

⎛ Z2 ⎞ + ⎜⎜ ⎟⎟ ⎝π ⎠

(4)

∞

∫ {a(q ) − 1}dq,

(5)

0

and

U elec − ion

2

∞ ⎧ ⎫ ⎧ 1 8πZ ⎫ ⎛ Z ⎞ 1 = lim⎨ V 0 (q ) + 2 ⎬ ⎜ ⎟ + − 1⎬q 4 dq. V 0 (q ) a(q )⎨ 3 ∫ q →0 q ⎭ ⎝ Ω ⎠ 16π 0 ⎩ ⎩ ε (q ) ⎭

{

Here, k F , N , a(q ) , WB (q ) and

}

(6)

ε (q ) are the Fermi wave vector, total number of

atoms, structure factor, bare ion pseudopotential [40] and the modified Hartree dielectric function [45]. The modified Hartree dielectric function

ε (q ) , which takes into account of the

conduction electrons interaction, is of the form [45],

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ε (q ) = 1 + {ε H (q ) − 1}{1 + f (q )}. In this expression

(7)

ε H (q ) is the static Hartree dielectric function [45] and f (q ) the

correction factor for the exchange and correlated motion of the conduction electrons. In the present study we consider the local field correction due to Hartree (HR) [45], Taylor (TY) [46], Ichimaru-Utsumi (IU) [47], Farid et al. (FR) [48] and Sarkar et al. (SR) [49] to investigate the relative influence of exchange and correlation effects. The Hartree (HR) [45] dielectric function does not include any exchange and correlation effect among the conduction electrons. Hence it is purely a static dielectric function, which is given by

f H (q ) = 0 .

(8)

The screening function of Taylor (TY) [46] is best justified at high densities. It covers the overall features of the various local field corrections function proposed before 1972. The correction factor of Taylor (TY) [46] is expressed as

⎛ q 2 ⎞ ⎡ 0.1534 ⎤ ⎟ 1+ fT (q ) = ⎜⎜ ⎥. 2 ⎟ ⎢ πk F ⎦ 4 k ⎝ F ⎠⎣

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(9)

Thermodynamics of Liquid Metals: A Variational Approach

199

The correction function of Ichimaru-Utsumi (IU) [47] is a fitting formula for the dielectric screening function of degenerate electron liquids at metallic and lower densities. It reproduce accurately the Monte Carlo results as well as those of microscopic calculations, which also satisfies self-consistency conditions in the compressibility sum rule and short range correlations. This function involves the logarithmic singularity at q = 2k F and the accompanying peak at q = 1.94k F . It can be expressed as

⎤ ⎡⎛ 4 − X4 ⎞ 2 + X ⎤ ⎡ ⎛ 8A⎞ ⎟⎟ ln fIU( X) = AX4 + BX2 + C + ⎢AX4 + ⎜ B + ⎟ X2 − C⎥ ⎢⎜⎜ ⎥. 3⎠ ⎝ ⎦ ⎣⎝ 4 X ⎠ 2 − X ⎦ ⎣

(10)

The correction function of Farid et al. (FR) [48] satisfies the exact asymptotic results for the short and long wavelength limits as determined in terms of some exact frequency moments of the density-density correlation function. The major ingredient of it is the quantum-Monte-Carlo results of Ceperley and Alder for the correlation energy in the paramagnetic state of the uniform electron gas. The equation is given by,

⎡⎛ 4 − X 4 ⎞ 2 + X ⎤ ⎟ ln f F ( X ) = AX 4 + BX 2 + C + AX 4 + D X 2 − C ⎢⎜⎜ ⎥. ⎟ ⎣⎢⎝ 4 X ⎠ 2 − X ⎥⎦

[

]

(11)

The screening function of Sarkar et al. (SR) [49] is latest one, which is derived in the same fashion as that of the Ichimaru-Utsumi (IU) [47] and Farid et al. (FR) [48], is given by,

[ (

)]

(12)

In Eqs. (10-12), X = q / kF.

The constants A , B , C and D involve in the above

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f S ( X ) = A 1 − 1 + BX 4 exp (− CX 2 ) .

expressions are well defined in the respective references [47-49]. The second most essential part in the computation of the Helmholtz free energy ( F ) is to calculate the entropy ( S ). The total entropy ( S ) is given by [3-5 15-18, 20]

S = S gas + Sη + Selec ,

(13)

with

S gas

1 ⎧ ⎫ ⎪ ⎛ 2πmk B T ⎞ 2 ⎪ ⎛5⎞ = ⎜ ⎟k B + k B ln ⎨Ω⎜ ⎟ ⎬, 2 ⎝2⎠ ⎠ ⎪ ⎪⎩ ⎝ h ⎭

Sη = k Bη (3η − 4)(1 − η ) , −2

and

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(14)

(15)

200

Aditya M. Vora

⎛ π 2 k B 2T ⎞ ⎟. Selec = ⎜⎜ 2 ⎟ k ⎝ F ⎠

(16)

Where, k B , m and h are the Boltzmann constant, the atomic mass and the Planck’s

constant, respectively. Using the information of packing fraction η , the entropy has been calculated. In the present study we have calculated η as a function of temperature [32].

η (T ) = AW exp (− BW T )

(17)

where T is the absolute temperature. The parameters AW and BW are given by Waseda [32].

3. RESULTS AND DISCUSSION The input parameters and constants used in the present investigation of the thermodynamic properties of the liquid metals are tabulated in Table 1. The computed values of the thermodynamic properties such as U elec , U ion , U elec −ion , U , various contributions to the entropy, S and F are tabulated in Tables 2-7. From the Tables 2 and 3, it is seen that the values of the U elec and U ion are negative while the U elec −ion has positive contribution to the internal energy. The present results of the

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U elec and U ion are found in qualitative agreement with those of theoretical results of the others [17, 18, 19, 20] and experimental data [33, 38]. There are large variations found in the presently computed results and others because of U elec and U ion terms are depended on the model potential and the structure factor of the liquid metals. It is seen from Tables 4 that, the local field correction functions affect the total internal energy ( U ) through U ion and U elec −ion . The effect is more visible on U elec −ion which involves the first and second order band structure energies. Presently computed results of the total internal energy ( U ) from FR-screening function are found lower while those from HRscreening function are found higher than the other local field correction functions. The comparisons of computed yielding with available theoretical [17, 18, 19, 20] and experimental [33, 38] data are highly encouraging and found qualitative agreement. The temperature dependent η is incorporated in the calculation of structure factor a(q ) . This

a(q ) is then used in the computation of U ion and U elec −ion for temperature dependency. It is

also observed that the magnitude of the internal energy increases with temperature.

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Table 1. The input parameters and constants used in the present calculations. Metals Li Na K Rb Cs Rh Ir Cu Ag Ni Pd Pt Ti V Cr Mn Fe Co Zr Sc Nb Mo Ta W Os Be Mg Ca Sr Ba Zn Cd Hg Au La Yb Ce Th Eu Al Ga In Tl Gd Tb Si Ge Sn Pb Sb Bi

Z 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 5.0 5.0

T 453.0 378.0 343.0 313.0 323.0 2236.0 2720.0 1423.0 1273.0 1773.0 1853.0 2053.0 2073.0 2273.0 2273.0 1633.0 1933.0 1923.0 2273.0 1933.0 2750.0 2895.0 3293.0 3695.0 3306.0 1562.0 1053.0 1223.0 1153.0 1103.0 823.0 723.0 393.0 1423.0 1243.0 1123.0 1143.0 2031.0 1203.0 1043.0 423.0 533.0 688.0 1703.0 1753.0 1733.0 1253.0 623.0 713.0 903.9 544.6

ΩO (au) 146.46 266.08 480.84 627.15 775.73 95.60 99.80 119.60 173.06 85.21 113.87 120.31 129.30 106.47 92.90 103.09 89.25 85.74 172.32 172.48 121.31 105.16 121.67 107.05 94.36 54.78 157.68 292.93 381.17 427.80 103.27 145.55 157.68 228.46 216.63 312.75 265.24 222.98 369.26 111.46 131.99 175.90 194.00 254.93 245.91 121.48 146.25 180.85 203.86 205.26 238.56

η 0.46 0.46 0.46 0.43 0.43 0.45 0.45 0.46 0.45 0.45 0.47 0.47 0.44 0.44 0.45 0.45 0.44 0.45 0.44 0.43 0.45 0.45 0.45 0.45 0.45 0.45 0.46 0.46 0.46 0.46 0.46 0.45 0.45 0.46 0.43 0.43 0.42 0.42 0.42 0.45 0.43 0.45 0.45 0.43 0.43 0.38 0.38 0.43 0.46 0.40 0.40

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rc (au) 0.97 1.60 2.00 2.05 2.07 0.71 0.72 0.69 0.94 0.59 0.68 0.68 0.84 0.78 0.73 0.69 0.65 0.62 0.92 0.91 0.87 0.83 0.90 0.85 0.76 0.75 1.36 1.61 1.87 2.38 1.28 1.01 1.57 0.74 0.84 0.80 0.81 0.88 0.95 1.12 1.17 1.28 1.45 0.79 1.46 0.97 1.01 1.04 1.41 1.33 1.40

202

Aditya M. Vora Table 2. U elec and U ion (in 10-3 au) of liquid metals.

U elec

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Metals

U ion

Present Results

Others [17, 18, 19, 20]

Present Results

Li

-76.03

-76.63

-324.75

Na K Rb Cs Rh Ir Cu Ag Ni Pd Pt Ti V Cr Mn Fe Co Zr Sc Nb Mo Ta W Os Be Mg Ca Sr Ba Zn Cd Hg Au La Yb Ce Th Eu Al Ga In Tl Gd Tb Si Ge Sn Pb Sb Bi

-81.60 -80.64 -78.87 -77.08 -75.05 -79.69 -89.49 -107.59 -64.23 -87.13 -91.12 -95.50 -83.41 -72.76 -79.73 -68.65 -65.05 -110.02 -109.01 -93.58 -84.20 -95.68 -88.33 -76.89 45.43 -117.68 -153.55 -160.66 -162.60 -71.99 -110.24 -116.96 -143.33 -140.00 -155.45 -149.55 -144.06 -160.13 -29.83 -72.67 -131.48 -147.92 -187.66 -183.60 41.49 -34.19 -105.58 -140.26 -93.66 -151.91

-81.71 -80.07 -78.61 -76.66 157.27 161.04 60.46 72.89 125.13 166.10 169.31 − − − − − − − − − − − − − − − − − − − − − 81.39 433.21 434.97 415.03 391.45 − − − − − − − − − − − − −

-266.08 -218.14 -194.64 -181.02 -731.99 -716.67 -686.33 -605.94 -765.84 -694.46 -679.62 -660.46 -704.06 -738.86 -719.01 -751.24 -762.79 -596.46 -598.46 -669.62 -702.17 -663.77 -690.25 -725.11 -1306.87 -911.08 -737.65 -675.35 -649.92 -1052.81 -947.40 -925.41 -800.47 -837.37 -740.68 -788.58 -827.75 -704.53 -2259.14 -2139.14 -1943.86 -1879.79 -1704.74 -1725.06 -4028.37 -3790.31 -3642.35 -3558.29 -5657.06 -5383.54

Others [17, 18, 19, 20] -262.37, -262.38, -262.39 -215.45, -215.46 -172.07 -161.03 -149.38 -835.67 -776.99 -379.17 -327.09 -887.28 -841.07 -831-77 − − − − − − − − − − − − − − − − − − − − − -328.28 -1081.46 -1145.82 -1084.30 -1113.29 − − − − − − − − − − − − −

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Table 3. U elec −ion of liquid metals.

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Metals Li Na K Rb Cs Rh Ir Cu Ag Ni Pd Pt Ti V Cr Mn Fe Co Zr Sc Nb Mo Ta W Os Be Mg Ca Sr Ba Zn Cd Hg Au La Yb Ce Th Eu Al Ga In Tl Gd Tb Si Ge Sn Pb Sb Bi

HR 5.41 42.08 37.32 21.83 13.20 -126.16 -125.78 -154.64 -87.18 -184.71 -147.39 -153.08 -103.23 -109.11 -111.06 -146.42 -161.87 -166.93 -101.98 -113.78 -73.06 -73.43 -59.39 -65.78 -96.13 -96.31 23.71 12.19 3.46 -142.03 16.91 -114.92 -38.76 -517.72 -423.52 -541.62 -511.23 -399.59 -446.73 -173.07 -222.43 -165.17 -230.14 -1509.34 -206.65 -1184.49 -1220.03 -1005.46 -570.31 -1569.92 -1533.04

TY 15.93 44.30 38.91 24.75 17.32 -46.58 -46.46 -54.34 -30.25 -71.12 -47.47 -49.61 -40.40 -42.83 -39.80 -55.28 -66.24 -63.88 -40.14 -48.96 -22.74 -22.40 -15.98 -18.67 -32.95 -44.20 62.53 36.28 50.32 89.33 106.69 -59.00 103.66 -201.37 -199.02 -233.00 -236.30 -199.27 -208.35 -71.29 -127.23 -82.33 -42.77 -856.14 -109.68 -1002.46 -1032.95 -799.71 -227.87 -974.45 -941.50

Present Results IU 17.32 44.71 39.21 25.26 17.95 -37.74 -37.63 -44.07 -23.38 -59.48 -37.63 -39.49 -32.76 -34.89 -31.68 -45.44 -55.72 -53.10 -32.60 -40.92 -16.48 -16.09 -10.37 -12.72 -25.54 -35.44 65.05 38.02 53.29 109.69 112.76 -49.27 114.60 -159.03 -167.06 -190.83 -197.96 -170.09 -174.47 -61.69 -118.23 -74.24 -22.40 -736.78 -100.34 -976.75 -1003.70 -762.22 -188.59 -912.37 -879.08

FR 17.81 44.76 39.24 25.34 18.11 -33.08 -33.01 -37.93 -20.33 -52.40 -31.36 -32.99 -29.39 -31.26 -27.59 -40.01 -50.06 -46.77 -29.31 -37.53 -13.83 -13.38 -8.18 -10.28 -22.00 -33.71 67.76 39.50 56.55 125.20 119.57 -47.33 124.86 -142.14 -155.98 -174.96 -184.17 -160.53 -162.76 -55.30 -112.54 -69.42 -10.77 -714.33 -94.60 -971.98 -999.93 -759.11 -168.08 -875.43 -842.85

SR 11.25 43.48 38.32 23.57 15.54 -87.46 -86.91 -104.88 -57.00 -131.64 -98.42 -102.03 -70.72 -75.77 -76.39 -101.80 -116.11 -118.51 -69.16 -79.26 -47.06 -47.58 -36.75 -41.74 -65.10 -73.72 42.76 25.15 27.58 -23.77 55.62 -84.34 28.34 -352.60 -304.52 -376.25 -364.10 -292.34 -317.09 -129.61 -178.34 -124.15 -141.12 -1150.94 -156.90 -1089.08 -1114.73 -885.75 -413.81 -1312.17 -1264.90

Others [17, 18, 19, 20] 32.07, 73.47, 87.12, 87.22, 102.40 65.07, 103.51, 117.87, 118.17, 139.48 56.67, 89.32, 103.03, 103.19, 126.03 35.52, 69.98, 85.05, 85.09, 108.24 18.42, 53.01, 68.73, 68.94, 91.50 -72.41 -76.16 -59.22 -74.59 -49.76 -77.29 -79.44 − − − − − − − − − − − − − − − − − − − − − -74.45 -147.64 -141.31 -143.47 -143.79 − − − − − − − − − − − − −

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204

Aditya M. Vora Table 4. Total internal energy ( U ) of liquid metals.

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Metals

HR

TY

Li

-395.37

-384.86

Na

-305.60

K Rb

Present Results IU

Expt. [33, 38]

FR

SR

-383.47

-382.97

-389.53

-254

-303.38

-302.97

-302.92

-304.21

-226

-261.46

-259.87

-259.57

-259.54

-260.46

-194

-251.68

-248.76

-248.25

-248.17

-249.94

-180

Cs

-244.91

-240.79

-240.15

-239.99

-242.56

-170

Rh Ir Cu Ag Ni Pd Pt Ti V Cr Mn Fe Co Zr Sc Nb Mo Ta W Os Be Mg Ca Sr Ba Zn Cd Hg Au La Yb Ce Th Eu Al Ga In Tl Gd Tb Si Ge Sn Pb Sb Bi

-933.20 -922.14 -930.46 -800.71 -1014.78 -928.98 -923.82 -859.18 -896.58 -922.69 -945.16 -981.76 -994.78 -808.47 -821.25 -836.26 -859.80 -818.83 -844.37 -898.13 -1357.74 -1005.05 -879.01 -832.55 -954.55 -1107.89 -1172.56 -1081.13 -1461.53 -1400.89 -1437.75 -1449.36 -1371.40 -1311.39 -2462.03 -2434.24 -2240.51 -2257.85 -3401.75 -2115.31 -5171.37 -5044.53 -4753.39 -4268.87 -7320.64 -7068.48

-853.62 -842.83 -830.16 -743.78 -901.19 -829.06 -820.35 -796.35 -830.30 -851.43 -854.03 -886.13 -891.73 -746.62 -756.43 -785.93 -808.78 -775.43 -797.26 -834.94 -1305.63 -966.23 -854.92 -785.69 -723.19 -1018.11 -1116.64 -938.71 -1145.18 -1176.40 -1129.13 -1174.43 -1171.08 -1073.00 -2360.26 -2339.04 -2157.67 -2070.48 -2748.55 -2018.33 -4989.34 -4857.45 -4547.63 -3926.42 -6725.17 -6476.95

-844.78 -834.00 -819.89 -736.92 -889.55 -819.21 -810.23 -788.72 -822.36 -843.31 -844.18 -875.61 -880.94 -739.08 -748.39 -779.67 -802.46 -769.82 -791.31 -827.53 -1296.87 -963.71 -853.18 -782.72 -702.83 -1012.04 -1106.91 -927.77 -1102.83 -1144.44 -1086.96 -1136.09 -1141.90 -1039.12 -2350.66 -2330.04 -2149.57 -2050.11 -2629.19 -2009.00 -4963.63 -4828.20 -4510.15 -3887.15 -6663.09 -6414.52

-840.12 -829.37 -813.75 -733.86 -882.47 -812.94 -803.73 -785.34 -818.73 -839.22 -838.75 -869.94 -874.61 -735.79 -745.00 -777.03 -799.76 -767.63 -788.86 -824.00 -1295.14 -961.00 -851.70 -779.46 -687.33 -1005.23 -1104.98 -917.51 -1085.94 -1133.35 -1071.09 -1122.30 -1132.34 -1027.42 -2344.27 -2324.35 -2144.76 -2038.48 -2606.74 -2003.25 -4958.86 -4824.43 -4507.04 -3866.63 -6626.15 -6378.29

-894.50 -883.27 -880.70 -770.53 -961.71 -880.01 -872.77 -826.68 -863.24 -888.02 -900.54 -936.00 -946.36 -775.64 -786.73 -810.26 -833.96 -796.19 -820.32 -867.09 -1335.15 -986.00 -866.05 -808.43 -836.30 -1069.18 -1141.98 -1014.02 -1296.41 -1281.89 -1272.37 -1302.23 -1264.15 -1181.74 -2418.57 -2390.15 -2199.49 -2168.83 -3043.34 -2065.56 -5075.96 -4939.23 -4633.67 -4112.36 -7062.89 -6800.34

− − − − − − − − − − − − − − − − − − − − − -879 − − − -1013.6 -1000 − − − − − − − -2035 − -2030 -2145 − − − − − − − −

Others [17, 18, 19, 20] -236.63, -251.80, -251.55, 265.55, -306.94 -157.69, -178.99, -179.30, 193.59, -232.08 -126.11, -148.95, -149.11, 162.82, -195.61 -131.39, -154.54, -154.58, 169.66, -204.11 -134.54, -157.10, -157.31, 173.03, -207.62 -750.11 -692.10 -377.93 -328.79 -811.91 -752.25 -744.90 − − − − − − − − − − − − − − -646.6 − − − -1064 -715.1 − -321.35 -795.88 -852.17 -812.73 -865.63 − -1646.10 − -2096.20 -1942.40 − − − − − -3733 − −

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Table 5. Various contributions to the entropy of liquid metals.

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Metals Li Na K Rb Cs Rh Ir Cu Ag Ni Pd Pt Ti V Cr Mn Fe Co Zr Sc Nb Mo Ta W Os Be Mg Ca Sr Ba Zn Cd Hg Au La Yb Ce Th Eu Al Ga In Tl Gd Tb Si Ge Sn Pb Sb Bi

S gas k B Present Results 8.7848 11.1784 12.5668 14.0055 14.8804 12.3884 13.3832 11.9039 13.0671 11.4458 12.6279 13.5923 11.5579 11.4559 11.3504 11.5370 11.4175 11.4579 12.8113 11.7510 12.4878 12.3931 13.4907 13.3865 13.3113 8.1929 10.7390 12.1083 13.5448 14.3343 11.7997 12.9558 13.9045 14.2480 13.6710 14.3677 13.8864 14.4695 14.3390 10.5484 12.1419 13.1770 14.1398 14.0198 13.9997 10.6949 12.3046 13.2545 14.2099 13.4193 14.3800

Others [17, 18, 19, 20] 8.82 11.22 12.66 14.04 14.92 15.82 17.80 14.26 15.39 14.40 15.66 16.74 − − − − − − − − − − − − − − − − − − − − − 16.44 16.40 16.44 16.19 17.77 − − − − − − − − − − − − −

Sη k B Present Results -4.1331 -4.1331 -4.1331 -3.5866 -3.5866 -3.9421 -3.9421 -4.1331 -3.9421 -3.9421 -4.3336 -4.3336 -3.7602 -3.7602 -3.9421 -3.9421 -3.7602 -3.9421 -3.7602 -3.5866 -3.9421 -3.9421 -3.9421 -3.9421 -3.9421 -3.9421 -4.1331 -4.1331 -4.1331 -4.1331 -4.1331 -3.9421 -3.9421 -4.1331 -3.5866 -3.5866 -3.4209 -3.4209 -3.4209 -3.9421 -3.5866 -3.9421 -3.9421 -3.5866 -3.5866 -2.8273 -2.8273 -3.5866 -4.1331 -3.1111 -3.1111

Others [17, 18, 19, 20] -3.78 -4.18 -3.59 -3.61 -3.60 -3.94 -3.94 -4.133 -5.000 -3.262 -4.333 -4.333 − − − − − − − − − − − − − − − − − − − − − -4.133 -3.586 -3.586 -2.966 -3.586 − − − − − − − − − − − − −

S elec k B Present Results 0.0411 0.0511 0.0688 0.0749 0.0891 0.1166 0.1459 0.0861 0.0986 0.0856 0.1085 0.1248 0.1322 0.1273 0.1163 0.0895 0.0963 0.0932 0.1755 0.1493 0.1680 0.1608 0.2016 0.2077 0.1709 0.0464 0.0633 0.1110 0.1248 0.1289 0.0373 0.0412 0.0236 0.1095 0.0923 0.1065 0.0971 0.1537 0.1274 0.0379 0.0172 0.0263 0.0362 0.1075 0.1081 0.0551 0.0451 0.0258 0.0320 0.0351 0.0234

Others [17, 18, 19, 20] 0.0431 0.0525 0.0734 0.0766 0.0925 0.114 0.141 0.085 0.106 0.077 0.099 0.111 − − − − − − − − − − − − − − − − − − − − − 0.118 0.102 0.084 0.088 0.153 − − − − − − − − − − − − −

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Aditya M. Vora

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In comparison with the presently computed results of the total internal energy ( U ) from static HR-function, the percentile influences for Li, Na, K, Rb, Cs, Rh, Ir, Cu, Ag, Ni, Pd, Pt, Ti, V, Cr, Mn, Fe, Co, Zr, Sc, Nb, Mo, Ta, W, Os, Be, Mg, Ca, Sr, Ba, Zn, Cd, Hg, Au, La, Yb, Ce, Th, Eu, Al, Ga, In, Tl, Gd, Tb, Si, Ge, Sn, Pb, Sb and Bi liquid metals of TY, IU, FR and SR-functions are of the order of 1.48%-3.14%, 0.45%-0.88%, 0.38%-0.73%, 0.69%1.39%, 0.96%-2.01%, 4.15%-9.97%, 4.22%-10.06%, 5.35%-12.54%, 3.77%-8.35%, 5.23%13.04%, 5.27%-12.49%, 5.53%-13.00%, 3.78%-8.59%, 3.72%-8.68%, 3.76%-9.05%, 4.72%11.26%, 4.66%-11.39%, 4.87%-12.08%, 4.06%-8.99%, 4.20%-9.28%, 3.11%-7.08%, 3.01%6.98%, 2.76%-6.25%, 2.85%-6.57%, 3.46%-8.25%, 1.66%-4.61%, 1.90%-4.38%, 1.47%3.11%, 2.90%-6.38%, 12.39%-27.99%, 3.49%-9.27%, 2.61%-5.76%, 6.21%-15.13%, 11.30%-25.70%, 8.49%-19.10%, 11.50%-25.50%, 10.15%-22.57%, 7.82%-17.43%, 9.89%21.65%, 1.77%-4.78%, 1.81%-4.51%, 1.83%-4.27%, 3.94%-9.72%, 10.54%-23.37%, 2.35%5.30%, 1.84%-4.11%, 2.09%-4.36%, 2.52%-5.18%, 3.67%-9.42%, 3.52%-9.49% and 3.79%9.76%, respectively. The calculated results of the total internal energy ( U ) for Li, Na, K, Rb, Cs, Rh, Ir, Cu, Ag, Ni, Pd, Pt, Mg, Zn, Cd, Au, La, Yb, Ce, Th, Al, In and Tl liquid metals deviate in the range of 33.76%-55.66%, 25.41%-35.22%, 25.26%-34.77%, 27.49%-39.82%, 29.21%44.06%, 11.21%-24.41%, 17.01%-33.24%, 53.90%-146.20%, 55.38%-143.53%, 8.73%24.99%, 8.07%-23.49%, 7.90%-24.02%, 8.79%-14.34%, 0.15%-9.30%, 9.66%-17.26%, 70.86%-354.81%, 30.46%-76.02%, 21.60%-68.72%, 28.46%-78.33%, 24.19%-58.43%, 13.43%-20.98%, 5.56%-10.37% and 1.11%-5.26% from the experimental data [33, 38], respectively. The various contributions to the entropy are given in Table 5. Among the three contributions, only Sη is structure dependent while other two i.e S gas and S elec depend only upon the density and are thus independent of the model pseudopotential. The major contribution to the total entropy comes from the term S gas . The present results of various contributions of the entropy are found in fair agreement with the others [17, 18, 19, 20]. Present results of the total entropy ( S ) are narrated in Table 6. The comparison of the calculated total entropy ( S ) with the available experimental [33, 38] finding shows that as the temperature of the liquid metals changes the deviation with experimental observation [33, 38] also increases. In general, the deviation is within 2.18% to 28.00%. Finally, using the total internal energy ( U ) and total entropy ( S ) we have generated the Helmholtz free energy ( F ) which are shown in Table 7. From the Table 7 it is noticed that, the present results of the Helmholtz free energy ( F ) of liquid metals are found in qualitative agreement with the available experimental [33, 38] or theoretical [17, 18, 19, 20] findings in the literature. Also, it is noted that, among the five employed local filed correction functions, the local field correction function due to HR (without exchange and correlation) gives the maximum numerical value of the Helmholtz free energy ( F ), while the local field correction function due to FR gives the minimum value.

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Table 6. Total entropy ( S k B ) of liquid metals. Metals

Present Results

Others [17, 18, 19, 20]

Expt. [33, 38]

Li Na K Rb Cs Rh Ir Cu Ag Ni Pd Pt Ti V Cr Mn Fe Co Zr Sc Nb Mo Ta W Os Be Mg Ca Sr Ba Zn Cd Hg Au La Yb Ce Th Eu Al Ga In Tl Gd Tb Si Ge Sn Pb Sb Bi

4.6928 7.0965 8.5025 10.4938 11.3828 8.5628 9.5870 7.8570 9.2235 7.5893 8.4029 9.3835 7.9298 7.8230 7.5245 7.6844 7.7535 7.6090 9.2266 8.3137 8.7137 8.6118 9.7502 9.6521 9.5400 4.2971 6.6691 8.0863 9.5365 10.3301 7.7040 9.0548 9.9860 10.2244 10.1766 10.8876 10.5626 11.2023 11.0455 6.6442 8.5725 9.2612 10.2339 10.5407 10.5211 7.9228 9.5224 9.6937 10.1089 10.3433 11.2923

5.08 7.10 9.15 10.51 11.40 11.99 13.27 9.87, 10.21 10.49, 10.84 11.21, 11.69, 12.76 11.43, 12.11 12.52, 12.93 − − − − − − − − − − − − − − 6.7856 − − − − 9.1354 − 12.04, 12.43 12.90, 13 12.94 13.32, 14.4 14.33 − 6.7787 − 9.2965 10.2760 − − − − − 10.1718 − −

5.74 7.84 9.14 10.27 11.14 − − 10.28 10.96 10.02 11.50 12.46 − − − − − − − − − − − − − − 9.02 − − − 9.18 9.94 − 12.00 12.87 − 14.67 − − 8.63 − 9.13 11.00 − − − − − 11.13 − −

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Aditya M. Vora Table 7. Helmholtz free energy ( F ) of liquid metals. HR

TY

Present Results IU

FR

SR

Li

-402.10

-391.59

-390.20

-389.71

-396.27

Na

-314.10

-311.88

-311.47

-311.42

-312.70

K

-270.70

-269.11

-268.81

-268.78

-269.70

Rb

-262.08

-259.16

-258.66

-258.57

-260.34

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Metals

Cs

-256.56

-252.43

-251.80

-251.64

-254.21

Rh Ir Cu Ag Ni Pd Pt Ti V Cr Mn Fe Co Zr Sc Nb Mo Ta W Os Be Mg Ca Sr Ba Zn Cd Hg Au La Yb Ce Th Eu Al Ga In Tl Gd Tb Si Ge Sn Pb Sb Bi

-993.86 -1004.76 -965.88 -837.91 -1057.41 -978.31 -984.85 -911.27 -952.92 -976.88 -984.92 -1029.24 -1041.14 -874.91 -872.16 -912.18 -938.79 -920.56 -957.36 -998.06 -1379.00 -1027.30 -910.34 -867.39 -990.65 -1127.98 -1193.30 -1093.57 -1507.62 -1440.97 -1476.49 -1487.61 -1443.49 -1353.49 -2483.99 -2445.73 -2256.15 -2280.16 -3458.62 -2173.74 -5214.87 -5082.34 -4772.53 -4291.70 -7350.26 -7087.97

-914.28 -925.45 -865.58 -780.98 -943.82 -878.39 -881.39 -848.44 -886.64 -905.61 -893.78 -933.61 -938.09 -813.07 -807.35 -861.86 -887.77 -877.15 -910.26 -934.87 -1326.90 -988.48 -886.25 -820.53 -759.29 -1038.20 -1137.39 -951.15 -1191.27 -1216.48 -1167.87 -1212.68 -1243.16 -1115.10 -2382.22 -2350.53 -2173.31 -2092.78 -2805.42 -2076.77 -5032.84 -4895.25 -4566.77 -3949.26 -6754.79 -6496.43

-905.44 -916.62 -855.32 -774.12 -932.18 -868.55 -871.26 -840.80 -878.70 -897.49 -883.94 -923.09 -927.30 -805.53 -799.31 -855.60 -881.45 -871.55 -904.30 -927.46 -1318.14 -985.96 -884.51 -817.56 -738.93 -1032.12 -1127.65 -940.20 -1148.93 -1184.51 -1125.70 -1174.34 -1213.98 -1081.22 -2372.61 -2341.52 -2165.21 -2072.42 -2686.06 -2067.43 -5007.13 -4866.01 -4529.28 -3909.99 -6692.72 -6434.01

-900.78 -911.99 -849.17 -771.06 -925.10 -862.28 -864.77 -837.42 -875.07 -893.40 -878.51 -917.43 -920.97 -802.24 -795.92 -852.95 -878.75 -869.35 -901.86 -923.92 -1316.40 -983.25 -883.03 -814.30 -723.43 -1025.32 -1125.72 -929.94 -1132.04 -1173.43 -1109.83 -1160.55 -1204.42 -1069.52 -2366.22 -2335.84 -2160.40 -2060.79 -2663.61 -2061.69 -5002.36 -4862.24 -4526.17 -3889.47 -6655.78 -6397.77

-955.16 -965.89 -916.12 -807.74 -1004.34 -929.34 -933.80 -878.76 -919.57 -942.21 -940.30 -983.49 -992.72 -842.09 -837.65 -886.18 -912.95 -897.92 -933.32 -967.02 -1356.41 -1008.25 -897.38 -843.27 -872.40 -1089.26 -1162.72 -1026.46 -1342.50 -1321.97 -1311.11 -1340.48 -1336.24 -1223.84 -2440.53 -2401.64 -2215.13 -2191.14 -3100.22 -2123.99 -5119.46 -4977.04 -4652.81 -4135.20 -7092.51 -6819.82

Others [17, 18, 19, 20] -244.09, -259.26, -259.36, -273.10, -314.40 -166.19, -187.50, -187.80, -201.09, -240.58 -136.05, -158.90, -159.06, -172.76, -205.42 -141.81, -165.00, -165.27, -180.34, -214.84 -145.49, -168.05, -168.26, -183.98, -218.56 -26810.25 -36120.38 -14528.26 -13358.15 -19882.65 -21172.98 -25704.52 − − − − − − − − − − − − − − -667.10 − − − -1032.70 -733.10 − -17681.91 -16041.56 -14534.44 -15221.91 -29111.63 − -1666.40 − -2108.90 -1915.50 − − − − − -3752.80 − −

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Expt. [33, 38] -265 -236 -202 -190 -182 − − − − − − − − − − − − − − − − − − − − − − − − − -1084 -1020 −

− -2035 − -2042 -2165 − − − − − -3656 − −

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Finally, using the total internal energy ( U ) and total entropy ( S ) we have generated the Helmholtz free energy ( F ) which are shown in Table 7. From the Table 7 it is noticed that, the present results of the Helmholtz free energy ( F ) of liquid metals are found in qualitative agreement with the available experimental [33, 38] or theoretical [17, 18, 19, 20] findings in the literature. Also, it is noted that, among the five employed local filed correction functions, the local field correction function due to HR (without exchange and correlation) gives the maximum numerical value of the Helmholtz free energy ( F ), while the local field correction function due to FR gives the minimum value. The percentile influences from static HR-function on the Helmholtz free energy ( F ) for Li, Na, K, Rb, Cs, Rh, Ir, Cu, Ag, Ni, Pd, Pt, Ti, V, Cr, Mn, Fe, Co, Zr, Sc, Nb, Mo, Ta, W, Os, Be, Mg, Ca, Sr, Ba, Zn, Cd, Hg, Au, La, Yb, Ce, Th, Eu, Al, Ga, In, Tl, Gd, Tb, Si, Ge, Sn, Pb, Sb and Bi liquid metals of various local field correction functions are found of the order of 1.45%-3.08%, 0.45%-0.85%, 0.37%-0.71%, 0.66%-1.34%, 0.92%-1.92%, 3.89%9.37%, 3.87%-9.23%, 5.15%-12.08%, 3.60%-7.98%, 5.02%-12.51%, 5.01%-11.86%, 5.18%12.19%, 3.57%-8.10%, 3.50%-8.17%, 3.55%-8.55%, 4.53%-10.80%, 4.45%-10.86%, 4.65%11.54%, 3.75%-8.31%, 3.96%-8.74%, 2.85%-6.49%, 2.75%-6.40%, 2.46%-5.56%, 2.51%5.80%, 3.11%-7.43%, 1.64%-4.54%, 1.85%-4.29%, 1.42%-3.00%, 2.78%-6.12%, 11.94%26.97%, 3.43%-9.10%, 2.56%-5.66%, 6.14%-14.96%, 10.95%-24.91%, 8.26%-18.57%, 11.20%-24.83%, 9.89%-21.99%, 7.43%-16.56%, 9.58%-20.98%, 1.75%-4.74%, 1.80%4.49%, 1.82%-4.24%, 3.90%-9.62%, 10.36%-22.99%, 2.29%-5.15%, 1.83%-4.08%, 2.07%4.33%, 2.51%-5.16%, 3.65%-9.37%, 3.51%-9.45% and 3.78%-9.74%, respectively. Hence the strong influence of local field correction on the Helmholtz free energy ( F ) is predicted. The computed findings of the Helmholtz free energy ( F ) for Li, Na, K, Rb, Cs, Zn, Cd, Al, In, Tl and Pb liquid metals deviate in the range of 32.09%-51.74%, 24.23%-33.09%, 24.85%34.01%, 26.54%-37.94%, 27.72%-40.97%, 0.49%-5.41%, 9.55%-16.99%, 14.23%-22.06%, 5.80%-10.49%, 1.21%-5.32% and 6.39%- 17.39% from the experimental data [33, 38], respectively. Thus it is confirmed from Tables 2-7 that, the presently calculated values of total internal energy ( U ) and total entropy ( S ) are in fair agreement with the experimental data [33, 38] and other reported data [17, 18, 19, 20]. This confirms the applicability of the EMC model potential in the investigation of thermodynamic properties of liquid metals of the different groups of the periodic table. Therefore, the present EMC model potential is equally useful in the calculation of various properties of liquid metals without any modification. Out of 51 liquid metals the theoretical or experimental data of only 17 liquid metals are available in the literature for comparisons. The numerical values of the thermodynamic properties of liquid metals are found to be quite sensitive to the selection of the local field correction function and showing a significant variation with the change in the function. Thus, the calculations of the thermodynamic properties are one of the sensitive tests for the proper assessment of the form factor of the model potential and in the absence of experimental information such calculations may be considered as one of the guidelines for further investigations either theoretical or experimental. In contrast with the reported studies, the present study spans the metallic elements of the different groups of the periodic table on a common platform of the model potential and common criteria for evaluating parameter of the model potential. This is very much essential for obtaining concrete conclusions.

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Aditya M. Vora

CONCLUSIONS Lastly we conclude that a proper choice of the local field correction function also plays a vital role for predicting thermodynamic properties of liquid metals. In comparison with the earlier reported investigation of thermodynamics of some liquid metals of the different groups of the periodic table, the present results are superior in qualitative as well as quantitative ways.

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[33] R. Hultgren, P. D. Desai, D. T. Howking, M. Gleiser, K. K. Kelly and D. D. Wagman, Selected values of the thermodynamic properties of the element and alloys, American Soc. Metals, Metal Park, Ohio (1973). [34] M. Shimoji, Liquid metals, Academic Press, London (1977). [35] T. E. Faber, An Introduction to the theory of liquid metals, Cambridge University Press, London (1972). [36] J. Hafner, Amorphous Solids and Liquid States, Plenum Press, New York (1985). [37] C. Hausleitner, G. Kahi and J. Hafner, J. Phys.: Condens. Matter 3 (1991) 1589. [38] M. M. G. Alemany, M. Calleja, C. Rey, L. J. Gallego, J. Casa and L. E. Gonzalez, J. Non-Cryst. Solids 250 (1999) 53. [39] N. W. Ashcroft and J. Lekner, Phys. Rev. 145 (1965) 83; Phys. Rev. 165 (1966) 83. [40] N. W. Ashcroft, Phys. Lett. 23 (1966) 48. [41] J. M. Wills and W. A. Harrison, Phys. Rev. B28 (1983) 4363. [42] K. C. Jain, N. Gupta and N. S. Saxena, Phys. Stat. Sol. (b) 160 (1990) 433. [43] A. Pratap, M. Rani, K. C. Jain, N. S. Saxena, Ind. J. Pure & Appl. Phys. 28 (1990) 657. [44] M. Rani, A. Pratap and N. S. Saxena, Ind. J. Pure & Appl. Phys. 27 (1989) 269. [45] W. A. Harrison, Pseudopotential in the Theory of Metals, W. A. Benjamin, Inc., New York (1966). [46] R. Taylor, J. Phys. F: Met. Phys. 8 (1978) 1699. [47] S. Ichimaru and K. Utsumi, Phys. Rev. B24 (1981) 7385. [48] B. Farid, V. Heine, G. Engel and I. J. Robertson, Phys. Rev. B48 (1993) 11602. [49] A. Sarkar, D. Sen, S. Haldar and D. Roy, Mod. Phys. Lett. B12 (1998) 639.

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Chapter 16

DOMINANT FEATURES OF INTERFACES IN NANOMETRIC DIELECTRICS O. P. Thakur* and Anjani Kumar Singh School of Applied Sciences, Netaji Subhas Institute of Technology, Sector-3, Dwarka, New Delhi, India

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ABSTRACT As the size of particle decreases to the nanometric scale, the number of molecules in contact with its surface increases sufficiently and as a result of this, the various shortrange, known and unknown, interactions affect the electromechanical and the electrochemical properties of nanocomposite at the interface. The interfacial regions with non-Centro symmetry and molecular organization, between two phases in contact are a nanometric system with properties in variably different from the bulk phases on either side. The presence of a large number of point defects even in equilibrium and the influence of the electrode’s wall on the orientation distributions of interfacial molecules play a vital role in describing the electrochemical and electromechanical properties of the structurally modified interface as compared to the bulk in continuum. In the space charge regions, the Debye- Huckel length, which is based on the bulk concentration, has been severely modified due to differential ionic concentration and segregation of charged clusters at the interface. Hence the simulation of diffuse double layer with the nanometric interface is not valid up to a great extent. It has also been observed that the interface cannot be realized as electromechanical transducer up to a large extent due to its anisotropic and fluctuating dielectric behavior.

Keywords: Nanometric Interface; Electromechanical coupling; Electric double layer; Dielectrics.

*

E-mail address: [email protected]

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1. INTRODUCTION

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In this paper we analytically discuss and review physical phenomena, which are applied to the interface by various researchers, along with their validity and applicability at the interface. The interface between two phases in contact can be categorized into three types i.e. solidsolid, solid-liquid and solid-gas interface with different interfacial local properties depending on the type of interactions and the nature of point defects along with the crystal structure of the interacting phases. The interface between two dissimilar phases, in which at least one dimensions is in the nanometric range, is a nanometric dielectric system with properties invariably different from those of the bulk phases in continuum on either side [1]. Due to substantial increase in the surface area to volume ratio of the nanoparticles, the numbers of molecules in its contact increases substantially and as a result of this, the extremely short range (hard core), quantum mechanical and repulsive forces dominate apart from several kinds of other forces acting at the interface. Several interactions are possible. The forces between ions are strongly coulombic and long range. When permanent dipoles are present, there will be weaker ion-dipole and dipole-dipole forces and induced dipole forces [2]. The interface between two phases becomes increasingly dominant as the particle size is reduced [3] and the size of the interacting particles appears to be more important than the material of the particles. As the solvent constitutes a major part of total volume of the solution (electrolyte), its contribution for the determination of electrochemical and electromechanical properties can’t be ignored. The orientation of dipoles, which varies from strong orientation at the electrode’s surface to the weak orientation in the electrolyte bulk through the diffuse double layers, depends mainly on the strength of the applied field and the polarity of the electrode. Field dependent nonlinear dielectric behaviour of water has been observed from the results [4] of simulations performed in presence of external electric field.

Figure 1. Hypothetical diffusion profile between two dissimilar phases in contact indicating inhomogeneous distribution of interacting particles at the interface. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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The interfacial properties and the width of interface between two interacting phases (Fig.1) depend primarily on the following parameters: i.

State of two interacting phases e.g. the width of solid-solid interface is smaller with high stability than the width of solid-gas interface with low stability. ii. Size & shape of interacting particles of two phases. iii. Shape and size of potential well or activation energy barrier. iv. Crystal structure of matrix phases particularly its packing fraction / density gradient. v. Thermal energy of the interacting particles. vi. Nature and concentration of point defects. vii. Lattice matching at the interface between two solid phases. viii. Strength of applied external fields and polarity on either side. The structural and chemical composition of the interface and its stability may differ from that of the bulk phases. In presence of charged species of an electric double layer, the local interface stoichiometry may also differ from that of the bulk. The electric double layer is formed due to segregation of charged species at the interface [5]. Due to anisotropic and nonlinear dielectric properties of interface, it is very difficult to realize the nanometric interface as an eletromechanical transducer up to large extent.

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2. POINT DEFECT CHEMISTRY AT THE INTERFACE The atoms or ions rearrange themselves during interaction in new positions of equilibrium, with large possibility of structural point defect. In many cases of the ionic solids, cations being smaller can occupy an interstitial close to their original site and thus cause no net changes in electrical charges. The electromechanical behavior of the material gets affected to varying degrees by the presence of these defects. The point defect chemistry and transport at the interface and its controlling parameters are of considerable significance in micro and nano structural aspects [5]. Even in local equilibrium and operational conditions, point defects are present and play an important role in electrochemical devices such as batteries, fuel cells, electrochemical sensors and electro chromic windows. The nature of point defects and its concentration severely modify the electrical properties of the interface which is structurally different from the bulk due redistribution of interacting particles. In the space charge regions, the defect concentration decays from the electrode’s surface to the bulk through the diffuse double layer and the defect profiles are determined for electro-ceramics hetero-structures [5]. The defect profile and its impact on interfacial properties are different for solid-liquid, solid-gas and solid-solid interface. The number, n of Frenkel defects has been derived [6] using Stirling’s approximations and is given by

n = ( NN i ) exp(− Ei 2 KT ) , 12

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(1)

216

Aditya M. Vora

where Ei is the energy required to displace an atom from a regular lattice site to an interstitial position, Ni is the total number of interstitial atoms and N is the number of atoms. The above equation shows that n is proportional to [NN i ] . 12

In equilibrium the interfacial point defects, structural vacancies and charge transfer clusters are stabilized at the interface by the chemical interaction between two interacting phases and by the crystal constraints at the interface. The concentration of the interface defects strongly depends on oxygen activity, temperature, composition and interface crystallography [7]. The coupling of electric field (surface charge) with the local charged species creates charged defects in the neighborhood due to redistribution of ionic charges in the space charged layer. The nature or character of space charged layer and the metallic charge transfer clusters at the interface depend on the interfacial defects clusters present at the interface. The lattice plane matching between two solid phases is an important parameter to determine the energy of the interface [7].

3. SEGREGATION OF INTERFACIAL CHARGED SPECIES

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The interaction zone between two dielectric phases in contact, say a metal electrode with a liquid electrolyte (Fig. 2) has a significant electromechanical property certainly different from the properties of the individual phases on either side in bulk. The highly structured plane cutting through the center of first counter ions balancing ions (to balance surface charge of metal) is called inner Helmholtz plane (IHP). At a distance of closest approach from IHP

Figure 2. (a) Electrode-electrolyte interface indicating the location of inner Helmholtz plane (IHP), Outer Helmholtz plane (OHP) and diffuse double layer with the dipole orientations varying from strong orientation on the surface to the poor orientation towards the bulk. (b) The schematic plot between the electric fields and the distance shows the oscillatory nature of the field with decreasing amplitude. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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(towards liquid’s phase), the ions having opposite charge to that of IHP form the outer Helmholtz plane (OHP). In continuity with the OHP there is diffuse layer in which the concentration of the opposite charged ions (opposite to surface charge) decreasing with distance from surface and having higher mobility of ions and the ions in the diffuse layer are always being exchanged at the surface. Due to finite size of the ions, the first ions of the Gouy-Chapman Diffuse layer cannot approach the metallic surface closer than a few nm but positioned at a distance (which is equal to the radius of counter ion) from the surface. Some of the mobile ions of diffuse layer are specifically adsorbed by the metallic surface forming stern layer as indicated in Fig 2. The potential will drop by some value over the molecular condenser (i.e. the Helmholtz plane) and by the Zeta potential. The double layer is formed in order to neutralize the charged surface and develops an electrokinetic potential. The potential drops off nonlinearly or roughly linearly from the surface to the stern layer, as the absorbed ions at the surface forming stern layer are not confined to a particular plane and then decays exponentially through the diffuse double layer. Due to electrophoresis, the charged particles will move with a fixed velocity in a voltage field and the particle’s mobility is related to dielectric constant, viscosity of the suspending liquids and the electrical potential at the boundary between particles and liquid. The charged layers of the interaction zone (Fig. 2) between a metal electrode and a liquid dielectric electrolyte determine the dielectric behavior of the interface along with other electromechanical properties and the change in concentration of the counter ions near the surface is governed by the Boltzmann distribution [1] as follows

ni = n0 exp(− z i eψ / KT ),

(2)

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where n0 is the concentration of ions in bulk electrolyte, z i is the ion valancy, K is Boltzmann constant, ψ(r) is the potential as function of distance r from OHP. The volume charge density of ions in the diffuse double layer can be given as

ρ i = ∑ z i eni

(3)

.

Combining Boltzmann distribution in equation (3) with the Poisson’s equation,

∇ 2ψ = − ρ

ε , where ε is the effective permittivity of the layer, we get

∇ 2ψ (r ) = −

e

∑z n e ε i

− zi eψ ( r )

0

KT

.

(4)

i

Using asymptotic method and other approximations, the solution to equation (4) has been given [8, 9] in the Debye-Huckel form as

ψ (r ) = ψ 0 e − Kr = ψ 0 e

−r

λ

,

where Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

(5)

218

Aditya M. Vora

⎛ 2e 2 K = = ⎜⎜ λ ⎝ εKT 1

1

⎞ 2 2 z n ∑i i 0 ⎟⎟ , ⎠

(6)

where ψ 0 is the potential at the IHP, and K

−1

or λ is the Debye length, which is the extent of

exponential decay of the diffuse double layer. The thickness of the diffuse double layer is 1

λdouble

⎡ ⎤ 2 = ⎢ε r KT , (2e 2 ∑ n0 z i2 )⎥⎦ ⎣

(7)

which indicates that the thickness of the double layer decreases with increasing valency and concentration. In other words if liquid phase (electrolyte) is highly conducting, the thickness decreases and shrinks to OHP. On the other hand, if the liquid phase is poorly conducting then the thickness is extended up to 10nm or more.

4. SIMULATION OF DIFFUSE DOUBLE LAYER WITH THE INTERFACE Keeping in mind the approximations of Helmholtz, Gouy-Chapman & Stern model (GCS theory), simulation of diffuse double layer with the interface is not valid up to a large extent due to following reasons:

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i.

The Poisson-Boltzmann theory, which is a continuum and mean field theory, can’t be applied at the interface having finite size of ions and fluctuations in the charged species due to segregations of charged clusters. ii. Only significant interactions are coulombic, which is certainly not true because there are other types of interactions available at the interface. iii. Electrical permittivity is assumed to be constant throughout the double layer; it is definitely a fatal flaw of the GCS theory while applying it at the interface having anisotropic and fluctuating dielectric behavior. iv. The solvent is considered to be uniform at atomic level, which is also not true due redistribution and reorganization of solvent molecules as a result of surface charge. v. The location of OHP is assumed to be constant irrespective of the ion size. And also the activity is not equal to the molar concentration in the interfacial region, so it is not logical to use Boltzmann distribution in this situation and the equation (2) is not suitable up to large extent and so it is not valid at the interfacial region. However, Boltzmann distribution might be suitable up to some extent in the case of bulk solutions in continuum. That is why Gouy-Champman theory is not completely accurate and the experimental value of the thickness of diffuse double layer is found to be generally greater than the theoretically calculated value. When the macro scale concepts are applied in the nanometric situation particularly in the field of interfacial electrochemistry or any other area like biology and material science, special attention has to be paid on its applicability at nanoscale.

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The ordering and layering (Fig.3) of water molecules at the electrode-electrolyte interface have been experimentally confirmed using x-ray scattering technique [10]. The orientation distribution of water dipoles on the solvation shell depends on the polarity of the electrode as indicated in (Fig. 4). It is also very important to consider the role of the polar solvent in the interaction zone. Due to its strong dipole moment (bulk dielectric constant for water is 78.5), it will definitely be influenced by the charge on the metallic surface (Fig. 3). For negatively charged metallic surface, water molecules will be oriented with the hydrogen atoms towards the metal surface (Fig. 4). In addition, the water would act as a first solvation shell for the metallic electrode and would exist in a fixed orientation (Bockris, Devanathan, and Muller model) and its effect cannot be neglected in the interaction zone.

Figure 3. Schematic diagram indicating ordering and layering of water molecules as a result of surface charge of the electrode.

Figure 4. Schematic diagram indicating the orientation of water molecules on the surface of electrode and depending on its polarity.

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5. FIELD DEPENDENT DIELECTRIC BEHAVIOR AT THE INTERFACE Several researchers [11, 12, 13] have studied the field dependent and distance dependent dielectric constant of water (considering hydration of ions) near the charged surface, and the biological system such as water in membrane spanning pores. The non-linear relation between the applied electric field and the dielectric constant has experimentally been confirmed [14] for the first time. A simple expression [15] for the field dependent dielectric constant (Fig.5) using Onsager and Kirkwood theories has been given as follows:

ε = n2 +

απN 0 (n 2 + 2) μ v ⎛ βμ v (n 2 + 2) E ⎞ E

L⎜⎜ ⎝

kT

⎟, ⎟ ⎠

(8)

μ v is

the dipole moment of the water molecule, E is the field strength, T is the absolute temperature, L(x) is the usual Langevin function and α & β are numerical factors. In the study of the field dependent dielectric properties of water using molecular dynamics computer simulations, the influence of the electrode’s surface on the orientational distributions of the water molecules of polar solvent (which is a major constituent of the electrolyte solution in most of the cases) is up to a distance of 100nm from the surface of the electrode, and an oscillating electric field with decreasing amplitude exists up to a distance of 100 nm from the electrode’s wall [16], which strongly indicates the anisotropic dielectric character of the solution as shown in Fig. 6. The same type of variation [16] has also been observed for density profile and charged clusters. The variation in density must influence the body force and the induced electrical strain and hence the dielectric tensor [17]. Thus the fluctuating dielectric permittivity is solely due to surface effect of electrode in addition to its role for producing electrostriction and Maxwell stress tensor [17] due to its electrical field. 90 80 70

Dielectric Constant (ε)

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where n is the optical refractive index, N0 is the number of molecules per unit volume,

60 50 40 30 20 10 0 0.0

0.1

0

E(V/A )

0.2

0.3

0.4

Figure 5. Schematic plot indicating field dependent nonlinear dielectric constant of water observed from theoretical [15] and experimental [4] studies. Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 6. Variation of electric fields with distance from the electrode’s surface, showing its oscillatory nature with decreasing amplitude. The same type of variation has also been observed for density profile and charged cluster [16].

6. ELECTROMECHANICAL COUPLING: INTERFACE AS PIEZOELECTRIC TRANSDUCER The interfacial electric field also induces mechanical stress apart from its contribution towards polarization and conduction process, as governed by Helmoholtz equation [17] as follows:

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F = ρE −

1 2 E ∇ε + f , 2

(9)

where the first term is a force generated by the charges in the dielectric, the second term is due to inhomogeniety in the permittivity ε and the third term f is a body force associated with electrostrictive strain. The various electrostatic properties and parameters, which are mainly influenced by the molecular structure, are determined by several researchers [17,18,19] and correlate deformation with polarizability of the dielectrics. The lateral displacement and the transverse coefficients become important for the exploration of interface as piezoelectric transducer [20]. The electromechanical coupling equations [17,18,19] represented by Sacerdote’s formulas for the electrically induced strains are

⎡ a + (1− 2σ )a2 ⎤ −1 1 2 S33 = − εE 2 ⎢(1+ 2σ ) − 1 ⎥Y = −q33E , εr 2 ⎣ ⎦

(10)

1 2 ⎡ σa1 + (1 − 2σ )a 2 ⎤ −1 2 εE ⎢1 − ⎥Y = q31 E , 2 ε r ⎣ ⎦

(11)

S11 = S 22 =

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where Y is Young modulus, σ is Poisson’s ratio, a1 & a 2 are electrostrictive strain parameters, q33 & q31 are electrostrictive coefficients, and S11 & S22 are tensile strains along X1 and X2 plane where as S33 is the compressive strain (Fig.7) along X3. If the field E in the DDL is changed to E 0 + ΔE , then the strain S33 becomes − q33 ( E 0 + ΔE ) , and the change in 2

strain S33 for ΔE 0 ⎠

(A14),

ν μ

⎞ ⎟ 0

∞

⎛ c2 ⎞ 2 − − − d λ exp b λ c λ ⎜ ⎟ j1 ( aλ ) = ∫ π −∞ 4b ⎠ ⎝ b

(B2)

⎛ c ⎞ 2b ⎛ a ⎞ ⎛ ac ⎞ π bc ⎡ ⎛ − a + ic ⎞ ⎛ a + ic ⎞ ⎤ − erf ⎜ ⎢ erf ⎟ ⎥ exp ⎜ − 4b ⎟ + a 2 exp ⎜ − 4b ⎟ sin ⎜ 2b ⎟ 2a 2 ⎣ ⎜⎝ 2 b ⎟⎠ ⎝ 2 b ⎠⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2

2

Re b > 0

Moreover the following recurrence relation is valid ∞

⎛ c2 ⎞ 2 − − − d λ exp b λ c λ ⎜ ⎟ jl +1 (aλ ) = ∫ π −∞ 4b ⎠ ⎝ c2 ⎤ ⎡ 4b ∂ ⎢ I l (a, b, c)e ⎥ 2 l 2l + 1 c 2l + 1 2b −4cb ⎣⎢ ⎦⎥ I l −1 (a, b, c) − I l (a, b, c) + e l +1 l +1 a l +1 a ∂c

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I l +1 (a, b, c) =

b

(B3)

Re b > 0

REFERENCES [1] M. Gavrila, J. Phys. B: At. Mol. Opt. Phys. 35, R147 (2002). [2] D. Daems, S. Guerin, H. R. Jauslin, A. Keller, O. Atabek, Phys. Rev. A 69, 033411 (2004). [3] M. Bachmann, H. Kleinert, A. Pelster, Phys. Rev. A 62, 052509 (2000). [4] J. S. Cohen, Phys. Rev. A 68, 033409 (2003). [5] L. A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986). [6] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985). [7] N. B. Delone, V. P. Krainov, Multiphoton Processes in Atoms: Springer, Heidelberg, (1994). [8] M. H. Mittleman, Introduction to the Theory of Laser-Atom Interactions: Plenum, New York, (1993). [9] M. Gavrila, Atoms in Intense Laser Fields; Academic Press: New York, (1992). [10] R. M.Potvliege, R. Shakeshaft, Atoms in Intense Laser Fields; Academic Press: New York, (1992).

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272

E. G. Thrapsaniotis

[11] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics; Cambridge University Press: New York, (1995). [12] D. Bauer, Phys. Rev. A 66 053411 (2002). [13] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, P. Agostini, Science 292 1689 (2001). [14] J. Wu, H. Zeng, Phys. Rev. A 68 15802 (2003). [15] W. Wasilewski, A. I. Lvovsky, K. Banaszek, C. Radzewicz, Phys. Rev. A 73 063819 (2006). [16] E. M. Daly, A. S.Bell, E. Riis, A. I. Ferguson, Phys. Rev. A 57 3127 (1998). [17] R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, J. Potasek, Phys. Rev. Lett. 59 2566 (1987). [18] G. Duchateau, E. Cormier, R. Gayet, Phys. Rev. A 66 023412 (2002). [19] E. Cormier, P. Lambropoulos, J. Phys. B: At. Mol. Opt. Phys. 30 77 (1997). [20] B. P. J. Bret, T. L. Sonnemans, T. W. Hijmans, Phys. Rev. A 68 023807 (2003). [21] R. Jason Jones, I. Thomann, J. Ye, Phys. Rev. A 69 051803(R) (2004). [22] J. M. Cervero, J. D. Lejarreta, J. Phys. A: Math. Gen. 29 7545 (1996). [23] C. Brif, A. Vourdas, A. J. Mann, Phys. A: Math. Gen. 29 5873 (1996). [24] M. Hillery, M. S. Zubairy, Phys. Rev. 26 451 (1982). [25] M. Matsumoto, J. Math. Phys. 37 3739 (1996). [26] E. G. Thrapsaniotis, J. Phys. A: Math. Gen. 30, 7967 (1997). [27] E. G. Thrapsaniotis, Eur. Phys. J. D. 15, 19 (2001). [28] E. G. Thrapsaniotis, Eur. Phys. J. D. 14, 43 (2001). [29] E. G. Thrapsaniotis, ArXiv:quant-ph/0401043 (2004). [30] E. G. Thrapsaniotis, Phys. Rev. A 70 033410 (2004). [31] E. G. Thrapsaniotis, J. Mod. Optics 53 1501 (2006). [32] E. G. Thrapsaniotis, Europhys. Lett. 63 479 (2003). [33] E. G. Thrapsaniotis, Phys. Lett. A 365 191 (2007). [34] E. G. Thrapsaniotis, Far East Journal of Dynamical Systems 10 23 (2008). [35] E. G. Thrapsaniotis, J. Phys. A: Math. Theor. 41 205202 (2008). [36] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets; World Scientific: Singapore (2004). [37] L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Non Relativistic Theory), Pergamon Press: Oxford (1977).

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ISBN: 978-1-61122-960-8 ©2011 Nova Science Publishers, Inc.

Chapter 20

A SCHEME FOR REMOTE PREPARATION OF MULTIPARTICLE ENTANGLED STATES Yu-Wu Wang1∗ and You-Bang Zhan2 1

College of Computer, Huaiyin Normal University, Jiangsu, P. R. China College of Physics, Huaiyin Normal University, Jiangsu, P. R. China

2

ABSTRACT

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In this paper a scheme for remote preparation of multiparticle entangled states is proposed. In this scheme, only two classical bits and one two-particle projective measurements are sufficient.

1. INTRODUCTION Quantum entanglement is one of the most striking features of quantum mechanics. Also multiparticle entanglement is a very important physical resource in quantum information protocols such as quantum teleportation [1], quantum telecloning [2], superdence coding[3], quantum cryptography [4] and so on. Quantum teleportation, proposed by Bennett et al. [5], is the process that transmit an unknown two-state particle, or a qubit from a sender (Alice) to a receiver (Bob) via a quantum channel with the help of some classical information. Some schemes of quantum teleportation of multiparticle entangled states via different channels have been presented [1,6-9]. Another important application of the entanglement, which correlates closely to teleportation, is remote state preparation (RSP) [10-12]. RSP is called “teleportation of a know state”. In RSP, Alice performs a measurement on her share of the entangled resource in a basis chosen in accordance with the state she wishes to help Bob in his laboratory to prepare. In the work of Lo [10], Pati [11], and Bennett et al. [12], they have shown that, for some special ensemble of qubits, the classical communication cost in RSP is less than that in ∗

E-mail address: [email protected]

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standard teleportation, but for general states, RSP requires as much classical communication cost as quantum teleportation. Lo [10] and Bennett et al. [12] have also studied the trade off between entanglement cost and classical communication cost in RSP. Recently, there have been many theoretical and experimental protocols for generalization of RSR [13-27]. For example, Shi et al. [16] have presented a scheme for RSP of N-particle pure state via (N+1)particle Greenberger-Horne-Zeilinger(GHZ) state [28] with one classical bit. More recently, Liu et al. [29] have proposed a scheme for preparing remotely an N-particle entangled state by N pairs of bipartite EPR states. In their scheme, one requires an N-particle orthonormal basis measurement and N bits of classical information. The purpose of the present paper is to give a scheme for remote preparation of multiparticle entangled states. We first discuss remote preparation of a 2N-particle entangled state via two (N+1)-particle GHZ states as the quantum channels. Then we investigate remote preparation of a (2N+1)-particle entangled state by using a (N+1)-particle GHZ state and a (N+2)-particle GHZ state as the quantum channels. We will show that, in the two cases, only two classical bits and one two-particle projective measurement are enough.

2. REMOTE PREPARATION OF MULTIPARTICLE ENTANGLED STATES Suppose that the sender Alice wishes to help the receiver Bob remotely prepare the 2Nparticle entangled state

| φ〉 = α | 0

0〉 + β | 1 1〉 , 2N

(1)

2N

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in which the parameter α is real and β is complex, and which is known completely to Alice but unknown to Bob. We also suppose that two (N+1)-particles GHZ states | φ 〉1 and | φ 〉 2 shared by Alice and Bob are in the following states, respectively

| φ 〉1 =

1 (| 0 2

0〉+ | 1 1〉 )1' 2 '

| φ〉2 =

1 (| 0 2

0〉+ | 1 1〉 )1" 2"

( N +1) '

,

( N +1) "

(2)

,

(3)

'

"

'

where Alice holds the particles 1 and 1 , Bob holds the particles 2 ,

2" ,

, N ' , ( N + 1) ' ，

, N " and ( N + 1)" . To achieve the RSP scheme, Alice needs to make a projective '

"

measurement on her two particles 1 and 1 . The measurement basis chosen by Alice is a set

{| ϕ 〉,| ϕ⊥ 〉,|ψ 〉,|ψ ⊥ 〉} computation basis vectors {| 00〉, | 01〉, | 10〉, | 11〉} by of mutually orthogonal basis vectors,

which are related to

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

A Scheme for Remote Preparation of Multiparticle Entangled States

275

| ϕ 〉 ij = α | 00〉 ij + β | 11〉 ij , | ϕ ⊥ 〉 ij = β * | 00〉 ij − α | 11〉 ij ,

| ψ 〉 ij = α | 01〉 ij + β | 10〉 ij , | ψ ⊥ 〉 ij = β * | 01〉 ij − α | 10〉 ij .

(4)

Then we have

| Φ〉 =| φ 〉1 ⊗ | φ 〉 2 1 = [| ϕ 〉1'1" (α | 0 0〉 2 '3' ( N +1) ' 2"3" ( N +1) " + β * | 1 1〉 2 '3' ( N +1) ' 2"3" ( N +1) " ) 2 2N 2N + | ϕ⊥ 〉1'1" ( β | 0 0〉 2 '3' ( N +1) ' 2"3" ( N +1)" − α | 1 1〉 2 '3' ( N +1) ' 2"3" ( N +1) " ) 2N

2N

+ | ψ 〉1'1" (α | 0

N

+ | ψ ⊥ 〉1'1" ( β | 0

01 1〉 2 '3' N

N

01 1〉 2 '3' N

( N +1) ' 2 "3"

( N +1) ' 2 "3"

+ β * | 1 10

( N +1) "

N

( N +1) "

N

− α | 1 10 N

N

0〉 2 '3'

( N +1) ' 2 "3"

( N +1) "

0〉 2 '3'

( N +1) ' 2 "3"

( N +1) "

) )].

(5)

' " Now Alice measures particles 1 and 1 . If the result of measurement is | ϕ ⊥ 〉1'1" , at '

Bob’s side the entangled state of particles 2 ,

, N ' , ( N + 1) ' ， 2" ,

, N " and ( N + 1)"

will collapse into the state

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β |0 2N

0〉 2'3'

( N +1) ' 2"3"

( N +1)"

− α | 1 1〉 2'3' 2N

( N +1) ' 2"3"

( N +1)"

with occurrence probability being 1/4. Then Alice informs Bob of her measurement result via a classical channel with just two classical bits. According to measurement result by Alice, Bob operates a unitary transformation

⊗ (| 0〉〈1 | + | 1〉〈 0 |) ( N +1) '

(| 0〉〈1 | + | 1〉〈 0 |) 2 ' ⊗

⊗ (| 1〉〈 0 | − | 0〉〈1 |) 2" ⊗ (| 1〉〈 0 | + | 0〉〈1 |)3"' ⊗

⊗ (| 1〉〈 0 | + | 0〉〈1 |) ( N +1)"

on the state

β |0 2N

0〉 2'3'

( N +1) ' 2"3"

into the original state | φ 〉 = α | 0 2N

( N +1)"

− α | 1 1〉 2'3' 2N

( N +1) ' 2"3"

( N +1)"

0〉 + β | 1 1〉 . Likewise, if Alice gets | ψ ⊥ 〉1'1" , at Bob’s 2N

'

side the entangled state of particles 2 ,

, N ' , ( N + 1)' ， 2" ,

, N " and ( N + 1)" will

collapse into the state Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

276

Yu-Wu Wang and You-Bang Zhan

β |0 N

01 1〉 2'3' N

( N +1) ' 2"3"

( N +1)"

− α | 1 10 N

N

0〉 2'3'

( N +1) ' 2"3"

( N +1)"

,

which can be transform into | φ 〉 by performing an appropriate rotation operation

(| 1〉〈 0 | − | 0〉〈1 |) 2 ' ⊗ (| 1〉〈 0 | + | 0〉〈1 |)3' ⊗

⊗ (| 1〉〈 0 | + | 0〉〈1 |)( N +1) ' ⊗ I 2" ⊗

⊗ I ( N +1)" .

So, Bob can construct the entangle state | φ 〉 in half time with only two classical bits broadcast from Alice. From Eq. (5), we find that if Alice’s outcome is | ϕ 〉1'1" in the twoparticle projective measurement, Bob will get the state

α |0 2N

0〉 2'3'

( N +1) ' 2"3"

( N +1)"

+ β * | 1 1〉 2'3' 2N

' if Alice finds | ψ 〉1'1" , then the state of particles 2 ,

( N +1) ' 2"3"

( N +1)"

.

, N ' , ( N + 1)' ， 2" ,

, N " and

( N + 1)" will be

α | 0 01 1〉 2 3

' '

N

N

( N +1) ' 2 " 3"

( N +1) "

+ β * | 1 10 N

N

0〉 2 '3'

( N +1) ' 2 "3"

( N +1) "

.

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Because Bob does not know α and β at all, so he cannot transform the state

α |0 2N

0〉 2'3'

( N +1) ' 2"3"

( N +1)"

+ β * | 1 1〉 2'3' 2N

( N +1) ' 2"3"

( N +1)"

and

α | 0 01 1〉 2 3

' '

N

N

( N +1) ' 2 "3"

( N +1) "

+ β * | 1 10 N

N

0〉 2 '3'

( N +1) ' 2 " 3"

( N +1) "

to the state | φ 〉 . However, if parameters α and β are real or the state

| φ〉 =

1 (| 0 0〉 + eiϕ | 1 1〉 ) , 2 2N 2N

and if Alice’s measurement result is | ϕ 〉1'1" or | ψ 〉1'1" , then Bob can construct theentangled state | φ 〉 by means of a suitable unitary operation with two classical bits from Alice. Similarly, we can present a scheme for preparing remotely (2N+1)-particle entangled state via two classical bits and a two-particle projective measurement. Suppose that the sender Alice wishes to help the receiver Bob remotely prepare a (2N+1)-particle entangled state

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

A Scheme for Remote Preparation of Multiparticle Entangled States

|ψ 〉 = α | 0

0〉 + β | 1 1〉

2 N +1

277 (6)

2 N +1

in which the parameter α is real and β is complex, and which is known completely to Alice but unknown to Bob. We assume that the two entangled states shared by Alice and Bob are an (N+1)-particle GHZ state | ψ 〉1 and an (N+2)-particle GHZ state | ψ 〉 2 , which are given by

| ψ 〉1 =

1 (| 0 2

0〉+ | 1 1〉 )1' 2 '

|ψ 〉 2 =

1 (| 0 2

0〉+ | 1 1〉 )1" 2"

( N +1) '

,

( N +1) " ( N + 2 ) "

(7)

.

(8)

'

"

We still assume where Alice holds the particles 1 and 1 , Bob holds the particles

2' ,

, N ' , ( N + 1) ' ， 2" ,

, N " , ( N + 1)" and ( N + 2)" . Alice can perform the two-particle '

"

projective measurement on particles 1 and 1 in the basis vectors described by Eq.(4). If Alice’s result of measurement is | ϕ ⊥ 〉1'1" or |ψ ⊥ 〉1'1" , Bob can prepare the entangle state | ψ 〉 with only two classical communication bits from Alice, the probability being 1/2. Likewise, if ' " the outcome of Alice’s measurement on the particles 1 and 1 is | ϕ 〉1'1" or | ψ 〉1'1" , since Bob

has no knowledge of α and β at all, he cannot construct the original state | ψ 〉 which Alice

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wishes to prepare remotely. However, if parameter α and β are real or the state is

|ψ 〉 =

1 (| 0 0〉 + eiϕ | 1 1〉 ) , 2 2 N +1 2 N +1

and if Alice’s measurement result is | ϕ 〉1'1" or | ψ 〉1'1" , Bob can construct the entangled state

| ψ 〉 by means of a suitable unitary operation with two classical bits from Alice.

CONCLUSION In summary, we have proposed a scheme for remote preparation of multiparticle entangled states. In the present scheme, a multiparticle entangled state is transmitted from a sender (Alice) to a receiver (Bob). Compared with protocols for RSP of multiparticle in Refs. [16] and [28], our scheme only needs two classical bits, which is far less then in Ref. [28]， and M/2(M=2N)-particle entangled quantum channel in our scheme is easier to be generated than the (M+1)-particle GHZ channel in Ref. [16].

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278

Yu-Wu Wang and You-Bang Zhan

ACKNOWLEDGMENTS This work was supported by Natural Science Foundation of Education Bureau of Jiangsu Province of China (Grant No 05KJD140035)

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REFERENCES [1] J.W.Pan, M.Daniell, S.Gasparoni, G.Weihs, A.Zeilinger, Phys.Rev.Lett. 86, (2001), 4435. [2] M.Murao, D.Jonathan, M.B.Plenio, V.Vedral, Phys.Rev. A 59, (1999), 156. [3] X.S.Lin, G.L.Long, D.M.Tong, F.Li, Phys.Rev. A 65, (2002), 022304. [4] J.Kempe, Phys.Rev. A 60, (1999), 910. [5] C.H.Bennett, G.Brassard, C.Crépeau, R.Jozsa, A.Pers, W.K.Woottess, Phys.Rev. Lett. 70, (1993), 1895. [6] J.Lee, H.Min, S.D.Oh, Phys.Rev. A 66, (2002), 052318. [7] C.P.Yang, G.C.Guo, Chin.Phys.Lett. 17, (2000), 162. [8] M.Gao, S.Q.Zhu, J.X.Fang, Commun.Theor. Phys. 41, (2004), 689. [9] Y.B.Zhan, Chin. Phys. 13, (2004), 1801. [10] H.K.Lo, Phys.Rev. A 62, (2000), 012313. [11] A.K.Pati, Phys.Rev. A 63, (2001), 014302. [12] C.H.Bennett, D.P.DiVincenzo, P.W.Shor, J.A.Smolin, B.M.Terhal, and W.K.Wootters, Phys.Rev.Lett. 87, (2001), 077902. [13] I.Devetak and T.Berger, Phys.Rev.Lett. 87, (2001), 197901. [14] B.Zeng and P.Zhang, Phys.Rev. A 65, (2002), 022316. [15] Y.Z.Zheng, Y.J.Gu and G.C.Guo, Chin.Phys.Lett. 19, (2002), 14. [16] B.S.Shi and A.Tomita, J.Opt. B: Quantum Semiclass.Opt. 4, (2002), 380. [17] A.Abeyesinghe and P.Hayden, Phys.Rev. A 68, (2003), 062319. [18] D.W.Berry and B.C.Sanders, Phys.Rev.Lett. 90, (2003), 057901. [19] Y.F.Yu, J.Feng and M.S.Zhan, Phys. Lett.A 310, (2003), 329. [20] D.W.Leung and P.W.Shor, Phys.Rev.Lett. 90, (2003), 127905. [21] A.Hayashi, T.Hashimoto and M.Horibe, Phys.Rev. A 67, (2003), 052302. [22] J.M.Liu and Y.Z.wang, Phys. Lett.A 316, (2003), 159. [23] M.G.A.Paris, M.Cola and R.Bonifacio, J.Opt. B: Quantum Semiclass.Opt. 5, (2003), S360. [24] S.A.Babichev, B.Brezger and A.I.Lvovsky, Phys.Rev.Lett. 92, (2004), 047903. [25] Y.B.Zhan, Commun.Theor. Phys. 43, (2005), 637. [26] X.Peng, X.Zhu, X.Fang, M.Liu and K.Gao, Phys. Lett.A 306, (2003), 271. [27] Y.X.Huang and M.S.Zhan, Phys. Lett.A 327, (2004), 404. [28] D.M.Greenberger, M.A.Horne, A.Shimony and A.Zeilinger, Am.J. Phys. 58, (1990), 1131. [29] J.M.Liu, J.R.Han and Y.Z.Wang, Commun.Theor. Phys. 42, (2004), 211.

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INDEX # 3  3d-transition metal binary alloys, vii

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A  absorption spectra, 110, 120, 124, 126 absorption with inversion (ADI), viii, 37 acoustics, x, 161 activation energy, 215 adsorption, 104 algorithm, 5, 166 amorphous phases, 150 amplification without inversion (AWI), vii, 15, 38, 39, 56 amplitude, 6, 16, 38, 74, 78, 82, 83, 168, 184, 216, 220, 221, 263, 267 annihilation, 60 assessment, 209 atmosphere, 104 atoms, viii, 1, 16, 18, 39, 40, 60, 61, 138, 151, 198, 215, 216, 249, 267 automobiles, 167

B  Bahrain, 227 bandwidth, 16, 17 Bangladesh, 191 bare state basis, viii, 37, 38, 39, 54 baryonic matter, 236 base, viii, 85, 97, 99, 115 batteries, 215 binding energy, 2, 3 Boltzmann constant, 23, 43, 200, 217 Boltzmann distribution, 217, 218 Bulgaria, 85

C  calculus, 179, 187 candidates, 235 capillary, 74 cation, 16 causality, 136, 187 charge density, 2, 217 chemical, ix, 103, 104, 215, 216, 224 chemical interaction, 216 chemical properties, 224 chemical reactions, ix, 103, 104 China, 273, 278 classification, 227 closure, 21, 26, 46, 61 clusters, xi, 213, 216, 218, 220, 241 CO2, 107 coding, 273 coherence, 16, 17, 18, 23, 32, 33, 38, 43, 44, 51, 53 coherent field Rabi frequency, vii, 15, 27, 38, 39, 54 common sense, 12 communication, 131, 273, 277 comparative analysis, 8 complications, 250 composites, 109 composition, 98, 100, 138, 150, 182, 215, 216 compounds, 194 compressibility, 140, 146, 153, 158, 199 computation, 145, 146, 147, 150, 152, 158, 159, 166, 168, 174, 196, 197, 199, 200, 274 computer, 131, 133, 220 computer simulations, 220 computing, 152, 193, 246 conception, 111, 122 concordance, 235 conduction, 145, 156, 193, 194, 198, 221 conductivity, 75 conductor, 150

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280

Index

configuration, viii, 37, 38, 53, 125, 126, 267 conjugation, xi, 94, 227, 228, 229 conservation, 105 construction, 167 consumption, 192 contour, 270 convention, 87 correlation, 132, 135, 138, 140, 146, 147, 150, 153, 158, 194, 197, 198, 199, 206, 209 correlation function, 135, 138, 199 correlations, ix, 131, 140, 147, 153, 158, 197, 199 Cosmological Redshift, xi, 235, 236, 244, 246 Coulomb interaction, 1, 6, 145, 156 Coulomb pseudopotential, ix, 137, 138, 139, 142, 145, 149, 150, 152, 156, 159 coupling constants, 20, 41 covering, 16, 39, 175 CPU, 167, 168, 173, 175 crystal growth, 73 crystal structure, 214 crystalline, 150 crystallites, 110 crystals, 116, 150

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D  damping, x, 34, 116, 179, 180, 183, 185, 187 Dark Energy, xi, 235, 236, 246 data set, 4 decay, vii, 15, 16, 17, 18, 19, 24, 25, 31, 32, 40, 46, 47, 51, 52, 53, 180, 182, 218 decomposition, 175, 228, 261 defects, 215, 216 deficiency, 86 deformation, 221 degenerate, 63, 66, 68, 140, 146, 153, 158, 199, 251, 252, 267 density matrix, vii, viii, 15, 17, 19, 20, 21, 27, 37, 39, 41, 42, 44, 45, 54 density matrix equations, vii, viii, 17, 20, 21, 27, 37, 42, 54 density-density correlation function, 199 depth, 11, 167 derivatives, 11, 96, 270 detection, 193 deuteron, vii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 deviation, x, 179, 206, 242 dielectric constant, 217, 219, 220, 223 dielectric permittivity, 220 dielectrics, vii, 221, 224 differential equations, viii, 73, 74, 104, 162, 228 diffraction, 6, 8, 9, 10, 11 diffusion, 104, 179, 214

Dirac and Klein-Gordon equations, ix Dirac equation, 85, 86, 87, 89, 90, 91, 94, 95, 99, 101 discretization, 162, 163 disorder, 157 dispersion, ix, 109, 110, 111, 115, 116, 117, 122, 124, 125, 127, 172, 173, 239 displacement, 221 distortions, 223 distributed load, 162 distribution, 2, 12, 112, 193, 214, 218, 219, 241, 265, 266 DOI, 178, 247 doppler, 18 dressed-state basis, viii

E  effective interaction strength, ix, 137, 138, 141, 142, 146, 149, 151, 154, 158, 159 elastic deuteron-nucleus scattering, vii electric charge, 87, 90 electric current, 194 electric field, 104, 116, 192, 214, 216, 220, 221, 222, 223, 224 electrical properties, 215 electrochemistry, 218, 223 electrodes, 193 electrolyte, 214, 216, 217, 218, 219, 220 electromagnetic, ix, 2, 4, 12, 87, 90, 94, 109, 111, 112, 113, 115, 120, 124, 126, 251, 257 electromagnetic fields, ix, 109 electron, ix, x, xi, 137, 138, 139, 140, 142, 143, 145, 146, 147, 149, 150, 151, 153, 155, 157, 158, 159, 191, 192, 193, 194, 196, 197, 199, 249, 250, 251, 254, 256, 257, 262, 265, 266, 267 electron state, 193 electron-phonon coupling, ix, 137, 138, 139, 142, 143, 145, 146, 147, 149, 150, 151, 155, 157, 159 electron-phonon coupling strength, ix, 137, 138, 139, 142, 143, 145, 147, 149, 150, 151, 155, 157, 159 electrons, 2, 138, 143, 145, 150, 156, 192, 193, 194, 198, 265 electrophoresis, 217 elementary particle, 102 emission, viii, 16, 37, 46, 245 energy, vii, viii, ix, x, xi, 1, 8, 9, 17, 18, 25, 26, 33, 40, 47, 53, 59, 62, 63, 64, 65, 66, 68, 70, 71, 72, 105, 109, 111, 112, 113, 115, 120, 126, 128, 140, 152, 157, 179, 183, 185, 193, 195, 196, 197, 199, 200, 204, 206, 209, 215, 216, 223, 235, 237, 238, 239, 249, 262, 263, 265, 266, 267 engineering, x, 161, 162, 163, 167, 179

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Index entropy, 196, 197, 199, 200, 205, 206, 207, 209 environment, 227, 241 EPR, 131, 133, 274 equilibrium, xi, 74, 197, 213, 215, 216, 223, 224 evolution, vii, viii, ix, 15, 25, 27, 29, 30, 31, 32, 33, 46, 48, 49, 50, 51, 52, 53, 54, 55, 73, 74, 85, 86, 87, 91, 92, 93, 94, 95, 96, 97, 102 excitation, ix, 17, 24, 25, 47, 109, 110, 111, 115, 120, 122, 127, 162, 174, 175, 176, 193

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F  federal government, 108 FEM, 161, 166, 169 ferromagnets, 194 fibre bundle formulation, viii, 85 field theory, 97, 101, 188, 218 films, 106 financial support, 108 finite element method, x, 161, 162, 168, 169, 171 flexibility, x, 16, 39, 161, 162, 163 fluctuations, 218, 257 fluid, x, 161, 162, 168, 177, 196 force, x, 10, 122, 179, 180, 183, 185, 187, 188, 220, 221 formation, 104, 108, 113, 224 formula, 5, 6, 7, 8, 12, 61, 62, 93, 140, 141, 146, 153, 154, 158, 169, 199 foundations, 180 Fractional derivative, x, 179, 187 fractional oscillators, vii free energy, 196, 197, 199, 206, 208, 209 freedom, x, 5, 161, 166, 167, 175, 191, 192, 193, 194 fuel cell, 215

G  galaxies, 241, 242, 243 Galileo, 235 General Relativity, 235, 236, 237, 239, 240, 247 geometry, 102, 119, 122, 166, 168, 171, 174, 175, 176 Gibbs–Bogoliubov (GB) variational method, x, 195 glass transition, 157 glass transition temperature, 157 glasses, 148, 149, 150, 151, 152, 154, 155, 156, 157, 158, 159 gravitation, 236 gravitational force, 242 gravity, xi, 157, 188, 235, 236, 242, 246 Greece, 249

281

growth, viii, 9, 10, 73, 74, 75, 77, 78, 79, 80, 81, 82, 107 growth rate, 107 guidelines, 209

H  Hamiltonian, x, 19, 20, 40, 41, 43, 44, 60, 90, 93, 96, 179, 180, 185, 186, 188, 196, 228, 250, 251, 252, 254, 255, 257, 261, 263, 264 harmony, 86 Hartree (HR), x, 195, 197, 198 heat capacity, 75 height, viii, 73, 74, 78, 79, 121, 167, 168, 169, 170 Hilbert space, 93 homogeneity, 241, 242, 243, 244, 245 Hubble Law, xi, 235, 236, 243, 246 Hungary, 73, 103 hybridization, 143, 156, 193, 196 hydrogen, xi, 1, 219, 249, 250, 261, 262, 263, 265, 267 hydrogen atoms, 219

I  Ichimaru-Utsumi (IU), x, 138, 140, 141, 146, 150, 152, 153, 158, 195, 197, 198, 199 identity, ix, 85, 86, 87, 88, 97, 98, 99, 182, 228, 229, 259 India, 15, 35, 37, 56, 137, 149, 195, 213 induction, ix, 103, 104, 106, 107, 108 induction period, ix, 103, 104, 106, 107, 108 inequality, ix, 76, 78, 131, 132, 136, 196 information processing, 192 ingredients, 235 initial state, 263 insertion, 167 integration, 4, 223, 236, 250, 254, 259, 264, 265 interface, xi, 77, 111, 113, 213, 214, 215, 216, 217, 218, 219, 221, 222, 223, 224 interference, viii, 16, 30, 37, 38, 39, 46, 49 internal field, 110 invariants, 238 inversion, vii, viii, 12, 15, 16, 17, 28, 33, 37, 38, 39, 48, 49, 51, 56 ionization, xi, 249, 250, 261, 262, 267 ions, 138, 150, 193, 214, 215, 216, 217, 218, 220, 224 Iran, 179 iron, 225 irradiation, 104, 107

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282

Index

isotope, ix, 137, 138, 141, 142, 146, 149, 150, 154, 157, 159 isotope effect exponent, ix, 137, 138, 141, 142, 146, 149, 151, 154, 157, 159 Italy, 235 iteration, 95

J  Japan, 71, 84, 194

K  kinetics, 104, 107 Klein-Gordon equation, 85, 86, 90, 91, 95, 96, 237 Kuwait, 227, 233

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L  laser radiation, ix, 103, 104, 107 lasers, 16, 17, 18, 26, 27, 28, 29, 32, 33, 34, 38, 39, 40, 48, 49, 51, 53, 249 lattices, 59 laws, 104, 111 layering, 219 lead, 38, 82, 109, 147, 159 Lie algebra, xi, 227, 228, 233 light, vii, x, 15, 16, 23, 38, 39, 43, 87, 110, 111, 116, 117, 119, 120, 122, 133, 191, 194, 228, 239, 240, 243, 245, 249, 250, 251, 252 Lippmann-Schwinger equation, ix, 109, 112, 127 liquid phase, 218 liquids, 140, 146, 153, 158, 199, 217 local electromagnetic fields, ix, 109 Local Hidden Variables model, ix, 131 localization, 114

M  machinery, 91 Mackintosh, 83 magnet, 72 magnetic field, 192, 194 magnetic moment, 192 magnetizations, 192 magnetoresistance, 194 magnitude, 4, 12, 15, 17, 18, 24, 25, 28, 32, 34, 47, 51, 53, 54, 146, 158, 159, 200, 222 manganese, 194 manifolds, 43 manipulation, 192

mapping, 88, 98 mass, 3, 7, 23, 43, 87, 90, 139, 151, 165, 200, 239 master equation, viii, 39, 54, 227 materials, x, 116, 157, 161, 170, 194 mathematical methods, 250 matrix, vii, viii, xi, 7, 15, 17, 19, 20, 21, 27, 34, 37, 38, 39, 41, 42, 44, 45, 54, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 99, 100, 101, 138, 151, 162, 165, 166, 167, 172, 175, 215, 227, 228, 229, 230, 231, 232, 233 matrix algebra, 166 matter, 2, 12, 60, 138, 150, 193, 237, 250 measurement, 131, 133, 134, 133, 134, 162, 171, 196, 273, 274, 275, 276, 277 mechanical stress, 221, 223 media, x, 162, 167, 179, 180, 188 melt, viii, 73, 74, 75, 77 melting, 75, 196 melting temperature, 196 memory, 187, 192 metal ion, 193, 194 metal oxides, 193 metals, vii, x, 138, 145, 146, 147, 148, 150, 151, 195, 196, 197, 200, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211 miniaturization, 109 Minkowski spacetime, 97 mission, 162, 170 Missouri, 103 mixing, 16, 38, 104 modelling, 162, 163 models, vii, x, 2, 16, 38, 71, 132, 161, 162, 168, 180, 223, 236 modifications, xi modulus, 222, 223 molecular dynamics, 196, 220, 223, 224 molecular structure, 221 molecules, viii, xi, 16, 17, 18, 23, 24, 25, 31, 34, 39, 43, 47, 51, 213, 214, 219, 220, 267 momentum, 4, 7, 24, 47, 59, 60, 61, 62, 68, 71, 105, 192, 239, 264 Moscow, 101, 102 multiparticle entangled states, xi, 273, 274, 277 multiplication, 44, 89, 93, 95, 96, 97, 100

N  nanometric dielectrics, vii, 224 nanometric range, 214 nanoparticles, 214 Netherlands, 84, 131 New Zealand, 225 Newtonian theory, 237, 240

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Index noble metals, 196 nodes, x, 61, 161, 168, 169, 172, 173 non-Centro symmetry, xi, 213 nonrelativistic deuteron wave function, vii, 1 novel materials, 138, 150 nuclear surface, 6 nuclei, 2, 8, 9, 11 nucleons, 1, 2, 4, 6, 10, 11, 12 nucleus, vii, 1, 2, 6, 7, 8, 9, 10 null, 244 numerical analysis, 166

O  one dimension, 59, 214, 261 operations, 166, 167, 175 optical properties, 110, 122 optoelectronics, 110 orbit, x, 191, 193, 194 oscillation, ix, 46, 82, 83, 109, 110, 114, 179, 180, 182, 183, 184 oxidation, 106, 107 oxygen, 193, 216

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P  PAA, ix, 137, 138, 142, 145, 149, 151, 154 pairing, 133 parallel, 117, 120, 122, 167, 192 partial differential equations, 104, 233 partition, 259 path integral methods, xi, 249, 250 percentile, 145, 156, 206, 209 Percus-Yevick (PY) hard sphere model, x, 195 permittivity, 110, 111, 117, 217, 218, 221, 224 phase transitions, 72 phonons, 143, 156 photonics, 110 photons, 40, 245, 267 physical phenomena, 214 physical properties, xi, 227, 228, 229, 232 physics, vii, x, 1, 2, 73, 88, 131, 138, 150, 179, 240, 241, 249 Planck constant, 87 platform, 209 point defects, xi, 213, 214, 215, 216, 223, 224 polar, 219, 220 polarity, 214, 215, 219 polarizability, 221 polarization, 120, 193, 221, 223, 257 polynomial functions, 162

283

population, 16, 17, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 38, 39, 40, 43, 46, 47, 48, 49, 51 preparation, xi, 273, 277 present value, 146, 158 probability, xi, 10, 25, 47, 112, 115, 193, 249, 250, 262, 265, 266, 267, 275, 277 probe, vii, viii, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 111 probe field Rabi frequency, vii, 15, 17, 26, 27, 34, 53 profit, 167 project, 20, 34, 42, 56, 115 propagation, 162, 177, 243 propagators, 86, 91, 92, 94, 95, 250, 261, 264 protons, 2 pseudo-alloy-atom (PAA) model, ix, 137, 138, 142, 145, 149, 151, 154

Q  quadrupole deuteron, vii, 1, 2 quantization, 251 quantum bits, 192 quantum computing, 193 quantum cryptography, 273 quantum dots, 110 quantum mechanics, viii, 1, 85, 86, 92, 97, 101, 102, 273 quantum objects, 87 quantum optics, 250 quantum theory, 131, 136 quantum well, 109 quarks, 1 qubits, 273

R  Rabi frequency, vii, 15, 17, 18, 19, 26, 27, 34, 38, 39, 40, 43, 53, 54 radiation, ix, 16, 38, 75, 109, 110, 111, 115, 119, 122, 124, 236, 240, 241, 249, 267 radius, 5, 6, 10, 171, 172, 173, 217 reactant, 104 reaction rate, ix, 103, 104, 105, 106, 107, 108 reaction time, 104 reactions, 104, 108 reality, x, 2, 4, 74, 191, 193 recall, 86, 97, 241 reception, 245 recurrence, 271 redistribution, 9, 10, 11, 215, 216, 218

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

284

Index

redshift, 110, 236, 246 reference frame, 241, 242 reference system, x, 195, 196, 197 refractive index, 220 relativistic quantum mechanics, viii, 85, 86, 92, 97, 101 relaxation, 24, 47, 179, 180, 188, 192, 250 reliability, 197 reproduction, 197 repulsion, 11, 193 requirements, x, 191, 192, 229 residues, 270 resistance, 170, 192, 194 resolution, 166 response, 23, 112, 113, 115, 116, 119, 135, 162, 193, 194, 224 restrictions, 25, 53, 240 Riemann tensor, 242 Romania, 73, 84 room temperature, 32, 34, 53, 106 routines, 166 rules, 147

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S  scalar field, xi, 235, 236, 237, 238, 239, 240, 246 scaling, 166 scatter, 10 scattering, vii, xi, 1, 2, 6, 7, 8, 9, 10, 11, 92, 123, 219, 249, 255, 261, 267, 268, 270 science, 179, 218 seed, 75, 104, 108 seeding, 104 segregation, xi, 213, 215, 224 self-consistency, 199 semiconductor, x, 116, 191, 192, 193 semiconductor spintronic devices, x, 191, 192 semiconductors, x, 191, 192, 193 sensors, 215 shape, 74, 110, 114, 116, 119, 126, 162, 164, 193, 194, 215 showing, xi, 146, 147, 158, 159, 209, 221, 235, 236, 246 signals, 241 signs, 135 silicon, viii, 73, 75, 78, 79, 82 simulation, xi, 133, 213, 218, 223, 224 simulations, 214 Singapore, 148, 188, 272 solid phase, 215, 216 solid state, 196, 224 solid surfaces, ix, 103, 104

solution, viii, x, 21, 63, 66, 67, 69, 73, 76, 77, 91, 92, 93, 94, 95, 104, 105, 112, 127, 128, 161, 162, 163, 164, 179, 181, 182, 184, 187, 197, 214, 217, 220, 233, 239, 244, 250, 267 solvation, 219 solvent molecules, 218 South Asia, 148, 160 spacetime, 87, 88, 93, 96, 97, 240 space-time, viii species, 215, 216, 218, 224 specific surface, 173 spectral finite element method, x, 161, 162, 171 speed of light, 244 spin, viii, x, 2, 24, 47, 59, 60, 61, 71, 72, 87, 91, 133, 134, 135, 191, 192, 193, 194 spintronic devices, x, 191, 192, 193 Spintronics, vi, x, 191, 192, 194 stability, 215, 225 stabilization, 249 stars, 241 steady-state limit, viii stoichiometry, 215 storage, 168, 192 stress, x, 191, 194, 220, 222, 223, 224, 225, 237 structure, 2, 3, 4, 5, 8, 9, 10, 11, 12, 16, 116, 157, 162, 177, 180, 186, 193, 194, 196, 198, 200, 206, 210, 215, 223, 241 substrate, 110, 113 superconducting state parameters (SSPs), ix, 137, 138, 149, 150 superconductivity, 138, 146, 150, 157, 159 superconductor, 147, 159 surface area, 214 surface layer, 111 surface properties, 196 susceptibility, 112, 113, 114, 115, 116, 117, 123, 124, 126, 127 Sweden, 161 symmetry, xi, 116, 117, 118, 122, 213 synchronization, 132 synchronize, 132, 133, 135

T  target, 6, 7, 8, 9, 10 Taylor (TY), x, 195, 197, 198 techniques, 161, 162 technology, x, 103, 110, 191, 192, 193 temperature, viii, ix, 23, 43, 73, 75, 77, 78, 82, 83, 104, 106, 107, 137, 138, 139, 145, 149, 150, 151, 157, 168, 196, 197, 200, 206, 216, 220, 227 tension, 75 testing, 163, 167

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Index thermodynamic properties, x, 195, 196, 197, 200, 209, 210, 211 thermodynamics, vii, 195, 197, 210 thermodynamics of liquid metals, vii torsion, 102 total energy, 183, 184, 186, 187 total internal reflection, 115 trade, 170, 274 transducer, xi, 213, 215, 221, 223 transformation, 44, 60, 61, 62, 115, 275 transformation matrix, 44 transformations, 233, 238, 239, 240, 242, 258 transistor, 193 transition metal, vii, ix, 137, 138, 139, 142, 143, 144, 145, 146, 147, 148, 149, 150, 156, 157, 158, 192, 196, 224 transition temperature, ix, 137, 138, 141, 142, 145, 149, 150, 152, 154, 156, 159 translation, 101, 102, 166 transmission, 115, 157, 162, 168, 169, 171, 173, 176, 177 transparency, 16, 38 transport, ix, 85, 86, 88, 89, 94, 97, 98, 102, 197, 215 transportation, ix, 85 treatment, 96, 104, 138, 151, 177, 250 trial, 162, 163, 166, 172 turbulence, 162, 167 tyre acoustics, x, 161

vacuum, 61, 74, 82, 87, 122, 235, 237, 257 valence, 139, 147, 157, 197 vanadium, 106, 107 variables, 132, 136, 180, 184, 250, 251, 252, 253, 254 variations, viii, 16, 39, 73, 74, 81, 82, 83, 200 vector, viii, 16, 61, 85, 86, 87, 92, 96, 97, 98, 101, 102, 116, 120, 122, 166, 197, 198 velocity, 23, 43, 87, 127, 166, 185, 217, 239, 243, 244 versatility, 163 vibration, 162 viscosity, 217

W  water, 214, 219, 220, 224 wave function, vii, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 175, 193 wave number, 139, 151, 169 wave propagation, 162, 167 wave vector, 7, 111, 116, 120, 124, 139, 151, 197, 198, 262 wavelengths, 26, 48, 110 wealth, 2 wires, 110, 157 Wisconsin, 103

X 

U  Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

285

x-rays, x, 191, 194

UK, 161, 224 Ukraine, 1, 13, 109 Universes, 236 USA, 103 Uzbekistan, 59

Y  yield, 152, 166

V  vacancies, 216

Perspectives in Theoretical Physics, edited by Thomas F. George, et al., Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,