Random Fields of Piezoelectricity and Piezomagnetism: Correlation Structures [1st ed.] 9783030600631, 9783030600648

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SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY  MATHEMATICAL METHODS

Anatoliy Malyarenko Martin Ostoja-Starzewski Amirhossein Amiri-Hezaveh

Random Fields of Piezoelectricity and Piezomagnetism Correlation Structures

SpringerBriefs in Applied Sciences and Technology Mathematical Methods

Series Editors Anna Marciniak-Czochra, Institute of Applied Mathematics, IWR, University of Heidelberg, Heidelberg, Germany Thomas Reichelt, Emmy-Noether Research Group, Universität Heidelberg, Heidelberg, Germany

More information about this subseries at http://www.springer.com/series/11219

Anatoliy Malyarenko Martin Ostoja-Starzewski Amirhossein Amiri-Hezaveh •

Random Fields of Piezoelectricity and Piezomagnetism Correlation Structures

123

•

Anatoliy Malyarenko Division of Mathematics and Physics Mälardalen University Västerås, Sweden Amirhossein Amiri-Hezaveh Department of Mechanical Sciences and Engineering University of Illinois at Urbana-Champaign Urbana, IL, USA

Martin Ostoja-Starzewski Department of Mechanical Science and Engineering Institute for Condensed Matter Theory Beckman Institute University of Illinois at Urbana-Champaign Urbana, IL, USA

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISSN 2365-0826 ISSN 2365-0834 (electronic) SpringerBriefs in Mathematical Methods ISBN 978-3-030-60063-1 ISBN 978-3-030-60064-8 (eBook) https://doi.org/10.1007/978-3-030-60064-8 Mathematics Subject Classification: 00A69, 74Axx, 60G60 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Spatially random materials pose mathematical challenges to theoretical physics and mechanics. These problems are exacerbated in the case of coupled field phenomena such as piezoelectricity and piezomagnetism, with the practical motivation coming from modern technology. The coupling of electrical/magnetic fields with elastic materials gives rise to rank 3 tensor-valued random fields, which have previously evaded description. The main purpose of this book is to provide an answer in terms of second-order wide-sense homogeneous and isotropic tensor-valued random fields. A complete description is given of such fields taking values in the three-dimensional linear space of piezoelectric tensors with at least D2 symmetry. Working from the standpoint of invariance of physical laws with respect to the choice of a coordinate system, spatial domain representations, as well as their spectra, are given in full detail for the orthotropic, tetragonal, and cubic crystal systems. Using group representation theory as the foundation, the derivations are done in terms of the complete description of the one- and two-point correlation tensors of the above class of fields as well as the spectral expansions of the fields in terms of stochastic integrals. Västerås, Sweden/Urbana, USA February 2020

Anatoliy Malyarenko Martin Ostoja-Starzewski Amirhossein Amiri-Hezaveh

v

Acknowledgements

The first named author is grateful to his colleagues at Mälardalen University for creating a friendly working and research environment. The second and third named authors were partially supported by the NSF under grants CMMI-1462749 and IP-1362146 (I/UCRC on Novel High Voltage/Temperature Materials and Structures) and the NIH under grant NIH R01EB029766.

vii

Contents

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1 1 3 9 12 13 15 19 19

...

21

2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Piezoelectricity Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Random Field Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27

1 The 1.1 1.2 1.3

Continuum Theory of Piezoelectricity and Piezomagnetism . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuum Electromagnetic Theory . . . . . . . . . . . . . . . . . . . . The Theory of Linear Piezoelectric Materials . . . . . . . . . . . . . 1.3.1 The Displacement Approach . . . . . . . . . . . . . . . . . . . . 1.3.2 The Stress Approach . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Random Piezoelectric Fields . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Determination of Mesoscale Properties Using a Homogenization Condition . . . . . . . . . . . . . .

3 The 3.1 3.2 3.3

Choice of a Basis in the Space VG . . . . Introduction . . . . . . . . . . . . . . . . . . . . . Example: The Piezoelectricity Class ½Z1  Example: The Piezoelectricity Class ½D2 

4 Correlation Structures . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . 4.2 Homogeneous Random Fields . . . 4.3 Conditions for Isotropy . . . . . . . . 4.4 The Structure of the Orbit Space . 4.5 Invariant Subspaces . . . . . . . . . . 4.6 Convex Compacta . . . . . . . . . . . 4.7 The “Spherical Bessel Functions” 4.8 Correlation Structures . . . . . . . . .

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41 41 41 42 46 50 57 61 67

ix

x

Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

List of Tables

Table 3.1 Table 4.1 Table 4.2

The bases of the space VD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . The stationary subgroups of the strata . . . . . . . . . . . . . . . . . . . . The convex compact sets Cm . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 50 62

xi

Chapter 1

The Continuum Theory of Piezoelectricity and Piezomagnetism

1.1 Introduction The focus of this book is on piezoelectricity and piezomagnetism in spatially random media. The piezoelectricity phenomenon is caused by the linear electromechanical interaction between the mechanical and electrical states in crystalline materials which lack inversion symmetry. In a piezoelectric material, an electrical charge is generated by the application of a mechanical force, while a mechanical deformation is caused by the application of an electrical field. Thus, from a thermodynamics standpoint, it is a reversible process. The situation is entirely analogous in a piezomagnetic material where, instead, the magnetic field plays the role of the electric field. Both piezoelectricity and piezomagnetism are caused by an absence of certain symmetries in a crystal structure. While the theory of deterministic homogeneous piezoelectricity and piezomagnetism is classical, this book develops tools for random field theories. First, we generalise the continuum physics equations of inhomogeneous media and then develop a random field description of piezoelectric and piezomagnetic properties. Since in both cases, the key role is played by the same type of rank 3 tensor, we conduct the mathematical developments in terms of piezoelectricity. The general approach in a continuum physics theory is that all interesting physical quantities are defined over a volume by the integration of density distributional functions, accounting for the effect of the same quantity defined for particles, which leads to the definition of each quantity with respect to spatial and time variables. These continuous physical quantities can then be employed to construct physical laws in terms of some integrals. Such a formulation enables us to fully characterise the macroscale behaviour of materials in the realm of thermodynamics. That is, one can develop a rational continuum thermodynamics in which the first and second laws of thermodynamics for a continuum matter are mathematically represented. As mentioned earlier, at first, these laws have integral forms since their satisfaction needs to be addressed in the whole domain of the problem, resulting in some volume © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Malyarenko et al., Random Fields of Piezoelectricity and Piezomagnetism, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-3-030-60064-8_1

1

2

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

and surface integrals. Next, by a localisation procedure, the aforementioned integral forms are reduced to a set of evolutionary partial differential equations (PDEs) along with some initial and boundary conditions. As the governing relations are not sufficient to fully characterise the physical quantities, more relations between the physical quantities are required. As we shall see below, this can be done through constitutive equations in which some of the dependent field variables are written in terms of the reset, so-called independent variables. Of course, we have various options in defining each variable to be dependent or independent, leading to different variants of governing equations. In addition, depending on whether the constitutive equations are linear or nonlinear—corresponding, respectively, to linear or nonlinear materials— we may arrive at linear or nonlinear PDEs. This is distinguished from another form of nonlinearity, called geometric nonlinearity, which is due to large deformations—a topic not treated in this book. Nevertheless, in practice, there are many situations in which the material behaviour is linear and the deformation gradient is small enough to justify the implementation of the linearised theory. Clearly, the theory of classical continuum thermodynamics discussed above lacks coupling effects between thermal and electrical or electrical and mechanical states. In the early stages of development of rational continuum thermodynamics these effects seemed to be inconsequential. However, this is not the case nowadays as several experimental results refute this point of view. These materials are called multi-functional materials or smart materials and have found applications in devices such as ultrasonic transducers and micro-actuators, thermal-imaging devices, health monitoring devices, biomedical devices, and in biomimetics (bionics) and energy harvesting [19, 22]. As a result, it is necessary to develop a consistent rational mathematical model to analyse the behaviour of such materials. One way to accomplish this is to develop a coupled continuum framework in which the laws of classical continuum mechanics are augmented by a contribution due to the presence of electromagnetic fields. This book includes material that has been developed after the publication of [20]. Therefore it extends it. In the above book, the first- and second-named authors elaborated the general theory of spectral expansions of random fields taking values in a linear space of tensors of a fixed rank and, eventually, with a fixed index symmetry. As an example, they included random fields taking values in some subspaces of the 18-dimensional space of piezoelectricity tensors. After the book was published, on the one hand, the second- and third-named authors developed a novel approach to the continuum theory of piezoelectricity and piezomagnetism which is applicable to random media. On the other hand, the firstnamed author found a simplified approach to spectral expansions of random fields that describe the above media. With this approach, a series of completely new examples of random fields that take values in the 3-dimensional space of the dihedral piezoelectricity class [D2 ], and which have various symmetries, from dihedral up to cubic, have been elaborated.

1.2 Continuum Electromagnetic Theory

3

1.2 Continuum Electromagnetic Theory This section summarises the foundations of continuum electromagnetic theory. We follow [8] both notionally and conceptually, referring the reader to the latter for more details. To begin with, consider a deformable body whose volume, denoted by V , is a simply connected, open and bounded subset of three-dimensional Euclidean space, with boundary ∂ V [12]. Furthermore, let V = V ∪ ∂ V . Two different coordinate systems, namely the material and spatial coordinate systems, are used in this section. To distinguish between them, similarly to [8], bold and lowercase indices respectively stand for the material and spatial coordinates. Also, in the remainder of the chapter, standard index notation for Cartesian coordinates is occasionally used. First, Maxwell’s equations in a continuous matter, in a fixed frame, are [8]: ∇ · B = 0 ,  = qe , ∇·D 1 ∂ B = 0 , c ∂t  1 ∂D 1 ∇ × H − = J , c ∂t c

∇ × E +

where

, H = B − M  = E + P , D

(1.1)

(1.2)

 P,  D,  B,  M,  H , J and qe are the electric field, volume electric polarisain which E, tion, electric displacement field, magnetic induction, volume magnetic polarisation, magnetic field strength, electric current and charge density, respectively. It is worth mentioning that the above governing equations can be obtained through corresponding microscopic scales by means of statistical averaging (see [8] for details). To generalise these equations for moving frames, which is needed for the balance laws of continuum mechanics in general, one must ensure that they are form-invariant under a Galilean transformation [8]: x˜ = x + V t,

t˜ = t .

(1.3)

Here V is the constant velocity of the moving frame with respect to the fixed frame. When the velocity of the motion of the continuum is much less than c, the speed of light in vacuum, the necessary and sufficient conditions can be written as [8]:

4

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

q˜e = qe ,  P = P ,  J = J + V qe ,   =M  − 1 V × P , M c 1  E = E − V × B , c 1  B = B + V × E , c      . H =B−M

(1.4)

Consequently, if one assumes that at each time, say t, the velocity of the matter in x , t) is V = −v , then we have: the moving frame RC ( qe = qe ,  P = P ,  J = J − vqe ,   =M  + 1 v × P , M c 1  E = E + v × B , c 1  B = B − v × E , c      . H =B−M

(1.5)

As a result, Maxwell’s equations in the moving frame read [8]:

where

∇ · B = 0 ,  = qe , ∇·D 1 ∇× E + B ∗ = 0 , c 1   ∗ = 1 J , ∇ × H − D c c

(1.6)

 · v) − ( B · ∇)v , B ∗ = B˙ + B(∇ ˙ + D(∇ ∗ = D  · v) − ( D  · ∇)v , D

(1.7)

are convective derivatives, which are frame indifferent, see [8, Definition (1.12.13)].

1.2 Continuum Electromagnetic Theory

5

To consider the electromagnetic effects in a thermodynamic continuum, the balance laws—i.e., the balance of mass, the balance of linear momentum, the balance of angular momentum, the first law of thermodynamics, and the second law of thermodynamics—need to include a contribution due to electromagnetic fields. The approach to account for such contributions is quite similar to the case where a continuum theory is built by passing from particle mechanics. In other words, the macroscopic contribution due to electromagnetic fields—i.e., electromagnetic force, electromagnetic couple, and electromagnetic power—are derived from microscopic contributions by some statistical averaging (see [8] for details). In this way, after quite long manipulations, the generalised local form of balance laws can be written as [8]: ρ˙ + ρvi,i = 0 , ρ v˙ i = σ ji, j + f i + f iem ,  [i P j] + B[i M  j] , σ[i j] = E

(1.8)

ρ e˙ = σi j v j,i − qi,i + ρh + Winem ,  i B˙ i + Ji E  i. − M  i ≥ 0, ˙ + σi j v j,i + θ −1 qi θ,i − Pi E −ρ(ψ˙ + ηθ) in which 1  ∗   ·M , fem = qe  J + P ) × B + ( P · ∇) E + (∇ B) E + ( c   · B˙ + J ·  E · π˙ − M E , Winem = fem .v + ρ 

(1.9)

P , ρ  · v) − ( P · ∇)v , P ∗ = P˙ + P(∇

π =

with electromagnetic jump or boundary conditions [8]:    = 0, n · D   n · B = 0 ,   n ×  E = 0 ,    = 0 , n × H

(1.10)

  with A ≡ A+ − A− , where A+ and A− , respectively, denote the values of A when approaching from the positive and negative sides of a surface characterized by the normal vector n [8]. Zero surface current, zero surface free charge, and zero surface em are, respecpolarisation are assumed in (1.10) [8]. Also, σi j , vi, j , qi , vi and ti(n)

6

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

tively, the components of Cauchy stress, gradient of velocity field, surface heat flux, velocity field, and electromagnetic surface traction. In addition, e, h, ψ, η and θ are the internal energy, rate of heat source per unit volume, Helmholtz free energy density, entropy, and temperature, respectively. Furthermore, σ[i j] denotes the skewsymmetric part of the Cauchy stress tensor, i.e., σ[i j] = 21 (σi j − σ ji ) . Clearly, the total stress in the presence of an electromagnetic field is no longer symmetric. Equations (1.1) and (1.8), together with boundary conditions, give the governing equations of a deformable body in the presence of electromagnetic fields. However, similar to the case of classical continuum mechanics, one needs to further define constitutive equations to successfully resolve the intrinsic indeterminate nature of the problem. In this regard, the constitutive equations need to satisfy several axioms that we, herein, just enumerate as follows [8]: (1) causality, (2) determinism, (3) equipresence, (4) objectivity, (5) time reversal, (6) material invariance, (7) neighbourhood, (8) memory, and (9) admissibility. The systematic approach to obtaining constitutive equations is to define a sufficient number of functionals, say α , that meet the aforementioned axioms. To this end, following [8], the dependent physical quantities are taken as follows:    , η, , M,  J , L = T E , Q,

(1.11)

while the independent ones are:    θ, B Y=  E,

(1.12)

along with spatial coordinates and a time parameter. Equations (1.11) and (1.12) are represented in material coordinates, which are related to spatial coordinates via the following relations [8]: ρ = J ρ0 , TIEJ = J X I,i X J, j σi j ,  i , JI = J X I,i Ji , J = det xi,I . Q I = J X I,i qi ,  I = J X I,i Pi , M I = J X I,i M (1.13) T E , in (1.11), is the symmetric tensor in the reference configuration corresponding to σ . Next, the first and second laws of thermodynamics in material coordinates can be formulated in the following form [8]: . 1 E ˙ I , TI J C I J + Q I,I − ρ0 h +  I  E I −M I B˙ I + JI E 2 (1.14) . 1 1 I ≥ 0. − ρ0 (ψ + θ˙ η) + TIEJ C˙ I J + Q I θ I −  I  E I −M I B˙ I + JI E 2 θ

˙ )= ρ0 (ψ + θ˙ η + ηθ

It is easy to see that 20 more functionals are required to have the same number of unknowns and equations. In doing so, we define the Helmholtz free energy for a deformable solid with heat, electrical, and mechanical coupling as follows [8]:

1.2 Continuum Electromagnetic Theory

7

˜ B, ˜ ∇ θ ; θ, X ) .  E, ˜ C, ρ0 ψ=ψ( R

(1.15)

By taking the derivative, we have: ˙ ρ0 ψ=

. . ∂ ψ˜ ˙ ∂ ψ˜  ∂ ψ˜  ∂ ψ˜ ∂ ψ˜ θ˙ . θ˙,I + CI J + EI + BI + ∂C I J ∂ BI ∂θ,I ∂θ ∂ E˜ I

(1.16)

˙ i and B˙ I are Substituting the result into (1.14) and considering that θ˙ , C˙ I J , θ˙,I , E independent variables, the inequality holds when the following relations are true [8]: η=− with [8]:

1 ∂ ψ˜ ∂ ψ˜ ∂ ψ˜ ∂ ψ˜ ∂ ψ˜ , TIEJ = 2 , = 0,  I = − , MI = − , (1.17) I ρ0 ∂θ ∂C I J ∂θ,I ∂E ∂ BI    B,  ∇ R θ ; θ, X ) , I ( C, QI = Q E,    B,  ∇ θ ; θ, X ) . E, J = J (C, I

I

(1.18)

R

Observe that (1.18) satisfies [8]: 1 I ≥ 0 . Q ,I θ,I + JI E θ

(1.19)

Up to this point, all the equations have been written in general form, i.e., no assumption has been made within the continuum framework. However, in many applications, a linearised theory is sufficient to find accurate results. To this end, let us assume that the natural state is free of fields, that is, it has the following form [21]: x 0 = X, 0I = 0, E 0I J = 0, E 0I = 0, B I0 = 0, . . . .

(1.20)

In addition, suppose that all the fields are small enough in the sense that [8, 21]       = δ , u = δ u,  E = δ E, B = δ B, θˆ − θˆ0 = δθ, δ θˆ 0, ∇ θˆ = δ∇θ . . . ,

(1.21)

where δ is a small number. Linearising nonlinear fields by expansion about the natural state and neglecting nonlinear terms, we find:

8

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

ρ u¨ i = σi j, j + f i , σi j n j = t(n)i , σ[i j] = 0 , ρ e˙ = σi j vi, j − qi,i + ρh + Winem , ∂ Pi ∂ Bi − Mi , Winem = Ji E i + E i ∂t ∂t −ρ(ψ˙ + ηθ˙ ) + σi j vi, j + θ −1 qi θ,i − Pi E˙ i − Mi B˙ i + Ji E i ≥ 0 ,

(1.22)

while Maxwell’s equations remain unchanged. Furthermore, due to the fact that all fields are small, a set of linear constitutive equations is sufficient. Hence, as a consequence of (1.16), ψ˜ has a quadratic form as follows [8]: 1 ψ˜ = ψ˜ 0 + ψ˜ I J ε I J + C I J K L ε I J ε K L − D IEJ K E I ε J K − D IBJ K B I ε J K − χ IE E I 2 1 E 1 B − χ I J E I E J − χ I B I − χ IBJ B I B J −  I J E I B J , 2 2 (1.23) in which ψ˜ 0 ,ψ˜ I J , C I J K L , D IEJ K , D IBJ K , χ IE , χ IEJ , χ IB , χ IBJ and  I J are functions of the X and θ . Here  I J denotes an infinitesimal strain tensor. By making use of the assumption given in (1.19), ψ˜ 0 ,ψ˜ I J , χ IE and χ IB in (1.23) can be expressed in terms of temperature in the following form [8]: ρ0 γ 2 )θ , ψ˜ I J = −B I J θ , 2θ0 χ IE = ω˜ I θ, χ IB =  I θ . ψ˜ 0 = −ρ0 η0 θ − (

(1.24)

Hence, based on (1.17), one can derive [8]: γ 1 θ + (B I J ε I J +  ω I E I + I BI ) , θ0 ρ0 = −B I J θ + C I J K L ε K L − D KE I J E K − D KB I J B K ,

η = η0 + TIEJ

I = MI =

(1.25)

 ω I θ + χ IEJ E J + D IEJ K ε J K +  I J B J ,  I θ + χ IBJ B J + D IBJ K ε J K +  J I E J .

Analogously, by taking into account the Clausius–Duhem inequality, (1.18) can be written in the form [8]: J , Q I = κ I J θ,J + κ IEJ E (1.26) θ J + σ I J θ,J , JI = σ I J E where κ I J and σ I J are symmetric and semi-positive definite. Also, let us assume that the material and spatial frames coincide. Therefore, we have: xi ≈ X J δi J ,

(1.27)

1.2 Continuum Electromagnetic Theory

9

where δi J = δi j = δ I j = δ I J . Hence, in linear theory, there is no difference between the material and the spatial frame and each description can be interchangeably used. Now, as was mentioned earlier, one can derive various constitutive equations based on independent and dependent variables chosen in the problem. In the linear case, nevertheless, by employing an appropriate Legendre transformation, it is straightforward to obtain all possible forms from one form. In the following, we shall consider these constitutive equations [27]: εi j = Si jkl σkl + dkiE j E k + dkiHj Hk + αi j θ , E Di = dikl σkl + χiEj E j + χiEj H H j + βiE θ , H Bi = dikl σkl + χ jiE H E j + χiHj H j + βiH θ ,

(1.28)

η = αi j σi j + βiE E i + βiH Hi + α s θ , and

σi j = Ci jkl εkl − ekiE j E k − ekiH j Hk + αi j θ , E Di = eikl εkl + κiEj E j + κiEj H H j + βiE θ , H Bi = eikl εkl + κ jiE H E j + κiHj H j + βiH θ ,

(1.29)

η = αi j εi j + βiE E i + βiH Hi + α s θ , in which, and hereafter, we use lowercase letters for indices. Clearly, each constitutive coefficient given in (1.28) and (1.29) has a physical meaning; Si jkl , Ci jkl , dkiE j , dkiHj , ekiE j , ekiH j , χiEj , χiHj , κiEj , κiHj , χiEj H and κiEj H denote components of the compliance tensor, stiffness tensor, direct piezoelectric tensor, direct piezomagnetic tensor, reverse piezoelectric tensor, reverse piezomagnetic tensor, permittivity under constant stress, permeability under constant stress, permittivity under constant strain, permeability under constant strain, magnetoelectric tensor under constant stress and magnetoelectric tensor under constant strain, respectively [3].

1.3 The Theory of Linear Piezoelectric Materials So far, we have summarised the continuum electromagnetic theory for both linear and nonlinear regimes. In the following, we focus on the theory of linear elastic dielectrics, i.e., materials that are not electrically conducting, while their magnetic property is negligible [8]. Also, for the sake of simplicity, we only consider the case of the isothermal condition. In this regard, analogous to the case of classical continuum mechanics, there shall be two approaches to analyse such materials: (i) the Displacement approach; and (ii) the Stress approach [25]. The reason behind each method lies mainly in the type of prescribed boundary conditions. As will be seen, to theoretically obtain a particular solution for a dielectric body, both the prescribed boundary conditions and the initial conditions are required. It turns out that

10

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

the displacement approach is a more suitable method when the prescribed boundary conditions are of a mixed-type, that is, when on some part of the boundary the displacement is prescribed, while on the remainder the traction is prescribed. Of course, having a displacement field prescribed over the entire boundary is a special case of the category mentioned above. On the other hand, when the mechanical boundary conditions are traction-type, the stress approach is more desirable as both governing equations and boundary conditions are written in terms of the stress field. As an example, from a computational physics perspective, by making use of the stress governing equations, one can develop a stress-type finite element method in which the exact satisfaction of boundary conditions is assured. Hence, in the remainder of the present chapter, we explain both methods, so that the reader can appreciate the regular types of PDEs we typically encounter in the analysis of linear piezoelectric materials. In addition, at the end of the chapter, we consider a special category of variational principles—namely, convolution variational forms—as an alternative means of constructing numerical methods, e.g., finite element methods, for the analysis of such materials. It should be mentioned that, from now on, body force per unit mass is used. Let us first list the governing equations relevant to the linear piezoelectric case under the isothermal condition. To do so, we assume the quasi-electrostatic condition, which is applicable in a number of situations, e.g., ultrasonics: ∇ · σ + ρ f = ρ u¨ on V × (0, ∞) ,  = qe on V × (0, ∞) , ∇·D ∇ × E = 0 on V × (0, ∞) .

(1.30)

It follows that the electric field can be expressed in terms of the gradient of some scalar field, say an electric potential function: E i = −ϕ,i ( x , t) on V × (0, ∞) .

(1.31)

Now, as a result of linearity, we have the following kinematic condition: εi j =

1 (u i, j + u j,i ) on V × (0, ∞) . 2

(1.32)

As there exist two sets of governing equations, we naturally have two types of boundary conditions: (i) mechanical boundary conditions; and (ii) electrical boundary conditions. Hence, partitioning the boundary ∂ V into the following complementary subsets: ∂ V = ∂ Vu ∪ ∂ Vσ , ∂ Vu ∩ ∂ Vσ = ∅ , (1.33) ∂ V = ∂ Vϕ ∪ ∂ VD , ∂ Vϕ ∩ ∂ VD = ∅ , we define the boundary conditions as follows:

1.3 The Theory of Linear Piezoelectric Materials

11



u i = uˆ i ( x , t) on ∂ Vu × [0, ∞) , ˆi ( n = t x , t) on ∂ Vσ × [0, ∞) , σ i j j

ϕ = ϕˆ ( x , t) on ∂ Vϕ × [0, ∞) , Electrical Boundary Conditions ˆ x , t) on ∂ VD × [0, ∞) , D j n j = d( (1.34)

Mechanical Boundary Conditions

ˆ x , t) are, respectively, the prescribed disin which uˆ i ( x , t), tˆi ( x , t), ϕ( ˆ x , t) and d( placement vector, traction vector, electric potential and electric displacement over the boundary. Moreover, the associated initial conditions are: u i ( x , 0) = u i0 ( x ), x ∈ V , u˙ i ( x , 0) = u˙ i0 ( x ), x ∈ V .

(1.35)

As mentioned earlier, in the linear regime, there are variants of constitutive equations that can be obtained from one another by employing the Legendre transformation. Since it is desired to obtain displacement and stress approaches, we utilise the following forms [3]: σi j = Ci jkl εkl − ekiE j E k on V × (0, ∞) , E Di = eikl εkl + κiEj E j on V × (0, ∞) ,

(1.36)

and alternatively εi j = Si jkl σkl + dkiE j E k on V × (0, ∞) , E Di = dikl σkl + χiEj E j on V × (0, ∞) .

(1.37)

The following symmetry conditions hold [3]: Ci jkl = Ckli j = C jikl = Ci jlk on V , ekiE j = ekEji on V , κiEj = κ jiE on V , Si jkl = Skli j = S jikl = Si jlk on V , dkiE j = dkEji χiEj

=

χ jiE

(1.38)

on V , on V ,

with the following relations: Si jkl Ci j pq = δkp δlq on V , E dkiE j = S pqi j ekpq on V ,

χiEj

=

S pqr s eiEpq e Ejr s

+

κiEj

(1.39) on V .

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1 The Continuum Theory of Piezoelectricity and Piezomagnetism

Also, as will be employed in the final section of this chapter, let us specifically define what we mean by an admissible piezoelectric process.  D,  ϕ] is called a piezoelectric Definition 1.1 An ordered array  = [ u , ε, σ , E, process provided that u i ∈ C 1,2 , εi j ∈ C 0,0 , σi j ∈ C 1,0 , Di ∈ C 1,0 , ϕ ∈ C 1,0 , E i ∈ C 0,0 , εi j = ε ji , σi j = σ ji ,

(1.40)

where by C M,N we mean the set of all functions whose spatial and time derivatives up to the order, respectively, M and N exist and are continuous. In this sense, an admissible process that satisfies (1.30), (1.31), (1.32), (1.34), (1.35) and (1.36) is a solution of a mixed initial boundary value problem governing the motion of a piezoelectric material under the aforementioned assumptions. To proceed further, let us assume the following continuity conditions [3]: i. ρ > 0 is continuously differentiable on V ;  dE , χ E are continuously differentiable on V and admit (1.38),  eE , κ E and S, ii. C, respectively; x ) is continuously differentiable on V ; iii. u0 ( x ) is continuously differentiable on V ; iv. v0 ( v. f and qe are continuously differentiable on V ; vi. uˆ and ϕˆ are continuous on ∂ Vu × [0, ∞) and ∂ VD × [0, ∞), respectively; vii. tˆ and dˆ are piecewise continuous on ∂ Vσ × [0, ∞) and ∂ VD × [0, ∞), respectively.

1.3.1 The Displacement Approach By making use of the above-mentioned equations, one can write the governing equations in terms of displacements and electric potential. Substituting (1.31) and (1.32) into (1.36), we find: σi j = Ci jkl u k,l + ekiE j ϕ,k , (1.41) E Di = eikl u k,l − κiEj ϕ, j , where the symmetry condition (1.38) has been employed. The following theorem represents a set of governing equations in terms of displacement and electric potential: Theorem 1.1 Let  = [ u , ϕ] denote an ordered array with u ∈ C 2,2 and ϕ ∈ C 2,0 . Then  is a solution of the mixed initial boundary value problem if and only if:

1.3 The Theory of Linear Piezoelectric Materials

13

(Ci jkl u k,l + ekiE j ϕ,k ), j + ρ f i = ρ u¨ i on V × (0, ∞) , E (eikl u k,l − κiEj ϕ, j ),i = qe on V × (0, ∞) ,

x , t) on ∂ Vu × [0, ∞) , u i = uˆ i ( (Ci jkl u k,l + ekiE j ϕ,k )n j = tˆi ( x , t) on ∂ Vσ × [0, ∞) , ϕ = ϕˆ ( x , t) on ∂ Vϕ × [0, ∞) ,

(1.42)

E ˆ x , t) on ∂ VD × [0, ∞) , (eikl u k,l − κiEj ϕ, j )n i = d( u i ( x , 0) = u i0 ( x ), x ∈ V¯ ,

u˙ i ( x , 0) = u˙ i0 ( x ), x ∈ V¯ . The previous theorem gives us the governing equations by which one can find a solution to the mixed initial boundary value problem, but no information as to whether or not the solution is unique. In the following statement, we give a sufficient condition for the uniqueness of the solution [8]: Theorem 1.2 Let V be a regular region. Let C and χ E be, respectively, a positive definite fourth-order and second-order tensor. Then there is at most one solution for the mixed initial-boundary value problem.

1.3.2 The Stress Approach In this section, we discuss the governing equations in terms of the stress field. This approach was originally employed as an alternative framework in [14, 15]. Proceeding in that vein, [11, 12] generalised the method, so it could be applied to the case of a mixed boundary condition, leading to a set of governing equations that are of integro-differential type. In addition, some special forms of the variational formulation have been derived in [11, 12]. These forms, in contrast to the classical variational forms, guarantee the systematic satisfaction of initial conditions. Subsequently, various studies have been carried out to explore the potential of the stress language (approach), see the review [25]. To obtain the governing equations, let us take the Laplace transform of (1.30)1 :  ρ[u i ] = ρ

u˙ 0 u i0 + 2i s s

+

[σi j, j + f i ] . s2

(1.43)

After some manipulation we obtain: ρu i = ρ(u i0 + t u˙ i0 ) + t ∗ (σi j, j + f i ) or 0 ρ u = t ∗ ∇. σ + t ∗ f + ρ( u 0 + t u˙ ) ,

(1.44)

14

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

in which ∗ denotes the convolution product of two functions in the sense of:

t f ∗ g( x , t) =

f ( x , t − λ)g( x , λ)dλ ( x , t) ∈ × [0, ∞) ,

(1.45)

0

where , in general, stands for a subset of Euclidean space. Thus, we have the following theorems [12]: Theorem 1.3 Let u i ∈ C 0,2 and σi j ∈ C 1,0 be a vector field and a second-order symmetric tensor field, respectively. Then u and σ meet (1.30)1 and the associated initial conditions if and only if 0 ρ u = t ∗ ∇. σ + t ∗ f + ρ( u 0 + t u˙ ) .

(1.46)

It is worth mentioning that the remaining results presented in this section are a reduction of those originally derived in [3] for the case of electro-magneto-elastic materials. Alternative conditions for obtaining a solution are thus:  D,  ϕ] is a solution of the Theorem 1.4 The admissible process  = [ u , ε, σ , E, initial mixed BVP if and only if it satisfies (1.30)2 , (1.31), (1.32), (1.34), (1.36) and (1.46). Now, suppose that ∂ Vu = ∅ . Let us alternatively have σi j ( x , 0) = σi0j ( x ), ϕ( x , 0) = ϕ 0 ( x ), x ∈ V¯ , 0 0 σ˙ i j ( x , 0) = σ˙ i j ( x ), ϕ( ˙ x , 0) = ϕ˙ ( x ), x ∈ V¯ ,

(1.47)

in which σi0j ( x ), σ˙ i0j ( x ), ϕ 0 ( x ), and ϕ˙ 0 ( x ) are the corresponding prescribed initial conditions for the problem being consistent in the sense of 1 0 (u + u 0j,i ), 2 i, j 1 Si jkl σ˙ kl0 ( x ) − dkiE j ϕ˙,k0 ( x ) = (u˙ i,0 j + u˙ 0j,i ), 2

x ) − dkiE j ϕ,k0 ( x) = Si jkl σkl0 (

x ∈ V¯ , x ∈ V¯ ,

(1.48)

and which, by using (1.34) and (1.47), can be uniquely obtained from (1.48). Lemma 1.1 Let u i ( x , t), εi j ( x , t), σi j ( x , t), E i ( x , t) and ϕ( x , t) satisfy (1.31), (1.32), (1.36)1 and (1.46) with σ ∈ C 2,2 on V¯ × [0, ∞), ϕ ∈ C 2,2 on V¯ × [0, ∞), σi j = σ ji . Then, we have:

 Si jkl σ¨ kl =

1 σik,k ρ

,j

 +

1 fi ρ

(1.49)

,j

+ dkiE j ϕ¨,k .

(1.50)

1.3 The Theory of Linear Piezoelectric Materials

15

Lemma 1.2 Suppose (1.49) holds. Define u i by (1.46), εi j by (1.37)1 , and E i by (1.31). Let σ , σ = σ T , and ϕ satisfy (1.50). Then the kinematic equation (1.32) holds true. Now, with the aid of Lemmas 1.1 and 1.2, one can obtain the following theorem: Theorem 1.5 Let σ ∈ C 2,2 , ϕ ∈ C 2,2 with σi j = σ ji . Then an ordered array  = [ σ , ϕ] is a solution of the initial mixed BVP if and only if: 1 1 Si jkl σ¨ kl = ( σik,k ), j + ( f i ), j + dkiE j ϕ¨,k on V × (0, ∞) , ρ ρ E (dikl σkl − χiEj ϕ, j ),i = qe on V × (0, ∞) , x , t) on ∂ V × [0, ∞) , σi j n j = tˆi ( ϕ = ϕˆ ( x , t) on ∂ Vϕ × [0, ∞) , E (dikl σkl − χiEj ϕ, j )n i = dˆ ( x , t) on ∂ VD × [0, ∞) , σi j ( x , 0) = σi0j ( x ), ϕ( x , 0) = ϕ 0 ( x ), x ∈ V¯ ,

(1.51)

σ˙ i j ( x , 0) = σ˙ i0j ( x ), ϕ( ˙ x , 0) = ϕ˙ 0 ( x ), x ∈ V¯ , u i ( x , 0) = u i0 ( x ), x ∈ V¯ , u˙ i ( x , 0) = u˙ i0 ( x ), x ∈ V¯ .

1.4 Variational Principles In this section, we provide convolutional variational principles concerning piezoelectric materials. These principles are interesting from the viewpoint of establishing numerical methods, e.g., finite element methods, to solve practical problems that often involve complicated geometry and boundary and initial conditions. In this regard, the following results are special cases of those of [4] for the case of electromagneto-elastic materials. To begin with, we have the following definition [4, 12] Definition 1.2 Let L be a linear space, K a subset of L, and (S) a functional on K . We define d ˜ (S + λ S) (1.52) δ S˜ (S) = dλ λ=0 for all real numbers λ , where S, S˜ ∈ L and S + λ S˜ ∈ K , and we say the variation of (S) is zero and write ‘δ (S) = 0 over L’ if and only if δ S˜ (S) exists and is equal to zero for all S˜ such that S, S˜ ∈ L and S + λ S˜ ∈ K . The following theorem represents a variational form in which no restriction on the desired quantities is assumed:

16

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

 D,  ϕ] Theorem 1.6 Let  be the set of all admissible processes. Let S = [ u , ε, σ , E, be an element of . Define the functional ϑt on  at each time, say t ∈ [0, ∞), by ϑt (S) =

1 2

  x , t)d x − Ci jkl ( x ) h ∗ εi j ∗ εkl (

V

V

E eki x) j (

−





x , t)d x − h ∗ εi j ∗ E k (

V

1 2

[h ∗ Di ∗ E i ]( x , t)d x − V

 h ∗ σi j ∗ εi j ( x , t)d x

  x , t)d x κiEj ( x ) h ∗ E i ∗ E j (

V

+



[h ∗ (Di,i − qe ) ∗ ϕ]( x , t)d x V

  1 x , t)d xn − ρ( x ) [u i ∗ u i ] ( (h ∗ σi j, j + f ib ) ∗ u i ( x , t)d x 2 V V

    h ∗ (ti − tˆi ) ∗ u i ( x , t)d x + h ∗ ti ∗ uˆ i ( x , t)d x +

+

∂ Vσ



+



h ∗ d ∗ ϕˆ ( x , t)d x +

∂ Vϕ



(1.53)

∂ Vu

 ˆ ∗ ϕ ( h ∗ (d − d) x , t)d x .

∂ V D

Then, S is a solution of the initial mixed-boundary value problem if and only if δϑt (S) = 0 over , within the time interval t ∈ [0, ∞). Next, let us write a specific variational form for an admissible piezoelectric process satisfying the strain-displacement relation: Theorem 1.7 Let  denote the set of all admissible processes which satisfy (1.32).  D,  ϕ] be an element of  and define the functional t on  Let S = [ u , ε, σ , E, at each time, say t ∈ [0, ∞), by

   1 h ∗ σi j ∗ εi j ( x , t)d x − x , t)d x Si jkl ( x ) h ∗ σi j ∗ σkl ( 2 V V

    1 − dkiE j ( x , t)d x − x , t)d x x ) h ∗ σi j ∗ E k ( χiEj ( x ) h ∗ E i ∗ E j ( 2 V V

+ [h ∗ Di ∗ E i ]( x , t)d x − [h ∗ (Di,i − qe ) ∗ ϕ]( x , t)d x

t (S) =



V

V



 1 x , t)d x − ρ( x ) [u i ∗ u i ] ( + f ib ∗ u i ( x , t)d x 2 V V

    x , t)d x − h ∗ ti ∗ (u i − uˆ i ) ( h ∗ tˆi ∗ u i ( x , t)d x − ∂ Vu

+ ∂ Vϕ





h ∗ d ∗ ϕˆ ( x , t)d x +



∂ Vσ

 ˆ ∗ ϕ ( h ∗ (d − d) x , t)d x .

∂ V D

(1.54)

1.4 Variational Principles

17

Then, S is a solution of the initial mixed-boundary value problem if and only if δ t (S) = 0 over , within the time interval t ∈ [0, ∞). Definition 1.3 An admissible piezoelectric process is called a kinematically admissible piezoelectric process if it satisfies the kinematic equations (1.31) and (1.32), the constitutive equations (1.36), and the essential boundary conditions, i.e., (1.34)1 and (1.34)3 . Consequently, the following variational form corresponds to a kinematically admissible piezoelectric process: Theorem 1.8 Let  denote the set of all kinematically admissible processes. Let  D,  ϕ] be an element of  and define the functional t (S) on  at S = [ u , ε, σ , E, each time, say t ∈ [0, ∞), by t (S) =

 1 h ∗ σi j ∗ εi j ( x , t) d x + ρ( x ) [u i ∗ u i ] ( x , t) d x 2 V V

 b  1 f i ∗ u i ( x , t) d x − x , t)d x − [h ∗ Di ∗ E i ] ( 2 V V

  h ∗ tˆi ∗ u i ( x , t)d x x , t)d x − + [h ∗ qe ∗ ϕ] (

1 2



V

−





(1.55)

∂ Vσ

h ∗ dˆ ∗ ϕ ( x , t)d x .

∂ VD

Then, S is a solution of the initial mixed-boundary value problem if and only if δ t (S) = 0 over , within the time interval t ∈ [0, ∞). Consequently, one can obtain a variational form in terms of the displacement and electric potential. To this end, the definitions of an admissible displacement-potential process and a kinematically admissible displacement-potential are required. Definition 1.4 An array S = [ u , ϕ] is called an admissible displacement-potential process if u ∈ C 1,2 , ϕ ∈ C 1,0 . Definition 1.5 An array S = [ u , ϕ] is called a kinematically admissible displacement-potential process if it is an admissible displacement-potential process satisfying (1.34)1 and (1.34)3 . Theorem 1.9 Let  denote the set of all kinematically admissible displacementpotential processes. Let S = [ u , ϕ] be an element of  and define the functional t on  at each time, say t ∈ [0, ∞), by

18

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

t (S) =

1 2



 h ∗ (Ci jkl u k,l + ekiE j ϕ,k ) ∗ u i, j ( x , t)d x

V

  b  1  E E h ∗ (eikl u k,l − κi j ϕ, j ) ∗ ϕ,i ( x , t)d x − f i ∗ u i ( x , t)d x + 2 V V

1 x , t)d x + ρ( x ) [u i ∗ u i ] ( + [h ∗ qe ∗ ϕ] ( x , t) d x 2 V V



   h ∗ dˆ ∗ ϕ ( x , t)d x . h ∗ tˆi ∗ u i ( x , t) d x − − ∂ Vσ

∂ VD

(1.56) Then, S = [ u , ϕ] is a solution of the initial mixed-boundary value problem if and only if δ t (S) = 0 over , within the time interval t ∈ [0, ∞). When mechanical boundary conditions are traction-type, it is more convenient to form variational principles in terms of stress and an electric potential, for which the exact satisfaction of those conditions are guaranteed. With this in mind, we first establish a functional in terms of stress and an electric potential function for mixed mechanical boundary conditions. Definition 1.6 An array [ σ , ϕ] in which σ is a second-order symmetric tensor and ˆ x , t) σ ∈ C 2,0 , ϕ ∈ C 2,0 is called an admissible stress piezoelectric process if ϕ = ϕ( on ∂ Vϕ × [0, ∞). Now, we have: Theorem 1.10 Let  denote the set of all admissible stress piezoelectric processes. Let S = [ σ , ϕ] be an element of  and define the functional ϒt on  at each time, say t ∈ [0, ∞), by 1 ϒt (S) = 2

 

 1 b h ∗ σi j, j ∗ σik,k ( f ∗ σi j ( x , t)d x x , t)d x − ρ ρ i ,j V V

1 E [Si jkl σi j ∗ σkl + χi j ϕ,i ∗ ϕ, j ]( x , t)d x + 2 V 

 b

fi E x , t)d x − uˆ i ∗ ti ( σkl ∗ ϕ,i − qe ∗ ϕ]( x , t)d x + + [−dikl ρ V ∂ Vu 



  h x , t)d x + + ∗ (tˆi − ti ) ∗ σi j, j ( dˆ ∗ ϕ ( x , t)d x . ρ



∂ Vσ

∂ VD

(1.57) Then, S is a solution of the initial mixed-boundary value problem if and only if δ ϒt (S) = 0 over , within the time interval t ∈ [0, ∞).

1.4 Variational Principles

19

Definition 1.7 An admissible stress piezoelectric process is called a dynamically x , t) on ∂ Vσ × [0, ∞). admissible stress piezoelectric process if σi j n j = tˆi ( Starting from ϒt (S), one may arrive at a variational form for a dynamically admissible stress piezoelectric process: Theorem 1.11 Let  be the set of all dynamically admissible stress piezoelectric processes. Let S = [ σ , ϕ] be an element of  and define the functional t on  at each time, say t ∈ [0, ∞), by t (S) =

1 2

 

 h 1 x , t)d x − ∗ σi j, j ∗ σik,k ( ( f ib ) ∗ σi j ( x , t)d x ρ ρ ,j V V

 1  E Si jkl σi j ∗ σkl + χi j ϕ,i ∗ ϕ, j ( x , t)d x + 2 V



   E σ ∗ ϕ ( dikl x , t)d x  − dˆ ∗ ϕ ( x , t)d x . − [q ∗ ϕ]( x , t)d x  + e kl ,i



V

∂ V D

V

(1.58) Then, S is a solution of the traction problem (i.e., ∂ Vu = ∅ ) if and only if δ t (S) = 0 over  within the time interval t ∈ [0, ∞).

1.5 Random Piezoelectric Fields 1.5.1 Basic Considerations This section provides a bridge to subsequent chapters on random fields. The dependent fields are stochastic because they depend on space and time, while the material property fields only depend on space, so they are just tensor-valued random fields (TRFs). First, given the fact that no material continuum is perfectly homogeneous and that mass density is one of the basic continuum properties, it should be taken as a random field (RF)   (1.59) ρ ( x , ω) ; x ∈ R3 , ω ∈  . Here x is the location in the material domain while ω indicates one realization of the mass density field ρ from a probability space. The mass density is defined through a straightforward integration over the mesoscale domain: x , ω) = ρδ (

1 Vδ

ρ ( x , ω) dV , Bδ

where Vδ is the volume of Bδ , while ρ ( x , ω) is the density at the finest measurable scale, the resolution of a given measurement technique. In the language of random

20

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

Fig. 1.1 The mesoscale domain Bδ of a disordered polycrystal (a) is the statistical volume element (SVE) of the random field. Source https://www.bing.com/images (b) Note that, the larger the SVE, the smoother the resulting random field

processes, this is akin to the so-called local averaging of random fields. The situation is illustrated in Fig. 1.1: mass density, being a continuum property, depends on the volume under consideration. The mesoscale domain Bδ of lengthscale L, for a microstructure of a typical grain size d, is parametrised by a dimensionless mesoscale δ :=

L . d

By analogy to (1.59), considering a linear elastic response of Bδ , the mechanical properties are described by an RF of the in-plane stiffness tensor 

 x , ω) ; x ∈ R2 , ω ∈  . Cδ (

The same goes for the electrical permittivity κ and the piezoelectric property eE . Thus, we have a four-component RF 

 x , ω) , Cδ ( x , ω) , κδE ( x , ω) , eδE ( x , ω) ; x ∈ R3 , ω ∈  ρδ (

(1.60)

in which the components are rank 0, rank 4, rank 2, and rank 3 tensor random fields. By mathematical analogy, the situation is identical in the case of piezomagnetic properties. This set of all the deterministic realisations defines a random material Bδ parametrised by a mesoscale δ, and occupying a domain Bδ ⊂ E2 :

1.5 Random Piezoelectric Fields

21

Bδ = {Bδ (ω); ω ∈ } . Essentially, the finest resolution L → 0 in (1.60) corresponds to δ → 0. With coarsening observation, the mesoscale δ increases and the randomness of properties decreases: the continuum RF model (1.60) is mesoscale-dependent. In the limit δ → ∞, the randomness is expected to die to zero, and the material property fields should become deterministic. For example, take x , ω, δ) = ρ eff , lim ρ (

δ→∞

lim Cδ ( x , ω) = Ceff ,

δ→∞

(1.61)

where, provided the RF is spatially homogeneous (a concept to be made rigorous in the next chapters), a conventional, deterministic piezoelectric continuum is obtained. In the above, we have introduced the effective material properties (such as stiffness, modulus, and Poisson ratio) which are typically employed in deterministic models of continuum mechanics. • How can one determine the SVE (mesoscale) properties? • How can one solve a macroscopic boundary value problem? • How can one proceed in the case of media with fractal structures and (also) long range effects? Note that the stiffness tensor cannot simply be locally averaged like the mass density in (1.59), lest we would obtain a Voigt-type (very stiff) estimate; averaging the compliance tensor would result in a Reuss-type (very soft) average. Overall, the mesoscale tensorial properties require a procedure eventually upscaling the mesoscale properties towards the effective ones such as Ceff in the second equation in (1.61).

1.5.2 Determination of Mesoscale Properties Using a Homogenization Condition In order to carry out the mesoscale upscaling and smoothing, we employ a scaledependent homogenization in the vein of the Hill–Mandel condition, e.g. [26]. The objective is to replace a heterogeneous linear elastic medium by a homogeneous linear elastic one. Since the mechanical work going into a volume Vδ is W = V2δ σi j εi j , the said condition requires that the volume average of the scalar product of the stress and strain fields equals the scalar product of their volume averages 1 σi j εi j = σ ε 2 ij ij energetic interpretation = mechanical interpretation, 1 2

(1.62)

where, for a function f (ω, x), the overbar  signifies the volume average over the mesoscale domain Bδ (ω), i.e. f (ω) = V1 Bδ (ω) f (ω, x)dV . In other words, the left

22

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

and right-hand sides of (1.29) express two different ways of looking at the work going into the material per unit of its volume. Turning to the piezoelectricity, we start from a more general expression W =

1 1 σi j εi j − E i Di , 2 2

which, upon substitution of the constitutive relations (1.36), i.e. σi j = Ci jkl εkl − eiEjk E k Di = eiEjk εkl + κiEj E j , becomes the total free energy (a so-called electric enthalpy)   1 1 H ε, E = εi j Ci jkl εkl − E k eiEjk εi j − E i κiEj E j . 2 2 At this point, recalling the inverse constitutive relations (1.37), we observe that the E , while the reverse piezoelectric tensor direct piezoelectric tensor is diEj p = Si jkl eklp E E is eklp = −Ckli j di j p . By analogy to (1.62), there are two ways of volume averaging W : 1 σi j εi j − 21 E i Di = σ ε − 21 E i Di 2 ij ij energetic interpretation = mechanical interpretation. 1 2

(1.63)

Now, consider the stress (σi j ) and strain (εi j ) fields as superpositions of their means (σi j and εi j ) with zero-mean fluctuations (σij and εi j ) σi j (ω, x) = σi j + σij (ω, x),

εi j (ω, x) = εi j + εi j (ω, x) .

Similarly, for the electrical (E i ) and electrical displacement (Di ) fields: E i (ω, x) = E i + E i (ω, x),

Di (ω, x) = Di + Di (ω, x) .

Substituting these into (1.63), we find for the volume average of the energy density over Bδ (ω), we obtain two conditions for volume-type cross-correlations σij εi j = 0, E i Di = 0 . This implies a spatial uncorrelatedness of fluctuations of stress and strain fields on one hand, and of electrical and electrical displacement on the other. These conditions can be satisfied by any one of three different types of uniform boundary conditions (BCs) for boundary value problems on the mesoscale domain.

1.5 Random Piezoelectric Fields

1. Uniform Dirichlet BC

and

23

 = εi0j x j u i (x)  = φ0, j x j φ (x)

∀ x ∈ ∂ Bδ , ∀ x ∈ ∂ Bδ .

In the above, εi0j is the constant strain, while φ 0 , j is the constant gradient of a scalar electric potential φ. [Note that, in general, E j = −φ, j .] 2. Uniform Neumann BC x ) = σi0j n j ti(n) ( and

D( x ) = D (n)0

∀ x ∈ ∂ Bδ ,

(1.64)

∀ x ∈ ∂ Bδ .

In the above, σi0j is the constant strain, while D (n)0 is a constant vector. 3. Uniform mixed-orthogonal BC    u i ( x ) − εi0j x j ti(n) ( x ) − σi0j n j = 0

∀ x ∈ ∂ Bδ .

With reference to [17, 26], it follows from the variational principles that the Dirichlet (respectively, Neumann) type BCs provide upper (lower) estimates on the mechanical and electrical responses of the mesoscale domain Bδ (ω). More specif(ε) ically, setting E i = Di = 0 results in a purely mechanical problem for Ckli j (δ, ω) σ) ( under the Dirichlet BC, and for Skli j(δ) under the Neumann BC. On the other hand, set-

ε) ting εi j = σi j = 0 results in a purely electrical problem for κiE( j(δ) under the Dirichlet

ε) BC, and for χiE( j(δ) under the Neumann BC. As always, the explicit dependence on δ indicates that these properties are mesoscale-dependent, while the dependence on ω that they are random. Upon computing, in a Monte Carlo sense, a number of stochastic boundary value problems for different realisations of the random mesoscale domains {Bδ (ω) ; ω ∈ }, one can assess the statistics of random tensors σ) E(ε ) E(ε) (ε) ( Ckli j(δ) (ω) , Skli j(δ) (ω) , κi j(δ) (ω) , χi j(δ) (ω) .

In the first place, upon statistical averaging, one obtains mesoscale bounds on the eff E,eff macroscale (effective) properties Ckli j and κi j : 

 −1  eff σ) ( (ε) Skli ≤ Ckli j ≤ Ckli j (δ) j (δ) ,   −1  E,eff ε) ε) χiE( ≤ κi j ≤ κiE( (δ) (δ) . j j

24

1 The Continuum Theory of Piezoelectricity and Piezomagnetism

Increasing (respectively, decreasing) the mesoscale domain size and repeating the eff E,eff procedure would result in tighter (wider) bounds on Ckli j and κi j : 

−1  −1     eff σ) σ) ( ( (ε) (ε )  Skli ≤ C ≤ S ≤ C ≤ C   kli j j(δ ) kli j(δ) kli j(δ) kli j(δ ) , δ < δ ,  −1  −1     E,eff ε) ε) ε) E(ε)  χiE( ≤ χiE( ≤ κi j ≤ κiE( j(δ  ) j(δ) j(δ) ≤ κi j(δ  ) , δ < δ .

(1.65)

These are hierarchies of mesoscale bounds describing the scaling from SVE towards the RVE of deterministic continuum physics. The trend towards the RVE is compactly described in terms of the so-called scaling functions [29, 30]. Note that the mixed-orthogonal BCs (1.36) yield intermediate responses (i.e., between the Dirichlet-type and Neumann-type bounds), which do not display clear scaling. In order to obtain the mesoscale coupled piezoelectric responses, we envisage two loading programs: Finding diEj p(δ) : (i) apply a stress-free BC (1.64) with a nonzero electric field E j = −φ 0 , j ; (ii) calculate the resulting volume averaged strain εi j ; −1  σ) εkl . (iii) calculate diEj p(δ) = Si(jkl(δ) Finding eiEj p(δ) : (i) apply a zero-strain BC (1.33) with a nonzero electric displacement field D j ; (ii) calculate the resulting volume averaged stress σi j ; −1  (ε ) σi j . (iii) calculate eiEj p(δ) = Ckli j(δ) Next, carrying out these loading programs in the Monte Carlo sense, taking ensemble averages, and noting the scale-dependent hierarchies (1.65), one should be able  to E,eff establish hierarchies of mesoscale bounds on the macroscale (effective) di j p and   E,eff ei j p , similar to those established for the thermal expansion coefficients in [18]. This research is presently being planned.

Chapter 2

Mathematical Preliminaries

2.1 Piezoelectricity Classes How many different classes of piezoelectric materials exist? To answer this question, we need a little bit more mathematics. Let O(3) be the group of all orthogonal 3 × 3 matrices. Let P be the linear space of all piezoelectric tensors diEjk with inner product (diEjk , fiEjk ) = diEjk fiEjk . The group O(3) acts on P by (gd E )i jk = gil g jm gkn elmn ,

g ∈ O(3), d E ∈ P .

(2.1)

For a given piezoelectric tensor d E , its orbit is defined as Od E = { gd E : g ∈ O(3) } . The set G d E of all g ∈ O(3) with gd E = d E is a closed subgroup of the group O(3) called the stationary subgroup of d E . It is easy to see that the stationary subgroup of a point gd E ∈ Od E is G gd E = { ghg −1 : h ∈ G d E } , that is, a group conjugate to G d E . As g runs over the group O(3), the point gd E runs over the orbit Od E , and the group G gd E runs over the conjugacy class [G d E ] of the group G d E , that is, over the set of all groups conjugate to G d E . We constructed a map from the set of all orbits of the group O(3) to the set of conjugacy classes of its closed subgroups. Mathematically, the conjugacy classes that belong to the image of the above map are called the piezoelectricity classes.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Malyarenko et al., Random Fields of Piezoelectricity and Piezomagnetism, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-3-030-60064-8_2

25

26

2 Mathematical Preliminaries

Physically, a piezoelectric material belongs to a given piezoelectricity class [G] if and only if its piezoelectric tensor d E belongs to the fixed point set of G: VG = { d E ∈ P : gd E = d E for all g ∈ G } . It turns out that there are 16 piezoelectricity classes, see [23]. To formulate the result, we need to assign names to all conjugacy classes of closed subgroups of the group O(3). Theorem 2.1 ([16, 24, 32]) Every closed subgroup of O(3) is conjugate to one of the groups in the following list. 1. { Z n : n ≥ 1 }, { Dn : n ≥ 2 }, T , O, I, SO(2), O(2), SO(3). 2. G × Z 2c , where G is one of the groups listed in item 1, and Z 2c = {δi j , −δi j }. − h : n ≥ 1 }, { Dnv : n ≥ 2 }, { D2n : n ≥ 2 }, O− , O(2)− . 3. { Z 2n To explain the introduced symbols, we follow [24]. Let the vectors e1 , e2 , and e3 constitute the standard basis of R3 . Denote by Q(v , ϑ) the rotation about v ∈ R3 by angle ϑ, and by σv the reflection through the plane normal to the u axis. Z n is the cyclic group of order n generated by Q(e3 , 2π/n). Dn is the dihedral group of order 2n generated by Z n and Q(e1 , π ). The tetrahedral group T of order 12 fixes a tetrahedron. The octahedral group O of order 24 fixes an octahedron or a cube. The icosahedral group I of order 60 fixes an icosahedron or a dodecahedron. SO(2) is the group of rotations Q(e3 , ϑ) with ϑ ∈ [0, 2π ). O(2) is the group generated by SO(2) and Q(e1 , π ). SO(3) is the group of orthogonal matrices with unit determinant. − with n ≥ 2 is the group of Z 2− is the order 2 reflection group generated by σe1 . Z 2n h 3 order 2n generated by Q(e , π/n) and σe3 . D2n is the prismatic group of order 4n − and Q(e1 , π ). Dnv is the pyramidal group of order 2n generated by generated by Z 2n Z n and σe1 . The group O− of order 24 has the same rotation axes as T but with six mirror planes, each through two 3-fold axes. O(2)− is the group generated by SO(2) and σe1 . See also [24, Appendix B], where the correspondence between the above notation and the classical crystallographic one is given. The 16 piezoelectricity classes are [Z 1 ], [Z 2 ], [Z 3 ], [D2v ], [D3v ], [Z 2− ], [Z 4− ], [D2 ], [D3 ], [D4h ], [D6h ], [SO(2)], [O(2)], [O(2)− ], [O− ] and [O(3)]. Consider the group G max = { g ∈ O(3) : gv ∈ VG for all v ∈ VG } . Obviously, G ⊂ G max . It turns out that G max is the normaliser of the group G in O(3): G max = { h ∈ O(3) : hGh −1 = G } . In what follows, we denote by K a closed subgroup of G max such that G is a closed subgroup of K . Note that the action of K on VG given by (2.1) has the following property: gd E is an orthogonal linear operator on VG . In other words, the map g → gv is an orthogonal representation of the group K in the space VG .

2.1 Piezoelectricity Classes

27

From now on, following [1], we denote a representation of a compact topological group by the same symbol as the finite-dimensional linear space where it acts. If it is necessary to consider the operators of a generic representation, we denote them by θ (g). If the operators have a standard notation, we use it instead.

2.2 A Random Field Approach At microscopic length scales, spatial randomness of the piezoelectric material needs x) to be taken into account. Mathematically, we consider the piezoelectric tensor d E ( as a single realisation of a random field. That is, for each position vector x inside x ) taking values in the a piezoelectric body B ⊂ R3 , there is a random tensor d E ( space VG . x ) is second-order, that is, E[d E ( x )2 ] < ∞ We assume that the random field d E ( E x ) is mean-square continuous, that for all x ∈ B. Moreover, assume that the field d ( is, for any x0 ∈ B we have lim

 x − x0 →0

E[d E ( x ) − d E ( x0 )2 ] = 0 .

Under a shift of the Cartesian coordinate system, for any positive integer n and for any distinct points x1 , …, xn ∈ B, the VnG -valued finite-dimensional distributions x1 ), . . . , d E ( xn )) do not change. In particular, the one-point correlation tensor (d E ( x) of the random field d E ( x ) = E[d E ( x )]

d E ( does not depend on x ∈ B, while its two-point correlation tensor x ), d E (y ) = E[(d E ( x ) − d E ( x )) ⊗ (d E (y ) − d E (y ))]

d E ( depends only on the difference y − x. Such a field is called wide-sense homogeneous. x ) : x ∈ B } as the restriction to B of It is convenient to consider a random field { d E ( a random field defined on all of R3 . What happens when one applies an orthogonal transformation g ∈ K to the Cartesian coordinate system? A position vector x ∈ R3 becomes the vector g x. The ranx ) becomes the tensor d E (g x) = gd E ( x ). The one- and two-point dom tensor d E ( correlation tensors of the transformed field must be equal to those of the original field: x ) ,

d E (g x) = g d E ( (2.2)

d E (g x), d E (g y) = (g ⊗ g) d E ( x ), d E (y ) . A random field satisfying these equations is called wide-sense K -isotropic, (or (K , θ )-isotropic, if necessary). In what follows, we will omit the words “wide-sense” and call such fields homogeneous and K -isotropic.

28

2 Mathematical Preliminaries

In this book, we calculate the correlation structure of homogeneous and K -isotropic random fields, that is, find the general form of their one- and two-point correlation tensors. Moreover, we find the spectral expansions of such fields in terms of stochastic integrals with respect to certain random measures. It turns out that the answer depends on the choice of a basis in the space VG . In the following chapter, we construct suitable bases in the above space.

Chapter 3

The Choice of a Basis in the Space V G

3.1 Introduction The space VG carries the orthogonal representation of the group K . A linear sub ∈ W for all g ∈ K and for all w  ∈ W. Any space W ⊆ VG is called invariant if g w orthogonal representation acting on VG has at least two invariant subspaces: {0} and VG . A representation is called irreducible if no other invariant subspaces exist. Let V1 and V2 be two real finite-dimensional orthogonal representations of a group K . A linear operator A : V1 → V2 is called intertwining if it commutes with the above representations: A(gv) = g(Av),

g ∈ K , v ∈ V1 .

The set of all intertwining operators is a real linear space. The representations are called equivalent if the above space contains at least one invertible operator. It is well known that any real finite-dimensional orthogonal representation VG of a compact topological group K is completely reducible. This means the following. Let Vi run over inequivalent irreducible orthogonal representations of K as i runs over some set of indices I . There exist unique nonnegative integers m i of which all but a finite number are zero such that VG is equivalent to the direct sum of m i copies of Vi . We choose a basis in the space VG in such a way that for any i with 1 ≤ i ≤ N , there is a subset of the constructed basis which is a basis of the space Vi . In the remaining part of this chapter, we explain the above construction, using examples.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Malyarenko et al., Random Fields of Piezoelectricity and Piezomagnetism, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-3-030-60064-8_3

29

3 The Choice of a Basis in the Space VG

30

3.2 Example: The Piezoelectricity Class [Z1 ] First, put K = Z 1 . We use information about conjugacy classes of closed subgroups of the group O(3) from [2]. The authors of this book use the Schoenflies notation. According to [24, Appendix B], the Schoenflies notation for Z 1 is C1 . The group C1 is described in [2, Table 1]. In particular, all irreducible orthogonal representations of the group K are equivalent to the trivial representation A(E) = 1, where E = δi j is the unique element of K . Here and below we use the notation for the group elements and group representations from [2]. It follows that the representation θ is the direct sum of 18 copies of the trivial representation A of the group K . Moreover, any one-dimensional subspace of the space VG carries the trivial presentation of K . It follows that the basis of VG can be chosen arbitrarily. We choose the standard basis described in [28]. For simplicity, in what follows we omit the upper index E in the notation for the piezoelectricity tensor. Specifically, the piezoelectric tensor is symmetric with respect to its second and third indices: dki j = dk ji . It follows that P = R3 ⊗ S2 (R3 ), where S2 (R3 ) is the 6-dimensional real linear space of all symmetric 3 × 3 matrices with real entries. On the other hand, by the second equation in (1.28), the piezoelectric tensor is a linear map acting from S2 (R3 ) to R3 . Define an orthogonal operator A : P → R3 ⊗ R6 as follows: A maps the 9 basis tensors ei ⊗ e j ⊗ e j of the space P to the basis tensors ei ⊗ f j of the space R3 ⊗ R6 , the 3 basis tensors √12 ei ⊗ (e2 ⊗ e3 + e3 ⊗ e2 ) to the basis tensors ei ⊗ f4 , the 3 basis tensors √1 ei ⊗ (e1 ⊗ e3 + e3 ⊗ e1 ) to the basis tensors ei ⊗ f5 , 2

and the 3 basis tensors √12 ei ⊗ (e1 ⊗ e2 + e2 ⊗ e1 ) to the basis tensors ei ⊗ f6 . In the above described compressed matrix notation, a piezoelectric tensor is described by a 3 × 6 matrix with real entries. In what follows, we calculate the elements of various bases in the above described basis. The normaliser of the group Z 1 is K = O(3). Its irreducible orthogonal representations are described as follows. The representation H is the linear space of real-valued homogeneous harmonic (with vanishing Laplacian) polynomials of degree  ≥ 0 in three variables x, y, z of dimension 2 + 1. The group K acts by gp( x ) = p(g −1 x) . The representation H∗ is the same linear space but with different action: gp( x ) = det gp(g −1 x) , see [10, 23]. Note that the elements of the space H are even (resp. odd) functions of the variable x whenever the number  is even (resp. odd). It follows that the representations H

3.2 Example: The Piezoelectricity Class [Z 1 ]

31

with even  and H∗ with odd  map the matrix −δi j to the identity operator I . The representations H with odd  and H∗ with even  map the matrix −δi j to the operator −I . In particular, the representation that maps g ∈ O(3) to itself is equivalent to H1 . The elements of the space R3 where this representation acts are pseudo-vectors (they change sign under reflection). In the irreducible components of the representation H1 ⊗ H1 , the matrix −δi j maps to I . The Clebsch–Gordan formula [10, 23] gives H1 ⊗ H1 = H0 ⊕ H1∗ ⊕ H2 . The one-dimensional irreducible component H0 acts on the space of scalars generated by the symmetric identity matrix δi j . There are two matrices of norm 1 in this space: √13 δi j and − √13 δi j . We choose the first matrix as one of the basis tensors of the space S2 (R3 ). The three-dimensional irreducible component H1∗ acts on the space of 3 × 3 skewsymmetric matrices. The elements of the above space are vectors. Finally, the five-dimensional irreducible component H2 acts on the space of traceless 3 × 3 symmetric matrices. The elements of this space are called deviators. The basis in the space of deviators (and, more generally, in the space where the representation H1 ⊗ H2 acts) was constructed in [9]. The matrices of this basis are k , where |1 − 2 | ≤  ≤ 1 + 2 and − ≤ k ≤ . We call them denoted by g[ 1 ,2 ] 0 the Godunov–Gordienko matrices. In particular, the (i, j)th entry of the matrix g0[1,1] 0[i, j] is g0[1,1] = √13 δi j , −1 ≤ i, j ≤ 1, exactly what we chose above. In general, the Godunov–Gordienko matrices can be calculated by the algok rithm proposed in [31]. The nonzero entries of the five symmetric matrices g2[1,1] , −2 ≤ k ≤ 2, that are located on and over the main diagonal, are as follows: 1 −2[−1,1] 2[1,1] = g2[1,1] = −√ , g2[1,1] 2 1 −1[−1,0] 1[0,1] 2[−1,−1] = g2[1,1] = g2[1,1] =√ , g2[1,1] 2 1 0[−1,−1] 0[1,1] g2[1,1] = g2[1,1] = − √ , 6 √ 2 0[0,0] = √ . g2[1,1] 3

(3.1)

The orthogonal representation (2.1) becomes H1 ⊗ (H0 ⊕ H2 ) = H1 ⊕ H1 ⊕ H2∗ ⊕ H3 . Note that this representation does not contain trivial components. In other words, VO(3) = {0}. In physical language, the piezoelectricity class [O(3)] describes materials that are not piezoelectric at all.

3 The Choice of a Basis in the Space VG

32

The first component consists of pseudo-vectors. Its basis tensors are 0 , d p = e p ⊗ g0[1,1]

1 ≤ p ≤ 3.

In the compressed matrix notation, the nonzero entries of the matrices d p become 1 p p p e p1 = e p2 = e p3 = √ . 3

(3.2)

The second component also consists of pseudo-vectors. Its basis tensors are dp =

2 1  

p−5[i, j]

j

g1[1,2] ei ⊗ g2[1,1] ,

4 ≤ p ≤ 6.

i=−1 j=−2 k[i, j]

Calculation of the Godunov–Gordienko matrices g1[1,2] , −1 ≤ k ≤ 1, gives the following results: √ √ √ 3 3 3 1 −1 −2 0 2 d4 = − √ e−1 ⊗ g2[1,1] + √ e−1 ⊗ g2[1,1] + √ e0 ⊗ g2[1,1] − √ e1 ⊗ g2[1,1] , 10 10 10 10 √ √ √ 3 2 3 −1 0 1 d5 = √ e−1 ⊗ g2[1,1] + √ e0 ⊗ g2[1,1] + √ e1 ⊗ g2[1,1] , 10 10 5 √ √ √ 3 3 3 1 −2 1 0 2 d6 = − √ e−1 ⊗ g2[1,1] + √ e0 ⊗ g2[1,1] − √ e1 ⊗ g2[1,1] − √ e1 ⊗ g2[1,1] . 10 10 10 10

We calculate the nonzero elements of the matrices d p , 4 ≤ p ≤ 6, using (3.1). We obtain √ 3 2 1 4 4 4 4 4 e11 = √ , e12 = e13 = − √ , e26 = e35 = √ , 10 15 15 √ 3 2 1 5 5 5 5 5 (3.3) = √ , e16 = e34 =√ , = e23 = −√ , e21 e22 10 15 15 √ 3 2 1 6 6 6 6 6 = e32 = −√ . e33 = √ , e15 = e24 = √ , e31 10 15 15 The third component consists of pseudo-deviators. Its basis tensors are dp =

1  2  i=−1 j=−2

p−9[i, j]

j

g2[1,2] ei ⊗ g2[1,1] ,

7 ≤ p ≤ 11 .

3.2 Example: The Piezoelectricity Class [Z 1 ]

33 k[i, j]

Calculation of the Godunov–Gordienko matrices g2[1,2] , −2 ≤ k ≤ 2, gives the following results: √ 2 1 −1 1 −1 2 1 − √ e1 ⊗ g2[1,1] , d = √ e ⊗ g2[1,1] − √ e0 ⊗ g2[1,1] 3 6 6 7

1 1 1 1 −2 1 0 2 − √ e0 ⊗ g2[1,1] + √ e1 ⊗ g2[1,1] − √ e1 ⊗ g2[1,1] , d8 = − √ e−1 ⊗ g2[1,1] 6 6 2 6 1 1 −1 1 − √ e1 ⊗ g2[1,1] , d9 = √ e−1 ⊗ g2[1,1] 2 2 1 1 1 1 −1 −2 0 2 − √ e−1 ⊗ g2[1,1] + √ e0 ⊗ g2[1,1] + √ e1 ⊗ g2[1,1] , d10 = − √ e−1 ⊗ g2[1,1] 2 6 6 6 √ 2 1 −1 1 −2 −1 11 1 + √ e1 ⊗ g2[1,1] . d = √ e ⊗ g2[1,1] + √ e0 ⊗ g2[1,1] 6 3 6 The nonzero elements of the matrices d p , 7 ≤ p ≤ 11, become 1 7 e16 =√ , 6

1 7 = −√ , e21 3

1 7 =√ , e23 3

1 7 = −√ , e34 6

1 8 =√ , e15 6

1 8 = −√ , e24 6

1 8 = −√ , e31 3

1 8 =√ , e32 3

1 9 =√ , e14 2

1 9 = −√ , e36 2

1 1 10 10 = − √ , e13 =√ , e12 3 3 √ 2 1 11 10 =√ , = −√ , e25 e14 6 3

(3.4) 1 10 =√ , e26 6

1 10 = −√ , e35 6

1 10 =√ . e36 6

The fourth component consists of 3rd order pseudo-deviators. Its basis tensors are 2 1   p−15[i, j] j g3[1,2] ei ⊗ g2[1,1] , 12 ≤ p ≤ 18 . dp = i=−1 j=−2 k[i, j]

Calculation of the Godunov–Gordienko matrices g3[1,2] , −3 ≤ k ≤ 3, gives the following results:

3 The Choice of a Basis in the Space VG

34

1 1 −2 2 d12 = − √ e−1 ⊗ g2[1,1] − √ e1 ⊗ g2[1,1] , 2 2 1 1 1 −2 −1 1 + √ e0 ⊗ g2[1,1] − √ e1 ⊗ g2[1,1] , d13 = − √ e−1 ⊗ g2[1,1] 3 3 3 √ √ 2 −1 1 −1 2 2 0 1 −1 −2 14 0 2 d = √ e ⊗ g2[1,1] − √ e ⊗ g2[1,1] + √ e ⊗ g2[1,1] + √ e1 ⊗ g2[1,1] , 30 30 5 15 √ 3 1 −1 1 −1 15 0 1 − √ e1 ⊗ g2[1,1] , d = − √ e ⊗ g2[1,1] + √ e0 ⊗ g2[1,1] 5 5 5 √ √ 2 1 −1 2 2 0 1 −2 16 1 0 2 d = √ e ⊗ g2[1,1] + √ e ⊗ g2[1,1] + √ e1 ⊗ g2[1,1] + √ e1 ⊗ g2[1,1] , 30 30 15 5 1 1 1 −1 2 1 d17 = √ e−1 ⊗ g2[1,1] + √ e0 ⊗ g2[1,1] − √ e1 ⊗ g2[1,1] , 3 3 3 1 1 −2 2 − √ e1 ⊗ g2[1,1] . d18 = √ e−1 ⊗ g2[1,1] 2 2

(3.5)

The nonzero elements of the matrices d p , 12 ≤ p ≤ 18, become 1 12 e11 =− , 2 1 13 = −√ , e14 3 √ 3 14 e11 =− √ , 2 5 1 15 e16 = −√ , 5

12 e13 =

1 , 2

1 12 e35 = √ , 2

1 13 e25 = −√ , 3

1 13 e36 = −√ , 3

2 14 = √ , e12 15

1 14 =− √ , e13 2 15 √ 2 15 e22 = √ , 5

1 16 = −√ , e15 30

1 15 = −√ , e21 10 √ 2 2 16 e24 = √ , 15

1 17 = √ , e16 3 1 18 = −√ , e15 2

√ 2 2 14 = √ , e26 15 1 15 = −√ , e23 10

1 16 =− √ , e31 2 15

2 16 = √ , e32 15

1 17 = √ , e21 6

1 17 = −√ , e23 6

1 17 = −√ , e34 3

1 18 e31 =− , 2

18 = e33

1 14 = −√ , e35 30 1 15 e34 = −√ , 5 √ 3 16 =− √ , e33 2 5

1 . 2

The description of the corresponding homogeneous and isotropic random fields is given in [20]. In the following section, we consider another example.

3.3 Example: The Piezoelectricity Class [D2 ]

35

3.3 Example: The Piezoelectricity Class [ D2 ] The first question we have to answer is the following: what is the dimension of the space V D2 ? Before giving the answer, we need to explain an important notion from representation theory. In what follows, we extensively use the book [2], which works with complex representations. Let V be a complex irreducible representation of a compact group K . According to [1], there are three mutually exclusive cases. 1. There is a map j : V → V such that g( jv) = j (gv), j (zv) = z( jv) (z ∈ C), and j 2 v = v. Such a representation is said to be of real type. In this case, we split V into the +1 and −1 eigenspaces of j; these are irreducible real representations which are isomorphic: the invertible intertwining operator is multiplication by i. We denote each of the obtained real representations by the same symbol V. 2. There is a map j : V → V which has all but the last property. The missed property is modified as follows: j 2 v = −v. Such a representation is said to be of quaternionic type and will never occur in our studies. 3. No such map exists. These representations are said to be of complex type. Let V be an irreducible representation of complex type, and suppose tV has the same underlying set and the same group action as V, but z ∈ C acts on tV as z used to act on V. The representation tV is called conjugate to V, it is irreducible and not equivalent to V. By definition, the representation r V has the same underlying space as V and the same action of K , but is regarded as a real linear space. This representation is irreducible. Define cH1 = C ⊗R H1 . This is a complex linear space where the scalar-vector multiplication is defined by z(z ⊗ v) = (zz ) ⊗ v. This is an irreducible complex representation of O(3) where the group acts by g(z ⊗ v) = z ⊗ (gv). Consider the restriction of the representation cH1 to the subgroup K = D2 . By [2, Table 22.10], it is equivalent to B1 ⊕ B2 ⊕ B3 . Unless otherwise stated, all representations are of real type. On which subspaces of R3 does each of the components act? [2, Table 3.1] contains the Euler angles of all elements E, C2x , C2y and C2z of  100

the group D2 . Using the definition of the Euler angles, we see that E = 0 1 0 , 001 1 0 0   −1 0 0   −1 0 0  C2x = 0 −1 0 , C2y = 0 1 0 , C2z = 0 −1 0 . We see that the restriction of 0 0 −1 0 0 −1 0 0 1 the representation B1 ⊕ B2 ⊕ B3 to the x-axis takes value 1 on E and C2x and −1 on C2y and C2z . By [2, Table 22.4], this representation is B3 . Similarly, B2 acts on the y-axis, and B1 on the z-axis. The irreducible components of the symmetric tensor square S2 (B1 ⊕ B2 ⊕ B3 ) are calculated using [2, Table 22.8]. We obtain: three copies of the trivial representation A act on the one-dimensional spaces generated by the matrices e1 ⊗ e1 , e2 ⊗ e2 and e3 ⊗ e3 . The representation B1 acts on the one-dimensional space generated by the matrix √12 (e1 ⊗ e2 + e2 ⊗ e1 ), the representation B2 on the space generated by

√1 ( e1 2

⊗ e3 + e3 ⊗ e1 ), and the representation B3 on the space gen-

3 The Choice of a Basis in the Space VG

36

erated by √12 (e2 ⊗ e3 + e3 ⊗ e2 ). Again by [2, Table 22.8] we have: the three copies of the trivial representation act on the one-dimensional spaces generated by the matrices √12 e1 ⊗ (e2 ⊗ e3 + e3 ⊗ e2 ), √12 e2 ⊗ (e1 ⊗ e3 + e3 ⊗ e1 ) and √12 e3 ⊗ (e1 ⊗ e2 + e2 ⊗ e1 ). In the compressed matrix notation, they become 3 × 6 matrices with the following nonzero elements: d116 = d225 = d334 = 1 .

(3.6)

In particular, dim V D2 = 3 and θ = 3A. The normaliser of the group D2 is O × Z c2 . Besides the value of K = D2 considered above, the group K can also take values D4 , D2 × Z 2c , D4h , T , D4 × Z 2c , O, T × Z 2c , O− and O × Z 2c . This follows from [2, Graph 11]. Put K = D2 × Z 2c . By [24, Appendix B], the Schoenflies notation for K is D2h . By [2, Table 31.5], the restriction of the representation cH1 of the group O(3) to the subgroup D2 × Z 2c is equivalent to the direct sum B1u ⊕ B2u ⊕ B3u , where B1u acts on the z-axis, B2u on the y-axis, and B3u on the x-axis. The representation S2 (B1u ⊕ B2u ⊕ B3u ) is equivalent to the direct sum 3A g ⊕ B1g ⊕ B2g ⊕ B3g , where the first copy of A g acts on the one-dimensional space generated by the matrix e1 ⊗ e1 , the second copy on the space generated by e2 ⊗ e2 , and the third copy on the space generated by e3 ⊗ e3 . The representation B1g acts on the space generated by the matrix √12 (e1 ⊗ e1 + e2 ⊗ e2 ), the representation B2g on the space generated by

√1 ( e1 ⊗ e3 + e3 ⊗ e1 ), 2 2 3 3 2

and the representation B3g on the space gen-

√1 ( e 2

⊗ e + e ⊗ e ). In the space P, the basis tensors of the subspace erated by V D2 are the tensors (3.6), where the one-dimensional space generated by d1 carries the representation B3u ⊗ B3g = Au , the space generated by d2 carries the representation B2u ⊗ B2g = Au , and the space generated by d3 carries the representation B1u ⊗ B1g = Au . In particular, θ = 3Au . Put K = D4 . We proceed in the same manner. By [2, Table 24.10], the restriction of the representation cH 1 of the group O(3) to the subgroup K is equivalent to A2 ⊕ E. [2, Table 24.5] shows that S2 (A2 ⊕ E) = 2 A1 ⊕ B1 ⊕ B2 ⊕ E, where the first copy of the trivial representation A1 acts on the one-dimensional space generated by the matrix √12 (e1 ⊗ e1 + e2 ⊗ e2 ), the second copy on the space generated by e3 ⊗ e3 . √1 ( e1 2 2 2

The representation B1 acts on the space generated by the matrix e2 ), the representation B2 on the space generated by

√1 ( e1 2

⊗ e + e ⊗ e1 ), and the

representation E on the two-dimensional space generated by √1 ( e2 2

⊗ e1 − e2 ⊗

√1 ( e1 2

⊗ e3 + e3 ⊗ e1 )

⊗ e + e ⊗ e ). We see also that the irreducible component A2 of the and representation A2 ⊕ E acts on the z-axis, while E acts on the (x, y)-plane. We investigate the structure of the representation (A2 ⊕ E) ⊗ (2 A1 ⊕ B1 ⊕ B2 ⊕ E), using [2, Table 24.8]. Using the above described technique, it is possible to write down the basis tensors in each irreducible component of the above 18-dimensional representation and find components acting on linear subspaces of the space V D2 . This is easy, because the basis tensors (3.6) of this space have already been calculated. 3

3

2

3.3 Example: The Piezoelectricity Class [D2 ]

37

It turns out that the above components are the components A1 and B1 of the tensor square E ⊗ E and the tensor product A2 ⊗ B2 = B1 . The last equality follows from [2, Table 24.8]. Indeed, the component A1 acts on the space generated by the tensor 1 1 [ e ⊗ (e2 ⊗ e3 + e3 ⊗ e2 ) + e2 ⊗ (e1 ⊗ e3 + e3 ⊗ e1 )], the first B1 component on 2 the space generated by 21 [e1 ⊗ (e2 ⊗ e3 + e3 ⊗ e2 ) − e2 ⊗ (e1 ⊗ e3 + e3 ⊗ e1 )], and the second B1 component on the space generated by √12 e3 ⊗ (e1 ⊗ e2 + e2 ⊗ e1 ). In the compressed matrix notation, they become 3 × 6 matrices with the following nonzero elements: 1 d116 = d125 = d216 = √ , 2

1 d225 = − √ , 2

d334 = 1 .

(3.7)

Moreover, θ = A1 ⊕ 2B1 . Put K = D4h . By [24, Appendix B], the Schoenflies notation for K is D2d . By [2, Table 42.5], the restriction of the representation cH1 of the group O(3) to the subgroup D4h is equivalent to the direct sum B2 ⊕ E, where B2 acts on the z-axis, and E acts on the (x, y)-plane. The symmetric tensor square S2 (B2 ⊕ E) is equivalent to the direct sum 2 A1 ⊕ B1 ⊕ B2 ⊕ E, where the first copy of A1 acts on the one-dimensional space generated by the matrix √12 (e1 ⊗ e1 + e2 ⊗ e2 ), the second copy on the space generated by e3 ⊗ e3 , the representation B1 on the space generated by e2 ⊗ e2 ), the representation B2 on the space generated by

√1 ( e1 2

√1 ( e1 ⊗ e1 2 2 1

−

⊗ e2 + e ⊗ e ), and

the representation E on the two-dimensional space generated by

√1 ( e1 2

⊗ e3 + e3 ⊗

e1 ) and √12 (e2 ⊗ e3 + e3 ⊗ e2 ). The irreducible components of the tensor product (B2 ⊕ E) ⊗ (2 A1 ⊕ B1 ⊕ B2 ⊕ E) which act on linear subspaces of the space V D2 are the component A1 of the tensor product E ⊗ E that acts on the space generated by the tensor d1 of Eq. (3.7), the component B1 of the same tensor product that acts on the space generated by the tensor d2 of Eq. (3.7), and the component B2 ⊗ B2 = A1 that acts on the space generated by the tensor d3 of the above equation. We have θ = 2 A1 ⊕ B1 . Put K = D4 × Z 2c . By [24, Appendix B], the Schoenflies notation for K is D4h . By [2, Table 33.5], the restriction of the representation cH1 of the group O(3) to the subgroup D4 × Z 2c is equivalent to the direct sum A2u ⊕ E u , where A2u acts on the z-axis, and E u acts on the (x, y)-plane. The symmetric tensor square S2 (A2u ⊕ E u ) is equivalent to the direct sum 2 A1g ⊕ B1g ⊕ B2g ⊕ E g , where the first copy of A1g acts on the one-dimensional space generated by the matrix √12 (e1 ⊗ e1 + e2 ⊗ e2 ), the second copy on the space generated by e3 ⊗ e3 , the representation B1g on the space generated by √12 (e1 ⊗ e1 − e2 ⊗ e2 ), the representation B2g on the space generated by

√1 ( e1 2

⊗ e2 + e2 ⊗ e1 ), and the representation E g on the

two-dimensional space generated by √12 (e1 ⊗ e3 + e3 ⊗ e1 ) and √12 (e2 ⊗ e3 + e3 ⊗ e2 ). The irreducible components of the tensor product (A2u ⊕ E u ) ⊗ (2 A1g ⊕ B1g ⊕ B2g ⊕ E g ) which act on linear subspaces of the space V D2 are the component A1u

3 The Choice of a Basis in the Space VG

38

of the tensor product E g ⊗ E u that acts on the space generated by the tensor d1 of Eq. (3.7), the component B1u of the same tensor product that acts on the space generated by the tensor d2 of Eq. (3.7), and the component A2u ⊗ B1g = B2u that acts on the space generated by the tensor d3 of the above equation. We have θ = A1u ⊕ B1u ⊕ B2u . Put K = T . By [2, Table 70.5], the restriction of the representation cH1 of the group O(3) to the subgroup T is T . By [2, Table 70.5], the symmetric tensor product S2 (T ) is equivalent to the direct sum A ⊕ 1 E ⊕ 2 E ⊕ T . The representations 1 E and 2 E are of complex type and mutually conjugate. Denote r 1 E by E. We obtain: S2 (T ) is equivalent to the direct sum A ⊕ E ⊕ T . The representation A acts on the one-dimensional space generated by the matrix √13 (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 ). The representation E acts on the two-dimensional space generated by the matrices √1 ( e1 ⊗ e1 − e2 ⊗ e2 ) and √16 (−e1 ⊗ e1 − e2 ⊗ e2 + 2e3 ⊗ e3 ). The representation 2 T acts on the three-dimensional space generated by

⊗ e3 + e3 ⊗ e2 ),

√1 ( e1 2

⊗

⊗ e + e ⊗ e ). The irreducible components of the tensor e + e ⊗ e ) and product T ⊗ (A ⊕ E ⊕ T ) which act on linear subspaces of the space V D2 are the components A and E of the tensor product E ⊗ E. Indeed, the component A acts on the one-dimensional space generated by the tensor √16 [e1 ⊗ (e2 ⊗ e3 + e3 ⊗ e2 ) + e2 ⊗ (e1 ⊗ e3 + e3 ⊗ e1 ) + e3 ⊗ (e1 ⊗ e2 + e2 ⊗ e1 )], while the component E acts on the two-dimensional space generated by the tensors 21 [e1 ⊗ (e2 ⊗ e3 + e3 ⊗ e2 ) − e2 ⊗ (e1 ⊗ e3 + e3 ⊗ e1 )] and 2√1 3 [−e1 ⊗ (e2 ⊗ e3 + e3 ⊗ e2 ) − e2 ⊗ (e1 ⊗ e3 + e3 ⊗ e1 ) + 2e3 ⊗ (e1 ⊗ e2 + e2 ⊗ e1 )]. In the compressed matrix notation, these tensors become the matrices with the following nonzero elements. 3

3

1

√1 ( e1 2

√1 ( e2 2

1 d116 = d125 = d134 = √ , 3 √ 2 d334 = √ . 3

2

2

1

1 1 1 d216 = √ , d225 = − √ , d316 = d325 = − √ , 2 2 6 (3.8)

We have θ = A ⊕ E. Put K = T × Z 2c . By [24, Appendix B], the Schoenflies notation for K is Th . By [2, Table 72.5], the restriction of the representation cH1 of the group O(3) to the subgroup T × Z 2c is Tu . The symmetric tensor square S2 (Tu ) is equivalent to the direct sum of complex irreducible representations A g ⊕ 1 E g ⊕ 2 E g ⊕ Tg or of real irreducible representations A g ⊕ E g ⊕ Tg , where E g = r 1 E g . The representation A g acts on the one-dimensional space generated by the matrix √13 (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 ). The representation E g acts on the two-dimensional space generated by the matrices √12 (e1 ⊗ e1 − e2 ⊗ e2 ) and √16 (−e1 ⊗ e1 − e2 ⊗ e2 + 2e3 ⊗ e3 ). The representation Tg acts on the three-dimensional space generated by

√1 ( e2 2

⊗ e3 +

e3 ⊗ e2 ), √12 (e1 ⊗ e3 + e3 ⊗ e1 ) and √12 (e1 ⊗ e2 + e2 ⊗ e1 ). The irreducible components of the tensor product Tu ⊗ (A g ⊕ E ⊕ Tg ) which act on linear subspaces of the space V D2 are the components Au and E u of the tensor product Tu ⊗ Tg , where

3.3 Example: The Piezoelectricity Class [D2 ]

39

E u is the result of applying the second construction to any of the irreducible unitary representations 1 E u or 2 E u . Indeed, the component Au acts on the one-dimensional space generated by the tensor d1 of (3.8), while the component E u acts on the twodimensional space generated by the tensors d2 and d3 of (3.8). We have θ = Au ⊕ E u . Put K = O. By [2, Table 69.5], the restriction of the representation cH1 of the group O(3) to the subgroup T is T1 . The symmetric tensor square S2 (T1 ) is equivalent to the direct sum A1 ⊕ E ⊕ T2 . The representation A1 acts on the one-dimensional space generated by the matrix √13 (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 ). The representation E acts on the two-dimensional space generated by the matrices √12 (e1 ⊗ e1 − e2 ⊗ e2 ) and √16 (−e1 ⊗ e1 − e2 ⊗ e2 + 2e3 ⊗ e3 ). The representation T2 acts on the three-dimensional space generated by √1 ( e1 2

√1 ( e2 2

⊗ e3 + e3 ⊗ e2 ),

⊗ e3 + e3 ⊗ e1 ) and √12 (e1 ⊗ e2 + e2 ⊗ e1 ). The irreducible components of the tensor product T1 ⊗ (A1 ⊕ E ⊕ T2 ) which act on linear subspaces of the space V D2 are the components A2 and E of the tensor product T1 ⊗ T2 . Indeed, the component A2 acts on the one-dimensional space generated by the tensor d1 of (3.8), while the component E acts on the two-dimensional space generated by the tensors d2 and d3 of (3.8). We have θ = A2 ⊕ E. Put K = O− . By [24, Appendix B], the Schoenflies notation for K is Td . By [2, Table 73.5], the restriction of the representation cH1 of the group O(3) to the subgroup O− is T2 . The symmetric tensor square S2 (T2 ) is equivalent to the direct sum A1 ⊕ E ⊕ T2 . The representation A1 acts on the one-dimensional space generated by the matrix √13 (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 ). The representation E acts on the two-dimensional space generated by the matrices and

√1 (− e1 6

⊗ e1 − e2 ⊗ e2 )

⊗ e1 − e2 ⊗ e2 + 2e3 ⊗ e3 ). The representation T2 acts on the three-

dimensional space generated by √1 ( e1 ⊗ e2 2 (A1 ⊕ E ⊕

√1 ( e1 2

√1 ( e2 2

⊗ e3 + e3 ⊗ e2 ),

√1 ( e1 2

⊗ e3 + e3 ⊗ e1 ) and

+ e ⊗ e ). The irreducible components of the tensor product T2 ⊗ T2 ) which act on linear subspaces of the space V D2 are the components A1 and E of the tensor product T2 ⊗ T2 . Indeed, the component A1 acts on the onedimensional space generated by the tensor d1 of (3.8), while the component E acts on the two-dimensional space generated by the tensors d2 and d3 of (3.8). We have θ = A1 ⊕ E. Finally, put K = O × Z 2c . By [24, Appendix B], the Schoenflies notation for K is Oh . By [2, Table 73.5], the restriction of the representation r H1 of the group O(3) to the subgroup O × Z 2c is T1u . The symmetric tensor square S2 (T1u ) is equivalent to the direct sum A1g ⊕ E g ⊕ T2g . The representation A1g acts on the one-dimensional space generated by the matrix √13 (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 ). The representation E g acts on the two-dimensional space generated by the matrices √12 (e1 ⊗ e1 − e2 ⊗ e2 ) and √16 (−e1 ⊗ e1 − e2 ⊗ e2 + 2e3 ⊗ e3 ). The represen2

1

tation T2g acts on the three-dimensional space generated by √1 ( e1 2

e1 ) and √12 (e1 ⊗ e2 + e2 T1u ⊗ (A1g ⊕ E g ⊕ T2g )

√1 ( e2 2

⊗ e3 + e3 ⊗ e2 ),

⊗ e3 + e3 ⊗ ⊗ e1 ). The irreducible components of which act on linear subspaces of the the tensor product space V D2 are the components A2u and E u of the tensor product T1u ⊗ T2g . Indeed,

3 The Choice of a Basis in the Space VG

40 Table 3.1 The bases of the space V D2 K

θ

The basis

D2 D2 × Z 2c D4 D4h D4 × Z 2c T T × Z 2c O O− O × Z 2c

3A 3Au A1 ⊕ 2B1 2 A1 ⊕ B1 A1u ⊕ B1u ⊕ B2u A⊕E Au ⊕ E u A2 ⊕ E A1 ⊕ E A2u ⊕ E u

(3.6) (3.6) (3.7) (3.7) (3.7) (3.8) (3.8) (3.8) (3.8) (3.8)

the component A2u acts on the one-dimensional space generated by the tensor d1 of (3.8), while the component E u acts on the two-dimensional space generated by the tensors d2 and d3 of (3.8). We have θ = A2u ⊕ E u . Our investigation is summed up in Table 3.1. In a similar way, it is possible to find the bases in the remaining spaces VG . For the reader’s convenience, we list the normalisers of the piezoelectricity classes not considered here. The normaliser of the groups Z 2 , Z 3 , Z 2− , Z 4− , SO(2), O(2), O(2)− is O(2) × Z 2c . The normaliser of D2v and D4h is D4 × Z 2c . The normaliser of D3 and D3v is D6 × Z 2c . The normaliser of Dh6 is D6 × Z 2c . Finally, the normaliser of O− is O × Z 2c . Note that the groups D2 and D2 × Z 2c belong to the orthotropic crystal system, the groups D4 , D4h and D4 × Z 2c to the tetragonal crystal system, and the rest of the groups to the cubic crystal system. It turns out that two groups share the same basis if and only if they belong to the same crystal system.

Chapter 4

Correlation Structures

4.1 Introduction In this chapter, we calculate the one- and two-point correlation tensors of a homogeneous and (K , θ )-isotropic random field for all 10 cases in Table 3.1. The spectral expansion of the field will also be calculated. The main idea is simple. We consider the correlation structure of a homogeneous random field and find the conditions under which it is isotropic.

4.2 Homogeneous Random Fields There exists no complete description of the correlation structure of a homogeneous random field taking values in a real finite-dimensional linear space. Instead, we have the following partial result. Let d1 , d2 , …, ddim VG be the basis of the space VG constructed in Chap. 3. Theorem 4.1 ([7]) A cVG -valued random field { d( x ) : x ∈ R3 } is homogeneous if and only if its two-point correlation tensor has the form  d( x ), d(y ) pq =

  . ei(k,y −x ) dF pq (k)

(4.1)

ˆ3 R

If the above random field takes values in VG , then F pq (−A) = Fq p (A),

ˆ 3) , A ∈ B(R

(4.2)

where −A = { −k : k ∈ A }. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Malyarenko et al., Random Fields of Piezoelectricity and Piezomagnetism, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-3-030-60064-8_4

41

42

4 Correlation Structures

ˆ 3 is the wavenumber domain, and F pq is a measure on the Borel In Theorem 4.1, R ˆ 3 ) taking values in the set of Hermitian nonnegative-definite matrices σ -field B(R with dim VG rows. Thus, if one applies condition (4.2), no VG -valued random fields are missing. Some of the remaining fields may still be cVG -valued and each case must be analysed separately.

4.3 Conditions for Isotropy Let d0 be the constant one-point correlation tensor of a homogeneous random field d( x ): d( x ) = d0 . Substitute this value into the first equation in (2.2). We obtain: a tensor d0 ∈ P is a one-point correlation tensor of a homogeneous and isotropic random field if and only if d0 = θ (g)d0 for all g ∈ K . In other words, if the representation θ does not contain any copies of the trivial representation θ 0 of the group K , then d( x ) = 0. Otherwise, d( x ) is an arbitrary tensor lying in the subspace V0G . Example 4.1 Consider the random fields for which we constructed a basis in the space VG in Chap. 3. When G = Z 1 and K = O(3), we have θ = 2θ 1 ⊕ θ 2 ⊕ θ 3 . No trivial representations are here. It follows that d( x ) = 0. The same is true when G = D2 and K ∈ {D2 × Z 2c , D4 × Z 2c , T × Z 2c , O, O × Z 2c }. When K = D2 , we have θ = 3A and d( x ) = C1 d1 + C2 d2 + C3 d3 ,

Ci ∈ R ,

where di are the tensors of the basis (3.6). When K ∈ {D4 , T , O− }, the representation θ contains one copy of the trivial representation of K , and we have d( x ) = C1 d1 ,

C1 ∈ R ,

where d1 is the tensor of Eq. (3.7) when K = D4 , and the tensor of Eq. (3.8) otherwise. Finally, when K = D4h , the representation θ contains two copies of the trivial representation of K , and we have d( x ) = C1 d1 + C2 d2 ,

Ci ∈ R ,

where di are the tensors of Eq. (3.7). To investigate the two-point correlation tensor, we substitute (4.1) into the second equation in (2.2). After simple algebraic manipulations, we obtain the following result: Eq. (4.1) describes the two-point correlation tensor of a cVG -valued homogeneous and isotropic random field if and only if

4.3 Conditions for Isotropy

43

F(g A) = (θ ⊗ θ )(g)F(A),

g ∈ K,

ˆ 3) , A ∈ B(R

(4.3)

for all g ∈ K . For details, see [20]. We have two conditions for the measure F = F pq : (4.2) and (4.3). Is it possible to replace them with one condition? The following lemma gives an affirmative answer to this question. To formulate the lemma, we need more notation. Consider the representation cVG ⊗ VG . By [1], it is a complex representation of real type. The +1-eigenspace of j is the real linear space of Hermitian operators in cVG and the restriction of the representation cVG ⊗ VG to this space is a copy of VG ⊗ VG . Lemma 4.1 There exists a group K˜ and its real orthogonal representation V˜ ⊆ VG ⊗ VG such that conditions (4.2) and (4.3) are equivalent to the following conditions. • The measure F takes values in the intersection of the set of all nonnegative-definite ˜ Hermitian linear operators in VG ⊗ VG with the space V. • ˆ 3) . (4.4) F(g˜ A) = g˜ F(A), g˜ ∈ K˜ , A ∈ B(R Proof This lemma was proved in [20]. Here we give a simplified proof. Let K be a closed subgroup of the group O(d), and let Z = {E, −E}, where E is the d × d identity matrix. Let VG ⊗ VGZ be the same set as VG ⊗ VG , but suppose the group Z acts trivially on the subspace S2 (VG ) of symmetric linear operators: g A = A for all g ∈ Z , and nontrivially on the subspace 2 (VG ) of skew-symmetric linear operators: −E A = −A. We observe that condition (4.2) takes the form F(g A) = g F(A),

g ∈ Z,

ˆ 3) . A ∈ B(R

(4.5)

− Assume −E ∈ K . Let VG ⊗ V+ G (resp. VG ⊗ VG ) be the subspace of the space VG ⊗ VG where −E acts trivially (resp. nontrivially). Put K˜ = K and − 2 2 V˜ = (VG ⊗ V+ G ∩ S (VG )) ⊕ (VG ⊗ VG ∩  (VG )) .

Indeed, this is the maximal subspace where the actions (4.3) and (4.5) coincide. Assume −E ∈ / K . Put K˜ = K ∪ { −Eg : g ∈ K }. It is easy to check that K˜ is a closed subgroup of O(d). Let V˜ be the same set as VG ⊗ VG , but define the action of the group K˜ by ⎧ ⎪ if g˜ = g ∈ K , ⎨g A, g˜ A = −E g˜ A, if g˜ ∈ / K , A ∈ S2 (VG ) , ⎪ ⎩ −g A, if g˜ ∈ / K , A ∈ 2 (VG ) .

44

4 Correlation Structures

It is easy to check that this formula defines an orthogonal representation of the group K˜ whose restriction to K is equal to VG ⊗ VG , while the restriction to Z is equal to VG ⊗ VGZ . Example 4.2 The groups O(3), D2 × Z 2c , D4 × Z 2c , T × Z 2c , and O × Z 2c contain −E. Moreover, θ (−E) = E, and V˜G = S2 (VG ). When K = O(3), we find V˜ = 5H0 ⊕ 8H1 ⊕ 10H2 ⊕ 7H3 ⊕ 5H4 ⊕ 2H5 ⊕ H6 .

(4.6)

When K = D2 × Z 2c , [2, Table 31.8] gives θ˜ = S2 (θ ) = S2 (3Au ) = 6A g .

(4.7)

When K = D4 × Z 2c , [2, Table 33.8] gives θ˜ = S2 (θ ) = S2 (A1u ⊕ B1u ⊕ B2u ) = 3A1g ⊕ A2g ⊕ B1g ⊕ B2g .

(4.8)

When K = T × Z 2c , [2, Table 72.8] gives θ˜ = S2 (θ ) = S2 (Au ⊕ E u ) = 2 A g ⊕ 2E g .

(4.9)

When K = O × Z 2c , [2, Table 71.8] gives θ˜ = S2 (θ ) = S2 (A2u ⊕ E u ) = 2 A1g ⊕ 2E g .

(4.10)

The groups D2 , D4 , T , and O are subgroups of the group SO(3). We have K˜ = K × Z c2 , where Z 2c = {E, −E}. Moreover, θ˜ = S2 (θ ) ⊗ A g ⊕ 2 (θ ) ⊗ Au , where A g (resp. Au ) is the trivial (resp. nontrivial) irreducible representation of Z 2c . When G = D2 , we obtain S2 (θ ) = 6A, 2 (θ ) = 3A, and θ˜ = 6A ⊗ A g ⊕ 3A ⊗ Au = 6A g ⊕ 3Au .

(4.11)

When G = D4 , [2, Table 24.8] gives S2 (θ ) = 4 A1 ⊕ 2B1 , 2 (θ ) = A1 ⊕ 2B1 , and θ˜ = (4 A1 ⊕ 2B1 ) ⊗ A g ⊕ (A1 ⊕ 2B1 ) ⊗ Au = 4 A1g ⊕ 2B1g ⊕ A1u ⊕ 2B1u . (4.12) When G = T , [2, Table 70.8] gives S2 (θ ) = 2 A ⊕ 2E, 2 (θ ) = A ⊕ E, and θ˜ = (2 A ⊕ 2E) ⊗ A g ⊕ (A ⊕ E) ⊗ Au = 2 A g ⊕ 2E g ⊕ Au ⊕ E u .

(4.13)

4.3 Conditions for Isotropy

45

When G = O, [2, Table 71.8] gives S2 (θ ) = 2 A1 ⊕ 2E, 2 (θ ) = A2 ⊕ E, and θ˜ = (2 A1 ⊕ 2E) ⊗ A g ⊕ (A2 ⊕ E) ⊗ Au = 2 A1g ⊕ 2E g ⊕ A2u ⊕ E u .

(4.14)

Finally, the groups D4h and O− are not subgroups of the group SO(3) and do not contain −E. Put K = D4h . We have K˜ = D4 × Z 2c . By [2, Table 41.8], we have S2 (θ ) = 4 A1 ⊕ 2B1 , 2 (θ ) = A1 ⊕ 2B1 . By [2, Table 41.4], under the isomorphism π : D4h → D4 given by  π(g) =

g, if g ∈ D4h ∩ D 4 , −g, otherwise

the representation A1 becomes A2 , and B1 becomes B2 . Then θ˜ = (4 A1 ⊕ 2B1 ) ⊗ A g ⊕ (A2 ⊕ 2B2 ) ⊗ Au = 4 A1g ⊕ 2B1g ⊕ A2u ⊕ 2B2u . (4.15) Put K = O− . We have K˜ = O × Z 2c . By [2, Table 69.8], we have S2 (θ ) = A1 ⊕ A2 ⊕ 2E, 2 (θ ) = A2 ⊕ E. By [2, Table 73.4], under the isomorphism π : O− → O given by  g, if g ∈ O− ∩ O , π(g) = −g, otherwise the representation A2 becomes A1 , and E does not change. Then θ˜ = (2 A1 ⊕ 2E) ⊗ A g ⊕ (A1 ⊕ E) ⊗ Au = 2 A1g ⊕ 2E g ⊕ A1u ⊕ E u .

(4.16)

Consider the measure ˆ 3) . A ∈ B(R

ν(A) = F pp (A),

It is well known that the measure F is absolutely continuous with respect to ν, that is, ˆ 3 taking values in the set of nonnegative-definite there exists a function f defined on R Hermitian matrices on VG with unit trace such that   dν(k)  , F pq (A) = f pq (k) A

see [5]. Equation (4.1) becomes  d( x ), d(y ) pq = ˆ3 R



 dν(k)  , ei(k,y −x ) f pq (k)

(4.17)

46

4 Correlation Structures

while Eq. (4.4) becomes

ν(g˜ A) = ν(A) ,  = g˜ f (k)  f (g˜ k)

(4.18)

ˆ 3 ), and k ∈ R ˆ 3. for all g˜ ∈ K˜ , A ∈ B(R

4.4 The Structure of the Orbit Space In this section, we analyse the first condition in (4.18). The description of all measures satisfying this condition is well known, see [6]. ˆ 3 by matrix-vector multiplication, the wavenumber Under the action of K˜ on R ˆ3 → ˆ 3 / K˜ be the set of all orbits, and let π : R domain is divided into orbits. Let R 3 ˜ 3 3 ˜ ˆ ˆ ˆ  R / K be the mapping that maps a point k ∈ R to its orbit. A set A ⊆ R / K is open if ˆ 3 . In what follows, for each group K˜ considered above and only if π −1 (A) is open in R 3 ˆ we construct a subset of R such that it intersects with each orbit at exactly one point, and the mapping that maps a point to the corresponding orbit is a homeomorphism. ˆ 3 / K˜ . Denote this subset by the same symbol R 3 ˜ ˆ It turns out that the set R / K is a finite stratified space. This means that the ˆ 3 / K˜ )m : 0 ≤ m ≤ M − 1 }. Each stratum above set is a union of disjoint strata { (R is a manifold, and the boundary of any stratum is a union of some other strata of a ˆ 3 / K˜ )m .  m a chart of the manifold (R smaller dimension. Denote by λ 3 ˜ ˆ  m ), and  The orbit of a point λm ∈ (R / K )m is also a manifold. Denote it by O(λ let σm be a chart on this orbit. It is well known that there exists a unique probabilistic  m ), denote it by dσm . A measure ν satisfies the first K˜ -invariant measure on O(λ condition in (4.18) if and only if there exist finite measures μm on Borel σ -fields ˆ 3 / K˜ )m ) is the product of the ˆ 3 / K˜ )m ) such that the restriction of ν to π −1 ((R B((R measures μm and dσm . Equation (4.17) becomes d( x ), d(y ) pq =

M−1  m=0





  dσm dμm (λ m ) . ei(k,y −x ) f pq (k)

(4.19)

m ) ˆ 3 / K˜ )m O(λ (R

ˆ 3 / K˜ is a union of two strata Example 4.3 Put K˜ = O(3). The stratified space R ˆ 3 /O(3))0 = {0},  (R

ˆ 3 /O(3))1 = { (0, 0, k3 ) : k3 > 0 } . (R

ˆ 3 / K˜ )m form a group K m ,  m ∈ (R The elements of the group O(3) that fix a point λ  m . In fact, K m depends only on m but not the stationary subgroup of the point λ  m . We have K 0 = O(3) and K 1 = O(2). Moreover, for any group K˜ we have on λ ˆ 3 / K˜ )0 = {0}  and K 0 = K˜ . (R ˆ 3 /D2 × Z c , we use To determine the structure of the stratified space R 2 [2, Table 31.1], where the Euler angles of each matrix of the group D2 are given.

4.4 The Structure of the Orbit Space

47

We calculate the matrices of all elements of the group D2 . Multiplying them by −1, we obtain the matrices for all remaining elements. It turns out that the group D2 × Z 2c contains 8 diagonal matrices, where each nonzero matrix entry is equal to either 1 or −1. ˆ 3 is an 8-point set if and only if It is easy to see that the orbit of a point k ∈ R k1 = 0, k2 = 0, k3 = 0. We choose ˆ 3 : k1 > 0, k2 > 0, k3 > 0 } . ˆ 3 /D2 × Z 2c )7 = { (k1 , k2 , k3 ) ∈ R (R We have K 7 = Z 1 . ˆ 3 is a 4-point set if and only if exactly one of the numbers The orbit of a point k ∈ R ˆ 3 /D2 × Z c )m , 4 ≤ m ≤ 6, must be subsets k1 , k2 and k3 is equal to 0. The strata (R 2 ˆ 3 /D2 × Z c )7 . The unique possible choice up to of the boundary of the stratum (R 2 enumeration is ˆ 3 /D2 × Z 2c )4 = { (k1 , k2 , 0) ∈ R ˆ 3 : k1 > 0, k2 > 0 } , (R ˆ 3 /D2 × Z 2c )5 = { (k1 , 0, k3 ) ∈ R ˆ 3 : k1 > 0, k3 > 0 } , (R ˆ 3 : k2 > 0, k3 > 0 } . ˆ 3 /D2 × Z 2c )6 = { (0, k2 , k3 ) ∈ R (R The stationary subgroups are K 4 = Z 2− (σz ), K 5 = Z 2− (σ y ) and K 6 = Z 2− (σx ). This notation means: K 4 is the unique subgroup of the group K˜ that is isomorphic to Z 2− and contains the matrix σz ∈ K˜ . ˆ 3 is a 2-point set if and only if exactly two of the numbers The orbit of a point k ∈ R k1 , k2 and k3 are equal to 0. By the same reasoning as before, the unique possible choice up to enumeration is ˆ 3 /D2 × Z 2c )1 = { (k1 , 0, 0) ∈ R ˆ 3 : k1 > 0 } , (R ˆ 3 : k2 > 0 } , ˆ 3 /D2 × Z 2c )2 = { (0, k2 , 0) ∈ R (R ˆ 3 : k3 > 0 } . ˆ 3 /D2 × Z 2c )3 = { (0, 0, k3 ) ∈ R (R The stationary subgroups are K 1 = D2v (C2x ), K 2 = D2v (C2y ) and K 3 = D2v (C2z ). Put K˜ = D4 × Z 2c . Using [2, Table 33.1], we calculate the matrices of all elements of this group. It turns out that besides the matrices of the subgroup D2 × Z 2c listed

0 10

before, the remaining 8 matrices are as follows:C4+ = −1 0 0 , S4− = −C4+ , C4− = 0 01 0 −1 0

0 −1 0

0 −1 0 + −    1 0 0 , S4 = −C 4 , C 21 = −1 0 0 , σd1 = −C 21 , C 22 = 1 0 0 , and σd2 = 0 0 1

0

0 −1

0 0 −1

 . −C22 ˆ 3 is a 16-point set if and only if k1 = 0, k2 = 0, k3 = 0 The orbit of a point k ∈ R and k1 = k2 . We choose

ˆ 3 : k1 > 0, k2 > 0, k3 > 0, k1 < k2 } . ˆ 3 /D4 × Z 2c )7 = { (k1 , k2 , k3 ) ∈ R (R We have K 7 = Z 1 .

48

4 Correlation Structures

ˆ 3 /D4 × Z c )7 . Consider the following manifold lying in the boundary of (R 2 ˆ 3 : 0 < k1 < k2 } . ˆ 3 /D4 × Z 2c )6 = { (k1 , k2 , 0) ∈ R (R The stationary subgroup of this stratum is K 6 = Z 2− (σh ). ˆ 3 /D4 × Z c )7 is The next two-dimensional stratum lying in the boundary of (R 2 ˆ 3 : k2 > 0, k3 > 0 } . ˆ 3 /D4 × Z 2c )5 = { (0, k2 , k3 ) ∈ R (R The stationary subgroup of this stratum is K 5 = Z 2− (σv1 ). ˆ 3 /D4 × Z c )7 is The last two-dimensional stratum lying in the boundary of (R 2 ˆ 3 : k1 = k2 > 0, k3 > 0 } . ˆ 3 /D4 × Z 2c )4 = { (k1 , k2 , k3 ) ∈ R (R The stationary subgroup of this stratum is K 4 = Z 2− (σd1 ). ˆ 3 /D4 × Z c )7 , The one-dimensional stratum lying in the common boundary of (R 2 ˆ 3 /D4 × Z c )5 is ˆ 3 /D4 × Z c )6 and (R (R 2 2 ˆ 3 : k2 > 0 } . ˆ 3 /D4 × Z 2c )3 = { (0, k2 , 0) ∈ R (R  , σv1 , σh ). The stationary subgroup is K 3 = D2v (C22 ˆ 3 /D4 × Z c )7 , The one-dimensional stratum lying in the common boundary of (R 2 c c 3 3 ˆ ˆ (R /D4 × Z 2 )6 and (R /D4 × Z 2 )4 is

ˆ 3 : k1 = k2 > 0 } . ˆ 3 /D4 × Z 2c )2 = { (k1 , k2 , 0) ∈ R (R  , σd1 , σh ). The stationary subgroup is K 2 = D2v (C22 Finally, the one-dimensional stratum lying in the common boundary of ˆ 3 /D4 × Z c )5 and (R ˆ 3 /D4 × Z c )4 is ˆ 3 /D4 × Z c )7 , (R (R 2 2 2

ˆ 3 : k3 > 0 } . ˆ 3 /D4 × Z 2c )1 = { (0, 0, k3 ) ∈ R (R The stationary subgroup is K 1 = D4v . Put K˜ = T × Z 2c . Using [2, Table 71.1], we calculate the matrices of all elements of this group. It turns out that besides the matrices of the group D2 × Z 2c 0 0 1 + calculated above, the remaining 16 matrices are as follows: C31 = −1 0 0 , 0 −1 0 0 0 −1

0 0 −1

− + + − + + − + + S61 = −C31 , C32 = −1 0 0 , S62 = −C32 , C33 = 1 0 0 , S63 = −C33 , C34 = 0 −1 0 0 1 0



0 0 1

0 −1 0 0 −1 0 − + − + − − + − 1 0 0 , S = −C 34 , C 31 = 0 0 −1 , S61 = −C 31 , C 32 = 0 0 1 , S62 = −C 32 , 0 1 0 64 −1 0 0 1 0 0



0 1 0 010 − + − − + − C33 = 0 0 −1 , S63 = −C33 , C34 = 0 0 1 , S64 = −C34 . −1 0 0

100

4.4 The Structure of the Orbit Space

49

Consider the stratum ˆ 3 : 0 < k3 < min{k1 , k2 } } . ˆ 3 /T × Z 2c )5 = { (k1 , k2 , k3 ) ∈ R (R The union of the orbits of the points of this set under the action of the group T × Z 2c ˆ 3 such that k1 = 0, k2 = 0, k3 = 0, and two of is the set of points (k1 , k2 , k3 ) ∈ R the three numbers |k1 |, |k2 | and |k3 | are greater than the remaining one. The above ˆ 3 . The orbit of a point union contains 24 connected components and is dense in R ˆ 3 /T × Z c )5 is a 24-point set, in particular, its stationary subgroup is K 5 = Z 1 . k ∈ (R 2 As we will see, there are more points with a 24-point orbit! ˆ 3 /T × Z c )5 : Indeed, consider the following stratum in the boundary of the set (R 2 ˆ 3 : 0 < k3 = k2 < k1 } . ˆ 3 /T × Z 2c )4 = { (k1 , k2 , k3 ) ∈ R (R + The stationary subgroup of this stratum is K 4 = Z 1 . In particular, the matrix C34  3 ˆ : 0 < k3 = k1 < k2 }, which is another part of maps it to the set { (k1 , k2 , k3 ) ∈ R ˆ 3 /T × Z c is not closed ˆ 3 /T × Z c )5 , thus the subset R the boundary of the stratum (R 2 2 ˆ 3! in R The points of the stratum

ˆ 3 : k1 > 0, k2 > 0 } ˆ 3 /T × Z 2c )3 = { (k1 , k2 , 0) ∈ R (R have 12-point orbits, and the stationary subgroup is K 3 = Z 2− (σz ). The points of the one-dimensional stratum ˆ 3 /T × Z 2c )2 = { (k1 , k2 , k3 ) ∈ R ˆ 3 : 0 < k1 = k2 = k3 } (R have 8-point orbits. The stationary subgroup is K 2 = Z 3 . Finally, the points of the stratum ˆ 3 /T × Z 2c )1 = { (k1 , 0, 0) ∈ R ˆ 3 : k1 > 0 } (R − − + + , S62 , C33 , or C34 , map have 6-point orbits. In particular, any of the matrices S61 c 3  3 ˆ /T × Z )1 to the set { (0, k2 , 0) ∈ R ˆ : k2 > 0 }, another part of the the stratum (R 2 ˆ 3 /T × Z c )5 . The stationary subgroup is K 1 = D v . boundary of the stratum (R 2 2 c ˜ Put K = O × Z 2 . Using [2, Table 71.1], we calculate the matrices of all elements of this group. It turns out that the 48 matrices of the group O × Z 2c have exactly one nonzero matrix entry in each row and in each column, and the nonzero entries may take values ±1. ˆ 3 is a 48-point set if and only if It is easy to check that the orbit of a point k ∈ R the set {|k1 |, |k2 |, |k3 |} contains 3 nonzero elements. We choose

ˆ 3 /O × Z 2c )7 = { (k1 , k2 , k3 ) ∈ R ˆ 3 : 0 < k1 < k2 < k3 } . (R The stationary subgroup of this stratum is K 7 = Z 1 .

50

4 Correlation Structures

Table 4.1 The stationary subgroups of the strata K˜ K1 K2 K3 K4 K5 K6 K7

D2 × Z 2c

D4 × Z 2c

T × Z 2c

O × Z 2c

D2v (C2x ) D2v (C2y ) D2v (C2z ) Z 2− (σz ) Z 2− (σ y ) Z 2− (σx )

D4v  , σ , σ ) D2v (C22 d1 h v  ,σ ,σ ) D2 (C22 v1 h Z 2− (σd1 ) Z 2− (σv1 ) Z 2− (σh )

D2v

Dv4 Dv3 D2v (C2 f , σd4 , σx )

Z1

Z1

Z3 Z 2− (σz ) Z1 Z1 – –

Z 2− (σd4 ) Z 2− (σd1 ) Z 2− (σx ) Z1

The three two-dimensional strata that are subsets of the boundary of the stratum ˆ 3 /O × Z c )7 are as follows: (R 2 ˆ 3 : 0 = k1 < k2 < k3 } , ˆ 3 /O × Z 2c )6 = { (k1 , k2 , k3 ) ∈ R (R ˆ 3 : 0 < k1 = k2 < k3 } , ˆ 3 /O × Z 2c )5 = { (k1 , k2 , k3 ) ∈ R (R ˆ 3 /O × Z 2c )4 = { (k1 , k2 , k3 ) ∈ R ˆ 3 : 0 < k1 < k2 = k3 } . (R Their stationary subgroups are K 6 = Z 2− (σx ), K 5 = Z 2− (σd1 ) and K 4 = Z 2− (σd4 ). The three one-dimensional strata that are subsets of the boundary of the stratum ˆ 3 /O × Z c )7 are as follows: (R 2 ˆ 3 : 0 = k1 < k2 = k3 } , ˆ 3 /O × Z 2c )3 = { (k1 , k2 , k3 ) ∈ R (R ˆ 3 : 0 < k1 = k2 = k3 } , ˆ 3 /O × Z 2c )2 = { (k1 , k2 , k3 ) ∈ R (R ˆ 3 : 0 = k1 = k2 < k3 } . ˆ 3 /O × Z 2c )1 = { (k1 , k2 , k3 ) ∈ R (R Their stationary subgroups are K 3 = D2v (C2 f , σd4 , σx ), K 2 = D3v , and K 1 = D4v . We summarise our findings in Table 4.1.

4.5 Invariant Subspaces In this section, we begin to analyse the second condition in (4.18). ˆ 3 / K˜ )m . By definition  to the stratum (R Consider the restriction of the function f (k) 3 ˜ ˆ  of the stationary subgroup, for a point λm ∈ (R / K )m we have m , m = λ g˜ λ

g˜ ∈ K m .

4.5 Invariant Subspaces

51

Substitute this equation into the second condition in (4.18). We obtain m ,  m ) = g˜ λ f (λ

g˜ ∈ K m .

(4.20)

Denote by θ˜m the restriction of the representation θ˜ to the subgroup K m . Equa m ) belongs to the invariant subspace V˜ m , where the repretion (4.20) means that f (λ sentation θ˜m acts trivially. In the following example, we find the basis tensors of the spaces V˜ m . Example 4.4 Let G = Z 1 , K = O(3). Put m = 0. Example 4.3 gives K 0 = O(3). By Eq. (4.6), dim V˜ 0 = 5. By definition of the Godunov–Gordienko coefficients, we obtain the basis of the space V˜ 0 in the form 1 f1 = √ (d1 ⊗ d1 + d2 ⊗ d2 + d3 ⊗ d3 ) , 3 1 f2 = √ (d4 ⊗ d4 + d5 ⊗ d5 + d6 ⊗ d6 ) , 3 1 (4.21) f3 = √ (d7 ⊗ d7 + · · · + d11 ⊗ d11 ) , 5 1 f4 = √ (d12 ⊗ d12 + · · · + d18 ⊗ d18 ) , 7 1 1 5 f = √ (d ⊗ d4 + d4 ⊗ d1 + d2 ⊗ d5 + d5 ⊗ d2 + d3 ⊗ d6 + d6 ⊗ d3 ) , 6 where d1 , …, d18 are the basis tensors (3.2), (3.3), (3.4), and (3.5). Put m = 1. Example 4.3 gives K 1 = O(2). It is known that the restriction of the representation θ  of the group O(3) to the subgroup O(2) contains a copy of the trivial representation of O(2) if and only if  is even. Equation (4.6) gives dim V˜ 1 = 5 + 10 + 5 + 1 = 21 . In addition to the tensors (4.21), we have, for example: f6 =

√ 2 1 1 0[i, j] g2[1,1] di+2 ⊗ d j+2 = − √ d1 ⊗ d1 + √ d2 ⊗ d2 − √ d3 ⊗ d3 , 6 3 6 i, j=−1 1 

and so on. Calculation of the tensors f7 , …, f21 may be left to the reader. See also [20]. In the remaining part of this example, G = D2 , and we use Table 4.1. Let K = D2 . Equation (4.11) gives θ˜ = 6A g ⊕ 3Au . If m = 0, we have K 0 = D2 × Z 2c . The invariant subspace V˜ 0 is the linear space of all symmetric 3 × 3 matrices with real matrix entries. We choose the following basis of this space:

52

4 Correlation Structures

f Ag ,i1 = di ⊗ di , 1 ≤ i ≤ 3, 1 f Ag ,41 = √ (d2 ⊗ d3 + d3 ⊗ d2 ), 2 1 1 A g ,51 = √ (d ⊗ d3 + d3 ⊗ d1 ), f 2 1 f Ag ,61 = √ (d1 ⊗ d2 + d2 ⊗ d1 ) , 2

(4.22a) (4.22b) (4.22c) (4.22d)

where di are the tensors of the basis (3.6). The same situation occurs when the subgroup K m is not a subgroup of the group SO(3), that is, for K 1 –K 6 . We have V˜ m = V˜ 0 for 1 ≤ m ≤ 6. On the other hand, K 7 = Z 1 , and V˜ 7 is the linear space of all Hermitian 3 × 3 matrices. In addition to the basis tensors of the subspace V˜ 0 described above, three new basis tensors appear: i f Au ,11 = √ (d2 ⊗ d3 − d3 ⊗ d2 ), 2 i f Au ,21 = √ (d1 ⊗ d3 − d3 ⊗ d1 ), 2 i f Au ,31 = √ (d1 ⊗ d2 − d2 ⊗ d1 ). 2

(4.23a) (4.23b) (4.23c)

Put K = D2 × Z 2c . Equation (4.7) gives θ˜ = 6A g . For any m, 0 ≤ m ≤ 7, the restriction of the trivial representation 6A g to the subgroup K m is itself trivial. The invariant subspace V˜ m is the linear space of all symmetric 3 × 3 matrices with real matrix entries, and we choose the basis tensors constructed above. Put K = D4 . Equation (4.12) gives θ˜ = 4 A1g ⊕ 2B1g ⊕ A1u ⊕ 2B1u . The 4 copies of the trivial representation A1g act on the one-dimensional spaces generated by the tensors (4.22a) and (4.22b), where di are the tensors of the basis (3.7). The two copies of the representation B1g act on the one-dimensional spaces generated by the tensors (4.22c) and (4.22d). The representation A1u acts on the one-dimensional space generated by the tensor (4.23a). The two copies of the representation B1u act on the one-dimensional spaces generated by the tensors (4.23b) and (4.23c). The restriction of the representation θ˜ to the subgroup K 0 = D4 × Z 2c contains 4 copies of the trivial representation A1g . Then, V˜ 0 is the 4-dimensional space generated by the tensors (4.22a) and (4.22b). By [2, Table 33.9], the same holds true for K 1 =  , σd1 , σh ), K 5 = Z 2− (σv1 ) and K 6 = Z 2− (σh ). We have V˜ m = V˜ 0 , D4v , K 2 = D2v (C22 m = 1, 2, 5, 6.  , σv1 , σh ) becomes trivial for The restriction of θ˜ to the subgroup K 3 = D2v (C22 ˜ the components 4 A1g and 2B1g . The space V3 is the 6-dimensional space generated by the tensors (4.22).

4.5 Invariant Subspaces

53

The restriction of θ˜ to the subgroup K 4 = Z 2− (σd1 ) becomes trivial for the components 4 A1g and 2B1u . The space V˜ 4 is the 6-dimensional space generated by the tensors (4.22a), (4.22b), (4.23b), and (4.23c). Finally, the restriction of θ˜ to the subgroup K 7 = Z 1 is trivial. The 9-dimensional space V˜ 7 is generated by the tensors (4.22) and (4.23). Put K = D4h . Equation (4.15) gives θ˜ = 4 A1g ⊕ 2B1g ⊕ A2u ⊕ 2B2u . Similarly to the case of K = D4 , the 4 copies of the trivial representation A1g act on the onedimensional spaces generated by the tensors (4.22a) and (4.22b), where di are the tensors of the basis (3.7). The two copies of the representation B1g act on the onedimensional spaces generated by the tensors (4.22c) and (4.22d). The representation A2u acts on the one-dimensional space generated by the tensor (4.23a). The two copies of the representation B2u act on the one-dimensional spaces generated by the tensors (4.23b) and (4.23c). We have: V˜ 0 is the 4-dimensional space generated by the tensors (4.22a) and (4.22b) by the same reasoning as before. By [2, Table 33.9], the same holds true for  , σd1 , σh ). We have V˜ 2 = V˜ 0 . K 2 = D2v (C22 The restriction of θ˜ to the subgroups K 1 = D4v and K 4 = Z 2− (σd1 ) becomes trivial for the components 4 A1g and A2u . The spaces V˜ 1 and V˜ 4 are 5-dimensional. They are generated by the tensors (4.22a), (4.22b) and (4.23a).  , σv1 , σh ) and K 6 = Z 2− (σh ) The restriction of θ˜ to the subgroups K 3 = D2v (C22 becomes trivial for the components 4 A1g and 2B1g . The spaces V˜ 3 and V˜ 6 are 6dimensional. They are generated by the tensors (4.22). Finally, the restriction of θ˜ to the subgroups K 5 = Z 2− (σv1 ) and K 7 = Z 1 is trivial. The spaces V˜ 5 and V˜ 7 are 9-dimensional. They are generated by the tensors (4.22) and (4.23). Put K = D4 × Z 2c . Equation (4.8) gives θ˜ = 3A1g ⊕ A2g ⊕ B1g ⊕ B2g . The 3 copies of the trivial representation A1g act on the one-dimensional spaces generated by the tensors (4.22a), where di are the tensors of the basis (3.7). The representation A2g acts on the one-dimensional space generated by the tensor (4.22b). The representation B2g acts on the one-dimensional space generated by the tensor (4.22c). The representation B1g acts on the one-dimensional space generated by the tensor (4.22d). V˜ 0 is the 3-dimensional space generated by the tensors (4.22a) by the same reasoning as before. By [2, Table 33.9], the same holds true for K 1 = D4v . We have V˜ 1 = V˜ 0 .  , σd1 , σh ) and K 4 = Z 2− (σd1 ) The restriction of θ˜ to the subgroups K 2 = D2v (C22 becomes trivial for the components 3A1g and B2g . The spaces V˜ 2 and V˜ 4 are 4dimensional. They are generated by the tensors (4.22a) and (4.22c).  , σv1 , σh ) and K 5 = Z 2− (σv1 ) The restriction of θ˜ to the subgroups K 3 = D2v (C22 becomes trivial for the components 3 A1g and B1g . The spaces V˜ 3 and V˜ 5 are 4dimensional. They are generated by the tensors (4.22a) and (4.22d). Finally, the restriction of θ˜ to the subgroups K 6 = Z 2− (σh ) and K 7 = Z 1 is trivial. The spaces V˜ 6 and V˜ 7 are 6-dimensional. They are generated by the tensors (4.22).

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4 Correlation Structures

Put K = T . Equation (4.13) gives θ˜ = 2 A g ⊕ 2E g ⊕ Au ⊕ E u . The 2 copies of the trivial representation A g act on the one-dimensional spaces generated by the tensors 1 f Ag ,21 = √ (d2 ⊗ d2 + d3 ⊗ d3 ) , (4.24) f Ag ,11 = d1 ⊗ d1 , 2 where di are the tensors given by (3.8). The first copy of the representation E g acts on the 2-dimensional space generated by the tensors 1 f E g ,11 = √ (−d2 ⊗ d2 + d3 ⊗ d3 ), 2

1 f E g ,12 = √ (d2 ⊗ d3 + d3 ⊗ d2 ) . 2 (4.25) The second copy of E g acts on the 2-dimensional space generated by the tensors 1 f E g ,21 = √ (d1 ⊗ d2 + d2 ⊗ d1 ), 2

1 f E g ,22 = √ (d1 ⊗ d3 + d3 ⊗ d1 ) . (4.26) 2

The representation Au acts on the one-dimensional space generated by the tensor i f Au ,11 = √ (d2 ⊗ d3 − d3 ⊗ d2 ) . 2

(4.27)

The representation E u acts on the 2-dimensional space generated by the tensors i f Eu ,11 = √ (d1 ⊗ d2 − d2 ⊗ d1 ), 2

i f Eu ,12 = √ (d1 ⊗ d3 − d3 ⊗ d1 ) . (4.28) 2

V˜ 0 is the 2-dimensional space generated by the tensors (4.24) by the same reasoning as before. By [2, Table 72.9], the restriction of θ˜ to the subgroups K 1 = D2v and K 3 = Z 2− (σz ) becomes trivial for the components 2 A g and 2E g . The spaces V˜ 1 and V˜ 3 are 6dimensional. They are generated by the tensors (4.24)–(4.26). The restriction of θ˜ to the subgroup K 2 = Z 3 becomes trivial for the components 2 A g and Au . The space V˜ 2 is 3-dimensional. It is generated by the tensors (4.24) and (4.27). The same space appears when K = O and m = 2. Finally, the restriction of θ˜ to the subgroups K 4 = K 5 = Z 1 is trivial. The spaces ˜ V4 and V˜ 5 are 9-dimensional. They are generated by the tensors (4.24)–(4.28). Put K = T × Z 2c . Equation (4.9) gives θ˜ = 2 A g ⊕ 2E g . V˜ 0 is the 2-dimensional space generated by the tensors (4.24) by the same reasoning as before. By [2, Table 72.9], the same holds true for K 2 = Z 3 . We have V˜ 2 = V˜ 0 . The spaces V˜ 1 and V˜ 3 are the same as in the case of K = T . The restriction of θ˜ to the subgroups K 4 = K 5 = Z 1 is trivial. The spaces V˜ 4 and ˜ V5 are 6-dimensional. They are generated by the tensors (4.24)–(4.26). Put K = O. Equation (4.14) gives θ˜ = 2 A1g ⊕ 2E g ⊕ A2u ⊕ E u . The 2 copies of the trivial representation A1g act on the one-dimensional spaces generated by the

4.5 Invariant Subspaces

55

tensors (4.24). The first copy of the representation E g acts on the 2-dimensional space generated by the tensors (4.25). The second copy of E g acts on the 2-dimensional space generated by the tensors (4.26). The representation A2u acts on the onedimensional space generated by the tensor (4.27). The representation E u acts on the 2-dimensional space generated by the tensors (4.28). The space V˜ 0 is the 2-dimensional space generated by the tensors (4.24), because the restriction of the trivial representation A1g to any subgroup is trivial. A new problem arises when we determine the basis of the space V˜ 1 . By [2, Table 71.9], the restriction of the representation A1g to the subgroup K 1 = D4v is trivial, and the above basis contains the tensors (4.24). The restrictions of the representations A2u and E u to K 1 do not have trivial components. The restriction of the representation E g to K 1 is equivalent to A1 ⊕ B1 . On which space does the trivial component A1 act? To solve this problem, we find the matrices of the representation E g for all 8 elements of the subgroup K 1 . A common eigenvector of all of them corresponding to the eigenvalue 1 generates the one-dimensional subspace on which the trivial component A1 acts. Unfortunately, [2, Table 69.7] contains the matrices of the unitary representation E g . To overcome this difficulty, we consider [2, Table 71.5]. According to this table, the restriction of the representation θ 1 of the group O(3) to the subgroup O × Z 2c is T1 . The matrix entries gi j of the orthogonal representation T1 in the standard basis { ei : 1 ≤ i ≤ 3 } have already been calculated (the 48 matrices mentioned before). By the above table, the tensor square T1 ⊗ T1 contains E g , and the basis in the space of this component is given by 3 

fi =

i[ j,k]

c E g [T1 ,T1 ] e j ⊗ ek ,

j,k=1 i[ j,k]

where the nonzero Clebsch–Gordan coefficients c E[T1 ,T1 ] are as follows: c1[1,1] E g [T1 ,T1 ] = 1[2,2] √1 , c 2 E g [T1 ,T1 ]

2[2,2] 2[3,3] √1 = − √12 , c2[1,1] E g [T1 ,T1 ] = c E g [T1 ,T1 ] = − 6 , and c E g [T1 ,T1 ] = the matrix entries of the representation E g become

(E g (g))i j =

3 

√ √2 . In this basis, 3

j[m,n]

ci[k,l] E g [T1 ,T1 ] (T1 (g))km (T1 (g))ln c E g [T1 ,T1 ] .

k,l,m,n=1

Calculations give the following result. The representation E g maps the matrices matrix, and the remaining elements of E, C2 , σv1 and σd2 to the 2 ×2 identity 0 . The trivial components A1 act on the onethe subgroup D4v to the matrix −1 0 1 dimensional spaces generated by the tensors f E g ,12 of (4.25) and f E g ,22 of (4.26). The space V˜ 1 is 4-dimensional. Consider the space V˜ 2 . By [2, Tables 71.9, 73.9], the restriction of the representations E g and E u to the subgroup K 2 = D3v is nontrivial, while the restriction of

56

4 Correlation Structures

the representations A1g and A2u to the same subgroup is trivial. The space V˜ 2 is 3-dimensional, and the basis is (4.24) and (4.27). Consider the space V˜ 3 . By [2, Table 71.4], the restriction of A2u to K 3 = D2v is nontrivial. The representation E g maps the elements E and σx of the subgroup D2v to the identity matrix, while the elements C2 f and σd4 are mapped to the matrix √

1 − 3 1 √ . The common eigenvector of the above matrices with unit norm corre2 − 3 −1 √ sponding to the eigenvalue 1 is 21 (1, − 3) . The basis tensors of the subspaces on which A1 acts are 

f3 =

1 E g ,11 √ E g ,12 − 3f ), (f 2



f4 =

1 E g ,21 √ E g ,22 − 3f ), (f 2

(4.29)

where f E g ,11 and f E g ,12 are given by (4.25), and f E g ,21 and f E g ,22 are given by (4.26). The representation E u maps the element E to the identity matrix, and the element σx to the negative identity matrix. None of the eigenvalues of E u (σx ) are equal to 1. The representation E u does not contain trivial components. The space V˜ 3 is 4-dimensional, and the basis is (4.24) and (4.29). Consider the space V˜ 4 . By [2, Table 71.4], the restriction of A2u to K 4 = Z 2− (σd4 ) is trivial. The matrix E g (σd4 ) has already been calculated, and the basis tensors are (4.29). We have E u (σd4 ) = −E g (σd4 ). No eigenvalues of this matrix are equal to 1. The space V˜ 4 is 5-dimensional, and the basis is (4.24), (4.27), and (4.29). Consider the space V˜ 5 . By [2, Table 71.4], the restriction of A2u to K 5 = Z 2− (σd1 ) is trivial. The matrix E g (σd1 ) has already been calculated. The trivial components A1 again act on the one-dimensional spaces generated by the tensors f E g ,12 of (4.25) and f E g ,22 of (4.26). We have E u (σd1 ) = −E g (σd1 ). The trivial component A1 acts on the space generated by the tensor f Eu ,11 of (4.28). The space V˜ 5 is 6-dimensional. Consider the space V˜ 6 . By [2, Table 71.4], the restriction of A2u to K 6 = Z 2− (σx ) is nontrivial. The matrix E g (σx ) has been calculated before and is equal to the identity matrix. The four copies of the trivial representation A1 act on the two 2-dimensional spaces with bases (4.25) and (4.26). We have E u (σx ) = −E g (σx ) and the restriction of E u to K 6 is nontrivial. The 6-dimensional space V˜ 6 is spanned by the tensors (4.24)–(4.26). Finally, the 9-dimensional space V˜ 6 is spanned by the tensors (4.24)–(4.28). Put K = O × Z 2c . Equation (4.10) gives θ˜ = 2 A1g ⊕ 2E g . The spaces V˜ 0 , …, V˜ 7 are intersections of the spaces of the previous case with the space of 3 × 3 symmetric matrices with real entries. That is, the tensors (4.24) constitute the basis of the spaces V˜ 0 and V˜ 2 , the tensors (4.24), f E g ,12 of (4.25), and f E g ,22 of (4.26) are the basis of V˜ 1 and V˜ 5 , the tensors (4.24) and (4.29) for V˜ 3 and V˜ 4 , the tensors (4.24)–(4.26) for V˜ 6 and V˜ 7 . Finally, put K = O− . Equation (4.16) gives θ˜ = 2 A1g ⊕ 2E g ⊕ A1u ⊕ E u . Again, all the spaces V˜ 0 , …, V˜ 7 contain the subspace V˜ 0 generated by the tensors (4.24). We only need to analyse the irreducible component A1u , because the rest of the components have already been analysed. By [2, Table 71.4], the restriction of A1u to

4.5 Invariant Subspaces

57

the groups K 1 –K 6 is nontrivial, while that to K 7 is trivial. To construct the spaces V˜ 1 –V˜ 6 , we take the spaces from the case of K = O with the same names, and delete the tensor (4.27), if it exists. The space V˜ 1 is spanned by the tensors (4.24), f E g ,12 of (4.25), and f E g ,22 of (4.26). V˜ 2 is spanned by (4.24), the spaces V˜ 3 and V˜ 4 by (4.24) and (4.29), the space V˜ 5 by (4.24), f E g ,12 of (4.25), f E g ,22 of (4.26), and f Eu ,11 of (4.28). The space V˜ 6 is generated by (4.24)–(4.26), and V˜ 7 by (4.24)–(4.28).

4.6 Convex Compacta  m ) in (4.20) is Hermitian, nonnegative-definite and has Recall that the matrix f (λ unit trace. The set of such matrices is convex and compact. The intersection of the above set with the linear space V˜ m is a convex compact set, call it Cm . The function ˆ 3 / K˜ )m to Cm .  m ) is an arbitrary measurable function acting from (R f (λ A point f ∈ Cm is called extreme if it does not lie in any open line segment joining two points of Cm . It is well known that any convex compact set in a real finite-dimensional space is the closed convex hull of the set of its own extreme points. In the following example, we describe each set Cm and give a geometrical description of its set of extreme points, if possible. Example 4.5 Let G = Z 1 , K = O(3). Put m = 0. The set C0 is the set of symmetric nonnegative-definite matrices with unit trace lying in the 5-dimensional space with basis (4.21). It turns out that C0 is the simplex with five vertices. For details, see [20]. When m = 1, the set C1 is the set of symmetric nonnegative-definite matrices with unit trace lying in some 21-dimensional space. The set of its extreme points has 3 connected components, see [20]. Until the end of Example 4.5 put G = D2 . If K = D2 and 0 ≤ m ≤ 6, then a point in Cm is a linear combination of the matrices (4.22), that is, ⎞ ν Ag ,11 √12 ν Ag ,61 √12 ν Ag ,51 ⎟ ⎜ = ⎝ √12 ν Ag ,61 ν Ag ,21 √12 ν Ag ,41 ⎠ , √1 ν A ,51 √1 ν A ,41 ν A ,31 g g g 2 2 ⎛

f=

6  i=1

ν Ag ,i1 f Ag ,i1

(4.30)

where the principal minors are nonnegative and the trace is equal to 1. The same convex compact set occurs when • K = D2 × Z 2c for all m; • K = D4 and m = 3, K = D4h and m ∈ {3, 6}, but ν Ag ,k1 are replaced with ν A1g ,k1 for 1 ≤ k ≤ 4, and with ν B1g ,k−4 1 for 5 ≤ k ≤ 6; • K = D4 × Z 2c and m ∈ {6, 7}, but ν Ag ,k1 are replaced with ν A1g ,k1 for 1 ≤ k ≤ 4, with ν A2g ,11 for k = 4, with ν B2g ,11 for k = 5, and with ν B1g ,11 for k = 6. The set C7 is even larger. It consists of the matrices

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4 Correlation Structures

⎛

ν Ag ,11 ⎜ √1 (ν f = ⎝ 2 Ag ,61 − iν Au ,31 ) √1 (ν A ,51 − iν A ,21 ) g u 2

√1 (ν A ,61 g 2

+ iν Au ,31 )

ν Ag ,21 1 √ (ν A ,41 − iν A ,11 ) g u 2

√1 (ν A ,51 g 2 √1 (ν A ,41 g 2

⎞ + iν Au ,21 ) + iν Au ,11 )⎟ ⎠

(4.31)

ν Ag ,31

with similar conditions. The same convex compact set occurs when • K = D4 and m = 7, but ν Ag ,k1 are replaced with ν A1g ,k1 for 1 ≤ k ≤ 4, and with ν B1g ,k−4 1 for 5 ≤ k ≤ 6, ν Au ,k1 is replaced with ν A1u ,k1 for k = 1, and with ν B1u ,k−1 1 for 2 ≤ k ≤ 3. • K = D4h and m ∈ {5, 7}, but ν Ag ,k1 are replaced with ν A1g ,k1 for 1 ≤ k ≤ 4, and with ν B1g ,k−4 1 for 5 ≤ k ≤ 6, ν Au ,k1 is replaced with ν A2u ,k1 for k = 1, and with ν B2u ,k−1 1 for 2 ≤ k ≤ 3. When K = D4 and m ∈ {0, 1, 2, 5, 6}, a point in Cm is a linear combination of the matrices (4.22a) and (4.22b), that is, ⎞ ⎛ ν A1g ,11 0 0 ⎜ ν A1g ,21 √12 ν A1g ,41 ⎟ f=⎝ 0 ⎠. 0 √12 ν A1g ,41 ν A1g ,31

(4.32)

The principal minors of this matrix are nonnegative and its trace is equal to 1 if and only if 1 −ν A1g ,21 + ν A1g ,11 ν A1g ,21 + ν 2A1g ,21 + ν 2A1g ,41 ≤ 0 . 2

ν A1g ,11 ≥ 0,

The set Cm is a closed cone. Its set of extreme points contains 2 components: the vertex with ν A1g ,11 = 1 and ν A1g ,21 = ν A1g ,31 = ν A1g ,41 = 0, and the circle with ν A1g ,11 = 0 and (2ν A1g ,21 − 1)2 + 2ν 2A1g ,41 = 1. The same convex compact set occurs when K = D4h and m ∈ {0, 2}. When K = D4 and m = 4, a point in C4 is a linear combination of the matrices (4.22a), (4.22b), (4.23b), and (4.23c), that is ⎞ √i ν A ,31 √i ν A ,21 ν A1g ,11 1u 1u 2 2 ⎟ ⎜ f = ⎝− √i2 iν A1u ,31 ν A1g ,21 √12 ν A1g ,41 ⎠ . − √i2 ν A1u ,21 √12 ν A1g ,41 ν A1g ,31 ⎛

(4.33)

This convex compact set is similar to the compact set (4.30), but its location in the 9-dimensional linear space of Hermitian 3 × 3 matrices is different. When K = D4h and m ∈ {1, 4}, a point in Cm is a linear combination of the matrices (4.22a), (4.22b), and (4.23a), that is ⎛

ν A1g ,11 ⎜ f=⎝ 0 0

⎞ 0 √1 (ν A ,41 + iν A ,11 )⎟ . ν A1g ,21 1g 2u ⎠ 2 1 √ (ν A ,41 − iν A ,11 ) ν A ,31 1g 2u 1g 2 0

(4.34)

4.6 Convex Compacta

59

The principal minors of this matrix are nonnegative and its trace is equal to 1 if and only if Cm is a cone. Its set of extreme points contains 2 components: the vertex with ν A1g ,11 = 1 and ν A1g ,i2 = ν A1g ,41 = ν A2u ,11 = 0, and the sphere with ν A1g ,11 = 0 and (2ν A1g ,21 − 1)2 + 2ν 2A1g ,41 + 2ν 2A2u ,11 = 1. When K = D4 × Z 2c and m ∈ {0, 1}, a point in Cm is a linear combination of the matrices (4.22a), that is, ⎞ ν A1g ,11 0 0 f = ⎝ 0 ν A1g ,21 0 ⎠ . 0 0 ν A1g ,31 ⎛

(4.35)

It is a triangle. When K = D4 × Z 2c and m ∈ {2, 4}, a point in Cm is a linear combination of the matrices (4.22a) and (4.22c), that is, ⎞ 0 √12 ν B2g ,11 ν A1g ,11 ⎟ ⎜ 0 ν A1g ,21 0 f=⎝ ⎠. 1 √ ν B ,11 0 ν A1g ,31 2g 2 ⎛

(4.36)

This convex compact set is similar to the compact set (4.32), but its location in the 9-dimensional linear space of Hermitian 3 × 3 matrices is different. Another incarnation of the convex compact set (4.32) occurs when K = D4 × Z 2c and m ∈ {3, 5}. A point in Cm is a linear combination of the matrices (4.22a) and (4.22d), that is, ⎛ ⎞ ν A1g ,11 √12 ν B1g ,11 0 ⎜ ⎟ f = ⎝ √12 ν B1g ,11 ν A1g ,21 (4.37) 0 ⎠. 0 0 ν A1g ,31 When K = T and m = 0, a point in C0 is a linear combination of the matrices (4.24), that is, ⎞ ⎛ 0 0 ν Ag ,11 1 ⎜ 0 ⎟ (4.38) f = ⎝ 0 √2 ν Ag ,21 ⎠. 1 √ ν A ,21 0 0 g 2 It is a closed interval. The same set appears when • K = T × Z 2c and m ∈ {0, 2}; • K = O and m = 0, K ∈ {O− , O × Z 2c } and m ∈ {0, 2}, but this time ν Ag ,i1 are replaced with ν A1g ,i1 . When K = T and m ∈ {1, 3}, a point in C0 is a linear combination of the matrices (4.24)–(4.26), that is

60

4 Correlation Structures

⎛

ν Ag ,11 ⎜ √1 ν f = ⎝ 2 E g ,21 √1 ν E ,22 g 2

√1 ν E ,21 g 2

√1 ν E ,22 g 2 √1 ν E ,12 g 2

√1 (ν A ,21 − ν E ,11 ) g g 2 √1 ν E ,12 √1 (ν A ,21 g g 2 2

+ ν E g ,11 )

⎞ ⎟ ⎠.

(4.39)

This is the same convex compact set as (4.30), but this time it is written in the coordinates (4.24)–(4.26). This set also appears when • K = T × Z 2c and m ∈ {1, 3, 4, 5}; • K ∈ {O, O− , O × Z 2c } and m = 6, K = O × Z 2c and m = 7, but ν Ag ,i1 is replaced with ν A1g ,i1 . When K = T and m = 2, a point in C2 is a linear combination of the matrices (4.24) and (4.27), that is, ⎛ ν Ag ,11 ⎜ 0 f=⎝ 0

0 √1 ν A ,21 g 2 √i ν A ,11 u 2

⎞ 0 √i ν A ,11 ⎟ . u ⎠ 2 √1 ν A ,21 g 2

(4.40)

This is not so easy to see, but it is a triangle similar to (4.35) with vertices ⎛

⎞ 100 ⎝0 0 0 ⎠ , 000

⎛ ⎞ ⎛ ⎞ 000 0 0 0 1⎝ 1 ⎝0 1 −i⎠ . 0 1 i ⎠ , and 2 0 i 1 2 0 −i 1

The same set appears when K ∈ {O, O × Z 2c } and m = 2. When K = T and m ∈ {4, 5}, a point in C4 is a linear combination of the matrices (4.24)–(4.28), that is ⎛

ν Ag ,11 ⎜ √1 ν f = ⎝ 2 E g ,21 − √i2 ν Eu ,11 √1 ν E ,22 − √i ν E ,12 g u 2 2

⎞

√1 ν E ,21 + √i ν E ,11 √1 ν E ,22 + √i ν E ,12 g u g u 2 2 2 2 √1 (ν A ,21 − ν E ,11 ) √1 ν E ,12 + √i ν A ,11 ⎟ ⎠ g g g u 2 2 2 √1 ν E ,11 − √i ν A ,11 √1 (ν A ,21 + ν E ,11 ) g u g g 2 2 2

. (4.41)

This is the same convex compact set as (4.31), but this time it is written in the coordinates (4.24)–(4.28). This set also appears when K ∈ {O, O− } and m = 7, but ν Ag ,i1 is replaced with ν A1g ,i1 and ν Au ,i1 is replaced with ν A1u ,11 . Another convex compact set appears when K = O and m = 1. A point in C1 is a linear combination of the matrices (4.24), f E g ,12 of (4.25) and f E g ,22 of (4.26), that is ⎛

ν Ag ,11 ⎜ 0 f=⎝ √1 ν E ,22 g 2

0 √1 ν A ,21 g 2 √1 ν E ,12 g 2

⎞

√1 ν E ,22 g 2 √1 ν E ,12 ⎟ ⎠ g 2 √1 ν A ,21 g 2

.

(4.42)

This set also appears when K ∈ {O− , O × Z 2c } and m = 1, but ν Ag ,i1 is replaced with ν A1g ,i1 .

4.6 Convex Compacta

61

When K = O and m = 3, a point in C3 is a linear combination of the matrices (4.24) and (4.29), that is, ⎛

ν Ag ,11 ⎜ √1 ν f = ⎝ 2 2 E g ,12 √ − 2√32 ν E g ,12

√

ν 2 2 E g ,12 1 1 √ ν A ,21 − √ ν g 2 2 2 E g ,11

− 2√32 ν E g ,12 √1 ν E ,12 g 2

1 √

√1 ν E ,12 g 2

√1 ν A ,21 g 2

+

⎞ ⎟ ⎠.

(4.43)

1 √ ν 2 2 E g ,11

This set also appears when K ∈ {O− , O × Z 2c } and m ∈ {3, 4}, K = O × Z 2c and m = 5, but ν Ag ,i1 is replaced with ν A1g ,i1 . The convex compact set from the case of K = O and m = 4 is a linear combination of the matrices (4.24), (4.27), and (4.29), that is, ⎛

ν Ag ,11 ⎜ √1 ν f = ⎝ 2 2 E g ,12 √ − 2√32 ν E g ,12

ν 2 2 E g ,12 1 √1 ν A ,21 − √ ν g 2 2 2 E g ,11 1 √

√1 ν E ,12 g 2

−

√ ⎞ − 2√32 ν E g ,12 √1 ν E ,12 + √i ν A ,11 ⎟ ⎠, g u 2 2

√i ν A ,11 √1 ν A ,11 u g 2 2

+

(4.44)

1 √ ν 2 2 E g ,11

while that from the case of K = O and m = 5 is a linear combination of the matrices (4.24), (4.27), f Eu ,11 of (4.28), and (4.29), that is, ⎛

ν Ag ,11 ⎜ √1 ν f = ⎝ 2 2 E g ,12 − √i2 ν Eu ,11 √ − 2√32 ν E g ,12

√ ⎞ 1 √ ν + √i2 ν Eu ,11 − 2√32 ν E g ,12 2 2 E g ,12 1 √1 ν A ,21 − √ √1 ν E ,12 + √i ν A ,11 ⎟ ν ⎠ g g u 2 2 2 E g ,11 2 2 1 √1 ν E ,12 − √i ν A ,11 √1 ν A ,21 + √ ν g u g 2 2 2 2 2 E g ,11

.

(4.45) In the case of K = O− and m = 5 it is a linear combination of the matrices (4.24), f Eu ,11 of (4.28), and (4.29), that is, ⎛

ν Ag ,11 ⎜ √1 ν f = ⎝ 2 2 E g ,12 − √i2 ν Eu ,11 √ − 2√32 ν E g ,12

√ ⎞ 1 √ ν + √i2 ν Eu ,11 − 2√32 ν E g ,12 2 2 E g ,12 ⎟ 1 √1 ν A ,21 − √ √1 ν E ,12 ν ⎠ g g 2 2 2 E g ,11 2 1 √1 ν E ,12 √1 ν A ,21 + √ ν g g 2 2 2 2 E g ,11

.

(4.46)

We summarise our findings in Table 4.2.

4.7 The “Spherical Bessel Functions” Rewrite Eq. (4.19) in the following form: d( x ), d(y ) pq =

M−1  m=0





ˆ 3 / K˜ )m K˜ (R



˜  m )) pq dg dμm (λ m ) , ei(gλm ,y −x ) (θ(g) f (λ

(4.47)

62

4 Correlation Structures

Table 4.2 The convex compact sets Cm K m 0 1 2 D2 D2 × Z 2c D4 D4h D4 × Z 2c T T × Z 2c O O− O × Z 2c

(4.30) (4.30) (4.32) (4.32) (4.35) (4.38) (4.38) (4.38) (4.38) (4.38)

(4.30) (4.30) (4.32) (4.34) (4.35) (4.39) (4.39) (4.42) (4.42) (4.42)

(4.30) (4.30) (4.32) (4.32) (4.36) (4.40) (4.38) (4.40) (4.38) (4.38)

3

4

5

6

7

(4.30) (4.30) (4.30) (4.30) (4.37) (4.39) (4.39) (4.43) (4.43) (4.43)

(4.30) (4.30) (4.33) (4.34) (4.36) (4.41) (4.39) (4.44) (4.43) (4.43)

(4.30) (4.30) (4.32) (4.31) (4.37) (4.41) (4.39) (4.45) (4.46) (4.43)

(4.30) (4.30) (4.32) (4.30) (4.30) – – (4.39) (4.39) (4.39)

(4.31) (4.30) (4.31) (4.31) (4.30) – – (4.41) (4.41) (4.39)

where dg is the probabilistic invariant measure on K˜ . In this section, we calculate the inner integral.  To perform this task, consider the function ei(gλm ,y −x ) as a function of g ∈ K˜  m , x, and y. This function is continuous. The Fine Structure Theorem under fixed λ [13] states the following. Let Kˆ be the set of equivalence classes of irreducible orthogonal representations of the group K˜ . Choose a basis in each space on which a representation θ ∈ Kˆ acts, and consider the corresponding matrix θkl (g). For each θ ∈ Kˆ there exists a nonempty subset Iθ ⊆ {1, 2, . . . , dim θ } such that the functions  { dim θkl (g) : θ ∈ Kˆ , 1 ≤ k ≤ dim , l ∈ Iθ } constitute an orthonormal basis in the Hilbert space of square-integrable functions on K˜ . Note that for irreducible unitary representations, we have Iθ = {1, 2, . . . , dim θ } for all θ . The corresponding result is known as the Peter–Weyl Theorem.  In particular, the continuous function ei(gλm ,y −x ) has the uniformly convergent Fourier expansion e

 m ,y − i(g λ x)

=



dim θ

θ∈ Kˆ

where  m , y − x) = jklθ (λ

dim θ



 m , y − x)θkl (g) , jklθ (λ

(4.48)

k=1 l∈Iθ





ei(gλm ,y −x ) θkl (g) dg .

K˜

 m ) may be written in the form How to use this result? The matrix f pq (λ

(4.49)

4.7 The “Spherical Bessel Functions”

m ) = f pq (λ

63

 n(θ, θ˜ ) dim θ θ∈ Kˆ i=1

 m )fθ,ik νθ,ik (λ pq ,

k=1

where n(θ, θ˜ ) is the number of copies of θ inside θ˜ . The representation θ˜ acts on this matrix as  n(θ, θ˜ ) dim θ dim θ  m )) pq =  m )fθ,ik (θ˜ (g) f (λ θkr (g)νθ,ir (λ (4.50) pq . θ∈ Kˆ i=1

k=1 r =1

Substitute (4.48) and (4.50) into (4.47) and use the Fine Structure Theorem. After simple algebraic calculations we obtain d( x ), d(y ) =

M−1  m=0



 dim θ  n(θ, θ˜ )

 m , y − x) jklθ (λ

k=1 l∈Iθ i=1 ˆ 3 / K˜ )m θ∈ Kˆ (R

(4.51)

 m )fθ,ik dμm (λ m ) . × νθ,il (λ  m , y − x). In the following example, we calculate the functions jklθ (λ Example 4.6 Let G = Z 1 , K = O(3). The Fourier expansion (4.48) becomes the Rayleigh expansion: 

ei(k,z ) = 4π

∞ 

i j (λr )

=0

 

Sm (ϑk , ϕk )Sm (ϑz , ϕz ) ,

m=−

where j is the spherical Bessel function, (r, ϑz , ϕz ) are the spherical coordinates in the space domain, (λ, ϑk , ϕk ) are the spherical coordinates in the wavenumber domain, Sm are the real-valued spherical harmonics, and z = x − y. This is the reason for the notation (4.49). We will call these functions the “spherical Bessel functions” in the remaining cases as well. Until the end of this example put G = D2 . Let K = D2 . It follows from (4.11) that n(A g , θ˜ ) = 6, n(Au , θ˜ ) = 3, and n(θ, θ˜ ) = 0 for the remaining irreducible representations of the group K˜ = D2 × Z 2c . We need to calculate only the spherical  m , z ) and j Au (λ  m , z ). Bessel functions j Ag (λ Let m = 7. By definition of the spherical Bessel function, we have  m , z ) = j A g (λ

1 8





ei(gk,z ) A g (g) .

g∈D2 ×Z 2c

With the help of [2, Table 31.4] we obtain  z ) = cos(k1 z 1 ) cos(k2 z 2 ) cos(k3 z 3 ) . j Ag (k,

(4.52)

64

4 Correlation Structures

Similarly,

 z ) = −i sin(k1 z 1 ) sin(k2 z 2 ) sin(k3 z 3 ) . j Au (k,

(4.53)

For the remaining values of m, the spherical Bessel functions are given by the same formula due to their continuity. When K = D2 × Z 2c , no new cases appear. Let K = D4 . It follows from (4.12) that n(A1g , θ˜ ) = 4, n(B1g , θ˜ ) = n(B1u , θ˜ ) = 2, n(A1u , θ˜ ) = 1. This time, we use [2, Table 33.4], and obtain  z ) = 1 cos(k3 z 3 )[cos(k1 z 1 ) cos(k2 z 2 ) + cos(k1 z 2 ) cos(k2 z 1 )] , j A1g (k, 2  z ) = 1 cos(k3 z 3 )[cos(k1 z 1 ) cos(k2 z 2 ) − cos(k1 z 2 ) cos(k2 z 1 )] , j B1g (k, 2 i A1u  j (k, z ) = − sin(k3 z 3 )[sin(k1 z 1 ) sin(k2 z 2 ) − sin(k1 z 2 ) sin(k2 z 1 )] , 2 i B1u  j (k, z ) = − sin(k3 z 3 )[sin(k1 z 1 ) sin(k2 z 2 ) + sin(k1 z 2 ) sin(k2 z 1 )] . 2

(4.54)

When K = D4h , two new cases appear.  z ) = − i sin(k3 z 3 )[sin(k1 z 1 ) sin(k2 z 2 ) + sin(k1 z 2 ) sin(k2 z 1 )] , j A2u (k, 2  z ) = − i sin(k3 z 3 )[sin(k1 z 1 ) sin(k2 z 2 ) − sin(k1 z 2 ) sin(k2 z 1 )] . j B2u (k, 2

(4.55)

When K = D4 × Z 2c , two new cases appear.  z ) = 1 cos(k3 z 3 )[− sin(k1 z 1 ) sin(k2 z 2 ) + sin(k1 z 2 ) sin(k2 z 1 )] , j A2g (k, 2 1 B2g  j (k, z ) = cos(k3 z 3 )[− sin(k1 z 1 ) sin(k2 z 2 ) − sin(k1 z 2 ) sin(k2 z 1 )] . 2

(4.56)

Let K = T . It follows from (4.13) that n(A g , θ˜ ) = n(E g , θ˜ ) = 2, n(Au , θ˜ ) = n(E u , θ˜ ) = 1. For the case of A g and Au , we use [2, Table 72.4] and obtain:  z ) = 1 [cos(k1 z 1 ) cos(k2 z 2 ) cos(k3 z 3 ) + cos(k1 z 2 ) cos(k2 z 3 ) cos(k3 z 1 ) j Ag (k, 3 + cos(k1 z 3 ) cos(k2 z 1 ) cos(k3 z 2 )] ,  z ) = i [sin(k1 z 1 ) cos(k2 z 2 ) cos(k3 z 3 ) + sin(k1 z 2 ) cos(k2 z 3 ) cos(k3 z 1 ) j Au (k, 3 + sin(k1 z 3 ) cos(k2 z 1 ) cos(k3 z 2 )] . (4.57) We calculate the matrix entries of the representation E g using the methods described in Example 4.4. The result is as follows: the representation E g maps the elements E,

4.7 The “Spherical Bessel Functions”

65

C2x , C2y , C2z , i, σx , σ y and σz of the normal subgroup D2 × Z 2c of the group T × Z 2c + + + + − − − − to the 2 × 2 identity matrix, the elements C31 , C32 , C33 , C34 , S61 , S62 , S63 and S64 to

√ − − − − + + + + −1 3 √ , and the elements C31 , C32 , C33 , C34 , S61 , S62 , S63 and S64 to − 3 −1 √

−1 − 3 . The matrix entries of the first column of the spherical Bessel the matrix 21 √ 3 −1 Eg 

the matrix

1 2

function j (k, z ) are

1 1 cos(k1 z 1 ) cos(k2 z 2 ) cos(k3 z 3 ) − cos(k1 z 2 ) cos(k2 z 3 ) cos(k3 z 1 ) 3 6 1 − cos(k1 z 3 ) cos(k2 z 1 ) cos(k3 z 2 ) , 6 1 Eg  z ) = − √ j21 (k, [cos(k1 z 2 ) cos(k2 z 3 ) cos(k3 z 1 ) 2 3 − cos(k1 z 3 ) cos(k2 z 1 ) cos(k3 z 2 )] . (4.58) Later we will see that only the first column of this matrix-valued function is important.  z ) = j11g (k, E

The representation E u is as follows:  E u (g) =

E g (g), if g ∈ T , −E g (g), otherwise .

 z ) is The spherical Bessel function j Eu (k, i i Eu  (k, z ) = sin(k1 z 1 ) cos(k2 z 2 ) cos(k3 z 3 ) − sin(k1 z 2 ) cos(k2 z 3 ) cos(k3 z 1 ) j11 3 6 i − sin(k1 z 3 ) cos(k2 z 1 ) cos(k3 z 2 ) , 6 i Eu  j21 (k, z ) = − √ [sin(k1 z 2 ) cos(k2 z 3 ) cos(k3 z 1 ) 2 3 − sin(k1 z 3 ) cos(k2 z 1 ) cos(k3 z 2 )] . (4.59) When K = T × Z 2c , no new cases appear. Let K = O. It follows from (4.14) that n(A1g , θ˜ ) = n(E g , θ˜ ) = 2, n(A2u , θ˜ ) = n(E u , θ˜ ) = 1. Using [2, Table 71.4], we obtain  z ) = 1 {cos(k1 z 1 )[cos(k2 z 2 ) cos(k3 z 3 ) + cos(k2 z 3 ) cos(k3 z 2 )] j A1g (k, 6 + cos(k2 z 1 )[cos(k1 z 3 ) cos(k3 z 2 ) + cos(k1 z 2 ) cos(k3 z 3 )] + cos(k3 z 1 )[cos(k1 z 2 ) cos(k2 z 3 ) + cos(k1 z 3 ) cos(k2 z 2 )]} , (4.60)  z ) = − i {sin(k1 z 1 )[sin(k2 z 2 ) sin(k3 z 3 ) + sin(k2 z 3 ) sin(k3 z 2 )] j A2u (k, 6 + sin(k2 z 1 )[sin(k1 z 3 ) sin(k3 z 2 ) + sin(k1 z 2 ) sin(k3 z 3 )] + sin(k3 z 1 )[sin(k1 z 2 ) sin(k2 z 3 ) + sin(k1 z 3 ) sin(k2 z 2 )]} .

66

4 Correlation Structures

We calculate the matrix entries of the representations E g and E u using the methods described in Example 4.4. The result is as follows: the representation E g maps the + matrix, the elements C31 , elements E, i, C2x , C2y , C2z , σx , σ y and σz to the identity

√ − − −1 3 √ , the elements C31 , C32 , 3 −1 − √

− − + + + + + − −1 − 3 , the elements C4x C33 , C34 , S61 , S62 , S63 and S64 to the matrix 21 √ , C4x , 3 −1√

− + + − 1 − 3 1  S4x , S4x , C2d , C2 f , σd4 and σd6 to the matrix 2 −√3 −1 , the elements C4y , C4y ,  0 − + + − −   , and the elements C4z , C2e , σd3 and σd5 to the matrix 01 −1 , C4z , S4z , S4y , S4y , C2c √

+   S4z , C2a , C2b , σd1 and σd2 to the matrix 21 √13 −13 . The representation E u maps the + + + − − − − , C33 , C34 , S61 , S62 , S63 and S64 to the matrix C32

1 2

elements E, C2x , C2y and C2z to the identity matrix, the elements i, σ x , σ y and σz to √  −1 0 + + + + 3 1 √ , the the matrix 0 −1 , the elements C31 , C32 , C33 and C34 to the matrix 2 −−1 3 −1 √

− − − − − − − elements S61 , S62 , S63 and S64 to the matrix 21 √13 −1 3 , the elements C31 , C32 , C33 √

− + + + + −1 − 3 , the elements S61 and C34 to the matrix 21 √ , S62 , S63 and S64 to the matrix 3 −1

√ √

+ − 1 3 1 − 3 1 1   √ √ , the elements C , the , C , C and C to the matrix 4x 4x 2d 2f 2 − 3 1 2 − 3 −1 √

− + + − −1 3  elements S4x , the elements C4y , S4x , σd4 and σd6 to the matrix 21 √ , C4y , C2c 3 1   − +  0 1 0 and C2e to the matrix −1 d5 to the matrix 0 −1 , 0 1 , the elements S4y , S4y , σ d3 and σ

√ + − −   the elements C4z , C4z , C2a and C2b to the matrix 21 √13 −13 , and the elements S4z , √

+ √ − 3 . S4z , σd1 and σd2 to the matrix 21 −−1 3 1 The spherical Bessel functions have the form 1 {cos(k3 z 1 )[cos(k1 z 3 ) cos(k2 z 2 ) − cos(k1 z 2 ) cos(k2 z 3 )] 12 + cos(k3 z 2 )[cos(k1 z 1 ) cos(k2 z 3 ) − cos(k1 z 3 ) cos(k2 z 1 )] + 2 cos(k3 z 3 )[cos(k1 z 1 ) cos(k2 z 2 ) + cos(k1 z 2 ) cos(k2 z 1 )]} , √ Eg  z ) = 3 {cos(k3 z 1 )[cos(k1 z 2 ) cos(k2 z 3 ) + cos(k1 z 3 ) cos(k2 z 2 )] j21 (k, 12 (4.61) − cos(k3 z 2 )[cos(k1 z 3 ) cos(k2 z 1 ) + cos(k1 z 1 ) cos(k2 z 3 )]} , i Eu  {sin(k3 z 1 )[sin(k1 z 2 ) sin(k2 z 3 ) − sin(k1 z 3 ) sin(k2 z 2 )] j11 (k, z ) = 12 + sin(k3 z 2 )[sin(k1 z 3 ) sin(k2 z 1 ) − sin(k1 z 1 ) sin(k2 z 3 )]

 z ) = j11g (k, E

+ 2 sin(k3 z 3 )[sin(k1 z 1 ) sin(k2 z 2 ) + sin(k1 z 2 ) sin(k2 z 1 )]} , √ i 3 Eu  {− sin(k3 z 1 )[sin(k1 z 2 ) sin(k2 z 3 ) − sin(k1 z 3 ) sin(k2 z 2 )] j21 (k, z ) = 12 + sin(k3 z 2 )[sin(k1 z 3 ) sin(k2 z 1 ) + sin(k1 z 1 ) sin(k2 z 3 )]} .

4.7 The “Spherical Bessel Functions”

67

Let K = O × Z 2c . No new cases appear. Let K = O− . It follows from (4.16) that n(A1g , θ˜ ) = n(E g , θ˜ ) = 2, n(A1u , θ˜ ) = n(E u , θ˜ ) = 1. Using [2, Table 71.4], we obtain i {sin(k1 z 1 )[sin(k2 z 2 ) sin(k3 z 3 ) − sin(k2 z 3 ) sin(k3 z 2 )] 6 + sin(k1 z 2 )[sin(k2 z 1 ) sin(k3 z 3 ) − sin(k2 z 3 ) sin(k3 z 1 )] + sin(k1 z 3 )[sin(k2 z 2 ) sin(k3 z 1 ) − sin(k2 z 1 ) sin(k3 z 2 )] .

 z ) = j A1u (k,

(4.62)

4.8 Correlation Structures To calculate the two-point correlation tensor of a homogeneous and isotropic random field, we use Eq. (4.51). ˆ 3 / K˜ )m and taking  defined on the set (R Note that a measurable function f K ,m (k) values in the set Cm can be uniquely written as the sum 

 = f K ,m (k)

 , f K ,m,θ (k)

θ∈ Kˆ : n(θ,θ˜ ) =0

 is calculated as follows: write down the function f K ,m (k)  where the function f K ,m,θ (k)   in the matrix notation, and replace all the entries νθ  ,il (k) with θ = θ by zeros. For  = f D2 ,7,Ag (k)  + f D2 ,7,Au (k),  where example, f D2 ,7 (k) ⎛

 ν Ag ,11 (k)

⎜   = ⎜ √1 ν Ag ,61 (k) f D2 ,7,Ag (k) ⎝ 2  √1 ν A ,51 (k) g 2

⎛

 √1 ν Ag ,51 (k) ⎞ √1 ν A ,61 (k) g 2 2 ⎟  √1 ν Ag ,41 (k) ⎟ ν Ag ,21 (k) ⎠ 2   √1 ν A ,41 (k) ν ( k) A g ,31 g 2

0

√i

⎜   = ⎜− √i ν Au ,31 (k) f D2 ,7,Au (k) ⎝ 2  − √i2 ν Au ,21 (k)

√i

,

 √i ν Au ,21 (k) ⎞ ν (k) 2 Au ,31 2 ⎟ ⎟ √i ν A ,11 (k) 0 u ⎠ 2 

ν (k) 2 Au ,11

,

0

see Eq. (4.31), referred to in Table 4.2 at the intersection of the row labelled “D2 ” and the column labelled “7”. Theorem 4.2 Let d( x ) be a homogeneous and (D2 , 3A)-isotropic random field. We have Ci ∈ R , d( x ) = C1 d1 + C2 d2 + C3 d3 , where di are the tensors of the basis (3.6). In the above basis, the two-point correlation tensor of the field has the form

68

4 Correlation Structures

d( x ), d(y ) =



7 

 y − x)f D2 ,m,Ag (k)  dμm (k)  j Ag (k,

m=0 3 ˆ /D2 ×Z c )m (R 2



(4.63)

 y − x)f D2 ,7,Au (k)  dμ7 (k)  , j (k,

+

Au

ˆ 3 /D2 ×Z c )7 (R 2

 y − x) (resp. j Au (k,  y − x)) is given by where the spherical Bessel function j Ag (k, Eq. (4.52) (resp. Eq. (4.53)). The field has the form d( x ) = C1 d + C2 d + C3 d + 1

2

3

7  3  8  i=1 m=0 n=1



 x) dZ imn (k)d  i, jn (k,

ˆ 3 /D2 ×Z c )m (R 2

(4.64)  x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k,  are centred real-valued random measures on k3 x3 , and where Z imn (k) ˆ 3 /D2 × Z c )m with the following nonzero correlations: when (m, m  ) = (7, 7), (R 2 then  D ,m,A g  dμm (k)  , fil 2 (k) E[Z imn (A)Z lm  n  (B)] = δmm  δnn  A∩B

otherwise  E[Z i7n (A)Z l7n (B)] =

D ,7,A g

fil 2

 dμm (k),  (k)

A∩B

E[Z i71 (A)Z l78 (B)] = −E[Z i78 (A)Z l71 (B)] = −E[Z i72 (A)Z l77 (B)] = E[Z i77 (A)Z l72 (B)] = −E[Z i73 (A)Z l76 (B)] = E[Z i76 (A)Z l73 (B)] = E[Z i74 (A)Z l75 (B)]  1  dμ7 (k)  . = −E[Z i75 (A)Z l74 (B)] = − filD2 ,7,Au (k) i A∩B

Proof Equation (4.63) is a particular case of Eq. (4.51). We enumerate the functions  x) in the lexicographic order, starting from jn (k,  x) = cos(k1 z 1 ) cos(k2 z 2 ) cos(k3 z 3 ) . j1 (k, Calculating the two-point correlation tensor of the random field (4.64) and taking into account the given correlation structure, we arrive at (4.63). In what follows, the random measures Z are always uncorrelated when the values of the index m are different.

4.8 Correlation Structures

69

Theorem 4.3 Let d( x ) be a homogeneous and (D2 × Z 2c , 3Au )-isotropic random field. We have d( x ) = 0. In the basis (3.6), the two-point correlation tensor of the field has the form 

 y − x)f D2 ,0,Ag (k)  dμ(k)  , j Ag (k,

d( x ), d(y ) = ˆ 3 /D2 ×Z c R 2

 y − x) is given by Eq. (4.52). The field where the spherical Bessel function j Ag (k, has the form  8 3    x) dZ in (k)d  i, jn (k, d( x) = i=1 n=1

ˆ 3 /D2 ×Z c R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, ˆ 3 /D2 × Z c  are centred real-valued random measures on R k3 x3 , and where Z in (k) 2 with the nonzero correlations:  D ,0,A  dμ(k)  . E[Z in (A)Z ln  (B)] = δnn  fil 2 g (k) A∩B

The proof is left to the reader. Theorem 4.4 Let d( x ) be a homogeneous and (D4 , A1 ⊕ 2B1 )-isotropic random field. We have d( x ) = Cd1 , where d1 is the first tensor of the basis (3.7). In the above basis, the two-point correlation tensor of the field has the form d( x ), d(y ) =



7 

 y − x)f D4 ,m,A1g (k)  dμm (k)  j A1g (k,

m=0 3 ˆ /D4 ×Z c )m (R

+





2

 y − x)f D4 ,m,A1u (k)  dμm (k)  j A1u (k,

m∈{3,4,7} 3 ˆ /D4 ×Z c )m (R 2



+



 y − x)f D4 ,7,B1g (k)  j B1g (k,

ˆ 3 /D4 ×Z c )7 (R 2

  y − x)f D4 ,7,B1u (k)  dμ7 (k)  , + j B1u (k, (4.65)

70

4 Correlation Structures

 y − x), …, j B1u (k,  y − x) are given by where the spherical Bessel functions j A1g (k, Eq. (4.54). The field has the form 

1  d( x ) = Cd1 + √ 2 i=1 m=0 n=1 3

7

16

 x) dZ imn (k)d  i, jn (k,

ˆ 3 /D4 ×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, and of cosines and sines of k1 x2 , k2 x1 , and k3 x3 for 9 ≤ n ≤ 16, ˆ 3 /D4 × Z c )m  are centred real-valued random measures on (R and where Z imn (k) 2 with the following nonzero correlations. For 0 ≤ m ≤ 6  E[Z imn (A)Z lmn (B)] =

D ,m,A1g

fil 4

 dμm (k)  . (k)

A∩B

For m ∈ {3, 4} and k ∈ {0, 1} E[Z im8k+1 (A)Z lm8k+8 (B)] = −E[Z im8k+8 (A)Z lm8k+1 (B)] = −E[Z im8k+2 (A)Z lm8k+7 (B)] = E[Z im8k+7 (A)Z lm8k+2 (B)] = −E[Z im8k+3 (A)Z lm8k+6 (B)] = E[Z im8k+6 (A)Z lm8k+3 (B)] = E[Z im8k+4 (A)Z lm8k+5 (B)] = −E[Z im8k+5 (A)Z lm8k+4 (B)]  1  dμm (k)  . =− filD4 ,m,A1u (k) i A∩B

For m = 7 and for k ∈ {0, 1} ⎧  D4 ,7,A1g  + filD4 ,7,B1g (k)]  dμm (k),  1 ≤ n ≤ 8, [fil (k) ⎪ ⎨ A∩B E[Z i7n (A)Z l7n (B)] =  D ,7,A ⎪  − filD4 ,7,B1g (k)]  dμm (k),  9 ≤ n ≤ 16 , ⎩ [fil 4 1g (k) A∩B

as well as

4.8 Correlation Structures

71

E[Z i7 8k+1 (A)Z l7 8k+8 (B)] = −E[Z i7 8k+8 (A)Z l7 8k+1 (B)] = −E[Z i7 8k+2 (A)Z l7 8k+7 (B)] = E[Z i7 8k+7 (A)Z l7 8k+2 (B)] = −E[Z i7 8k+3 (A)Z l7 8k+6 (B)] = E[Z i7 8k+6 (A)Z l7 8k+3 (B)] = E[Z i7 8k+4 (A)Z l7 8k+5 (B)] = −E[Z i7 8k+5 (A)Z l7 8k+4 (B)]  1  + (−1)k filD4 ,7,B1u (k)]  dμ7 (k)  . =− [filD2 ,7,A1u (k) i A∩B

In the first line of Eq. (4.65), the terms that correspond to m ∈ {0, 1, 2, 5, 6} may be further simplified. Indeed, in this case the cone Cm is a union of closed intervals, or one-dimensional simplexes. One extreme point of each interval, say f1 , is the vertex  runs over the disk with ν A1g ,11 = 1 and ν A1g ,21 = ν A1g ,41 = 0, another one, say f2 (k), 2 2 D4 ,m,A1g  (k) ∈ Cm may with ν A1g ,11 = 0 and (2ν A1g ,21 − 1) + 2ν A1g ,41 ≤ 1. A point f be represented as  = λ(k)f  1 + (1 − λ( p))f2 (k)  . f D4 ,m,A1g (k) Introduce the measures   dμm (k), 

m1 (A) = λ(k)

m2 (A) = μm (A) − m1 (A) .

A

Then we have 

 y − x)f D4 ,m,A1g (k)  dμm (k)  j A1g (k,

ˆ 3 /D4 ×Z c )m (R 2



 y − x)f1 d m1 (k)  j A1g (k,

= ˆ 3 /D4 ×Z c )m (R 2



+

 y − x)f2 (k)  d m2 (k)  . j A1g (k,

ˆ 3 /D4 ×Z c )m (R 2

The “arbitrary function” f1 of the first integral on the right-hand side takes values in  of the second the singleton. Therefore it is a constant. The arbitrary function f2 (k) integral takes values in the two-dimensional disk, not in the three-dimensional cone, as before.

72

4 Correlation Structures

Another way to simplify the correlation structure of a homogeneous and isotropic random field is as follows. Inscribe a simplex into the convex compact set Cm . Allow the function f to take values only in the above simplex rather than in all of Cm . The corresponding integral becomes the sum of as many integrals as vertices in the simplex. Each integral does not contain arbitrary functions. The resulting formula describes only a sufficient condition for its right-hand side to be the twopoint correlation tensor of a homogeneous and isotropic random field. The greater the Lebesgue measure of the inscribed simplex in comparison with that of Cm , the closer the sufficient condition is to a necessary condition. See the examples in [20]. Theorem 4.5 Let d( x ) be a homogeneous and (D4h , 2 A1 ⊕ B1 )-isotropic random field. We have d( x ) = C1 d1 + C2 d2 , where d1 and d2 are the tensors of the basis (3.7). In the above basis, the two-point correlation tensor of the field has the form d( x ), d(y ) =

7 



 y − x)f D4h ,m,A1g (k)  dμm (k)  j A1g (k,

m=0 3 ˆ /D4 ×Z c )m (R 2

+





m∈{1,4,5,7} 3 ˆ /D4 ×Z c )m (R

+





2



  y − x)f D4h ,m,A2u (k)  dμm (k)  j A2u (k,



  y − x)f D4h ,m,B1g (k)  dμm (k)  j B1g (k,

m∈{3,5,6,7} 3 ˆ /D4 ×Z c )m (R 2

+







  y − x)f D4h ,m,B2u (k)  dμm (k)  , j B2u (k,

m∈{5,7} 3 ˆ /D4 ×Z c )m (R 2

 y − x) and j B1g (k,  y − x) are given by where the spherical Bessel functions j A1g (k, A2u  B2u  Eq. (4.54), and j (k, y − x) and j (k, y − x) are given by Eq. (4.55). The field has the form 1  d( x ) = C1 d1 + C2 d2 + √ 2 i=1 m=0 n=1 3

7

16



 x) dZ imn (k)d  i, jn (k,

ˆ 3 /D4 ×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, and of cosines and sines of k1 x2 , k2 x1 , and k3 x3 for 9 ≤ n ≤ 16, ˆ 3 /D4 × Z c )m  are centred real-valued random measures on (R and where Z imn (k) 2 with the following nonzero correlations. For m ∈ {0, 1, 2, 4}

4.8 Correlation Structures

73

 E[Z imn (A)Z lmn (B)] =

D h ,m,A1g

fil 4

 dμm (k)  . (k)

A∩B

For m ∈ {3, 5, 6, 7} ⎧  D h ,m,A 1g  + filD4 ,m,B1g (k)]  dμm (k),  1 ≤ n ≤ 8, ⎪ [fil 4 (k) ⎨ A∩B E[Z imn (A)Z lmn (B)] =  D h ,m,A1g ⎪  − filD4 ,m,B1g (k)]  dμm (k),  otherwise. [fil 4 (k) ⎩ A∩B

For m ∈ {1, 4} and k ∈ {0, 1} E[Z im8k+1 (A)Z lm8k+8 (B)] = − E[Z im8k+8 (A)Z lm8k+1 (B)] = −E[Z im8k+2 (A)Z lm8k+7 (B)] = E[Z im8k+7 (A)Z lm8k+2 (B)] = −E[Z im8k+3 (A)Z lm8k+6 (B)] = E[Z im8k+6 (A)Z lm8k+3 (B)] = E[Z im8k+4 (A)Z lm8k+5 (B)] = −E[Z im8k+5 (A)Z lm8k+4 (B)]  1 D h ,m,A2u  dμm (k)  . =− fil 4 (k) i A∩B

For m ∈ {5, 7} and k ∈ {0, 1} E[Z im8k+1 (A)Z lm8k+8 (B)] = − E[Z im8k+8 (A)Z lm8k+1 (B)] = −E[Z im8k+2 (A)Z lm8k+7 (B)] = E[Z im8k+7 (A)Z lm8k+2 (B)] = −E[Z im8k+3 (A)Z lm8k+6 (B)] = E[Z im8k+6 (A)Z lm8k+3 (B)] = E[Z im8k+4 (A)Z lm8k+5 (B)] = −E[Z im8k+5 (A)Z lm8k+4 (B)]  1 D h ,m,A2u  =− [fil 4 (k) i A∩B D h ,m,B2u

+ (−1)k fil 4

 dμm (k)  . (k)]

Theorem 4.6 Let d( x ) be a homogeneous and (D4 × Z 2c , A1u ⊕ B1u ⊕ B2u ) -isotropic random field. We have d( x ) = 0 .

74

4 Correlation Structures

In the basis (3.7), the two-point correlation tensor of the field has the form d( x ), d(y ) =

7 



 y − x)f D4 ×Z 2c ,m,A1g (k)  dμm (k)  j A1g (k,

m=0 3 ˆ /D4 ×Z c )m (R 2

+





 y − x)f D4 ×Z 2c ,m,B1g (k)  j A2g (k,

m∈{6,7} 3 ˆ /D4 ×Z c )m (R

+

 2



 y − x)f D4 ×Z 2 ,m,B1g (k)  dμm (k)  j B1g (k, c

m∈{3,5,6,7} 3 ˆ /D4 ×Z c )m (R 2

+





 y − x)f D4 ×Z 2c ,m,B2g (k)  dμm (k)  , j B2g (k,

m∈{2,4,6,7} 3 ˆ /D4 ×Z c )m (R 2

 y − x) and j B1g (k,  y − x) are given by where the spherical Bessel functions j A1g (k, A2g  B2g  Eq. (4.54), j (k, y − x) and j (k, y − x) are given by Eq. (4.56). The field has the form 

1  d( x) = √ 2 i=1 m=0 n=1 3

7

16

 x) dZ imn (k)d  i, jn (k,

ˆ 3 /D4 ×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, and of cosines and sines of k1 x2 , k2 x1 , and k3 x3 for 9 ≤ n ≤ 16, ˆ 3 /D4 × Z c )m  are centred real-valued random measures on (R and where Z imn (k) 2 with the following nonzero correlations. For m ∈ {0, 1, 2, 4}  E[Z imn (A)Z lmn (B)] =

D ×Z 2c ,m,A1g

fil 4

 dμm (k)  . (k)

A∩B

For m ∈ {3, 5, 6, 7} E[Z imn (A)Z lmn (B)] c ⎧  D4 ×Z 2c ,m,A1g  + filD4 ×Z 2 ,m,B1g (k)]  dμm (k),  1 ≤ n ≤ 8, [fil (k) ⎪ ⎨ A∩B =  D ×Z c ,m,A c 1g ⎪  − filD4 ×Z 2 ,m,B1g (k)]  dμm (k),  9 ≤ n ≤ 16 . ⎩ [fil 4 2 (k) A∩B

4.8 Correlation Structures

75

For m ∈ {2, 4} and k ∈ {0, 1} −E[Z im8k+1 (A)Z lm8k+7 (B)] = −E[Z im8k+7 (A)Z lm8k+1 (B)] = −E[Z im8k+2 (A)Z lm8k+8 (B)] = −E[Z im8k+8 (A)Z lm8k+2 (B)] = E[Z im8k+3 (A)Z lm8k+5 (B)] = E[Z im8k+5 (A)Z lm8k+3 (B)] = E[Z im8k+4 (A)Z lm8k+6 (B)] = E[Z im8k+6 (A)Z lm8k+4 (B)]  D ×Z c ,m,B2g  dμm (k)  . = fil 4 2 (k) A∩B

For m ∈ {6, 7} and k ∈ {0, 1} −E[Z im8k+1 (A)Z lm8k+7 (B)] = −E[Z im8k+7 (A)Z lm8k+1 (B)] = −E[Z im8k+2 (A)Z lm8k+8 (B)] = −E[Z im8k+8 (A)Z lm8k+2 (B)] = E[Z im8k+3 (A)Z lm8k+5 (B)] = E[Z im8k+5 (A)Z lm8k+3 (B)] = E[Z im8k+4 (A)Z lm8k+6 (B)] = E[Z im8k+6 (A)Z lm8k+4 (B)]   D ×Z c ,m,A2g  fil 4 2 = (k) A∩B D ×Z 2c ,m,B2g

+ (−1)k fil 4

  dμm (k)  . (k)

Theorem 4.7 Let d( x ) be a homogeneous and (T , A ⊕ E)-isotropic random field. We have d( x ) = C1 d1 , where d1 is the first tensor of the basis (3.8). In the above basis, the two-point correlation tensor of the field has the form

76

4 Correlation Structures

d( x ), d(y ) =



5 

 y − x)fT ,m,A g (k)  dμm (k)  j A g (k,

m=0 ˆ 3 (R /T ×Z 2c )m



 y − x)fT ,2,Au (k)  dμ2 (k)  j Au (k,

+ ˆ 3 /T ×Z c )2 (R 2

+





2 2   Eg   E g ,qk dμm (k)  jk1 (k, y − x) ν E g ,q1 (k)f q=1

k=1 m∈{1,3,4,5} ˆ 3 (R /T ×Z 2c )m

+

5 



 ν E u ,11 (k)

m=4 ˆ 3 (R /T ×Z 2c )m

2  E   , jk1u (k, y − x)f E u ,1k dμm (k) k=1

 y − x), …, j Eu (k,  y − x) are given by where the spherical Bessel functions j Ag (k, E g ,ik E u ,1k and f are given by Eqs. (4.25), Eqs. (4.57)–(4.59), and the basis tensors f (4.26), and (4.28). The field has the form 1  d( x ) = C1 d1 + √ 3 i=1 m=0 n=1 5

3

24



 x) dZ imn (k)f  i, jn (k,

ˆ 3 /T ×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, of cosines and sines of k1 x2 , k2 x3 , and k3 x1 for 9 ≤ n ≤ 16, and  are centred realof cosines and sines of k1 x3 , k2 x1 , and k3 x2 for 17 ≤ n ≤ 24. Z mn (k) c 3 ˆ valued random measures on (R /T × Z 2 )m with the following nonzero correlations. For m ∈ {0, 2}  T ,m,A g  dμm (k)  . fil (k) E[Z imn (A)Z lmn (B)] = A∩B

For m ∈ {1, 3, 4, 5} and 1 ≤ n ≤ 8  E[Z imn (A)Z lmn (B)] = A∩B

⎡  + ⎣filT ,m,Ag (k)

2  q=1

⎤ E ,q1

 il g ν E g ,q1 (k)f

 . ⎦ dμm (k)

4.8 Correlation Structures

77

For m ∈ {1, 3, 4, 5} and 9 ≤ n ≤ 16    fT ,m,Ag (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤ √  − 1 filE g ,q1 − 3 filE g ,q2 ⎦ dμm (k)  . + ν E g ,q1 (k) 2 2 q=1 2 



For m ∈ {1, 3, 4, 5} and 17 ≤ n ≤ 24    fT ,m,Ag (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤  − 1 filE g ,q1 + 3 filE g ,q2 ⎦ dμm (k)  . + ν E g ,q1 (k) 2 2 q=1 2 



√

For m = 2 and 0 ≤ k ≤ 2 E[Z i2 8k+1 (A)Z l2 8k+5 (B)] = −E[Z i2 8k+5 (A)Z l2 8k+1 (B)] = E[Z i2 8k+2 (A)Z l2 8k+6 (B)] = −E[Z i2 8k+6 (A)Z l2 8k+2 (B)] = E[Z i2 8k+3 (A)Z l2 8k+7 (B)] = −E[Z i2 8k+7 (A)Z l2 8k+3 (B)] = E[Z i2 8k+4 (A)Z l2 8k+8 (B)] = −E[Z i2 8k+8 (A)Z l2 8k+4 (B)]   dμ2 (k)  . = filT ,2,Au (k) A∩B

For m ∈ {4, 5} and 1 ≤ n ≤ 8 E[Z im1 (A)Z lm5 (B)] = −E[Z im5 (A)Z lm1 (B)] = E[Z im2 (A)Z lm6 (B)] = −E[Z im6 (A)Z lm2 (B)] = E[Z im3 (A)Z lm7 (B)] = −E[Z im7 (A)Z lm3 (B)] = E[Z im4 (A)Z lm8 (B)]   ilEu ,11 dμm (k)  . = −E[Z im8 (A)Z lm4 (B)] = ν Eu ,11 (k)f A∩B

78

4 Correlation Structures

For m ∈ {4, 5} and 9 ≤ n ≤ 16 E[Z im9 (A)Z lm13 (B)] = −E[Z im13 (A)Z lm9 (B)] = E[Z im10 (A)Z lm14 (B)] = −E[Z im14 (A)Z lm10 (B)] = E[Z im11 (A)Z lm15 (B)] = −E[Z im15 (A)Z lm11 (B)] = E[Z im12 (A)Z lm16 (B)] = −E[Z im16 (A)Z lm12 (B)]   √  3 Eu ,12 1 Eu ,11   . f ν Eu ,11 (k) − fil − = dμm (k) 2 2 il A∩B

For m ∈ {4, 5} and 17 ≤ n ≤ 24 E[Z im17 (A)Z lm21 (B)] = −E[Z im21 (A)Z lm17 (B)] = E[Z im18 (A)Z lm22 (B)] = −E[Z im22 (A)Z lm18 (B)] = E[Z im19 (A)Z lm23 (B)] = −E[Z im23 (A)Z lm19 (B)] = E[Z im20 (A)Z lm24 (B)] = −E[Z im24 (A)Z lm20 (B)]   √  3 1 E ,11 E ,12  − fil u +  . f u ν Eu ,11 (k) = dμm (k) 2 2 il A∩B

Proof Only one detail requires additional explanation. We choose I E g = I Eu = {1}. Theorem 4.8 Let d( x ) be a homogeneous and (T × Z 2c , Au ⊕ E u )-isotropic random field. We have d( x ) = 0 . In the basis (3.8), the two-point correlation tensor of the field has the form d( x ), d(y ) =

5 



 y − x)fT ×Z 2c ,m,Ag (k)  dμm (k)  j Ag (k,

m=0 3 ˆ /T ×Z c )m (R 2

+





2 

k=1 m∈{1,3,4,5} 3 ˆ /T ×Z c )m (R 2

 y − x) jk1g (k, E

2  q=1

 E g ,qk dμm (k)  , ν E g ,q1 (k)f

4.8 Correlation Structures

79

 y − x) and j E g (k,  y − x) are given by where the spherical Bessel functions j Ag (k, Eqs. (4.57) and (4.58), and the basis tensors f E g ,ik are given by Eqs. (4.25) and (4.26). The field has the form 1  d( x) = √ 3 i=1 m=0 n=1 3

24

5



 x) dZ imn (k)f  i, jn (k,

ˆ 3 /T ×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, of cosines and sines of k1 x2 , k2 x3 , and k3 x1 for 9 ≤ n ≤ 16, and of  are centred realcosines and sines of k1 x3 , k2 x1 , and k3 x2 for 17 ≤ n ≤ 24. Z imn (k) ˆ 3 /T × Z c )m with the following nonzero correlations. valued random measures on (R 2 For m ∈ {0, 2}  T ×Z c ,m,A g  dμm (k)  . fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

For m ∈ {1, 3, 4, 5} and 1 ≤ n ≤ 8 ⎡



T ×Z 2c ,m,A g

⎣fil

E[Z imn (A)Z lmn (B)] =

 + (k)

2 

⎤  ilE g ,q1 ⎦ dμm (k)  . ν E g ,q1 (k)f

q=1

A∩B

For m ∈ {1, 3, 4, 5} and 9 ≤ n ≤ 16   T ×Z c ,m,A g  fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤ √ 3 1 E ,q2  − filE g ,q1 −  . ⎦ dμm (k) fil g + ν E g ,q1 (k) 2 2 q=1 2 



For m ∈ {1, 3, 4, 5} and 17 ≤ n ≤ 24   T ×Z c ,m,A g  E[Z imn (A)Z lmn (B)] = fil 2 (k) A∩B

⎤ √ 3 1 E ,q2  − filE g ,q1 +  . ⎦ dμm (k) fil g + ν E g ,q1 (k) 2 2 q=1 2 



80

4 Correlation Structures

Theorem 4.9 Let d( x ) be a homogeneous and (O, A2 ⊕ E)-isotropic random field. We have d( x ) = 0 . In the basis (3.8), the two-point correlation tensor of the field has the form d( x ), d(y ) =

7 



 y − x)fO×Z 2 ,m,A1g (k)  dμm (k)  j A1g (k, c

m=0 3 ˆ /O×Z c )m (R 2

+





2 

 y − x) jk1g (k, E

+

 E g ,qk dμm (k)  ν E g ,q1 (k)f

q=1

k=1 m∈{1,3,4,5,6,7} 3 ˆ /O×Z c )m (R 2





2 

 y − x)fO×Z 2c ,m,A2u (k)  dμm (k)  j A2u (k,

m∈{2,4,5,7} 3 ˆ /O×Z c )m (R 2

+





 ν Eu ,11 (k)

2 

Eu   , jk1 (k, y − x)f Eu ,1k dμm (k)

k=1

m∈{5,7} 3 ˆ /O×Z c )m (R 2

 y − x), …, j Eu (k,  y − x) are given by where the spherical Bessel functions j A1g (k, E g ,ik E u ,ik and f are given by Eqs. (4.25), Eqs. (4.60) and (4.61), and the basis tensors f (4.26), and (4.28). The field has the form 1  d( x) = √ 6 i=1 m=0 n=1 3

7

48



 x) dZ mn (k)f  i, jn (k,

ˆ 3 /O×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, of k1 x1 , k2 x3 , and k3 x2 for 9 ≤ n ≤ 16, of k1 x3 , k2 x1 , and k3 x2 for 17 ≤ n ≤ 24, of k1 x2 , k2 x3 , and k3 x1 for 25 ≤ n ≤ 32, of k1 x3 , k2 x1 , and k3 x2 for  are centred real33 ≤ n ≤ 40, and of k1 x3 , k2 x2 , and k3 x1 for 41 ≤ n ≤ 48. Z imn (k) c 3 ˆ valued random measures on (R /T × Z 2 )m with the following nonzero correlations. For m ∈ {0, 2}  O×Z c ,m,A g  dμm (k)  . fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

4.8 Correlation Structures

81

For m ∈ {1, 3, 4, 5, 6, 7} and 1 ≤ n ≤ 8, 17 ≤ n ≤ 24 ⎡



O×Z 2c ,m,A g

⎣fil

E[Z imn (A)Z lmn (B)] =

 + (k)

2 

⎤  ilE g ,q1 ⎦ dμm (k)  . ν E g ,q1 (k)f

q=1

A∩B

For m ∈ {1, 3, 4, 5} and 9 ≤ n ≤ 16   O×Z c ,m,A g  fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤ √ 1 3 E ,q1 E ,q2   . ⎦ dμm (k) fil g − fil g + ν E g ,q1 (k) 2 2 q=1 2 



For m ∈ {1, 3, 4, 5} and 25 ≤ n ≤ 32   O×Z c ,m,A g  fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤  − 1 filE g ,q1 + 3 filE g ,q2 ⎦ dμm (k)  . + ν E g ,q1 (k) 2 2 q=1 2 



√

For m ∈ {1, 3, 4, 5} and 33 ≤ n ≤ 40   O×Z c ,m,A g  fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤  − 1 filE g ,q1 − 3 filE g ,q2 ⎦ dμm (k)  . + ν E g ,q1 (k) 2 2 q=1 2 



√

For m ∈ {1, 3, 4, 5} and 41 ≤ n ≤ 48   O×Z c ,m,A g  fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤ √  1 filE g ,q1 + 3 filE g ,q2 ⎦ dμm (k)  . + ν E g ,q1 (k) 2 2 q=1 2 



82

4 Correlation Structures

For m ∈ {2, 4} and 0 ≤ k ≤ 5 E[Z im8k+1 (A)Z lm8k+8 (B)] = −E[Z im8k+8 (A)Z lm8k+1 (B)] = −E[Z im8k+2 (A)Z lm8k+7 (B)] = E[Z im8k+7 (A)Z lm8k+2 (B)] = −E[Z im8k+3 (A)Z lm8k+6 (B)] = E[Z im8k+6 (A)Z lm8k+3 (B)] = E[Z im8k+4 (A)Z lm8k+5 (B)] = −E[Z im8k+5 (A)Z lm8k+4 (B)]  O×Z c ,m,A2u  dμm (k)  . = fil 2 (k) A∩B

For m ∈ {5, 7} and k ∈ {0, 2} E[Z im8k+1 (A)Z lm8k+8 (B)] = −E[Z im8k+8 (A)Z lm8k+1 (B)] = −E[Z im8k+2 (A)Z lm8k+7 (B)] = E[Z im8k+7 (A)Z lm8k+2 (B)] = −E[Z im8k+3 (A)Z lm8k+6 (B)] = E[Z im8k+6 (A)Z lm8k+3 (B)] = E[Z im8k+4 (A)Z lm8k+5 (B)] = −E[Z im8k+5 (A)Z lm8k+4 (B)]    O×Z c ,m,A2u  + iν Eu ,11 (k)f  ilEu ,11 dμm (k)  . = fil 2 (k) A∩B

For m ∈ {5, 7} and 9 ≤ n ≤ 16 E[Z im9 (A)Z lm16 (B)] = −E[Z im16 (A)Z lm9 (B)] = −E[Z im10 (A)Z lm15 (B)] = E[Z im15 (A)Z lm10 (B)] = −E[Z im11 (A)Z lm14 (B)] = E[Z im14 (A)Z lm11 (B)] = E[Z im12 (A)Z lm13 (B)] = −E[Z im13 (A)Z lm12 (B)]   √   3 Eu ,12 1 Eu ,11 O×Z 2c ,m,A2u    . f (k) + iν Eu ,11 (k) − fil + = fil dμm (k) 2 2 il A∩B

4.8 Correlation Structures

83

For m ∈ {5, 7} and 25 ≤ n ≤ 32 E[Z im25 (A)Z lm32 (B)] = −E[Z im32 (A)Z lm25 (B)] = −E[Z im26 (A)Z lm31 (B)] = E[Z im31 (A)Z lm26 (B)] = −E[Z im27 (A)Z lm30 (B)] = E[Z im30 (A)Z lm27 (B)] = E[Z im28 (A)Z lm29 (B)] = −E[Z im29 (A)Z lm28 (B)]   √   3 1 O×Z 2c ,m,A2u E ,11 E ,12  + iν Eu ,11 (k)   . f u − f u (k) = fil dμm (k) 2 il 2 il A∩B

For m ∈ {5, 7} and 33 ≤ n ≤ 40 E[Z im33 (A)Z lm40 (B)] = −E[Z im40 (A)Z lm33 (B)] = −E[Z im34 (A)Z lm39 (B)] = E[Z im39 (A)Z lm34 (B)] = −E[Z im35 (A)Z lm38 (B)] = E[Z im38 (A)Z lm35 (B)] = E[Z im36 (A)Z lm37 (B)] = −E[Z im37 (A)Z lm36 (B)]   √   1 Eu ,11 3 Eu ,12 O×Z 2c ,m,A2u    . f f (k) + iν Eu ,11 (k) + = fil dμm (k) 2 il 2 il A∩B

For m ∈ {5, 7} and 41 ≤ n ≤ 48 E[Z im41 (A)Z lm48 (B)] = −E[Z im48 (A)Z lm41 (B)] = −E[Z im42 (A)Z lm47 (B)] = E[Z im47 (A)Z lm42 (B)] = −E[Z im43 (A)Z lm46 (B)] = E[Z im46 (A)Z lm43 (B)] = E[Z im44 (A)Z lm45 (B)] = −E[Z im45 (A)Z lm44 (B)]   √   3 1 O×Z 2c ,m,A2u E ,11 E ,12 u u  + iν Eu ,11 (k)  − fil  . f (k) − = fil dμm (k) 2 2 il A∩B

Theorem 4.10 Let d( x ) be a homogeneous and (O− , A1 ⊕ E)-isotropic random field. We have d( x ) = C1 d1 , where d1 is the first vector of the basis (3.8). In the above basis, the two-point correlation tensor of the field has the form

84

4 Correlation Structures d( x ), d(y ) =



7 

 y − x)fO− ,m,A1g (k)  dμm (k)  j A1g (k,

m=0 ˆ 3 (R /O×Z 2c )m





+

2 

 y − x) jk1g (k, E

 E g ,qk dμm (k)  ν E g ,q1 (k)f

q=1

k=1 m∈{1,3,4,5,6,7} ˆ 3 (R /O×Z 2c )m



2 

 y − x)fO− ,m,A1u (k)  dμm (k)  j A1u (k,

+ ˆ 3 /O×Z c )7 (R 2

+





 ν E u ,11 (k)

2 

Eu   , jk1 (k, y − x)f E u ,1k dμm (k)

k=1

m∈{5,7} ˆ 3 (R /O×Z 2c )m

 y − x), …, j Eu (k,  y − x) are given by where the spherical Bessel functions j A1g (k, Eqs. (4.60), (4.61), and (4.62), and the basis tensors f E g ,ik and f Eu ,ik are given by Eqs. (4.25), (4.26), and (4.28). The field has the form 

1  d( x ) = C1 d + √ 6 i=1 m=0 n=1 3

7

48

 x) dZ imn (k)f  i, jn (k,

1

ˆ 3 /O×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, of k1 x1 , k2 x3 , and k3 x2 for 9 ≤ n ≤ 16, of k1 x3 , k2 x1 , and k3 x2 for 17 ≤ n ≤ 24, of k1 x2 , k2 x3 , and k3 x1 for 25 ≤ n ≤ 32, of k1 x3 , k2 x1 , and k3 x2 for  are centred real33 ≤ n ≤ 40, and of k1 x3 , k2 x2 , and k3 x1 for 41 ≤ n ≤ 48. Z imn (k) ˆ 3 /T × Z c )m with the following nonzero correlations. valued random measures on (R 2 For m ∈ {0, 2}  O− ,m,A g  dμm (k)  . fil (k) E[Z imn (A)Z lmn (B)] = A∩B

For m ∈ {1, 3, 4, 5, 6, 7} and 1 ≤ n ≤ 8, 17 ≤ n ≤ 24  E[Z imn (A)Z lmn (B)] =

⎡ ⎣filO

−

,m,A g

 + (k)

2 

⎤  ilE g ,q1 ⎦ dμm (k)  . ν E g ,q1 (k)f

q=1

A∩B

For m ∈ {1, 3, 4, 5} and 9 ≤ n ≤ 16   O− ,m,A g  fil (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤ √ 3 1 E ,q1 E ,q2   . ⎦ dμm (k) fil g − fil g + ν E g ,q1 (k) 2 2 q=1 2 



4.8 Correlation Structures

85

For m ∈ {1, 3, 4, 5} and 25 ≤ n ≤ 32   O− ,m,A g  fil (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤  − 1 filE g ,q1 + 3 filE g ,q2 ⎦ dμm (k)  . + ν E g ,q1 (k) 2 2 q=1 

2 

√

For m ∈ {1, 3, 4, 5} and 33 ≤ n ≤ 40   O− ,m,A g  fil (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤  − 1 filE g ,q1 − 3 filE g ,q2 ⎦ dμm (k)  . + ν E g ,q1 (k) 2 2 q=1 

2 

√

For m ∈ {1, 3, 4, 5} and 41 ≤ n ≤ 48   O− ,m,A g  fil (k) E[Z imn (A)Z lmn (B)] = A∩B

⎤ √ 1 3 E g ,q1 E g ,q2   . ⎦ dμm (k) f f + ν E g ,q1 (k) + 2 il 2 il q=1 2 



For m = 7 and k ∈ {0, 2} E[Z i7 8k+1 (A)Z l7 8k+8 (B)] = −E[Z i7 8k+8 (A)Z l7 8k+1 (B)] = −E[Z i7 8k+2 (A)Z l7 8k+7 (B)] = E[Z i7 8k+7 (A)Z l7 8k+2 (B)] = −E[Z i7 8k+3 (A)Z l7 8k+6 (B)] = E[Z i7 8k+6 (A)Z l7 8k+3 (B)] = E[Z i7 8k+4 (A)Z l7 8k+5 (B)] = −E[Z i7 8k+5 (A)Z l7 8k+4 (B)]    −  + iν Eu ,11 (k)f  ilEu ,11 dμ7 (k)  . = filO ,7,A1u (k) A∩B

86

4 Correlation Structures

For m = 7 and 9 ≤ n ≤ 16 E[Z i7 9 (A)Z l7 16 (B)] = −E[Z i7 16 (A)Z l7 9 (B)] = −E[Z i7 10 (A)Z l7 15 (B)] = E[Z i7 15 (A)Z l7 10 (B)] = −E[Z i7 11 (A)Z l7 14 (B)] = E[Z i7 14 (A)Z l7 11 (B)] = E[Z i7 12 (A)Z l7 13 (B)] = −E[Z i7 13 (A)Z l7 12 (B)]   √   3 1 O− ,7,A1u  E ,11 E ,12  − fil u +  . f u (k) + iν Eu ,11 (k) = −fil dμ7 (k) 2 2 il A∩B

For m = 7 and 25 ≤ n ≤ 32 E[Z i7 25 (A)Z l7 32 (B)] = −E[Z i7 32 (A)Z l7 25 (B)] = −E[Z i7 26 (A)Z l7 31 (B)] = E[Z i7 31 (A)Z l7 26 (B)] = −E[Z i7 27 (A)Z l7 30 (B)] = E[Z i7 30 (A)Z l7 27 (B)] = E[Z i7 28 (A)Z l7 29 (B)] = −E[Z i7 29 (A)Z l7 28 (B)]   √   3 1 O− ,7,A1u  E ,11 E ,12 u u   . f f (k) + iν Eu ,11 (k) − = −fil dμ7 (k) 2 il 2 il A∩B

For m = 7 and 33 ≤ n ≤ 40 E[Z i7 33 (A)Z l7 40 (B)] = −E[Z i7 40 (A)Z l7 33 (B)] = −E[Z i7 34 (A)Z l7 39 (B)] = E[Z i7 39 (A)Z l7 34 (B)] = −E[Z i7 35 (A)Z l7 38 (B)] = E[Z i7 38 (A)Z l7 35 (B)] = E[Z i7 36 (A)Z l7 37 (B)] = −E[Z i7 37 (A)Z l7 36 (B)]   √   1 3 O− ,7,A1u  E ,11 E ,12   . f u + f u = (k) + iν Eu ,11 (k) −fil dμ7 (k) 2 il 2 il A∩B

For m = 7 and 41 ≤ n ≤ 48 E[Z i7 41 (A)Z l7 48 (B)] = −E[Z i7 48 (A)Z l7 41 (B)] = −E[Z i7 42 (A)Z l7 47 (B)] = E[Z i7 47 (A)Z l7 42 (B)] = −E[Z i7 43 (A)Z l7 46 (B)] = E[Z i7 46 (A)Z l7 43 (B)] = E[Z i7 44 (A)Z l7 45 (B)] = −E[Z i7 45 (A)Z l7 44 (B)]   √   3 Eu ,12 1 Eu ,11 O− ,7,A1u    . f (k) + iν Eu ,11 (k) − fil − = fil dμ7 (k) 2 2 il A∩B

4.8 Correlation Structures

87

For m = 5 and k ∈ {0, 2} E[Z i5 8k+1 (A)Z l5 8k+8 (B)] = −E[Z i5 8k+8 (A)Z l5 8k+1 (B)] = −E[Z i5 8k+2 (A)Z l5 8k+7 (B)] = E[Z i5 8k+7 (A)Z l5 8k+2 (B)] = −E[Z i5 8k+3 (A)Z l5 8k+6 (B)] = E[Z i5 8k+6 (A)Z l5 8k+3 (B)] = E[Z i5 8k+4 (A)Z l5 8k+5 (B)] = −E[Z i5 8k+5 (A)Z l5 8k+4 (B)]   E u ,11 dμ5 (k)  . = iν E u ,11 (k)f il A∩B

For m = 5 and 9 ≤ n ≤ 16 E[Z i5 9 (A)Z l5 16 (B)] = −E[Z i5 16 (A)Z l5 9 (B)] = −E[Z i5 10 (A)Z l5 15 (B)] = E[Z i5 15 (A)Z l5 10 (B)] = −E[Z i5 11 (A)Z l5 14 (B)] = E[Z i5 14 (A)Z l5 11 (B)] = E[Z i5 12 (A)Z l5 13 (B)] = −E[Z i5 13 (A)Z l5 12 (B)]   √   3 1 E ,11 E ,12 u u  − fil  . f + = iν Eu ,11 (k) dμ5 (k) 2 2 il A∩B

For m = 5 and 25 ≤ n ≤ 32 E[Z i5 25 (A)Z l5 32 (B)] = −E[Z i5 32 (A)Z l5 25 (B)] = −E[Z i5 26 (A)Z l5 31 (B)] = E[Z i5 31 (A)Z l5 26 (B)] = −E[Z i5 27 (A)Z l5 30 (B)] = E[Z i5 30 (A)Z l5 27 (B)] = E[Z i5 28 (A)Z l5 29 (B)] = −E[Z i5 29 (A)Z l5 28 (B)]   √   1 3 E ,11 E ,12   . f u − f u = iν Eu ,11 (k) dμ5 (k) 2 il 2 il A∩B

For m = 5 and 33 ≤ n ≤ 40 E[Z i5 33 (A)Z l5 40 (B)] = −E[Z i5 40 (A)Z l5 33 (B)] = −E[Z i5 34 (A)Z l5 39 (B)] = E[Z i5 39 (A)Z l5 34 (B)] = −E[Z i5 35 (A)Z l5 38 (B)] = E[Z i5 38 (A)Z l5 35 (B)] = E[Z i5 36 (A)Z l5 37 (B)] = −E[Z i5 37 (A)Z l5 36 (B)]   √   1 3 E ,11 E ,12 u u   . f f + = iν Eu ,11 (k) dμ5 (k) 2 il 2 il A∩B

88

4 Correlation Structures

For m = 5 and 41 ≤ n ≤ 48 E[Z i5 41 (A)Z l5 48 (B)] = −E[Z i5 48 (A)Z l5 41 (B)] = −E[Z i5 42 (A)Z l5 47 (B)] = E[Z i5 47 (A)Z l5 42 (B)] = −E[Z i5 43 (A)Z l5 46 (B)] = E[Z i5 46 (A)Z l5 43 (B)] = E[Z i5 44 (A)Z l5 45 (B)] = −E[Z i5 45 (A)Z l5 44 (B)]   √   3 1 E ,11 E ,12  − fil u −  . f u = iν Eu ,11 (k) dμ5 (k) 2 2 il A∩B

Theorem 4.11 Let d( x ) be a homogeneous and (O × Z 2c , A2u ⊕ E u )-isotropic random field. We have d( x ) = 0 . In the basis (3.8), the two-point correlation tensor of the field has the form d( x ), d(y ) =

7 



c

 y − x)fO×Z 2 ,m,A1g (k)  dμm (k)  j A1g (k,

m=0 ˆ 3 (R /O×Z 2c )m

+





2 

Eg  jk1 (k, ˆ 3 /O×Z c )m ( R 2 k=1 m∈{1,3,4,5,6,7}

y − x)

2 

 E g ,qk dμm (k)  , ν E g ,q1 (k)f

q=1

 y − x) and j E g (k,  y − x) are given by where the spherical Bessel functions j A1g (k, E g ,ik E u ,ik and f are given by Eqs. (4.25), Eqs. (4.60) and (4.61), and the basis tensors f (4.26), and (4.28). The field has the form 1  d( x) = √ 6 i=1 m=0 n=1 3

7

48



 x) dZ imn (k)f  i, jn (k,

ˆ 3 /O×Z c )m (R 2

 x) are 8 different combinations of cosines and sines of k1 x1 , k2 x2 , and where jn (k, k3 x3 for 1 ≤ n ≤ 8, of k1 x1 , k2 x3 , and k3 x2 for 9 ≤ n ≤ 16, of k1 x3 , k2 x1 , and k3 x2 for 17 ≤ n ≤ 24, of k1 x2 , k2 x3 , and k3 x1 for 25 ≤ n ≤ 32, of k1 x3 , k2 x1 , and k3 x2 for  are centred real33 ≤ n ≤ 40, and of k1 x3 , k2 x2 , and k3 x1 for 41 ≤ n ≤ 48. Z imn (k) c 3 ˆ valued random measures on (R /T × Z 2 )m with the following nonzero correlations. For m ∈ {0, 2}  O×Z c ,m,A g  dμm (k)  . fil 2 (k) E[Z imn (A)Z lmn (B)] = A∩B

4.8 Correlation Structures

89

For m ∈ {1, 3, 4, 5, 6, 7} and 1 ≤ n ≤ 8, 17 ≤ n ≤ 24 ⎡



O×Z 2c ,m,A g

⎣fil

E[Z imn (A)Z lmn (B)] =

 + (k)

2 

⎤  E g ,q1 ⎦ dμm (k)  . ν E g ,q1 (k)f

q=1

A∩B

For m ∈ {1, 3, 4, 5} and 9 ≤ n ≤ 16 



E[Z imn (A)Z lmn (B)] =

O×Z 2c ,m,A g

fil A∩B

+

2  q=1

 (k)

⎤ √ 1 3 E ,q1 E ,q2 g g   . ⎦ dμm (k) f f ν E g ,q1 (k) − 2 il 2 il 

For m ∈ {1, 3, 4, 5} and 25 ≤ n ≤ 32 

 E[Z imn (A)Z lmn (B)] =

A∩B

+

2  q=1

O×Z 2c ,m,A g

fil

 (k)

⎤ √ 3 1 E ,q1 E ,q2 g g  − f  . ⎦ dμm (k) ν E g ,q1 (k) + f 2 il 2 il 

For m ∈ {1, 3, 4, 5} and 33 ≤ n ≤ 40 

 E[Z imn (A)Z lmn (B)] =

A∩B

+

2  q=1

O×Z 2c ,m,A g

fil

 (k)

⎤ √ 3 1 E ,q1 E ,q2 g  − f g  . ⎦ dμm (k) f ν E g ,q1 (k) − 2 il 2 il 

For m ∈ {1, 3, 4, 5} and 41 ≤ n ≤ 48   O×Z c ,m,A g  fil 2 (k) E[Z imn (A)Z lmn (B)] =

⎤ √ 1 3 E g ,q1 E g ,q2   . ⎦ dμm (k) fil fil + ν E g ,q1 (k) + 2 2 q=1 A∩B

2 



Example 4.7 Let d( x ) be a homogeneous and (O × Z 2c , A2u ⊕ E u )-isotropic random field. Assume that the spectral measure of the homogeneous random field d( x) is absolutely continuous with respect to the Lebesgue measure on the wavenumber ˆ 3 . In that case, the measures μm , 1 ≤ m ≤ 7, are zero measures, because domain R ˆ 3 /O × Z c )m whose dimensions are less than 3. The they are supported by the sets (R 2 c 3 ˆ 3 : 0 ≤ k1 ≤ k2 ≤ k3 }. The ˆ closure of the set (R /O × Z 2 )0 is the set  = { k ∈ R measure μ0 has the form

90

4 Correlation Structures



 dk,  u(k)

μ0 (A) = A

 :  → [0, ∞) is a measurable function with where u(k)   dk < ∞, u(k) 

and where A is an arbitrary Borel subset of . Theorem 4.11 gives d( x ) = 0. Equation (4.60) gives  1 {cos(k1 z 1 )[cos(k2 z 2 ) cos(k3 z 3 ) + cos(k2 z 3 ) cos(k3 z 2 )] 6  + cos(k2 z 1 )[cos(k1 z 3 ) cos(k3 z 2 ) + cos(k1 z 2 ) cos(k3 z 3 )] + cos(k3 z 1 )[cos(k1 z 2 ) cos(k2 z 3 ) + cos(k1 z 3 ) cos(k2 z 2 )]} c  k)  dk,  × fO×Z 2 ,0,A1g (k)u(

d( x ), d(y ) =

 is a measurable function on  that takes values in the set (4.38). where fO×Z 2 ,0,A1g (k) This set is the closed interval with extreme points f Ag ,11 and √12 f Ag ,21 , where the above matrices are given by (4.24). We obtain c

 = λ(k)f  Ag ,11 + fO×Z 2 ,0,A1g (k) c

 A ,21 1 − λ(k) f g , √ 2

 :  → [0, 1] is an arbitrary measurable function. The two-point correlawhere λ(k) tion tensor of the random field d( x ) in the basis (3.8) takes the form  d( x ), d(y ) = {cos(k1 z 1 )[cos(k2 z 2 ) cos(k3 z 3 ) + cos(k2 z 3 ) cos(k3 z 2 )] 

+ cos(k2 z 1 )[cos(k1 z 3 ) cos(k3 z 2 ) + cos(k1 z 2 ) cos(k3 z 3 )] + cos(k3 z 1 )[cos(k1 z 2 ) cos(k2 z 3 ) + cos(k1 z 3 ) cos(k2 z 2 )]} 1 0 0

 dk × 0 0 0 u 1 (k) 000 + {cos(k1 z 1 )[cos(k2 z 2 ) cos(k3 z 3 ) + cos(k2 z 3 ) cos(k3 z 2 )] 

+ cos(k2 z 1 )[cos(k1 z 3 ) cos(k3 z 2 ) + cos(k1 z 2 ) cos(k3 z 3 )] + cos(k3 z 1 )[cos(k1 z 2 ) cos(k2 z 3 ) + cos(k1 z 3 ) cos(k2 z 2 )]} 0 0 0

 dk,  × 0 1 0 u 2 (k) 001

where

   = λ(k) u(k), u 1 (k) 6

 λ(k)  = 1 −√  u 2 (k) u(k). 12 2

4.8 Correlation Structures

91

The random field d( x ) has the form d( x) =

3  48   i=1 n=1



 x) dZ n(i) (k)f  i, jn (k,

 x) are described in Theorem 4.11, the tensors fi are given where the functions jn (k, (i) by (3.8), and Z n are centred real-valued random measures on  with nonzero correlations  (1) (1)  dk , E[Z n (A)Z n (B)] = u 1 (k) A∩B  (2) (2)  dk , E[Z n (A)Z n (B)] = u 2 (k) A∩B   dk . E[Z n(3) (A)Z n(3) (B)] = u 2 (k) A∩B

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94

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Index

B Balance laws, 5 Boundary condition uniform Dirichlet, 23 uniform mixed-orthogonal, 23 uniform Neumann, 23

C Cauchy stress, 6 Charge density, 3 Conjugacy class, 25 Continuity conditions, 12 Convective derivative, 4 Convolution product, 14 Correlation tensor one-point, 27 two-point, 27 Crystal system cubic, 40 orthotropic, 40 tetragonal, 40

D Deviator, 31 Displacement-potential process admissible, 17 kinematically admissible, 17

E Electric current, 3 Electric displacement, 11 Electric displacement field, 3 Electric enthalpy, 22

Electric field, 3 Electric potential, 11 Electric potential function, 10 Electromagnetic surface traction, 6 Entropy, 6 Equivalent representations, 29

F Fine Structure Theorem, 62 Fixed point set, 26 Fourier expansion, 62

G Galilean transformation, 3 Godunov–Gordienko matrix, 31 Governing equations, 12 Group − Z 2n , 26 O− , 26 O(2)− , 26 O(3), 25 SO(3), 26 conjugate, 25 cyclic, 26 dihedral, 26 icosahedral, 26 octahedral, 26 prismatic, 26 pyramidal, 26 reflection, 26 tetrahedral, 26

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Malyarenko et al., Random Fields of Piezoelectricity and Piezomagnetism, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-3-030-60064-8

95

96 H Helmholtz free energy density, 6

I Intertwining operator, 29 Invariant subspace, 29

K Kinematic condition, 10

M Magnetic field strength, 3 Magnetic induction, 3 Maxwell’s equations, 3 in the moving frame, 4 Mesoscale, 20

N Normaliser, 26

O Orbit, 25

P Permeability under constant strain, 9 under constant stress, 9 Permittivity under constant strain, 9 under constant stress, 9 Peter–Weyl Theorem, 62 Piezoelectric process, 12 admissible stress, 18 dynamically admissible stress, 19 kinematically admissible, 17 Piezoelectricity class, 25 Pseudo-vector, 31

R Random field, 27 mean-square continuous, 27 second-order, 27 wide-sense K -isotropic, 27 wide-sense homogeneous, 27 Random material, 20 Random tensor, 27 Rate of heat source, 6

Index Rayleigh expansion, 63 Representation completely reducible, 29 conjugate, 35 irreducible, 29 of O A1 , 39 A2 , 39 E, 39 T1 , 39 T2 , 39 of O × Z 2c A1g , 39 A2u , 39 E g , 39 E u , 39 T1u , 39 T2g , 39 of O− A1 , 39 E, 39 T2 , 39 of T A, 38 E, 38 T , 38 1 E, 38 2 E, 38 of T × Z 2c A g , 38 Au , 38 E g , 38 E u , 38 Tg , 38 Tu , 38 1 E , 38 g 2 E , 38 g of O(3) H∗ , 30 H , 30 of D2 A, 35 B1 , 35 B2 , 35 B3 , 35 of D2 × Z 2c A g , 36 Au , 36 B1g , 36 B1u , 36 B2g , 36 B2u , 36 B3g , 36

Index B3u , 36 of D4 A1 , 36 A2 , 36 B1 , 36 B2 , 36 E, 36 of D4 × Z 2c A1g , 37 A1u , 37 A2u , 37 B1g , 37 B1u , 38 B2g , 37 B2u , 38 E g , 37 E u , 37 of D4h A1 , 37 B1 , 37 B2 , 37 E, 37 of Z 1 A, 30 of complex type, 35 of quaternionic type, 35 of real type, 35 orthogonal, 26 S Scalar, 31 Spherical Bessel function for O, 66 for O− , 67 for T , 64, 65 for D2 , 63, 64 for D4 , 64 for D4 × Z 2c , 64

97 for D4h , 64 Stationary subgroup, 25 Strata ˆ 3 /O(3), 46 of R ˆ of R3 /O × Z 2c , 49, 50 ˆ 3 /T × Z c , 49 of R 2 ˆ of R3 /D2 × Z 2c , 47 ˆ 3 /D4 × Z c , 47, 48 of R 2 Stratified space, 46 Surface heat flux, 6 Symmetry conditions, 11

T Temperature, 6 Tensor compliance, 9 direct piezoelectric, 9 direct piezomagnetic, 9 magnetoelectric, 9 reverse piezoelectric, 9 reverse piezomagnetic, 9 stiffness, 9

V Vector displacement, 11 traction, 11 Velocity field, 6 Volume average, 21 Volume electric polarisation, 3 Volume magnetic polarisation, 3

W Wavenumber domain, 42