Nuclear Quadrupole Resonance Spectroscopy [1964 ed.] 9780126077612

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NUCLEAR RESONANCE

QUADRUPOLE SPECTROSCOPY

SOLID STATE PHYSICS Advances in Research and Applications Editors FREDERICK

SEITZ

DAVID

Department of Physics University of Illinois Urbana, Illinois

General Electric Research Laboratory Schenectady, New York

SUPPLEMENT T. P. DAS

TURNBULL

1

AND E. L. HAHN: NUCLEAR QUADRUPOLE RESONANCE SPECTROSCOPY

ACADEMIC NEW

PRESS YORK

INC.,

PUBLISHERS

« LONDON

« 1958

NUCLEAR QUADRUPOLE RESONANCE SPECTROSCOPY

T.

P.

DAS

Saha Institute of Nuclear Physics, Calcutta, India

E. Department of Physics,

L. HAHN

University of California, Berkeley, California

ACADEMIC PRESS INC., PUBLISHERS NEW YORK e LONDON « 1958

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Preface

Nuclear quadrupole resonance is one of the special branches of radiofrequency resonance physics that is being used more and more as an analytical tool for solid state studies and structural chemistry. A great deal of material on the analysis of compounds and crystals exists in the literature that considers the point of view of purely magnetic interactions. The nuclear electric quadrupole moment plays a minor role, or is absent, and the nuclear magnetic moment interactions are chiefly important. This review emphasizes the pure nuclear quadrupole interactions with crystalline and molecular electric fields, and magnetic interactions play a minor role. We have attempted to relate the theory, applications, and important examples of nuclear quadrupole studies and to present this information in a systematic fashion as a useful source of reference for the general investigator. We wish to thank the U. 8. Office of Naval Research and the U.S. National Security Agency for their support during the preparation

of this review at Berkeley. One of the authors (T. P. D.) would like to express his indebtedness to Professor R. Bersohn of Cornell University for introducing him to the subject of nuclear quadrupole resonance, and to the U. S. Atomic Energy Commission for support of early work on this review while at Cornell. Several suggestions made by Professor Bersohn have been incorporated in the review. We are grateful to Professor M. Cohen and Professor F. Reif of The University of Chicago for valuable suggestions and correspondence, and to Dr. R.

Livingston

of the Oak

Ridge

National

Laboratory

and

Dr.

D.

C.

Douglass of Cornell University for informing us of several of their results prior to publication. We wish to thank Mrs. E. Thornhill for typing the entire text and for assistance in proofreading. We would also like to thank Mr. W. E. Blumberg, Mr. Marvin Weber, and Mr. E. G. Wikner, of the University of California, for helpful comments

and criticism of the text. T. E. Calcutta, India Berkeley, California March, 1958

P. Das L. Haun

Contents

Preface.

.

Introduction

.

I. Theory 1. Frequencies and Intensities of Pure Quadrupole Spectra . . a. Hamiltonian. . . b. Pure Quadrupole Spectra for ‘Axially Symmetric Field Gradients c Zeeman Splitting of the Quadrupole Spectra. Case of Axially Symmetric Field Gradient. . Loe d. Pure Quadrupole Spectra with Nonaxial Field Gradients. e. Zeeman Splitting for the Nonaxial Case. . . .... 2. Static Splitting and Broadening of Quadrupole Spectra. a. Splitting and Broadening of Spectra Due to Magnetic Interactions b. Electrical Sources of Broadening. 3. Effects of Internal Motions in Molecular Solids on Nuclear Quadrupole Resonance . . . a. Effects of Torsional Motions of “Molecules . b. Effects of Hindered Internal Rotations on Pure ‘Quadrupole Resonance . . Le 4.

Theory

of Transient

Experiments

in i Quadrupole Resonance.

a. Pure Quadrupole Spin Echo and Free Induction Signals . . b. Spin Echo and Free Induction Signals in the Presence of a Weak Magnetic Field. . c. Relative Advantages of Steady-State and Transient Experiments Il. Instrumentation

64 70 71 77

83

. .

. Regenerative

. Super-Regenerative Method of Detection . . Circuits for High-Frequency Bromine and Iodine Resonances. . Circuitry for Nuclear Quadrupole Echoes and Free Induction

Continuous

Wave Detectors

Ill. Applications of Interest to the Solid State. 6. Information on the Constitution of the a. Number of Chemically Inequivalent b. Interpretation of Zeeman Splittings ing Orientations of Molecules in the vii

83 83 84

.

TD

. .

Coma

5. Apparatus . . General Considerations

.

.

91 93

97

Crystalline Unit Cell . . . 98 Lattice Sites in the Unit Cell 98 and Their Use in DeterminUnit Cell . . 102

Vili

CONTENTS c. Study of Phase Transitions by Quadrupole Resonance. . 7. Interpretation of Nuclear Quadrupole Coupling Data in Terms of Electron Distribution in Free Molecules . a. Townes and Dailey Theory for Interpretation of Nuclear Quadrupole Coupling Constants in Free Molecules. . b. Relation between Asymmetry Parameter and Double Bond Characters of Halogen Bonds in Planar Molecules. . . . 8. Intermolecular Binding in the Solid State. a. Relative Importance of Direct and Indirect Effects of Neighboring Molecules on the Nuclear Quadrupole Interaction. b. Differences in Ionic Characters of Chemical Bonds in the Gaseous and Solid States . ce. Bond Switching. Lo d. Interpretation of Quadrupole Resonance Spectra iin ‘Solid Halogens in Terms of the Intermolecular Binding. . . . 9. Quadrupole Resonance Studies of Nuclei Other Than Halogens . a. Compounds of Group I Elements of the Periodic Table . b. Compounds of Second Group Elements . Lone c. Compounds of Third Group Elements. d. Compounds of Fifth Group Elements . . 10. Broadening of Quadrupole Resonance Lines by Impurities . a. General Considerations . b. Quadrupole Resonance in Irradiated Specimens. c. Quadrupole Resonance in Solid Solutions

Author

Index

.

Subject

Index

.

113 119 120 154 164 164 166 170

TABLES

Taste I. Secular Equations for Pure Quadrupole Interaction. . TaBLeE

II. The

Splitting of Quadrupole

Lines from

Cl*> Nuclei

.

13

at Chemically

Inequivalent Sites in Molecules . . TasiE III. Multiplicity of Cl Quadrupole Lines Due to Chemical Inequivalence of Chlorine Sites in Crystals . TaB.eE IV. Multiplicity of Cl’* Quadrupole Resonances in 1 Crystals Containing H-Bonded Structures. . Tas.e V. List of Substances for Which Zeeman Measurements Have Been Performed to Date Tasie VI. Orientation of p--Dichlorobenzene “Molecules According to ‘Hendricks. . 2... TasLe VII. Orientation of ‘p-Dibromobenzene “Molecules “According to Hendricks. . 2. 2 1). 2 . Taste VIII. Orientation of p-Dichlorobenzene Molecules According to Dean and Pound. | . TaBLeE IX. Values of b, Q, and qat for Pp Electrons i inn Some Atoms. TasBLe X. Value of gat for p Electrons in Some Atoms from Optical Fine Structure Data .

99 100 101 103 116 116 117 126 131

TABLES

TaBLE XI. Properties of AX Br,I.... 2... Taste XII. Asymmetry

Bonds Where A

= C, Si, Ge, Sn; and X = Cl, 148

Parameter

in

a | Number

of ‘Substituted

Organic

159 Chlorine Compounds Compared with Bersohn’s Calculations. Tape XIII. Resonance Frequencies for Cl*> Nuclei in Chlorobenzenes as Functions of the Number of Orthochlorine Neighbors. . 161 TasBLe XIV. A List of Compounds Whose Halogen Quadrupole ‘Coupling Constants Have Been Studied for Both the Gaseous and Solid States. . 167 Taste XV. Data on Nuclear Quadrupole Interaction in Solid Halogens and the Relative Orientations of Molecules from

X-Ray

Diffraction

.

Taste XVI. Quadrupole Interaction Data for the Nuclei of Some Group I Elements. . . 181 TasLe XVII. Quadrupole Interaction Data for the Nuclei of Some Group Il Elements... . . 184 TasLe

XVIII.

Quadrupole

Coupling ‘Constants

of Nuclei

inn Compounds of

Group III Elements . Lo TaBLE XIX. Quadrupole Coupling Constants in ‘Compounds Elements... .. Be

186 of ‘Group

Vv . 194

Introduction

The fundamental analysis and description of nuclear electric quadrupole interactions have been given in a previous article! by Cohen and Reif. They discuss the quantum-mechanical Hamiltonian describing the interaction of the nuclear quadrupole moment with the surrounding charge distribution, and treat the case where the electric quadrupole interaction energy is a small perturbation upon the interaction between the magnetic dipole moment of the nucleus and a large applied magnetic field. We shall deal with the converse case where the quadrupole interaction energy, even in the absence of any applied magnetic field, is strong enough to give rise to resonance frequencies in the radio-frequency range.?* The lines resulting from transitions between these quadrupole levels will be termed quadrupole spectra. Any additional perturbation caused by the magnetic interaction with a small applied magnetic field will produce Zeeman splittings of the quadrupole spectra. The aim of this review is to explain the features of the quadrupole spectra, and to show how they are interpreted in terms of the electron distributions that produce the necessary electric field gradients at nuclei. Furthermore, the spectra will provide information concerning positions of molecules and ions in the crystalline unit cell and their motional behavior. In many cases, the study of quadrupole spectra has revealed new information regarding these features of the solid state. The technique used in experiment is generally that of the field of nuclear magnetic resonance. We shall attempt

to review most of the phenomena studied to date, and to outline the scope of the quadrupole resonance technique from the point of view of its usefulness for investigations on the nature of the solid state. 1M. H. Cohen and F. Reif, Nuclear quadrupole effects in nuclear magnetic resonance. Solid State Phys. 5, 321 (1957). 2 Zero field quadrupole resonance was first observed for Na? in sodium halide molecules by the molecular beam technique [W. Nierenberg, N. F. Ramsey, and 8. B. Brody, Phys. Rev. 70, 773 (1946); W. Nierenberg and N. F. Ramsey, ibid. 72, 1075 (1947)]. § The theory of nuclear quadrupole interactions in solids was first given by R. V Pound [Phys. Rev. 79, 685 (1950)]. The first successful pure quadrupole resonance experiment in solids was performed by H. G. Dehmelt and H. Kriiger for Cl%* nuclei in transdichloroethylene [Naturwiss. 37, 111 (1950); 38, 921 (1951)]. 1

2

INTRODUCTION

The review is divided broadly into three parts. Part I will deal with the phenomenological theory for the frequencies of the quadrupole spectra, the nature of their Zeeman patterns, the line shapes and line

widths,

and

the relaxation

times.

Some

results

will be mentioned

occasionally in this part to illustrate the theory. Part II will deal with the required experimental apparatus for observing the different aspects of the quadrupole spectra. Part III will deal with interpretation of experimental results. Results obtained to date in imperfect and impure solids will also be discussed.

|. Theory

1. FREQUENCIES

AND

INTENSITIES

OF

PURE

QUADRUPOLE

SPECTRA

a. Hamiltonian

As discussed in the article by Cohen and Reif,’ the Hamiltonian for the interaction of the quadrupole moment of a nucleus with the field gradient at its position due to surrounding charges is given by the tensor-scalar product,

HR = Q: VE = ) Qu(VE)a-™

(1.1)

where Q is the tensor defining the quadrupole charge distribution in the nucleus. Its irreducible components in terms of coordinates 2, y, z are given by

Q = srapay Gl! - 1) Qe! = graf MO ia, + il) + Us tilt) Qe 42

(12)

V/6 eQ . VOY ar — 1) WI, + tl,)?.2

The scalar quadrupole moment Q of the nucleus is defined by Casimir‘ as eQ = fp.r2(3 cos? 6:7 —

1)dr;

(1.3)

where p; is the charge density in a small volume element dr; inside the nucleus at a distance r; from the center, and 6,7 is the angle which the radius

vector

r; makes

with

the

nuclear

at the nucleus is defined by the tensor —V,; in Cartesian coordinates, where

spin

axis.

(VE’),

The

having

field gradient

9 components

aV

Yu = onde, and V

(x,

t=

X,Y; 2)

is the electrostatic potential at the nucleus due to the surround-

4H, B. G. Casimir, Interaction between atomic Tweede Genootschap Haarlem 11, 36 (1936). 3

nuclei

and

electrons.

Teyler’s

4

I,

THEORY

ing charges. If the Laplace equation gives Var

+

Vuy

+

Viz

=0

meaning that the electric field at the nucleus is produced entirely by charges wholly external to the nucleus, then (V#’) is a symmetric traceless tensor.! Therefore we can represent the tensor by its five irreducible components as follows: (VE")o

= Vez

1

(VE") 41 (

— —=

+ tVy.)

(V2

iVv6

(1.4)

1 VE") )42 49 = —— — Vyyyy + 2tVay). 2 Vb (Vax ( y)

If one transforms the tensor to a set of principal axes X, resulting irreducible components in the new system are:

Y, Z, the

(VE)o = aV2z = 3eq (VE)a1

=

(1.5)

(VE)i2 = 3>

(Vxx — Vyy) = >t

Conventionally e?¢Q is termed the quadrupole coupling constant for the nucleus in the particular environment under consideration. The asymmetry parameter for the field gradient tensor 7 is defined by the relation Vxx — Vyy 1.6 (1.6) =Vez Throughout our discussions we shall assume that® the condition

(1.7)

[Vxx| < |Vyy| < |Vzz|

applies in the principal axis system. Therefore 7 can vary from 0 to 1, where 7 = 0 corresponds to axial symmetry around Z: Vxx

=

Vyy

Vaz

=

eq

=

—3eq

(1.8)

and 7 = 1 corresponds to the condition:

The components

Vxx

=

0

Vyy

=

—Vaz.

(1.9)

of the field gradient tensor in a laboratory system of

5 C. Dean, Phys. Rev. 96, 1053 (1954); Thesis, Harvard University

(1952).

1.

FREQUENCIES

AND

INTENSITIES

5

coordinates can be obtained from those in the principal axis system given by (1.5), if the Eulerian angles (a, 8, y) describing their relative orientations are known.® The field gradient tensor can be specified

completely in terms of five quantities, namely q, 7, a, 8, and y. Of course we cannot get q from the spectra unless the quadrupole moment

Q of the nucleus is known. Therefore the statement should be amended to read: the features of the nuclear quadrupole interaction tensor are

determined completely by the five quantities e?qQ, 7, a, 8, and y. In the following sections the theory for the determination of these quantities from the observed quadrupole spectra will be described. b. Pure Quadrupole Spectra for Axially Symmetric Field Gradients The features of pure quadrupole spectra and their Zeeman effects have been discussed by a number of authors.!:?57 The convention for the choice of XYZ axis is that given by (1.8). It is then found from (1.1), (1.2), and (1.5) that, if m and m’ represent the magnetic quantum numbers for the nuclear spin in different states, then the matrix elements of 3Cg are

= TOIe*qQ— 1) [3m?2 — I(T +1)] 5mm

(1.10)

where 6mm’ is the Krénecker 4, i.e.

Snmt = 0 =]

msm!

(1.11)

m=rm,

We choose a representation in which the eigenvalues of J, are diagonal. Then the energy levels are given by

E, = A[Bm? — I(I + 1)]

(1.12)

where

_

__€qQ

A = ger — iy

(1.13)

This result holds for both half-integral and integral spins. The energy levels are thus doubly degenerate in m, since the two states Yim have the same energy. For half-integral spins there are J + 4 energy levels

all doubly degenerate, while for integral spins there are J + 1 energy levels where J of these are doubly degenerate, and only one of them with m = 0 is nondegenerate. 5 R. Bersohn, J. Chem. Phys. 20, 1505 (1952). TR. Livingston, Science 118, 61 (1953).

6

Il. THEORY

Transitions between the energy levels given by (1.12) are produced by the application of an oscillating magnetic field that interacts with the magnetic dipole moment of the nucleus, thereby producing a timedependent perturbation. Electric fields are not used because the nucleus® does not have an electric dipole moment. Conceivably, transitions could be brought about by applying oscillating inhomogenous electric fields with which the quadruple moment of the nucleus

would undergo a time-dependent interaction. But, in order to induce a sufficient number of transitions by this procedure, electric field gradients too large for generation in the laboratory would be required. The time-dependent Hamiltonian representing the interaction of the nucleus with the applied radio-frequency magnetic field is given by

H(t) = —yh[Hx(OIx + Hy@ly + H2(t)Iz) where y is the magnetogyric ratio H(t) are the components of the magnetic field 2H, cos wt. Since the momentum of the nuclei are given

for the nucleus, and Hx(t), Hy(t), linearly polarized radio-frequency matrix elements of the spin angular by

=

Mb mm!

= [(I + m(1 Fmt only transitions Am

=

+1

(1.14)

DB onzim

(1.15)

can be induced by the X and Y components

of the rf field; the Z component

produces Am = 0 transitions. The

Am = 0 transitions do not interest us here, since they do not involve any change in energy. The transition probabilities for the Am = +1 transitions, according to Eqs. (1.14) and (1.15), involve the component of the rf field perpendicular to the axis of symmetry of the field gradient. By the conventional time-dependent perturbation theory it can be shown, as in the case of magnetic resonance,’ that maximum probability for the transition between two levels m and m+ 1 results when the rf field frequency w satisfies the Bohr condition w = wn, where Ein4i

=

—_

En

(1.16)

Using (1.12),

wn = 3a (2\m| + 1). There

will be J — } distinct

transition

frequencies

(1.17) for half-integral

8. N. F. Ramsey, ‘‘ Nuclear Moments,” p. 8. McGraw-Hill, New York, 1953. 9G. E. Pake, Nuclear magnetic resonance. Solid State Phys. 2, 24 (1956).

1. FREQUENCIES

AND

INTENSITIES

7

spins and J of them for integral spins. The quantity A, and therefore e’?@Q, can be obtained very accurately by measuring the resonance

frequencies, if assignment of the various m values pertaining to the observed frequencies can be made. The measurement of A is thus possible when two or more of the J — 4 frequencies are observed so that the two m values that satisfy the relation

wr _ 2|mi] + 1 We

2|me|

+

(1.18)

1

can be uniquely determined. For spin J = $ (for example, Cl** and Br”) and J = 1 (N14), the problem of assignment does not arise, because there is only one resonance frequency in each case. For spin

I = $,° no nuclear quadrupole moment is found, and the question of quadrupole resonance frequencies does not arise. The orientation of the axis of symmetry of the field gradient tensor could be found by studying the dependence of the intensity of the quadrupole resonance lines upon the orientation of the rf field with respect to axes fixed in the crystal in which the nuclei are contained. We note (1) that the intensity vanishes when the rf field is along the symmetry axis; and (2) the intensity becomes a maximum when the field is applied in a plane perpendicular to the symmetry axis. However, intensity measurements are neither very convenient nor precise. They are apt to be rather complicated when several lattice sites exist in the unit cell. These sites may have different directions for the axes of symmetry (Cl*> resonance in NaClO; for example, to be discussed in Section 6b). In practice, the Zeeman splittings in a weak constant magnetic field (to be described in Section 1c) are used to locate the symmetry axis. Nevertheless the directional dependence of the intensity may be used as a check on the result obtained from the Zeeman study.

c. Zeeman Splitting of the Quadrupole Spectra.

Case of Axially Sym-

metric Field Gradient When a constant magnetic field Ho is applied at an angle 6 with respect to the symmetry axis, the net Hamiltonian is given by 5K = Kg + Huy, where XQ is given by (1.10), and

Ku

= —hQU, cos 6+

J, sin 6 cos ¢ + J, sin @sin ¢).

(1.19)

The azimuthal angle for H» is given by ¢ for a particular choice of X and Y axis of the principal system, and 2 = yH». The resulting Zeeman pattern is independent of ¢. The magnetic field is considered

8

I.

THEORY

to be weak, i.e., hQ «< e?gQ. We shall consider the cases of half-integral and integral spins separately. (1) Half-integral spins. In the absence of Ho there are J + 4 doubly degenerate energy levels Hm. The Zeeman field removes this degeneracy, and for m > % there are two energy levels, Egm

= A[8m?

— I(T + 1)] F mhQ cos 0

(1.20)

corresponding to the states yi, and Ym, respectively. Actually there is some mixing between the adjacent states y,, and Yn»-1 due to the ZI, and J, terms in (1.19). However, this mixing is only of the first order in Q/A and may be neglected. The case of the y 4; states is somewhat special. They are of the same energy in the absence of the field, but give finite off-diagonal elements for Hy. Therefore, the prescription of the degenerate perturbation theory?° has to be used, and the Hamiltonian 3y must be diagonalized for these states. This leads to zero-order mixing of the states yy, and yy to form new states y, and y_ with energies given by

Ry where

4[$—1a+0| + Fno cos 8

(1.21)

f=[1+ (+4? tan? 6}

and v4 = 4; sin a + py cosa y_ = p_y sin a — p44 COS a

(1.22)

tana = [(f + 1)/(f — D]}t The arrangement of the energy levels is shown schematically in Fig. 1. When the rf field [described by the Hamiltonian 3¢’(é) in (1.14)] is applied, transitions occur between the different energy levels. As

pointed out in Section 1b, the Am = 0 transitions are of no interest, since they involve no change of energy for levels with m > 4. For the

mixed states y, a transition corresponds to the smaller arrow in Fig. 1 and involves a frequency much smaller than the quadrupole resonance

frequency. For most nuclei of interest, with Zeeman fields of the order of 100 gauss, these transitions correspond

to frequencies only of the

order of a hundred kilocycles or less and shall not be considered. With m > +4, the Am = +1 transitions between the +m and +(m-+ 1) levels lead to two frequencies wmt and wm, which replace the single 10L. I. Schiff, “Quantum

Mechanics,” Chapter 7. McGraw-Hill,

New

York, 1955.

1.

FREQUENCIES

AND

INTENSITIES

9

pure quadrupole frequency wm in (1.16). The new frequencies are

ont = 8A (2|m| + 1) + 2 cos 8.

(1.23)

These are symmetric about wm, and each of them may be shown to have -I +I

+ (mt) +(m+1)

Wq'|

3)

Wal

|wg

1

! !

\ 1 I 1

1 Om

Wm

+

Bg

Om

Wij2

Wq

we’

Wa’

Fia. 1. Arrangement of energy levels for half-integral spins for axially quadrupole interaction in the presence of a Zeeman field. The order of levels has been chosen so as to correspond to a positive y. The heights the different spectral lines have no bearing to their actual intensities,

symmetric the energy shown for except for

the aa’8p’ pattern where the 8§’ lines are drawn smaller than the aa’ lines to show

that they are in general weaker.

half the intensity

of the

original

line.

Their

intensities

involve

the

matrix elements | |?, and are seen to be independent of the orientation of the Zeeman field. The Am = +1 transitions between mixed states Y, given by (1.22)

10

I.

THEORY

and the states y,; give four lines, as shown in Fig. 1, with frequencies wo, =

8S

9 cos 6

wp = 91 3 FT 6A 3-f Wa’ = 4 + —z~

9 00s 0

wy =

0 cos 0,

A

4 BET

(1.24) 2 cos 6

The four lines are symmetric about the pure quadrupole frequency «, = 6A/h. The inner pair of lines a, a’ have equal intensities propor-

tional to | |*, while the intensities of the outer pair of 6, 6’ are proportional to ||? Using Eq. (1.22) for the states vy, and y_ and Eq. (1.14) for 3’(¢), the following conclusions may be drawn. a. For a particular orientation of the Zeeman field, both pairs of

components have the maximum intensity when the rf field is at right angles to the symmetry axis. b. The intensities of the two pairs involve f and hence depend on the orientation of the Zeeman field. c. The intensity ratio of the outer to the inner pair is (f — 1)/ (f + 1), which shows that the inner pair is always stronger. This dependence of the intensities of a and 8 components on the Zeeman field indicates that the orientation of the symmetry axis can be determined from intensity measurements, but it is much more convenient from the experimental point of view to study the directional dependence of the Zeeman splittings. It is clear from (1.23) that for spins greater than 3 the Zeeman splitting 2Q cos @ of a pair of lines arising from +m— +(m-+ 1)(m > $) is a maximum for 6 = 0, and zero for 6 = 90°. The splitting is independent of the azimuthal direction of the Zeeman field H> in the principal axis system and is constant for orientations of Ho lying on the surface of a cone with its axis coinciding with the symmetry axis. The splitting is zero when the cone collapses into a plane perpendicular to the symmetry axis. Therefore the symmetry axis may be located by making stereoscopic plots of the Zeeman splitting. The orientational dependence of the splitting of the aa’ and 6§’ lines is somewhat more complicated. We shall now consider the splittings for some special orientations of the

Zeeman field. Henceforth the term locus of constant (or zero) splitting

1.

FREQUENCIES

AND

INTENSITIES

11

shall refer to all the orientations of the magnetic field which produce a given splitting.

a. For 6 = 0°, i.e., f = 1 with the Zeeman field parallel to the symmetry axis, the splittings of the a components to be

are found from (1.24)

Aw(a) = 22.

(1.25)

The 8 components from (1.24) appear to have exactly twice the splitting of the a components. But in this case, with f = 1, the intensities of the 8 components vanish according to (1.22), because there is no mixing between the pure ~4, and y_, states, and 89’ transitions in Fig. 1 are no longer allowed. b. For 6 = 90°, the splittings are given by

Aw(a) = Aw(8) =

UI + 3)2

(1.26)

and again a single pair of lines is obtained. As seen from Eq. (1.23), this is due to the fact that the +3 levels coincide for this orientation of the Zeeman field. ce. For 6 = tan- [2 »/2/(J + 4)], the splittings are given by Aw(a)

= 0

Aw(B) = 6Q.

ll

The lines aa’ coalesce into a single pair at frequency. Equations (1.21) and (1.23) between the energy levels #4, and H_;, equal for this orientation. For the cases of 6 are 0) = tan! (+/2) =

and

0) = tan-! (+/8)

(1.27) the original pure show that the and between Hy of J = $ and $, 54°44’

quadrupole separations and F_ are the values

(1.28)

= 43°19’

respectively. The locus of no splitting of the aa’ components will be defined, therefore, by Ho lying in a right circular cone with an axis coinciding with the symmetry axis and having a semivertical angle 4. This cone of no-splitting is easy to locate because a strong line is obtained at the original unperturbed frequency. The symmetry axis is then easily obtained. The behavior of the splittings in the other orientations described above can be used to confirm the result. When different nonequivalent directions of symmetry axes are present, one can still unravel the loci of no-splitting for each of the axes from stereoscopic plots of the Zeeman pattern, and thus determine the orientations of these axes with respect to axes fixed in the erystal.

12

I.

THEORY

(2) Integral spins. For integral spins, the lowest state yo is nondegenerate even in the absence of a magnetic field. The Zeeman

splittings for all the pure quadrupole lines involved are then given by the formula (1.23), and the analysis for locating the symmetry axis is the same as for the lines arising from the m > $ levels in the case of half-integral spins.

d. Pure Quadrupole Spectra with Nonaxial Field Gradients The analysis of pure quadrupole spectra and their Zeeman effects is more complex when the field gradient does not have axial symmetry. This is actually the general case. In many instances the nucleus under study is situated in axially symmetric positions in the free molecule, but this symmetry is destroyed within the crystal lattice due to intermolecular interactions. Furthermore, even in single molecules the double-bond character of the chemical bonds between atoms often leads to lack of axial symmetry in the field gradient. All the five quantities q, 7, a, 8, and y discussed in Section la now

have to be determined in order to describe the field gradient

com-

pletely. Not only is the field gradient tensor in the principal axis system to be determined, but also the orientations of the three principal axes X, Y, and Z relative to the laboratory system must be found. (1) Half-integral spins. The pure quadrupole spectra now will be discussed for the case of half-integral spin. 3Cg is given by (1.1) which contains the field gradient components given by (1.5). Let m represent the magnetic quantum number with respect to the Z axis of the principal system. Then the matrix elements of 3Cg will be

,

= A[3m? — I(T + 1)] A

¥)

22 9

3e*qQ “10h

3e°qQ

11 54

allowed

(1.36)

5

h4(6 + 5) =— 30h (: + 20)

For nuclei having J = $ (for example, Cl®* and Br7), both e?qQ and 7 13 R, Livingston and H. Zeldes, Tables of eigenvalues for pure quadrupole spectra, spin 5/2. Oak Ridge National Laboratory Report ORNL-1913 (1955).

1.

FREQUENCIES

AND

INTENSITIES

15

cannot be obtained from the pure quadrupole spectrum alone, since only one frequency is present. In such cases Zeeman studies of single crystals are necessary to obtain 7. Of course, when previous knowledge of the molecular and crystal structures permits the assumption of a very small value of 7, then, according to (1.35), one can take e?¢Q (frequency units) as approximately twice the observed frequency.

For J = $ (I'’, for example), two frequencies are available from the pure quadrupole spectra. If both of these frequencies are determined experimentally, Eqs. (1.36) may be used for small 7 in order to

obtain both e?gQ and 7. In particular, any departure of the observed ratio w/w, from the value 2 immediately suggests that 7 is finite. If n is found from (1.36) to be greater than 0.1, then the more accurate Eq. (1.34) has to be used in conjunction with the tabulated eigenvalues

of Cohen” or Livingston and Zeldes.!* For higher spins, e.g., $ and $, there is a maximum of three and four frequencies possible, respectively.

If all of these are determined 7 can be obtained quite accurately, since there are more e’qQ and 7.

than

two

equations

to

obtain

the

two

unknowns

The intensities of the pure quadrupole lines will now be considered. Dean® and Cohen”? have discussed these for various values of J. For any spectral line arising from transitions between the degenerate levels +m and +m’, there will be contributions from all the four transitions shown in Fig. 2. Therefore the transition probabilities for all these four

transitions have to be added to the total intensity Wim’ of the line in question. Therefore

Winm © ||? + ||? J nm! = .-

(1.38)

16

I,

THEORY

The quantities D, G, and J may be evaluated using the angular momentum matrix elements (1.15) and the expansions (1.30) for the states Am, Bim’, etc. Cohen! has tabulated D, G, and J for transitions Am = +1 and Am = +2 in the cases of spins $, $, and $. In the case of spin $, the energy levels are given exactly by (1.32), and analytic expressions may be obtained for D, G, and J directly in terms of 7. In all cases it is found that for the allowed transitions (Am = +1) D, G, and J are positive, and D is small compared to G. Therefore, from (1.38), the maximum intensity would be expected for 6; = 90° and $1 = 0°, which corresponds to the situation where H, is applied along the X axis of the principal system. This result may be compared with the corresponding result for the axially symmetric case, obtained in Section la where 6 = 90° for maximum intensity. Since there was

then no distinction

between

found. For powders, (1.38) reduces to

one has to average W

nm’

X and

x

$Dnm

Y axes, no

+

Wim

¢@ dependence

over 6; and

3Gmm’.

was

¢1, so that (1.39)

When 6; = 7/2, ¢: = 0 in a single crystal, every line m = m’ has its maximum possible intensity, namely, W mm!

=

(Ginm!

+

2S mm!)

In the typical case of J = $, 7 = 0.2, and mm’ = $, $, Cohen’s tables give Dinm' = 0.07, Gam? = 7.51, and Umm = 0.9. This gives Winn = 4.65. The corresponding intensity for a powder is, from (1.39), only about

2.5, so that there is a reduction in intensity by a factor $ from the maximum value for the single crystal. Volkoff and Lamarche noted this point in connection with their calculations on Al?’ resonance in spodumene.!4 Cohen! has shown that some of the forbidden transi-

tions (|Am| > 1) would become visible for 7 > 0.5. A typical case is the § = 4 transition for J = $, where the intensity approaches about zs of the allowed transition $ — # in the neighborhood of yn = 1. (2) Integral spins. We shall now consider the case of integral spins, which are much less abundant than half-integral spins. Integral spin nuclei (I # 0) are rare because they require an odd number of protons and an odd number of neutrons. This combination usually is not favored by nuclear binding energy considerations. Some light nuclei which have integral spins are ;H?, ;Li®, ;B!°, and ;N!*. The deuterium nucleus does not have a large enough quadrupole moment to be of interest to us in our present discussions. The abundance ratio of 3;Li® relative 4G.

M. Volkoff and G. Lamarche, Can. J. Phys. 32, 493 (1954).

1.

FREQUENCIES.

AND

INTENSITIES

17

to 3Li’ is about 1:13 and that of ;B! relative to ;B!! is about 1:5; 7N"

has an abundance of about 99.6%. If one is interested in studying the molecular and crystal structure of lithium and boron compounds by quadrupole resonance, it is convenient to deal with the more abundant isotopes 3Li7 and ;B". This is reasonable because the field gradient at the position of the nucleus in any compound is hardly affected by changing from one isotope!® to another. The case of spin 1 will now be considered in some detail because of its importance for nitrogen compounds. If (1.5) and (1.6) are used, the Hamiltonian 3Cg in the principal axes can be written as

ge = (1,2A’ + 1,2B’ + 1,2C’)

(1.40)

where

A’ =wor Bi =

Vi = 2A ay Vy = —A(L — 1)

C= saggyVe = AG +0) and A and » are defined in Eqs. (1.13) and (1.6). Bersohn® has pointed out that (1.40) resembles the Hamiltonian for the rotation of an asymmetric-top molecule with 2A’, 2B’, and 2C”’ replacing the reciprocal moments of inertia, while 7,, 7,, and 7,, which are integral in the present case, replace the components J:, J,, J, of rotational angular momentum of the molecule. The asymmetric top rotational spectra have been studied in great detail,!® and therefore the features of the quadrupole spectra of integral spins may be obtained by analogy with results of microwave studies. The eigenvalues of the Hamiltonian (1.40) can be written directly by comparison with Eq. (4-4) of Townes and Schawlow’s!* book (denoted henceforth by T and 8).

p= Pty

41) + ( - Bae) w.

(141)

Secular equations giving solutions w for several values of J have been

tabulated in Eq. (4-5) of T and 8. For J = 1, three values for w are 16 §. Geschwind, G. R. Gunther-Mohr, and C. H. Townes [Phys. Rev. 81, 288 (1951)] found small differences in g with different isotopes. This was later explained by T. C. Wang as arising from vibrational effects [Phys. Rev. 99, 566 (1955)], to be discussed in Section 3.1. 16C. H. Townes and A. L. Schawlow, ‘Microwave Spectroscopy,’’ Chapter 4. McGraw-Hill, New York, 1955.

18

I. THEORY

given, namely, w=0

and

w=1+b0

(1.42)

where b = 7/3. For J = 3, there are four equations for w, three quadratic and one linear, which we shall not reproduce here. For small ), one obtains for all integral spins w=

where

mM?

+

cib

+

cob?

m is the magnetic quantum

+

e3b3

+

number

c4b4

+

see

(1.438)

with respect to the prin-

cipal Z. axis of the field gradient tensor, and ¢, c2, c3; have been tabulated in Appendix III of T and 8. It is to be noted that, since 7 can range from 0 to 1, b varies from 0 to $. The relation (1.43) can be con-

sidered, therefore, as approximately valid over the entire range of 7. For spin J =

1, Eqs.

(1.42) give three energy levels,

Eo = (B+

C’) = —-A’ = -2A

and Ey,= (B’ + C’) +(4'-8 aa

+b) =A(1 +7). (1.44)

The subscripts of H denote the magnetic quantum numbers that apply to the pure states in the limit that 7 = 0. The eigenstates are correspondingly

Ao = Wo

and

Agi =

oe

(1.45)

A radio-frequency field applied in any arbitrary direction gives rise to three transitions, namely, Ay1@ Ao, Ao= A-1, and Ayi @ A-4. The first two form a doublet with frequencies

wo, = (4 _B + °) (1+) = Bee ( + 2)

(1.46)

The third transition (A,; @ A_1;) gives rise to a line at a very low frequency and is not of interest here. The separation of the doublet is

Aw _= ?'qQn hk 2 Equations (1.46) and 7 from pure

(1.47)

and (1.47) give enough conditions to obtain e?g¢Q quadrupole measurements alone. This case differs

from that of spin J = $ where study of the Zeeman pattern is essential to obtain 7.

to

The intensities of the two members of the doublet are proportional ||? and ||? respectively, where

1.

FREQUENCIES

AND

INTENSITIES

19

x’ (t) is given by (1.14). The higher frequency line has a maximum intensity when the rf field lies in the X direction, and the low-frequency line then has zero intensity. The opposite is true with the rf field in the Y direction. There will be two lines of equal intensity in a powder, since there is an isotropic distribution in the orientations of the principal axes. e. Zeeman Splitting for the Nonaxial Case (1) Half-integral spins. The Zeeman splitting in the nonaxial case (including the axially symmetric case as a special condition, 7 = 0)

was first treated by Bersohn,® using a perturbation procedure. Dean subsequently

treated

the

theory

of Zeeman

splitting rigorously

and

discussed the case of J = # in detail, since the problem in this case can

be

solved

exactly.

Cohen!?

has

given

numerical

results for the

splitting of the quadrupole spectra for spin J = $, $, and $. When a constant magnetic field H is applied to the nucleus, the net Hamiltonian

is

I = Kg + Ry where 3g is given by (1.1) and 3Cy is given by (1.19). Again, 0, ¢ give the orientation of the steady field in the principal axis system, but now

there is a dependence of the energy eigenvalues upon ¢. The magnetic field is again regarded as weak, i.e., hQ K e?gQ, as in Section 3. Therefore Cy may be regarded as a perturbation on Hg. With A,» and Bim [Eq. (1.30)] as zero-order states, the matrix elements of 3Cy are given by

= — = $2a, cos 6 = $2 sin O(bme* + cnet?)

(1.48)

where Am

=

Pe =

(Chim?

—

(J

$)C,, mn?

+

8Cy,m?

2f1(4) C4, mCym

+

+ +

5C yn?

—

st)

2f1(#) Cj, mC}, m

+

2fr(H)C;,

+

mC ym

2A GC,

mC p,m

+

7

‘]

(1.49)

vo

In these equations the coefficients Cin,’ are the same as given in (1.30), and fr(m) is defined by (1.29). A quadratic equation is obtained, therefore, for the energies of an originally degenerate pair of states Am, Bm. The solutions are Ens

= En(O)

+ ne

[Gm? cos? 6 +

(bm? + Cm? + 2bmCm COS 2p) sin? 6].

(1.50)

20

I.

THEORY

The degenerate energy values /’,,(0) in the absence of the field are given by Eqs. (1.82) and (1.383) for J = $ and $, respectively. The

quantities dm’, bm, and Cm can be obtained from Eq. (1.49) with Cmm given by (1.30). Dean® has obtained an, b,, and Cn, correct to the first power in 7, for any spin J as follows: am = 2m

by = 1+ 2), bn = 0

(for m > 4)

(1.51)

y= y= -T-HU FHT HDG Cm = 0

(for m > #).

For spin $, where exact solutions for Cmm are obtainable, a, = b; =

—1— 2/p 1—1/p ¢; = n/p

a, = by = =

Dean® gets

—1+ 2/p 1+ 1/p

(1.52)

—c4

p = (1 + 7°/3)4. For small y, only first powers of 7 are related, and terms involving the

Eqs. (1.52) reduce to those in (1.51) for J = $. Cohen’? has tabulated values Of Gm?, Dm, Cm?, and bmCm for spins $, $, and $, 7 varying from 0.1 to 1.0 in intervals of 0.1. For small 7, the approximate expressions given by (1.51) may be used without much error. For n = 0, one gets Eqs. (1.20) and (1.21) from Eqs. (1.50) and (1.51). It is to be noted from (1.51) that bj, is zero-order in 7, while b, for m > $ involves powers of » higher than the first. Also both c, and c; are of first order in 7. Since b» and cm contribute to the mixing between the degenerate states A,, and B_,», it is clear from Eq. (1.51) that zero-order mixing occurs only between the states A, and B_;. For large n, however,” there is also appreciable mixing of states A, and B_,» for m > 4. Radio-frequency transitions are treated by using the time-dependent interaction Hamiltonian (1.14). The three transitions Am = 0, +1 are possible for any general orientation of the radio-frequency field. For large 7, all four transitions shown in Fig. 2 are possible. From (1.50), these transitions correspond to four different frequencies,

w = we(m = m2) + F ((mni] + [m,])

(1.53)

where

[m] = +[am? cos? 6 + (bm? + Cm? + 2bmCm Cos 2) sin? 6] and we(m; = m2) is the single quadrupole

(1.54)

frequency given by (1.34).

1.

FREQUENCIES

AND

INTENSITIES

21

In analogy with the notation for the axial case, we shall call the pair with the larger frequency separation Q([m,] + [m.]) the 6 pair, and the pair with smaller frequency separation Q({m1] — [me]) the @ pair. For small 7, and m, and m, both greater than 4, the a pair is strong and the 6 pair extremely weak. This follows from considerations of the mixing of A, and B_,, states discussed above and from the considerations about allowed and forbidden transitions in Section 4. For m, = 4, the magnetic field causes a zero-order mixing of the A, and B_, states. Therefore the intensity W() of the 6 lines is compara-

ble in order of magnitude to W(a), but W(8) < W(a). The difference W(a) — W(@) depends upon the relative orientations of the rf field and Zeeman field with respect to the principal axes. For a detailed discussion of these points the reader is referred to Dean’s paper.® As shown for the case of axial symmetry, a consideration of the locus of no splitting for the a lines is again very useful. The zero splitting of the a lines occurs when [m1] = [me]. Using Eq. (1.54), the equation for the locus (00, ¢o) of no splitting of a components is sin? 0) = An,”

(Ain.? — Dm? = Cm?)

—_—

Am,

.

— (Am? — Bing? — Cm,”) +

(Dinglmy — Ymylm,) COS 20 (1.55)

For small values of », Eq. (1.51) may be used for dm, Dm, and Cn, SO that mM,

=

4

:

Me

=

|

sin’

0

=

ie

S

s+(r+3) —

m

m=

(1.56)

-*(1+3)

('-

Sy(u+ 8) cos 20

3

H| sin? 0) =

%

mi,mM, > 3

wt alll ee 2) .

n?

]

sin? @) =

1.

+3)|

5

(1.57)

(1.58)

These equations show the following features. 1. For the $ = $ transitions, the locus of no splitting is independent of ¢, 1.e. it forms a right circular cone around the Z principal axis with semivertical angle given by (1.57).

22

I,

THEORY

2. For the $ = # transitions, the locus of no splitting is a function of both 6) and ¢o, and forms an elliptic cone around the Z axis. The angle of inclination 60 of the field to the Z axis is a maximum in the XZ plane and a minimum in the YZ plane. It is possible, therefore, to obtain the X, Y, and Z axes by determining the locus of no splitting. In addition, a knowledge of the values of 6) in the XZ and YZ planes, denoted by 60(¢ = 0°) and 6.(@ = 90°), respectively, allows a determination of the asymmetry parameter of y from the following relation:

_

7”

3(8 + U + 3)7]

{=

(i+ 42d — BU + 8)

which is obtained from (1.56). The

An additional relation derived used to determine 7. This is _

7 =

4)(0°) — sin? @5(90°)

[sin? 60(0°) 4 sin? 06(90°)

o

first bracket reduces to 3 for spin $.

from Awy

(1.59)

—

(1.53) Awx

Nor

and

(1.54)

may

also be (1.60)

where Awx and Awy are the separations between the a components when the magnetic field is in the X and Y directions, respectively. This relation may be used once the X and Y axes have been obtained by studying the locus of zero-splitting. However, Dean has pointed out that the measurement of n, using Eq. (1.57), is more reliable than (1.60), because the former involves only measurements of angles while the latter involves small frequency splittings more liable to error. Equation (1.58), of course, can be used as a rough check on the value of 7 obtained by Eq. (1.59). The procedure that is used in applying the results of the last four sections in determining the quadrupole interaction tensor of a nucleus from observed quadrupole spectra will now be summarized.

1. First, the pure quadrupole spectrum is studied. In measuring only the frequency, either a powdered sample or a single crystal may be used, although higher intensities can be obtained with a single crystal, as discussed in Section 4. For J = $, only a single frequency

is present, and it is not possible to obtain both egQ and 7. For higher spins, J — 4, widely different frequencies are obtained which allow a separate determination of egQ and 7. The presence of two or more closely placed frequencies indicates more than one chemically inequivalent site in the unit cell. A lattice site is termed chemically inequivalent to another lattice site if the principal field gradient tensor components at the respective sites are different. Physical inequiva-

1.

FREQUENCIES

AND

INTENSITIES

23

lence shall apply to lattice sites which have at the respective sites the same magnitudes for the principal components of the field gradient,

but whose principal axes are in different directions.

One physically

inequivalent lattice site can be obtained from another by a symmetry operation of the space group of the crystal. Two chemically equivalent sites cannot be obtained from one another by any such operation. Pure quadrupole frequency measurements alone cannot distinguish between physically inequivalent lattice sites. 2. In single crystals, studies may be made of the dependence of the intensity of the pure quadrupole lines on the orientation of the rf field with respect to axes fixed in the crystal. Use of Eq. (1.38) would then enable a rough assignment of the orientations of the principal axes of the field gradient with respect to the axes in the crystal. 3. Zeeman measurements are made on the single crystals. A powder cannot be used, because a distribution in 6 and ¢ leads to a distribution in Zeeman lines, according to (1.53) and (1.54), and produces a broadening. When a Zeeman field H» is applied in a direction (6, ¢) in the principal axis system in a crystal, one sees two pairs of symmetrical lines a and @ in place of the unsplit $ = 4 pure quadrupole lines. The frequencies of the a and @ lines are given by (1.53) and (1.54). A stereographic plot of the splitting of the a2 components for different orientations of the magnetic field are then made with respect to axes fixed in the crystal. As seen from Eqs. (1.25) and (1.26), a two-line pattern (a and 8 patterns coinciding) is obtained when the field gradient has axial symmetry and the field is in a plane perpendicular to the axis of symmetry. Failure to find any such orientation indicates that the field gradient is not axially symmetric. The locus of no splitting of a components is then traced, and the directions of the principal axes with respect to the axes fixed in the crystal are obtained from Eq. (1.56). The asymmetry parameter 7 is then obtained from the locus using Eq. (1.59). It may be confirmed from Eq. (1.60) by measuring the splittings with the Zeeman field along the X and Y principal axes. When there are physically inequivalent sites in the unit cell, a superposition of different Zeeman patterns results, because the applied magnetic field has a different orientation with respect to the different axes. Since the locus of no splitting of a components involves a superposition of two strong lines, such loci can be traced without much difficulty for each of the different sites, and the respective principal axes at the different sites therefore can be obtained separately.

(2) Integral spins. Only spin J = 1 will be considered, since it is of importance in the case of N'* nuclei. The discussion which follows

24

I.

may

be

generalized

easily

for

THEORY

higher

integral

spins.

When

a weak

Zeeman field is applied, the net Hamiltonian is H = 3g + Ky, where Haq and Ky

are given by (1.1) and

(1.19). There

can be two extreme

cases: a. The perturbation in the energy caused by 3C

is small compared

to the separation 2An between the Ax states defined in Eq. (1.45). b. The

perturbation

caused

by 3Cy is large with respect to 2A7.

We shall deal with the case when 3Cy and off-diagonal elements of 3g are comparable. This would include both (a) and (b) as particular cases. The zero-order levels correspond to Ax: and Apo given in (1.45). The energies corresponding to states Ay: or A_1 and A» are widely different, so they do not mix. The energy corresponding to Ao has the

same value given in (1.44), because A_, states, however, give the secular equation

= 0. The Ay: and

(E — A)? — h2Q? cos? 6 — A%? =

(1.61)

which yields the energy levels

Es = A + (h?0? cos? 6 + A%n?).

(1.62)

The eigenstates are

CiAi + C2A_1 and

—C2A, + CiA-1 where C,

=

1

hQ cos 6

[2(h?2Q? cos? 6 + A2n?)4}(h20? cos? @ + A2y?)t — An]

and

(1.63)

Cs = —

(h?Q? cos? 6 + A2n?)t — An

“oGia? cos? 6 + A®?)#]

For the condition An > hQ cos 6, the energy levels reduce to Es

= A

+ hQ cos 6

(1.64)

and the eigenstates yi: and y_1 become the unmixed states for magnetic quantum numbers +1. In the general case, two resonance lines occur with splitting

bw = ; (h2.Q2 cos? 6 + A?)4.

(1.65)

1.

FREQUENCIES

AND

INTENSITIES

25

The splitting dw reduces to 2An for very weak fields (hQ cos 6 An) and to 240 cos @ for strong fields (AQ cos 6 >> An). No Zeeman studies have been reported on N#* quadrupole resonance. The quadrupole frequencies obtained in a few cases!’ indicate that the value of A is in the region 1 to 2 me. For N"™ the frequency Q is about 300 ke per kilogauss of applied field. Hence, with an asymmetry of 7 = 0.1, An is of the order of 100 ke, and a field of about 300 gauss is necessary to satisfy the condition AQ cos 6 ~ An. The additional splitting of the lines due to the field is a maximum when @ = 0°. The Z axis of asymmetry can be determined from stereographic study of the Zeeman splitting, but, since there is no dependence of the splitting on ¢, the azimuthal angle for the Zeeman field in the principal axis system, the

X and Y principal axes cannot be obtained from Zeeman study. The dependence of intensity of the pure quadrupole lines on the orientation of the rf field could be used for the ¢ determination, as discussed in Section 4. (3) Intermediate fields. In the treatment of the Zeeman effect

adopted in this section and in Section 3, the Zeeman field is regarded as a weak perturbation on the pure quadrupole energy levels derived in Sections 2 and 4. In the case when Ky is comparable to He, a perturba-

tion procedure is no longer justified. A secular equation must then be solved for the entire Hamiltonian 3Cg + Ky in order to obtain the correct energy levels. Parker!® has carried out such calculations for I = 1andI = $ in the case of axial symmetry, and has tabulated the energy levels over the entire range 0 to © of the parameter A = A/2hQ. In their article,! Cohen and Reif discuss the region in the neighborhood of \ = 0, while the region in the neighborhood of \ = o is that considered by us in Sections 2 and 3. Lamarche and Volkoff!® have dealt with the case of J = $ over the entire range \ = 0 to for a finite value of 7. The reader is referred to these papers for details. It should be noted that in most molecular solids, barring compounds of a few elements like boron, aluminum, nitrogen, or in the case of halogen compounds with a strong ionic character, the condition \ = 1 requires very large magnetic fields. If one is interested only in studying quadrupole resonance in molecular solids with a view to obtaining information about structure and not about details necessary for confirmation of quadrupole spectra theory, then investigations in the weak Zeeman region are sufficient for this purpose. 7G,

D. Watkins and R. V. Pound, Phys. Rev. 85, 1062 (1952),

18 P, M. Parker, J. Chem. Phys. 24, 1096 (1956).

19 G, Lamarche and G. M. Volkoff, Can. J. Phys. 31, 1010 (1950).

26

T.

2. Static

SPLITTING

The discussion of only to isolated nuclei magnetic interactions are perturbations due

AND

THEORY

BROADENING

OF QUADRUPOLE

SPECTRA

quadrupole spectra in the last section applies situated in a perfect and rigid lattice. Actually can occur among neighboring nuclei. Also there to lattice impurities, dislocations, order-disorder

effects, and vibrational and rotational motions of the molecules.

In

this and the following Section 3, the effects of these perturbations will be considered. It will be shown that they destroy the monochromatic character of lines in the quadrupole spectra and lead to additional splittings, shifts, and distributions in frequencies. Of course there may exist multiplicities of the lines due to the presence of inequivalent lattice sites, as discussed in Section le, but these splittings are usually an order of magnitude greater than the splittings and broadenings to be discussed here. If in special cases the splittings due to inequivalent sites are comparable to splittings resulting from the above perturbations, then the former will be difficult to discern in the

observed spectra. a. Splitting and Broadening of Spectra Due to Magnetic Interactions There are two types of magnetic interactions between the nucleus

whose spectrum is being studied (hereafter described as the resonant nucleus) and other surrounding nuclei of the same or different species (nonresonant nuclei). First, a direct magnetic dipole-dipole interaction between neighboring nuclei?® may exist. When the resonant nucleus has one very close neighbor, then the magnetic dipole-dipole interaction

with it predominates over that of distant neighbors, and a

fine splitting

of the quadrupole spectra is produced, which is analogous to the Pake splitting in nuclear magnetic resonance.*! A close neighbor may lie in the same molecule in which the resonant nucleus is contained or in an adjacent molecule. Not many such examples of observed dipolar splittings of quadrupole spectra have been reported. In fact, the one case2? in which it has been established conclusively is HIO; (iodic acid), where a splitting occurs due to interaction between the proton and the I'27 nucleus. The magnetic dipole-dipole interaction between the resonant nucleus and other distant neighboring nuclei, other than the very close neighbor, broadens the fine components produced by the action of the close neighbor. When there is no particularly close 20 See, for example, p. 39 in ref. 9. 21G, E. Pake, J. Chem. Phys. 16, 327 (1948). 22 R, Livingston and H. Zeldes, J. Chem. Phys. 26, 351 (1957).

2.

STATIC

SPLITTING

AND

BROADENING

27

neighbor, only a general broadening of the quadrupole and Zeeman splitting spectra occurs. The second type of magnetic interaction that may be present is the

indirect spin-spin interaction between

two nuclei

(hereafter termed

the J interaction), which originates from the magnetic interactions of two nuclei with the electrons that form the chemical bond between them.?3 Such interactions also can occur between nuclei in atoms not directly bonded to one another but through other atoms. In such cases, however,?? it will be weak and its effects on the quadrupole spectra may not be detectable. The J interaction, therefore, can lead to an appreciable fine splitting of the quadrupole spectra, if its short-range effect is sufficiently strong to overcome the direct dipole-dipole interaction. A broadening*4 due to J interaction can take place only if it occurs with equivalent strength between a resonant nucleus and a number of neighbors. This can happen in metals or in ionic crystals,4 but not in molecular solids where usually very little direct bonding between molecules exists. Hence, there can only be some splitting due

to J interaction between nuclei within molecules. Some examples of splitting of pure quadrupole spectra due to J interaction are given in the literature.?*:?6 These occur between the two I!” nuclei in iodine

molecules, between Br’? nuclei in bromine molecules in the solid, and between I'!?7 and Cl*> nuclei in solid ICI. (1) Indirect spin-spin J-interactions. There are two cases to consider, namely, indirect spin-spin interaction of a resonant nucleus with another of the same species, or with another nucleus of a different species. These two cases are illustrated by solid iodine and BCl; respectively, mentioned previously. Interactions between like nuclei will be considered first. For simplicity, only the case of axially symmetric quadrupole interaction of the resonant nuclei is considered. For a pair of interacting nuclei 1 and 2, the total quadrupolar Hamiltonian is given from (1.10) by

Re = Al(38l.,2 — 1,7) + (31,2 — 1.7)] where A

= e?qQ/4/(2I

—

(2.1)

1), and J refers to the equal spins of the two

23 See p. 62 in ref. 9; also N. F. Ramsey, Phys. Rev. 91, 303 (1953); H. McConnell, J. Chem. Phys. 24, 460 (1956). *4 Refer to N. Bloembergen and T. J. Rowland, Phys. Rev. 97, 1679 (1955). 258. Kojima and K. Tsukada, J. Phys. Soc. Japan 10, 591 (1955). 26G.

D. Watkins

and

R.

M.

Walker,

Bull. Am.

Phys.

Soc.

[2] 1, 11

(1956).

28

I.

THEORY

nuclei. The J interaction is given by a small perturbation Hamiltonian, Ky

where

=

J indicates

JI,

I,

=

J (Lieloe

the

strength

+

Lislos

of spin-spin

+

LyT oy)

interaction.2*

(2.2)

The

two

nuclei will be considered to have spins $, which is the simplest possible case. In order to obtain the energy levels and eigenstates of the problem, a 16 by 16 determinant would have to be solved if one started out with a system given by the eigenstates Yn,Wm, of the Hamiltonian (2.1). Here yn, and Yn», refer to the eigenstates defined by the magnetic quantum numbers m; and mz of the two spins. But if one considers the states Yyw [where J = (I, + Iz), and J(J + 1) and M represent the eigenvalues of J? and Jz, respectively], only linear and quadratic equations have to be solved. These secular equations each are characterized by the eigenvalues M of Jz, since Jz commutes with both Hq and 3,7. Quadratic secular equations arise from mixing of the J = 3 and J = 1 states for both M = 1 and M = 0, and from mixing of

J = 2

and J = 0

states for M = 0. The details of the algebra will

not be given here. The frequencies that result in place of the single frequency w, given by (1.16) are w, — 2.57, w, — 1.5J, wo, wo + 1.5J, and w, + 2J. These represent five components at separations —2.5,/,

—1.5J/, 0, 1.5/7, and 2/ from the central component. These splittings are not in very good agreement with the pattern for iodine observed

by Kojima

e¢ al.,25 who

obtained a five-line pattern for the $@4

transition with separations —24 kc, —8 ke, 0, +12 kc, and +26 ke from the center. However, only spin } nuclei have been considered here, and the predicted pattern for these may be different from those for spin $ nuclei. Besides, we have neglected a small tensor?* part of the J interaction which is eliminated in liquids due to random motion but persists in the solid state. Some splitting could occur due to direct magnetic dipole-dipole interaction between the iodine nuclei; how-

ever, it will be shown in Section 2a(2) that this cannot lead to splittings greater than

1 kc. Kojima

and his group?’

mention

that they

have observed splittings in the Br’? resonance spectrum of solid bromine, which may be ascribed to indirect spin-spin interaction between

Br’® nuclei in the molecule. Detailed results have not been published yet.

The splitting due to J interaction between a resonant nucleus and another of a different species will now be discussed. Since we are not

interested in transitions of the nonresonant nucleus, only the quadrupole

interaction

of the resonant

one need

27 Refer to the concluding paragraph of ref. 25.

be considered,

together

2.

STATIC

SPLITTING

AND

BROADENING

29

with the perturbation due to the spin-spin interaction with the nonresonant neighbor. Thus HR =KHagt Ks

(2.3)

where

Ke = A(3l.,? — Ih’) and ay

=

J (L121

22

+

TizDoz

+

L,I

ey)

(2.4)

where 1 refers to the resonant nucleus and 2 to the nonresonant one.

Spins $ for both nuclei will be considered first. The handling of the Hamiltonian 3 in (2.3) is again somewhat complicated due to degeneracies in the yi, states of the resonant spin. We will again omit the details of the algebra and mention the results. The Fz, and HE, levels, given by (1.12) for the Hamiltonian 3g only, are each split by 3, into four doubly degenerate levels characterized by mixtures of states WmWm,

Where

m,

and

m,

refer

to the

magnetic

numbers

of the

two

spins. Transitions induced by an rf field at the resonance frequency of the resonant nuclei then lead to six lines with frequency separations —4J, —2.55J, —1.05J, 0, 1.25/, 2.757 from the pure quadrupole

frequency given by (1.17). For interaction with nonresonant nuclei with spin IZ, = 4, two lines with separation —1.5/J and 0.5J from the pure quadrupole frequency (1.17) are obtained. No detailed reports concerning the splitting of pure quadrupole spectra due to indirect

spin-spin interaction between unlike nuclei are available in the literature. Kojima?’ and his co-workers mention the case of IC]. Here we have two unlike nuclei with spins 7; = $3 and J, = 3. The theoretical pattern to be expected may be obtained exactly by a procedure adopted for the simpler cases discussed above. But we shall not analyze it here, since no detailed description of the experimental pattern has yet been published. It has been pointed out that the only instance where indirect spin-

spin interaction between unlike nuclei occurs conclusively is in the case

of BCl;.*!

Livingston

the Cl*® spectrum.

Crystal

noticed

two

structure

lines

data

separated

indicate

by

that

5 kc

in

there

is

only one molecule in the unit cell with all three chlorines equivalent, so the splitting cannot be attributed to inequivalent sites. Recently

Douglass*! has re-examined the Cl** spectrum in BCl; and finds six lines. This result agrees with the prediction that the splitting is due to J interaction. The direct magnetic dipole interaction between the B!! and Cl*5 nuclei, discussed in the next section, can lead to a splitting of only 1 ke. Douglass, however, points out that his line-shape pat-

30

I. THEORY

tern can be explained also by the dipole-dipole interaction between B" and Cl* nuclei, assuming a large asymmetry 7 in the field gradient at the Cl** nucleus. In this section we have discussed only the case of axial symmetry. Exact values of J are difficult to calculate owing to lack of knowledge of the proper molecular wave functions. But from considerations based upon the mechanism of the indirect spin-spin interaction,?* Bloembergen and Rowland*! propose that its importance is measured by viv2(A1A2/AE), where y1 and v2 are magnetogyric ratios of the two nuclei, and A; and A, ** are hyperfine interaction constants for the outermost S electrons of the atoms. Here AH is the excitation energy from the ground singlet state of bonding electrons to the lowest triplet state. The hyperfine interaction constant A increases from atom to atom roughly as the cube of the effective nuclear charge that the valence electrons see,?* while AF is relatively constant. The spinspin interaction, therefore, would be expected to be of increasing importance for bonds between heavier atoms. Thus, one may expect fairly large spin-spin interactions in the As, Sb, and Bi halides, particularly in the bromides and iodides. The halogen quadrupole spectra in some of these compounds have been studied®° with super-regenerative spectrometers, which are not, however, very satisfactory for studying the fine splittings and line shapes of quadrupole spectra. For measurement of effects of indirect spin-spin interaction, it would be interesting to restudy the halogen spectra in these compounds with a regenerative spectrometer.*! The bromides and iodides of boron would also be interesting, since the Br’ and I!?7 nuclei would be expected to have larger spin-spin interactions with B™ than Cl**, In hydrogen halides, the spin-spin interaction should be small, since the hydrogen atom does not have a very large hyperfine interaction. Moreover it will be seen that the direct magnetic dipole-dipole interaction is expected to be relatively more important in these cases. (2) Magnetic dipole-dipole interaction: splitting effects. A rough measure of the magnetic dipole-dipole interaction of a nucleus with its neighbor is obtained from the energy of interaction of its dipole moment with the magnetic field due to its neighbor. In frequency 28, Fermi, Z. Physik 60, 320 (1930). 29 See p. 142 in ref. 16. 30H. G. Dehmelt and H. Kriiger, Z. Physik 180, 385 (1951); H. G. Robinson, Phys. Rev. 100, 1731 (1955). 31T. C. Wang, Phys. Rev. 99, 566 (1955); D. C. Douglass, Thesis, versity (1957); R. Livingston, J. Phys. Chem. 57, 496 (1953).

Cornell

Uni-

2.

STATIC

SPLITTING

AND

BROADENING

31

units, this is given by

Av = pney/(r* X 2m) where y is the magnetogyric ratio of the resonant nucleus, tne is the magnetic moment of the neighboring nucleus, and r is the distance between the two nuclei. An estimate of Av for a few cases will now be made. 1. For the interaction between the I!?7 nuclei in the iodine molecule, y = 0.54 X 104, poo = 2.809un (uv = nuclear magneton), r ~ 2.66 A and Av = 0.7 ke. 2. For the interaction between B!! and Cl*® nuclei y = 0.264 X 104, une = 2.689un, andr = 1.65 A, and

in

BCls,

Av = 1.2 ke. 3. For the interaction between the proton and Cl* nuclei in the HCl molecule, y = 0.26 X 104, ua. = 2.793un, r ~ 1.29 A, and Av = 2.5 ke.

The observed splittings of Cl’ and I!?7 resonances in BCl; and iodine, respectively, have been discussed in Section 2a(1). Thus the observed splittings are considerably larger than the orders of magnitude of the splitting calculated from the magnetic dipole-dipole interaction. From Bloembergen and Rowland’s*‘ approximate expression, the J interaction for HCl appears to be comparable to the expected dipole-dipole

interaction. But in this compound

Meal and Allen*®? have noticed a

very broad quadrupole resonance line for Cl* at liquid nitrogen temperature (77°K). This temperature is only 12° below the ) point of 89°K in HCl. Near and above the transition temperature the lattice is disordered,*? and the immediate intermolecular neighbors of the Cl* nuclei are at random positions, which would lead to a broadening.

This

broadening

will

obscure

effects

of

dipole-dipole

interaction

between the proton and chlorine.* In order to observe the effects of the 32H. C. Meal and H. Allen, Jr., Phys. Rev. 90, 348 (1953). 33G.

L.

Hiebert

and

D.

F.

Hornig,

J.

Chem.

Phys.

20,

918

(1952);

J.

Frenkel,

“Kinetic Theory of Liquids,” p. 80. Dover Publications, New York, 1955. «That the broadening of the Cl%> resonance lines in these nuclei is due to the presence of disorder in the lattice and not due to dipole-dipole interaction between proton and Cl*5 is confirmed by the fact that the Cl®* line in DCl is also broad at liquid nitrogen temperature although the deuteron has a magnetic moment smaller than that of the proton by a factor of more than 3.

32

I. THEORY

magnetic dipole-dipole and J interactions, the sample must be cooled far below the A point in order to reduce the broadening due to the

disorder in the lattice. Thus in most cases involving directly dipole-dipole interaction is weak compared in case the resonant nucleus has a neighbor, with a large magnetic moment is the effect

action expected to (HIOs;) studied by molecule is shown directly bonded to

bonded atoms, the to the J interaction. not directly bonded of the dipole-dipole

direct Only to it, inter-

be pronounced. Such a situation occurs in iodic acid Livingston and Zeldes.?? The structure of the HIO; in Fig. 32 in Section 6b. The hydrogen atom is not the iodine but is at a distance of 2.407 A, as revealed

from neutron diffraction®® measurements. Dipole-dipole interaction between the proton and iodine nucleus, therefore, is expected to lead to splittings of the order of a few kilocycles in the I!?” resonance, while the effect of the J interaction is much smaller. The splitting of the I)?” resonance has been observed by Livingston and Zeldes and, as discussed in Section 6b, there is good agreement with the splitting to be 0 H On 0 expected from the dipole-dipole interac-

So7

Sc~"SH X

tion. X

Fig. 3. Schematic diagrams

If chloric

assumed

and bromic acids are

to have the same structure as

iodic acid, one would expect similar splittings in them as well. Another set of compounds in which a proton may come

for structure of o-halobenzalde-

loge

hydes and o-halobenzoic acids.

directly

iodine.

acids or o-halobenzaldehydes

X denotes chlorine, bromine,

,

or

to y

a halogen

atom,

though

not

bonded to it, is the o-halobenzoic ,

shown

in

Fig. 3. A study of the chlorine quadrupole resonance in o-chlorobenzoic acid by Bray,*® and later by Douglass, has shown the absence of any splitting due to the magnetic dipole-dipole interaction. A probable explanation of the negative result in this case will be taken up later in this section. Since the distance between nonbonded atoms depends on the sum of their van der Waal’s radii,?7 only proton neighbors can be close enough to give rise to effective magnetic dipole-dipole interaction. Therefore the theory for the splitting of the quadrupole spectra by magnetic dipole-dipole interaction shall be considered only for spin 4. The total Hamiltonian is given by 3C = Hg + Hie, where Wg is the 35 B.S. Garrett, Oak Ridge National Laboratory Report ORNI-1745 36 P, J. Bray, J. Chem. Phys. 28, 220 (1955). 37 C, Coulson, “ Valence,’ p. 270. Oxford U. P., New York, 1952.

(1954)

2.

STATIC

SPLITTING

AND

BROADENING

33

pure quadrupole Hamiltonian as in Eq. (2.3), and?° Hie = Cli: Ie — 32,1022)(3 cos? 012 — 1) _

$(Lailee

+

I 42D21)

SiN

012 COS

O;2e7*4!?

_

$(_il 22

+

I_2

sin

612

Oye tb ib12

21)

COS

— $sin? 6:2([4:042e7%

+ [_J_2et?##)].

(2.5)

Here J, = (Uz + t1,), C = dyrye2h?ri2—3, and 612 and ¢12 represent the colatitude and azimuth of the vector riz joining the proton and the resonant nucleus (chlorine, for example) in the principal axis system

of coordinates for the field gradient tensor. The components

of the

proton spin are denoted by the suffix 2 and the chlorine by 1. In the absence of any magnetic field, the eigenstates yz, and Ws, of the chlorine, and the states y,, of the proton, are doubly degenerate. Of course for finite 7 the pure quadrupole states of chlorine are no longer Ws, and y4,, but instead are given by the A, and B_, states of Eq. (1.30). For simplicity we shall consider only the case of axial symmetry. The case of finite asymmetry can be dealt with by an exactly analogous procedure. The net eigenstates of the two-spin system without 312

will be Wm,Wm,, Where m, and mz refer to the magnetic quantum numbers of the two spins. Upon consideration of the secular equation for i in this representation, it can be seen that the original energy levels Ey, and Ey, given by (1.12) are split up by the action of the perturbation Ki2. The £4, level splits up into a pair of energy levels each of

which is doubly degenerate while the H,, level splits up into four nondegenerate levels. Instead of the single pure quadrupole frequency w, given in (1.17), therefore, one gets four pairs of lines symmetrically

disposed about «;. Each of these lines will be broadened by magnetic interaction of the Cl®> with distant protons and the individual components of the pattern may not be resolved. Douglass*! has studied the resonance carefully with a regenerative spectrometer and found only a single broad line. This broad line is probably due to a smearing of the 8-line pattern discussed above. In the presence of a weak Zeeman field applied to a single crystal, the splitting of the Zeeman components by the magnetic dipole-dipole interaction is considerably simpler than the splitting for the pure quadrupole lines described above. This happens because the interaction of the magnetic moment of the resonant nucleus (Cl*® in the case of o-chlorobenzoic acid) with the Zeeman field removes the degeneracy between y,, states leading to the nondegenerate y states of Eq. (1.22).

The magnetic field also removes the degeneracy of the ¥4, states of the

34

I. THEORY

adjacent proton. The energy levels of the Cl** nucleus given by (1.20) or (1.21) are now nondegenerate, and the nondegenerate perturbation theory can be applied to obtain the additional splitting due to the dipole-dipole Hamiltonian 312. Thus, in the axially symmetric case the energies of the y, state of the Cl** nuclei are given by = EB,

Ey,

Ey

=

Ey ms

me|3C 198,

mM2>

proton axis with respect to the plane containing the axis of symmetry and the Zeeman field. The three angles 6, $12, 612 are shown in Fig. 4. With an asymmetric field gradient, the equation for splitting of the Zeeman components will be more complicated than (2.6). It will involve one more angle, namely, the azimuthal angle ¢ for the Zeeman field in the principal axis system. Livingston and Zeldes’ measurements”? on iodic acid yield values of Aw for the Zeeman components of I!?” due to H-I dipole-dipole interaction. By use of an equation like (2.6) for various orientations of the Zeeman

field

(and

2.

STATIC

SPLITTING

AND

hence

of a),

C, 612, and

BROADENING

35

¢12 for the I-H

vector can be

obtained. Their exact results will be discussed in Section 6b. (8) Magnetic dipole-dipole interactions: broadening effect. The magnetic dipole-dipole interaction between a resonant nucleus and all the neighboring nuclei leads to a broadening of its quadrupole spectrum. This is most easily seen as follows. Consider, for example, a Cl5 nucleus interacting with a set of protons at different distances Z

Fig. 4. Orientations of H and ry. in the principal axis system of the field gradient tensor. Angle ¢ is of no importance for axial symmetry about Z.

and directions from the Cl*>. Each proton produces its own eight-line pattern considered in Section 2a(2). The pattern due to the different neighboring protons is different because of different distances and

directions of the lines joining their positions to the halogen nuclei, as is clear from Eq. (2.6). The superposition of these patterns from the different neighbors leads to only a single broad line. The breadth of the line is measured by its second moment defined by

= fr

(w — wo)?g(w) dw

where wo is the central frequency given by (1.17) or (1.34). The spectral intensity of the line at a frequency w is given by g(w). Actually the net breadth of a line is a superposition of broadenings due to various

sources

as discussed

previously.

The

net line-shape

function

g(w)

36

I. THEORY

resulting from the broadening due to the various sources is rather complicated and cannot be obtained rigorously. For an estimate, the second moments due to the various causes are usually determined separately, and their sum is taken to be representative of the observed second moment. The procedure for calculating the second moment due to the magnetic dipole-dipole interaction between neighboring nuclei was given first by Van Vleck for nuclear magnetic resonance and electron paramagnetic resonance.** It has been extended to the broadening of quadrupole spectra by Abragam and Kambe.*® This procedure uses the relation

=

— Tr {[&, J,]?

fe Tr (2)

(2.7)

where represents the mean square frequency of the quadrupole spectrum. The second moment is then obtained from the rela-

tion Here &

— wo? =

,

where

w)

is the

central

frequency.

is given by

= 2,

+ 2, Rizr +

y

Bie

+ 2,

ow

(2.8)

v>k’

where 3Cg; for each tion is carried out give the magnetic Thus 3Cy, is given resonant species; nuclei of different the third term on

resonant nucleus is given by (1.1), and the summaover all the resonant nuclei. The other four terms dipole-dipole interaction energy of all the nuclei. by (2.5), where 7 and k represent two nuclei of the 3C,;, represents the magnetic interaction between species. There are as many summations given by the right side of (2.8) as there are species. The

magnetic dipole-dipole interaction between

the resonant

nuclei and

all the nuclei of different species is represented by 3Cy. For example, in fluorochloromethane, CH,CIF, the first term on the right-hand side of (2.8) would refer to the quadrupole interactions of the Cl nuclei; the second summation refers to the magnetic dipole interaction among Cl*5 nuclei; the third term refers to magnetic dipole interactions between the protons and fluorines; and the fourth term refers to the magnetic-dipole interaction of Cl*® nuclei with the protons and fluorine nuclei. The net spin vector of the system will be

1=)ut+) a

(2.9) kK

38 J, H. Van Vleck, Phys. Rev. 74, 1168 (1948). 39 A, Abragam and K. Kambe, Phys. Rev. 91, 894 (1953).

2.

STATIC

SPLITTING

AND

BROADENING

37

where I; refers to the spin of the 7th Cl®> nucleus, and I, refers to spin of the k’th nonresonant nucleus. This summation is carried over all nuclei. The trace is taken over all quantum states of the spin I. The commutator of 3¢ and J, is given by [8¢, J,]. Abragam

Kambe*®

give the details of the procedure involving calculation

the out net and

of

the traces in (2.7). They deal with two situations: (1) resonant spins 1

or $ interacting among themselves; and (2) resonant spins interacting with unlike nuclei of any general spin. For the interaction among resonant nuclei they obtain:

(I = 1):

(I = $):

+)

ther

i

[51 — Bj") + 91 — yin?)? — 2(1 — 385%?) (aj? — Byx?)];

= vs) yhPrie [2071 — Byix")? k

+ 1512yi2(1 — yx?) + 45911 — yyn?)? — 108(1 — 3y%2?) (aj? — By?)].

(2.10)

For the interaction of resonant nuclei of spins 1 or $ with unlike nuclei of arbitrary spin, they obtain

= dy%yHT'(L! +1) Y rel + F(DG))( = Bye")? Z + 9G’) + 2FD)) vin?

— vie®) + 9FCD)EM)(L — ysw)?].

(2.11)

The direction cosines of the radius vector r,, between nuclei j and k in the principal axis system are given by ax, Bj, and yj. Here I’ and y’

refer to the spins and magnetogyric ratios of the nonresonant nuclei, while

F(1) = 0, F@) =4% G(I’) = 1, for any arbitrary nonresonant spin without quadrupole coupling

GJ’)

=

3 (21’ + 1)

87’

+1)

for any

half-integral nonresonant

spin with

quadrupole coupling and GJ’)

When

= 0, for any integral coupling.

both resonant

and

nonresonant

nonresonant

spin

with

spins are present,

quadrupole

the total

second moment is given by the sum of (2.10) and (2.11). Of the three

38

I, THEORY

cases of nonresonant

spins listed above,

the first applies to spin 4

nuclei such as F!9, Ag?7, Ag! 'T]203, or T12°5 which do not have quadrupole moments. The second and third apply to other nuclei with finite electric quadrupole moments.

Ayant?® has calculated the second moment to be expected for Cl*5 resonance in paradichlorobenzene (whose crystal structure is known‘!) using expression (2.11) for magnetic dipole interaction between the Cl} nuclei and protons. He finds a root-mean-second moment of 2.1 ke. Wang*! obtains half-widths of the order of 1.43 to 2.74 ke between temperatures 27.4°C to —195°C. This variation of line width with temperature shows that not all the width can be explained by magnetic dipolar splitting. An increase in line width with decrease of temperature may be expected from lattice contraction, leading to decrease in the rj, in (2.10) and (2.11); however, an increase by a factor of two cannot be explained by this mechanism. This observed increase in broadening with decrease of temperature is probably due to strains developed in the crystal as the sample is cooled. (4) Broadening due to the earth’s magnetic field. In Sections le and le the variation in Zeeman splittings of quadrupole lines with the

orientation of the magnetic field was discussed. In a powder a distribution in orientation of crystalline axes occurs with respect to an applied

magnetic

field, and

hence

there is a corresponding

distribution

in

orientation of the principal axes. This produces a superposition of a number of Zeeman patterns and leads to a single broadened line. In particular, for a spin $ nucleus with an axially symmetric field gradient,

the spread in the a components,

for @ in the range 0° to 180°, will

produce a frequency spread Aw = 2yH where

H is the applied

field. The

observed

line width

may

be even

larger due to the spread in the 8 components. The earth’s field, which is usually near 0.1 to 0.5 gauss, may lead to a similar broadening. Hence for Cl®5 and Br’, with magnetogyric ratios 0.26 X 104 and 0.67 X 104, respectively, line breadths of about 0.2 ke and 0.6 ke, respectively, are

expected for a field of 0.5 gauss. Koi et al.4? report some broadening due to this cause for Br? resonance in NaBrO3;. They minimized this broadening by using a small permanent magnet to cancel out the earth’s field. 40 Y, Ayant, Compt. rend. 288, 949 (1951). 41U. Croatto, S. Bezzi, and E. Bua, Acta Cryst. 5, 825 (1952). 42,Y, Koi, A. Tsujimura, and T. Fuke, J. Chem. Phys. 28, 1346 (1955).

3.

EFFECTS

OF

INTERNAL

MOTIONS

IN

SOLIDS

39

b. Electrical Sources of Broadening Dislocations and strains in the crystal or powder grains!”:*! have been found to broaden the quadrupole spectra. These effects are not quantitatively understood as yet. For such an understanding a rigorous determination of intermolecular interactions and how they affect electric field gradients at the nuclei would be required. Effects of interactions between adjacent molecules have not been quantitatively explained even for perfect solids, as pointed out later in Section 8. The broadening due to strains and dislocations arises from random distortions of intermolecular interactions, and a corresponding random distribution of field gradients at nuclear sites. Strains may result from the crystal growth process, cold work, or from cooling a sample under study. Wang?! has detected an increase in line width in antimony and chlorine compounds with decrease of temperature. Watkins and Pound?’ have also observed similar effects in some nitrogen compounds while observing the N!4 pure quadrupole resonance. A similar distribution in field gradients is produced by the disorder in the crystal lattice.*? The example of HCl has been mentioned in Section 2a(2). In this case, above the \ point, HCl molecules undergo rotation about axes perpendicular to the line joining them, so that the closest intermolecular neighbors of the different Cl** nuclei may be either chlorines or hydrogens. Therefore different Cl*> nuclei may see different environments, and a broadening of the Cl*> pure quadrupole line results. Other instances of such broadening and even disappearance of resonance lines have been reported.*3 Presence of impurities also causes a random distribution in environments around different nuclei, and hence leads to a broadening of the pure quadrupole lines. Some systematic studies with intentionally added impurities have been made by Dean$ and others.‘4 3. Errects or INTERNAL NUCLEAR QUADRUPOLE

Motions IN RESONANCE

MoLecuLar

SOLIDS

ON

The effects of molecular motion upon the frequency, line shape, and intensity of pure quadrupole spectra are now to be considered. Present theory for effects of the internal motions upon quadrupole spectra is by no means rigorous. Our review, as based upon the available literature,*> will be at best semiquantitative and mostly qualitative. 48H. Allen, Jr., J. Am. 44C.

Dean,

J.

Chem.

Chem. Soc. 74, 6074 (1952). Phys.

28,

1734

(1955);

J. Duchesne

and

A.

Monfils,

Compt.

rend. 238, 1801 (1954). 45 In a very recently published article, T. Kushida, G. B. Benedek, and N. Bloem-

40

I. THEORY

The two types of molecular motion considered to have appreciable effects on quadrupole resonance patterns are the torsional and the hindered rotational motions of the molecules about definite axes in

the crystal. The frequencies of these motions are determined by the interactions between electrons in adjacent molecules, difficult to predict theoretically.4* These frequencies mined?’ approximately from studies on the infrared and and from dielectric measurements in the solids. First

and are rather can be deterRaman spectra the effects of

torsional motions will be taken up, and later the effects of hindered rotational motions will be considered. Very few investigations of the effects of hindered rotational motion of molecules**:4° have been made. a. Effects of Torsional Motions of Molecules Infrared and Raman effect studies show that the torsional motions of molecules inside crystals occur at frequencies corresponding to tens or hundreds of wave numbers (cm~!). These frequencies are large compared to the pure quadrupole frequencies, which are at best in the region of 1000 mc or less, corresponding in wave numbers to 0.1 cm or less. The nuclei are being agitated, therefore, by the torsional motion at a rate very fast compared to their resonance frequencies. Bayer has shown®® that two effects result from this condition: (1) there is a shift in the resonance frequency, since the nucleus sees a net temperature-dependent averaged field gradient which is different from what it would see if the molecule were stationary; and (2) there is a temperature-dependent contribution to the relaxation processes affecting the quadrupole resonance. A phenomenological discussion of the idea of relaxation times for quadrupole resonance will be given bergen

[Phys. Rev.

104,

13865

(1956)] have

discussed

the theory of the effects of

temperature and pressure on quadrupole resonance frequencies in detail and presented extensive experimental results in p-dichlorobenzene, KClO;3, and Cu.0. H. 8. Gutowsky and G. A. Williams [Phys. Rev. 105, 464 (1957)] have also recently reported the results of their measurements on effects of pressure on quadrupole coupling constants of Na? in NaClO; and NaBrQO3;. This review was completed before these papers appeared. Therefore we have not been able to incorporate their arguments in this section. 46 See, for example, E. N. Lassettre and C. Dean [J. Chem. Phys. 17, 31 (1949)] for shapes of potential barriers in free molecules, and J. S. Koehler and D. M. Dennison [Phys. Rev. 57, 1006 (1940)] for calculation of energy levels. 478. Mizushima, ‘Structure of Molecules and Internal Rotation.’’ Academic Press, New York, 1954; E. B. Wilson, J. C. Decius, and P. C. Cross, ‘“‘ Molecular Vibrations.”’ McGraw-Hill, New York, 1955. 48M. Buyle-Bodin, Ann. phys. [12] 10, 533 (1955). 49H. W. Dodgen and J. L. Ragle, J. Chem. Phys. 25, 376 (1956). 50 H. Bayer, Z. Physik 180, 227 (1951).

3.

EFFECTS

OF

INTERNAL

MOTIONS

IN

SOLIDS

41

in Section 3a(3).°! The torsional motions are interrupted by interaction with other vibrational and rotational degrees of freedom of the molecules. This leads to random fluctuations in the field gradient seen by the nucleus (in addition to the coherent fluctuations at the torsional frequencies). If this fluctuation has a Fourier frequency component in the neighborhood of a quadrupole resonance frequency of the nucleus, it will induce transitions between the corresponding quadrupole energy levels and produce relaxation effects as in nuclear magnetic resonance. These are the main features of Bayer’s theory for the

effects of torsional motions on quadrupole resonance spectra. In the four parts (1), (2), (3), (4), of this subsection, we shall deal with the theory in some mental results.

(1) Shift

detail and

of quadrupole

point

out how

far it agrees

resonance frequency

due

with

experi-

to the effects of

molecular torsional motions. The treatment given here is based on Bayer’s®° article and partly on later work by Wang?*! and Dean.* The case considered will be a nucleus with spin 3 in a field gradient without axial symmetry. The molecule containing the nucleus will be assumed to undergo torsional motions about the three principal axes of the field gradient. This corresponds exactly to the effects of torsional motions on Cl*> resonance in paradichloro-benzene, where Saksena®? has found, from Raman effect studies, three frequencies, 94 cm—!, 27 cm—!, and 54 cm7! associated respectively with the torsional motions around the Z, Y, and X directions shown in Fig. 5. The analysis of other specific cases of higher spins, or of cases where torsional motion does not occur about the principal axes of the field gradient, may be carried out by a suitable modification of the present procedure. In Fig. 5 the axes fixed in space are denoted by primes, while axes fixed in the molecule are unprimed. For small rotations 6x, 6y, 6z about the three axes X, Y, and Z, it may then be shown that the

primed and unprimed

components

of the field gradient tensors are

related by the following equations: Vex Vyry: Vag Vxry Vy Va

= = = = = =

(1 — Oy? — 62?) Vxx + 02?°V yy + Oy°Vizz 02°Vxx + (1 — Ox? — Oy?)Vyy + Ox°Vxx Oy*Vxx + Ox*Vyy + (1 — Ox? — Oy?) Vaz O0zVxx + (0x0¥ — 02)Vyy — OxOyVzz —Oy0zVxx + OxVyy + (O6¥0z — Ox) Vaz —OyVxx — OxOzVyy + (0x0z + Oy) Vaz.

(3.1)

51 For the meaning of relaxation times as applied to nuclear magnetic resonance, refer to Cohen and Reif,! or the earlier article by Pake® in the Solid State Physics series.

52 B, D. Saksena, J. Chem. Phys. 18, 1653 (1950).

42

J. THEORY

x’

Fia. 5. Space-fixed and moving principal axes for field gradient tensor at the Cl35 nucleus in p-Cl, benzene.

Only terms up to the second power of the small displacement have been retained. Thus the instantaneous field gradient tensor for the oscillating molecule is both time-dependent and nondiagonal. But, as discussed a little earlier, the observed quadrupole frequencies involve averages over the torsional motion. Since the averages of 6x, 0y, and 6z are zero, i.e.

=

=

=0

it follows from (3.1) that the averaged field gradient tensor is diagonal, and is given by the same Eq. (1.5), but with the q and 7 for the stationary molecule replaced respectively by

q = afi _ 3 (

+ ) — 5 ( — ) +36 — 7)

Wang,*!

and

others

who

have

studied

the

quadrupole resonance frequencies for halogen nuclei in a number of compounds at liquid air and room temperatures. They find the expected decrease in frequency with increase of temperature. Dean® has tried to examine Bayer’s theory quantitatively for Cl** resonance in a number of chlorine derivatives of benzene. In Fig. 7 his experimental frequency versus temperature curves are shown for a number of these compounds, together with a theoretical curve for paradichlorobenzene obtained from (3.9). In obtaining the theoretical curve, the torsional

frequencies employed, i.e. vx, vy, and vz, were taken from Saksena’s data from Raman spectra; and the moments of inertia Ax, Ay, and Az were calculated from the dimensions of the molecule as obtained by Croatto, Bezzi, and Bua‘! from electron diffraction measurements. The experimental curve for paradichlorobenzene is seen to have a smaller slope than the theoretical one. Dean has pointed out that better agreement might be obtained if the effects of the intramolecular stretching vibration of the C—Cl bond were considered.®? Support for this statement comes from the large observed changes in the chlorine quadrupole coupling constants in the gaseous diatomic alkali

halide molecules with their vibrational states.5°57 Both increase and decrease of q with vibrational quantum numbers have been detected. These changes have been attributed by Duchesne to changes in electron density at the Cl®* nuclei with the vibrational state. This may well be the case for the C—Cl stretching vibrations in p-Cl, benzene. The theory for this effect could be worked out by a procedure similar

to that of Bayer

discussed

above.

The

distance FR in place of the 0x, 6y, and 54 55 56 57

Pp. R. J. N.

variable now

is the C—Cl

6z of Bayer’s theory.

gq and 7

J. Bray and R. G. Barnes, J. Chem. Phys. 28, 407 (1955). Livingston, J. Phys. Chem. 57, 496 (1953). Duchesne, J. Chem. Phys. 25, 368 (1956). F. Ramsey, “Molecular Beams,” p. 311. “Oxford U. P., New York, 1956.

46

I. THEORY

must be known as functions of # in order to obtain their time dependence due to oscillations in R. It will be obvious from Section 7 that this R dependence cannot be obtained definitely from the theory for interpretation of quadrupole coupling constants in terms of the electronic structure of molecules in its present state.

37.57 37.0 o-NOo¢Cl

36.536.0 oO

—E >

2¢ 355+ o 3

mw

Ss

oS

oLlee

35.0

~~

33.5 73

1

1

123

173

l ! 223 273 Temperature (°K)

Fic. 7. Temperature dependence of Cl** number of compounds. The solid curves are and Pound [J. Chem. Phys. 20, 195 (1952)], for p-Clz¢ calculated from (3.8), neglecting

L

323

j

373

quadrupole resonance frequencies in a experimental ones obtained by Dean and dashed curve is a theoretical one the terms involving 7.

The breaks in some of the experimental curves in Fig. 7 correspond to phase transitions which lead to changes quadrupole frequencies and also can lead to changes in the torsional frequencies wx, wy, and wz expressed by Eqs. (3.8) and (3.9). Dean has also plotted the temperature derivatives of the curves of Fig. 7 versus the temperature. Discontinuities in the slopes of these, as seen from (3.9), do not involve abrupt changes in the rigid molecule quadrupole resonance frequency Wm, but only in the characteristics of the torsional motions, i.e., in

3.

EFFECTS

OF

INTERNAL

MOTIONS

IN

SOLIDS

47

wx, wy, and wz. Dean finds no discontinuity in wx, wy, and wz for paradichlorobenzene through the temperature of the phase transition shown in Fig. 7. The results on phase transitions will be discussed in Section 6, particularly in the case of paradichlorobenzene, which has been studied in detail by Dean and others.*® In Eq. (8.7) for ’ there is a term, independent of the asymmetry, arising from the different frequencies of torsional oscillations about

X and Y. For paradichlorobenzene near its melting point (53°C), this would lead to a contribution to 7’ of the order of 0.01, which may be compared with the observed value of 7’ = 0.08 + 0.03 at room temperature. Although in this case the contribution to 7’ from the torsional motion is within the experimental error, it may be comparatively more important in other cases. Dean has pointed out that this

effect should be corrected for in applying the theory for the prediction of double-bond characters (of the bonds made by halogen atoms with their adjacent atoms in molecules) from observed asymmetry parameters (see Section 7b). It would be interesting to study 7’ as a function of temperature in other halogen substituted benzenes in order to determine the importance of this contribution to 7’. Douglass,®°® Casabella, and Bray® have observed an interesting

splitting of quadrupole resonance lines due to vibrational motions of molecules in molecular crystals. In BCl; Livingston®® first reported that there were two components in the Cl** resonance. Douglass subsequently made a careful recording of the Cl*® resonance line shape and found two prominent lines. One was weak and narrow and the other

strong and broad. The latter component itself appeared to consist of at least four lines. The separation between the prominent lines was found to be 2.8 ke. Crystal structure data®! showed that the splitting could not be explained by the presence of two inequivalent Cl** nuclei in the unit cell. Dipole-dipole interaction also appeared to be too small to explain the observed splitting between the two prominent lines, and the net observed pattern was also not in keeping with theoretical predictions in the presence of indirect spin-spin interaction [see Section 2a(2)]. Douglass has explained the splitting as originating from the effect of internal vibrations within the molecule. BCl; has two isotopes B'® and B!!. The frequencies and amplitudes of the various 58 For a list of references on various methods by which the phase transitions in paradichlorobenzene have been observed, see Dean‘ and also B. C. Lutz, J. Chem. Phys. 22, 1618 (1955). 59 See Chapter 4 in ref. 31. 80 P. A. Casabella and P. J. Bray, private communication to R. Bersohn. 61M. A. Rollier and A. Riva, Gazz. chim. ital. 77, 361 (1957).

48

I. THEORY

modes of internal vibrations involving motion of the boron atom will therefore be different for B!°Clz; and B''Cl; molecules. Douglass has

made a detailed analysis of the problem in a manner analogous to the treatment of Bayer [see Section 8a(1)] by transforming the field gradient tensor from a moving coordinate system fixed in the vibrating molecule to a stationary frame in the laboratory. He has made use of

the force constants

for the molecule

tabulated

obtains

of 2 kc

Cl**

BC];

a difference

between

by Herzberg®?

resonance

and

frequencies

in

and B!!Cl;. The additional splitting of the higher frequency

line can possibly be explained by dipole-dipole interaction or indirect

spin-spin interaction

between

the boron

and

chlorine

nuclei.

That

Douglass’ explanation is the correct one is indicated by the fact that it provides a proper explanation of all the observed features of the Cl** resonance line shape. First of all, it predicts that the Cl®* resonance frequency will be lower in B!°Cl; than in B"Cl;. The lower frequency line will therefore be expected to be weaker than the high-frequency line because B!° is the less abundant isotope, in accordance with observation. Also, since B!° has a smaller magnetic moment than B", the splitting or broadening of the Cl®* line due to dipole-dipole or indirect spin-spin interaction will be smaller in B!°C]; than in B!Cl,, so the weaker line should appear narrower than the stronger line (refer to Sections 2a-c), again agreeing with observations. More recently Casabella and Bray have studied Br’® resonance in BBr; and find it to consist of two lines as well. The separation is now much larger, namely about 380 kc. This is in agreement with Douglass’

explanation where the difference in frequency is expected to be proportional to the quadrupole resonance frequency wm in the stationary

molecule [see also Eq. (3.8)]. Casabella and Bray find by a quantitative calculation, analogous to that of Douglass for BCl;, that the observed splitting in BBr; is in keeping with expectations. One would expect this kind of splitting due to different vibrational

motion in molecules with different isotopic compositions to be more prominent in molecules containing protons and deuterons, since the proportionate difference in masses for these two isotopes is the largest, namely 2:1. Duchesne and co-workers®® have studied halogen reso-

nances in the pairs of compounds CH;Br7? and CD;Br7*, CH3Br®! and CD;Br*!, and CH3I!?7 and CD,I?’, and do find large differences in 62 G. Herzberg, ‘Molecular spectra and molecular structure,’ II, in ‘Infrared and Raman Spectra of Polyatomic Molecules,” p. 155. Van Nostrand, New York, 1951. 63 J. Duchesne, A. Monfils, and J. Garsou, Physica 22, 817 (1956).

3.

EFFECTS

OF

INTERNAL

MOTIONS

IN

frequency, but in a direction opposite to that analysis analogous to that of Douglass, i.e., the frequencies in the hydrogen compound are found the deuterated compound. It is likely that the

SOLIDS

49

expected from an halogen resonance to be larger than in amplitudes of the

vibrational modes of the molecules involving motion of the proton or deuteron are large, and so the accompanying changes in electronic structures for these molecules of the type pointed out by Duchesne®®

are large.

These

resonance

frequency

changes with

could lead to an increase in the halogen increasing

amplitudes

of vibrational

mo-

tions, which would counteract the effects of the reorientations of the field gradient tensor considered by Douglass.

(2) Transition probabilities between the pure quadrupole levels due to molecular torsional motions. In the last subsection, the shifts in quadrupole resonance frequencies due to the torsional motions of molecules in molecular crystals were discussed. In deriving this shift in frequency, the components of the field gradient tensor in the stationary system of coordinates given by Eq. (3.1) were averaged in time to give average values of qg’ and »’. These values when substituted in Eq. (1.6), and then in Eq. (1.1), gave a quasi-stationary Hamiltonian with energy levels and eigenstates resembling those obtained in Section 1. The terms in Eq. (3.1) which drop out on averaging,

however,

appear

Hamiltonian.

They

instantaneously are off-diagonal

as time-dependent in the eigenstates

terms

in the

of the quasi-

stationary part of the Hamiltonian and hence induce transitions. For simplicity, axial symmetry is assumed for the field gradient in the rigid molecule (7 = 0), and the oscillations are assumed to take place about an axis perpendicular to the axis of symmetry. Then the off-

diagonal elements in the Hamiltonian are, from Eq. (3.1),

ear = M8 (Le + iL). + Le + ily] Hie = —

hug

. 6°(t)(Iz + t,)?

(3.10)

where wg is defined in (1.17). These terms can produce transitions between pure quadrupole states differing in m by +1 and +2, respectively. The following picture was used by Bayer in calculating the transition probabilities due to such terms. The molecules may be regarded as oscillating in a nearly parabolic potential well, so that the

energy levels for the torsional motion resemble those for a harmonic oscillator. At any instant there is a distribution in population among the various torsional energy levels. This distribution is not stationary.

50

I, THEORY

because the molecule is also undergoing other types of motion and the torsional motion interacts with these. This gives rise to a fluctuating distribution of population among the torsional energy levels. This, in turn, appears as a fluctuation in the amplitude of the torsional oscillations. If these fluctuations have Fourier components at the quadrupole resonance frequency or at twice this frequency, then they will be effective in producing transitions in the pure quadrupole states

through the terms 3,1 and 3C,2 in Eq. (3.10). Using (3.10), with the time-dependent perturbation theory,*® one gets probabilities for transitions m—> m + 2 or m— m + 1 to be, respectively:

Wat = @M | om + 2IL43)m>|*Fo(200) 2

*

Wy"

(16)! 3

= (eo)

. (3.11)

l |2J1(we).

Following Bloembergen et al.,®° we define J(we)

= |

K(r) exp (twer) dr

with

(3.12)

Y Ki(7) = Lye

: /

a(t)6(t + 7) dt

for W417

and

(3.13) v

Kir) = Lys . / 62(t)62(t + 7) dt for Was? Expressions motion.

K(r)

are the

“correlation”

functions

The superscript 7 is used in Eqs.

for the torsional

(8.13) to signify that these

transition probabilities are due to the torsional motion as contrasted to those produced written as

by the rf field given by

(3.32).

6(t) = D(Et) cos we.

Now

6(t)

can

be

(3.14)

When the molecule is in the rth torsional energy level in the potential

well, the amplitude D,(é) is given by®® ZALD,?2(b) w??

=

(2r

+

1)hox

(3.15)

where D,(t)

= [2h(2r + 1)/Aw:]?.

64 See Chapter 8 in ref. 10. 85 N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 78, 679 (1948). 86 A, refers to the moment of inertia about the axis of oscillation and should not be confused with the A of (1.13).

3.

EFFECTS

OF

INTERNAL

MOTIONS

IN

SOLIDS

51

The average value of D,?(¢) over all levels r during long intervals of

time ¢ is given by twice the value in Eq. (3.5), upon using a Boltzmann distribution over the levels. For short intervals 7, the average value of and over long times ¢ will depend, however, on 7 because of the fluctuations discussed earlier in this section. From (8.18) and (8.14), one obtains

K(r) = 4(1 + $e0s wr)

for Am = +2

and

(3.16) K(r)

= $cos

wr

for Am

=

+1.

To get and , the extent of interaction of the torsional mode in question with other degrees of freedom

of the molecule would have to be known. This is difficult to calculate generally, so Bayer has introduced the following simple assumptions.

(a) The lifetime 7) of a molecule in the ground torsional state is different from lifetimes in the higher states. All lifetimes in the excited

state are assumed to have the same value 74, and 7, and 7) are assumed to be temperature-independent. (b) All transitions from excited torsional levels occur through the ground state; in other words, a molecule always returns first to the ground state from an excited state before going over to another excited state. Actually these assumptions may not be justified in all practical cases, but even so the conclusions reached probably will be valid

qualitatively. These assumptions lead to a relation between 7,4 and 7 by

the

principle

of detailed

balance;

namely,

that

the

number

of

transitions to and from the ground state must be equal. Thus, if n, represents the population in the rth torsional state, then n,/no = exp (—rx), and therefore Ta/To = 1/(e* — 1) (3.17) where x = hw:/kT, and T denotes the absolute temperature. The evaluation of under these assumptions then

proceeds as follows. Let S,(t) denote S,(t) = Dt)

— D?(t)

(3.18)

with an average over different r being signified by dropping the suffixes r on both left and right. From Eq. (3.18) we obtain

=

+ ||%

(3.19)

52

I. THEORY

Using Bayer’s assumptions (a) and (b),

= [|*™ exp (— !) + y Substituting for D,(t) from (3.16) and (3.17), one gets h2

K(t) = Aloe for the Am Ws"

=

=

(1

_—

Eq.

|? = exp ((3.15)

e-*)ettl/70

and +

(—

(2

making cosh

x

1?

+2 transitions. This gives from Eq.

5 (hse)?

—

tl),

use

(3.20)

of relations

L)eW!t/r0

(3.21)

(3.11)

aay? qt

2(cosh x — 1)/[1 + we?ra2(e? — 1)?] +

(= — 1)?

(2 cosh x — 1)/(1 + wo?ra?)

+A+B.

(3.22)

The two terms A + B inside the braces involve terms of the same form

as the two written down but with wa replaced respectively by (w: + we) and (a, — we). It was pointed out in the introduction to Section 3a that the torsional motions have frequencies very large compared to wg, the pure quadrupole frequency, in almost all cases. Therefore W427 can be written as Wa"

=

83

>Ceosh

hig)?

a aoe?

x — 1)/[1 + we?ra2(e? — 1)?] + (2 cosh x — 1)/(1 + wa?ra?)

(er — 1)? 4 2(cosh x — 1)/{1 + w2ra2(e? — 1)7] + (2 cosh x — 1)/A +

*ra?) |

(e7 — 1)? (3.23) For the Am

=

+1 transitions, it may be shown by an exactly analogous

procedure that Wai

ra? 43

(hwe)

T 2 Ta hAw,

E

+

W477 4?(e*

—_

1

3 — ee

2(cosh x — 1) 1)?

+

ee

—1

, 1

+

al

(3.24)

3.

These

EFFECTS

expressions

OF

may

INTERNAL

MOTIONS

IN

be further simplified

SOLIDS

53

by noting

certain in-

equalities. Thus,

w?ra? > 1

(3.25)

because the lifetime in the excited states is definitely large compared to the period of the torsional motion. In addition there are two extreme regions of interest. The first is defined by the equations weta K 1,

and

weta(e? — 1) K1

(3.26)

which signify that the lifetime in the excited states is small.

Under

these circumstances, Eq. (3.23) reduces to Wat

5 (hese)

2

4 cosh x — 3 3(e — 1)?ee

aay?

(3.27)

The second extreme condition is

wate >> 1,

and

weta(e* — 1) > 1

(3.28)

corresponding to a long lifetime for the molecule in its excited torsional states. Then W +. reduces to 3

1

Waal Tw= 2?76 hwo)” 2. (Ag)___"re 2(cosh x — 1)/(e* — 1)? + (2 cosh x — 1)

(e* — 1)°

(3.29)

The transition probability for the Am = +1 transitions is only affected by the condition (3.25), and by none of the conditions (3.26) or (3.28). Therefore, Eq. (3.24) reduces to

Wai? =

3 (hog)? 2

hu

Ate

e7 — 2

—_

1

(3.30)

et

Bayer has discussed the validity of these equations by applying them to some measurements on Cl*> resonance in transdichloroethylene (CHCI=CHCl) made by Dehmelt and Kriiger.*’ These authors find a temperature coefficient for the frequency of the order of 2.5 ke per

degree in the neighborhood

of 70°K,

and

a quadrupole

resonance

frequency of 35.48 mc/sec. Using Eq. (3.9), Bayer finds a torsional frequency of w ~ 27 X 1.45 X 10!? sec~!. With this value of w, he has plotted (2W427)-! and (2W4:7)-}, as given by Eqs. (3.23) and

(3.24), versus rt, at T = 70°K. These curves are shown in Fig. 8 over the range tT, = 107}? to 10-‘ sec. It is apparent from them that (2W417)-! 67 H. G. Dehmelt and H. Kriiger, Z. Physik 129, 401 (1951).

54

I. THEORY

shows no minimum

over this range of 7, while

(2W.7)-!

shows a

minimum at a value of 74, determined by weta

~

1,

(3.31)

(2W7)-1 (sec)

This condition is analogous to the corresponding one obtained for nuclear magnetic resonance by Bloembergen e al.** Also over the range of r, shown, the value of W417 throughout is smaller than W427.

10-2

10-4 10-12

10-10

10-8

10-6

10-4

Tq (sec) Fia. 8. Theoretical plots for (2W27)-! and (2W417)-! against ra in transdichloroethylene at temperature 7 = 70°K and » = 1.45 X 10!% sec~!. [H. Bayer, Z. Physik 180, 227 (1951).]

In particular, in the region of the minimum of (W427)-!, W417 is 10° times smaller than W 27. Actually, from the complete Eqs. (3.23)

and (8.24), it may be shown that both W417 and W427 also exhibit minima at a value of ra given by wita = 1, but this is of no practical interest since the lifetime 7,4 in the excited torsional states is always greater than the lifetime of the torsional period. Also the line width for Cl’® resonance in transdichloroethylene was found by Dehmelt

and Kriiger to be about 10 ke near 70°K. From Eq. (3.40) of Section 3a(3), which relates the spin-lattice relaxation time 7; to W427, and from Eq. (3.52), which shows the contribution to the line breadth from the relaxation mechanism, an upper limit of 10‘ sec-* is found for

W27.

Actually, most of the broadening is due to the static causes

discussed in Section 2. In the absence of a direct measurement of 7,

Bayer assumed a value 10 for W427. It is of the same order as the values

for

W427

obtained

by

Dean

in other

organic

chlorine

com-

pounds, and is probably a reasonable choice. One then gets a value of ta = 5 X 107" sec from Fig. 8. With this value of 7, Bayer has plotted

(2W.27)-! and (2W417)—! as a function of temperature. The two curves

3.

are shown

EFFECTS

in Fig.

OF

INTERNAL

MOTIONS

9. For the entire range

IN

SOLIDS

of temperatures

55

between

O°K and 150°K, W417 is more than a factor of 100 smaller than W427. Therefore its influence on the populations of the pure quadrupole states may be neglected. Dean® has made measurements on spin-lattice relaxation times 7; in a number of substituted chlorobenzenes. Using Eq. (3.40) of Sec-

tion 3a(3), Wa27 for parachlorophenol, parachloroaniline, and paradichlorobenzene at temperature 77°K are found to be of about the same order, namely

5 sec~!. From

Eq.

(3.23), this indicates that the

102

(2W7)-} (sec)

10!

10°

1071

10-2

0

50

100

150

Temperature (°K)

Fia. 9. Theoretical plots for (2W27)-! and (2W4:7)-! versus temperature in transdichloroethylene. » = 1.45 X 10!2 sec-! and ra = 5 X 1071! sec.

values of 7. for these compounds are of the same order as that obtained by Bayer for transdichloroethylene. Since no independent estimates

of 7, are available, of the values of 7, qualitatively. It is C—Cl bond are of

all that can be said is that the order of magnitude found are reasonable and support Bayer’s theory also possible that the stretching vibrations of the importance in contributing to Ws2. and W41. No

estimates of their contributions have been made. The influence of the thermal transitions between the pure quadrupole levels, discussed in

this subsection, on the spin-relaxation time 71, and hence on signal strength and signal width, will be taken up in the next subsection. Equations (3.23) and (3.24) have been deduced for spin $, but it is clear that they can be easily modified to apply to any spin J. This can

be done by replacing the numerical factors $ and ~ by other relevant factors involving J, m, and m’ (m and m’ refer to the initial and final

56

I. THEORY

states, respectively). A notation W427(m, m’) will be used in the future

for these general expressions analogous to (3.23) and (3.24). (3) Consequences of transitions induced by torsional motions: relaxation times. In Section 1d, | |? was taken as a measure of the relative intensities of the spectral lines arising from transitions between initial and final quadrupole states y; and yy. 3C: is the perturbing

Hamiltonian due to the action of the radio-frequency field on the magnetic dipole moment of the nucleus. The absolute intensities also involve the influence of other factors, such as time-dependent relaxation phenomena, which will now be considered. The usual method of detecting quadrupole

resonance

in a

solid

substance, to be discussed in detail in Section 5, is to place the sample in the tank coil of a Fig. 17. The oscillator quadrupole spectrum, resonance is noted. In

radio-frequency oscillator circuit as shown in circuit is then tuned to the frequency of the and the change in level of oscillations at exact actual practice, a modulation of the oscillator

frequency at an audio rate is used to obtain a visual reproduction of the signal. Of particular interest here is the change in level of oscillations

at exact

resonance.

This

happens

because

power

is absorbed

from the oscillator at the quadrupole resonance frequency due to the transitions induced in the system of resonant nuclei by the rf magnetic field in the coil. Let us consider the case of axially symmetric quad-

rupole interaction. The results deduced for it can be extended to the nonaxial case, and also to the situation in the presence of a Zeeman field. For an axially symmetric case, the energy levels (1.12) are char-

acterized by the magnetic quantum numbers m. Thus, if the frequency w of the oscillator is equal to w,, given by Eq. (1.17), it may be shown,

using first-order time-dependent perturbation theory® and Eqs. (1.14) and (1.15), that the probability for a transition between states Wn and m+1 is Wom”

=

ty7H

|

|?g (om)

(3.32)

where H;, is the amplitude of the rf field. The normalized line shape function is g(w) [( l - g(w) dw = 1) | which describes the distribution in resonance frequency due to various broadening effects. There is an additional broadening of g(w) due to thermal transitions between

the quadrupole energy levels. Consider now the spin populations in different energy levels under the combined influence of the rf transitions and thermal transitions. Since Wnt

tome”

=

mom

3.

EFFECTS

OF

INTERNAL

SOLIDS

57

(for e?@Q = positive®*)

is given by

(Niner — Nm)tVH VL + m)(I — m + 1)g(wm)

(3.33)

the rate of absorptive transitions

MOTIONS

IN

where N,, and N41 are the spin populations in energy states EZ, and En+1, respectively. The power absorbed per second is, therefore, Prom

= ho(Nmgi

— Nn)tVH vd

+m)

Due to rf absorption, the populations NV, their equilibrium Boltzmann values N,,°

2N

=

rei

— m+

1)g(wm).

(3.84)

and Nn+1 are disturbed from

(—En/kT)

(3.35) Ny?

=

ey y

1 exp

(- Emyi/kT).

The factor 2 on the right comes from the fact that m and the same energy level #,,, and N is the total number

nuclei. At Boltzmann

—™ refer to of resonant

equilibrium the net difference in population

between levels ZF, and Em41 is

_

2N

~ OT +1

(2

_ En)

_ 6N(2m + 1)A

kT

~

“Ql -+ D)kT

(3.36)

with A given by Eq. (1.13). When the rf field is absent and there is Boltzmann equilibrium, the rate of upward and downward transitions between two levels #,, and EH, must be equal. Hence the upward and downward thermal transition probabilities between these states have to be properly weighted. Therefore Wom

N. m°?

=

Wm?

N. im?

and

(3.37) Wom?

|W

m'smt

=

Nm®/Nn®

=

exp

[((Em

—~

En')/KkT}.

As discussed in Section 3a(2), only the Am = +2 transitions are important. Then with®® W?(m, m2) given by the generalization of 68 For e?@Q = negative, Nn4i and Nm are interchanged to obtain the number of absorptive transitions. 6° Notice the distinction between W7(m, m + 2) and Wmsma2? and Wma2sm’. The latter two represent the correct thermal transition probabilities after the proper weighting condition (3.37) is applied.

58

I. THEORY

Eq. (3.29), it is found that T Wm—ms2

=

W

_— (m,

m

2

£ 2) 2)

(

—

Ema

—

Em =)

kT

and

(3.38) Wmt23m

WT(m,

T=

mt

2)

— 22

is valid

Emx2 (:

provided

_-

En

ee kT

+

Equation

(3.38)

resonance

at 30 me, this is certainly true down

)

(Ems: — Em)/kT «1.

For

Cl*

to a temperature

of

10-*°K, and for I}? close to 1000 me, this is true down to a temperature of at least 1°K. The rate of change of population density in various levels due to the transition probabilities in Eqs. (3.38) and (3.32) may be written down in general as I

we! = » [Waa® Nw — Won’? Nw’) n’=—I +

(Wrrasm™N

a!

-_

W

mtn ™N

m')

5 n’nOm'm|-

(3.39)

The symbols 6n’n and Sm’m imply that the rf field has a frequency equal to the resonance frequency between the levels Hy, and Ey, only. Equation (3.29) has a characteristic difference from the corresponding equation for the ideal nuclear magnetic resonance experiment. In the

latter,”° the levels m, m+

1, m+2:--,

etc., are equally spaced

and the radio-frequency field induces transitions between every pair of successive levels. This is not the case in quadrupole resonance

experiments.7!

As

an

illustration,

consider

I???

nuclei

which

are

irradiated by a radio-frequency magnetic field at a frequency equal to the resonance frequency between +4 and + levels. The rf field induces transitions between the +4 and + levels, but the torsional motions produce the two Am = +2 transitions +4 ¥% and +4 1/6.

(4.4)

and

% E. L. Hahn, Phys. Rev. 80, 580 (1950). 93 See pp. 1-91 in ref. 9; see also E. L. Hahn, Physics Today 6, No. 11, 4 (1953). %¢T, P. Das and A. K. Saha, Phys. Rev. 98, 516 (1955).

72

I. THEORY

Equation (4.3) ensures that, although the spins possess a distribution in resonance frequencies, they behave identically during the rf pulse.

The significance of (4.4) will be clear later. Equations (4.3) and (4.4) will include the condition 7 > ty, i.e., the pulse interval is very much larger than the width of the pulses.

‘] B

cr D

0 ty

E

tT Ttly

Fia. 13. Applied rf pulses to obtain free induction and echo signals.

The wave function for a certain nuclear spin can be written at any instant as i

y=

>

Cn(£)WmeiEmt/n

(4.5)

m=—}3

with 3

Y [Cn(i)|? = 1. m= —}

The functions y,, refer to the eigenstates of the pure quadrupole Hamiltonian (1.10), and correspond to the HZ, energy levels given by

(1.12). At time ¢ = 0, there will be thermal equilibrium and |C,(0)|? =

|C_,(0)|? = te-Aoo'/2e7

|C,(0)|? = |C_,(O)|? = Fetter'/2#7, The

time-dependent

Schrédinger

equation

(4.6)

for a nuclear spin with a

resonance frequency wo’ in the presence of a pulse will be, therefore, thy = Rp = (Ko + Rip

(4.7)

Ho = fee (31,2 — I)

(4.8)

KH, =

(4.9)

where

and —2yhI,H,

cos wot.

When the pulse is removed the Schrédinger equation reduces to

thy = Roy.

(4.10)

4.

THEORY

OF

TRANSIENT

EXPERIMENTS

73

Using (4.5), the solutions of (4.10) and (4.7) may be shown to be8%4 respectively,

3

W(t + to) =

»

VmC'm(to) exp (—tE mt/h)

(4.11)

WnCm(t + to) exp (—tE mt/h)

(4.12)

m=—}

and i V(t + to) =

» m=—3

hor)’

where Hi; = — 33

hu’

Es, = + ae

.

:

and the coefficients C,,(¢ + to) in

(4.12) are related to coefficients C,,(to.) by the matrix relation®®

C(t + to) = RCn(to); 0

COS V3 ont

cos V8

0

R=

nt

i sin Vi ost

0

0

isin Vi unt

sin V8 ws

0

os vB ou

0

7 sin view

0

(4.13)

0 cos V5

of

We define w: = yH; where the amplitude of the rf field 2H; is assumed to be uniform over the whole sample. For a pulse of width t,, it is convenient to use the relation

V3 wily = &.

(4.14)

In order to obtain the expectation values of the magnetization parallel and perpendicular to the rf coil at a time t, we successively

apply the solutions (4.11) and (4.12) in the absence and presence of pulses, respectively, and relate the final y,,(¢) to the initial y,,(0) determined by Eq. (4.6). The expectation values of J, and J, are then obtained

using

y,,(t).

Proceeding

in this

manner

one

obtains

for a

95 Notice the difference on the right-hand sides of Eqs. (4.11) and (4.12). In place of Cn(to) in Eq. (4.11), we have Cn(¢ + to) in (4.12). Cn(é + to) is related to all the Cn(to) for different values of m by the matrix relation (4.13). This mixing of the Cy,(to) in Eq. (4.12) is a result of transitions produced by the rf field.

74

I. THEORY

time t >

ty

_ V3

=0

=0.

=

woh

aap

.

.

’

Sin & sin wot (4.15)

This equation shows that there are no components of magnetization along the symmetry axis of the field gradient, or perpendicular both to the symmetry axis and averaging over the Gaussian distribution in g(wo’), component of the magnetization for the sample is M,

V3 Nwoh?y

= —aRP

.

along the direction the rf field. Upon given by (4.1), the x given by

5t2\

SID — exp | — “9 ) sin wot

(4.16)

where N is the total number of spins. The voltage induced in a coil with n turns and area A perpendicular to the x direction is proportional to

AV,

= An

dM, de

_

AnNyucrh? . _ 6 V3 “aR Sin — cos wot exp ( “).

(4.17)

Due to the exponential term in ?? the amplitude of the induced rf voltage drops down rapidly at a rate determined by the inhomogeneity 6. It represents the free induction signal following the first pulse. The rf voltage induced in a coil perpendicular to the rf field is zero, so that one cannot do a crossed coil experiment®® to observe the response of the nuclei in zero magnetic field. This is also true of steadystate experiments in pure quadrupole resonance as pointed out by Dean.® Only a single coil experiment®> can be performed in which the

nuclei induce a voltage in the same coil which produces the rf field. It will be seen in the next subsection presence of a Zeeman field which states Ym. Then both single coil and used. After two pulses (Fig. 13, £ > by the spins is given by AV,

_ V3 AnN yw rh? | sin|.

= “aap

that this removes cross coil 7 + ¢t,),

is no longer true in the the degeneracy of the techniques can then be the rf voltage produced

E

52?

— cos? 5 C08 wot exp { — >

+ sin £ cos € cos wo(t — 7) exp(- re) — sin £ sin? ; COS w(t — 27) exp(-

me

20"),

96 F. Bloch, W. Hansen, and M. Packard, Phys. Rev. 70, 474 (1946).

(4.18)

4.

THEORY

OF

TRANSIENT

EXPERIMENTS

75

The first of the terms in the brackets of (4.18) on the right represents the tail of the free induction signal following the first pulse. It will die down before the second pulse if the condition (4.4) on 7 is satisfied. The second term gives the free induction signal following the second

pulse, which has a maximum at ¢ = +r. The third term gives the echo which leads to pulse equal to echo for a fixed condition (4.4)

a maximum the interval 7 is also seen on 7 were not

at ¢ = 27, at an interval after the second between the pulses. The damping of the to depend on 6. Also it is seen that if the satisfied, there would be overlap between

I D Fia. 14. Free induction and echo signals for Cl*> applied at B and D. The modulations on the free due to the presence of a small Zeeman field of 5.3 sweep is 2.4 milliseconds and the pulse width ¢, and

B.

Herzog, Phys.

Rev. 98, 639

in NaClO;. The two pulses are induction and echo signals are gauss. The total length of the satisfies § = 7/2. [E. L. Hahn

(1954).]

the tail of the free induction signal following the second pulse and the echo. Echo and free induction signals for Cl*> in NaClO; and Br* nuclei in NaBrO; are shown in Figs. 14 and 15. In Fig. 14 the modulations on the signals are due to the effects of an applied weak Zeeman field,

which will be discussed in the next subsection. The four signals, shown in Fig. 15, following a third pulse are termed secondary echoes.*® From the rate of decay of the echo and free induction signals one can measure 6. The expressions (4.17) and (4.18) for the response of the nuclei following the rf pulse can be extended to any general spin J by replacing

V/3 by [I — |m|)Z + |m| + 1)}! in Eqs. (4.13) and (4.14). For the case of nonvanishing 7, one has to work with the wave functions A, and B_, in Eq. (1.30), instead of the ¥,, for axial symmetry used here.

76

I. THEORY

No echo expressions or reports of experiments for field gradients without axial symmetry are available in the literature.

The effects of the thermal motions

on free induction and echo

amplitudes have not been worked out explicitly. The following procedure, however, would be suitable to take into account the random Bayer transitions discussed in Section 3a(2). The transient signals can first be calculated (see above) as though thermal effects were completely absent. Thermal relaxation effects are then separately con-

sidered by writing down differential equations for the elements of the

atedbes B

D

E

F

Fic. 15. Echo signals for Br’! in powdered NaBrO3;. The first two pulses are shown at B and D, E is the spin echo signal due to them. At F a third pulse is applied, leading to four “‘secondary” spin echoes. The sweep length is 2 milliseconds and the intervals between the first and second, and first and third pulses are 0.27 and 0.71 millisecond, respectively. [M. Bloom, E. L. Hahn, and B. Herzog, Phys. Rev. 97, 1700 (1955).]

density the spin Section Bloch”

matrix o [¢mm' = Cn*Cm, where the C,, are as in Eq. (4.5)] for system due to the effect of the Bayer transitions considered in 3a(2). Such equations have been discussed by Wangsness and for nuclear magnetic resonance. The detailed balance equations

(3.39) are actually the equations for the diagonal element of the density

matrix.®* The expectation values , , and of the components of the spins, as calculated using the solutions of these density matrix equations, would then appear as combinations of exponential functions

in time

involving

the transition

W"

of Eqs.

97 R, K. Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953). 98A somewhat analogous procedure in some special cases of nuclear

magnetic

resonance are described by I. Solomon,

Phys.

probabilities

Rev. 99, 559

(1955).

4.

THEORY

OF

TRANSIENT

EXPERIMENTS

77

(3.29) and (3.30). The net expressions for the free induction and echo signals

would

then

have

factors

showing

damping

effects

due

to

relaxation. From Eq. (4.18), the amplitude of the echo at the position of its maximum,

i.e., at ¢ = 27, appears

to be independent

of 7, the

interval between the pulses. This is no longer true when relaxation effects are considered, as is apparent from Fig. 16, where amplitudes of echo signals for Cl** nuclei in NaClO; are plotted as a function of

the

pulse

interval

r.

In

addition

to

a modulation

effect®®

Fig. 16. Echo envelope giving echo amplitude for Cl5 in NaClO; function of the interval r between two pulses. The first pulse is at B and pulse is applied at varying intervals 7; its positions are not visible in because of its large height and sharp drop. The magnetic field H is 11.8 gauss and is applied along the [100] direction of the cubic unit cell. E. L. Hahn, and B. Herzog, Phys. Rev. 97, 1699 (1955).]

that

crystal as a the second the picture of strength [M. Bloom,

appears in Fig. 16, due to the effect of a Zeeman field, there is a distinct damping of the echo envelope as a function of 7. Measurement

of the

envelopes of the echo and free induction rate of damping signals, as functions of the pulse interval r, permits a determination of the spinlattice relaxation time 7';.8°.%°

b. Spin Echo and Free Induction Signals in the Presence of a Weak Magnetic Field The transient response of nuclei following rf pulses in the presence of a Zeeman magnetic field will now be discussed. A spin $ in an axially symmetric field gradient is considered. The Zeeman field H is

78

I. THEORY

applied at an angle 6 to the axis of symmetry. The z-axis is chosen in the plane of the applied field and the axis of symmetry, and the rf field is applied along the 2 axis. The frequency of the rf field is chosen to equal wo, the resonance frequency in the absence of the Zeeman field. We apply the Schrédinger equation (4.7) now with

Hy = fey (31,2 — 12) — hQcos OJ, — QI, sin @

—(4.19)

where Q = yH Ry,

=

—2yhI.H,

cos wot.

(4.20)

In place of the wave functions y,, used in the absence of the Zeeman field, it is now preferable to use the eigenstates of 3Cy in Eq. (4.19), namely ~,; and ¥s given by Eq. (1.22). The equations for the spin states in the presence and absence of the rf field are formally the same as those given in (4.11) and (4.12). But Cz, and Y,,; are now replaced by Cx and ¥4, and the phase terms, instead of containing the eigenvalues (1.12) of the pure quadrupole Hamiltonian, now involve the eigenvalues (1.20) and (1.21) of the net Hamiltonian (4.19) in the presence of the weak field. In place of (4.13), the matrix R will now be given by é

won

&

be

0

0

isin § sin a £

isin § cos a

t

Cos 5 £

2 sin 5 cos a

7 sin 5 sin a

COs 5

0

0

COs 5

R= ran

t sin 5 cos a

_nen

cos 5

c

ba

isin 5 sin a

..

€

7 sin 5 COs a

7 sin 5 sin a

(4.21) g

where sin @ and cos a, related to the angle 6 between the axis of symmetry and the Zeeman field, is defined in Eq. (1.22). The procedure for calculating the free induction signal and echo is then just the same as in Section 4a; namely, that the equations corresponding to (4.11) and (4.12) are applied to relate the net wave function y(¢) at time ¢ to ¥(0) at ¢ = 0. The expectation values of and are then calculated and averaged over wo’ using Eq. (4.1). The voltages AV, and AV, induced in coils in the x and y directions, respectively, are

4.

n oe

THEORY

OF

TRANSIENT

= COS wot {ws

a) AV,

= An cat

EXPERIMENTS

Antoun

+ Ge

+ tt

cos | Q cos @

cos (0 cos StI.) 6

= COS wot [vB Aereotn

5 ef.)

sin £ é

79

sin (0 cos

sin — é

yr

coon}

sin (0 cos 6

§@ —=— 3 + f ‘)| coerl,

(4.22)

Equations (4.22), when compared with (4.17), reveal the following features. First, the voltage induced in a coil perpendicular to the rf field is no longer zero, which means that a crossed coil experiment is feasible. The only difference between the voltages AV, and AV, is that their phases of modulation differ. Second, there are now modulations superimposed on the decay of the free induction signal. The fre-

quencies of the modulations correspond to the splitting of the steadystate pure quadrupole line due to the magnetic

field. Modulations

of

this type are seen in Fig. 14, where the free induction signal for Cl** in NaClO; in the presence of a weak magnetic field is shown. In a similar manner, the transient induced voltages following the two pulses in Fig. 13 may be obtained. Only the terms representing

the echo signal will bé given. Thus _

AV. =

V/3 ANnywo?h?

4k

AGP

sin & sin? § cos wo(t — 27) exp | -

F y co

f-1

+(G) Pa | + Op

a(t —5 an

5 |" cos 0(3 — f)(t — 2]

5

E cos 6(3 FDe—

cos |

|

3Q cos 0 fQeos6, _ cos ———— t cos—7 (t — 2r)

+ cog — 2% 008 § (é — 27) cos 12 =

Oy — cos3 F608

Oy cost? =

|].

(4.23) Equation

(4.23) shows that there are also modulations

appearing

on

the echo, as shown in Fig. 14. But these modulations on the echo or free induction signals are damped by the decay term due to inhomogeneities. If 6 is large, the transient signals are too heavily damped to reveal any modulation effects. However, from Eq. (4.23), the amplitude

80

I. THEORY

of the echo signal at its maximum

(i.e., at ¢ = 2r) is seen to involve

the factor

oe

E cos @ +l,

cos s) cos €

=

Qr cos s)

_ 5 608 ( tf Qr cos s) - 5 cos € >t Qr cos a) |

(4.24)

This indicates that the echo amplitude plotted as a function of 7, the interval between the two pulses, will also exhibit modulations with frequencies corresponding to the Zeeman splittings in steady-state experiments. Figure 16 illustrates this modulation effect for Br*! resonance in NaBrO; in the presence of a weak magnetic field. The

over-all damping of the pattern is due to relaxation effects discussed qualitatively in the last subsection. It is clear that the damping of the echo envelope is much less pronounced than the damping of the modulations on the free induction and echo signals due to the static inhomogeneity 6. Thus, the “envelope modulation” can be seen even when the modulations on the echo and free induction signals are damped down due to static inhomogeneities (see ref. 89 by M. Bloom). It can be shown by a procedure analogous to that used in this subsection that the line splittings due to magnetic dipolar or J interactions between a resonant nucleus and its immediate neighbor (discussed in Section 2) will also cause modulations of the transient signals. Even in cases where the static inhomogeneities obscure the splittings in steady experiments, these splittings will still show up as envelope modulations in an echo experiment, unless the relaxation time 7’; is shorter than the periods of the modulations that are to be

detected. The restriction due to 71, however, is not serious, because even if 7, is as low as réo of a second, splittings greater than 0.1 ke could still be observed. c. Relative Advantages of Steady-State and Transient Experiments The relative advantages of the steady-state and transient techniques in yielding information of value in analysis of molecular and crystal structures will now be listed. 1. The transient echo technique in general requires more elaborate equipment, particularly for pulse techniques, than the steady-state methods. If one is merely interested in locating the frequencies of quadrupole spectral lines, the echo technique offers no advantage over the steady-state experiments, and in fact renders data which are more

difficult to interpret.

4.

THEORY

OF

TRANSIENT

EXPERIMENTS

81

2. If, on the other hand, accurate location of principal axes and measurements of 7 are required, the echo technique definitely offers better accuracy. It will be seen in Part III that the properties of the field gradient tensor provide very useful information on molecular

and crystal structures. This information is often limited by the inaccuracies of experimental data on 7 and the principal axes. To cite a few examples, the interpretation of the structure of solid iodine requires an accurate knowledge of the angle between the axes of the individual molecules in the crystal and their respective field gradient z axes. Current theories®® on the structure require a value of this angle in the neighborhood of half a degree, which is within the limits of error of

the recent experimental data of Tsukada.’ The limit of error arises because the location of the locus of ‘‘no-splitting”’

(see Section

le) is

rendered somewhat inaccurate because of the widths of the Zeeman lines. However,

as discussed in Section 4b, the envelope

modulation

of the echo is not affected by the width of the frequency distribution, and would therefore permit a much more accurate location of the nosplitting locus. Another example is the Cl** resonance in Ba(ClO3)-H2O studied recently by Zeldes and Livingston.?? Here, the crystal structure obtained from x-ray data!®! would favor an appreciable value of 7. Zeldes and Livingston could only set an upper limit of 0.005 on 7. A more definite determination was not possible because of the large line width of 5 ke that arises in part from the magnetic dipole interaction of the Cl*> nuclei with the protons of the water molecules. An echo measurement of the Zeeman effect would make possible a more accurate estimate of 7 that might provide a stronger basis for questioning the x-ray data. 8. Use of the echo technique definitely would facilitate the study of small splittings of quadrupole resonance lines due to strong magnetic dipole-dipole or indirect spin-spin interactions between neighboring

nuclei.

This

point

has

already

been

mentioned

in the

concluding

paragraph of Section 4b.

4. Relaxation measurements by the steady-state technique require a measurement of rf field so that the of T, from this measurements of 99 See R. 100 K, 101G.

the signal strength as a function of amplitude of the relations (3.45) and (3.47) may be used. Evaluation requires a knowledge of both H; and T,*. These H, can be obviated by comparison with a nucleus

Section 8d; also C. H. Townes and B. P. Bersohn, private communication, to be Tsukada, J. Phys. Soc. Japan 11, 956 Kartha, Proc. Ind. Acad. Sct. 86A, 501

Dailey, J. Chem. Phys. 20, 35 (1952); published. (1956). (1952).

82

I. THEORY

whose 7; is known. Thus, if H,’ and H” refer to the rf fields required to give a maximum signal for the two nuclei (known and unknown T,’s) then VHT

YT 3*

HTT

y*"

=1

i.e.,

TY Ty

_

yA TF V?

Ay”?

To*”"

where the single primes refer to the nucleus whose 7’; is not known and the double primes to the one for which 7; is known; H,’’*/H,’? can be estimated from the ratios of the power levels of the rf oscillator for the strongest signals in the two cases. But determination of the ratio T.*”/T.*' involves a study of the line shapes in the two cases (refer to Section 3a), nor can it be uniquely defined in all solids. This cannot be done accurately with a super-regenerative circuit (see Section 5c). A regenerative circuit would give a better measurement of the line shape, provided the signal is not near saturation. The echo technique, on the other hand, only requires the observation of the damping of the

envelopes of the free induction and echo signal®*-% as functions of the pulse interval 7, as pointed out in Section 4a. Therefore, it would lead to better accuracy in relaxation measurements, and at the same time the use of a large continuous wave power level, required in the steadystate technique for small 7;, would not be necessary. 5. Echo envelope decay measurements also provide a unique method for the determination of spin-spin coupling effects which arise in experiments involving double resonance.*®® The echo amplitude of a given nucleus A is sensitive to fluctuations of magnetic dipolar fields caused by neighboring unlike nuclei B. Upon exciting the cw quadrupole resonance of B, it is possible to determine this quite accurately by observing a maximum change in the echo relaxation of A. This is caused by modifications in the dipolar field fluctuations at the A sites. In such experiments A is usually a quadrupole coupled system also, which is easily observed by the echo method.

Il. Instrumentation

5.

APPARATUS

a. General

Considerations

The detection of quadrupole resonance absorption requires application of a radio-frequency magnetic field at a fixed frequency, and at a suitable intensity. The rf field H, should not be so large that y?H TT >* > 1 [see Section 3a(3)]. If a proper line shape measurement is desired, the rf field should have such a value that y?H?7iT,* one needs

Hi

~ 35 gauss and

for Br’79,

H, ~ 14 gauss for this purpose, if the pulse width is taken to be 10 usec.

=

A00SZ 9 OBE}OA IH

40}D9UU0D

00

yul| N\

vT2te

o7

=

0

30

MOL

Oz? 19

a

ae

4w OT

9129 TA

A00S—

‘OUI OE 7B SoOyda soUBUOSaL ajodnipeNb oj 10}BI[I0s0 pas[nd oul-yE “Fz ‘DIY VE AOT

=/

srt] Og-p

Z109-1d HeMpseg

09 j}p-—t

\L

ANG

s00°0

Sul] [eIxXe-09 pisiy S008 € £A

ajdwes

OA1.t“ vl# suiny $7

Z1# Susn} $g Wale yysua]

atu

$d 121 PPM E Cals WFTw suey apim

(aajesuases ‘7 oF pajdnos yuly) -q’y ,, 2

82-Z ayWwYyO

Z edt E a


71, the interval between successive pulse pairs.!!® An RC network integrates the signal voltage over a long period of time. The average

amplitude of the echo signal, over the interval 6 of time for which the data gate is applied, meter.

is then read in terms

of the deflection of a de

14D), Holcomb and R. E. Norberg, Phys. Rev. 98, 1074 (1955). 115 t; is the interval between successive similar pairs of pulses of the type shown in Fig. 4.13, and should not be confused with 7, the pulse interval within a pair.

lll. Applications of Interest to the Solid State

Parts

I and

frequencies

and

II of this article have

dealt with the theory

line shapes of quadrupole

of the

spectra and the experi-

mental techniques for studying them. Part III will deal with the rela-

tion between the quadrupole spectra and the structures of the molecules and crystals in which the nuclei are contained. The field gradient

tensor at a nucleus depends upon the charge distribution in the neighborhood of the nucleus. The charge distribution is determined by the electrical forces in a molecular crystal that (1) bind the atoms within

the molecules, and (2) bind neighboring molecules to form The quadrupole spectra will depend, therefore, upon the extent of both intra- and intermolecular bindings in the molecular crystals there is evidence that intramolecular much

stronger than intermolecular

binding,

the crystal. nature and crystal. In binding is

and that the properties

of molecules in the solid state do not differ very much from the properties of free molecules in the gaseous state.!!® Thus infrared and Raman spectra show nearly similar force constants in both the solid

and gas phases. Also the latent heats of melting, which depend upon the bindings of the molecules in the solid state, are often small compared to the heats of dissociation that are determined by the intramolecular binding. Therefore, in obtaining information about the nature of the intramolecular binding from quadrupole resonance data in the solid state alone, the approximation

is often made!”

that the

electron distribution within the molecule in the solid state differs very little from that for the free molecule. But in those cases where the quadrupole coupling data in the free molecule are available from microwave spectra,!!” the quadrupole spectra in the solid state can be used to obtain information about the alteration of the electron distribution within the molecule when it is under the influence of electrical forces 6 Refer, for example, to J. C. Slater, “Introduction Chapters 16 and 24, McGraw-Hill, New York, 1938. 117 See Appendix in ref. 16 for a list. 97

to

Chemical

Physics,’

98

III.

APPLICATIONS

from neighboring molecules in the solid.®1!81!9

The interesting case

of the solid halogens falls in this category and shall be dealt with in some detail. Section

6 will deal only with the gross data on crystal structure

that can be obtained from quadrupole spectra. Such information pertains to the number and orientation of the molecules in the unit cell, the chemical equivalence or inequivalence of different sites for the same

nucleus in the unit cell, and the changes in the orientation of molecules that take place when phase transitions occur. In Sections 6, 7, and 8 the discussion will be mainly confined to halogen quadrupole spectra, since these have been extensively studied. Section 9 will deal with the quadrupole spectra in nuclei other than halogens. Section 10 will deal with studies on quadrupole spectra in imperfect crystals. 6.

INFORMATION

Unit

ON

THE

CONSTITUTION

OF

THE

CRYSTALLINE

CELL

a. Number of Chemically Inequivalent Lattice Sites in the Unit Cell The terms “physically” and ‘chemically” inequivalent lattice sites were introduced in Section le. It was mentioned there that physi-

cal inequivalence of sites cannot be detected without studying Zeeman splitting

of the

quadrupole

spectra

in

single

crystals.

Chemically

inequivalent sites, on the other hand, lead to multiple lines in the pure quadrupole spectrum itself. There are two types of chemical inequiva-

lence to be considered. In the first case, the atoms containing the nuclei may be in chemically different positions in the free molecule to start with; in the second case, the atoms are in equivalent positions in the

free

molecule,

but

when

introduced

in the

solid,

the

interaction

between neighboring molecules destroys their equivalence. The splittings produced in the first case are expected to be larger in general

than in the second case, because of the comparative weakness of intermolecular forces as compared to the forces that exist inside the molecules. As examples of the first type of chemical inequivalence, chlorine (Cl#5) resonances in some compounds studied by Bray!?° and Allen‘? are tabulated in Table II. The structural formulas for these molecules are shown schematically in Fig. 26. Each of the first three

compounds has two inequivalent groups of chlorine sites: those belonging to a CCl;,

CCleH,

or CH2Cl,

respectively,

or to a COCI

group.

u8 H. G. Robinson, H. G. Dehmelt, and W. Gordy, J. Chem. Phys. 22, 511 (1953). 19 K, W. H. Stevens, Tech. Report No. 197, Cruft Laboratory, Harvard University, 1954. 120 See p. 704 in ref. 36.

Tasie

II.

6. INFORMATION

ON

Toe

QuapRUPOLE

Spurting

CHEMICALLY

or

CRYSTALLINE

INEQUIVALENT

Compound

Lines

SiTES

Chloroacetyl chloride? CH.CICOCI Oxalyl chloride’ CICOCOCI Chem.

From

Cxi**

Soc. 74, 6074

99 NucLEI

aT

IN MOLECULES (mc) at 77°K

33.721 40.132 40.473 40.613 32.147 32.962 38.353 38.521 39.189 39.386 30.437 37.517 30.217

Dichloroacety] chloride’ CHCl.COCI

Allen, Jr., J. Am.

CELL

Frequency

Trichloroacety! chlorides CCl;COCl

“H.

UNIT

(1952).

+P. J. Bray, J. Chem. Phys. 28, 704 (1955).

One therefore would expect at least two different frequencies for the Cl®5 resonance in these compounds, as shown in Table II. For chloroacetyl chloride there are two lines separated by a little more than 7 me. Dichloroacetyl chloride shows four closely spaced lines and two at a frequency nearly 6 me lower. Trichloroacety] chloride shows three

Ne cl

0

NG

¢l

er

NG

ee Ne NG oN 0

Cl

Fia. 26. Schematic structural formulas of the acid chlorides in Table II.

closely spaced lines and one more about 6 mc lower. The separations of the last three lines in trichloroacetyl chloride reflect the effect of intermolecular forces which render the three chlorine atoms of the CCl; group somewhat inequivalent. The observation of two lowfrequency lines and four lines of higher frequency in dichloroacetyl chloride, may indicate at least two inequivalent molecules in the unit cell. These frequency differences caused by the influence of intermolecular forces are seen to be small compared to differences arising out of the inequivalence of two Cl** positions in the free molecule.

The Cl*> frequency in oxalyl chloride (COCI): is tabulated as a refer-

100

III.

APPLICATIONS

ence. Here, the two chlorine atoms are equivalent in the free molecule, and a single line is observed showing that the intermolecular forces do not distort this equivalence appreciably. The frequency of the Cl*5 resonance in (COC]l)2 indicates that the lower frequencies in the first

three compounds belong to the Cl®* of the COC] group, and that the higher ones belong to the CHCl, CHCl:, and CCl; groups. A number of other examples, where chemical inequivalence of nuclear positions in the free molecule leads to multiplicity of nuclear

quadrupole spectra, are cited in the literature. A very significant set of the measurements on the various chlorobenzenes!”! (see Section 7), which provide a test of the Townes and Dailey!”? theory for interpreting quadrupole coupling constants, is particularly significant.

We will consider now some examples of the other type of inequivalence where nuclear positions that are equivalent in the free molecule are rendered inequivalent in the crystalline lattice due to intermolecu-

lar forces. The splittings of the Cl** resonances belonging to the CCl; and CHCl, groups in trichloro- and dichloroacety! chlorides discussed

earlier result from this effect. More striking are the halogen resonances Tasie

III.

Muutiericrry or Ci*® QuapRUPOLE Lines DuE TO INEQUIVALENCE OF CHLORINE SITES IN CRYSTALS

Compound

CHEMICAL

Resonance frequencies at 77°K (mc)

CCl, + CBr,?

More

than 4 lines around

268 mc

SiCl, + SiBr,? GeCl, *

147.511,

147.576

GeBr,?°

175.584, 175.949

SnClh, 2

SnBr,° SnI,?

162.032, 165.208 165.270, 165.401 207 .682, 209.127

@ R. Livingston, J. Phys. Chem. 57, 496 (1953). +A. L. Schawlow, J. Chem. Phys. 22, 1211 (1954).

in the tetrahalides of the fourth group elements C, Si, Ge, and Sn, tabulated in Table III. Thus, in SiBr,, GeBr,, Gel4, and SnIy, Schawlow!!! finds two lines for the $ — 4 transition of the halogen nucleus in each case. In CBr, studied by Schawlow, and in SiCl,, GeCli, and 121P, J. Bray, R. G. Barnes, and R. Bersohn, J. Chem. Phys. 26, 813 (1956). 122 C, H. Townes and B. P. Dailey, J. Chem. Phys. 17, 782 (1949).

6.

INFORMATION

ON

CRYSTALLINE

UNIT

CELL

101

SnCl, studied by Livingston,®> four lines are found in each case. In CCl, Livingston finds evidence for at least fifteen and probably sixteen lines. The structures of these molecules have been studied by electron diffraction.!2* In the vapor state they are found to consist of regular tetrahedra of halogen atoms with the fourth group atom at the center. On the other hand, x-ray data for the solid state,!?4 available for a few

of these compounds, indicate that the molecules have distorted tetrahedral structures. This is in agreement with the multiplicities of the halogen quadrupole resonances observed in the solid state. However, no detailed assignments of the observed frequencies to the different halogen positions in the lattice have been made. Still another interesting group of compounds that illustrates the multiplicity of quadrupole spectra due to intermolecular effects is shown in Table IV. TasLe

IV.

Muttipuiciry or Ci#* QUADRUPOLE RESONANCES Containing H-BonpED StTrRucTURES

IN CRYSTALS

Compound

Resonance frequency (mc) at 77°K

CCl;CH(OH): ¢ chloral hydrate CCl;CH(OH)(OC2H;)* chloral alcoholate CCl;COOH *° trichloroacetic acid

38.190, 39.429, 39.515 38.516, 38.705, 39.14 39.967, 40.165, 40.240

@H, Allen, Jr., J. Am. Chem. Soc. 74, 6074 (1952). +P, J. Bray and P. J. Ring, J. Chem. Phys. 21, 2226 (1953).

The crystal structure of chloral hydrate has been studied by Kondo

and Nitta,!25 who show that the three C—Cl distances in this molecule are respectively 1.79, 1.78, and

1.72 A. The two longer distances are

attributed to H bonding}** of two of the chlorines with hydrogens of the hydroxyl groups in neighboring molecules. The quadrupole spectra, as seen from Table IV, contain two closely spaced lines, which therefore have to be assigned to the two hydrogen-bonded chlorines,

while a third line further away is ascribed to the chlorine, which does not form H bonds. The crystal structure of chloral alcoholate is not known, but since there is one OH group per molecule, one of the three chlorines in the CCl; group would be expected to form a hydrogen 123 Pp, W. Allen and L. E. Sutton, Acta Cryst. 8, 46 (1950). 124 W, Hiickel, ‘Structural Chemistry of Inorganic Compounds,” p. 471. Elsevier, New York, 1951. 125 §, Kondo and I. Nitta, X Sen 6, 53 (1950). 1267, Pauling, ‘‘Nature of Chemical Bond,”’ Chapter 9. Cornell University Press, Ithaca, N. Y., 1948.

102 bond

III. with

an OH

group

APPLICATIONS

of a neighboring

molecule.

By

comparison

with chloral hydrate, two of the Cl®® resonances would be expected to be close to each other in frequency, and the third one,should occur at a higher frequency. This is seen to be the case from Table IV, but

the alteration in frequency due to H-bond formation is less marked than in chloral hydrate.

In trichloroacetic acid the chlorine atoms do

not form H bonds. The OH group in the molecule forms an H bond with the oxygen atom in the COOH

group of a neighboring molecule.

The three Cl** resonances are seen, from Table IV, to be close to each other in frequency as expected.

b. Interpretation of Zeeman Splittings and Their Orientations of Molecules in the Unit Cell

Use in Determining

The Zeeman patterns for quadrupole spectra in single crystals enable a determination of the asymmetry parameter 7 and the orientation of the principal axes of the field gradient tensor with respect to axes fixed in the crystal. The direction of these principal axes with respect to axes in the individual molecules can be obtained approxi-

mately from considerations of the molecular structure (see Section 7). The Zeeman measurements, therefore, can determine the orientations of the molecules in the crystalline lattice.!2”7 Study of Zeeman splittings also helps to distinguish between physically inequivalent nuclear sites in the unit cell. If these sites belong to different molecules in the crys-

tal, the Zeeman

measurements

then determine the relative orienta-

tions of the molecules with respect to each other. Zeeman measurements of pure quadrupole spectra have been carried out in only a few compounds. A list including most of these compounds is given in Table V. The Cl** resonance in NaClO; has been studied by a number of observers. The latest measurements are

by Zeldes and Livingston.?? In Fig. 27 the crystal structure of NaClO; is shown schematically. The unit cell is cubic, with the length of each

side equal to 6.57 A, and contains 4 molecules. Each molecule consists of a Na+ ion and a ClO;~ ion. The ClO;— ions form pyramids with the chlorine atom at the apex. Each of the Nat ions and associated ClO;— pyramids lie along a direction parallel to a body diagonal. Because of the threefold symmetry around each Cl*> nucleus, the asymmetry parameter 7 is zero. Thus there are four axially symmetric field gradient

tensors (numbered 1, 2, 3, 4 in Fig. 27) with the same q but with four different

directions

for the

axes

of symmetry,

different body diagonals. The pure quadrupole

parallel

to

the

four

spectrum for all the

127 For a good example, see H. C. Meal, J. Chem. Phys. 24, 1011

(1956).

6.

Tasie

INFORMATION

ON

CRYSTALLINE

UNIT

V. List or SusBsTaANCES FOR WHICH ZEEMAN Have Been PERFORMED TO DaTE

Substance NaClO; Sodium chlorate

Nucleus studied Cl

Author BR. Livingston, Science 118, 61 (1953). Y. Ting, E. R. Manring, and D. Williams, Phys. Rev. 96, 408 (1954). T. C. Wang, Phys. Rev. 99, 566 (1955). H. Zeldes and R. Livingston, J. Chem. Phys. 26, 351

NaBrO; Sodium bromate

KCI1O; Potassium chlorate

Br7??

cls

103

CELL

MEASUREMENTS

Crystal structure data Cubic X-ray data by W. H. Zachariasen, Z. Krist. 71, 517 (1929).

(1957).

TT. Kushida, Y. Koi, and Y. Imaeda, J. Phys. Soc. Japan 9, 809 (1954).

Cubic X-ray data Hamilton,

H.

Monoclinic X-ray data

Zeldes and R. Livingston, J. Chem. Phys. 26, 351

by Z.

100A, 104 (1938).

Zachariasen,

(1957).

by

J. E. Krist.

W. Z.

H.

Krist.

71, 501 (1929). Ba(ClOs;)

2 HO

C35

H.

Barium chlorate

Zeldes and R. Livingston, J. Chem. Phys. 26, 351

Solid bromine

Br?®!

(1957).

S. Kojima, K. Tsukada, A. Shimauchi, and Y. Hinaga, J. Phys. Soc. Japan 9, 795 (1954).

Monoclinic X-ray data by G. Kartha, Proc. Indian Acad. Sci. 836A, 501 (1952). Orthorhombic X-ray

data by

B. Vonne-

gut and B. E. Warren, J. Am. Chem. Soc. 58, 2459

(1936).

Solid iodine

[!27

S. Kojima and K. Tsukada, J. Phys. Soc. Japan 10, 591 (1955).

Orthorhombic P. M. Harris, E. Mack, and F. C. Blake, J. Am. Chem. Soc. 50, 1583 (1928).

HIO; lodic acid

q!27

R.

Orthorhombic

Livingston and Zeldes, J. Chem. 26, 351 (1957).

H. Phys.

(1)

X-ray

data by

M.

T.

Rogers and L. Helmholtz, J. Am. Chem. Soc. 68, 278 (1941).

104

III. Tasie

APPLICATIONS

V. List or SuBsTANCES FOR Have Been PERFORMED

Substance

Nucleus studied

WHICH ZEEMAN MEASUREMENTS TO Date (Continued)

Author

Crystal structure data (2) Neutron-diffraction by B.S. Garrett, Oak Ridge National Laboratory Report ORNL1745

Na2B,07:-4H:0

Bu

R.

Kernite

R. Haering and G. M. Volkoff, Can. J. Phys. 34, 577

(1954).

Monoclinic X-ray work

by

Mallada 5, 3 7, 21 (1948). As2O3 Arsenic trioxide

Asi5

H.

Kriiger and U. MeyerBerkhout, Z. Physik 182, 221 (1952).

L.

Por-

toles, Hstud. Geol. Inst.

(1956).

(1947);

Low-temperature phase arsenolite is cubic. High-temperature phase claudetite is mono-

clinic. (1) Arsenolite has been studied using x-ray diffraction by R. M. Bozorth, J. Am. Chem. Soc. 45, 1621 (1923). (2) Claudetite has been studied by A. J. Frueh, Jr., Am. Mineralogist 36, 833

(1951).

C.H.Cle p-Dichlorobenzene

Cl

C. Dean and R. V. Pound, J. Chem. Phys. 20, 195 (1952). B. C. Lutz, J. Chem. Phys. 22, 1618

(1954).

Low-temperature a phase—monoclinic. (1) X-ray work by S. B. Hendricks, Z. Krist. 84, 85 (1932). (2)

Electron

diffraction

by U. Croatto, 8. Bezzi, and E. Bua, Acta Cryst. 5, 825. (1952). High-temperature 6 phase—triclinic. X-ray work by G. A. Jeffrey and W. J. McVeagh, J. Chem. Phys. 23, 1165 (1955).

6.

TasLe

INFORMATION

ON

V. List or SussTances Have

Been

Nucleus studied

Substance

CRYSTALLINE

ror

PERFORMED

WHICH TO

UNIT

ZEEMAN

Date

105

CELL

MEASUREMENTS

(Continued)

Author

Crystal structure data

8. Kojima, K. Tsukada, and Y. Hinaga, J. Phys. Soc. Japan 10, 498 (1955).

Monoclinic X-ray work by Hendricks, Z. 84, 85 (1932).

Cl

H. C. Meal, J. Chem. Phys.

Crystal structure not known from other sources.

CsH;(CH2CI)Cl = Cl* p-Benzoylchloride

H. C. Meal, J. Chem. Phys.

Crystal structure not known from other sources.

C.H,Br2 p- Dibromobenzene

Br?

CoHs(N H2)Cl

24, 1011 (1956).

p-Chloroaniline

24, 1011 (1956).

S. B. Krist.

four chlorines therefore will be equivalent, but on applying a magnetic field, each will show its own Zeeman pattern depending on the direca”

oR [|

co -

-

- -

Ax

Fic. 27. Crystal structure of NaClO; (schematic). A, orientations of the magnetic fields discussed in the text.

B,

C refer to special

tion of its axis of symmetry. The Zeeman pattern for axially symmetric quadrupole interaction was discussed in Section 1c. The frequencies

of the Zeeman components are given by Eqs. (1.24). Figure 28 shows

106

III.

APPLICATIONS

a theoretical plot of the frequencies of the various resonance

compo-

nents aa’p’ versus the angle @ between the magnetic field and the axis of symmetry. The plot indicates the features of the Zeeman pattern for special orientations of the magnetic field that were discussed in Section lc. It is seen from Fig. 28 that the a and 8 components for

6 < 90° interchange their roles.in the region clear from Eqs. (1.21) and (1.24). 2

6 > 90°. This is also

B

1 a

c

e

8©

>

0°

c

4

1

10°

30°

£

1

50)

1

1

70°

90°

1

110°

54°44’

30°

1

1

150°

170°

125°16'

& Py

-1

-2

8

Angle in degrees

Fia. 28. Theoretical plot of Zeeman separations of various components aa’Bp’ for different orientations of applied magnetic field, for an axially symmetric field gradient. The Zeeman separations are in units of 2 = yH.

When the Zeeman field is applied along the direction denoted by A

in Fig. 27, i.e. along a fourfold axis in the unit cell, it makes the same angle, 54°44’, with respect to each of the directions 1, 2, 3, and 4. In this case, from Fig. 28 and Eqs. (1.27), one would expect a three-line

pattern with the aa’ components for all the four chlorine atoms superimposed at the position of the unsplit line, while the 68’ components will be situated symmetrically about this line. Figure 29(a) shows the

signal pattern obtained by Zeldes and Livingston”? which agrees with expectations. When the Zeeman field is along the direction B, i.e.,

6. INFORMATION

ON

CRYSTALLINE

UNIT

CELL

107

parallel to a face diagonal (which is also a twofold axis of the unit cell), it makes angles of 90° with two of the field gradient directions, 1 and 4 in Fig. 27, and angles of 35°16’ with the two other directions 2 and 3.

--- ¢------29.93 me---

(a)

6 =35.27°

(b)

@=90° Fia. 29. (a) Signal pattern for Cl** in NaClO; when the magnetic field is applied along direction A in Fig. 27. (b) Signal pattern when magnetic field is applied in direction B in Fig. 27. The second derivatives of the actual line shapes are shown, as recorded by R. Livingston and H. Zeldes. [Oak Ridge National Laboratory Report ORNL—1913, Table IV (1955).]

From Fig. 28 and Eqs. (1.26) and (1.24), one would expect a superimposed two-line pattern from chlorine atoms 1 and 4, and a super-

imposed four-line pattern from chlorine atoms 2 and 3, leading to a net six-line pattern. This is indeed observed, as is shown in Fig. 29(b), taken from Zeldes and Livingston’s paper.?? When the field is along the

direction C, it makes an angle of 0° with the field gradient direction 1, and equal angles of 70°32’ with the other three directions 2, 3, 4. From Fig. 28, one would then expect a six-line pattern, a doublet

108

III.

from

chlorine atom

APPLICATIONS

1, and four-line patterns from the other three

chlorine atoms superimposed

on each other. Such a pattern has been

reported by Livingston. In addition, Ting, Manring, and Williams!?” have studied the separation of the doublet due to the chlorine atom 1 in this position as a function of the magnitude of the applied field. Strict proportionality is to be expected since Eq. (1.25) is rigorous,

independent of the ratio hw/egQ, while the accuracy of Eqs. (1.24) for @> 0 depends upon the smallness of hw/egQ. They confirmed the expected proportionality for fields as strong as 8000 gauss, where hw

is of the order of % of e?gQ. NaBrO; has the same structure as NaClO;,

with the Na+

and

BrO;~ groups placed at somewhat different positions along the original directions 1, 2, 3, 4 due to the different sizes of Na+ and BrO;7 ions.

The

Zeeman

Kushida,

patterns

Koi,

and

of Br7®

Imaeda,}2”

resonance are

in NaBrO3;,

entirely

analogous

obtained

by

to those

for

NaClO3.

Zeldes and Livingston have studied the Zeeman splittings for Cl® resonance in monoclinic KClO3; and BaClO;3-H.O crystals. These structures are known from x-ray measurements (Table V). The orientations of the magnetic field were obtained from the known Zeeman pattern of a NaClO; single crystal, which was cemented to their sample. They found good agreement between the determined

Z axis of the field gradient tensors at the Cl** nuclei and those expected from the reported crystal structures. However, the asymmetry parameters 7 were found to be rather small, viz., 0.006 and < 0.05, respectively, for the potassium

and barium

salts. This does not agree well

with the structures determined from x-ray measurements, which show that the ClO;- pyramids are considerably distorted from the regular

shape that they have in NaClOs. In the case of iodic acid, the x-ray work by Rogers and Helmholz?2”

reveals that there are four molecules in the unit cell whose positions are

obtained

from

one

another

by

symmetry

operations

involving

screw axes. The molecules were shown to contain distorted IO; pyramids,

as shown

schematically

in Fig.

30,

with

the

hydrogen

atom

attached to On, which is the oxygen atom at the greatest distance from the iodine atom. The positions of the H atoms could not be determined from x-ray measurements. If it is assumed that the OnH group forms hydrogen bonds with two oxygen atoms of two neighbor-

ing molecules (leading to the distortion of the pyramid IO;), and that the bond angle IOnH 1274 See Table V.

and OnH

distance are the same as in the water

6. INFORMATION

ON

CRYSTALLINE

UNIT

CELL

109

molecule (namely about 109° and 1.01 A, respectively), an IH distance of 2.48 A is obtained. The distorted nature of the IO; pyramid was confirmed by the Zeeman measurements by Livingston and Zeldes which indicated a large value of 7 of 0.4505 at the I!?7 nuclei. Since 7 was so large, the approximate Eq. (1.50) cannot be used to obtain the energy levels in the presence of the magnetic field. Therefore Livingston and Zeldes made use of Cohen’s!” tables to calculate 7. They found

that the Z axis of the field gradient at each of the I?” nuclei is not perpendicular to the plane of the three oxygens, but is inclined at an angle 13°12’ with respect to this direction. The magnetic dipole interaction between the proton and the I”? nucleus splits the Zeeman lines

On

Om

Fic. 30. Schematic diagram of an iodic acid molecular unit according to Roger and Helmholz’s structural assignment. (Distances shown are in angstroms.)

of I}#” into doublets. A typical set of curves obtained by Livingston and Zeldes for the frequencies of the various Zeeman components

versus orientation

of the Zeeman

field is shown in Fig. 31. All the

directions of the magnetic field considered in Fig. 31 lie in the be plane. An examination of the crystal structure shows that when the magnetic

field lies in the bc plane, the four iodine nuclei in the unit cell divide into two pairs, and two members of each pair have the same orientation of magnetic field in their respective principal axis systems.

This

is why eight Zeeman components are shown in Fig. 31 instead of the sixteen to be expected for any general orientation of the magnetic field. The doublet splittings are prominent only for the 6 components

and depend strongly upon the orientation of the field, as predicted from Eq. (2.6). Using an equation analogous to (2.6), but suitable for

spin J = $ and finite asymmetry 7, the IH distance can be determined

110

III.

APPLICATIONS

200

LL

160; 120

Zeeman separation (kc)

80

- 120 -160c-axis lL

Ll

-40°

1

—-20°

0

b-axis 1

ed

20°

40°

1

1

g0°

l

80° 90°

Angle in degrees

Fig. 31. Zeeman

separations

(in kc) for the various

components

of the I}?7

resonance lines ($— 4) in crystalline HIO;. The field of 64.6 gauss was applied in the be plane. Doublet splitting of the 6 components due to I-H dipole-dipole interaction is perceptible when the magnetic field is applied in a direction close to the b axis. [Refer to R. Livingston and H. Zeldes, Oak Ridge National Laboratory Report ORNL—1913, Table V (1955).]

from the doublet splitting. Livingston and Zeldes?? find an IH distance of 2.33 + 0.02 A. This is different from that expected from the structure proposed by Roger and Helmholz, but close to the value 2.407

+ 0.013 A 128 obtained by Garrett and Levy from neutron diffraction measurements. 128 Professor M. H. Cohen has pointed out to us that Livingston and Zeldes used a somewhat inaccurate formula for the doublet splitting of the Zeeman lines. A more rigorous treatment based on an obvious modification of (2.6) for finite 7 and I = § would probably improve the agreement between the IH distances found by Garrett and Levy, and Livingston and Zeldes.

6.

INFORMATION

ON

CRYSTALLINE

UNIT

CELL

111

The results of Zeeman measurements for Br7® nuclei in solid bromine and I!?7 nuclei in solid iodine will be discussed in Section 8. The As7* resonance in As2Q3 will be discussed in Section 9.

Parachloroaniline,

studied

by

Meal,!?”7 is a good

example

of a

crystal for which no information on crystal structure is available from other sources. Here quadrupole coupling data are used to obtain information on aspects of the crystal structure. The structure!?® of the

p-chloroaniline

molecule is as shown

schematically in Fig. 32. The

chlorine and nitrogen atoms are in the plane of the benzene ririg. The positions of the hydrogens of the NHgz are not known. They may or may not be in the plane of the benzene ring. No information on this point can be obtained from quadrupole resonance data for Cl*>, but the quadrupole data for the N*" nuclei would be helpful for this purpose. Only the Cl*> quadrupole resonance has been studied by Meal. In order to determine the constitution of the unit cell, there are two questions to answer: (1) how many molecules are there in the unit cell; and (2) how are the molecules oriented relative to each other? One would expect the field gradient tensor at the Cl®> nucleus to have the principal system of

coordinates defined by the Z axis along the C—Cl bond, the X axis defined perpendicular to the plane

of the benzene ring, and the Y axis defined mutually perpendicular to both X and Z. This assignment of

Cl

N H

H

Fia. 32. Molecular structure of parachloroaniline (schematic). Hydrogen atoms on the benzene ring are not shown. The positions of the hydrogen atoms in the NH», group are not

known. They may

nowwein the plane

X, Y, and Z directions satisfies the convention given ring. by Eq. (1.7), as will be clear from the considerations in Section 7b. Meal used a single crystal that was cut cylindrically. Any orientation of the Zeeman field can be characterized by the angle it makes with the axis of the cylinder and two axes in mutually perpen-

dicular directions chosen at right angles to the latter. The procedure for locating the principal axes and asymmetry 7 of the field gradient tensor has already been outlined in Section le. Following this procedure, Meal found that there were two physically inequivalent sites for the Cl®> nuclei in the crystal, both with 7 = 0.06 + 0.03. The

Z principal axes for the two sites were inclined to each other at 79 + 1°. Two methods involving use of Eqs. (1.57) and (1.58) respectively were used

to determine

the

Y axes.

The

orientations

obtained

for the

Y

axis by these two procedures differed by 20°. The mean value was 129 See pp. 153 and 139 in ref. 126.

112

III.

aAppLicaTIOoNs

taken, which indicated that the Y axes for both sites were mutually parallel to the axis of the cylindrical crystal. A morphological exami-

nation of the crystal showed that it had an orthorhombic unit cell with axial ratio a:b:c = 0.939:1:0.800. The a axis was found to be parallel to the axis of the cylindrical crystal, and hence parallel to the Y axes of the two field gradient tensors. The relative orientations of the various axes are shown in Fig. 33. c

X, (90,129)

Z2 (90,141)

Z, 1 (90,39) ’

Fig. 33. Relative orientations of the crystalline axes and field gradient principal axes at the two physically inequivalent lattice sites in the unit cell of p-CINH2CeH, crystal. The angles in the brackets refer to the colatitude @ and azimuth ¢ respectively. 6 is measured from the a axis and ¢ from the b axis in the be plane. Circle

is drawn to indicate that the directions X1, Z:, X2, Ze all lie in the bc plane. [Refer to H. C. Meal, J. Chem. Phys. 24, 1011 (1956).]

At this stage it becomes important to answer question (1) posed above, namely, what is the number of molecules per unit cell? This question Meal answers by pointing out that the two Zeeman patterns due to the two physically different sites exhibit equal intensities, which

leads one to conclude that the numbers of molecules per each of the two sites are equal. The exact number in the unit cell cannot be stated definitely, since there may be two or more molecules of each type oriented parallel to one another. The assignment of the X and Y directions of the field gradient tensor at the Cl** nucleus relative to the plane of the benzene ring was stated at the beginning of this paragraph. Use of this assignment together with the information in Fig. 33 then determines the relative orientations of molecules belong-

6.

INFORMATION

ON

CRYSTALLINE

UNIT

CELL

113

ing to the two different sites. This relative orientation is shown in Fig. 34. Since the field gradient tensors are symmetric with respect to inversion of their principal axes, the relative orientation of the molecules, as shown in Fig. 34, and one where either of the molecules is inverted about the center of its benzene ring, cannot be distinguished by means

of the Zeeman

data. Meal}?’ has also carried out a similar

Zeeman study of p-chlorobenzoyl chloride and found evidence for at

Fig. 34. Relative arrangements of the two p-CINH.CeH, molecules in the unit cell. [Refer to H. C. Meal, J. Chem. Phys. 24, 1011 (1956).]

least four molecules per unit cell. Results of Zeeman studies on p-dichlorobenzene and p-dibromobenzene are discussed in the next section, which deals with the study of phase transitions. c. Study of Phase Transitions by Quadrupole Resonance Sections 3a and 3b dealt with the changes in the frequencies, line widths, and 7’; as functions of the temperature, and how they could

be related to the torsional and rotational motions of the molecules. The present section will deal with the effects of phase changes on the

quadrupole spectra. With first-order phase changes, involving change of crystal structure,!*° the quadrupole spectra can be observed both above and below the transition temperature. Resonance frequencies undergo abrupt changes at the transition temperatures. These changes

in frequency

occur

because

the

electron

distributions

within

the

molecules are affected by intermolecular forces, and therefore depend

quite sensitively upon environment undergoes due to the change in abrupt change occurs in Another

quantity

that

the environment around each molecule. This an abrupt change during a phase transition crystal structure, and hence a corresponding the electron distribution within each molecule. may

undergo

a

sudden

change

at a phase

130 The term “‘first-order phase transitions” is used here specifically to distinguish from order-disorder transitions that smear the resonance beyond recognition. as pointed out in Section 2b.

114

III.

APpPLicaTIONs

transition is the temperature coefficient of the frequency. But in some cases, as for example in p-dichlorobenzene,'*! practically no such

change is observed.

This may

well indicate that the intermolecular

torsional oscillations do not change appreciably at the phase transition. It may also be true, as pointed out in Section 3a, that the temperature coefficient depends more upon the intramolecular vibrations, which do not change very much during the phase transition. If the phase change involves a rearrangement of the molecules in

the unit cell, it would lead to a change in the orientations of the field gradient principal axes. This change could be observed accurately by Zeeman measurements on a single crystal. The results of such measurements by Dean and Pound,?*! and others (see Lutz®*), will be

considered later. From the considerations in Section 3a, both the line width and 7’; would also be expected to be affected by changes, if any, in the torsional motions of the molecules during the change of phase. If a rearrangement of the molecules occurs, the line width would be altered, in addition, by a change in the magnetic dipolar term in

(3.26). No measurements on changes in 7

or line width through first-

order phase transitions have been reported.

Allen‘? has detected phase transitions in some compounds by studying the frequencies of their

organic chlorine Cl** quadrupole

spectra. Cotts and Knight1*? have observed the phase transition in KNbO; by studying the Nb** quadrupole spectrum. Their results have been discussed by Cohen and Reif.! Dean and Pound!*! have studied the phase transitions in a number of compounds containing chlorine atoms attached to a benzene ring. In Fig. 7 a number of their curves for the Cl® quadrupole resonance frequencies versus temperature are shown.

They have made the most careful measurement on p-dichlorobenzene. Later this substance was also studied by Lutz.®8 The phase transition in p-dichlorobenzene has been observed by use of a number of different techniques, among them analysis of the cooling curve from the melt, Raman spectra, and x-ray diffraction. The results of these measurements, which have been discussed in

detail by Dean, of Dean

and

seemed

Pound

rather contradictory.

and later investigation

However,

the studies

of the high-temperature

phase by Lutz, together with careful electron and x-ray diffraction work by Croatto e¢ al.4! and Jeffrey and McVeagh,'* respectively, have helped to resolve the contradictions. These results shall be sum131 C, Dean and R. V. Pound, J. Chem. Phys. 20, 195 (1952); also ref. 5. 132 R. M. Cotts and W. D. Knight, Phys. Rev. 96, 1285 (1954). 133 G, A. Jeffrey and W. J. McVeagh, J. Chem. Phys. 28, 1165 (1955).

6.

INFORMATION

ON

CRYSTALLINE

UNIT

CELL

115

marized briefly to show the role that quadrupole resonance can play in investigations on phase transitions, and how far it can supplement

observations by other techniques. The cooling measurements of Beck and Ebbinghaus,'** Sirkar and Gupta,!8> and Campbell and Prodan!** gave very confusing results.

Beck and Ebbinghaus and Campbell and Prodan determined a transition temperature at 39.5°C at which a contraction occurred on cooling. Sirkar and Gupta located the transition at a much lower temperature, namely 32°C. Raman spectra measurements also gave somewhat contradictory results. Vuks,!*7 from his measurements on low-frequency Raman spectra, had concluded that a phase transition centered at 32°C did occur, but that it could not be initiated in the forward direction (heating) unless the sample was heated above 37°C, and in the backward direction (cooling) unless cooled to 22°C. Saksena®? confirmed these conclusions and postulated a free rotation of the molecules about an axis perpendicular to the benzene ring in the high

temperature phase.

Sirkar and Gupta!*® found no definite evidence

for any change in the low-frequency Raman spectra due to the phase transition. Sirkar!® has criticized Saksena’s postulate of free rotation, pointing out that the Raman spectra measurements did not really confirm it. The x-ray measurements, prior to Dean and Pound’s work, were performed by Hendricks!*® and Sirkar and Gupta.14° Hendricks’ measurements, presumably on the low-temperature phase

(not stated in his paper), showed that the unit cell was monoclinic and contained two C!—C] directions. These were related to each other by a 180° rotation about the b axis. The molecules were assigned the orientations shown in Table VI. The Z and X axes refer respectively to the Cl—Cl direction and the perpendicular to the plane of the benzene ring. The angle @ gives the angle of inclination of any direction with the b axis, and ¢ is its azimuth measured from a particular direction chosen by Hendricks, in a plane perpendicular to the h axis. An alternative arrangement shown in Table VII could also fit this x-ray diffraction data. Hendricks assigned this arrangement to p-dibromobenzene which gave the same x-ray scattering data. The 134K, Beck and K. Ebbinghaus, Chem. Ber. 39, 3870 (1906). 185 §, C. Sirkar and J. Gupta, Ind. J. Phys. 10, 473 (1936). 136 A. N. Campbell

and L. A. Prodan,

J. Am.

137 M, Vuks, Compt. rend. acad. sci. URSS 138 §. C. Sirkar, J. Chem.

Phys.

20, 742

Chem.

Soc. 70, 553

1, 73 (1936).

(1952).

139 §. B. Hendricks, Z. Krist. 84, 85 (1935). 140, C. Sirkar and J. Gupta, Indian J. Phys. 11, 283 (1936).

(1948).

116

III.

aAprLicaTIons

uncertainty in the orientations of the molecules arose from the fact that the other atoms, besides the chlorines, namely carbons and

hydrogens, could not be located due to their low scattering powers. Sirkar and Gupta!*? made x-ray measurements of the high temperature 8 phase by heating the sample to 51°C for half an hour and Tasie

VI.

ORIENTATION

OF p-DICHLOROBENZENE To HENDRICKS

Mo.LecuLes

Z axis Molecule

1 2

ACCORDING

X axis

0

o

0

’

49° 49°

3°42’ 183°42’

90° 90°

93°42’ 273°42’

photographing the x-ray pattern at 37°C. Their results did not differ from Hendricks’ scattering data for the low-temperature phase (a phase). They postulated that during the forward transition, the molecules switched from the configuration that Hendricks assigned to them in Table VI for the a phase to that (@ phase) described in

Table VII. Their measurements are not consistent with the free rotation postulate of Saksena, which would require a disordered structure and thus an x-ray pattern distinctly different from that for the a phase. Tasie

VII.

ORIENTATION

OF p-DIBROMOBENZENE to HENDRICKS

MOLECULES

Z axis Molecule

1 2

ACCORDING

X axis

0

d

0

o

51°54’ 51°54’

53°12’ 233°12’

90° 90°

143°12’ 323°12’

At this stage, Dean and Pound?*! made their careful investigations on the Cl®* quadrupole resonance. As is clear from Fig. 7, they found a change in the Cl*> resonance frequency in the neighborhood of the transition temperature 32°C found by Sirkar and Guptz!** and Vuks. }37

Also the rate of the transition was found to be rather slow, and dependent

on the previous

with Vuks’

heat

conclusions.

treatment

When

of the

the sample

specimen,

in agreement

was freshly transformed

from the 6 phase to a phase (within a quarter of an hour), the forward transition a— 6 occurred in only about five minutes on heating to

6. INFORMATION

about

35°C.

This

ON

transition,

CRYSTALLINE

however,

took

UNIT

twelve

CELL

117

to twenty-four

hours when the sample was maintained at 32°C. The backward transition (8 > a) occurred slowly upon cooling from the melt to 27°C, but could not be accelerated by cooling to temperatures below 22°C. It is clear from the curve in Fig. 7, and also from a plot of the temperature coefficient of the frequency versus temperature, that no marked change in the temperature coefficient occurs during the phase transition. As pointed out in the beginning of this section, this does not prove definitely that no change occurred in the characteristics of the torsional motions of the molecules, but it definitely rules out the free rotation in the 6 phase postulated by Saksena.®? Such a rotation would cause a very large change in the temperature coefficient of the frequency in the neighborhood of the transition temperature. Dean and Pound also carried out a Zeeman study of the Cl** quadrupole spectrum on a single crystal of p-dichlorobenzene. Using

the procedure discussed in Section le, they determined the asymmetry parameters and principal axes at the Cl*® nuclei in the two molecules in the unit cell at the a phase. They found yn = 0.08 + 0.03 for the Cl®> nuclei in both molecules, and assuming that the X axis of the field gradient tensor at each Cl** nucleus is perpendicular to the plane of the benzene ring, they obtained the orientations shown in Table VIII for the two molecules in the unit cell. The notations in Table VIII, TasBie

VIII.

ORIENTATION

OF

p-DICHLOROBENZENE

to

Dran

AND

MOLECULES

Z axis (Cl—C] direction)

Molecule 1 2

ACCORDING

PouNnD

X axis

0

o

6

’

52°30’ 52°30’

5°30’ 185°30’

60° 60°

121° 301°

viz. 0, ¢, X, and Z, are the same as in Tables VI and VII. Comparing Table VIII with Table VI, it is seen that the Cl—Cl or C—Cl directions in both molecules are close to those found by Hendricks. But

the planes of the molecules are found

(from the orientations of the

X axes) to be inclined at about 30° to the axis of symmetry (b axis) instead of being nearly parallel to it, as expected from Hendricks’

assignment

in Table

VI.

Croatto

tron diffraction measurement

et al.t! later performed

an elec-

on the a phase. A picture of the mono-

118

III.

APPLICATIONS

clinic unit cell obtained by them

is shown

between the plane z of one molecule in cule in the unit cell is obtained by through 180°) and the crystallographic (010) = 63°56’, and (100) = 86°20’. tions of the molecules in the unit cell

in Fig. 35. The

angles

the unit cell (the other molea rotation about the b axis planes are 7(001) = 31°52’, This shows that the orientadefinitely disagree with Hen-

drick’s assignment in Table VI, which would have placed them almost parallel

to the

(001)

plane.

On

the

other

hand,

their results

agree

better with Dean and Pound’s assignment in Table VIII. In addition, Croatto

and

Bezzi!#!

have

shown

that

Hendricks’

assignment

of

different structures to p-dichlorobenzene and p-dibromobenzene is not

Fig. 35. Arrangement of p-Cl2¢ molecules in the monoclinic lattice (a, phase). [U. Croatto, S. Bezzi, and E. Bua, Acta Cryst. 5, 825 (1952).]

correct, and that the two are isostructural with unit cells of the forms shown in Fig. 35. Dean and Pound obtained very little data on the 6 phase of paradichlorobenzene. These data showed that 7 was about the same in the 8 phase as in the a phase and also that only a single Cl—Cl

direction existed in the unit cell. This direction was different from either of the directions described by Tables VII or VIII. They suggested that the Zeeman patterns due to one of the molecules in the unit cell possibly had been missed and that the structure probably

remained monoclinic in the 8 phase, with the same direction for the b axis as in the a phase. However, the orientation of the molecules probably changes as proposed by Sirkar and Gupta.!*549 This suggestion was proved wrong by the later Cl** quadrupole resonance measurements of Lutz,*® which agree with recent x-ray work by Jeffrey and McVeagh.!*3 141 U, Croatto

and S. Bezzi, Gazz. chim. ital. 79, 240

(1949).

7.

INTERPRETATION

OF

NUCLEAR

QUADRUPOLE

COUPLING

DATA

119

Lutz’s Zeeman data on the 6 phase showed that there is definitely only one orientation for the Z principal axes of the field gradient tensors at the Cl** nuclei in the unit cell. This direction could not be related

in a simple manner to the two Cl—Cl directions in the a phase found by Dean and Pound. Jeffrey and McVeagh!* find by careful x-ray work on the 8 phase that the unit cell is triclinic, and has nearly half the volume of the monoclinic unit cell with only one molecule within it. No evidence of the disorder postulated by Saksena®? was found. More recently Dean and Lindstrand!*? have observed a third phase for p-dichlorobenzene, termed the y phase. This is obtained by adding a few mole per cent of xylene or benzene to p-dichlorobenzene. Throughout the range over which quadrupole resonance has been observed, namely —76° to +10°C, the Cl®* nuclei during the y phase

have a resonance frequency 430 ke higher than the a phase at the same temperature. Benedek e¢ al.'43 have also obtained a new phase at 24.8°C by subjecting the a phase to a pressure of 1600 atmospheres. The phase they observed is presumed to be the y phase also, because the Cl®5 frequency in this phase is also about 430 ke higher than in the a phase at the same temperature.

The Zeeman splitting of Br7® resonance in p-dibromobenzene has been studied

by

Kojima

eé al.!44 at room

temperature.

They

found

good agreement between their results for the orientation of the principal axes of the field gradient tensor at the Br7® nuclei and the crystal structure found by Croatto and Bezzi.!4! This confirms the conclusion of Croatto et al.4! that p dibromobenzene is isostructural with the

a phase of p dichlorobenzene. 7. INTERPRETATION OF NUCLEAR QUADRUPOLE CouPLING Data IN TreRMS OF ELECTRON DISTRIBUTION IN FREE MOLECULES Nuclear quadrupole spectra provide information concerning the structure of molecules in the solid state when they are no longer free but under the influence of neighboring molecules. In order to infer the interaction between the molecules in the solid, it is first necessary to understand the nuclear quadrupole interaction in free molecules in terms of the electron distributions within them. Data on nuclear

quadrupole interactions in free molecules have been obtained principally from the fine structure caused by such interactions in the rota142 C, Dean and E. Lindstrand, J. Chem. Phys. 24, 1114 (1956). 143 G. B. Benedek, N. Bloembergen, and T. Kushida, Bull. Am. Phys. Soc. [2] 1, 11 (1956).

1448. Kojima, K. Tsukada, and Y. Hinaga, J. Phys. Soc. Japan 10, 498 (1955).

120

III.

APPLICATIONS

tional microwave spectra‘*® of molecules. However, microwave spectroscopy!® can be applied only to molecules that possess permanent electric dipole moments. Important examples of molecules that do not possess electric dipole moments in the free state are the halogens

(fluorine, chlorine, bromine, and iodine) and the tetrahalides of elements

of

germanium,

the

fourth

group

of

the

and tin). The method

periodic

table

(carbon,

of molecular beam

silicon,

spectroscopy

could be applied to such molecules, but it has been applied only to the study of quadrupole interactions in a few diatomic molecules. !4° In discussing the theory of the origin of nuclear quadrupole inter-

actions in free molecules, we shall make use of data obtained from microwave and molecular beam spectroscopy wherever available. In the case of those molecules where such data are not available, results of nuclear quadrupole resonance measurements on the solid substance will be considered as approximately representative of free molecules. The basis for believing that this approximation is not far from correct was pointed out in the beginning of Part III. In Section 8 it will

be shown that for those molecules which have been studied by the methods of both microwave spectroscopy and nuclear quadrupole resonance, the small differences in quadrupole interaction in gaseous and solid states may be interpreted in terms of the intermolecular interactions in the solid state. a.

Townes and Dailey Theory for Interpretation of Nuclear Quadrupole Coupling Constants in Free Molecules

The fundamental quantities determined by quadrupole resonance, microwave, and molecular beam spectroscopy measurements are e?qQ and 7. The asymmetry parameter 7 does not involve any properties of the nucleus, and as will be seen in Section 7b, it can be interpreted directly in terms of the electron distribution in molecules. The interpretation of e?@Q shall be considered in this section. This quantity involves a nuclear property Q and a molecular property g. A knowledge of one, therefore, can determine the other.14* The quantity q could be estimated if the charge distribution over the molecule were known. This involves a knowledge of the wave functions for the electrons in the molecule. Accurate wave functions are at present available only for the hydrogen molecule and somewhat less accurate SCFMO 145 See Chapter 6 and Appendix VI in ref. 16; Chapter 11 and Table XI in ref. 57. 146 Q can be roughly estimated on the basis of some kind of model for the nucleus. [See, for example, the article by R. J. Blinstoyle, Revs. Modern Phys. 28, 76 (1956).]

7.

INTERPRETATION

wave

functions

OF

NUCLEAR

QUADRUPOLE

are available for a few

other

COUPLING

molecules.!47

DATA

121

Accurate

calculations of q, similar to those of Nordsieck, Ishiguro, and Newell'48 for the hydrogen deuteride molecule (HD), cannot be performed, therefore, for other molecules at the present time. One has to rely upon simpler semi-empirical considerations in order to evaluate g, which is the approach of the Townes and Dailey theory.!??4° Townes and

Dailey express the value of eq in the molecule in terms of eqat, defined as the field gradient due to an electron in the lowest p state outside the inner closed shells of the free atom. The electron produces a maximum field gradient when it is in this state,®° because (1) the electrons in the closed shells do not as a whole contribute anything to the q for the atom; (2) the contribution from electrons in S states (1 = 0) outside the closed shells is zero; and (3) the contributions from other states decrease rapidly with increasing n and 1. Townes and Dailey

EXPTESS Yinol AS

Qmot = fut

(7.1)

where f is a quantity depending on the electronic structure of the molecule. The quantity f, as we shall see shortly, can be obtained from physical and chemical considerations. Since the quadrupole moment Q does not depend upon the electronic charge distribution in the molecule or the atom (refer to Section 3a), one can write

emo) = fe*qa.

(7.2)

It will be seen presently that both e?q.Q and qa can be determined in many cases from hyperfine splitting of molecular beam spectra!®! which involve transitions between the energy levels of the free atom in a magnetic field. These quantities are also less accurately determined from the hyperfine splitting of optical spectra. In such cases, use of Eq. (3.2) helps in assessing the correctness of the predicted f for the molecule. In some other cases e?g..Q cannot be obtained directly from molecular beam or optical spectra. It is then necessary to estimate 47H. M. James and A. S. Coolidge, J. Chem. Phys. 1, 825 (1933). For a list of references on molecules to which self-consistent molecular orbital treatments have been applied, see R. S. Mulliken, J. Chem. Phys. 28, 1836 (1955), Section 4. 8.C.F.M.O. stands for ‘Self-Consistent Field Molecular Orbital. 148 A. Nordsieck, Phys. Rev. 68, 310 (1940); G. F. Newell, Phys. Rev. 78, 711 (1950); E. Ishiguro, J. Phys. Soc. Japan 8, 129, 133 (1948). 149 See Chapter 9 in ref. 16. 150 Refer to Table 9-2 in ref. 16 for a numerical demonstration of this point. 151 See Chapter 2 in ref. 57.

122

III.

apPpLicaTIONs

gat from other sources,!*? and use Eq. (3.2) together with the f predicted by the Townes and Dailey theory in order to obtain Q. The correctness of the Q for a nucleus obtained by such a procedure can only be proved if approximately the same value is obtained for it in a number of different molecules. A brief discussion of the methods for obtaining gat is useful at this stage. (1) Quadrupole coupling constant in atoms. Casimir!®* has calculated the perturbation energy AF for an electron outside a closed shell

due to its interaction with the magnetic dipole and electric quadrupole moments

of the nucleus as

-Id+1)JVU+ AE = Jog + pack C+1) BIQI — 12 QF — 1)

1)

|

(7.3)

In Eq. (38.3)

C=FF+1)-JU +1) -11 +1). The spin angular momentum of the nucleus is equal to J/ and the total

angular momentum of the electron is given by Jh, where J =1 +4 and Jf is the orbital angular momentum of the electron. The total angular momentum of the whole system, nucleus plus electron, is given by Fh, where

F=I+J,[+J-1,-+-I-J or

J+LhJ+1I-1,:::J-I1

depending upon whether IJ > J or J > J, respectively. The quantities

a and b are given by a=_

1 wByh2i(l + 1) nuclei wave spectra chlorine atoms for their outer

have been determined for this molecule from its microby Townes, Merritt, and Wright.!®° The iodine and have the configurations 5s?5p® and 3s73p', respectively, electrons. Both atoms lack one electron for completion

of a closed shell. The closed shell would possess spherical symmetry and produce zero field gradient at the nucleus. The free chlorine and 166 C. H. Townes, (1948).

F. R.

Merritt, and B. D. Wright, Phys. Rev. 78, 1334, 1249A

132

III.

APPLICATIONS

iodine atoms, therefore, have field gradients equal in magnitude to the field gradients due to the 3p and 5p electrons respectively, but are opposite in sign (see Table IX). When the two atoms combine to form the molecule ICI, their electron orbitals overlap, leading to a net reduction in energy of the molecule with respect to the total energy for the two free atoms. This reduction in energy explains the stability of the molecule.!** The principal overlap occurs between the outermost pz electrons directed along the internuclear axis, and produces a a bond between the two atoms. There may be overlaps between the p,

and p, electrons on the two atoms leading to IT bonds, but their overlaps will be neglected at present for reasons of simplicity. The two electrons forming the « bond move in a molecular orbital, which is usually chosen as a linear combination of the atomic orbitals on the two atoms.'*7 This molecular orbital may be written as y=

Va + ty

G+ ® + 28)!

where y; and Yo refer to atomic atoms, and S is given by

orbitals for the iodine and

S = fla dr.

(7.11) chlorine

(7.12)

The quantity 7 is a measure of the relative importance of the roles played by the atomic orbitals ya and y; in the molecular orbital y. The quantity!®*

I

_1-?

“TF?

(7.13)

is usually referred to in the literature of chemical valence theory as the

ionic character of the bond. Thus, if 7 = 1, then J = 0, and the bond is completely covalent; for 7 = 0, J = 1, and the bond is 100% ionic. In the ionic case the Cl ion forms a closed shell and the I+ ion lacks two p, electrons. The actual situation is somewhat intermediate between the two extremes. The method for predicting approximate values of the ionic character will be discussed later in this section. The orbitals Yo. and y; in Eq. (7.11) will not necessarily represent pure p, atomic states. Actually, it is well known from valence theories!* that a stronger o bond is produced if the bonding electrons are assumed to move in orbitals made up of mixtures of s and p, states of the atoms 166 167 168 169

See See See See

p. 78 in ref. 37. Chapter IV in ref. 37. p. 148 in ref. 37. p. 77 and Chapter VIII in ref. 126.

7.

INTERPRETATION

OF

NUCLEAR

QUADRUPOLE

COUPLING

DATA

133

instead of pure p, states. This mixing of s and p atomic states leading to strongly directed atomic orbitals is known as hybridization. In the

ICI molecule there can be hybridization for both atoms. In general, other states with appropriate symmetry can also mix with the p, state. Thus d hybridization, which involves some mixing of the d,, state!®*:17° with the p, state, should also be considered. The normalized atomic orbitals you and y which take part in o-bond formation therefore can be written as Ya

=

(1

—

3

— d?)

> .c1 +

sear

+

dwa,.c1

and

(7.14)

vr = (1 — 8? — d? Naar + 8'bar + d'Van where

s? and

d? give the amounts

of s and

d hybridization

at the

chlorine atom, and s’? and d’ represent the corresponding quantities at the I atom. Different contributions to the field gradient q,,°' along the Z axis at the Cl** nucleus can now be listed. These are (1) the two electrons in the molecular orbital y in Eq. (7.11); (2) the two pairs of 3p, and 3p, electrons on the chlorine atom; (3) the electrons which were in the 3s? shell in the free atom; (4) the electrons in the inner shells, namely 1s?2p® for chlorine; and (5) the nucleus of the iodine atom and all its electrons except for the one that takes part in the molecular orbital y. The last source may be regarded approximately as a unit positive charge at the position of the iodine nucleus. These different sources of

field gradients will now be considered separately. 1. The electrons in the molecular orbital y in (7.11) give rise to a field gradient equo, where quo =

29. — cos? (8 Ba Bes — 1)1) dt

—2 Jv

To

2

~

TF

+724

=

[vee

28

(da +

2

and

qa

[ve

—_—

Boot ta =) cl

(3 cos

2

Bex

dr

+

.

2tqou) (7.15)

1) dr

—

l

a

.

qi

.

To 2

dc

=

J

vive, 2208

fo 3 To

_ 1) dr.

170 The notation for the atomic states follows that of Eyring et al. (p. 227 in ref. 11); d,, is used in place of d,2 for convenience in adding an additional suffix to indicate the atom to which this state belongs.

134

III.

APppLicaTIONS

From Eq. (7.14) we obtain

ga = (1 — 8? — d*)qp,0 + A7Qa,,0. + 28dqea,,c1In

Eq.

(7.16)

the

notations

eg ,,

ege,,c: Stand

respectively

(7.16) for the

field gradients due to electrons in the p, or d,, atomic orbitals on the free chlorine atom. Here qa,,c1 is given by

2 Aq — 1 /

Veo 3 cos? Ga

ro?

= 1 Wa,.01

dr

and is the only cross term that occurs in Eq. (7.16). The other possible cross terms are seen to vanish on the basis of parity considerations. 2. Each of the 3p, and 3p, electron pairs on the chlorine atom produces a field gradient Gr.c1 =

—49p,01

according to (7.10). The net contribution to the field gradient from these w electrons (as the electrons moving in orbitals perpendicular to

the o bond are called) is, therefore, Qx =

—2Qp,c1.

(7.17)

3. The electrons in the 3s? shell do not contribute anything to the

field gradient in the free atom. In the ICI] molecule the atomic orbital on

the

chlorine,

which

participates

in the

o bond,

involves

some

s hybridization. Therefore, the lone pair of electrons which moved in the 3s? state in the free atom, now move in an orbital crt orthogonal to the 3s-3p hybrid Yo given in (7.14), Yat

=

(1 —

8?)Wvecr +

These two electrons contribute an amount given by gat = —28"qp,c.

spp,c1.

(7.18)

got to the field gradient, (7.19)

4. We shall now consider the role of the inner shell electrons (1s?2p°) of the chlorine atom. It was pointed out in Section 7a(1) that the field gradients due to an externul charge are affected by the inner

shell electrons in the atom. The field gradients ga, gcr and qr, gx, and qci* discussed above, therefore, are altered from their respective values

obtained from Eqs. (7.15), (7.16) (7.17), and (7.19). The alterations in ga, Gort, and q, are taken care of by employing for q»,c: in Eqs. (7.16), (7.17), and (7.19) the experimental value q,, obtained from the atomic beam measurements

on the chlorine atom, without correcting for the

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135

Sternheimer effect (refer to Table IX). Small corrections due to the Sternheimer effect are necessary for the terms d?qq,,c. and sdQsa,,01 in Eq. (7.16). However, it will be seen later in this section that these terms, including gc, are small for most molecules, so the influence of

small corrections to them may be neglected. The value of qi as given by (7.16) corresponds to the field gradient due to an electron moving in the s-p hybrid orbital on the iodine atom given by (7.14). This field gradient is affected by the perturbation of both inner shell electrons (1s?2p°) of the chlorine atom, as well as the outer electrons in

the orbitals Ya, Yat, 3p,, and 3p, of the chlorine atom. Thus the actual field gradient due to the electron on the iodine orbital y; will be given by q(1 — Ri), where —R, takes account of the antishielding effects due to the electrons on the chlorine atom. FR, will differ somewhat from the antishielding R, experienced by a charge entirely external to a neutral chlorine atom. This follows because y overlaps Yau, and an electron in the orbital y; therefore partially penetrates into the chlorine atom. 5. The effective positive charge at the position of the iodine nucleus

produces a

field gradient at the chlorine nucleus equal to 2 qt = Ri (1 — Re)

(7.20)

where F is the IC] distance. From (7.15), (7.16), (7.17), (7.19), and (7.20), the net field gradient gz"! along the Z axis of the chlorine atom is therefore qa"

=

2

T+ a +

is [qara(l

—

8? —

d?) +

qa,

+

2.dqca,.cil

“9 =

2Gac1 +

287Garcr

+

lz

(1 —

Re)

+

_

at

2

|

4tgon +

T+

2+

Qis

(7.21)

In Eq. (7.21), the terms in the first curly brackets represent the contributions to q.,°' from electrons on the chlorine atom itself. Those terms in the second bracket represent the contribution from the iodine atom,

and the last term gives the contribution to q.' from the ‘‘overlap”’ electron density for the proximations to simplify The term involving may be neglected, since

¢ bond. Townes and Dailey made some apEq. (7.21). These will now be considered. qa,,c. in the first curly brackets in Eq. (7.21) qa,,a, the field gradient due to an electron in

136

III.

aprpLicatTions

the 3d., state, is insignificant compared to qp,a, the field gradient due to a 8p electron. From this it does not follow that the term 2sdqja,,c1 can be neglected. However, it involves the product of both s and d hybrid characters, and the quantity sd is usually much smaller than s?, since the d character is usually found to be relatively unimportant in most molecules. This point will be discussed later in this section. Schatz!” has shown for the cases of HCl and CH;Cl, that q..“' given by (7.21) will be too small, if Slater orbitals are used for the different atomic orbitals involved in Eq. (7.14). This underestimation occurs because the orbitals Yc: and yx in (7.14) for the IC] molecule would be somewhat different from the atomic orbitals for the free atom. A consideration of wave functions for the hydrogen molecule!”? suggests that the effective Z (nuclear charge) for the atomic orbitals which takes

part in the o bond within the molecule would be larger than the Slater values for the free atom.'7* A larger effective Z corresponds to a smaller size of the atomic orbital. Therefore gaia in Eq. (7.21) would be larger than the value for a free atom. This increase is counteracted by the overlap term s in the denominator. The term s has been tabulated for Slater orbitals by Mulliken e¢ al.!74 and varies between zero and unity

for various bonds. As Gordy!”> has pointed out, there is little error in assigning the free atom value to gaa in (7.21) and dropping the overlap term s in the denominator. There is no rigorous justification for this procedure, but the fairly consistent interpretation of quadrupole data that is obtained by it shows that it is not far from being correct. The second term is usually neglected from a consideration of

the magnitude of the distance®®:!?2176 R, but the effects of the large factors (1 — Re) and (1 — R,) are not considered explicitly. Actually, for an external charge acting on the Cl** nucleus of a Cl- ion (1s?2s?2p°3s?3p°), Sternheimer and Foley!® obtain a value of R equal 171 P. Schatz, J. Chem. Phys. 22, 755 (1954). Similar conclusions have been obtained from recent calculations on B! coupling constants in some boron compounds [T. P. Das, J. Chem. Phys. 6, 763 (1957)]. 172 For a list of references on the various wave functions that have been calculated for the Hz molecule, see p. 119, Table VII in ref. 37. Also refer to T. P. Das |Bull. Am.

Phys.

Soc.

[2] 1, 313

(1956)] for some

conclusions

on the values ob-

tained for the field gradient at the deuteron in HD molecule using these wave functions. 173 See p. 163 in ref. 11. 174R. 8. Mulliken, C. A. Rieke, D. Orloff, and H. Orloff, J. Chem. Phys. 17, 1248 (1949).

115 W. Gordy, Discussions Faraday Soc. 19, 14 (1955). 176 C, H. Townes and B. P. Dailey, J. Chem. Phys. 28, 118 (1955).

7.

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to —57.7. In Eq. (7.21) Re is approximately equal to the antishielding experienced by a charge outside a neutral chlorine atom. Therefore R, and R, will be smaller than R, since the negative ion is more deformable than the neutral atom. For the sake of argument, a value of —30 for both (1 — R,) and (1 — R,) will be assumed. With this assumption, a unit positive charge on an adjacent nucleus about 2 A away would

produce a field gradient at the Cl** nucleus equal to 3.75 X 10-4 e/cm’. This is about ten per cent of gaci tabulated in Table IX. But there is also the second term in the second brackets, arising from the iodine p, orbital, which has a sign opposite to the first term [see Eq. (7.15)]. If

i = 1, i.e., the bond I—Cl is almost completely covalent, the sum of the two terms in the second brackets, representing the contribution of the nuclear charge and electrons on the iodine atom to the field gradient at the chlorine nucleus, is nearly zero. This is also clear physically, since the chlorine nucleus sees a nearly neutral iodine

atom. However, in the case of ionic molecules like NaCl or KCl, the Nat or K+ ion appears as a positive charge to the chlorine nucleus, and would make a significant contribution to q,.'. This contribution would in such cases be relatively more important, since the electrons on the chlorine themselves would have a configuration close to that for the Cl- ion, thus making very little contribution to q..“'. This is even more true of the iodides and bromides, since the I~ and Br~ ions are expected to be more deformable than Cl-. The last term of Eq. (7.21), which may be termed the overlap density term, has been shown by Townes and Dailey}”? to be less than wy of datc: when Slater orbitals are used for the I and Cl atoms in ICI. This term would be even smaller in molecules in which the ionic

character is greater than in ICI. For the lighter atom boron, which has a smaller covalent bond radius in its compounds, recent calculations!”! show that the last term in (7.21) is not quite so insignificant as in the case of the halogens. However, in this case, for nearly completely covalent bonds, the net contribution from the distant atom and this overlap density term is very small. For molecules which are not too ionic in character, one obtains, therefore, the approximate expression 1-

qu

=

|-

2

as

rk

Gace. + 2(1

—

2dua

(7.22)

Using Eq. (7.13) for the ionic character J, Eq. (7.22) may be written in the form

qe! = [(1 — 8? + a? — I) + (8? + d?)]qatci.

(7.23)

138

III.

appLicaTIONS

When the p, and p, electrons on both atoms also take part in bond formation between the two atoms, there will be a further decrease in

qe from the value given by (7.22). Portions of the electron cloud from the 2p, and 2p, states of the atoms move into the region between the

atoms in order to overlap with each other and produce II bonds perpendicular to the « bond. The double-bond character due to either the

pz OY p, electrons is defined by the fraction IJ, after proper normalization, as II = [Electron density in the pz (or p,) state of the free chlorine atom (or free iodine atom)] — [electron density in the pz (or p,) state of the bound chlorine atom (or bound iodine atom)].

The

II bonds

reduce

the contribution

to the field gradient at the

chlorine nucleus from the 3p, and 3p, electrons on chlorine from the value given in (7.17) to

qn = —2 (

II

- r) atc

(7.24)

causing a reduction of Iga. In addition to this reduction, there is also a negligible alteration in the contribution from the iodine atom which we will not consider. Equation (7.23) for q,.“! therefore is altered to

qf = [1-8 + d®@-T-M + 1s + a)]quc. In a similar manner, be shown to be

the field gradient q..' at the iodine nucleus can

qe! = [1 — s? + a? + I — 0) — I(s? + d?)]qacs. Equations

(7.25) (7.26)

(7.25) and (7.26) hold not only for the chlorine and iodine

atoms in IC] molecule, but in general for a halogen atom bonded to any other atom. Equations (7.25) and (7.26) apply respectively when the halogen is electronegative or electropositive with respect to the other atom. In addition, these equations may be written in a general

form which applies also to atoms that form multiple bonds; for example, N14 and As’7° in NH3, AsH;, and AsF3. This will be clear from the following discussion.

Except in rare cases where both J and (s? + d?) are large, the term I(s? + d?) can be neglected. Equations (7.25) and (7.26) then reduce to

qa and

= (1- +a

— I — Da@ata

(7.27)

Qe! = (1 — 8? +d? +I — TW) quer.

(7.28)

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139

These expressions for q;.°' and q,.' can be given simple physical interpretations in terms of the electron density on the chlorine atom. Let us consider q,,"' first. If the chlorine atom forms a purely covalent p bond, it would have a configuration 3s*3p.73p,?sp. for its outer electrons. There would be a net deficiency of one electron in the p. state from the closed shell configuration. A field gradient results at the Cl* nucleus equal to the negative of that due to a p, electron. On the other hand, if the bond is 100% ionic, there should be a Cl- ion with the configuration 3873p273p,73p.7. With the net number of electrons in the

Dz, Py, and p, states equal, there would be spherical symmetry at the Cl35

nucleus

and

the

field gradient

would

be

zero.

With

the

ionic

character I for the o bond, the fractional importance of the 3s’3p* and 3s°3p® configurations for the outer electrons of chlorine would be (1 — J) and J, respectively. This leads to a net population (1 + J) in the 3p, state. If there is some s hybridization, Eqs. (7.14) and (7.19) show that the electron originally in the 3p, state loses an amount s?

of p character, while the two electrons in the 3s? state each gain an amount

s? of p character.

The

net population in the p, state of the

chlorine atom is therefore increased by s?. On the other hand, since the 3d, state is originally empty, d hybridization leads to a loss p character by an amount d?. Also the 2p, and 2p, states each lose amount II of their populations when there is double bond character this amount. If U,, U,, and U, represent the electron populations

of an of of

the 3p2, 3py,, and 3p, states of the chlorine atom, it is evident therefore that U,=(1+s?-@+4+1) U, = (2 — I) U, = (2 — Tl). (7.29) Since q>,c. = —4Qp.c1, therefore

the net field gradient

in the p, direction

2+ U gu = (= 0, + FEU) (gue,

is

(7.30)

Upon substitution from Eq. (7.29), Eq. (7.30) is seen to reproduce Eq. (7.27), which was obtained from more detailed considerations.

These detailed considerations were presented in order to bring out the approximations involved in Eq. (7.27). The negative of the quantity f in (7.2) is thus equal to [—U, + (Uz + U,)/2], and is termed the

“y-electron defect.” If U, happens to be greater than (U, + U,)/2, the

quantity f is termed

‘“p-electron

excess,’

and

the

quadrupole

140

III.

appLicaTions

coupling constant e?gQ then has the same sign as e7q..Q for a p, electron. At the I'*’ nucleus in ICI, the fractional importance of the ionic character is given by J for the ionic state J+ (outer configuration 3873p,73p,”). The p-electron defect given in (7.30) is increased, which leads to an increase in q..', as seen from Eq. (7.28). Equation (7.30) gives the general formula applying to all atoms, whether they appear at the end of a bond or form bonds with a number of other atoms. The choice of x, y, and z axes is dictated by the consideration as to which two of the three p states (pz, py, pz) differ least in population. The direction of the third one is then chosen as the z axis. The choice between z and y directions is made in accordance with the convention (1.7) in Section la. This point will become clear from examples discussed both in the present and the succeeding two chapters. The example chosen, namely the ICI molecule, has cylindrical symmetry about the ICI direction, so that

Uz = U,.

(7.31)

Therefore, considering the p-electron excesses in the p, and p, directions, we have Qrx!

= | U, —

—

+2 Us

Qacr

= | Uy oat: — 2

and so n = 0 at the Cl** nucleus. There are many compounds, such in which the chemical bonds do about the bond axis (see Section

Quct = Qw"' — (7.82)

The same is true for the I!?7 nucleus. as chlorobenzenes or chloroethylenes, not have this cylindrical symmetry 7b). In such cases

U, 4

U,

(7.33)

and there is a finite 7 given by n=_ Qex ro”~ Ww

=_

_3(Uz

a

z

— Uy) U.+ U, —

(7.34)

Qo

For small n Eq. (7.34) may be put in a more convenient form. Equations (1.85) and (1.386) for the quadrupole resonance frequencies (I = $, $) involve powers in yn? which may be neglected if » is small. Hence using (1.35), (7.30), and (7.84), we have for spin 3,

3(U,y — Uy (42) 7

=

den

(7.35)

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141

For spin $

3(Uy — Ue) (2) n=

oa

(7.36)

In these equations, «, refers to the frequency of them = +3—2m = +3 transition. Therefore n can serve as a measure of the population difference in the p, and p, states. Equations (7.35) and (7.36) have been applied by Bersohn!”’ and others!”° to analyze the double bond character of carbon-halogen bonds in organic compounds. This will be discussed in Section 7b. (3) Rules for assigning hybrid, ionic, and double bond characters of chemical bonds. In making use of Eq. (7.27), (7.28), or (7.30) to

analyze quadrupole coupling constants in free molecules, some methods are necessary for predicting the hybrid, ionic, and double bond characters of the chemical bonds formed by an atom with other atoms. Semiempirical rules for this purpose are available in the literature on valence theory. They are summarized and discussed in a recent article by Dailey and Townes.!79 We shall now consider these rules in some detail. Hybridization: When the atom of interest forms bonds with a

number of other atoms, information as to the s-hybrid characters of its atomic orbitals taking part in these bonds may be obtained from a knowledge of the angles between the various bonds. These bond angles may be determined from rotational microwave spectra or electron diffraction data for the free molecule and x-ray or electron diffraction data on the solid. Let two bonds be described by the following two s-p hybrids on the central atom [see Eq. (7.14)]: (1

~

sv)

Wo,

ql —- 82°) Wo

+

Sis

+

sos.

(7.37)

The wave functions yp, and y,, refer to p states of the atom, symmetric about the two bond directions, where y, refers to the s state. Since the two bonds must be independent of each other, the two states shown in (7.37) must be orthogonal to each other. If @ is the angle between the two bonds, this condition requires that

siso + [C1 — si2)(1 — 82”)]! cos 6 = 0. 177 R. Bersohn, J. Chem. Phys. 22, 2078 (1954). 178 J. H. Goldstein and J. K. Bragg, Phys. Rev. 75, 1453 (1949). 179 See Dailey and Townes!’* and references therein.

(7.38)

142

III.

APPLICATIONS

If the two bonds are assumed to be similar so that s1 = s, = s, then Eq. (7.36) yields

s? = cos 6/(cos 6 — 1)

(7.39)

which gives the amount of s character on each bond. For example, in methane (CH,) the four CH bonds are tetrahedrally disposed and the angles between successive bonds are all equal to 109°28’. From Eq. (7.39) this leads to s-hybrid character of 0.25 for the carbon atomic orbital in each CH bond. In ethylene (C2H,) the angle between two

CH bonds is 120°, leading to an s character of about 0.33. In NH; and H,O, the bond angles!®°18! are, respectively, 106°47’ and 104.5°, corresponding to s characters of 0.18 and 0.20, respectively. There are two possible sources of error in the values of s? deduced from Eq. (7.39). First, there may be some d hybridization present, so that the wave functions (7.37) do not give accurate descriptions of the atomic orbitals on the central atom. This would be more important for heavier atoms. For example, Bird and Townes!®? find some disagreement

between the p characters of sulfur orbitals in H.S, as obtained from Eq. (7.39) and from the analysis of the S** quadrupole coupling data. They suggest that d hybridization on the sulfur atom may be a possible cause for the discrepancy. The second source!* of error is the repulsion between the atoms bonded to the atom of interest. This effect would depend on the sizes of the attached atoms as well as on the net charges on them in the molecule. Gordy'* suggests that the observed difference of 8° in bond angles between AsH; and AsF; could arise from the larger repulsion between the more electronegative fluorine atoms. In spite of these small errors, the relation (7.37) gives some measure of the s characters of the bonds. No such measure is available when the atom in question is chemically bound to only one other atom. Such a situation occurs in most halogen compounds, for example, in the IC] molecule already discussed. In these cases, Townes and Dailey??? have proposed the following rule: if the halogen atom is bonded to an atom having an electronegativity smaller than its own by 0.25 unit, then there will be 15% s hybridization at the halogen atom. Otherwise the s hybridization is zero. The d hybridization is always smaller than 5%. 180 181 182 183 184

See p. 209 in ref. 37. See p. 238 in ref. 16. G, R. Bird and C. H. Townes, Phys. Rev. 94, 1203 (1954). See p. 210 in ref. 37. W. Gordy, W. V. Smith, and R. F. Trambarulo, ‘‘ Microwave p. 315. Wiley, New York, 1953.

Spectroscopy,”

7.

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Electronegativity'!®* is a measure of the electron-seeking power of an atom. For example, a halogen atom lacks a p electron to complete

its shell. An alkali atom, on the other hand, has a tendency to part with the single electron outside of its closed shell. Therefore, the halogen atom is regarded as being electronegative, while the alkali atom is electropositive. A measure of the electronegativity x is provided by

the relation

x = 0.18(1 + E) where

J is. the ionization

energy

of the atom

and

£ is its electron

affinity, both measured in electron volts. The electron affinity corresponds to the ionization energy of the negative ion. Tables of electronegativities for various atoms are available in standard texts on valence theory.'86187 The most recent table is due to Huggins!88 who

has compiled his table from spectroscopic data.

the most

recent

thermodynamic

and

Townes and Dailey arrived at their rule for the hybridizations at halogen atoms from the following qualitative considerations. First, neighboring elements like oxygen and sulfur in Group VI of the periodic table, or nitrogen and arsenic in Group V, are known to have about 20% s hybridization in many of their bonds with other atoms. So the s hybridization for the halogen atom would be somewhat smaller and not necessarily zero. Second, the s hybridization involves a net promotion of electrons to the p state, as discussed in Section 7a(1). This promotion energy would be smaller for a negative halogen ion than for a neutral halogen atom or positive ion. This consideration leads to the restriction that the attached atom be less electronegative by 0.25 unit than the halogen atom. And finally, the d hybridization is assumed to be small in most cases because, as seen from Eq. (7.27), the s and d characters oppose each other in their contributions to the field gradient. Therefore, if some d character occurred, a larger amount of

s character would be required to explain the observed quadrupole coupling constants. This would indicate large promotions from s to p state and from p to d state, which would be very unfavorable from an energetic standpoint. While the assignment of exact numbers 0.25 for the electronegativity and 0.15 for the hybridization cannot be regarded as rigorous, the Townes and Dailey rule is at least qualita185 186 187 188

See See See M.

p. p. p. L.

132 in ref. 37. 64 in ref. 126. 134 in ref. 37. Huggins, J. Am. Chem. Soc. 76, 4123 (1953).

144

IIL.

appLicaTIONs

tively justified from these considerations. Also it leads to a fairly consistent interpretation of halogen quadrupole data as will be pointed

out later. Tonic character: The validity of earlier relations!**:!°° between the ionic character J of a bond between atoms A and B and their difference in electronegativity z, — vp has been discounted by Dailey and Townes.!”° The dipole moment of a molecule is made up of contribu-

tions from various sources which are comparable in magnitude. The various contributions are the positive charges on the nuclei, the distribution in electron density of the bonding electrons, and the deformation of the inner shells of the different atoms by the bonding electrons and distant nuclear charges. Accurate estimates of these various

contributions are not available, and therefore reliable values of the ionic characters cannot be deduced from dipole moment data. In contrast to this, except for strongly ionic molecules, the contribution to the field gradient at a nucleus is determined mainly by the p-electron density in its atom. Dailey and Townes have used Eqs. (7.27) and (7.28) to analyze halogen quadrupole coupling data, obtained from microwave spectroscopy and molecular beam experiments, for a num-

ber of gaseous diatomic molecules. Using the criterion of bond length (refer to Section 8c) to be discussed later in this section, the double bond character in most of the molecules was found to be small. The d hybridization is neglected and s hybridization is obtained from the Townes and Dailey rule mentioned earlier. The solid curve in Fig. 36 shows the best fit that can be made with the observed data, and gives the empirical relation between electronegativity difference and ionic character. From the discussion in Section 7a(2) on antishielding of the inner shell electrons and its influence on the quadrupole coupling constant of ionic molecules, one would expect that the solid curve in Fig. 36 is more reliable in the region of low rather than high electronegativity differences. The dotted curve is due to Gordy, who has proposed that hybridization effects are small or nearly zero at the halogen atoms. His curve for the ionic character is represented approximately by |t, — xp|

[=

5

for |va — xs| < 1.5. Above this value of the electronegativity difference it nearly coincides with Townes and Dailey’s curve. It is not possible 189 See Eq. (12.1) in ref. 126. 190 N, B. Hannay and C. P. Smyth, J. Am. Chem. Soc. 68, 171

(1946).

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145

from the data shown in Fig. 36 to draw conclusions on whether Gordy’s or Townes and Dailey’s rule is more correct. Essentially both rules are based on plausible choices for the values of the three quantities

s, d, and I in Eqs. (7.27) and (7.28). Since there is only one quantity known

for

most

of the

diatomic

molecules

100

considered,

Kr KBr °

90+

namely

the

KCI

RbCloCsCl

LiBr

Lile

°NaBr oNal

°TICl

80+

lonic character

70

y

60;

/

50F

,

40+

/

30+

1

20F

10

Tcl 1

0

04

O08

1

i:

1

1

1.2 16 #20 24 Electronegativity difference

1

2.8

Fig. 36. Curves showing the variations in ionic character with electronegativity difference. The continuous curve is due to Dailey and Townes and the dotted curve is due to Gordy. The data points indicated refer to ionic characters derived from microwave and molecular beam nuclear quadrupole interaction data using the Townes and Dailey hybridization rule for halogen atoms. The Dailey and Townes curve is from J. Chem. Phys. 28, 118 (1955) and the Gordy curve is from his recent paper

[Discussions Faraday

Soc.

19, 14 (1955)].

e’qQ at the halogen nucleus, one could make

alternative choices of I

and (s — d) and still match the data fairly well with Eqs. (7.27) and (7.28). In the case of the diatomic molecules [Cl and BrCl, the quadrupole coupling constants for the nuclei at both ends of the molecules are available. One can therefore deduce the ionic character J from the

146

III.

APpPLicaTIONS

data for either nuclei. In the case of ICI, the Townes and Dailey rule leads to 15%

s hybridization at the chlorine atom and zero s hybridi-

zation at the iodine atom. Using Eqs. (7.25) and (7.26), one then gets I = 0.115 from the Cl** data and J = 0.229 from the I!?? data, as shown in Fig. 36. On the other hand, following Gordy, if one makes the assumption that there is zero hybridization at both atoms, the

value of J from the Cl** data is changed to 0.248. In BrCl, the electronegativity

difference

between

the

bromine

and

chlorine

atoms

is

only 0.2. Both Townes and Dailey and Gordy’s rules therefore lead to zero s hybridization at either atom. Using Eqs. (7.25) and (7.26), one then finds that the values of J come out to be 0.056 and 0.110 from the Cl]** and Br’? data respectively, as indicated in Fig. 36. The chlorine quadrupole coupling constant in Cl, molecule is interesting in this connection. Livingston’s!*! measurements show that, in the solid state, the observed coupling constant (108.95 me) is practically equal

to e’qQ for an outer p electron in chlorine atom (see Table IX). Although the data are from the solid, there are reasons (see Section 8c) to believe that the quadrupole coupling constant in the free chlorine molecule is not very much different. In the case of the chlorine mole-

cule, both Townes and Dailey’s and Gordy’s rules predict zero hybridization and zero ionic character, which agrees with Livingston’s result. No such comparison is possible in the case of the observed quadrupole coupling constants for solid bromine and iodine because the data in these cases are not representative of the free molecule. The relative accuracies of the two sets of rules for ionic and hybrid characters can only be settled when accurate molecular wave functions are available for these diatomic molecules. In our subsequent discus-

sions we shall use Townes and Dailey’s rules. One other point will be mentioned here. In more complicated molecules, where the two atoms

A and B are bonded to each other as well as to other atoms, the ionic character of the bond between A and B would be close to the value given by the solid curve in Fig. 36 for the electronegativity difference between the atoms A and B. But small departures would be expected

when additional atoms are attached to A and B, and this would affect quadrupole coupling data markedly. This point is illustrated very well by Livingston’s!®? data on chloromethanes.

Double bond character: For the case of planar molecules, double bond character may be ascribed entirely to the conjugation of p, or p, electrons,

and

would

therefore

lead

to an

191 R, Livingston, J. Chem. Phys. 19, 803 (1951). 192 R, Livingston, J. Chem. Phys. 19, 1434 (1951).

asymmetry

parameter

7

7. INTERPRETATION

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147

given by (7.34). But in those molecules where there is cylindrical symmetry about the line joining an atom to its neighbor, no distinction is possible between the p, and p, electrons. Hence, no measure of the double bond character is available from y, which has a value of zero. In such cases a semiempirical rule proposed by Pauling! is used. According to this rule, if Ri and Rz represent the bond distances for single and double bonds between the two atoms, and R is the observed distance, then

R=

(ai; + 8a2R2)/(a1 + 32)

(7.40)

where a; and a: are the fractional importance of the single and double bond structures. Thus, a2 corresponds to II and a; = 1 — IJ, so that

ll = (R, — R)/(Ri + 2Ro — 3R2).

(7.41)

The distances ,; and R,; may be obtained from the singly and doubly bonded covalent radii of atoms that have been tabulated by Pauling. !%* Thus, if Ris and Ry, refer to the single bond covalent radii of atoms A and B, then Ry

=

Ris

+

Rip

_

0.09|r4

_

Xgl.

(7.42)

The last term is a correction term due to Stevenson and Schomaker,!** with x, — x, representing the electronegativity difference between the two atoms. A similar relation holds for the double bond radii of the atoms.

(4) Illustrations of Townes and Dailey’s theory. Schawlow!! has analyzed the halogen quadrupole coupling constants in a number of fourth group tetrahalides that have been studied by him and others. Although some of the quadrupole spectra show splittings due to intermolecular effects, these splittings are small compared to the frequency differences observed as we pass from one compound to another. In an analysis of the change in the properties of a certain type of bond in passing from one molecule to another in a series, one is justified therefore in using the averages of the frequencies of the different lines for each compound. This average is assumed to be representative of the quadrupole coupling constant in the free molecule. Schawlow has used Eq. (7.27), together with hybrid and double bond characters obtained by the rules outlined above,'** and the observed halogen quadrupole coupling constants, to obtain the ionic characters of the AX bonds in 193 See p. 164 in ref. 126.

194.V, Schomaker and D. P. Stevenson, J. Am. Chem. Soc. 68, 37 (1941). 195 Schawlow makes use of Townes and Dailey’s rather than Gordy’s rule for the hybridization at the halogen atom.

148

II].

appLicaTIONs

the different fourth group tetrahalides. Here A refers to a fourth group element C, Si, Ge or Sn, and X refers to the halogens. For the bond distances R in Eq. (7.41) he used the values given in tables by Allen and Sutton!** based on the electron diffraction measurements on the compounds in the gaseous state. His results are tabulated in Table XI, together with the corresponding electronegativity differences from Huggins’ tables.!8* The results show clearly the increasing TasLe

XI.

Properties

or AX

Bonps

AND

= Cu,

X

WHERE Br,

Per-

R(A)

A

= C, S1, Gu,

Sn;

I Percentage

centage

hybrid

Per-

Electro-

double

character

centage

negativity

bond

athalogen

ionic

difference

character

atom

character

0 0 45 +412 60415 41 +4 16

15 15 15 15 15

19 8 33 44 1647 25 +8

R,(A)

R2(A)

C—Cl C—Br Si—Cl Si—Br Ge—Cl

1.76 1.91 2.16 2.31 2.21

1.56 1.70 1.96 2.11 2.01

1.765 1.91 2.02 2.15 2.08

Ge—Br

2.36

2.16

2.29

+0.02

16

+6

15

27

+3

1.05

Ge—I

2.55

2.35

2.50

+0.03

I1 +7

15

19 +3

0.75

Sn—Cl Sn—Br Sn—I

2.39 2.54 2.73

2.19 2.34 2.53

2.30 2.44 2.64

+4 0.03 +0.02 +0.04

22 + 11 25 +8 2141

15 15 15

38 28 17

1.25 1.05 0.75

+ 0.015 +0.02 +0.02 +0.02 +0.03

+5 +44 +5

(X

— A)

0.55 0.35 1.25 1.05 1.25

ionic character as the electronegativity difference increases. The numerical agreement with Dailey and Townes’ curve in Fig. 36, however, is not very good. Only the data points obtained from Gelj, SnIy, CCl, and CBr, lie close to the Dailey and Townes’ curve. The rest lie somewhat below this curve. The data depart from Gordy’s

curve even more. It is to be noted that in the free tetrahalide molecule, there is threefold symmetry about each halogen bond. Hence, although there is appreciable double bond character in most cases, as seen from Table XI, there would be no difference between Ux and Uy in Eq. (7.384), and therefore no 7. But in the solid, intermolecular forces may destroy the threefold symmetry, and a finite 7 would result (see Sections 8b and 8c). Another interesting set of halogen compounds has been studied by Livingston.!*? These are the chloromethanes and some bromo- and

7.

INTERPRETATION

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DATA

149

fluorochloromethanes. The Cl** resonances of the solids were studied, but again the mean resonance frequency for each compound is regarded as representative of the free molecules. Livingston made measurements at three temperatures, namely, 77°K, 20°K, and 4°K. The temperature variations in frequency were found to be negligible compared to the frequency differences between different compounds. The asymmetry parameters were not studied, but they were assumed

80;

CFCI30

_

CF3Cl

=E

8 CFoBrCl

OM

CFoCl rahe

Cc

x

w

6

75-

Vo

oO

&

a 3B

oO oO wo

%

CHF>CI 70+ CH3CI

65

i

l

1

4

1

2

3

4

Number of chlorine substituents

Fic.

37.

Livingston,

Quadrupole J. Chem. Phys.

coupling 19, 803

constants

in substituted

chloromethanes.

[R.

(1951).]

to be small enough so that n? in the relation (1.35) could be neglected. The quadrupole coupling constants e?gQ (in mc) could then be obtained by doubling the observed quadrupole resonance frequencies. In Fig. 37 the observed e?¢Q at 20°K are plotted against the number of chlorine substituents in the different molecules. The results on the chlorinated methanes indicate that the value of e’qQ for the Cl** nucleus increases with increase in the number of chlorine substituents. By Townes and Dailey’s rule, the hybridization at the chlorine atoms will remain near 15% in all the four compounds.

150

III.

APPLICATIONS

The e?¢Q’s in Fig. 37 therefore indicate that the ionic character decreases with increase in the number of chlorine substituents. This can be explained by noting that the chlorine atom is more electronegative

than carbon and tends to extract electrons from it, leading to a certain ionic character I for the C—Cl bond. When it has other chlorine atoms to compete with, it cannot acquire the same ionic character as it could if only hydrogen atoms were attached to the carbon atom. A comparison between the pairs of compounds CH;Cl and CHF,Cl, CH2Cl, and CHFCl:, and CHCl; and CFCl; (see Fig. 37) is also interesting. In the latter two of these pairs, a fluorine atom is replaced by a hydrogen atom, and in the first pair, two fluorines are replaced by two hydrogens. In each case, the Cl*® quadrupole coupling constant is larger in the fluorine compound, which indicates that the ionic character of the C—Cl bonds is lessened by fluorine substitution. This is as expected, since the fluorine atom is more electronegative than hydrogen, and so provides greater competition for the chlorine atoms in extracting electrons out of carbon. The e?gQ for Cl®> in CF,BrCl is smaller than in CFCl;. This results because bromine is less electronegative than either fluorine or chlorine, so that the C—Cl bond in CF,BrCl has a larger ionic character than in CFCs.

However, from considerations of ionic character, the quadrupole coupling

constant

for Cl*® in the set of compounds

CF;Cl,

CF:2Cl.,

CFCl;, and CCly, should increase with the number of fluorine substituents. Actually it decreases as shown

in Fig. 37. Livingston has pro-

posed that this reversal of order might be due to an increase in double bond character for the CCl bond with substitutents. This would be expected since such double bonded structures, Fig. 38, would require positive formal

cl FC==cl*

cl

The

increasing number of fluorine from physical considerations, represented schematically in charges on the chlorine atom.

double

bond

character

would

be

4 F-c==cr \

expected to be a maximum when the number of fluorine and chlorine substituents are equal, namely in CF.Cle, be-

Fic. 38. Double-bonded structures in fluorochloro-

cause in this situation, maximum charge transfer can occur between the electro-

4

Cl methanes.

F

.

.

.

negative fluorine atoms and the relatively electropositive chlorine atoms. It would be smaller in both CFCl; and CF;Cl as compared to CF2Cl». This tendency is manifested in Fig. 37, where the Cl** coupling constant in CF;Cl is seen to be slightly larger than in CF,Cl.. The competition between these ionic and double bond

influences of the fluorine atoms on the C—Cl bonds is also noticeable

7.

INTERPRETATION

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COUPLING

DATA

151

in the cases of CF;Cl and CF.BrCl where, if only the ionic influence were effective, the Cl*® coupling constants would be expected to differ more than shown in Fig. 37.

All the compounds shown in Fig. 37, except CClu, lack threefold symmetry around the C—Cl bond in the free molecule. Hence finite values of 7 are to be expected for the field gradient tensors at the Cl*> nuclei. Superimposed upon this will be any asymmetry due to the intermolecular effects in the solid. As a last set of examples, we consider a few cases where the atom of interest is bonded to a number of other atoms in order to illustrate the rule (7.36) for obtaining the hybrid characters in A

such cases. The examples to be considered belong to a general class of molecules with the structure shown in Fig. 39. In Fig. 39, B contains the nucleus of interest and three bonds BC are equivalent and different from bond BA. In particular cases there may be no atom at A, and the hybrid state directed along BA may either be empty or contain a “lone pair’’°° of electrons. It is assumed that no d hybridization is present. Evidently Fig. 39 includes as

ae

particular cases the tetrahedral sp* and planar sp?

\here the bonds

hybrid structures characteristic of carbon in organic compounds. Let the angle between the direction BA

BC are equivalent and equally

and any of directions BC be @, and let $2and.$,?repre-

inclined

sent the s characters of the bonds BA and BC. The

"4

B J c Fia. n TG. 39. Sche-

to

the

BA.

z axis is taken along BA and the x axis in a plane containing the bond BA and any one of the bonds BC. Using Eq. (7.38), the atomic wave functions for the B atom in the different hybrids may be written By,

+

Biv. +

(1

~ iB?) yp,

(1 —

$17)F(Yp, cos 0 — Hp, sin 4)

V3

Bw. + (1 — $1?)3 (v,. cos 6 + Yee sin 6 — -5= ¥p, sin s) Sw. + (1 — $,2)3 (v., cos 6 + Pa sin

6+ M3,

(7.43)

sin a)

with $, = [(sin? 6 — 2 cos? 6)/(2 — 2 cos? 6 + sin? 6)]# and

_

1 — $;?

B= la — $1") cos? 6+ =| 196 See p. 106 in ref. 37.

;

cos 6.

(7.44)

152

III.

APPLICATIONS

Four equivalent tetrahedral bonds: In this case, cos 6 = — 4 and one gets from Eqs. (7.43) and (7.44) U,=

U,=

U.=

(1-1)

(7.45)

where J is the ionic character which is chosen positive for electron loss from B. Hence from (7.30) it follows that (mois = O in this case. Three equivalent

planar bonds

BC:

(7.46) If there is a lone pair in the

hybrid state BA, then Ux = Uy

=(1-J)

and

(7.47)

Uz = 2. If there is any double bond character in the BC bonds there will be electron loss from the hybrid BA, and Uz would have a value 2 — II in Eq. (7.47), so that 7 dmoinQ

=

(1

+

I

— The?qarnQ.

(7.48)

The field gradient is axially symmetric about BA, which is along the

threefold axis of the molecule. This situation would correspond to the quadrupole

interaction

of N'4 in NO;~

ion, which

has not yet been

studied. In the B(CHs3)3 molecule the carbon atoms and boron form three equivalent planar bonds BC, and there is no lone pair on the boron atom. At the boron atom in the molecule, therefore, Uy

=Uy=(1-J)

and

(7.49) Uz = 0.

Hence the field gradient is again along the threefold axis of the molecule, and the quadrupole coupling constant for B!! in the molecule would be given by (€7qmo1)

pn

=

-(

~

T) (e?qatQ)

au.

(7.50)

Now ¢?¢Q..Q/h for B!! is equal to 5.39 me (see Table IX), while the electronegativity difference of 0.6 between boron and carbon gives I = 0.09. Therefore (e7qmiQ/h) su =

4.92 me.

7.

INTERPRETATION

OF

NUCLEAR

QUADRUPOLE

COUPLING

DATA

153

Dehmelt!*? obtained coupling constants of 4.969 me and 4.873 mc for the two phases below and above —190°C, respectively, of this compound. Pyramidal configuration of three bonds: With an angle @ between the hybrids BA and BC, and with a lone pair in the hybrid state BA in Fig. 39, Ux, Uy, and Uz are given by Uz = 201 — $7) + A — Y[38d — $,2) cos? 6] Uy = (1 — Jd — $,”) sin’ 6 Uy = #11 — YU — $1’) sin? ¢.

(7.51)

The ionic character J of the BC bonds is taken as positive if the atom B appears as positive in the molecule. Then from (7.30) we get for the nucleus of atom B

e*qmoi = [2(1 — $*) + 3(1 — 1) — $1’) cos? 0 — $1 — D(A — $12) sin? d]e’quQ.

(7.52)

This is an extreme case in which the BC bonds have no double bond character. It applies to the ammonia molecule NHs;, since the hydrogen atom has a single electron and cannot produce any double bond character in the NH bond. For NH3, microwave data show an e?qQ of — 4.084 me for N", and the angle @ is nearly 112°. The electronegativity difference between N and H atoms is 0.9; the ionic character J of the NH bond is found to be 0.33 from Fig. 36. Using (7.51) and (7.52), we then get

Cm

= e*quQ X 0.45.

The observed value of e?@miQ then leads to a value of e?quQ = —1.838 me for N!4. Using the value of gts in Table X, the quadrupole moment of Q(N!4) is found to be about +0.015 & 10-*4 cm?. In view of the approximations involved in obtaining e?q..Q from the data for the

molecule, and also because of uncertainties!®? involved in deducing qa from optical data, this value of Q(N!) is not as reliable as the values for other nuclei in Table IX, for which atomic hyperfine structure data are available. Consider the extreme case where there are no electrons in the hybrid state BA. In place of Eq. (7.52) we now have for the B nucleus in Fig. 39

edna = 811 — 1)(1— $1?) (cos 197 H. G. Dehmelt, Z. Physik 188, 528 (1952).

6 —

sin? 6

») e*qu.Q.

(7.53)

154

III.

aprpLicaTIONs

In NaClO; the observed bond distances!®* for the ClO3~ pyramids give

cos? @ = 0.0951 sin? 6 = 0.9049

(1 — $?) = 0.7938 (1 — $1?) = 0.7312.

In the extreme case of a lone pair on the chlorine atom this would give,

from (7.52), C7 maQ

~ 0.8(1 + De?quQ.

Using Livingston and Zeldes’!*® Cl*> resonance frequency of 30.632 mc at —196°C and the value of b = $(e?quQ)ous = 54.873 me from Table IX, we obtain: 1+

J = 0.65, or J =

—0.35

(7.54)

which indicates that in this case the chlorine atom would carry some negative charge. In the other extreme, with no lone pair electrons,

Eq. (7.53) gives I = 0.35

(7.544)

ie., the chlorine atom would carry some positive charge. From an analysis of the quadrupole coupling constant of the Na?* nuclei in NaClO; by the method outlined by Reif and Cohen,! Bersohn!*4 obtains a charge of +0.114e on the chlorine atom and —0.295e on the oxygen atoms. This shows that the actual situation is intermediate

between the two extremes considered above. Our treatment here for the Cl®> coupling constant neglected the direct contributions from the positive and negative ions in the crystal, which may be appreciable.

Other structures, involving different arrangements of the BC and BA bonds, can be analyzed by the same procedure as that followed here, making use of (7.30) and (7.38) and the Dailey and Townes curve of Fig. 36. b. Relation between Asymmetry Parameter and Double Bond Characters of Halogen Bonds in Planar Molecules For planar molecules a relation may be established between the asymmetry parameter for halogen nuclei and the double bond character of the bond in which the halogen atom takes part. Such a relation, which is based on Eqs. (7.35) and (7.36), was first proposed by Goldstein and Bragg!”8 from an analysis of microwave data on nuclear quadrupole interaction in free molecules. It was analyzed in some detail 198 See p. 245 in ref. 126.

199 R. Livingston and H. Zeldes, private communication.

7.

INTERPRETATION

OF

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QUADRUPOLE

COUPLING

DATA

155

by Bersohn®” for carbon-chlorine bonds in a number of aromatic compounds. Recently Goldstein?” also applied this relation to analyze the electron distribution in vinyl] chloride. Duchesne and Monfils,*°? and Bray, Barnes, and Bersohn!”! have studied the chlorine quadrupole coupling constants of various chlorobenzenes. The asymmetry parameters for the chlorine nucleus in these compounds have not been measured. However Bray, Barnes, and Bersohn!! have shown that the quadrupole coupling constants can be given a consistent interpretation using Townes and Dailey’s!?? theory in conjunction with Bersohn’s analysis of the double bond character for the C—C] bond. These results on chlorobenzenes will be discussed briefly in the second part of the present section. First, we shall briefly Cl Cl

Ne C==C. AZ

Na

" H

3

H

H

Fic. 40. Structural formulae for vinyl chloride and chlorobenzene molecules. In the valence bond picture, these represent one each of the various possible resonance structures in either case.

review

Bersohn’s

analysis

of the double

bond

character

of C—Cl

bonds in planar carbon compounds in order to illustrate the general procedure for all planar molecules. (1) Bersohn’s analysis of double bond character in C—Cl bonds. Consider either of the planar molecules vinyl] chloride or chlorobenzene (Fig. 40). According to the molecular orbital picture,?°* each carbon atom has three sp? hybrids inclined to each other at 120°.2°4 Each of these three hybrids takes part in a o bond with a neighboring atom. In ethylene, two of the hybrids at each carbon atom are bonded to hydrogens and one to the neighboring carbon atom. In addition, there is one 2pr electron (perpendicular to the plane of the molecule) on each carbon atom. These two electrons form a a bond between the carbon atoms leading to a double bond. In benzene, in addition to the o bonds 200 R. Bersohn, J. Chem. Phys. 22, 2078 (1954). 201 J. H. Goldstein,

J. Chem. Phys.

24, 106

(1956).

202 J, Duchesne and A. Monfils, J. Chem. Phys. 22, 562 (1954). 203 See Chapters 8 and 9 in ref. 37. 204 When the carbon atom is not bonded to three atoms of the same kind, the three hybrids may be somewhat different from one another, and may also be inclined to each other at somewhat unequal angles.

156

III.

APPLICATIONS

formed by each carbon atom with a hydrogen atom and neighboring carbon atoms, there are six z electrons. The 7 electrons are not confined to the carbon atoms but instead are distributed over the whole ring. According to the valence bond pictures these six electrons resonate between the two Kekule pairing schemes?” shown in Fig. 41. When the chlorine atom replaces CO O} a hydrogen, a chlorine electron in a 3s-3p, hybrid [with 15% s character according to the Townes _Fia. 41. Kekule and Dailey rule of Section 7a(3)], forms a o bond pairing schemes for with the neighboring carbon atom. Of the two pairs

zene.

The

CH

Of 3p, and 3p,? electrons on the chlorine atom, the

bonds are omitted

38pz

for brevity.

benzene ring or the vinyl chloride molecule have the proper symmetry to conjugate with the 7

electrons

perpendicular

to

the

plane

of

the

orbitals on the carbon atoms. Hence there will be a net electron loss from the 3p; states of the chlorine atom with the result that some double bond character will appear in the C—Cl bond. In the valence bond theory the chlorobenzene molecule is regarded as resonating between the single and double bonded chlorine structures shown in Fig. 42(a). The case of vinyl chloride is shown in Fig. 42(b).

i

TT

‘

H

4”

cit

oH

Nu

HZ

cl

Nu

(b)

(a)

Fig. 42. (a) Resonance structures in chlorobenzene. in vinyl chloride.

(b) Resonance structures

The double bond character II of the C—Cl bond in each case is defined as the fractional importance of the resonance structure in which the

C—Cl

bond is double bonded.

In the molecular orbital theory, it is

defined as the z-electron loss from the chlorine atom, according to Eq. (7.24). The two definitions are closely analogous, and calculations of the double bond character by both methods would give identical results when carried to the ultimate limit of accuracy.?°6 In his calculation of II, Bersohn used a simple molecular orbital approach based on 206 In addition to these Kekule structures there may be some Dewar structures in which two nonadjacent carbon atoms may be bonded. These are of much smaller importance than the Kekule structures.

206 See p. 140 in ref. 37.

(See Coulson, p. 229 in ref. 37.)

7.

INTERPRETATION

OF

NUCLEAR

QUADRUPOLE

COUPLING

DATA

157

the general procedure of Coulson and Longuet-Higgins®” for conjugated m-electron systems. Since the 3p, electrons exist as a lone pair on the chlorine atom, it follows from Eq. (7.35) that n = SIle*quiQ/hw,.

(7.55)

In the Hiickel?°’ approximation for the benzene molecule and its derivative, the electrons are regarded as moving in the averaged field of the nuclear charges and the o electrons, and so may be dealt with separately from the o-electron system. Let a general molecular orbital wave function for the z electrons be denoted by y,. This can be written as a linear combination of a electron (pz) states ¢, on all the carbon and chlorine atoms:

bi = >) Crd

(7.56)

The number of electrons on the rth atom is then given by

hy = y Cri];

(7.57)

where the summation is carried over all the possible molecular orbitals (equal in number to the number of atoms taking part in the conjugation) with n; = 0 or 2, depending upon whether the 7th orbital is occupied or unoccupied. From Eq. (7.24) it then follows that Il=2-

Ny

=

2-

>

[Coul?nj.

(7.58)

Since ) |C,;|? = 1, Eq. (7.58) can be rewritten as

n=) |Cavl? where obtain tonian orbital,

(7.59)

7’ now refers only to the unoccupied molecular orbitals. To the Co; and Cay, a secular equation involving the net Hamilfor the whole z-electron system has to be solved. For the ith one then gets, by the use of the variation principle,? the set

207 C. Coulson 208, Hiickel, 209 |, Pauling New York,

and H. C. Longuet-Higgins, Proc. Roy. Soc. A192, 16 (1947). Z. Physik 76, 628 (1932). and KE. B. Wilson, “Quantum Mechanics,” p. 187. McGraw-Hill, 1935.

158

III.

apPLicaTIONS

of equations

Y Crily — Hib] = 0. This leads to the determinantal equation Crs

—

ESre||

= 0

(7.60)

for the energies E;, where 5,, = . There are four classes of terms 3C,, to consider, viz., Ieci, Kee, Io,c,, and Koa. The first two are termed Coulomb integrals, and the latter two are resonance integrals. The two Coulomb integrals can be related by Coulson’s?”

prescription that Roc

where

xq and

— Keo

=

(ar

—

Ic)B

2¢ are the electronegativities

(7.61)

of chlorine

and

carbon

atoms respectively, and 8 = 3o,c,. It may be argued?!! for Hoc that the magnitude of the resonance integral will depend to a certain extent upon the overlaps (Sca =

)

of the x electrons on the carbon

and chlorine atoms. From tables!®> of overlap integrals, Bersohn obtained Sco = 0.272, and Sco = 0.128 by interpolation, which suggested that ica

< 8/2. Bersohn?™” assumed 3Ccco. = 48, where d was an

unknown quantity smaller than 4, and was to be evaluated from the asymmetry parameter data for various molecules. Following Coulson and Longuet-Higgins’?"’ procedure, it can be shown from the secular

equation (7.60) that?!? = T=

A*B? 3.142

° l

(ac.

Acre ™ dy — ty)? Aca™

(7.62)

where r refers to the carbon atom that is the immediate neighbor of the chlorine atom in the molecule. The notation Ajg,..., pg...“ refers to the determinant in Eq. (7.60) with the energy / replaced by zy and

the pth, qth, rth, etc., rows and columns dropped.

For small values

of \, the conjugation of the w electrons of the chlorine atom with the w-electron system of the rest of the molecule may be regarded as a perturbation. Using perturbation theory Bersohn?” obtains the follow210 See p. 242 in ref. 37. 211 H, C. Longuet-Higgins and M. de V. Roberts, Proc. Roy. Soc. A230, 110 (1955). 212 To prevent confusion with the double bond character z, the numerical constant w is written as 3.142.

7.

INTERPRETATION

OF

NUCLEAR

QUADRUPOLE

COUPLING

DATA

159

ing equation: II = 2?

IC 262 ) 7" mf (Hercr — €')?

(7.63)

t

where the summation is over the unoccupied states of the r-electron system without the chlorine atom. The term C,,, represents the amplitude of the atomic wave function

¢, in the 7th molecular

orbital for

this system, and e, represents the energy of the 7th orbital. The rth carbon atom is again the one that is bonded to the chlorine atom. One

more result was deduced by Bersohn, which is useful in interpreting the results of Bray e¢ al. on chlorobenzenes. This refers to the change

(All,) in double bond character of the bond between a chlorine atom and the rth carbon atom in the molecule when another chlorine atom is attached to the ¢th carbon atom: \Cv,|?|Cve?

AN, = —6

~

(Kaa

—

v4.

(7.64)

«)4

v

The summation over 7’ refers to the unoccupied orbitals of the carbon system. The change in II is seen to be fourth order in A, and when)

< 3,

it is very small compared to the original double bond character II. In Table XII the observed asymmetry parameters 7 in a number of organic chlorine compounds are shown, together with Bersohn’s TaBLeE

XII. ASYMMETRY PARAMETER IN A NUMBER OF SUBSTITUTED ORGANIC CHLORINE ComMPpouUNDS COMPARED WITH BERSOHN’S CALCULATIONS

Compound

Temperature*

Resonance frequency

0

TI (Calculated)

CH.=CHCl+ CFCI=CH, p-C.H.Cl.¢ p-CsH,CICH.Cl¢ p-CoH.CIN He? COCI,*4

20° gas 77° 77° 77° 77°

33.61 36.65 34.78 34.57 34.15 35.58-36.23

0.07 + 0.03 0.085 + 0.015 0.08 + 0.02 0.07 +0.02 0.06 + 0.03 0.25 + 0.03

0.444r2 0.428d2 0.350A2 0.350X2 0.2642 1.158n?

*The temperatures quoted are absolute temperatures (°K). « R. Livingston, J. Phys. Chem. 67, 496 (1952); J. Chem. Phys. 19, 1613 (1951). ’ J. K. Bragg, T. C. Madison, and A. H. Sharbaugh, Phys. Rev. 77, 148 (1950); the asymmetry in CFCICH; is obtained from microwave spectra. ¢H. C. Meal, J. Am. Chem. Soc. 74, 6121 (1952). 4G. W. Robinson, J. Chem. Phys. 21, 1741 (1953); microwave work,

160

III.

APPLICATIONS

calculated values of the double bond character of the C—Cl bonds in terms of the parameter X. The data in Table XII

support the prediction

of (7.63) that 7 is

proportional to the double bond character. The experimental data for the first four compounds indicate that the double bond characters for the C—Cl bonds in them are nearly equal, as is borne out by the entries in the fifth column of the table. The agreement between theory and experimental data for p-CsH,CIN Hs is not so satisfactory. However, 7 for COC], is large and Bersohn’s calculations also predict a

relatively large II]. The

experimental

data indicate that a value of

\ = + is reasonable for the C—Cl bond in most cases. Exact numerical agreement between experiment and theory is not to be expected because of solid state intermolecular contributions to 7 (to be discussed in Section 8), and because of the contribution to » from torsional motions of the molecules in the solid state pointed out in Section 3a. Recent microwave measurements by Goldstein®®! indicate that 7 for vinyl chloride is about 0.14 in the free molecule. From comparison of this with the data for the solid (see Table XII), it appears that the intermolecular effects reduce the asymmetry from the value for the free molecule. Moreover, the microwave value of » gives somewhat better agreement with Bersohn’s?” calculated II in Table XII than does Livingston’s value for the solid. This analysis for the C-—-Cl bonds also holds in principle for C—Br and C—I bonds. The overlaps between carbon 2p, orbitals and bromine 4pz or iodine 5p, orbitals are not very different from that for chlorine (Sa = 0.134, as compared to Sco: = 0.128), so that the value of X in these cases would also be in the neighborhood of 4. Bersohn has pointed out that the small values of A, which indicate weak conjugation between a electrons on carbon and halogen atoms, are consistent.

with the chemical properties of halogen substituted aromatic compounds. Among such properties are (1) the weak ortho- and paradirective property of the halogen substituents in the benzene ring, as compared to other substituents such as NH: and OH, and (2) the equality of heats of hydrogenation of vinyl chloride and ethylene (to produce the saturated compounds ethyl chloride and ethane, respectively).

(2) Chlorine quadrupole resonance frequencies in chlorobenzenes. We shall conclude this section by a discussion of the Cl®* quadrupole resonance frequencies of various chlorobenzenes which have been tabulated and analyzed recently by Bray et al.!*! These resonances

were observed by Meal,?!’ Dean 213 H.C.

Meal,

J. Am.

Chem.

(thesis®), Duchesne

Soc. 74, 6121

(1952).

and Monfils,?°?

7.

INTERPRETATION

OF

NUCLEAR

QUADRUPOLE

COUPLING

DATA

161

and Bray et al.'*! Usually the specimens were in powdered form, so the asymmetry parameters are generally not available for interpretation. The lines in many of the resonances were multiple due to the presence of intermolecular effects, which give rise to chemically inequivalent lattice sites. Wherever such splittings occur, the means of the multiple line frequencies have been taken as representative of the frequencies associated with the free molecules. The resonance frequencies associated with different chlorine positions in the various chlorobenzenes are tabulated in Table XIII taken from Bray, Barnes, and Bersohn’s paper.'?! Figure 43 gives a plot of the chlorine resonance frequencies versus the number of substituent chlorine atoms in the TasLe

XIII. Resonance AS FUNCTIONS OF

Data point in figure

FREQUENCIES THE NUMBER

Compound

FOR CL*® NucLEI IN CHLOROBENZENES OF ORTHOCHLORINE NEIGHBORS

Number of orthochlorine neighbors

Averaged frequency at 77°K (mc)

0 1

C1C.H; m-ClCeH,

0 0

34.622 34.936

2 3

p-CleCeHs 1,3,5 Cl,CeH3

0 0

34.779 35.851

0

36.285

4

5

1,2,4 ClhCoHs

0

(chlorine 4)

1,2,3,5 CliCsHe

35.403

(chlorine 5) 6

o-Cl.C.H4

1

35.729

7

1,2,4 Cl;CeHs (chlorines 1 and 2) 1,2,3 Cl;CeH; (chlorines 1 and 3) 1,2,3,4 CliCgHe

1

36.394

1

36.335

1

37.013

(chlorines 1 and 1,2,3,5 CliCeHe (chlorines 1 and

1

36.903

1

36.803

]

37.485

2

37.002

2

37.506

8 9 10

I 12 13

14

1,2,4,5 ClCH»

Cl;C.H (chlorines 2 and 1,2,3 Cl;CsH, (chlorine 2)

4) 3)

6)

1,2,3,4 CliCoH»

(chlorines 2 and 3)

15

1,2,3,5 ClhCeHe

2

37.480

16

Cl;C.H (chlorines 3, 4, 5)

2

37.944

17

Cl.Ce

2

38.442

162

III.

appLicaTIONsS

benzene ring. In Table XIII the first column gives the positions of the different resonance frequencies in Fig. 43. The second

column

gives

the compound and the particular position for the chlorine atom in the molecule to which the row refers. The third column gives the number

of ortho-neighbors for the chlorine atom in question, and the fourth column gives the resonance frequency. Figure 43 shows that the data

38.5

7

38.0F

Ci®> resonance frequencies (mc)

37.5

37.0

36.55

36.0F

35.5

4 35.0F °2 34.5 1

1 2

L ! i 1 3 4 5 6 Number of substituent chlorine atoms

Fia. 43. Plot of the Cl** resonance frequency in the chlorobenzenes versus the number of substituent chlorine atoms. [See P. J. Bray, R. G. Barnes, and R. Bersohn,

J. Chem. Phys. 26, 813 (1956).]

divide into three distinct curves which are nearly straight lines. The three straight lines refer to the three sets of chlorine atoms, having respectively 0, 1, and 2 ortho-neighbors. For each line the Cl** resonance frequency increases with the number of substituent, chlorine atoms.

Bray, Barnes, and Bersohn!?! have analyzed the behavior of the Cl*® resonance frequencies shown in Fig. 43 as follows. They point out that it is easy to explain the increase in resonance frequency for a Cl*

nucleus by addition of another substituent chlorine atom in a position

7.

INTERPRETATION

OF

NUCLEAR

QUADRUPOLE

COUPLING

DATA

163

that is not ortho with respect to the original chlorine atom. From Eq. (7.64), it is clear that addition of the second substituent chlorine atom does not affect the double bond character of the original C—Cl

bond appreciably. On the other hand, the two chlorine atoms face competition from each other insofar as extraction of o electrons from the

carbons

is concerned

[see Section

7a(3)].

There

is, therefore,

a

drop of ionic character at the first chlorine atom from its original value. According to the Townes and Dailey theory and Eq. (7.27), it is seen that this would lead to an increase of e?gQ and hence an increase

in resonance frequency. This argument explains the upward slope of each of the three lines in Fig. 43 with increase in the number of chlorine

substituents. The increase in the Cl*> coupling constant in passing from

Fic. 44. Relative orientations of « and x orbitals in the chlorine atom and adjacent carbon atom when the former is pushed out of the plane of the benzene ring.

chlorine atoms without ortho-neighbors to those with one and two ortho-neighbors is somewhat more complicated. Bastiansen and Hassel?!4 have observed that when a chlorine atom has an orthoneighbor, the two adjacent C—Cl

bonds are pushed respectively 18°

above and below the benzene ring because of repulsion between the adjacent chlorine atoms. Duchesne and Monfils?°? had pointed out earlier that this would lead to a decrease in the double bond character of each of the C—Cl bonds due to decrease in overlap between the pz atomic orbitals on the carbon and chlorine atoms. However, Bray

et al.!?! have shown that the out-of-plane bending of C—Cl actually leads to an increase in double bond character. In Fig. dotted lines refer to the orientations of the x(3p.) and o (3s-3p, orbitals on the chlorine atom (in the bent position of the C—Cl and the solid lines refer to the corresponding orbitals on the 214, Bastiansen and O, Hassel, Acta Chem, Scand. 1, 489 (1947),

bonds 44, the hybrid) bond), carbon

164

III.

appLicATIONS

atom. The overlap of the z electron of the chlorine with the electrons on the carbon atom is when the axes of symmetry of the states in the two atoms are parallel; while in the situation shown in Fig. 44, the overlap is

cos?

18°

+

sin?

18°

which is greater than . This increase of double bond character due to the influence of an orthohalogen neighbor has been confirmed by Bray, Barnes, and Bersohn by their measurements on I!?7 resonance in para- and ortho-di-iodobenzene. They found asymmetry parameters n equal to 4.1 + 0.5% and 8.5 + 1.5%, respectively, in the two compounds. In the chlorobenzenes, the influence of an ortho-chlorine neighbor, which leads to an increase of the double bond character of the C—Cl bond, gives rise to a decrease in e?¢Q according to Eq. (7.27). But in addition to this there will be the ionic effect via the o electrons discussed earlier. The ortho-chlorine neighbors compete even more strongly for electrons from the adjacent carbons than in the case of chlorine atoms attached to nonadjacent carbons. This leads to an

increase in e?¢Q, which more than counteracts the decrease due to the ortho-influence on double bond character. A Cl*® nucleus with an ortho-neighbor would therefore possess a larger coupling constant than one without such a neighbor. By a similar argument a Cl*> nucleus with two ortho-neighbors would possess an even larger e?¢Q than one with only a single ortho-neighbor. These arguments qualitatively explain the trends of the three lines in Fig. 43. A quantitative explana-

tion would require an actual numerical evaluation of the changes in electron densities in C—Cl bonds due to the influence of neighboring chlorine atoms on the benzene ring. 8.

INTERMOLECULAR

a. Relative Importance

BINDING

IN

THE

SoLip

STATE

of Direct and Indirect Effects of Neighboring

Molecules on the Nuclear Quadrupole Interaction The nuclear quadrupole spectra of solids show the influence of neighboring molecules on the field gradients at the nuclei in any particular molecule. There are two ways in which the field gradient at a nucleus can be influenced by the neighboring molecules. First, there is the direct effect, caused by the field gradient at the nucleus due to charge distributions in neighboring molecules. In molecular crystals?! 215 See p. 260 in ref. 37.

8.

INTERMOLECULAR

BINDING

IN

THE

SOLID

STATE

165

individual molecules are held in their equilibrium positions by electro-

static (Madelung)

and intermolecular (van der Waals) forces. Due to

the large intermolecular distances, a nucleus in any particular molecule will see all the non-nearest neighbor molecules as neutral charges confined to a space of relatively small dimensions compared to their distances from the nucleus. Hence the direct contribution to the nuclear quadrupole interaction from the distant molecules will be practically zero. An example will help in illustrating the order of magnitude of the direct contributions from the nearest neighbor molecules. Let us consider the case of solid IBr, whose crystal structure is known from x-ray diffraction measurements.*!§ The molecules are arranged in parallel planar sheets as in solid iodine (see Fig. 49). The bromine atom has as its nearest intermolecular neighbors two iodine atoms on the same plane, each at a distance of about 3.38 A. The difference in electronegativity!®* between iodine and bromine atoms is 0.3. This would lead to an ionic character of about 5% (see Fig. 36), corresponding to an effective charge of +0.05 on the iodine atoms. Taking into account Sternheimer and Foley’s (see Das and Bersohn!*!) value of 70.7 for the antishielding factor for Rb*+ ion, a value 100 seems reasonable for the bromine atom. A field gradient

g = 0°12 X 10% cm-3 is then obtained at the bromine

nucleus due to each of the nearest

intermolecular iodine neighbors. The magnitude of this is less than 0.2% of the field gradient due to a =: electron on the bromine atom. Thus the direct influence of the nearest intermolecular neighbors on the field gradient can also be neglected. On the other hand, in ionic crystals like NaClO; and KCIlO; for example, there are unit positive charges on the positive ions (Nat, K+) and unit negative charges on the (ClO3)~ ion, and the direct influence of neighboring ions on the field gradient can be appreciable. Therefore a large fraction of the differences in Cl®> quadrupole coupling constants in the ClO3;~ ion in the three salts?!7 NaClO;, Ba(ClO3)2;H2O, and KCIOs; (e?qQ/h equals respectively to 59.86, 58.76 mc, 56.20 me at room temperature) may be due to the direct effects of the charges on neighboring ions. Aside from their direct contributions to the field gradient, the neighboring molecules can also affect the field gradient in an indirect 216. S. Heavens and G. H. Cheesman, Acta Cryst. 3, 197 (1950). 217 The values quoted for NaClO; and KCIO; are from Livingston

and

Zeldes’

latest paper.?? For Ba(ClO;)2 the quadrupole coupling constant is that obtained by K. Torizuka [J. Phys. Soc. Japan 9, 645 (1954)].

166

III.

manner.

The

energy

of binding

crystals is made

up

Madelung

represent

terms

APPLICATIONS

of the molecules

of Madelung

and

van

contributions

in the molecular

der Waals

from

terms.®®

the direct

The

Coulomb

interactions between the charge distributions on the various molecules to the cohesive energy of the solid. The van der Waals terms represent the contribution of the polarization and dispersion forces between the molecules. A combination of the energies due to these two sources may be such as to favor an alteration in the ionic characters of the

intramolecular chemical bonds in the solid from those for free molecules. In addition, the individual molecules in the solid may be close enough to give rise to intermolecular bonds which would lead to a weakening of the intramolecular bonds within the molecules. There also may be hydrogen bonds in the solid that would affect the intramolecular bonds. The evidence for all these indirect effects, which is available from quadrupole resonance data, will now be presented. b. Differences in Ionic Characters of Chemical Bonds in the Gaseous and Solid States Table XIV presents a list of compounds for which the halogen quadrupole coupling constants have been obtained in both the gaseous and solid states from microwave and quadrupole resonance spectroscopic

measurements. The references for the data on the solids are listed in the fifth column. The data for the gases are taken from Townes and Schawlow.?!8 The Cl*5, Br7*, and I!27 quadrupole coupling constants of the first

six compounds in Table XIV

are smaller for the solid than for the

corresponding gas. Since the halogens are negatively ionic in the C—X bonds (X = Cl, Br, I), the reductions in the quadrupole coupling constants upon vaporization indicate that the ionic character of the

C—X

bonds is greater in the solid than in the gaseous

Eq. (7.27)]. This increase in ionic because it leads to an enhancement ing energy of the crystal. In IC] the is larger in the solid than in the gas.

state [see

character of the solid is favored in the Madelung term in the bindI??”7 quadrupole coupling constant From Eq. (7.28) this also indicates

an increase in the ionic character over the gaseous molecule,

since

iodine is positively ionic in this compound. The same increase in ionic character is reflected by the decrease in the Cl* coupling constant in

solid ICI. The ratios e7GmiQ/e’gaQ of the first six compounds

in Table XIV

are plotted in Fig. 45. This figure clearly shows the general increase in 218 See Appendix VI in ref. 16.

8.

INTERMOLECULAR

BINDING

IN

THE

SOLID

STATE

Taste XIV. A List or Compounps WHosE HALOGEN Courtine Constants Have Breen Stupiep ror Botu AND Souip STaTEs

167

QUADRUPOLE THE GASEOUS

e*qQ/h (me) Molecule Nucleus

Gas

Solid

CH;Cl

Css

—74.74

—68.4*

CF;Cl

Clss

—78.05

—77.58

CH;Br

Br79

577.15

529

Reference for quadrupole resonance work R. Livingston, 1434 (1951). R. Livingston,

J. Chem.

Phys.

19,

J. Chem.

Phys.

19,

1434 (1951). H. G. Dehmelt,

Naturwiss.

37, 398

(1950). CF;Br

Br79

619

604

H.

Zeldes

and

R.

Livingston,

J.

Chem. Phys. 21, 1418 (1953). CH;I

[127

— 1929

— 1766

CF,I

y127

—2143.8

—2069

ICl

[127

—2930

—3037

H.

G.

Robinson,

22, 512

C135 ICN

[127

—82.5 —2420

—74.4 — 2549

H.

G.

Dehmelt,

and W. Gordy, J. Chem. Phys. 22, 512 (1954). H. G. Robinson, H. G. Dehmelt, and W. Gordy, J. Chem. Phys. (1954).

H. G. Dehmelt, Naturwiss. 37, 398 (1950). P.J. Bray, J. Chem. Phys. 28, 704 (1955). H. G. Dehmelt, Naturwiss. 37, 398 (1950).

* Sign of the quadrupole coupling constants for the solids cannot be obtained from the quadrupole data. The signs indicated are written down by comparison with the signs for the gaseous molecules.

ionic

character

of the

carbon-halogen

bond

in

all the

compounds

upon transition from the gaseous to the solid states. In addition, the differences in the halogen

(X = Cl, Br, I) quadrupole

coupling con-

stant in the gaseous and solid states for each of the compounds

of

the type CF;X is significantly less than the corresponding difference in CH;X. This indicates that the C—X bond becomes more ionic upon solidification for the compounds CH3X than for CF3X. A qualitative reason for this may be as follows. The carbon atom!88 is more strongly electropositive with respect to the fluorine atom than the

other halogens, while with respect to hydrogen it is slightiy electronegative. Upon solidification both the C—X and C—F bonds tend to increase in ionic character. This means that the fluorine and the other halogen atoms compete in extracting electrons out of the carbon atom in CF;X as it condenses to the solid. On the other hand, in the

168

III.

APPLICATIONS

CH;X molecules, the CH bond, in tending to be more ionic, makes an increased donation of electron density to the carbon atom and so aids the halogen atoms in extracting electrons from the carbon atom. 1.0

0.9F

Ve S|.

of

e]

©

0.8

o7F

CF3X

8

CH3X 3

o Gas e Solid

0.6

Fia.

CF;X.

45.

Plot

[See H.

511 (1954).]

1

1

X=Cl

X=Br

of e%gmoiQ/e%qatQ

G.

Robinson,

H. G.

for gaseous

and

1

X= solid

compounds,

CH;X

and

Dehmelt, and W. Gordy, J. Chem. Phys. 22,

The C—X bond can therefore undergo a larger change in ionic character upon solidification of the vapor in CH;X than in CF;X. This explains the observed trend in the change of halogen quadrupole

coupling constants shown in Fig. 45. Alternatively Robinson et al.'48 attribute the greater difference in e?¢n.Q between gaseous and solid CH;Cl to the action of the dipole moment of the CH bond. This bond

8. INTERMOLECULAR

can approach

the halogen

BINDING

IN THE

SOLID

nucleus in a neighboring

STATE

169

molecule

more

closely than can a CF bond. However, this explanation does not seem

tenable because of the small dipole moment

of the CH

bond?!® (0.4

debye), and the relative unimportance of direct intermolecular effects in the field gradient in molecular crystals (see Section 8a).

The change in the quadrupole coupling constant for the I)?’ nucleus in ICN has been discussed by Townes and Dailey.®® The structure

of the free ICN

molecule

in the gaseous state is shown

in

Fig. 46a. The iodine atom forms a single o bond, and the orbital on the iodine atom takes part in the bond represented by a hybrid of the form (7.28). Since iodine and carbon have nearly equal electronegativity, a

change in the ionic character of the I—C o bond could come about by a resonance between structures (a) and (b), shown in Fig. 46, without I

1*:

Cc (a)

N

Cc

N7

(7) sizs==N

Jt.

C=

N

I*.

C==:=::

(c)

Fia. 46. Resonance structures involved in molecular chains of ICN.

the formation of any intermolecular bonds in the solid. The structure (b) involves a loss of a w electron from the iodine atom. This would

lead to a decrease in the coupling constant of the I!?”7 nucleus, in contrast to the observed increase upon solidification, as seen from Table XIV. From x-ray data the crystal structure of ICN is known??? to correspond

to an end to end arrangement

cules.

I—N

The

of parallel chains of mole-

separation between neighboring molecules in these

chains is found to be 2.78 A, which is smaller than the normal separa-

tion

of 3.65

Townes

and

A between Dailey®®

nonbonded

iodine

therefore postulated

and

nitrogen

additional

bonds

atoms.??! between

the iodine and nitrogen atoms of neighboring molecules, as shown in the structure c in Fig. 46. In structure c there is a loss of one electron

from the lone pair on the sp, hybrid [of the form (7.18)] on the iodine atom. Therefore, there is a net loss of p, electron density (increase of vacancy in the p, state) on the iodine atom as compared to structure a. 219 ©, Coulson, Trans. Faraday Soc. 38, 433 (1942). 220 R, W. G. Wyckoff, ‘Crystal Structures,”’ Vol. I. Interscience, New York, 1948. 221 See p. 189 in ref. 126.

170

III.

APPLICATIONS

Structure c therefore would involve a larger coupling constant for I)?’ than

gas

structure

and

the

a. From

solid,

the observed

Townes

and

I!?? coupling

Dailey

constants

of the

have ascribed a fractional

importance of 10% to structure c in the solid state. In the next chapter it will be seen that the postulate of an additional bond between iodine and nitrogen atoms is in qualitative agreement with

the observed quadrupole coupling constant for the N' nucleus in the gaseous and solid phases of ICN. As shown in Table IV, Section 6, the quadrupole spectra of chloral hydrate CCl;CH(OH)». consists of three lines. Two of them are close to each other, and the third one is at a somewhat lower frequency. The crystal structure data mentioned in Section 6a show that, in the solid state, two of the CCl bonds in the CCl; group are somewhat longer

than the third. These two bonds originate from the two chlorine atoms that form hydrogen bonds with hydroxy] groups of neighboring molecules. Duchesne?”? has pointed out that an increase in the length of the C—Cl bond leads to a decrease in the ionic character. Therefore,

in chloral hydrate one would expect the two chlorine atoms, which form hydrogen bonds, to exhibit larger Cl*® quadrupole coupling constants than the one that does not form a hydrogen bond. This is seen to be the case from Table IV. In chloral aleoholate [CCl;CHOH(OC2H;s)], since there is only one hydroxyl group per molecule, only one of the three chlorines of the CCl; would form a hydrogen bond and therefore show a larger Cl®* coupling constant than the other two, as seen from Table IV. These examples show that the formation of a hydrogen bond

by a halogen atom can affect the field gradient at the halogen nucleus appreciably. c. Bond Switching

Before

discussing

the

quadrupole

coupling

constants

of

solid

halogens, it is helpful to consider the phenomenon of “bond switching”

which has been observed to have quite marked effects on the nuclear quadrupole

interaction

in solid

compounds.

Such

effects have

been

demonstrated22?.224 in the I!27 resonance in AsI; and Bils, and should also occur in SbI3 because of its structural similarity?®® to the former two compounds. 222 J, Duchesne, J. Chem. Phys. 20, 1804 (1952). 223 S, Kojima, K. Tsukada, S. Ogawa, A. Shimauchi, and Y. Abe, J. Phys. Soc. Japan 9, 805 (1954). 224 R, G. Barnes and P. J. Bray, J. Chem. Phys. 28, 1177 (1955). 225A, F, Wells, ‘Structural Inorganic Chemistry,’ p. 278. Oxford University Press, New York, 1950.

8. INTERMOLECULAR

BINDING

IN THE

SOLID

STATE

171

From electron diffraction measurements, it appears that in the vapor state AsI3, SbI3;, and Bil; consist of regular pyramidal molecules with the arsenic, antimony, or bismuth atom at the apex. X-ray data show that the unit cells of the solid are rhombohedral, and the molecules are arranged in such a manner that there are parallel planar sheets

of As, Sb, or Bi atoms with parallel sheets of iodine atoms on either side, as shown in Fig. 47. Each As (or Sb or Bi) atom has six nearest iodine neighbors, with three each on either side of its own plane. The iodine atoms make up an octahedral arrangement (not exactly regular)

around the As (or Sb or Bi) atom. In AsI; the distance between the arsenic atom and each of its six nearest iodine neighbors is 2.98 A. The

Fig. 47. Arrangement of iodine and arsenic atoms in AsI; crystal. The unit cell is hexagonal with aj = 7.187 A and co = 21.39 A. The complete circles denote iodine atoms above the plane of the As atoms shown and the dotted circles those below. There is a nearly regular (not exactly) octahedral coordination around the As atoms. Each I atom has two close As neighbors 2.98 A away. Angle As-I-As = 88°.

iodine atom has two arsenic neighbors each at a distance 2.98 A, with

the angle As—I—As equal to 88°. The I}?7 quadrupole resonance in AsI; has been studied by Kojima et al.??? These authors found that the $— 4 transition gave a single

line up to 110°C and thereafter a triplet, indicating that a change of structure occurs at 110°C. The Br7? resonance in AsBrs was found to be a triplet at all temperatures. Kojima and his group concluded from these observations that the high temperature phase of AsI; had the same crystal structure as AsBr;. This point will be discussed in Section 9d in connection with the As7> quadrupole resonance in arsenic halides. Kojima and his group found that the quadrupole coupling constant for the I!?7 nuclei in AsI3 at 83°K is 1328.24 mc, and the asymmetry parameter is 18.4%. This large asymmetry parameter

would be difficult to explain if the iodine atom formed a single bond with an As atom

as in a free pyramidal

molecule,

since the charge

distribution due to the « bond between arsenic and iodine is symmetric

172

III.

aApPLicaTIONS

around the As—I line. There may be some multiple bond character, of course, with the contributions of the x, and 7, bonds perpendicular to the « bond being unequal, because there is no threefold or higher symmetry in the molecule about an I—As bond (see Section 7a). But an unreasonably large multiple bond character would be required to account for all the observed asymmetry. Kojima and his group have proposed that in the solid state each iodine atom forms two equal bonds with its two equivalent nearest neighbor As atoms. Each of these bonds is assumed to have half the strength of the As—I bond in the free

molecule.

In

‘‘valence-bond”’ language, there is a resonance between alternative structures in which

the iodine atom is bonded to either of the two equivalent arsenic neighbors. This situation may also be described as a switching of the bond to the iodine atom from one of the two equivalent arsenic atoms to the other. Let g repre-

Fig. 48. Immediate environ-

ment of each iodine atom in viel and I—As,

yen and

I—As,

of prinshown

sent the field gradient at I!” along the As—I direction in the free molecule. For simplicity, the field gradient is as-

gymed to be axially symmetric. In the solid, we then have axially symmetric field gradients eq/2 along the directions in Fig.

48.

Consider now the new system

of axes shown in Fig. 48 with the X and Y axes chosen in the plane of the iodine atom and its two arsenic neighbors as shown in Fig. 48. The following components are then obtained for the field gradient tensor at the I!?7 nucleus on transforming the two axially symmetric tensors along I—As, and I—As, to the XYZ gxx = 4 (1 —

3 cos

system: 0)

qvy = ‘(l + 3 cos 6) qzz

and

(8.1)

-_—% = 3

qxz = qyz = gxy = 0. This shows

that the XYZ

system is the

principal axis system for the field gradient tensor at the I}? nucleus. Therefore

= (XX — WY 0

qzz

_ 3 co8 8,

(8.2)

8.

INTERMOLECULAR

BINDING

IN

THE

SOLID

STATE

173

Using the observed value of 18.4% for 7 at 83°K, one gets a value of 6 = 86°30’. If a small part of the asymmetry were ascribed to multiple bond character, a larger value of @ would be obtained, in better agreement with the observed value of 88° from x-ray data. Also since qzz in the solid is equal to —q/2, the observed value of qgzz shows that the I27 coupling constant e?¢Q for the free molecule is about 2656 mc. This is seen from Table IX to be larger than e?g¢,..Q (2292 mc). From Eq. (7.27) and consideration of ionic or multiple bond characters and hybridization, a value for e*qmiQ/h smaller than 2292 me would be expected. The large disagreement with the experimental value of 2656 me results either from large amounts of d hybridization at the iodine atom or a significant departure from Townes and Dailey’s approximations discussed in Section 7a. Barnes and Bray?*4 have found an asymmetry parameter 7 of 0.258 in Bil;. From (8.2), this value of 7 leads to a Bi-I-Bi angle of

85.1°, as compared to the value 90.7° from x-ray measurements. A better agreement with the latter value would again be obtained if some of the asymmetry were ascribed to the multiple bond character of the Bi—TI bond. Only the $— 4 spectral line for I!?” in SbI; has been observed by Barnes and Bray. Therefore 7 is not known experimentally. The arsenic, bismuth, and antimony tri-iodides represent extreme cases of bond-switching where an intermolecular bond having the same strength as the intramolecular bond occurs in the solid. However,

in

some cases, for example the solid halogens, weaker intermolecular bonds may be found. The intermolecular bonds would lead to finite asymmetry parameters in these cases as well, but would affect e?qzzQ less drastically than for the cases discussed in Section 8a. d. Interpretation

of Quadrupole

Resonance

Spectra in Solid Halogens

in Terms of the Intermolecular Binding The available nuclear quadrupole interaction data for the solid halogens are tabulated in Table XV. The nuclear quadrupole interaction data for the free molecules are not available. Since the free halogen molecules do not have electric dipole moments, their microwave

spectra

cannot

be

observed.

Although

the

molecular

beam

magnetic resonance method is feasible for studying the quadrupole interaction in the gaseous state, it has not yet been applied to the halogens. The origin of the large asymmetry parameters for solid iodine and bromine has been discussed by a number of authors. 9% 118:119,226 Ag 226 §, Kojima, K. Tsukada, 795 (1954).

A. Shimauchi,

and Y. Hinaga, J. Phys. Soc. Japan 9,

174

III. TaBLe

XV.

HALOGENS

Data AND

oN THE

ApPLicaTIONS

NUCLEAR RELATIVE

X-Ray

QUADRUPOLE ORIENTATIONS

OF

Bromine

n*

—0.20¢

e?Qqzz

—

Ota

(mc) (mc)

Temperature of quadrupole resonance measurement Angle between Z principal axes of the two rows of molecules in the unit cell Angle between molecular axes of the two rows

IN SOLID

MOLECULES

FROM

DIFFRACTION Chlorine

2

INTERACTION

—0.16 a

108 .95¢

765 .86

109.74

769.756

20°K

Todine

2292 .712

253°Ke 83°K?4

4°K 63°16’

70° + 5°

64° + 12°

+ 0.01

2156¢

+ 2%

64° + 12%

of molecules from

x-ray data’ * The value of 7 is quoted as negative, which indicates that the directions assumed for X and Y axes should be interchanged, if the convention that » shall always lie between

0 and

1 (Section

1) is to hold.

However,

no erroneous

conclusions

will be obtained if we consistently make a particular choice of X and Y axes throughout the ensuing discussion. ¢§. Kojima, K. Tsukada, A. Shimauchi, and Y. Hinaga, J. Phys. Soc. Japan 9, 795 (1954). ®’K. Tsukada,

J. Phys. Soc. Japan

11, 956

(1956).

¢R. Livingston, J. Chem. Phys. 19, 1484 (1951). 4H. G. Dehmelt, Z. Physik 180, 480 (1951). ¢ Calculated by R. Bersohn from R. V. Pound’s observed frequencies in Phys. Rev. 82, 343 (1951), using Tables of eigenvalues of pure quadrupole spectra spin $, Oak Ridge National Laboratory Report ORNL—1913 (1955). Private communication with R. Bersohn. ‘ The estimated errors in the x-ray data were provided by Dr. R. L. Collin in a private communication with Dr. R. Bersohn. 9R. L. Collin, Acta Cryst. 5, 431 (1952). * B. Vonnegut and B. E. Warren, J. Am. Chem. Soc. 68, 2459 (1936). *P. M. Harris, E. Mack, and F. C. Blake, J. Am. Chem. Soc. 60, 1583 (1928).

we shall see, some conclusions may be drawn about the nature of the binding between the adjacent molecules from a combined knowledge of the asymmetry parameter and orientations of the principal axes of the field gradient tensor at the halogen nucleus. The halogens all crystallize to isomorphic orthorhombic structures

with the molecules arranged in planar sheets parallel to the ac plane, as shown in Fig. 49(a). There are two rows of molecules in the ac plane which can be obtained from one another by a reflection about the ab

8.

INTERMOLECULAR

BINDING

IN

THE

SOLID

STATE

175

plane, followed by a translation of a/2 parallel to the a axis. The angle between the two rows of molecules as determined from x-ray data is shown for each halogen in the last row of Table XV. In the case of iodine, the arrangement of the immediate neighbors of each iodine atom A is as shown in Fig. 49(b). The molecular partner B is 2.70 A

ON

LA

ory

A t

c

f

3.54 77

c

s

7

169°7' ,” 2.70

7

A.

105°25'

‘\ 3.54

B

‘\_D

(d)

~s

Fic. 49. (a) Arrangement of halogen molecules in their orthorhombic unit cells. The projections of two adjacent unit cells on the ac plane are shown. The solid lines indicate molecules lying on the plane of the paper, while the dotted lines indicate molecules in parallel planes above and below the plane of the paper. The iodine atoms which are denoted by A, B, C, and D in (b) are indicated here by corresponding letters. [See K. Tsukada, J. Phys. Soc. Japan 11, 957 (1956).] (b) Immediate environment of an iodine atom at A in (a).

away, which is larger than the distance 2.67 A in the free molecule obtained by electron diffraction measurements of the vapor.!?3 This suggests that the intramolecular bond is somewhat weaker in the solid than in the free molecule. The nearest intermolecular neighbors of A in the same plane are C and D, which are each 3.54 A away; the nearest neighbors in different planes are more than 4.30 A away. The van der

Waals radius??’ of each iodine atom is about 2.15 A, so that two non227 See p. 189 in ref. 126.

176

III.

APPLICATIONS

bonded iodine atoms would be more than 4.30 A away from each other. Townes and Dailey®® proposed that there were weak bonds between

the iodine atom at A and its two closest neighbors C and D. This would

explain the slight weakening

of the intramolecular

bond

AB.

Since the atoms on the parallel planes above and below A were further than 4.30 A away from A, it was assumed that the adjacent planes are held together by van der Waals forces. Townes and Dailey consid-

ered the two bonds AC and AD to be of equal strength, but Bersohn® has later generalized Bersohn’s treatment, as a particular case, Assume that the

their analysis assuming unequal bond strengths. from which Townes and Dailey’s analysis follows will be considered first. three bonds AB, AC, and AD are orthogonal.

Three independent axially symmetric field gradients g, Kq, and Aq can then be considered to act along the three directions AB, AC, and AD

respectively at the I!?” nucleus at A. The quantity q is slightly different from the field gradient in the free molecule by virtue of the slight lengthening of the AB bond in the solid. It will be interesting to com-

pare the value of the field gradient in the free molecule, when available, with the value of g obtained from the analysis of the solid iodine data.

A system of coordinates x y z is considered with z along the molecular axis AB, and with the z axis perpendicular to the plane of the molecule

AB, i.e., parallel to the b axis of the crystal. From (1.4) it then follows that the components of the field-gradient tensor in this system are given by Qzz

dex

q [:

K

+

5

=4y-—K~ = 5 [1 K

,.

(3

cos?

Opac

_

1)

+

(3

cos?

Opap

_ 1)

|

A]

Quy =

~Qez — Vex

Qry =

Yr

(8.3)

Qe =

3q . — 9 |K 8iN Ogac COS Ogac + A SiN Ogan COS Opanl).

= O

and

Diagonalizing this tensor to eliminate the off-diagonal element q,z, it

may be shown in the principal axis system XYZ of the tensor that a= |hatK+n

+E] q

(8.4)

8.

INTERMOLECULAR

BINDING

IN

THE

SOLID

STATE

177

where R=((1+

K +d)? — 4K sin? @pac — 4A sin? Onan — 4K (cos? Onac + Cos? Osan — 2 COs? Ozac COS? Onan —

2 sin Osan COS Opap SiN Opac COS Opac)]

(8.5)

and

1 _~

Qxx — Qyy

Qaz

_

~#1

+ K

+A)

+ ih

RL EK +2) + 3R

(8.6)

The X direction is still parallel to the b direction but the Z axis makes an angle a with the molecular axis given by a = tan“ r 1 1+ © (3 cos" Oaac — 1) + 5 ( cos? Osan — 1) ~gQd+kKt+

A) - ae

${K sin Ogac COS Opac + A SIN Opap COS Opan}

, (8.7)

Since there are two rows of molecules whose directions are obtainable from one another by a reflection in the ab plane (Fig. 49a), there will be two sets of principal axes in the crystal. The Z axes for the two sets are inclined to each other at an angle which differs from the angle between the axes of the molecules by 2a. The latest value for the angle between the two Z axes is due to Tsukada and is shown in the fourth row of Table XV. It is clear from Eqs. (8.6) and (8.7) that, having a and 7, K and d

can be determined. These values of K and X, together with the observed value

of e?qzzQ/h

and

Eq.

(8.4), would

yield the coupling

constant

e?qQ/h. This, due to reasons mentioned earlier, should be somewhat different than the corresponding

value for the free iodine

molecule.

However, the value of a cannot be obtained from the data given in Table XV because of the inaccuracies in both the listed x-ray and the

quadrupole data. It is expected that a would be about a degree or less, which is smaller than the uncertainties in both the x-ray and quadrupole resonance data. Definite conclusions on the magnitude of K and therefore

must

await

more

accurate

x-ray

and

quadrupole

spectro-

scopic measurements. The accuracy of the latter could be improved by employing the echo technique discussed in Section 4c. the

Townes and Dailey®® assumed K = X in Eqs. (8.4)-(8.6). From observed values of e?gzzQ/h and n, they obtained 2480 mc for

e’9@Q/h and K = \ = 0.091. On the basis of Eq. (8.7), this value of K would lead to a value of a = 26’. The coupling constant for the

178

III.

aprLicaTIoNns

free molecule obtained by this method is greater than the value of e'quQ/h in Table IX. The assumption K = 2X is not valid, because the intermolecular bonds AC and AD are not equivalent; that is, the angles (@sac and @gap) they make with the main bond AB within

the molecule are widely different. Also, tensor addition of the axially symmetric field gradients along AB, AC, and AD, according to (8.3), is made on the assumption that the bonds AB, AC, and AD are orthogonal. It is clear that two bonds of equal strength along AC and AD cannot be orthogonal to the main bond along AB when the angles BAC and BAD are so widely different. One therefore has to make use of the general equations (8.4)—(8.7) deduced by Bersohn. However, a more accurate experimental value of the angle a has to be determined before these equations can be used to determine K, \, and e?qQ. Robinson et al.!!® and later Stevens!!® have tried to explain the additional bonds by considering d hybridization at the iodine atoms. Stevens’ theory is particularly interesting. He points out that a hybridized atomic orbital formed by a combination of 6s, 5p., and 5d,, states on the iodine atom at A shows maxima in the radial electron density at angles 0° and about 105° with the molecular axis AB and a minimum at 180°. The angles 180° and 105° are close to those of the auxiliary bonds of the Townes and Dailey theory. Stevens then shows that a mixing of this sp.d., hybrid and a p,d,z hybrid (y axis in the plane of the molecule) can occur due to the van der Waals forces between adjacent iodine molecules. This moves the extremum direc-

tions for the electron density closer to the directions of the auxiliary bonds of the Townes and Dailey’s theory. Bersohn’s analysis for solid iodine, discussed

above,

is equally

applicable to solid bromine and chlorine. In the case of bromine there appears to be some inconsistency in the Br7® quadrupole data shown in Table XV. The small departure of the observed e?qzzQ/h from the value for a pure p bond would indicate that K and } in (8.4) are small.

On the other hand, 7 is larger than in iodine, pointing to larger K and X. A careful redetermination of 7 as well as a with a regenerative appara-

tus or using the echo technique would be helpful in resolving this apparent discrepancy. For solid chlorine no value of 7 is available, but

the value of e?¢zzQ/h indicates that the intermolecular bonds are weak. If, as proposed by Stevens, d hybridization van der Waals forces are responsible for the

bonds,

the intermolecular

bonds

should

at the atoms and additional covalent

get weaker

in going from

iodine to chlorine for two reasons. First, the energy necessary to promote a 7: electron to a d state is least in iodine and largest in chlorine;

9.

STUDIES

OF

NUCLEI

OTHER

THAN

HALOGENS

179

second, the sizes and deformabilities of the outermost atomic orbitals are least in chlorine and greatest in iodine. Therefore van der Waals forces between neighboring molecules would be the strongest for iodine atoms and weakest for chlorine. Kojima et al.??* have found y to be only 0.033 at the I'?7 nucleus in the interhalogen compound ICI. By analogy with IBr, a structure known from x-ray data,?!® ICI would be expected to have a structure isomorphous with the solid halogens. Thus the low observed value of y for I!?’ indicates that the intermolecular bonds between the iodine and chlorine atoms of neighboring molecules are much weaker than the bonds between iodine and iodine atoms in solid iodine. This is to be expected on the basis of the above arguments, since the chlorine atom would show a much smaller tendency to form intermolecular bonds than an iodine atom. 9. QUADRUPOLE HALOGENS

In

the

RESONANCE

earlier

sections,

Stupiges

the

oF

Nucuer

quadrupole

OrHer

coupling

THAN

constants

of

halogen nuclei were discussed. Similar considerations are employed for interpreting the quadrupole interactions of other nuclei. The halogen atom has only one electron less than the number required to produce a closed shell. In other atoms, like those of the third and fifth group elements, there are in each case more than one electron outside

of the closed shell (three and five respectively for the two examples cited) that can take part in chemical binding. Thus, while the halogen atom is usually bonded to a single atom, other atoms are bonded to more than one atom in most of their compounds. The general procedure for interpreting the quadrupole coupling constants in atoms bonded to more than one atom has been discussed already in Section 7a(3). In the present section, we shall review the available quadrupole coupling data for nuclei other than the halogens. No detailed numerical interpretation of the data, such as was given in the case of halogens in Sections 7 and 8, will be attempted. Only a qualitative correlation between the quadrupole data and molecular and crystal structures, wherever available, will be presented. The interpretation of quadrupole interaction data in ionic crystals, where the field gradient arises mainly from the charges external to the ion, has been considered already by Cohen and Reif.! We again shall confine ourselves only to the data for molecular crystals, where the major contribu228 S, Kojima, (1955).

K. Tsukada,

S. Ogawa and A. Shimauchi, J. Chem. Phys. 23, 1963

180

III.

apPLicaTIons

tions to the field gradients at the nuclei come from the electrons on their respective atoms. In paramagnetic ions, the field gradients at the nuclei also arise from the atomic electrons. But in such cases, d-

and f-electron states, which

are often very much

modified

by the

crystalline potential, have to be considered. These will not be considered here. However, as in the atomic beam experiments, a comparison of hyperfine structures of paramagnetic resonance spectra due to nuclear magnetic dipole and electric quadrupole interactions, respec-

tively,

permits

an

approximate

determination

of the

quadrupole

moment Q by use of equations analogous to (7.6) and (7.8). By such a procedure, Bleaney, Bowers, and Pryce?*® calculated the quadrupole

moments of Cu® and Cu® nuclei (—0.159 and —0.147 barn, respectively) from paramagnetic resonance measurements on diluted copper Tutton salts. a. Compounds of Group I Elements of the Periodic Table

The quadrupole interaction data for nuclei of some of the first group elements are given in Table XVI. The data for the first seven

Fia. 50. Arrangement of copper and oxygen atoms in Cu,0.

compounds may be interpreted by the ionic model calculations of Bersohn, which have been discussed in the article by Cohen and Reif.! The quadrupole interaction data for the Cu.O can be treated either by

considering the crystal to be composed of Cu+ and O-~ ions, or that the copper and oxygen atoms are covalently bonded. Bersohn!* has performed calculations for an ionic model. The unit cell of CuzO is cubic?*° and contains two molecules. Figure 50 shows the framework of 229 B. Bleaney, K. D. Bowers, and M. H. L. Pryce, Proc. Roy. Soc. A228, 166 (1955). 280 See p. 361 in ref. 225.

181 HALOGENS THAN OTHER NUCLEI OF STUDIES 9.

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b. Compounds of Second Group Elements Table XVII lists the observed nuclear quadrupole coupling constants for Be? and Hg?” nuclei. The quadrupole interaction of Be® in beryl possibly can be interpreted on the basis of an ionic model. The asymmetry parameter is finite, so that five components qzz, Qyy, Qzz;

Wye, and qzy of the field gradient would have to be calculated in any system of coordinates xyz. Then the calculated tensor would have to be diagonalized to obtain the components qgxx, gyy, and qzz in the principal system XYZ. Such a calculation would be lengthy, since the 231 See p. 541 in ref. 225. 332 H. Kriiger and U. Meyer-Berkhout, Z. Physik 182, 171 (1952).

APPLICATIONS III.

184

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9.

STUDIES

OF

NUCLEI

OTHER

THAN

HALOGENS

185

crystal structure of beryl is complicated. In beryllium metal, which has a hexagonal close-packed structure, Kittel?** has estimated the contribution to the field gradient along the c axis from the Bet“ cores. Using Knight’s data in Table XVII, he finds the Be*® quadrupole moment Q = 0.02 barn. However, there may be a finite contribution to the field gradient from the conduction electrons, which has not been

estimated. The crystal structure of HgCl, is known to be orthorhombic with distinct linear Cl—Hg—Cl molecules arranged in planar sheets.*%4 This structure somewhat resembles that of solid iodine shown in Fig. 49a. The two chlorine atoms, however, have slightly different

surroundings and cannot be obtained from each other by a symmetry operation of the space group of the crystal. Therefore, they are chemically inequivalent. This is manifested by the two slightly different pure quadrupole frequencies observed by Dehmelt, Robinson, and

Gordy for the Cl** nuclei, viz., 22.2303 + 0.0008 mc and 22.0505 + 0.007

mc.

The

Hg?!

preted by Dehmelt, two Hg—Cl

bonds

quadrupole

Robinson, are made

coupling

and Gordy.

constant

has

been

inter-

They assumed that the

up of 6s-6p, hybrids

on the Hg

atom,

each bond having a p, character of 50%. The ionic characters of the bonds were estimated from the observed quadrupole coupling constants for the chlorine nuclei (using Gordy’s rules for hybridization, not Townes and Dailey’s).?** Using the value of the field gradient per

p electron tabulated in Table X, they obtained 0.6 barn for the value of the quadrupole moment of Hg**! This value may be compared with the value of —0.559 barn obtained by Schiiler and Schmidt?** from the hyperfine structure in the optical spectra. The sign of the quadrupole moment cannot be determined from the quadrupole resonance

measurements. The agreement between the values of Q for Hg?! obtained by the two methods is quite satisfactory, considering the fact that Robinson, Dehmelt, and Gordy’s value was obtained using Gordy’s rules for interpreting the Cl** quadrupole coupling constant. A smaller value for Q would have been obtained if Townes and Dailey’s rules had been employed. c. Compounds of Third Group Elements

In contrast to most of the first and second group elements, which mainly form ionic compounds, the elements of the third group show a 238 284 235 236

C, See See H.

Kittel, quoted by W. D. Knight in Phys. Rev. 92, 539 (1953). p. 119 in ref, 225. Section 7a(1). Schiller and Th. Schmidt, Z. Physik 98, 239 (1935).

appLicaTIONS III.

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188

III.

APPLICATIONS

greater tendency to form covalent bonds. Actually the binding in their compounds is intermediate between ionic and covalent. The ionic character increases as one goes further down the periodic table so that in compounds boron is the least and indium the most ionic in the group. The observed quadrupole coupling constants for compounds of elements in this group are tabulated in Table XVIII. The interpretation of the quadrupole coupling constant of B™ in

B(CHs)3 has already been discussed in Section 7a(3). The near equivalence of the B!! quadrupole coupling constants in B(CsH;)3 and B(CHs)3 shows that in both these molecules the field gradient at the B!! nucleus is mainly determined by the electron distribution in the BC bonds. The field gradient is axially symmetric and is expected to have its principal Z axis along the threefold symmetry direction

perpendicular to the three BC crystal could verify this point. The structures of kernite,?*”

bonds.

A Zeeman

euclase,?**

study of a single

spodumene,

and

beryl?*?

are too complicated to describe here. In kernite, the field gradients at the B" nuclei arise from electrons in covalent bonds between boron and oxygen atoms

(Das!7!).

In spodumene,

euclase, and

beryl, how-

ever, the crystal structures indicate that aluminum is present as Al+++ ions rather than as covalently bonded atoms. The field gradient at the Al?’ nucleus then arises directly from the charges on neighboring

ions. The crystal structure of KBF, at room temperature is known to be orthorhombic, where the BF,~ ion has a tetrahedral arrangement of BF bonds. There can thus be no contribution to the field gradient at the B!! nucleus from the bonding electrons between boron and fluorine.**° This is indicated by the small B!! quadrupole coupling constant

in KBF, as compared to that in B(CH3;)3. The field gradient at the boron nucleus therefore arises directly from the charges on the neighboring K+ and BF, ions. At temperatures above 300°C, KBF, exhibits a simple cubic structure,?*! with the K+ and BF,~ ions arranged in a NaCl lattice. Therefore in this phase the field gradient at the B!

nucleus should vanish. The resonance in the higher phase has not yet been studied. NaBH,

is known to have a NaC!

structure**#! and Bray

237 “Structure Reports,” Vol. 10, p. 179, 1949. 238 J. Warren

and

B.

E.

Biscoe, Z. Krist. 86, 516

(1933).

239 See pages 577 and 579 in ref. 225 for structures of beryl and spodumene respectively. 240 See Section 7a(3). 241 R, W. G. Wyckoff, ‘Crystal Structures,” Interscience Publishers, New York, 1951.

9,

STUDIES

OF

NUCLEI

OTHER

THAN

HALOGENS

189

and Silver,?#? as would be expected, obtained no quadrupole splitting of the B"™ magnetic resonance line. Al,O3 has a rhombohedral unit cell containing two Al.0O; molecules.*43 The immediate environment of each aluminium atom is as shown in Fig. 51. There are six oxygen neighbors arranged in the form of an octahedron (not regular). These oxygens form two parallel

equilateral triangles on either side of the aluminum atom. Each of the three members on a triangle is at the same distance from the Al atom, which is different from the corresponding distance of the oxygen atoms

Fia. 51. Arrangement of oxygen atoms around aluminum in Al,0;. The dark circles represent oxygen atoms above the plane of the paper and the lighter ones below. [See A. F. Wells, ‘Structural Inorganic Chemistry,” p. 364, Fig. 113. Oxford University Press, New York, 1950.]

on the other triangle adjacent to the aluminum atom. There is there-

fore threefold symmetry at the aluminum atom about the direction perpendicular to the two oxygen triangles, and 7 should be zero. Bersohn'* has calculated the field gradient at the Al?’ nuclei assuming the crystal to consist entirely of Al+++ and O-~ ions. Using the recently

calculated (Das and Bersohn'*!) antishielding factor Ry = —2.59 for Al*++ ion and the quadrupole moment of Al?’ in Table IX, he obtains

a quadrupole coupling constant which is 65% of the observed value. This

indicates

that

there

is

some

covalent

bonding

between

the

aluminum and its neighboring oxygen atoms. The crystal structure of Ga,.O; is isomorphous with Al,O3 while In,O3; has a slightly different 242 P, J. Bray and A. H. Silver, Bull. Am. Phys. Soc. [2] 1, 323 (1956). 243 See p. 364 in ref. 225.

190

III.

APPLICATIONS

structure.*44 Since the spin of In"™® nucleus is $ [see Eq. (1.17)], four quadrupole

1:2:3:4.

resonance

The

frequencies

quadrupole

would

be

expected

in

the

coupling constants for Ga®® and

ratio

In!

in

Ga.O; and In20; have not yet been studied. It would be useful to obtain them, since they would indicate whether or not there is less covalent binding in Ga2O3 and In.O; than in Al,O3. If these compounds

happen to be less covalent, a better agreement

would be obtained

between an ionic model calculation and experiment than that found by Bersohn for Al.O3. The In!"* resonance”** in InI; is of interest in view of recent results

on the halogen

resonances*4*—?47

in the compounds

of the general

formula AI; (where A = Al, Ga, and In), AlBrz and GaCl;. In all these compounds the halogen resonances show characteristic triplet structures with two higher frequencies close to each other, and a low-

frequency InI; at 173.00 in I!*7, halogen

component

further away.

For example,

Ludwig

finds in

room temperature the three frequencies 173.46 + 0.01 me, + 0.01 me, and 122.56 + 0.04 me for the +$— +4 transition Barnes and Segel*‘® ascribe this characteristic pattern for the resonances to the presence of dimeric molecular units in the

unit cell. Palmer and Elliott?48 and Brode?*® have made electron diffraction measurements on the vapors of the trihalides of aluminum, gallium, and indium. It has been proved conclusively that the aluminum trihalides exist as dimeric molecules, as shown in Fig. 52, with

the distances AX;

and

AX,

Stevenson

and Schomaker?®®

diffraction

data

somewhat

different from

have pointed

do not prove

conclusively

each

other.

out that Brode’s electron

that there

are dimeric

molecules in the vapors of gallium and indium trihalides. Brode’s data on Gal; vapor show a monomeric unit with three iodine atoms arranged in an equilateral triangle coplanar with the gallium atom. Only the crystal structures of AlCl; and AlBrs are known. Renes and McGilvary?*! have shown by x-ray diffraction that the crystal structure of anhydrous AlBr; contains dimeric molecular units as in Fig. 52. On the other hand, the crystal structure of anhydrous AlCl; has been 244 See p. 359 in ref. 225. 245 G, W. Ludwig, J. Chem. Phys. 25, 159 (1956). 246 R. G. Barnes and S. L. Segel, J. Chem. Phys. 25, 180, 578 (1956). 247 H. G. Dehmelt, Phys. Rev. 92, 1240 (1953). 248K. J. Palmer and N. Elliott, J. Am. Chem. Soc. 60, 1852 (1938). 249 H, Brode, Ann. Physik 87, 344 (1940). 250 T), P. Stevenson and V. Schomaker, J. Am. Chem. Soc. 64, 2514 (1942). 251 P, A. Renes and C. H. McGilvary, Rec. trav. chim. 68, 283 (1944).

9.

STUDIES

OF

NUCLEI

OTHER

THAN

HALOGENS

191

shown by Ketelaar, McGilvary, and Renes** to consist of parallel planar sheets of Cl~ ions with parallel sheets of Al++*+ ions between them, as shown in Fig. 53. The arrangement of the chlorine ions around each Al*t++ ion is similar to that for Al.O3 (see Fig. 51), except that now the Al*+* ion is at the center of the octahedron of Cl- ions.

Fig. 52. Dimeric units of Al trihalide molecules in the vapor. [A. F. Wells, “Structural Inorganic Chemistry,” p. 286, Fig. 87. Oxford University Press, New York, 1950.] oN

ry

ry

7

“fs

'

’

“ov ¢~

Wo

y

ov"orvo e

an

TN

or Sb!! nuclei about the C; axes of is expected that the asymmetry paramcases, which is seen to be the case from

Table XIX. In the case of As,Os, Dehmelt and Kriiger* have proved by a Zeeman study that 7 for the As’> nuclei (J = $) is zero. From the ratio of the two frequencies corresponding to the $— 4 and 3— #3 transitions Barnes and Bray** have concluded that 7 is zero at the Sb!?! nucleus (J = $) in Sb4Os¢. Claudetite and valentinite are high-temperature phases*®* of As2O3 and Sb2Os, respectively, which become unstable below the tem-

peratures — 13°C and 70°C. Claudetite is monoclinic while valentinite is orthorhombic.

These

high-temperature

oO —_

Sb

4

oO

0 \ Sb c

So 0

No

0

Per

No

\

can be supercooled

0 \ Sb a

4

\

phases

\

4 0

\

ON 0 YON

No

Fia. 55. Arrangement of Sb and O atoms in the high-temperature phase (valentinite) of Sb.O;. [A. F. Wells, ‘Structural Inorganic Chemistry,” p. 496. Oxford University Press, New York, 1950.]

considerably below the transition temperatures, and, as seen from Table XIX, Bray, O’Keefe, and Barnes have worked with claudetite at temperatures as low as 77°K. The crystal structure of claudetite has been shown by Frueh?*’ to be entirely different from arsenolite. It

resembles valentinite for which the arrangement of Sb and O atoms is as shown schematically in Fig. 55. There are infinite zigzag chains of alternate arsenic and oxygen atoms along the c axis of the monoclinic

unit cell. A pair of adjacent chains are linked by an intermediate row of oxygen atoms leading to irregular sheets perpendicular to the b axis. The O—As—O bond angles at the arsenic atoms are all near 102°, as compared to 100° in arsenolite. There is no longer any axial symmetry

around the arsenic atom and so a finite 7 is to be expected. Bray, O’Keefe, and Barnes obtained indirect evidence for this. They find that the temperature coefficient of the As’® quadrupole resonance frequency in claudetite is about four times larger than in arsenolite. They ascribe this large coefficient to the temperature variation of 7»? 287 A. J. Frueh, Jr., Am. Mineralogist 36, 833 (1951).

9. STUDIES

OF NUCLEI

OTHER

THAN

HALOGENS

201

[see Eq. (1.35)], as determined by (8.7). Since the As7® nuclei in claudetite have finite y, their resonance frequencies are affected by

oscillations about the Z axis of the field gradient. In arsenolite, with n = 0, such an oscillation does not affect e?¢Q/h [see (3.6)]; hence there is no effect on the frequency. Valentinite, shown schematically

in Fig. 55, exhibits an arrangement of antimony and oxygen atoms similar to that of arsenic and oxygen in claudetite. However, the three bond angles at the antimony atom depart much more from the corresponding values of the low-temperature senarmontite phase and

are unequal (81°, 93°, and 99°). A considerably larger asymmetry is

Fia. 56. Arrangement of double chains in Sb.S; and Bi,S; (schematic). Shaded atoms refer to one chain and unshaded to the other. [See A. F. Wells, “Structural Inorganic Chemistry,” p. 400, Fig. 124. Oxford University Press, New York, 1950.]

therefore to be expected at the Sb!*! nucleus than that at the As’ nucleus in claudetite. No quadrupole resonance measurements in valentinite have yet been reported. Bi.zO3; is known from x-ray data to exhibit three and possibly four phases?** (monoclinic, tetragonal, and cubic a, 8). Bi?’ quadrupole

resonance measurements on Bi.O3; would be interesting therefore but have not been reported Among the sulfides, has been studied. The isomorphous and consist

yet for any of the phases. only the Sb!?! quadrupole resonance in Sb.2S3 crystal structures?** of BixS3; and Sb.S;3 are of parallel double chains as shown in Fig. 56.

Each antimony (or bismuth) atom is covalently bonded to three sulfur 258 See p. 400 in ref. 225.

202

III.

APPLICATIONS

atoms which lie on a plane parallel to the chain axis. However, there are two

chemically different types of antimony

(or bismuth)

atoms,

designated by M, and M; in Fig. 56. Here M, has three equidistant sulfur neighbors at a distance 2.50 A, and none at a distance less than 3.14 A away; WM; has one neighbor at 2.37 A, two at 2.67 A, and two, belonging to the adjacent chain, a distance 2.83 A away. This leads to the possibility that there are additional weak bonds between M,

and the two adjacent sulfur atoms in the neighboring chain (shown by dotted lines in Fig. 56). One would expect on this basis two different field gradients at the two sets of Sb!*! nuclei. The asymmetry param-

eter of the field gradient at the 1, sites would be smaller than that at the M; sites. Wang*! has observed only single Sb!*! lines for the $ — 4

and $— % transitions. The frequencies are consistent with the very small 7 shown in Table XIX. Also, from intensity measurements on a single crystal (see Section 1d), he shows that the Z axis of the field gradient is perpendicular to the chain length. This is expected at the M, sites because the symmetry at M, sites is approximately threefold.

A different resonance frequency for the M, sites was not observed. This implies either that the reported crystal structure is not correct, or that the frequency at the M, sites is very much different from that at M,, and may have been missed by Wang. No measurements of Bi? resonance in BioS; have yet been reported. The crystal structure of As2S3 is not known nor are any quadrupole data on As’ in As.Ss available. Electron diffraction measurements have shown that molecules of

the chlorides, bromides, and iodides of arsenic, antimony, and bismuth are regular pyramids?® in the vapor state. In the solid state the struc-

tures of the iodides of all three elements are known?® to be similar to the structure of AsI; discussed in Section 8c. They all have hexagonal

unit cells consisting of the layer arrangements of alternate arsenic and iodine atoms, shown in Fig. 47. Among the other compounds of the set only the crystal structures?*! of AsBr3 and SbBr; are known. They contain distinct pyramidal units of AsBr; and SbBr; molecules.

Raman spectra measurements indicate that arsenic trichloride crystals also consists

of distinct

molecular

units.

The

quadrupole

resonance

spectra should reflect these differences in structure among the different halides. In this connection the analysis of the available data in Table 259 See p. 458 in ref. 225. 260 See p. 278 in ref. 225. 261 N, V. Sidgwick, ‘‘Chemical Elements and Their Compounds,” 793. Oxford, Univ. Press, London and New York, 1950.

Vol. I, pp. 792-

9.

STUDIES

OF

NUCLEI

OTHER

THAN

HALOGENS

203

XIX together with the natures of the observed halogen resonances in these compounds is interesting, and will now be discussed briefly.

Among the iodides, only the As75 resonance in AsI; has been studied [by Barnes and Bray (see Table XIX)]. The asymmetry parameter for the field gradient at the As’® nucleus was not obtained. However, it is expected, from the symmetry of the crystal structure, that it is zero; and the Z axis of the axially symmetric field gradient is expected to be along the threefold axis at the arsenic atom. The I!’ resonance in this compound has already been discussed in Section 8c. The transition in the I!27 resonance?’ from a singlet to a triplet structure above 110°C suggests that there is a phase transition from the layer-like structure to one involving distinct pyramidal molecules as in AsBr3 and SbBrs. The As resonance in the trichloride and tribromide have been studied by Barnes and Bray. The As7> quadrupole coupling constants are (see Table XIX) quite different from that in AsI;. This is what would be expected in view of the distinctly different crystal structures. Since AsBr; and AsCl; consist of pyramidal molecules and the As—Br bond is less ionic than As—Cl, it is expected on the basis of an analysis similar to that for NH; in Section 7a(3) that the As’® nucleus should have a smaller coupling constant in AsBr;. This is indeed the case (see

Table XIX).

The Cl*> and Br*! resonances in these two compounds

have also been studied’** and both exhibit triplets. This indicates that the halogen atoms in the molecules are inequivalent, possibly because of intermolecular bonds. In view of this inequivalence of the halogen atoms, it would be interesting to measure the asymmetry parameters for the As’ using single crystals. Sb!?! resonances in antimony tribromide and trichloride exhibit features similar to those of the As7> resonance in the corresponding arsenic compounds. The Sb!*! resonance again occurs at a larger frequency in the chloride than in the bromide (Table XIX). This would be expected, on the basis of arguments similar to those for the corresponding arsenic compounds, if both compounds consist of distinct pyramidal molecules. In view of the large observed asymmetry parameters, the pyramids evidently are considerably distorted. Wang has found that the Cl®* resonance in the chloride consists of a pair of triplets. This is consistent with the idea that distorted pyramidal SbCl; molecules exist in the crystals so that the chlorine atoms in each pyramid are chemically inequivalent. However, the presence of two Cl*® triplets suggests that there may be two inequivalent molecules in the unit cell. An x-ray investigation of the structure would be helpful.

204

III.

aPPLicaTIons

The Br*! resonance in SbBrs; was found by Schawlow!!! to be single at 24°C. This is somewhat surprising in view of the appreciable asymmetry in the field gradient at the Sb!?! nucleus, which suggests the presence of distorted SbBr; pyramids, as in SbCl;. A reinvestigation with a regenerative circuit is essential to find out if any Br*! lines were overlooked.

Of the bismuth trihalides, only in the case of BiCl; has the quadrupole resonance been studied (see Table XIX). The Bi?’ nuclei (J = $) produce four lines whose frequencies depart considerably from the ratio 1:2:3:4 expected for axial quadrupole interaction. This is expected in view of the large asymmetry parameter 7 = 0.583. Robinson has found the Cl** resonance to be a doublet at 83°K. This is to be compared with the corresponding findings of Livingston’? and Wang on AsCl; and

SbCl; mentioned earlier, that the Cl®* resonances exhibit one and two triplets, respectively. It cannot be said definitely from these data whether the unit cell contains distinct distorted BiCl; molecules. No x-ray data are available. The crystal structures of the triphenyls of arsenic, antimony, and bismuth are not known. Electron diffraction data on As(CH3)3 indicate a pyramidal arrangement of the arsenic and carbon atoms with the arsenic atom at the apex. It is possible that solid triphenyls involve a similar pyramidal arrangement. If this is the case, then the finite asymmetry parameters for the Sb! and Bi? resonances in these compounds (Table XIX) indicate that the pyramids are somewhat distorted. Since the spin of As7>is $, Bray and Barnes* could not obtain from their measurements in powdered As(C¢H;)3. The doublet natures of the As75 and Sb!”! resonances suggest the presence of two inequivalent arsenic and antimony sites in the unit cell. On the other hand, the Bi**® resonance in the bismuth compound suggests the presence of only a single bismuth site. The Sb!#! resonance pattern obtained by Barnes and Bray indicates a Sb(C3H7O)3 structure. Also the results presented in Table XIX suggest the presence of distorted pyramidal molecules in the crystal. X-ray determination of the crystal structure is necessary before a more complete analysis of the quadrupole data can be made. In concluding this chapter, mention should be made of Dehmelt’s!* interesting measurement of 8%? quadrupole resonance in natural rhombic sulfur. The S** nucleus has an abundance of only 0.74%. Dehmelt therefore used a large sample of volume nearly 3000 cc. At room temperature he located 4 closely spaced resonances at 22.801, 22.866, 22.896, and 22.964 mc. Therefore, since the S** nucleus has a

10.

BROADENING

OF

LINES

BY

IMPURITIES

205

spin of $, his results indicate the presence of four chemically inequivalent sulfur sites in the unit cell. This is in agreement with Warren and Burwell’s?®? x-ray measurements of rhombic sulfur which indicate molecular units of Ss in the crystal. Each unit had a puckered arrange-

ment of four pairs of inequivalent sulfur atoms indicated by I, IT, ITI, and IV in Fig. 57. The S—S bond distances are all close to 2.12 A, while the bond angles at the various sites are as indicated in the figure. From a consideration of the bond angles, Dehmelt drew conclusions as to the s-p hybrid structure of the sulfur atomic orbitals taking part in the S—S bonds. Using the Townes and Dailey arguments outlined Il

Ill

IV

I

II

Fic. 57. Schematic diagram of molecular units of Ss in rhombic B. E. Warren and J. J. Burwell, J. Chem. Phys. 3, 6 (1935).]

sulfur.

[See

in Section 7, he obtained 0.053 barn for the magnitude of the S** quadrupole moment. This agrees well with Townes and Geschwind’s?®*

value

of

—0.055

barn

from

the

microwave

spectra

of the

OCS*

molecule. 10.

BROADENING IMPURITIES

OF

QUADRUPOLE

RESONANCE

LINES

BY

a. General Considerations It was pointed out in Section 2b that presence of impurities in the sample containing the resonant nuclei leads to a broadening of the

quadrupole

resonance

and may

smear it beyond

detection.

Bloem-

bergen”! has discussed the influence of impurities in metals on the magnetic resonance of nuclei with quadrupole moments. Cohen and Reif}?64 have considered the same problem for ionic crystals. The 262 B. EK. Warren and J. T. Burwell, J. Chem. Phys. 3, 6 (1935). 263 C. H. Townes and S. Geschwind, Phys. Rev. 74, 626 (1948). 264 F. Reif, Phys. Rev. 100, 1597 (1955).

206

III.

APPLICATIONS

broadening influence of the impurities is regarded as being the result of two effects. First there is a “charge effect,’’ which arises if the

impurity core in the metal (e.g., zinc impurity added to copper) or the impurity ion in the ionic crystal (e.g., Cd++ in AgBr) has a different charge from the cores and ions respectively of the pure sample. The additional field gradient at a nucleus is caused by the extra charge on the impurity. Those nuclei that have an impurity center in their neighborhood see a different field gradient from those that do not, and the field gradients at nuclei situated at different distances from impurity atoms will be different. This would lead to a distribution in field gradients over the various nuclei in the sample and therefore cause a broadening of the resonance signal. The second effect is the “size effect.”’ This effect is significant if the foreign ion cores in metals or foreign ions in ionic crystals have sizes that are appreciably different from that of the original ion cores or ions in the sample. A distortion of the electron shells of the ions containing the resonant nuclei then occurs, leading to changes in the field gradients at the nuclei. Such an effect has been found (Bloembergen”!) in metallic aluminum with zinc or magnesium impurity. In molecular crystals, both the charge and size effects will be present. Charged impurities may be introduced by x-ray or y-ray irradiation when electrons and charged molecular ions (free radicals) are produced. These will give rise to the “charge effect”’ by producing distributions in the field gradients at the nuclei, similar to the case of metals and ionic crystals discussed in the preceding paragraph. But

in addition, the electrons and free radicals can lead to a broadening due to their magnetic interaction with the nuclear magnetic dipole moments. The broadening due to the magnetic and charge effects would be comparable for nuclei with smaller quadrupole moments

such as B11, N!4, or Cl*> nuclei. For Br7*, I1?7, As75, and other heavier nuclei, the charge effect will predominate. The “size effect’? can occur in molecular crystals if a neutral impurity molecule is introduced. An impurity molecule produces a change in the field gradient at the nuclei of a neighboring molecule by

the two processes, direct and indirect, on the intermolecular contributions to the field gradient (see Section 8). The direct effect would be small for a neutral impurity molecule. The indirect effect comes about because of the change produced in the bonds within neighboring molecules by the presence of the impurity molecule. The change in the field gradient at the nucleus due to this indirect effect of a neighboring impurity molecule is of the same order of magnitude as the inter-

10.

BROADENING

OF

LINES

BY

IMPURITIES

207

molecular contributions to the field gradients in pure solids. These (see Sections 6 and 8) vary from fractions of megacycles to a few megacycles in various cases. It is clear therefore that a large percentage of impurity molecules can smear the resonance beyond detection. The broadening effect due to this so-called ‘size effect’? would depend not only upon the size of the impurity molecules, but upon their chemical constitution as well, since the intermolecular forces between an impurity molecule and a neighboring molecule depend on the dipole and quadrupole moments of the electron distribution in both the molecules.?** Thus, two impurity molecules of the same size may produce widely different broadening effects if their electric dipole and quadrupole moments are widely different. The term ‘‘size effect’’ is therefore not very adequate. We shall, however, continue to use it in the subsequent sections as the. broadening effect due to neutral impurities. In the next two sections we shall consider the results of the few systematic studies on the charge and size effects that have been made by various authors.

b. Quadrupole Resonance in Irradiated Specimens Duchesne, Monfils, and Garsou?*® irradiated paradichlorobenzene with Co® y rays at 20°C and found thirty-seven per cent decrease in intensity of the Cl®* resonance for irradiations of 2.7 X 108 roentgens. The intensity recovered its normal value after a few hours. The broadening and consequent decrease in the intensity of the resonance is probably due to the production of electrons and free radicals by the rays. However, Duchesne et al., suggest the possibility of production of new unstable molecular species by the irradiation, which could also broaden the Cl*> resonance by the size effect. In addition, any mechanical damage produced by the irradiation would lead to strains in the crystal lattice and a consequent broadening of the resonance. However, this last broadening mechanism was not significant as shown by the recovery of the signal intensity to its original value on passage of time. A study of the paramagnetic resonance in the free radicals produced by the irradiation, and their broadening effect on the proton resonance in the paradichlorobenzene molecules would be interesting. Such measurements could give an indication of the density of the free radicals, and permit an estimation of the importance of this broadening influence on the Cl* resonance relative to the size effect due to unstable molecular species proposed by Duchesne, Monfils, and Garsou. 265 See Chapter 18, sec. 18a in ref. 11. 266 J. Duchesne, M. Monfils and J. Garsou, J. Chem. Phys. 28, 1969 (1955).

208

III.

APPLICATIONS

c. Quadrupole Resonance in Solid Solutions The early observations on broadening of nuclear quadrupole resonance due to impurities were made by Dean!*! and by Duchesne and Monfils.44 They found that the Cl* resonance in p-dichlorobenzene is broadened by the presence of small amounts of p-dibromobenzene in solid solution. Dean* later made a systematic study of the broadening of Cl®> resonance in various solid solutions of p-Cle¢, p-Bro¢, and p-BrCl¢. In pure p-BrCl¢ he could not detect a Cl®> resonance, but he could detect weak Cl** resonances in solid solutions of p-BrCl¢ in p-Brod. The sharpest Cl* signal was obtained in a solution containing two molar per cent of p-BrCl¢ in p-Br2¢. Below this concentration the Cl** signal was too weak to detect because of the small number of Cl nuclei. The resonance progressively increased in breadth as the p-chlorobromobenzene concentration was increased. Dean also found that the Cl*> resonance line widths in solid solutions of either p-BrClé

or p-Cl2¢ in p-Bro¢ were equal if the relative concentrations of chlorine and bromine atoms were the same in both cases. These observations by Duchesne and Monfils and Dean are satisfactorily explained by the size effects at the Cl** nuclei due to neighboring bromine atoms. The crystal structures of p-Cle¢ and p-Bred were discussed in Section 6c. The crystal!-?67 structures of p-BrCl¢ and

solid solutions of p-Cl.¢ and p-Bre¢ have been shown by x-ray studies to resemble the crystal structures of p-Cle¢ and p-Bre¢. The chlorine and bromine atoms are distributed randomly over the halogen positions in the unit cell. Therefore, in pure p-BrCl¢ there is equal probability that the neighbor of a Cl** nucleus will be either a chlorine or bromine atom. So the size effect smears the Cl*® resonance beyond

detection. For solutions of p-BrCl¢ in p-Bra¢, the environments of Cl* nuclei

contain

more

bromine

than

chlorine

atoms.

Therefore,

the

breadth of Cl*® resonance decreases as the p-BrCl¢ concentration is decreased and the resonance becomes detectable. Dean’s observation of the equality of the breadth of the Cl** resonance in solid solutions of either p-BrCl¢ or p-Clo¢ in p-Bre¢ for the same bromine-chlorine atomic concentrations is also understandable. Because of the similarity in the crystal structures of the three compounds, the randomness of the environments of Cl** nuclei depends only on the bromine-chlorine concentration ratio. Segel and Lutz?®8 observed that the intensity of Cl®> resonance in p-Cle¢ is reduced 85% upon the addition of 2.5% of p-Br2¢. The same 267 A. Klug, Nature 160, 570 (1947). 268 S. L. Segel and B. C. Lutz, Phys. Rev. 98, 1183 (1955).

10.

BROADENING

OF LINES

BY

IMPURITIES

209

concentration of p-I.¢ caused a reduction in intensity to 20% of the original value. The p-I2¢ molecule is larger than that of p-Br2¢, and would thus produce a larger change in the field gradient at the Cl* nuclei in neighboring p-Clze¢@ molecules. The crystal structure of p-In¢ is different from p-Clo¢. Therefore the p-I2¢ impurities are much more segregated than p-Bro¢, since the bromine atoms in the latter can replace the chlorine atoms at random. Thus the environment of the Cl** nuclei is less random for p-I2¢ impurity than for p-Bre¢ impurity. This would explain the smaller impurity broadening in the case of p-I2¢ impurity observed by Segel and Lutz. These authors pointed out that as the impurity concentration is increased the Cl*> resonance broadens, but the total integrated area under the resonance curve remains constant. This observation is contradicted by the results of

Kraus,

Michel,

and

Tantilla?®®

on

the

Cl®>

resonance

in various

chlorine compounds. These consisted of the chloromethanes, chlorobenzene, and t-butyl-chloride containing a variety of non-chlorinated

impurities, such as benzene, hexane, and methyl alcohol. A decrease of the area under the resonance curve with increase in impurity concentration is in accord with general expectations, since this area depends on the total number of Cl**> nuclei in the sample. 269Q, Kraus, (1956).

R. E. Michel, and W.

H. Tanttila, Bull. Am. Phys. Soc. [2] 1, 215

Author

Index

The numbers in parentheses are footnote numbers and are inserted to enable the reader to locate a cross reference when the author’s name does not appear at the point of reference in the text.

A

Bray, P. J., 32, 45, 98, 100, 155, 159 (121), 160, 161, 162, 163, 167, 170, 173, 186, 189, 195, 196, 197, 200, 203 (54), 204 Brode, H., 190 Brody, S. B., 1, 4(2), 5(2), 70(2)

Abe, Y., 28, 29, 91(25), 93, 170, 171(223), 203 (223) Abragam, A., 36, 37

Allen, H., Jr., 31, 39, 45, 98, 114 Allen, P. W., 101, 148, 175(123)

Brown,

L. C., 184,

B Barnes, R. G., 45, 100, 122, 131, 153 (152), 155, 159(121), 160(121),

Cc

161, 162, 163(121), 170, 173, 190, 191, 192(246), 195, 196, 197, 200,

Campbell, A. N., 115 Casimir, H. B. G., 3, 122, 123 Chang, C. E., 64 Cheesman, G. H., 165 Cohen, M. H., 1, 3, 5(1), 18, 14, 15,

203 (54), 204 Bastiansen, O., 163 Bayer, H:, 40, 41 Beck, K., 115 Beers, Y., 70 Benedek, G. B., 39, 119 Bersohn, R., 5, 7(6), 13, 17, 19, 100,

16, 19, 20, 25, 41, 62, 109, 114, 129,

154, 179, 180, 205 Collin, R. L., 174 Coolidge, A. S., 121 Cotts, R. M., 114 Coulson, C., 32, 132, 133(168), 142,

141, 155, 158, 159(121), 160, 161,

162, 163(121), 165, 174, 183, 189 Bezzi, S., 38, 45, 104, 114(41), 117

143, 151, 155, 156, 157, 158, 164,

(41), 118, 119 Bird, G. R., 142 Biscoe, B. E., 188 Blake, F. C., 103, Bleaney, B., 180, Blinstoyle, R. J., Bloch, F., 59, 74, Bloembergen, N.,

187

Bua, E., 38, 45, 104, 114(41), 117(41), 119(41) Buck, P., 126 Burrell, J. T., 205 Buyle-Bodin, M., 40, 68

Andrew, E. R., 70, 85, 88(105) Ayant, Y., 38

169 Cranna, N. G., 181, 187

Croatto, U., 38, 45, 104, 114, 117, 118,

174 183 120 76, 83(96) 27, 30, 31, 39, 50,

119 Cross, P. C., 40

D

54, 58, 59, 62, 63, 74(65), 84(65),

Dailey, B. P., 81, 98(99), 100, 121,

131, 136, 137, 141, 142, 144, 155,

119, 205, 206 Bloom, M., 70, 71, 73(89), 75(89), 77 (89), 82(89), 83(89), 93(89) Bowers, K. D., 180, 183 Bozorth, R. M., 104 Bragg, J. K., 141, 154

169, 173(99), 176, 177 Daly, R. T., Jr., 127 Das, T. P., 71, 73(94), 129, 136, 137 (171), 165, 183, 188, 189 Davis, L., 123 211

212

AUTHOR

Dean, C., 4, 5(5), 13, 15, 19, 20, 21, 39, 40, 41, 43, 45, 47, 55, 68, 74, 83(5), 89, 104, 114, 116, 119, 208 Decius, J. C., 40 Dehmelt, H. G., 1, 30, 53, 88, 98, 126, 158, 167, 168(118), 173(118), 174, 178(118), 184, 186, 190, 191, 196, 197, 200, 204 DeLaunay, J., 64 Dennison, D. M., 40, 68 de V. Roberts, M., 158 Dodgen, H. W., 40, 67 Douglass, D. C., 29(31), 30, 33, 39 (81), 47 Duchesne, J., 39, 45, 48, 49, 155, 160, 163, 170, 207, 208

Eades, R. G., 187 Ebbinghaus, K., 115 Eck, T. G., 127 Elliott, N., 190 Eyring, H., 13, 14(11), 48, 123, 133, 136, 207

Fermi, E., 30 Foley, H. M., 129, 130, 136 Frenkel, J., 31 Frueh, A. J., Jr., 104, 200 Fuke, T., 38

G Garrett, B. S., 32, 104 Garsou, J., 48, 207 Geller, 8., 198 Geschwind, §., 17, 45, 205

Goldstein, J. H., 141, 154, 155, 160 Gordy, W., 98, 136, 142, 168(118), 173(118), 178(118), 184, 197 Gunther-Mohr, G. R., 17, 45 Gupta, J., 115, 116, 118

Gutowsky, H. 8., 39, 64, 65, 68 H Haering, R. R., 104, 186 Hagiwara, S., 28, 29, 91(25)

INDEX Hahn, E. L., 70, 71(89), 73(89), 75 (89), 77(89), 82(89, 92), 83(89), 182 Hamilton, J. E., 103 Hannay, N. B., 144 Hansen, W., 74, 83(96) Harris, P. M., 103, 174 Hassel, O., 163 Heavens, O. S., 165 Helmholtz, L., 103 Hendricks, 8. B., 104, 105, 115, 208 (139) Herzberg, G., 48 Herzog, B., 70, 71(89), 73(89), 75(89), 77 (89), 82(89), 83(89) Hiebert, G. L., 31 Hinaga, Y., 108, 105, 119, 173, 174 Hirahara, E., 181 Holcomb, D., 96 Holloway, J. H., 127 Hornig, D. F., 31

Hiickel, W., 101, 157

Huggins, M. L., 148, 148, 165(188), 167 (188)

Imaeda, Y., 103 Ishiguro, E., 121

Itoh, J., 181, 182 Tutsis, A. P., 125

J Jaccarino, V., 127, 128 James, H. M., 121 Jeffrey, G. A., 104, 114, 118, 119

K Kambe, K., 36, 37 Kaplan, D., 182 Kartha, G., 81, 103 Ketelaar, J. A. A., 191 Kimball, G. E., 18, 14(11), 48, 123, 133, 136, 207 King, J. G., 127, 128 Kittel, C., 185 Klug, A., 208 Knight, W. D., 84, 114, 184 Koehler, J. S., 40, 68 Koi, Y., 38, 103

AUTHOR

Kojima, §., 27, 28, 29, 91, 108, 105,

119, 170, 171, 173, 174, 179,

203 (223) Kondo, §., 101 Koster, G., 124 Kraus, O., 209 Krénig, R., 61 Kriiger, H., 1, 30, 53, 104, 182, 183, 195, 196, 200 Kusaka, R., 181, 182 Kusch, P., 127 Kushida, T., 39, 103, 119

L Lamarche, G., 16, 25 Lassettre, E. N., 40, 68 Lew, H., 125, 127 Lindstrand, E., 119 Lipscomb, W. N., 192 Livingston, R., 5, 14, 15, 26, 29(31), 30, 32, 34, 39(31), 45, 47, 81, 85, 88, 101, 102, 103, 106, 107, 146, 148, 165, 167, 174, 203(7), 204 Lloyd, J. P., 59 Longuet-Higgins, H. C., 157, 158 Ludwig, G. W., 187, 190, 191 Lutz, B. C., 47, 104, 114, 118, 208

M McConnell, H., 27, 28, 30(23) McGilvary, C. H., 190, 191 Mack, E., 103, 174

MeVeagh, W. J., 104, 114, 118, 119 Manring, E. R., 103 Meal, H. C . 31, 39(32), 102, 105, 111, 113. 160 Merritt, F. R., 131 Meyer-Berkhout, U., 104, 182, 188, 195 Michel, R. E., 209 Mizushima, 8., 40 Monfils, A., 39, 48, 155, 160, 163, 207, 208 Mulliken, R. S., 121, 136 Murakami, M., 181

N Newell, G. F., 121 Nierenberg, W., 1, 4(2), 5(2), 70

213

INDEX

Nitta, I., 101 Norberg, R. E., 70, 96 Nordsieck, A., 121

O° Ogawa, S., 170, 171(223), 179, 208 (223) O’ Keefe, C., 195 Orloff, D., 136 Orloff, H., 136 P

Packard, M., 74, 83(96) Pake, G. E., 6, 26, 27, 28, 30(23), 33 (20), 41, 59, 60, 62, 64, 65, 68, 71, 123(9) Palmer, K. J., 190 Parker, P. M., 25

Pauling, L., 101, 111, 132, 143, 144,

147, 154, 157, 169, 175 Perl, M., 126 Petch, H. E., 181, 187 Portis, A. M., 59 Portoles, L., 104 Pound, R. V., 1, 25, 39(17), 50, 54, 58, 59, 62, 74(65), 84, 104, 114, 116, 181, 186, 194, 208(131) Powles, J. G., 64 Proctor, W. G., 64, 70, 77(90), 82(90,

91), 84

Prodan, L. A., 115 Pryce, M. H. L., 180, 183 Purcell, E. M., 50, 54, 58, 59, 62, 74 (65), 84(65), 88

R Rabi, I. I., 126 Ragle, J. L., 40, 67 Ramsey,

N. F., 1, 4(2), 5(2), 6, 27,

28, 30(23), 45, Redfield, A. G., 59 Reif, F., 1, 3, 5(1), 114, 129, 154, Renes, P, A., 190, Rieke, C. A., 136 Riva, A., 47 Roberts, A., 84

70, 120, 121 25, 41, 58, 62, 88, 179, 180, 205 191

214

AUTHOR

Robinson, H. G., 30, 98, 167, 168, 173 (118), 178, 184, 196, 197 Robinson, W. A., 70, 82(91) Rogers, M. T., 103 Rollier, M. A., 47 Rowland, T. J., 27, 30, 31

INDEX

Trambarulo, R. F., 142 Tsiunaitis, G. K., 125 Tsujimura, A., 38 Tsukada, K., 27, 81, 103, 105, 119, 170,

171(223), 173, 174, 179, 203(223) Vv

S Saha, A. K., 71, 73(94)

Saksena, B. D., 41, 45(52), 115, 117, 119

Van Vleck, J. H., 36, 62 Volkoff, G. M., 16, 25, 104, 181, 186, 187 Vonnegut, B., 103, 174

Vuks, M., 115, 116

Satten, R. H., 128

Schatz, P., 136, 137(171) W

Schawlow, A. L., 17, 30, 91, 97, 100,

120, 121, 142, 147, 166, 179(218),

198, 204 Schiff, L. I., 8, 50, 56 Schmidt, Th., 185 Schomaker, V., 147, 190 Schiiler, H., 185 Segel, S. L., 190, 191, 192(246), 208 Senitzky, B., 126

Shimauchi, A., 28, 29, 91(25), 103, 170, 171(223), 173, 174, 179, 203 (223) Sidgwick, N. V., 202 Silver, A. H., 186, 189 Sirkar, S. C., 115, 116, 118 Slater, J. C., 97

Smith, W. V., 122, 131, 142, 153(152) Smyth, C. P., 144 Solomon, I., 76 Sternheimer, R. M., 125, 129, 130, 136 Sterzer, F., 70 Stevens, K. W. H., 98, 173(119), 178 Stevenson, D. P., 147, 190 Stroke, H. H., 128

Walker, R. M., 27

Walter, J., 13, 14(11), 43, 123, 133, 136, 207 Wang, T. C., 17, 29(31), 30, 38, 39

(31), 41, 45, 83, 85, 86, 88, 103,

195, 196 Wangsness, R. K., 76 Warren, B. E., 103, 174, 205 Warren, J., 188 Waterman, H. H., 186

Watkins, G. D., 25, 27, 39(17), 194 Weisskopf, V., 61 Wells, A. F., 170, 180, 183, 185, 188,

189, 190, 198, 201, 202

Wessel, G., 125, 127 Wigner,

E., 61

Williams, D., 103, 184, 187 Williams, G. A., 39 Wilson, E. B., 40, 157 Wright, B. D., 131 Wyckoff, R. W. G., 169, 188

Sutton, L. E., 101, 148, 175(123)

Y

T Yamagata, Tanttila, (90), Ting, Y., Torizuka, Townes,

W. H., 64, 70, 77(90), 82 209 103 K., 165 C. H., 17, 30, 45, 81, 97, 98

(99), 100, 120, 121, 131, 136, 137, 141, 142, 144, 155, 166, 169, 173 (99), 176, 177, 179(218), 205

Y., 182 Z

Zabel, C. W., 123 Zacharias, J. R., 123 Zachariasen, W. H., 103 Zeldes, H., 14, 15, 26, 32, 34, 81, 102, 103, 106, 107, 165, 167

Subject Index A separate Index

to Chemical

Compounds

A

will be found

at the end of this index.

Acid chlorides, 99

Arsenic trioxide, 104, 195, 200 Arsenic triphenyl, 196, 204 Arsenic-antimony oxide, 199

Aluminum,

Arsenolite,

Al*++ in spodumene, euclase and beryl,

188

configuration in, 125 impurity broadening in, 206 Aluminum bromide, 190 Aluminum chloride, 190 structure of, 191

Aluminum iodide, 190 Aluminum

oxide,

189

quadrupole coupling of Al*+t++ in, 189 structure of, 189

Aluminum Aluminum

trihalides, structure of, 191 trioxide, 186

Ammonia,

quadrupole

coupling con-

stant in, 153 Antimony compounds, quadrupole resonance of Sb!2! in, 201 Antimony isopropoxide, 197 Antimony sulfide, 201 Antimony tribromide, 195, 202 Antimony trichloride, 195 Antimony triiodide, 170 Antimony triphenyl, 196 Antimony trisulfide, 195 Anti-shielding in cuprous oxide, 183 Arsenic compounds, quadrupole resonance of As’5, 199 in halides, 171 Arsenic oxide, 200 Arsenic tribromide,

195, 202

Br’ resonance in, 171 Arsenic trichloride, 195, 202 Arsenic trifluoride, 142 Arsenic triiodide, 195, 202, 203 bond switching in, 170 I?? quadrupole resonance in, 171 structure of, 171

104,

195

Arsine, bond character in, 142 Asymmetric top rotational spectra, similarity 7 = 1 in non-axial case, 17, 18 Asymmetry parameter 7, in antimony compounds, 202 appearance of due to deviation from 3-fold symmetry, 148 definition of, 4 effect of in torsional motions, 43 evaluation of in Townes and Dailey theory, 140 measurement by echo envelope, 81 of substituted organic chlorides, 159 Atomic wave functions for different hybrids, 151 Axially symmetric electric fields, pure quadrupole

spectra in, 5

resonance intensities in zero magnetic field, 7 transition frequencies, 6

Barium chlorate, 103, 108, 165 Bayer theory of quadrupole resonance shift, 41 Beryl, 184, 187, 188 Beryllium metal, 184 quadrupole moment and field gradient in, 185 Bismuth compounds, quadrupole resonance of Bi? in, 201 Bismuth oxide, 201 Bismuth

sulfide, 201

Bismuth triiodide, asymmetry parameter for I!27 in, 173

216

SUBJECT

INDEX quadrupole resonance frequency in,

bond switching in, 170 Bismuth triphenyl, 197 Bond switching, 170 Bond vibrations, effect of on quadrupole resonance, 45 Boron tribromide, 48 Boron trichloride, 31, 48, 196 internal vibrations in, 47 J interaction in, 29 Boron triethyl, 186 Boron trimethyl, 186 Box car integrator, 96 Broadening, due to the earth’s magnetic field, 38 due to impurities, 39 due to strains and dislocations, 39 of quadrupole resonance lines by impurities, 205 of quadrupole spectra, calculation of Abragam and Kambe, 36 of spectra due to magnetic interactions, 26 Bromine chloride, 145 Bromine cyanide, 194 N"™ quadrupole resonance in, 198 Bromine, solid, 103

for quadrupole echoes and free induction, 93 Claudetite, 104, 195, 200 Coaxial cavity spectrometer, 92 Configuration interaction, 124 Continuous wave feedback oscillator, principle of, 85 Continuous wave quadrupole resonance detectors, 84 Correlation functions for torsional motion, 50 Coulson and Longuet-Higgins procedure, 157 Crossed coil experiment in spin echoes, 79 in steady state experiments, 83 Cuprous oxide, 40, 182 anti-shielding in, 183 structure of, 180

Cc

D

Carbon tetrabromide, 100 Carbon tetrachloride, 100 Charge effect in impurity broadening, 206 Chemical bonds, ionic characters of, 166

Chloral alcoholate, 101 quadrupole spectrum of, 170 Chloral hydrate, 101, 102 quadrupole spectrum of, 170 Chlorine*,

quadrupole coupling in HgClz, 185 quadrupole lines, multiplicity of, in crystals, 100 Chlorine quadrupole resonance frequencies in chlorobenzenes, 160 Chloroacetyl chloride, 99 Chlorobenzenes, 161 double bond character in, 155

160

1-Chloro-4-chloromethy] benzene, 159 Chloromethanes, quadrupole coupling constants in, 148, 149 Circuits, for high frequency Br and I cw resonances,

91

d Hybridization, 143 Damped oscillator treatment of resonance breadth, 61 Degenerate perturbation theory, Zeeman splitting for half integral spins, 8 Deuterated methyl bromide, 48 Deuterated methyl iodide, 48 Deuterium chloride, 31 Dichloracetyl chloride, 99 Double bond character, Bersohn’s molecular orbital method, 156

in C—Cl bonds, 155 definition of, 138 in halogen-substituted methanes, 150 rules for assigning, 141 Double resonance experiments in quadrupole

systems,

82, 182

SUBJECT

Echo envelope measurement, 77 Echo modulations due to weak Zeeman field, 75 Echo signal “box car”’ integrator, 96 Eigenvalues, for axial case, 5

for J = 34, non-axial case, 13 for I = 54, non-axial case, 13 Zeeman splitting in axial case, 7 Zeeman splitting in non-axial case, 19 in presence of magnetic dipole-dipole interaction, 34 Electric field gradient, definition in Townes and Dailey theory, 133 tensor, 4

in various groups of periodic table, 130 Electric shielding, Sternheimer factor of, 125 Electronegativity, definition of, 143 Elements, compounds of, Group I, 180 Group II, quadrupole coupling in, 183 Group III, quadrupole coupling in, 185 Group V, 192 Ethylene, bond character in, 142 Euclase, 187, 188 Eulerian angles, definition, 5 Experimental apparatus, general requirements of, 83 F

Feedback oscillator detector of Wang, 87 Fluochloromethane, 36 1,2-Fluoroethylene chloride, 159 Free nuclear induction decay, 74 Frequency shift due to molecular torsional motions, 41

G Gallium, configuration in, 125 Gallium oxide, 189

217

INDEX

Gallium trichloride, 190 Gallium triiodide, 190 Germanium tetrabromide, 100 Germanium tetrachloride, 100

H Half integral spins, pure quadrupole with axially symmetric fields, 5 pure quadrupole spectra with nonaxial field gradients, 12 Zeeman splitting in axially symmetric fields, 8 Zeeman splitting in non-axial case, 19

Halogen molecules, structure in solids, 175 Halogen quadrupole coupling constants, Group IV tetrahalides, 147 Halogens, calculation of quadrupole constants in solids, 176 quadrupole spectra in solids of, 173 Hamiltonian, for magnetic dipole-dipole interaction, 33 pure quadrupole, 3 radiofrequency part, 6 Hexachlorobenzene, 161 Hexamethylene tetramine, 194 quadrupole spectrum of N!4 in, 193 structure of, 199

Hg?°!, quadrupole coupling of in HgClz, 185 Hindered internal rotations, effect on pure quadrupole resonance, 64 Hiickel approximation, 157 Hybrid character of chemical bonds, comparison of Gordy and Townes and Dailey theories, 146 rules for assigning, 141 Hybrid states, atomic wave functions for, 152 Hybridization, definition of, 133 Hydrochloric acid, broadening of Cl resonance in, 39 magnetic dipole-dipole splitting effects in, 31 Hyperfine structure, effect on level spacing,

58

218

SUBJECT

use of to determine quadrupole coupling constants, 180

INDEX

description and curves for, 144, 145 difference in gaseous and solid states, 166

Impurities, broadening by, 205 Indirect spin-spin interaction (J interaction), 27 Indium, configuration in, 125 Indium oxide, 189 Indium triodide, 187 In''5 resonance in, 190 Inequivalence, chemical and physical, definition, 22 Infra-red and Raman spectra, 40 Integral spins, pure quadrupole with axially symmetric fields, 5 pure quadrupole non-axial case, 16 Zeeman splitting in axially symmetric electric fields, 12 (J = 1), Zeeman splitting in nonaxial case, 23 Intermediate fields, Zeeman coupling comparable with electric coupling, 25 Intermolecular binding, 164 Iodic acid, 32, 103, 108 magnetic dipole-dipole interaction in, 26 structure of, 109

Zeeman separations of, 110 Iodine bromide, 179 quadrupole interaction in, 165 Iodine chloride, 27, 145, 167, 179 bond character in, 142 quadrupole coupling in solid and gas compared, 166 Iodine cyanide, 167, 194 I'27 quadrupole coupling constant in,

rules for assigning, 141 Irradiated specimens, quadrupole resonance in, 207 Irradiation, effect on Cl3> resonance in parachlorobenzene, 207

J J interaction, effect of in echo measurements,

80

in HCl, 31 indirect spin-spin interaction, 27 splittings due to, 28

K Kekule structure, 156 Kernite, 104, 186 structure of, 188

L Lecher line super-regenerative detector, 92 Line shape, due to magnetic

dipolar interaction,

61 due to torsional vibrations, 61 variation with temperature, 63 Lithium carbonate, 181 Lithium sulfate, 181 Locus of constant splitting, 10 Locus of no splitting,

Zeeman

splitting

for non-axial case, half integral spins, 21

M

169

N!4 resonance in, 193, 198 Iodine molecule, magnetic dipole-dipole splitting effects, 31 Iodine, solid, 103 Ionic character of chemical bonds, comparison of Gordy and Townes and Dailey theories, 146

Madelung forces, 165, 166 Magnetic dipole-dipole interactions, broadening effect, 35 in halobenzene compounds, 32 Hamiltonian, 33 in iodic acid, 26 splitting effects, 30, 58

SUBJECT

Magnetic interactions, splitting, broadening, 26 Mercuric chloride, 184 quadrupole coupling in for Cl#> and

Hg2"!, 185

structure and quadrupole resonance spectra in for Cl®> and Hg?"!, 185 Meta-chlorobenzene, 161 Methane, bond character in, 142 Methyl bromide, 48, 167 Methyl chloride, 167 quadrupole coupling in solid and gas compared, 168 Methyl iodide, 48, 167 Molecular crystals, charge and size effects in due to impurities, 206 Molecular orbital, definition of, 132

N Narrow banding quadrupole resonance detection, 88 Nitrogen, N14, quadrupole resonance in BrCN, 198 quadrupole resonance in hexamethylene tetramine, 193 quadrupole resonance in ICN, 193 Non-axial case, pure quadrupole resonance intensities, 15 Zeeman splitting in, for half integral -spins, 19 Non-axial field gradients, pure quadrupole spectra, 12 Nuclear quadrupole moment, definition, 3

O° Orbitals, ¢ and 7 in the C—Cl chlorobenzenes,

bond of

163

Organic chlorides, substituted, asymmetry parameter

7 in, 159

Orthochlorobenzene, 161 Orthodiiodobenzene, 164 Orthohalobenzaldehyde, 32 Orthohalobenzoic acid, 32 Oxaly] chloride, 99

219

INDEX

Pp p Electron excess, definition of, 139 Parabenzoylchloride, 105 Parabromochlorobenzene, 208 Parachloroaniline,

105,

111,

159,

160

crystalline and field gradient axes, 112 Paradibromobenzene,

105, 208

Paradichlorobenzene, 38, 40, 47, 104, 159, 161, 207, 208 effect of irradiation on Cl resonance, 207 phase transitions in, 114, 117 torsional motions in, 41, 42 Zeeman

studies in, 117

Paradiiodobenzene, 164 Paramagnetic impurities, effect of on thermal relaxation, 63 Pentachlorobenzene, 161 Periodic table, electric field gradient as function of, 130 Phase transitions, study of, 113 Phosgene, 159 Pi (x) electrons,

conjugated systems, definition of, 134

157

Planar molecules, asymmetry parameter

and double bond character in, 154 Potassium boron tetrafluoride, structure of, 188

Potassium chlorate, 40, 103, 108, 165, 182

Potassium cuprous cyanide, 182, 183 Potassium fluoborate, 186 Potassium niobate, phase transitions in, 114 Potassium silver cyanide, 183 Potential barrier for hindered rotation, case of transdichloroethane, 65 Potential barrier heights, in transdichloroethane, trichloroacetic acid, 69 Principle

axis system,

4

for C185 in paradichlorobenzene, 42 Pulsed oscillator circuit for 30 me quadrupole

resonance,

94

for 150 me quadrupole resonance, 95 Pure quadrupole interaction for nonaxial cases, secular equations,

13

220

SUBJECT

INDEX

R

Q Quadrupole coupling constants, in atoms, 122, 126 of Ga® and In!45, 190 in gaseous molecules, 45 obtained from optical fine structure data, 131 in substituted chloromethanes, 148, 149

in terms of electron distribution, 119

Townes and Dailey theory of, 120 Quadrupole Hamiltonian, pure, 3 Quadrupole interaction, in molecules, Townes and Dailey theory of, 131 tensor determination,

summary,

22

Quadrupole moment tensor, 3 Quadrupole resonance frequencies, of arsenic, antimony, and bismuth compounds, 199 broadening due to torsional motions, 60 in chlorobenzenes, 160 in irradiated specimens, 207 of nuclei other than halogens, 179 pure, for axially symmetric field, half integral and integral spins, 5 with non-axial field gradients, half integral spins, 12 in solid halogens, 173 in solids, 208 splitting due to chemically inequivalent sites, 98 to vibrational motions, 47 temperature dependence of, 44 Zeeman splitting of in axial case, 7 non-axial case,

19

Quadrupole resonance, pure, in alkali halide vapors, 70 effect of hindered internal rotations, 64

effects of pressure and temperature on, 40

impossibility in liquids, 69 intensities, non-axial case, 15 non-axial case, integral spins, 16

Radiofrequency time dependent Hamiltonian, 6 Raman processes for thermal relaxa-

tion, 64

Relaxation times resulting from torsional motions, 56 Resonance

intensities,

for axially symmetric electric fields in small Zeeman field, 9, 10, 11 in zero magnetic field, 7 for non-axial fields in small Zeeman field, 19 in zero magnetic field, 12 Resonance saturation method, assumptions of, 59 Rotation transformation matrix due to rf pulsing, pure quadrupole spin echoes, 73 quadrupole spins in weak magnetic field, 78

S Second moment, definition, 35 due to dipole-dipole broadening, 36 Secular equations, pure quadrupole interaction for non-axial

cases,

13

Self-quenched oscillator method, 89, 90 Senarmontite, 195 Sigma (c) bond, definition of, 132 Signal-to-noise ratio in super-regenerative detectors, 91 Silicon tetrabromide, 100 Silicon tetrachloride, 100 Size effect in impurity broadening, 206 Sodium boron tetrahydride, quadrupole resonance of B!! in, 188 Sodium bromate, 40, 64, 103, 108 spin echo signals in, 75 Sodium chlorate, 7, 40, 64, 102, 103,

165, 181

Cl coupling constant in, 154 crystal structure, 105 signal pattern for Cl#* in, 107 spin echoes in, 75 theoretical plot of Zeeman separations, 106

SUBJECT

Sodium nitrate, 181 Sodium thiosulfate, 182 Solid solutions, quadrupole resonance in, 208 Spin 1, case of non-axial asymmetry, pure quadrupole, 17 Zeeman splittting, 23 Spin echoes, feasibility of crossed coil technique, 79 free induction signals in weak magnetic field, 77 in pure quadrupole resonance, 71 Spin-lattice relaxation, definition, 59 direct and indirect processes, 64 effect of paramagnetic impurities, 63 Van Krannendonk theory of, 64 Spin-lattice relaxation times in substituted chlorobenzenes, 55 Spin-relaxation measurements by echo method, 81 Spin-spin relaxation time, definition of, 60 Splitting of spectra due to magnetic interactions, 26 Spodumene, 181, 187, 188 Stannic tetrabromide, 100 Stannic tetrachloride, 100 Stannic tetraiodide, 100 Sternheimer electric shielding factor, 125

Sulfides of bismuth and antimony, structure of, 201 Sulfur, quadrupole resonance of S*3 in rhombic state, 204 rhomic structure of, 205 Super-regenerative, circuit, 90 Lecher line type, 92 method of detection, 88

T Temperature dependence of quadrupole resonance in transdichloroethane, effect of rotation, 67 Temperature dependence of quadrupole resonance frequencies, 44

221

INDEX

Temperature-dependent averaged field gradient, 40 1,2,3,4-Tetrachlorobenzene, 161 1,2,3,5-Tetrachlorobenzene, 161 1,2,4,5-Tetrachlorobenzene, 161 Tetrahalides, C, Si, Ge, Sn, 148 Tetramethyl boron, field gradient in, 152 Thermal relaxation, direct and indirect processes in, 64 Torsional motions of molecules, effect of, 40 relaxation times due to, 56 transition probabilities due to, 49 transitions induced by, 58 Townes and Dailey, and Gordy curves for ionic character, 145 Townes and Dailey theory, for interactions in molecules, 131 for quadrupole coupling constants, 120 Transdichloroethane, 66, 67 Transdichloroethylene, 53, 54 Transient experiments, advantages and disadvantages of, 80 in quadrupole resonance, 70 Transition frequencies, in axially symmetric electric fields, 5, 6 Transition probabilities due to molecular torsional motions, 49 Transition rates due to torsional motions,

theoretical plots of, 54

Transitions induced by torsional motion, 58 1,3,5-Trichlorobenzene, 161 1,2,4-Trichlorobenzene, 161 1,2,3-Trichlorobenzene, 161 Trichloroacetic acid, 69, 101 Trichloroacetyl chloride, 99 Trifluorobromomethane, 167 Trifluorochloromethane, 167 Trifluoroiodomethane, 167 Trimethyl and triethyl boride, quadrupole coupling constant in, 188 Tutton salts, 180

U Ultrasonic irradiation experiments, 64 Unequally spaced levels, significance of, 59

222

SUBJECT

INDEX

Zeeman separations, for axially electric symmetric field in NaClO; (Cl Valentinite, 200 van der Waals forces, 165, 166 Van Krannendonk spin-lattice relaxation theory,

resonance), 106 Zeeman splitting, axial symmetry

64

integral spins,

Vinyl chloride, 159 double bond character in, 155

case,

general Hamiltonian, 7 half integral spins, 8 12

Zeeman splitting, non-axial case, half integral spins, 19 integral spins (I = 1), 23 Zeeman splitting, use in determining

Z

molecular and crystal orientations,

Zeeman lines in powders, 23 Zeeman modulations in spin echo experiments, 79 InpEx

AIBr3, 190 AICls, 190, 191 Alls, 190 Al.Os, 186, 189 AsBr;, 171, 195, 202 As(CH;)s, 204 As(CeHs)s, 196, 204 AsCls, 195, 202 AsF;, 142 AsH;, 142 AsIs, 170, 171, 195, 202, 203 As20s, 104, 195, 200 As,Os, 200 As,Oo(Sb,Os), 199

To

102 Zero point vibrations, effect of on quadrupole

CHEMICAL

48

CBr,, 100

CCl;,CH(OH)., 101, 102, 170 CCl;COCI, 99

CCl;COOH, 69, 101 CCl;CH(OH)(OC2Hs),

101, 170

CCl,, 100 CeClo, 161

B(CHs3)s3,

152,

186,

188

B(C2Hs)s, 186 BCl;, 29, 31, 47, 196 B'°Cl;, 48 B'UCl;, 48 Ba(ClO3)e-H20, 103, 108, 165 Be3Al2(SiOs).6, 184, 187, 188 Bi(CeHs)s, 197 Bil;, 170, 173 BieS;, 201 p-BrCeH,Cl, 208

167

CF;Cl,

167

CeHCl;,

161

1,2,3,4-CsH2Cla, 161 1,2,3,5-CsHeCl,, 161 1,2,4,5-CeH2Cls, 161 1,3,5-CsHsCls, 161 1,2,4-CsH;Cls, 161 1,2,3-CsHsCls, 161 p-CsH,Br2, 105, 208 m-Ce6H,Cle,

161

o-CsH,Clo, 46, 161 159

CFI, 167 CHCI=CHCI, 53, 54 CHC1.COCI, 99 CH.=CHCI, 155, 159

CH,CI—CH.Cl, 65, 66, 67 CH2CICOCI, CH.CIF, 36

CH;Cl, 167, 168 CH,I'2", 48, 167 CH,, 142 C2H,, 142

p-CsH,CICH,Cl, 159

CD;Br’®, 48 CD;Br*!, 48 CD17, 48 CF;Br,

43

CoMPOUNDS

BrCN, 194, 198 BrCl, 145

CFCI=CHz, BBrs,

resonance,

99

(CH,)sNu, 193, 194, 199 CH;Br”, 48, 167 CH;Br*!, 48, 167 CH,Cl, 167, 168

o-CsH,NO:;-Cl,

46

p-CsHsClz, 38, 40, 41, 42, 46, 47, 104, 114, 117, 159, 161, 207, 208

p-CsH,Cl-OH, 46 o-CeHale,

164

p-CoHaly,

164

p-CoHs(CH.Cl)Cl, 105 CcH,Cl, 155, 160, 161 COCICOCI, COCl:, 159

99

p-ClCsH,(NHz), 105, 111, 112, 159, 160 Cu.0, 40, 180, 182, 183

SUBJECT

ICI, 27, 142, 145, 166,

DCI, 31

167, 179

InIs, 187, 190 GaCl;, 190 Gal;, 190 Ga20;,

189

Ga2Oz, 189 GeBr,, 100 GeCl,, 100

In2Os,

223

INDEX

NaBrO;, 64, 75, 103, 108 NaClOs, 7, 40, 64, 75,

102, 103, 105, 106, 107, 154, 165, 181

189

NaNO,

KAg(CN)s, 183 KBF,, 186, 188 KCI0;, 40, 103, 108, 165, 182 KCu(CN)., 182, 183 KNbO;, 114

HBeAISiOs, 187, 188 HCl, 31, 39 HIOs, 26, 32, 103, 108, 109, 110 HgCl,, 184, 185

LiAl(SiOs)2, 181, 187, 188 LizCO;, 181 Li,SO,-H20, 181

IBr, 165, 179 ICN, 167, 169, 193, 194, 198

NH, 153 NaBH, 188 NasB,07-4H,0, 104, 186, 188

181

Na2S820;-5H.20,

Sb(C3H;0)s, 197 Sb20;, 195, 200 Sb.S;, 201 SbBrs, 195, 202 SbCls, 195 SbI;, 170 Sb(Ce6Hs)3, 196 Sb.S8s, 195, 201 SiBr,, 100 SiCl,, 100 SnBr,,

100

SnCl,, 100 SnIy,

100

182