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NBS MONOGRAPH 19

Atomic Energy Levels

In Crystals

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UNITED STATES DEPARTMENT OF COMMERCE • Luther H, Hodges, Secretary NATIONAL BUREAU OF STANDARDS • A. V. Astln, Director

Atomic Energy Levels in Crystals

John L. Prather

National Bureau of Standards Monograph 19

Issued February 24, 1961

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 2S, D.Ci - Price 60 cents

Preface

This present work had its inception in 1954, shortly after Professor G. H.

Dieke of The Johns Hopkins University, "with the support of the U.S. Atomic

Energy Commission, initiated a program of experimental studies of the sharp

line absorption spectra of crystals. The effort was directed principally to rare earth salt crystals, but some work was also done on the ruby and m'anyl

salts. Dr. Dieke suggested to the author that a theoretical study to accompany

the experimental program would be desirable and encouraged the undertaking

of such a study.

It rapidly became apparent on surveying the problem that the techniques

of group theory would be of central importance in such a study. Early in 1955,

Dr. Charles ]M. Herzfeld, then of the Naval Research Laboratory in Washington,

addressed the Spectroscopy Seminar at Hopkins on the techniques of group

theory with particular application to the paramagnetic resonance of salts of

the iron group. This was the beginning of a continuing association with Dr.

Herzfeld in the study of group theory, an association which has been a somce

of great personal and professional satisfaction to the author to the present time.

In August of 1955, through the courtesy of Dr. F. G. Brickwedde, then

Chief of the Heat and Power Division of the National Bureau of Standards,

and Dr. R. P. Hudson, Chief of the Cryogenic Physics Section of the Bureau,

the author was appointed a Guest Worker at the Bmeau, and office space was

provided and the extensive hbrary facilities of the Bureau made available.

This appointment continued when Dr. Herzfeld was appointed Chief of the

Heat Division.

The present work is largely an exhaustive collection of the results of group

theory which apply to the theory of atomic energy levels in crystals, together

with a careful, and detailed, but not especially abstract development of the

general theory to permit an intelligent application of the results to experimental

problems. Dr. Herzfeld, in collaboration with Professor Paid H. Meijer of

The Catholic University of America, has written a review on the general subject

"Group Theory and Crystal Field Theory", soon to be published, treating the

subject from a more abstract viewpoint of modern algebra and concentrating on the fundamental aspects of the theory. This latter work and the present one form a complementary pair of works on the general topic. It is hoped

that, together, they will meet the needs of most workers in the field.

Ill
Contents

Page

1. Introduction 1

2. The electrostatic field 3

2.1. Expansion in tesseral harmonics 3

2.2. Symmetry 6

2.3. Preferred coordinate systems 9

2.4. Cubic point groups 12

3. Groups, characters, and representations 14

3.1. Representations and quantum mechanics 14

3.2. Rotations in three dimensions -. 14

3.3. Group theory and the free atom or ion 18

3.4. Group theory and the perturbed ion 20

3.5. Group characters and the rep Dj 21

3.6. Noncubic crystallographic groups 24

3.7. Noncubic double groups 25

3.8. Reps and characters 27

3.9. Infinite and cubic groups 29

3.10. Table of group characters 30

4. Degeneracy, selection rules, and polarization 37

4.1. Splitting of free ion levels 37

4.2. Kramers degeneracy 50

4.3. Orthogonal states and vanishing matrix elements 52

4.4. Selection rules and polarization 53

5. Matrix elements 65

5.1. The reduced matrix element 65

5.2. Wigner coefiicients 66

5.3. L-S coupling and the one-electron approximation 72

5.4. Rare earth ground terms 73

6. Applications 74

6.1. General considerations 74

6.2. A crystal field 77

6.3. The fluorescence spectrum of the terbium ion 78

7. Bibliography 82

List of Tables

1. Normalized Legendre functions 4

2. Normalized tesseral harmonics 5

3. Structure of noncubic point groups 8

4. Symmetry elements in tesseral harmonics 9

5. Potential coefficients present for various symmetries 10

6. Transformation of tesseral harmonics 11

7. A character table for the reps Dj 24

8. A table of group characters 31

9. Type and number of perturbed levels 38

10. Correlation tables 46

11. Reduction of the cubic groups 48

12. Multipole moment transformations 56

13. Selection rules ; 57

14. Angular dependence of multipole components 64

15. Wigner coefficients 67

16. Reduced matrix elements 75

17. A representative crystal field 77

18. Fluorescence spectrum of TbClj -61120 79

iv
Atomic Energy Levels in Crystals '

John L. Prather^

Discrete energy levels observed within certain crystals are treated as due to perturbations

of the energy leveli of the free ion by an electrostatic field arising from the crystal lattice. The analytic procedures for determining the field from the charge configuration are given,

and the resulting fields are classified according to their symmetry. After a general survey

of group-theoretical ideas, the applicable groups are analyzed in detail, and characters appropriate for both integral and half-integral angular momenta of the free ion are tabulated. These are applied to the determination of the number and type of levels arising from a free ion level with /<8. The results of this analysis are tabulated, as are the selection rules for electric dipole, magnetic dipole, and electric quadrupole transitions. Calculation of the perturbation matrix elements by the use of Wigner and Racah coefficients is discussed. Examples of the application of these several techniques to specific problems are given.

1. Introduction

In contrast to the sharp lines or resolvable

bands characteristic of the optical absorption

spectra of free atomic and. raolecular systems,

such spectra for soUds consist for the most part

of broad absorption regions or bands vrithout any

resolvable structm-e. These spectra are therefore

of Httle used for analysis of the details of the solids

for which they are observed, revealing only general

information such as the presence or absence of

certain molecular groupings. Nevertheless, it was

early recognized in the development of optical

spectroscopy that certain naturally occurring

minerals did in fact show a characteristic line

absorption spectrum. The earliest observations were on the crystals of salts of the iron group

elements and of the rare earth elements, though

later observations have shown similar spectra in crystals of all elements characterized by a partially filled inner shell of electrons. In this

present work we shaU be concerned primarily

with the rare-earth spectra, though we shaU

sketch the application of the development to other possible cases. The earliest data on these spectra, covering

about 50 years before 1905 are reviewed by

Kayser (see bibliography). Subsequent to the

discovery of the Zeeman effect in 1895, J. Becquerel

initiated an extensive investigation on the effects of magnetic fields on the absorption lines of

natural rare earth crystals, their polarization, and

particularly their rotation of the plane of polarization of light (Faraday effect). These measurements, carried out both at room temperature and

at liquid air temperature, are summarized in his

paper of 1908. In this same year, Becquerel and

Onnes reported on the spectra and magnetic

effects at liquid hydrogen temperatm-es, and

measurements at liquid helium temperatures were

finally reported in 1926 bv Becquerel, Onnes, and

de Haas, du Bois and Elias (1908, 1911) also

' Based on a dissertation submitted to the Faculty of Philosophy of The

Johns Hopkins University in partial ftdffllment of the requirements for the

degree Doctor of Philosophy.

' Supported in part by the U.S. Air Force, through the Air Force Office of

Scientific Research of the Air Research and Development Command, under

Contract No. CSO and A 6S0-56-21.

' Guest worker, formerly from The George Washington University, now

with Beloit College, Beloit, Wise

reported work on chromium, cobalt, and uranimn

crystals as well as on the rare earths. The year 1929 marked the beginning of an

extensive program of investigation of these spectra

at various temperatures by Spedding, Freed, and

collaborators. The principal experimental ad- vance over earlier work lay in the use of synthetic

crystals of more definite composition than the

natural crystals, which usually consisted of a mixture of rare earths of varying composition.

Hence there had been little if any rehabUity in intercomparison of the results obtained from

various crystals, even of the same nominal type.

This was a considerable advance over the previous

attempts at a systematic experimental approach

to these spectra, though by today's standards even these crystals were not of satisfactory

purity. These measurements were also much

aided by continuing developments in the theory

of these spectra, vv'hich we shaU trace below.

All of these spectra show the same general

pattern and behavior upon reducing the temperature. The absorption lines for various elements

faU into groups in several regions of the spectrum

the general wavelength regions where the lines

fall depend on the rare earth involved, while

details concerning the number of lines, the exact

position, intervals, Zeeman effect, etc., depend

on the negative ions involved and on the details of the crystal structure. In comparison with room temperature measurements, those at re- duced temperatm-es show generally sharper lines,

together vnth a slight overall shift of the groups

to the red and increased separation of the lines within a group (Spedding and Bear, 1932). The

ntimber of lines also changes; certain "high temperature lines" disappear, while others appear or become more intense at lower temperatures.

Such lines have also been observed in solutions of the ions, though these lines are generally broader

than in the crystals.

Although there was some theoretical speculation on these spectra, no theoretically satisfactory

contributions save the derivation of empirical

energy level diagrams coidd be made prior to the

advent of quantum mechanics. The fij'st major

1
contribution directly related, to the rare earth

spectra was the calculation by Hund (1925) of the room temperature paramagnetic susceptibility of the rare earth crystals on the assumption that the

crystals consisted of an independent statistical assemblage of ions, characterized by an unfilled

shell of 4/ electrons which became increasingly

filled as one proceeded up the periodic table. The ground states of these ions were calculated on the assumption that Hund's rules for free atoms were valid, i.e., that of all the terms arising

from the given 4/" configuration, that with the

greatest multiplicity had the lowest energy, and

of those of this greatest multiplicity, the largest

L value was lowest. For a given L-S term, the

resulting multiplet was taken as normal in the

first half of the series (the 4/ shell being less than

half filled) and as inverted in the last half of the

series. The agreement with experiment was

quite good, save in the cases of europium and

samarium, and these two discrepancies were

corrected by the second order calculations of Van

Vleck and Frank (1929), Hund having considered

only the lowest level of the ground m^ultiplet as contributing to the susceptibility. This agreement with experiment not only gave evidence for

the electronic structure of the rare earth ions in the crystals, but also suggested that the effects on

the magnetic properties of the ion due to inclusion

in the crystal were small in comparison with kT

at 300° K and also small in comparison with the

multiplet splitting of the ground L-S term.

It was perhaps natural to assume that the line spectra observed were similarly due to the spectra

of the free ions, modified to some slight degree by

their inclusion in the crystal, but the first suggestion to this effect was made by Brunetti and

OUano (1929), who considered the crystal as a perturbing electrostatic field on the free ion

spectrum. This idea was also developed by

Freed and Spedding (1929) in initiating their

program of experimental observations noted

above. The classic paper in these early years of

the theory is that by Bethe (1929). Bethe

pointed out that the detaUs of the theoretical

treatment depended on the strength of this

external crystal field compared to the other interactions present (principally the Coulomb interaction of the electrons and the spin-orbit interaction), and that the symmetry alone of the

external field, independent of its magnitude,

would determine the number and type of levels arising from a given level of the free ion. These

were specifically computed by Bethe for certain

symmetries, emphasizing in the process the

importance of group-theoretical ideas for such

calculations, and the properties of one-electron wave functions under these same crystal fields were also calculated. These ideas were later extended to treat the Zeeman effect in crystals

(Bethe, 1930). At about the same time, Kramers

(1929, 1930) considering principally the Faraday

effect, formulated the fundamental requirement

that these electrostatic fields arising from the

crystal must leave every level of an odd-electron

system at least twofold degenerate.

Although it seemed clear that the idea of an

electrostatic perturbation acting upon the levels of the free ion was the key to the problem, it was

not at all certain what transitions were responsible

for the lines actually observed. The situation was

complicated by utter ignorance of the actual free ion spectrum, and even as recently as 1958 only

that of CelV is known (Lang, 1936). The earliest

suggestions were that the transition involved a

4/ electron excited to either or 6s (Spedding,

1931; Spedding, 1933). These were based principally on analogy with the selection rules applicable

to free ions. However, the sharpness of the lines indicates a freedom from interaction with other

ions which would not be expected from such levels (involving 5d or 6s) having a fair probability of

being at appreciable distances from the nucleus.

Furthermore, the general intensity of the lines relative to the high concentration of ions in a

solid yields transition probabilities corresponding

to transitions forbidden in the usual type of

atomic spectra (Van Vleck, 1937) . For this reason,

it is today generally considered that the lines come from transitions between terms arising

from the gi'ound 4/" configuration. This view is strengthened by the simplicity of such sharp line spectra in the salts of cerium and ytterbiiun

(Dieke and Crosswhite, 1956) both of which have

only one excited level from the ground configurations 4/^ and 4/^^, respectively. Van Vleck also made some estimates of the relative probability of transitions due to electric quadrupole radiation,

m^agnetic dipole radiation, electric dipole radiation

arising from absence of a static center of symmetry

in the crystal field (which destroys the characterization of levels as odd or even), and electric dipole

radiation arising from destruction of a static center

of symmetry by lattice vibrations. It was con- cluded that there was no clear quantitative reason

to reject completely any of these as being not

responsible for the lines observed.

During this period (the 1930's) detailed calculations were made by Penney and Schlapp (1932,

Schlapp and Penney) based on the assumption of a cubic crystal field, adjusting the parameters to

fit observed magnetic susceptibility data. Although good agreement with experiment in this regard was obtained, it is in marked disagTcement

with the spectral data. This may be ascribed to the insensitivity of susceptibility data (a statistical average over all ions present) to details of the

crystal field. The same objection also applies to attempts to derive the crystal field from specific

heat data (Ahlberg et al., 1937). It is now known

that the symmetry of the cubic system is too high

to account for the spectroscopic observations.

Apart from theoretical considerations of the

rare earth spectra alone, several general theoretical

contributions have been made which are applicable

in major part to the determination of the energy

2
levels of rare earth ions in crystals. Since these

will be referred to in detail later in this work, we

shall merely mention them for general background

at this point. The general applicabiUty of group

theory to problems of atomic structure has been

emphasized by Weyl (1928), clarified by Eckart

(1930), and treated at length by Wigner (1931).

The calculations of Bethe (1929) for certain symmetry groups were extended by Wigner (1930) to other symmetries, though this was in connection

with problems of molecular vibrations. The

monimiental work of Condon and Shortley (1935)

is invaluable for the study of the free ion spectrum.

While experimental data on many different

crystals were collected by many observers, using

many different techniques (absorption spectra,

Zeeman effect, fluorescence spectra, magnetic

measiu-ements, specific heat data) diu-ing the fol- lowing years, the next major advance was the application of paramagnetic resonance techniques to the rare earth crystals. A general review of this work may be found in the paper by Bleaney and

Stevens (1953), while additional data are given in

the supplementary article by Bowers and Owen

(1955). This method has the advantage of giving

data of high acciiracy and resolution—its major

limitation is in the fact that with rare exceptions

only data on the ground level in the crystal are obtained.

The next major advance in the general theory was the work of Racah (1942, 1942a,b) on complex

spectra, which provided very powerful general

techniques for the solution of problems in manyelectron systems. His results have been made

applicable to specific nmnerical calculations by

the recent appearance of numerical tables by

Biedenham (1952), Simon (1954), and Simon et al. (1954). While the procedures of Racah have

been used in several fields, and their applicability

to the problem of rare earth spectra in crystals has

been mentioned by Elliott and Stevens (1953), no widespread use of them has been made in crystal spectra problems. The general work of

Bethe (1929) on the applications of group theory

to the problem has subsequently been applied

by various authors to special cases, but no general

overall survey of the possibilities has been given.

The results have been obtained by Hellwege

(1948, 1948a-d) without the explicit use of group

theory, but much elegance and conciseness is lost thereby, and the resiilts are presented in a rather

cumbersome form. A general survey of these

and many other aspects of the problem by Fick

and Joos (1957) has recently appeared.

Recent renewed interest in these spectra and

the associated properties of the ions has generated

a requirement for a comprehensive survey and

development of the theoretical techniques required

for an analysis of such spectra. While the theory

in its most general terms is known, its applications

in the past have either been concerned with a

particular salt or ion, or have used only a portion

of the theoretical equipment available, are widely

scattered through the literature, and their useful- ness is limited by differences in conventions of

sign, phase, normalization, notation, etc. It is the purpose of this paper to provide a comprehensive theoretical approach to the analysis of

these spectra in a form particularly adapted to the requirements of the experimental workers.

In the interest of meeting this aim, mathematical

rigor and long purely algebraic manipulations

have for the most part been omitted, except

where necessary for an understanding of the

concepts and procedures involved. It is hoped

that it will be of value to experimentalists in the

field who have had only a general background in

the ordinary theory of the atomic spectra of free atomic systems.

Since the general problem is that of computing

the effect on the free ion of the electrostatic

perturbations arising from the crystal lattice, we

begin by considering in section 2 the development

of this potential in a power series of tesseral harmonics, in explicit terms of the charge and position

of the ions giving rise to the field. An important

property of such fields is their symmetry, and

this will be discussed in detail. In section 3, we

shaU consider the connection between the theory

of groups and quantiun mechanics. We begin

by sketching the group-theoretical ideas associated

with the problem of the free atom. This leads

naturally into those modifications caused by the

inclusion of the ion in the crystal lattice. The

properties of the groups involved in the present

problem are then considered in detail. In section

4, we present explicitly the results of the considerations of section 3 to the degeneracy of levels,

selection rules and polarization of lines in the

spectrum of the crystal. In section 5, we consider

the calculation of the matrix elements of the

pertm-bing electrostatic potential. Finally, we

conclude in section 6 by a review of the various

factors influencing the experimentally observed

spectra, correlated with the spectra of the free

ion, and some examples of the application of the

techniques to specific problems.

2. The Electrostatic Field

2.1. Expansion in Tesseral Harmonics

Let us consider, in a given frame of reference, an ion with a charge q located at the point described

in spherical coordinates as (R, a, jS). The potential

at the point (r, d, 4>) may be expressed as a series of Legendre polynomials in the variable cos j,

where 7 is the angle between the directions given

by a, |8, and 6, (p. The unit vectors in the two di- rections are sin a cos iSi+sui a sin jSj+cos ok, and

sin </) cos <^i+sin 6 sin 0j +cos ^k, respectively.

Thus, cos 7 is the scalar product of the two unit vectors and cos 7=sin 6 sin a cos ^ cos (^+sin 9 sin a

sin j8 sin ^+cos 6 cos a=sin a sin 0 cos (<^— /3)-|-

3
COS 6 cos a. The potential is given, for r<R, by

V=a±-^,P:,icosy). (2.1)

7v=o rC

The total potential is the sum of similar contributions from each ion of the crystal lattice. In the

above expression, the contribution of each ion is described with reference to a different axis, and

it is convenient to refer each contribution to the same axis, which we choose to be the 2-axis of

the given frame of reference.

This may be accomplished through the use of

the addition theorem of spherical harmonics, which

expresses Pn(cos 7) in terms of a, jS, 6, and 4) (Stratton, 1941). ^ (N—M)^

P;v(cos7)=P^r(cosa)P^(cos^) +2^^|^^-^

P^(cOSa)Pilf(cOS e) COS Mi4>-^). (2.2)

The potential due to the i-th charge can then be

be put into the form

CO yA^ p Vi=ai^ -snm (cos at) Pm (cos e) ^ (N—MV

W+S) (cos aO P'^ (cos e)

(cos M(t) cos Mj8j+sin sin Mft)

J.

(2.3)

The total potential may then be obtained by sum- ming over all charges in the lattice. As used above, the P^(cos 6) are not normalized

in the quantum-mechanical sense, since

fPf (cos d) Pk (cos e) d (cos d)

=8'N,K Om.L 2 (iV+M)!

2N+1 (iV-M)!

and it is consequently convenient to define

/2iV+l {N-M)l

(2.4)

Tabulated functions for N up to and including

6 are given in table 1. Similarly, the functions

-p sin M4>, cos AI^, and constitute an

orthonormal set of functions over the interval

[0, 27r] in the variable cj). We shall define the

following tesseral harmonics, which are functions

of position on the surface of the unit sphere, and

may be evaluated either from d, </>, or from x, y,

Table 1. Normalized Legendre functions

yi

2

e;=f cos

VlO

4

Vl5

(3 cos2 d- 1)

sin e cos d

3^Vl4 (5 cos' 6»-3

' 4 cos e)

,^V42 (5 cos2 e-l) '8 sin 0

V105

,

4

Vto

sm- B cos

e| =-^sin3 6

e2= 3V2

16

3VT0

(35 cos« 0-30 cos2 e+3)

(7 cos' e—2> cos 6) sin d

3V5

(7 cos2 d—l) sin2 e

3V7O

cos 6 sin' 6

3V35 . ,^

16i,

_V22 (63 cos5 0-70 cos' 6'+ 15

' 16 cos e)

Vl65 (21 COS* 9-14 cos2 S+l)

16 sin e VM55 (3 cos' 0— cos e) sin2 0

-v/TTO

61=

32

3V385

16

3VT54

32

(9^cos2 0-1) sin' 0

cos 0 sin* 0

sin^ 0

/26

e? =-^(231 cos" 0-315 cos* 0+105 cos^ 0-5)

/273 16

V2,730

64

V2,730

32

3a/9T

(33 cos5 0-30 cos' 0+5 cos 0) sin 0

(33 cos* 0-18 cos2 0+1) sin2 0

(11 cos' 0-3 cos 0) sin' 0

32

372^002

32

(11 cos2 0-1) sin* 0

cos 0 sin^ 0

64 sm^ 0

and z. They are given as functions of x, y, and • z in table 2:

CM oM sinM(^ co_

CM r\

, a=^el (2.5)
Table 2. Normalized tesseral harmonics

QQA 6 ^ z

" 2 'v^V

" 2 \r;V

VlO J_,2£-^-y^

' 4'"^

_,_\15 1 zz

_i_\15 J_ yz -~ 2 '^-r^

VTB 1 x^—y'

^2 — —

Vlo 1 2X!/

yi4^ 1 2g3-3z2z-3;/2g

V42 J_ 422x-a:3_a.y2

_ V42 1 4:Z-y—x^y—y^

S5 =

Si=

8

'•^

Vl05 1 zx^—zy'^

4~'7P ^ \ IT

'

VlOB 2xyz

VTO z-^-3a:y2

^ 8 "v;;:' r3

VtO J_ Zx'^y-f

8 'V; r3

ro=3V2 J_ 82^ + + 3y^ - 24^222 - 242/222 + egSyZ

' 16 \

2^"

SVlO 1 (422 -3z2- 37/2) (X2)

Si

Q

Si

C5

SI

CI

SJ

8 3\^ 1 (422 -3a;2-32/2) (2/2) 8

3\'5 1 (622- 8

3V5 1 (622--x2- 2/2) (2x2/) 8 Vt

3^70 1 (z2--32/2) (X2) ° -V TT 3\^ 1 (3x2 -2/^) (2/2) 8 r4

_3V35 1 x^- 6x22/2 + 2/* 16

3V35 J_ 4x2/ (x2- 2/2)

^ 16 'V^" 1'

-„_V22 J_ 825-4023(a;2 + 2/2) + 15(x2 + 2/^)22

' 16 ^27'

Q=V165 1 (82^+x^ +2/*+2x22/2- 12X222- 122/222)X 16 Vx

SI _Vl65 1 (Sz*+x*+y*+ 2x22/2 — 1 2x222 - 1 22/232)

Se=

q=

s?

S;

C;

Vl,155 J_ (222-x2-2/2)(x2-y2)2

8 Vx' '-^

Vl,155 _1_ (222 -X2- 2/2) (2x2/2) 8 Vx'

VtTO J_ (822-x2-2/2)(x3-3x;/2)

32 Vx"

VTTQ J_ (822 -x2- 2/2) (3x22/ -2/3)

" 32 Vx' _ 3V385 1 (x«- 6x22/2 + 2/^)2 16

3\/385 1 (X2--2/2) (4x2/2) 16 Vx

3\/l54 1 X5- 10x^2/2 + 5x2/* 32 Vx

3VT54 1 2/5- 102/5x2 + 52/0;* 32 j*5

V26 1 1 6 26 - 1 20 (x2 + 2/2) 2« + 90 (x2 + 2/2) 2 22 - 5 (x2 + 2/2) 3

' 32 r8 OV273 _]_ [82S-2023(x2 + 2/2)+52(x2 + 2/2)2]x

" 16 V^'

ci_ v273 _1_ [82S-2023(x2 +2/2)+52(x2+2/2)2]2/ ^~ 16 'V^*

C5 =

C2_ J6

V2,730 J_ [I62* - 16 (x2 + 2/2) 22+ (x2 + 2/2)2](x2 - 2/^) 64 'y-' r6

V2,730 [162«- 16(x2+ 2/2) 22+ (x2 +2/2)2](2x2/) 64 r6

»-6 —_ V2,730 (822 - 3x2 - 3^2) (a;2 _ 3y2) (3;^)

^6 00 /—

32

V2,730 1 (822-3x2 -32/2) (3x2 -2/2) (2/2) 32

3V9T 1 (1022-X2- 2/2) (x<- 6x22/2 + 2/*) 32

3V9I 1 (1022-X2- 2/2) (4x3?/ — 4x2/3)

»-6 —

^8 —

32

3-v/2,002 J_ (xs- 10x32/2 + 5x2/*) 2 32 V^'

3V2,002 (j/ii- 102/3x2+ 52/x*)

2

32 'V;;'

Ve.QOe J_ x"- 15x«2/2+15x22/* — 2/° 64 r^

V6,0U6 J_ (3x2-2/2) (x2- 32/2) (2x2/) 64 V^"

16

551664—61 2
With these substitutions, the potential may now

be written in the form

N=0 M=0

(2.6)

where the coefficients and S^^ depend on the

size and location of the neighboring ions and are given explicitly by

^'^-JNVi ?

^^-"^^^

'^^=2:^ ?

^

(2.7b)

We recall here that the i-ih ion, charge g,, is located in spherical coordinates at (Ri,a{,l3i).

The terms C^{ai,l3i) and S^(Q;j,jSi) represent the

quantities (2.5) evaluated for the several pau's of

angles involved in the summation.

We shall also have occasion to utilize the complex

form of (2.5), defining

YMN —

and

V2

(2.8)

This will similarly require, in lieu of (2.7a, b) the

coefficients

(2.9)

This expression (2.6) for the potential diverges for r>it. As we shall see later, however, we shall require only terms for up to and including 6, and we shall assume that the wave function of the

free ion vanishes sufficiently rapidly for large r that negligible error is made by integrating to

infinite distances.

2.2. Symmetry

The effects of the electrostatic field on the free ion spectrum may be separated into two categories:

first, those depending on the qualitative nature of

the field (its shape or symmetry), which is ex- pressed through the mere presence or absence of

certain of the Cm or S^; and second, the quantitative details which depend on the sign and magnitude of the Cn and S^- It is convenient to consider separately these two characteristics of the

field. There are several standard and equivalent

techniques for describing the symmetry of the

electrostatic field. For example, given an arbitrary point (x, y, z) of the field, one may indicate

the coordinates of all points having the same

value of the potential, such as {x, —y, z), {—x, y, z), (—X, —y, z). Another widely used scheme is to

indicate by a symbolic notation the operations

which convert a given initial point into one of

equal potential. Thus, in the example just given,

the operations are respectively a reflection in the

x-z plane, a reflection in the y-z plane, and a

rotation of tt about the z axis, and may be designated as ffxz, <^vz, and Co. Finally, for a given

point {x, y, z) one may give those 3X3 matrices

which transform the original point into one of

equal potential. If one is thoroughly familiar

with the notation and properties of the various

rotations, reflections, and the inversion, there is no reason to prefer one of these to the other. If one does not have this familiarity, the use of the

matrices has the advantage that the effect of two

successive operations can be computed by matrix

multiplication without recourse to geometric

arguments. Thus, if r'=Ar and r" = Br for all r, then r"'=ABr is also a point of equal potential

as is r"" = BAr. For the problem to be considered

here, there are only a few basic matrices which

must be considered. These, together with their

symbolic notation (Schonflies, 1923) are: r

cos 4> —sin cj) 0

C4=

sin (p

0

0 -1

cos 4>

0

1

0

— 1

0

0

r

C3 =

E=

12

V3

2

0

12

V3

2

0

0

1 0

2

0 1

V3 0

1 -2 0

i 0 1

0

0

-1

-

0

0

Thus, the point (or vector) r={x, y, z) under

the operation becomes

r'= (x cos (p—y sin <^, x sin 0+2/ cos (j), z). We see that this is equivalent to a rotation of the

vector about the g-axis by an angle counterclockwise (i.e., by +0 in terms of the right-hand

6
screw convention) when looking toward the origin

. along the positive z-axis. This may also be interi preted as a new description of the old point {x, y,

z) in terms of a new coordinate system where the

coordinate system has been rotated by the angle

4> in the opposite sense. In figure 1, we have ini- tially (in two dimensions only) the vector r in the

Y Y

/ f

///

/ ^ X

/

/

/

Figure 1

X, y frame. The operation C„ will then give r' ia the x, y frame. However, this is clearly the same as r described in the rotated frame x' , y'

.

We shall use both interpretations in the present

work.

The designation C„ represents a r<,-fold axis of

symmetry (?i=2,3,4,6), since the matrix is obtained from C„, by restricting </> to 27r/w. Each

rotation by 2ivln yields an equivalent point in the

potential field and, including the original one,

thei-e are n such altogether. The element C2 is a

twofold axis of symmetry about the y-axis—it is characteristic of the collections of these symmetry

operations to be emphasized here (the noncubic

point groups) that there is at most one axis having more than twofold symmetry, and hence rotations

about the y-axis can be restricted to those of order

two. The element I, the inversion, is noteworthy

in the above as being the only one with a determinant —1, all others being +1. Viewed as an

operation on a coordinate system, the inversion

represents the transformation to left-handed

rather than the usual right-handed system.

In addition to the symmetry operations given

above, there are operations which may be repre- sented as the product of the element I with those

rotations given above. If we are considering a

collection of these symmetry operations containing both I and the rotation, no fundamentally

new information is obtained by considering the

products, but there are certain collections of

symmetry operations which include some of these

product operations but not the factor operations

separately. These product operations, together

with their symbolic notation, are as follows:

^1 0 or ^1 0 0^

IC2= 0 1 0 0 —1 0

.0 0 0

^12

3

2 0

^0 —1 0^

3

2

12 0 iq= 1 0 0

.0 0 -1. .0 0

There are only a finite niunber of possible combinations of these basic operations which are of

interest in the present work. These are diagrammed in table 3 and are known as crystallographic point groups, since each of the combinations constitutes a mathematical group, the

elements of which leave the potential of a point

unchanged. In interpreting table 3, the starting

point is the column headed C„. The groups C„

for the values of n for which we are interested,

consist of the operation C„ and its n distinct

powers, so a total of n symmetry operations are represented in the group C„. This set of operations may be enlarged by adding the element I (moving along the diagonal to the left), as is indicated under the heading C„Xl. This notation represents the fact that I commutes with aU

powers of C„. The group may also be enlarged

by adding the element C2 (moving along the diagonal to the right). Here, the heading {CnCz}

represents that the various operations do not now

necessarily commute, but that all possible combinations of them and their powers are included.

Both of these enlarged groups have 2n elements.

To the latter set, I may now be added (moving

farther to the right), obtaining 47i operations.

From these groups containing I, in some cases gToups of lower order may be obtained by suppressing half the elements, including the element I

itself but not all products containing I. For

example, let us consider the group C2, which contains the elements C2 and E. If we add I, we

7
Table 3. Structure of noncubic point groups

obtain C2A, containing C2, E, Col, and I. From

this we obtain Cj by considering only E and

C2l=o-ft. Similarly, by adding C2 to C2, we obtain

D2, which involves E, C2, C2, and C2C2. By the

further inclusion of I, we obtain D2ft with E, C2,

C2, C2C2, I, IC2, IC2, and IC2C2. By suppressing

I, IG2, C2, and C2C2, the remaining fom* elements

E, C2, IC2 ( = trc) and C2<^v constitute the gxoup C2S. There are a few points which should be made

clear in this connection. First, in the abstract

mathematical sense, these groups are not all distinct. For example, the group C2 contains the

elements C2 and E, while the group I contains

the elements I and E, and they both have the same multiplication table. On the other hand,

there is some degree of distinction which should

be made between the E=C2 (which we might call the identity in rotation space) and the E=I2

(which we might call the identity in inversion

space). Strictly speaking, the identity element of

the groups such as C„XI is the product of these

two separate identity elements.

We shall now investigate, term by term, the

symmetry possessed by the tesseral harmonics

tabulated above. The overall symmetry of a given potential will then be at least that of those

symmetry elements common to each term of the

potential expansion. Conversely, if the S3mimetry

is known from other considerations, we shall Imow

what terms must be included to describe such a

condition.

C^: For this element to be present, the potential

must be independent of 4), a condition which is

met only by terms with M=0 for all N.

C„: For invariance here, y(0) =y(0+27r/7i).

Since the terms involve sin M<f> or cos M^, it is clear that n=M, M/2, M/S, for integral values

only, etc. We shall specify n=M, since the other

cases are covered by the powers of C„ which are

also present in any group.

C2 : A point with the coordinates (r, 6, <f>) under

the transformation C2 becomes (r, ir—d, tt— <^). Since Qf^ is a polynomial of degree N—M in cos 6 and degree M in sin 6, and cos (tt— &) = —cos 6, sin (tt— e)=sin d, then 0^ changes as (— l)^"-^.

However, sin M(n- —<^) = sm M^T cos Ad^cj) —cos ]Sd.ir sin M0=:(— 1)^+1 sin M4> and cos M(x— 0) =

cos Mir cos M^+sin Mir sm M^=(— 1)^ cos M4>.

Hence C terms have this symmetry for A^" even and S terms have it for N odd.

I: Under this operation, the point (x, y, z) becomes (—x, —y, —z), and it is clear from

inspection of table 2 that terms with even N have

this symmetry, while terms with odd N do not.

o-fti This operation results, in rectangular coordinates, in the substitution oi —z for 0, and, in

spherical coordinates, in the substitution of ir—d

for 6. This is the same as the d substitution in

C2 and supplies a factor (— 1)^"*^. Hence, N

and M must both be even or both odd for this

element.

a„: A vertical plane of symmetry exists at

(^=;8 if V{r, e, i3+a) = F(r, 6, 13— a) for all values

of a. For S terms, we have sin M(/3+a) =

sinM;8cos Ma+cos M/S sin Ma and sin AI(l3—a) =

sin MiS cos Ma—cos M/S sin Ma. This yields the

requirement that cos M/S sin Ma=—cos M/3

sin Ma=0. For this to be independent of a, cos M/3 =0, M|3=7r/2, 37r/2, 5x/2 ... and planes

of symmetry exist at |8=7r/2M, 37r/2M . . . (2M— l)7r/2M. Similarly for C terms, we obtain

the requirement that sin MjS sin Ma=0, M)3=0,

TT, 27r, Stt. . . and planes of symmetry exist at 13=0, tt/M, 27r/M,_ (M-l)7r/M. We shall find it convenient to distinguish these sets of planes, describing those arising from C terms as a„ and those

from S terms as aa. Note that the set c7„ for a given even Mincludes both and aa for M/2, and

that any vertical plane is a plane of symmetry if M=0. Let us emphasize that the x— 2 plane (/3=0)

is a plane of symmetry for all C terms, while the

y—3 plane (^^—^ is a plane of symmetry for C

terms if Mis even, for S terms if M is odd.

S4: This operation transforms (r, 6, </>) into

(r, TT—d, ^+(7r/2)) and 6^ transforms as (—1)^-^.

S terms involve sin M(<^)+ (7r/2)) =sin M4> cos M7r/2

+COS M(/) sin M7r/2. Modd yields a cos Mf term

which we reject, and Meven yields (—1)^''^ sin M4>.

Thus S terms transform as (— for M

even. For C terms, the 6 contribution is the same as for S terms and cos M(</)+(7r/2))=cos M^

cos M7r/2— sin M4> sin Mir12. As for S terms, we

reject M odd and in </> obtain (—1)-''^''^ cos M(^ for M even. C terms then transform as (—1)^"*-''*^,

8
I just as S terms. The overall requirement for S4

is then that both M and N—^M must be even.

85: Since this element is defined as C3I, it is necessary and sufficient that both C3 and I be

present for this element to be present, and it need

not be considered separately.

Based on the above analysis and discussion, the

symmetry elements present in each of the tesseral harmonics of table 2 have been given in table 4. An analysis of table 4 will indicate the potential

coefficients (2.7) which must be included to repre- sent a potential appropriate to the groups of

table 3, and the results of this analysis are given

in table 5. Two points are worthy of note in the

! use of table 5. In each case where, for a given

' M and N both C and S are shown, the fu-st S

appearing is shown in parentheses. This represents the fact that, though both are consistent with

the required symmetry, the S term may be suppressed by a suitable rotation about the s-axis.

ij This may be done, of course, for any one C, S

pair, but only for one. All other indicated combinations of C and S must then be allowed, and this one S omitted. Second, with the potential terms

considered, it will be noted that the groups De

and Ces cannot be distinguished from the group

, Ds;, of higher synometry, nor can Ce be distinguished

from Car. Further analysis of the potential terms

shows that the coefficients S^, allowed in De, and

SI in Csn, neither of which is allowed in De^, are the lowest order terms separating these three

groups. Similarly, 87 in Cs will separate it from

Cer, where this coefficient is forbidden. Finally,

it will be noted that the groups D„, C^n, and C„

have been omitted from the table. These sjmmetries do not arise from the electrostatic fields considered here.

2.3. Preferred Coordinate Systems

1 In considering the expansion of the potential

i (2.6) and the calculation of the coefficients (2.7),

it is clear that the coefficients (2.7) will depend

upon the choice of the coordinate system used.

On the other hand, it is also clear that all expansions of the field (2.6) must be physically equivalent, and that a choice between two differing descriptions of the same field is merely a matter of

choosing the simplest description. Following universal convention, the principal axis of symmetry

in the preceding considerations has been taken to be the s-axis, and in the calculation of the coeffi- cients (2.7) this will have to be determined by an

inspection of the given configuration of charges,

as will also the location of any ^-axis (C2). Any

other choice of axis will, in general, yield a more

complicated expansion, and the symmetry will be

apparently (but not actually) lower.

It would be well to discuss further the point

mentioned in the preceding subsection concerning

the planes of symmetry. For a given N and M,

both C terms and S terms describe planes of symmetry, differing merely in their orientation with

respect to the vertical coordinate planes. Clearly,

any linear combination of these two terms also

represents a set of planes of symmetry, at some

intermediate angle. If only one N and M (other

than M=0) is present, the linear combination will represent no more information than wiU either one

alone. The general preference arises from the fact

that the actual calculation of matrLx elements in

section 5 wUl be done through the Y terms (2.8)

rather than the C and S terms. It will be seen from (2.9) that the suppression of S terms (as far as possible) will yield real coefficients for the

matrix elements. A conventional preference for C terms then implies the x-s plane as the preferred vertical plane of symmetry, and the y-axis

for the operation C'2, since IC2 is then a reflection

in this preferred plane. Sometimes (e.g., 'Dsn), we

shaU admit S terms and keep C2 as the y-axis

rather than adopt some other axis in the plane

as C2. Once an expansion has been obtained in a given

coordinate system, that in any other coordinate

system may in principle be obtained from the fact

that each of the terms (2.5) or (2.8) in the new

coordinate system will be expressible as a sum of

Table 4. Symmetry elements in tesseral harmonics

M = 0 1 2 3 4 5 6

N=l

2

3 C

4

5

6

CodIC20'iS40' h

CcxiIC20'oS40'A CcoC7,

C0C1IC2O' jSiffA

aha,

<r JC2

C2IC20' hOv

C20't S4

C2lC2a'i ffu C2a'rS4

C2lC'2aAa' 5

C3IC2O' n So

Caff ha V C3lC2<Tc Se

C4lC2crAO'!iS4

C4CT „ C4lC2ffA a

G^aha V C5lC2ff 5 CelCjaACTiSe

12

3 S

4

56

an ffdCi

ffdl ah adC2

C2ltr had

C2C20' d S4

C2Ta';i ad C2C20' d S4 Qi^Oh ad

C3C2 0';i ad CjIadSe

CaCoO"/! (Id CalffdSo

Cilohaa^i

GiCiad

QilahOd^i

9
\

CO CQ OQ CO CO iX2 00 CO 000 0 0 0 000 0 0 0 0 0 0 b

'I CO 02 CO &2 CO CO 02 CO =Q U 000 0 0 0 000 0 0 0 0 0 0

^co 02 CO 02 02 CO CO 02 00000 000 0 u

gc/jtyjco

0000 0 u 000000 00000 000 0 Q 000 OcoOc^O Oc^O 0 g ^ 02

oo 000 0 0 0 0 g CO 02 CO ^ £^ u 000 000 0 0 0

(J gc^co 000000 000 u 000000 0000 0 Q 000 ooOc^O 0 g CO CO u 000 0 0 0 g CO 000 0 0 0 U g 000000 0 u

000000 000

Q 000 Oc^O

CO ^ u 000 0 ^ 000 CO CO 0 0 Q 000 000 0 0 0 Q 000 0 0 Q 000 0 0 0 Q 000 CO 02 0 u 000000 0 u 000 0 Q 000 0 Q 000 0 8 U 000000 8P 000

i-H (N >0 CO rH (M CO Tt< 10 0 IM CO 0 CO CO Ttno CO -TtH 10 CO 10 CO CO 0 i-l iM CO 10 CD

10
Table 6a. Transformation of tesseral harmonics

\ 0

0' \

\

Q CI SI

0 1 0

>-i 0 0 I

s\ 1 0 0

Table 6b. Transformation of tesseral harmonics

0' \

\

c° r2 S^ C3

02

Q

12 0 V3

2 0 0 Q 0 0 0 0 1 Q

V3

2 0

12

0 0

SI 0 1 0 0 0

Si 0 0 0 1 0

the 2A'^+1 terms of the same value of N but of

different M in the old coordinate system. WhUe

general formulas have been derived for such rela- tions (Wigner, 1931), a detailed study has been

carried out for a second coordinate system 0'

obtained by a rotation of 0 by —120° about the

(i+j+k) axis. This is equivalent to a cyclic

permutation a; of the variables; i.e., the

functions {y,z,x) and Siv iy,^,^) were evaluated

in terms of C§ {x,y,z) and S/^ {x,y,z). The results of this evaluation are presented as the matrices of

table 6. It should be noted that these matrices are orthogonal.

Table 6c. Transformation of tesseral harmonics

\

\ 'J 0' \

03 C2 C3

S>3

0

4 0 Vio

4

0 0 0

0 0

1 0 Vl5 Q 0 0

4 4

ct 0 Vio

0 0 0 0

4 4 Q 0 0 0 0 Vl5

4

0

14

SI Ve 0 Vio 0 0 0 0

4 4

Si 0 0 0 0 0 1 0

SI _Vio 0 _V6 0 0 0 0

4 4

Table 6d. Transformation of tesseral harmonics

0' \

Q Q CI Q Si s^ SI St

a

3

8

0

4

0 V35

8

0 0 0 0 G 0 0 0 0 0 0 4

0 Vii

4 Q

Q

V5

4

0

0

0

12

0

0

0

V7

4

0

0

0

0 Vl4

4

0

0

0

4 Q

V35

8

0 V7

4

0

18

0 0 0 0

Si 0

34

0

4

0 0 0 0 0

Si 0 0 0 0 0 V2

4 0 Vii

4

0

SI 0

4 0

34

0 0 0 0 0

St 0 0 0 0 0

4 0 _vl

4

0

11
Table 6e. Transformation of tesseral harmonics

Q Q Q Q Q Q SI SI SI St SI

0

0

0

0

0

0

Vl5

8

0

Vto

16

0

3Vl4

16

Vl5

8

0

4

V2I

0

0

0

0

0

0

4

0

_V6

8

0

V30

Vto

16

0

_V6

8

0

9V2

16

0

0

0

0

0

0

0

0

0

0

0

V21

8

0

9V2

16

0

3Vl4

16

0

V30

8

0

Vio

16

0

0

0

0

0

0

18

0 V|2

16

0 V2IQ

16

0

0

0

0

_1

2

0

2

0

0 V|2

16

0

1316

0

3^^516

0

V3

2

0 V2TO

16

0 3V5

16

0

1

2.4. Cubic Point Groups

In addition to the 27 noncubic point groups of

table 3, there are 5 point groups Imown as the

cubic groups. While these are not known to

arise in natural rare earth crystals, they are important in the consideration of crystals of other types;

e.g., the iron group, and some data on rare earth

ions have been obtained by the inclusion of the

ion in a foreign lattice of cubic symmetry (Low,

1958) . We shall therefore complete the discussion

by considering these groups.

The five groups are assigned the symbols O,

0,= (0XI), T, T;,= (TXI), and T,^, but only four

distinct abstract groups are involved since Ta and

O are isomorphic. The group O has 24 elements

and represents the pm"ely rotational symmetry

of the cube (or octahedron), while O^, 48 elements,

represents the full symmetry of the cube. T is of order 12 and represents the pure rotational

symmetry of the tetrahedron, while T^, a subgroup

of Oil with 24 elements, represents the full symmetry of the regular tetrahedron. is also of

order 24. These groups represent a higher degree of symmetry than our previous considerations have suggested. In particular, O includes fourfold axes along X, y, and s, threefold axes along the principal

diagonal of each octant, and twofold axes along

the diagonals of the coordinate planes. T has

4 threefold axes, of which at most one can be

along a coordinate axis, but has also three mutually

perpendicular twofold axes. Kotations of these

groups wUl be generally designated C„ for rotations

2ir —' and they may be further distinguished by

primes. However, the distinctive conventions of

section 2.2 which are appropriate for D^^ and its subgroups, cannot be adhered to for these high

symmetries. These symmetries cannot arise from

considering the symmetry elements common to a sum of terms as in the previous cases, but

depend upon certain fLxed ratios in the coefi&cients. In other words, these may be viewed as higher

symmetries arising from "accidental" values of

the coefficients in a case of lower symmetry.

As a particular example, let us consider the

group D4, representing the rotational symmetry

of a rectangular prism, with potential coefficients

(table 5) ClC'l CI Q, St, and CI The x and y

axes are equivalent, but the long dimension, the

3 axis, will generally be different. If the prism

actually becomes a cube, the x, y, and 0 axes will

all be equivalent. Hence we expect that any

combination of these terms which is invariant

under a substitution x-^y-^s—^x will represent a

higher symmetry, in this case O. This is just the

substitution considered in table 6. We see, for

12
Table 6f. Transformation of tesseral harmonics

\

\ 0

0' \

\

Q Q Q s^ Si Oe •36 ^6 C6^6

Q

516

0 V2I0

32

0 _3V7 0 V462 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 0 Vio

16

0 V3

4

0 3V22

16

V-6 V210

32

0 1732

0 _V30 0 3V55 0 0 0 0 0 0

0 0 0 0 0 0 0 0

916

0 V30

8

0 V55

16

r*

3^/716

0 V30

32

0

13ID

0 0 0 0 0 0 0

^6 0 0 0 0 0 0 0 0 V165

16

0 V22

8 0 V316

r« V462

32

0 3V55

32

0

_-v'66 0 _ 1 0 0 0 0 0 0

Jo 0

58

0 3aT0

16 0

^/6616

0 0 0 0 0 0 0

Si 0 0 0 0 0 0 0 Vio

1 A 0

916

0 VT65

16 0

Si 0 sVTo

16

0

1 16 0 _VT65

X u 0 0 0 0 0 0 0

sj 0 0 0 0 0 0 0 Vf

4 0 V30

8 0 V228 0

SI 0 V66

16 0 VT65

16 0

516 0 0 0 0 0 0 0

s? 0 0 0 0 0 0 0 3V22

16 0 V5516 0 V3-16 0

example, that Q becomes a combination of and

Ci- Since Ci is not allowed in D4, we suppress the

second degree terms and proceed to those of the

fourth degree. Although both Q and Q yield

CI, if_the coefficients are in the ratio 0°: Ct=

V5,_the Q will cancel and the combination

Q-r-^^ a will be invariant under the operation

of table 6. SimUarly, if CI:CI=-1: V7, Q and

Ce arising from the transformation will vanish

and Q—^J7 Q will also be invariant. S| will admix Sf, forbidden in D4, so St must also vanish.

We conclude then that under these conditions, we have but two coefficients, and left to fix the strength of the field, and the symmetry is O

rather than D4. In fact, all of the terms ahowed

also contain I as a symmetry element, and the

symmetry is even higher, i.e., O^. The distinction between O and O/, will arise from potential

terms of higher order.

On the other hand, the cube may be considered

as standing on a comer with a threefold axis along

the z axis, arising as a special case of D3 symmetry.

The allowed coefficients will be generally 0°; SI;

a, CI; SI; CI CI, and CI Since we do not have

data corresponding to table 6 to apply our previous

method, we shall set up a charge configiu-ation

having the desired symmetry and orientation,

evaluate the coefficients (2.7) and determine the

necessary relations among them. This has been

done in detail as an example in section 6—we

report merely the result at this point. The

coefficients C", -S'l, and SI must vanish, while the

ratios among the nonvanishing coefficients are Cl:Cl= -7:2^5 and (7g:(7i:(7^=24:2Vl05:V462.

in this case as well, C° and Cq are the only adjustable parameters to fix the strength of the field. The group T may lU^ewise be considered either as a special case of D2, where the x, y, and z axes

all become equivalent, or as a special case of C3

vath the principal threefold axis of the figure

vertical. In the former case, the allowed coeffi- cients are: (7^; (71; SI; CI, CI, C\; SI, SI; and CI

CI, Ct, and Ct A consideration of these terms

and table 6 shows that the following conditions

13
must be met:

(7°=Ci=0 *Si is arbitrary (71=0 ^|=^|=0

C°:Ct=V7.V5 Cl^.Ct=-l.^|^

and Ci:(7^=-ll:V55.

The constants SI, CI, CI and CI can be considered

the variable coefficients describing the strength of the field. The symmetry becomes Tn if the only one of these terms lacking the element I, SI, is suppressed. If we start from the case of T)ya symmetry instead of merely D2, we obtain

symmetry but must suppress the coefficient CI,

leaving SI, CI, and CI

3. Groups, Characters

3.1. Representations and Quantum

Mechanics

Let us consider a Hamiltonian H, and yj/i (i=

1,2, . . . n) an 7? -fold degenerate solution of the

Schroedinger equation H\(/i=E\l/i. Let us further

consider a group of transformations (generally

rotations or the inversion) which may be applied

to this equation. If an element of the group is P,

with inverse P~\ then PHi/'i=P£V'i) which may be

rewi'itten as

(PHP-')(Pf,)=£'(P;Ai). (3.1)

This we interpret as yielding a transformed

Hamiltonian (PHP"') and a transformed solution

of the Schroedinger equation. Let us now

assxmie that the Hamiltonian is invariant under

the group of operations; i.e., that PHP~' =H or PH=HP, and that 0 is another operation of the

group. Then (3.1) becomes ll{P\Pi) =E{P^i), and

Fi/i must be a solution of the original wave equation with the same energy; i.e., it must be a linear

combination of the original wave functions

F4^i =^Pii^P (3.2)

Similarly 0'/',: =SQoV'^, and (OP)^. =Z;(QP)o'/'y.

i . i Thus the effect of each operation of the group may

be expressed as a ny^n unitary matrix. The

importance of the concept becomes most evident

when we consider the product QP as the operation

P followed by the operation 0-

=j:^QaP>cj^j. (3.3)

j k

Thus, the matrix for (QP) is the product of the

matrix for P by that for 0- These unitary

matrices are called a "representation of the group"

because any relation between elements of the

group is also a relation between the matrices corre- sponding to the group elements.

In evaluating the coefficients for the threefold

axis vertical, we have again had recourse to the

establishment of a specific configuration, namely,

that of the regular tetrahedron itself. This will yield terms for rather than T or T/,. Configurations for these latter symmetries are considerably more complicated, and have not been

evaluated for this orientation. The nonvanishing

coefficients must meet the following conditions:

(7°:(7|=V5:2 C'i.Cl=-^|^5:5 and

(7°:Ci:C«=4V2310:77V5:35V22. i

and Representations

Clearly, the precise form of the matrices depends on the particular choice of original wave functions,

and any other set, related to the original set by a unitary transformation may be selected and will yield a transformed representation of the group.

If it is possible to find a transformation such that

the group operations cause only m of the n wave

functions to transform among themselves, and the

remaining n-m functions among themselves, the

representation is said to be (fully) reducible,

otherwise irreducible. Following Melvin (1956),

we shall for brevity speak of an irreducible representation as a "rep". If the representation is a

rep, the degeneracy is said to be essential, since

the n wave functions may be transformed into

each other purely by symmetry operations. If the representation can be reduced into two or more reps, the degeneracy is essential between wave functions belong to the same rep, but "accidental" when wave functions belonging to different

reps are concerned, since wave functions belonging

to different reps are not related to each other

purely by symmetry operations. Accidental degeneracy is either due to a purely fortuitous con- sequence of the numerical parameters of the sys- tem under consideration, or to the presence of

additional symmetry not previously considered j (sometimes called "excess degeneracy"). An ex- ample of this latter will be considered in connection with Kramers' theorem (section 4.2), which

states that all levels of an atomic system with an odd number of electrons wiU be at least twofold

degenerate under the influence of external fields of purely electrostatic character. In this case, it will be seen that wave functions belonging to

different reps of the rotation-reflection group may

actually be related by symmetry with respect to time reversal.

3.2. Rotations in Three Dimensions

One of the transformations in which we shall be

particularly interested is that of an arbitrary rotation of the coordinate axes in three dimensions, so we shall now examine this in some detail. Let us

14
consider a fixed {x, y, z) reference frame, and a movable frame (^, -q, f), the origin of the latr- ter being fixed at the origin of the {x, y, z)

frame. We assmne initially that the ^, 17, and f axes coincide with the x, y, and z axes, respectively. Any rotation will move the (|, 17, f) frame

to some new orientation which is uniquely determined by the rotation performed. We are in- terested in the various ways of specifying the

orientation of the 17, f) frame and its relation to the rotation involved.

It is clear that there will be thi-ee independent

quantities to be specified, two to fix the direction

of the axis of rotation and one to fix the angle of

rotation about this axis. Consider a vector r

fixed in the (^, 17, f) frame rotated through an

angle co about an axis fixed in {x, y, z) containing

Di(u,co) =

and r,r' are the column vectors (f, 77, or {x, y, 2), and {x', y' , z'), respectively.

Since r is fixed in the 17, f) frame, its components in that frame are the same before and

after the rotation j they are the same as its coor- dinates {x, y, z) before the rotation in the fixed

frame.

The direction of the (^, r}, f ) frame may be determined then by applying this relation to the vectors

(1, 0, 0), (0, 1, 0), and (0, 0, 1) in turn. The set of aU three dimensional rotations constitutes a group which we shall designate as dts- The set of real matrices Di (u, w) is orthogonal, hence

unitary, and meets the requirements of section

3.1 for a representation of the group. (Note that

Di is a particular element of A-) It is important

that the group of matrices Di be not confused

with the group of abstract operations 9?3, but

this requires a clear understanding of the differ- ence between a group and any particular repre- sentation of the group. In elementary work, a

three-dimensional rotation is invariably thought

of as an operation on a vector. This is adequate

in these elementary cases, since this leads, as we

have seen, to a particularly simple faithful repre- sentation (a unique matrix for each operation) of

the group. However, we have already mentioned

in section 2.-3 that an element of dlz will induce

not only the linear substitution Di among the

three components of a vector, but also a linear

substitution among the 2N-{-l components of

the tesseral harmonics of degree A'^ (table 2) All

these substitutions are equally well representations of 9?3, and they are faithful for N^-O.

Geometrically, a vector may be resolved into its components, which are the tesseral harmonics of

degree 1, which lie along the three coordinate

axes, so that a linear substitution among the components of the tesseral harmonics of degree 1

the unit vector u with direction cosines (a, /3, 7).

Observe that r may be resolved into a component

parallel to u, (r-u)u, and a component perpendicular to u, (r-(r-u)u), and that the vector (u X r)

is perpendicular to both and has the same

magnitude r sin (u,r) as the latter. The component of r along u, (r-u)u will be

unchanged, while the perpendicular component

r— (r-u)u will be rotated through the angle w

into (r— (r-u)u) cos w+(uXr) sin w. Thus,

r'==r cos co+(l —cos w)(r-u)u+ (uXr) sin co. (3.4)

This may be written as the matrix equation

r'=Di(u,co)r, where

(3.5)

necessarily implies a corresponding linear substitution among the components of a vector,

which we interpret as a rotation of the vector. Likewise, any linear combination of the 2iV+l

tesseral harmonics of degree (or any quantity

having similar transformation properties) may

be viewed as a vector in a space of 2A^+ 1 dimensions, mth the 2^"+! tesseral harmonics serving

as the unit vectors. An element of diz may also be represented by a rotation matrix in this 2N-i-l

dimensional space. Our choice of properly nor- malized tesseral harmonics ensures that the

resulting matrices are orthogonal. In terms of

this concept, the matrices of table 6 are the

matrices D.v for A''= 1,2,3,4,5,6 and

»=J|(Hj+k),»=|. . A general rotation may also be described by a

point along the direction of u and at a distance w

from the origin. AU rotations are thus represented

as points \vithin or on the surface of a sphere of

radius tt. More precisely, the points represent the

results of the rotation, while the details of the

rotation are fixed by prescribing a path from the

origin (representing the original orientation) to this particular end point. It is important to notice

that aU points of the closed sphere represent distinct orientations, save those on the sm-face where

diametrically opposed pairs of points represent the same orientation, being attained by a rotation tt about oppositely directed axes. Consequently,

there are two essentially different types of paths

by which one can go from the origin to another

point within the sphere. Thus, if it is desired to reach the point ^0; ^; 0 ^

in the (ll co) space, this

could be accomplished by a single rotation ^ about

''cos CO-}- (1— cos co)a^

(1 —cos a))a/3+7 sin w

^(1 —COS w)a7— jS sin CO

(1 —cos oija^—y sin co

COS co-(- (1 —COS co)/3^

(1 —COS wjy^-l-a sin co

(1 —COS co)a7+/3 sin co"^

(1 —COS cci)l3y—a sin co

COS co-f- (1 —cos co)7^ ^

15
the y-axis, and the resulting path is a straight Hne

from (0, 0, 0) to ^Oj ^> oj- On the other hand, a

rotation by tt about the —y axis will be repre- sented by a straight line from (0, 0, 0) to (0, — tt, 0) . Since the latter point is equivalent to (0, tt, 0)

,

a further rotation - will result in a path from

(0, TT, 0) to (^0, ^> 0^. We have thus reached

^0; 0^ by two different paths. These are essentially different since there is no continuous

deformation of the paths, keeping the end points

fixed, which will cause them to coincide. Paths

which can thus be made to coincide are said to be of the same homotopy class, otherwise they

are of a different homotopy class. There are only two homotopy classes for rotation in three

dimensions. We shall call them Po (for paths

involving no points on the surface of the sphere, or an even number of pairs of such points) and

Pi (paths involving an odd number of pairs of

points on the surface of the sphere)

It is frequently more convenient to describe

directly the orientation of the (|, i?, f) frame than

to describe the rotation as above; i.e., we give

directly the angles between the two frames. The

general technique is due to Euler, and the three

angles which must be specified are known as Euler's angles. There are a great number of

different choices which can be made, and indeed

have been made by various authors, a freedom

which complicates considerably comparisons between different authors. We shall follow here

what seems to be the most frequent choice by

recent American writers on the subject (Edmonds,

1957; Rose, 1957). The position of the f axis will be specified by 6, the angle between f and s, and

by </>, the angle between x and the projection of f

in the x-y plane. These are just the usual spherical

coordinates of the unit vector along the f axis in

the (x, y, z) system. For the thu-d angle, we shaU

use ip, the angle between tj and the intersection of

the x-y and the ^-rj planes. Experimental

workers (Dieke and Crosswhite, 1956) have found

convenient an alternate choice x, the angle between ^ and the intersection of the x-z and I-t? planes. These angles are related by the equation

X=Xo+'/'. where tan xo=cos 6 tan ^. If 0=90°,

then x='A- A major advantage of Euler's angles is that any

given orientation of the (^, >j, f) frame can be

obtained by three rotations in succession about the

coordinate axes. For these rotations, the matrices

(3.5) take on a particularly simple form, and their

product, expressing the general form of (3.5) in

terms of Euler's angles, may be computed directly.

The three rotations are:

1. A rotation by 4> about the z,^ axes (which

initially coincide). The ^-17 plane remains in

the x-y plane, but the ^-f and the rj-f planes are rotated by the angle ^.

2. A rotation by 6 about the 1? axis. This will

bring the f axis into its final position, as specified

by 4) and 6. The ^— 17 plane will no longer be in

the x-y plane, but their intersection is the 17 axis.

3. A rotation by ^ about the f axis. This wiU

bring the ^ and -q axes to their final orientation.

In performing these rotations, it wiU be noted that

each one is carried out in the (|, n], f) frame, which

is generally in a different position each time as a

result of the preceding rotation in the sequence.

The same final configuration can be obtained by

rotating about the {x, y, z) frame axes, provided,

that the sequence of the angles is reversed. In

other words, a rotation of the (^, ij, f) frame first about the s-axis by the angle yj/, then about the

y axis by d, and finally about the s-axis again by

the angle </>, will yield exactly the same final orientation of the (^, 17, f) frame. This may be

established either by purely geometric considerations, or by specific multiplication of the matrices

involved. Using this latter sequence of angles, we

obtain from (3.5)

ifu=(0, 0, 1), u=4> or oru=(0, 1, 0), u=d

^cos (j) —sin <i> 0"^

sin <j) cos 0 0

,0 0 u

Rfl

cos 6 0 sin

0 1 0 —sin 0 0 cosd^

''cos \f/ —sin \^ 0"^

sin ^ cos \f/ 0

L 0 0 u

and their product R0R9R;^ yields the matrices of the representation Di of dls, now in terms of the three

Euler angles, ^cos COS Q cos i/'— sin 4> sin ^ —cos </> cos B sin t/'— sin <^ cos ^ cos <^ sin ff^

sin 0 cos Q cos ;/' +cos 0 sin —sin 0 cos B sin ;/'+cos <^ cos ^ sin <^ sin Q

-sin Q cos sin Q sin i/' cos Q

(3.5a)

16
Since these matrices are orthogonal, the inverse of a given matrLx is its transpose. In terms of u,co, this amounts to reversing either the vector u or the sign of the angle of rotation. In terms of the

Euler angles, the inverse amounts to reversing the

sign of the angles and also their sequence. In

general, we shaU not indicate the variables for Di (or for Dm), since either set may be used.

A third extremely important description of a

three-dimensional rotation arises from the CayleyKlein parameters. Let us consider the general

2X2 matrix with complex elements, which we

shall write in the form

he'

de

where a, b, c, and d are real nonnegative numbers,

and a, 0, y, and 5 are real numbers. This matrix

contains eight arbitrary parameters. If we require

that it be a unitary matrix U, four conditions of

restraint on these parameters are imposed, and

there remain but four independent quantities.

These four conditions are: a=d; b=c; a^+6^=l;

and 2Q;+5=2)3+T+Tr. The fm-ther condition

that det U=-|-l requires that 2a+5=0, leaving

three independent quantities. We can then

write

ae U=| I, be~'^ ae"'"/

remembering that a^+6^=l. The independent

parameters are then, essentially, a, a, and /3. Fxu-ther, let R=

^—iv

-r

be a Hermitian matrix with a trace of zero, |, rj, and f being arbitrary real numbers. A unitary

transformation

URU-i =R' (3.6)

wiU leave invariant the Hermitian property, the

trace, and the determinant, so R' can be written

R'=

z x—iy>

x+iy —2 I

Furthermore, det R'= — (x^+y^+s^) =det R' =

~ (^^+17^+ f^). If we interpret the sets of numbers

?j, f) and (x, ?/, z) as the coordinates of a point in the movable (^, 17, f) system and the fixed (x, y, z)

system, respectively, we see that the transformation leaves unchanged the distance of the point

from the origin, so that the transformation can be

interpreted as a rotation. The matrix U can be

explicitly calculated in terms of the Euler angles

by finding the matrices U,^, U9, and U^, so that

U^U^UsU^. Thus,

cos ^ —sm -

sm cos ^2 2^

U=

-i(<t>-'P) e cos

2

+i(<t>->l') 2 • o

e sm

2

sm

cos-

(3.7)

Just as in the previous discussion distinguishing

between 9?3 and Di, we have here an abstract

group of unitary unimodular transformations in a complex space of two dimensions, and a particular representation of the group through the

matrices (3.7) . The abstract group we shall designate as U2, the pai'ticular representation by the

collection of U matrices (3.7) as Z)i/2. The ele- ments of the representation D112 may be given

either in terms of the Euler angles as in (3.7), or

in terms of the unit vector u and the angle co, though we shall not give the latter form here explicitly. Since it will not generally be necessary

to indicate the variables, we shaU generally write

the matrices merely as D1/2 (instead of U) to parallel our earlier distinction between Di and Di.

Let us consider the implication of this transformation in more detail. We first observe that the

matrix R can be written in the form

R=?

0 1\ /O -i\ /I 0

1 0/ \i 0/ \0 -1

where the 2X2 matrices are the Pauli matrices

corresponding to the x, y, and s-components, re- spectively, of the spin angular momentum operator. If we let (^, 77, f) be a unit vector, the

matrix R is the operator corresponding to the projection of the spin angular momentum in the

direction given by the unit vector ($, rj, f). Let

us consider for simplicity U for d=\l/=0. R' =

URU"^ may similarly be written

R' = (^ cos (})—ri sin ^)

a 0

/O -i\ fl

-]-{r] COS 4>+^sm<f) \ +rl

0/ \o

This corresponds to our interpretation that the vector with components ($, 17, f), fixed in the

(?, n, f) frame, is given a rotation about the f

axis, positive in terms of the right hand screw

17
convention. R' is now the operator corresponding to the pi-ojection of spin in the new du'ection,

still described in the movable system by rj, f), but described in the fLxed (x, y, z) system by

3;=! cos ^—?? sin <^ ^=77 cos <^+| sin 3=f.

Note that the Pauli matrices retain their form, i.e., they are expressed in the fixed (JC, y, z) frame.

However, we can also write

0

where ^, >?, and f retain their original values, and

the Pauli spin matrices have been transformed to the form appropriate to the new |, ??, f axes. Recalling that this is a rotation by this should

be equivalent to a rotation of the physical system,

the fixed x, y, z axes, and the spin functions described therein, by — In other words, generally, a transformation of the spin operators a, by U is equivalent to a transformation of the spin functions themselves by U"^ More formally, consider <^=R 1^, where ^ and are spin wave functions. This expression, under the transformation

U, yields

U<i=URU-'W or </.' = RV'

which represents merely the transformation of each side of the expression to a new frame of

reference; in other words, essentially the same

angular momentum measurement. From the

standpoint of a simple rotation of R to measure

angular momentum in the new direction fixed by

R' as considered in the original frame of reference, we are interested in the quantity R'i/'=URU~V =X which is generally different from ^. This new measurement may be transformed by U""^ to give

U-iX=U-i[URU-^]UU-V=-R[U-V].

In other words, the rotation of R to R', URU-'

gives the same result as the inverse transformation

of the spin wave functions by Since we are usually interested in transforming wave functions, we shall write Di4=U"^ rather than Di^=U.

The presence of the half-angle functions in this

transformation reflects the very special properties

of the spin transformations. In particular, the

rotation q!)' =<^>+27r, e' =0-f27r, ;/'' ='A+27r clearly

jdelds an orientation of axes identical to that designated by <^>, \p. On the other hand, it may be

seen that By^{(j)' , 6', f) = -DyX(p, 9, xP). Thus,

for each change of orientation in ordinary threedimensional space, there are two matrices corre- sponding to this in the two-dimensional spin space. For this reason, the set of matrices Dy^ does not

constitute a representation of the three-dimensional rotation group in the stricted mathematical

sense of the term. It has become customary to

refer to D14 as a "two-valued" or "ambiguous"

representation, and the abstract group U2 as the

double rotation group. If a rotation is specified

by giving the initial and final configuration of

axes, the matrices Dy^ are fixed apart from the

sign. The sign may also be fixed if the detailed course of the rotation is also prescribed. Clearly,

the matrix Dy is a continuous function of the

variables (f>, d, ^ and becomes the unit matrix for no rotation at all. In other words, if the initial and final axes are the same, we know only that

the appropriate Dy^ matrix is either the unit 2X2

matrix or its negative, while the detailed knowledge that no rotation at all has occurred (or one

involving a path in the (uw) space of homotopy

class Pq) enables us to fix upon the unit matrix as

the appropriate one. Generally, if two axes are

related by the Euler angles 4>, d, xp we will select Di^ (cj), 6, xp) if the transformation is of homotopy

class Pq, and —Di^(0, d, if of class Pj.

3.3. Group Theory and the Free Atom or Ion

The Hamiltonian for a free atom contains the

following types of terms:

1 1. Kinetic energy terms -—

2. Central field potential terms \-y'{'"i)

3. Electrostatic repulsion terms V(r^

4. Spin-orbit interaction terms ^{ri)\i•^^.

In the above, the Vir^ is selected so that the

overall effect of the third term is minimized. In

addition to the terms of the free ion Hamiltonian, we shall also be interested in the effects of the

crystal lattice potential V discussed in the pre- ceding section and of an external magnetic

field B. Consequently, we shall be interested in

the operations which leave invariant these various

terms in the Hamiltonian.

Let us first consider terms (1) and (2) only.

In this case, each electron is considered to move

independently of the detailed motion of the other

electrons present, their mutual interaction appearing in an averaged form in the V(yi) term. Term

(1) gives the Laplacian operator, which may be

viewed as a scalar product of two vector operators

and is therefore invariant under an arbitrary rotation of coordinate axes about the force center. Since second derivatives are involved, it is also invariant under the inversion. Similarly, the

second term involves only the scalar magnitude

of r which is invariant under the same two operations. The solutions of the Schrodinger equation

for one electron for these two terms only are

18
characterized by the quantum numbers n, I, rrii, and m., and have a degeneracy equal to 2(2Z+1),

there being 2l-\-l values of rrii and 2 values of

all belonging to the same energy and all related to

i each other by the particular choice of coordinate

axis. These 2l-r \ functions are just the functions

I (2.8). Thus, a rotation of the coordinate axes will cause each of the 2^+1 functions in the new set of axes to be expressed in terms of all 2l-\-_\ functions

referred to the old axes, yielding a 2Z+ 1 dimensional

representation of the rotation as in section 3.2. It is shown in the general theory of groups that

for aU nonnegative integral values of I these representations are in fact reps. General formulas for

the matrix elements in terms of Euler's angles

have been given by TTigner (1931). Specific ex- amples of these matrices using a real basis (2.5)

rather than the complex basis (2.8) have abeady

been given in table 6. It is further to be noted

that, for Z>0, the representation is faithful, i.e., there is a unique matrix corresponding to each

rotation. We shall designate the rep by Di, its elements by D;. For a single electron, as in hydrogen or in the

alkali-type atoms or ions, term (3) in the HamUtonian vanishes. So long as term (4) is neglected,

the transformation of the 2Z+1 values of m; and

the two values of are completely independent

of each other, and may even be described mth

respect to different coordinate axes. The general

transformation may be represented by a matrix

of 2(2^+1) dimensions, the direct product of Di^

and D(. Upon the introduction of term (4), the

independent transformation of spin and orbit no

longer leaves the Hamiltonian invariant. The in- variance of the scalar product of two operators

(e.g., 1-s) is founded upon the assmnption that

each is referred to the same set of axes. The

group of allowed transformations is now less general, since both spin and orbit must simultaneously undergo the same rotation. The 2(2^+1)

dimensionsal matrix is still an element of a repre- sentation of the group of allowed transformations,

but it is no longer u-reducible, and it may be

transformed by a new choice of wave functions

into a diagonal mati'ix of sub-matrices of dimensionality 2(Z+i) + l and 2(Z— 1) + 1. The correct choice of wave fimctions is precisely that dictated

by the usual transformation from a m,, to j,

raj representation in one-electron spectra. The

wave functions are determined by the coefficients

known variously as Wigner, Clebsch-Gordon, or vector addition coefficients, and yield allowed

values of the total angular momentum or A similar argument is apphcable to the case of

two electrons. Without terms (3) and (4) in the

Hamiltonian, aU transformations are independent

and most generally yield a matrix of 2^(2Zi + l)'* (24+1) dimensions. The introduction of term

(3) now excludes those transformations affecting

differently the space parts of the two one-electron wave functions, and the space portion of the

matrix can be reduced into sub-matrices of

dimensionality 2Z+1, where L ranges from

Z1+Z2 to 1^1— Z2 1- The restrictions of the Pauli

principle require that the foin--dimensional spin

matrix be simultaneously reduced, yielding a

three-dimensional submatrix Di and the unit

1 by 1 matrix Dq. This reduction of the spin

transformation matrix corresponds to the wellknown existence of triplet and singlet states in

two-electron spectra. For our immediate piu-- poses, perhaps the most noteworthy point is the

disappearance of the ambiguity in sign of the spin

transformations. This is a general characteristic

of atoms or ions with an even nmnber of electrons. The reps of U2 are infinite in number and are of

both even and odd dimensionality. Those of even dimensionahty include and others designated

as D3I2, D0/2, etc., the dimensionalities being

2, 4, 6, . . ., respectively. These representations

are all faithful, and hence all are double-valued

representations of the three-dimensional rotation

group. The odd dimensional reps are the

^0,1,2,3 • • • previously discussed as reps of the

three-dimensional rotation group, but these latter are not faithful. Aside from the identity rep Du,

any element Dj, J integi-al, corresponds to J half-integral. This is a point of greatest importance when considering the representations of

the point groups, as we shall see shortly.

The introduction of term (4) into the Hamiltonian now requires the reduction of the (2*S'+1)-

(2i+l) dimensional direct product into a diagonal

array of sub-matrices D^, where J=L-\-S,

L-\-S-l, ... \L-S\. Here, the integral _ ,S yields integral J and the spin ambiguity remains

suppressed. In certain atoms, the magnitude of

the various terms in the Hamiltonian requires

that term (4) be introduced before (3). The re- duction of the various matrices must then be

accomplished in a different order, corresponding

to the j—j coupling scheme rather than the L—S

scheme discussed above. For 3 or more electrons,

the details can become increasingly complicated,

but the general features remain the same. In

particidar, integral J's without spin ambiguity

appear for systems with an even number of

electrons, while half-integral J's corresponding to double-valued representations of pure rotations

appear for systems with an odd number of

electrons.

There is another symmetry which must be

considered at this point, that of the inversion.

Terms (1), (2), (3) in the Hamiltonian depend only on the length of a vector or vector operator and

hence are invariant under the transformation from

a right-handed to a left-handed coordinate system.

The angular momentum operator, orbital or spin,

has the transformation properties of a cross product of two vectors, i.e., an axial vector or pseudovector. Although the two vectors will change sign on inversion, their product will not and the vector

operators (pseudovectors) 1 and s as well as the

Hamiltonian term 1-s are also invariant. It is

19
important to note at this point the distinction

between a vector and its description. Let us consider the vector r=(x, y, z) and t={u, v, w), with

rXt= (yw— sw, zu—xw,xv—yu). If we now switch

to a left-handed system, the vector r remains the

same, but it is now described by {—x, —y, —z).

Likewise, t remains the same but its description

becomes the negative of the original. On the

other hand, the product rXt retains the same

description, but is consequently a different vector

since it is now in a left-handed system. The dis- tinction between a quantity and its description is a fundamental one which must on occasion be

made very carefully. If we consider a one-elec- tron wave function \l/{x, y, z, s), the description of

this state in the inverted frame is ^(—x, —y, —z,

s), the description of an angular momentum being

invariant. \f/ is a, homogeneous polynomial of degree I in the variables x, y, z and the transformation under the inversion is that of (—1)' and

that of a w-electron system (—1)^'*. Strictly

speaking, of course, 2Z« will not have a definite

numerical value when terms (3) and (4) in the

Hamiltonian are introduced, but only down

through term (2). Nevertheless, the introduction

of (3) and (4), invariant under the inversion, will admix only wave functions of the same value of

(—1)^'' which may be used to describe states of

the free atom or ion. In addition to J, the states may be designated even or odd, {g or u, respectively) depending on (— l)^'*.

3.4. Group Theory and the Perturbed Ion

It is the primary purpose of this paper to consider in detail the effects of terms in the Hamiltonian arising from the crystal lattice perturbation V and from the external magnetic field B. There are two techniques which may be used in the solution of the problem, both of which must

be used in a complementary fashion for a complete

discussion. One technique is to apply further the

theory of groups, reducing the representation of

the state of the free ion according to the reps of

the appropriate electrostatic-magnetic field symmetry group. This technique will yield the number of levels, their description in terms of their

behavior under the allowed symmetry operations,

and selection and polarization rules for various

types of transitions. The method will not yield

purely numerical details, such as the relative positions of levels, their variation with electric and

magnetic field intensities, or transition probabilities (save for the vanishing forbidden transitions).

The other technique is to calculate to some degree

of approximation the matrix elements of the per- turbing potentials in terms of the free ion wave

functions, interpreting the results in the framework of the usual perturbation theory. In the

final analysis, this latter procedtire will yield not

only the numerical details not obtained from the

former technique, but can be so interpreted as to yield all of the results that the former technique

will give. Nevertheless, the group-theoretical approach is one of great generality and provides a very concise systematic framework for stating the ' results obtained. Furthermore, the principles of

group theory (or essentially similar arguments)

are used to determine what matrix elements of the ; perturbing potentials vanish if the second approach | is used, and to calculate the magnitude of those { which do not vanish. Hence, the two approaches ; are not really so divergent as might at first sight

appear to be the case. In this work, the contributions to the problem which can be made with the ' use of group theory will be studied in detail before ? proceeding to the alternate considerations of calcu-

lations involving specific matrix elements. While

it is not the intent of this paper to furnish a general

treatise on group theory, and a general acquaint

ance with at least the basic definitions wiU be ') presumed, certain concepts will be examined in ! detail and applied as the necessity arises. i; Let us consider the very simple group C3, con-ii sisting of the elements C3, Cf, and E. We shall' describe the group in terms of the effect of its ! operations on the basis triangle of figure 2a, il taking G3 as a counterclockwise rotation of the

triangle by 120°. In the present configuration, ' the operation G3 will cause the vertex (1) to:^ appear at the point previously occupied by. (2),^

(2) at that occupied by (3), and also (3)-^(l).i

We may shorten this to C3: (l)->(2)-^(3). We

may also describe the operations in terms of their 1 effect on the triangle in figure 2b, where we see. that we also obtain C3: (l)-^(2)^(3). Notice 1 that figure 2b differs from figure 2a only in that

C3 has been applied to the basis triangle before i we use it to describe the group. We could alsoi

have applied Ci to figure 2a before describing the

operation, and again would have obtained Cz'i

(l)->(2)->(3). A study of the operation Ci will? show that, using any of these basis triangles,! we obtain Q: (1)^(3)^(2) as its description.

Thus, if we restrict ourselves to transformationsi

within the group under discussion, there is no^

transformation which will cause Ci and C3 to have-; the same description in terms of their effect on ! the basis triangle. Let us now consider the group

D3, where we have added to our original set of

thi'ee elements three rotations of 180° about axes

in the x-y plane, described in terms again of figure

2a as C2: (2)^(3); C^: (l)-^(2); and C'^: (1)^(3).

Still restricting ourselves to transformations mthin

the group, let us consider the effect of C3 in terms

of the basis of figure 2c, derived from figure 2a by

the application of C2. Here we see that we obtain

C3: (1)—^(3)->(2), but this is the same as Ci

(2) (I) (3)

(3) (2) (2)

a b , c Figure 2
described in terms of figure 2a. We thus see that

. in our first case C3 and Cf were essentially different,

j in that none of our allowed transformations permit

! them to hare the same description, while in the

second case at least part of this essential difference

has been lost. This difference is expressed by the

, notion of class. In the first case C3 and Cf are

: said to be in a different class, while in the second

I case they are said to be in the same class. More generally, if we consider the general trans- formation, P'=TPT~^ and hmit oiu-selves to

' transformations T belonging to the group of P,

• we see that P' is also in the group of P, and is defined as being in the class of P. For each T of

; the group a P' is obtained, not necessarily dis-

• tinct, and the complete set of P' constitutes the

^ class of P. Clearly, if P and T commute, P' =P.

; In an AbeHan group, where all elements commute,

every element is in a class by itself. Conversely,

if P' is distinct from P, then P and T must not

! commute. The separation of a group into classes may be accompHshed either by geometric considerations such as outhned above, or by formal

computation. The formal computation is aided

by the following considerations. Let N be the

set of group elements which commute with P.

It may be shown that this set of elements is in

fact a subgroup (possibly the entire group) called

the normalizer of P. Any element of N as the

transforming element j^elds a P' =P. Let K be

a group element not in N (K does not exist if the

normalizer is the whole group). Then K as the

transforming element T pelds a P' different from

P, but any element in the complex KN yields the same P' as does K. Similarly, if L is a group

element not in N or KN, it will yield still another

P", but all elements in LN will yield the same P".

Generally, if the group order is g and the order of

the normalizer is n, the ratio g/n, an integer, is called the index of N, and the group may be spht

into g/n complexes, one of which is the subgroup

N. All of the elements of the class of P may then

be obtained by transforming P with an element

from each of these g/n complexes. Hence the

class of P has g/n distinct elements.

These considerations may be applied to the

group D3 discussed above as an example. The

group order is 6, and the elements which com- mute -^^-ith C3 are E, C3, C3. Hence the index of

the normalizer is 2, and there are two elements in

the class of C3, one of which is C3 itself, the other

of which may be obtained by transforming C3 \vith

any one of Co, C2, or C2. To accomplish this latter we need the multiplication table for the

group elements, usually summarized as C3=C2= E,

C3C2 = C2C3. The other element in the class of C3

is C2C3C2~^=C2C3C2=C2C2C3=Ci,as was geometrically shown above. C2 commutes only with itseK

and E, so the order of its normahzer is 2, its index

is 3, and there are three elements in the class of C2. These are C2 and the products of C2 with C3 and Cf.

If we have a set of g unitary matrices forming a

representation of a group of order g, the set of

traces of these g matrices is said to be the character of the representation. The importance of

the concept lies in the invariance of the trace xmder

a unitary transformation, Tr P' =Tr UPU"^=

Tr P. We see immediately that if U is a matrix

of the representation, then P' is an element of the

class of P and aU elements of the same class have

the same trace. If U is a matrix reducing the

representation into two or more reps, the character

of the representation is the sum of the characters

of the component reps. This latter fact will allow us to determine what reps may be present in a given representation without actually finding the

particular U necessary to accomphsh the reduction.

This latter procedure is precisely what we have

outlined above in sketching the appHcation of group-theoretical ideas to the description of states

of free atoms or ions, and what we wish to examine

in detail when considering the effect of the introduction of the crystal field V and the external

magnetic field B. Before the introduction of the

external fields, the Hamiltonian is invariant under

arbitrary rotation (of the simple or double group)

and inversion, and the wave functions are trans- formed by Dj and (— l)^'-, respectively. After

the introduction of the external fields, the Hamiltonian is now invariant only with respect to subgroups of the original group of transformations.

The original rep of the complete group will not in

general be a rep of this smaller group but merely

a representation, reducible into reps of the smaller

group. This is the expression of the fact that the

original 2j-\- 1 fold degenerate level will under the

infiuence of these external fields split into several

levels of lower degeneracy. The character of the

original representation is the sum of the characters

of the reps into which the state splits. We shaU

now outline the method whereby this determination of the component reps may be accomphshed.

3.5. Group Characters and the Rep-Dj

Let us consider a finite group of g elements,

with the elements separated into classes. There

will, in general, be several reps for this group.

Let us write the trace in the i-th. rep and j-th class as xn- For a given i, j will assume values from

1 to n, where n is the number of classes, and

will be a set of niunbers, generally complex and

not necessarily aU distinct. Strictly speaking,

the set of numbers Xij for all j and a given i constitutes the character, while the number xtj for a

particular i and j is merely the trace of a matrix,

and one component of the character. This may

be emphasized by considering the character as a

vector Xi in a space of n dimensions, with the n

components, Xij- This is a distinction which is usually either overlooked or ignored. We shaU

now introduce a weighted trace, defined as

-y/^ Xij, where Nj is the number of elements in

the class j, and, correspondingly, a weighted character ^1. Fundamental to our further considera21
tions will be the orthogonality theorem for these

weighted characters, which we shall state without

proof, referring to Wigner (1931) for details. This theorem states the following:

1. The weighted character for a given ^ is a normalized vector in the Hermitian sense in a space of n dimensions. Symbolically

J = l (3.8)

2. The weighted characters belonging to dif- ferent reps are orthogonal in the Hermitian

sense.

(3.9)

3. The set of weighted characters ?j for all i forms a com.plete set in the space of n dimensions,

in terms of which the weighted character of any

representation may be expanded. If S is such a

weighted character, with components "Ej^-d —^ X^,

V 9

and ai=^rSi=l (3.10)

Two important conclusions may be immediately

drawn from this theorem. We observe that, since

the weighted characters form a complete set in a

space of n dimensions, there must be n of them.

In other words, the index i as well as^" must assume

values from 1 to n, and the number of reps is equal

to the number of classes of the group. The

orthogonality theorem suggests that we consider

these weighted characters as the elements of a ny,n matrix. The orthonormality (3.8) and (3.9)

in the rows (i.e., in the index i) implies a similar

relation in the columns (i.e., in the index j). In

n particular, we have S ^iB^fs=l

i=l where j=-E' refers to the class of the unit element.

The unit element is always in a class by itself, so Ne=1- The matrix corresponding to the unit

element in any rep is the unit matrix of the di- d

mensionality di of the rep, so XiB=dt and ^^e—-^-

Consequently,

2!j k'tsiiE — 1 implies

1=1 (3.11)

For the finite groups under consideration, there

will be a unique solution to this relation in terms

of integral values of df The dimensionality of

the reps (and hence the maximum allowable

essential degeneracy) will be determined by the

order and class structure of the group. Thus, for the group D3 discussed above, with six elements

and three classes, we have 2^+1 + 1 =6, and the

maximum essential degeneracy will be 2 We shall begin our discussion of the character

of specific reps by considering the reps Dj, initially

limiting J to 1. The ideas of class, character,

and orthogonality just discussed were with specific

reference to finite groups, but are applicable to

infinite groups with but minor modification, the

most far-reaching being in the orthogonality relations. In this case the finite sums cited are re- placed by integrals over parameters defining the

elements of the group. We shall not require the

theorem for infinite groups to be discussed here.

A matrix Dj expresses a transformation of a

vector r (^, 7], f) into a vector r'(x, y, z). The

transformation is orthogonal, leaving the length

of the vector unchanged, and may be viewed as a

rotation about an axis along rXr' with an angle of r r' rotation w such that cos co= . .. . To resolve the!

|r||r'|

group elements into classes note that we can first rotate the vector r into the x-y plane and, if necessary, perform a rotation about r so that r' will also lie in the x-y plane. We thus transform

Di into a rotation about the z axis where it will,

from (3.5) for U=(0, 0, 1) assume the form

^ cos CO —sin CO 0^

Ri(a,) = sin CO cos CO 0

.0 0 1>

We see that all rotations with the same angle of rotation are in the same class, and that the trace

is 1 +2 cos CO. It is important to observe that the

fundamental interval of co and the complete range

of the character, here a vector with an infinite number of components, may be obtained with co between the limits 0 and tt. This reflects the fact

that a rotation, -co about a given axis is equivalent to a rotation co about an oppositely directed

axis. Thus every rotation is in the same class as

its inverse.

Preparatory to considering other integral values

of J, let us transform Ri(co) to the purely diagonal

form RJ(co) given by

0 0^

1 0

0 g-"

This is the matrix describing the effect of a rotation about the z axis on the three states ( + 1, 0, — 1) of a ^ electron. For a d electron, we know

22
0 0 0 0

^

0 0 0 0

0 0 1 0 0

0 0 0 0

.0 0 0 0

]| that such a rotation may be described by the

matrix

Since there is a unique 1 to 1 correspondence between the matrices Di and D2, it follows that any

matrix D2 may be brought into the form R2(w) by

the transformation corresponding to that bringing

Di into R^(w). Here the trace is 1+2 cos w+2

cos 2u. A similar argmnent holds for other in- tegral J. A general expression xj-(w) is then given

J

by S This is a geometrical progression in

which the first term is e"*-''", the common ratio

is e*", and which contains 2j-\-\ terms. The

general formula for the sum of such a progression

gives

which may be simplified to

sin (2J+1)

(3.12) sm- We see from an inspection of this formula that

the relations considered for hold generally for

integral J, namely, x(w) = x(— w), and that w on

the interval (0, tt) covers the allowed range.

Let us now consider the group for J halfintegral, initiall}^ limiting oiu-selves to J=^'

Any unitary matrix may be diagonahzed, and

(a 0

Dj will assimie the particular form I\0 a*. where |a| is -fl. This diagonahzed form of can be written as

e+'2 0

0

By equating the trace of this matrix to that of the

nondiagonahzed matrix (3.7), we obtain

2 cos -^=2 cos cos

a relation between the angle of rotation w and the

Euler angles 4), 6, We see that in this group, as weU as in that with integral J, every element

and its inverse are in the same class, since the

character does not depend on the sign of w and

that the classes are determined by the parameter

0). The character of this rep is 2 cos co/2. The

range of the parameter w/2 in defining the group

element is (0, tt) corresponding to w in three dimensions on the interval (0, 2Tr), again confii-ming

the double natiu-e of this group in describing real rotations. Let oiq be the angle, less than tt, re- lating two orientations of three-dimensional axes, the corresponding diagonahzed two-dimensional

matrix being

R^('^o)=

with character 2 cos coo/2. The other twodimensional matrix corresponding to the same

orientation of three-dimensional axes is then

given by

R|(a;o+2x) =

0

0

0

-R|(wo).

The character of R|(wo+2t)=2 cos (coo/2 + 7r) =

—2 cos a>o/2. Hence these two elements of

corresponding to the same three-dimensional rotation have different traces and belong to different

classes, unless the trace is zero. This latter occurs

for coo=ir, at Avhich point the two matrices become

/i 0\ /-i 0> R.(7r)= and R^(37r)=(

\0 -iJ \0 ^y For real rotations of tt, the two elements are in the same class, because they are then reciprocal

elements.

For higher half-integral values of J, with a unique 1 to 1 correspondence with the elements of

the group Z)i, a similar reduction to diagonal form

must be obtainable, yielding

+''^'- 0 . . . 0 0

0 0 0

0 0

The trace of this matrix is a geometrical progression of the same form as the earlier one for integral

J, so the general formula (3.12) will still be valid.

Values of xA<^) have been calculated for co and

J of interest and are presented in table 7. In

further confirmation of previous discussion, for integral J, co/2 need only range from 0 to 7r/2, since

the numerator is an odd polynomial of the denominator, and the numerator and denominator

will change sign at the same time. For half

23
Table 7. A character table for the reps Dj

\ 0° 60° 90° 120° 180° 240° 270° 300° 360° J\\

27r/6 27r/4 27r/3 IT 4n-/3 67r/4 107r/6 2ir

0 1 1 1 1 1 1 1 1 1

1/2 2 V3 1 0 -1 -V2 -V3 -2

1 3 2 1 0 -1 0 1 2 3

3/2 4 V3 0 -1 0 1 0 -V3 -4

2 5 1 -1 -1 1 -1 -1 1 5

5/2 6 0 0 0 0 V2 0 -6

3 7 -1 -1 1 -1 1 -1 -1 7

7/2 8 -V3 0 1 0 -1 0 V3 -8

4 9 -2 1 0 1 0 1 -2 9

9/2 10 -V3 V2V — 1 0 1 -V2V *^ a/3 — 10

5 11 -1 1 -1 -1 -1 1 -1 11

11/2 12 0 0 0 0 0 0 0 -12

6 13 1 -1 1 1 1 -1 1 13

13/2 14 V3 -V2 1 0 -1 ,/2 -V3 -14

7 15 2 -1 0 -1 0 -1 2 15

15/2 16 V3 0 -1 0 1 0 -V3 -16

8 17 1 1 -1 1 -1 1 1 17

integral J, the numerator will be a product of an

odd function of sin co/2 and an odd function of cos co/2. No new values are obtained by going to negative values of w, since again the numerator

and denominator will both change sign together

at co=0. However, the odd terms in cos aj/2

change sign at co/2 = 7r/2, so that the range (0, 2tv) in w will give distinct traces in this latter case. This is due, of course, to the fact that the division

between the two homotopy classes of Di occurs

at w=ir rather than at co=0.

3.6. Noncubic Crystallographic Groups

We shall now consider the details of the subgroups of Dj in which we shall be interested.

Upon the introduction of the crystal field and/or

the external magnetic field the general rotationinversion invariance of the free ion Hamiltonian

is destroyed and only those transformations which

leave invariant terms representing these external

fields are admissible. These were defined in

section 2 in terms of the elements DiXl. Because

of the unique 1 to 1 correspondence between

DiXl and D^Xl for integral t/>0, the structure

as a group of the allowed transformations for all integral J'>0 is the same as those of the perturbing

fields. This correspondence does not hold for J

half-integral. In this case we again select out of

Dj-Xl those transformations which correspond in

terms of DiXl to invariance of the pertiu"bing

fields. However, there are two elements of

DjXl, J half-integral, corresponding to the same

transformation in terms of DjXl, so there Avill be

twice as many elements in the corresponding subgroups. For this reason the structure of the subgroups for half-integral J wiU be different from

those for integral J and must be considered

separately. We shall initially limit ourselves to those subgroups appropriate for integral J. We shall examine more closely the detailed structure of those groups listed in table 3, ex- cluding temporarily the infinite groups listed at

the bottom of the table and the cubic point groups.

These we divide into four general types

1. The cj^cHc groups C„

2. The dihedral groups D„

3. The holohedric groups C„XI, D„XI

4. The hemihedric groups, which are the subgroups of the holohedric groups which do not

contain I alone but only in products with rotations.

The cyclic groups C„ are Abehan, of order n. The elements of the group are the various powers

of C„ with Cl= E. Every element is in a class by

itself; consequently, there are nreps for the group.

The dihedral groups D„, ?i>l, are of order 2n,

and include the n powers of C„ and products of

these n elements with the elements C2'. Products

of C„ and Cg are related by C2C„=C~„'C2. Hence

the group in non-Abelian for ?i>2. If n=2,

every element is in a class by itself, otherwise the

class structure is more complicated. For w]>2,

the element C* and its inverse C^~* are in the same class. If n is odd, there are n-1 powers of

Cn (eliminating the unit element in a class by

7h 1 itself) which are grouped by pairs into classes.

24
The n products of C2 with the powers of C„ are all in the same class, so we have a total of

71— l_rH-3

classes. For n>2 and even, the element C"'^

is its own inverse and like E is in a class by

^ 2

itseK. The other n-2 powers of C„ fall into

^

classes, each with two elements. The n products

of the form C*C2 will in this case fall into two

classes, one with k even, the other with k odd.

Here the total number of classes is given by

The case for n=3 has been discussed above

(section 3.4) in detail in connection with the

definition of class. In the present work, this is the only case we shall be concerned with, though

a little geometric consideration on the symmetry

of the pentagon, heptagon, etc, will confirm the

above discussion of odd n. Since n is odd, an

axis of the form C'2 will pass through a vertex and

the opposite side of a n-gon. For n even, such

axes will join either opposite pairs of vertices or

sides, and no operation of the group either represents a rotation about a line joining a vertex and a

side, or will transform a rotation about one type

of axis into one about the other.

The holohedric groups D„Xl, C„XI contain

twice as many elements as D„, C„ and hence are

of order 4n or 2n, respectively. Since I commutes

\\-ith all elements, the group C„X I is Abelian. It

is interesting to note that for odd n it is also cyclic

since the group can be written in terms of the

powers of C„I, with (IC„)"=I. In the case of

D„X I, if A, B, and X are elements of D„ such that

A=XBX~\ the element IX will similarly trans- form B into A. Thus the addition of I to the

group elements does not affect the class structure

of those elements not containing I. On the other

hand IA=X(IB)X-i=(IX) (IB) (IX)-; so the

class structure of those elements containing I exactly reproduces that of those elements not containing I. Let us consider the relations between D3, D3XI

(=Dza) and the hemihedric subgroup Qj. D3

contains six elements in three classes as follows:

E; C3, Ci; C2, C3C2, C3C2. Dza contains these three

classesand threemore : I ; IC3, ICi ; IC2, IC3C2, ICiCa.

We will now consider which elements must be

suppressed to eHminate I as an element of the

group without reverting to D3. Clearly, I must

itself be eliminated, as weU as the class of C3I,

since (C3l)^=I. If the class of IC2 is ehminated we are back to D3, so we retain these three ele- ments, noting that we cannot obtain I from powers

of these elements as was done with IC3 since the

square of any of them is the unit element. In

looking at the classes not containing I, we must

retain E to have a group, and the class of C3 must

be retained, since C§= (IC2) (ICiC2) and both of the latter are retained. On the other hand, C2

(IC2) =I, so we conclude that the class of C2 must

also be eliminated, so we obtain for the class structure of C3,: E; C3, Cf; IC2, IC3C2, ICiG2 which

is the same as that of D3 (i.e., the unit element,

two elements of order three, and three elements of

order two). Furthermore, the multipHcative rela- tions between the elements of C^^ are the same as

those between corresponding elements of D3. The

product of two elements of Q;,, neither containing

I or both containing I clearly follows the parallel

relations in D3. If only one contains I, say IC3C2,

its products with the elements E, C3, Cf yield I times the products of E, C3, Ci with the element

C3C2 of D3. We shall see later the importance of

the fact that not only is the class structure the

same, but that the corresponding multiplication

table is the same (though the latter implies the

former) for the two groups.

We may generahze these considerations, and

assert that the structure of any hemihedric group

is the same as that of the pure rotation group

from which it is derived through the intermediary

holohedric group of double order. Such groups

are said to be isomorphic. The only difference,

mathematically superficial, is in the geometric

designation of the operations involved. The primary significance of this for the present is that the

mmiber of actually independent groups to be considered is considerably less than the 27 finite groups

enumerated in table 3, and that only the struc- tures of the pure rotation groups and the holohedric groups have to be considered.

3.7. Noncubic Double Groups

We now turn our attention to the double groups,

which we shall designate as nQo, iiDes, etc., depending on the corresponding subgroup of DiXl.

This notation will be used when it is desued to particularly emphasize the double group. Usually

we shaU merely imply these groups through the use of the symbols C^^, D^n, etc., in connection with

half-integral J. Some of these groups were first considered by Bethe (1929). The problem was

again considered by Opechowski (1940), who

clarified many of the ideas involved. We shall begin with the groups nC„. The element of

DiA for <t>=2Tr/n, e=i^=0 is, from (3.7), [Dyi=l]-']

the matrix

e " 0

Lo e » J

This is obviously the element of a cyclic group of

order 2n. We shall caU this iiC„, observing that

iiC" is given by the matrix I J which

V 0 .-1/

we usually designate by R. The unit element is then R^. As in the case of C„, each element of

iiC„ is in a class by itself. The groups and

iiCn are isomorphic. The inverse of hC* is C2n-k TJ cn-k

The double dihedral groups consist of the powers

25
of iiG„, and the powers and products of the matrix

/ 0 +1\

Di^(0, TT, 0) =nCi=l j with nC„ and

\-l 0/

its powers. The element is of order 4; its square is R. We note from dii^ect multiplication

of the corresponding matrices that (11CI2) (iiC„) =

(iiC^^)(iiG2). Groups such as this are called

dicyclic groups, and have the defining relations as abstract groups A"=B2=R, R2=E, A-iB=BA,

a group of order 4n. The class structure may be

determined from the matrices themselves, or from

the defining relations of the abstract group.

A pictorial argument is here inapplicable ab initio,

but we shall investigate the relations between the

structure of D„ and iiD„. We shall use the

defining relations.

The elements which commute with A clearly

include the 2n powers of A, but no element in- volving B will commute with A (unless A~^=A,

which implies that A=R and that n=l, which is excluded, since n>2). Thus the order of the

normalizer of A is 2n, its index is two, and there

are two elements in the class of A, which we see immediately from the defining relations are A

and A~^ Generally, A* and A~* will be in the same class, except when k=n, since A"=R com- mutes w^ith all elements and is in a class by itself. The normalizer of B consists of B, R, B^ and

E, of order 4 and index n, so there are n elements

in the class of B, obtained by transforming B

with the first n powers of A. We have ABA~^ =

A^B:, A2BA"^=A^B; etc., or generally the even powers of A, times B. We have left over the

class of AB, with normalizer AB, R, (AB)^, E. The n elements in the class of AB may likewise be

obtained by transforming by the first n powers of

A, yielding generally the odd powers of A times B.

Summarizing, there is the class of E with one ele- ment, the class of R with one element, the {n—l)

classes of A* with two elements, the class of B

with n elements, and the class of AB with n

elements. Altogether thei^e are w+3 classes. We notice that the class structure is the same for

both odd and even n.

It will now be interesting to compare the class structure of uD,, with that of the group D„ from

which it is derived. Let us write the defining

relations of D„ in terms of the abstract elements

A and B as A"=S2=E; AB=BA-\ so that we

distinguish corresponding elements by italics. Let us further, in iiD„, write powers of A

greater than n in terms of products of powers of

A less than n and the element R. Thus, A""'"*'= RA'', and A~*=A^'*~'^=RA"'~*. Let us consider

the class of A, which is {A, A~^). There are four

elements in iiD„ corresponding to these two,

A and RA to the first, A-i=RA"-i and A"-i to the second. These four elements are in separate

classes by pairs (A, RA""^) and (RA, A"~i). If n

is even, these are both classes of order 2n. If n

is odd, A is of order 2n, but RA is of order n. Similarly, the class of A^, A^~" will generally

split into (A^ RA'^-*) and (A"-*, RA*) . If w is even,

tlaese are of the same order; if n is odd, these are

of the same order for even k, and of different

order for odd A;. The sole exception to this will occur if k=n/2 (which will require n to be even),

where there is but one element in the class {A"''^)

and the two elements A"'^, RA"^^ are in the same

class. Let us now consider the classes of B and B.

If n is odd, the class of B contains A^B, A*B,

. . . A"-iB, RAB, RA^B . . . RA'^-^B, but it does not contain RB, nor does it contain AB.

The class of AB also contains A=^B, . . . A^-^B,

RB, RA^B . . . RA^-iR. On the other hand, the

class of B contains B, AB, A^B, . . . A^-^B, so the class of B will split into two classes, one containing B and RAB, the other containing AB and

RB (and other elements as indicated above). If n is even the class of B will still include only the even powers of A times B, but A''=R will now be

included, so B and RB are now in the same class. The class of B now involves only B and the even powers of A times B, so for n even the class of B

will not split, but will merely contain twice as many elements.

We may summarize this behavior by the fol- lowing diagram, where we give in one column the

typical classes of D„, in the next column the cor- responding classes of nD^, for both odd and

even n.

n odd

iiD„

E (E) (R)

(A'i, RA"-*)

(A-'-'s RA*)

(B, AB, Am . .) (B, RAB, A2B . .) (RB, AB, RA2B . .) n even

(£) (E) (R)

(A*, RA"-*)

(A"-*, RA*)

(A"/2, RA'-zs) {B, Am, Am . .) (B, RB, A2B, RA2B . . .) {AB, Am, Am .) (AB, RAB, A^B, RA'B, . .) We see that the class structure of nD„ foUows the ii same pattern for both odd and even n; the pattern

varies m the case of D„, in particular, for classes » of elements of order 2. Results of the comparison i may be summarized into the following rules, first given by Opechowski: i! 1. For each class of D„ of order other than 2, ) there are two classes of u each having the same [ number of elements as the class of D„. j 2. If there is but one class in D„ of order 2 (w is odd), this win in nDn follow rule (1). 3. If there are two or more classes in of

order 2, {n is even) i.e., there are two or more

mutually perpendicular two-fold axes, these classes

26
will not split in iiD„ but will contain twice as many elements. These rules also hold for the

cubic groups, essentially special cases of D, or D4. Finally, we may observe that we deduced the

I existence of n+3 classes in nD„. This should be

considered in the light of the above rules with the

I (ri+3)/2 classes of D„ (n odd), all of which split

! in nD„ or 3 +«/2 classes for n even, where n/2

classes split in iiD„.

i The relations between the double pm-e rotation

groups and the associated holohedric and hemihedric groups is the same as between the single

rotation groups and their associated holohedric

and hemihedric groups, and therefore need not be

discussed separately.

3.8. Reps and Characters

We shall now examine briefly the notions of an

invariant subgroup and of a factor group. We

shaU see that these, together with the orthogonalitj' relation for group characters, will permit us

to derive the characters for most of the finite groups in which we shall be interested. While

these group characters have been derived by very

general considerations, a detailed study will prove

to be very instructive. Let us consider a subgroup S of G, of order s and index n {ns=g). If X is anv element of G, S is an invariant subgroup

if XSX-i=S, or XS=SX. This does not imply

that every element S, of S commutes with X, but

onh- that XSi = SjX, or that the set of elements

XS is the same as the set SX, differing only in the

order if at all. li n=2, we may ex-pand G into

the sum S+ST, where T is an element of G not

in S, or into the sum S+TS. We see that ST=TS,

or that any subgroup of index 2 is an invariant

subgroup. It is clear that S is either a class of G

or a sum of classes. Let us associate with each element X of G the

complex SX. While there are g X's, there are only n distinct complexes SX. Let P and 0 be

elements of G with product PO=R- The associated complexes, similarlv multiplied, give (SP)

(SO)=SSPO=S(PO) = (SR), smce S commutes

with any group element, and SS = S. We thus see that a relation PO=R holds either for the group

elements themselves or for the associated complexes SP, SO, and SR. These n complexes

constitute a factor group of order n, the unit ele- ment being S. The importance of the factor group

for our purposes lies in the fact that any rep of the

factor group will likewise be a rep of the original

group. Since the factor groups are of lower order,

their reps may frequently be found by inspection.

As an illustration of these ideas, let us consider

the group D, with elements E, C2, C2, C2C2. Since

C2, C2, and C2C2 are aU of order 2, E and any one

of them will constitute a subgroup of index 2, and

hence an invariant subgi'oup. There are thus

three "proper" factor groups (of order >1 and

<4):

1: (E, C2) (C2, C2C2) 2: (E, CO (C2,^C2CD

3: (E, C2C2) (C2, C2).

We observe that the set of numbers 1, 1 and 1,-1

constitute reps for the factor groups. We may

then obtain four reps for the whole group by

associating with each of the three factor gxoups

the two reps. This would generally yield sLx reps, but those arising from the rep 1, 1 are the same

for aU three factor groups. These reps are tabulated below. Those numbered 1-3 are obtained

from the corresponding factor gToup by the association 1, —1 of the factor group, while (4) arises

from 1, 1. E C2 C2 G2C2

1 1 -1 — 1 1 -1 1 — 1 1 — 1 -1 1 1 1 1 1 We are now prepared to discuss the characters

of the reps of specific groups, to tabulate their

niunerical values, and to systematize the description of states in terms of their behavior under the

group operations, just as the quantum numbers

L, S, and J, and AIl, Ms, and Mj describe the

behavior of states under general rotations. In the

original work of Bethe (1929), the various reps were designated generally by r, and distinguished

by numerical subscripts such as Ts, Ts, etc. This

notation has persisted in this field, though it has no systematic meaning and does not suggest the

actual behavior of the state under consideration.

An alternate notation has been highly developed in connection with the theory of groups as applied to molecular vibrations (Herzberg, 1945; MuUiken,

1955), and it is a modification of this that we shall employ. While we shaU discuss the notation applicable to each group as it is discussed in detail

below, it seems desirable to cover the notation

generally at this time, particularly to point out

variations from the notation appropriate to molecular spectra.

We shaU consider first the single groups, i.e., those appropriate to integral J. We shaU find

that the reps are of dimensionality three, two, or

one. Those of dimensionality three, which arise

only in the cubic groups, are designated by the

symbol T. If the inversion is a group element,

the symbol will carry the subscript "gf" or "u", as appropriate, and, if necessary, they may be further

distinguished, essentially by convention, by subscripts "1" or "2". Two-dimensional reps will be

designated as E, again with a or if appropriate. The two-dimensional reps of D^^ and

C„„, infinite in number, will be found to corre- spond to the pair of states ±Mj of the free ion;

they will thus be distinguished by Mj appended

as a subscript. For these two gi'oups, molecular

spectroscopists use the symbols 11, A, etc. It

is believed that the notation of the present work

is more adaptable to crystal spectra, since D^^

and C^^ are only useful approximations to a case

of lower symmetry, where the notation is also used in molecular spectroscopy.

One-dimensional reps have as elements numbers,

real or complex, of magnitude 1. If all elements

27
are real, i.e., +1 or —1, they are designated as "A" if the element for rotation about the 2-axis

is +1, "B" if the element is —1. They may be

further distinguished by subscripts 1, 2 according

as the element for rotation about the y-axis is + 1 or —1; g or u for the inversion; or (') or (")

for +1, —1, respectively, for reflection in the x-y

plane. While this is the general policy to which

the molecular notation conforms, it is usually

abandoned in the case of the group Da or Da;,. This is because the x, y, and z axes are distinguishable only by convention, and the reps are called

Bi, B2, Bi. This does not seem to be warranted; even the assignment of subscripts in the 5's must

be done by convention, so one may as well con- form to the general rule. If the elements are not

all real, we use the symbol C/t, according to the

quantity {^Y- The general procedures of molecular spectroscopy merge a complex rep with its conjugate, which must also appear, as a two-dimensional rep designated as "E'\ We are interested

in distinguishing these reps because of their separation when a magnetic field is applied.

The double groups do not appear in problems

of molecular vibrations. Their reps are of dimensionality four, two, and one. That one of di- mensionality four is the appropriate collection of

the elements of the rep 1)3/2 of II2, so we designate

it as D3/2 here also. Some of those of dimensionality two which appear are similarly collections of

elements of A^, which we continue to designate

as Dy^; those two dimensional reps not so designated are assigned the symbol oSk, where "*S"' represents the idea of "spin'' or ''specific", the

anterior subscript for a two-dimensional rep, and

the posterior one (if necessary) by convention.

Similarly, the one-dimensional reps are designated

"(S"', with a distinguishing subscript according

to the root of 1 involved in the rep. The reps of the cyclic groups C„ are all onedimensional. It is clear that just as the group

consists of the n powers of C„ so also wUl the n

powers of Vl serve as a rep. It is similarly clear that the n powers of {^lY wUl also serve as a

rep, where k may be 2, 3, 4, . . . n. If k=n, we

designate it by yl, a general notation for a rep

where every rotation about the 0-axis is the unit

one-dimensional matrix. If n is even, for k=n/2

we use B, similarly a general notation for a rep

where the element of the rep corresponding to the

basic group element C„ is the 1X1 matrix (—1).

In other words, ^ is a symmetric rep, B an antisjrmmetric rep. Note that Cn-ic is the complex

conjugate rep to Ck- The double cyclic groups iiC„ are isomorphous

with the group Can, and hence the reps are the

various powers of CVl)*- For k even, we designate the rep by Ckn or A or B since it wiU then

be also the corresponding rep of C„. For k odd,

we shall call the rep {S for spin or specific) to emphasize that it is peculiar to the double group.

Sk and S2„.k are complex conjugate reps. We note

that, if ijl =o:,o:\l then yi = -u>\ '^^/l = -i<jo.

The dihedral groups are less easily analyzed, i! The reps (and characters) for Da were obtained J above in connection with the illustration of the ! application of the factor group concept. In terms i of numbering there used, we shall describe (4) as ;i Ai, (1) as A2, (2) as Bi, and (3) as B2. Here ! again, A is used for reps which are totally symmetric for rotations about the z-axis, being dis- tinguished by subscripts according to the behavior ; for rotations about the secondary y-axis. B is j used where the basic rotation about the 2-axis is antisymmetric, and again distinguishing various , behavior for rotations about the y-axis.

The group D3 contains the invariant subgroup

(E, C3, C3), so two reps are obtained as the rep | (1, 1) and (1, —1) of the factor group. Since I (3.11) admits of the solution 2^+V + V=6, the

j other rep is two-dimensional and its character i may be deduced from the orthogonality relation ; for group characters. The one dimensional reps

are designated as Ai and A2, while two-dimensional

f reps generally are designated as E.

For D4, 8 elements in 5 classes, eq (3.11) admits

the solution 22 + 12 + 12+12+12=8^ go there are four 1-dimensional reps and one 2-dunensional

rep. Since (E, C4, Q, C4) is an invariant subgroup, two reps are again the reps of the factor

group (1, 1) and (1, —1), designated Ai and ^a-i

We may also verify that (E, Q) is an invariant

subgroup, whose factor group is isomorphous with

Da, having the structure (E, C|) (C4, CI) (Ca, QCO

(C4C2, C4C2). Ai and A2 of Da give no new reps, but Bi and B2 of Da will give us new reps, which we shall designate as Bi and B2 in this case also. The two-dimensional rep E may now be found

from the orthogonality condition.

In the case of Dg, 12 elements in 6 classes, = 22+22+12+12+12+12=12. The group Ce is an

invariant subgroup, and the (1, 1) and (1, —1) ofi

its factor group C3 are Ai and A2 respectively of

Dg. The group C3 is also an invariant subgroup,

with factor group Da, whose reps Bi and B2 give

Bi and B2 of Dg. Finally, the group (E, Co) is am

invariant subgroup with fa,'ctor group D3, whose

rep E will yield a new rep E2 of Dg. We findi from the orthogonality condition the rep Ei..

The reps are numbered according to the reps ofi D„ h with which they may be correlated (table 10). 1 For the double groups generally, cyclic or di-' cyclic, (E, R) is an invariant subgroup whose'

factor group is the corresponding single group, ;

| so the reps of the single group are also reps of the

double group, as we have already seen in the case I of the cyclic groups. For any half-integral J, I [ the traces of D^(E) and Dj(R) are (2J+1)

and — (2J"+1), respectively. Hence we see thatj t we cannot resolve a faithful representation for! 1 half-integral J into a sum of reps, any one ofi 0 which has the same trace for E and R. This) 0 latter is clearly the case for those reps derived 1 1, from the invariant subgroup (E, R), namely those,' ; which are also reps of the single group, so we con- 1 i; elude that they will not be useful for describing

28
i states arising from an odd number of electrons,

i: Those reps which are peculiar to the double

'\ groups we shall call specific reps (the basis for the

i. "S" notation in the cyclic groups used earlier);

li we include the non-specific reps for the completei ness required by the orthogonality theorem and

; for the treatment of selection rules.

II D2 contains 8 elements in 5 classes, so

J 22-M2+i2+i24-i2==8. The four 1-dimensional

'I reps are the reps of D2, so the only specific rep is two-dimensional. It is obvious that this must be

'j equivalent to the appropriate elements of Dy,, or

in other words, the representation is its own rep.

r We shaU emphasize this by designating the rep

f as Dy^.

I 11 D3 has 12 elements in 6 classes, whence

!| 22+22+12+124-12+12^12. Half of these are

! the reps of D3, so there are two specific one dii mensional reps and one specific two-dimensional

j! rep. We observe that (E, nCi, R11C3) is an int variant subgroup. Its associated complexes

(cosets) are (aC;, nClnCj, RiiC3nC;), (nG3, R,

RnCl) and (nCanCs, RnCj, RiiClnCj) yielding a

factor group isomorphous with C4. The reps A

i and B of C4 yield no new reps, but Ci and C3 will

yield specific reps which we shall distinguish as

Si and S3. Because of these two 1-dimensional

reps into which the representation from Dy could

perhaps split, we cannot conclude that the repre- sentation from is also a rep here, but this

proves to be actually the case if the two-dimensional character is calculated from the orthogonality relations. As before, we call this rep Dy.

II D4 has 16 elements in 7 classes. Since

22+22+22+12+12+12+12=16, we find on subtracting the reps of D4 that there are but two

specific reps, both two-dimensional. One of these

! must be D^; the other, derived from the orthoI gonality relations, we shall call 2*S'.

j II De has 24 elements in 9 classes. Equation

I (3.11) gives us 22+22+22+22+22+l2+i2-|-i24_

! 12=24. Eliminating the reps of Dq, we have

I three specific reps, aU two-dimensional. One of

[ these must be Dy. Unfortunately, we cannot

! obtain either of the other two characters by the

methods used so far, and an alternate procedure

must be used. We obtain from table 7 the chari acter of the representation arising from D3/2, which

is four-dimensional and hence reducible. By application of the expansion theorem (3.10) we see that Dy, is included once in the representation.

I The other rep involved we shall call 2S1. The

I final rep may now be obtained from the orthogonality theorem as 2'S'2. The liolohedric groups, single or double, are

: defined as the direct product of the group (E, I) = I with a pure rotation group. The reps may be

obtained as products of the reps (1,1) and (1, —1)

of I with those of the pure rotation group. The

, reps are designated like those of the pure rotation

i group, with the additional subscript "g" if (1, 1)

is used and "u" if (1, —1) is used.

It has already been pointed out that the hemihedric groups are isomorphous with the corre- sponding (in the sense of table 3) pure rotation

group and hence have the same reps. Most of

these hemihedric reps are described just like the

corresponding pure rotation reps. The exceptions are the groups C3;,, Cj, and S4. For the

first three of these, the reps are classified according to behavior under the operation IC2 = ah, symmetric reps carrying a (') while antisymmetric

reps have (")• Dss is further subclassified according to C2. This clifference m notation is peculiar

to the single groups—the specific reps of all hemihedric double groups will be designated like those

of the corresponding pure rotation group. In the

case of S4, isomorphic with C4, the generating ele- ment is taken as the operation IC4 = C4lC2 = C4cr;,, which is usually considered as the fundamental

rotation-reflection operation. This is why the rep

Ci of S4 corresponds to C3 of C4 rather than to

(7i of C4.

3.9. Infinite and Cubic Groups

We have finally to discuss the infinite groups

involving infinitesimal rotations about the 2-axis,

namely C„, C^,,, D„, D„r, C^,, and the cubic

groups. In C„, the group operation consists of

arbitrary rotations C{4)) such that C(4)i) C(<^9) =

C(02) C(0i) = C(0i +^2), and that C(27r) = C(0).

If k is any number, e'*-* will meet the first requirement, but the second limits k to a real integer,

positive or negative, including zero. If y!: =0, we

shall call the rep A, otherwise C^. For the group

iiC<„, the first of the above conditions holds, but

the second is replaced by C(27r) = —C(0). Here

k must be half-integral, positive or negative, and we designate the reps by *S'2j: {2k is an odd integer).

C„7i is C„XI, so the reps for this group will carry

the subscript "g" or "u".

In D^, the group elements consist of the rotations C(^) and the rotation C2, such that C2C(0) =

C{—<j))C'2. Considering as an invariant subgroup, the reps (1, 1) and (1, —1) of the factor

group C2 yield reps A^ and A2 of D<„. As discussed

under C„, e'**" will serve as a rep insofar as C(^)

is concerned, but the operation C2 makes e'**' and

equivalent. Except when k=0 (which gives

Ai and A2) our rep for C(<^) must have the twodimensional form

/e''^ 0 \

and C2 will assume the form

which will transform C(<A) into C(— </>). It may

be shown that there are no other reps. These reps

we shall call E^, where k is a positive integer. In

the case of iiD<„, the reps become

0 \ / 0 1\

iiC(0)= I iiCH

\o e-'^'^'y \-i 0/

551664—61 3 29
and we shall designate them as 2S1C (k odd). Keps

of D^T, will have the additional subscripts "g" or "u", whUe reps of the hemihedric group C^, wUl

be designated like those of D„.

In the gTOup O, there are 24 elements in 5

classes. These are the class of E, 1 element; the

class of C4, 6 elements; the class of C2, 3 elements;

the class of C3, the body diagonal axes, 8 elements;

and the class of C2 (here, C'2 is not along the y-axis),

6 elements. (3.11) gives 3^+32+22+12 + 12=24.

The set of elements (E, 3C2, 8C3) is an invariant

subgroup with factor group of order 2. Its reps

(1,1) and (1, —1) are the reps Ai and A2 of O. If we consider Dj for J=l, we find that neither ^1

nor A2 is in the representation. Since it cannot

be reduced to Ai-\-E or A2-\-E {E being the twodimensional rep), it must itself be one of the threedimensional reps. These are generally designated

as T—this particular rep we shall call Ti. Dj,

J=2, contains neither Ai, A2, or Ti, so it must

reduce to E-\-T2, but we cannot separate them.

On going to J=3, we find from this seven-dimensional representation, Ai and Ti accounting for

four. The remaining three must then be T2

which, subtracted from the J=2 case leaves E.

The groups and Ta follow our previous considerations on pure rotation, holohedric, and

hemihedric groups. The reps of parallel

those of O.

The double group II O has 48 elements. The

splitting into classes, as compared with the group

O, follows the same rules as for the dihedral

groups given in section 3.7, so the class structure

is: E (1); R (1); (6); RnQ (6); (6);

(8);_ RiiQ (8); 11C411C; (12). (3.11), after eliminating the reps of O, indicates three reps for nO,

two 2-dunensional and one 4-dimensional. As

usual, Dy, is one of these reps. in this case

is also a rep, the four-dimensional one. The

remaining rep obtained from orthogonality we

call 2S. The group T, 12 elements, may be separated

into the class E (1); C2 (3); Q (4); and C| (4). It is interesting to note that, in spite of the rela- tively high symmetry, a rotation C3 and its inverse Cz^ = Cl are in different classes. (3.11)

yields 32+12+12+12=12. The elements (E,3C2)

constitute an invariant subgroup, with a factor

group of order 3, C3. The reps A, C\ and C2 of

C3 are thus the one-dunensional reps A, Ci, and C2

of T. The rep T may then be obtained from

orthogonality and completeness considerations.

The group iiT, 24 elements divides (section 3.7)

into E (1); R (1); ^Cz (6); 11C3 (4); R„C3 (4); iiQ (4); and RnCI (4). Subtracting off the

reps of T, we have 22+22+2^=12, indicating reps we shall call D^, 2S1, and 282- As the notation

suggests, is its own rep. Unfortunately, our

previous techniques for finding reps fail in this case, since the reps 2'S'i and 2*S'2 always appear

together and cannot be separated. We must

therefore proceed to a more general method.

The product of any two classes of a group [Ca]

and [Cfi] with Na and Nf, elements, respectively,

wUl be a total of NaNb elements, not necessarily

all distinct, which may themselves be written as

a sum of classes [GJ, or [CJ[C6] =l]A^a,6.JCJ,

g with Na.-b:q a small nonnegative integer. A particular 2 may not appear at all, or may appear more than once. It may be shown (Murnaghan,

1938) that this expansion implies the following

relation between the traces of the i-th rep

NaN,X

9 where XiE is the trace of the unit element, i.e., the

dimension of the rep. By developing in detail

the multiplication table of a group, sufficient

relations may be found to calculate the complete

character table of any group. In the present

case, we already have all but two of the seven

possibilities for nT. A solution of these equations for XiE=2 will then give the desired

characters.

j 3.10. Table of Group Characters I The foregoing considerations have been sum- j marized in table 8, a table of group characters for

both the single and double groups in which we

shall be interested. Each portion of the table is constructed around one of the single holohedric

groups. Typical elements of the various classes of the group, together with an indication of the

total number of elements in the class, are shown

to the right of the group designation, while the

designations of the several reps are shown under

the group designation. For convenience, we also give the notation of Bethe (1929). The characters j for the various classes and reps are the entries in the table. The classes, reps, and characters for

the corresponding pure rotation group may be

obtained by dropping all classes containing I and

the subscripts "g" and "u" on the reps. Above and to the left of each basic character

table are shown the classes and reps of the corre- sponding hemihedric groups. The appropriate

group designations are given upward and to the

left of that of the basic holohedric group, while

the class structure is given along the same row

and the several reps in the same column, so that

both classes and reps correspond to those of the

holohedric group. A careful study of these will indicate very clearly the relations between a holohedric group and its hemihedric derivatives.

The characters and reps for the specific reps of

the double groups are given in table 8 immediately

below the reps of the single groups. Because of

the frequent splitting of the classes of the single

group into two classes discussed above, there will in such cases be two characters for each class of the single group. If A is a class element of the

single group, the first double group character wUl

be that for the class of nA, the second for that

of RiiA. Upon inspection, it will be seen that

such pairs of characters have opposite signs, while

those arising from classes of the single group which

do not split are all zero.

30
3 II •« r r -t" -t- T T T T ^<;j <»'^ 'U'ii

-e- ©-

i I I I I I I

I 1 i

I I I I I

I I I

-SII II

u "2 MS ce ^= t,3 ceO J.I > I^J . - - I.I . e _ I . I OO g C3Cj bo CiCi oq&J oqsq c>2 0q

I I oo oo oo oo oo oo

5

(M IM IM (M

IM N s s CO CO s s

CO CO CO CO CO CO CO to M CO CO to o o O O o o o o o o O O

o o o u o o o o o o o a

(N (N (N (M IM IM IM IM (M IM

n

<! H

(M IM IM (M C<1 iM

I I I IM IM (M IM IM IM

I I 1 IM IM IM IM IM IM

I I I

I 1—I 1—I T—I O O

I I oo oo oo CO oo

IM IM

-0.-6- ©e- IM IM

IM IM s s

m CO «3 CO to m

o o O O o o o o

o o o o

(M IM IM IM IM iM IM (M

CO CO o o

o a

IM IM IM (M

-©6-

CO CO

CO CO o o

'A ,-1 IM IM IM <M C<1

IM IM IM IM IM II M II

(M IM IM IM IM

Cqtq ^Eq lijBq 02 CQ

U

CO

X!

31
oO

00

u

u

CO

1 1 1 1 OOO O o o o o o o

u

CO

1—( I—

i 1 77" o o o o o o o o o o

7 1

CI d (M d

1 1 o o o o o o

U

CI

1 1 1 1 1 1

T-H T-^ CI CI

1 1

>-l T-H CI d

1 1

I-H >—

1

1-H T—

1

U

1 1 1 1 1 1

l^ko o o

1

I^ICO o o

1

1

ICO Ico

1

ICO 1^

1 d CI CI d d d

1 1

1 1

1 1

CI C? CI CI

1 1

1 1 d d d d

1 1 d d1

u

CO u1—

fO

ISu

CO

T—

1 1 7

1—1 1—1 1—

1 o o o o o o o o o o

u

CO O1—

CO u

CO

I—1 I—1 T—

1 1

T-( r-H i-H

1 1 o o o o o o o o o o

U U U

T—

I—I 1—t 1—

7 777

CI CI CI CI

1 1 o o o o o o

^ T-< d d

7 7

o

CI u O

CI

1 1 1 1

i 1

T-H ^ d d

1 1

U u U

1 1 1 1 1 1

ICO O O

1 1

[CO ICO O O

(CO Lco

(CO Ico

i 1 d d d d d d w 1-H T—1 I—

i-H 1—1 1—

(

CI CI CI o

1 1 1 1 d d d d

i 1 d d

bo 3^ 3 5 § ^ s

o

Q

32
I—

I

(N

o

o

I—

(

I ! I I

^^rt ^^^rt oo o o oo

II II

rt^^ ^^^^ <N(MOO oo II III

^^^^ oo^^ 1

'

' ' l(NI(M liNliM

I I

IM <M <M (N

I I

I I I I I (N (N IM CO

I I

rtrt—1,-H OOOO OO

II II

,_(^^ ^rt^rt OOOO oo

II II

,H>-(,-i ^ ^ ^ ^. IMIMOO oo

I I

|IMIC<1 |IM|(N

,-1 >-l ,-1 l-H T-H oo

I I

(M IM IM (M ^, , c^c^ I I II

IM (M IM IM

Be 3 ?! ee 3 b

•.s> '^i. 'S' ^ o

I I I

I I I

i-H r-l 1—I T-l IM (N

I I

(M IM

I

*^s> -fSi •'S' '(S* o O

I I

( y—^ r-i O O

II '-s. --s- o o

I I

>-l ^ T-< r-l (M ^ .-^ IM IM II MM

-^1 ^ Cq CO &a Q

33
00

IS 7""7 o o

o ^77^ "77" oo

c5

1 1 1 1

I—1 T—( 1—< 1—

(

1 1

I—1 .—1 1-H i-H

1 1

1

1

61—

U

U II M

u y II II

u

' u

1 II 1

tn

1 1

a

OO a 64 3 ^ ^ "^j

M » W 3 ojcqcqcq

60 3

cq cq

3 3'3'3

•(S- 'Oi "c* 3 3

iS> *<St '-s* y 3 3

1

"3^3

1 1 3 3

1

1

'3'^3

1

1

1

1 1

."3 ."3 T

"

" T

1 1

.3.3 J J

T

"

" T

1 1

.^.^.3.3 r

"

" r

"3 "3

1 3 31

i—i t—

11

1

1 3 3

1

1

11

3 3rH,-l

1 1 3 3 1—1 1—

1 1

^3^3 3 3

1 1

3 3 3 3

1 1

1

1

1—1 1—1 3 3

1 1

u 1

1 1 1 1 1

1

1

1 1

1 1

1 1

1 1

1 1

1

U 3 3

1

^3 ^3

1

1

1—1 i—H

1

1 3 3

1

"3 "3

1

1

1

^3 ^3 ^ ^

1 1

3 3 '-^

1 1

3 3

"3 '^3

1 1

3 3 3 3

1 1

-^7 3 3

1 1

T-H l-H 3 3

1 1

3 3 '3^3 3 3 "3 "3 M

3 '3

1

3 3

1 7" "3^3

1 3 31 "7

1 1

•e^

1 1

1 1 J

."3. 3.

3

1 1

'c*

1 1

11 II II

"3 "3

-r* 3 3'3'3

•iS< "c-i tS' 3 3•?v> -iSi "Ti u y 3 3

1 1 77

3 3 °3 °3

1 1

I—1 1—1

1 1

'"S ^3

cSi "oi 'csi

1 1

1 1

.3. 3."3 ^3 r T ~

1 1

.^.^.3.3

r r

w

"3 '3 3 3 ^3^3 3 3

3 S"-"^

1 1 1 1

3 3 rH ,-H

^3^3 3 3

1 1 1 1

"3 '3 3 3

^ ^3 "3

1

1 1

rH rH ^3 "3

u

1 1 1 1 1 1

1 1

1 1

1 1

1 1

1 1

1 1

u c5 3 3 3 3 ^3 "3 l-H i-H

"3 l-H rH ^ "3 i-H .-H II II

3 3

"3

'3 3 3 "3 '3

1 1 1 1

3 3 rHrH 3 3

1 1 1 1

.3.3.^.^ ."l J.3.3

"3°3

-<S. 't^

to

l-H u

"3 '3

1 1

3 3 i-H l-H

1 1

"3 "3 3 3

1 1

^

T r

3 3

1

"

" r r

"3^3 3 3

1

1 1 3^3

tS> 'r^

1 1

r-rl

•c O 00

6a 3 oTcq"

6c 3 » 3 w 3 ^ 3 6e 3 60 3 02 02 02 CO

0 0 02 CO ^ 02

34
- T T T

- 1

1 1 i 1

1 1 1 1 •„11 II

o

3 3 ^3 '3 "3 3 3 '-' ^ 1 1

1 III 3 3 "3 "3

1 1 1 J ^^^^ ^V-;=-S»

1 1 1 1 |-«|-«|-«|-«

'

1 ' 1 U

^3 ^3 3 3 3 3 "3 "-I 1 1

1

1 1 1 . 3^3 --I 3 3

1 1 1 II II

II II M

1 1 1

u U 1 1 1 1 [^l^l^l^

T T 1 1

1 1 1

u

3 3 'a "3 -3^3 3 3 -- II 1 ' II

3 3 1-1 — ^3 ^3 o U ^^^^ ^^^^ II II

II II ..... II II u

'3 "3 -1 3 3

3 3 ^3 — — „ „

3 3 --^ -H 3 3 II II II u

tii

M

CO «3 ^»

00 00 COCO C0£>3 CC'C^ J

"£•.3 ^i"a "^"4 «3»3

^ Ck; OO'"^"^ tiQ^OQCc OQCoc^ao?

00

35
Table 8. A table of group characters—Continued

c. E IC,

E C2 ' I IC2

^R

s 1 _ I i — i Bu 11 -1 — i i A' A, 1 1 1 1 Au 1 1 -1 — 1

S\g 1 -1 i — i 1 -1 i —i

Q 1 -1 t — I -1 1 — i i Sig 1 -1 — i i 1 -1 — i i

1 -1 — i i -1 1 i — i

Au

A,

4:A2

E

T2

D

2^

D3/2

Aig(Ti)

A2g{r-^

A2U

Eg(Tz)

T2g{ Ts)

DngiTe)

2SAT7)

Dv2g(T,

D.J2U

2 -2

2 -2

-2

-2

-4

-4

6IC4

ec^

11 -1

-1

0

0

1

1 -1

-1

^|2 -V2

V2 -V2

-V2

-V2

0

0

V2~ V2

00

3C2

sc.

1

111

2

2 -1

-1

-1

-1

0

0

0

0

0

0

6IC^

6C2

8C3'

8C3'

1

1

1

1 -1

-1

0

00

0

1

-

1

-

1

-

1

-

1 -1

1 -1

1 -1

2 -2

3 -3

3 -3

2

- -2

2

- -24

-4

6IC4

1 -1

-1

1

0

0

1 -1

-1

1 V2 -V2

-V2 ^|2

-V2 V2

0

0

V2

-V2

0

0

3IC2

1 -1

1 -1

2 -2

-1

1 -1

1

0

0

0

000

6IC^ 8IC^

1 -1

1 -1

-1

10

0

0

0

1

-

• 1

1

-

-1

1

1

-

T,, E 3 Co 4C3 4 CI I 3IC2 4IC3 4ICi

Ag 1 1 1 1 1 1 1 An 1 1 1 1 -1 -1 -1

C\g 1 1 01 co^ 1 CO Co2 1 1 0) co^ -1 — CO -Co2

C2g 1 1 w- CO 1 Co2 CO C2U 1 1 co- CO -1 -co2 — CO

Tg 3 -1 0 0 3 -1 0 0 Ta 3 -1 0 0 -3 1 0 0

2 -2 0 1 -1 -1 1 2 -2 0 1 -1 -1 1 2 -2 0 1 -1 -1 1 -2 2 0 -1 1 1 -1

2 -2 0 CO — to — co- C02 2 — 2 0 10 — CO — co^ co- 2'Slu 2 -2 0 01 CO -C02 Co2 -2 2 0 CO CO co* — CO*

2 -2 0 CO" — co^ CO CO 2 -2 0 03^ CO- — CO CO 2^2 „ 2 -2 0 CO- — CO- CO CO -2 2 0 — CO- CO- CO — CO

36
4. Degeneracy, Selection Rules, and Polarization

4.1. Splitting of Free Ion Levels

i We are now in a position to apply the results of our previous considerations to the problem of

degeneracy in the presence of a perturbing external

field. The general procedure has been covered

i in the previous section; we consider here the spe-

! eific example of a free ion level described as 4a(t/=4,(— 1)^'*= — 1) in a field of C^^ symmetry. From table 8 we obtain the class structure

of C6„ as E, 2C6,_ 2C3, Co, 3IC2,_ 3IC3C2. The

traces of the matrices of the rotation group J—4: corresponding to the first four of these classes may be obtained directly from table 7. Those for

the last two classes will be —1 times the traces

of the corresponding pure rotation. We obtain

the following for the character table of a reducible

representation of C55. We also include for immediate convenience the reps Ai and A2.

E 2C6 2C3 C2 3IC2' 3IC3C2'

9 -2 0 1 -1 -1

1 1 1 1 1 1 A2 1 1 1 1 -1 -1

The expansion theorem for group characters

(3.10) was given in terms of the weighted traces

and

but it is usually more convenient to use the

traces directly. Thus ai =^i-E (3.10) becomes gat =y^,NiT(*iKi. Using this latter form, we obtain for

^x/l2a^ =l(l)(9) + 2(l)(-2) + 2(l)(0) + l(l)(l)

+ 3(1)(-1)-1-3(1)(-1) =0, so Ai will not appear in the reduction. Likewise, 12aA2=l(l)(9)

+2(1) (-2)+2(l) (0)+l(l) (l)+_3(-l) (-1)

+ 3(— 1)(— 1) =12, so ^2 wUl appear in the reduction once. The results of this and similar calculation for all the subgroups of Dj of present interest are given

in table 9. As in table 8, the results for the pure

rotation groups are not given separately, and may

be obtained from the associated holohedric group

by omitting the "g"- "u" distinction. Only the

group I has been omitted; this has no sjrmmetry

apart from the inversion, and states are either

Ag or Au. The integral range of J is from 0 to 8, while half-integral J ranges from to If only

electric fields constitute the perturbation on the

free ion, the appropriate group may be obtained

from table 5 or from crystallographic considerations. For a pure magnetic field, we shall consider

only a uniform field with S3rmmetry C^;,. In case both fields are present, the appropriate group is the group of those symmetry operations common

to the electric and magnetic fields. Unless the

magnetic field is parallel to one of the symmetry

axes of the crystal field, there wUl in general be no symmetry common to both perturbations.

In accordance with our previous discussions,

the physical implication of this reduction is that

all one dimensional reps appearing in the reduction

represent states arising from the free ion level J,

which are not related by symmetry to any of the

other states so arising. These states, then, are

either nondegenerate or "accidentally" degenerate,

the latter being due to numerical values of the

parameters hence very improbable in any actual

case. Any two-dimensional rep represents a pair

of states from the original 2t7+l which are

essentially degenerate, being related by symmetry

operations, but not degenerate with the other

2J—1 states. Similarly for any reps of higher

dimension. If a particular rep appears only once

in the reduction the state or states it represents

may be uniquely associated with a particular

combination of the original 2t/+l states. If the

rep appears more than once there will be two or more states of the original 2J+ 1 having the same symmetry properties, and the appropriate states

in the crystal may be a linear combination of

these. This can only be determined from the

numerical parameters involved. These considerations are modified by Klramers degeneracy to be

considered in detail in section 4.2. Certain pairs

of reps are noted in table 9 as being Kramers

conjugate states. These states, even though

belonging to different reps, are degenerate if the

pertiirbing field is purely electrostatic.

It is often of considerable interest to follow the

behavior of a given level during the introduction

of perturbing fields of increasingly lower symmetry.

For example, one may wish to see the effect on the

levels under D^a symmetry of a reduction of symmetry to Csp. This may be accomplished by the

correlation table, table 10. Since most of the

groups with which we are presently concerned are subgroups of D„s, we start \vith the reps of D^^.

For each group of lower symmetry, the designation

of the level or levels is given in the appropriate

column. Again, the pure rotation groups are not

given separately, requiring only the suppression

of "g"-"u" in the associated holohedi'ic group.

The procedure involved here is the application to the characters of the reps of D^^ precisely the same procedures as were applied to the rotationinversion group characters in constructing Table 9. Notice that here, in many cases, the rep of D^s is

still a rep of the subgroup, since the levels fre- quently remain twofold degenerate.

For an example, let us consider the state ^„

under the influence of a crystal field of symmetry

551664—61 4 37
Table 9. Type and number of perturbed levels

0 1 2 3 4 5 6 7 8

g u 9 u g u g u g u g u g u g u g u

1 1 1 1 1

A 1 1 1 1 1 A,o 1 1 1 1 A2U 1 1 1

1 1 1 ] 1 1 1

1 1 1 1 1 1 1 1 1 1 1 ^coh E.u 1 1 1 1 1

1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1

Eso 1 ] 1 1

1 1 E^g 1 1 E^u 1 1

ElQ ^ 1

1 Eyu

Eso E%u

A, 1 1 1 1 1 1 1 1 1 Ao 1 1 1 1 1 1 1 1

E, 1 1 1 1 1 1 1 1 1 1 1 1 1 Ceo V E2 1 1 1 1 1 1 1 1 1 1 1 1 1

E3 1 1 1 1 1 1 1 E, 1 1 1 1 1 1 1

E, 1 1 1 1 1 E^ ; ; 1

El 1 Es 1

Ao 1 1 1 1 1 1 1 1

A 1 1 1 1 1 1 1 C±lg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

C±2e 1 1 1 1 1 1

ri 1 1 1 1 1 1 1 1 1 1 1 Coo h 1 1 1 1 1

1 1 1 1 n 1 1 1 C±5o 1 1 1 1

1 1 1

1 1 n 1 C±7ii 1 1 C±Tu

^±8« 1

Note: The states Cng, C-„g, and C„u, C_„„, are Kramers conjugate states.

38
Table 9. Type and number of percurbed levels—Continued

0 1 2 3 4 5 6 7 5

g u a y n n

y u n

y XL y g u 9

n

y li

1 1 1 2 1 2

1 1 1 2 1 2

1 1 1 1 2

1 1 1 1 2 1

Big 1 1 1 1 1 1 Bin 1 1 1 1 1 D

^2 a 1 1 1

1i 1 1

1 1 1 1 1

El a

1±1

J. 11

1i 0 9 Q

1 1 1 1 2 2 3 3 E20 1 1 2 2 2 2 3

1X 9 9 9 9 0

A 1 1 1 1 1 2 1 1 2 2 1 A2 1i 1 1 1 1 1 1 9 2 1 1 2

r> 1 1 1 1 1 1 1 1 1 1 1 1 B2 11

11

1i

1i1i 111 i111i11

E, 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 E2 1

1i i 9 9 9 9 9 9 9 9Li

•}

A

' A\ 1 1 1 1 1 2 1 1 1 2 1 A2 1 1 } 1 1 1

11 1 2 1 1 1

1 1 1 1 1 1 1 2 1 1 1 2

111i1±1i

1 1 1

J. 11 9Li 1

J. 1J.

E' 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 E" 11

1111

11

1i Li 9Li 9Li 9 9Li 0 9Li 0Q0

A 1 1 2 1 2 1 3 Alu I I 2 \ 2 1 3

A 1 1 1 2 1 2 2 A2U X \ 1 2 1 2 2

-r\ P

"iff 1 1 1 1 2 2 2 Blu 1 1 1 1 9 2 2

•O20 1 1 1 1 2 2 2 B2U 1i

-11

111i 9 9Li 9Li Eg 1 1 2 2 3 3 4 4 Eu 1i 1 0 Q 3

A A

AAi i 1 1 1 2 1 1 2

9 1i 1 2 3 2 A2 1 1 1 1 1 2 2 1 1 2

0 1 9 Q

C4. B, i i

-11 1-11

1i

1 9 9 9 9 9 2

1 1 1 1 1 1 1 1 2 2 2 2 2 2 E 1 1 1 0 z

0 9 q0Q0 3 3

/I /I 4

AAx 1 1 1 1 2 1 1 1 2 2 1 2 3 2 A2 1 1

1 1 1 1 9 i 1 2 9Li 9 9 9LI

Dad B, 1 1 1 1 1 2 1 1 2 2 2 1 2 3 B2 1 1 1 1 1 1 1 2 2 1 2 2 2 2 E 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

39
Table 9. Type and number of perturbed levels—Continued

0 1 2 S 4 5 6 7 5

U 9 u g U g u ? g u u ^? u

1 1 1 2 1 3 2 3 A , 1 1 1 2 1 i> 2 3

1 2 1 2 2 3 2

11 i 9 200

9

Eo 1 2 2 3 4 4 5 6

TP 1 2 2 3 4 4 5 6

A 1 1 1 1 2 2 1 1 2 3 2 2 3 3 2 A. 1 1 1 2 1 1 2 2 1 2 3 3 2 2 3

Mil 1i 1 9 9 9 00 q A A K c0

G AU

Aio 1 2 1 3 2 4 3 5 A , 1 2 1 3 2 4 3 5

1 1 2 2 3 3 4 4

1 1 2 2 3

q0 4 4

Big 1 1 9 2 3

q0 4 4

11 i 9 9 q0 3 J. A

Big 11

11 9 9 0 3

/l T> 1 1 2 2 3 3 4 4

A, 1 1i 9Li

11 1 9 0 9 9 0

A 0q 4. 4 At 1 1 1 2 2 1 2 3 3 2 3 4 4 3 4 5

C2. Br 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

111i1i

1i 9 9 9 9 Q q0

q0

A J.

1 1 1 1 1 1 3 3 3 A 1 1 ]^ ]^ 3 3 3 Bg 2 2 2 2 2 2 R•"a 2 2 2 2 2 2

1 1 1 1 2 2 3 3

r». \ ]^ 2 2 3 3

1 1 2 2 2 2 3

<^2m 1 1 2 2 2 2 3

1 1 2 2 2 2 3

C'4a 1 1 2 2 2 2 3

1 1 1 1 2 2 3 3 Csa 1 1 1 1 2 2 3 3

Note: The states (Cio, C50 )i (C20, (Cza, Cda) are Kramers conjugate states.

1 1 1 1 2 1 2 1 2 3 2 3 2 3 2 A" 1 1 1 2 1 2 1 2 1 2 3 2 3 2 3

c; 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 cr 1 1 1 1 1 1 2 2 2 2 2 3 2 3 3

c; 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3

1 1 1 1 1 1 2 2 2 2 2 3 2 3 3

Note: The states (CJ, C2) and (CJ', C2') are Kramers conjugate states.

40
Table 9. Type and number of 'perturbed levels—Continued

0 1 2 S 4 5 6 7 8

g u g u g u g u g g ? u g u

1

1

1

1

1

1 3

3

3

3

3 3 5

5

5

5 5 5

Cin

1

1 22 22

3 3 4

4

4 4

5

5

6 6

C2g C2U1

1 22 22

3

3 44 44

5

5

6 6

Note: The states (Cu, C20) and (Cia, CjO are Kramers conjugate states.

Ao 1 1 1 1 3 3 3 3 5 Au 1 1 1 1 3 3 3 3 5 Bo 2 2 2 2 4 4 4

C4i Bu 2 2 2 2 4 4 4

Clg 1 1 2 2 3 3 4 4 Clu 1 1 2 2 3 3 4 4

1 1 2 2 3 3 4 4

1 1 2 2 3 3 4 4

Note: The states (Cu, C30) and (Ciu, Ca,.) are Kramers conjugate states.

1 1 1 2 1 2 3 2 3 2 3 4 3 4 5 4

S4 1 1 2 1 2 1 2 3 2 3 4 3 4 3 4 5

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

C3 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

Note: The states Ci and C3 are Kramers conjugate states.

^0 1 1 3 3 5 5 7 7 9

1 1 3 3 5 5 7 7 9

2 2 4 4 6 6 8 8 i: 2 2 4 4 6 6 8 8

c. 1 1 2 3 2 3 4 5 4 5 6 7 6 7 8 9 8 ^" 1 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8 9

1 1 1 1 4i« 1 1 1 1

A20 1 1 1 A2U 1 1 1 Eo 1 1 1 1 1 2 Eu 1 1 1 1 1 2

1 1 1 2 1 2 2

1 1 1 2 1 2 2

T20 1 1 1 1 2 2 2

T2U 1 1 1 1 2 2 2

41
Table 9. Type and number of perturbed levels—Continued

0 1 2 3 4 6 6 7 8

g u g u g u g u g u g u g u g u g u

Ai

1

1 1

1 1 1

1 1 11 1

\ E 1 1 1 1 1 1 1 1 1 1 2 2

T, 1

1 1

1 1

1

1111

1

1 21

12122

1 22

2

2 22 22

A,

Au

1

1

1

1

1 1 2

2

1

1

1

1

C\u

C20

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

1

1

1

1 2 2 22

3

3

3

3 4

4 4

4

Note: The states (Ci^, C2a) and (Cm, C2u) are Kramers conjugate states.

1 S 5 7 9 11 IS 15 2 2 2 2 2 2 2 2

u g u g u 9 u g u g u g u

2510 1 1 1 1 1

2'Slu 1 1 1 1 1

2'S3t, 1 1 1 1

2'S3!, 1 1 1 1

Doo/i 2'S5o 1 1 1

2'S5u 1 1 1

2'S7£i 1 1 2^7^ 1 1

2^9(7 1

2'S9m 1

2*51110 1

2*511!, 1

2'Sl3o 1

1 1 2S15U 1

2*81 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2^3 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

21S7 1 1 1 1 1 1 1 1 1

2S9 1 1 1 1 1 1 1

21811 1 1 1 1 1

2'S'13 1 1 1

2'S'15 1

42
Table 9, Type and number of perturbed levels—Continued

Silo

<Sdb3(;

Sztog

Coo A "Si.ju S±nu

S± 13 u

S

2 52

72

9

2

112 13

2 15

2

g u g u g u g u g u g u 0 u g u

1

1

1

1 1

1

1

1

1

1

1

1

1 1

1

1

1 1

1

1

1

1

1

1

1

Note: The states (S„„, S.„i,) and (S„u, S_„J are Kramers conjugate states.

2^2 u

1

1

1

1

1

1

1

1

1 1

1

1

1

1

1

1

2

2

1

1

2

2

2 2

2

2

2 2

2

2

3

3

2

2

2

2

3

3

3

3

2 2

2^2

1 1 1 1 1 11 1

1 1

1 1

1 1 1 1 2 2

1 1 2 2

2 2

2 2

2 2

2 2

3 3 2 2

2 2

3 3

3 3

2 2

Dh

D,H 2^, 2S2

1

1

11 1

1

1 1

1 1

1 1

1 2

1 1 2 1

1 2

2 2

2 1

2 2

2 2

2 2

3 2

2 2

2 3

3 2

3 3

2 3

Da.

2Su

1 11

1 1

1

1

1 2

2

2

2

2

2

3

3

2

2

3

3

3

3

3

3 4

4

4 4

4 4

Cj, Da

2S

1 1 1 1

1 1

1 1 2 22 2

2 2

3 3 2 2 3 3

3 3 3 3 4 4

4 4

4 4

Da

2S

1 11 1 1 1

1 2

2 1 2 22 2

3 2 2 33 3

3 3

3 4 4 3 4 4

4 4

Dao

Dau

D3d

Si.

Ssu

1

1

1

1

1

1 1

1

2 2

1

1 1 1

3

3

1

1 1 1

3

3 2

2

2

2

4 4

2 2

2 2

5

5

2

2

2

2

5 5

3

3 3

3

Note: The states (Sio, S30) and (Sm, S3J are Kramers conjugate states.

43
Table 9. Type and number of 'perturbed levels—-Continued

3

2 52

7

2

5 i7 13

2 J5

2

g u g u g u g u g u

1 1 i 1

1 1 1 1 1 1

9 9

1 1 1 1

0 0

1 1

1 1

0 0

2 2

2 2 2 2 2 2

0 0 2 2 2 2

0 0

3 3

3 3

Note: The states (Si, S3) are Kramers conjugate states.

Slo s,„

S3„ Sin

C4;, Sjp

1

1

1

1

1

1 1

1 1

1 1

1

1 1 2

2

2

2

1

1

2

2 2

2

2

2

2

2

3

3

2

2

2

2

3 3

3

3

3

3

3 3

3 3

3 3 4

4

4 4

3

3

4 4

4

4

4 4

4 4

Note: The states (Sio, Syg), (S^, Stu\ {Ssg, S^g), (Ssu, S^u) are Kramers conjugate states.

Sis,

Si S5

Sy

1 1

1

1

1 1

1 1

1 1

1 1

2 1

1 2

1 2

2 1

2 2

2 2

2 2 2 2

2 3 3 2

3 2

2 3

3 3

3 3

3 3

3 3

4 3 3 4

3 4 4 3

4 4

4 4

4 4

4 4

Note: The states (Si, S7) and (S2, S5) are Kramers conjugate states.

Do;, A<» 1 1 2

2

3 3 4

4

5

5

6

6 77 8

8

Sis s,„

S30 S3„

Cf, S51, S5„

Sju

Sgg

Ssa

Siij

Sllj,

1 1

1

1

1

1 1

1

1

1 1

1

1 1

1 1

1

1 1

1

1

1 1

1

1

1

1

1

2

2

2

2

1

1

1

1

1 1 2 2

2

2

2

2

2 2

1

1

2

2 2

2

2

2

2

2

2 2

2

2

3 3

2

2

2

2

2

2

2

2

3

3

3 3

3 3

2 2

2 2

3 3

3 3

Note: The states (Si^, Sii„\ (Si,,, SuJ, {S30, Sg^^ {Ssu, SgJ, (S5„, Sj^), (Ss^, S7J are Kramers conjugate states.

SiS3

C3fc S5

S7

S9

Sii

1

1

1

1

11 1

1

1

1 1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 2

1 1

2 1 2 1

1 1 1 2

1 2

2 2

2 1 2 1

2 2

1 2

2 2

2 2

2 2

2 2

2 2

2 2

3 2

2 2

2 3

2 3

2 2

3 2

3 2

3 3

2 3

2 3

3 3

3 2

Note: The states (Si, Sn), (S3, Ss), (S5, 5?) are Kramers conjugate states.

44
Table 9. Type and number of perturbed levels—Continued

Slu

Sg Sju

Sso

12

32

6

2

72

92112i.5

9 u g u g u g u 9 u 9 u ? u

1

1

1

1

1

1

2

2

1

1

2

2

2

2

2 2

3

3

2

2

3

3

3

3

4 4

3

3

4 4

4

4

4 4

5

5

4

4

5

5

5

5

6 6

5

5

Note: The states (Sig, S^g), (Sm, /S5J are Kramers conjugate states. The states (S30, Ssu) appear an even number of times as Kramers conjugate pairs.

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8

Slg

Sig

1 1 1

1

2

2 2

2

3

3 3

3

4 4

4

4

5 5

5

5

6

6

6

6

7

7 7

7

8

8

8

8

Note: The states (S^, S^g) and (Sm, S^u) are Kramers conjugate states.

D^g

Ok D%g

iSg

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

2

1 1

2 2

1

1

1

1

2

2

2

2

1 1

3

3

2 2

Dh

2S

1

1

1 1

1

1 1 1

1 1

1 1 1 1

12 2

1

1 1 2 2

1 1

1 2

2 2

2 1

1 2

3 3

2 1

iSlg

2^20 2S2U

1

1

1

1

1 1

1

1

1 1

1 1

2 2

1

1

1 1

1 1

2 2

2

2

2

2

2

2

2

2

3 3

2

2

2 2

2 2

3 3

3

3

Note: The states (2<Si„ 2821) and {2S1U, 2S2U) are Kramers conjugate pairs of states.

45
(J tH rjH -gH

o'

60 60 3 3 cq cq CQ oq CQr-^ 03r-^ CQ COr-^Wi M 3 3 QO 00 CC CD60 Ba 3 3 60 60 3 3 Qo>-<j o?i >-J^ OnKl^ OTi Si^

BO 60 3 3

u' cq cq cq cq Dq cq'cq CD QO 00INI^ 03Ki^ ^

o?

q:) cq ^ ^ CQ cq CQ CQ Tr" 0?1 00

60 60 3 3 60 60 3 360 6a 3 3 60 60 3 3 60 60 3 3 60 60 3 3 ^ N.^ N-/

U cq ^ Eq Eq

u "^N Cj O Cj o

q'

Qq cq QqDq

60 60 3 3 C2*ca'"H cqcq

Be 6e 3 3 cq oqGq cq'

60 60 3 3 03 CD tt^ 0:S' h<J

Be 60 3 3

Q Dq CQ Eq Ec; Eq Eq DQ Cq Eq Eq TT' ^ 0*' K<, 0^

U

60 3

6* 60 3 3 03 02 C2 CC60 60 3 3 60 e« 3 3 03 02 CD CD

60 6o' 3 3

U ta Eq CD l><j or!' Co* Kl^ CO1^ Eq Eq TMS "*H ~H Eq to "'^^

Q Eq" Eq" Eq

W <M 3 3 "^ Eq Eq"

60 60 3 3

Eq" Cd

Q

Eq E3 Eq ^

U tI -qj

60 « 3 3 60 60 3 3

CCi CC\ CC\ ""N

60 60 3 3 60 60 3 3 60 60 3 3 60 6o 3 3

U

to Eq la Bq 03 CD* CC 03 cq Eq Eq ^ Eq Eq

Q

60 3

h<j Ed Eq'

0* W S 3

60 3 Eq Ed

BO 60 3 3 00 0!!! 00*^ CD Ed Ed°

60 60 3 3

"'TJ

Q Eq Eq

Qc 3 6t « 3 3 CD 03t-*^ On1*4^ cci Ed' Eq' Eq Eq

60 60 3 3

^T* '^T^ Eq Eq

8 O 60 3

BO 3 60 3 Bo 3 ^ T ,2 f

Cj o o o

60 3 ^ T ^ ?

8 O Ed Ed Cd

8 Q

ca cq E^ Eq Eq Eq Eq Eq Eq

t~ 3 BO 3 Eq Eq

46
arjOQOQiyj sstyjonc/} cqc<!V3C^ ojc^ojc^ c^c<j&q&q tr^c/^CQCQ &oc<J&3&3 tyjcotyjoi

BO Be 3 9 6fl ft« a a CO CO CO CO

<M Qa 3 3 COCOC*QCO

«• « 3 3 &3 CO CO CO

Q Q Q Q Cs Q Q Q

COCiQCOC^ CO CO CO c^ CO CO CO 02

e« at a ;i ijQ ^

e« 3 3 CO CO CQ CO

*C 6< 3 3 cot^coc^' CO CO 02

w M 3 3 CO OQ CO CO

q

XX

q o205coc>2 q

xxxx

q q q coco&qc^' q

xxxx

q q q oJoqcooJ

ccc<;coco coOQCoco OQ£<}Coco cocococq OQCociQCo cotyjcooi t^cococo C0C00206 a«;3 beS W3 ecS eoS eo 3 ed3 XX XX XX XX XX XX XX XX qq qq qq qq qq qq qq qq

q^. ^q q q^ q=3, ^q ^q q*^

CO 02 OQ CO cyjCoOQoJ 05 02 CO 02 to CO CO oj 03 Co CO Co Co 02 02 CO OQ CO 02 02 CoOqCooJ

q q ^ ^ ^. ^ qq qq '^'^ qq

q q oQoJoQco q q

« 3 60 3 6a 3 X X ^c^'^'' X X X X 606033

q q coCQOQOQ q q q q cocooJoq

q^'^ '^'^ '^iq ^!q^ q^^ i^*^

«o_:=J, 6ose33 606033 666o33 6o6o33 _6o^3 Bo„3j:; Boeo33

OjcooJ^O coofojof 0Q0Q0Q02 C00202CO OqCOCOCO CoCococo cococoCO C0020202

q'q '^i^ '^'^ qq qq

q

' q ^. ^. ^, q q q q ^ ^ q q

60 3 60 3

Xq

CQco oqco cqcq' oqco qq qcf '^'^

- I - I 02 CO 02 OQ "Tm? "T^T "T^T ^TIT sT-T tsTsT

O2O2OQO2 OQO2O2O2 OQOQCOOQ O2O2O2OQ COCOCOOQ COCOCOCO CO 02 CO CQ

CQCO OQOQ OQOQ C0O2 COCO COCO COOJ 0203

COOQ 02 02 02 02 coco OQOQ OQCO 02 02 CO02

47
but its behavior under lower symmetry does not ; depend upon the rep of D^s from which it arose. ) In a manner of speaking, the columns of table 10 .j are first differences of the entries of table 9.

j Table 1 1 shows the reduction of the reps of the il cubic group into the reps of their own cubic sub- ! groups, and into reps of the noncubic groups.

Here, we have considered the two possibilities ; which may arise in destroying cubic symmetry. J Thus, the group O may be reduced to D4 by i destroying the threefold axes, or to D3 by destroy- } ing the fourfold axes. In either case, the principal J axis remaining is considered to be the principal

| axis of symmetry. J While our discussion has been that appropriate j to the weak-field case (sections 3.3, 3.4), the;

results are immediately applicable to both the j intermediate field case and the strong field case, i, In the former, the external perturbation is stronger f than the spin-orbit interaction. Consequently, • instead of reducing the representation D^XDs

into those reps of the group leaving invariant the

spin-orbit term in the Hamiltonian, i.e., Dj, we

reduce the representation into those reps of the

group leaving invariant the crystal field potential V. Since the spin is not included in the Hamil- [ tonian at this stage of the approximation, we

reduce Di, only into reps Xj. This may be

done by using table 9 for L instead of J

(liere, the double groups will not appear, since

L must be integral). The rep for the wave

function, including spin, will then be XtXDs.

We shall parallel spectroscopic notation for such

a state by using the appropriate symbol from

table 8 for the rep Xi, with an anterior superscript

2»S'+1 for Ds, the whole being placed in paren- I; theses. If the spin-orbit interaction is now intro- 5

Table 11. Reduction of the cubic groups I

0, ^2h T D2d C3,

A, 4, Au Au Ag Au Ag Ai A Ai Ai

A^ A2 A^u Au A,u Au A2 A Bi A2 A, A2 Big Aig C\g Au Cu E QC2 AiBi, E

Au 4i A2u Au, Qu T: T A2E A.E

CigCig E A\gBu E Au C2g T2 T B2E AiE

Ey, E A\uBiu Eu Aiu Dy Dh Dy Dy

2S Dy 2S Dy

Tu T EgA2g EgA2g T A2gBuB2g A gCigCig Dil2 2S1 2S2 Dy 2S Dy^SiSs

T E^Aiu Tu AiuBiuBiu AuC,uC2u

T r. EgBig EgAu Dy^g Dyg SigSsg Tu EuBiu EuAiu Dyu Dhu SluS^u

lyy. Dng 2^1 s Dyg Si gSz g 2S Dhu Dyu SiuSsu

iSg .8 2<Sy Dv2e 2821 Dy,g SjgS^g

Dhu 2S2U Dy,u

2'S'lg 2'S'2g Bv2 Du^g Su Ssg

D3/2U 2'SlM 2Siu Din Dyu 2'S'u

where table 5 shows that potential coefficients

ClCtCl CICICI; act and CI are present, and

table 9 shows that there are two states Ai^ and

one each of A2U, Biu, and B2U, indicating that the

original fivefold degeneracy of the state 2„ has

been destroyed. If the terms C2, (7°, and CI are more significant than the rest, an initial approximation using these terms only leads to D„;j symmetry, with states Am, E^^, and Eiu- The inclusion of CI will reduce the symmetry to De^,, but the

states retain their designation. On the other

hand, the inclusion of C\ will reduce the symmetry to Din, leave the Aiu level unaffected, leave

the Ely, level unaffected save for a redesignation to Eu, and cause the E^u level to split to B^u and

Biu- If we now include all terms, we see that Eu

of Dift splits to Biu and B2U of D2;,, and that the

states Bin and B2U of D4ft become Aiu and A2U,

respectively, in Don- This Aiu state can interact

with the Aiu state from the original D^^ approximation, so that a rigid correlation with the states of D„;, has been lost. Table 10 has been so constructed that the same

process may be followed for still lower order

groups. For every column in the table, its subgroups are found further to the right in the table

(or m the same column without the "(7" —"w"), and

the designation of any level under increasingly

lower symmetry may be followed across the table. Entries of the same line have significance only

with respect to the allowed subgroups of any given

group. Thus, there is no correlation between

parallel entries in Ds^ and D4/i, though either may

be correlated with t)2n- All splittings of doubledegenerate reps have been correlated to ensure

consistency. Since the basic entries of the table are the reps of D^^, any desired rep of some other

group must be found from the table by inspection

48
Iduced, the rep XiXDs is now merely a representation, so that it must be reduced according to the

^ overall symmetry of the problem, which is still that of the crystal field V. If Xi is one-dimenisional, the splitting which arises is that of the

• spin states only through Ds, and the angular mo- mentum is said to be quenched; ii Xi is of dimensionality greater than 1 but less than 2Z+1, the

orbital angular momentum is partially quenched.

This verbaUy depicts the effect on the orbital

angular momentum of the strong crystal potential

field V, which destroys (Xi one-dimensional) or

limits {l<CXiB<C^L-\-l) any further influence on

the angular momentum either by spin-orbit inter-

' action or by a moderate external magnetic field.

' Still another intuitive view is that the crystal po-

' tential partially or completely fixes the orientation

: of the angular momentum in the lattice, so that

these other influences have no further effect. As an example, let us consider the ^i^a level of

' the europium trivalent ion, the symbol indicating

a reduction of D3XD3 to D2 by spin-orbit interaction. A weak perturbing field D^^ vvOI (table

9) split this level into states described as QmAig] CF)B,,; CF)B2g; and CF)Eg. An mtermediate case field will split the ''F term (table

9 for Sg) into (M2,); CB,g); {^B^g); and 2{'Eg), a

total of (2S+1)(2Z+1)=49 states. The introduction of the spin-orbit coupling will now cause each of these to be split further. Consider the

level (^A-yg). The rep A2g in D^n has the character

(1, 1, 1, -1, —1, 1, 1, 1, —1, —1), while the

representation from Dz has the character (parity even, since it is a spin representation) (7, —1, — 1, — 1, —1, 7, —1, —1, —1, —1), so the representation AzgXDz has the character (7, —1, —1, 1, 1, 7, —1, —1, 1, 1) and is reducible into Aig\ Big\

Big] and 2Eg. The final states may be designated

{'A2g)A,g- CA2g)B,g; {'A2g)B-2,; and 2(M,,)£',.

This notation parallels that of the usual L-S

coupling scheme. The labels inside the parentheses indicate the approximate transformation

properties of the state for transformations of spin

and orbit separately which leave invariant the

HamUtonian, while the final designation is the accurate specification of the behavior of the state under the simultaneous transformation of spin

and orbit. In the strong field case, we introduce the crystal

field in our sequence of approximations (section

3.3) before considering the details of the electrostatic interaction of the electrons, retaining only

the initial central field average V'{ri). We now

specify our initial energy by giving only the configuration, and the reduction of the representations begins at this point. As a matter of notation,

following MuUiken (1955), we shall designate the

reps and characters for a single electron by lower

case letters (e.g., flag)- If there is but one electron,

there is no distinction between the intermediate

field and strong field cases, and we proceed as outlined for that case. We shall illustrate the procedure for the case of

2 nonequivalent electrons, and then consider the

restrictions imposed on equivalent electrons by the

operation of the Pauli principle. We shall take

the configuration in a field of C25 symmetry.

The threefold degeneracy of a p state (apart from

spin) is completely lifted, and we obtain (table

9 for ly) ai] bi] and 62 for each electron, so the configuration breaks up into (ai)(ai); (ai)(62); (^2) (ai); etc., a total of 9 possibilities corresponding

to the original 9 orbital possibilities in pp. The

spin transformation will be Dy^XDy^- The introduction of the details of electrostatic interaction

cannot cause any further splitting of the orbital wave function, since all the states are one dimensional, though a relabeling may occur. The re- quirement for totally antisymmetric wave functions will require that the spin transformation

Dy^y^Dy^ be reduced to Do and Di, giving the usual

singlet and triplet states, respectively. Thus, (ai)

(ai) becomes (ai) (a^) (^Ai) and (aj (aj) (^^1) ; (61)

(62) becomes {b,){b.2){'A2) and (6i)_(62)('^2); etc. Note that this latter state is distinct from (62)

(61) (^^2), which is also an allowed state in the

reduction. The further introduction of spin-orbit

interaction now requires the reduction of XiXDo

for the singlet states, XiXDi for the triplet states, in accordance with the overall symmetry of the

problem, which is still that of the external crystal

field potential V. Consider (60 (^2) ('^2). The

character of ^2 is (1, 1, —1, —1), while the character of Di in (even parity for a spin transformation) is (from table 7) (3, —1, —1, —1), so the

character to be reduced is (3, —1, 1, 1), giving

Ai, Bi, and Bo. We may write the final states, then, as p^? (60 (^2) ('^2)^1, or pp{b{){b2){^A2)Bi,

for example.

If the electrons are equivalent, i.e., we write

{aiY instead of (ai)(ai), and we cannot distinguish

(ai)(62) from (62) (ai). Furthermore, the Pauli

principle requires the spin state of opposing .spins for the orbital states {aiY; {biY; and (62)^ so that

these appear as singlets only. The remainder of

the orbital states may appear either as singlets or

as triplets. Apart from these considerations, the

development follows that of the nonequivalent

electrons. A possible final state would be written

p'^ib.fQAMi.

We have selected here a particularly simple case

in that there is no orbital degeneracy present in the reduction of Di for a p electron. The requirements of the Pauli principle are then particularly

easy to satisfy. If the reduction yields a twodimensional rep, or remains three dimensional,

the development must be followed more carefully.

We introduce the procedure by considering, in the

concepts of the present work, the familiar operation of the Pauli principle in labeling the levels of

the configuration p'^ itself. We imagine a uniform

magnetic field such that the resulting perturbation in the HamUtonian, —M-B, has the symmetry C^n- The degeneracy of each p electron

is lifted, and we have (table 9) the states a^, Cm

and c-iu for each electron. Because of the Pauli

49
principle, the state {CiuYC2o occurs with antiparallel spins only; i.e., a singlet state. We see that no state for n gi'eater than 2 arises. We then infer from table 9 that there exists a

singlet level L=2 in the configuration when the

magnetic field is removed. Similarly, we deduce

a triplet L=l and a singlet L=0.

A similar process can be used if the crystal field leaves degeneracy in the 1 -electron wave functions.

The differences are two : first, it is generally more

convenient to remove the degeneracy by the

imaginary application of an electrostatic potential

of a symmetry which is a subgroup of that actually

present; second, the correlation of states in this

additionally perturbed problem with those of the

primary perturbation is not unique, and several

alternatives must be considered.

Let us further consider p'^, this time under the

symmetry Q,, where 1^ reduces to ai and e. Our

problem really lies in the study of (e)^, since

(ai)(e) can yield either triplet or singlet, and

(aiY is necessarily a singlet. Without the considerations of the Pauli principle, {eY yields Ai,

A2, Bi, and B2, both triplet and singlet, and the

Pauli principle will suppress certain of these terms. Since (e), including spin, represents 4 states, we

4.3 expect ~2~~^ states altogether. Let us reduce

the symmetry to C21,; (e) becomes 61+62, and the

allowed orbital states are (61)^; (62)^; and (61) (62)- The first two are both Ai and necessarily singlet,

while the third is A2, either singlet or triplet, so we have our total of six states. If we examine

table 10 for the relations between these states of

C2V and states of €4^, we see that Ai in the former

implies either Ai or Bi in the latter. Since we

have Ai twice in C2» and cannot have either Ai or Bi more than once from (eY in C^^, we conclude

that both are present in €4^ as singlet terms. However, the singlet, or the triplet, A2 implies

either A2 or B2, and we cannot decide which is the

correct choice. If we reduce the symmetry to C4

instead of C2V, (e) reduces to (ci) and (ca), yielding

singlets (ciY and (csY, and singlet and triplet

{ci){cz). The first two give B states, while the

third gives A states. Examination of table 10

shows that B in C4 implies either Bi or B2 in Ci^. Again, since we have two of the former, and are allowed at most one of each of the latter, we

conclude the presence of singlet Bi and singlet B2,

together with the singlet Ai previously deduced.

The triplet state in C28 implies either A2 or B2 in

C45, while the triplet state in C4 implies either Ai

or A2. Obviously, the only choice consistent with

both is the triplet A2. We have, thus, ^Ai; ^A^;

^Bi] and ^^2. The balance of the problem then

follows previous discussion. In general, each

problem must be considered as an individual case.

4,2. Kramers Degeneracy

There is one very important symmetry operation, in addition to spatial rotations and the

inversion, which should be considered at this time. This is the symmetry of "time-reversal".

This was initially considered by Kramers (1929,

1930) in connection with the rotation of the

plane of polarization of polarized light. The

matter has been thoroughly considered by Wigner i (1932) in the nonrelativistic case in which we are

here interested, and more recently by lOein (1952), i We shall outline here the principal concepts in- volved, referring the reader to the original papers i for additional details. '1 Classically, the reversal of time in a mechanical

system is accomplished by the reversal of the

direction of all velocities (or momenta) by the

substitution r'=r, p' = —p. Quantum mechanically, we assume that this is also time reversal, if a similar substitution s' = —s is made for any spin { angular momentum involved. In the case of a ^ free atom, the Hamiltonian H(r, p, s) involves,

p and s in kinetic energy terms with p^ and»

spin-orbit interaction terms with p-s, both of which j; are invariant under the time-reversal operation, j This invariance of the Hamiltonian remains under ) the influence of external perturbations of a purely ^ electrostatic character, in which case the vector

potential A may be set equal to zero and the ; scalar potential V{r) introduced. This invariancei

is destroyed by the application of magnetic fields—• t a term p-A arises in the Hamiltonian, which is!: linear in p and changes sign on time reversal.

In general, then, we shall seek an operator s K such that KH(r,p,s)K-i=H(r,-p,-s), and K^l

{r,s,t)=^-^{r,-s,-t). Since the operator K repre-l sents a transformation of our wave functions, we|; shall require it to be a unitary operator. If we

iEt consider the spin-free theory, with ^(r,i) =\J'(r)e

' we see that the operation —t is accomplished . merely by taking the complex conjugate, an opera-

"

tion we shall call Kq. Insofar as the spin-free I theory is concerned, this is sufficient; the p operator —-ifiV becomes—p= +'if(-Vupon taking the complex

conjugate. Let us now consider the Pauli spin

I theory for one electron. The Pauli spin matrices

-c:) <-) -CJ)!

c are both unitary and Hermitian. The trans- li formation KoSj,Ko~' = —Sj, since Sj, is purely,

imaginary, but and Sj, both real, are invariant^

under Kq. Hence, in order to effect the reversal,

of Sj: and s^, we seek yet another operator, which J must also be a unitary operator U, such that it commutes with r, p, Sj^, and Kq, but anticommutes

with Sx and S2 so that U SjU~' = —s^r and similarly for S2. This latter requirement is met by Sy

itself, or by some product of Sy with a complex

number a of modulus 1. To ensure commutation ' with Ko, the product aSy must be real, which I follows if a=i, so that U='iS;, and K=KotSj,.

j Note that the operator U is still unitary but no

j 50
(j longer Hennitian. For n electrons,

U=(i)"(S,)i(S^)2(S„)3 . . .

'} Phj-sically, it is clear that time reversal and any

' rotation-inversion operation commute. Mathematically, the inclusion of the operator K in an

, extended group of the form DjXiXK cannot be

accomplished since K is not a linear operator.

The basic rules for the operator K were given by

Wigner as:

' 1. KV=UKoUKo\J'=UV=(-l)"'/', smce U is real, K^=l, and s^ = l.

I 2. K {a^Pfb(j>)=a*KxP+b*K(t>. This shows the

' nonlioearity of K.

3. (</,,,A) = (KoV',Ko<A)=(UKoi/',UKo<^,)=(K^,K<^).

Applying (3) and (1), we have (K\{/,\p) = (K\p,

' KKrP)=^{KrP, KV) = (-l)"(K^,i/'). We see that

I the Hermitian scalar product must be zero if n

• is odd, or that yf/ and K\p are then orthogonal and

' hence linearly independent, yet have the same

energy if KH=HK. This is the content of

Kramers Theorem as usually stated, namely,

that every level for an odd-electron system is at

least twofold degenerate under the influence of

external perturbations of a purely electrostatic

character. We must examine further the inclusion of the operator K into our previous considerations on group representations.

We shall consider the following three cases (Frobenius and Schur, 1906; Wigner 1932):

1. The rep R is real, or may be put into some

equivalent form which is wholly real. This refers

to the elements R of the rep not merely the traces.

2. R cannot be made real, and R* is not

equivalent to it. 3. R cannot be made real, but R* is equivalent

to R.

For the finite groups a calculation of - XI x(R^) will 9 R

yield 1, 0, or —1 in cases (1), (2), or (3), respectively. Case (1) occurs for integral J in. Dj and

ia all its subgroups save those one-dimensional

reps where complex numbers appear explicitly

in the character table (table 8). The only such

specific reps of the double groups are S^g, S^u of

iiSa and 5*3 of nQ- Case (2) holds in those one- dimensional reps where there are obviously complex characters. The only two-dimensional reps

in this category are the pairs (2S'i(„25'2ff), (2*S'i„, 2S2U), and (281,282) in the groups nT^ and nT.

Case (3) does not hold for any reps of the single

groups, but holds for all those of the double

groups not in (1) or (2). In case (1) let us consider a group operation 0

such that Qxl^i^l^QijiPjandKQ^i=QK4^i=i:QijKyPj,

so the function Kt/' will transform like ^

under the operations of the group. Let us further

consider 4>i=4'i+'^'Pi and Q(=i(i/'<— K^i)- Clearly

0( and fij transform only among themselves under

the group operations. Furthermore, if there is an

even number of electrons, K0i=Ki/'iH-KV<=Ki^i

-\-\pi=4>i and Kfii=Kii/'j— KiKr/'j=— 'iKi/'i+'i^i =i2i so the two sets of functions 4>i and are not related by time reversal or otherwise, and any

degeneracy present is presumably only an accidental degeneracy. Practically speaking, this means that our set of wave functions may be

chosen to be real, and that in such a case time

reversal yields no connections among the set of

real functions that are not already present as a

result of the rotation-inversion symmetry. If there is an odd number of electrons, K<^i=K)/'i—

\pi9^4>i and we cannot so separate the (f> and sets of wave functions as we did above. Hence the

rep must appear twice, once for xp and once for orthogonal to it. The only actual such cases are the reps 83g,83u of Ss and 83 of C3 and it may

be verified fi'om table 9 for Sg that they do in fact

appear only an even number of times in any

reduction of a group of half-integral J. In case (2), Q^i^'EQijxp) yields KQxPi=QKxPi =XQ*jK^j. Since the rep ^* is not equivalent to R,

\p and Kr/' belong to different reps of the group and

hence are orthogonal (section 4.3), but have the same energy. Hence a separation of a rep such

as E of Css into Ci and C2 of C3 by an electrostatic

field does not imply a removal of the degeneracy.

It is complex conjugate reps such as these which

are usually grouped together into a single twodimensional rep by workers in the field of molecidar spectroscopy.

The equivalence of R and 7?* in case (3) requires

the existence of a unitary matrix S such that

SRS-i=R* or S*R*(S-i)*=R, so S*SRS-HS-i)* =R or S*SR=RS*S. We see that S*S

commutes with every R and hence must be a multiple of the unit matrLx, say cE, with c of

absolute value 1. With S*S=cE, we have S*=

cS~^ =c(S)*, where S is the transposed matrix of

S. This is a consequence of the unitary property

of S. The conjugate of this relation is S=c*S,

and the transpose of this gives S=c*S, finally

giving S =c*(c*)S, so (c*)^=l or c=±l. If c=4-l, we have S with +1 along the minor

diagonal and zero elsewhere—in this case R=R*

and case (1) applies, so we take c= — 1. Then S

will consist of alternate -fl and —1 along the

minor (nonprincipal) diagonal. Incidentally, this requires that the dimensionality of S be even. Let us put S in the form such that +1 is in the

upper right corner, and let us label the rows and

columns of S and the R matrices by the numbers

j, . . . —j, where j is half-integral.

Then

8r„={-iy-'Ki-,) and {8-')a,= {-iy-'+'K,-i>y

The transformation SRS~' is then as follows:

{R8-')a, =i:.Ral.{8-'),,= lZIiau{-iy-^^'h<-^^

k k = (-l)^+*+^i?a(-»,

51
and terms of the form

k = (-l)^^'+^-^^-^i?(-c)(-.)=(-l)''-^i?(-.)(-«.

This latter term must then also be {R*)cd, since R

and R* are equivalent by the transformation S. We now consider

b and

=SQ*a.K;A.=S(-l)*-'^Q(-.,(-.)KV',

b b or 0(-lWa=2:Q(-a)(-.)(-l)*K^..

If we compare this with the transformation on

ip-a obtaining Q4'~a=12Q(-a)(.~b)i^-b we see that

-b (—lyKypa will transform like under the group

operations, and that(— 1)~°K;/'_a will transform like

^a- The linear combination (/)a=\J'a+ (— l)~''K;^_a

will transform in a similar fashion under the group

opera-tions. Further, we observe that

K4>a= K^<z+ (- 1 )"KV-a= ^4^a-i- W-a

= -{- im^a+ ( - 1 )'^K^a] = - (- m^a,

since our convention has "a" half-integral, (— 1)"

is purely imaginary and Ko(— 1)"= (— 1)~"= — (— 1)", and K^iAa=— '/'a- The important point

is that the set of functions <^a (a =j, J — 1, . . . —j)

transform among themselves under the group

operations, and that the operator K merely yields a wave function which, apart from a phase

factor, is already in the set. Similarly, if ^a=i{'^a— i^'i-Y^^'-a) the Qa wUl also transform

among themselves under the group operations and

time reversal, but will not involve the 4>a. Hence

we conclude that the introduction of time reversal

does not require any additional essential degeneracy in case (3). For emphasis, we shall summarize the cases where we find that the symmetry of time reversal

(with external electrostatic fields only) gives us degeneracy beyond that expected from previous

considerations.

1. States belonging to complex conjugate reps

are degenerate. This is the case for both an odd

or even number of electrons.

2. The reps S^g, Ssu of Se and Sz of C3, arising

from odd-electron systems will always occur twice, once for yp and once for its orthogonal but degenerate conjugate K;^. Finally, let us consider

K=(i)"(S,)i(S,)2 . . . (S,)„Ko

operating on a wave function consisting of 2"

Since

and

K;A=/*(r)(-l)2^ '^i{-S,)U-S,). . . U-S„).

It will be noted that K will commute with L^,

and J^, but not with L„ Sj or J^. If /(r) belongs

to the eigenvalue L and M^, /*(r) will belong to L, —Ml if the usual phase factor is supplied. Similarly, the original spin terms,

belonging to S, Ms will belong to S,—Ms with the

—-\-M . phase factor (—1) 2 ^. Thus

Kyl.{L,ML,S,Ms)

n = (-l)M,(^-l)Ms(-iy^(^L-ML,S,-Ms).

The transformation from L,Ml,S,Ms to L,S,J,

Mj is accomplished by the Clebsch-Gordan

coefficients (section 5.2), which have phase factors

such that

/-yL S J / ^ ^L+S-J L S J

In a state described by J,Mj, we must also introduce this factor, and we obtain finally

lLyp{L,S,J,Mj) = (_l)M,(_l)Ms(_l)l(-l)^+S-^V'(i/V,-W = ( _ 1) /-M, ( _ 1) ^+^+1

^ (L,S,J, -Mj).\

L+S+- ' The factor (—1) ^ we shall neglect, but the j

term is of some importance, since we 5 shall be considering states which are a linear ] combination of states with the same J and different i Mj. There will be a physically significant shift

of relative phases among the components of such ' states when passing to the time-reversed state. i

4.3. Orthogonal States and Vanishing Matrix

Elements , Before proceeding to the consideration of

j selection rules and polarization, we shall digress , here briefly to consider certain aspects of the ^ calculation of matrLx elements of the form

Ja-W. which wm be required at tUs toe,

deferring to the next section additional details. Let us assume initially that X=\. Let 4^a and

belong to reps (1) and (2) respectively of the

appropriate symmetry group, so that any group

52
operation 0 will yield Q4'a=I2Q'lli^'k and

k Qh=^QfiiJ, where and Qfi are the same

if the two reps (1) and (2) are the same, otherwise

they are distinct. The operation 0 on i/'fi/'a then

gives j:QTQiUTn'=i:Q,a.j.rr^'r The

matrix with elements Qt,a jk is called the direct product of and 0"', written Q'"'*^"'. If these

latter are of dimensionality Ui and ^2, then the di- rect product is a nin2Xnin2 matrix. Obviously,

Q(2)*xa) jg element of a representation of the

group—its trace is 2 Qm.m={I2 QT) (S Q'ii),

ab b a the product of the traces of the two "factor"

matrices.

In general, (J(2)*x(i) y^r]\[ \yQ either a rep or a sum

of reps, in which latter case it may be reduced by

the methods of the preceding section. In either

case, we obtain ^

Qi^a.^k^'T^l' =12 Q.fl^'^J +

j,k d Zl Qe/'/' /'

• • where i^T,^/', etc., are the combinations of lA'f *'/''" appropriate to the reduced

representation. Our original integral then be- comes The basic argument we shaU use here is that our

integral, i.e., the matrix element, represents a

physically significant quantity and hence cannot

depend on the choice of any symmetrically equivalent reference frame. Consequently, the terms

of the form Z Qcajip^^dr must either actually be

independent of 0 or must vanish. If the rep (3)

is actually independent of 0, it is then in fact the

completely symmetricor identity rep of the group,

and we must ascertain the conditions under which

the rep is present in the representation Q(2)*x(i)_

The identity rep has the character + 1 for every

group operation, so the orthogonality theorem for group characters tells us that - X) 2 Qia ba is the

g Q ab number of times the symmetric rep occurs in the

reduction. But we recall that Z S Qba ba=

0 ab 2 (X)^66*) (2 Q^^), and this latter expression is 0 6 a merely the orthogonality theorem applied to the

characters of the reps (1) and (2), which is zero unless (1) = (2). Hence we conclude the very

important result, that states belonging to different

reps of the appropriate symmetry group are orthogonal.

The example just considered took ^=1. If X

itself, either a function or operator, transforms as a rep or sum of reps of the symmetry group, the

product X\p^l^ must first be reduced. If this reduction does not contain the rep (2), then the

integral must vanish.

4.4. Selection Rules and Polarization

Selection rules and polarization rules are in- timately connected, and we shall consider them

together. We shall consider in detail the effect of an incident perturbing electromagnetic radiation superimposed upon the static electric and/or

magnetic field already present. In a region of

space not including the sources of the field, the

perturbing radiation is usually described by its associated vector potential A, with the auxiliary

requirement V-A=0, which will allow us to set the

scalar potential V=0. The fields E and B are

1 dA

then given by E= ^ and B=VXA. This

c ot vector potential modifies the classical Hamiltonian

term for one electron, p^, to

2^ (P-^ ^y=^2^ (S~c ^-^--c ^0"

Neglecting the term in A^, we obtain as the perturbation term

H' = -

2mc

(p.A+A.p)

Since p is, apart from a constant factor, the vector operator V, the term (p-A) vanishes and our final H' = —^ A-p. This acts on each electron indimc

vidually, and the total perturbation requires that

this be added for all electrons in the atom. If a

system is initially in a state '^a='^a^ the

probability CjC* that the system under the in- fluence of the time-dependent perturbation H' will at a later time be in the state ^6 is determined by

so we shall investigate this integral.

The components of A are functions of (x, y, z, t). We shall assume that the variation of A with t is harmonic. We shall not require explicitly the

harmonic factor e'"' in the following development

and shall generally omit further reference to it; i.e., A(^;, y, z, t) =A(x,y, z)e''''. For the wavelengths

of light in which we are interested, X is of the order

of 10~* cm. In comparison with the order of size of the atom, 10~* cm, to a first approximation the

variation of the components of A with (x, y, z) may frequently be neglected. We shall see that

this approximation may be inadequate for our

purposes, and we will improve this approximation

by developing the components of the vector po53
tential in a Taylor's series about the origin. This

will yield

where the derivatives are evaluated at the origin.

Similar relations hold for Ay and A^. To this approximation, H' becomes

+(t)-+(l^)-}

The first three of these terms will involve matrix

elements of the components of p only, since A°

is a constant. The general quantum-mechanical

equations of motion allow J^irtp^adr to be written

{Ei—Ea)Ji*r^adT. Note that our integral

im

now involves the time-independent wave functions. Thus, these three terms can be written in terms of the components of the vector r rather

than of p. Since er is the classical electric dipole,

transitions induced by these three terms are

called electric dipole transitions. Let us assume

temporarily that there is no external perturbation

apart from the radiation, so that J and parity

remain good quantum numbers. Let us consider an initial state transforming as Jg, J^O. Each

of the vector components x, y, and z will transform

eis lu (i.e., as the components of an odd state J=l). Applying our previous considerations,

the character of the rep Jg is XI e'^* for both proper

and improper rotations, while that for lu is

! + for proper rotations and (—1) times

this for improper rotations. The product of the

characters for pure rotation yields

/J \ j+\

\-j / -(j+i)

-J -(.J-i)

and (—1) times this for the improper rotations.

It is immediately clear that the direct product

luXJff yields {J—l)u+Ju+iJ+})u, and (from

section 4.3) our final state arising from the

transition must be one of these. This is equivalent to the well-known selection rule AJ=0, ±1,

and a particular example of Laporte's rules, where

tion B-L to H'. Since the quantity—::::— L

2mc 2mc

is the magnetic moment operator, transitions

arising from these terms are called magnetic

dipole transitions. The components of L (a

pseudovector) transform as in contrast to the

electric dipole's lu, so here even states combine

with even, or odd with odd. We may note at

this point that we have not yet included any

terms in our radiative perturbation representing

the interaction of the radiation with the spin

of the electron. This may be accomplished at

this point (in a more or less ad hoc manner)

even must combine with odd terms. If J=0,

lu'X.Og=lu, and we obtain the restriction that

J=0 does not combine with J=0.

The other set of 9 terms in H' does not itself transform under a single rep of the rotation group,

and must be separated into terms, each one of , which does so transform. The 9 terms consist of I the general products of the 3 components of the . vector r with those of the vector p. It is well

known that three linearly independent terms

arise from the vector product rXp, and a fourth

arises from the scalar product r-p, leaving five terms, constituting the components of an irre- ducible tensor of rank 2 (in contrast to the original

set of 9 components, which is also a tensor but

not irreducible). In the terminology of the pre- ceding paragraph, wehhYe luXlu—^g+lg-'rOg- In

order to exhibit these more explicitly, let us

consider (^)^^^'^ + (^) '^P^'' adding

and subtracting (^^^ ^Pv + (^^^^

pression may be brought in to the form

+2 \-^-^) ^'p^-yp>

and a similar relation may be derived for the other

2 pairs of similar form. This process amounts to finding the antisymmetric portion of the original

9 components, viewed as a 3X3 matrix. We

observe that the first factor of the second term is the z component of vXA=B and that the second

factor is the z component of rXp= L, so, including

the other 2 pairs of terms, we have a contribuby using 2mc (L+2S) as the interaction with

the magnetic part of the incident radiation, though

this will not change the transformation properties

of the term, since spin is also an angular momentum operator with the transformation Ig. It will be recalled from section 3.2 that the transformation (3.6) is equivalent to a rotation among the

components of the spin angular momentum

54
loperator (then called R) and hence equivalent to

|a transformation like the three components of

ja state J=l.

I In considering the other six linearly independent

iterms, we again have recourse to the general

quantum mechanical equations of motion. In

particular, we obtain

jand

jjlf we further simplify the notation by writing

A„ for (M) A., tor (^+^) our re- maining perturbation can be written in terms of

the matrix elements of

In lieu of x^, y^, and z^, let us consider the linearly

independent combinations

Z2 =222 — —?/2

with the inverse relations

x'^l p'+l r'-i

„,2 — 1 ^2_1 72__1 J2, y — 3' 6^ 2 P

Substituting, we have as our perturbation the

matrix elements of

r\lA,,+iA,,+iA^^) ^Z\\A,- \A,,~ \Ayy)

+p^(Mi2— \Ayy) +Axyxy+Aj:,xz+Ay,yz.

Of these six terms, that in is the Og term, in- variant under rotation and inversion, and here

actually zero, since A^^+Ayy+A,,=di\ A=0.

Notice that the terms Z^, xy, etc., are, apart from

normalization factors, the tesseral harmonics of

degree 2 (table 2). The remaining five terms are the components

of the quadrupole moment, and transitions aris- ing from them are called electric quadrupole

transitions. The general selection rule is here

given by a consideration of the 2g transformation

properties of the quadrupole moment. If J>2,

^.X J,= (J+2),+ (J+l),+J,+ (J- 1),+ (J-2)„

so the general selection rule is Aj=±2, ±1, 0. If J=l, 2,Xlg=Sg^2g-\-lg, so J=l does not

combine with J=0. Likewise, 2gX0g=2g, so J=0 does not combine with J=0 or J=l.

We have up to this point considered that the

symmetry present was that of the full three-dimensional rotation-inversion group. If this is not

the case, the 3 electric dipole components, the 3 magnetic dipole components, and the 5 electric quadrupole components wiU belong to different

reps of the symmetry group. These may be determined from table 9 for lu, Ig, and 2g, respectively, and the detailed correlation may in most

cases be obtained from table 10. The use of

table 10 after an initial application of table 9 for the group D„;, (for noncubic fields) is perhaps

most convenient. The reduction to D„;j will separate both lu and Ig into one 1-dimensional rep

and one 2-dimensional rep, while 2g yields one

1-dimensional rep and two 2-dimensional reps. These may be followed to lower symmetries with

table 10. Details may be obtained from table 8. For example, L, belonging to Ig, yields, in 'D^h,L^

belonging to A2g, while and Ly both belong to

Eig. Upon reducing the symmetry to ^48, now

becomes A2 while and Ly now both belong to E. If we consider the reduction from H^j^ to

CooA, we see that Eig splits into Cig and C^ig. However, these do not represent the transformation of

either or Ly, but of L^-^iLy and L^—iLy, re- spectively. In this case, and also for the subgroups of C^!i, we shall consider these latter, along

with ij, as the three linearly independent components, each of which involves only one rep of

the symmetry group. Similarly, we shall resolve

the electric dipole term r into x, y, and z for most

groups, using x-\-iy, x—iy, and z for the subgroups

of C^n, and the electric quadrupole components

into Z^, xz, yz, p^, xy, or, alternatively, Z^, z{x-\riy),

z(x—iy), (x-\-iyy, and {x—iyY. These resolutions are presented in table 12. As usual, there are two possibilities for the

cubic groups, one in which the 0-axis is an even-fold

axis and one in which it is an odd-fold axis. This

makes a difference only for the quadrupole components, since the electric and magnetic dipole

components remain indistinguishable under these

symmetries. If we consider the 2-axis to be an

even-fold axis, i.e., C4 for O and O,,, S4 for T^, C2

for Tft and T, the five real quadrupole components

remain a satisfactory basis for the resolution in all except for two components in and T,

where we require the normalized components

^[Q+^a] and ;^[C^-^Ci]

transforming as Cig and dg respectively.

If we take the 2-axis to be the C3 axis in these

cases, we find that these quadrupole components,

the tesseral harmonics of degree 2, do not serve as a basis for the reduction of the 2g representation

55
Table 12. Multipole moment transformations

a; y z - Z2 xz yz p' xy

DooA

Ceo V E, Eiu

El AouAi

Eig El

Eig El A2g

A2

Aig

Ai

Eig ElEig

El E2S E2

Eig Ei Oh T,u T,

Tiu T2

Tiu T,

Tu

Ti

Tig

T,

Tig

Ti

Eg

E

T2g T2T2g

T2

Eg

E

Tig

T2

Tu

E,u Eiu Tu T,

Eig

T

Eig

T

A2g

(^) T

Eig

Tg

Eig Eig

Tg

E2g

Eu

El Eu

El AiuAi

Eg

El Eg

El A2g A2

Aig

Ai

Eg

El Eg

El

Big E2

B2g Ei E'Eu

E'Eu

A',' A2U

E"

Eg

E"

Eg

A', A2gA[

Aig

E"

Eg

E"

Eg

E'EgE'Eg

C4p

E

E

E

E Ai

E

E

E

E

A2

Ai

Ai

Ai

E

E

E

E

Bi

Bi Bi

Bi

D2A

C3, B2U E

Biu

E

A2u

Ai

B2g

E

Big

E

A2g

A2

Aig

Ai

Big

E

B2g

E

Aig

E

Aig

E

B,B

B2

J-J R u Ai

Au

Bo0f>

s BiR

A2

A AiA Bi B2R

Ai

A Ai

A

c. A' A' A" A" A" A' A' A" A" A' A'

x + iy X— iy z Lx-\-iLy ix + iy) z {x — iy)z {x-\-iy)^ {x—iyY

Coo A Ciu C-lu Au aI g 0—1 g Ag Ag C] g C~ig C2 g C~2g

CeA

Cih Ciu Au

u Cig

Cig Czg Ag

Ag

Ag

Ag

Cig

Ci g

Cig

Czg

C2g

Bg

Cig

Bg

Czh

Se C[

Ciu A"

A u

C['

^ Is c. A'

A A'A

C[' C','

'~^2g '~'2g C[

"-•18

S4 C3 B Ci A A Cz Ci B B

^ See text.

of the quadrupole interaction. For example, the

reduction of 2g under yields the following as a

basis:

^:[_|c.+|ci];[-|sj+|si]

n: [f q+fCi] {fs;+f si] ; [cs]

While this, of course, yields the same polarization

results in the physical sense, the trigonal aspect

will not be considered further due to its formal

mathematical complexities.

Because of the different transformation properties of these components, they will most generally

permit transitions between different states of the

perturbed ion. The nonvanishing matrix elements

may be deduced by the procedures outhned above

(section 4.3). These are the selection rules

which are applicable in a case of definite symmetry.

Furthermore, the relative amount of any particular

component of the multipole interaction present

will depend upon the vector potential, and in

particular, upon the direction of propagation and

state of polarization of the incident radiation. A

detailed examination of this latter feature leads \ to the polarization rules for absorption of radiation.

The selection rules for most of the groups in

which we shall be interested (omitting only the

very simplest) are given in table 13. For each

rep in the left column, the states to which a

particular component of the multipole interaction

will permit transitions is shown in the column

under that multipole component. Generally, the

table parallels table 8, in that closely related

groups are shown with but minor modifications on the same table. Results for pure rotation

may be obtained by dropping the "g"—"u" dis- tinction in the tables. In a few cases, the changes

in notation and convention are sufficiently far- reaching to require a separate table. It is particularly interesting to note the table for

CooTi, appropriate to a uniform magnetic field. If we denote the general rep by Cmp (Co=A), we

see that electric dipole transitions have AM=0,

56
Table 13. Selection rules

z L^,Ly,xz,ijz Z2 p'',xy

A2U Eiu Aig E\g Aig E2g

Aig E\g A2U Eiu Aiu Eiu

Aiu Eiu Aig Eig A2g Eig

A2u All Aiu Eiu Aiu Eiu

Big Eiu E2uAiuA2u E\g E2gAigA2g Eig EigEig

Ei„ Eig EigAigAog Eiu E2uAiuA2u Eiu EiuEiu

Ezg E2u EiuEiu E2g E2g EigAigA2g

E2U Eo gE\ g E2u E',-u El^iu-^lu^2u A, A-,

E3. Ezu EiuE2u Eig EigE2g Eig ^—'Og-'—' 1 g Eig EigE2g Eiu EiuE2u Eiu E-ouEiu

(Pattern repeats for higher states)

2*514 2'Slu 2<Slu 2'S3u iSig iSlg 2'S3s iSig iSig iSog

2<Slu iSig 2>Sig 2<S3g 2'Slu 2'Slu 2'S3u 2'Slu iSiu iSsu

2S3J 2^3 u 2'Sla iSbu 2'S3g 2*514 2'S5g 2^38 2Slg iSlg

oOi a '>0 =; a 2'S3u 2'Slu 2'S5u i,'^ 1 U i."^ i tl

2'Si5u 2S3U 287 -u 2'S5j 2'S3g o'Sts iSig iSig iSgg

2/S3J 2'S7j 2S,u 2'S3u '"Stu 2'Sou 2'Slu 2»S9u

(Pattern repeats for higher states)

For Ceo 5, delete and "u" and read the upper left corner as: Ceo D z

Ai Ai

A2 A2

z L:^,Ly,xz,yz Z2

Aig Aiu Eiu Aog Eig Aig Eig

Aiu Aig Eig A2u Eiu Aiu Eiu Ail Aiu Eiu Aig Eig Aig Eig

A2U Aig Eig Aiu Eiu Aiu Eiu

Big B.u Eiu B2g E2g Big Eig

Bin Big Eig Biu Eiu Biu Eiu

Big Biu Eiu Big Eig Big Eig

Bin Big Eig Biu Eiu Biu Eiu

Eig Eiu AluA2uE2u Eig AigA2gE2g Eig BigBigEig

Ei„ Eig AigAigEig Eiu AiuA2uE2u Eiu BiuBiuEiu

Eig E2U BluB2uElu Eig BigBigEig Eig AigAigEig

Eig Eig BigB2gEig Eiu BiuB2uElu Eiu AiuAiuEiu

D^u iSiuDy^u Dng iSigDy^g Dy^g iSlg 2'S2g Dy.g iSigDyig Dy^u iSiuDy^u Dy.u 2'Slu 2'S2u

2'Slg 2»Slu 2'S2uZ)hu iSig iSigDy^g iSig iSigDy^g

2'Slu 2'Slg 2S2gDiig 2'Slu iSiuDf^u iSlu iSiuDyu

iSig 2'S2u 2'Slu 2^2u 2'S2g 2'Slg. 2'S2g iSig iSigDi^g

2S2U 2'S2« iSlg 2'S2s 2*52 u 2'Slu 2'S2u iSiu iSiuDy^u

For Cbc, drop "g"—"w" and amend upper left corner to read: Ce, z

Ai Ai

Ai Ai

Bi Bi

Bi Bi

57
Table 13. Selection rules—Continued

z Lx,Ly,xz,yz Z2 xy

A2u Eu A2g Eg Aig Big Big

A\u A2g Eg A2U Eu Aiu Bin Biu

Ais Alu Eu Aig Eg A2g B2g Big

Aiu Aig Eg Am Eu A2U Biu Biu

Big B2U Eu B2g Eg Big Aig Mg

Bi. B2g Eg Biu Eu B\u Aiu A2u

Big B\u Eu Big Eg B2g A2g Aig

Bin Bu Eg B,u Eu B2U A2u Am

Eg Eu AiuAiuBiuBiu Eg A\gA2gBigB2g Eg Eg Eg

Eu Eg AigA2gBigB2g Eu A\uA2uB\uB2u Eu Eu Eu

D}4g 2SuDi^u Dy,g 2S gDi^g iSg 2Sg

2SgD]^g Dy^u iSuDy^u B>y2u 2'S«

2S g iSuDy^u iSg iSgDyg iSg Dy^g

iSg 2SgDy^g iSu tSuDy^u Dy^u

For C4,: Drop "g" and "u" and read 2 in column. For D2d- Drop "g" and "u" and read z inxy column.

D3;. z x,y,p'',xy LiLyXZ,yz Z2

A[ a;' E' a; E" A\

A'2 A[' E' A[ E" a;

A\- A', E" a;- E' Ai

A',' A[ E" Ai E' A-;

E' B" A.1A.2E E' AiA'^'E" E' E" E' A1A2E" E" A'lA'^E' E"

2S2 2S1 2S2 Dy^ 2S1 Dh

2S1 2^2 2S2 Dy^ iSi

2'S2 Dh Di^ 2S1 282 2S1 2S2 2S2

±1, with change in parity, magnetic dipole

transitions have AM=0, ±1, with no change in

parity, while electric quadrupole transitions have

AM=0, ±1, ±2, with no change in parity.

In order to examine in detail the polarization

effects, let us consider a set of orthogonal unit vectors in spherical coordinates.

ro=sin d cos 0i+sm 9 sm </)j+cos dk

dr

0o=-^°=cos 6 cos </)i+cos 6 sin </)j— sin

-sin </)i+cos

1 dro

sin d d0

58
Table 13. Selection rules—Continued

Da z xz,yz Z2

Alu Eu A2g Eg Aig

A-, E,^g A-, R A

,

^lu

A2t Alu Eu Aig Eg Aig A o

.

A,, E A, ^u

Eg Eu Aiu Aiu Eu Eg Aig Aig Eg Eg

E„ E,^g ^1 g ^2g ^ g E A,^lu •^2u A-, -'-'« E J^u E

Slg Szu Slg

St. .Si ^3 u S,f^lu

Sz. Slu Slg D^g Ssg

Slg Slu S,u Dh. Slu Szu Di4u Slg Sig Di^^g D^g

Slg Ssg Dyg D^u SluS3uD,,.iu

For CsjI Drop "g" and "u" and read z in column.

z X y L„xy Ly,XZ

Aig Aiu Biu Biu A2g Bog Big Aig

Aiu Aig Big Big Aiu Biu Biu Aiu

A2g Aiu Biu Biu Aig Big Big Aig A2u Aig Big Big Aiu Biu Biu Aiu

Big Biu Aiu Aiu Big Aig Aig Big

Biu Big Aig Aig Biu Aiu Aiu Biu

B2g Biu Aiu A2u Big Aig A2g Big

Biu Big Aig Aig Biu Aiu Aiu Biu

Dm. D^u D^g Dy2g Dng Diig

Dj^u Dy^u Dy^u

y,Lx yz

XjLy

XZ

Lz,xy z,Z\p^

Co,: Omit "g"—"u" and read bottom of column.

x + iy x—iy z {x+iy)z

(x — iy)''

(x — iy)z — iLy

{x+iyY

L,,Z^

Ag Clu Ciu A u Cig Cig Ag

Au Cig Cig Ag Clu Ciu Au

Cig Ciu Au Clu Cig Ag Cig Clu Cig Ag Cig Ciu Au Clu

Cig Au Clu Ciu Ag Cig Cig Ciu Ag Cig Cig Au Clu Ciu

Slg Slu Sou Slu Slg S,g S\g Slu Szg Ssg Slg Slu Sou Slu

Szg Ssu Slu Szu Sse Slg Slg Szu Ssg Slg Slg Siu Slu Slu

S:g Slu Szu Ssu Slg Slg Ssg Siu Sis Slg Sog Slu Slu Ssu

59
Table 13. Selection rules—Continued

Ceo k x+iy z Lx+iLy Lx — iLy (x+iyy ix- iij)^ ix + iy)z {x — iy)z

A, C-lu Au Cig C-ig Ag Cig C-ig

Cig C-ig Ag Ciu C-lu Au Ciu C-iu

Cig Au Ciu Cig Ag Cig Cig C-ig

n. A g c,^\g Co A^ u Co^iu ' lu C-ig C-iu C-lu Ag C-ig C-ig Cig C-3g C-lu Ag C-2g C-ig Au C-iu C-lu Ciu C-iu

C3u Ciu Ciu C3g Cig Cig Cig Ag

c^Ig Co Ci^£u c,^lu Co^iu c.^iu C-2g C-,u C-3u C-iu C-ig C-3g C-ig Ag C-ig C-2u C-ig C-3g C-ig C-lu C-3u C-iu Au C-iu

Czg Ciu Cig Cig C-ig Cig Cie

Cig C,g C3g Ciu Ciu C3U Ciu Ciu C-3g C-i-j. C-3u C-ig C-ig C-3g C-ig C-ig

C-3u C-2g C-is C-3g C-ig C-ig C-3u C-lu C-iu

(Pattern for higher C^ns.u repeats)

Sig Slu S-lu Slu Ssg S-ig Sis Sig S-3g

Ssg S-ig Sig S3U S-lu Slu Siu S-3u

Slu S-3u S-lu SlB S-3g S-ig Sig S-ig

Sig S~3g S-ig Slu S-3u S-lu Siu S-iu

Sou Slu S3U Sig Sig Sig Sig S-ig

S,g Sig Ssg Siu Slu Siu Slu S-lu

S-lu S-5u S-3u S-ig S-ig S-3g Sig S-ig S-3u S-ig S~ig S-3g S-lu S-ou S-3u Slu S-lu

Ssg Syu S3u Siu Sig Sig Sig Sig Sig

Sig S3g Sig Slu Ssu Ssu Sou Slu S-sg S-3u S-7u S-5u S-3e S-7g S-ig S-ig S-gg S-5u S-3g S-7s S-ig S-3u S-lu S-iu S-lu S-9u

(Pattern for higher S^ng.u repeats)

z X + iy x—iy Z/j-f- iLy Lx — iLy (x+iy)^ (X- iy)^ (x+iy)z {x — iy)z

Ag Au Ciu Ciu Ag Cig Cig C2g Cig

A g Cig Cig Au Ciu Ciu C2U Ciu

C\g C\u Ciu Au Cig Cig Ag Bg

C\g Cig Ag Ciu Ciu Au Bu

Cig Ciu Bu Ciu Cig Bg Cig Cig Ag

Ciu Cig Bg Cig Ciu Bu Ciu Ciu A u Bg Bu Ciu Ciu Bg Cig Cig Cig Cig

Bu Bg Cig Cig Bu Ciu Ciu Ciu Ciu

Ciu Ciu Bu Cig Cig Bg Ag Cig

Cig Cig Bg Ciu Ciu Bu Au Ciu C^g C^u Au Ciu Cig Ag Cig Cig Bg

Csu Cig Ag Cig Ciu Au Ciu Ciu Bu

Sig Slu Ssu Sllu Sig Ssg Siig Sig Sgg Sig Szg Siig Slu Ssu Sllu Siu S'ju Ssg S3U Siu Slu Sig Sig Sig Sig Siig

S3g Sig Sig Szu Siu Slu Slu Sllu

Ssg SflU S,u S3U Sig Sig Ssg Sig Sig

Sig Sig Ssg Siu Slu Siu Siju Slu

Sig Slu S,u Siu Sig S^g Sig Siig Sig

Sig Ssg Sig Slu S^u Siu Sllu Siu

Sou Sllu Slu Siig Sig Sig Sig

Sig Siig Sig S^u Sllu Slu Slu Siu Sug Sllu Slu Sgu Siig Sig Sgg Sig Sig

Sllu Sug Sig S^g Sllu Slu Sciu Siu Slu

60
Table 13. Selection rule^—Continued

Ag

C\g

Bu

Bg

Czg

Slu

Sig

S^g

Sou

Ssg

Stu

Sts

x + iy

Cig

Bu

Bg

Csg A u Ag

S,u

Ssg

So^i Sag

S-u Sjg

Slu

Sig

x—iy

Czu

Csg Au

Ag

Clu

Cig

Bu

Bg

Stu

S,g

Slu

Sig

Ssu

Ssg

Sou

Soe

Ag

Au

Clu

Czg Czu

Sig Slu

Szg Szu

Sss S,u

Syg

S^u

{x +iy)z

Cig Clu

Bu

Czg Czu Ag

Au

Szg Szu

Ssg Siu

Srg Stu

Sig Slu

Lx —iLy

{x—iy)z

Czg Czu Ag

Au

Cig Clu Bg

Bu

Sig Stu

Sig Slu

Szg Szu

S-og Sou

{x — iyY

{x + iyy

Bg

Bu

Czg Czu Ag

Au

Cig Clu

Sig Ssu

Sig Slu

Sig Slu

Szg Szu

S4

{x + iyyz

{x — iyy

{x +iy) {x—iy)z

Lx +iLy

(x—iy)

{x + iy)z

A B Ci Cz A

C'l Cz B A Ci B A Cz Ci B

Cz Ci A B Cz

Si Sr. Sz Si Si Sz . s. Ss Si Sz

S-o Si s, Sz S3

S7 Sz Si Ss S7

Czh z x+iy x—iy Lx+iLy Lx iLy

{x—iyY {x +iyy {x + iy)z {x — iy)z

A' A" C'l A' C'l' C',' A" A' C'l cV A" C'l c.

C'l c. A' C'l C", A"

cr C'l C," A" C'l' c. A' cr A' C'l c. A" C'l' Ci' A" C'l' C',' A' C'l

Si Si s. So Si Sz Sii Sz s. Sii Si Sz s. Si

s, Sii Si s, Ss Si Sz S7 Si Sz Sn Si s. s.

s. Sz So Si s, Sn Si

Sii s. Si Sz Sii Si s.

551664—61 5 61
Table 13. Selection rules—Continued

C2A x,y z xz,yz

A, A, Bg As

A. u B, As Bu Au

Au Bu Ag Bg

B„ A, Bs Au Bu

Slu Sis Slu Sig S,u Slu

Sis Sig Sss Slu Sz.

Oh r L xz,yz,xij

Ais Tiu Tis Eg Tis Alu Tis Tiu Eu Tiu

Aog T.u Tig Es Tis Aou Tis T2u Eu Tiu Eg Tiu T'u Tig T-is Ais Aig Eg Tig Tis Eu Tis Tu T,u Tiu Alu Aiu Eu Tiu Tiu

Tis Alu EuTiu T,^ Aig Eg Tig T-is Tis Tig Ais Eg Tig Tig Tiu Aig Eg Tig T^s Alu Eu Tiu Tiu Tiu Tiu Aiu Eu Tin Tiu

T2S Aiu Eu Tiu Tiu Ail Eg Tig Tig Tis Tig Aig Eg Tig Tig Tiu A,s Eg Tig T-ig AiuEu Tiu Tiu Tiu Tiu Alu Eu Tiu Tiu Dhs Dhs T>z/2s Dzns iSg Dz/2 g

5 -03/2 8 Dy^u Dz/iu Dz/2u zSu Dzn

2'Sg 2Su T>iiiu iSg Dz/ig T>il2s T>3/2s 2'Sm 2Sg T>il2g 2(Sa 2)3/2

u

Diliu •Dh" -03/2 u Dws 2Su iSg Di^g Dz/ig iSs Dy^g Dz/2g iSs Dy^g Dz/2g D3/2U iSg Di^s T>3/2s 2<Su Dy^u Dz/iu iSu DkiL Dz/2u iSu Dy^iI Dzliu

and the inverse relations

i=sin 6 cos </)ro+cos 6 cos 4)9^— sin

j =sin 6 sin (^1*0+ cos 6 sin </>0o+cos 4>^q

k=cos 0ro— sin dO^.

Let us further consider A as a plane polarized

wave, incident toward the origin along the vector

To [fq points to the soui'ce of A], and having components along 00 and ^0 perpendicular to the

direction of propagation.

A= {A cos 4/6q-^A sin yp^Q) exp i(cr[r-ro]+c<;0-

In the terminology of the Euler angles of section

3.2, the plane polarized light is incident along the f axis with A parallel to the ^ axis. At the origin,

A° may be resolved into components as foUows

-4°=^e'"'(cos 4' cos 6 cos </>— sin xp sin </>) Al^Ae'^^cos \p cos 6 sin (/)+sui ^ cos 0)

^5=yle'"'(— cos xp sin 6). The vector A changes in space only along the

direction Tq, and hence a variation along the x-axis, for example, will change A according to the

component of the variation along Tq. Formally,

dA^ dA d[r-ro] dx d[r-ro] dx

Letting

62 I
Table 13. Selection rules—Continued

r L

xz^yZjXy

A, Tu C\g 2« Tu c lu C2U Tu T, C2g A,

c\l T, T ^2u Au

^ Is Tu T, Ag

T Tu Au Clu T, A~ u u ^2u T^u 9 1 a 0 0 J. a Tg T^g Ag Cig C2g Tg ^u^lu u ^

u

Tu T

r3 s -^J^u 2*5^11* 2S2U / / 1 / _ OlJt IT 909 0 2'S'u 2«S2 Dyie 2<Slg 2'S2« D}^u 2S1U 2S2U 2SIU 2'S2u

2*^28 iSie 2^28 Diiu 2S1U 2S2U 2^2. Dhu

2821 -DH" i^lu 2S2U Due 2<Slj 2S2g 2'Slg lS2u Dyjg 2^11 2S2g D^u 2S1U 2S2U Dviu 2S1U

r L 72 .2 " ,P xz vz xv

^1 T2 Ti E

A2 T: T2 E

E Ti T2 Ti To A, A[2 E

T, A2 E Ti T2 Ai E Ti T2 T, T2

T2 At, E Ti T2 Ai E Ti T2 Ti T2 2S D3/2 T>ii D3/2 D3/2

2S D^i Dz/2 2S D3/2 D3/2

D3/2 Dh 2^ D3/2 Dyj 2S D3/2 Dh 2 S D3/2

we have

5A

Obviously, the resolution of A' into components

is parallel to that of A, so we obtain the following

9 terms involving the derivatives of A

:

-^=A' (cos \f/ cos 6 cos 0—sin rp sin 0) (sin 6 cos </>)

-A' (cos 4' cos 6 cos 0—sin \p sin 4>) (sin 6 sin <^>)

-^^=A' (cos ^ cos 0 cos 0—sin ^ sin 0) (cos d)

-^=A' (cos ^ cos 0 sin </)+sin ^ cos 0) (sin 0 cos ($)

-=x4' (cos 1^ cos ^ sin 0+sin i/- cos 0) (sin Q sin 0)

^^=A' (cos i/- COS (9 sin (ji+sin ii cos 0) (cos 0)

du4 -^=-4' (—COS i/' sin Q') (sin 9 cos </>)

d2

=A' (—cos \i' sin ^) (sin Q sin

(—cos ^ sin ^) (cos Q).

These may now be combined as previously defined

(page 55) to yield

A^nc=^A' (cos i/ sin 20 cos' (^— sin i/- sin Q sin 20)

Ayj,=2-^' (<^*^s "A 20 sin^ </)+sin sin 0 sin 20)

A' (—cos sin 20)

yl' (cos ^ sin 20 sin 20+2 sin sin 0 cos 20)

^12=^' (cos i/ cos 20 cos 0 —sin ^ cos 0 sin 0)

Ay2=A' (cos 1/' cos 20 sin 0 +sin 1/' cos 0 cos 0)

5j:=—A' (cos sin 0+sin i/- cos 0 cos 0)

52/=-4' (cos -^z cos 0—sin i/- cos 0 sin 0)

5^=^' (sin v^ sin 0). We may note that A^x-VAyy-\-A^^ is in fact zero, and that the angular factors for the components of

B are the same as those for A with a rotation of 90°

in showing the mutual perpendicularity of B and

A. In order to obtain our final five quadrupole

components, there is a question of normalization

which must be considered. Strictly speaking, the

appropriate "base vectors" for the quadrupole

components are the normalized tesseral harmonics of table 2 for N=2. Since we shall be

interested in relative intensities, we may omit a common factor, which we shall here choose to be

15

^^^=:- With the omission of this factor, xy, xz, and

2V^

yz will still be the proper basis, but for the other 2, where we have heretofore used x^—y'^ and 22^

63
—x'^—y'^, we must now include factors of \ and

respectively, so the balance of the term with these

two bases will become multiplied by the reciprocals

of these factors. We shall continue to denote

these two renormalized bases as p- and Z^. Finally, we must consider the resolution of the

multipole interactions into the (complex) base

vectors appropriate to the group C„,, and its subgroups with complex reps. The terms in z, L^,

and will be unaffected by this change. In lieu

of X and y, we shall use -^{x-^iy) and -^{x—iy),

for Lx and Ly a similar combination, and for and

xy, (p^-^-ixy) and -p (p^—ixy). This requu'es

the use of the following linear combinations of

the previous terms:

A'. V2 A

o_V2 {Al +iAD

^2+=-V (A,2-iA,y) A<,^=\ {Afi^iA^y).

These are all summarized in table 14. Let us now consider some of the conclusions of

this analysis. We shall discuss in detail six cases.

1. Only one component of the eleven considered

is effective in producing the transition. In this case, the rate of absorption will be proportional to the Hermitian square of the matrix element, and

hence proportional to the Hermitian square of the

appropriate angular dependence factor of table 14. Such a case is exemplified by the B^g—i?2g transition

in D47,, which are joined only by Zj. The angular

dependence is thus (sin sin

2. Two components only of the eleven join the

states, these two belonging to a doubly degenerate

rep and consequently arising from the same type

of multipole interaction (e.g., in De;,, A^g is joined

to E-ig by both xy and p^. Then the total matrix

element will involve the sum of the two angular

factors for the two multipole components, and the

angular dependence will be proportional to the

Hermitian square of the sum of these angular

factors.

3. Two components only join the states mvolved, arising from different multipole interactions (e.g., A] of Ces will join itself bj^ either z or

Z^). The matrix element will then mclude the

sum of the angular factors for the two components,

but one will be multiplied by a factor "q" determined by the numerical ratio of the strength of

Table 14. Angular dependence of multipole components

1. Electric Dipole ^° = A( — cos \p sin e) A° = A{cos rp cos 0 cos — sin sin 0) A2= A'(cos_i/' cos 6 sin 0 + sin i/- cos 0) A+=a(^^^^ (cos 1^ cos 6 — i sin 4/)e~^*

(cos \p cos d + i sin </')e'*

2. Magnetic Dipole

Bj = A'(sin sin e) B^ = A'( — cos \p sin 0 —sin cos B cos 4>) By— A' {cos ip cos <#> —sin \p cos 9 sin <p) B+ = A'^^~^ ( — sin \p cos 9 — i cos i/')e~'*

B- = A'(^Y^ ( — sin tp cos d + i cos i/')e»*

3. Electric Quaclrupole

A,2=A'(^:^^ (_cos <p sin 26) Ai2 = A'(cos \p cos 2d cos ^ —sin tp cos 6 sin 4>) Ayz — A' (cos ip cos 2(9 sin </) + sin \p cos 6 cos 4>) Ap'^ = ^A' (cos tp sin 28 cos 2</) — 2 sin \p sin d sin 2<^>) Axu = iA' (cos tp sin 2d sin 2^ + 2 sin tp sin 6 cos 20) =4'^^^ (cos tp cos 20— i sin tp cos 0)e~'*

A,~ =A'^j^^ (cos tp cos 2d + i sin tp cos e)e'*

A2+ = A'(^~-^ (cos tp sin 2e-2i sin tp sin e)e-2'*

A2- = /l'(^— ^ (cos tp sin 2e + 2i sin tp sin (9)e2''*

the two types of interaction, and is fixed by explicit

calculations and not by general symmetry considerations. The angular variation will then be proportional to the Hermitian square of this sum, and

will contain q and g^. If one type of interaction

is very much stronger than the other, then q wQl

be small and it may be neglected. In the present

problem of crystal spectra, this is not necessarily

the case, and each case must be examined in

detail.

4. Two components join the states in question,

arising from the same multipole interaction, but

the components belong to one-dimensional reps

(as distinguished from (2)). For example, in

D27J, A2g is joined to itself by both Z^ and p^

The details here are very similar to case (3). The

matrix element will include the sum of the angular

factors. However, in contrast to (1), the term Z^

is not related by sjonmetry to p^, so one angular

factor will again be multiplied by a factor q, expressing the relative strength of the interaction,

and the Hermitian square will again contain both

q and Generally, one may only expect that q

will in this case be nearer 1 than in case (3).

5. Only one component joins the states, but

another component joins another pau' of states,

both of which are respectively degenerate with

the first pair. This is a frequent situation in

64
j C„;j and its subgi'oups. Here the transitions

! proceed independently, and the total absorption

I will be proportional to the siun of the Hermitian

i squares of the angular factors, rather than to the

i Hermitian square of the sum as in previous cases,

i 6. If the states are joined by thi-ee components,

I all belonging to the same triply degenerate rep,

I as is the case for both electric and mag-netic

j dipole transitions in the cubic groups, there are

no polarization effects. We shall not discuss in detail more complicated

cases. In general, the angular dependence will include factors expressing the relative strengths

of the transitions arising from different dipole

interactions, as Vv^ell as from nondegenerate interactions of the same order. In specific cases, these

relative strengths may be computed from the

methods of the following section.

5. Matrix Elements

5.1. The Reduced Matrix Element

;!

I We now turn our attention to the procedm-e

: alternate to that which has occupied our attention

, in the preceding two sections, namely, the calculation of the matrix elements of the perturbing

potentials of the crystal field and their interpretation. These potentials were discussed in detail

in section 2. We have akeady touched upon

certain general featiues of the problem in section

4.3 —we must now examine this situation in more detail, and also determine the simplifying

relationships w^hich exist among the nonvanishing

I elements.

So far in our discussion of the transformation

properties of the several wave functions belonging

to a degenerate rep of a group, w^e have emphasized

the equivalence of these several wave functions,

• as well as the properties of the rep (notably the

character and class structui-e) which were invariant under a unitary transformation. We shall now depart from this procedure. We are now

interested in picking out a particular set of linearly

independent wave functions, in terms of which

specific matrices representing the operations of

the group may be written do"\vn. This is equivalent to the introduction of a specific system of

base vectors in ordinary vector analysis.

Since we start with the wave functions of the

free ion, described by the quantum number J,

with 2j-{-l linearly independent wave functions, we select for our basis, functions distinguished by

the quantum number M (or Mj will sometimes be

used for clarity) where M=J, J— I, «/— 2, . . . —J, and such that the matrix representing rotations about the 2-axis is diagonal with elements

Me^^* .We have seen from the preceding section

that wave functions belonging to different reps

(here, with different J values) are orthogonal—we

see that these 2^7+1 functions themselves belong

to different reps of the group C„, and hence are mutually orthogonal within a given J. These wave functions we can write as |aJM>- (in the

notation of Dirac), where a represents all quantum

numbers other than J and M. To emphasize that

such a state is one component of a 2J'+1 dimension rep of the rotation group, we may also write

in the form (aJ)|JM>. A different choice of axes will leave the (scalar) portion of the function

unaltered, but will convert the normalized component of the rep |JM> into a combination of

the same J and dift'erent M.

For these calculations, it mil be convenient to use the complex potentials Y? (2.9) rather than

the real Q and S? (2.5) as om- perturbation. We

note here that these perturbing potentials actually

act upon the space coordinates of the individual

electrons, so the total perturbation is a sum of

such over all electrons. Clearly, if these potentials obey a certain transformation law in the

coordinates of one electron, then- sum will similarly

transform when all the coordinates of all electrons

imdergo the simultaneous transformations implied

by the quantum number J, or when only the

space coordinates of all electrons undergo the

simultaneous transformation implied by the quantum number L. Our problem has therefore

become that of computing the sum of terms of

the form

<a'J'M'\r''Y'AaJMy (5.1)

We shall for the time bemg disregard the factor r". According to section 4.3, we Imow that the

matrix element will vanish unless J' is one of the

values J" arising from the reduction of the product

DjXDy, nsiinely, J+v, J-\-v—l, . . . \J—v\. Let

us expand the product Y!^|a;JM> into a sum of

terms

i:CZ^^iYMaJ)\J'^M"> (5.2)

M"

where the C-coefficients are the Wigner coefficients to be discussed further in section 5.2, the term

(Y^llaJ) represents the portion of the product in- variant under a rotation of the axes, and \ J"M"'^

represents the M"-ih component of the

normalized rep Dj,> of the rotation group. We

now know that the remainmg term of (5.1),

<Ca'J'M'\ = (a'J'XJ'Ad'\ is orthogonal to ah

states except |J'M'>, i.e., <J'M'\J"M">=8j,jndM'M" (5.3)

so we obtain, from (5.2) and (5.3),

<a' J'M' I Y? |aJM>= Ci,;i,: (a' J' 1

1 Y. | |aJ) • (5 .4)

In this expression, parity is included in the a,a'.

If V is even, a and a must be of the same parity;

if odd, of opposite parity.

65
The importance of this step is that we have

placed into the term GmIm, the dependence of

the matrix element on M, M', and /x, and the

"reduced matrix element" {aJ'\\y^\\aJ) contains

those portions of the interaction not dependent on

these quantum numbers. If the matrix element

is evaluated for one particular Ad, M', and /z, then

the reduced matrix element can be obtained, and

from it the matrix element for any other M, M',

and fx. This relation was initially given by Eckart

(1930) and again by Wigner (1931) and Kacah

(1942a). There is some divergence in the litera- ture on normalization and phase factors for these

reduced matrix elements. We have here followed

Eckart, Wigner, Rose, and others, while Racah,

Biedenharn (1952), and Simon et al. (1954) have

used as reduced matrix element a quantity greater

by a factor yj2j'-\-l.

Eckart and Wigner have based their derivation

of this result on the ideas of the representations of

the rotation group, while Racah has developed the

algebraic application of the angular momentum

operators Jx, Jy, and J^, which may be mterpreted

as differential rotations about the x, y, or s-axes, respectively. Since the vectorial character of a vector operator T with components T^, T°, is fixed by the general commutation rules [J2,T"] =

nT" and [Jr±iJy, r"]=V(l T?i)(l±w+1) r"±i, a

generalized irreducible tensor operator of degree

k has been defined as one having 2^+1 components and satisfying the commutation rules

\Jz, lt\=nli and [J, d= iJy, T,"] =^l{kTn){k±n+l)

1"=*=^ Our quantity Yf^ is a specific example of

such an operator.

Stevens (1952) has emphasized a geometrical

interpretation of this result, and has given what

amounts to a technique for calculating certain of

the Wigner coefficients in the special case J'=J.

The behavior under rotation of any quantity

which may be expressed in terms of variously

oriented coordinate systems, may be described by

one or more components along a set of mutually

orthogonal base vectors in an mfinite number of

dimensions. Subspaces of dimension 1, 2, 3,

. . . 2t/+l determine a particular Dj, while the

individual vectors withm the subspaces correspond

to the various rows of the Dj. A product of two

components, each of which lies along one of these

base vectors (such as the Y!f|a'e/M> being considered) may itself be resolved into components

by means of the Wigner coefficients; all save the one parallel to the third quantity <aVM'j will vanish in the matrix element. Stevens also pointed out that it is possible to form irreducible tensors j'i from the noncommuting components of

J by taking the 2i'+l totally symmetric products

of degree v. This is analogous to the process we

have already used several times in connection

with the reduction of the general product of

two vectors. We then have

<ajM'mccjM>=ci,i^, miwj),

since this reduced matrix element is diagonal in both a and J, and independent of a. These matrix elements may themselves be calculated, giving

a set of numbers proportional to the Wigner

coefficients. Certain tables based on this procedure have been given by Stevens (1952), by Elliott

and Stevens (1953a), and by Judd (1955).

5.2. Wigner Coefficients

The Wigner coefficients Cm\mIm, frequently writ- ten {JiJ2JM\JiJ2M)Al2) are the elements of the unitary matrix C which will reduce the (2t7]-j-l)X

(2t/2+l) dunensional direct product of the components of Dj^ and Dj^ and which will provide the

proper linear combinations of the products of

these components IJiMi^ and !t72^2> appropriate to the reduced component |t/M>. The

columns of the C matrix are distinguished by the

double indices (Mj, Mo), vhUe the rows bear the

indices (J, M). The phase conventions implied

in (2.8) ensure that the matrix elements may be

chosen to be real. The inverse is merely the

transposed matrix. The orthogonality relations

for a real unitary (=orthogonal) matrix then give usS {JxJ2MM2\JMM) {J,J2MM2\JiJtJ'M') =8rr>d. jj'Omm' (5.5a)

}

J,M

=8m^m[8m^'^- (5.5b)

The product of \JiMi^ and |J2Af2> will, under

a rotation about the s-axis through 4>, be multi- - plied by e^^i'* e^'^^^^^e'^'^'i^^'^^^ and hence M=

Mi+Mi, and all coefficients not meeting this ^ requirement are zero. The index M is frequently

suppressed when written in the form CMlufBecause of this relation between the Mi and M2,

the sum in (5.5a) is in effect a sum over ikfi only

(or M2 only).

A general formula for these coefficients has been j given by Wigner, who carried out in detail the

process outlmed above of reducing the direct

product matrix. Condon and Shortley have approached the problem from the standpoint of

transforming eigenstates of two commuting angular momenta J, and J2 with quantum numbers . t/iM] and J2M2 into eigenstates of J =Jt+.T2 with ' quantum nmnbers J and M. They pointed out

that the vector addition coefficients may in principle be obtained from the initial condition-

{JiJtJ^J-^J^J'iJi-\-J2\\J^^J2\) = '^ and from a successive application of the Jj—iJy and Jxz operators, but that a general formula is difficult to obtain. Racah (1942a) utilizing as well the

operator Jx-\-iJy has provided a purely algebraic

derivation of a general formula for the coefficients { which is relatively convenient and symmetric in the various parameters. Based upon this for66
are

mula, Simon (1954) has provided a numerical

table of these coefficients, but no J greater than

^2 is included. Condon and Shortley have given

formulas for the cases where J2=,^^, 1, and 2, while Falkoff et al. (1952) gis^e ^2=3.

j For actual computations, a numerical table of

[Wigner coefficients is required. The only such

'table known to the UTiter is that of Simon (1954).

;Unfortunately, this covers an madequate range of

'values for rare-earth spectra, and relies on symmetry properties of the Wigner coefficients, in1 / 2^7+1 \''-

(ivolving factors of the form ( 2X+I/ ' ^^^^^

[usually somewhat awkward. To meet partially

khis need for the present application, table 15,

containing the coefficients Cii'/ to 6 decimals for J'<8, f=2, 4, 6 and m =0, 2, 3, 4, 6 consistent with

the V involved has been computed. These wdl

permit calculation of matrix elements diagonal in and between states of the same parity.

I In the case /x=0, 3, and 6, the values for

'!Af=e7—/i were calculated from the formula of

Sacah and those for other values of M were cal- culated by the use of the tables of Stevens and

others noted above (section 5.1). For ^=4 and

y=4, M=2, factors corresponding to those of

Stevens were computed and used in a similar process. For M=2, J' =6 this became too miwieldlj^

and the formula of Racah was used directly for

all entries. Coefficients were aU checked for

obedience to the requirement

and selected entries were checked for

M

2J+1

'2f+l

M

It is believed that errors do not exceed 2 in the

sixth decimal. In a very few entries a seventh

decimal was caiTied. It is to be noted that only positive values of jj. are given, and only entries for M>—^. To obtain other values required, the following special cases of the general symmetry relations are re- quired (diagonal in J and v even)

:

(5.6a)

(5.6b)

C-,,U,//=i-iyCZ-'. (5.6c)

The relation (5.6c) is especially noteworthv: we

observe that (for half-integral J)

and hence must be zero. This is intimately asso- ciated with the essential degeneracy of liramers

discussed earlier.

Table 15. Wigner coefficients

\mj\ 0 1 2 3 4 5 6 7 8 \

12

3

-0. 632456 -. 534522 -. 516398

0. 316228 -. 267261 -. 387298

0. 534522

0 0. 645497

4

5

6

i

-. 509647 -. 506369 -. 504525

-. 433200 -. 455732

468488

-. 203859 -. 303822 -. 360375

. 178376 -. 050637

180188

0. 713506

. 303822

. 072075

0. 759554

. 396412 0. 792825

7

8 -. 503382 -. 502625 -. 476415 -. 481682 -. 395515 -. 418854 -. 260680 -. 314140

071912

167542

. 170790

. 020943

. 467426

. 251312

0. 817996

. 523568 0. 837708

I -\ 1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2

3/2

5/2

7/2

-0. 447214 -. 478091 -. 487950

0. 447214 -. 119523 -. 292770

0. 597614

. 097590 0. 683130

9/2 11/2

13/2

-. 492366 -. 494728 -. 496139

-. 369274 -. 409917 -. 434122-. 123092 -. 240296 -. 310087

. 246183

. 014135 -. 124035

0. 738549

. 353377

. 124035

0. 777429

. 434122 0. 806228

15/2 -. 497050 -. 449712 -. 355036 -. 213022 -. 023669 . 213022 . 497050 0. 828417

67
Table 15. Wigner coefficients—Continued

\\MJ\

\

0 1 2 3 4 5 6 7 8

2

34

0. 534523

. 426402

. 40.i^91

-0. 356348

. 071067

. 201146

0. 089087i -. 497469 — . 245844

0. 213201 —. 469340 0. 312893

5

6

7

. 392232

. 386953

. 383807

. 261488

. 294821

. 315270

065372

. 050672

. 127428

-. 392232 -. 248756 -. 126413-. 392232 -. 442232 -. 357408

0. 392232

304035 -. 441175

0. 456052

217795 0. 508189

8 . 381771 . 328747 . 180281 -. 031814 -. 254514 -. 413585 -. 413585 137862 0. 551447

\

1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2

5/2 7/2

9/2

0. 308606

. 341882

. 354787

-0. 462909 -. 113961

. 059131

0. 154303 -. 493829 -. 335077

0. 265908 -. 433629 0. 354787

11/2

13/2

- 15/2

. 361298

. 365073

. 367467

. 154842

. 212959

. 250811

167745 -. 043944

. 044718

425815

310988

196371

348394 -. 446201

390798

0. 425815 -. 260284 -. 429684

0. 483384 -. 176929 0. 530786

\

•'\

0 1 2 3 4 5 6 7 8

3

4

5

-0. 482804 -. 373979 -. 347368

0. 362103

. 018699

104210

-0. 144841

. 411377

. 312631

0. 0241402 317882

. 251842

0. 0747958 -. 416842 0. 130263

6

78

-. 335531 -. 329015 ~. 324985

167765

205635

230198

. 184542

. 082254

. 005416

. 360696

. 324080

. 251864

. 067106

. 289534

. 346651

461355

090479

. 176034

0. 184542

470492

211240

0. 235246 -. 457688 0. 281664

\\ M 1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2

7/2 9/2 11/2

-0. 241402 -. 273115 -. 286829

0. 434524

. 204836

. 057366

-0. 241402

. 341393

. 358536

0. 0482806 -. 376533

. 157756

0. 102418 -. 444585 0. 157756

13/2

15/2 -. 294280 -. 298843

. 036785 -. 099614

. 272209

. 179308

. 334007

. 346658

-. 016185

. 235090

469376 -. 1553980. 210410

466195 0. 258997

68
I

I

I Table 15. Wigner coefficients—Continued

\

\

-1 0 1 2 3 4 5 6

12

3

0. 774597

. 654653

. 632455

0. 534522

. 577350 0. 408248

4

5

6

. 624188

. 617915

. 592156

. 603023

. 495434

. 558291

0. 330289

. 483494

A 977^ p;n \j. £i t i oou

. 377965 0. 239046

7

8

. 616515

. 615587

. 605405

. 606977

. 572054

. 581136

. 516377

. 538028

. 438160

. 477567

. 336817

. 399561

0. 210042

. 303489 0. 187317

\

\

-1/2 1/2 3/2 5/2 7/2 9/2 11/2

3/2

5/2 7/2

0. 632456

. 621059

. 617213

0. 462910

. 534522 0. 365148

y/z 11/2 13/2

c-i CA r7

. 614509

. 613941

. od4U7o

. 579365

. 588349

. 4DU5b6

. 508862

. 537087

0. 301511

. 402291

. 459933

0. 256776

. 356263 0. 223607

15/2 . 613572 . 594088 . 555088 . 496486 . 418079 . 319313 0. 198030

C'.i ^\

-1 0 1 2 3 4 5 6

23

4

-0. 563436 -. 449467 -. 424052

0. 345033 -. 123091 -. 245844

0. 522233

. 186989 0. 560968

5

6

7

-. 413449 -. 407884 -. 404568-. 299572 -. 328395 -. 345779

0-. 110558 -. 180745. 358057

. 183513

. 056476

0. 554700

. 449089

. 308827

0. 532554

. 495260 0. 505389

8 -. 402422 -. 357114 -. 227940 -. 035172 . 187317 . 391802 . 515831 0. 477567

\

-1/2 1/2 3/2 5/2 7/2 9/2 11/2

5/2

7/2 9/2

-0. 345033 -. 372194 -. 381690

0. 462910

. 053722 -. 1166080. 550482

. 285631 0. 560968

11/2

13/2

15/2

-. 386244 -. 388809 -. 390406

-. 204837 -. 256681 -. 289807

. 099950 -. 022676 -. 105958

. 410891

. 252439

. 126363

0. 544705

. 476312

. 354689

0. 519238

. 507935 . 491412

551664—61 6 69
Table 15. Wigner coefficients—Continued

f J6J

/\

-1 0 1 2 3 4 5 6

34

5

0. 494727

. 383214

. 355945

-0. 361298

0

. 12/813

0. 127738 -. 434524 —. 302079

0. 289683 — . 308257 0. 397959

67

8

. 343817

. 337140

. 333010

. 184542

. 220710

. 243919

-. 163087 -. 058394

. 017964

-. 376632 -. 316265 — . 232843-. 147214 -. 335450 — 361685

0. 465530

0— . 247027

0. 505389

0 526926

\\M

J\

\

-1/2 1/2 3/2 5/2 7/2 9/2 11/2

7/2 9/2

11/2

. 255476

. 285631

. 298081

-. 442498

186989 — . 035129

. 215917 -. 381691 — . 359967

. 349825 —. 227663 . 435942

13/2

15/2

. 304672

. 308643

. 058394

. 119537 -. 255871 -. 156366 -. 365192 -. 349645 -. 070719 -. 294427

. 488252

. 064249 . 517994

\ -1 0 1 2 3 4 5

2

34

-0. 527046 -. 265908 -. 186989

-0. 564076 -. 469339 -0. 494727

567

-. 146176 -. 120654

102998

-. 392232 -. 335111 -. 291972

-. 506369 -. 472034 -. 431342-0. 423659 -. 490098

493065

-0. 363803 -. 457141 -0. 315165

8 -. 089984 -. 258460 -. 393234 -. 474495 -. 486664 -. 420595 -0. 275723

J \^

-1/2 1/2 3/2 5/2 7/2 9/2

5/2 7/2

9/2

-0. 577350

402015

311649

-0. 531816 -. 498639 -0. 458029

11/2 13/2

15/2

-. 255476 -. 216867 -. 188584-. 442497 -. 391883 -. 349644

501745 -. 487398 -. 457792

-0. 392232

474594 -. 492112

-0. 338200 -. 438906 -0. 294427

\

'\

-1 0 1 2 3 4 5

3 0. 469340 -0. 221249

4 . 270973 . 323875 -0. 409673

5 . 201802 . 376036 . 058255 -0. 487398

6 . 163087 . 360696 . 248129 -. 147214 -0. 509963

7 . 137637 . 332940 . 322787 . 079066 258078 -0. 505389

8 . 119407 . 304863 . 347875 . 209882 -. 071755 -. 372080 -0. 487838

70
Table 15. Wigner coefficients—Continued

\

^\\

-1/2 1/2 3/2 5/2 7/2 9/2

7/2 9/2 11/2

13/2

15/2

0. 442498

. 373979

. 316163

. 272507

. 239074

-0. 334497

. 186990

. 324509

. 357008

. 354605

-0. 458029 -. 053661

. 163319

. 270834

-0. 503382 -. 223634

0 -0. 509963 -. 333849 -0. 497673

^ Mi \\ M

\

-2 -1 0 1 2 3 4

2

34

5

67

8

0. 745356

. 594588

. 560968

. 546941

. 539583

. 535193

. 532354

0. 460566

. 494727

. 506369

. 511894

. 514990

. 516920

0. 312893

. 392232

. 432629

. 456490

. 471881

0. 226455

. 313114

. 366006

. 401022

0. 171499

. 253968

. 310630

0. 134387

. 209427 0. 108148

\, M

J\

\

-3/2 -1/2 1/2 3/2 5/2 7/2

5/2

7/2

9/2

11/2

13/2

15/2

0. 577350

. 550482

. 539792

. 534366

. 531214

. 529213

0. 376051

. 440738

. 469339

. 484930

. 494472

0. 264443

. 349825

. 397958

. 428225

0. 196116

. 281399

. 336926

0. 1512475

. 230164 0. 120199

\ M

J\

\

-2 -1 0 1 2 3 4

34

5

678

-0. 541947 -. 419790 -. 389919 -. 376632 -. 369318 -. 364795

0. 349825 -. 079333 -. 195539 -. 248128 -. 277194 -. 295182

0. 501745

. 233021

. 067106 -. 037801 -. 107785

0. 504505

. 382472

. 247517

. 137400

0. 465530

. 441640

. 354765

0. 417311

. 454447 0. 370540

\s M

J\

\

-3/2 -1/2 1/2 3/2 5/2 7/2

7/2 9/2 11/2

13/2

15/2

-0. 312893 -. 334497 -. 341495 -. 344698 -. 346452

0. 458029

. 102418 -. 055544 -. 141599 -. 194225

0. 512092

. 322921

. 167849

. 056068

0. 487398

. 419971

. 308798

0. 441641

. 452034 0. 393445

71
Table 15. Wigner coefficients—Continued

\\ M

J\\

-3 -2 -1 0 1 2

34

5

0. 733799

. 568399

. 527952

0. 429669

. 457220 0. 273241

6

7

8

. 509963

. 500060

. 493934

. 465530

. 469098

. 470948

. 345246

. 383017

. 405906

0. 184542

. 260982

. 310016

0. 130491

. 200114 0. 095673o

\ M

J\ -5/2 -3/2 -1/2 1/2 3/2

7/2 9/2

11/2

0. 554700

. 518874

. 503381

0. 339683

. 397958 0. 223100

13/2

15/2

. 495034

. 489956

. 423922

. 438230

. 299758

. 344798

0. 154399

. 228062 0. 111283

5.3. L-S Coupling and the One-Electron

Approximation

It is well to emphasize at this point that the

relation (5.4) involves no approximations —these we shall introduce in evaluating the reduced

matrix element. The first of these is the approximation of L-S coupling, i.e., the state |at/M>

may be written \aJMy =Y^\0LSMlMs'>

Ml

{LSMiMs\LSJM) , and consequently,

<aJ'M'\y>i\aJM>= S (L'S'J'M'lL'S'MLM's)

<l3'L'S'MiM's\Y'i\^LSMiMs> {LSMiMs\LSJm

(5.7)

We now recall that does not act upon the spin

coordinates of the system, that it is thus a scalar

quantity with respect to S, and therefore

<^'L'S'MiM's\i'i\^LSMrMs> =hs' 5m,m-

<^'L'SMlMs\Yt\^LSMrMs>.

This will reduce (5.7) to the equation

(5.8)

<a'J'M'\Yl\aJM> =8ss' 2

{L'SJ'M'\L'SMiMs)

<^'L'SMlMs [Yt] &LSMiMs>{LSMiMs\LSJM).

(5.9)

All other matrix elements vanish —in particular we see that there are no matrix elements between

terms of different multiplicities. Also we see that

(5.4) is again applicable, and obtain

<^'L'SMlMs\Y'.\^LSMiMs> = {LvL'ML\LvMLti)

i^'USWY.W^LS). (5.10)

Hence, we can write (5.9) in the form

<:^'L'SJ'M'\Yl\^LSJMy= Zl

{L'SJ'M' \L'SMiMs) (LvL'MLlLvMLti)

(LSMrMslLSJM) {&'L'S\ \Y,\\&LS), (5.11)

and this expression is also (5.4). If we multiply

both expressions by {JvJ'M'\JvMfx), introduce

the requirement that Mi=ML-^ ix, and sum over M, the expression on the right of (5.4) wiU reduce

to {a'J'\\Yy\\aJ) by (5.5a) while the right side of

(5.11) becomes

2 {JvJ'W \JvMii) (L'SJ'M'\L'SiMi,+ix)Ms)

[LvL'[Mr^+ix]\LvMj^ii)

(LSMzMslLSJM) {I3'L'S\ |Y,| \I3LS). (5.12)

Such a sum of the product of 4 Wigner coefii- cients appears recurringly in the theory of complex

spectra. The properties of such expressions were

first studied extensively by Racah (1942a), and

they have consequently become known as Racah

coefficients. In the present application the essential significance of (5.11) is that the two states JM and J'M' are "uncoupled" into LSMjMs

and L'S'M'iM's so that the effect of the

acting only on the L part (i.e., the space coordinates) may be determined. The Racah coeffi72
cients appear generally in connection with such

uncoupling or recoupling of angular momenta.

The Kacah coefficient in (5.12) depends on six parameters JJ', LL', S, and v, in spite of the

explicit appearance of the projection quantum

numbers M, M', Ml, Ms, and /x, they are eliminated in the summation.

Although a general algebraic formula for these

coefficients is quite unwieldy, numerical tables

have been computed by Biedenharn (1952) and

by Simon et al. (1954). The first of these carries

the entries in a closed fractional form, the other

has a ten-place decimal form. In order to clarify

difference in notation, let us rewrite (5.12) as

M,

Ci,^ul{^'L'S\\\M^LS). (5.13)

Omitting algebraic details, the use of the symmetry relations for the Wigner coefficients will permit this to be transformed into

(-

/ (2J+1)(2L- + 1) V (2S+1)(2.+ 1) ^ ^ J L Sp S L' J' AIM ^^^L ^ <.M+MOm'-M-Mi)

c

Li V J V (H'L'SWY.mS)

and a'direct comparison with the notation of this

with'eq (1), page x of Simon et al. will give

(a' J'

1 1 Y,| laJ) = (- l)^+^-^'-V(2J+l)(2i'+l)

W(JLJ'L';Sv)(l3'L'S\\Yy\\0LS). (5.14)

The six-parameter expression W is tabulated in Sunon et al. for J,J'< 15/2; L,L' < 9/2 (though of course a half-integral L has no significance in the

present application) 5'<3 and i'<8, though the

useful range of values may be extended through

the use of the syrometry relations given in the

introduction to the tables. We are now in a position to make comparisons

between quantities which depend only on the

ratios of the matrix elements for a given ^'L'S,

combination, but for additional details we

must evaluate the reduced matrix element

{1^'L'SWY^W^LS) . In principle, this maybe accomplished in a way simQar to the derivation of (5.13)

i.e., we now assume that the state \LSMiMs^ may

be m-itten as a sum of antisymmetrized products

of one-electron wave functions \nlmims^ and

the matrix element (5.10) can be expressed in terms of one-electron matrix elements such as

<ji'l'm\m's\'Yv\nLmi'msy'.

Unfortunately, this procedure has several limitations. First, there is the additional assumption

that the state involves only one configuration of

the free ion. In the present case, it is expected

that configiu-ations other than 4/" will interact

with the surrounduigs and other rare earth ions

to such an extent that they will no longer be characterized by sharp energy levels, but by broad

levels or bands. The sharpness of the observed

lines suggests that the contribution of other

configurations to the wave functions of the free ion is, at least as far as the energy is concerned,

negligible. It is quite possible that contributions

from other configurations are responsible for

features such as intensities and polarization; this

will be discussed further in the following section.

An additional disadvantage is that, unlike the

transformation from a JAI representation to an LS representation, there is no general formula for passage from the LSMlMs representation of a

state to its one-electron representation. The

general method of Gray and Wills (Condon and

Shortley 5^) may be used, but each configiu-ation

must be considered by itself, and no general

formula can be derived.

5.4. The Rare Earth Ground Terms

In the evaluation of the matrix element (5.10) we are free to select any consistent set of the

projection quantum numbers; we shall therefore

consider the specffic element [M'l=L' ;Ms=S\

<^'L'SL'S\Y^\0LSLS> = {LvL'L'\ LvU)

{^'US\\Y.\I3LS). (5.15)

By selecting the maximum values of the projection quantum numbers, we shall find that the

transformation to the one-electron representations

is thereby simplified considerably.

For any particular rare earth, the analysis is best carried forward in a systematic manner,

beginning with the ground state. In accordance

with Hund's rules, for 4/" we select the highest

allowed multiplicity, and then the highest L

consistent with this multiplicity. The corresponding antisymmetrized products of one-electron wave

functions are usually written {mim.im2 • • • I; for example, in this notation we start with 132 1| for (++++ + +1 (+-+++ + + +1 Nd, 13210-1-2} for Eu, 133210-1-2-3} for Tb, etc. While the analysis may be carried out

for any configuration, we shall consider the specific

example of Nd.

Let us begin with {32 ll=|^/ 6|>- The application of the synametrie operator J/^-=L^—iLy

(Condon and Shortley 5^3) yields {320}^ l«/6|>

A second application of yields 2V3 I32-1I

+V10 l31ol=V22|*/4|>- The state 10

73
!32-tl-2V3l31ol] is orthogonal to |'/4 |> and

Li is also a quartet with M£=4 and hence is the

state 1*6^4 Repeated application of

wUl yield the statesl^T'' 3 | >,|*Z> 2 ^ >, and|^^0 1>,

all of which arise from ^f' (Condon and Shortley

ableF). Similarly, beginning with |332| = |2X8->

Li we can obtain the state 7

Li There are

a total of four states with Mi,=6, Ms=^ namely

\^L^\y\m^\y,\n<cXy and|*/6^>. Thefirst

Li Li Li Li two are obtained by a further application of

^ 3 while the latter is obtained from |*/ 6 by the

Li operator ^^^S^—iS^. The fourth state is then

In endeavoring to continue this process to states where 71/^ =5, we find that in addition to those

arising from higher values of L or S, that there are two additional states with Mi=5, Ms=\. They

must be therefore both described as \^'H 5|>,and

additional labels are necessary to distinguish them.

The usual procedure is to explicitly diagonalize the

submatrix of electrostatic interaction between the

two states (Condon and Shortley 7^), but further

work of Racah (1949) has shown that a more

abstract classification of states based on group

theory is possible. The former technique has

been applied by Judd (1955) to the specific case

of Eu.

In any perturbation calculation, the matrix

elements first to be considered are those between

different states of a degenerate level. In the

rare earth ions, and in crystal spectra generally,

the one most important level is the ground level. While the great significance of optical spectroscopy, in contrast to magnetic resonance measurements, lies in data provided concerning excited

levels, a general ion-by-ion analysis of the excited

levels is a major undertaking, and only the

explicit details for the ground multiplet will be

developed here. As a specific example, we shall

f\u-ther consider the case of Nd, where (5.15)

becomes

<4f */6|Y°|4f */6|>

321

I

Y° |32 1 |> (5.16)

We shall here insert again the factor r" in the

potential, omitted since (5.1) since it is invariant

under the transformations we have been considering up to this point. Again recalling that r^YS is an electrostatic perturbation upon the individual

electrons, we see that it is a spin-free quantity of

Condon and Shortley type F (6'^9), and (5.16)

3 +

becomes 2!3 <C4/ m

TO=1

+

where <E>°= 1_

27r and 6° is given in table 1. The spin terms of the

matrix element give +1, the (j> terms give merely

and the 6 integral is

j:

07626? sin e de

+1 C (3 m, 3 m)

where c" is given in table 1* of Condon and Shortley. For the time being, we shall write the r" integral as <C?'''>. By evaluating (5.14) and (5.15) for all the rare

earths, the reduced matrix elements for the

ground levels apart from the term -^^r'^ have

been computed and presented in table 16. The

reduced L-S matrix elements are included principally for information, though they are directly

useful in cases where the electrostatic perturbation

is greater than the spin-orbit interaction, or where

the electrostatic and spin-orbit perturbations must

be considered simultaneously.

6. Applications

6.1. General Considerations

In this concluding section, we shall discuss

further the general significance of the preceding

sections in the interpretation of crystal spectra,

the connection mentioned at the beginning of sec- tion 3.4 between the group-theory technique of sections 3 and 4, and the matrix element calculations of section 5, and shall illustrate the application of these general techniques to specific

problems.

Generally, a spectrum may be considered as understood if the spacing of the energy levels is known, the effect upon them of a magnetic field,

selection and polarization rules and relative intensities of lines, and all these are correlated with the

structure of the atom responsible for the spectrum.

These may all in principle be obtained from a knowledge of the initial and final states of the

transition involved in the spectral line. Unfortunately, our knowledge of these initial and final states is largely a matter of approximation which

is not always entirely satisfactory. This is true both for the free ions and for the perturbed ions

in the crystal. Let us review briefly the situation

for the free ion. The good quantum numbers for the wave functions of the free ion (section 3.3) are the parity

74
Table 16. Reduced matrix elements

Element L S v = 2 f = 4 ;'=6

Ce+++

Yb+++

3 1/2

5/2 7/2

— 0. 325732

301569

-. 307788

. 261168

. 289329

u. ^yiuDo

0-. 245532 Pr+++ Tm+++

5 1

6

54

-. 276822

265205 — 249840 -. 260040

-. 261529 -. 224931 — . 174353

~'. 208628

. 455018

. 321183

. 261268

Nd+++

Er+++

6 3/2 -. 106082 -. 168698 642369

9/2 11/2

13/2

-. 098820 — 091864 -. 093887

-. 131239 -. 106106

-. 356506 -. 169019

15/2 101524 -. 144946 -. 457704

Pm+++

Ho+++

6 2

4

5

6

. 106082

. 095469

. 084208

083901

. 168698

. 115742

. 054866

. 642369

. 260356 -. 090686

. \jo ( Otjyf

78

. 089965

. 100398

. 088311

. 139515

. 062990

. 420886

Sin+++

Dv+++

5 5/2

5/2 7/2 9/2

. 276822

. 217802

. 160051

. 155288

. 261529

. 102885 -. 033771 -. 038079

455018

0

. 252972

. 220158

11/2

13/2

15/2

. 174559

. 208638

. 253811

-. 007942

. 070737

. 193261

. 210017

. 112678 -. 228851

Eu+++

Tb+++

3 3

0

12

. 325732

0-. 199469 -. 012363

360856

0

0 150786

. 491065

00

0

34

5

6

— 054289

. 032147

. 138409

. zboiOi

• ofim-1 4.^O

. 171347

. 130764 —. 168697

081844

'. 259348 -. 227510

. 064237

Gd+++ 0 7/2 0 0 0

Note 1: Where no / value is given, the tabulated quantity is (/3L*S||Y,||/3L<S), otherwise (^LSJ\\Yy\\0LSJ). Note 2: For the second ion of the above pairs, all signs change.

transitions. Thus, consider a transition probability for electric dipole radiation involving the

matrix element L'SJ' M' \eY\^LSJM). If J'=J-\-\ so the transition is allowed in J, but

L'=L\-2, it will be forbidden in L, and will be

nonzero only if the spin-orbit interaction has mixed

terms with spatial angular momentum Z+1 in

either L or U or both. The application of a magnetic field splits a state into its 2J-\-l components with a splitting determined to a first approximation by the matrix element of (L+2S) • B,

yielding the usual expression for the Lande

gf-factor. Selection rules for M now become sig-

"^f" or "u", the total angular momentum J and

its component M. The strict selection rules AJ=0, ±1 (0+^0) for electric and magnetic dipole

radiation, and AJ=0, ±1, ±2 (O-f^O, 0+^1) for

electric quadrupole radiation, with change in

parity in the first case, no change in the last two, were calculated in section 4.4. There is no physical significance to a selection rule on M in the

absence of a nonspherical perturbation. If the

spin-orbit interaction is small, L and S approximately describe a state and we obtain selection

rules in L similar to those in J, while in S we obtain A(S'=0 for electric dipole or quadrupole transitions, AiS'=0, ±1 (0-h>0) for magnetic dipole - 75
nificant, and we have AM=0, ±1 for either type

of dipole radiation, AM=0, ±1, ±2 for quadrupole transitions, and their associated polarization

effects. The perturbing potential F (2.6) will influence a

state \aJM^ in general by four types of matrix

elements, the effects of which we shall examine in

detail. These are:

1. Matrix elements diagonal in a, J, M.

2. Matrix elements diagonal in a, J.

3. Matrix elements diagonal in a.

4. Matrix elements nondiagonal in a, though

perhaps diagonal in J and M. These matrices

(or rather, approximations to them) may be

calculated by the methods of section 5, though

the scope of table 15 itself will suffice only for elements of types (1), (2), and those of (4) diagonal

in parity and J. We will now examine the effect on the original level of these several matrix

elements.

Matrix elements of type (1) arise from potential

terms with v even, ^=0. Their effect is to split

the level into pairs of states ±M (and the single

level M=0 for integral J) in the ratios indicated

by table 15. The inclusion of these matrix

elements only results in an initial axial field approximation which is sometimes useful. The

selection rules on J and parity, as well as any

approximate rules on L and S, remain valid, as do

those on M corresponding to the free ion in a magnetic field. This case corresponds closely to a reduction in symmetry from Dj to D^h, which

differs thi'ough the inclusion of matrix elements of

types (3) and (4) which are diagonal in M.

Elements of type (2) arise from potential

terms with v even but mt^O, still with the ratios

determined by table 15 and of the same order of

magnitude as those of type (1). The solution of

the secular determinant for elements of (1) and

(2) together will yield the first approximation to the energies of the perturbed states as well as the

zero-order linear combination of states with

various M required for higher order perturbation

calculations. States with AI values differing by

ju will interact and M will no longer be a good

quantum number for these states. However, if 2J is less than the lowest n involved, there will be no matrix elements of this type and the rules for

(1) only will stiU be valid. This situation will occur for levels of low J in fields of high symmetry. Generally, the selection rules for J and a

will still hold. Selection rules on AI will apply to those states which do not involve a combination

of Mvalues, and can be applied to the components

of those states which do involve a mixing of two

or more M states. Type (3) matrix elements stUl involve only

potential terms with v even, since a includes the

parity quantum number. The addition of these

matrix elements to our scheme destroys the

validity of t/ as a quantum number which has

held for (1) and (2). These elements are those

between various J values of a given L-S term,

since they are still diagonal in a. In order of

magnitude, these elements are as large as those of

types (1) and (2), since all three come from the same reduced L-S matrix element (table 16) and

differ only in the associated Racah and Wigner

coefficients (5.14, 5.4). Their effect is numerically

somewhat smaller, since terms they contribute in

perturbation theory are divided by the energy ii differences between the unperturbed levels. A

state of given J=Jo will be mixed directly with

other J values such that \ J—Jo\< v. The selec- j tion rules for the free ion J are now either poor \ or completely worthless, depending on the numerical parameters. Since both the upper and lower ]_ states of a transition can involve a range of J,

the effective selection rule is merely AJ<2j' + 1. Matrix elements of type (4) will generally be

numerically the smallest of the four types. They

arise from all terms in the potential (2.6), this

being the only one of the four where the terms

with odd V are included. Such elements, at least as far as the ground state is concerned, involve

relatively distant levels, so the influence on the

splitting of the ground state will be small and lie perhaps more in the overall position of the level with respect to the unperturbed level. Perhaps

the most important effect of such elements is on

the selection rules and the associated polarization

effects, and in the intensity calculations. Just as elements of type (3) destroyed «/ as a quantum

number, so will these destroy a as a quantum

number, a includes the specification of the configuration and consequently the parity of a state. In the most general case, there will be matrix

elements for even v between 4/" and higher even

configurations, such as 4/"~^5(i^, and for odd v between 4/" and 4f"-~^5d. These latter elements

destroy the parity of the state and lift the restric- tions of Laporte's rule for electric dipole transitions. Intensities corresponding to electric dipole

transitions between states principally 4/" in character must, therefore, be computed from these ex- tremely small inter-configurational matrix elements

and are therefore extremely weak; nevertheless, all crystal spectra are in any absolute sense quite

weak and such transitions must be considered in the case of any nonholohedric perturbing potential.

The further effect of a uniform magnetic field will depend to a large extent on the degeneracy

allowed by the electrostatic perturbation. The

magnetic field does not allow any remaining degeneracy; hence, levels degenerate in the electrostatic field, including the Kramers conjugate

states, will be split by a term linear in the magnitude of the field B which can be computed from

a solution of the associated secular determinant.

The average magnetic moment of any nondegenerate state is zero (Van Vleck, 1932; Klein, 1952)

so these states will show no first order effect for weak magnetic fields. If the crystal field leaves

two or more levels relatively close together, a case

of intermediate or strong magnetic field approximation can be observed and the level will show a

76
finite displacement from its B=0 position. Unless the magnetic field and crystal field symmetry

axes are parallel, there will generally be no symmetry and no selection rules. Marked variation

of intensities of lines has been observed in the

crystal Zeeman effect. A more specific example

of some of these considerations Tvill be given in

section 6.2.

6.2. A Crystal Field

The direct approach to a crj^stal problem is to obtain the crystal structm^e by diffraction measurements, geometric considerations, or other crystallographic techniques. These techniques do not

locate the lighter ions with sufficient accuracy,

particularly water of crystallization, which contribute greatly to the crystal field. As an illustra- tion of the procedme, let us consider a possible

short range structure for rare-earth chlorides of

the form A'Cls-GHjO (H. S. Robertson, informal

communication) where 6 Cl~ (three from adjoining

X) are in two horizontal planes equidistant from

X, forming an equilateral triangle in each plane,

the two triangles being rotated 60° to each other

about a vertical axis. The water molecules [considered as dipoles] are then located along the

X-C\ lines. While detailed calculations were not carried out

due to uncertainties in the various distance parameters, certain features of considerable interest

may be obtained through partial application of

(2.7). The above charge arra}^ ma}^ be described

by

0i-6=-^ /^T-g — 010-12 =2

013-15 =71"

P16-I8

~Y

23/!— 23n+l ——^ 23n+2=6

and we shall take 0 as a variable parameter. There

are a total of 18 charges in this array. Those

above the x—y plane have the indices 1-9, while

those below have 10-18. Indices of the form 3n

refer to Cl~, S/i-fl refers to 0H~ and 3n-\-2 refers

to H+. The angular portions of the potential were calculated over a range from 6=0° to 90°. The angular factors (2.4) were obtained from the

tables of Tallqvist (1908). This table is not

widely available, but no other table covering its range has been prepared to the knowledge of the

writer. The National Bureau of Standards table

(1945) covers values of M only up to and including 4. In performing the summation (2.7), the only

generally nonvanishing coefficients are {N<Q)

CI CI CI, CI, CI and Q, though this may be deduced directly from table 5 without explicitly per- forming the smnmation. The values of these,

apart from a radial factor of the form

are given in table 17. Certain general qualitative

features may be noted by inspection. For ^=0

(corresponding to aU charges being located along

the vertical axis), the symmetry is D„h (table 5). For other angles, the symmetry is generally Dsa, while at 90° (a planar configuration) it is Deft. A

particular exception occurs for 0=54°44' where

the symmetry is "accidentally" O^, belonging to the cubic point groups (section 2.4). For this

value of 6, the three pairs of charge axes are mutually perpendicular and the C2 term vanishes.

For other values of 6 (e.g., 31°), other coefficients

vanish, but there is no general theoretical signifi- cance to this. It is to be noted that for ^<10°,

the terms responsible for departure from D^^

sjonmetry are small, while in the vicinity of 0=90°

the terms causing departure from Dqh symmetry

are large and quite sensitive to small angular

variations.

Most features of even this simple problem re- quire the specification of the radial distances of the ions, as well as the average radial integrals

{r"") of the wave fimctions. On the other hand,

aspects of the Zeeman effect can be understood

without further knowledge of these parameters,

and these aspects are most important for understanding more complicated cases. Let us assume

Table 17. A representative crystal field

e CI CJ CI Ct

0° 9. 5120 7. 0898 5. 8991 0 0 0

10° 9. 0818 6. 0491 4. 1557 0. 0765 0. 2113 0. 0001

20° 7. 8430 3. 3675 0. 4242 . 5575 1. 3485 . 0063

30° 5. 9450 0. 1662 -2. 2064 1. 6053 3. 0365 . 0619

40° 3. 6168 -2. 2617 -1. 9088 3. 0170 3. 7557 . 2795

45° 2. 3780 -2. 8802 -0. 8756 3. 7074 3. 3393 . 4953

50° 1. 1392 -3. 0311 . 3326 4. 2850 2. 3851 . 8007

'Oh 0 -2. 7571 1. 3109 4. 6604 0. 7915 1. 1740

60° -1. 1890 -2. 0494 1. 9068 4. 8160 -. 4338 1. 6716

70° -3. 0870 -0. 0269 1. 2322 4. 2086 -2. 5978 2. 7282

80° -4. 3258 1. 8852 -0. 7794 2. 4595 -2. 3645 3. 6146

90° -4 7560 2. 6587 -1. 8435 0 0 3. 9624

" See text. 77
that we have the case ^=0° of the preceding table

17, i.e., D„A symmetry. We see from table 9

that the 2J+ 1 states are separated into doubly

degenerate pairs corresponding to ±M, plus (for

integral J only) a nondegenerate state M=0.

The application of a uniform magnetic field parallel to the e-axis will reduce the symmetry to C^^

and all levels are now nondegenerate. The splitting

is given by twice the diagonal matrix element

{M\L,+2S^\M}B, since the -M level is depressed by this same amount and there are no

nondiagonal elements, Lz-\-2Sz having no dependence on 4>. If the field is applied parallel to the y axis, the only symmetry remaining is C27,, but

this is referred to a horizontal axis, rather than a

vertical one. The secular determinant will now

involve matrix elements of L^-{-2Sy. This has

a 4> dependence of the form e=^''*, and will have

nonvanishing matrix elements between states dif- fering in M by 1. Thus, of all the ±M pairs, a transverse magnetic field will have no diagonal

elements and will have nondiagonal elements only

in the case of ±1- The magnitude of this ele- ment _ will be W(^+l) (J— 5 + 1) g^B and the

splitting will be twice that or (e/+|) g^B.

If we now proceed to the planar case Dg^, we

5 see from table 10 that for states remain

as in the D^;, case and the Zeeman effect for these lower J values is unchanged. For J=3, the

original pair M= ±3(£'3g,3a) now splits into two

nondegenerate states Big_iu and B2g,2u, which are the states

1^ V2

vi

+3> +

V2

-3> and ^

V2

+ 3> —3>, respectively. This arises from the e^^^*

in the potential with nonzero matrix elements

between terms where AM=±6. These states

will show no first-order Zeeman effect, either

longitudinal or transverse. For t/=4, we see that

the states coming from ± 2 and ± 4 have the same symmetry {E2g,2u), and while we still have degenerate pairs, their properties will depend upon the

solution of the associated secular determinant.

One pair will be given by an expression of the

form a|±4> + (l —a^)^| =F2>- while the other will be of the form (l-a^)^ | ±4>-a| T 2>. The

coefficient a will be fixed by the magnitude of CI. For these terms, the longitudinal Zeeman splitting

win be 2[4a2-2(l-a2)] = 12a2-4 and 2[4.{l-a') —2a2] = 12a^— 8, respectively, while the trans- verse effect remains zero. This longitudinal

effect thus depends explicitly on the crystal field. For J—^i a similar phenomenon takes place

7 through a combination of ± ^ > with =f|>,

and the longitudinal effect will again depend

explicitly on the crystal field. A major difference

will occur in the transverse effect—here there are nonvanishing off-diagonal matrix elements of the

form < 2-^!/" -2Sy in the secular determinant

and there will be a transverse Zeeman splitting, also depending explicitly on the crystal field. Under, still lower symmetries than De,,, the

Zeeman efi^ect in odd-electron systems becomes

increasingly dependent on the crystal field, since

Kramers degeneracy remains in all cases, or vanishes in first order in most even-electron

systems where the crystal field states generaUy

become nondegenerate.

6.3. The Fluorescence Spectrum of the

Terbium Ion

Because of the inexact knowledge of the radial

parameters necessary for fiu-ther work on the

direct approach to a solution of a given problem,

the more common procedure is to utilize the quantities <j'^'y-C^N and Kj'^^Su as adjustable parameters to fit as best as possible the observed data.

This leaves the radial wave function integrals

combined with the crystal field parameters—they

are frequently left in this form since further information about either one of them separately is not

readily available. We shall illustrate this procedure by some considerations of the fluorescence

spectrum of TbCl3-6H20.

The experimentally observed fluorescence lines

(Singh, J. H. U., Dissertation, 1957), observed both

at 4.2°K and 2.2°K, are given in table 18. The

exciting light, unpolarized, is incident along the

crystal axis (the z axis). For both temperatures,

the spacing of each level from the lowest of its group is given, the average of the two temperature

values, and the spacing of this (to the nearest cm~^) from the mean of the group. The overall

mean for each group is also given. The fluorescent

light is observed on the side of the crystal opposite

the incident radiation, also along the 2-axis. The fluorescent lines are relatively weak and are generally observed without regard to polarization;

however, a few plates were taken at 4.2°K showing polarization (electric vector along x or y).

These data were made available by private communication from Dr. Singh and are also included

in table 18. In the interpretation of fluorescence

spectra it should be remembered that the spectra

may originate near the surface of the crystal, and

that local fields, different from those in the main

body of the crystal, may significantly affect the

regularities expected.

The Tb ion has the configuration 4/^. According to Hund's rule, the ground state is a "^F multiplet with t/=6 lowest, with J from 5 to 0 above

the lowest. The fluorescence spectrum is caused

by transitions from a higher level (probably

but this is not important for the present considerations) to various levels of the ground multiplet.

Qualitatively, the agreement with experiment is excellent. The groups of lines Z, F, X, W, V, U,

T, correspond to J=6, 5, 4, 3,2, 1, and 0, in that

order. Within each group, there are 2J'+1 lines,

indicating that the crystal field has removed all degeneracy. This is confirmed by the absence of

any first order Zeeman effect.

78
Table 18. Fluorescence spectrum of TbCly6HiO

Line

(4.2°K) (2.2°K)

V V A;. X y

z, 20538. 81 20503. 66

0. 00

35. 15 20538. 77

503. 72

0. 00

35. 05

0. 00. 00 -167

-132

z,

z.

498. 20

461. 26

40. 61

77. 55

498. 33

461. 59

40. 44

77. 18

40. 77 52oO -127

-90

No

z,z. 380. 70 158. 11 454 97

379. 70

83. 80

159. 07

00. 158. 59 -83

-8 Z = 20372

z-Zi369. 16

359. 49

169. 60

179. 32

370. 19

359. 27

168. 58

179. 50

ioy.

179.

1 9

41 212

z,

^10 306. 14 232. 67

347. 85

305. 73

190. 92

233. 04

1 on

232. 86 24

66 Zn

7

239. 08

193. 84

299. 73 344 95

238. 33

192. 86

300. 44

345. 91

ouu. 345. Uo

43

133

178

Z\i 179. 61 359. 20 179. 33 359. 44 00 V. 0^^9 192

18433. 00

413. 02

0. 00

19. 98

18433. 04

413. 02

0. 00

20. 00

0.

11 Qy. 00QQyy -127

-107

s w

s w

F3

Y,

400. 92

359. 26

32. 08

73. 74

400. 99

359. 41

32. 05

73. 63

^9

73. 68 -95

-53 w s

s w

Y,

Y,

341. 00 92. 00 341. 09

333. 69

91. 95

99. 35

y i. 99. yo

35 -35

-27

s w

s w

X XOtJUKJ

F7

•J 8

308. 29 294 96 124 71

138. 04

308. 57

295. 90 124 47137. 14

1 9d

137. oy59 -2

11

s w

s w

-f 10

250. 87

124. 83

182. 13

308. 17 124 82 308. 22

1 ft9 308.

i 02055

181 ? T

Yn 107. 90 325. 10 107. 81 325. 23 i 0 198 ? ? X,

X2

17193. 56 0. 00 17193. 49

188. 20

0. 00

5. 29

u.

5. no29 -74

-69 w w

X,

Xi

184. 65 8. 91 184 76

135. 74

8. 73

57. 75

00. 57. R9oZ

75 -65

-16

ss w X=17120

Xi

^6

131. 91 61. 65 131. 98 104 88

61. 51

88. 61

Di.

88. Oo61 -12

15 vw

X7

Y

067. 31 126. 25 067. 52

056. 26

125. 97

137. 23

1 9fi 137.

1 1 1 1 23 5263 X, 013. 94 179. 62 014 48 178. 81 179. 22 105 u

16162. 72 0. 00 16162. 85

080. 68

0. 00

82. 17

0. 82. 00

17 -98 — 15 w w

s

Wi

070. 26

043. 44

92. 46

119. 28

070. 33

043. 44

92. 52

119. 41

92. 119. 4934 -5

22 w w w s W=16065

w,

037. 91 n^9 QQ

124 81 037. 95 n^9 87 124 90 124

129. 8686 27

32 w ws w

UZo. U/ io'i. DO 104:. \JV 134. 69 37 s w

V,

^ 2 15499. 89

430. 03

297. 53

292. 94

0. 00

69. 86

202. 36

206. 95

-145 — 7557

62 "=15355

zoo, ly 1 P^9^^ 1 7 101 vw s u,U2

15001. 25 14994 680. 00

6. 57 -23

-17

s vw vw s 17=14978

u,T

938. 41

14691. 73

62. 84 40 r= 14692

79
Let us consider the quantitative details. For

best results, our theory requires L-S coupling

and a crystal field splitting small compared to the

multiplet separation. We may check the validity

of L-S coupling by the interval rule, taking the

separation of the mean of adjacent groups and

dividing by the higher J value. We obtain, in cm"\ the following:

J=6 344 J=4 264 J=2 158. 5 ,7=5 237 J=3 237 J=l 286

This is not especially impressive for very good

obedience to L-S coupling requirements. Let us

also calculate the total spread of a group, divided

by the separation of the group mean from that

for the next lower J. We obtain:

J=6 0. 174 J=4 0. 170 J=2 0. 653

J=5 .274 J=3 .190 J=l .220

This indicates that the influence of adjacent levels may be rather significant. The irregularity here

for f7=2 is quite interesting; it is only partially

due to the irregularity shown in the interval rule

check, since the total spread of J=2 is 247 cm"^

whUe that of ^=3 is only 179 cm^^

In beginning an analysis, we must first obtain some idea of the symmetry. From table 9, we

see that the only symmetries giving no degeneracy

for an even-electron system are D2n, C2e, C2h, and Cj. The overall crystal symmetry suggests

either C2,, or Co^. From table 5, we see that

D2ft requires the following potential coefficients: CI CI Ct CI CI CI C\ Ct CI The lower symmetry

Civ is obtained by the inclusion of odd potential

terms, which will influence the energy splittings

by a very small amount (section 6.1), while

will include even S terms. These latter will strongly iafluence the energy splittings. In order

to reduce the number of independent parameters, we shaU assume 02^, which will be energetically

equivalent to C2,,. In deducing the field parameters, we need to know not only the energies of the levels but also the free ion M values from which they come. Where the crystal field does not remove all degeneracy, this may be partially obtained from

the Zeeman splittings. For the present case, the

only source of such information is the polarization data.

We begin our detailed analysis with the simplest

nontrivial case ^7=1. We have three levels at —23, —17, and +40. The only potential terms

of Daft contributing to matrix elements within

e7=l are CI and C|. It is a natural first guess to consider that the CI term causes a splitting of

the state |0> to +40, and | + 1> and | — 1> to —20, with a small splitting of the latter two to — 17 and —23 by C|. This is not necessarily the

case, and must be in agreement with the polarization data.

From table 14, where radiation in the z direction corresponds to 0=0=0°, ^=0° for x polarization and 1^=90° for y polarization, we see that

a;-polarized radiation may be produced by either

Al, By, or Ax2, while ^/-polarized radiation may be

produced by Al, Bj, or Ay^. These are the electric dipole, magnetic dipole, or electric quadrupole

terms, respectively. We shall assmne that the

transitions are magnetic dipole in character. We

do not know the symmetry of the upper state in the fluorescent spectrum, but there are only 4 possibilities in a Daft symmetry. From table 13 we

deduce the allowed lower state for each possible

upper state as follows:

Upper

state

B.{U.2)

B-ig Big

B\g B2g

Big A.g Aig

B2g Au Aig

The identification of Z7i and XJi of table 18

ideally requires that they be present in only one

of the two polarizations, rather than "very weak,"

but the actual behavior is quite close to this ideal. We now turn to table 9, and find that the lower

state Ig yields in Das symmetry the states A^g,

Big, and Big. Since Aig is not present in the

lower group, we conclude that the upper state is either Aig or Aog, that the lower states at —23 I and —17 are Big and Big, while the other state from J=l is Aig. We cannot uniquely assign

Big and Big to a particular one of the —23, —17

levels. The upper level is presiunably ^Di and a

4g level under Dj,, gives both Aig and Aig levels. Interestingly enough, the same resiilts, Avithout

"g" and "u" subscripts, hold for Ci^ symmetry.

We now see from table 10 that Big and Big arise

from the state Eig of D^^, i.e., they come from

M= ± 1 and our earlier guess is confirmed.

In order to achieve maximum simplicity in the

calculations, it is in general desirable to use zero- order wave functions which belong to reps of the

symmetry group. In the present case, the state |0> belongs to Aig, but the states | + 1> and | — 1>

are not appropriate, and we must use the (essentially real) combinations Big-. -^| — 1> +

V2

-^| + l>andi52,:-i|-l> —p| + l>- We shall

\l2 V2 y2

frequently abbreviate these as |l + > and |1 — >,

respectively. We now introduce the perturbation

V=r\C°iQ+ClQ). Since the potential V is symmetric vmder Das and our three zero order wave functions belong to difl^erent reps, all non- ; diagonal matrix elements vanish, and the energies are merely <0|F|0>, <l + |F|l + > and

<1 — |F|1 — >. It is a general characteristic of

80
crystal field problems that one must be freely able

to shift from complex to real forms of wave functions and vice versa, as the symmetry demands.

In the present case, we have from (2.9) ¥1 = 02

and Tl^Y2'=^Cl Hence <0|F|0> =

-v2 ial\\Y2\Wl)Cim<r'> =4:0 cm-i or, from tables

15 and 16, (0.199469)(-0. 632456) Yl<r^>=4:0

cm-\ so that FK?'^> = -317.1 cm-^. Similarly,

<l-|-|F|l + >=(al||Y2||«l)[|(C?§^ + C_Po^)r°

+liCAf + Ci_r) ri]<r-2> = - 17 cm-^

RecaUing (5.6) and using the value of just computed above, we obtain (0.199469) (0.774597)

3^<r2> = +3, or 71=19.4 cm-^. The sign of F| is ambiguous, since the state |l + > could

equally well have been assigned to —23 cm~^

These results, while interesting, are really not too profound, since we have only two independent

energies relative to the mean of the group, and we had two adjustable coefficients at our disposal.

When we proceed to the J=2 case, we see that

the allowed levels (table 9) are (2) Aig, A2g, Big,

and B2g. In contrast to the J—l case, the Big

and B2g are |1— > and ll +>. By using table 10, we see that Aig and A2g arise from 1±2>. In

particular is |2-ty > while A2g is |2— >. These

may most conveniently be deduced from the

transformation rules of table 8, in comparison

with the tesseral harmonics of table 2, keeping in mind the phase convention for odd positive M of

(2.8). The only nondiagonal matrix elements

in the resulting secular determinant will be

<0lF|2 +>, so that we have only one second

degree equation to solve for the energies.

The polarization data are available only for the

state at +101 cm"-'. This has polarization hke

the state Big or Bog of J=l, and hence is presumably also |l + > or |1 — We do not have any

polarization data on the other levels, so we do

not know which states are which. We are therefore forced to a trial-and-error process. The

additional potential parameters significant for J=2 are CI CI CI If we select |l + > as 101

cm"' and anj^ one other as |1 — >, this Avill fix CI and CI, so that only C| is left to fit the other

three levels. In attempting this process, the work is some- what simplified by observing that (a2||Y2||Q!2) is very small, and that second degree terms will have only a small contribution to the matrix

elements. Consequently, they were disregarded

for the trial and error work, and were included

later for a more exact calculation. The required

matrix elements are, omitting the factor {a2 ] | Y4I \oQ)

common to aU,

<o!F|o>=cro^rKr*>

<0|F|2 + >=^;^ {ClJi+CJ2f)Yl<r'> =~ C4m<r'>

<2+|F|2+>=| (C?ir+CJ«) YKr^y

+1 (C_if+Ci_f) Yt<r'>

^Cim<r'>+ CJrYt<r'>

<1+ 1 v\i+>=cim<r'>+ cJifY\<T'y

<\-\V\\-y=Cf^Y\<^r'y-CJ{fY\<r'~>

<2- 1 F|2->= Ci^2yo^r*>- t7_2 42F|<r*>

These may be simplified by observing that C?o^: Gfi : Cro'=6:-4:l and that (7ir=-Vp GJif- If we introduce the following abbreviations,

<0|F|0>=Fo <-2|F|+2>=^

<-l|F|-l>=5

the matrix elements may be simplified to

<0|F|0>=Fo

<2+|y|2+>=iyo+^

<2-|F|2->=|Fo-/3

<1 + |V!1 + >=-|Fo+5

<0|F|2+> =-^|(V2)5=-iV35

<1-|F|1->=-|Fo-5

The form of these matrix elements suggests the

physical situation. The C% term splits the 2«7+l

2

levels into one at Fo, a pair at —^Fo and a pair

at -^Fq. The first pair is then further split by

C\ into states separated by 25. The second pair

is also split into two separated by 2/3 through G\,

but one of these interacts further (via Cf) with

|0> through the nondiagonal matrix element

The trial and error fit was actually tried for aU

possible pairs of states as 1 1 + > and 1 1 — The

best fit was obtained for 101 (in agreement with

the polarization data) and —75. With this pre- liminary information, the second degree terms were included in the matrix elements, the fourth

81
degree coefficients were ctiosen for a least squares

fit in the energy, and ttie following were obtained:

JJJ \KJkJiSJ

101 108. 0 |l + >

62 65. 9 |2->

57 45. 4 0. 820 |0> + 0. 572 |2+>

-75 -85. 8 |l->

-145 -133. 5 0. 820 |2+>-0. 572 |0>

<r4>ro=- 187.8 cm-i <r*>r| = 1,141 cm-i

<r*>Ft= -626.2 cm-\

The fit is only fair. There are three possible sources of inaccuracy in the treatment. The first of these is the possibility of deviation from the

L-S coupling approximation. This would appear

essentially in the values of the reduced matrix

elements. Purely from the empirical standpoint,

they coiild be treated as additional parameters to provide a better fit to the data. A second appreciable source of discrepancy is the interactions

with other states, principally those within the

ground multiplet. The nondiagonal Wigner coefficients and Racah coefficients are both generally

of the same order of magnitude as the diagonal

ones, and the reduced L-S matrix elements of

table 15 are appropriate for all combinations within the ground multiplet. Nondiagonal matrix ele- ments can therefore be of appreciable magnitude

and, in view of the relatively small multiplet

splitting, could become quite important in a more

refined theory. Finally, there is also the possibility that the symmetry is such as to require the

inclusion of terms. This almost doubles the

total nvmaber of parameters. This will, of course, permit an adjustment to the data, but there is no

clear requirement for the inclusion of these terms.

It may be noted here again that the earlier check on miiltiplet separation showed a distinct anomaly

for the J=2 group.

Let us now turn our attention to J=3. Our

allowed states are as follows:

^2g: |0>andl2 + > A,: \2->

Big-. |3 + >and|l + > B2,:\l-> and |3->.

If we compute the levels from the second and

fourth degree terms only, we obtain a set of levels which show a rather sm-prisingly similar trend to the actually observed levels, which include the

effect of sixth degree terms. Unfortunately, the

polarization data both here and for still higher J

does not appear to follow either the rules appropriate for D2/, symmetry found adequate for lower

J, or any other pattern easily amenable to analysis.

In particular, the states given do not especially even suggest the reported polarization data.

E (obs) E (calc) State

37 45. 1 0. 948 l + > -0. 319 3+>

32 36. 9 . 756 3-> -.654 l->

27 21. 8 . 784 0> -. 620 2+>

22 8. 6 . 620 0> +. 784 2+>

-5 -14. 8 . 319 l + > +.948 3+>

-15 -28. 0 |2- >

-98 -69. 6 0. 756 l-> +.654 3->

There is obviously a considerable amount of work

yet to be done before the Tb spectrum is understood. Experimentally, there is need for more

attention to polarization data for fixing the character of the states involved. Theoretically, there are

several lines of endeavor to be pursued. The

problems become numerically so tedious that

machine calculation is probably required for fur- ther success. The first of these is a systematic

machine calculation of nondiagonal Wigner coefficients. The published table of Simon (1954)

is inadequate in range, and the extension of table

15 by desk calculations is almost out of the question. For the most part, the tables of the Racah

coefficients (Simon et al., 1954) are adequate, so that the availability of Wigner coefficients would

permit a study of nondiagonal contributions to the

energies. Furthermore, it appears that in problems of low symmetry, such as the present one, there will be a requirement for trial-and-error

calculations. These are well adapted to machine

calculations, but are again, except in very simple cases, almost out of the question for desk

calculators.

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84 U.S. GOVERNMENT PRINTING OFFICEl 1961
THE NATIONAL BUREAU OF STANDARDS

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Radiation Physics. Neutron Physics. Radiation Theory. Radioactivity. X-ray. High Energy Radiation. Nucleonic Instrumentation. Radiological Equipment.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry. Inorganic

Chemistry. Electrodeposition. Molecular Structure and Properties of Gases. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances. Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics. Mass and Scale. Capacity, Density, and Fluid Meters. Combustion Controls. Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer

Structure. Plastics. Dental Research. Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion. Metal Physics.

Mineral Products. Engineering Ceramics. Glass. Refractories. Enameled Metals. Constitution and Microstructure. Building Technology. Structural Engineering. Fire Protection. Air Conditioning, Heating, and Refrigeration.

Floor, Roof, and Wall Coverings. Codes and Safety Standards. Heat Transfer. Concreting Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics.

Data Processing Systems. SEAC Engineering Group. Components and Techniques. Digital Circuitry. Digital

Systems. Analog Systems. Applications Engineering.

• Office of Basic Instrumentation. • Office of Weights and Measures. BOULDER, COLORADO

Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Gas Liquefaction.

Radio Propagation Physics. Upper Atmosphere Research. Ionosphere Research. Regular Prediction Services. Sun-Earth Relationships. VHF Research. Radio Warning Services. Airglow and Aurora. Radio Astronomy and

Arctic Propagation.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Standards. High-Frequency Electrical Standards. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time Standards. Electronic Calibration Center. Microwave Circuit Standards. Radio Communication and Systems. Low Frequency and Very Low Frequency Research. High Frequency and

Very High Frequency Research. Modulation Systems. Antenna Research. Navigation Systems. Systems Analysis. Field Operations.