Selection rules for the space group of cuprite, the symmetry points and lines

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Selection rules for the space group of cuprite, the symmetry points and lines

K. Olbrychski, R. Kolodziejski, M. Suffczyński, H. Kunert

To cite this version:

K. Olbrychski, R. Kolodziejski, M. Suffczyński, H. Kunert. Selection rules for the space group of cuprite, the symmetry points and lines. Journal de Physique, 1976, 37 (12), pp.1483-1491.

�10.1051/jphys:0197600370120148300�. �jpa-00208551�

SELECTION RULES FOR THE SPACE GROUP OF CUPRITE,

THE SYMMETRY POINTS AND LINES

K. OLBRYCHSKI

(*),

^R. KOLODZIEJSKI

(*),

^M.

SUFFCZY0143SKI (*)

Institute of

Physics,

^Polish

Academy

^of

Sciences, Warsaw,

^Poland and H. KUNERT

(**)

Institute of

Physics,

^Technical

University, Pozna0144,

^Poland

(Reçu

^le

2 juillet 1976, accepté

^{le 23 aout}

1976)

Résumé. ²⁰¹⁴ Les produits directs de représentations irréductibles du groupe double

O4h

^de cuprite

sont réduits. Un facteur correspond ^au point ^de symétrie, l’autre ^{à la} ligne ^de symétrie ^{dans la zone}

de Brillouin.

Abstract. ²⁰¹⁴ The paper presents selection rules for the double space group

O4h

^of cuprite, ^i.e.

decompositions ^{of direct} products of the irreducible representations ^{in which one} factor refers to a

high symmetry point and the other to a symmetry line of the Brillouin zone.

Classification Physics ^Abstracts

7.152 ^- 8.800

. 1. Introduction. ^- The

present

paper is ^a conti- nuation of the _paper

[1],

^it lists further selection rules for the space group of

cuprite.

^{Selection rules are useful}

in the

investigation

of the electron band

symmetries, optical transitions,

infrared lattice

absorption,

^elec-

tron

scattering

^and

tunnelling,

^neutron

scattering,

magnon

sidebands,

^etc.

[2]. Analysis

^of

scattering

processes

involving photons, phonons

^or magnons in

crystalline

^solids

generally requires

^the

appropriate

selection rules to be worked out.

Recently

^{much attention has} been directed to the calculation of the Clebsch-Gordan coefficients of the space group

representations [3-9].

^In

particular,

Birman and coworkers

[8, 9, 10]

have shown that the elements of the

first-order, one-excitation, scattering

tensor are

precisely

certain Clebsch-Gordan coeffi- cients or

prescribed

^linear

combinations;

the elements of the

second-order, two-excitation,

process ^{are a}

particular

^{sum of}

products

of Clebsch-Gordan coeffi- cients.

The factorization of a matrix element or a

scattering

tensor element into a Clebsch-Gordan coefficient and

a reduced matrix element

yields

^a maximum realiza- tion of the

simplifications

^{due to the} symmetry ^{of a}

problem.

(*) ^{Address :} Instytut Fizyki PAN, Al. Lotnik6w 32/46, ^02-668 Warszawa, ^Poland.

(**) ^{Address :} Instytut Fizyki Politechniki Poznanskiej, ^ul.

Piotrowo 3, ^60-965 Poznan, ^Poland.

The Clebsch-Gordan

coefficients

for the irreducible

representations

^{of the}

crystal

space groups ^or the

crystal point

groups ^are useful for an

analysis

^{of the}

Brillouin

scattering

tensor,

scattering

^{tensors for}

morphic effects, two-photon absorption

^{matrix ele-}

ments,

scattering

^{tensors for}

multipole-dipole-reso-

nance Raman

scattering, higher-order

^moment expan- sions in infrared

absorption

^and

diagonalization

^of

the

dynamical

^{matrix of}

crystal

vibrations.

For a calculation of the Clebsch-Gordan coefficients

or

scattering

^{tensors an} elaboration of the selection rules is a first _necessary

step.

^In

fact,

^{a reduction of}

products

of the irreducible

representations

^{of the}

relevant

crystal

group

gives

^the

frequency

^{of occur-}

rence of each irreducible

representation

^{in a}

product

and thus a survey of the matrix elements which vanish

by

symmetry alone and of those

remaining

^{for which}

the calculation of the Clebsch-Gordan coefficients is

required.

A

listing

of the selection rules for the irreducible

representations

^{of the} space group of

cuprite, Cu20,

seems of

particular

^{interest in} view of wide _scope of

experimental

and theoretical

investigations performed

on this

crystal [11-28].

^{Results of extensive}

optical experiments

ⁱⁿ

CU20, including

^the

two-photon absorption [19]

^{and the Raman}

scattering [20, 21, 24]

are available.

It is established that the extrema of the electronic bands in

cuprite

^{are at the center of the Brillouin zone.}

In

cuprite

^the

similarity

of the calculated

density

^of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370120148300

1484

states

[17]

^{and the}

experimental absorption

^and

reflection spectra in the ultraviolet suggests

[18]

^that

the maxima on the lower _energy side could corres-

pond

^to transitions to the symmetry

points

^{M and}

X,

and that the

peaks

ⁱⁿ the violet can

originate

^from

transitions to the conduction bands MFX. For the

phonon

spectra ⁱⁿ

cuprite

less information is avai- lable

[25-28],

^the occurrence of the extrema of the

pho-

non branches _away from the zone center cannot be excluded. Therefore a list of selection rules

including

all symmetry

points

^and

symmetry

^{lines seems}

appropriate.

2.

Decomposition

^{formula. - The} transition

ampli-

tude of an electron from the state

§§l

^{to the state}

^§f

due to an interaction described

by

^the operator

t/J

^is

proportional

^{to the}

integral

The

integral

vanishes unless the

representation Dh

^is

contained in the

product D’

^x

D’.

^{Thus the selection}

rules are obtained from

decomposition

^{of the Kro-}

necker

product

^{of two} irreducible

representations

^into

irreducible ones. The irreducible

representations

^are

labelled

by

^{the wave vectors}

k,

m, h and the indices p, q, r,

respectively. C;::;h will

be coefficients

expressing

the

frequency

^of occurrence of the

representation Dh Dk

^x

Dm

^q

For the space group G with the elements

{ a I v }

^and

with the characters

xp, Xr;, i

of the irreducible _repre- sentations

Dp, Dq , D’, respectively,

^the

frequencies

of occurrence are

given by

Here

the

^vectors

^k,

^{m, h}

^belong

^{to the}

^suitably

^cho-

sen

1/1 Ul I part

of the Brillouin _zone, the so-called

representation domain 0, containing

^{one arm from}

each star.

The ^integer I U I

is the order of the

single point

group G of the _group G. An

example

^{of the}

representation

^{domain of the} space group

Oh

^{is shown}

in

figure 1, by heavy

^lines.

Eq. (3)

^{can be}

expressed

ⁱⁿ

terms of the characters

ql’

of the small representa- tions

d’

which induce the

representations D k

^of

G

[29, 30] :

Here the sum indexed

by

^{a is taken over the rele-}

FIG. 1. ^- The first Brillouin zone and the representation ^domain for the space group 0: ^of cuprite. b1, b2, b3 ^{are the basic vectors}

of the reciprocal ^lattice.

vant

leading

^{wave vector} selection rules

(LWVSRs),

see Lewis

[30],

^{i.e. over the}

elements

^determined

by

^the

expansion

^{of the}

point

group G into double cosets

([29],

^p.

208),

where

G’

is the

point

^{group of the wave vector}

group Gh of h and

G’

is the

point

group of

Gk.

The index

P(a)

^means

that fl

^is

^dependent

^{on a ; it is an}

arbitrary

element of G

satisfying

where ⁼ means

equality

^{modulo a vector} of the reci-

procal

^{lattice of the} group G. The

symbol If

^{is to}

remind

^us

^that,

^{if for}

^{given k,}

^{m, h and a no}

^{element fl}

of G

satisfying

eq.

(5) exists,

^{then we have zero}

instead of the sum over S.

La

^{is the}

^point

^{group of}

La

⁼

^G

^A

Gb,

the intersection of the _group

Gak

of the vector ak and the _group

Gh

of h. The _{is, ta} and rp

are

fixed,

e.g. the

simplest

fractional translations associated with

S,

^{a and}

p, respectively,

^{in the}

group’G.

t/J:p

^and

t/J’q

^are

^{given by}

^the relations of the type

For the small

representation d’

^{of the unbarred}

primitive

translation

{ E t}

^{we assume the convention}

i.e. we choose the +

sign

^{in the} exponent ^{on the}

right-hand

^side.

¹

is the unit matrix of dimension of the

representation d’.

^In the above considerations G

can be a

single

^{or a double} space group. Corres-

pondingly

^{all the} groups considered are

single

groups ^or double _groups,

respectively.

^{If G is a}

double space group, ^{we use} the same

symbols

^{for the}

point operations

^{of a double} group ^as for

correspond-

ing operations

^{of the}

single

group. For

given k,

^{m from} the

representation

domain 0 all the vectors h for which the coefficient

(4)

may be different from zero can be found as follows

[1] :

^we consider the vectors

ki

from the star of k and mj ^{from the star of m. We}

construct the vectors

ki

⁺ mj ^{with one of the vectors}

ki,

mj fixed and the second varied. In this _way, on

account

of eq. (5),

^{we obtain} representants ^{h =}

ki

⁺ mj of the star of the vector h for which the coefficient

(4)

may be

nonvanishing.

, TABLE I

3.

Description

of tables. ^- Table I list coordinates of the

symmetry points,

^{lines and}

planes

^{of the} repre- sentation domain 0 for

cuprite.

^{In table II we}

give

the characters of the small

representations

for all the symmetry

planes

^{of the}

representation

domain of the group

01 :

the labels I of the

operations h,

^{are those}

of Miller and Love

(M-L) ([31],

p.

123).

Tables III-XII present ^the

decompositions

^{of the}

Kronecker

products

of the irreducible

representations

of the space group

Oh

into irreducible

representa- tions,

eq.

(2),

^{where k runs over the four}

symmetry points

^{and m-over the six} symmetry ^{axes of the} Brillouin zone. The irreducible

representations

^{of the}

group

0’

^{are labelled}

by

labels of the

corresponding

small

representations

of Miller and Love

[31].

^Powers

mean

frequency

^of occurrence c; of the

given

^irre-

ducible

representation [32].

In some cases there is a need to reduce the wave

vectors to the

representation

domain 0 : if in the

TABLE II

Symbol.d ⁱⁿ the left side margin ^refers only ^to the selection rules r x A ⁼ Zci A j.

Symbol.d ^{in the} right ^side margin ^refers only ^to the selection rules R x A ⁼ Eci Ti’.

TABLE IV

1486

TABLE V

Symbol I in the left side margin ^refers only ^to the selection rules for F x E = 2;ci fi.

Symbol 2; ^{in the} right ^side margin ^refers only ^to the selection rules for R x 2; = Eci S’*

TABLE VI

TABLE VII

TABLE VIII

TABLE IX

TABLE X

selection

rules,

eq.

(2),

^{a vector h goes out of the}

representation

domain 0 for certain values of k and _m, then a vector h’ can be found in 0 such that

where a E G and K is a vector of the

reciprocal

^lattice.

E.g.

^for

and

we have

and

The latter vector lies in the

representation

^{domain 0}

for

0 , ii -1

^but

for 2

q 5 ^{1 it} goes ^out of 0.

The vector

for ’ 2 I I

^lies

just

^{in 0.}

In

general,

the relation between

representation dh

and a

representation d"’

which induces the same full group

representation

^as

dh

^{is ,}

where

I# lvlc-G h, I LX I r..}is

^an element of G with the rotational part ^a,

qlh

^and

§ti

^are characters of the

representation dh

^and

d"’, respectively.

A closer examination of the tables reveals that a

reduction of the wave vectors h to the

representation

domain 0 is

required only

^{in tables}

VIII,

^{IX and XI}

1488

TABLE XI

TABLE XII

where the relations

(8)

^{are written out for each case.}

Note that our

representation

^{domain 0 can be defined}

by

the coordinates of its

point (kx, ky, ^{k) 7r/a}

^which

satisfy

^the

inequalities

In the tables the wave vectors h

parametrized

^{in a} way different from that of table I are

distinguished by primes

and their coordinates are

specified

^{in the}

appropriate

table. The same

symbols

label the vec-

tors h and their

representations,

e.g. in table VIII : io eis the vector

and

2°

ei (i

⁼

^{1, 2, 3,} 4)

^{are full} group

representation corresponding

^{to wave vector}

and

In table XIII we summarize the notations of the

single-valued

^{and the}

spinor, separated by

space,

representations

^for

symmetry points r, R,

^{M and}

X, according

^to Miller and Love

[31],

Zak et al.

[33],

Kovalev

[34]

^and

Bradley

and Cracknell

[29].

^Labels

of the irreducible

representations

^{for the} symmetry lines are

given

^{in table XIV.}

While

establishing

^the

correspondence

^between

representations

^one should notice different definitions of the small

representations dp

^{of the}

^primitive

TABLE XIII

TABLE XIV

TABLE II A

1490

translations

{ E I R,, I

^{where E is the rotation}

through 0° :

Kovalev and

Bradley

^have

whereas Miller and Love

and’Zak

assume

(1 is

^{the unit}

matrix).

We follow Miller and Love and Zak and we compare their

representations

^{with the}

complex conjugate

repre- sentations of Kovalev and of

Bradley

and Cracknell.

In the tables II A-VI A we

supplement

the selection rules of ref.

[1] :

^{we list the}

decompositions

^{of Kro-}

necker

products

^of

single-valued by

the double-valued

representations

^{at four} symmetry

points

^{in the}

Brillouin zone.

Numbers with the minus

sign

^above

correspond

to the odd

representations,

^numbered

by

^{M-L with}

-

sign;

those without _any

sign correspond

^{to even}

representations,

^numbered

by

^{M-L with +}

sign.

TABLES III A AND IV A

TABLE V A

TABLE VI A

4.

Examples.

^{- Numerous}

examples

^{of the}

appli-

cation of the selection rules ^at the center of the Brillouin zone of

cuprite

^{have been}

given

^{for the multi-}

pole

radiation transitions in ref.

[13]

^and

compared

with

experiment

^{in ref.}

[16],

^{and for the Raman}

scattering

^{tensor in ref.}

[9]

^and

compared

^with

experi-

ment in ref.

[24].

In

cuprite

^{the Is} exciton of the

yellow

^{series is}

electric-dipole forbidden,

^{and we have to consider}

the next term

beyond electric-dipole (ED)

^{in the}

electron-photon

interaction

[9].

^{In a cubic}

crystal

of the symmetry ^class

Oh

^{the term reduces to Electric-}

Quadrupole EQ(r 25 +)

⁺

Electric-Quadrupole EQ(Fl2+)

⁺

Magnetic-Dipole MD(Fl5+)-

^{When the}

incoming photon

^{is in resonance with an}

EQ(F251)

transition and the

outgoing photon corresponds

^to

an

ED(F,,-)

transition the

phonon symmetries

which can arise are

given by

For in

photon

^{in resonance with an}

EQ(Fl2l)

transition and out

photon

^{at ED} transition the

phonon symmetries

^{which can arise are}

given by

For in

photon

^{in resonance with a}

MD(FL 5 1)

transition and out

photon

^{at ED} transition the

phonon symmetries

^{which can arise are}

given by

All these selection rules can be read from table

II,

with

help

of the table

VII,

ⁱⁿ paper

[1].

The selection rules

T x R, T x M, T x X, r x 11, FxA,

Fx2:, ^etc.

have

analogous meaning.

In our papers ^we derive and list the selection rules

at all symmetry

points

^{and lines in} the Brillouin zone

of

cuprite.

^{While at}

present experimental

^{data have}

not been

yet

accumulated for detailed

comparison,

these

general

selection rules are

important, particularly interesting

^{is the}

isomorphism

of the selection rules

at the r and the R

point.

^The

isomorphism

^{is seen}

in table II of _paper

[1],

^and

examples

of similarities

can

^{be seen}

in

^tables

III, IV, V,

^{VI of the} present paper.

Furthermore,

ⁱⁿ

cuprite

^the

gradient

of the electron and the

phonon

^{energy versus} wave-vector

dispersion

curves may vanish at the off-center symmetry

points [17, ^{18, 26,} 28].

Ascertainment of their

possible

contribution to the interband and the exciton

phonon-

assisted

optical

transitions

requires

^{further inves-}

tigation.

Appendix.

^-

Comparison

of the irreducible _repre- sentations of Miller and Love

(M-L) [31],

^{Zak et al.}

[33], Bradley

and Cracknell

(B-C) [29]

with those of Kovalev

[34]

is facilitated

by noticing

^the

following

relations between their

spinor

^matrices

D1/2.

^We

denote the matrices of the

two-dimensional,

^double-

valued

representation

of the full

orthogonal

group

by D112

^{with the}

symbol

^{of the}

respective

^authors.

1° The relation between the matrices

D112

^{of Miller}

and Love and of Kovalev is

([31],

p.

14, 123, [34],

p.

24-26)

where the

sign -

holds for the Miller and Love’s

rotations j

⁼

3, 6, 7, 8, 17, 24, 27, 30, 31, 32, 41,

⁴⁸

(or

^Kovalev’s

operations hi

⁼

^{h3, h65 h7, hs, hl7g}

h249 h27, h3o, h31, h32, h41, h4s);

; for the

remaining

rotations the matrices

D2L(j)

^{are the same as}

K 2 (h j)

^.

where R

{

0,

n }

^is the active rotation

through

^the

angle 4?

^{about the axis with the versor n and}

6/2

is the Pauli

spin

^{vector. The}

angles

of the successive rotations are :

qf(O 4/

²

7r)

^{about the z}

axis, 8(0 K

⁰

n)

^{about the new x}

axis,

T(O

T 2

7r)

^{about the z} axis which is obtained after the two

previous

^rotations.

20 The relation between the matrices of

Bradley

and Cracknell

([29],

p.

418)

and of Kovalev is

with

where the

angles

of the active rotation R are

([29],

p.

52) :

and the matrix S is of the form

30 The relation between the matrices of

Bradley

and Cracknell and of Zak

([33],

^p.

5)

^is

with

where the

angles

of the successive rotations are :

4/(0 ik

²

7r)

^{about the z}

axis, e(o

^{e ,}

7c)

^{about the} new y

axis,

T(O , T

²

7r)

^{about the z} axis which is obtained after the two

previous rotations,

and the matrix S is of the form

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