Selection rules for second order infrared and raman processes. I. Caesium chloride structure and interpretation of the second order Raman spectra of CsBr and Csi

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Selection rules for second order infrared and raman processes. I. Caesium chloride structure and interpretation of the second order Raman spectra of

CsBr and Csi

S. Ganesan, E. Burstein, A.M. Karo, J.R. Hardy

To cite this version:

S. Ganesan, E. Burstein, A.M. Karo, J.R. Hardy. Selection rules for second order infrared and raman processes. I. Caesium chloride structure and interpretation of the second order Raman spectra of CsBr and Csi. Journal de Physique, 1965, 26 (11), pp.639-644. �10.1051/jphys:019650026011063900�.

�jpa-00206328�

SELECTION RULES FOR SECOND ORDER INFRARED AND RAMAN PROCESSES.

I. CAESIUM CHLORIDE STRUCTURE

AND INTERPRETATION OF THE SECOND ORDER RAMAN SPECTRA OF CsBr AND CsI.

By

^{S. GANESAN}

(1),

^{E. BURSTEIN}

(2),

Department ^of Physics ^and

Laboratory

for Research on the Structure of Matter,

University

^of

Pennsylvania,

^{U. S. A.}

and A. M.

KARO,

Lawrence Radiation Lab.,

Chemistry

Division, Livermore, California, ^{U. S. A.}

and

J. R. HARDY,

Atomic Energy ^Research Establishment, Harwell, Berkshire, England.

Résumé. 2014 On donne les règles ^de sélection relatives à deux phonons pour les processus Raman et infrarouges ^dans les cristaux

possédant

la structure du chlorure de césium. Les règles ^{sont tout}

à fait analogues ^à celles relatives à la structure du sel gemme, favorables ^aux processus de diffusion Raman du second ordre, ^et défavorables pour ^ceux du même ordre concernant

l’infrarouge.

^On montre, ^à l’aide des courbes de dispersion théoriques, ^{que les} spectres ^Raman du second ordre de CsBr et CsI peuvent ^être interprétés par des paires ^de phonons ^aux

points

^de symétrie ^{de la}

zone de Brillouin. On trouve que chaque ^maximum observé est dû à

plusieurs paires

^de phonons ^aux

différents points ^de symétrie. ^Les

multiples

contributions aux maximums observés, ainsi que la

grande

polarisabilité

des ions césium et halogènes,

expliquent

l’intensité relativement forte du

spectre. La limite du spectre d’absorption infrarouge ^{de CsBr}

s’explique

par l’existence de

paires

de phonons ^aux points ^de

symétrie

A et 03A3.

Abstract. 2014 Two phonon selection rules for Raman and infrared processes ^are given ^for

crystals

having ^{the caesium} chloride structure. The selection rules are quite ^{similar to} that of the rocksalt structure, favourable for second order Raman scattering processes and quite ^severe for the second order infrared processes. With the help ^{of the} theoretical dispersion ^curves it is shown that the second order Raman spectra of CsBr and CsI can be accounted for in terms of phonon pairs ^at symmetry points ^{in the s. c. Brillouin zone. Each} of the observed peaks ^{is found to arise}

from several phonon pairs ^at the different symmetry points. The many contributions ^to the observed peaks together ^{with the} high

polarizability

^{of the} caesium and halogen ions account for the

relatively

^high

intensity

^{of the} spectrum. The limited structure in the infrared

absorption

spectrum ^{of CsBr can} be accounted for in terms of

phonon

pairs ^{at A and 03A3} symmetry points.

PHYSIQUE 26, 1965,

1. Introduction. ^- The second order

(two phonon)

^Raman

spectra

of the alkali halides exhi- bit a wealth of structure. The

corresponding

second order infrared

absorption spectra

^exhibits

relatively

^little structure. In the case of the alkali halides

having

^{the rocksalt structure this}

striking

difference in the character of the two

types

^of

second order

spectra

^is

primarily

^{due to the fact}

that the selection rules for second order Raman processes are

relatively

^lenient whereas the selec- tion rules for second order infrared processes ^are

fairly

restricted.

Using

^the selection rules for second order Raman processes in the rocksalt

structure, Burstein, Johnson,

and Loudon

[1]

^were

able to

interpret

^{the observed structure in the}

Raman

spectrum

^of

Nad, KBr,

^{and NaI in terms}

of

phonon pairs

^at

X, L,

^{and A}

symmetry points

of the

(f.

^c.

c.)

^{Brillouin zone}

corresponding

^to

(1) Supported by

the Advanced Research Projects Agency.

(s) Supported

ⁱⁿ

part

by ^{the U. S.}

Army

^Research Office, ^Durham.

maxima in the combined

density

^{of states curves,}

In the case of NaCI the theoretical ^curves of

Hardy

^{and Karo}

[2]

^{were used as a}

guide

ⁱⁿ

making

the

assignments.

^The

analysis

indicated that

phonons

^at

(or near)

^{the X}

symmetry point played

a

major

role in the

scattering,

^{and it was, in}

fact, possible

^{to establish a} consistent set of

frequencies

for the

phonon

^{at the X}

point

from the Raman data. In the case of KBr and

NaI,

the observed structure in the Raman

spectra

^{could be}

directly

correlated with the neutron

scattering phonon

^dis-

persion

^{data of}

Woods, Brockhouse, Cowley,

^and

Cochran

[3]

^{for these two materials.}

The second order Raman

spectra

^{of CsBr and} CsI which have the caesium chloride structure also exhibit considerable structure and an

analysis

^of

the selection rules for the second order infrared

absorption

and Raman

spectra

^{for the caesium}

chloride structure

by

^Ganesan

[4]

^{shows that the}

situation is similar to that for

the’rocksalt

^struc-

ture, i.e.,

^the selection rules for the second order Raman

spectrum

^are

quite favourable,

^{where as}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011063900

640

those for the second order infrared

absorption

^are

relatively

restricted. In the

present

paper ^we show that the

peaks

^{in the Raman}

spectra

^{of CsBr}

and CsI can be accounted for

by

combinations of

phonon frequencies

^at almost all the

symmetry points

^{in the zone. The}

analysis

^{of the structure}

in the Raman

spectra

^{was carried out}

using

^the

theoretical

phonon dispersion

^{curves of Karo and}

Hardy [5]

^{as a}

guide

^{for the}

assignment

^of

phonon frequencies.

2.

Symmetry

^{of the}

phonons

and selection rules.

- The caesium chloride structure

belongs

^{to the}

space group

01.

^{It is a set of two}

interpene- trating simple

^cubic

lattices,

^{with two atoms} per unit

primitive

^{cell. The Brillouin zone is}

again simple

^{cubic and it is shown in}

figure

1 with all the

symmetry points

indicated. We determine

FIG. 1. ^- Brillouin zone for caesium chloride structures with the symmetry points.

the

symmetries

of the various

phonons

at ^these

points

^{in the} zone,

by placing displacement

^vectors

on each atom and

finding

their transformation pro-

perties.

^At

r,

^{the centre of the} zone, both the acoustic and

optical phonons

transform like

r 15.

We

use here the notation of

Bouckaert,

^Smolu-

chowski,

^and

Wigner [6]. Along

^the

symmetry

directions

[100], [111],

^and

[110]

^the

assignment

of

phonon symmetries making

^{use of the}

compati- bility

^{relations is}

straightforward.

^{At the zone}

boundaries, however,

^{there are two}

assignments possible

^{for each}

phonon,

^each

assignment having

a definite

parity.

Since each atom is a centre of

symmetry,

^we

study

^the

displacement,

x, of similar

particles

ⁱⁿ

adjacent

^{cells to} establish the

parity

of the

phonons.

^The

phase

of the motion varies from cell to cell _{as . x} ^q

t ^{( 1)}

^where

^{q is}

^{q the wave}

vector, 1

is the cell index and K is the

particle

^index.

We find that at X and R the

phonons

^have

even

parity

^{and at M the}

phonons

^{have odd}

parity.

This is further checked

by looking

^at the compa-

tibility

^{relations between} the various

assignments along

various directions. ^The

phonon symmetries

are

given

^{in Table 1. A similar}

analysis

^{of the}

parity

^{of the}

phonons

^{at the L}

point

^{for the NaCI}

structure indicates that both the

optical

^{and acous-}

tical

phonons

^{have even}

parity.

This result differs from that

given by Burstein, Johnson,

^and

Loudon

[1]

^who

assign

^odd

parity

^{for the}

optical phonons

^{and even}

parity for the

acoustical

phonons.

The selection rules for the two

phonon

^Raman

and infrared processes ^are worked out in the usual way

[7], [8].

^{These are}

given

^{in Tables 2 and 3.}

We

find

that all combinations are allowed for second order Raman processes. For second order infrared processes ^{overtones are}

strictly

^forbidden

because of the center of

symmetry [9],

^{and no}

combinations are allowed at the

r, X, M,

^{and R}

points.

^These

points

exhibit the

complementary

nature of the selection rules for the two

types

^of processes.

TABLE 1

SYMMETRY POINTS AND PHONON SPECIES IN CAESIUM CHLORIDE STRUCTURE

at

Parity ^well

defined only

^at r, X, ^{M and R.}

TABLE 2

TWO PHONON INFRARED PROCESSES IN CAESIUM CHLORIDE TYPE STRUCTURES

(3) These exhibit

complementarity

^{in the} selection rules between Raman and infrared processes.

TABLE 3

TWO PHONON RAMAN ACTIVE PROCESSES IN CAESIUM CHLORIDE TYPE STRUCTURES

, ,

(4) ^{These exhibit}

complementarity

^{in the} selection rules between Raman and infrared processes.

The selection rules also

give

^some information about the nature of

polarization

of the scattered radiation. As

pointed

^out

by

^Kleinman

[10],

^the

concept

^of

polarization

of the scattered radiation is not

meaningful

unless the direction of incidence and direction of observation are

specified along

with the

polarization

of the incident

light.

^{In the}

absence of such information for the second order Raman

spectra

^{of Cs Br and Cs}

I,

^{we have not}

given

the reduction of the Kronecker

products

^for

the

phonon

combinations.

In the case of

simple

^cubic

lattices,

^Rosenstock

[11]

^has

shown

^{that most of} the critical

points

^lie

at the _corners,

along

^the

edges

^{and on the faces}

of the cube.

Phillips [12],

in his critical

point analysis,

shows that for this case

r, R, M,

^{and X}

must be

ordinary

^critical

points

in all three modes.

r and R are three-fold

degenerate ; M

^{and X are}

two-fold

degenerate

and lie in

planes

^of

symmetry.

Further his

analysis yields

^those

points

^{for which}

some

components

^{of the}

gradient

vanish and these

occur at the faces of the cube. The

unique

^feature

about the caesium chloride structure is

that, except

^{for the}

1Y, A,

^{and Xl}

points,

all the other

points

^{occur on the same} face of the zone.

Analy- zing

the critical

points required by symmetry

^one

can

get

^a

symmetry

^set and from this

using

^Morsels

topological

^thorem

(the

existence of some critical

points

necessitates the existence of

others)

^{one can}

get

^{a minimal set which} is the smallest ^set satis-

fying

these relations and

containing

^the

symmetry

set.

Phillips’ analysis gives

^the

following

^results.

For short range forces the

symmetry

^{set contains}

all the critical

points.

Introduction of second

neighbour

forces and hence

long

range

forces,

pro- duces more critical

points, however,

^{the minimal}

set still contains all the additional critical

points

introduced

by

these forces. From his Tables we see that critical

points

appear ^at

r, R,

^and

X,

and M for first

neighbour

^{forces and additional}

critical

points

^{appear at}

^{A, S,}

^and

^{E, etc...,}

^when

second

neighbour

^{forces are included. This can be}

seen in the

Karo, Hardy phonon dispersion

^curves

FIG. 2a. - Frequency ^{versus wave vector curves} along [100] ^{for CsBr and CsI.} (After ^A. M. Karo and J. R. Hardy, ^{to be}

published.)

FIG. 2b. ^- Frequency versus wave vector curves along [111] for CsBr and CsI. (After A. M. Karo and J. R. Hardy, ^{to be}

published.)

FIG. 2c. ^- Frequency ^{versus wave vector curves} along [110] for CsBr and CsI. (After ^A. M. Karo and J. R. Hardy, ^{to be}

published.)

for CsBr and CsI

given

ⁱⁿ

figures 2(a, b, c).

^Accor-

dingly

^{for a} proper

analysis

^{of the observed}

peaks

642

in the second order

spectra,

^we consider all the

points

^{in the zone.}

3.

Interpretation

^{of the} second order

spectra.

^-

3.1. CAESIUM BROMIDE. ^- The first measurement of the second order Raman

spectrum

^{was carried}

out

by Narayanan [13]

^who

reported

⁵

peaks.

^A

later measurement

by

^{Stekhanov and Koral’kov}

[14]

^{showed 9}

peaks.

^The

peaks

^{in the CsBr}

spectrum (and

also in the CsI

spectrum)

^{are relati-}

vely

^{intense in} contrast to the other alkali halides.

Stekhanov and Koral’kov attribute this ^to the

high polarizability

of both the caesium and halo- gen ions. Another feature of interest is the fact that the

spectrum

^{of CsBr}

(and

^{also of}

CsI)

^is

completely depolarized

^{which is also in} contrast to the

spectra

^{of NaCI -}

type crystals

^{which are}

appreciably polarized.

^The

spectrum

^{is shown in}

figure

^3.

FIG. 3. ^-

Microphotometer

record of the Raman spec- trum of CsBr. (A. ^I. Stekhanov and A. P. Korol’kov [14].)

Since the selection rules for the two

phonon

Raman

scattering

^are very

lenient,

^{as in the case}

of NaCI -

type lattices,

many

phonon pairs

^are

active in the second order

spectra.

^{From the}

theoretical

dispersion

^curves of Karo and

Hardy [5]

shown in

figure 2,

^{we see that the}

spread

^{of the}

spectrum

^{is small} and that critical

points

^{occur also}

at A and E. ^As critical

points

^{also occur at}

^S, E,

^{and T}

(not

^{shown in}

figures 2a, b,

^and

^c)

^we

also consider

phonon

combinations at these

points.

TABLE 4

SECOND ORDER RAMAN SPECTRUM IN CsBr

The branches at T, ^{S and Z are called}

arbitrarily LO,

y0i, TOa, LAi, TA1 ^and T A z in ^{the order of} decreasing

energy.

Taking

^the

corresponding frequencies

from the dis-

persion

curves, ^we show in Table 4 that ^{we can}

reasonably

^{account for all the} lines. The inte-

resting

feature is that many combinations of

phonon pairs

contribute to a

given peak.

^An

experimental polarization study

may be of ^some

help

^but

again

^{due to the}

possibility

^{of so many}

combinations,

it may ^not resolve the

ambiguity

ⁱⁿ

the

assignment.

The infrared

spectrum

^{of CsBr was measured}

by

^Geick

[14]

ⁱⁿ

1961,

^{and is}

given

ⁱⁿ

figure

^4.

It shows

considerably

^{less structure than the}

FIG. 4. ^- Infrared absorption spectrum ^{of CsBr.}

(After ^{R. Geick} [15].)

Raman

spectrum

^{as is to be}

expected

^{from the}

rigour

of the selection rules. We find that we can

account for all the

peaks

^{in the}

spectrum

^{in terms}

of

phonon pairs

^{at the A and 2}

points.

^The

assignments

^are

given

^{in Table 5.}

TABLE 5

INFRARED SPECTRUM IN CsBr

(15) Quite ^{broad -}

probably

^{extra structure.}

3.2. CAESIUM ^{IODIDE. - The} second order Raman

spectrum

^{of CsI was measured}

recently by

Krishnan and

Krishnamurthy [15] (fig. ^5). They

observed 12 lines very

closely spaced, extending

from 19 cm-1 to 181 em-1. ^The extent of the

’

FIG. 5. ^-

Microphotometer

^record of the Raman spectrum

of CsI. (R. S. Krishnan and N.

Krishnamurthy

[16].) whole

spectrum

is much less than in the case for CsBr. In common with CsBr the lines are found to be

sharp

and intense.

From the

dispersion

^curves for CsI shown in

figure 2,

^{we see that the}

spread

^of

frequencies

^is

much less than that for CsBr. The small

spread

ⁱⁿ

frequency

^{is due to the}

nearly equal

^{masses of the}

caesium and iodine atoms. It is most

striking

^at

the R

point

^{where the}

optical

and the acoustical

phonon

^{branches are}

nearly degenerate.

^{As in the}

case of CsBr we can account for the

peaks

^{in this}

spectra

^{in terms of}

phonon pairs arising

^{from the}

various

symmetry points

in the Brillouin zone.

The

phonon pair assignment

^{for the observed}

peaks

are

given

in Table 6.

TABLE 6

SECOND ORDER RAMAN SPECTRUM IN CSI

644

TABLE 6

(continued)

Conclusion. ^- The selection rules for the CsCI structures are

quite

^{similar to} those for the NaCI structures and it is

possible

^{with the}

help

^{of the}

theoretical

dispersion

^{curves of Karo and}

Hardy

to account for the observed structure in the Raman

spectra

of CsBr and CsI in terms of

phonon pairs

^{at various}

symmetry points

^{in the s. c.}

Brillouin zone

(s).

Each of the observed

peaks

^is

found to arise from several

phonon pairs

^{at the}

symmetry points.

This may be

attributed,

^on

the one hand to the

high symmetry

of the CsC]

structure and the near

degeneracy

^{of the}

phonon

(1) ^{we can also} expect phonon pair contribution to both Raman and infrared spectra from branches having equal

or

also associated with ^{opposi le}

slopes particularly

peaks in the combined ^{A and Z points}

density

^{which areof}

states. These are not included in our

analysis,

^because

of the uncertainty ^{in the wave vector} at which the

slopes

are

equal

^or opposite ^{and the} corresponding

large

^uncer-

tainty

^{in the} phonon pair ^{suni or} difference

frequency.

branches at the zone

boundary,

^{and on the other}

hand to the fact that all the

symmetry points

^are

located on the same face of the ^zone and ^are

possible

^critical

points.

^{The small}

frequency spread

of the second order Raman

spectra

^{of CsBr}

and CsI and the many contributions ^to each of the observed

peaks, together

^{with the}

higher

^defor-

mability

^of the caesium and

halogen

^{ions account}

for the

relatively high intensity

of the observed Raman

spectra.

In view of the many

phonon pair assignments

for each of the observed

peaks,

^{it is not}

possible

^to

use the Raman

spectra

^{to derive}

phonon

^fre-

quencies

^at any of the

symmetry points (7).

^Since

the selection rules allow all

phonon pair

^combi-

nations and overtones at the

symmetry points,

^one

can

expect

the combined

density

^{of states derived}

from the theoretical

dispersion

^{curves to}

provide

^a

reasonably

^close

representation

of the observed

spectra (8). Among

^other

things

^this

approach provides

^the

possibility

^of

obtaining

^a verification of the theoretical

phonon dispersion

^{curves and}

under

optimum

circumstances _{it may}

provide

^infor-

mation about the second order matrix elements.

The

relatively

small number of second order

absorption

bands observed in the room

tempe-

rature infrared

spectra

of CsBr is a consequence of the very restricted selection rules. Our

analysis

indicates that the

major

contribution to the bands

comes from

phonon pairs

^at the A and Xl sym-

metry points.

^The

absorption

^{bands are}

quite

broad and _may

actually

^{be made} up of a super-

position

of several bands.

Absorption

^measu-

rements at low

temperature

may

possibly yield

additional structure. If _so, this would allow us to make more definite

phonon pair assignments

^for

the observed

peaks.

(’) ^An examination of the phonon pair symmetry repre- sentation indicates that

polarization

measurements will

only

^{have limited}

applicability

ⁱⁿ establishing unique phonon pair assignments for the observed peaks.

(8) A detailed treatment of the combined density ^of

states and the Raman spectra of alkali halides is given by Hardy ^{and Karo} in another paper

published

in this pro-

ceedings (J. Physique, 1965, 26,

000).

,

REFERENCES

[1] ^BURSTEIN (E.), ^JOHNSON (F. A.) and LOUDON

(R.),

Phys. Rev., 1965, 139, ^{A 1240.}

[2] ^KARO (A. M.) ^{and HARDY} (J. R.), Phys., Rev., 1963, 129, 2024.

[3] ^WOODS (A. ^D. B.), BROCKHOUSE (B. N.), ^COWLEY (R. A.) and COCHRAN (W.), Phys. Rev.,

1963,

131,

1025.

[4] ^GANESAN (S.), ^{Bull. Amer.} Phys. Soc., 1965, 10, ^389.

[5] ^KARO (A. M.) ^{and HARDY} (J. R.), ^{To be}

published.

[6] ^BOUCKAERT (L. P.), SMOLUCHOWSKI (R.) ^{and WIGNER} (E.), Phys. Rev.. 1936, 50, ^58.

[7] BIRMAN

(J

L.), Phys. Rev., 1963, 131, 1489.

[8] ^LOUDON (R.) and JOHNSON (F. A.), ^Proc. Roy. Soc., London, 1964, A 281, 274.

[9] ^LOUDON (R.), Phys. Rev., 1965, 137, 1784.

[10] ^KLEINMAN (L.), ^{Solid State} Comm., 1965, 3, 41.

[11] ROSENSTOCK (H. B.), Phys. Rev., 1955, 97, 290.

[12]

^PHILLIPS (J. C.), Phys. Rev., 1956, 104, 1263.

[13]

^NARAYANAN (P. S.), ^Proc.

Ind. Acad.

Sc., 1955, ^A 42,

303.

[14] ^STEKHANOV (A. I.) ^{and KOROL’KOV} (A. P.), ^Soviet Phys., ^{Solid State,} 1963, 4, ^2311.

[15] ^GEICK (R.), ^Z. Physik, 1961, 163, 499.

[16] ^KRISHNAN (R. ^{S.) and} KRISHNAMURTHY

(N.),

^Ind.

Jour. Pure and Appl. Phys., 1963, ^{1, 239.}