Raman scattering of light from H— centers in CaF2

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Raman scattering of light from H- centers in CaF2

M. Ashkin

To cite this version:

M. Ashkin. Raman scattering of light from H- centers in CaF2. Journal de Physique, 1965, 26 (11),

pp.709-716. �10.1051/jphys:019650026011070900�. �jpa-00206339�

RAMAN SCATTERING OF LIGHT FROM H2014 CENTERS IN

CaF2 By

^M.

ASHKIN,

Westinghouse ^Research Laboratories, Pittsburgh, ^Pa.

Résumé. 2014 L’auteur a étudié la diffusion Raman de la lumière par les vibrations localisées de monocristaux de CaF2, contenant de faibles concentrations d’ions H- substitués aux F-. Il donne les règles ^de sélection. En considérant le défaut comme une particule vibrant dans un puits ^anhar- monique, ^{il calcule les} énergies ^{de l’état} fondamental, ^{et des} premier ^{et deuxième} harmoniques ^du

mode localisé. A l’aide de ce modèle, il calcule les intensités pour les transitions ^de l’état fonda- mental aux composantes ^{Raman actives de ces} niveaux. Ces intensités contiennent les coefficients de développement ^{de la} polarisabilité, qui ^{sont conservés comme} paramètres, ^à l’exception ^du

coefficient du premier ^ordre Pxy,z, qui ^{a été} calculé par la méthode des variations. La valeur obtenue est Pxy,z ⁼ 0,163 Å2. On ne peut ^{donner de résultats} expérimentaux concluants, ^{à cause de diffi-}

cultés dans la préparation des échantillons : ceux qui contenaient une concentration suffisante en

ions H- donnaient une large bande d’émission dans la région ^{où l’on} prévoyait la raie Raman.

Abstract. - The Raman scattering ^of light ^{from the} localized vibrations of single crystals ^of CaF2 containing low concentrations of H- substituted for fluorine ions has been studied. Selec- tion rules are given. ^{For a model of the defect as a} particle vibrating ^{in an} anharmonic well, ^the energies ^{of the} ground state, fundamental, and the first and second harmonic of the localized mode are calculated. Intensities are calculated with this model for transitions from the ground

state to the Raman active components of these levels. These intensities contain the expansion

coefficients of the polarizability ^{which are left as} parameters ^{with the} exception of the first order coefficient Pxy,z. ^This coefficient has been calculated variationally ; ^the calculated value is

Pxy,z ^{= .163 Å2. No definite} experimental ^{results can be} reported because of trouble with

sample preparation. Samples ^{with a} sufficient concentration of H- show a broad emission band in the region ^{where the} Raman line is expected.

PHYSIQUE 26, 1965,

1. Introduction. ^- Raman

scattering

^{from a -}

crystal

^{is the inelastic}

scattering

^of

light by

^the

lattice vibration of the

crystal.

The initial and final states differ

by

^the lattice modes excited.

These modes are restricted in a

perfect crystal by

two

general

considerations if one is

using

^radiation

which is

essentially

^{of zero wave} number. In the first

place

^the translation

symmetry

^{of the}

perfect

lattice and energy conservation of the

scattering

process

requires

^{the sum of the wave vectors of the}

participating

^{modes to be}

effectively

^{zero. The}

second consideration concerns the

symmetry

^{of the}

allowed modes that can

participate

in the Raman effect. In the case of the first order Raman effect

a mode must transform like a second rank tensor under the

operation

^{of the}

crystal point

group in order to be Raman active.

When defects are

present

^{in a}

crystal

^{the Raman}

spectra

^{is altered.} The removal of lattice trans- lational

symmetry

breaks the zero wave number selection rule which can

change

^line

spectra

^to

band

spectra

and introduce additional bands which

were not

present

in the absence of defects. The defect may, in ^some case, also alter

drastically

^some

of the

modes

of the

perfect crystal.

^{The case of a}

light

^{mass defect is an}

example

^{where modes from}

top

^{of a lattice band}

split

^{off to} form discrete

(possible degenerate)

^{levels. If these new levels}

are Raman active then

they

show up ^{as new} lines in the

spectra.

Similar considerations hold for the infrared

absorption

^of

light by crystals

contain defects.

An

impurity-induced

first order electric moment

was invoked

by

Lax and Burstein

[1]

^{to account}

for

part

^{of the}

absorption

^{in diamond.}

Schafer [2],

and Fritz

[3]

have observed localized vibrations in the infrared

spectra

^of alkali-halide

crystals

^con-

taining

substitutional

hydrogen

^ions.

Hayes

^et

al.

[4]

^have

observed,

in the infrared

spectra

^of substitutional H- and D- in

CaF2,

^{the funda-}

mental and the second and third harmonic of the localized vibration. In this last

system

^{the infra-}

red active fundamental and overtones of the loca- lized vibration are also Raman active but there are

additional overtones which

though

infrared inac- tive are Raman active.

Stekhanov and

Eliashberg [5]

have studied the Raman

spectra

^{of KCI}

crystals containing I, Br,

^and Li. These defect

systems give

^{rise to}

resonance modes

which,

with their

overtones,

appear in the observed

spectra.

^The

frequency

^of

these resonance modes lie within the bands of lattice

frequencies

^{of the}

perfect crystal.

Experiments

^{to measure the Raman}

scattering

of

light

^{from the} localized vibrations of H- substi-

tuting

^{for some} of the F in

GaF2

^{are in} progress

[6].

These

experiments

^{use a} He-Ne laser as a

light

source. Difficulties in

preparing samples

^have

occurred and

samples

^{with a} sufficient concen-

tration of H" have ^so far had a broad emission

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011070900

710

band in the

region

where the Raman lines due to the localized mode are

expected.

Direct obser- vation of the

spectra

^{has not}

yielded

any conclu- sive evidence of the localized mode. The first derivative of the Raman

spectra

^has

yielded

^results

of a very

preliminary

^{nature and the}

experiments

need to be carried further before any definitive conclusions will be drawn.

Therefore,

^{most of}

this paper will deal with the theoretical

aspects

^of

the

problem.

^The fundamental and harmonics of the localized mode will be classified

according

^to

their transformation

properties

^{under the}

point

group ^at the F site and the selection rules which follow are stated. When the

crystal

^is

vibrating

in the localized

mode,

^a model of the defect as a

particle vibrating

^{in an} anharmonic well will be used to calculate the location of the overtones and the Raman intensities for a

given experimental

situation. This model was used

by

^Sennett

[7]

to discuss the infrared

experiments

^of

Hayes

^et

al.

[4].

^{This latter}

experiment provides enough

information to determine the

parameters

^{of the}

model : the vibrational

frequency

^of the harmonic localized mode and the three anharmonic force constants. The electronic

polarizability

^{in the}

Raman case is introduced

phenomenologically

^and

is

expanded

^to second order in the

displacements

of the defect atom with some of the coefficients left

as

parameters.

The first order

polarizability

^has

been calculated

variationally

^{for a}

point-ion-model

of the lattice.

II. Selection rules. ^-- If a

point

^{defect is intro-}

duced

substitutionally

^{into a}

perfect crystal

^and

the force constants of the

perturbed crystal

^are

different from those of the

perfect crystal only

ⁱⁿ

the

vicinity

^of the defect then in the harmonic

approximation

^the Hamiltonian of the

crystal

^with

defect can be

diagonalized exactly [8].

^{When the}

mass of the defect atom is much less than the mass

of the substituted host atom, there may ^occur vibrational modes of the

lattice,

^{the localized}

modes,

^with

amplitudes highly

localized about the defect site and

frequencies

outside the bands of the

perfect crystai.

When

anharmonicity

is included in the

descrip-

tion of the

lattice,

the modes of the harmonic lat- tice are

coupled together.

^This

coupling

^{of the}

harmonic modes leads in a familiar way ^to finite

widths,

^{shifts of} levels and

partial

removal of

dege-

neracies of overtone and combination levels.

The

description

of the defect

crystal vibrating

in the localized modes as a

particle

^{in an anhar-}

monic well is

essentially equivalent

^to

retaining

^the

terms of the Hamiltonian

expressed

in the normal modes ot the harmonic defect

crystal

^{which contain}

only

^{localized mode}

operators.

^Sennett

[7]

^has

considered the effects of

coupling

^{to the band}

modes. His calculations were

necessarily

^some-

what

qualitative

^{due to} the lack of inforrnation about the

pertinent

^{atomic force constants fo-}

these interactions. We will not consider this cour

pling.

^The

approximation

^of

using

^{the mass of H-}

as the

equivalent

^{mass for the}

particle

in the well is also made.

In this

description

^{the harmonic} oscillator

(h. o.)

states for the

particle

^{will be used as basis states}

and the

anharmonicity

^treated

by perturbation theory.

We ^use the notation of

Wilson, Decius,

^and

Cross

[9]

^tor the irreducible

representations

^{of the}

symmetry

groups. The fluorine in

CaF2

^{is at a site}

of Td

symmetry.

^{The H-} defect will be assumed to possess this

symmetry, i.e.,

^{under the}

operations

of the group

Td,

^{the well}

potential

^transforms

according

^to

A 1 (the

^invariant

representation).

The wave functions of the anharmonic well can be

expressed

^{as a} linear combination of h. o. func- tions. Since the anharmonic

potential

^can

only

mix functions of like

symmetry,

^{the full wave}

function is constructed from h. o. functions of the

same

symmetry.

^For

symmetry

considerations it is sufficient to

discuss

the

symmetry

of the unper- turbed h. o. function for each level.

The

symmetry

^{of the localized modes}

(1. m.)

^are characterized

by

^the

displacements

of the defect atom and therefore the 1. m. transforms

according

to

F2

^the

polar

^vector

representation

^{of Td. There}

are three

degenerate modes,

^{which can be viewed}

as vibrations along the three

mutually

perpen- dicular _{x, y,} z-axis. A normalized h. ^{o. wave} function for a

given

vibrational level will be deno- ted

by nx ny nz

> where nx ny nz are the number of

quanta

^{in the mode}

vibrating along

^{the x, y,}

and z-axis

respectively.

^{The nth level of the}

harmonic oscillator has n ⁼ nx + nv + ^n,,.

The

intensity

of the Raman scattered

light

per unit solid

angle

per ^{scatterer can} be written

[9.0].

were

Qo

^{is the}

frequency

^of the incident

light,

m ⁼

00

+ ^{Q is the}

frequency

of the scattered

light,

^nk is the unit vector in the direction of the electric vector of one

linearly polarized

^compo-

nent of the scattered

radiation,

and E+ and E- =

(E+)*

^{are the}

components

of the incident electric field when the field is written

The coefficient icxy,(3À ^is

given

by

where .En is the energy of ^a vibrational

state In

^>

of the

system,

^{Z is the} vibrational

partition

^func-

tion,

^and

P-1

^{is the}

temperature

^times Boltzmann’s constant.

Only

transitions ^{from the}

ground

^state

of the oscillator will be considered and the thermal average will be

omitted, Pap

^{is the}

operator

^{for the}

electronic

polarizability

^tensor.

Before

discussing specific

^{levels the}

general

^selec-

tion rules which follow from

symmetry

^conside-

rations will be stated. P transforms like a second rank tensor under the

operations

^{of the group Td}

and P therefore transforms like

A 1

^{-f- E +}

F 2,

^or

more

specifically

^the

following assignment

^{can be}

made :

Recalling

^{that the}

ground

^{state n = 0}

belongs

to

A 1,

^the allowed Raman transitions from the

ground

^{state are}

Transitions to

A 2

^and

Flare

^forbidden.

To conclude this section we

give

^the

decompo-

sition of the levels n ⁼

0, 1, 2, 3,

^{into the irredu-}

cible

representations

of the group

Td,

^{the norma-}

lized wave functions that

belong

^{to the} represen- tations which are Rarnan

active,

and list the

dege-

neracy of the ^{h. o.} level :

n ⁼ 0

(ground state), A 1, non-degenerate

A1 : ^{’¥(A1) =} 1000 ^>

n ⁼ 1

(fundamental), F2,

^3-fold

degenerate

The

superscript

^on the lv’s labels the irreducible

representation

^{to which it}

belongs,

^the

subscript

labels the row of the

representation

^{to which T}

belongs.

^{For n =}

3,

^{the two}

F2 representations

are

distinguished by primes.

III.

Energy

^{shifts. ---} In the absence of anhar-

monicity

the levels of the now

isotropic

^{three di-}

mensional harmonic oscillator well are

degenerate.

The

degeneracies

^{for n =}

0, 1, 2,

^{3 have, been}

listed

previously.

^{When the}

anharmonicity

^is

included the

degeneracy

^is

partially removed,

states

belonging

^to different irreducible represen-

tations, apart

from accidental

degeneracy,

^have

different

energies

^{and when an} irreducible repre- sentation occurs more than once for a

giv en n

^sets

of

partners

^{have different}

energies.

The Hamiltonian for ^a

particle

^{in an anhar-}

monic well

including

^terms of fourth order in the

displacements

^is

where

where p is the momentum and _x, y, z ^{are the} dis-

placements along the x,

y,

z-axis, respectively,

^of

a

particle

^{of mass} m,

and coo

^{is the}

frequency

^of

the t,hree

degenerate

^harmonic localized modes.

V3

^and

V4

^are constructed from all the terms cubic and

quadric

^{in the}

displacements, respectively,

which

belong

^to

A1

^{in Td. The}

coefficients, L, M1,

and

M2

^{are related to} the Fourier coefficients of the atomic force constants

appropriate

^to interactions

only involving

^{the localized mode. Values for}

L, M 1, M 2’

^and coo ^were obtained

by

Sennett from a

comparison

of the observed energy levels in the infrared

spectrum

with theoretical

expressions

^for

the

F2

levels. Our theoretical

expressions

^differ

in one case from Sennett’s results but the calcu- lated values of

L, M1, M2,

^and wo ^are insensitive to this difference and we use Sennett’s values.

The

change

^{in t,he} energy ^{of a}

state In

> be-

longing

^an irreducible

representation

^{which occurs}

once in the

decomposition

^{of a level is}

712

where

E:::)

^{is the}

unperturbed

energy of the ^state

m

> and Em ^{is the}

perturbed

^{energy to}

Ü(V4)

⁼

0 ( V3).

^{WheIl an} irreducible _represen- tation occurs more than _once, as for the n =.

3F2 levels,

^{it is} necessary ^{to use}

degenerate pertur-

bation

theory.

The

perturbed n =1, 2,

and 3 levels are consi- dered and the calculation of energy

splitting

^are

carried to second order in the third order

potential

and to first order in the fourth order anharmonic

potential.

The

energies

^a.re

expressed

^{in terms of}

À ==

A/2mcoo

^{the mean} square

particle displace-

ment. The calculated _energy shifts are

where

The numerical values are based on the

following

values for the model

parameters given by

Sennett

[7 j :

Figure 1

shows the level scheme for the Raman active n ⁼⁼

0, 1, 2, 3

^levels.

FIG. 1. ^- Level scheme for Raman active levels.

IV. Intensities of the Raman scattered

light.

^-

Equations (2.1)

^and

(2.2)

express the

intensity

^of

the Raman scattered

light

^{in terms of matrix ele-}

ments of the

polarizability operator

between diffe- rent vibrational states. Intensities will be obtain- ed for transitions between the

ground

^{state and}

states which reduce ^to one of the n ---

1, 2,

^or

3 harmonic oscillator states in the absence of

anharmonicity.

^{In these}

calculations.,

^wavefunc-

tons correct to second order in the cubic

anharmonicity [O(V’)

⁼

O(V4)]

will be used and terms linear and

quadratic

^{in the}

displacement

of the defect will be included in an

expansion

of the

polarizability.

The calculations will be limited to the

special geometry

^of

experiments

ⁱⁿ

progress with ^a He-Ne laser.

This

expansion

will be written where

Pg§

is the static

polarizability,

audits ^the «-Cartesian

component

^{of the}

displa-

cement of the defect. P1°> is

responsible

^for

Rayleigh scattering,

^{P(l) and P(2)}

gives

^{the domi-}

nant contribution to the first and second order Raman

effect, respectively.

The

expansion

coefficients in

Eq. (4.2)

^and

(4.3) display

^{the site}

symmetry

^{of the} defect and it is therefore convenient to let each of

the x,

y, z-axis be

parallel

^{to the two-fold axes of the}

CaF2 crystal.

Under a

symmetry operation

of the group

Td,

^the

expansion

coefficients

obey

^the

following

^tensor

transformation laws :

where s is the matrix of the

symmetry operation

in

equation.

^{When we}

apply

^these

expressions

^we

find,

among others

(cx, P

== x, y,

z)

The

remaining

^non-zero

Pao,g

^are obtained froni

E cf. (4.7)

^{with the use of}

These relations show that

are the

independent

second order

expansion

^coef-

ficients and

are the

independent components

^{of oc(3,y8.}

It is convenient to introduce the

following

ope- rators

and combinations of coefficients

and to set

In terms of this

notation,

^we may write

The

operator A1

^{connects the}

A1 component

^of

thc same or different states.

&,L(82)

^{connects the}

A1 colnponent

^{of a state to the first}

(second)

^{row of}

the E

component

^{of a}

state ; P(2)

^{connects the}

A 1 component

^{of a state to the third row of the}

F2 component

^{of a state.}

In the

present experiments

^{with a He-Ne}

laser,

in some cases the

CaF2 sample

^{is inside the laser}

cavity

^and

precautions

^{are taken not to}

destroy

^the

laser action. This amounts to

making

^the

angle

pp between the inward normal of the entrance face of the

sample [100]

and the direction of propaga-

tion of the incident

light equal

^to the brewster

angle.

^For

CaF2

pp ss 56°. In the

present

expe- riments the normal and

propagation

^{direction are}

in the

(001) plane

^and the scattered

light

^{is viewed}

along

^the

[001]

direction. The

angle

^{between the}

refracted incident beam and

[100;

?,, ^ss 34°. In

figure

^{2 the relevant} directions are shown.

FIG. 2. ^- Sample orientation relative to incident light.

The

polarisation

directions are

parallel

^{to the}

(001)

^{face of the}

crystal

^{and we can therefore}

write.

The

intensity

scattered into unit solid

angle

ⁱⁿ

the direction of the z-axis for the

geometry

^of

figure

^{2 is}

where

Under proper conditions the

separate

^contri-

butions from each

polarization

direction i ⁼

1,

or 2 could be measured. In the

present experi-

ments the detector

accepts light

^of

arbitrary pola-

rization directions in the

xy-plane

^and

Eq. (4.27)

will be summed over all these

possible states.

^We

find

,o-

714

The calculation of 3zr,>z ^and Jxv,xy ^are

straight-

forward and we obtain the

following

^{results :}

The

quantities ^{a7 and aT}

^are

Here

,ZV-:r

^and

BT-

^{enter in} the proper zeroth order wavef unctions for the

degenerate n

⁼⁼

3, F2

^levels

(3Td row)

and are obtained from the

following expressions :

The numerical values are based on values for the

model

parameters

found in Sennett’s thesis

[7]

^and

given

ⁱⁿ

Eq. (3.6).

V.

Polarizability.

^- The coefficients

Paev.z,

^...,

Paev,aev

^{have been left as}

parameters

^{which can be}

determined from

intensity

measurements. We

now calculate

variationally

^the first order coef- ficient

Prv,z

^{for a}

point-ion-lattice

model. All

dynamical

effects due to the motion of the ion will be

neglected

^{and the}

polarizability

^{will be}

obtained from the energy of ^an H- ion substituted for a F- ion

displaced

^an infinitesimal distance from its

equlibrium position

^{in a} uniform electric field. The

remaining

^{atoms of the}

CaF2

^{lattice are}

treated as

point-ions

^{and remain at their}

equili-

brium

positions.

To obtain

Pzv,z

^{we can let the}

displacement

^{u of}

the

proton

^{of H- be}

along

^{the z-axis}

(x,

y, and z

are two-fold axes of the

crystal)

and the electric field F be in the

xy-plane

If u and F are treated as

small,

^{then the}

energyE

of the

system

^{can be}

expand

in powers of ^u and F

as follows :

Prv,z is identified as the coefficient of

(Fx

Fv

u) /2

ⁱⁿ

this

expansion.

The energy

(5.3)

^{is obtained}

by minimizing

^the

expectation

^{value of the} Hamiltonian with

respect

to variations in the

parameters

^{of some}

suitably

chosen wavefunction. A similar calculation

by Gourary [11]

^{of the} energy and wavefunctions ^of U-centers in the alkali halides without the dis-

placement

^{and the electric field} gave results which

were in fair

agreement

^with

experiment.

The calculation is

performed

ⁱⁿ

steps

^to

simplify

the work. Atomic units

(a. u.)

will be used and the H anxiltoni an will be written

w h e r,-,

and

The

origin

of the coordinate

system

^{has been}

chosen as the

equilibrium position

of H-. The

715 kinetic energy of the

proton

^{of H- has been}

neglec-

ted and the sum in

(5.7)

^{is over all}

points

^{of the}

CaF 21

^lattice

except

^the

origin

^{with Z = - 1 for}

F and + 2 for Ca. The interaction of the

proton

with the electric field F.u in the

present problem

vanishes

by Eqs. (5.1)

^and

(5.2).

The

quantity

is first minimized. The variational function

Uo

^is

chosen as

, ""r , 9.

where ^a

and P

^{are the} variational

parameters

and

No

^{is a} normalization factor. This function

multiplied by (1.

⁺

cr12)

^{was used}

by

^Chan-

drasekhar

[12, 13]

for the free ion

(in

^free

space)

^and

as it stands

by Gourary

^{in the} calculation cited

[11].

Omitting

this correlation _{term crl2}

gives

^{a value}

of the ionization

potential

^for the free ion which is smaller than the measured value. This ionization

potential

^{is small}

compared

^{with the} total energy of both the free ion and the ion in the

crystal.

In view of this and of the

degree

of accuracy desired in this

calculation,

^{we have}

neglected

^the

correlation term.

From the form of

Ho(u)

^of

Eq. (5.5),

^{we can set}

u ⁼ 0 to obtain a

and P

^from

(5.8).

^{Then the}

solution of the

equations

gives

the wavefunction

’Y¿.

The lattice sums which occur in

Eq. (5.8)

^{can be}

reduced to sums over a

Cscl-type

lattice of lattice

parameter a j2

^{where a is} the lattice

parameter

of

CaF2.

__

We also use the

Madelung

^{constant for CsCI}

The minimization is

performed

^{on a}

computer

and ^we find

-

With these values

The wavefunction

Uo

^{has the same character as}

that of a free

ion ;

^one

tightly

^{bound site and}

one

loosely

bound site. Like the alkali halide case, the free ion is "

larger "

than the ion in a

crystal ;

^{a = 1.. 03925}

and p

⁼ 0.28309 for the free ion. The values of a

and P given

ⁱⁿ

Eqs. (5.13)

and

(5.14)

will be fixed in the

remaining parts

^of

the calculation.

We next include

simple

tetrahedral terms in the wavefunction. It is useful to define

for an

operator

^{0 and a} variational function X.

For the variational function we take

and minimize

H,(O)

>T. ^with

respect

^{to w and}

T.

The final

step

in the variational

procedure

^{is to}

rninimize H >- with

where

with

_{y, g} the

only

variational

parameters.

^The

form

(5.18)

^is

suggested by perturbation theory

with H’ the

perturbation.

^{We make the}

following approximation

ⁱⁿ

evaluating

^H >it :

With this

approximation

^the

quantity

^{to mi-}

nimize is

..

where

We include the terms

explicitly appearing

ⁱⁿ

Eq. {5.3)

ⁱⁿ

evaluating (5.20).

We will ^not write out

explicitly

^{the terms of}

HO(O)

>’Yo ^with

Eq. (5.16)

^and

Eq. (5.20)

^with

Eqs. (5.18)

^and

(5.19).

^Before

presenting

^the

results of the calculation we remark that the work

was shortened

by

^use of the "

gradient

^formulas

[14]

^{and tables of}

3-j symbols [16].

When the minimization is

performed

^{we find}

that (1) and ^{T N} 10-4 and the tetrahedral terms in the wavefunction _appear to be

negligible.

^These

two

parameters

^{will set}

equal

^{to zero in the latter}

parts

^{of the} calculation.

716

For small u and F we ca.n write

Eq. (5.20)

ⁱⁿ

the form

where

A,

..., G are known coefficients which we

list later and E(O)’ is

independent

^{of F. The}

values of _y

and u

^{that minimize E are}

The order of these

quantities

^are consistent with

our calculation of E. When these

quantities

^are

substituted in

Eq. {5.24}

^{we find}

and

The results of a numerical calculation

give

Therefore,

Schwartz

[16]

calculated the static

polarizability

for H- in free space and obtained P(O) = 26.8

A3.

From the

compressing

^{of the} wavefunction we

expect

and indeed find the value of P(O) in the

crystal

^{to be} smaller than the free space value.

When better

samples

^are

obtained,

it may be

possible

^to compare the calculated value of

Pxv,x

with a value obtained from absolute

intensity

measurements

[8].

Acknowledgments.

^- The author has benefited from many discussions ^on the

experimental aspects

of this

problem

^{with Dr. D.} ’w. Feldman and Dr. J. H.

Parker,

^{Jr. Dr. J.}

Murphy

has assisted in the first

part

of this pa.per. The numerical work was

ably performed by

Miss Brenda

Kagle.

Discussion

M. COWLEY. - The lack of tetrahedral distortion in _your wave function _appears

surprising

^{in view of}

Williss measurement of the

anisotropic Debye-

Waller factor of

CaF2.

M. RUSSELL. - At

Royal

^Radar

Establishment,

we have looked for the H- localised mode in

CaF2

at 77 oK and 20 oK. It is known that the line width is

considerably

^{smaller at low}

temperatures

than at room

temperature.

We have observed a

fluorescence in our

crystal

^which

prevented

any observation of the localised mode.

REFERENCES [1] ^LAX (M.) ^{and BURSTEIN} (E.), Phys. Rev., 1955, 97, 39.

[2] ^SCHAFER (G.), Phys. ^Chem. Solids, 1960, 12, 233.

[3] ^FRITZ (B.), Phys. ^Chem. Solids, 1962, 23, ^375.

[4] ^HAYES (W.), ^JONES (G. D.), ^ELLIOTT (R. J.) ^and

SENNETT (C. T.), ^Lattice Dynamics, Proc. Int.

Conf. held at Copenhagen, ^ed. by ^{R. F.} Wallis, Pergamon Press, ^New York, 1965, p. 475.

[5] ^STEKHANOV (A. I.) and ELIASHBERG (M. B.), ^Soviet Physics, ^Solid State, 1964, 5, ^{2185. Soviet} Physics,

Solid State, 1965, 6, ^2718.

[6] ^These experiments ^are being performed by D. W. FELD-

MAN and J. H. PARKER at the Westinghouse Research Laboratories.

[7] ^SENNETT (C. T.), ^{Some Problems in the} Theory ^of Solids, Thesis, ^{U. of} Oxford, unpublished, ^1964.

[8] ^MARADUDIN (A. A.), ^MONTROLL (E. W.) ^{and WEISS} (G. H.), Theory of Lattice Dynamics in the Har-

monic Approximation, ^Academic Press, ^New York,

1963.

[9] ^WILSON (E. B., Jr.), ^DECIUS (J. C.) ^{and CROSS} (P. C.),

Molecular Vibrations, McGraw-Hill, New York,

1955.

[10] ^BORN (M.) and HUANG (K.), Dynamical Theory ^of Crystal Lattices, Oxford, University Press, ^1954.

[11] ^GOURARY (B. S.), Phys. Rev., 1958, 112, 337.

[12] CHANDRASEKHAR (S.), Astrophys. J., 1944, 100, 176.

[13] ^BETHE (H. A.) and SALPETER (E. E.), Handbuch der

Physik, ^Vol. XXXV, Springer-Verlag, ^{Berlin. 1957.}

[14] ^ROSE (M. E.), Elementary Theory ^of Angular ^Momen- tum, ^John Wiley, ^{New York, 1957.}

[15] ^ROTENBERG (M.), ^BIVINS (R.), METROPOLIS (N.) ^and

WOOTEN (J. K., Jr.), ^The 3-j ^and 6-j Symbols,

The Technology Press, Cambridge, Mass., 1959.

[16] ^SCHWARTZ (C.), Phys. Rev., 1961, ^{123. 1700.}