Electronic polarizabilities of ions in uniaxial crystals. II. Tetragonal system

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Electronic polarizabilities of ions in uniaxial crystals. II.

Tetragonal system

E. Uzan, V. Chandrasekharan

To cite this version:

E. Uzan, V. Chandrasekharan. Electronic polarizabilities of ions in uniaxial crystals. II. Tetragonal

system. Journal de Physique, 1973, 34 (8-9), pp.733-739. �10.1051/jphys:01973003408-9073300�. �jpa-

00207435�

ELECTRONIC POLARIZABILITIES OF IONS IN UNIAXIAL CRYSTALS.

II. TETRAGONAL SYSTEM

E. UZAN and V. CHANDRASEKHARAN

Laboratoire des Interactions Moléculaires et des Hautes

Pressions, CNRS, Bellevue,

⁹²¹⁹⁰

Meudon,

^France

(Reçu

^{le 17}

janvier 1973,

^{révisé le 19 mars}

1973)

Résumé. ²⁰¹⁴ Nous avons récemment introduit un modèle

théorique

pour le calcul des

pola-

risabilités

électroniques

des ions dans des cristaux uniaxes à

partir

des indices de réfractions ordinaire et extraordinaire. Dans le

présent article,

^{le modèle est}

appliqué

à des cristaux appar- tenant au

système tétragonal.

^En

particulier,

^les

polarisabilités électroniques

^de Mg2+,

Mn2+, Co2+,

Zn2+ et F- dans des fluorures et de Hg2+ ^{et I- dans}

HgI2,

^sont calculées. Les difficultés soulevées _par

l’application

^{de la méthode aux cas de}

TiO2,

GeO2 et SnO2 sont discutées.

Abstract. ²⁰¹⁴ We have

recently

introduced a theoretical model for the calculation of electronic

polarizabilities

of ions in uniaxial

crystals

from their

ordinary

^and

extraordinary

refractive indices.

In this _paper, this model has been

applied

^to

tetragonal crystals.

^In

particular,

the electronic

polarizabilities

^of Mg2+, Mn2+,

Co2+,

Zn2+ and F- in fluorides and Hg2+ and I- in

HgI2

^have

been obtained. The difficulties of this method in the case of

TiO2,

GeO2 and SnO2 ^are discussed.

Classification Physics ^Abstracts

18.10

1. Introduction. ^- In the

previous

paper

[1 ],

^which

will be

designated

^{as Part}

I,

^we deduced the electro- nic

polarizabilities of (Ca 2 +, 02-)

^and

(Mg", 02-), uniquely

^{from the}

extraordinary

^and

ordinary

indices of calcite and

magnesite, assuming

^that

the

polarizability

of each ion is characterized

by

a

single parameter

^even in the presence of the local

anisotropic

^field and that the static

approximation

for the

dipole

interaction is

good

^{at small q vectors}

involved in the

optical region.

^It amounts to

comple- tely ignoring

the distortion on the electronic clouds of ions and

attributing

^the

birefringence

^of

crystals

to the local

anisotropic

^{field at each ion}

produced by

the other ions

[2].

These values of

polarizabilities

were of the same order of

magnitude

^{as those derived}

by

Tessman et al.

(TKS) [3]

^{and more}

recently by

Pirenne and Kartheuser

(PK) [4]

for ions in cubic

crystals,

^{which are}

optically isotropic (Oh, Td

^and

Th classes).

^{But no}

acceptable

^{values were found}

for

Na+

and

02 -

in sodium nitrate. This

might

^be

due to, ^either

ignoring

^the

polarizability

of the central ion of the

negative

^radical

N03 ,

^or the distortion of the electronic clouds

resulting

from covalent effects.

So,

^{it was} considered worthwhile to take _up, in this paper,

simpler

^uniaxial

crystals

^with

only

^two

types

of ions such ^as fluorides and oxides which

crystallize

with a rutile structure

[5] belonging

^{to the}

tetragonal system.

The former are

practically ionic,

^{like the}

alkali halides. Hence the values of

polarizability

of F- in these

crystals

^{deduced from}

birefringence correspond

^to that based ^on the

additivity

^of

pola- rizability

^for

optically isotropic crystals [3], [4].

Furthermore,

^we

attempted

^{to take into account local}

electronic distortion

by introducing

^two

anisotropic parameters

^to characterize the

negative

^ion

polari- zability

^{as was first} considered for the _oxygen ion

by

Bolton et al.

[6].

^{But our method is much more}

straightforward

^and

simpler ;

furthermore ^we propose two

hypotheses

^{for this}

anisotropy.

This paper also includes a discussion ^on

HgI2

^which

belongs

^{to another}

space group of the

tetragonal system

and contains two

large

^ions.

The two fundamental

equations

of Part 1 for the dielectric constant e ^at

optical frequencies

^and

E(s),

the local electric

field,

^are

given

^{below :}

where E is the external field and

a(k)

^the

polarizability

of the ions k. As before we evaluate the lattice

dipole

sums

T’(ks) by

the Ewald method and use group

theory

^{to block}

diagonalise

the interaction matrix.

Eliminating

^E between eq.

(1)

^and

(2) permits

^{us to}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003408-9073300

734

obtain the solutions of

a(k)

for known values of _E,

at

optical frequencies.

’°

2. Lattice

geometry.

^{- The}

symmetry

^{of rutile struc-}

ture is described

by

the space group

D 14 -P 42/mnm.

The

positive

^{ions are located at}

(0, 0, 0)

^and

(t, t, t)

and the

negative

^{ions at}

(u,

^u,

0), ( -

u, - u,

0), (-L + u, 1 - u, -1) and (-L - u, -1 + u, u is nearly

0.306 in all

crystals

^{of this}

type.

^{The unit cell data}

[7]

and the

optical

^data

[8], [9]

^are

presented

in table I.

A group theoretical

analysis

of the 18 electric field

components

^at the 6 ions for zero wave

vector,

^clas- sifies them into

As the external field E in the

crystal

has the character

(A2u

⁺

Eu)

^{there are two}

equations

for the internal fields when E is

parallel

^{to the}

tetragonal

^{axis Z}

(corresponding

^{to the}

extraordinary ^index, ne)

^and

four when E is

perpendicular

^{to it}

(corresponding

^to

the

ordinary

^index

no).

^{These two sets of}

equations

connect the

polarizability

^{of the}

positive ^ion,

^the

polarizability

^{oc of the}

negative ion,

the lattice constants a, e and u and the refractive

indices ne

^{and no.}

By

^an

iterative

procedure, they

may solved

for fl

^{and a.}

Of the four

possible solutions,

^the

physically

accep- table ^one is chosen.

HgI2 belongs

^to the space group

D 4h 15 -P 42/nmc

with 6 ions _per unit cell located at

(0, 0, 0), (2, i, 2),

(0, t, M), (t, 0, - M), (0, t, 2

⁺

u)

^and

(t, 0, t - u).

The electric field

components

^{at these}

points

^{may be}

grouped

^{as a sum of}

Hence there ^are two

equations

associated

with ne

and three with _no, which

yield

three solutions

fors

and a and these will be discussed later on.

The six

independent

^tensors for the rutile structure and the seven for

HgI2

^are listed in tables II and III

respectively.

Tables IV and V

display

the interaction matrices of

symmetry A2. and Eu

^{for the two struc-}

tures. The term k in the tensor

T’(kk)

^is

simply

^related

to the Lorentz factor L

calculated _by

Mueller

[12]

and _many others

[13] by

^the

following

^{relation :}

since we defined our tensors such that

ET/i =

⁰

and

adopted

^the rationalized

system

of units. It affords an excellent check on our numerical _compu- tations.

3. Discussion of results. ^- 3.1 FLUORIDES. ^- The

birefringence

of fluorides is small

compared

^{to that}

of the carbonates dealt with in Part I. In

spite

^of

this,

we

get

^a

physically acceptable

^set of values for the

polarizabilities

^{of the two} ions in each fluoride consi-

TABLE 1

Crystal

^data

TABLE Il

Independent

^tensors

for

^{the rutile structure}

TABLE III

Independent

^tensors

for HgI2

dered which are

presented

ⁱⁿ table VI. This is

quite satisfactory

for it fulfils ^our

expectation

that the scalar

polarizability

^is

already

^a

good approximation

^for

fluorides.

Furthermore,

the values for F- ions in different

crystals

^{are close to the TKS}

[3]

^{and PK}

[4]

values derived from alkali halide data and to the theo- retical values

[14], [15], [16].

^This

signifies

^{that the}

birefringence

^is

mainly

^{due to the}

anisotropy

^{of the}

dipole

field and that the ionic distortion is

relatively

small in contrast to the oxides discussed later on.

Only

^{the value of}

a(F-)

ⁱⁿ

CoF2

^is

quite large

^and

that of

f3(C02+)

is low. We presume that this arises from errors in the refractive index data and most

probably

^{from the}

birefringence

^{which is too small.}

So we have recalculated the values

of fl

^{and a}

assuming

^{the same order of}

magnitude

^of

birefringence (An

⁼

0.038)

^as for other fluorides whose

positive

ions have 3 d shells and these values are also

given

in table VI.

Next we studied how the values calculated were

sensitive to the

changes

in the lattice

parameters.

Any

variation in a without

changing

the dimensionless

quantity e (= cla)

^{or u,}

simply

alters the values

of f3

and a

by

^{a factor}

^a3. Similarly,

^the ratio e has

only

a

slight

^{effect on} the values. But small variations in u

produced relatively important changes in fl

^{and a,}

as shown in

figure

1. The values of

a(F-)

^{for u =} u,,

(crystallographic value)

^are

quite

consistent with those in literature and are very ^{near to} the

asymptotic

maximum reached as u

increases,

unlike for

a(O’-)

in the oxides discussed

later

^on. But the values

of f3

for the

positive ions, having

^{3 d shell}

electrons,

decrease more with

increasing

^u.

(The

^{curves of a}

and f3 plotted

^for

CoF2

^{refer to} the literature values of the

indices,

^{but with our} assumed values :

FIG. 1. ^- Polarizabilities of ions in fluorides as a function of u. u, is the crystallographic ^{value of u.}

no =

1.514,

ne ⁼ 1.552

they

^{would be more in line}

with those of a

and fl

^for

ZnF2

^and

MnF2.)

^{It shows}

that the effect of

neglecting

electronic distortion in

our model

begins

^{to be}

appreciable

^{for the}

positive

ion even in these

highly

^ionic

crystals.

^{In the}

figure 1,

the value of

(2

^{oc +}

fi)

^{is also}

plotted

^{with an}

arbitrary origin

for the ordinate. This

quantity

varies much less with ^u and _may be

regarded

^{as a kind of molar}

polarizability

corrected for

anisotropy.

As in the case of calcite

[1 ],

^{we also calculated}

the variation

of fi

^{and oc with}

wavelength.

^{In the}

736

ture

tru

IV TAp 1 at

m

u

^à

Int

N H

z

l,

ri

W 1 BL

£ .2

u

¹

j§

TABLE VI

Ionic Polarizabilities

of fluorides

(*)

With assumed values of indices : _{no -}

1.514,

ne ⁼ 1.552.

case of

MgF2, fi

^{and oc show}

slight dispersion

^increas-

ing

^with

decreasing wavelength

^to about 1 800

Á.

But below this

wavelength,

^no measured values for the refractive indices exist in literature but

only precise

^{values of the}

birefringence

^measured

by

Chandrasekharan and

Damany [17], [18].

^{The bire-}

fringence

is anomalous in that it reaches ^a maximum at about 1 300

Á

_{and goes} to zero at 1 194

Á.

It would be of

great

^{interest to} understand this in terms of the

polarizability.

3.2 OXIDES. ^- The

birefringence

of the oxides is

quite large

and that of rutile is even

greater

^than that of the carbonates

[1 ].

The values

of ne

^for

Ge02

and

Sn02

^{are not} known but have been estimated from a fit to infrared

reflectivity [19]. However,

^it

is certain that their refractive indices and

probably

their

birefringence

^{is much} smaller than that of rutile.

This

already

indicates that the

polarizability

^of

^{02 -}

is not a constant in the three

crystals.

Also the volume v of the unit cell of rutile is

greater

^{than that of}

Ge02 although

^the

^Ge4+

^{ion is}

larger

^than

Ti4+,

^further

lending support

^to the variation of the size and the

polarizability

^{of the}

^{02 -}

^{ion. In contrast to the case}

of

fluorides,

^{however none} of the four

possible

solutions

for fl

^{and oc} in all three oxides were

physically acceptable.

^For

example,

the smallest

positive

^value

(36.3 Â3)

^of

P(Ti4 1)

^{was an order of}

magnitude higher

than that

quoted by

^TKS and this observation _agrees with the results of Bolton et al. In the case of

Ge4+

and

Sn4+, this

^value

of fl

^{was too low. So we tried to}

see the effect of

varying

^the

parameter

^{u on the values}

of fl

^{and a}

(Fig. 2).

^This

produces

^much

larger

^varia-

tions than in the case of fluorides and the values of a

at u ⁼ Uc ^are

quite

far from the

asymptotic

^maxima.

This indicates

clearly

that it is no

longer possible

^to

ignore

the covalent

effect,

^{but as in the case of fluorides}

a and

(2

^{a +}

fl) (with

^an

arbitrary origin

^{for the}

ordinate)

^{tend to}

asymptotic

maxima. The value of

a(02-)

^at this limit is

reasonably

within the range of values

given by

other authors

[3], [20].

^{Thus we}

are, in a way,

compensating

the effect of the covalent

bonding by assuming

^a

larger

^{value of u.}

However,

the values for the

positive

^ions

^Ge4+

^and

^Sn4+

^is

small. The variation of the

parameter

^{u to fit the}

pola- rizability

^{values is to be}

compared

^to the method of

fitting

the interatomic distance rp of diatomic molecules to the

anisotropy

of molecular

polarizability [1].

FIG. 2. ^- Polarizabilities of ions in oxides as a function of u.

rp ^is

always greater

^{than the}

spectroscopic value re just

^{as u has to be}

greater

^{than the}

crystallographic

value.

When the

temperature

^{of the}

crystal increases,

^all

the three

parameters

a, e and u could vary. But ^as

u is

approximately

^{the same for all}

crystals

^{with rutile}

structure under

study,

while a and e vary

considerably,

it is

probable

that the value of u for any

crystal

varies

hardly

^with

temperature.

But if it

does,

^the effect on the refractive indices and

especially

^on

the

birefringence

^{would be}

large.

Next, assuming

^{still a}

single

^scalar

parameter f3

for the

positive ion,

^{we tried an}

ellipsoidal

^model

for the

0?’

ion with two

parameters ail’ parallel

to the axis of revolution and _al,

perpendicular

^to

this axis. The

negative

ion in rutile structure

(02 -

^or

F-)

^{is almost at the center of an}

equilateral triangle

^of

positive

ions. So if the distortion of the electronic cloud is due to the nearest

neighbours (covalent effect)

^{then the axis of} revolution is

perpendicular

to this

triangle (Model

^{1 indicated}

by subscript (1»).

In the unit

cell,

^{two of the}

negative

^ions

(3

^and

4)

have this axis

along -

^{450 to} the X-axis and the other two at + 450 to it.

The Z-axis

corresponds

^to a-L. In this model

1,

we have

To solve these

equations,

^{it is better to refer to a}

system

of coordinates

X’,

^{Y’ rotated 45° with}

respect

738

to

X,

^Y.

Only

this model 1 has been considered

by

Bolton et al. In their convention the terms

parallel

and

perpendicular

^are interverted with

respect

^to

ours.

In the second model

(model 2),

^{we consider the}

distortion of the oxygen ion ^as

being produced by

the field of the other oxygen ions and in this case

(XII

refers to the Z-axis and _al ^to X and Y axes. Then we

have :

We chose for

f3

the value

given by

^{TKS or that}

roughly

calculated for

Ti4l [6].

^{The results for two}

models are

presented

^{in table VII. Our results for}

TABLE VII

Polarizabilities

of

^{ions in} oxides with a

scalar f3 for the positive

^{ion and an}

ellipsoidal

^model

for ^02-

(*)

^This value is taken from table IV of reference

[3]

whereas the value

given

^{in table VI is in error.}

(**)

This value is estimated

[6]

^{while the value}

given

^{in reference}

[3]

^{cannot be used since it is derived}

from the data of rutile

neglecting birefringence.

model 1 _agree with those of Bolton et al. in the case

of

Ti02 ;

^{it is seen} that the difference

(oc(l) - cf,l»)

is much

larger

^{than the} difference

Il - ^a12’)

ⁱⁿ

model 2. A much

larger anisotropy

^of

^{02 -}

^{is thus}

needed to account for the

birefringence

^{when the}

influence of the first

neighbours (positive ions)

^is

considered

important.

^{But if the second}

neighbours

and other

negative

^{ions are}

responsible

^for

distortion,

much smaller

anisotropy

^is sufncient. Thus the model 2 is

preferable.

^{In the case of}

Ge02

^{the values of}

a(2)(02 -)

^and

a (2)(02 -)

^are much smaller and in the case of

Sn02 they

^{are even} smaller. This

clearly

indicates that even with a model of

anisotropic pola- rizability,

^{there is} considerable variation of the

values f3

and a as well as

anisotropy

ⁱⁿ the different oxides.

One

possible

^{cause for this}

might

^{be that in all cases}

we have

ignored

^the

anisotropy

^{of the}

positive

^ions.

This is

certainly

untenable when their

polarizability

is

equal ^to,

^{or even}

greater than,

^{that of the}

negative

ion.

Secondly,

^{as mentioned}

earlier,

^{the size of the} unit cell

might

^{have an influence on the} values. This

type

^of

correlation,

^which is very much

smaller,

^has

already

been found for cubic

crystals by

^PK

[4].

In the case of

fluorides,

both models lead to a

small

anisotropy

^{for the F ion. This is}

quite

^under-

standable for their

birefringence

^{is small and}

they

^are

highly

^ionic.

3.3

HgI2.

^- Unlike the

positive

^uniaxial

crystals

discussed in the

foregoing section, HgI2

^exhibits

strong negative birefringence.

This arises

primarily

from its

layer

^{lattice structure with the ratio e}

greater

than 1 in contrast to the others with e less than 1

[12].

^{The two ions}

Hg2+

^{and I" are}

quite large

^and

the contribution of the inner shell electrons 5 d is

appreciable

^for

Hg 21 [14].

^{So we}

anticipated

^that

even the scalar model

might work,

^{and this is indeed}

borne out. Three solutions were obtained for

(f3, a) = (141, 45.6), (54.5, 75.8), (- 6.8, 119) (A3).

Evidently,

^{the last is}

physically

unrealistic and the first is

rejected

^{as the value}

of fl

^{is too}

large.

^{The second}

solution is in fair

agreement

with other

experimental

values

for, oc(I-) = (80.8-81.9) [3], [4].

^{The value}

calculated for

f3(Hg2+)

^of the free ion

[14]

^{is about}

twice as

large

^{as our value. This seems}

quite

^reaso-

nable for the calculations are

expected

^{to be}

good

^to

a factor of 1.5 to 2. Also the theoretical value is

generally larger

^{for a}

positive

^ion

(cf. f3 ^{(cal)/f3 (exp) -}

²

for

Cs+

and

Rb+ [14]).

In

figure 3,

^we

represent P(Hg2’)

^and

a(I-)

^as

functions of u. The

anisotropic

^{values of}

af,2)(I-)

and

oc (2) (1-)

^{at u =} Uc for model 2 are shown in the inset of this

figure.

^{It is to be noted that the}

anisotropy

is

quite

^{small as}

anticipated

^{in Part 1.}

FIG. 3. ^- Polarizabilities of Hg2+ ^{and I- in} HgI2 ^{as a function} of u.

4. Conclusion. ^-

Starting

^{with the}

simple dipole

model

giving

the refractive indices as functions of

polarizabilities

of the ions we have been able to solve

the inverse

problem

^to

yield

the values of the

pola-

rizabilities of the two ions in each

crystal.

^{Our values}

are

complementary

^{to those} derived from a least squares fit of

experimental

refraction data of alkali halides

using simple additivity.

For fluorides and the iodide

studied,

^the

simple

^scalar

polarizability

^model

suffices.

But,

^{in the case of}

oxides,

it is essential to take into account the distortion of the ions

by

^ten-

sorial models. Reasonable

anisotropic

values for the oxygen ion ^are obtained. It would be

interesting

^to

apply

^{this model to} the carbonates and sodium nitrate studied in Part I. Most

probably

^this

might

^{lead to}

acceptable

^values

(ail’ ^(XjJ)

^for the nitrate whereas all solutions were unrealistic in the scalar

approximation.

In the

simple

^classical

theory of piezo-optic

coefficients

of cubic

crystal,

Mueller had

already

introduced the idea of

anisotropic

deformation of the

polarizabilities.

These calculations _may be

pursued

^{in the case of}

birefringent crystals, particularly

^{for the} fluorides and molecular

crystals.

The

rapid

variation of the

polarizabilities

^{with u}

for a

given optical

dielectric constant E,

clearly implies

a

large change

^{of E if u varies}

during

^a lattice vibration.

So it would be

interesting

^to

calculate,

^{with our}

model,

the intensities of Raman lines.

Acknowledgments.

^- The authors wish to express their sincere thanks to Dr. Poulet for his keen interest in our work and for

drawing

^{our attention}

especially

to references

[4]

^and

[6].

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