A unified study of the specific heat, ionic conductivity and thermal expansion of SrCl2 incorporating two-level systems

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A unified study of the specific heat, ionic conductivity

and thermal expansion of SrCl2 incorporating two-level

systems

Mohua Makur, Sujata Ghosh

To cite this version:

A unified

_study

of the

_specific

_heat,

ionic

_conductivity

and

thermal

_expansion

of

_{SrCl2 incorporating}

two-level

_systems

Mohua Makur and

_Sujata

Ghosh

Physics Department, Jadavpur University,

Calcutta, 700032

(Reçu

le 12

janvier

1988, révisé le 6

_{juillet, accepté}

le 16

_septembre

₁₉₈₈₎

Résumé. 2014 Nous

présentons

un modèle tenant _compte des

_systèmes

à 2 niveaux

_qui

_reproduit

qualitativement

la variation

_thermique

de la chaleur

_spécifique,

la conductivité

_ionique

et la dilatation

_thermique

de

_SrCl2,

un chlorure

_{superionique.}

Les

_systèmes

à 2 niveaux sont identifies à des

_agrégats

de défauts,

qui

sont trouvés stables

d’après

des calculs

_{d’énergie statique largement}

utilisés pour

expliquer

de

_façon

satisfaisante les

_expériences

récentes de diffusion de neutrons.

Abstract. 2014 We _present a model

_{incorporating}

two-level _systems which

_{reproduces qualitatively}

the _temperature variation of the

specific

heat, ionic

conductivity

and thermal

expansion

of

SrCl2

a

_superionic

chloride. The two level _{systems may be identified with defect} clusters, which have been found stable from static energy calculations and are

_{widely employed}

to

_explain

successfully

recent neutron

_{scattering experiments.}

Classification

Physics

Abstracts

66.30Dn

_65.40Hg

- 61.70Bv

1. Introduction.

The

_superionic

chlorides

_(CaFz,

_{SrF2, BaF2,}

_SrCl2

and

_PbF2)

have the

_following

remarkable

properties

-

(i)

The

_specific

heat shows a

_{pronounced Schottky-type peak}

at a

_temperature

Tc

[1],

the ionic

_conductivity

increases

_rapidly

but

_continuously

in the

_region

of

7c

[2],

the bulk modulus vs. T curve

[3]

shows a

_change

in

_slope

near

_7c

and it has

_recently

been

_reported

that the thermal

_expansion

_[4]

also has a

_{pronounced peak}

near

_Tc.

All these _{features appear in the} same

_temperature

_{range and have been observed for}

ail

members of the fluorite group. It is

expected

that all these

_{peculiarities}

are manifestations of the same effect. But so far there has been no

_attempt

to

_develop

a unified

_microscopic

or even

phenomenological theory

to account for all these

_phenomena.

Earlier

attempts

through

« sublattice

_{melting »}

theories

_suggested

that

the entire anion sublattice breaks down at

_Tc,

but recent

_experimental

and theoretical work

_[5,

_6,

_7]

shows that the concentration of defects near

_T,

is

_likely

to be much lower than

_{(0.5-10 %),}

and it is

possible

to

_explain

the

_high

ionic

_conductivity

with such low defect concentrations.

In the

_present

_paper a model

_{incorporating}

two level

_systems

_(TLS)

is

_proposed

and

_applied

to

_SrC12.

This model

_reproduces

the

_specific

heat

_{anomaly quite}

well and with

certain

simplifying assumptions

the

_qualitative

features of the ionic

_{conductivity (Ic)}

vs. T curve and

also the thermal

_expansion

_{(a )}

vs. T and bulk modulus

_f3

(T)

vs. T curves can be

_explained.

432

It is

_possible

to

_envisage

a

_{physical picture}

of the TLS which is somewhat similar to the defect clusters

_suggested

to

_explain

neutron

_{scattering experiments}

_[5].

The next section

_gives

a

description

of the model and its

_application

to

_SrCl2

and in the last

section the

_{physical implications}

of the model are discussed. We also

_briefly

discuss results of the model

_applied

to

_CaF2,

_BaF2

and

_{PbF2 crystals.}

2. The model.

This model is based on the idea that when a Frenkel defect is created the

_surrounding

_{group of} ions relax into

_positions

such that a « defect-cluster » with a _{stable minimum energy}

configuration

is formed. Similar defect clusters have been

_proposed

to

_interpret

_{quasi-elastic}

neutron

_scattering

results

_[5],

_{and also from static energy considerations}

_[8].

Let us _{suppose that the solid consists of groups of}

_Nc

_{molecules each. The group of ions} can

be in either of the two states

_(i)

in

_{perfect crystal configuration}

_{without any}

defect,

we take the energy of this state as the zero-level and

_(ii)

with a

_specific

number

_(usually

one or

_two)

of Frenkel defects per

cluster,

and the

_neighbours

relaxed to

_{configurations}

such that the total energy E is minimised. Of course, several different

types

of clusters are

_possible,

but we assume for

_simplicity

that

_only

one

_type

of stable defect-cluster

_{configuration}

occurs here.

It seems reasonable to assume that the clusters of the same

_type

are not

_exactly

identical. The extent of relaxation of the

_neighbours

of the _{interstitial may vary within certain}

limits,

so

the energy E _{of the upper level has} a certain

_{spread varying}

from

_El

to

_E2.

NA

So one _{gram mole of the solid consists of}

_NT

= N A

_[NA

denotes the

_Avogadro

_number]

Nc

groups of ions which behave like two level

systems

(TLS).

At low

_temperatures

all TLS are in the zero _{energy state,} but as the

_temperature

rises Frenkel

_pairs

start to form and some of the TLS are « excited ».

We assume that the

_peculiar

features of the fluorites arise from the excitation of these TLS and

_proceed

to

_study

the effect of this excitation on different

_properties.

The contribution due

to TLS is of course to be added with the usual lattice contribution. To compare with

experimental

results we estimate the lattice contribution

_by

an

_{extrapolation}

from low

temperature

and add the TLS

_part.

The

_partition

function for

_{NT non-interacting}

TLS is

where

_{g (E)}

is the

_density

of states _{for the energy levels.}

Different

_properties

are calculated from

_(1).

(i)

SPECIFIC HEAT. -

Assuming

for

_simplicity

that

_g(E)

=

c, a _constant, the

_expression

for the constant volume

_specific

heat

_(Cy)

calculated from

₍₁₎

is

Fig.

1. -

Specific

heat data

_C,

for

_{SrCl2 :}

- derived from

experimental

data

_[1]

as

_explained

in the

text, --- calculated from

present model, -

extrapolated

from low

_temperature.

(2)

shows a

_{Schottky-type peak}

as observed

_{experimentally.}

Schrôter and

_Nôlting

_[1]

_give

the

_{Cp -}

T curve for

_SrCl2.

We have converted this to the

«

experimental » Cv -

T curve shown in

(Fig. 1), using

the

_expression

.

where

a v thermal

expansion

coefficient

V volume

K isothermal

_{compressibility.}

These data are taken from Moore

[9]

and Roberts and White

[4].

The best fit to the

experimental

curve is obtained from

(2)

with the

_following

set of

_parameters

Now this same set of

_parameters

are used to calculate the contribution of

TILTS -

to other

properties.

(ii)

IONIC CONDUCTIVITY. - We estimate the contribution of TLS to the ionic

434

(1 c)

in an indirect way to see whether the

_qualitative

features of the

_experimental

1 c -

T curve

_[2]

are

_reproduced.

The exact mechanism of ionic conduction is

_yet

unknown. Earlier theories which

_proposed

very

high

defect concentrations or « Sublattice

_{melting »}

at

_T,

have been

_discarded,

since other

_experiments

show that the defect concentration around

_Tc

is not more than - 10 %.

However from the simultaneous occurrence of anomalies in

_specific

heat

_{quasi-elastic}

neutron

diffraction

cross-section,

and

_{1 c}

in all

fluorites,

it is obvious that there is a connection between

these effects. It has been

_suggested

that cluster formation

_helps

ionic conduction in some

indirect way

[5, 10].

Here we assume that the defect clusters i.e. excited TLS

_provide

an _easy

_path

for the motion of the mobile anions.

Thus,

excited TLS are considered to be

_{conducting regions}

embedded in a matrix of

_ground

state TLS which are

_{non-conducting regions,}

and the situation simulates a

_{percolation problem.}

Though

we cannot

_justify

this

_assumption

from

_microscopic

considerations it is clear from other

_experiments

_[11]

that distortion in the lattice may

give

rise to enhance

_conductivity

locally.

At low

_temperatures

_{conducting regions}

are isolated and

_conductivity

of the

_crystal

is zero.

As

_temperature

rises

_{conducting regions}

increase and conduction starts as soon as a

continuous

_path

is

_established,

i.e. at the

_percolation

threshold. After that the

_conductivity

rises

_rapidly

and saturates when the whole solid becomes

_conducting.

The size of each TLS

_according

to our set of

_parameters

is -

_{90 aô}

₍₂

_ao = lattice

constant).

Assuming

for

_simplicity

that the TLS are

_spherical

_{their diameter is - 6 ao ;}

_{sufficiently larger}

than the lattice

_spacings

that a continuum model can be

used,

assuming

the TLS cannot see

the

_background

lattice.

With these very

simplified assumptions

we

_proceed

to calculate

_{1 c.}

The fraction of excited

TLS,

f eX ( T )

is

_{given by}

The volume fraction of

_{conducting spheres}

According

to the continuum model in

_{percolation theory}

_{[12, 13]}

the

_percolation

threshold is reached

at v

= vc =

0.25,

and

beyond

vc the

conductivity

rises as

The

_{I, - T}

curve calculated from these results is shown in

_{(Fig. 2)}

_together

with the

experimental

curve

_[2].

One additional

_parameter

_Io

is fitted to the

_conductivity

at

T = 1 100 K. The result is discussed in the last section.

It is to be noted that in the case of

_Ic

the normal lattice contribution is

negligible,

and the

whole

effect is ascribed to the TLS.

(iii)

THERMAL EXPANSION AND BULK MODULES. - The creation of Frenkel defects causes a

change

in volume of the

solid,

the Frenkel defect formation volume. So the excited TLS should have a volume

_slightly

_larger

than the defect-free TLS.

Introducing

an additional

parameter

which

_represents

the fractional increase in volume of the TLS on

excitation,

we

Fig.

2. - Ionic

conductivity

data

_Ic

for

_{SrCl2 :}

- from

experiment

[2],

--- calculated from

present

model, -.-.

extrapolated

from low _temperature.

Supposing

that the volume of the TLS in

_ground

state is _vo and on excitation increases to

vo + Av.

Volume of the solid at T is

N ex

is the number of excited TLS.

So,

The volume

_expansion

coefficient is then

fex

is

_{given by}

₍₄₎

and x =

L1v

is the increase in volume/unit volume of the TLS. The linear vo

expansion

coefficient a 1

= 3

et v . a 1 calculated in this way is

compared

with the results of

Moore

_[9].

_{A very}

reasonable value of the

_{parameter x}

_{gives good}

_agreement

with the

experimental

curve. The best’ fit is obtained for x = 1.25 %. An estimate of the defect

formation volume for

_{CaF2 [14]}

corroborates that x should be of the order - 1 %. The results in references

_[4]

and

_[9]

are however in

_disagreement

with the work of Shand et al.

[15]

which

shows no

_anomaly

in the thermal

_expansion.

The bulk modulus is obtained

_{by differentiating}

the volume

₍₇₎

with

_respect

to _the

press-ure P

436

If is

_neglected

we

_have,

where 03B20

represents

the bulk modulus of the defect-free

_crystal.

The additional contribution

to ,03B20

obtained,

though

in the

_right

direction is much too small.

9/ex

...

The

_neglect

of

ap

is however

_unjustified

as shown

_by

the

_strong

_pressure

_dependence

of ap

Ic

exhibited in the

_experiments

of Oberschmidt and Lazarus

_[16].

Inclusion of a

_parameter

Fig.

3. - Linear

expansion

data a

1 for SrCl2 :

- from

experiment

[9],

---

calculated from present

model, -.-.

extrapolated

from low temperature.

Fig.

4. - Bulk modulus data

j8 for

_SrC12:

from

_experiment

_[3],

.... calculated from

_{equation (11),}

a

_feX

_a In

_Ic

afex

which

_gives

the correct

_magnitude

of

(

3P

as in

_[16]

_reproduces

the

_experimental

ap

B ap )

13 (T)

curve _{very well} as shown in

_figure

4. Details of this calculation will be

_given

elsewhere. Is calculated

_using

_03B2(T)

obtained

_{by extrapolating}

the

_/3

_(T)

curve from low

_{temperatures.}

The

_experimental

results in

_figure

4 are from Cao Xuan An

_[3].

Discussion.

Let us now examine

_critically

the

_{physical implications}

of the

_present

model.

In the

_partition

function there are four

_parameters

_El,

_{Ez, NT}

and c which are determined

from the

_specific

heat data.

El

and

_E2

represent

the lower and upper limits to the _{energy of the TLS} on excitation

referred to the

_ground

state _energy. c is the

_density

of states in the interval

_El

to

E2

assumed to be constant.

_NT

is the number of TLS in one mole which

_gives

the size of each TLS.

We should see whether the

_parameters

used are of

_physically

reasonable

_magnitude

and

are of the same order of

_magnitude

as the Frenkel defect formation

enthalpy

(3.168

x

10-12

_erg

_s)

_{and cluster formation energy} as obtained from different

_experimental

and

theoretical work

_[5].

So we

_get

a

perfectly

reasonable value for the energy difference between

the defect-free and defect-cluster

_{configurations}

of the TLS.

The value of

_NT

we have used

_gives

_{N c}

=

NA/NT

= 35. So 35 molecules constitute each

TLS. This is also consistent with sizes

_{given by}

standard cluster models

_{[5, 8].}

c, the

density

of states comes out -

1026

erg-1

from the

_present

calculation. This order of

magnitude

is

_quite

reasonable. Calculations for

_amorphous

materials based on TLS

_give

similar results

_[17].

The

_ground

state of a TLS i.e. the

_perfect

_crystal

state has a

_unique

configuration.

But the upper

level,

i.e. the defect cluster appears to not to have such a definite

configuration. Slight

differences in the distance of the

_positions

of the relaxed nearest and

second nearest

_neighbours

to the interstitial

_(relaxation

around a _{vacancy is}

_usually

_neglected)

cause

_changes

in the energy of the cluster.

So the variation of the cluster energy within certain limits

corresponds

to a

_change

in

position

of the relaxed ions

_along

_specific

directions. Our results indicate that a _very

_large

number of cluster

_{configurations}

are

_possible

with very

_{closely spaced}

_{energy levels.}

In

_figure

2 it is seen that the calculated curve starts

_abruptly

from

_{Ic = 0,}

at the threshold where T = 980 K. Whereas the

experimental

_curve tails off

slowly

to _zero. The conduction in

the low

temperature

region

is

_probably

due to small amounts of

_impurities

in the

_sample.

Experiments

on

_{doped samples}

[18]

show

_clearly

that for low

_temperatures

conduction

_by

extraneous defects

_predominates

and for

_{increasing impurity}

concentrations a set of

_parallel

curves are obtained. These merge in the

high

_temperature

region

where conduction

by

intrinsic defects

_{predominates.}

So it is more

_meaningful

to construct a curve for the

hypothetical

pure

crystal by extrapolating

from the

high

temperature

region,

as shown in the

figure.

The threshold in this case is 973 K.

The

_qualitative

agreement

between the two curves is

_{quite striking considering}

the gross

approximations

that have been made

further,

the threshold

_depends

_{upon the}

_shape

of the units. Ballav et al.

_[12]

have shown that for

_spheroidal

units in

_general

the threshold

vc is

highest

for a

_spherical

unit.

_Distorting

the

_sphere

towards a needle or disc

_shape

lowers

438

experiment.

Of course we have avoided the central

_question

of the mechanism of conduction

through

defect clusters. This has to be

_investigated

from a

_{microscopic approach.}

_Further,

we

calculate conduction without

_considering

the number of Frenkel defects in each cluster and hence the true Frenkel defect concentration

_{nc (T ).}

_{Here ne}

is included in the

_parameter

I o

which is fitted from

_le

at T = 1 100 K.

Let us see whether the order

_{of nc}

from the

_present

_{model agrees with other estimates. For} this we have to

_assign

a number of defects to each excited

TLS,

this is

_usually

one or two

according

to standard cluster models so, for one defect per

cluster, ne at

_Tc

= 0.55 % and at

1 100 K where

_nearly

96 % of TLS are

_{excited nc -}

1 %. Twice these values will be obtained if the clusters have two defects each. This result is in

_agreement

with recent defect concentration results which show

_{that nc}

= 0.5-5 % in the

_temperature

_{range 1 000-1 100 K.}

We have

_completed

a

_preliminary

calculation of these

_properties

for

_CaF2,

_BaF2

and

PbF2

which have similar

_properties.

Similar fits for

_{C,, Ic, a}

and f3

are obtained and the

parameter

values show

_encouraging

trends.

The formation

_energies

increase in the order

_PbF2,

_{SrCl2, CaF2}

and the cluster sizes increase in order

_{PbF2, CaF2,}

_SrCl2

as is observed elsewhere

_[5].

Details of these calculation

will be

_reported

later.

In

_conclusion,

_{it may be said that}

_though

refinement of several

_aspects

is

_required.

The TLS model seems to be a

_possible

alternative

_approach

to the

_existing

_{theories which may lead} to a

better

_{understanding}

of the different

_{interesting properties}

of

_superionic

fluorites.

Acknowledgment.

The authors are

_grateful

to Prof. A. N. Basu and Prof. D.

_Roy

for

_illuminating

discussions and

_particularly

to Dr. T. K. Ballav for some very

helpful suggestions.

M. M. is

_grateful

to

UGC for

_grant

of a research

_fellowship.

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