Indian Journal of Pure & Applied Physics
Vol. 43, September 2005, pp. 660-663
Temperature dependence of lattice energy of fluorite type AB2 crystals, alkaline
earth oxides and heavy metal halides – Evaluation from sound velocity data
M Subrahmanyama, E Rajagopal & N Manohara Murthy
Department of Physics, Sri Kriashnadevaraya University, Anantapur 515 003
a
Department of Physics, V R College, Nellore 524 001
Received 29 November 2004; accepted 10 May 2005
Lattice energies of CaF2, SrF2, BaF2, CdF2, EuF2, MgO, SrCl, AgCl and TlBr at different temperatures have been
evaluated making use of single crystal elastic constant data and employing Kudriavtsev’s theory which relates the lattice
energy of the crystal, U, with mean sound velocity, um, in the crystal. The lattice energies of both MgO and SrO decrease
with increase in temperature and the variations are parabolic and similar. The lattice energies of AgCl and TlBr vary with
temperature parabolically up to 80 K and thereafter linearly up to 300 K. The results are explained in terms of the structure
of the crystals, mutual interaction of the ions and anharmonic effects associated with as a function of temperature.
Keywords: Lattice energy, Kudriavtsev’s theory, Elastic constants, Ultrasonic velocity
IPC Code: C08F2/56
1
Introduction
Lattice energy is an important parameter of ionic
crystals as it is useful in understanding the binding
forces and the potential function responsible for
binding the ions in crystals. The elastic constants and
their pressure derivatives are indicative of the short
range contributions to the lattice energy. Experimental
determination of lattice energy based on Born-Haber
cycle is not always possible for all types of ionic
crystals1 and one has to apply theoretical methods2-11
to evaluate lattice energy. Kudriavtsev’s theory12 has
been applied successfully to evaluate lattice energies
of a variety of ionic crystals based on sound velocity
data13-15. The success of the theory lies in the fact that
it could predict accurately the lattice energies of ionic
crystals even if there is enough covalency present like
in the case of divalent metal chalcogenides14. The
authors have evaluated the temperature dependence of
lattice energy of alkali halides16 and ammonium
chloride and ammonium bromide17 based on
Kudrievtsev’s theroy12 and useful information could
be obtained about the phase changes as a function of
temperature from these studies. As an extension of
these studies the authors have evaluated the lattice
energies of CaF2, SrF2, BaF2, CdF2, EuF2, MgO, SrO,
AgCl and TlBr as a function of temperature and the
results are reported in this paper.
2
Theoretical Details
According to Kudriavtsev’s theory12, the lattice
energy of an ionic crystal, U, is related to the mean
sound velocity, um, in the crystal by the relation:
U=
9 Mu m2
γ n1
… (1)
where M and γ represent molecular weight and ratio
of specific heats of the crystal respectively. For solids,
γ is taken as equal to unity. n1 is a constant which
depends on the lattice structure. Expressing M in kg
mol-1 and um in ms-1, one can get U in J mol-1.
Depending on the binding present in the crystal, that
is, pure ionic or partly ionic and partly covalent n1
may take on values like 2, 5, 7, 9 and 10. For pure
ionic crystals like alkali halides n1=5 gives lattice
energy data in agreement with experimental data13,14.
The crystals investigated in the present study are
cubic in nature. The mean sound velocity, um is given
by the relation:
⎡1 ⎛ 1
2 ⎞⎤
um = ⎢ ⎜⎜ 3 + 3 ⎟⎟⎥
⎢⎣ 3 ⎝ u l u t ⎠⎥⎦
−1 / 3
… (2)
where ul and ut represent the longitudinal and transverse sound wave velocities in the polycrystalline
SUBRAHMANYAM et al.: TEMPERATURE DEPENDENCE OF LATTICE ENERGY OF AB2 CRYSTALS
aggregates of the crystals respectively. ul and ut have
been evaluated from the single crystal elastic
constants data using Voigt-Reuss-Hill approximation18.
3
Results and Discussion
To evaluate the temperature dependence of lattice
energy, um as a function of temperature is required.
The values of um have been evaluated from single
crystal elastic constant data. For CaF2, SrF2, BaF2 and
EuF2, the single crystals elastic constant data from
literature19-23 have been utilized. For MgO and SrO,
single crystal elastic constant data of Marklund and
Mahmoud24 and Johnsten et al.25 respectively have
been used. For AgCl and TlBr, the single crystal
elastic constant data of Vallin26 and Vallin et al.27
respectively have been used.
The calculated values of lattice energies of CaF2,
SrF2, BaF2, CaF2, EuF2, MgO, SrO3 and AgCl, TlBr
as a function of temperature are shown graphically in
Figs 1-3 respectively.
As can be seen from Fig. 1, lattice energy varies
with temperature with a maximum around 40 K and a
minimum around 260 K for CaF2. For SrF2 the
variation of lattice energy with temperature is almost
linear with a clear maximum around 20 K. For BaF2,
lattice energy varies linearly with temperature. For
EuF2, the lattice energy varies quadratically from 80
Fig. 1⎯Variation of lattice energy U with temperature T for CaF2,
BaF2, SrF2, CdF2 and EuF2 O→ CaF2,● → SrF2, ∆→BaF2,
▲→CdF2 and □→EuF2
661
to 200 K and thereafter it varies linearly with the
temperature. For CdF2, the lattice energy decreases
smoothly with rise in temperature. Though Gerlich28
determined the temperature dependence of elastic
constants of BaF2 from 4.2 to 300 K, the same data
could not be used owing to the paucity of density data
for BaF2 with temperature. Hence, single crystal
elastic constant data of Wong and Schuele23 have
been used to estimate the temperature dependence of
lattice energy of BaF2. Even though two points on a
graph are insufficient to decide the nature of the
variation, yet the variation can be considered to be
linear in view of the fact that BaF2 differs much in its
lattice properties from the other two members of the
series namely CaF2 and SrF2.
Although, SrF2 occupies an intermediate position
between CaF2 and BaF2, the variation of lattice energy
with temperature is smaller in the case of SrF2 than
for both CaF2 and BaF2. Barium fluoride has the
interesting property of being a mechanically
isotropic crystal. The value of anisotropy factor
A=[2C44/(C11-C12)], where C11 and C12 are elastic
stiffness coefficients, has a value close to unity over
the temperature range 4.2-300 K for BaF2. The rigid
ion model, which is applicable to CaF2 is less
applicable to BaF2. The temperature variation of
elastic constants is larger for BaF2 than in the case of
CaF2 and SrF2 indicating that anharmonicity effects
are more pronounced in the case of BaF2.
The lattice energy of CdF2 varies smoothly with
temperature. Of all the cubic fluorides, CdF2 has the
smallest compressibility, the next to the smallest
anisotropy and the largest deviation for the Cauchy
condition. Since the Cauchy condition C12 = C44
results from the assumption of point charges, the
failure of the Cauchy condition indicates that CdF2
has the largest amount of covalent binding in the
normally ionic cubic fluorides. Cauchy condition is
also not satisfied in the case of EuF2.
As can be seen from Fig. 2, the lattice energies of
both MgO and SrO decrease with increase in
temperature, the nature of variation being quite
parabolic and similar. The temperature dependence of
lattice energy of these, two NaCl-structured alkaline
earth oxides can very well be compared with those of
alkali fluorides16. As the size of the alkali metal ion is
increasing, the variation of lattice energy with
temperature is smooth. Here also, as the size of the
metal ion and thus the ionic radius is increasing, the
lattice energy is decreasing in a regular manner. This
662
INDIAN J PURE & APPL PHYS, VOL 43, SEPTEMBER 2005
Fig. 2⎯Variation of lattice energy U with temperature T, for
MgO and SrO O→MgO, ●→ SrO
is a consequence of the decreasing electrostatic force
with increasing charge separation. Considering any of
the oxide-fluoride pairs such as MgO-NaF, CaO-KF
and SrO-RbF for the sake of comparison, the lattice
energy values are always larger for the oxides,
because of tighter electrostatic binding due to the
doubly charged ions.
As can be seen from Fig. 3, the lattice energy of
AgCl decreases with temperature parabolically up to
80 K and linearly from 80 to 300 K. Although the
noble metal halides such as AgCl, are of the same
crystal structure as the alkali halides, the outer
electrons of the cations are not in closed shells as in
the case of alkali halides. Hence, it is not surprising
that differences in elastic properties occur between the
two types of materials and cannot be explained with
simple Born-type models29. The infrared dispersion
frequency of AgCl is found to decrease almost
linearly from 4.2 K to room temperature30. But lattice
energy of AgCl is found to vary parabolically up to 80
K and thereafter linearly with temperature. It may be
mentioned here that Barber et al.31 studied the
temperature dependence of elastic constants of AgBr,
but because of lack of density data, the temperature
dependence of lattice energy could not be evaluated.
However, Barber et al.31 contended that the changes
in elastic constants of AgBr with temperature are
identical26 to those reported for AgCl. On this basis,
one can conclude that the variation of lattice energy of
AgBr with temperature will also be similar to that of
AgCl.
The lattice energy of TlBr decreases with
temperature, parabolically up to 80 K and linearly
from 80 to 300 K, behaviour similar to that of AgCl
Fig. 3⎯Variation of lattice energy U with temperature T for AgCl
and TlBr O→AgCl, ●→TlBr
(Fig. 3). But, the variation of the infrared dispersion
frequency with temperature for TlBr shows a different
trend. Jones et al.30 reported that the infrared
dispersion frequency shows a positive temperature
coefficient in the low temperature region for TlBr,
whereas for AgCl and the alkali halides, it shows a
negative temperature coefficient. This discrepancy
has been attributed to the different crystal structure to
TlBr as compared to AgCl and the alkali halides. The
structural difference between the CsCl-type structure
of TlBr and NaCl-type structure for AgCl may also be
understood by another feature. The anisotropy factor
A=[2C44/(C11–C12)] has a positive temperature
coefficient for NaCl type compounds and a negative
temperature coefficient for CsCl-type compounds.
The elastic constants of TlCl as a function of
temperature were represented by Gluyas et al32., but
due to non-availability of density data, the temperature dependence of lattice energy of TlCl could not be
studied.
4
Conclusions
The present study highlights the application of
Kudriavtsev’s theory12 in estimating the lattice
energies of fluorite type AB2 crystals, alkaline earth
oxides and heavy metal halides. In particular, evaluation of lattice energy as a function of temperature,
when experimental data are scanty, is the most
important feature of this type of study. Studies on the
temperature dependence of lattice energy are useful in
understanding the anharmonic nature of lattice vibrations and mutual interaction of the anions and cations
of the crystals.
SUBRAHMANYAM et al.: TEMPERATURE DEPENDENCE OF LATTICE ENERGY OF AB2 CRYSTALS
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