**********************************************************
CHAPTER 2
LATTICE ENERGY OF IONIC SOLIDS
**********************************************************
61
2.1
INTRODUCTION
One of the fundamental problems in the
theory of solids
is the calculation of the binding energy of the crystal.
an
important role in understanding
forces
and
properties.
compound
their
effects
The
crystal
may be
on
the nature
thermal,
energy or
defined as
of
elastic
lattice
the energy required
group
deal
crystals,
for
to
which
the
with
in
this
theoretical
treatment
interaction
study.
of
diatomic
potential
calculation
several
of lattice
of
reviews
Sherman
6
It
stable
present.
on
the
ionic
The
has
been a
ionic
lattice
evaluation
of
theoretical
energies
of
ionic
subject of extensive
models
dealing
with
the
apart
from
the
crystals
experimental method based on the Bom-Haber cycle
hensive
The simplest
crystals using appropriate forms of
functions,
There are
ionic
to separate a
respect are
energy was first made by Bom and Lande1 ’ 2 .
lattice energies
anharmonic
energy of an
ions by infinite distances.
crystals
the interionic
and
formula weight of its
of
It plays
lattice
energy
3-5
calculations
.
are
Compre­
given
by
7
and Wad ding ton .
is
well
equilibrium
Hence,
known that it
in
ionic
is not possible to establish a
crystals
unless
other
forces
are
in order to account for the stability of ionic
crystals it is necessary to introduce forces between the ions that
are noncoulombic.
The history of the development of the theory
of lattice energies
is
largely
on account of the
development of
62
the ideas
about
expressions
for
involved these
Haber
4
these noncoulombic
the
evaluation
forces.
of
the
There
lattice
energy,
8 , Born and Lande 12
Born
’ ,
forces.
are several
which
Madelung 9
and
developed the theory of ionic crystals and devised formulae
which permit
crystal.
the
The
by considering
calculation of
the
lattice energy
Born equation was improved by
the repulsive term on
wave mechanics.
ionic
Born and Mayer
the basis
5
of the ideas of
A simpler approach to lattice energy calculations
was suggested by the Russian Chemist Kapustinskii
pression
of an
requires no
Madelung
of the crystal structure.
compressibility
constant and
Ladd and Lee
12
’
.
His ex­
hence no knowledge
used
parameters
coefficients, cubical expansion
like
coefficients
and
the zero-point energy terms for the evaluation of lattice energies
of alkali halides.
Kudriavtsev
13
developed
an expression relating
the mean sound velocity to the lattice energy of ionic
crystals.
The quantum mechanical calculations of the crystal binding of ionic
crystals were given by Hylleraas
14
15
and Landshoff .
The
details
of the various equations mentioned above are discussed in detail
in Section 2.2.
The
is
difficult
Lattice
direct
and
energies
experimental
has
can,
been
determination
carried
however,
be
out
only
related
to
of
lattice
energy
in
a
cases.
other
few
measurable
energy quantities by means of a thermochemical cycle due to Born
and Haber^’4 (the
by the
Born-Haber
Born-Haber
cycle).
cycle is an
The
experimental
lattice energy given
lattice
energy and
63
is not dependent upon the nature of the assumptions made about
the
bonding
theoretical
in
the
crystal.
approaches
for
A
brief review
the evaluation
of
the
different
of lattice energies
ionic crystals is presented in the following section.
of
64
2.2
THEORIES OF LATTICE ENERGIES
Bom's Theory of Lattice Energy
The
be
crystal,
or lattice energy
of an ionic compound
definedas the energy required to
of its ions
by
infinite distances.
may
separate a formula weight
Conversely,
it is also the
energy released when a formula weight of a substance is produced
by bringing together from infinite distances the necessary number
of positive and negative gaseous ions.
The
crystalline state
properties
of
ionic solids
mainly
depend upon the nature of the chemical bond
between the atoms
and
ionic
the
interaction
energies.
Whether
an
or
covalent
compound is formed in the chemical union of atoms is dependent,
not
only
required
upon
the
difference
between
for ionization, but also upon
the
amount
of
energy
the lattice energy
of the
salt produced.
Solid
ionic
molecules.
deposition
compounds
Instead,
of
a
the
are
crystal
regular array of
not
made
lattice
ions
in
is
up
of
built
which
individual
up
each
by
ion
surrounded by a definite number of ions of opposite charge.
the
is
For
example a crystal of sodium chloride may be regarded as a huge
cluster of ions in which six chloride ions are grouped spatially
around each sodium ion and six sodium ions are grouped around
each chloride ion.
The number of ions which may be grouped
around a central ion is determined by the radii of the respective
65
ions
involved.
In the vapour
state,
however,
are essentially molecular in composition.
ionic compounds
Such molecules,
as for
example, NaCl, are polar in their structure and can be considered
as
ion
pairs,
in
which the portion of a molecule occupied
by
the sodium atoms is positively charged and that part containing
the chlorine atom is negatively
a
gaseous
molecule
of
an
charged.
ionic
The pair of ions in
compound
are
probably
held
together at an equilibrium distance from each other as a result
of a
balance
between
the
forces of attraction
(between the two
oppositely charged ions) and the forces of repulsion which come
into play at short distances.
Consider a single molecule consisting of a pair of charges
Z^e and Z2e.
opposit.ely
The electrostatic potential energy between these two
charged ions may be expressed as
_
pp
ZlZ2e
electrostatic
„ 1
r
the Z values represent the integral
changes on the two ions,
e
is the charge on the electron; and r is the distance between the
centres of the two ions.
the
electrostatic
crystal,
forces
energy
However, equation 2.1 represents only
between
two
ions,
whereas,
within
a
the same two ions are subjected to additional coulombic
from
other
ions.
Furthermore
ionic
crystals
are
not
composed of simple ion pairs, but of ionic clusters in which each
cation is surrounded by a definite number of anions and each anion
is likewise encompassed by a fixed number of cations.
in
order
to
calculate
the
coulombic
energy
of any
Therefore,
single ion,
66
allowance
must
neighbouring
be
ions
made
of
for
both
the
number
signs.
and
Because
arrangement
of
of
electrostatic
interactions with surrounding ions the coulomb energy of a single
ion is greater than indicated in equation 2.1 by a factor typical
of
each
kind
Madelung
of
crystal
constant and
electrostatic
structure.
This
designated as A,
contributions
of
factor,
known
as
is the summation of the
neighbouring
ions
the type of lattice in which the ion occurs.
and
depends
on
As a result of this
summation, the coulombic energy of a single ion may be indicated
as -A.e/r.
The electrostatic potential energy per ion pair is
A Z
pp
—
r electrostatic
~
_
Z
e2
.„.^
o
r
9
***
where A is the Madelung constant and the negative sign indicates
that
energy
is
released
when
the
two
ions
station
themselves
within the crystal.
The
care
of
Madelung
additional
constant
(A)
electrostatic
is a
forces
correction factor to take
exerted
by
neighbouring
ions upon an ion pair and is entirely dependent upon the geometry
of
the
crystal.
The
following
are
the
values
constants for a few crystal lattices.
for
sodium chloride lattices,
A
= 1.747558
for
cesium chloride lattices,
A
= 1.762670
for
sphalerite, ZnS lattices,
A
= 1.638060
of
Madelung
67
In addition to the coulombic
forces which
operate between
ions there are repulsive forces which arise to prevent ions from
approaching
each other
a result of a general
atoms and
too closely.
These forces originate as
mutual repulsion of the electron clouds of
ions whenever these clouds penetrate each other and
the electrons do not form a bond
with a common electron pair.
As a result of the repulsive forces opposing those of coulombic
attraction,
oppositely
charged ions come to equilibria at definite
distances from each other.
when
the
electron
decrease rapidly
shells
with
The forces of repulsion become strong
of
two
ions are in
distance.
The
factor
close contact
of
repulsion
difficult to calculate and is usually represented by Born03
where
B is a constant and n is
the power of
but
is
as B/rn ,
the distance to
which r (the distance between ions) must be raised to give the
correct repulsive force.
The total lattice energy per ion pair within a crystal of
the MX type may be indicated as
U MX
A Z1 Z2 6
.
..................................
+
Be2
2.3
n
and the total energy per formula weight of MX is
N A Z1 Z2 6
U
+
N. Be2
n
wtere Uo is the lattice energy and N is Avagadro's number.
2.4
68
In those crystal types where r and n can be determined
experimentally,
coefficient
involved,
B,
a
it
is
and
more
if
possible
we
omit
to
some
calculate
of
the
general Born equation for
the
repulsive
mathematical
steps
lattice energy
may
...
2.5
be stated as:
-N A Z
U
=
o
Z
e2
------------—-----r
(1 - -)
n
Bom-Mayer Equation
Born and Mayerg improved
equation
(2.5)
by
considering
the repulsive term on the basis of the ideas of wave mechanics,
which
with
indicate
distance
that
from
the electron
the
nucleus.
repulsive term in the form be
to
the
B
in equation
density falls off exponentially
—r/p
(2.4),
Accordingly,
, where
and p
b is a
they
wrote
the
constant similar
is a constant
for a
given
species; thus the expression for lattice energy becomes
U
■N A Z. Z„ e2
,
--------- ----------- ♦ b e‘r/p
2.6
Equation (2.6) gives U = 0 and r = » , i.e., the potential
energy of the molecule
each other.
is zero when the ions are separated from
As the ions are brought close to each other the
attraction and repulsion potentials increase in the
schematically by the dotted curves in Fig. 2.1.
manner shown
69
The net potential
energy
and attraction potentials,
U
is the sum
and is the solid curve in the figure.
The latter curve shows a minimum at r = r
correspond
to
the molecule.
of the repulsion
the equilibrium
and U = U ,
which
o
separation and binding energy of
o
For this minimum we must have
and hence differentiation of equation 2.6 with respect to r gives
-r0/P
Z1 Z2 6
b e
or
P Z1 Z2 e
-r /P
b e o
2.7
On substitution of Eq. 2.7 into Eq. 2.6, we obtain
2
--------1—=-----ro
-N Z
U
The values
of
Z
e
0
(1 - ±-)
2.8
ro
P (calculated from compressibility measurements)
are found to be the same for most crystals, being equal to 0.345 X.
If the number of ions in the stoichiometric formula of the
crystalline substance is
then the number in a gram mole is N v.
The Born-Mayer equation 2.8 may
as
then be conveniently rewritten
Repulsion (+3
<=
A ttraction (-3
Distance of Separation - r
Fig. 2.1.
Potential energy diagram for ionic bonding.
70
N e2 Z
,NV
A
U
)
(-
1
2
,
2.9
(NV/2)
and writing the structural coefficient for one ion, a =
this becomes
U
av
N e zi z2
-(-y) (---------- (1 - p
=
Here a
p
2.10
has different values for different structure types
and is independent of the Madelung factor A.
Kapustinskii Equation
10 11
Kapustinskii ’
suggested
energy calculations.
structure
a
simpler
approach
to
lattice
He found that on passing from one type of
to another,
the change in a
was proportional
to the
change in the interatomic distance r for the differing coordination
numbers.
Each crystal may then be supposed to transform into
a sodium chloride lattice (coordination number 6)
of lattice
energy
(i.e.,
stability),
if
without change
the coefficients a and
v
are simultaneously modified so as to have the values corresponding
to a sodium chloride lattice; thus the following substitutions are
made:
a.
r = ra + V the
coordination
most ions.
sum
number
6;
of
the
these
ionic
values
radii
are
(in
now
A)
known
for
for
71
b.
a
=
—z------
=
(f, >
1.74756, the value of the Madelung factor
for the NaCl lattice
c.
p
d.
N e
o
0.345 A .
=
2
= 332.0 K cal, the product of fundamental constants.
Thus Eq. 2.10 becomes
-290.1
Z± Z2v
[1
U
(r
+
a
0.345
-
r )
c
(r
2.11
+ r }
a
c
where ra and r£ are respectively the radii of the anion and cation,
v is
the
total number of ions in
the stoichiometric unit
which
specifies one mole of the substance, e.g. 2 for NaCl, 3 for CaF2,
5 for A1203Kapustinskii' s
quantities.
effect
ions
of
in
The
equation is
attractive
the
(negative)
the
Coulombic,
point-charge
the
lattice.
The
non-Coulombic
interaction
algebraic
term
two
net
interactions among all
the
term
the
represents
of
the
repulsive
between
sum
arises
electron
from
the
clouds
on
neighbouring ions.
The usefulness of the Kapustinskii's equation lies in the
fact that
crystal,
it
even
is possible to calculate the lattice energy of any
if
its
structure
is
unknown,
by
assuming
it
to
possess a sodium chloride lattice and using the appropriate values
of the ionic radii.
72
Bom-Haber Cycle
The lattice energy of an ionic compound is the energy set
free when the crystal lattice is formed from free gaseous positive
and
negative ions.
The process
of
gaseous ions
condensing
to
form ionic crystals is of rare occurrance and is therefore largely
of theoretical interest only.
The direct determination of the lattice energy of an ionic
crystal has been carried out for only a few compounds.
majority
of
cases
it is not
directly;
however a
possible
to
measure
In the
this
cyclic process has been devised
energy
by Born
3
4
and Haber which relates the crystal energy to other thermochemical
quantities.
Where
the energy of formation of a crystal from
component elements is known,
into
the
energies
of a
its
it is possible to split this value
number
of
processes,
which
may
be
postulated as constituting the intermediate steps in the formation
of the crystal.
By
means
of
a
circuitous
procedure
it
is
possible
to
calculate algebraically the theoretical lattice energy of a crystal.
For
example,
consider
the
formation
of
sodium
chloride
elementary solid sodium and elementary gaseous chlorine.
from
It may
be assumed that the sodium metal is evaporated and the diatomic
chlorine is dissociated;
the electrons
then the alkali atoms are
so obtained
are
transferred
to
the
ionized,
and
halogen atoms
so that positive sodium ions and negative chloride ions are left
in the gaseous phase.
The various steps involved in this cycle
NaCl(S)
> Na (g) + Cl (g)
(i)
■KJ
(ii)
<iv)
Na(S) + ^Cl2<g)4
T
(ill)
+U
+ I—E
Na(g) + Cl(g)
+ igD
c
2.2.
Measurement
of
cyclic process,
the
lattice
(+) and
absorbed respectively.
(-)
energy
using
the
Born-Haber
indicate energy released and
73
are
i.
decomposition of the solid into the constituent ions,
ii.
the formation of neutral atoms from ions,
iii.
the formation
atoms, and
iv.
the formation of
standard states.
of
standard
the
states
solid
This is illustrated in Fig.
crystal.
The
algebraic
sum
(i) to (iv) is equal to zero.
which is
of
from
of
elements
its
elements
in
the
their
2.2 for the case of the NaCl
the energies
in
The energy UQ for
the cohesive energy
from
the
the processes
process
of the ionic lattice,
(i),
is therefore
calculated from the known energies for the processess (ii),
(iii)
and (iv).
In process (ii) the energies involved are the ionization
energy,
of the sodium atom and
I
the chlorine atom.
metal
the electron affinity,
E,
of
In (iii) sodium atoms combine to form sodium
while chlorine atoms combine
to
form
chlorine
so that the energies involved are the cohesive energy U
(Cl2)
gas
of
the
sodium metal and half the dissociation energy D of the Cl2 molecule.
Finally the heat of formation, Q is involved in process (iv) when
NaCl is formed.
U
=
o
The lattice energy Uo is therefore given by
Q
w
+ U 4d+I - E
c
2
...
2.12
For NaCl in units of kilocalories per mole, Q = 98.6, U
=
c
26.0, D = 57.9, I = 83.3 and so the lattice energy, UQ = 189.2 K cal/
mole.
This value agrees with that obtained by direct measurement
74
to well within the limits of the experimental errors.
Lattice
(Equation
energies
2.12)
are
derived
experimental
from
the
Born-Haber
values
which
are
cycle
independent
of any assumption about the cohesive forces in the crystal.
the
otherhand,
does
assume
the
the
derivation
cohesive
of equations
forces
to
be
2.5,
2.8,
completely
On
and
2.11
electrostatic
in nature.
Kudriavtsev's Relation
Based on Kudriavtsev's theory
13
an
expression
which
relates the mean sound velocity U
with the lattice energy U
m
o
22
23
crystal can be derived ’
and is as follows:
(
UD)
2.13
MU
rn
where
of a
Y , M and U
molecular
represent
the ratio
of
specific heats,
m
weight of the crystal and mean sound velocity in the
system respectively,
n^ is
a
constant
which
lattice structure and the values of n^ that
5, 7, 9 and 10.
The mean sound velocity,
can
depends
be
can
upon
the
used
are
3,
be
obtained
from the experimental determination of the velocity of longitudinal
and shear wave propagation in polycrystalline solids or from the
single crystal elastic constants data making use of the Voigt-Reuss24
Hill approximation .
75
Quantum Mechanical Prediction of Lattice Energies
14
Hylleraas
applied a general quantum mechanical treatment
to the calculation of lattice energies.
functions of the hydrogenic type
the
entire
computation
could
be
He used one-electron wave
with nuclear screening so that
performed
analytically.
The
approximate wave functions used by him were
* =
exp t-(Z - |g) |-]
h
with Z = 1 for hydrogen and 3 for lithium, a^ is
o
distance (0.58 A).
the
...
2.14
Bohr
unit
76
Lattice energy calculations based on interionic potentials
A
have
large
been
number
suggested
of
by
interaction
different
potential
workers
energy
from
functions
time
to
time.
These potentials have been used to compute the various properties
of ionic crystals.
These
new
approaches differ from one another
in the form of the repulsion energy considered.
A few of
such
studies are outlined here.
The
lattice
energies
of
all
calculated by Prakash and Behari
20
the
alkali
using
halides
have
been
the following expression:
2
<b
y
=
The
first
well-known
energy,
2nd
A log
on
the RHS
energy;
term
(1 + ~) - ■—
9
6
r
r
e
the
is the
in
the
last
+
U
...
above equation
term
logarithmic
is
the
form for
2.15
is
the
zero-point
the
overlap
A, B and C are parameters.
Dass et al
the
term
Madelung
the
repulsion.
+
r
potential
114
have
energy
given
function
the
which
following
takes
complete
into account
form
of
all
the
forces of interaction between the ions at a separation distance.
2
<f>(r)
2
-—— + A exp (—}
r
P
_2
Z
e
2 ,
2 r'
v
+ <22)
C_
r
D_
6
r
8
n „2 2
2 2 e aia2
r
7
Y
2.16
77
where
Z
and
is
the valency
o<2 are
the
the
polarizability
A, C, D are constants.
represent
of
ions,
e
values
the electronic charge,
of
the
two
ions
and
In the above equation the first two terms
the electrostatic and
overlap
interactions respectively,
while the third and fourth terms are due to van der Waals forces.
To
account for the polarization forces, Rittner
fifth and sixth terms.
113
introduced
the
The covalent energy has been represented
by a constant Y.
Using the modified Bom model, Gupta and Sharma
92
has pro­
posed the expression for lattice energy taking into account several
interactions.
iJj
where
(r)
2.17
e
a is the Madelung constant, e the electronic charge, r the
equilibrium interionic distance,
dipole-dipole
interaction
interaction parameter,
e
of the repulsive term.
Gupta and Sharma
92
X the repulsive parameter,
parameter,
D
the
C the
dipole-quadrupole
the zero-point energy and n the index
The above equation has
been used
by
to evaluate lattice energies of several heavier
halides.
A
been
generalised
logarithmic
form
suggested by Thakur and Pandey
the lattice energies.
75
of
’
repulsion
115-117
to
energy
has
calculate
78
. .
r4
B ,
A l08 <5 * -H>
r
U
2.18
where A and B are potential parameters and n = 1,2,4...
A new type of repulsive term in the interaction potential
has been suggested by Usha Puri4^,
which
is
found
to
yield
better results than those obtained by logarithmic form (Eq. 2.18}
for the repulsive term of interaction potential.
°
21 Z2 °2
cj>(r)
Z b -Ar
----- e
r
r
2.19
f(r) is the potential energy per unit cell,
a is Madelung constant,
Z^e and Z^e are the charges on the ions, r is the ionic separation.
Z is the number of nearest neighbours of any ion.
The constants
^ and
as
b
in
the
exponential
term
are
known
potential
parameters.
Thakur and Sinha44 have proposed a modified form of the
Kapustinskii-Yatsimirskii equation
08
using
logarithmic
form
of
lattice potential energy and is given by
1201.61 2 n Z% Z2
U
[1 - log1(J (1 +
=
(r
where r
e
+' r }
a
-)]
(r
+ r }'
c
a
+
10.5 2
n
Z2
,.
2.20
and r are the cationic and ionic radii; p is a constant
c
a
equal to 2^ , In is the total number of ions present in the
79
molecular formula of the compound;
The
above
equation
has
been
and 2^ are the ionic charges.
applied
to
calculate
the
lattice
energies of alkali halides and alkaline earth chalcogenides.
The
the
use
Born-Mayer
of
the
by Thakur et al.
calculations,
equation
compressibility
78
in
its
term
generalised
has
been
form
without
further modified
Their empirical expression for lattice energy
based upon the logarithmic form of potential energy
is given by
N A e2 Z
U
Z
=--------- v
1o
11 " 1O810 (a + P r0 )]
where a and p an parameters.
•*'
2,21
For alkali halides, a = 1.015 and
p = 1.60416 X2 .
Taking into consideration the repulsive interaction between
nearest neighbours
as
et alt118 has obtained
well as next nearest neighbours,
Shanker
the following expression for the evaluation
of lattice energies of alkali and silver halides.
w .
VV M
r
<♦«*♦_>
r
...
2.22
where the first term on the right of above equation represents
the Madelung energy (aM the Madelung constant, Ze the ionic charge,
r
the
the
interionic
separation).
van der Waals
dipole-dipole
The second and
and
third
dipole-quadrupole
terms are
energies.
80
The fourth and the last terms represent the short range overlap
repulsive interactions operative between nearest neighbours (unlike
ions)
and also between next nearest neighbours
(like ions).
M
and M' are the numbers of first and second neighbours.
A brief review of work on the lattice energy calculations
of ionic crystals is presented in the following section.
81
2.3
BRIEF
REVIEW
OF
EARLIER
WORK
ON
THE
EVALUATION
OF
LATTICE ENERGIES OF IONIC CRYSTALS
Studies on the theory of lattice energies of ionic crystals
have been undertaken as early as in 1918.
large
number
119
crystals
.
of
attempts
to
evaluate
There have been a
lattice
energies
of
ionic
The quantitative theory of lattice energy has been
8
developed mainly through the pioneering works of Born ,
Born and
12
9
4
Lande 1 , Madelung and Haber .
The various lattice energy
calculations were reviewed comprehensively
by
Sherman6
and
7
Waddmgton .
Of
all
the ionic crystals,
alkali halides are relatively
easy to subject to theoretical treatment since they have simple
crystal structures and are bound
forces
between
calculated
approaches.
the
the
ions.
lattice
The
A
energies
principal
by the well-understood Coulomb
large
of
number
alkali
calculations
of
halides
by
the
theory have been made by Mayer and Helmholtz^2 6 ,
deBoer
27
, Huggins
28
29
and by Seitz .
workers
with
different
classical
Verwey
12
Ladd and Lee
have
ionic
and
have used a
method to compute lattice energies eliminating the need for 'basic
radii'.
Kapustinskii10 ’ 11 has
also
but his values are rather low.
calculated
lattice
The theory of Born and Mayer
15
has been extended by the work of Landshoff
using
of quantum
mechanics.
energies,
In addition
to
the
the
methods
correction terms of
Born and Mayer, Landshoff 15 has incorporated additional interactions
82
related to the superposition of the electron clouds, the attraction
between electrons and nuclei and the mutual repulsion of electrons.
16
Lowdin
has calculated the lattice energies of LiCl, NaCl, KC1
and NaF.
Lattice energies
by Cubbiccotti
17
, Saxena and Kachhava
and Prakash and Behari
potential
of alkali halides were also estimated
to estimate
20
.
Pande
21
18
, Pandey
has
IQ
used
lattice energies.
and
the
Bom-Mayer
An exponential
repulsive energy has been suggested by Dixit and Sharma
Pandey
23
24
Sharma
has
.
used
the
Prakash
Rydberg
form
22
of
and
potential
function.
A logarithmic form of repulsive potential has been used by Misra
et al
25
.
A
modified electrongas
26
by Cohen and Gordon .
basis
of
the
effective
polarization.
has
27
Calais
has reviewed
28
and Sho
calculations. Kai Wen
treatment
havemade
nuclear
40
charge
and
et al
37
,
50
Singh ,
Nirwal
Andzelm
47
,
Singh
and
54
41
of
48
,
Dedkov
Agrawal
have
51
Saxena
, Sinha and Thakur
Singh and Nirwal45 ,
Lister
Shanker
and
rate
the
cation
Many workers ^ have
38
Piela ,
and
, Zhadonov and Polyakov
Shankar
on
the lattice energy calculations of alkali halides.
Thakur and Sinha44 ,
and
the lattice energy
A semi-empirical free model with atomic constants
also attempted
et al
employed
calculations
as parameters has been used by Shorezyk^.
Puri
been
,
and
Temrokov
the
,
49
Garg
43
, Usha Rani
Shankar et al46 ,
52
Islam ,
obtained
42
39
,
Shanker
Nirwal
Singh
and
53
et al. and
...
lattice energies using
different approaches.
Dissociation
energy
studies
have
been
used
for
the
55
estimation of lattice energies of alkali halides by Jha and Thakur ,
83
Yadav
56
57
and Kaur et al .
Singh et al
58
has obtained lattice
energies from a relation between the lattice energy and coefficient
of thermal expansion.
Lattice energies have also been obtained
58
59
60
by Yamashita and Asano , Shukal et al » , Kaur et al , Rehman
62
61
and Shams
and Reddy et al .
The
lattice
energies
tellurides
of alkaline
have
been
earth
calculated
oxides,
by
sulphides,
selenides
and
a
workers.
6
Sherman has used the simple Born formula for the eval­
uation of lattice energies of alkaline earth oxides.
and Maltbie
energies,
63
number
of
Later Mayer
have used the Born-Mayer expression for the lattice
calculating
the
London
polarizabilities of the free ions.
dispersion
energies
deBoer and Verwey
64
from
has
the
recal­
culated the lattice energies of the alkaline earth oxides, because
the Mayer and Maltbie interatomic distances differed significantly
from those obtained from the X-ray data.
Other calculations on
65
66
these crystals have been made by Van Arkel and deBoer , Fowler
68
and by Kapustinskii. Kapustinskii and Yatsimirskii , and Huggins
69
and Sakamoto
have recalculated
the
lattice energies of all the
alkaline-earth chalcogenide crystals.
Saxena et al
70
, Kumari Kha et al
71
72
, Pandey and Pant
obtained lattice energies of some alkaline earth oxides.
have
Son and
73
Bartels
have obtained lattice energies of MgO, CaO and SrO using
a simple Born model.
Cantor
74
75
, Thakur
, Upadhyaya and Singh
76
,
Thakur and Sinha 44 , Thakur777 , Thakur et al78 , Singh and Shanker47
and Mackrodt and Steward 79 have also estimated lattice energies of
84
oxides.
More
recently,
Islam
52
,
80
et al .
and
Shanker
Singh
81
et al . have also calculated the lattice energies of alkaline earth
oxides.
The
lattice energies
calculated by Hylleraas
Lundquist
82
calculated
a
quantum
hydrides have
mechanical
been
approach.
the same approach.
energy of
Bom-Lande expression.
lattice
also
using
metal
lattice energy of LiH using
83
Bichowskii and Rossini derived the lattice
the
has
14
of alkali
LiH using
energies
KazarnovskiiP^ has
are
used
available
from
these values
to
in the Born repulsion coefficient.
lattice energies of all
the
Accurate values of
thermochemical
calculate
Waddington
7
.
data.
the exponent
has
evaluated
the alkali metal hydrides using a simple
Bom-Mayer expression, ignoring van der Waals terms.
The lattice energies of alkali metal chalcide crystals viz.
L^O, Na^O, K^O, Rb20 and CS2O have been calculated by Morris
85
,
who considers these salts to be more ionic than the alkaline earth
oxides.
Waddington
7
has tabulated calculated values from the work
of Sherman^ and West^®.
The
(other
lattice
energies
than alkali metals)
of
viz.
the
halides
of
the argentous,
univalent
the thallous and
the cuprous halides were calculated by Sherman .
The
energies in these solids were recalculated by Mayer
88
Mayer and Levy .
Later Ladd and Lee
lattice energies of the silver and
89
have
metals
87
lattice
and
recalculated
by
the
thallous halides by a method
avoiding the use of the Huggins basic radii,
which are difficult
85
to fix for these salts.
The lattice energies of these heavy metal
halide crystals have also been evaluated by Saxena et al
an
Q1
Q?
and Trivedi , Murthy and Murti , Gupta and Sharma ,
^ , 94,95 „ , . . t ,96,97
, ,78
Thakur
, Bakshi et al
, Thakur et al
Shanker et al
qq
, Shanker and Agrawal
1 [in
70
,
Gohel
Qq
Sharma
, .
98
, Jam and Shanker
and Singh and Khare
1 n-i
.
The lattice energies of the divalent metal halides have been
evaluated by Sherman6 using the Born-Lande equation.
Morris'1102
has
the
extended
Lande
the
equation and
McClure and Holmes
data
theoretical calculation,
for
has
103
recalculated
have
also
lattice
of
the
data.
thermochemical
transition metal halides.
the
Born-
The
divalent transition metal
104
105
halides have been further studied by Orgel
, Griffith and Orgel
and Hush and Pryce'*'^.
energies
thermochemical
recalculated
lattice energies of the
thermochemical
the
again using
86
2.4
AIM AND SCOPE OF THE PRESENT WORK
The
study
of
lattice
energies
of
ionic
crystals play
an
important role in understanding a variety of phenomena of physical
and
chemical
interest.
Studies
on
the
lattice
energy of
ionic
crystals have been reviewed in the preceeding section in detail.
The review work presented in this chapter reveals that a large
number
of
suggested
workers
interaction
potential
functions
have
been
within the frame work of the Born model by different
from
time
to
time
properties of the crystals.
phonon
energy
frequencies
of
in
order
While workers
ionic
crystals
to
study
the
various
43 120 121
’
*
concerned with
have used
the exponential
form of Bom-Mayer equation for the repulsive part of the lattice
energy,
logarithmic forms for the same have been used by some
workers94 ’ 115-117 interested
of
crystals
in
the
such
as
atomization
Previous
attempts
to
calculate
of
interionic
such
interaction
study
of chemical properties
energy,
the
electron
lattice
potentials,
energy
gave
affinity
etc.
on
basis
values
the
which
in
several cases fall off the accurate data obtained from experimental
and
quantum
instances
in
mechanical
which
calculations.
There
these models are found to
are
also
other
be inadequate in
predicting correctly the elastic and dielectric properties of ionic
crystals.
resulting
and
The Born-Mayer theory has later
improvement
experimental
in
lattice
the
agreement
energies
for
been refined,
between
ionic
the
with
calculated
crystals.
The
87
refinements
considered
or
corrections
less)
the
small
to
(a few kilo calories per mole
the
lattice
energy
arising
from
van der Waals interactions and zero-point vibrational energy.
The detailed calculation of the lattice energy of an ionic
crystal
according
to
interionic
potentials
requires
knowledge
of
a number of input data, such as crystal geometry, compressibility
at
0 K,
interionic
estimations
sometimes
of
distance,
these
various
unpredictable.
parameters one
etc.
cannot
and
are
rather
difficult
absence
of
of
terms
In
obtain
Measurements
the
the
lattice
any
correct
and
these
energy values by these
potentials.
The
Kapustinskii' s
energy values
equation
which uses only
for
the
calculation of
lattice
ionic radii as input data,
gives
lattice energies which fall on the low side of the experimental
7
values .
The
Kapustinskii's
equation
has
also
been
criticized
for emphasizing the sum of ionic radii, r C + r3 , whereas
in
many
crystals the interionic distances obtained predominantly by anionanion contacts
crystal
42
.
binding
elaborate.
The
of
Such
ionic
quantum
mechanical
crystals
calculations
crystals like LiF and NaF 38 which
are
calculations
tedious
and
have
therefore
been
are
composed
of
+
mechanical
calculations
crystals of heavier ions.
ionic
solids
have
been
are
This is
not
easily
very
the
much
limited
lighter
+
containing only 2 electrons (Li ) and 10 electrons (Na
quantum
of
to
ions
—
and F ).
extendable
The
to
why most of the studies on
phenomenological
in
nature
where
the
88
potential parameters are fitted from the crystal data on compressibility and interionic distance, as
Such
a
fitting
complicated
of
when
neighbours
as
potential
the
well
pointed
parameters
repulsive
as
out fay Sharma et al 118
becomes
interactions
between
next
much
between
nearest
more
nearest
neighbours
are
considered.
Keeping
proposed
in
some
view
simple
the
and
above
limitations,
straight
empirical
the
relations
111
evaluation of lattice energies of alkali halides
,
112
chalcogenides
, alkali metal hydrides, alkali
halides
and
energies
are
distance.
compared
some
heavy
directly
The
with
calculated
calculated
the
the
proposed
energies
has
values
experimental
agreement is considered
of
metal
halide
from
for
author
alkaline
metal
crystals.
the
of
ionic
thermochemical
the
earth
divalent
The
values
these
for
has
lattice
interionic
crystals
data.
are
The
to be support for the essential validity
relations.
A
been presented
by
suitable
test
calculating
of
these
lattice
the valence electron
plasmon energy and hence the other opto-electronic properties namely
the Penn gap, Fermi energy, So-parameter
and
the
polarizability in the case of alkali halide crystals,
earth chalcogenide crystals.
the lattice energy and
For the purpose,
electronic
and alkaline
relations between
the plasmon energy are proposed for the
above ionic solids.
For an ionic crystal, the atomization energy (AH )
is
of
stability
than
a
importance as
it
gives
a
better
idea
of
crystal
89
lattice
energy.
The
atomisation
energy
can
be
calculated
in
general, from the lattice energy through the relation:
E
a
=
U - I
+ E
p
in which E is the electron affinity of the anion and I the first
ionisation energy of the cation).
In the present study simple relations are proposed between
the lattice energies
formation,
the heats
the results of
experimental
discussed
existing
and
data.
in
the
theories
relations. The
The
of atomisation and heats of
which are compared with the available
present values
light
of
those
viz.
Kapustinskii,
of
lattice
calculated
energies
from
the
Born-Mayer,
are
different
Kudriavtsev's
details of the opto-electronic properties and
their
expressions are described elsewhere in Chapter 1.
i.
New
Relations
proposed
between
the
Lattice
Energy and
the Interionic Distance
Simple empirical relations are proposed for the evaluation
of lattice energies based on the assumption that a linear relation
exists
between
within
a
the
molecular
lattice
group
energy
of
and
compounds.
the
interionic
These
distance
relations
are
applied to the alkali metal halides, alkaline earth chalcogenides,
alkali
metal
hydrides,
higher
metal
halides
alkali
and
some
metal
chalcides,
divalent
halide
some
univalent
crystals.
The
following are the empirical relations for various molecular group
of compounds:
90
U
=
324.72 - 49.51 (rQ)
U
=
1047.6
U
=
U
=
U
=
U
=
322.31
...
242.4 (rQ)
67.05 (rQ)
1027.11 - 171.69 (r )
901.00 - 144.3 (rQ)
...
286.54 - 47.75 (rQ)
where
U
and rQ are
...
the
Alkali halides
2.23
Alkaline earth
chalcogenides
2.24
Alkali metal
hydrides
2.25
Alkali metal
chalcides
2.26
Alkaline-earth
divalent halides
...
2.27
Ga, In, T1 halides
...
2.28
and
the
lattice energy
interionic separation (X) respectively.
(kcal/mole)
The constants in the above
relations are unique in the sense
that they represent the best
fit with the experimental data.
ii
Relations between the lattice energy and the plasmon energy
New
and
the
relations
are
valence-electron
established
plasma
between
energy
assumption that a linear relationship
the
based
lattice energy
on
the
simple
between the two parameters
is valid in the case of molecular group of compounds under study.
U
=
6.70 (few ) + 81.13
...
Alkali halides
...
2.29
U =
24.3 Cho) ) + 381.9
^
...
Alkaline-earth
chalcogenides
...
2.30
U
9.47 Chtd ) + 96.69
Alkaline metal
hydrides
2.31
73522
91
U
=
19.13 (tio) ) + 293.84
p
...
Alkali metal
chaleides
...
2.32
U
=
15.045 (Ixo ) + 282.87
p
...
Alkaline-earth
divalent halides
...
2.33
The
the
result
numerical
of
a
constants
fit
of
the
in
the above equations are also
experimental
data,
similarly
to
equations 2.23 to 2.28.
In
the
above
equations,
lia)
is
the
valence
electron
r
129
plasma energy (eV) which is given by Jackson
on
the
basis
of
the plasma oscillations theory of solids as follows.
28.8 (|p)1/2
tico
2.34
where Z is the total number of valence electrons given by
Z. + Z
A
(ZA and Zg are
B
the
principal
valence
states
_3
composing the crystal). P is the density (g.cm )
of
the
and
elements
M
is
the
atomic or molecular weight (gr.) of the solid.
The
expressions
for
energy (Ep) in terms of tiw
*
the
r*
Penn-gap
(E )
and
the
Fermi
are given by
tl(0
2.35
eV
- U1/2
and
5.K. UNI VERS! I Y LIBRARY,
ANANTAPUR-515 063
0.2948 (tuo )4/3
P
eV
2.36
92
3
The expression for electronic polarizability (a) (cm )
Ravindra and Srivastava
128
,
which
is
derived
on
the
due to
basis
of
the well known Clausius-Mossotti relation and the Penn-like models
is given by:
(tiw )
S
p
a =
(tiw
n
x M
— x 0.396
x m“24
10
cm 3
d
o
)'
3 E
P
...
2.37
where M and d are the molecular weight (gr.) and density (g.cm
respectively.
-3
}
SQ is a constant for a particular compound and is
given by
E
L
iii
Relations
4E
re i
1
-3
+
the
2.38
4E_
p
EJ
between
2
Lattice
Energy
and
the
Heats
of
Formation and Heats of Atomisation
The
between
the
following
are
lattice
energy,
the
linear
the heat
relationships
established
of formation and
heat
of
atomisation.
U
=
f
-1.65 (AHq) +11.43
U
=
1.387 (AH ) - 30.32
...
U
=
2.414 (Ah ) - 44.56
a
...
3
...
Alkali halides
Alkali halides
Alkali metal
hydrides
AHf
is the heat of formation and AH
o
a
atomisation or atomisation energy in kcal/mole.
where
...
2.39
...
2.40
...
is
2.41
the heat
of
93
RESULTS AND DISCUSSION
Lattice energies
halides,
alkali
alkaline-earth
metal
higher
chalcogenides,
chalcides,
metal
for several ionic solids viz.,
alkali
alkaline-earth
halides
are
results
of
calculated
the calculations are
hydrides,
divalent halides and some
from
expressions taking interionic distance (rQ)
the
metal
alkali
simple
empirical
as the input data and
presented
in
Tables
2.1
to
2.8 and Figures 2.1 to 2.6.
The
calculated
lattice energies of these ionic
solids
are utilized to evaluate plasma energies from simple correlations
proposed
between lattice
energy
and
plasma
calculated
plasma
values
are,
in
estimation
of
gap,
Fermi
energy
certain
energy,
polarizability,
SQ-parameter
An
the
energies
properties
and
experimental
may
check
be
on
those
from
obtained
obtained
from
approach
experimental
like
in
the
the
Penn
electronic
Born-Mayer,
values
for
than
calculated
from
the
are well compared with
Born-Lande,
solids
those
of
experimental
KnpiiRtinRkii
The lattice energies obtained
all
values
The lattice energy values
the proposed correlations
Kudriavtsev's relations.
present
used
hence the
the
method based on the Born-Haber cycle.
obtained
turn,
The
in order to test the applicability of the proposed
relations.
lattice
opto-electronic
energy.
agree
calculated
closely
from
other
and
from the
with
the
theories.
94
Figures
2.1
approach
various
to 2.6
over
demonstrate the superiority of the present
other
theories
theories.
employed
The
in
the
basic
differences
comparisons
are
between
critically
analysed below:
As it is well-known the principal interactions in ionic
lattices
are
the
static Coulombian
to Madelung's energy,
force.
Though
established,
van der Waals*
the
no
interactions
first
simple
two
which
rise
interaction and the overlap
interaction
expression
give
for
the
terms
third
are
well
follows
from
the theoretical considerations.
As more acceptable forms of the
overlap
two
energy,
the
following
expressions
are
in
frequent
use:
(i)
The
adjustable
parameters
On
the
Born
form'*'b/r11,
and
vary
basis of quantum
this form
is not
from
correct,
b
substance
mechanics,
rigoursly
where
Seitz
20
although
has
it
and
to
n
are
substance.
suggested
may
be
that
a fair
approximation for a shorter range fo r.
(ii)
Born ' and
exponential
prediction.
form
a
Mayer 5
exp
They took
proposed
(-r/P),
based
a
repulsive
upon
quantum
term
of
mechanical
P =0.345 x 10" cm quite arbitrarily,
for
all types of ions and assumed that the constant a has ionic radii
dependence.
values
of
Because
lattice
of
the
energies
uncertainity
calculated
different models differ largely,
case
e.g.
by
they
of
these
different
differ by
assumptions,
workers
on
25% in the
131
of Rb+ and 62% in the case of Li+ as given by Goldschmidt
95
and
132
Huggins and Mayer.
potential
is
finite
and
Moreover,
in this form,
electrostatic
potential
the repulsive
- oo at
is
r=0.
Hence,
the net interaction remains negative, i.e. attractive, which
looks
unphysical.
modified
this
expression
But
Mayer
term
and
q 7
and
added
coworkers ’ ’
another
term
q
subsequently
which
made
the
more cumbersome from the calculation point of view.
the
new expression
did
not
exclude
the
ionic
radii
dependence and the above limitation in the vicinity of r=0.
(iii)
was
Another
suggested
where
b
nearest
by
and
n
unlike
mechanically,
repulsive
term
of
Born-Lande1
in
the
are
constants
ions
for
it may
be a
a
and
r
inverse
power
form
ionic interaction
is
the
br n
potential
distance
between
But
quantum
givencrystal.
fair approximation for a short range
of r.
(iv)
The
lattice
energies
can
be
related
to
other
measurable energy quantities by means of a thermochemical cycle
due to Born and Haber3,4.
Born-Haber
dependent
bonding
cycle
upon
in
the
is
the
dissociation
Bom-Haber cycle.
exprimental
nature
of
crystal.
dissociation energy,
of
an
The lattice energy obtained by the
to
ionization
evaluate
the
One
lattice
energy
assumptions
requires
and
made
energy
not
about
the
sublimation
energy, electron affinity
lattice
is
of
a
energy,
and heat
solid
by
96
(v)
The Kapustinskii's equation proposed a general method
for the calculation of lattice energy values which uses only ionic
radii
as
input
equation
values.
are
7
data.
often
The
Thelattice
fall
on
the
Kapustinskii
low side
equation
has
of
also
the
been
this
experimental
critisized
for
whereas
in
many
crystals the
interionic distancesare detained predominantly
by
anion-
(vi)
energy UQ
o
MU =
m
U
parameters
constants
10
of
7,
one
to
To
The
mean
calculate the
value
the
of
term
9 and 10.
and
solids or
single
is
a
sound
of
sound
determined
transverse
crystal
constant and
the
velocity
mean
experimentally
longitudinal
n^
relating
has
wave
elastic
the values
Depending upon the suitability of different
one
has
predict correct
alkali halides,
expression
with the
requires
velocities
structures,
an
a crystal
inpolycrystalline
data.
5,
crystal
,
of
propagation
of 3,
developed
(Yn.U )/9.
1 o
velocity
to
42
Kudriavtsev
lattice
radii, r +r ,
by
the sum
contacts.
ionic
obtained
emphasizing
anion
of
energies
to
change
lattice
the suitable
the
energies
value
e.g.
value of n^ = 5;
of
for all
nJ
from
3
NaCl-type
for CsCl-type cesium
halides the suitable value of n^=7.
(vii)
the
As
lattice
potentials
as
already
energy
requires
crystal
of
a
geometry,
pointed
an
ionic
knowledge
out,
the
crystal
of
a
compressibility,
detailed
calculation
according
number
of
interionic
to
of
interionic
input data
distance
such
etc.
2.1.
■H
O'
(-*
a>
c
0)
0)
4J
CO
u
•r-p
u
o
Ref.124
Eg.2.48
Ref.78
Refs.107,108
- 86.0
- 99.9
201.7
183.7
206.4
191.0
170.5
160.9
188.5
162.7
170.5
159.6
189.2
163.2
156.6
171.8
160.1
189.2
164.2
157.5
178.5
167.0
170.1
163.2
177.1
164.8
2.981
’H
u.
0)
Z
■—*
u
eg
z
U
CO
eg
z
«
z
u.
sc
NO
PA
On
PA
u
180.6
157.7
180.6
163.7
157.2
162.1
154.7
3.285
NO
157.9
PA
fcA
185.9
CD
185.3
NO
r-
2.815
rCO
150.1
O'
sc
U.
JD
sr
*~4
CJ
JD
X
<r
PA
•Cf
PA
PA
<r
tr
u.
U)
CJ
ON
O'
PA
CM
<r
fcA
iA
ON
CO
o
<r
a
PA
NO
lA
A'
PA
PA
o
iA
PA
7.79
<r
140.8
a
157.5
159.9
- 78.1
- 82.7
154.7
ON
134.9
I
178.1
130.2
166.6
164.2
- 80.5
- 94.3
128.9
148.7
139.8
151.0
168.7
169.5
155.5
138.7
125.9
125.2
138.4
130.1
-126.9
O
1.70
NO
160.5
165.8
170.4
152.2
154.6
160.9
124.1
- 86.4
171.9
174.5
140.7
125.3
- 93.5
127.9
119.5
138.5
120.2
149.5
176.5
- 79.0
-102.9
On
147.7
- 93.7
- 90.9
ON
PA
142.9
148.4
M3
170.6
CO
159.9
- 90.6
- 83.6
CM
On
r**
1
iA
PA
CM
129.2
ON
M3
125.1
130.7
139.7
lA
3.950
r~~
177.1
162.5
140.9
lA
175.9
lA
ON
3.005
PA
143.0
CO
»A
143.4
CD
163.9
r-
149.7
CO
CD
164.8
PA
158.2
154.7
-104.2
- 95.2
s
177.0
204.0
167.1
184.5
-134.5
-109.4
as
3.663
146.7
197.1
-
r“
PA
T
PA
PA
(M
Average
percentage
deviation
PA
PA
NO
PA
CO
rt
*A
lA
a
tA
Csl
CM
O'
PA
PA
lA
lA
CM
CsBr
*—
PA
PA
M3
3.526
CM
MS
f-
CO
f-*
CsCl
PA
1
CM
PA
RbBr
CD
CM
PA
«f
3.293
CM
O'
O'
KBr
<*
CO
CD
*A
2.664
CD
PA
167.2
1
179.3
M3
CO
192.8
155.5
- 98.2
CD
<r
o
CO
210.1
M3
68.8
152.7
153.1
173.6
-136.3
-120.0
-104.9
221.2
179.9
180.6
187.1
185.4
(xi
CO
•H
-J
CM
211.5
179.2
213.6
219.5
210.5
2.307
2.814
<?
M3
128.7
- 98.6
177.1
170.4
169.5
169.7
180.4
175.0
3.025
134.1
-109.6
192.2
201.8
189.4
181.9
184.5
194.3
188.7
163.9
148.8
2.747
157.9
203.3
210.4
192.1
197.7
2.566
LiCl
202.7
165.1
227.7
192,1
cn
»
238.9
lA
CM
195.5
PA
238.9
CM
203.3
00
i
- 83.7
2
- 97.7
lA
-129.0
r-
o
-112.3
Lartae
Born-
202.7
Ref.123
Eq.2.49
<r
CO
246.6
7
.
Exptl
data
Present
study
PA
225.3
Ref.
.
data
Exptl
Present
study
Thakur
et al.
—<
CO
o
Kudriavtsev
<40>
0C
2.009
Ref .6
3
Ref.78
ra
Kapustinskii
E
Ref.89
N
<~4
u
BornLande
0)
o
BornMayer
V<
Exptl.
data
o
Eg.2.23
formation (AH°) and heats of atomisation toHg) of alkali halides at 298 K
'
Ref.122
Present
study
lattice energies (U), heats of
< s
tn
fA
LiF
Alkali
halide
Table
:
I
I
I
4) 1
-H 1
O 1
£ 1
>s. II
—<
CO }
o »
X 1
1
CD 1
r j
rCM
lA
i
j
o
p-»
PA
PA
r-
a
a
CM
NO
<r
280
calculated
240
200
160
120
120
160
200
240
280
experimental
Fig. 2.1. Plot showing the relationship between experi­
mental lattice energies (kcal/mole) of alkali
halides and th
calculated from Eq.2.23,
Born-Mayer equaton,
Kapustinskii equation
and Kudriavtsev equation. (Data taken from
Table 2.1). The line is of unit slope.
m indicates Eq.2.23, 3}f indicates Born-Mayer
equation,Q indicates Kapustinskii equation,
A indicates Kudriavtsev equation.
Table
•H D
W 13
CO • H
XX -H
0
lx.
cc
CD
1
•H
u.
CO
u
z
Z
CO
M
CD
CO
z
fA
CM
fA
Z
«
U1Z
Os
SO
r-
sD
CD
U
SZ iZ
in
GO
CM
fA
fA
fA
sO
so
fA
£3
QC
7.97
6.67
12.91
6.60
6.40
10.08
8.18
7.19
5.76
0.71
0.72
0.69
0.70
0.71
0.72
0.72
0.74
0.71
0.68
4.391
2.100
4.913
6.928
5,731
7.962
Known
R e f .126
0.887
2.976
4.120
R avindra
R e f. 125
1.0329
3.2690
1
3.4976
1.8851
6.406
1.991
6.4636
2.6256
4.9012
5.9167
8.1330
2.537
4.626
5.770
5.7466
6.7960
4.3726
5.6938
CO
in
<*
sO
fA
A-
O
tA
fA
A*
fA
8.7680
3.601
5.690
8.859
sO
0.73
1.509
5.761
0.989
2.697
3.804
6.521
fA
8.92
7.17
12.54
11.77
10.73
13.41
12.55
9.16
0.69
0.74
0.76
0.77
0.714
sO
CM
sO
154.1
•o
10.06
9 .2 4
7 .1 0
10.53
ON
lA
CM
159.9
9.29
15.92
10.91
15.03
12.40
ON
15.55
12.09
10.98
sO
fA
SO
fA
CD
3.950
fA
A*
3
CO
<r
175.9
148.5
140.9
129.2
fA
rA
sO
A"
CM
Os
A*
o
fA
VO
CO
<r
1.6100
1.6700
1.7876
sO
CO
CO
A-
CM
177.1
185.3
162.1
154.7
1.3980
1.4930
1.5530
11.91
sO
10.06
18.06
CM
13.29
12.38
15.27
11.45
10.25
9.96
0.72
0.77
0.79
0.80
fA
12.03
10.29
21.80
13.29
11.00
8 .5 2
17.64
13.29
E q.2.48
<r
fA
a
t
A*
3.005
3.560
185.9
163.7
157.2
148.6
150.1
M3
16.67
12.49
13.11
20.11
15.68
»
A-
r~
fA
fA
AAON
fA
3.4340
194.3
170.1
163.2
153.6
CO
192.8
19.31
15.56
<*
1.3610
1.4900
1.5590
1.6770
177.1
187.1
178.5
CM
fA
167.0
2.815
a
CM
tn
Os
2.664
3.139
3.293
3.526
iI
219.6
I
fA
CO
A'
fA
A*
Csl
cr
CO
CO
tA
CsBr
03
sO
fA
o
in
■st
lA
ON
A*
CsCl
u-
r~
CO
CM
CM
CM
ON
CsF
w
T3T1
<Z Xz
oz
T“
■sf
CM
CM
RbBr
(-4 o
t»
lA
CM
CM
>Cfr
Os
tA
a
CD
a
RbCl
i
fA
fA
sO
CD
fA
RbF
C
1A
CM
t—
M3
CM
r~
CO
•o
KCt
4- CM
"M.
sO
<r
CM
210.5
185.4
03
fA
CM
sO
A*
fA
CM
■
1.3360
1.5400
1.6412
1.7745
o_
r~
fA
lA
o
sO
2.307
2.814
2.981
a.
or
uJ
CM
14.01
3
<t
203.3
194.3
180.4
£
CM
197.7
188.7
175.0
c
C
Q.
a> uj
a. •
CT
UJ
22.24
R e f. 2.47
P resen t
study
E le c tro n ic p o la riz a b ility ( a )
»*
lA
O
o
CO
CM
<f
LiBr
>■03
tA
(si•
25.96
17.99
16.27
13.16
H CU
U_ U w
U.
21.52
17.40
16.06
>> 4>
1,3915
1.6620
1.7840
U U
S3
2.009
2.566
2.747
3.025
3>
CD
S tan d ard
v alu e
E q .2 .4 3
o
in
R ef. 89
Ce
O
study
E q.2.29
«
P resen t
CO
CP
E x p tl. d ata
(k cal/m o le)
•H
£
P resen t
stu d y
E q .2 .2 3
U
2 .2 . L a ttic e e n e rg ie s and o p to -e le c tro n ic p ro p e rtie s o f a lk a li h a lid e s
CM
'o
tA
lA
CO
o
<r
CO
>>
cn
Ul
<u
cCD
CM
I
I
1
1
$
i
1
1
f
1
1
♦
!
I
i
98
i
I
t
i
i
I
I
i
i
i
l
I
I
l
I
i
i
»
i
*
iI
t
»
!I
1
1
1
1
1
1
1
1
I
»
11
i»
Ii
«l
1
l
l
1
I
I
i
f
»I
l
i
I
I
l
l
»
l
I
I
»
i
i
»
li
*i
i
f
i
i
I
«
1
»
J
1
»
1
I
sO
1
1
1
1
1
1
I
1
1
}
1
1
1
1
1
1
1
1
1
1
I
j
!
t
i
i
i
i
99
Measurements
rather
and
difficult
correct estimations of these various
and
sometimes
unpredictable.
So
terms are
one
cannot
obtain lattice energies by these potentials in the absence of any
of these parameters.
Keeping
in
view
the
above
limitations,
the
author
has proposed a simple of means of obtaining lattice energies for
nearly
100
ionic solids from
interionic distance.
The results in
each case are compared with the experimental data.
Table
2.1
presents
the
lattice
energies
for
alkali
halide crystals calculated from equation 2.23 and the experimental
values
obtained
comparison,
methods
the
the
values
viz.
Born-Haber
of
equation
in the
same table.
the experimental
energies
present
values
and
obtained
For
from
the
other
theoretical
Bom-Lande
Kudriavtsev's
sake of
equation,
equation
are
also
To highlight the comparisons made
plot
(Figure 2.1)
the values
is drawn between
obtained
from
different
A glance at the Figure 2.1 reveals that the lattice
calculated
a
much
thermochemical
obtained
by
equally
good
predicting
a
cycle.
equation,
and
between various theories,
approaches.
U
Born-Mayer
Kapustinskii's
included
from
in
the
closer
cycle
present
study
agreement
with
values
than
the
the other theories.
the
performance
correct
by
values
for
alkali
the
the
for
Born-Mayer
alkali
experimental
corresponding
The figure also
halides.
halides
values
presents an
equation
The
in
average
100
percentage deviation between the calculated and
values
for
the present
to be only 1.7%
while
it is maximum of 8% in the case of Kudriavtsev's equation.
This
clearly shows
method
the relative
is
found
the experimental
superiority
of the author's
relation
over others' in predicting the accurate values of lattice energies.
Equation 2.29 gives the correlation between the lattice
energy
U and
halide
crystals.
rQ values
the plasma energy (
The
are utilized
lattice
in
) in
energy values
equation
2.29
energies in . alkali halide crystals and
in
Table 2.2
plasma
along
with
oscillations theory
further
utilized
So-parameter
and
the
the plasma
the values are presented
It is
from
the
readily seen
from
good agreement between the predicted
The computed values of plasma energy
to
evaluate
opto-electronic properties viz.,
(Ef),
to evaluate
of solids.
the standard ones.
are
obtained from
the standard values obtained
the table that there is a
and
the case of alkali
the
some
of
the
the Penn gap (Ep),
electronic
important
Fermi energy
polarizability
(Of)
using
the appropriate relations as described at length in the previous
chapter.
It is encouraging to observe that the
(X -values tally
closely with the corresponding experimental values.
This clearly
corroborates the validity of our relation in describing the various
opto-electronic
properties.
present
lattice property
study
like
demonstrates
the
the
usefulness
of a
in linking
with several opto-electronic properties through simple
linear relations.
simple
The
lattice energy
921
773
741
688
826
716
691
932
800
798
752
839
764
726
937
937
786
766
865
758
730
2.436
2.105
2.602
2.731
2.770
2.405
2.846
2.962
3.179
MgO
MgS
MgSe
MgTe
CaO
CaS
CaSe
CaTe
SrO
SrS
Average percentage
deviation
1.75
599
647
3-72
610
636
597
642
610
642
629
3.302
3.500
Bate
746
655
BaSe
673
753
3.194
BaS
662
2.770
BaO
666
640
675
733
640
3.331
SrTe
699
647
624
640
691
3.122
SrSe
790
686
799
662
660
675
718
3.010
668
721
698
706
841
771
939
Ref. 6
Sherman
739
843
724
769
799
934
807
863
900
1074
Modified
KapustinskiiYatsimirskii
Ref .44
753
682
799
696
822
2.580
776
778
679
650
766
.677
857
908
825
BeTe
863
934
2.225
1044
1085
BeSe
937
1048
Ref. 78
Ref. 116
1.649
Born-Mayer
equation
Born-Haber
cycle
2.105
Eq. 2.24
Present
study
Lattice energies (U) (kcal/mole)
BeS
Ref. 78
CA)
a
r
BeO
Crystal
<(r*
o
calculated
Fig. 2.2. Plot showing the relationship between experi­
mental lattice
energies
(kcal/mole)
of
alkaline-earth
chalcogenides,
and
those
calculated from Eq.2.24 and Born-Mayer equa­
tion. (The data is taken from Table 2.3).
The lion is of unit slope.
indicates Eq. 2.24,5^ Born-Mayer equation.
c
CE
CD
r*.
03
cc
in
2.225
U53
BeSe
€
CD
M5
M3
K\
•it
CM
as 03
CO
o in
£ £ £
13.98
8.25
7.05
5.26
15.16
15.43
716
in
M3
CD
PA
iA
PA
<t
in
lA
776
5.8
4.0
2.770
W
u
m
P—
o
NO
PA
U
in
CD
CD
o
O'
PA
T—
CRC
!
1
1
CM
NO
NO
CM
•o
NO
t"*
NO
NO
.
Handbook, 55th edn., Ohio (1975).
1
and d values from
1
ta k in g
t
2.53 x 1024 a
I
d
11.34
7.71
0.82
I
(e_+2)
8.93
6.22
■
(f^ -D
597
12.41
I
C la u s iu s -M o s s o tti re la tio n :
3
10.91
O'
CM
r-»
629
r-
4.81
12.07
8.07
8.51
10.86
6.19
7.79
7.05
8.18
in 00
Obtained from th e
PA
lA
r-
610
5
599
NO
15.68
lA
PA
<*
o
3.500
IA
tA
13.09
<t
CD
NO
O o
BaTe
16.19
<#
CM
NO
11.97
6.86
PA
•o
10.10
co
CM
CO
3.302
ON
NO
12.21
On
PA
CO
BaSe
CO
10.60
tA
CO
<3
PA
BaS
O'
ON
f-
r-
13.21
2.97
6.15
7 .6 3
8 .7 0
2.62
6.02
11.05
CO
PA
675
8.20
10.21
5.67
10.62
CO
<t
12.69
17.35
18.08
13.80
778
682
IA
tA
822
r-
<t
3.122
13.89
12.12
650
O'
<t
SrSe
14.11
14.28
691
726
679
677
<t
3.010
CM
<t
tA
2.580
NO
tA
SrO
a
<t
NO
730
IA
CM
PA
6.3
1.76
1.53
0.80
0.85
0.86
0.88
r*
in
tA
(A
3.179
5.76
8.28
8.10
PA
On
CM
NO
CaTe
Known
0.91
0.89
CO
d
758
688
NO
2.962
M
0.81
15.81
13.07
19.76
19.82
826
752
O'
PA
00
776
NO
tA
CaSe
8.3
M
0.81
12.07
12.98
16.58
798
NO
2.846
lA
2.770
On
2.731
932
CO
CM
CaS
s
Mt
786
937
(N
817
x
0.83
0.84
0.85
13.77
7.59
18.28
17.87
r->
PA
ON
2.95
•
5.10
tA
ON
2.105
CM
ON
15.72
3v
cm )
CM
CM
■O
CaO
19.06
16.33
24.26
22.81
921
CO
CO
CD
765
in
2.602
o
19.06
<J
CM
17.75
15.59
O a
CO
tA
857
O
tA
7.89
6.00
19.35
Eq.2.46
Present study
M n -24
a lio
E le c tro n ic p o la r iz a b ility
•st
BeTe
18.05
21.62
19.52
825
o
908
108.5
Eq.2.47
E q.2.45
r-
R e f. 127
PA
Ar-
((eV)
V
<t
c y c le
Standard
value
Fermi
19.52
01
o
'ft
4J
4->
CO
_J
28.35
c
0)
On
r»
CM
22.81
(fl
CL3
•ft
O'
(-4
01
3.0
C3
7.1
r-t
(0
O
-52
MD
PA
CT
U
R e f. 78
03
CM
R ef .78
O
£
1.649
CL
2.105
cd
lA
•Jj
E q.2.24
£
§
CT
LJ
BeS
£
(0
Present
study
CP
u
0)
PA
lA
Haber
6
O'
O'
BornHay er
eg.
33
fA
CO
Born-
>
<D
lA
Present
study
Penn
gap
o p to -e le c tro n ic p ro p e rtie s o f a lk a lin e -e a rth chalcogenide c ry s ta ls
<}
<5
CM
PA
CM
8e0
and
>>
CD
CM
<t
lA
969
u
® u. >
tA
vD
<J
099
u3
O'
CM
008
C rystal
L a ttic e energies
U1 3
IA
C98
Table 2 -4 -
*
102
CM
103
Similarly
equations
2.48
and
2.49
give
the
linear
f
the heat of formation (AHq)
relations between the lattice energy,
and the heat of atomization
(AH 1 in the case of alkali halide
a
crystals.
Table 2.1 lists the values of A H* and AH for alkali
o
a
halides obtained from the latt ice energies.
It may be observed
that
these
values
agree
satisfactorily
with
the
experimental
the
alkaline
values.
The
lattice
chalcogenides
(11-V1
evaluatedusing
Table 2.3.
energies
group
equation 2.24,,
for
semiconducting
and
compounds)
are
Figure 2:2 demonstrates the performance of each of
pattern as was observed
be evident from
compounds
that the
the values
(cyclic) values.
It
of the same order as
may
the present values of U are
obtained
from
It may
computed values for these
are fairly accurate and nearly
that
It shows the same
in the case of alkali halides.
the figure
experimental
to note
112
the values are tabulated in
the relations employed in the present study.
the
-earth
also
be
of interest
better than those of
the Born-Mayer equation and
from
the
.52
modified Kapustinskii-Yatsmirskii' s equation.
The
chalcogenides
plasma
are
energies
estimated
(
)
directly
from
energies obtained in the present study.
for
the values
The
is
compared
found
with
between
the
literature
the estimated and
values.
of lattice
values computed
from equation 2.30 are reported in Table 2.4,
been
alkaline-earth
where they have
A
good
literature values,
agreement
and
this
u
OlC
o
4J
-P
CO
-J
•H
0
21.14
20.51
16.67
13.12
12.47
16.02
12.84
9.28
8.16
R e f. 78
709.4
614.1
547.0
465.7
602.6
542.2
486.3
470.7
r—
n
R ef. 7
695.0
601.7
532.8
513.0
I
1
516.0
565.0
446.0
i
614.1
525.0
504.7
446.6
520.2
465.9
R e f. 78
e q u a tio n
NO
451.1
432.7
543.0
477.7
457.1
3.20
3.32
<r
r—
—
ON
t
<r
NO
in
CM
oCM
oCM
.0
cn
oCM
CO
CM
CO
CM
CO
CO
CM
CO
CM
-O
CO
CM
CD
CD
CJ
-J
9.76
10.42
12.56
15.48
10.00
ON
580.2
CO
525.8
2.82
Z
601.3
_J
2.48
CO
536.1
oCM
2.86
•H
NO
2.92
oCM
12.78
r~
2.79
lA
548.1
c
0
613.3
u
0
Standard
value
E q.2.43
CD
P re se n t
stu d y
E q.2.41
CT>
Thakur
Plasma ener
>-.
Sherman
o
2.41
0
♦H
cn
Born-M ayer
o f a lk a li m etal c h a lc id e c ry s ta ls
CO
683.7
P resent
stu d y
Eq .2 .2 6
plasma e n e rg ie s
rH
R e f. 78
(A)
K
J
L a ttic e e n e rg ie s ,
0
rH
o
B
2.00
C ry s ta l
2 .5 .
3a
ON
tA
99*9
>
0)
6 ’ 169
Table
104
cn
t
ON
CM
KN
CJ
calculated
experimental
Fig. 2.3. Plot showing the relationship between experi­
mental
lattice
energies
(kcal/mole)
of
alkali metal chalcides, and those calculated
from Eq.2.26.
(Data taken from Table 2.5).
The line is of unit slope.
indicates Eq.2.26.
156.1
154.9
3.194
CsH
164.4
163.8
3.024
RbH
170.7
172.8
2.854
KH
193.5
2.440
NaH
216.5
BornHaber
cycle
Ref.35
194.5
215.5
2.042
LiH
Present
study
Eg.2.25
o
(A)
(fi
162.0
168.6
177.2
202.2
234.0
Ref.7
BornMayer
160.5
170.4
178.2
6 21
7.05
7.96
10.43
12.47
228.6
201.3
Eg.2.40
Present
study
Ref.42
Kapustinskii
(
AH
6.51
7.05
7.82
9.84
12.92
Eg.2.43
Standard
value
p
Plasma energy(tiu} )(eV
ui), and heats of atomisation
Lattice energies (U) (kcal)/mole
Ref.42
Crystal
hydrides at 298.15 K
Lattice energies (U), plasma energies
H
Table 2.6.
)
)
82.8
86.2
89.8
99.5
107.5
Eg.2.50
Present
study
a
82.0
92.0
88.0
91.7
112.1
Ref.123
Standard
valaue
A H (kcal/mole)
of alkali metal
105
calculated
experimental
Fig. 2.4. Plot showing the relationship between experi­
mental
lattice
energies
(kcal/mole)
of
alkali metal hydrides, and those calculated
from
Eq.2.25,
Born-Mayer
equation
and
Kapustinski equation. (Data taken from Table
2.6). The line is of unit slope.
■ indicates Eq.2.25,indicates Born-Mayer
<£ equation,Qindicates Kapustinskii equation.
2 -7 -
D iv a le n t
h a lid e
T a b le
la t t ic e e n e rg ie s (U ),
CD
C3
u
o
649
625
598
sO
o
1.91
2.10
1.77
t—
Cv i-4
U
Lu
Q}
0)
CO
U.
GJ
H-1
0)
CO
CD
CD
X
co
CS
u
CS
CM
sfl
cn
0)
o
CN
Cs1 rM
o
CT>
2
lA
23.25
2 1 .0 0
17.02
628
624
CM
CO
Cn
U
*—
X
X
cr
CM
CJ*
CN
U
CJ
CD
00
CD
CJ
CJ
tA
T—
lA
Os
sO
3.03
r-4
<*
2.87
CN
Lu
CD
O
SO
CO
2.67
fA
2.20
lA
2.88
Os
CM
t—
CD
u
CO
CN
SO
r—4
a
CN
in
CD
u
CN
CM
U
u
C/J
cn
3.20
CN
CN <-(
CJ
Lu
CO
CD
CD
OD
Os
470
<r
2.99
14.05
12.04
18.42
14.03
12.44
10.54
557
485
Os
492
588
506
•
1
BaBr
15.50
509
SO
SO
so
560
20.02
592
rA
10.85
15.03
15.98
r~
2.32
13.97
15.50
13.50
491
CO
CO
so
2.82
14.92
CM
514
PA
16.03
14.54
16.05
547
SO
598
c
<u
iA
538
CO
2.52
rKS
lA
LA
2.67
14.78
16.50
16.37
£
N
'—1
CD
O
J*
18.65
CD
575
c
CD
18.00
0)
20.21
700
>s
O)
t*
Os
Os
516
I
<*
2.51
1
LA
CM
2.10
SO
SO
PA
587
20.95
r1
2.18
CM
22.74
CN
17.76
rA
695
CM
692
SO
720
CM
839
v a lu e
E q .2 .4 3
S tandard
sO
SrF
_J
1.75
0)
-
699
£fl
0}
•I'i
on
u
R e f .130
P re se n t
s tu d y
E g . 2.42
f-H
Q-
R e f. 7
01
r—i
O
C a r lin
B
c y c le
a lk a lin e - e a r th d iv a le n t h a lid e s .
CO
CD
B o rn -H a b e r
)
3
-*=
R e f. 130
■H
4-J
4J
P re s s e n t
s tu d y
E g . 2 .2 7
P
CD
(A )
plasm a e n e rg ie s (fiui
>
r*
Os
15.78
CM
iA
sO
sO
13.22
<r
fA
<t
1
CM
UH
CD
CD
o
900
800
700
600
500
400
experimental
ig. 2.5. Plot showing the relationship between experi­
mental
lattice
energies
(kcal/mole)
of
alkaline-earth divalent halides, and those
calculated from Eq.2.27. {Data taken from
Table 2.7). The line is of unit slope.
** indicates Eq.2.27.
Crystal
Table
r—H
1
C3
C_)
m
halides.
2.403
Ref. 96
CM
169.1
178.9
166.6
158.7
\0
p"
sO
Til
I
nBr
161.4
CM
A
<3A
2.621
v-
TlBr
r-*
172.8
0s
168.7
1
170.9
i
167.8
1
2.487
1
T1C1
i
195.0
i
186.9
1
2.087
1
TIE
i
154.9
I
2.757
1
Ini
1
165.0
i
161 .0
2.577
130
Ref. 108
1
170.1
163.5
CO
PA
nCl
174.1
Ref.
Exptl.
Carlin
1
iA
I
.
Present study
Eq 2.28
t
181.5
2.200
(A)
-
Lattice energies (kcal/mole)
Lattice energies for Ga/In/Tl
1
U"S
Gal
GaBr
2.8.
PA
A
C“*
162.1
106-A
I
1
CM
200
190
180
170
160
150
160
170
180
190
200
experimental
2.6. Plot showing the relationship between experi­
mental
lattice
energies
(kcal/mole)
of
Ga/In/Tl-halides, and those calculated from
Eg.2.28. (Data taken from Table 2.8). The
line is of unit slope.
•Jfc indicates Eq.
2.28.
106-B
has allowed the author to use these values in evaluating the opto­
electronic
properties
in
these
compounds.
The
electronic
polarizabilities for these compounds computed from the fioujp values
(Table 2.4 )
compare
fairly well
with
those obtained
from
the
well-known Clausius-Mossotti relation.
The lattice energies for alkali metal hydrides,
metal
chalcides,
halides
are
alkaline-earth
estimated
divalent
from the
revalues
equations 2.25 to 2.28 respectively.
are tabulated in Tables 2.5,
their
corresponding
theoretical
values
and
with
Ga/In/Tl-
the
help
of
The values of U so obtained
2.6 and 2.7 respectively along with
thermocyclic
taken
halides
alkali
from
(experimental)
different
values.
works
have
included in the respective tables for comparison.
Other
also
been
It is obvious
from the tables that the lattice energies computed from equations
2.5 to 2.8 agree in most cases with the experiment.
evident from
present
approach
However,
marked
Figures
in
the
deviation,
2.3
It is also
to 2.6 that the agreement between
and
the
case
of
it
may
experimental
data
is
quite
certain compounds where
be
attributed
to
there
the
the
good.
is
a
uncertainity
involved in measurements of r .
The evaluated lattice energies are utilized to estimate
plasma energies for alkali metal hydrides, alkali metal chalcides,
and
alkaline-earth
and
2.33
divalent
respectively.
halides
The
plasma
from
equations
energies
so
2.31,
2.32
obtained
are
106-C
presented
in
Table 2.5
to
2.7 respectively,
and
compared with
the standard values.
In the case of alkali metal hydrides,
heats
(atomisation
and
of
atomisation
the
results
are
reported
energy)
in
have
Table
data.
These
been
2.6
along
values
do
the
evaluated
with
agree
the
corresponding
experimental
with
each other.
The heats of formation and the heats of atomisation
are not evaluated for other crystals for which the experimental
data are not available.
An
the
author
distance,
important
has
feature
established
lattice
energy,
of
the
correlations
plasma
energy,
present
work
between
the
heat
formation and
of
is
that
interionic
heat of atomisation through a set of simple and straight empirical
relations.
ionic
rQ
Lattice energies for over 100 diatomic and
crystals
as
the
evaluated
have
only
are
gap*
been estimated using the interionic distance
input
further
opto-electronic
triatomic
parameter.
utilised
properties
to
viz.,
The
lattice
determine
the
plasma
energies
so
certain
important
energy,
the
Penn
Fermi energy, S - parameter and the electronic polarizability,
thus establishing a
link
between the lattice properties and the
opto-electronic properties in ionic solids.
It
is evident from the analysis of the present study
that the proposed linear relations,
despite their simplicity,
are
quite capable of obtaining reasonably reliable estimate of lattice
energies and
plasma energies in the ionic crystals under study.
106-D
The
in
proposed
ionic
relations not only
crystals
but
also
yield
their
most satisfactory
comparison
with
results
experimental
and other theoretical values provides a direct and precise check
of
the
Further,
the
appropriateness
the
present
and
suitability
of
linear
relations.
agreement with experimental values suggests
approach
reduces
the
number
significantly as compared to the previous models,
of
that
parameters
and hence this
may be considered in preference to the cumbersome models having
involved tedious procedural calculations.
107
REFERENCES
1
M. Born and A. Lande
Verhandl. deut. Physik. Ges. 20 (1918) 210.
2
M. Bom and A. Lande
Ber. Preass. Akad. Wiss, Berlin, 45 (1918) 1048.
3
M. Born
Verhandl. deut. Physik. Ges. 21 (1919) 13.
4
F. Haber
Verhandl. deut. Physik. Ges. 21 (1919) 750.
5
M. Born and J. Mayer
Z. Phys., 75 (1932) 1..
6
J. Sherman
Chem. Revs., 11 (1932) 93.
7
T.C. Waddington
Advances in Inorganic Chemistry and Radio Chemistry, Vol.
I, Academic Press, New York, (1959).
8
M. Born
Atomtheorie des Festen Zustandes. Teubner, Leipzig, (1923),
and Handbueh d. Physik., 24 (1927) 420.
9
E. Madelung
Physik. Z., 19 (1918) 524.
10
A. F. Kapustinskii
Z. Physik. Chem., B22 (1933) 257.
11
A. F. Kapus tinskii
J. Phys. Chem. (USSR), 5 (1943) 59.
12
M.F.C. Ladd and W.H. Lee
Trans. Faraday Soc., 54 (1958) 34.
13
B. B. Kudriavtsev
Sov. Phys. Acoust, 36 (1956) 172.
14
E.A. Hylleraas
Z. Physik., 63 (1956) 172.
15
R. Landshoff
Z. Physik., 102 (1936) 201.
108
16
P.O. Low din,
Arkiv. Mat. Astr. Fysik., 35A (1947) No. 9 and No. 30.
17
D. Cubbiccotti
J. Chem. Phys.,
(1961) 2189.
31
(1959)
1646;
33
(1960)
1579 and
18
S.C. Saxena and C.M. Kachhawa
Indian J. Pure Appl. Phys., 2 (1964) 336.
19
J.D. Pandey
Indian J. Phys., 42 (1968) 447.
20
S. Prakash and J. Behari
Indian J. Pure Appl. Phys., 7 (1969) 709.
21
H.C. Pande
Lett. Nuovo Cimento Soc. Ital. Fis., 10 (1974) 657.
22
V.K. Dixit and M.N. Sharma
Indian J. Phys., 44 (1970) 419.
23
J.D. Pandey
Z. Phys. Chem., 243 (1970) 221.
24
M.N. Sharma
J. Chem. Phys., 52 (1970) 3846.
25
K. D. Mishra, K.P. Pandey and M.N. Sharma
J. Phys. Soc. Japan, 31 (1971) 977.
26
A. J. Cohen and R.G. Gordon
Phys. Rev., B12 (1975) 3228.
27
J. L, .Calais
Int. J. Quant. Chem. Symp., 9 (1975) 497.
28
Yuan-Kai Wen and Jun Sho",
Ti Ch'iu Hua Hsuch, 1 (1975) 35.
29
R. Shorczyk
Acta Crystallogr., A32 (1976) 447.
30
S. C. Goyal and J. Shanker
Curr. Sci., 41 (1972) 859.
31
A.K. Sarkar and S. Sengupta
Phys. Stat. Sol. B58 (1973) 775.
32
M.J.L. Sangster
J. Phys. Chem. Solids, 35 (1974) 195.
34
109
33
M.J.L. Sangster
Solid State Commun., 15 (1974) 471.
34
M.W. Lister
Thermochim. Acta, 8 (1974) 341 and 10 (1974) 325.
35
K.P. Thakur
Austr. J. Phys., 29 (1976) 46.
36
E.F. Ivanova, L.A. Petrenko, YU. A.
Kosenko
Russian J. Phys. Chem., 50 (1976) 330.
37
D.S. Puri, V.K. Garg and M.P. Verma
Phys. Stat. Sol., B78 (1976) 113.
38
J. Andzelm and L. Piela
J. Phys. C., 10 (1977) 2269.
39
K. N. Saxena
Chem. Phys. Lett., 48 (1977) 366.
40
V.K. Garg, D.S. Puri and M.P. Verma
Phys. Stat. Sol. B80 (1977) 63.
41
V.A. Zhadonov and V.V. Polyakov,
Sov. Phys. Cryst., 22 (1977) 488.
42
A.K. Sinha and L. Thakur
Indian J. Pure Appl. Phys., 15 (1977) 115.
43
Usha Puri
Indian J. Pure Appl. Phys., 15 (1977) 387.
44
L. Thakur and A.K. Sinha,
Indian J. Pure Appl. Phys., 15 (1977) 794.
45
R.K. Singh and V.V.S. Nirwal,
Lett. Nuovo Cimento Soc. Ital. Fis., 22 (1978) 559.
46
J. Shanker, G.G. Agrawal and R.P. Singh, J. Chem. Phys.,
69 (1978) 670.
47
R.P. Singh and J. Shanker,
Phys. Stat. Sol. B93 (1979) 373.
48
M. W. Lister
Thermochim. Acta, 31 (1979) 297.
49
G.V. Dedkov and A.I. Temrokov
Fiz. Tverd. Tela, 21 (1979) 1218.
Petrenko and A.A.
110
50
V.V.S. Nirwal and R.K. Singh,
Phys. Rev., B20 (1979) 5379.
51
J. Shanker and D.P. Agrawal
Phys. Stat. Sol., B98 (1980) 535.
52
A.K.M.A. Islam
Phys. Stat. Sol. B100 (1980) K5.
53
J. Shanker, V. K. Jain and L.P. Sharma
Indian J. Pure Appl. Phys., 18 (1980) 314.
54
V.V.S. Nirwal and R.K. Singh
Physica B + C, 103 (1981) 187.
55
S.S. Jha and L. Thakur
Indian J. Pure Appl. Phys., 19 (1981) 366.
56
R. B. Yadav
Solid State Commun., 46 (1983) 341.
57
A. J. Kaur, P.C. Jain, P.S. Bakshi and J. Shanker,
Indian J. Chem., 22A (1983 ) 595.
58
J. Yamashita and S. Asano
J. Phys. Soc. Jpn., 53 (1984) 3112.
59
D.D. Shukla, Y.S. Sainger, R.M. Misra,
Misra and M.N. Sharma
Indian J. Pure Appl. Phys., 22 (1984) 210.
60
A.J. Kaur, W.N. Bhende, P.S. Bakshi and J. Shanker
B123 (1984) 71.
61
S. M. M. Rehman and N. Shams
J. Fiz. Malays, 6 (1985) 49.
62
R.R. Reddy, T.V.R. Rao and R. Viswanath
Proc. Indian Natn. Sci. Acad., 55A (1989) 901.
63
J.E. Mayer and M. Maltbie
Z. Physik, 75 (1932) 748.
64
J.H.de.Boer and E.J. W. Verwey
Rect. trav. Chim., 55 (1936) 443.
65
A.E. Van Arkel and J.H. de Boer
Chemische Binding als Electrostatisch
Leipzig, (1931).
R.
Singh,
Erscheinung,
B.P.
Hirzel,
111
66
R. H. Fowler
Statistical Mechanics, 2nd edn., Cambridge
London and New York (1936), p. 330.
67
A. F. Kapustinskii
Acta Physicochim. (USSR), 18 (1943) 370.
68
A.K. Kapustinskii and K.B. Yatsimirskii
Zhur. Obshchei Khim., 26 (1956) 941.
69
M.L. Huggins and Y. Sakamoto
J. Phys. Soc. Jpn., 12 (1957) 241.
70
S. C. Saxena, C.M. Kachhava and S. Mathur
Indian J. Pure Appl. Phys., 2 (1964 ) 338.
71
D. Kumari Jha, K.P, Thakur and B.N. Jha,
Z.
Krystallogr,
Kristalgeom.
Kristalphys.
(1971) 35.
Univ.
Press,
Kristalchem.
72
J.D. Pandey and Uma Rani Pant
Vignana Parishad Anusandhana Patrika, 18 (1971) 81.
73
P.R. Son and R.A. Bartels
J. Phys. Chem. Solids, 33 (1972) 819.
74
S. Cantor
J. Chem. Phys., 59 (1973) 5189.
75
K. P. Thakur
J. Inorg. Nucl. Chem., 36 (1974) 2171.
76
K. S. Upadhyaya and R.K. Singh
J. Phys. Chem. Solids, 36 (1975) 293.
77
K. P. Thakur
Austr. J. Phys., 30 (1977) 325.
78
L. Thakur, A.K. Sinha and K.P. Thakur
Indian J. Phys., 52A (1978) 521.
79
W.C. Mackrodt and R.F. Steward,
J. Phys. C., 12 (1979) 431.
80
J. Shanker, V.K. Jain and R.S. Singh
Solid State Commun., 44 (1982) 293.
81
R. S. Singh, V.K. Jain and J. Shanker
Phys. Stat. Sol. Bill (1982) 519.
82
S. O. Lundquist
Arikiv. Fysik., 8 (1954) 177.
112
83
F.R. Bichowskii and F.D. Rossini
Thermochemistry of Chemical Substances,
York (1936).
Reinhold,
84
J. Kazarnovskii,
Z. Physik., 61 (1930) 236.
85
D.F.C. Morris
Proc. Roy. Soc. A242 (1957) 116.
86
C.D. West
J. Phys. Chem., 39 (1935) 493.
87
J.E. Mayer
J. Chem. Phys. 1 (1933) 327.
88
J.E. Mayer and R.B. Levy
J. Chem. Phys., 1 (1933) 647.
89
M.F.C. Ladd, J. Chem. Phys., 60 (1974) 1954.
90
V.B. Gohel and M.D. Trivedi
Indian J. Pure Appl. Phys., 5 (1967) 265.
91
C. S.N. Murthy and Y.V.G.S. Murti
J. Phys. C., 3 (1970) L22.
92
D. C. GUpta and M.N. Sharma
Indian J. Phys., 44 (1971) 495.
93
M.N. Sharma
Z. Physik. Chem. Neue Folge., 85 (1973) 86.
94
K. P. Thakur
Acta Phys. Pol., A48 (1975) 419.
95
K. P. Thakur
J. Inorg. Nucl. Chem., 38 (1976) 1433.
96
P.S. Bakshi, S.C. Goyal and J. Shanker
J. Inorg. Nucl. Chem., 39 (1977) 548.
97
L. P. Sharma, P.S. Bakshi and J. Shanker,
Indian J. Pure Appl. Phys., 16 (1978) 858.
98
V.C. Jain and J. Shanker
Pramana - J. Phys., 13 (1979) 31.
99
J. Shanker, P.S. Bakshi and L.P. Sharma
J. Inorg. Nucl. Chem., 41 (1979) 1285.
New
113
100
J. Shanker and D.P. Agrawal
Phys. Stat. Sol., B102 (1980) 495.
101
R.K. Singh and P. Khare
Phys. Stat. Sol., B103 (1981) 337.
102
D.F.C Morris
J. Inorg. Nucl. Chem., 4 (1957) 8.
103
D.S. McClure and 0. G. Holmes,............... , . .
O.N.R. Rept., Contract N6 Ori. 211-111, June (1955).
104
L.E. Orgel
Proc. 10th Solvay Conf. Chem., Bruseels, (1956) p. 289.
105
J.S. Griffith and L.E. Orgel,...
Quart. Revs. (London) 11 (1957) 381.
106
N.S. Hush and M.N.L. Pryce,
J. Chem. Phys., 28 (1958) 244.
107
K.V. Siva Kumar and E. Rajagopal
Acoust. Lett., 7 (1983) 87.
108
M. Subrahmanyam, E. Rajagopal and N. Manohara Murthy
Acustica, 60 (1986) 182.
109
W.P. Mason
Physical Acoustics, Vol. Ill, Part B, Academic Press (1965)
p. 50.
110
J. Frenkel
Theory of Solids, Leningrad, (1924).
111
R.R. Reddy, M. Ravi Kumar and T.V.R. Rao
Cryst. Res. Technol. (1993).
112
R.R. Reddy, M. Ravi Kumar and T.V.R. Rao
J. Phys. Chem. Solids , 54 (1993) 603.
113
E.S. Rittner
J. Chem. Phys., 19 (1951) 1030.
114
Lachman Dass, S.C. Saxena and C.M. Kachhava
Indian J.Pure Appl. Phys., 3 (1965) 178.
115.
K.P. Thakur
Indian J. Pure Appl. Phys.,
116
K.P. Thakur
Indian J. Chem., 12 (1974) 376.
11 (1973) 549.
114
117
K.P. Thakur and J.D. Pandey
J. Chem. Phys., 71 (1974) 850.
118
L.P. Sharma, H.P. Sharma, P.S. Bakshi and J. Shanker
Indian J. Phys., 54A (1980) 318.
119
M.P. Tosi
Solid State Physics, Academic Press, New York, 16 (1964) 1.
120
A.M. Karo and J.R. Hardy
J. Chem. Phys., 48 (1978) 3173.
121
A. VI. Overhauser and B.J. Dick
J. Phys. Rev., 112 (1958) 90.
122
A.C. Wells,
Structural Inorganic Chemistry,
p. 357.
123
124
R.T. Sanderson
Chemical Bonds and
(1971) p 133-175.
Bond
R.T. Sanderson
Inorganic Chemistry,
New Delhi, (1971).
Affliated
Oxford,
Energy,
3rd edn.,
Academic Press,
East
West
125
A. Kumar, N.M. Ravindra and R. Rath
J. Phys. Chem. Solids, 40 (1979) 1141.
126
C. Kittel
Introduction to Solid State Physics,
Wastern, New Delhi (1974) p. 227.
London
Press Pvt.
Ltd.,
4th
Wiley
edn.,
127
N.M. Ravindra, R.P. Bhardwaj,
Srivastava,
Infrared Phys., 21 (1981) 369.
128
N.M. Ravindra and V.K. Srivastava
Infrared Phys., 20 (1980) 67.
129
J.D. Jackson
'Classical Electrodynamics', Wiley Eastern, New Delhi (1978).
130
R.L. Carlin
'Transition Metal Chemistry', Vol. 3, Marcel Dekkar Inc.,
New York (1966).
131
V-M. Goldschmidt
Skr.narske Viclensk - Akad , (1926,1927)
132
M.L.Hu$£ins and XE.Mayer
J.Chem. Phys. 1 (1933) 643.
K.
(1962)
Sunil Kumar and V.K.