CHPATER VI
LATTICE ENERGIES, BAND GAPS AND OPTO ELECTRONIC
PROPERTIES OF SOME IONIC CRYSTALS
266
6.1
Introduction
The
study
of
fundamental
importance
inter-ionic
forces
thermal
and
by
and
Lande.
energy
may
of
other
ral
to
crystals
apart
Madelung
of
methods
need
to
for
ents
and
lattice
attempts
lattice
Ladd 9
'
14-20
have
and
ionic
been
was
energy
initiated
of
interms
Seve-
Born-Meyer
of
the
ionic
is
based
method
distance
in
adopted
'
used
the
different
eliminating
for
expansion
the
halides.
evaluate
crystals.
made
to
the
coeffici­
evaluation
Based
on
Kudrivstev's theory"*-*,
to
case
parameters like com­
cubical
alkali
methods
and other
theoretical
with
energies
energies
Hyllerass*1*
numerous
alkali
Others
of
elastic,
Kapustinskii^^ interrelated
coefficients,
energies
of
laboratory.
starting
internuclear
point
nature
determination
experimental
cycle^.
of
indirectly
the
lattice
the
radii.
zero
methods of
of
the
evaluate
basic
in
1-3
compounds.
pressibility
are
methods
constant,
ionic
calculated
parameters
the Born-Haber
energy
is
on
The
Experimental
from
the
effects
lattice
be
evaluate
solids
properties.
ionic
related
theoretical
model
on
the
in
understand
their
anharmonic
of
lattice
to
and
treatment
Born
cohesion
lattice
of
the
there
energies
Recently numerous
calculate
the
lattice
267
energies
of
alkali
Dirac method.
deterimined
energies
The
All
involving
straight
main
relation
halides,
using
hybrid/
Thomos-Fermi-
these methods require experimentally
physical
author's
alkali
halides
parameters
to
arrive
tedious
mathematical
aim
to
to
is
evaluate
alkali
propose
the
divalent
at
lattice
calculations.
a
lattice
halides
simple
and
energies
of
and
Ga,
In,
TL halides.
The most important single parameter of a semi­
conductor
mental
is
its
methods
energy
of
gap, and
determining
consequently
gap
investigated extensively.
However,
difficult
quantity
is
to
measure
remarkable
edge
fact
effective
conductors
samples
are
approximate
experiments
a
energy
function
values
will
better
transport
employed
it
of
gap
provide
for many
the
to
as
in
establish
widths,
means
it
semi­
Measurements
temperature
frequently
and
some of the band
known
gaps.
been
is usually quite
phenomenon such
of
have
accurately,
the values of
are
their
simple electrical
as
that
masses
than
effect
this
widths
experi­
the
Hall
intrinsic
at
but
to
of
least
optical
estimate
gap
widths more direct and accurate.
Numerous
correlations
have
been proposed bet­
ween band gaps
and various other quantities e.g., single
bond
heats
energies,
of
formation
per
molecule,
atomic
268
numbers,
Welker's
binding
gap
30
Postulate,
energy
energy
that
the
related
directly
to
or
energy
or
posed
linear
relation
in
bond
the
diamond-type
energy,
a
lattice
two
of
length so that
of
the
bond
a
length
atomic
energy
or
and
gap
by
of
butions
this
to
proposed
a
relating
the
between
refractive
An
diamond
results
homopolar
with
the
as
Suchet
to
pro­
energy
covalent
at
the pure
structure,
a
function
28
relates
of
the
formulae allowing
the
and
and
the value
heteropolar
Kumar
the well
index
21
the
electrons
type
.
Recently
of
refractive
et
contri-
al.2 2,
known Moss
the
energy
have
formula
gap
for
22
Realising the success of Kumar et al.,‘
have
gaps and
proposed
refractive
excellent review
index
24
The
the reciprocal
relationship
strength
value.
al2~*'26
band
halides.
the
fact,
to
the
parameters
Goodman
shared
empirical
modification
alkali halides.
et
bond
of
the gap energy of
similar
means of
In
on
length and
related
of
structural
computation
Reddy
is
semiconductors
exhibit
energy.
to
atoms.
connected
equally
bond distance
should
depends
number
between bond
the minimum
covalent
great
structure.
between
the bond
energy
indirectly
binding
a
gap
According
and the effective charge of
is
are
gap
21-30
etc.,
,
electronegativities
and
energy
a
linear
indices of
relation
the
alkali
on the relations between
gap of
semi-conductors has
269
by Moss.*^
given
and
gap
Different
refractive
observations
index
were
between
by
made
energy
Ravindra
32-34
et al
6.2
Lattice Energy
The
compound
is
energy
the
of
the
energy
crystal
lattice of
released when
an
ionic
ions come together
from infinite separation to form a crystal
The
theoretical
was
initiated
of
an
ion
MX
+
M ( 9)
treatment
by
pair
electrostatic
Born
M+,
and
X
energy
(i)
(S;
of
the
Lande
2.
separated
of
ionic
lattice
Consider
by
a
attraction
is
energy
the
distance
energy
'r'
obtained
the
from
Coulomb's law.
Uo = Z1 Z2 e2/4»cor
In
the
crystalline
actions.
The
lattice,
summation
•••
there
of
all
will
the
be
more
(2)
inter­
interactions
is
known as Madelung constant M.
U
There
are
o
M Z-
various
lattice energy,
Z2
74
expressions
(3)
Qr
for
the
which are presented here.
calculation
of
270
Born-Lande (or) Born Equations
6.2.1
2
„
2,
(Na h Zx Z2 S ^
(Na B/r )
Uo *
=
Avagadro
ions,
number,
r
B is a constant.
is
If
the
(4 )
distance between
( 5 U/ 6r)r_r _q
unlike
at
the
o
equilibrium distance r^ of the ion is utilised then
A
U
'r'
is
data.
by
the
For
Born
M Z. Z _ e
12
exponent
majority
Pauling
are
successful
in
(1
obtained
calculations
sufficient.
(5)
-
from
'n'
compressibility
values
suggested
The above equation
predicting
accurate
is quite
lattice
energy
values.
The Born-Haber Cycle
6.2.2
According
reaction
is
one
is
or
the
to
same
several
Hoss's
whether
steps.
law
the
The
the
enthalpy
reaction
Born-Haber
takes
cycle
represented for the formation of ionic crystal.
AH
M(g)
I .E
-»
M+(g)
A*
AH
&HE.A
AM
Ax(g) ----- —» X-(g)
AX.
M ( s)
+
AH
+l>X2(g}-- ----- » MX(s)
U
of
a
place
may
be
271
AH. = ah
+
f
AM
AH. v +
AX
The terms
A H,„ and
zation
the
For
of
gaseous
sociation
cule.
AH, _
I.E
metal
and
the
AH^
energy
Ionization
.
E.A
+
(6 )
U
A H,„ are the enthalpies of atomi-
non-metals
(Bond
+ A H
is
plus
energy
non-metal
the
RT) . of
I.E./
respectively.
enthalphy
of dis­
the diatomic mole­
and
electron
affinity
E.A./ may be obtained from literature.
6.2.3
Kapustinskii Method
Kapustinskii
constant/
5-7
has
internuclear
formula
of
an
ionic
crystal
structure
a
interrelated
distance
and
compound.
In
reasonable
Madelung
the
the
empirical
absence
estimation
of
of
lattice
energy can be obtained from the equation
120200 V Z1 Z2
(1 _ 34^5
(7)
where 'V'
is
compound and
the number
'r
values obtained
'
is
of
ions per
estimated
f rom
from this method
'molecule'
of
ionic radii.
fall
on
the
the
The
low side
of the acceptable lattice energies.
6.2.4
The
Quantum
Mechanical
Prediction
of
Lattice
Energies
Hylleraas^
used
one
electron
wave
functions
272
of
the
hydrogenic
entire
type with
computation
could
(Z
be
screening
performed
so that
analytically.
5/16)r
Exp
=
nuclear
...
ah
Z = 1 for hydrogen;
3 for lithium etc
a,
(8 )
is the Bohr
unit distance.
Kudrivtsev 's Theory
6.2.5
. 11
Kudriatsev ' s
elastic
sound
to
constants
velocity
1
U
m
,
theory
get
utilises
lattice
the
the
data
The
energy.
th e.lattice energy
rel ating
on
of
the
is represented by
crystal,
u2
=
nmQ r + (rRT/M)
o
m
and m
where n
are
function,
r,
principal
specific
absolute
R,
—
constants defining potenti al
T,
M
and
healt s /
energy
represent the ratio of
o
t h e universal gas constant,
Q
molecular
temperat ure,
(9)
wei ght
of
the
approximations,
the
substance and the potential energy.
Incorporating
proper
eg. (9) can be written as
U2
=
(rn./a)Q.
m
where
and
n^
Q.i
is
are
(10)
ii
the
lattice
constants,
U
energy
m
is
of
the
evaluated
crystal,
using
'a*
elastic
273
constants.
Kudriatsev's
method
predicts
approximate
value of lattice energies.
6.2.6
between
within
New Relation developed
by the Author
Assuming
that
a
lattice
energies
a
obtained
there
molecular
the
is
and
group
following
linear
relationship
inter-ionic distances
halides,
relations
for
the
various
author
25
molecular
group halides are as follows:
U
=
-49.51(r ) + 324.72
o
(Alkali halides)
...
U
=
-144.30(r ) + 901.00
(11)
(12)
o
(Divalent halides)
and
U
where U
the
=
and
-47.75(r ) + 286.54
o
(Ga, In, T1 halides)
rQ
are
lattice
inter-ionic distance
above
relations
represent
the
are
best
intern-ionic distance
has
the
been
Carlin
35
taken
(A°).
unique
fit
(A°)
from
with
(13)
energy
The
in
the
(kcal/mole)
constants
sense
experimental
related
Boswarva
to
and
the
in
and
the
that
they
data.
The
above
Murthy
20
work
and
274
6.3
Band
Gap#
Bond
Energy
and
Electronic
Polari­
zability of Ionic Crystals
The following methods are used for the
deter­
mination of band gaps.
6.3.1
Experimental Methods
Long
experimental
gaps
of
of
the
3S
37
Bube
and
methods
for
insulators
band gaps
commonly
measured
the
and
of
have
described
determination
semiconductors.
any
the
of
one
of
The
band
values
semiconductors are
insulators and
by
the
following
six
simple methods.
1.
dependence
Temperature
in
an
intrinsic
temperature range
of
the
conductivity
material
(or)
in
a
in which intrinsic conducti­
vity dominates.
2.
Temperature
dependence
of
R
intrinsic
material
in
an
temperature range
the
Hall
constant
(or)
in
a
in which intrinsic conducti­
vity dominates.
3.
Variation
of
absorption
as
a
function
of
wavelength.
4.
Variation
of
with wavelength.
photoconductivity
excitation
275
5.
Variation
of
luminescence
excitation
with
wavelength.
6.
Variation ol
resulting
length,
of
luminescence emission
free
electrons
recombination
of
the
and
the
recombination
free holes.
free
holes is observed,
ition,
from
with wave­
When
electrons
and
the
free
through a radiative trans­
maximum
of
the
emission
spectrum
corresponds to the band gap.
Values of
methods
are
band gaps,
summarized
for
a
determined by different
number
of
materials
by
the meaning
of
the band gap
n , 37
, r
36
Bube
and Long
6.3.2
Bube's Method
Keeping
for a compound,
ion
between
acterizing
in
one can expect a fairly close connect­
the
a
mind
electronegativity
compound
and
the
difference
band
gap
char­
of
that
found
that
compound.
From
compounds
and
with
those
it
is
observations
ionic
bonding
compounds
smallest values.
vity
the
with
Using
possible
to
have
it
is
largest
covalent
binding
the concept of
construct
a
band
gaps
have
electronegati-
simple
formula37
276
=
where
E^
is
N„
X - Nm
A,, + A.,
M
X
C
the
band gap,
(14)
C
is
a
the
number
of
valence
electrons
of
number
of
valence
electrons
of
constant,
the
atomic number of the cation and A
anion,
N
N
cation,
is
the
is
the
is
the
X
m
am
M
is the atomic number
A
of
the
of 43/
anion.
If C
then E^(eV)
6.3.3
is
empirically assigned
the value
is calculated by the above equation.
Scalar Method
f
A
empirical
energy
systematic
practical
gap
ledge ' of
of
the
their
search
has
analysis
III-V
been
which
can
semiconductors
covalent
and
ionic
made
for
predict
from
radii,
a
an
the
know­
in
the
absence of an adequate theoretical treatment.
This search has yielded the expression
B(R
Eg = —
a
logir,
^ 10
38
V) R
V
Cov.
Cov______
B(R,
III) T,
III
ion
ion
(15)
where
E
g
is
electron volt
equal
valent
to
40
radii
the
at
and
energy
Qk;
R.
ion
(in A°)
gap
(1/a)
and
of
of
is a
the
semiconductor
constant
in
approximately
are the ionic and co­
cov
the group III and group V
R
277
elements
respectively.
B(R.
.Ill)
lon
valent
and
The
are
atomic
ionic
atoms
quantities
^
radii
B(R
constants
obtained
from
for
the
.V)
and
the
co­
cov
zero
inter­
cepts of the curves.
6.3.4
Modified Ravindra Method
Nagabhushana
for
a
the
energy
theoretical
Rao
gaps
of
has
obtained
compound
approach.
40
Ravindra et al.,
have
39
For
relation
semiconductors
a wide
proposed
a
an
from
range of materials
empirical
relation­
ship of linear form.
n = 4.084 - 0.62 E
...
(16)
s?
A
slight
readjustment
of
Lorentz-Lorentz
relation yields
2
B + 2a
n
where
B
=
=
(17)
B~
(3XV/4ttn),
V
being
molecular
volume
(m/p )
and a = polarizability.
A
readjustment
of
the
terms
in
eq.(17)
yields
(B/a)
=
l(n2 +
Z)/(n2 - 1)]
(n2 - 1)
a
(n2 + 2)
B
1
3
0 ^ 2
2 + n
(18)
278
Substituting
the value of
'n'
from Ravindra's relation
and on simplication value of a is given by
a =
B
1 - ------------^-----------1 + 8.3395 (1 - 0.1518119 E^)
y
(19)
...
The above
equation
is
used
for
the
calcu­
lation of band energy gap.
6.3.5
New Relations Developed by the Author
The
semiconductor
cations
in
filters
index
of
refractive
considerable
modulators
a
design.
is
of
material
Among
etc./
is
a
the various
where
key
parameters
the
a
role.
Interrelations
are
lowest
necessary
the
a
appli­
for
refractive
parameter
index/
parameters
of
importance for
refractive
dominant
indices
integrated optical devices such as switches/
and
of
evaluation
energy
for
controlling
band
gap
connecting
the
device
plays
these
estimation
the
two>
of
the
following
re­
refractive index of newly developed materials.
The
author
25
lations
for
the
energies
and
electronic
halides as follows:
has
deduced
determination
of
the
band
polarizability
of
gaps/
the
bond
alkali
279
Method I
The
energy
gap
refractive
by
the
well
index
known
is
related
to
relation 41
Moss
the
and
is
given by
E„ n
=
G
where
£_
and
Kumar
et
22
al.,
G
n
(20)
173
are
energy
have
been
gap
and
given
refractive
the
index.
following modified
equation
=
Eg n
In
the
present
...
52
study,
the
author
25
(21)
proposes
a
relation, which is similar to eq.(21)
E„ n
=
G
Again
it
between
cular
is
(22)
44.8
assumed
that
refractive index
group
following
halides.
equation
a
linear relationship exists
and energy gap within a mole­
Based
is
on
proposed
this
25
assumption,
by
the
author
(in
E
G
for
'n'
eV) .
and
E
refractive
Experimental
from eq.(23).
the
are
evaluation
values
(eg)
23)
—9.12n + 22.18
where
the
index
of
n
and energy
are
used
to
gap
get
The E„ values thus derived are used
G
of
electronic
polarizabilities
(a)
280
of the alkali halides utilizing the equation
./44.8/E -1
0.396 x 10
a
-24
M
d
G
y4T.87iJ>2
The
symbols
usual
of
on
involved
significance
in
the
and
are
above
equation
similar
to
22
Kumar et al. The above equation
Lorentz-Lorentz
presented
by
equation.
Kumar
The
22
et al», ,
have
the
their
equation
is derived based
experimental
have
above equations to solve for n,
(24)
been
E„ and
used
data
in
the
a values.
G
Method II
Manca23
connects
indices
the
of
and
band
alkali
gap,
Moss41
relations
bond
halides.
energies
Reddy
et
are
and
different
relations
for
band
energies
of
halides.
The
author 43
the
following
relations, suitable to
and bond energies for alkali halides.
refractive
al.,2^3'26'42
proposed
alkali
directly
gaps
and
have
bond
arrived
at
evaluate band gaps
They are:
EG = Es + 3’5
(25)
Eg = 3.3 + 1.08 (Ax2
(26)
Es = 2 + 0.54 (AX2)
20
(27)
(28)
281
where
E„,
E_,
O
O
energy,
AX
Pauling
corresponding
vely.
index.
atoms
are
and
the band gap,
proposed
connects
relation
of
index
a
energy
is
single bond
difference
refractive
et al.,22
which
The
n
electronegativity
Kumar
relation,
and
and
form
the
respecti-
modified
gap
the
of
Moss
refractive
(for
alkali
halides only)
Eg n4
In
analogy,
equation
=
the
which
52
author
relates
proposes
between
43
another
single
bond
form
of
energy
and
index
(n)
refractive index
E
en
=
20
S
For
the
evaluation
of
refractive
and electronic polarizability (a) of the alkali halides,
the author proposes the following relations
n
= 2.049 -0.11 E.
...
(29)
O
0.396 x 10
-24
0.012 E^ - 0.451 E„ + 3.2 „
S&__________________________
0.012 E^ - 0.451 Ec + 6.2 d
O
’
O
...
In
the
above
equations
E
C?
andE„
are
o
in
eV.
(30)
Here M
M
282
and
d
are
molecular
halides.
The
electronegativites
weight
a
and
b
are
approximately
single
bond
refractive
indices
and
may
ing
a
fairly
in
similar
scaled
a
to
good
down
In
is taken
values
of
other
para-
22
the
2
the
as constant,
one
Manca 23
has
G
.
fundamental
all
energy
4
n .
or
present
A III B V semiconduct­
for
medium
factor
energies,
and Kumar et al.
2.
on
dielectric
alkali
E_ = a(E
- b), where
G
s
with
the
value
of
'a'
relation
Based
of
23
constants
compounds.
that
Manca
24
if b = 0.4653
achieve
proposed
to
equal
investigation
density
necessary
meters are taken from Pauling
According
and
principle
levels
41
Moss
are
by
a
proposed
n4/X1
=
77
...
(31)
=
95
...
(32)
refractive
index
the relation
corresponding to
E
n4
G
where
and
E
e
is
the
the
energy
dielectric
Relations
approximation
and
diamond
average,
for
gap,
constant
(31) and
some
n
the
of
the
(32)
materials
structure but Ravindra
of
semiconductor.
are
the
in
good
zinc
blende
find
that,
on
it under estimates the value of the refractive
283
•
index,
with
the
PbTe.
In
order
maximum
to
error
being
minimise
the
for
the
errors,
case
of
Ravindra
33
proposed a modified Moss relation, namely
4
E„ n
Relation
(33)
conductors
maximum
=
108
holds
fairly
irrespective
error
in
of
the
for most
well
their
of
structure
calculated
value
...
(33)
the
semi-
with
the
of
the
refractive index, being ±24%.
On
the
same
lines
described
above,
the
author
proposes another form of equation which relates between
single bond energy and refractive index.
n4(E
- 1.0097) = 96.9
...
(34)
s
The
the
above
refractive
equation
index
is
used
of
in
evaluating
various
compound
semiconductors.
Elaborate
and
presented in
clear
discussion
on
the previous chapter,
bond
as
energies
is
such the details
are not presented here.
6.4
Results and Discussions
The relevant data from the literature 20 ' 22 3S
are utilised
ionic
to
crystals.
evaluate
The
the
lattice
evaluated
energies
lattice
for
energies
some
for
284
alkali
halides,
divalent,
Ga,
In and Tl halides,,
using
eq.(ll) to (13) are presented in Tables 1, 2 and 3.
A keen observation of Tables 1,
2 and 3 reveal
that a linear relation between lattice energy and inter­
ionic
distance
halides
holds
reported
good
in
in
the
the
case
present
Ga,
with
In
and
Tl
halides
are
the experimental values.
average
percentage
sented.
Kapustinskn
predict
a
compared
values.
are
The
reason
for
is
and
and
Csl)
also pre-
11
theories
different
Merits
results
demerits
demonstrated
to
the
simplified
In
the
case
deviation
and other
of
and
for
divergence
are
comparable
divalent
beryllium
in
the
with
halides.
case of
of
of
by
is attri­
involved
halides,
predicts
the
Kudrievstev's
theories
assumptions
energies computed with eq.(12)
except
agreement
LiF
values
clearly
the
from experimental
theory 12 ' 13.
sults
for alkali
excellent
Kudr iavstev1 s
present
work
(13)
lattice
44
Waddington
buted
and
significantly
to
The
(excepting
rather
Kapustinskii's
results
in
experimental
5-7
to
types of
In the present study the
deviation
between calculated and
all
study.
energies computed with equations (11)
and
of
the
lattice
fairly good re­
experimental
The main
in
reason
beryllium halides may
values
for the
be due
to the lack of reliable experimental data on inter-ionic
227.70
192.40
181.60
169.60
200.70
173.20
163.70
241.80
210.10
201.70
183.70
206.40
191.00
184.50
167.10
211.50
179.90
170.50
160.90
188.50
162.70
161.30
146.70
211.10
186.60
178.40
166.50
193.60
170.40
163.10
151.90
218 70
185 90
176 70
165 40
194 40
169 40
162 40
153.00
NaF
NaCl
NaBr
Nal
KF
KC1
KBr
KI
156.70
a
,
171.60
170.40
178.60
176 80
Lil
196.94
201.80
189.40
190.40
190 70
LiBr
207.93
198.90
202 20
210.40
method
192.10
12
19
245.60
method
Yadav
259.30
method^ 4
Kudriavstev 1s
LiCl
(11)
Kapustinskii
227.70
Eq.
25
for alkali halides
225.90
49
Present method
cal/mole)
246.80
value
Most probable
(k
LiF
Salt
TABLE 1 - Lattice energies
in
co
CM
286
03
r—1
m
T3
m
to
>
<u
X
to
>
td
T3
3
S*S
1
'S
03
•jj
Q)
CTi
&
CM
r~H
o
•
H
o
M-•
ID
rH
o
in
VO•
in
rH
o
to
•
CO
o
03
o
in
CM
to
rH
03
1—4
ro
CO
•
CM
CO
*H
o
ro
in
in
rH
o
co
03•
'rH
M*
o
to
CO
C3
rH
i
i
i
i
o
03
o
o
o
o
03
o
O'
o
03
ro
rH
rH
X)
o
JZ
O
o
a>
E
o
CM
o
o
f-*
rrH
'U
o
r~-
o
CM
o
03
03
rH
rH
o
ro
ID
rH
CO
rH
CM
-H
•r|
Q,
10
!*3
o
sz
<0
g
rH
00
rH
in
-0<
r-
rH
<-i
o
rH
o
03
m
n
r-
•^r
CO
rH
rH
in ,
o
o
03
rH
o
38
to
£«•
4J
CO
•r|
03
o
ai
•
*H
rH
in
CM
T3
O
x:
4J
a>
rH
Eq.
-U
0)
(0
(V
o
VO
co
03
M3
00
ro
in
in
rH
rH
M3
rH
rH
rH
■tr
CM
rH
u
a,
a»
TJ
<U
3
C
•H
4J
c
o
u
rH
X
<0
X5
O
a
to
o
rH
tu
03
03
0)
3
o
03
m
>
m
r-i
o
o•
O
m
CO
ID
in
rH
rH
rH
•
•
o
o
CM
o
rH
•
n
in
•
co
•
00
r-
rH
rH
rH
rH
rH
o
•
C^*
in
o
r"
•
CO
rH
s
<0
■U
c
0)
o
u
0)
ac
o
(U *H
cn tj
0)
rH
4J
XI
03
rH
<d
to
rH
tu
X
a
o
X
ns
V4
m
X
ns
H
X
as
rH
U
fa
10
u
CQ
u
u
to
to
u
(0 <0
U *H
M
0}
u
ttl >
> Ql
< *o
287
TABLE 2 - Lattice energies (k cal/mole) for divalent halides
Crystal
Present
method^”*
Carlin
Experi-
Others
mental —
,
44
Ref. 13
values
Eq. (12)
BeF2
699
839
BeCl2
649
BeBr2
Ref. 45
-
-
-
720
-
-
-
625
692
-
-
_
BeI 2
698
665
-
-
-
MgF2
646
700
-
-
-
MgCl2
587
599
-
-
-
MgBr2
564
575
-
-
-
Mg I
538
547
-
-
-
CaF2
598
628
624
532
622
CaCl2
539
537
-
-
-
CaBr2
516
514
-
-
-
Cal2
486
491
-
-
-
SrF2
584
592
588
536
592
SrCl
516
509
506
395
496
SrBr,
494
488
Srl2
464
463
BaF2
566
557
566
479
561
BaCl,
494
485
BaBr.
470
466
Bal
440
439
288
TABLE 3 - Lattice energies (k cal/mole) for Ga, In, TL halides
Crystal
Present study^~*
Carlin‘S
Eq. (13)
Others
Ref. 13
Ref. 48
GaCl
181.30
-
_
-
GaBr
174.10
-
-
-
Gal
163.50
-
-
-
InCl
171.80
170.10
-
-
InBr
165.00
161.00
-
-
Ini
154.90
153.90
-
-
T1F
186.90
195.00
-
-
T 1C 1
167.80
170.90
168.70
172.80
TlBr
161.40
166.60
178.90
169.10
Til
161.60
162.10
289
distance and lattice energies.
halides
of
CaF^,
BaF^^
SrF^
In the case of divalent
and
SrCl^
the
44,45
in good agreement with experimental values
to Kudriavstev's values
The
are
the
two
solid.
of
is
an
sublimation
quantities
kcal/mole
kcal/mole.
produce
of
to Katzin 46 ,
54.3
energy
are
compared
13
important
According
NaCl
187.0
heat
results
and
both
the
and
any
evolution
of
for
heat
the
Therefore
lattice
the
solution
of
energy
process
order
and
sublimation
lattice
solution
energy
of
50
is
must
to
60
kcal/mole for the process, which can be written as
(M+ X"),
.
(vap)
=
(M+ X~),
,
.
(soln . )
This equation relates both for solid and solution.
heats of
sublimation
increasing
halogen
and
The
lattice energies decrease with
Z.
For
alkali
metal
halides
for
which vapour inter-atomic distance data and heat of sub­
limation data are available,
the M+ - X
puted
a
for
measured
M
the
vapour
-
distances
X
are
true for the fluorides of
the
fluorides
distance
computed
M - X spacing,
expected.
of
the
is
in
little
the
again
larger
vapour.
the alkali
alkali
distances com­
The
the
same
earth metals.
earths,
greater
than
the
than
M
the
2+
is
For
-
X
measured
consistent with the electrostatic picture
From
magnesium
up,
however,
the
M*2 +
Cl
290
spacing
drops
slightly
below
that
of
the
measured M
-
X distance and the discrepancy increases with increasing
Z of the halide.
For
ween
the
spacing
the
beryllium
electrostatic
becomes
additional
very
heat
salts,
the discrepancy
computation
large.
One
contribution
and
might
from
the
measured
expect
the
bet­
that
formation
an
of
covalent bonds over that of expected from purely electro­
static
small
approach
spacing
of
the
results
ions,
from
because
a
of
the
large value of
apparent
(U
-
AH).
Here the divergence is attributed with the above spacings
and
the contribution of
Moreover,
is
estimated lattice energies in the present work
mainly
dependent
Covalency plays
salts.
have
Katzin
the
organic
46
has
an
indication
phase.
In a
greater
the
covalency
in
He
of
the
out
that
concluded
may
heat of
covalency
covalency
energies
of
the
Ga,
In+
that
marked
the
heat
In
in
the vapour
one can say that the
in
sublimation.
the case of
a very
salts
be regarded as partially
degree of
degree of
A°).
beryllium
deviation, and
also
correlation
distance,
the case of beryllium
rather different way,
lower the
was drawn
r(inter-atomic
pointed
solubility.
sublimation
lattice
on
a dominant role in
largest
of
the
the formation of covalent bond.
the vapour phase
The same conclusion
and Tl
halides
is
are
salts.
only
If
equal
the
to
P.91
those
out
of
salts,
the
computed
M
-
X
distances
smaller compared to the measured ones .
halides
and
K
for
which
lattice
values
energy
are
of
both
heat
available,
come
The thallous
of
show
sublimation
even
stronger
covalency effects.
In
is
to
the
evaluate
experimental
by
the
author.
alkali
n
halides
4.
These
E
the
electronic
estimated.
obtained
results
(shown
in
cules).
In
the
only
The
indicates
equation(24) .
evaluate
E
errors.
the
5.58,
evaluated
for
simple
are
are
presented
and
are
of
in
Table
utilized, hence
alkali
halides
are
polarizabilities
thus
in
with
good
agreement
(excluding
whereas
the
proposed
gaps
study
reliability,
the
percentage deviations
cases
average
on
and
are
The
eq.(23)
aim
equation
obtained
5)
present
Thus
is
G
Table
based
main
author's
electronic
minimum
the
thus
values.
both
22
the
author's
using
polarizabilities
in
others.
indices,
and
The
is
values
G
energy
experimental
' a '
E
values,
presented
of
the
Utilising
values
G
study,
reliable
refractive
experimental
the
present
the
it
is
10.9
more
reported
deviation,
applicability
proposed
mole­
average %deviation
percentage
and
fluorine
are
new
accurate
of
relation
by
5.58
the
to
with minimal
O
H
in
fN
rH
rH
rH
0
on
•
in
in
N•
o
id
rH
in
rH
in
•
rH
lO
•
rH
rH
CM
lO
CO
CN
in
<n
•
iH
10
rH
o
o
in
o
0
t"
m
o
o
o
o
t
o
in
•
CO
(N
rg
<N
<f
•
rH
4-1
<U
05
CO
rg
t'-
rnO’
CM
ro
m
10
O
i0
rH
rH
rH
rH
id
rg
in
rH
in
•
00
if
m
•
rH
00
10
•
o
rri0
CO
10
m
rg
rH
rH
rH
*H
rH
00
N*
•
rH
rH
m
•
ro
in
in
•
rH
rH
10
O’
r-
1-1
01
m
o
rH
o
r«
rH
for alkali halides
m
(N
'O
3
<N
W
IN
C
•
o
iH
Q)
W
(U
Vi
CU
rH
10
•
•
rH
rH
0
rH
O’
M
cn
rg
(0
4J
(eV)
4 - Refractive indices and energy gaps
TABLE
4J
0
c
Cl)
E
*H
«
o
i—i
0
CO
00
in
in
rH
rH
»H
rH
<H
o
o
o
o
o
rH
o
rH
o
o
m
•
o
10
•
id
0
CO
•=f
0
in
in
rH
rH
r-l
rH
rH
o
o
o
00
O
10
o
o
10
ai
00
r-
10
o
o
o
o
o
in
o
rH
Vi
<U
a
x
«
in
CN
o
u
>1
-o
3
4->
m
n
CN
-u
c
(1)
m
a)
0
^f
O’
(N
00
0
00
m
Vc
cu
•
CM
rH
i
rH
i
o
i
o
01
in
A\
o
*
00
o
r•
rH
A\
CO
0
•
n
•
00
00
•
r-
<N
•
10
•H
A\
A>
<y
i—i
3
O
<U
i—I
o
E
Cm
•H
>4
iH
o
Vi
CO
•H
►J
M
*H
i4
Cm
CO
2
rH
Vi
o
m
z
cq
m
M
m
z
z
rH
fa
o
C*5
Vi
CQ
!*5
M
Con td
292
O
O
•
00
00
•
o
A\
A'
Average percentage deviation
6.30
kq
Csl
CO
CsBr
A\
CsCl
10.00
7.70
00
CsF
Rbl
7.70
00
RbBr
8.20
CTi
RbCl
10.40
•
O’
Id
50
40
(23)
1.787
1.695
5.13
5.24
1.650
1.633
1.563
1.596
1.538
1.610
50
1.670
1.510
1.708
1.553
1.586
1.529
1.647
1.495
Ref. 22
1.441
( 22 )
£.Z>
1.612
Eq.
Present study
1.553
m*4
22
9*4
1.553
1.493
1.398
^
1.455
•
1.483
.. .
Experimental
60
20
06
m
in o
W
Present study
in
CM
RbF
J
k
(
E r»
c
06
Molecule
<
00
Table 4 (continued)
l
in
Eq.
(25)
Present study
eg
halides
c
o
e
•H
ai
a
X
-24
of alkali
3.497
4.422
6.463
2.625
4.901
5.916
3.311
4.224
5.926
2.556
4.457
5.376
3.237
6.406
1.991
5.224
4.40
o
o
o
i-H
8.10
7.20
Lil
kj
NaCl
NaBr
LiCl
7.90
KBr
KI
8.60
08*9
KCl
KF
Nal
08*6
fO
2
00*9
t-'-
7.249
Contd
8.133
1.885
1.764
1.148
o
o
•M1
7.155
6.010
6.414
6.145
o
C0
o
LiF
3.269
(24)
)
3.661
Eq.
3
25
cm
Present study
(x 10
4.120
w
CO
cn
5.90
o
t'-
LiBr
CO
2.438
C0
2.735
ifl
4J
2.976
CM
1.032
25
Energy gaps (eV) and electronic polarizabilities
o
1.095
Molecule
-
•
4-1
<1>
a:
05*6
TABLE 5
<N
CM
.
rH
u
u
7.50
5.90
in
<N
'O
3
in
cu
Average percentage deviation
Csl
CsBr
7.20
00
CsF
9.40
o
o*
Rbl
RbBr
«
RbCl
cr
2
RbF
03
,
06*9
Molecule
(continued)
c
b
<u w
6.795
9.244
4.372
5.693
6.458
8.768
6.108
7.902
4.358
5.432
6.435
8.434
5.770
6.834
8.859
5.690
3.601
10.91
5.746
5.133
Ref. 22
4.626
(24)
25
3.393
Eq.
,
Present study
3.281
22
2.537
Experimental
m
cn
r"
09*8
5
■U
«
OS *8
Table
295
00
in
m
296
The
calculated
relations
(25)
and
band
gaps
(26)
are
Noting the success of Manca
proposed
(26)]
the
for
related
above
alkali
to
and Vijh
29
formulae
halides.
The
alkali
same
is
E„
values
as
such
Table
6.
(25)
that
are
and
E_
is
G
estimated
o
cited
the
from
, Reddy et al. ,
known
through Pauling electronegativities.
values
in
[eqns.
It
o
Eq
halides
presented
23
simple
E_.
for
in
literature
author
25
22
proposes
The experimental
are
an
only
approximate,
empirical
relation
for the evaluation of EG involving accurate experimental
refractive
indices.
The
proposed
relation
25
is of
the
form
E_ = -9.12n + 22.184
G
The
estimated
known
Moss
41
E
G
and
values
are
substituted
Lorentz-Lorentz
in
the
relation
well
resulting
electronic polarizabilities values of alkali halides. The
average
percentage
5.58.
Presuming
deviation
these
E_
m
the
values
above
are
average
eq. (26) is
percentage
8.67
deviation
(excluding
25
accurate,
average percentage deviations are presented
The
study
the
in Table 6.
derived
Fluoride
is
from
molecules).
Since the agreement between different methods is good with
an
error
author *s
of
less
equations
than
±1%,
(25)
and
it
(26)
is
concluded
can
be
used
that
the
for
the
297
TABLE 6 - Single band -- gap energies (eV) of alkali halides
22
Alkali
halide
eg
ci
43
Calculated E G values
Eq.
(25)
E25
i3
Eq. (26)
9.41
13.02
9.49
LiCl
~10.00
8.34
7.62
7.02
LiBr
~
8.50
7.82
6.79
5.91
Lil
>
5.90
7.17
5.73
4.38
NaF
>10.50
8.83
13.67
9.99
NaCl
8.60
7.73
8.06
8.13
NaBr
7.70
7.24
7.19
7.21
5.80
6.86
6.06
6.00
KF
10.90
8.57
14.35
9.77
KC1
8.50
7.84
8.52
8.59
KBr
7.80
7.41
7.62
7.96
6.20
6.85
6.42
6.88
10.40
8.50
14.35
9.43
RbC 1
8.20
7.84
8.52
8.56
RbOr
7.70
7.40
7.62
0.02
6.10
6.80
6.40
7.15
10.00
8.65
15.06
8.65
o
o
8.08
9.01
7.50
7.80
7.66
8.06
6.95
6.30
7.06
6.79
5.88
9.69
8.67
>
Nal
^
KI
RbF
Rbl
^
CsF
A
N
CsCl
CsBr
Csl
Average
percentag<a
deviation
£
•
~12.00
CO
LiF
298
estimation of reliable and accurate
mated
refractive
using
eq.(34)
are
in good
with Moss
41
indices
are
for
presented
agreement with
(28)
in
are
7.
experimental
These values
values
compared
presented
in
utilising equations
Table
8.
The
(27)
results
thus
in good agreement with those of Pauling 24
average
excluding
Table
compounds
33
and Ravmdra et al.,
obtained are
The
The esti­
semiconducting
Single bond energies,
and
values.
percentage
deviation
fluorides.
A
using
slight
eq.(27)
deviation
is
7.02
has
been
observed for the single bond energies of lithium halides
using
for
eq.(28).
these
Several
deviations
halides"^'.
from
the case of
ation(28)
relation.
sense that
is
fluorides
Normally
group differs
In
of
the
The
the
first
remaining
fluorine
entirely
explanations
a
constant
it represents
it
have
from
been
that
of
in
other
member
of
a
members
of
the group.
is more pronounced
new
given
one
similar
eq.(28)
is
the over all
to
periodic
52
.
Equ-
the Moss^
unique
in
the
best fit with the
experimental data.
Utilising
eq.(29),
the
refractive
evaluated and presented in Table 8.
relation
and
polarizability
eq.(29),
can
be
indices
are
From Lorentz-Lorentz
the
evaluation
made.
The
of
relation
electronic
(30)
is
2 99
TABLE 7 - Calculated values of n for compound semiconductors
Semiconductor
Refractive index (n)
Bond
energy
Known
(eV)
Present
study
Eq. (34)
Moss
41
1Ravindra
BN
4.15
2.10
2.35
2.13
2.20
BP
3.66
-
2.45
-
-
BAs
3.01
2.63
-
-
BSb
2.80
AIN
4.63
2.16
2.71
2.27
2.24
2.31
A1P
3.11
2.75
2.60
2.37
2.45
AlAs
2.52
3.00
2.83
2.57
2.66
AlSb
2.20
3.19
3.00
2.78
2.87
GaN
4.21
2.40
2.34
2.33
2.40
GaP
2.62
2.90
2.78
2.55
2.64
GaAs
2.12
3.30
3.05
2.90
3.00
GaSb
1.83
3.79
3.29
3.29
3.40
InN
3.72
2.42
2.05
3.10
2.94
-
InP
3.10
3.04
I nAs
1.64
3.50
3.52
4.03
4.16
InSb
1.39
3.95
3.99
4.79
4.95
SiC
3.12
2.59
2.60
2.35
2.43
ZnS
4.39
2.27
2.31
2.27
2.35
ZnSe
3.83
2.43
2.42
2.46
2.54
ZnTe
3.54
2.70
2.48
2.54
2.63
CdS
3.68
2.38
2.45
2.51
2.59
CdSe
3.20
2.49
2.57
2.73
2.82
CdTe
2.93
2.70
2.66
2.85
2.94
4.28
8.68
8.74
Average percentage
deviation
_
33
M
53
0
RbF
C sl
1 .4 8 2 2
1 .5 4 4 9
1 .5 9 1 1
1 .6 5 7 1
5 .5 3
3 .9 9
3 .7 6
3 .3 4
7 .0 2
4 .8 5
4 .3 8
3 .7 4
9 .2 2
1 .4 9 8 7
1 .5 7 1 3
1 .6 1 9 7
1 .6 8 5 7
4 .9 4
4 .4 9
4 .2 3
3 .8 5
7 .5 2
4 .6 7
4 .1 6
3 .5 6
1 .9 9 1
4 .0 8 0
5 .2 2 4
7 .2 4 9
2 .5 3 7
4 .6 2 6
5 .7 7 0
3 .6 0 1
2 .6 2 5
4 .9 0 1
5 .9 1 6
8 .1 3 3
3 .3 9 3
5 .7 4 6
6 .7 9 5
9 .2 4 4
4 .3 7 2
5 .6 9 3
6 .4 5 8
S .7 6 8
7 .9 4 6
3 .4 1 8
5 .6 2 8
6 .8 7 2
9 .0 1 9
4 .1 7 6
5 .2 9 2
6 .4 2 2
2 .6 9 1
4 .8 9 7
6 .0 1 5
6 .1 0 9
CO
1 2 .5 5
O
Vi
4i
a
ai
01
u
w
to
<
>
d e v ia tio n
4 .1 6
3 .5 6
5 .1 5
4 .5 8
5 .0 0
4 .3 4
3 .9 0
3 .3 0
1 .1 4 8
3 .2 3 7
4 .3 8 1
6 .4 0 6
3 .2 6 9
1 0 .9 0
in
CTl
r"
r-'
CsF
CsCl
CsBr
Rbl
RbCl
RbBr
1 .4 9 1 0
1 .5 7 1 3
1 .6 1 8 6
1 .6 8 0 2
5 .1 2
4 .5 0
4 .2 0
3 .7 7
7 .5 2
4 .6 1
4 .1 6
3 .5 6
1 .0 3 2
1 .8 8 5
3 .4 9 7
4 .4 2 2
6 .4 6 3
1 .7 9 1
3 .5 7 9
ID
5 .0 7
4 .3 4
3 .9 1
3 .3 5
7 .1 8
4 .3 8
3 .9 4
3 .3 8
Kumar
2 .9 7 6
4 .1 2 0
6 .1 4 5
0 .9 4 4
2 .4 5 6
3 .2 7 2
5 .5 0 5
04
KCl
KBr
KI
CO
O*
KF
1 .4 6 2 4
1 .5 8 3 4
1 .6 3 7 3
1 .6 7 8 0
5 .2 5
4 .2 8
3 .8 7
3 .3 9
io
5 .3 3
4 .2 3
3 .7 4
3 .3 7
CO
CO
CM
o
r-j
O
CO
NaF
NaCl
NaBr
t0
1 .3 9 8 6
1 .5 1 6 3
1 .5 7 2 4
1 .6 4 5 0
cu
CO
CO
i£>
4 .1 6
4 .7 4
3 .2 1
E q. (3 0 )
e le c tro n ic p o l a r i z a b i l i t i e s ( a )
04
L il
m
>
t"
00
00
o
4 .3 3
3 .6 7
0
E q. (2 8 )
rn
4 .9 7
3 .7 9
3 .3 5
2 .8 3
E q . (2 7 )
E q . (2 9 )
U
5 .9 1
( P a u lin g )
E s tim a te d
- S in g le b o n d e n e r g ie s (e V ), r e f r a c t i v e in d ic e s (n ) and e l e c t r o n i c p o l a r i z a b i l i t i e s
(x lO -2 4 cm 3 ) o f a l k a l i h a lid e s 13
<D
■U
•H
l-l
L iF
L iC l
L iB r
A lk a li
h a lid e
TABLE 8
SO
J
6 .8 3 4
8 .8 5 9
069*5
toQ)
qe
Cn
W
4
'• j e
04
ZZ
c
u
D
<u
Q)
tn
*j
c
<u
o
cc
UJ
301
interesting
the
electronic
energy.
in
and
The
significant
mainly
polarizability
evaluated
because
directly
electronic
to
it
single
polarizabilities
22
fair agreement with Kumar et al.,
values.
ation
(30)
single bond
is
useful
energy
of
and
reliable
alkali
electronic polarizability.
relates
halide
in
are
The equ­
evaluating
directly
bond
from
the
its
30 2
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