Indian Journal ot' Pure & Ap pli ed Ph ys ics
Vol. 38. Jul y 20()O. pp. 495-49X
Lattice energy of mixed alkali halide crystals:
Evaluation from sound velocity studies
M Subrahma nyam
Department or Ph ys ics. VR Coll ege. Nellore 524 00 I
and
E Rajagopal & N Man ohara Murth y*
Department or Ph ysics, Sri Kri shn adev araya University. Anantapur 5 15 003
Rece i ved 19 January 2000: revised 22 M ay 2000; accepted 8 June 2000
Based un single-crys tal elasticcunstant data and employ ing Ku dri avtsev's th eory, w hi ch relates the latti ce energy o rth ecrystal
U to th c mca n soun d ve locity Lim in the crys tal. th e latti ce energies or NaCI-KC I. NaBr- KBr. KI -KBr anc! KCI -Rh CI mi xed
crystals ha ve heen ev aluated. In general. th e latti ce energies o r these mi xed crystal have been rounc!to decrease with increase
ill the cOIl ce ntrati (lIl Or the secolld (;()m flOIl enl. T he flresent study high li ghts the appli cati on or Ku cl ri avtsev's theory in predictin g
th e latti ce energies or mi xed iOlli c crystals making usc or sou nd ve locity measurements.
1Introd uction
Crystalline state properti es of toni c solids mainly
depend upon th e nature of th e chemi cal bond betwee n
the ions and the interactio n energies. The elast ic constants and their press ure deri va ti ves are indi cative of th e
short range co ntributi ons to th e co hesive energy. Lattice
energy of ion ic crysta ls is an important parameter whi ch
is direc tl y co nnec ted to th e binding forces in ioni c so lids
and in turn to the elas ti c co nstants. Rece ntl y Subrahmall ya m e f 0/. 1-.1 have appli ed Kudri av tsev's(' theory for the
evaluation of lattice energies of a few ioni c and mi xed
ioni c c rystals and the temperature depe nd ence of latt ice
energy. The prese nt wo rk deal s wi th the evaluat ion of
lattice energy of mix ed alkali halide crystals NaCl-KCI ,
NaBr-KBr, Kf + KBr and KC I + RbCI using Kudri avtsev' S6 th eo ry and elast ic constant data from literature.
A mi xed crystal has properti es analogoll s to those of
th e host crystal s. The prese nce of a small impurity
co ntent mi ght change the strength of the repul sive forces
whil e maintaining th e crys tal sY lllmctry. A sur vey of the
lite rature shows that th ere are a few th eo reti ca l stud ies 7- lo ca rri ed out on th e static and elasti c prope rti es or
ll
mix ed crys tals. Argy ri ou and Howa rd mad e an attem pt
to ca lculat e the elec trostatic potenti als, Madelun g co nstants and interacti on ene rgies in ioni c crystals. Sri nil
vasa Rao and Sa nya l " studi ed th e press ure induced
stru ctural phase tran siti ons and elast ic co nstants at hi gh
press ure for sodium halides usin g a mod el potenti al that
includes many body inte ractions due to electron shell
l3
overlap. Reddy and Suryanaraya na observed a linea r
decrease in C II with temperature and a steady increase
in C44 with temperature in KCI- SrCI" mixed crysta ls. it
is reasonab le to ex pect that the vari ati ons takin g place
in the ph ys ical properties of pure and mi xed ionic crystals with vari ati on in temperature, press ure and chemical
co mpos ition will depend on th e lattice energy. In view
of th e above observation s, it is of interes t to evaluate the
latti ce energy of mix ed alkali halide crystals fro m their
elastic properties .
2 Theoretical Considerations
The sound ve locity U ill is related to the lattice energy
U of an ioni c crystal. acco rdin g to Kudri av tsev ' s th eI> by
ory,
... ( I )
where y represents th e ratio or spec ifi c heats whi ch
ca n be taken to be equal to unity and M represents the
molec ul ar wei ght of the crysta l, 11. 1 is a constant whi ch
depends upon the latti ce structure. The detailed deri vati on is gi ven in earl ier co mmuni cati ons of the aut hors 1.-1.
Ex pressing M in kg 111 0 1- 1 and £1 m in I11 S- 1 one ca n get U
l
in J mor , III may take on va lues like 3,5 , 7,9 and 10
dependin g upon th e nat ure of bondin g present in the
cry stals. For pure ioni c crystal s 11 1 = 5 gives latti ce
INDIAN J PURE & APPL PHYS, VOL 38, JULY 2000
496
l
The calculated values of lattice energy of NaCI- KCI,
energy data in agreement with experimental data ,2 and
hence II I =5 is taken in the present study to evaluate the
lattice energy of mixed ionic crystals,
NaBr-KBr, KI-KBr and KCI -RbCI as a function of the
The crystals in the present study are cubic in nature,
The mean sound velocity LIlli is given by the relation
in Figs I, 2, 3 and 4 respectively , The lattice energy
U
III
[* (-4
=
- ul
+ 2, ] - 1/ :1
",(2)
mol % of the second component are shown graphically
values of these systems reported in literature are also
included in the above figures for the sake of comparison.
880r---------------------~
It,
The longitudinal and tran sverse wave velocities LI I
and U I of the pol yc rystalline aggregates of these cubic
crystals can be evaluated from the single crystal elastic
constant data using the Voigt-Reuss-Hill approximal4
tion ,
840
-
I"
0
800
E
3 Results and Discussion
The mean sound velocities of the NaCI-KCI , NaBrKBr, KI-KBr and KCI-RbCI mixed crystals have been
evaluated making use of single crystal elastic constant
data taken from literature"-I ?, To evaluate the sound
velocities, the density data of the mixed crystals are
necessary, For NaCl-KCI, NaBr-KBr, KI-KBr and KCIRbCl mixed crystals, no such c1ata are available in literature for concentrations for which the elastic constants
are reported , The density values for the mixed systems
of the present study have been evaluated using Vegarcl's
law l ~ , which predicts a linear composition dependence,
The same procedure was followed by Nathan et aL. 19 for
20
the case of KBr-KI system , Sirdeshmukh and Srinivas
have also outlined the utility by Vegard's law for the
estimation of densities of mixed crystal systems, Barrett
and Wallace 21 have determined the densities of the
NaCI-KCI system , For the NaCI-KCI system, the density value evaluated in the present work for a concentration of 89 mol cYD KCI in NaCI works out to 2,{)()5 X 103
kg m-' , This value very well compares with the density
value of 1,9964 x 10' kg m-' reported by Barrett and
21
Wallace for a concentration of 89,9 mol % KCI in
NaCL Similarly a den sity value of 2,034 x 103 kg m-3 is
evaluated for a concentration of73 mol % KCI in NaCI
.
71
whereas the value g iven by Barrett and Wallace- for 70
mol % KCI in NaCl is 2.0117 X 103 kg m- 3 . These two
sets of values are in excellent agreement with the experimental values, th e deviation being less than I %. For
KI-KBr system, the density data have been reported at
different concentrations at which elastic data have been
determined . The application of Vegarcl's law lH yields
density data accurate to about 4 % indicating a good
agreement between experimental and predicted values
of densities for mixed crystals.
-,
•
~
......,
760
=>
720
680
0
20
60
1.0
80
100
mol % ;KCI
Fig. I - Variation of lattice energy (U) with mol% KCI or NaelKel mixed crystal rCA) Present work, (B) Data evaluated by
22
Belomestnyka el al. using Born' s equation}). (C) Data evaluated
24
by Belomestnyka el a!. 22 using Nemilov 's equat ion j
860.---------------------~
820
~
1-
780
o
E
A
700
660~--~--~----~--~----~
o
20
1.0
60
80
100
mol "I. K Br
Fig. 2 - Variati on 01' lattice energy (U) wi th mol % KBr of NaBrKBr mixed crystal [(A) Present work. (8) Data evaluated by
Belomestnyka el at. 22 using Born's equation2:l . (el Data eV::l luated
22
by Belomestnyka el al. using Nemilov 's equation 24 j
SUBRAHMANYAM
et
SOOr-------------------~
760~--------------------1
7eo
71.0
760
720
A
700
-,_
0
o
E
497
al.:LATrICE ENERGY OF CRYSTALS
.,E
6eo
71.0
720
-"
::>
700
660
680
640
0-0
B
660t=~c~~~~~~~~~~
o
600
580L-__~____L-~-L____~__~
o
20
00
100
20
L.O
60
mol % Rb(;I
80
100
Fig. 4 - Variation of lattice energy (U) with mol % RbC! of
KCI-RbCI mixed er~stal [(A) Present work. (B) Data evaluated hy
Zwadovskaya el a/. _'i using theoretical values of r", (C) Data
evaluated by Zavadovskaya el a{25 using experimental values
of rol
mol% KI
Fig. 3 -
Variation of lauice energy (U) with mol % KI in
KI-KBr mixed crystals [(A) Prcsent work. (B) Data evaluated hy
Zavadovskaya et a[2 5 using theoretical valucs of ro , (C) Data
·
evaluated by Zavadovskaya et ({ I_.?'i
-. usmg
ex pen. menta I va Iues 0 t'
ro]
It is clear from Figs I and 2 that the lattice energy of
NaCI-KCI and NaBr-KBr mixed crystals decreases nonlinearly as the composition of the second component is
increased. For these two mixed systems, the maximum
negative deviation in the lattice energy with linear values occurs at 73 mol % of KCI in NaCI and 80 mol %
of KBr in NaBr, respectively . Similar trend of variation
of lattice energy with the increase in the concentration
of the second component in these mixed crystals was
22
reported by Belomestnykh and SlIkhllshin . They have
evaluated the lattice energies of these two mixed crystal
.
?l
.
f rom
systems uSll1g
Born ,s-an d Neml'1 ov ,?4
s- equatIons
single crystal elastic constant data. This negative deviation in the lattice energy from linear values may be
attributed to the presence of defects in the lattice. The
number of defects increases with the decreasing chemical stabi Iity of the mixed crystal.
As the concentration of the second component increases, the number of weakly bound ions in the lattice
also increases thereby paving the way for the destabili-
zation of the lattice. It is to be mentioned here that among
the alkali halide mixed crystal systems, the NaCI-KCI
system has the least stability and hence maximum deviation has taken place in the lattice energy values.
As can be seen from Fig. 3, the lattice energy of the
system KI-KBr varies in a non-additive manner with
composition. The maximum deviation in lattice energy
occurs around 63 mol % of KI in KBr. This trend of
variation is similar to the results given by Zavadovskaya
et ai .25 who have evaluated the lattice energy of KI-KBr
mixed crystals on the basis of Durham and Hawkins
26
method . In these calculation Zavadovskaya used two
different values for the equilibrium distance r" between
neighbouring ions. These results are also presented in
Fig. 3 for comparison.
For the KCI-RbCI mixed system, it is evident from
Fig. 4 that the lattice energy decreases in a nonlinear
manner with increase in the composition of RbCI-KCI.
The maximum deviation from linear values of lattice
energy in this system corresponds to an equimolar concentration of 50 mol % RbCI in KCI. This deviation from
linear values increases with the decreasing resistance of
the mixed crystal to decompose . Zavadovskaya ef al. 2,)
have also evaluated the lattice energy of the KCI-RbCl
26
system using the Durham and Hawkins procedure . The
results of the present work are similar to their studies.
49X
INDIAN J PURE & APPL PIIYS , VOL 38, JULY 2000
)
As can be see n from Figs 3 and 4 the lattice energy
data obtained in th e prese nt work are hi gherthan the data
reported in literature 2) for th e syste ms KI-KBr and
RbCI-KCI. rt may be menti oned here that th e lattice
energy of mixed crystal s has been determined onl y to a
rough approximation in whi c h no allowance has been
made fo r the energy of th e elasti c di stortion . Moreove r,
th e lattice energy values calculated under the assumption s made by Zavadovskaya e l 01.2) are incapable of
ex plaining observed changes in the ph ys ical properti es
of mi xed crystal s with co mpos ition. ]n the present work,
du e to lack of density data of th e mi xed crystal s for the
co ncentrati ons at which elasti c constants have been
reported in literature, the auth ors have made use of
Vegard's law 'x for calculating the density of the mix ed
c rysta l ~; . Any deviation of the density data calculated
from th e real values might affect the latti ce energy data
ca lcul ated. The dev iatio ns in the calculated lattice energy data of th e prese nt work with that of Zavadovskay
el 0 1.25 for the systems KI-KBr and RbC I-KCI may be
att ribut ed to th e un certainti es in vo lved in th e input data .
The negat ive dev iati on in th e latti ce energy from
linear va lues of these mi xed crystal s may be attributed
to the prese nce of defects in th e mi xed crys tal lattice.
The number o f defects increase with th e dec reasin g
chemi cal stability of the mi xed crys tal. As th e conce ntrat ion of th e seco nd co mponent increases. th e number
ofwea kly bound ions in the latti ce also increases thereby
pav ing the way for destabili zation of the lattice. Co nsequ entl y. th e latti ce energy, dependin g on the co mp os iti on of the mi xed crystal. vari es over th e co mpos iti on
ran ge wit h nega ti ve additi vit y. The dev iati on will be
more for th e leas t stabl e system and less fo r the mos t
stable system. The behav iour indi cates th e weakenin g
of the binding forces between the ions in the mi xed
crysta l latti ce co mpared with th e latti ce of th e hos t
crys tal.
In an earlier paper-l, the auth ors ha ve reported the
variati on of latti ce energy of NaCI-NaBr, KCI-KBr and
AgC I-AgB r mi xed crys tal s. In the present wo rk , studies
are made on the NaC I-KCI , NaBr-KBr, KI-KBr and
KC I-RbCI mi xed sys tems. Fro m both the studi es it ca n
be co ncluded that th e var iati on in the latti ce energy as a
fun cti on of co mpos iti on is more or less the same wheth er
one has a cati on or an ani on impurity. In ge neral , th e
co mpos iti on dependence of latti ce energy is hi ghly nonlinea r. This was also emphasized by Sirdes hmukh and
Srini vas ") in the ir rev iew of th e ph ys ical properti es of
alka li ha lide mix ed crys tals.
4 Conclusions
The res ults orthe present work hi ghli ght the applicati on of Kudriavtsev's6 theo ry in estimat ing th e lattice
energies of mixecl crystal s, makin g Lise of ultrasonic
ve locity data. In view of the fac;: that the experimental
data on the lattice energies of mixed ionic crystals are
sparse, the prese nt study beco mei; useful and significant.
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