A comment on the Born model treatment of the polymorphic transitions of the alkali halides

LETTERS

TO

A comment on the Born model treatment of the polymorphic transitions of the alkali halides*

THE

EDITOR

SO67

Equations (2) and (3), which apply separately to the two crystal structures, contain as unknown parameters the interionic distance at zero pressure re for the C&l structure, and the parameter p for (Remiued 27 March 1963) both structures. A choice of values for these three parsmeters allows one also to determine the transiIN A recent Born-model treatment of the pressure tion pressure, by finding the intersection of the transition of the potassium and rubidium halides from the NaCl to the CsCl structure@) it was Gibbs free energy curves of the two structures as functions of the pressure.@ shown that it is possible to obtain a fit of the L&VDIN@~ assumed a common value of p for pressure-volume relationship of the two phases of the two structures, and determined this value for each salt around the transition point and to account each salt from the compressibility of the NaCl for the observed values of the mechanical work phase at atmospheric pressure. This determinaaccompanying the transition, only by allowing for tion implies a rather good agreement with the a difference in the parameters entering the Born experimental pressure-volume relationship of this repulsive energy of the two phases. A subsequent phase up to fairly high pressures. He then evalupaper of SCHUMACHER@) seems to contradict this ated ro for the C&l structure from the value of conclusion, insofar as it reports values of the that the pretransition pressure, and of the volume change of rs for the NaCl phase by bung exponential parameter B in equation (1) be also the NaCl phase up to the transition, in potassium common to the two structures. As shown in Table chloride and rubidium iodide which are computed 1, for the salts in question this choice of repulsive by assuming equal hardness of the Born repulsive for the CsCl structure yields fair energy in the two crystal structures, and are in parameters agreement with the experimentsl pressure-volume good agreement with experiment. We should like relationship of this phase, but leads to a gross to point out that, while we were unable to reprooverestimate of the mechanical work ptdot induce the Schumacher values for these quantities, volved in the transition. In fact, an approximation the Schumacher choice of repulsive parameters in which this work is simply taken equal to the for the CsCl structure implies a disagreement with difference in lattice energy of the two structures at experiment in the pressure-volume relations~p zero pressure(J) yields values of pthq which are of this phase, and does not account consistently only about 10 per cent farger. for the observed values of the mechanical work s~U~C~R(z) assumes also a common value involved in the transition. Schumacher follows the early work of L~IDIN@) of p for the two structures, taking for it the somewhat arbitrary value 0.345 A. However, he chooses in writing the lattice energy of an alkali halide to determine ra for the CsCi structure by simply crystal in the form adding the amount 0.08 A to the value of YQfor the exp(-r/p) W&r) = -G-l-MB (1) NaCl phase.@) This choice yields values of rs for the CsCl structure which are too low: in the The parameter B is then eliminated by means of potassium and rubidium salts, they are respectively the equilibrium equation of the static crystal at only slightly larger than, and practically equal to, zero pressure, to obtain the following expressions the experimental values of the interionic distance for the pressure and the Gibbs free energy of the of this phase at the transition point.6 7) This static crystal as functions of the interionic distance: assumption is equivalent to an underestimate of the parameter B in the CsCl structure, and involves then an overestimate of the absolute value of its Gibbs free energy. This in turn causes a considerable decrease of the computed transition pressure, accompanied by an overestimate of the volume change involved in the transition. The theoretical results for &ha+ are fo~itously some* Based on work performed under the auspices of the U.S. Atomic Energy Commission. what improved, but the pressure-volume relation-

1068

LETTERS

Table 1. Parameters

ptdwt (Kcaljmole)

TO

THE

of the pressure transition

EDITOR

of the potassium

and rubidium h&,&s*

KC1

KBr

KI

RbCl

RbBr

RbI

2.0 3.3 2.1

1.9 3-l 1.9

1-Q 2-9 1.8

0.7 3.7 1.9

0.7 3.1 1.7

0.7 2.7 1.4

o(NaC1) I-------vo(NaC1)

0.085 0.085 0.085

0.088 0.092 0.092

0.105 0,108 0.108

0.030 0.029 0.029

0.033 O-033 0.033

0.035 0.035 0.035

v(CsC1) l--------co(NaC1)

0.197 0,207 0.230

0,193 0.213 0.238

o-190 0.228 0.253

0.170 0.166 0.193

0.166 0.167 0.198

0.161 0.177 0.202

* For each quantity, the first row reports the experimental values, taken from BRIDGMAN,@) and the second and third row report the values computed using respectively the choice of LSwdin and of Schumacher for rs in the CsCl structure. The volume changes refer to the experimental transition pressure. The theoretical results have been obtained using the values of p reported by Lowdin, which are 0*324,0~335,0~350,0~338,0~352 and 0.351 E\for the six saltsThese agree rather closely with more recent data.

ship of the CsCl phase is misrepresented, as shown in Table 1. The results reported here have been obtained with the Lowdin values for the parameter p, but this does not affect the preceding discussion. Finally, the comparison of the theoretical results reported in Table 1 with the corresponding experimental values displays clearly the opposite shifts of the computed ptAv$ and of the computed pressure-volume relationship in the CsCl phase, which accompany a change in the pre-exponential parameter B when the hardness p of the Born repulsive energy is maintained common to the two crystal phases. These results provide then an independent proof of the fact that a proper Bornmodel treatment of polymorphic transitions in ionic crystals requires the use of different repulsive parameters for the two crystal structures, to be determined from data on both phases. Acknowledgements-The

author is indebted to Dr. D. P.

SCHUMACHERfor informative correspondence on his work, and to Dr. F. G. FUMI for helpful discussions.

Argonne rational Argonne, Illinois.

Labora~o~,

MARIO P.

TOSI

References 1. TOSI M. P. and FUMI F. G., J. Phys. Chem. Sotids 23, 359 (1962). 2. SCHUMACHERD. P., Phys. Reo. 126, 1679 (1962). 3. Lijwnr~ P. O., A Tkeoretica~ ~~~est~~ut~oninto Some Properties of ionic Crystals, pp. 99-103 and Figs 7 and 8. Uppsala (1948); Phil. Mug. Supjol. 5, 140 (1956). 4. BORN M. and HUANG K., Dynamical Theory of Crystal Lattices, Section 13. Oxford University Press, London (1954). 5. ZACHAHIASENXV. H., unpublished work reported by KITTEL C., Introduction to Solid State Physics, Table 3.5. Wiley, New York (1956). 6. BRIDGMAN P. W., Proc. Amer. Acad. Arts Sci. 76,

1(1945). 7. JACOBSR. B., Phys. Rev. 54,468

(1938).