- Email: [email protected]
The structures of the ammonium halides
J Phys. Chrm. Solrds Vol. 44, No. 7, pp. 633 638, 1983
@X2-3697/83 $3.00+.00 Q 1983Pergamon Press Ltd.
Prmted inGreatBritain
THE STRUCTURES OF THE AMMONIUM
HALIDES
G. RAGHURAMA
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
and RAMESH
NARAYAN
Raman Research Institute, Bangalore 560 080, India (Received 12 April 1982; accepted in revised form 4 August 1982) Abstract-A
comparison with the alkali halides suggests that all the ammonium halides should occur in the NaCl centre-of-mass structure. Experimentally, at room temperature and atmospheric pressure, only NH,1 crystallizes in this structure, while NH,F is found in the ZnO structure, and NH&I and NH,Br occur in the CsCl structure. We show that a distributed charge on the NH,+ ion can explain these structures. Taking charges of + 0.2e on each of the five atoms in NH,+, as suggested by other studies, we have recomputed the Madelung energy in the cases of interest. A full ionic theory including electrostatic, van der Waals and repulsive interactions then explains the centre-of-mass structures of all the four ammonium halides. The thermal and pressure transitions are also explained reasonably well. The calculated phase diagram of NH,F compares well with experiment. Barring the poorly understood NH,F(II) phase, which is beyond the scope of this work, the other features are in qualitative agreement. In particular, the theory correctly predicts a pressure transition at room temperature from the ZnO structure directly to the CsCl structure without an intermediate NaCl phase. A feature of our approach is that we do not need to invoke hydrogen bonding in NH,F.
1.INTRODUCDON TO THE PROBLEM
The ammonium halides crystallize in simple lattices despite the presence of the complex NH,+ ion. They have therefore attracted much theoretical interest. After Bleick’s[l] initial work in estimating the cohesive energies of these crystals, there have been repeated attempts[2-51 to improve the results. Theoretical attempts to understand the structures of these crystals have concentrated almost exclusively on the orientational order (see, for instance, Vaks and Schneider[6] and references therein). To our knowledge the more fundamental problem of explaining the lattice or centre-of-mass structures of the ammonium halides has remained essentially unexplored. This is the question we consider in this paper. The ammonium halides present an intriguing problem in the matter of their centre-of-mass structures. Narayan [7] has shown that the crystal structure of an ionic crystal of the formula type AB is primarily determined by two parameters: (a) The radius ratio r + /r _ , where r+ and r- are the radii of the cation and the anion respectively; and (b) The strength of the van der Waals interaction among the ions in the crystals. In Table 1 we compare the above characteristics for the ammonium and rubidium halides. The van der Waals coefficients have been taken from Bleick[l] and Hajj[8] while the radius ratios have been calculated using the univalent radii of Narayan[9]. Though the van der Waals dipoledipole coefficients C in the ammonium halides are slightly greater than those in the rubidium halides, the radius ratios are always smaller, These two effects should therefore cancel each other. Since all the rubidium halides crystallize in the NaCl structure, it is reasonable to expect that a theory of the ammonium halides which treats them on the same 633
footing as the alkali halides would predict only the NaCl structure. Column 6 of Table 5 confirms that this is so. On the other hand, the observed structures of the ammonium halides show a rich variety. At room temperature and atmospheric pressure NH,F crystallizes in the wurtzite (i.e. ZnO) structure, NH&l and NH,Br in the CsCl structure and it is only NH,1 which forms the NaCl structure. In this paper we attempt to resolve the above paradox. We begin by pointing out the significant fact that the tetrahedral symmetry of the NH,+ ion matches well with the 4 coordinated and 8 coordinated environments in the ZnO and CsCl structures but does not suit the octahedral symmetry of the cation site in the NaCl structure. The electrostatic interaction should be sensitive to the symmetry matching between the NH,+ ion and its environment for the following reason. It is widely agreed[S, l&12] that the NH,+ ion has a distributed charge with a part of the positive charge residing on the four H-atoms. It is then clear that the electrostatic binding will be significantly increased in those structures (like the ZnO and CsCl structures) where nearest neighbour anions are available right opposite all the four H-atoms of the NH,+ ion. On the other hand, the electrostatic binding in the NaCl structure will be hardly altered. We show in this paper that with the introduction of modified Madelung constants which take into account the distributed charge on the NH, + ion, the centre-of-mass structures of all the ammonium halides can be explained. We also obtain encouraging results on the pressure and thermal structural transitions. The low temperature order-disorder transitions are beyond the scope of this work and have, in any case, been considered in earlier work[6].
G.
634
RAGHURAMA
md
coefficients, WEi,represents the repulsive interaction, Wzp the zero point energy and W,, the thermal free energy. V is the volume per molecule. We have estimated the different terms in (1) as follows:
Table 1. Radius ratios and van der Waals dipol+dipole coefficients for the ammonium and rubidium halides Crystal
Van der Waals Coefficient c in 10-60 ergs.cm6 (Ref. 1, 8)
Radius ratio r+/r_ (Ref.9)
NH4Cl
850
0.899
RbCl
691
0.939
NR4BK
1074
0.836
RbBr
924
0.874
NH41
1511
0.766
RbI
1330
0.800
The distributed charge model of the NH4+ ion has been invoked in earlier studies[S, 10-l 11, but mainly for cohesive energy calculations. In Section 2 we introduce the theoretical framework of our studies, describing the Madelung constant calculations in some detail. We have adopted a completely ionic treatment of the ammonium halides in this paper. In particular, we do not need to invoke any hydrogen bonding, even in NH.,F. Section 3 describes the results of our calculations. We show that the distributed charge model considerably improves the predicted results compared to the point charge model of the NH,+ ion. Section 4 then discusses some conclusions to be drawn from the present study. In what follows, the work “structure” refers to the centre-of-mass structure only, unless otherwise specified. 2. THEORY
The Gibbs free energy per molecule of an ionic crystal of the type A + B - at temperature T and pressure P can be written as G(r.P,T)=---;-;+W,+Pv+W, r
+ wtll (1) where r is the nearest neighbour distance, CLis the Madelung constant, C and D are the van der Waals dipole-dipole and dipolequadrupole interaction
Table 2. Madelung
constants
Crystal structure
R. NARAYAN
2.1 Electrostatic energy In the point charge model of the ammonium halides, the + e charge is assumed to reside entirely on the central nitrogen atom of the NH.,+ ion. The Madelung constants are then exactly known for the ZnO, NaCl and CsCl structures and take the values shown in Table 2. However, based on the electronegativities of the constituent atoms in the NH, + ion, Pauling[l2] proposed a distributed charge for the ion. It is not yet clear how much charge should be placed on each of the four tetrahedrally arranged hydrogen atoms. We have divided the unit positive charge equally among all the five atoms in NH4+ giving a protonic charge qH = + 0.2e. This is the value suggested by Pauling on the basis of electronegativity values[l2]. It is also in close agreement with the study of Goodlife et al. [5], who quote an average value of qH = + 0.194e and is within the limits suggested by Levy and Peterson[l3] on the basis of torsional frequency calculations. A recent X-ray study [ 141 quotes a qH of + 0.09e but is questionable since it departs significantly from the other studies. We have calculated the Madelung constants for the various ammonium halides in different crystal structures using the distributed charges, employing the method of Bertaut [15] as modified by Jones and Templeton[l6]. In all cases, we have assumed that the NH,+ ion is perfectly tetrahedral with a partial charge + 0.2e on each of the five atoms. We have also assumed a fixed N-H bond length of 1.02 8, (independent of structure, pressure, temperature, etc.) as given by electron diffraction studies in NH,Cl(l7]. The orientation of the NH4+ ion in the various crystals has been taken as follows: (a) CsCl structure. The CsCl phases that border the NaCl phase in the ammonium halides have the tetragonal NH, + ions oriented at random with respect to the two equivalent positions in the unit cell. We have treated this static disorder by considering eight “half hydrogens”, with f qH charge on each, disposed towards the corners of the cubic unit cell
in the ammonium halides at room temperature and distributed charge models Lattice ParEmet rs+ In li =4.439 -7.164
and pressure
Madelung Constants Distributed Percent Change Point Charge Charge Model Model 1.64108
1.7471
6.46
CSCl
rN_c1=3.35
1.76268
1.7850
1.27
NH4Br
CsCl
rN_Br=3.516
1.76268
1.7809
1.03
NH41
N&l
rN_I -3.63
1.74756
1.7518
o-.24
ZIIO
m4F NH4c1
+data
;
for point charge
for NH&F from ref. (18). Others from ref.(29).
The structures of the ammonium halides with the nitrogen atom at the body centre. Table 2 shows our computed Madelung constants in NH&l and NH,Br at room temperature and pressure. The calculations were also repeated for the “ferroordered” lattice where all the NH,+ ions are oriented identically. The results changed by only - 0.05%. This is significant and shows that the major effect comes from the use of the distributed charge model which alters the Madelung constant by - 1.2% while different orientational models make only negligible corrections to this. We invoke this result below in the discussion of the NaCl structure. (b) NaCl structure. In the NaCl structure, there are different possible orientations for the NH,+ ion. A study of a few of these possibilities led to a likely minimum potential energy configuration where one of the four N-H bonds is directed at an anion and another is in the plane defined by two other anions. For instance, the NH,+ ion at the origin of the cubic cell was given the fractional coordinates: N : (0.0,0.0,O.O)
(2)
H, :(x, 0.0,O.O) H,:(-
xI,yI, 0.0)
H,:(-x,, H4:(-
-.YY,,z) x,,
-YY,,
-z)
where x, x,, y, and z were adjusted to give N-H bond lengths of 1.02 A. Our model of the NH,+ orientation is supported by spectral absorption studies[l9]. Also the analysis of neutron diffraction data for ND,Br [20] shows maxima in the direction of bromine ions as would be expected with our model in the presence of static disorder. However, we have not considered the random orientation of the complex ions in neighbouring cells in our calculations. This should not affect the final results for the following reason. As Table 2 shows, the difference in the Madelung constant in NH,1 in the NaCl structure between the point charge and distributed charge models is only 0.24%, whereas it is about five times larger in the CsCl phase. As previously shown, inclusion of disorder makes a negligible contribution to the results even in the CsCl phase. Thus we can confidently expect that our simplistic treatment of the orientational disorder in the NaCl structure will be more than adequate. (c) ZnO structure. Here, the hydrogens in the NH4+ ion are taken to be oriented towards the four nearest anions with no orientational disorder. We always consider a structure with an ideal c/a ratio = ,/8/3 except for the case of NH,F at room temperature and pressure where experimentally measured data[l8] have been used. The Madelung constant for this case is shown in Table 2. Table 2 shows that the effect of the distributed charge model is largest in the ZnO structure and least in the NaCl structure. This is precisely what one expects (as discussed in the Introduction) and is
635
clearly in the right direction to explain the experimentally observed structures. It should be noted that the Madelung constant is now a function of r and therefore has to be computed afresh for each crystal at each pressure and temperature. 2.2 Van der Waals interactions The dipole-dipole and dipole*uadrupole coefficients C and D can be calculated in the desired structures using the relations C = s, _c+ - + s, +C+ + + s_ _c-
(3)
D=T+_d+_+T++d+++T_~d__
(4)
where cij and d, are ion-dependent interaction coefficients for which we have used the values reported by Bleick[l]. S, and TUare lattice sums taken from Tosi[21]. For the ZnO structure, we have used the values corresponding to the zincblende (ZnS) structure, which is reasonable considering the close similarity between the two structures. 2.3 Repulsive energy For the repulsive interaction we have employed the theory of compressible ions developed by Narayan and Ramaseshan[22,23]. In this theory, the ions are treated as compressible space filling polyhedral cells and the repulsion arises from the increased compression energy at the cell faces. The compression energy of the cation, for instance, is written -as B, W rep,+ = -
2n ss
exp [- r’(s)b+l do
(5)
where d.s is an area element on the surface of the ionic polyhedron at a distance r’(s) from the centre (for further details, see Ref. [22]). B, and Q, are parameters to be determined from crystal data. A similar expression to (5) is used for the anion. The parameters B _ and u _ of the halogen ions are available from an earlier study of the alkali halides[22,23] and have been used here. For the ammonium ion, we refined B, and a+ as described in Section 3. The above theory of repulsion has been rather successful in explaining the structures of the alkali halides[22,23] and is a natural choice for the present study. 2.4 Zero point energy and thermal free energy We have estimated these contributions in terms of the Debye theory as follows: w, = 9k8,/4
(6)
W,, = 6kT In (0,/T)
(7)
where k is the Boltzman constant and 0, is the Debye temperature. B. was estimated for each crystal from the calculated compressibility as suggested in Ref. [23]. Equation (7) is used only as a high temperature (T 2 0,) approximation.
636
G.
and R. NARAYAN
RAGHURAMA
Table 3. Repulsion parameters B and (r for the ions of interest B (er~s/cm2)
10n
12 i) 1.037 x 1012 ii) 1.098 x 10
mt
0
(X)
O.OB412 0.08412
F-
1.126 x lo7
0.1713
Cl-
3.633 x lo7
0.1866
BK-
4.046 x lo7
0.1980
I-
4.304 x lo7
0.2144
i) Point charge model ii) Distributed charge model.
3. RESULTS 3.1 ReJinement of ammonium ion parameters To determine B, and Q+ for the NH,+ ion, we
employed the least squares method of Narayan and Ramaseshan[23]. Using a trial set of values, the equilibrium r of the four ammonium halides at room temperature and zero pressure, as well as high pressures, were computed by minimising the free energy (1). The second derivative of the lattice energy, d2WJdr2, was also computed. By comparing the calculated r and d2WL/dr2 with the experimental values, a misfit factor R was computed in terms of a weighted sum of squares of differences[24]. R was then minimized by varying B, and 0 + . This exercise was done with the point charge model as well as the distributed charge model. The refined parameters of the ammonium ion along with the parameters employed for the halogen ions are listed in Table 3. The rms errors in r and d2WL/dr2 are 0.734% and 12.4x, respectively for the distributed charge model. The detailed results are shown in Table 4. 3.2 Structural stability The Gibbs free energy at room temperature and atmospheric pressure was computed at the calculated equilibrium r for the four ammonium halides, each in
the three competing structures, viz. ZnO, NaCl and CsCl structures. The repulsion energy in the ZnO structure was computed using the formulae given by Narayan and Ramaseshan[23] for the ZnS structure. The error due to this approximation is expected to be negligible. Table 5 gives the calculated free energies and compares the predicted structures with that experimentally observed. As expected, the point charge model over-estimates the stability of the NaCl structure and predicts the NaCl structure for all the four crystals. On the other hand, the distributed charge model is quite successful and predicts the correct structure in all cases. It is particularly significant that the theory correctly predicts the ZnO structure for NH,F without invoking any extra hydrogen bonding. This is discussed further in Section 4. 3.3 Pressure and thermal transitions Encouraged by the above results, we have attempted a study of the polymorphic transitions in the ammonium halides at high temperatures and pressures. The theoretical calculations lead to the transition parameters listed in Table 6. For comparison the experimental data are also given. It. is seen that the thermal transition in NH,Br (CsCl phase +NaCl phase) is satisfactorily explained. The transition in NH,Cl is also predicted, though at rather high temperatures. A pressure transition in NH,1 is predicted at low pressures in agreement with experiment. Heynes[25] has recently observed a new phase at high pressures in some of the ammonium halides. However, since very little is known about the structural details of this phase, we have not considered it in the present study. The pressure transition to the CsCl phase in NH,F is predicted quite well. It is interesting that the theory correctly predicts the absence of a NaCl phase in between. Figure 1 compares the experimental and calculated phase diagrams of NH,F. Barring the NH,F(II) phase, which has been given a tetragonal
Table 4. Comparison with experiment of the calculated r and d*W,/dr* values, using the distributed charge model Crystal
structure
PKWSliE
Tell&Be-
Nearest neighbour distance (x) Expt. Calc. Error(%)
d2WI,/dr2(lo5 erg cm-')
(Kbars)
rature (OK)
0
300
2.707 2.685 -0.813
1.55
-
1.469 1.078
r+(P)
Expt.
CAC.
Error (%)
rn4F
ZllO
NH4C1
CsCl
0
300
3.350 3.321 -0.875
1.64
0.855
Nn4c1
CsCl
80
300
3.107 3.108 +0.019
1.58
2.344 2.438
Nn4c1
NaCl
0
523
3.260 3.295 +1.08
1.62
NH4Br
CsCl
0
300
3.516 3.511 -0.142
1.65
0.816 0.888
+ 8.82
NH4Br
CSCl
80
300
3.234
3.247 +0.411
1.58
2.160 2.247
+ 4.03
NH4Br
N&l
0
523
3.450 3.475 +0.727
1.63
NB4I
NaCl
0
300
3.630 3.682 +1.43
1.64
rn4I
CsCl
5
300
3.725 3.731 +0.156
1.67
0.896 0.803
NH41
CsCl
80
300 3.435 3.432 -0.073 rms error 0.734
1.59
2.059 2.069
-
0.953
-
0.816
-
0.749
+26.1 + 4.01 -
-10.4 5.34 12.4
The structures
of the ammonium
halides
637
PRESSURE
Fig.
1. Experimental
PRESSURE
KILCBARS
(left) and predicted (right) phase diagrams diagram is from Ref. [30].
for NH,F.
KlLC@J=
The experimental
phase
Table 5. Structural stability at room temperature and atmospheric pressure--comparison between experiment and theory with the point charge and distributed charge models. J and x refer to correct and wrong
Crystal
NH4F
NH4Cl
NH4Br
NH41
Observed structure
Zn0
CsCl
CsCl
NaCl
predictions
Calculated free energy (Kcal./mole ) Structure Point Charge Distributed Model Charge Model NaCl
-190.36
-192.32
CsCl
-187.11
-192.95
ZnO
-182.88
-196.41
NaCl
-166.81
-167.07
CSCl
-166.19
-168.03
zno
-159.76
-166.23
NaCl
-158.86
-159.03
CsCl
-157.58
-159.05
zno
-152.20
-157.93
NaCl
-149.51
-149.60
CsCl
-147.10
-148.16
ZllO
-143.25
-148.02
structure[26] and is therefore beyond the scope of the present study, there is reasonable agreement in the broad features. We note that the triangular region of stability in the NaCl phase at high temperatures is well brought out by the calculations. The transition
Table 6. Thermal between experiment
Crystal and transformations
and pressure transitions-comparison and theory with the distributed charge model Transition Parameters Calculated Experimental (Ref.30,31) 23 Kbar at 100°C
13.1 Kbar at 100°C
NH Cl CsCl4NaCl
62O'C at 0 Kbar
184.3'C at 0 Kbar
NH4BT CsCl*NaCl
759 at 0
137.8OC at 0 Kbar
NH41 NaCl+CsCl
6.2 Kbar at 27'C
NH4F zno + CSCl
Kbar
respectively
0.57 Kbar at 27'C
Predicted Structure Point Charge Distributed Model Charge Model
N&lx
J ZnO
N&lx
*/ CSCl
X NaCl
4 CsCl
4 NaCl
4 NaCl
temperature and pressure possible reason for this section. 4.
are however is discussed
too high. A in the next
DISCUSSION
The highlight of the present work is that, for the first time, we have a theory which satisfactorily explains the observed centre-of-mass structures of all the ammonium halides. The key feature of our approach is the distributed charge model of the ammonium ion which causes important changes in the relative stabilities of the ZnO, NaCl and CsCl structures. A point charge model predicts only the NaCl structure for all the crystals considered. With the distributed charge model, the Madelung constant is a function of the nearest neighbour distance r. This is easily understood since, for a fixed N-H bond length, the distance between the positive charge on the H-atom and the negative charge on the halogen ion is a function of r. Because of this effect, the modification introduced by the distributed charge is
638
G. RAGHURAMA and
maximum in NH,F and reduces progressively for larger anions. Consequently, NH,F takes up the “anomalous” ZnO structure. NH&I and NH,Br both prefer the CsCl structure over the NaCl structure but NH,Cl is stable in this structure for a larger temperature range. On the other hand, NH,1 continues in the NaCl structure at room temperature and requires a small pressure to convert it into the CsCl structure. The pressure and thermal transitions among the centre-of-mass structures in the ammonium halides are all qualitatively explained by our theory, barring the NH,F(II) phase which we have ignored, lacking details. It is significant that our theory correctly predicts that there is no NaCl phase in NH,F at room temperatures. Quantitatively, our calculated transition pressures are typically 5-10 Kbars higher than the experimental values and our transition temperatures are a few hundred degrees too high. To appreciate the significance of the discrepancies, it should be noted that even the earlier successful study of the alkali halides, e.g. [22,23] does not make much better predictions. Garland and Jones [ 1l] have suggested that dq,/d V is negative in the ammonium halides. This means that qH in these crystals will decrease with temperature and increase with pressure. This effect will tend to decrease the transition temperature in NH,F and NH&I as well as the transition pressure in NH,1 thus reducing the discrepancy between theory and experiment. Another possibility in NH&I could be that the ion may rotate relatively freely at higher temperatures[27], thus reducing the electrostatic binding in the CsCl structure. With these added effects, it might be possible to obtain better quantitative agreement for the thermal and pressure transitions. The ZnO structure of NH,F needs special mention. In the normal picture of ionic crystals, this structure is believed to become competitive only at very low radius ratios (6 0.4). For NH,F, the radius ratio is 0.88. For this reason, hydrogen bonding has usually been invoked in this crystal. The nearest neighbour distance in NH,F has also been considered anomalously low, thus further prompting the introduction of hydrogen bonding. In our calculations, we obtain the ZnO structure, as well as good agreement in r (see Table 4) using only an ionic theory with no hydrogen bonding. A major part of the result is contributed by the distributed charge on the NH,+ ion which probably models hydrogen bonding through the increased electrostatic binding. However, one should also note the arguments of Narayan[28] who showed that the
ZnO and ZnS structures could occur in purely ionic crystals even at fairly high radius ratios, provided the ions are reasonably compressible. This is because the equilibrium value of r in a 4 coordinated structure can be much less than that in the NaCl structure
R. NARAYAN
(particularly for compressible ions) and can thus compensate for the reduction in the Madelung constant. Finally we note that the complex nature of the NH,+ ion has been introduced only for the electrostatic energy calculations. We have computed the repulsive energy in the same way as for simple ions. The success of such a simplified approach is noteworthy. Acknowledgements-We thank Prof. S. Ramaseshan for his continued interest in the present investigation. We also thank the referee for pointing out some of the recent literature
on orientational
order in the ammonium
halides.
REFERENCES
I. Bleick W. E., J. Chem. Phy. 2, 160 (1934). 2. May A., Phy. Rev. 52, 339 (1937). 3. Ladd M. F. C., Trans. Faraday Sot. 66, 1592 (1970). 4. Murthv C. S. N. and Murti Y. V. G. S.. J. Phvs. Chem. Solids 31, 1485 (1970). 5. Goodlife A. L.. Jenkins H. D. B.. Martin S. V. and Waddington T.C., Mol. Phys. 21, 761 (1971). 6. Vaks V. G. and Schneider V. E., Phys. Stat. Solidi A35, 61 (1976). 7. Narayan R., Pramana 17, 13 (1981). 8. Hajj F., J. Chem. Phys. 44, 4618 (1966). 9. Narayan R., Prumana 13, 559 (1979). T., Proc. Phys. Math. Sot. Japan 24, 135 10. Nagamiya (1942). 11. Garland C. and Jones J. S., J. Chem. Phys. 41, 165 (1964). 12. Pauling L., The Nature of Chemical Bond, 2nd Edn, p. 72. Oxford and IBM Publishing (1948). 13. Levy H. A. and Peterson S. W., J. Am. Chem. Sot. 75, 1540 (1953). 14. Sylvia L. M., Acta Cryst. (Denmark) A34, 656 (1978). 15. Bertaut E. F.. J. Phvs. Rad. 13. 499 (1952). 16. Jones R. E. and Templeton D.‘H., i Chem. Phys. 25, 1062 (1956). 17. Schigeya Kuwabara, J. Phys. Sot. Japan 14,1205 (1959). 18. Adrian H. W. W. and Feil D., Acfa Crysr. A25, 438 (1969). 19. Plumb R. C. and Homig D. F., J. Chem. Phys. 21, 366 (1953). 20. Press W., Acta Crysr. A29, 257 (1973). 21. Tosi M. P., Solid State Physics (Edited by F. Seitz and D. Turnbull), Vol. 16, p. 1. Academic Press, New York ,.^_.. (IYfJ4).
22. Narayan R. and Ramaseshan S., Phy. Rev. Lert. 42,992 (1979). 23. Narayan R. and Ramaseshan S., Pramana 13, 581 (1979). 24. Narayan R. and Ramaseshan S., J. Phy. Chem. Solids 37, 395 (1976). 25. Heynes A. M., J. Phy. Chem. Solidr 41, 769 (1980). 26. Morosin B. and Schriber J. E., J. Chem. Phys. 42, 1389 (1965). 27. Gutowsky H. S., Pake G. E. and Bersohn R., J. Chem. Phys. 22, 643 (1954). 28. Narayan R., Prumuno 13, 571 (1979). 29. Wyckoff R. W. G., Crystal Structures, 2nd Edn, Interscience, New York (1965). 30. Pistorius C. W. F. T., J. Chem. Phys. 50, 1436 (1969). 31. Stevenson R., J. Chem. Phys. 34, 1757 (1961).
@X2-3697/83 $3.00+.00 Q 1983Pergamon Press Ltd.
Prmted inGreatBritain
THE STRUCTURES OF THE AMMONIUM
HALIDES
G. RAGHURAMA
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
and RAMESH
NARAYAN
Raman Research Institute, Bangalore 560 080, India (Received 12 April 1982; accepted in revised form 4 August 1982) Abstract-A
comparison with the alkali halides suggests that all the ammonium halides should occur in the NaCl centre-of-mass structure. Experimentally, at room temperature and atmospheric pressure, only NH,1 crystallizes in this structure, while NH,F is found in the ZnO structure, and NH&I and NH,Br occur in the CsCl structure. We show that a distributed charge on the NH,+ ion can explain these structures. Taking charges of + 0.2e on each of the five atoms in NH,+, as suggested by other studies, we have recomputed the Madelung energy in the cases of interest. A full ionic theory including electrostatic, van der Waals and repulsive interactions then explains the centre-of-mass structures of all the four ammonium halides. The thermal and pressure transitions are also explained reasonably well. The calculated phase diagram of NH,F compares well with experiment. Barring the poorly understood NH,F(II) phase, which is beyond the scope of this work, the other features are in qualitative agreement. In particular, the theory correctly predicts a pressure transition at room temperature from the ZnO structure directly to the CsCl structure without an intermediate NaCl phase. A feature of our approach is that we do not need to invoke hydrogen bonding in NH,F.
1.INTRODUCDON TO THE PROBLEM
The ammonium halides crystallize in simple lattices despite the presence of the complex NH,+ ion. They have therefore attracted much theoretical interest. After Bleick’s[l] initial work in estimating the cohesive energies of these crystals, there have been repeated attempts[2-51 to improve the results. Theoretical attempts to understand the structures of these crystals have concentrated almost exclusively on the orientational order (see, for instance, Vaks and Schneider[6] and references therein). To our knowledge the more fundamental problem of explaining the lattice or centre-of-mass structures of the ammonium halides has remained essentially unexplored. This is the question we consider in this paper. The ammonium halides present an intriguing problem in the matter of their centre-of-mass structures. Narayan [7] has shown that the crystal structure of an ionic crystal of the formula type AB is primarily determined by two parameters: (a) The radius ratio r + /r _ , where r+ and r- are the radii of the cation and the anion respectively; and (b) The strength of the van der Waals interaction among the ions in the crystals. In Table 1 we compare the above characteristics for the ammonium and rubidium halides. The van der Waals coefficients have been taken from Bleick[l] and Hajj[8] while the radius ratios have been calculated using the univalent radii of Narayan[9]. Though the van der Waals dipoledipole coefficients C in the ammonium halides are slightly greater than those in the rubidium halides, the radius ratios are always smaller, These two effects should therefore cancel each other. Since all the rubidium halides crystallize in the NaCl structure, it is reasonable to expect that a theory of the ammonium halides which treats them on the same 633
footing as the alkali halides would predict only the NaCl structure. Column 6 of Table 5 confirms that this is so. On the other hand, the observed structures of the ammonium halides show a rich variety. At room temperature and atmospheric pressure NH,F crystallizes in the wurtzite (i.e. ZnO) structure, NH&l and NH,Br in the CsCl structure and it is only NH,1 which forms the NaCl structure. In this paper we attempt to resolve the above paradox. We begin by pointing out the significant fact that the tetrahedral symmetry of the NH,+ ion matches well with the 4 coordinated and 8 coordinated environments in the ZnO and CsCl structures but does not suit the octahedral symmetry of the cation site in the NaCl structure. The electrostatic interaction should be sensitive to the symmetry matching between the NH,+ ion and its environment for the following reason. It is widely agreed[S, l&12] that the NH,+ ion has a distributed charge with a part of the positive charge residing on the four H-atoms. It is then clear that the electrostatic binding will be significantly increased in those structures (like the ZnO and CsCl structures) where nearest neighbour anions are available right opposite all the four H-atoms of the NH,+ ion. On the other hand, the electrostatic binding in the NaCl structure will be hardly altered. We show in this paper that with the introduction of modified Madelung constants which take into account the distributed charge on the NH, + ion, the centre-of-mass structures of all the ammonium halides can be explained. We also obtain encouraging results on the pressure and thermal structural transitions. The low temperature order-disorder transitions are beyond the scope of this work and have, in any case, been considered in earlier work[6].
G.
634
RAGHURAMA
md
coefficients, WEi,represents the repulsive interaction, Wzp the zero point energy and W,, the thermal free energy. V is the volume per molecule. We have estimated the different terms in (1) as follows:
Table 1. Radius ratios and van der Waals dipol+dipole coefficients for the ammonium and rubidium halides Crystal
Van der Waals Coefficient c in 10-60 ergs.cm6 (Ref. 1, 8)
Radius ratio r+/r_ (Ref.9)
NH4Cl
850
0.899
RbCl
691
0.939
NR4BK
1074
0.836
RbBr
924
0.874
NH41
1511
0.766
RbI
1330
0.800
The distributed charge model of the NH4+ ion has been invoked in earlier studies[S, 10-l 11, but mainly for cohesive energy calculations. In Section 2 we introduce the theoretical framework of our studies, describing the Madelung constant calculations in some detail. We have adopted a completely ionic treatment of the ammonium halides in this paper. In particular, we do not need to invoke any hydrogen bonding, even in NH.,F. Section 3 describes the results of our calculations. We show that the distributed charge model considerably improves the predicted results compared to the point charge model of the NH,+ ion. Section 4 then discusses some conclusions to be drawn from the present study. In what follows, the work “structure” refers to the centre-of-mass structure only, unless otherwise specified. 2. THEORY
The Gibbs free energy per molecule of an ionic crystal of the type A + B - at temperature T and pressure P can be written as G(r.P,T)=---;-;+W,+Pv+W, r
+ wtll (1) where r is the nearest neighbour distance, CLis the Madelung constant, C and D are the van der Waals dipole-dipole and dipolequadrupole interaction
Table 2. Madelung
constants
Crystal structure
R. NARAYAN
2.1 Electrostatic energy In the point charge model of the ammonium halides, the + e charge is assumed to reside entirely on the central nitrogen atom of the NH.,+ ion. The Madelung constants are then exactly known for the ZnO, NaCl and CsCl structures and take the values shown in Table 2. However, based on the electronegativities of the constituent atoms in the NH, + ion, Pauling[l2] proposed a distributed charge for the ion. It is not yet clear how much charge should be placed on each of the four tetrahedrally arranged hydrogen atoms. We have divided the unit positive charge equally among all the five atoms in NH4+ giving a protonic charge qH = + 0.2e. This is the value suggested by Pauling on the basis of electronegativity values[l2]. It is also in close agreement with the study of Goodlife et al. [5], who quote an average value of qH = + 0.194e and is within the limits suggested by Levy and Peterson[l3] on the basis of torsional frequency calculations. A recent X-ray study [ 141 quotes a qH of + 0.09e but is questionable since it departs significantly from the other studies. We have calculated the Madelung constants for the various ammonium halides in different crystal structures using the distributed charges, employing the method of Bertaut [15] as modified by Jones and Templeton[l6]. In all cases, we have assumed that the NH,+ ion is perfectly tetrahedral with a partial charge + 0.2e on each of the five atoms. We have also assumed a fixed N-H bond length of 1.02 8, (independent of structure, pressure, temperature, etc.) as given by electron diffraction studies in NH,Cl(l7]. The orientation of the NH4+ ion in the various crystals has been taken as follows: (a) CsCl structure. The CsCl phases that border the NaCl phase in the ammonium halides have the tetragonal NH, + ions oriented at random with respect to the two equivalent positions in the unit cell. We have treated this static disorder by considering eight “half hydrogens”, with f qH charge on each, disposed towards the corners of the cubic unit cell
in the ammonium halides at room temperature and distributed charge models Lattice ParEmet rs+ In li =4.439 -7.164
and pressure
Madelung Constants Distributed Percent Change Point Charge Charge Model Model 1.64108
1.7471
6.46
CSCl
rN_c1=3.35
1.76268
1.7850
1.27
NH4Br
CsCl
rN_Br=3.516
1.76268
1.7809
1.03
NH41
N&l
rN_I -3.63
1.74756
1.7518
o-.24
ZIIO
m4F NH4c1
+data
;
for point charge
for NH&F from ref. (18). Others from ref.(29).
The structures of the ammonium halides with the nitrogen atom at the body centre. Table 2 shows our computed Madelung constants in NH&l and NH,Br at room temperature and pressure. The calculations were also repeated for the “ferroordered” lattice where all the NH,+ ions are oriented identically. The results changed by only - 0.05%. This is significant and shows that the major effect comes from the use of the distributed charge model which alters the Madelung constant by - 1.2% while different orientational models make only negligible corrections to this. We invoke this result below in the discussion of the NaCl structure. (b) NaCl structure. In the NaCl structure, there are different possible orientations for the NH,+ ion. A study of a few of these possibilities led to a likely minimum potential energy configuration where one of the four N-H bonds is directed at an anion and another is in the plane defined by two other anions. For instance, the NH,+ ion at the origin of the cubic cell was given the fractional coordinates: N : (0.0,0.0,O.O)
(2)
H, :(x, 0.0,O.O) H,:(-
xI,yI, 0.0)
H,:(-x,, H4:(-
-.YY,,z) x,,
-YY,,
-z)
where x, x,, y, and z were adjusted to give N-H bond lengths of 1.02 A. Our model of the NH,+ orientation is supported by spectral absorption studies[l9]. Also the analysis of neutron diffraction data for ND,Br [20] shows maxima in the direction of bromine ions as would be expected with our model in the presence of static disorder. However, we have not considered the random orientation of the complex ions in neighbouring cells in our calculations. This should not affect the final results for the following reason. As Table 2 shows, the difference in the Madelung constant in NH,1 in the NaCl structure between the point charge and distributed charge models is only 0.24%, whereas it is about five times larger in the CsCl phase. As previously shown, inclusion of disorder makes a negligible contribution to the results even in the CsCl phase. Thus we can confidently expect that our simplistic treatment of the orientational disorder in the NaCl structure will be more than adequate. (c) ZnO structure. Here, the hydrogens in the NH4+ ion are taken to be oriented towards the four nearest anions with no orientational disorder. We always consider a structure with an ideal c/a ratio = ,/8/3 except for the case of NH,F at room temperature and pressure where experimentally measured data[l8] have been used. The Madelung constant for this case is shown in Table 2. Table 2 shows that the effect of the distributed charge model is largest in the ZnO structure and least in the NaCl structure. This is precisely what one expects (as discussed in the Introduction) and is
635
clearly in the right direction to explain the experimentally observed structures. It should be noted that the Madelung constant is now a function of r and therefore has to be computed afresh for each crystal at each pressure and temperature. 2.2 Van der Waals interactions The dipole-dipole and dipole*uadrupole coefficients C and D can be calculated in the desired structures using the relations C = s, _c+ - + s, +C+ + + s_ _c-
(3)
D=T+_d+_+T++d+++T_~d__
(4)
where cij and d, are ion-dependent interaction coefficients for which we have used the values reported by Bleick[l]. S, and TUare lattice sums taken from Tosi[21]. For the ZnO structure, we have used the values corresponding to the zincblende (ZnS) structure, which is reasonable considering the close similarity between the two structures. 2.3 Repulsive energy For the repulsive interaction we have employed the theory of compressible ions developed by Narayan and Ramaseshan[22,23]. In this theory, the ions are treated as compressible space filling polyhedral cells and the repulsion arises from the increased compression energy at the cell faces. The compression energy of the cation, for instance, is written -as B, W rep,+ = -
2n ss
exp [- r’(s)b+l do
(5)
where d.s is an area element on the surface of the ionic polyhedron at a distance r’(s) from the centre (for further details, see Ref. [22]). B, and Q, are parameters to be determined from crystal data. A similar expression to (5) is used for the anion. The parameters B _ and u _ of the halogen ions are available from an earlier study of the alkali halides[22,23] and have been used here. For the ammonium ion, we refined B, and a+ as described in Section 3. The above theory of repulsion has been rather successful in explaining the structures of the alkali halides[22,23] and is a natural choice for the present study. 2.4 Zero point energy and thermal free energy We have estimated these contributions in terms of the Debye theory as follows: w, = 9k8,/4
(6)
W,, = 6kT In (0,/T)
(7)
where k is the Boltzman constant and 0, is the Debye temperature. B. was estimated for each crystal from the calculated compressibility as suggested in Ref. [23]. Equation (7) is used only as a high temperature (T 2 0,) approximation.
636
G.
and R. NARAYAN
RAGHURAMA
Table 3. Repulsion parameters B and (r for the ions of interest B (er~s/cm2)
10n
12 i) 1.037 x 1012 ii) 1.098 x 10
mt
0
(X)
O.OB412 0.08412
F-
1.126 x lo7
0.1713
Cl-
3.633 x lo7
0.1866
BK-
4.046 x lo7
0.1980
I-
4.304 x lo7
0.2144
i) Point charge model ii) Distributed charge model.
3. RESULTS 3.1 ReJinement of ammonium ion parameters To determine B, and Q+ for the NH,+ ion, we
employed the least squares method of Narayan and Ramaseshan[23]. Using a trial set of values, the equilibrium r of the four ammonium halides at room temperature and zero pressure, as well as high pressures, were computed by minimising the free energy (1). The second derivative of the lattice energy, d2WJdr2, was also computed. By comparing the calculated r and d2WL/dr2 with the experimental values, a misfit factor R was computed in terms of a weighted sum of squares of differences[24]. R was then minimized by varying B, and 0 + . This exercise was done with the point charge model as well as the distributed charge model. The refined parameters of the ammonium ion along with the parameters employed for the halogen ions are listed in Table 3. The rms errors in r and d2WL/dr2 are 0.734% and 12.4x, respectively for the distributed charge model. The detailed results are shown in Table 4. 3.2 Structural stability The Gibbs free energy at room temperature and atmospheric pressure was computed at the calculated equilibrium r for the four ammonium halides, each in
the three competing structures, viz. ZnO, NaCl and CsCl structures. The repulsion energy in the ZnO structure was computed using the formulae given by Narayan and Ramaseshan[23] for the ZnS structure. The error due to this approximation is expected to be negligible. Table 5 gives the calculated free energies and compares the predicted structures with that experimentally observed. As expected, the point charge model over-estimates the stability of the NaCl structure and predicts the NaCl structure for all the four crystals. On the other hand, the distributed charge model is quite successful and predicts the correct structure in all cases. It is particularly significant that the theory correctly predicts the ZnO structure for NH,F without invoking any extra hydrogen bonding. This is discussed further in Section 4. 3.3 Pressure and thermal transitions Encouraged by the above results, we have attempted a study of the polymorphic transitions in the ammonium halides at high temperatures and pressures. The theoretical calculations lead to the transition parameters listed in Table 6. For comparison the experimental data are also given. It. is seen that the thermal transition in NH,Br (CsCl phase +NaCl phase) is satisfactorily explained. The transition in NH,Cl is also predicted, though at rather high temperatures. A pressure transition in NH,1 is predicted at low pressures in agreement with experiment. Heynes[25] has recently observed a new phase at high pressures in some of the ammonium halides. However, since very little is known about the structural details of this phase, we have not considered it in the present study. The pressure transition to the CsCl phase in NH,F is predicted quite well. It is interesting that the theory correctly predicts the absence of a NaCl phase in between. Figure 1 compares the experimental and calculated phase diagrams of NH,F. Barring the NH,F(II) phase, which has been given a tetragonal
Table 4. Comparison with experiment of the calculated r and d*W,/dr* values, using the distributed charge model Crystal
structure
PKWSliE
Tell&Be-
Nearest neighbour distance (x) Expt. Calc. Error(%)
d2WI,/dr2(lo5 erg cm-')
(Kbars)
rature (OK)
0
300
2.707 2.685 -0.813
1.55
-
1.469 1.078
r+(P)
Expt.
CAC.
Error (%)
rn4F
ZllO
NH4C1
CsCl
0
300
3.350 3.321 -0.875
1.64
0.855
Nn4c1
CsCl
80
300
3.107 3.108 +0.019
1.58
2.344 2.438
Nn4c1
NaCl
0
523
3.260 3.295 +1.08
1.62
NH4Br
CsCl
0
300
3.516 3.511 -0.142
1.65
0.816 0.888
+ 8.82
NH4Br
CSCl
80
300
3.234
3.247 +0.411
1.58
2.160 2.247
+ 4.03
NH4Br
N&l
0
523
3.450 3.475 +0.727
1.63
NB4I
NaCl
0
300
3.630 3.682 +1.43
1.64
rn4I
CsCl
5
300
3.725 3.731 +0.156
1.67
0.896 0.803
NH41
CsCl
80
300 3.435 3.432 -0.073 rms error 0.734
1.59
2.059 2.069
-
0.953
-
0.816
-
0.749
+26.1 + 4.01 -
-10.4 5.34 12.4
The structures
of the ammonium
halides
637
PRESSURE
Fig.
1. Experimental
PRESSURE
KILCBARS
(left) and predicted (right) phase diagrams diagram is from Ref. [30].
for NH,F.
KlLC@J=
The experimental
phase
Table 5. Structural stability at room temperature and atmospheric pressure--comparison between experiment and theory with the point charge and distributed charge models. J and x refer to correct and wrong
Crystal
NH4F
NH4Cl
NH4Br
NH41
Observed structure
Zn0
CsCl
CsCl
NaCl
predictions
Calculated free energy (Kcal./mole ) Structure Point Charge Distributed Model Charge Model NaCl
-190.36
-192.32
CsCl
-187.11
-192.95
ZnO
-182.88
-196.41
NaCl
-166.81
-167.07
CSCl
-166.19
-168.03
zno
-159.76
-166.23
NaCl
-158.86
-159.03
CsCl
-157.58
-159.05
zno
-152.20
-157.93
NaCl
-149.51
-149.60
CsCl
-147.10
-148.16
ZllO
-143.25
-148.02
structure[26] and is therefore beyond the scope of the present study, there is reasonable agreement in the broad features. We note that the triangular region of stability in the NaCl phase at high temperatures is well brought out by the calculations. The transition
Table 6. Thermal between experiment
Crystal and transformations
and pressure transitions-comparison and theory with the distributed charge model Transition Parameters Calculated Experimental (Ref.30,31) 23 Kbar at 100°C
13.1 Kbar at 100°C
NH Cl CsCl4NaCl
62O'C at 0 Kbar
184.3'C at 0 Kbar
NH4BT CsCl*NaCl
759 at 0
137.8OC at 0 Kbar
NH41 NaCl+CsCl
6.2 Kbar at 27'C
NH4F zno + CSCl
Kbar
respectively
0.57 Kbar at 27'C
Predicted Structure Point Charge Distributed Model Charge Model
N&lx
J ZnO
N&lx
*/ CSCl
X NaCl
4 CsCl
4 NaCl
4 NaCl
temperature and pressure possible reason for this section. 4.
are however is discussed
too high. A in the next
DISCUSSION
The highlight of the present work is that, for the first time, we have a theory which satisfactorily explains the observed centre-of-mass structures of all the ammonium halides. The key feature of our approach is the distributed charge model of the ammonium ion which causes important changes in the relative stabilities of the ZnO, NaCl and CsCl structures. A point charge model predicts only the NaCl structure for all the crystals considered. With the distributed charge model, the Madelung constant is a function of the nearest neighbour distance r. This is easily understood since, for a fixed N-H bond length, the distance between the positive charge on the H-atom and the negative charge on the halogen ion is a function of r. Because of this effect, the modification introduced by the distributed charge is
638
G. RAGHURAMA and
maximum in NH,F and reduces progressively for larger anions. Consequently, NH,F takes up the “anomalous” ZnO structure. NH&I and NH,Br both prefer the CsCl structure over the NaCl structure but NH,Cl is stable in this structure for a larger temperature range. On the other hand, NH,1 continues in the NaCl structure at room temperature and requires a small pressure to convert it into the CsCl structure. The pressure and thermal transitions among the centre-of-mass structures in the ammonium halides are all qualitatively explained by our theory, barring the NH,F(II) phase which we have ignored, lacking details. It is significant that our theory correctly predicts that there is no NaCl phase in NH,F at room temperatures. Quantitatively, our calculated transition pressures are typically 5-10 Kbars higher than the experimental values and our transition temperatures are a few hundred degrees too high. To appreciate the significance of the discrepancies, it should be noted that even the earlier successful study of the alkali halides, e.g. [22,23] does not make much better predictions. Garland and Jones [ 1l] have suggested that dq,/d V is negative in the ammonium halides. This means that qH in these crystals will decrease with temperature and increase with pressure. This effect will tend to decrease the transition temperature in NH,F and NH&I as well as the transition pressure in NH,1 thus reducing the discrepancy between theory and experiment. Another possibility in NH&I could be that the ion may rotate relatively freely at higher temperatures[27], thus reducing the electrostatic binding in the CsCl structure. With these added effects, it might be possible to obtain better quantitative agreement for the thermal and pressure transitions. The ZnO structure of NH,F needs special mention. In the normal picture of ionic crystals, this structure is believed to become competitive only at very low radius ratios (6 0.4). For NH,F, the radius ratio is 0.88. For this reason, hydrogen bonding has usually been invoked in this crystal. The nearest neighbour distance in NH,F has also been considered anomalously low, thus further prompting the introduction of hydrogen bonding. In our calculations, we obtain the ZnO structure, as well as good agreement in r (see Table 4) using only an ionic theory with no hydrogen bonding. A major part of the result is contributed by the distributed charge on the NH,+ ion which probably models hydrogen bonding through the increased electrostatic binding. However, one should also note the arguments of Narayan[28] who showed that the
ZnO and ZnS structures could occur in purely ionic crystals even at fairly high radius ratios, provided the ions are reasonably compressible. This is because the equilibrium value of r in a 4 coordinated structure can be much less than that in the NaCl structure
R. NARAYAN
(particularly for compressible ions) and can thus compensate for the reduction in the Madelung constant. Finally we note that the complex nature of the NH,+ ion has been introduced only for the electrostatic energy calculations. We have computed the repulsive energy in the same way as for simple ions. The success of such a simplified approach is noteworthy. Acknowledgements-We thank Prof. S. Ramaseshan for his continued interest in the present investigation. We also thank the referee for pointing out some of the recent literature
on orientational
order in the ammonium
halides.
REFERENCES
I. Bleick W. E., J. Chem. Phy. 2, 160 (1934). 2. May A., Phy. Rev. 52, 339 (1937). 3. Ladd M. F. C., Trans. Faraday Sot. 66, 1592 (1970). 4. Murthv C. S. N. and Murti Y. V. G. S.. J. Phvs. Chem. Solids 31, 1485 (1970). 5. Goodlife A. L.. Jenkins H. D. B.. Martin S. V. and Waddington T.C., Mol. Phys. 21, 761 (1971). 6. Vaks V. G. and Schneider V. E., Phys. Stat. Solidi A35, 61 (1976). 7. Narayan R., Pramana 17, 13 (1981). 8. Hajj F., J. Chem. Phys. 44, 4618 (1966). 9. Narayan R., Prumana 13, 559 (1979). T., Proc. Phys. Math. Sot. Japan 24, 135 10. Nagamiya (1942). 11. Garland C. and Jones J. S., J. Chem. Phys. 41, 165 (1964). 12. Pauling L., The Nature of Chemical Bond, 2nd Edn, p. 72. Oxford and IBM Publishing (1948). 13. Levy H. A. and Peterson S. W., J. Am. Chem. Sot. 75, 1540 (1953). 14. Sylvia L. M., Acta Cryst. (Denmark) A34, 656 (1978). 15. Bertaut E. F.. J. Phvs. Rad. 13. 499 (1952). 16. Jones R. E. and Templeton D.‘H., i Chem. Phys. 25, 1062 (1956). 17. Schigeya Kuwabara, J. Phys. Sot. Japan 14,1205 (1959). 18. Adrian H. W. W. and Feil D., Acfa Crysr. A25, 438 (1969). 19. Plumb R. C. and Homig D. F., J. Chem. Phys. 21, 366 (1953). 20. Press W., Acta Crysr. A29, 257 (1973). 21. Tosi M. P., Solid State Physics (Edited by F. Seitz and D. Turnbull), Vol. 16, p. 1. Academic Press, New York ,.^_.. (IYfJ4).
22. Narayan R. and Ramaseshan S., Phy. Rev. Lert. 42,992 (1979). 23. Narayan R. and Ramaseshan S., Pramana 13, 581 (1979). 24. Narayan R. and Ramaseshan S., J. Phy. Chem. Solids 37, 395 (1976). 25. Heynes A. M., J. Phy. Chem. Solidr 41, 769 (1980). 26. Morosin B. and Schriber J. E., J. Chem. Phys. 42, 1389 (1965). 27. Gutowsky H. S., Pake G. E. and Bersohn R., J. Chem. Phys. 22, 643 (1954). 28. Narayan R., Prumuno 13, 571 (1979). 29. Wyckoff R. W. G., Crystal Structures, 2nd Edn, Interscience, New York (1965). 30. Pistorius C. W. F. T., J. Chem. Phys. 50, 1436 (1969). 31. Stevenson R., J. Chem. Phys. 34, 1757 (1961).