Lattice dynamics of hexagonal ice

Volume 4, number 1

15 September 1969

CliE&IlCAL PHYSICS LETTERS

LATTICE

DYNAMICS

OF

HEXAGONAL

ICE

T. M. HAIUDASAN and J. GGV’INDAHAJAN Department

of Physics,

Indian Imtikte India

of Science,

Sungalore-12,

Rereived 18 July 1969

The lattice dynamics of hexagonal ice is worked out with the force constants deduced from the experimental elastic constants.

The elastic constants of hexagonal ice were worked out by Penny [l] on a lattice dynamical basis using Barnes’ model [2] for the structure with the nearest neighbour interactions. She assumed that the hydrogen tetrahedron around each oxygen is regular and hence could express the six force constants in terms of two independent force constants which were evaluated from Poisson’s ratio and Young’s modulus. The experimental results became available a few years 1. :er and it was found that only two elastic constants agreed well with her predictions and these two values happened to be those which depend entirely on the two independent force constants Q! and P. This suggested that the relations of the other force constants in terms of these two are not very valid. As such, we thought is would be desirable to express the elastic constants in terms of all the six force constants and to deduce the values of the force constants from the known elastic constants [3]. A similar work was attempted by Forslind [4] without taking the hydrogen positions into consideration. These force constants could be used to compute the normal mode frequencies for the wavevector tending to zero and the result can be compared with those of Raman and infrared data [5]. The most general expressions for the elastic constants in the same notation as Penny are 2accll=-

2=c44

2(o-8)2 UYfB

+6@+2a-3r

c2 6((~+8)5 = 2 (6o + S/3 + 5)

1266

a&c33

2flCb13+C&

=

3&(2a+2/3-

Z&$ll

5) t 3fie(y-8) 6yi29

fC12) = - $$-

-

J5E

I

+4@-(Y)

where a and c are the lattice constants. These expressions are very general and we can get back Penny’s expressions by substituting for y, 8, 6, E and 5, t‘neir corresponding values in terms of (Y and p. These equations were soLved numerically using the following values of elastic constants. Cl1 = 1.333,

cl2 = 0.603,

~33 = 1.428,

c44 = 0.326

(all in units of 1011 dynes/cm2). The values of the force‘constants dynes/cm are as follows CY= 0.083, 6 = E = -0.086,

Cl3 = 0.566,

t3 = o.iO4, 4 = 0.336,

in units of i04 y = 0.694, 6 = 4.027.

The elements of the dynamical matrix which is Hermitian and is of order 36 are given in Penny’s work. The matrix was diagonalised in a CDC 3600 computer installation at TIF’H, Bombay, for q = 0. The results together with the experimental values are given in table I. A perusal of the results in table 1 indicates that the agreement is fairly satisfactory taking into 11

V&me

CHEMICAL PHYSICS LETTERS

4, number 1

Table 1 The norm& mode frequencies

. -1 ‘ Computed values

of hexagw&

(in cm-l)

Raman (41

spectry.

Infr;;;

:+j

ice.

Neutron [?I scattering ~

3120

117? 997

900.

971 458

516

477

468

347

373 356

‘;= 300

330

= 300

900

simple valence force field. Thus it was decided to compute the normal mode frequencies for the

84 wave vectors of the Brillouin zone, to work out the freqrzency distribution and to compute the various thermodynamic properties associated with it. The frequency distribution obtained recently by neutron scattering 171will throw more light on the validity of the model and the force’ constants by a comparison with the computed phonon spectrum. This work is in progress and the results of the investigation will be published elsewhere. The authors are grateful to Professor R. S. Krishnan for his guidance and encouragement. One of us (TM&i) thanks the University Grants Commission for the award of a Fellowship.

344 342 335

15 September 1969

334 254

252

231

232

217

212

275 229

220 = 200

196

193

190

N 190

145

122

140

149

57

53

63 53

consider&ion the various assumptions made in the model. Gur resuIts also agree with those given by Bertie [6] calculated on the basis of z

12

REFERENCES [I] A. H. A. Penny, Proc. Cambridge Phil. Sot. 44 (L948) 423. 121W. H. Barnes, Proc. Roy. Sac, Al26 (1929) 670. (31 R. E. Green and L. Mackinnon, 6. Acoust.Soc. Am. 28 f1956) 1292. [4] E. Forslind, Proc. Wed. Cement Concrete Res. Inst. Roy. Inst. Technol., Stockholm. No. 21 (1954). [5j N. Ockmac, Advan. Phys. 7 @X58) 199. 161J. E. Bertie and E. Wballey, J. Chem. Phys. 46 (1967) 1271. [7f H. Prask, H.Boutin and S-Yip, J. Chem. Phys. 48 (1968) 3367,