Materials Science and Engineering, 57 (1983) L3-L4 L3

Letter

O n L i n d e m a n n ' s m e l t i n g c r i t e r i o n

O. P. GUPTA

Physics Department, J. Christian College, Allahabad 211006 (India)

(Accepted September 4, 1982)

1. INTRODUCTION

L i n d e m a n n ' s c r i t e r i o n for m e l t i n g has b e e n the s u b j e c t o f n u m e r o u s ear l ie r pape r s [ 1 - 7 ] . I n these paper s , d i f f e r e n t c r i t e r i a o f m e l t i n g

are u s e d a n d w i d e d ivers i f i ed i n t e r p r e t a t i o n s o f t he p h e n o m e n o n are p r o v i d e d . L i n d e m a n n [ 1] f i rs t sugges ted t h a t a so l id m e l t s w h e n t he r a t i o X r eaches a c o n s t a n t va lue . F o r a s i m p l e D e b y e m o d e l o f solids, L i n d e m a n n ' s h y p o t h e s i s leads to a r e l a t i o n s h i p b e t w e e n t he m e l t i n g

TABLE 1

Values of X for cubic metals

t e m p e r a t u r e TM, t h e D e b y e t e m p e r a t u r e (9, t he a t o m i c w e i g h t A a n d t he a t o m i c v o l u m e V

o f the f o r m

TM = ( 9 2 A V 2 / 3 / C 2 (1)

w he r e C is a n u m e r i c a l c o n s t a n t w h i c h is t h e same for all t h e c rys ta l s [8] . I n the p r e s e n t c o m m u n i c a t i o n we e x a m i n e L i n d e m a n n ' s h y p o t h e s i s e m p l o y i n g o u r r e c e n t m o d e l o f l a t t i ce d y n a m i c s . T h e m o d e l has e x p l a i n e d succes s fu l ly t h e l a t t i ce d y n a m i c s o f c u b i c

m e t a l s [ 9 - 1 2 ] .

2. THEORY

T h e ra t io X can be w r i t t e n as [13]

X _ (SRi 2) ~.. (Eq,p)

Ro 2 - q, . MNRo2cOq, 2 (2)

Metal Group T M Lattice (K) constant

(A)

Elaslic constants (X 1011 dyn cm -2) X

Cl l C12 C44

Li Ia 452.0 3.842 Na Ia 370.0 4.240 K Ia 337.0 5.226 Rb Ia 312.0 5.698 Cs Ia 302.0 6.050 Cu Ib 1356.0 3.614 Ag Ib 1233.0 4.080 Au Ib 1336.0 4.078 Ca IIa 850.0 5.580 Sr IIa 768.0 6.080 A1 IIIa 932.0 4.049 Th IIIb 2023.0 5.084 Pb IVb 600.0 4.950 V Vb 1983.0 3.034 Nb Vb 2773.0 3.300 Ta Vb 3269.0 3.302 Cr VIb 1890.0 2.879 Mo VIb 2900.0 3.146 W VIb 3660.0 3.165 (~-Fe VIII 1808.0 2.866 F.c.c. Co VIII 1768.0 3.550 Ir VIII 2727.0 3.833 Ni VIII 1723.0 3.523 Pd VIII 1822.4 3.890 Pt VIII 2042.0 3.923

1.440 1.211 1.094 0.0252 0.808 0.664 0.586 0.0222 0.416 0.341 0.286 0.0211 0.296 0.244 0.160 0.0211 0.245 0.208 0.159 0.0229

16.839 12.142 7.539 0.0139 12.399 9.367 4.612 0.0119 19.234 16.314 4.195 0.0129

2.064 1.454 0.611 0.0159 1.468 0.988 0.574 0.0160

10.678 6.074 2.821 0.0119 7.530 4.890 4.780 0.0160 4.953 4.229 1.490 0.0099

22.795 11.870 4.255 0.0179 24.600 13.400 2.870 0.0169 26.091 15.743 8.182 0.0180 35.000 6.780 10.080 0.0081 44.077 17.243 12.165 0.0084 52.327 20.453 16.072 0.0088 23.310 13.544 11.783 0.0110 22.100 14.700 12.400 0.0120 59.600 25.200 27.000 0.0109 25.080 15.000 12.350 0.0120 22.700 17.590 7.170 0.0119 34.670 25.070 7.650 0.0118

0025-5416/83/0000-0000/$03.00 © Elsevier Sequoia/Printed in The Netherlands

L4

where M is the mass of the a tom, N the n u m b e r o f a t o m s per un i t vo lume , ¢Oq, p the lat t ice f r e q u e n c y of wavevec to r q and po la r i za t ion p , and (Eq, p ) the average energy of the p h o n o n in the m o d e q, p . A t TM, (Eq,p) = kTM ; hence eqn. (2} reduces to

hTM ~ cOq, p -2 (3) X - M N R o ~ 2 q,p

where h is the B o l t z m a n n cons tan t .

3. RESULTS AND DISCUSSION

F o r the ca lcula t ion of X f r o m eqn. (3) in 5he p resen t work , the in tegra t ion was per- f o r m e d numer ica l ly using a m od i f i ed version of H o u s t o n ' s m e t h o d descr ibed previous ly [14 -16 ] .

The calcula ted values of X for a n u m b e r of cubic meta ls are p resen ted in Tab le 1, t oge the r wi th the inpu t p a r a m e t e r s f r o m ref. 14. I t can be seen f rom Tab le 1 t h a t the values of X are near ly the same fo r e l ements in the same group o f the per iodic table bu t tha~ there is a wide var ia t ion in X f r o m one group to another . The var ia t ion is cons i s ten t

wi th the na tu re of b inding in metals . The resul ts o f the p resen t s tudy suggest t h a t L i n d e m a n n ' s hypo thes i s o f a cons t an t X fo r all solids is inadequa te .

REFERENCES

1 F.A. Lindemann, Z. Phys., 11 (1910) 609. 2 J .J . Gilvarry, Phys. Rev., 102 (1956) 308. 3 D. Kuhlmann-Wilsdorf, Phys. Rev. A, 140 (1965)

1599. 4 Y. Ida, Phys. Rev., 187 (1969} 951. 5 T. Gorecki, Z. Metallkd., 67 (1976) 269. 6 P.R. Couchman, Philos. Mag., 35 (1977) 787. 7 S.A. Cho, Z. Metallkd., 71 (1980) 47. 8 M. Blackmann, Handb. Phys., 7, Part I (1955) 325. 9 O.P. Gupta, Z. Naturforsch., 36a (1981) 1242.

10 O.P. Gupta, J. Phys. (Paris), Colloq. C6, (1981) 380.

11 0. P. Gupta, Solid State Commun., 42 (1982) 31. 12 O.P. Gupta, Nuovo Cimento D, in the press. 13 D. Pines, Elementary Excitations in Solids,

Benjamin, New York, 1963. 14 O.P. Gupta, D.Phil. Thesis, Allahabad University,

1978. 15 O.P. Gupta and M. P. Hemkar, Nuovo Cimento B,

45 (1978) 255. 16 O.P. Gupta and H. L. Kharoo, J. Chem. Phys., 74

(1981) 3577.