Theory of melting of alkali metals

Volume 103A, number 1,2

PHYSICS LETTERS

18 June 1984

THEORY OF MELTING OF ALKALI METALS A.M. BRATKOVSKY, V.G. VAKS and A.V. T R E F I L O V 1, V. Kurchatov Institute o f Atomic Energy, Moscow 123182, USSR Received 4 January 1984 Revised manuscript received 20 April 1984

We show that the use of simple but adequate pseudopotential models and the Weeks-Chandler-Andersen approach to the liquid theory enables us to calculate the melting curves Tm(p) for simple metals such as Na, K, Rb and Cs with a high precision when the thermodynamic properties of both solid and liquid phases are evaluated consistently. The agreement with experiment in the volume jumps in fusion is also well for small pressures p ~ 0 but worsens when p rises which seems to indicate an inadequate description of the equilibrium lattice defect thermodynamic contribution.

The problem o f developing microscopic theories o f melting attracts much attention in condensed state physics, see e.g. refs. [ 1 - 9 ] . Until now attempts of a quantitative approach to the problem have been connected mainly with computer simulation methods [3,4] which have limited resources and accuracies. Analytical approaches to simple metal melting based on the pseudopotential theory have also been attempted [ 5 - 9 ] . However, the pseudopotential models used and methods for the description o f solid and (or) liquid phases were usually rather crude (see the discussion in refs. [7,8,10]) and only the recent results o f Pelissier for Na [8] can be regarded as quantitative. Earlier (see refs. [ 1 0 - 1 4 ] , and references therein) we have shown that the simple but adequate pseudopotential model [11,14] describes very accurately the inter4onic interactions and all the thermodynamic properties o f alkali metals (except for Li) in wide ranges o f the pressure p and temperature T from T = 0 up to the critical-point vicinity. Consistent methods for the thermodynamic calculations in the pseudopotential theory for both the solid and liquid phases have also been described. Here we show that the use o f the methods and model permits us to evaluate the melting curves T m ( p ) for Na, K, Rb and Cs with a high precision. Our results provide us also with detailed information on various contributions to the thermodynamics o f melting, in particular, on the so-called pre-melting anomalies [2]. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

The melting temperatures T m ( p ) are determined from the equations

G~(p, T ) = Gs(p, T); a~, s =F~, s +pa~, s .

(1)

Here F i s the free energy per atom, the indices t~ and s correspond to the liquid and solid phases and ~2 = ~2(p, T) is the atomic volume found from the equation 0F(a, T)/~a = -p. For the liquid F = F~ is calculated using the thermodynamic perturbation theory (TPT) in the form proposed by Weeks, Chandler and Andersen (WCA) [15,10]. In all the versions of the TPT the inter-ionic potential ~(r) is divided into that o f ~0(r) for some reference system and the perturbation v(r) = ~(r) - 90(r). The expansion o f F in powers of v has the form [161 F = F 0 + F 1 + . . . = F 0 +½(v) 0 + . . . .

(2)

Here F 0 is the reference system free energy and ( )0 means the averaging with the radial distribution function ( R D F ) g o ( r ) of the reference system. The simplest and most widely used TPT version (referred to as the HSV method) is the use o f the hardsphere (HS) reference system with the diameter dHs = dHS V determined from the variational principle for F: 3F/3dHs = 0. Only the terms F 0 and F 1 are kept in expansion (2) and it is written as

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FHSV = Fsi + FHS + ½(~fl}HS "

(3)

Here Fsi is the structure-independent (electronic) term characteristic of metals [ 10] and FHS is F for the HS system [16]. In the WCA approach [15,16] ~o0(r) = [ ~ r ) - ~0(rm)] O(rm - r) where r m is the ¢0(r) first minimum position, O(x)equals 1 for x > 0 and zero for x < 0. It has been shown in ref. [10] that the expansion (2) for F w c A can be written in the following form convenient for the discussions and calculations 1

FWCA = Fsi + FHS + ~(~)HS + FI~ •

rmMHS

dx xZgHs(X, r/)~P0(XdHs) .

(5)

- T exp(Slv - 13Ely -/3pf21v),

G s = E s t ( ~ ) + F h ( ~ , T) + p ~ + Fan + F e + Gde f .

' "

Here Est is the static energy; Fh, the harmonic phonon free energy; Fan , F e and Gdef, the contributions of the phonon anharmonicity, of the electron thermal excitations and of the equilibrium lattice defects (of the "pre-melting anomalies" [2]), respectively. We

/

/,

T[K)

'

N

o

300 ~

Cs

200

t00

u~[O/oj 0

(6)

(7)

where/3 = lIT. The vacancy formation energy Ely and entropy Sly at p = 0 have been estimated in ref. [13] from the experimental specific heat data supposing the Slv values not too large, Sly ~ 1. The vacancy volume ~21v(P = 0, Tin) "~ 0.792(0, Tin) and the pressure dependence of the f21v and Ely have been taken from the calculations [17], and the variation of Sly along the melting curve has been neglected. We have made detailed calculations of the thermodynamic properties near T m and melting characteristics 500

Here r~ = rrd3Hs/6~2is the packing fraction, x = r/dHs and gHS is the RDF of the HS system. The higher-order terms in ~ and v neglected in (4) have been estimated as 8Fwc A ~ 12r/~2T m [10]. For alkalis near Tm~ ~ 0.1, r~ ~ 0.5 and F w c a "" 0.05 T m . The comparison with experiment in ref. [10] and below implies that the actual error of approximation (4) is still smaller. We have also indicated in ref. [10] that at large compressions u = 1 - ~2(p, T)]~2(O, Tm)~>0.15-0.2 the accuracy of the WCA method begins to drop, especially for the volume derivatives o f F , since the packing fractions 77= r/WCA become too large in this case. For the solid phase G s is calculated as (see refs. [11-13])

76

calculate ~2 = ~2s(p, T) and Gs(P, T) using the "quasiharmonic perturbation theory" (QHPT) [ 12,13], i.e. the consistent expansion in powers of u 1 = ptJBst, where Ph = --~Fh/Of2 is the harmonic phonon pressure and Bst = - ~ 2 ~ 2 E s t / ~ 2 is the static bulk modulus. For alkalis near T m u 1 ~ 0.06 and the terms up to u~ in G s are kept. The expressions for Fan and F e were given in ref. [13]. Gde f is commonly assumed to be due to the vacancy creation and we adopt the assumption: G d e f = Gva c =

(4)

Here the first three terms have the same form as that in (3) with the value dHS = d w c A determined from some special condition of WCA [ 15,16] (dwc h > dHSV). The term FI~ describes the effects of the difference between ~o0(r) and ~0HS(r), of the ~00(r) "softness", in first order in the parameter of the softness ~ (which has been defined and evaluated in ref. [101), Fa~ = - 1 2 f 1

18 June 1984

0

,b

10

15

Fig. 1. The melting curves Tm versus the experimental compression ueXp(p), see the text. For clarity the results for each heavier metal are displaced below those for the lighter one by AT = 50 K. Values of Tm on the ordinate axis are given for Na; for K they must be read as (Tm)K = Tm + 50 K, etc. The experimental points are from refs. [1,18].

Volume 103A, number 1,2

18 June 1984

PHYSICS LETTERS

Table 1 Comparison of the calculated and observed (in parentheses) characteristics of alkali metals along the melting curves. Metal Na

p (kbar) 0

u exp (%)

Tm(P) (K)

Ss

S~

0

370 (371) 437 (433) 504 (493)

6.74 (6.93) 6.84 (7.02) 6.95

AS

~2s (au)

fz~ (au)

AI2/fZs (%)

rL (%)

7.72 (7.78) 7.86 (7.84) 7.99

0.98 (0.844) 1.02 (0.823) 1.04 (0.801)

271.0 (270.6) 241.7 (242.3) 216.3 (217)

278.5 (277.6) 248.5 (246.8) 219.4 (219.8)

2.8 (2.55) 2.8 (1.84) 1.4 (1.29)

17.7

8.8

10.5

21.7

19.8

Na(HSV)

0

0

641

9.69

10.78

1.09

281.7

317.3

11.2

K

0

0

338 (337)

8.11 (8.22)

9.10 (9.06)

0.99 (0.844)

520.5 (515.3)

534.4 (528.5)

2.7 (2.55)

17.7

Rb

0

0

310 (312) 375 (373) 443 (433)

9.33 (9.41) 9.49 (9.54) 9.61 (9.56)

10.31 (10.25) 10.46 (10.34) 10.54 (10.34)

0.98 (0.828) 0.97 (0.803) 0.94 (0.781)

630.4 (631.8) 560.1 (562.6) 494.8 (495.6)

646.9 (648.1) 573.7 (573.0) 497.7 (501.0)

2.6 (2.57) 2.4 (1.67) 0.6 (1.09)

17.4

291 (302)

10.16 (10.28)

11.14 (11.115)

0.98 (0.846)

796.4 (786.7)

817.1 (806.0)

2.6 (2.45)

Cs

3.8

11.0

10.0

21.6

0

0

for Na, K, Rb and Cs. Some o f the results are presented and compared with experimental data [1,18,19] in table 1 and figs. 1,2. For convenience the experimental compression"u sexp = 1 - ~s(p)/~2s(0) along the melting curve [1,18] corresponding to the given p is presented in figs. 1 and 2 instead o f p . Table 1 and fig. 1 show, firstly, that for all moderate u s ~ 0.15 the calculated T i n ( p ) a g r e e with the observed ones within several kelvin, i.e. 1-2%. Such a precision confirms the adequacy o f both expressions (4) and (6) for G~, G s and the pseudopotentia] model o f interactions used. A more detailed analysis o f the results [20] shows that the accuracy of the model for the liquid and solid phases separately is not as high as that in Tm(P ). For example, for Na a t p = 0, T = T exp the calculated values of G*(T) = G(T) - G(0) are G~* = 1494, G s* = 1493, while G *exp = 1540 K [19] ; for Rb the values are 1976, 1974 and 1986 K, respectively. However, the accuracy o f expressions (4), (6) for a given model o f interactions appears to be much higher than that o f the model used for the metals considered. Sdch differential quantities as melting characteristics do not seem very sensitive to slight inaccuracies o f the model, and the corresponding errors 6G~ and fiG s in eq. (1) for Tin(p) are almost can-

17.9 18.1

17.6 17.5 17.5

celled; to a large extent this is also true for the errors 6S~, 5S s in AS' = S~ - S s and 6~2~, 5 ~ s in A~2 = ~2~ - ~2s, see table 1 and below. Therefore, the use o f the WCA and QHPT approaches (4), (6) seems very promising for quantitative evaluations of the melting characteristics when the used models of interactions are not too crude. Note also that since ~(G s - G~)/aT = AS ~ 1 (see table 1) the error in Tm(P ) according to (1) is 6T m 6G~ - fiG s. Thus, the allowance made for relatively small contributions to G~ or G s affects noticeably the accuracy o f the T m ( p ) calculations. For example, in solid Na at p = 0, T = T m the abovementioned terms ,.,..U 2 ' ~ U l3 , Fan , F e* and Gde f in (6) are negative and have values o f 6 G i = 39, 4, 17, 9 and 3 K, respectively. Omission of any o f them would have resulted in underestimating T m by ISG il. In a liquid the term FI~ in (4) is very important. Thus, in liquid Na at p = 0, T = TmFI~ = - 4 9 2 K, and the neglect o f this term in the HSV approximation (3) results in considerable overestimating T m [see the line Na(HSV) in the table]. Therefore, the simple HSV method is insufficient for a quantitative treatment of melting, while most o f the previous approaches were based on this method, see refs. [5,6] and a review in ref. [10]. 77

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the vacancy creation and must therefore have some other origin - if the generally accepted moderate values of Sly ~< 1 [21,13] are supposed. The quantitative calculations of the Sly and Ely values might help to resolve the problem. In the last column of the table we present Lindeman's ratio r L = 2(x2)l/2/dnn (see ref. [12]). For all the considered metals and compressions the rL value is practically the same: r L ~ 0.18. Therefore, Lindeman's melting rule holds here rather accurately.

\ f

The authors are greatly indebted to S.M. Stishov and A.M. Nikolaenko for the valuable discussions and information on the experimental data. Ib

'

2O

Fig. 2. The relative volume jumps A~/~2s along the melting curves versus the compression u exp. The dashed curve means the results for Rb obtained for the "pressure-independent defect volume", see the text. The experimental points [1,18]: e-Na,

X -K,A-

Rb,•-Cs.

Let us discuss the results for the volume jumps A ~ given in the table and fig. 2. A t p = 0 the theoretical A~2th agree well with the observed A~~exp. However, when p rises the disagreement increases sharply and for 0.03 ~ u s <~ 0.15 the A~2th exceed A~ex p noticeably while for u s > 0.15 the A~2th fall off with p too rapidly. The latter is obviously due to the mentioned inapplicability of the WCA method for large u s which is also displayed in fig. 1 in a nonphysical variation of Ttmh(us) for u s ~ 0.2. For small u s the unrealistic rise of A~'~th above A~'~exp is entirely due to the assumption (7) of the "vacancy" origin of Gde f. When p rises, Gvac(P) (7) and ~ d e f = OGdef/OP tend rapidly to zero and become negligible at u s ~ 0.05. However, if we suppose that the defect volume ~-~def: (i) at p = 0 is close numerically to the estimated above f~vac, (ii) does not actually vary significantly with p: ~2def(p ) ~'~def(0), then the calculated A ~ ( p ) will agree well with the A~ex p for all u s <~ 0.15. This is illustrated in fig. 2 for Rb. The analogous (though less pronounced) improvement occurs in the fusion entropy/XSth if we suppose the defect contribution to S s to be similarly pressure independent. Thus, our results give another indication (in addition to the direct measurements of the vacancy concentration [ 13]) that the pre-melting anomalies in thermodynamics cannot be explained by

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18 June 1984

References [1] S.M. Stishov, Usp. Fiz. Nauk 114 (1974) 3. [2] A.R. Ubellohde, The molten state of matter (WileyInterscience, New York, 1978). [31 J.P. Hansen and L. Verlet, Phys. Rev. 184 (1969) 151. [4] B.L. Holian, G.K. Straub, R.E. Swanson and D.C. Wallace, Phys. Rev. B27 (1983) 2873. [5] D. Stroud and N.W. Ashcroft, Phys. Rev. B5 (1972) 371. [6] H.D. Jones, Phys. Rev. A8 (1973) 3215. [7] A. Angelie and J.L. Pelissier, Physica 121A (1983) 207. [8] J.L. Pelissier, Physica 121A (1983) 217. [9] D.A. Young and M. Ross, Phys. Rev. B29 (1984) 682. [10] A.M. Bratkovsky, V.G. Vaks and A.V. Trefilov, J. Phys. F13 (1983) 2517. [11] V.G. Vaks and A.V. Trefilov, Fiz. Tverd. Tela 19 (1977) 244. [12] V.G. Vaks, E.V. Zarochentsev, S.P. Kravchuk, V.P. Safronov and A.V. Trefilov, Phys. Stat. Sol. 85b (1978) 63, 749. [13] V.G. Vaks, S.P. Kravchuk and A.V. Trefilov, J. Phys. F10 (1980) 2325. [14] A.M. Bratkovsky, V.G. Vaks and A.V. Trefilov, J. Phys. F12 (1982) 1293. [15] J.D. Weeks, D. Chandler and H.C. Andersen, J. Chem. Phys. 54 (1971) 5237; 55 (1971) 5222. [16] J.A. Barker and D. Henderson, Rev. Mod. Phys. 48 (1976) 588. [17] G. Jacucci and R. Taylor, J. Phys. F9 (1979) 1489. [18] A.M. Nikolaenko, I.N. Makarenko and S.M. Stishov, Solid State Commun. 27 (1978) 475. [19] E.E. Spielrein, K.A. Yakimovitch, E.E. Totsky, D.P. Timrot and V.A. Fomin, Thermophysical properties of alkali metals (Standards, Moscow, 1970). [20] A.M. Bratkovsky, V.G. Vaks and A.V. Trefilov, JETP (1984), to be published. [21] A. Seeger, J. Phys. F3 (1973) 248.