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On thermodynamic properties of crystals in the metastable region
Pergamon
Solid State Communications, Vol. 91, No. 12, pp. 941-944, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098(94)$7.00 +. 00 0038- i 098(94)00506-0
ON THERMODYNAMIC P R O P E R T I E S OF CRYSTALS IN T H E METASTABLE REGION V.I. ZUBOV*, A.A. Caparica & N.P. Tretiakov* Universidade Federal de Golhs, Instituto de Matem~tica e Fisica, C,P. 131 74001-970, Goi~nia, Goi~m, BRAZIL J.F. Sanchez Ortiz Department of Theoretical Physics, People's Friendship University 117198, Moscow, RUSSIA
( R e c e ~ 16 March, 1994 by C.E.T. Goncalves da Silva)
The possibility of superheating crystals h a s been firmly established but experimental investigations of their metastable states p r e s e n t great difficulties. We have u n d e r t a k e n a theoretical s t u d y of t h e i r t h e r m o d y n a m i c properties n e a r the limit of stability. Using the correlative m e t h o d of u n s y m m e t r i z e d self-consistent field, we have explored the properties of Van der Waals crystals with the facecentered cubic lattice. The exponents governing their peculiarities in the vicinity of the unstable point have been evaluated and discussed.
Keywords: D. a n h a r m o n i c i t y , D. crystal binding a n d e q u a t i o n of state, D. p h a s e transition, D. thermal expansion, D. t h e r m o d y n a m i c properties
liquid phase. Well-prepared monocrystals can be superheated when heated from the inside, for instance by high-frequency electromagnetic field 7. Metallic samples can be superheated by a highpowered current pulse 8. Recently, the laserinduced superheating of the Pb ( I 1 I) surface considerably above the bulk melting temperature has been observed 9.
It is well known that in first-kind phase transitions there exist metastable states. They correspond to local rather t h a n absolute m i n i m u m of the Gibbs free energy of a system and can occur u n d e r s u c h conditions when a n o t h e r phase is the most stable having the absolute m i n i m u m ~a. Metastable states of fluid systems have long been seen and to date m u c h studied a'5. Supercooling and superheating are common for p o l l m o r p h i c t r a n s i t l o n s 8. Investigations of crystals superheated above the melting temperatures have been hampered by occurence of lattice defects, especially at surfaces, which are liable to be nuclei of the
On the other hand, a theoretical s t u d y of a metastable crystal is complicated by the fact that at temperatures close to and especially above the melting point, the anharmonicity of the lattice vibrations, being strong, may not be treated as a small p e r t u r b a t i o n . The q u a s i - h a r m o n i c approximation gives values for temperatures of thermodynamic instability of crystals which are
• On leave from Peoples' Friendship University, Moscow, Russia 941
THERMODYNAMIC PROPERTIES OF CRYSTALS
942
n e a r or below t h e i r m e l t i n g p o i n t s . This c o u l d signify the i m p o s s i b i l i t y of s u p e r h e a t i n g crystals. However, a c o n s i s t e n t i n c l u s i o n of a n h a r m o n l c terms provides both d y n a m i c a l j° a n d t h e r m o d y n a m i c ~'12 relative s t a b i l i t y of c r y s t a l s above t h e i r m e l t i n g t e m p e r a t u r e s . The i n f e r e n c e a b o u t a n o c c u r e n e e of t h e m e t a s t a b l e region of a c r y s t a l b e y o n d the solid-liquid c o e x i s t a n c e curve Is c o n f i r m e d b y c o m p u t e r s i m u l a t i o n s ~3. So, the possibility of s u p e r h e a t i n g c r y s t a l s above t h e i r m e l t i n g p o i n t s h a s b e e n firmly e s t a b l i s h e d b u t e x p e r i m e n t a l i n v e s t i g a t i o n s of their m e t a s t a b l e s t a t e s p r e s e n t great difficulties. This s t i m u l a t e s a n i n t e r e s t in theoretical s t u d i e s . To p e r f o r m c a l c u l a t i o n s we u s e the correlative m e t h o d of u n s y m m e t r i z e d s e l f - c o n s i s t e n t field (CUSF/L14-m. It e n a b l e s one to t a k e i n t o a c c o u n t t h e m a i n a n h a r m o n i c t e r m s w i t h o u t r e s o r t i n g to perturbation techniques. I n c l u d i n g a n h a r m o n i c t e r m s u p to the f o u r t h order i n t o the zeroth a p p r o x i m a t i o n , the e q u a t i o n of s t a t e for a s t r o n g l y a n h a r m o n i c c u b i c c r y s t a l a t the t e m p e r a t u r e T = 0 / k a n d e x t e r n a l p r e s s u r e P is of the form a 1 dKo + [30 dK2 P = --~v[-2 da 2 K 2 da
+ (3-0)0 dK4] 4K 4 da
D_2. s ( X + ~ ) = 3X.
(2)
D-I
,s(~'+ 5~(X) "-6X )
where Dr(z) are the parabolic cylinder functions. Corrections to the zeroth approximations (1 ) take into account the interatomic correlations 15, higher-order anharrnonic terms 'I a n d q u a n t u m effects 16.,8. It is k n o w n t h a t the c o n d i t i o n s of t h e r m o d y n a m i c stability of c u b i c crystals are: Br > 0,
C ~ i > 0,
~o,
T -~ •o
(3)
w h e r e B~ is the i s o t h e r m a l b u l k m o d u l u s , CTIjthe i s o t h e r m a l elastic coefficients a n d Cv t h e isochoric specific heat. Violating one of t h e s e i n e q u a l i t i e s implies a loss in t h e r m o d y n a m i c stability. The c o r r e s p o n d i n g e q u a t i o n t o g e t h e r with e q u a t i o n s of s t a t e define the limit of stability. For V a n - d e r - W a a l s crystals with the facec e n t e r e d c u b i c lattice to w h i c h we restrict o u r c o n s i d e r a t i o n here, the t h e r m o d y n a m i c s t a b i l i t y is lost w h e n the i s o t h e r m a l b u l k m o d u l u s goes to zero, i.e. t h e first of i n e q u a l i t i e s b r e a k s down, w h e r e a s o t h e r coefficients of s t a b i l i t y r e m a i n positive. Fig. 1 shows the theoretical limit of s t a b i l i t y for solid a r g o n a n d its e x p e r i m e n t a l m e l t i n g curve 19 with the l o w - t e m p e r a t u r e e x t r a p o l a t i o n 2°. The width of the m e t a s t a b l e region i n c r e a s e s with p r e s s u r e . This is c o n s i s t e n t with the i n f e r e n c e 3'19 a b o u t the a b s e n c e of critical p o i n t s o n melting. Both curves come close t o g e t h e r at negative p r e s s u r e s w h e r e two p h a s e s are m e t a s t a b l e a n d could have the c o m m o n horizontal tangent at absolute zero of t e m p e r a t u r e b u t below the p o i n t L, s t r e t c h e d liquid is u n s t a b l e 2~.
(i)
Here a is the l n t e r a t o m l c d i s t a n c e , v(a) t h e v o l u m e of the u n i t cell, Ko(a)/2 t h e static e n e r g y per atom, K2 a n d K4 are the s e c o n/ dh . .-- - . . .a. . .n_ d fourthorder force coefficients a n d [3(KJ3/0K4") is the s o l u t i o n of the t r a n s c e n d e n t a l e q u a t i o n
(X)
cl,
Vol. 91, No. 12
C ~ i - c r l z > 0,
In p h a s e t r a n s i t i o n s a n d critical p h e n o m e n a , one of t h e m o s t i n t e r e s t i n g a n d c o m p l i c a t e d p r o b l e m s is the critical e x p o n e n t s w h i c h govern the p e c u l i a r i t i e s of t h e r m o d y n a m i c properties of a s y s t e m in the vicinity of the critical p o i n t 2. A l t h o u g h m e l t i n g curves have n o critical p o i n t s , s o m e t h e r m o d y n a m i c properties of crystals p o s s e s s s i n g u l a r i t i e s in the m e t a s t a b l e region n e a r the limit of stability (curve 2 in Fig. 1). Of course, the s a m e is t r u e for a n y m e t a s t a b l e p h a s e . CUSF e n a b l e s one to e v a l u a t e s u c h e x p o n e n t s for crystal p h a s e s .
For the p u r p o s e of e v a l u a t i n g u n s t a b l e e x p o n e n t s we have solved the e q u a t i o n of s t a t e (1) with all corrections to a high degree of a c c u r a c y . A n h a r m o n i c t e r m s u p to the sixth o r d e r have b e e n i n c l u d e d . At this p o i n t we have o b t a i n e d the c o o r d i n a t e s Ts, a s of the u n s t a b l e p o i n t of a n isobar. In the case of Ar, utilizing the L e n n a r d - J o n e s pairwise p o t e n t i a l together with
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THERMODYNAMIC PROPERTIES OF CRYSTALS
943
12 I
S
1.00
2
8
C 0.90 y. n
4 O. 80
0
I
I
-2
I
I
I
070
I
-0.07
0.'0 (0-%)/%
0,07
T(K)
Fig. 1. The e x p e r i m e n t a l m e l t i n g c u r v e of a r g o n 19 w i t h the l o w - t e m p e r a t u r e e x t r a p o l a t i o n 2° (I) a n d its t h e o r e t i c a l T M limit of s t a b i l i t y (2); L is the u n s t a b l e p o i n t of s t r e t c h e d liquid a r g o n 2~.
Fig.2. The s t a b l e ( I ) a n d u n s t a b l e (2) b r a n c h e s of t h e zero i s o b a r of solid a r g o n i n t h e vicinity of its limit of s t a b i l i t y S.
W e h a v e u s e d five p a i r s of p o i n t s T,, T 2 a n d
derived t h e Axilrod-Teller t h r e e - b o d y p o t e n t i a l we have a t zero p r e s s u r e Ts=0.930614~,
-
1
(7)
2
as=1.223741o(4)
w i t h a n a c c u r a c y of 0.4%. h e r e e a n d o are t h e p a r a m e t e r s of t h e L e n n a r d J o n e s p o t e n t i a l 22. This t e m p e r a t u r e is a b o u t 42 p e r c e n t above t h e m e l t i n g point. Note t h a t m o r e realistic B a r k e r p o t e n t i a l gives t h e v a l u e of "Is w h i c h is s o m e w h a t below (4). Thereafter, we h a v e solved e q u a t i o n of s t a t e a l o n g t h e s t a b l e b r a n c h of t h e i s o b a r i n the i m m e d i a t e vicinity of t h e p o i n t (4), see Fig.2. Finally, p r e s u m i n g as it u s u a l l y is, t h a t at (Ts-T)/Ts << 1 t h e r e is t h e power dependence a s - a - (Ts-T)P
in ( a"-al ) as-a2
i
Br_(Ts_T)];
-! ~t,C_(Ts_T)
2
(a)
(5)
we h a v e c o m p u t e d t h e c o r r e s p o n d i n g e x p o n e n t
-
We also have c a l c u l a t e d t h e r m o d y n a m i c p r o p e r t i e s i n t h e n e l g h b o u r h o o d of t h e u n s t a b l e p o i n t (4). The Isochoric specific h e a t a n d c o m p o n e n t s of the elastic t e n s o r r e m a i n finite a t t h i s point, w h e r e a s t h e i s o t h e r m a l b u l k m o d u l u s goes t o zero a n d t h e t h e r m a l e x p a n s i o n coefficient a n d t h e isobaric specific h e a t t e n d to i n f i n i t y with t h e f o r m u l a
(6)
%-T~ in (T---~-~2)
w h e r e ah = a(T) a r e t h e s o l u t i o n s of t h e e q u a t i o n of state.
It s h o u l d be e m p h a s i z e d t h a t the r e s u l t s o b t a i n e d are far from trivial s i n c e e q u a t i o n of s t a t e (1) c o n t a i n s h i g h e r t r a n s c e n d e n t a l f u n c t i o n s w h i c h i n n o w a y b e a r a r e s e m b l a n c e to the d e p e n d e n c e (5), (7), (8). It is i n t e r e s t i n g also t h a t the q u a n t u m correction a n d higher-order a n h a r m o n i c i t y w h i c h (especially t h e latter) s i g n i f i c a n t l y affect the w i d t h of the m e t a s t a b l e region, h a v e practically n o effect o n t h e u n s t a b l e exponents.
944
THERMODYNAMIC PROPERTIES OF CRYSTALS
So, using CUSF we have u n d e r t a k e n calculations of unstable exponents for Van der Waals crystals with FCC lattice. Note that their thermodynamic instability Is not directly related to so-called soft modes. The role of soft modes in the first-order phase transitions has been discussed recently 2a. Since the stability limit in one-component crystals is experimentally unattainable we are hopeful of arousing interest In calculations of such properties using other theoretical methods or computer simulations for comparison with our results. S u c h a limit has been observed In some alloys in which the process of phase transition occurs just through splnodal decomposition 24.
Vol. 91, No. 12
In conclusion we note that for other crystal lattices, the thermodynamic Instability m a y be related not to a violation of the first condition (3) but to those of other conditions, for Instance, to the vanishing of the shearing components Ca4 or Cll - Ct2.
Acknowledgement
Two of the authors (V.I.Z. and N.P.T) are financially supported by Conselho Nacional de Desenvolvlmento Clentifico e Tecnol6gico - CNPq (Brazil).
R e f e r e n c e s
I . R . B r o u t , Phase Transitions, Benjamin, New York (1965). 2. H.E.Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford ( 197 I). 3. L.D.Landau & E.M.Lffshitz, Statistical Physics, Nauka, Moscow (1976). 4. V.P.Skripov, Metastable Liquids, J o h n Willey, New York (1974). 5. V.P.Skripov, Non-Equilibr. Thermodyn. 17, 193 (1993). 6. N.N.Sirota, Crystal Res. and Technol. 17, 661 (1982). 7. Yu.B.Rumer & M.Sh.Ryvkin, Thermodynamics, Statistical Physics and Kinetics, Nauka, Moscow (1977). 8. M.M.Martynyuk, Termochim. Acta 206, 55 (1992). 9. J.W.Herman & H.E.Elsayed-Ali, Phys. Rev. Lett. 69, 8, 1228 (1992). 10.N.M.Plakida & T.Siklos, Acta Phys. Hung. 45, 37 (1978). I 1.V.I.Zubov, Phys. Stat. Sol (b) 87, 385; 88, 43 (1978).
12.L.K.Moleko & H.R.Glyde, Phys. Rev. B30, 4215 (1984). 13.V.G.Baidakov, A.E.Galashev & V.P.Skripov, Fiz. Tverdogo Tela (USSR) 22, 2681 (1980). 14.V.I.Zubov, Ann. Phys. (Leipzig) 31, 33 (1974). 15.V.I.Zubov,Phys.Stat.Sol.(b) 72,7 l, 483 (1975). 16.V.I.Yukalov & V.I.Zubov, Fortschr. Phys. 31, 627 (1983). 17.V.I.Zubov & V.B.Magallnskll, in Teploflzlcheskie Svoistva Metastabilnykh Sistem (Thermophysical Properties of Metastable Systems), Sverdlovsk, p.42 (1984). 18.V.I.Zubov, Int.J.Mod.Phys. B6, 367 (1992). 19.S.M.Stishov, Uspekhl Fiz.Nauk (USSR) 114, 3 (1974). 20.V.P.Skripov, Teploflzika Vysokhih Temperatur (USSR) 19, 85 (1981). 2 l.V.P.Skrlpov & M.Z.Faizullin, see Ref. [ 17], p.8. 22.G.G.Chell & I.J.Zucker, J.Phys. CI, 35 (1968). 23.J.A.Krumhansl, Solid State Commun. 84, 251 (1992). 24.J.W.Cahn, Trans. Metall. Soc. AIME 242, 166 (1968).
Solid State Communications, Vol. 91, No. 12, pp. 941-944, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098(94)$7.00 +. 00 0038- i 098(94)00506-0
ON THERMODYNAMIC P R O P E R T I E S OF CRYSTALS IN T H E METASTABLE REGION V.I. ZUBOV*, A.A. Caparica & N.P. Tretiakov* Universidade Federal de Golhs, Instituto de Matem~tica e Fisica, C,P. 131 74001-970, Goi~nia, Goi~m, BRAZIL J.F. Sanchez Ortiz Department of Theoretical Physics, People's Friendship University 117198, Moscow, RUSSIA
( R e c e ~ 16 March, 1994 by C.E.T. Goncalves da Silva)
The possibility of superheating crystals h a s been firmly established but experimental investigations of their metastable states p r e s e n t great difficulties. We have u n d e r t a k e n a theoretical s t u d y of t h e i r t h e r m o d y n a m i c properties n e a r the limit of stability. Using the correlative m e t h o d of u n s y m m e t r i z e d self-consistent field, we have explored the properties of Van der Waals crystals with the facecentered cubic lattice. The exponents governing their peculiarities in the vicinity of the unstable point have been evaluated and discussed.
Keywords: D. a n h a r m o n i c i t y , D. crystal binding a n d e q u a t i o n of state, D. p h a s e transition, D. thermal expansion, D. t h e r m o d y n a m i c properties
liquid phase. Well-prepared monocrystals can be superheated when heated from the inside, for instance by high-frequency electromagnetic field 7. Metallic samples can be superheated by a highpowered current pulse 8. Recently, the laserinduced superheating of the Pb ( I 1 I) surface considerably above the bulk melting temperature has been observed 9.
It is well known that in first-kind phase transitions there exist metastable states. They correspond to local rather t h a n absolute m i n i m u m of the Gibbs free energy of a system and can occur u n d e r s u c h conditions when a n o t h e r phase is the most stable having the absolute m i n i m u m ~a. Metastable states of fluid systems have long been seen and to date m u c h studied a'5. Supercooling and superheating are common for p o l l m o r p h i c t r a n s i t l o n s 8. Investigations of crystals superheated above the melting temperatures have been hampered by occurence of lattice defects, especially at surfaces, which are liable to be nuclei of the
On the other hand, a theoretical s t u d y of a metastable crystal is complicated by the fact that at temperatures close to and especially above the melting point, the anharmonicity of the lattice vibrations, being strong, may not be treated as a small p e r t u r b a t i o n . The q u a s i - h a r m o n i c approximation gives values for temperatures of thermodynamic instability of crystals which are
• On leave from Peoples' Friendship University, Moscow, Russia 941
THERMODYNAMIC PROPERTIES OF CRYSTALS
942
n e a r or below t h e i r m e l t i n g p o i n t s . This c o u l d signify the i m p o s s i b i l i t y of s u p e r h e a t i n g crystals. However, a c o n s i s t e n t i n c l u s i o n of a n h a r m o n l c terms provides both d y n a m i c a l j° a n d t h e r m o d y n a m i c ~'12 relative s t a b i l i t y of c r y s t a l s above t h e i r m e l t i n g t e m p e r a t u r e s . The i n f e r e n c e a b o u t a n o c c u r e n e e of t h e m e t a s t a b l e region of a c r y s t a l b e y o n d the solid-liquid c o e x i s t a n c e curve Is c o n f i r m e d b y c o m p u t e r s i m u l a t i o n s ~3. So, the possibility of s u p e r h e a t i n g c r y s t a l s above t h e i r m e l t i n g p o i n t s h a s b e e n firmly e s t a b l i s h e d b u t e x p e r i m e n t a l i n v e s t i g a t i o n s of their m e t a s t a b l e s t a t e s p r e s e n t great difficulties. This s t i m u l a t e s a n i n t e r e s t in theoretical s t u d i e s . To p e r f o r m c a l c u l a t i o n s we u s e the correlative m e t h o d of u n s y m m e t r i z e d s e l f - c o n s i s t e n t field (CUSF/L14-m. It e n a b l e s one to t a k e i n t o a c c o u n t t h e m a i n a n h a r m o n i c t e r m s w i t h o u t r e s o r t i n g to perturbation techniques. I n c l u d i n g a n h a r m o n i c t e r m s u p to the f o u r t h order i n t o the zeroth a p p r o x i m a t i o n , the e q u a t i o n of s t a t e for a s t r o n g l y a n h a r m o n i c c u b i c c r y s t a l a t the t e m p e r a t u r e T = 0 / k a n d e x t e r n a l p r e s s u r e P is of the form a 1 dKo + [30 dK2 P = --~v[-2 da 2 K 2 da
+ (3-0)0 dK4] 4K 4 da
D_2. s ( X + ~ ) = 3X.
(2)
D-I
,s(~'+ 5~(X) "-6X )
where Dr(z) are the parabolic cylinder functions. Corrections to the zeroth approximations (1 ) take into account the interatomic correlations 15, higher-order anharrnonic terms 'I a n d q u a n t u m effects 16.,8. It is k n o w n t h a t the c o n d i t i o n s of t h e r m o d y n a m i c stability of c u b i c crystals are: Br > 0,
C ~ i > 0,
~o,
T -~ •o
(3)
w h e r e B~ is the i s o t h e r m a l b u l k m o d u l u s , CTIjthe i s o t h e r m a l elastic coefficients a n d Cv t h e isochoric specific heat. Violating one of t h e s e i n e q u a l i t i e s implies a loss in t h e r m o d y n a m i c stability. The c o r r e s p o n d i n g e q u a t i o n t o g e t h e r with e q u a t i o n s of s t a t e define the limit of stability. For V a n - d e r - W a a l s crystals with the facec e n t e r e d c u b i c lattice to w h i c h we restrict o u r c o n s i d e r a t i o n here, the t h e r m o d y n a m i c s t a b i l i t y is lost w h e n the i s o t h e r m a l b u l k m o d u l u s goes to zero, i.e. t h e first of i n e q u a l i t i e s b r e a k s down, w h e r e a s o t h e r coefficients of s t a b i l i t y r e m a i n positive. Fig. 1 shows the theoretical limit of s t a b i l i t y for solid a r g o n a n d its e x p e r i m e n t a l m e l t i n g curve 19 with the l o w - t e m p e r a t u r e e x t r a p o l a t i o n 2°. The width of the m e t a s t a b l e region i n c r e a s e s with p r e s s u r e . This is c o n s i s t e n t with the i n f e r e n c e 3'19 a b o u t the a b s e n c e of critical p o i n t s o n melting. Both curves come close t o g e t h e r at negative p r e s s u r e s w h e r e two p h a s e s are m e t a s t a b l e a n d could have the c o m m o n horizontal tangent at absolute zero of t e m p e r a t u r e b u t below the p o i n t L, s t r e t c h e d liquid is u n s t a b l e 2~.
(i)
Here a is the l n t e r a t o m l c d i s t a n c e , v(a) t h e v o l u m e of the u n i t cell, Ko(a)/2 t h e static e n e r g y per atom, K2 a n d K4 are the s e c o n/ dh . .-- - . . .a. . .n_ d fourthorder force coefficients a n d [3(KJ3/0K4") is the s o l u t i o n of the t r a n s c e n d e n t a l e q u a t i o n
(X)
cl,
Vol. 91, No. 12
C ~ i - c r l z > 0,
In p h a s e t r a n s i t i o n s a n d critical p h e n o m e n a , one of t h e m o s t i n t e r e s t i n g a n d c o m p l i c a t e d p r o b l e m s is the critical e x p o n e n t s w h i c h govern the p e c u l i a r i t i e s of t h e r m o d y n a m i c properties of a s y s t e m in the vicinity of the critical p o i n t 2. A l t h o u g h m e l t i n g curves have n o critical p o i n t s , s o m e t h e r m o d y n a m i c properties of crystals p o s s e s s s i n g u l a r i t i e s in the m e t a s t a b l e region n e a r the limit of stability (curve 2 in Fig. 1). Of course, the s a m e is t r u e for a n y m e t a s t a b l e p h a s e . CUSF e n a b l e s one to e v a l u a t e s u c h e x p o n e n t s for crystal p h a s e s .
For the p u r p o s e of e v a l u a t i n g u n s t a b l e e x p o n e n t s we have solved the e q u a t i o n of s t a t e (1) with all corrections to a high degree of a c c u r a c y . A n h a r m o n i c t e r m s u p to the sixth o r d e r have b e e n i n c l u d e d . At this p o i n t we have o b t a i n e d the c o o r d i n a t e s Ts, a s of the u n s t a b l e p o i n t of a n isobar. In the case of Ar, utilizing the L e n n a r d - J o n e s pairwise p o t e n t i a l together with
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THERMODYNAMIC PROPERTIES OF CRYSTALS
943
12 I
S
1.00
2
8
C 0.90 y. n
4 O. 80
0
I
I
-2
I
I
I
070
I
-0.07
0.'0 (0-%)/%
0,07
T(K)
Fig. 1. The e x p e r i m e n t a l m e l t i n g c u r v e of a r g o n 19 w i t h the l o w - t e m p e r a t u r e e x t r a p o l a t i o n 2° (I) a n d its t h e o r e t i c a l T M limit of s t a b i l i t y (2); L is the u n s t a b l e p o i n t of s t r e t c h e d liquid a r g o n 2~.
Fig.2. The s t a b l e ( I ) a n d u n s t a b l e (2) b r a n c h e s of t h e zero i s o b a r of solid a r g o n i n t h e vicinity of its limit of s t a b i l i t y S.
W e h a v e u s e d five p a i r s of p o i n t s T,, T 2 a n d
derived t h e Axilrod-Teller t h r e e - b o d y p o t e n t i a l we have a t zero p r e s s u r e Ts=0.930614~,
-
1
(7)
2
as=1.223741o(4)
w i t h a n a c c u r a c y of 0.4%. h e r e e a n d o are t h e p a r a m e t e r s of t h e L e n n a r d J o n e s p o t e n t i a l 22. This t e m p e r a t u r e is a b o u t 42 p e r c e n t above t h e m e l t i n g point. Note t h a t m o r e realistic B a r k e r p o t e n t i a l gives t h e v a l u e of "Is w h i c h is s o m e w h a t below (4). Thereafter, we h a v e solved e q u a t i o n of s t a t e a l o n g t h e s t a b l e b r a n c h of t h e i s o b a r i n the i m m e d i a t e vicinity of t h e p o i n t (4), see Fig.2. Finally, p r e s u m i n g as it u s u a l l y is, t h a t at (Ts-T)/Ts << 1 t h e r e is t h e power dependence a s - a - (Ts-T)P
in ( a"-al ) as-a2
i
Br_(Ts_T)];
-! ~t,C_(Ts_T)
2
(a)
(5)
we h a v e c o m p u t e d t h e c o r r e s p o n d i n g e x p o n e n t
-
We also have c a l c u l a t e d t h e r m o d y n a m i c p r o p e r t i e s i n t h e n e l g h b o u r h o o d of t h e u n s t a b l e p o i n t (4). The Isochoric specific h e a t a n d c o m p o n e n t s of the elastic t e n s o r r e m a i n finite a t t h i s point, w h e r e a s t h e i s o t h e r m a l b u l k m o d u l u s goes t o zero a n d t h e t h e r m a l e x p a n s i o n coefficient a n d t h e isobaric specific h e a t t e n d to i n f i n i t y with t h e f o r m u l a
(6)
%-T~ in (T---~-~2)
w h e r e ah = a(T) a r e t h e s o l u t i o n s of t h e e q u a t i o n of state.
It s h o u l d be e m p h a s i z e d t h a t the r e s u l t s o b t a i n e d are far from trivial s i n c e e q u a t i o n of s t a t e (1) c o n t a i n s h i g h e r t r a n s c e n d e n t a l f u n c t i o n s w h i c h i n n o w a y b e a r a r e s e m b l a n c e to the d e p e n d e n c e (5), (7), (8). It is i n t e r e s t i n g also t h a t the q u a n t u m correction a n d higher-order a n h a r m o n i c i t y w h i c h (especially t h e latter) s i g n i f i c a n t l y affect the w i d t h of the m e t a s t a b l e region, h a v e practically n o effect o n t h e u n s t a b l e exponents.
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THERMODYNAMIC PROPERTIES OF CRYSTALS
So, using CUSF we have u n d e r t a k e n calculations of unstable exponents for Van der Waals crystals with FCC lattice. Note that their thermodynamic instability Is not directly related to so-called soft modes. The role of soft modes in the first-order phase transitions has been discussed recently 2a. Since the stability limit in one-component crystals is experimentally unattainable we are hopeful of arousing interest In calculations of such properties using other theoretical methods or computer simulations for comparison with our results. S u c h a limit has been observed In some alloys in which the process of phase transition occurs just through splnodal decomposition 24.
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In conclusion we note that for other crystal lattices, the thermodynamic Instability m a y be related not to a violation of the first condition (3) but to those of other conditions, for Instance, to the vanishing of the shearing components Ca4 or Cll - Ct2.
Acknowledgement
Two of the authors (V.I.Z. and N.P.T) are financially supported by Conselho Nacional de Desenvolvlmento Clentifico e Tecnol6gico - CNPq (Brazil).
R e f e r e n c e s
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12.L.K.Moleko & H.R.Glyde, Phys. Rev. B30, 4215 (1984). 13.V.G.Baidakov, A.E.Galashev & V.P.Skripov, Fiz. Tverdogo Tela (USSR) 22, 2681 (1980). 14.V.I.Zubov, Ann. Phys. (Leipzig) 31, 33 (1974). 15.V.I.Zubov,Phys.Stat.Sol.(b) 72,7 l, 483 (1975). 16.V.I.Yukalov & V.I.Zubov, Fortschr. Phys. 31, 627 (1983). 17.V.I.Zubov & V.B.Magallnskll, in Teploflzlcheskie Svoistva Metastabilnykh Sistem (Thermophysical Properties of Metastable Systems), Sverdlovsk, p.42 (1984). 18.V.I.Zubov, Int.J.Mod.Phys. B6, 367 (1992). 19.S.M.Stishov, Uspekhl Fiz.Nauk (USSR) 114, 3 (1974). 20.V.P.Skripov, Teploflzika Vysokhih Temperatur (USSR) 19, 85 (1981). 2 l.V.P.Skrlpov & M.Z.Faizullin, see Ref. [ 17], p.8. 22.G.G.Chell & I.J.Zucker, J.Phys. CI, 35 (1968). 23.J.A.Krumhansl, Solid State Commun. 84, 251 (1992). 24.J.W.Cahn, Trans. Metall. Soc. AIME 242, 166 (1968).