Melting Temperature of Lithium, Sodium and Potassium at High Pressures

Transcription

1 International Journal of Applied Science and Technology Vol. No. 4; April 1 Melting Teperature of Lithiu, Sodiu and Potassiu at High Pressures Chuanhui Nie Shangyong Huang Wei Huang School of Science, Beijing University of Civil Engineering and Architecture Beijing, 144, China Abstract Based on Lindeann s law and the ebye odel, the elting teperature T of etals has been argued. It s found that the elting teperature appears to be decided finally by the second Grüneisen paraeter q,which is assued to have a power for of volue in the preset work. Hence we can obtain a new expression of T different fro that of the prior works. Good agreeent between theory and experient is found. PACS: F; 64.3; 65.; 65.9 Keywords: Metals; High pressure; elting teperature 1. Introduction eterination of the pressure dependence of the elting teperature is of great iportance for understanding structures of solids (Schlosser et al 1989; Fang and Chen 1994; Kechin 1995; Anderson 1998; Fang 1998; Kaver and Jeanloz 1998; Boehler et al 1; Kechin 1; Wang et al 1; Chauhan and Singh ; Zubov et al 3; Liu and Chen 5; Asanua et al 1; Chauhan et al 11). According to Lindeann s law, the relationship between the elting teperature T and the ebye teperature 3 is given by T CV, where C is a constant which depends on the aterial, V is the volue at T. In the ebye odel of a solid, the Grüneisen ln ln paraeter.in the eantie, the second Grüneisen paraete q ( ) T. ln V ln V Both Schlosser et al (1989) and Fang et al (1994) assue that, for etals,. V const, so q 1.But q itself is often a function of volue (Jeanloz 1989; Anderson et al 199; Nie ). In this paper, we shall choose q as q 3n qx (1) V for the alkali etals, where n 3 is a positive constant, x ( ), q and V are the second Grüneisen V paraeter and volue at zero pressure and roo teperature T, respectively, and V is the volue at arbitrary elting teperature T.Then we can get a new expression for T with pressure. We shall copare this pressure dependence of elting teperature T with experiental data. 1. Theory Fro the above analyses, we can get q x 6 ( 1) q n 3 T T ( ) exp n x [1 ( ) n 3n ] () 9

2 Centre for Prooting Ideas, USA 1 V where is the Grüneisen paraeter at zero pressure roo teperature, 3 ( ) and T is elting V teperature at zero pressure. In order to find the relation between T and the pressure P T, V ), we shall search for the relation between ( P( T, V ) and x. According to Birch equation concerning theral effects (Nie and Chen 1999): P ( T, V ) P( T, V ) (, T ) B(, T )( T T ) (3) 3B(, T ) P( T ( ) 1 [, V ) x x B (, T ) 4]( x 1) (4) 4 where (, T ), B(, T ) and B (, T ) are the volue theral expansion coefficient, the isotheral bulk odulus and its first order pressure derivative at zero pressure and roo teperature, respectively. in Eq.() can get fro Eqs. (3) and (4) while letting P( T, V ). 3. esults and iscussions With the ethod entioned above, we fitted the experiental data on the elting teperature under a high pressure for alkali etals (Li, Na and K). The values of the paraeters (, T ), B(, T ), B(, T ) at T (Boehler 1983) are listed in Table 1, the values of, q and n obtained by the fitting are listed in Table. To copare the fitting accuracy, we calculate the elting teperature T of Li, Na and K versus P with Eqs. () and (3) using the paraeters listed in Table. The calculated results in this paper (Eq.()) Eq.(5) by Fang (1998), and the experiental data (Leudeann and Kennedy 1968) are all listed in Table 3 (MS stands for the root ean square deviations of T ). (1) Fro Table 3, we can find that Eq.() is one of the accurate representations for the pressure dependence of the elting teperature of alkali etals, which can be obtained by cobining Lindeann s law with the ebye odel and the Birch EOS, with the assuption that q is a function of volue. The results agree well with the experient. () Fang (1998) has used another ethod to obtain the pressure dependence of elting teperature of the alkali etals : CT T P ln 1 C1( T T ) T (5) He copared the results with Sion s epirical relation T c P a[( ) 1] (6) T and elting relation presented by Schlosser et al.(1989) T T x ( 1 P (7) ) All these relations yield good agreeent for Li, Na and K.at low pressures. When the pressure is larger than 15Gpa, they deviate fro each other. And Eq.(5) is ore close to experient than Eqs.(6) and (7). In Table 3, we calculate values of the root ean square deviations (MS) of T for Li, Na and K by Eqs.() and (5),we can see that results fro Eq.() tend to reality ore than Eq.(5). Besides, in deducing Eq.(5),Fang (1998) has used the linear relationship for the change of volue V along the elting curve V V 1 C ( T T )] (8) [ 1 It s pointed that Eq.(8) ight be iperfect, especially at very high pressures (Fang 1998). (3) While assuing /V is a constant, Fang and Chen (1994) has presented another expression of T for etals as 93

3 International Journal of Applied Science and Technology Vol. No. 4; April 1 x 3 x 3 T T ( ) exp [1 ( ) ] (9) We can get Eq.(9) easily while choosing q n =1 in Eq. (). But fro Table we see that q n 1. Therefore Eq.(9) is just obtained as a approxiation. We prefer q to be a function of volue as Eq.(1). That ight be a new ethod to predict the high teperature second Grüneisen paraeter q for alkali etals. (4) Kechin (1995; 1) has predicted that the elting curve equation at high pressure is T b T (1 P / a) exp( c ) (1) P where P P P. Eq.(1) supports Taann s hypothesis that all aterials have a axiu elting teperature at high pressures. The present Eq.() is just the right agreeent with this point of view, while Eq.(5) isn t satisfied with Taann s hypothesis (see Fig.1). Moreover, ass (1995) has studied the elting axiu in alkali etals. It s found that the elting axiu occurs at 86, 377 and 7 kbar for Li, Na and K, respectively. Fro Fig.1, we can see that our results are in good agreeent with these conclusions. Table 1. Values of (, T ), B(, T ), B(, T ) at T =98K ( Boehler 1983) Solid (, )(1 3 1 K ) B(, T )(1 8 Pa) B, T ) T Li Na K Table. Fitting values of, q and n T ( K q n solid ) [19] Li Na K Table 3. Calculated values of T through Eqs. (), (5) and the experiental data (Leudeann and Kennedy 1968) for Li, Na and K ( P Li Na K (1 8 Pa) () (5) [19] () (5) [19] () (5) [19] MS

4 Centre for Prooting Ideas, USA Fig.1. Melting curves of Li, Na and K. The dashed line: Eq.(5), the solid line: the present work, and the solid circles ( ) : the experiental data (Leudeann and Kennedy 1968). 95

5 International Journal of Applied Science and Technology Vol. No. 4; April 1 Acknowledgeents Thanks for supports fro Funding of Scientific esearch of Beijing Municipality (51167), People s epublic of China. This work is also supported by Funding of Scientific esearch of Beijing University of Civil Engineering and Architecture (19367), People s epublic of China. eferences Anderson,O.L. (1998), The Grüneisen paraeter for iron at outer core conditions and the resulting conductive heat and power in the core. Physics of the Earth and Planetary Interiors,19, Anderson,O. L., Isaak,., Oda, H. (199). High-teperature elastic constant data on inerals relevant to geophysics. eviews of Geophysics, 3, Asanua, H., Ohtani, E., Sakai, T.,Terasaki,H.,Kaada,S.,Kondo,T.,Kikegawa,T.(1), Melting of iron-silicon alloy up to the core-antle boundary pressure: iplications to the theral structure of the Earth s core, Physics and Cheistry of Minerals,37, Boehler,. (1983), Melting teperature, adiabats, and Grüneisen paraeter of lithiu, sodiu and potassiu versus pressure. Physical eview B, 7, Boehler,. oss, M. Soderlind, P., Boercker,.B. (1), High-pressure elting curves of Argon, Krypton, and Xeon: eviation fro corresponding states theory. Physical eview Letters, 86, Chauhan,.S., Singh,C.P. (). Analysis of elting for alkali halides based on the potential energy curve. Physica B, 34, Chauhan,.S., Snehlata, K., Singh,C.P. (11). efects induced elting in alkali halides. Physica B, 46, ass, N. (1995), Melting axiu in alkali etals. Physical eview B, 5, Fang, Z. H. (1998), A new universal relation for the pressure dependence of the elting teperature of solids. Physics Status Solidi (b), 7, Fang, Z. H., Chen, L.. (1994), A siplified treatent to calculate the elting teperature of etals under a high pressure. Journal of Physics: Condensed Matter, 6, Jeanloz,. (1989), Shock wave equation of state and finite strain theory. Journal of Geophysical esearch, 94, Kaver, A., Jeanloz,. (1998), High-pressure elting curve of platinu, Journal of Applied Physics, 83, Kechin V.V. (1995), Therodynaically based elting-curve equation, Journal of Physics: Condensed Matter, 7, Kechin,V.V. (1), Melting curve equations at high pressure. Physics eview B, 65, 51. Leudeann,H.., Kennedy, G.C. (1968), Melting curves of lithiu, sodiu and potassiu, and ubidiu to 8 kilobars. J.Geophys. es 73, Liu, Q. Chen, L.. (5). A siple elting theory applied to alkali halide, alkaline-earth chalcogenide, and alkali chalcogenide copounds.canadian Journal of Physics, 83, Nie, C. H. (), Volue and teperature dependence of the second Grüneisen paraeter of NaCl. Physics Status Solidi (b), 19, Nie,C.H., Chen, L.. (1999),Coparison of three teperature dependent equtions of state of condensed atter. Physics Status Solidi (b), 15, Schlosser, H., Vinet, P., Ferrante, J. (1989), Pressure dependence of the elting teperature of etals. Physical eview B, 4, Wang, Z.W., Lazor, P., Saxena, S.K. (1), A siple odel for assessing the high pressure elting of etals: nickel, aluinu and platinu, Physica B 93, Zubov,V.I., odrigues,c.g., Zubov, I.V. (3), On a possible elting curve of C 6 fullerite, Physics Status Solidi (b), 38,