S tatic structure factor for simple liquid metals

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1 MOLECULAR PHYSICS, 999, VOL. 96, NO. 5, 835 ± 847 S tatic structure factor for simple liquid metals J. N. HERRERA² Department of Chemical Engineering, 49 Dougherty Engineering Building, University of Tennessee, Knoxville, TN , USA P. T. CUMMINGS Chemical Engineering, 49 Dougherty Engineering Building, University of Tennessee, Knoxville, TN , USA and Chemical Technology Division, Oak R idge National Laboratory, Oak R idge, TN , USA and H. RUI Â Z-ESTRADA Facultad de Ciencias Fõ Â sico-matemaâ ticas, BenemeÂ rita Universidad AutoÂ noma de Puebla, Apdo. Postal 5, Puebla, Pue., C. P. 700 MeÂ xico (Received 9 D ecem ber 997; revised version accepted 6 August 998) The structure factors S k for simple liquid metals are calculated using a model uid composed of hard spheres of Y ukawa-type in the mean spherical approximation. The solution of the Ornstein± Zernike equation for the direct correlation function is exact and the analytic expressions are obtained for the structure factor. These expressions are then used to predict structure factors for a number of simple liquid metals, leading to good agreement with experimental data.. Introduction There is an increasing need for understanding the various properties of metals, oxides and salts in the liquid state, because of their important role in metallurgical processes [, ]. Progress in the modelling for metals has been slow [, ]. Interatomic potentials of the modern genre began to appear about 40 years ago when many di erent pseudopotentials were proposed. In fact, even today, there are great uncertainties associated with the ab initio calculations of liquid metal interactions. Thus, although we can borrow the techniques of classical statistical mechanics, we are always, up to some extent, unsure of the potential to which they are to be applied. The hard spherical (HS) system is used as a general reference system [, ]; however, an even more general reference system is obtained if one allows for a screening of the Coulombic interactions between the charged spheres. For this purpose, the hard-sphere Y ukawa ( HSY) uid in the mean spherical approximation ( MSA) has been solved by Waisman [3] and Hayter and Penfold [4]. The HSY reference system was rst ² Permanent address: Facultad de Ciencias Fõ Â sico- MatemaÂ ticas, BenemeÂ rita Universidad A utoâ noma de Puebla, A pdo. Postal 5, Puebla, Pue., C. P. 700, MeÂ xico. applied by Hayter et al. [5] to t the structure factor of the liquid alkali metals. These authors concluded that the most important property of the pair potential for the prediction of the structure factor, S k, is its slope at a potential energy of the order of k B T. Thus the main advantage of the HSY over the HS reference uid is its ability to account not only for the e ective size of the atoms, but also for the softness of the interatomic interaction at close contact. Thus the reference pair potential is a hard sphere ( HS), being perturbed with an attractive ( or repulsive) tail. The analytic solution of this model in the mean spherical approximation ( MSA) was introduced by Waisman [3], however this gave rise to a complex set of equations to solve. A second, simpler, formulation for a Y ukawa uid was given by Blum and Hù ye [6] and Cummings and Smith [7, 8]. More recently Ginoza [9] revised the Blum± Hù ye solution and proposed a new MSA version for a Yukawa uid. In Ginoza s formulation, the direct correlation function has a very simple structure, involves a fundamental parameter g, and provides all the thermodynamic properties ( see [0, ]). In this new version the free energy is being minimized with respect to this parameter [0]. The objective of this work is to determine the static structure factor for some simple liquid metals described 006± 8976/99 $. 00 Ñ 999 Taylor & Francis Ltd.

2 h 836 J. N. Herrera et al. g as HSY uids in terms of, rst in the long-wavelength limit and then, its corresponding structure factor and its k -derivative. In this work we present the S k of simple liquid metals using both cases, an attractive Y ukawa tail ( A HSY) with the parameters obtained by McLaughin and Y oung [,, 3], and a repulsive Y ukawa tail (RHSY) with the parameters obtained by Hayter et al. [5] and Hausleitner and Hafner [4]. In section we describe the MSA approximation and determine the structure factor S k, the long-wavelength limit S k 0, and assess the reliability of the A HSY-MSA to describe the experimental structure factor at k 0. In section 3 we determine the structure factor S k and its k -derivative for some liquid metals. Finally, we summarize the main results in section 4.. Theoretical results The MSA is de ned in terms of the radial distribution function g r, which in the case of liquid metals, is proportional to the probability density of nding an ion at distance r from a given ion; and the direct correlation function c r which is de ned by the Ornstein± Zernike equation [5]: h r c r q c s h r s ds. In equation ( ), the total correlation function h r is de ned to be the deviation of g r from its ideal-gas value of ( i.e. h r g r ). The MSA for hard core systems is then given by the exact core condition g r 0, for r < s, combined with the approximation [6]: c r u r k B T, for r > s, 3 s where k B is Boltzmann s constant, T is the absolute temperature and is the diameter of the hard spheres. Equation ( 3) represents the extrapolation of the large separation asymptotic behaviour of c r to short ranges [6] which, without the tail, becomes the hard sphere Percus± Y evick result. The pair potential, u r /s, of the hard-sphere Yukawa uid is de ned as u y k B T, y <, ² /k B T exp z y /y, y >, where z s, is the inverse screening length, ² is the strength of the Y ukawa interaction at hard contact and y r /s. This spherical potential model can be studied completely in the mean spherical approximation. Using the 4 Baxter± Wertheim formalism [7, 8], the nal solution consists of solving a fourth-order polynomial equation in terms of a single parameter g. Since this parameter satis es the MSA closure, it leads to the expression [9, 0]: where z 0 z and g g z g z 0 3h h u 0 z h h z z u 0 z h h w z z u 0 z exp z z z ² /k B T, 5 w z z 3h h, 6 w z, 7, 8 exp z /z 3 9 p 6 q s 3, 0 where q is the number density. Baxter s factorization of the Ornstein± Zernike relation produces an expression for the direct correlation function in the Fourier space, based on the Weiner± Hopf technique [0]: p ~ c k ~ Q k ~ Q k, where k is the wave number given in AÊ. Baxter showed that the ~ Q k function satis es the expression: ~ Q k p 0 dy Q y exp ik y. For a Yukawa uid the factor correlation function was obtained by Blum and Hù ye [6] and rewritten more recently by Ginoza [0]. In terms of Ginoza s notation, the factor correlation function Q y, is given by: Q p s y where A y y B y C exp z y exp z D exp z y, y <, D exp z y, y >. 3

3 h q q Static structure factor for sim ple liquid m etals 837 A B C D with and h h g 3h h h g g 4g 4z g g, 4 h h z, 5 exp z z a z g exp z z 0 a 0, 6 g exp z 3h a 0 z g z 0, 7 z h h z h, 8 a z z. 9 In this work we evaluate the static structure factor for a Y ukawa uid, as de ned by equation ( 4). The isothermal compressibility can be written in terms of the structure factor in the long-wavelength limit as: c T /c 0 T S k 0, 0 0 where c T /k B T q, is the ideal gas compressibility. The isothermal compressibility, in terms of the correlation function, is given by: S k 0 p ~ c k 0 ~ Q k 0, Substituting equation ( 3) into equation ( ), we obtain: ~ Q k 0 h a 6b 6c 6d, with a A /, b B /, and c C exp z d z z 3 C D. 4 Thus, the structure factor in the long-wavelength limit becomes (equation ( )): p ~ c k 0 h a a b c d h h b 3h b c d h c 3h c d d. Note that this expression is exact in MSA. McLaughlin obtained the exact structure factor in the long-wavelength limit using the mean density approximation []. In the limit ² /k B 0, the static structure factor for a hard sphere system is recovered, i.e. 5 S HS k 0 p ~ c k 0 h h 4. 6 The general expression for the structure factor S k is: S k q ~ ~ ~ h k Q k Q k ~ Q k and its explicit expression for our case is given in the appendix. Experimentally, it is possible to measure the density variation of the static structure factor [9]. Cummings and co-workers [0, ] studied the theoretical form of this quantity whose principal result can be stated as follows. If we assume that the density dependence of the Fermi wave number behaves as F /3, then the radial distribution function g r can be taken as a universal function ^g x with x r q /3, so that S k becomes a universal function ^S y with ^y k q /3. Therefore, S k q 3 k S k k 7. 8 This model is referred to as the uniform uid model (UFM) and was rst introduced by Egelsta and coworkers [9]. This last expression was also studied by Hayter et al. [5], and for more details we recommend reading the original references. Note that equation ( 8) is the rst-order approximation of the exact expression given by Hayter et al., i.e. S q k S z k z k q where the z k are physical variables [5]. 0, 9 3. Static structure factor for simple liquid metals Given the set of HSY parameters (h, ² / k B T and z s ), g is calculated by solving equation (5). For an attractive tail (² < 0) there are always two real solutions, g < g < 0, g being the physical solution in the sticky limit (² /k B T, z, and ² / z k B T c o n s t a n t), but there is no real solution [] in a certain density range [3]. For ² > 0, two positive solutions exist for a su ciently strong screened potential, and again the lower root is the physically meaningful one. In gure, we show a typical curve obtained from equation (5) in the case of attractive (AHSY) and repulsive (RHSY) tails. The numerical values for some simple liquid metals were collected by MacLaughlin, Hayter et al. and Hausleitner and Hafner [5,, 4]. In table we display the values of the parameters used to calculate the structure factor S k 0 for some simple liquid metals. The static structure factors for simple liquid metals were calculated in the long-wavelength limit (equation (5)) for an AHSY uid, and for a HS uid ( equation ( 6)). The values obtained using a

4 838 J. N. Herrera et al. C h Figure. Parameter zg for a rubidium liquid metal which is plotted as a function of reduced density, taking an A HSY and an R HSY respectively. Table. Yukawa uid parameters. Metal T /K h ² / k B T z s s /au Na K R b A l Mg Pb Table. Static structure factor S k 0 S 0 for an AHSY. Metal S 0 (HS) S 0 (MDA) S 0 (MSA) S 0 (EX P) Na a K a R b a A l Mg a,b 0.07 a,b Pb b a b Chaturvedi et al. ( 98) [5]. McA lister et al. (974) [6]. mean density approximation (MDA) were obtained by MacLaughlin []Ð the parameter s ( ², z and h ) could be adjusted to t the experimental values [, 4]. Numerical results of S k 0 are contained in table. These results depend on the parameters. Note that the A HSY model predicts accurately the static structure factor, in the long-wavelength limit, for ve liquid metals, except for liquid Pb. However, this metal is a special case because the other liquid metals have S 0 E X P > S 0 H S, and consequenty Pb should be better described by a RHSY as we will see later. The static structure factors of various liquid metals are plotted in gures ± 7 using the A HSY interaction potential. Note that in the long-wavelength limit the prediction of this model is very accurate in that it agrees well with the experimental results, even though for all liquid metals studied here the agreement diminishes for k > 0. The solution of the MSA for the repulsive Y ukawa uid was studied by Hayter et al. and by Hausleitner and Hafner [5, 4]. Their results were obtained by tting the three parameters ( ², and s ) to the experimental structure factor. However the values used in both references are di erent for the same liquid metal. In table 3 we show the MSA prediction for S k 0 for simple liquid metals using a repulsive potential. Note that in this case the theoretical prediction is accurate only for liquid Pb. The static structure factor for Na, K, Al, Mg, Pb, and Rb, described as RHSY uids, are shown in gures 8±

5 Static structure factor for sim ple liquid m etals 839 Figure. The static structure factor of Na liquid metal, which was calculated using an A HSY in MSA with z 33.6, ² / k B T.08, h and T 378 K (full line) [], the dots represent the experimental results collected by Waseda [7]. Figure 3. S k for K liquid metal using an A HSY, with z 3.54, ² / k B T 0.980, h and T 343 K (full line) [], the dots are the experimental results from [7].

6 840 J. N. Herrera et al. Figure 4. S k for A l liquid metal using an AHSY, with z 35.79, ² / k B T.87, h and T 943 K (full line) [], the dots are the experimental results from [7]. Figure 5. S k for Mg liquid metal using an A HSY, with z 3.4, ² / k B T 0.96, h and T 953 K (full line) [], the dots are the experimental results from [7].

7 Static structure factor for sim ple liquid m etals 84 Figure 6. S k for Pb liquid metal using an A HSY, with z 7.0, ² / k B T , h 0.40 and T 63 K (full line) [], the dots are the experimental results from [7]. Figure 7. Structure factor for rubidium continuum line is from an A HSY- uid (full line) [], with z 33.80, ² / k B T.057, h 0.468, and T 33 K, and the dots are the experimental results from [7].

8 84 J. N. Herrera et al. Figure 8. S k for Na liquid metal using a R HSY, with z 7.0, ² / k B T 9.5, h and T 378 K (full line) [5], the dots are the experimental results from [7]. Figure 9. S k for K liquid metal using an R HSY, with z 6.74, ² / k B T 0.098, h and T 343 K (full line) [5], the dots are the experimental results from [7].

9 Static structure factor for sim ple liquid m etals 843 Figure 0. S k for A l liquid metal using a RHSY, with z 5.08, ² / k B T.336, h and T 943 K (full line) [4], the dots are the experimental results from [7]. Figure. S k for Mg liquid metal using a R HSY, with z 8.65, ² / k B T.630, h and T 953 K (full line) [4], the dots are the experimental results from [7].

10 844 J. N. Herrera et al. Figure. S k for Pb liquid metal using a R HSY, with z 7.803, ² / k B T 8.784, h and T 63 K (full line) [8], the dots are the experimental results from [7]. Figure 3. The structure factor for a rubidium liquid metal obtained with an R HSY - uid (full line) [5], with z 8.9, ² / k B T 7.3, h 0.380, and T 33 K, the dots are the experimental results from [7].

11 Static structure factor for sim ple liquid m etals 845 Figure 4. The wavenumber derivative of the structure factor for an Rb liquid metal using A HSY ( full line), using the parameters z 33.0, ² / k B T.008, h 0.460, and T 38 K and the circles are the experimental obtained by Egeslsta et al. [9]. Figure 5. The wavenumber derivative of the structure factor for an Rb liquid metal using R HSY ( full line), using the parameters z 8.9, ² / k B T 7.3, h 0.370, and T 38 K and the circles are the experimental results from [9].

12 846 J. N. Herrera et al. Table 3. Parameters and S k 0 for the R HSY uid. Metal T /K h ² / k B T z s s /au S 0 (HS) S 0 (MSA) Na a Na b Na a K a K b K a Rb a Rb b Al b Mg b Pb c a b c Hayter et al. [5]. Hausleiter and Hafner [4]. Li et al. [8].. Note that in all cases the MSA for RHSY appears to be an accurate approximation for the prediction of the static structure factor of these liquid metals. Liquid rubidium has been studied extensively [5, 0, 4]; in this work we evaluated the static structure factor and its wavenumber derivative in the MSA. The static structure factor obtained for rubidium using both the A HSY and the R HSY models are compared with the experimental results in gures 7 and 3 respectively. The RHSY model predicts a principal peak for higher k and its plot is notably di erent from AHSY uid for all k, i.e. there is a phase shift between the RHSY and AHSY uids. These plots were obtained from S k given in the appendix for which the parameters are given in tables and 3, respectively. The k -derivative of the static structure factor for liquid Rb is shown in gures 4 and 5. Again we nd that R HSY is a reasonable representation for a liquid metal in MSA. 4. Conclusions Our results indicate that the new version of the MSA for a reference pair potential u r u HS r u Y r, can be considered as a good reference system in the theoretical study of liquid metals. It is important to note that when we use an A HSY, its application is limited in accuracy to the long-wavelength limit, while we nd the opposite situation for an R HSY potential. Based on these results we believe that the hard-sphere Y ukawa potential can be a good reference system for liquid metals. The natural extension to this model consists of improving this reference by introducing a linear combination of Y ukawa tails, and is the subject of future work. J. N. Herrera and H. Ruõ Â z-estrada were supported by CONA CY T of MeÂ xico (Grants 806P-E and 3648-E). This work was supported by the division of Chemical Sciences, O ce of Basic Energy Sciences, US Department of Energy. The authors are grateful to A. A. Chialvo and A ra. Mendieta for very helpful observations. Appendix The structure factor S k is given by: where ~ Q k S k ~ Q k, k 4 z k k k z z cos k k sin k a z a / k z sin k k cos k b z / a /k c exp z k c d b a / b z c exp z z z b / k z cos k k sin k a / a z /k z c exp z z sin k k cos k a z / a b z k k z k 4 c d b c exp z k a b z z z exp z z a / b.

13 Static structure factor for sim ple liquid m etals 847 The derivative of S k with respect to k is straightforward but it requires a lengthy mathematical manipulation. References [] Young, W. H., 99, Phys., 55, 769. [] Hafner, J., 987, From Ham iltonians to Phase D iagram s ( Berlin: Springer). [3] Waisman, E., 973, Mol. Phys., 5, 45. [4] Hayter, J. B., and Penfold, J., 98, Mol. Phys., 4, 09. [5] Hayter, J. B., Pynn, R., and Suck, J.-B., 983, J. Phys. F, 3, L. [6] Blum, L., and Høye, J. S., 978, J. Stat. Phys., 9, 37. [7] Cummings, P. T., and Smith, E. R., 979, Mol. Phys., 38, 997. [8] Cummings, P. T., and Smith, E. R., 979, Chem. Phys., 4, 4. [9] Ginoz a, M., 986, J. Phys. Soc. Japan, 55, 95. [0] Ginoz a,m., 990, Mol. Phys., 7, 45. [] Herrera, J. N., Blum, L., and Garciía-Llanos, E., 996, J. chem. Phys., 05, 988. [] McLaughlin, I. L., 997, Mol. Phys., 9, 377. [3] McLaughlin, I. L., and Young, W. H., 98, J. Phys. F,, 45. [4] Hausleitner, C., and Hafner, J., 988, J. Phys. F, 8, 03. [5] Ornstein, L. S., and Zernike, F., 94, Proc. Akad. Sci., 7, 637. [6] Høye, J. S., and Stell, G., 977, J. chem. Phys., 67, 439. [7] Baxter, R. J., 968, Aust. J. Phys.,, 563. [8] Wertheim, M. S., 963, Phys. Rev. L ett., 0, 450. [9] Egelstaff, P. A., Page, D. I., and Heard, C. R. T., 97, J. Phys. C, 4, 453; Egelstaff, P. A., Suck, J.-B., Glaser, W., McPeherson, R., and Tesistma, A., 980, J. Phys. Coll., 4, C8,. [0] Cummings, P. T., 979, J. Phys. F, 9, 477; Cummings, P. T., and Stell, G., 98, Mol. Phys., 43, 67. [] Arlinghaus, R. T., and Cummings, P. T., 987, J. Phys. F, 7, 797. [] Mier y Teraín, L, Corvera, E., and Gonz alez, A. E., 989, Phys. Rev. A, 39, 37. [3] Garciía-Llanos, E., 996, MSc thesis, BenemeÂ rita Universidad A utoâ noma de Puebla ( unpublished). [4] McLaughlin, I. L., and Kanna, K. N., 994, Phys. Chem. L iq., 7, 99. [5] Charturvedi, D. K., Rovere, M., Senatore, G., and Tosi, M. P., 98, Physica B,,. [6] McAlister, S. P., Croz ier, E. D., and Cochran, J. F., 974, Can. J. Phys., 5, 847. [7] Waseda, Y., 980, T he Structure of Non-Crystalline Materials (New York: McGraw-Hill). [8] Li, D. H., Moore, R. A., and Wang, S., 986, Phys. L ett., 8A, 405.

Influence of the Friedel oscillations on the structure of liquid rubidium

J. Phys. F: Met. Phys.. 12(1982)233-44 Printed in Great Britain Influence of the Friedel oscillations on the structure of liquid rubidium P T Cummingst and P A Egelstaff Department of Physics, University