Static structure factor of molten titanium

Volume 59A, number 4

PHYSICS LETTERS

13 December 1976

STATIC STRUCTURE FACTOR OF MOLTEN TITANIUM J.R. TODD

*

**

Hughes Aircraft Company, Fullerton, California, USA

and J.S. BROWN Department of Physics, University of Vermont, Burlington, Vermont, USA Received 3 May 1976 A comparison between the measured liquid titanium static structure factor of Waseda and Tarnaki (1975) at 1700°Cand the theoretical structure factor of a system of hard spheres in the Percus-Yevick approximation is presented. Both the Ashcroft-Lekner hard sphere PY model and the recently proposed Sharma-Sharma modification are used and discussed.

In connection with some calculations of the transport properties of liquid titanium, we have recently had occasion to use some model calculations of the static structure factor S(k). The simplest model which fits the experimental results of X-ray scattering analyses of liquid metals over the last few years is the Percus-Yevick hard sphere S(k) which was first presented by Ashcroft and Lekner [1], based on the Wertheim-Thiele [2] solution of the PY equations. In this model, hereafter referred to as the AL model, the structure factor S(k) is given by S(k) = lf[l —C(k)j (1) where C(k)

—

~

[ax3 (sin x

—

x cos x)

+ /3x2(2x

sin x

X —

[x2 —21 cos x

—

[x4

—

—

2) + y([4x3

1 2x~+ 24] cos x

+

24)]

—

24x} sin ~ (2)

with x = ku and the constants a, 13 and y are given by a °(l+2fl)2/(1 _fl)4, p = _6fl(l+~)2/(1 ~)4 (3) y = ~ri(1 +2~)2/(l_n)4 *

**

Research supported under Contract No. ERDA E(1 11) 3551 by the U.S. Ener~’Research and Development Administration. Based on work submitted to Univ. of Vermont in partial fulfillment of the M.S. degree.

302

in terms of the packing factor ii. This factor is related to the hard sphere diameter a by r~= irn u~/6with n the number density of atoms in the molten metal. Rcently Sharma and Sharma [311, hereafter referred to as SS, have presented a modification of the hards sphere AL results, which satisfies the Carnahan-Starling [4] equation of state. The only modification of the S(k) formula is in the constants a, 13, andy, which in the SS PY model are given by

(~ 4)}/(l 2fl)2 + n~ = [l8+20i~— ~ +7 4 7~]/(l—fl) i 2 3 / 4 y°~{(l+2i~) +~ (r~—4)},(l--r~) a= ~

+

—~

(4)

In our liquid Ti transport calculations, to be published elsewhere, we have used the AL model for S(k), and in addition have compared it to results gotten by using the experimental S(k) results for Ti at 1700 C. These results were measured by X-ray means recently by Waseda and Tamaki [5] hereafter referred to as WT. Therefore, in view of the recent appearance of the paper of Sharma-Sharma, it may be of interest to present our calculations for molten Ti. For this 3d transition metal WT quote the packing factor to be 0.44 and the hard-sphere diameter a to be 2.53 A, which they claim reproduces the peak height well for the first shell of neighbors. The density was taken as 41.5 g/cm3 at 1700 C. In fig. I we have reproduced a computer plot of S(k) containing the

Volume 59A, number 4

PHYSICS LEYFERS

13 December 1976

STRUCTURE FACTORS OF LIQUID Ti 3.0 X-ray data of WT ——

2

~,

—

X-ray data of WT

Sharma ~ Sharma Ashcroft B Lekner

2.4

1 I

k (A

Fig. 1. Static structure factors of molten Ti at 1700°Cfrom X-ray data compared to the AL and SS hard-sphere models.

experimental points from WT and the PY hard-sphere results of AL and SS for the above choice of hardsphere parameters. Two points are immediately evident from this comparison. The first is that the AL and SS models do not produce very much of a difference for all k values from 0 up to 12 A—1. Secondly it is clear that neither the AL nor the SS hard-sphere models fits the data very well in terms of their ability to locate the peaks, although the magnitude of the peaks is generally reproduced well, To rectify this deficiency, we used an alternate set of hard-sphere parameters suggested by WT. These were 77 = 0.58 and a = 2.77 A, claimed to reproduce the peak positions more accurately. It was found that they both did this, but at the expense of badly misrepresenting the magnitude of S(k) throughout the full range. Waseda and Tamaki [5] have suggested that Ti does not appear to satisfy the hard sphere model well. As suggested by Schiff [6], two criteria for the softness or hardness of the potential may be cited: (a) the damping of successive peaks in S(k), and (b) the shift in the main peak of S(k) with increasing temperature. A perusal of structure factor data (Waseda [7]) shows that both criteria suggest that Ti is in fact a soft-core metal. Without embarking on an analysis of why this might be the case, we note that if the packing fraction

I

— --

Ashcroft-Lekner Sharma - Sharma

Fig. 2. Structure factor compared to AL and SS models for = 0.44 and a = 2.76 A (i.e., not satisfying the hard-sphere relation).

~ and the hard-sphere diameter a are treated as independent parameters, and we hold 77 = 0.44 and vary a, considerable improvement between theory and experiment may be obtained for a 2.76 A. In fig. 2 we have presented a plot of S(k) for the AL and SS models with 77 0.44 and a = 2.76 A. In passing we note that these values do not satisfy the hard-sphere packing relation ~ = irna3/6, and in some sense describe the packing of deformable or “squshy” spheres, However, in a purely ad hoc way the fit of the AL model with this choice of parameters is very good, at least for low k. Note the value of 2kf in fig. 2. Calculations of electronic transport in liquid metals involve the structure factor 5(k) up to 2kf. For such applications the AL model gives an excellent fit. However, we see that the peaks fall progressively out of agreement with experiment as k increases, and in addition damp out more rapidly. This would suggest that, even if liquid Ti is a soft-sphere liquid, the modifications of the hardsphere model used here do not correctly describe its properties for large k. For this purpose it would be desirable to have a molecular dynamics calculation, which is independent of the details of the particular statistical approximation used. Such calculations have been performed by Hansen and Schiff [10] for a numher of simple metals, using core potentials of the form r~with n taking on a number of values from °° to about 2, describing varying degrees of hardness. To 303

Volume 59A, number 4

PHYSICS LETTERS

knowledge such calculations are presently unavailable for liquid Ti. The packing fraction parameter 77 can be obtained for simple metals by expressing the free energy Fof the hard-sphere system in terms of 77 and then miniour

mizingF. Calculations of this type have been per-

References 111 N.W. Ashcroft and J.

F. Thiele, J. Chem. Phys. 39(1963)474. [3] R.V. Sharina and K.C. Sharma, Phys. Lett. 56A (1976)

141

We wish to thank the University of Vermont Computer Center for providing the necessary computing time on the Xerox Sigma 6. Also we acknowledge generous support from the U.S. Energy Research and Development Administration under Contract No. ERDA E(l 1-l)-355l. Finally we acknowledge the expert help of Mr. Mark Cruise and Mr. George Adams in producing the computer plots reproduced here.

17 181 191

304

Lekner, Phys. Rev. 145 (1966) 83.

[21 MS. Wertheim, Phys. Rev. Lett. 10(1963)321;

simple metals by Stroud and Ashcroft [8] and Edwards and Jarzynski [9]. However, to our knowledge no one has yet been able to extend the approach to transition metals. formed for

13 December 1976

107.

N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 107.

15] Y. Waseda and S. Tamaki, Phil. Mag. 32 (1975) 273. [61 D. Schiff, The properties of liquid metals, Proc. Second Intern. Conf. at Tokyo, September 1972, ed. S. Takeuchi (John Wiley and Sons, New York 1973), Y. Waseda, private communication. D. Stroud and N.W. Ashcroft, Phys. Rev. B8 (1972) 371. D.J. Edwards and 1. Jarzynski, J.. Phys. C, Solid St. Phys.

5 (1972) 1745. p10] J.P. Hansen and D. Schiff, Mol. Phys. 25 (1973) 1281.