1. Density of cubic substitutional solid solution alloys Y.A. Chen aT*, C.R.M. da Silva a3’,D.M. Rowe b a Institute de Pesquisas Espaciais-INPE, 12227-900, SZo Josk dos Campos, SP, Brazil b School of Engineering, University of Wales CF2 IXH, Cardi& Wales, UK 指導老師 : 戴子堯 報告者 : 王日隆 Institute of Mechanical Engineering Date ﹕ 2011/12/26

2. Introduction A general expression for the density of substitutional solid solution alloys as a function of composition has been used to calculate alloy composition profiles of several metallic and semiconductor compounds. Good agreement is obtained with experimental data available from lattice parameter measurements.

3. The density of a metallic alloy or a semiconductor compound is an important physical constant which is frequently required in a programme of material development. One of its useful applications is in the determination of the longitudinal composition profile of an alloy or single crystal, as obtained from density measurements on slices transverse to the growth axis. A general expression for the density of substitutional solid solution alloys,as a function of composition has been derived using the mixture rule and considering the mass and volume “additivity” of the two end-members of the alloy [ 1 ]:

4. where and are respectively, the density and atomic weight of the element (or compound) M or N. In applying the additive rule it is assumed that the solution volume does not change upon mixing. Consequently, the requirement is that both M and N have similar properties, viz the same crystal structure,similar atomic sizes, electronegativities and valences. In this work, the density values calculated using Eq. (1) were compared to the experimental data of several cubic solid solution alloys. In the cubic-system alloy, the lattice constant,is related to the density by the expression :

5. where n is the number of atoms or molecules in the unit cell, and is Avogadro’s number. By combining Eqs. (1) and (2), the lattice parameter can be expressed as: Eq. (3) is equivalent to the one derived by Zen [2,3]to explain the deviation from the linear dependence between the unit cell parameter and composition in acubic solid solution alloy:

6. Eq. (4) converges to Vegard’s rule only if the endmember molar volumes and are approximately equal, since it is the volumes and not the lattice parameters which are additive. Similarly, Vegard’s rule can also be applied to the density profiles, as given by Eq. (1). In general, the density exhibits a slightly parabolic dependence on composition. However, when, the denominator in Eq. (1) approaches unity and the relationship becomeslinear.

7. In Fig. 1, Figs. 2 and 3 the density profiles calculated using Eq. (1) (solid line) are compared with the data obtained from X-ray diffraction patterns(symbols), for different metallic cubic solidsolution alloys [4], [5] [6] [7] semiconductor compounds.

8. Values of the lattice parameters and the molar volume deviation (ΔV%) between the endmembers of alloys are listed in Table 1. The last column shows the density deviation (Δp%) between the calculated and the experimental values for x =0.5, when the departure from linear behaviour, and the deviation Δ p% reach a broad maximum. Within the error limits, the values of the calculated and measured densities are in agreement, with the maximum deviation varying from 0.01% to 1.76% for the Fe-Cr and Cr-Mo systems respectively. The variation in density of a substitutional solid solution alloy with composition is usually greater than that of the lattice parameter. Consequently, the general expression for the density of an alloy as a function of its composition reported in this paper facilitates the determination of accurate alloy composition profiles from density measurements.

9. References [l] Y.A. Chen, IN. Bandeira, D.M. Rowe and G. Min. J. Mater. Sci. Lett. 13 (19941 1051. [2] E.A. Zen, Am. Mineralogist 41 (1956) 523. [3] C.P. Kempter, Phys. Stat. Solid 18 (1966) K117. [4] W.B. Pearson, in: A Handbook of Lattice Spacings and Structures of Metals and Alloys, ed. G.V. Raynor, International Series of Monographs on Metal, Physics and Physical Metallurgy, Vol. 4 (Pergamon Press, London, 19581. [5] J.P. Dismukes, L. Ekstrom and J. Paff, J. Phys. Chem. 68 (1964) 3021. 161 N.R. Short, Brit. J. Appl. Phys. (J. Phys. D) 1 (19681 129. [7] J. Blair and R. Newnham, in: Metallurgy of Elemental and Compound Semiconductors, Vol

10. Thank you for your attention!