Vegard’s law has been used to estimate the lattice parameters, from the XRD data, of alloyed samples [28-29]. The straightforward mathematical expression of Vegard’s law, for calculating the lattice parameter, of a binary compound (A1-xBx) can be given as; a(sub.)al= a(sub.)A (1-x)+a(sub)B (x) . Here, a(sub)al, a(sub.)A, a(sub.)B, and x are the lattice parameters of alloyed sample, lattice parameters of element A, lattice parameters of element B and fractional concentration of B element in the host A, respectively. However, there are further amendments in the Vegard's low for the precise understanding of the alloy level in a certain compound [30]. By applying the Vegard’s law and calculated lattice parameters of Pt/PtNi NPs, the degree of alloying (Da) was calculated as described by Salgado et al [31]; D(sub.)a = (a -a(sub)0 / (a(sub)c - a (sub)0), where a is the experimental lattice parameter of a particular sample, a(sub)c is the lattice parameter assuming that all the nickel is alloyed (i.e., theoretically calculated from the Vegard’s law) and a(sub)0 is the lattice parameter of platinum NPs. The Da is found to 65.23 %and 67.21 % for the Pt3Ni1 and Pt1Ni1 NPs, respectively. {32}

28. L. Vegard: Z. Phys. 5 (1921) 17.

29. L. Vegard: Z. Cryst. 67 (1928) 239.

30. A. R. Denton and N. W. Ashcroft, Phys. Rev. B, 1991, 43, 3161-3164.

31. J. R. C. Salgado, E. Antolini, and E. R. Gonzalez, J. Phys. Chem. B, 2004, 108, 17767-17774.

32. Journal of Physics and Chemistry of Solids 110 (2017) 187–194

Calibration curve of known alloy compositions. BTW, it’s Vegard’s rule (not law) as Aditya correctly spelled above.

Vegard's law or rule is obbey for matterials having single phase and the same crystal structure. In this case the lattice dependence is linear so that it's value could directly represent the alloyed composition. For materials having different crystal structure the dependence of lattice parameter not linear in the two ends but still having a sysrematec form and then can be used for determining the alloyed composition as seen in the attached picture

Aditya Sharma: thanks for the explanation, the system I'm using is tungsten oxide and alloying Ni into it. If I put the parameters into that equation I'm getting x values more than one. Considering the lattice parameter (a ) of alloyed sample 7.3409 and initial tungsten oxide 7.3289 A. The nickel oxide lattice parameter cubic is 4.1705 A.

Also, where do I get theoretical values of complete alloyed lattice parameters of nickel , to calculate degree of alloying.

you can do a simulation