China
Africa
Explore chapters and articles related to this topic
Solid State Background
Published in P.J. Gellings, H.J.M. Bouwmeester, Electrochemistry, 2019
Isaac Abrahams, Peter G. Bruce
In an ideal solid solution, AxByX, if ion A is replaced by a larger ion B, then a linear expansion in the unit cell dimensions is predicted, and, conversely, if ion B is smaller than ion A, a linear contraction is predicted. This is known as Vegard’s law (Figure 3.46). Vegard’s law states that in a solid solution unit cell parameters should change linearly with composition. For example, the cation interstitial system Li1+xTi2−xGa(PO4)3 shows a linear expansion in the rhombohedral cell dimensions a and c (hexagonal axes) up to x = 0.3 (Figure 3.47).52 Above x = 0.3 a phase transition to a triple cell superstructure occurs, and is indicated by a discontinuity in the cell expansion.
Systems Based on InAs
Published in Vasyl Tomashyk, Quaternary Alloys Based on III-V Semiconductors, 2018
InAs–ZnGeAs2. The solid solutions based on InAs contained up to 80 mol% ZnGeAs2 and crystallize in the cubic structure (Giesecke and Pfister 1961; Voitsehovskiy and Goryunova 1962). The solid solutions with higher concentration of ZnGeAs2 exist in the chalcopyrite-type structure. The lattice parameters for both solid solutions obey Vegard’s law.
First-principles calculations to investigate thermodynamic and mechanical behaviors of molybdenum-lanthanum alloy
Published in Journal of Nuclear Science and Technology, 2023
Lu Wang, Kun Jie Yang, Chenguang Liu, Yue-Lin Liu
As shown in Figure 8, the lattice constant alteration of Mo1-xLax (0 < x < 0.1) binary alloy is significantly dependent on the La concentration. There is a substantial linear relationship between La concentration and lattice constant. In metallurgy, Vegard’s law is an approximate empirical rule which holds that a linear relation exists between the crystal lattice parameter of an alloy and the concentrations of the constituent elements. It can be seen that the present Mo1-xLax alloy system indeed follows Vegard’s law. At the same time, the computed lattice constant of Mo1-xLax binary alloy increases with the increasing La content, suggesting that La can enlarge the lattice structure of Mo. Such a changing trend can be well understood from that the atomic radius of La atom is 1.88 Å, obviously larger than that (1.40 Å) of host Mo atom.
Element migration during stress rafting of γ′-Co3(Al, W) precipitates
Published in Philosophical Magazine Letters, 2020
Dong Wang, Yongsheng Li, Shujing Shi, Xinwen Tong
The stress-free strain is given bywhere is the Kronecker-Delta function. The lattice parameter and the average lattice parameter can be approximated by Vegard’s law.
Experimental determination of effective atomic radii of constituent elements in CrMnFeCoNi high-entropy alloy
Published in Philosophical Magazine Letters, 2022
Takeshi Teramoto, Momoko Narasaki, Katsushi Tanaka
Figure 2 presents the XRD patterns of all specimens listed in Table 1. The lattice parameters are determined from the five peaks hatched in gray. Peaks associated with the standard sample (LaB6) and the specimens are shown with indices. Figure 3(a) presents the variations in the lattice parameters for each diffraction peak, while Figure 3(b) presents the determined lattice parameters for all the constituents. The columns headed by + and – in Figure 3(b) present the lattice constants of the constituents from Cr+ to Ni+ and Cr- to Ni-, respectively, with the corresponding standard deviations. The maximum standard deviation of the lattice parameter is approximately 0.1 pm. The composition dependence of the lattice parameter is assumed to have a linear relationship based on the Vegard’s law. Though the lattice parameter of a multi-element alloy shows a non-linear change with composition, the approximation to Vegard’s law is reasonable as long as the applied composition range is narrow and no strong non-linearity exists in the composition dependence of the lattice parameter. On the assumption of a linear composition dependence, the determined lattice parameters have been subjected to a multiple regression analysis, and the composition dependence of the effective lattice parameters thereby estimated. The effective lattice parameter (aeff) is expressed using Eq. 1 based on the multiple regression analysis as Here, CCr, CMn, CFe, CCo, and CNi denote the molar concentrations of Cr, Mn, Fe, Co, and Ni, respectively. The value of the coefficient of determination R2 is 0.999. As this is close to unity, the predicted lattice parameters show reasonable agreement with the experimental ones. The relationship between the experimentally determined lattice parameters and the lattice parameters predicted from the regression equation is presented in Figure 4, where perfect correlation is denoted as a line. The effective lattice parameters of a virtual FCC structure consisting of pure elements are derived from Eq 1. For example, the effective lattice parameter of the FCC lattice consisting of pure Cr is 357.82 pm, which is obtained by substituting the composition of Cr to be 100% in Eq. 1. The effective atomic radius (reff) is estimated from the effective lattice parameter of the constituent elements by applying a rigid-sphere model to the FCC structure. Then the relationship between the effective atomic radius and effective lattice parameter is expressed using the following equation [31]: The effective lattice parameters and effective atomic radii of the constituent elements in CrMnFeCoNi high-entropy alloy at 22.50 °C are summarized in Table 2. The predicted data agree with the experimental ones with the standard error of the effective atomic radii being approximately 0.4 pm. These results indicated that the effective atomic radii are estimated with optimal accuracy.