Bain path

Figure 2. Total energy for FeaNi along the Bain path with V = const (a). Binding energy versus volume of the unit cell for the ferromagnetic (FM) bet (circle) and fee (square) states. Diamonds results for the nonmagnetic (NM) fee phase.

Figure 11. The difference between the free-energy densities of fee and bcc phases of particles interacting through a Yukawa potential, as a function of temperature, determined through the FG methods discussed in Section V.C. The error bars reflect the difference between the upper and lower bounds provided by FG switches between the phases (along the Bain path [85]) in the two directions.

For a volume-conserving Bain path, 5F= 0, the elastic constant C is obtained from the curvature of i/coh at the bcc and fee structures, respectively. 

Figure 2. The energy along a Bain path connecting the bcc and fee structures of tungsten. 

Similarly, other deformations of a lattice can give other elastic constants, or combinations of them. The deformation in Fig. 1, leading to Eq. (3), is the most common version of a Bain path. In the volume-conserving approximation, the volume of the system is kept constant, while da is varied. A more accurate calculation allows also the volume to vary, so that the energy is minimised for each da. 

Potassium, rubidium, cesium. All three elements have the bcc stmcture. Sliwko et al. (1996) obtained C for K and Rb from a calculation of the tetragonal Bain path. Jianjun Xie et al. (2000) calculated C and C44 for Cs. Grimvall and Ebbsjd (1975) found fee K to be dynamically stable. 

Nickel, palladium, platinum. All three metals have the fee structure. Craievich et al. (1994) calculated C for bcc Ni and obtained a negative value. The Bain path calculation by Wills et al. (1992) for bcc Pt shows that C = 0. 

Alippi, P., Marcus, P. M., and Scheffler, M., 1997, Strained tetragonal states and Bain paths in metals, Phys. Rev. Lett. 78, 3892-3895. 

The strict solution for the problem of the resistance to the motion of a small sphere moving through gas has been obtained by Baines et al. (1965). They considered both specular and diffuse reflection of the molecules at the surface of the sphere mass of which is large in comparison with the mean mass of gas molecules and the radius to be small compared with the mean free path of gas molecules. All these assumptions are applicable for circumstellar outflows. Fadeyev and Henning (1987) used these solutions for calculation of momentum transfer from silicate dust grains to gas molecules in cool 0-rich red giants... 

We consider two simple transformation paths connecting cubic structures. They are the bcc-fcc transformation path via tetragonal deformation corresponding to extension along the [001] axis (the usual Bain s path) and the trigonal deformation path that corresponds to uniaxial deformation along the [111] axis (Figs. 1 and 2). 

Craievich et ah have shown that some energy extrema on constant-volume transformation paths are dictated by the symmetry. Namely, most of the structures encountered along the transformation paths between some higher-symmetry structures, say between bcc and fee at the Bain s path, have a symmetry that is lower than cubic. At those points of the transformation path where the symmetry of the structure is higher, the... 

---