Inside fropho

Calculating forces, we need to introduce finite displacements to each atom in unit cell. In principle, we need number of displacements of freedom. In the case that target crystal model has crystal symmetry operations more than two, the number of displacements are reduced. Using sufficient forces and displacements, we will create force constants. In fropho, force constants are calculated as follow,

$\displaystyle \Phi_{\alpha\beta} \begin{pmatrix}jj' \\ ll' \\ \end{pmatrix} = - \frac{F_{\alpha}(jl)}{\Delta u_{\beta}(j'l')},$ (3.1)

where $ \Phi$ is force constant, $ \alpha$ and $ \beta$ are independent directions which are arbitrarily chosen, $ F_{\alpha}$ is force of $ \alpha$ direction and $ \Delta u_{\beta}$ is displacement of $ \beta$ direction. $ j$ denotes an atom in the $ l$-th unit cell. Symmetry operations also symmetrize the force constants. Therefore a force constant matrix should be connected to another one. To do so, group theory is employed for sophisticated symmetrization. In fropho, which is not sophisticated, symmetry operations derived from crystal symmetry are used for symmetrize them 'numerically'.

Usually force constants are treated as $ 3 \times 3$ matrix. In fropho code, when generating force constants, this is treated as $ 9 \times 1$ matrix. This treatment has a good point that we can know all the force constants at the same time by solving linear equation of a matrix. In the following sections, I will describe how to construct matrices and how to solve it.

togo 2009-02-12
Get fropho at Fast, secure and Free Open Source software downloads