Crystal Field Phonon Interaction

Here some formulas are listed. Starting point is the phonon part, written in more general terms.

The index $i$ now denotes all atomic positions in a crystal, the momentum vector is $\vec p_i$ and the displacement vector is $\vec U_i$. Fig. 24 shows the Born-van-Karman model definition of longitudinal springs $c_L$ and transversal springs $c_T$ used in the following derivation.

$$H_{\rm ph} = \sum_{i} \frac{a_0^2 \vec p_i^2}{2 m_i} + \frac{1}{2}\sum_{ij} \f... ...)}{2\vert\vec R_{ij}\vert^2} (\vec U_j . \vec R_{ij}- \vec U_i . \vec R_{ij})^2$$

$$+ \frac{c_T(ij)}{2} (\vec U_j - \vec U_i )^2$$ (75)

Note that the sum over indices $i,j$ counts each spring twice, thus a factor of $1/2$ has been added to the Hamiltonian before the sum.

Note that the difference in undisplaced lattice positions is denoted by $\vec R_{ij}=\vec R_{j}-\vec R_{i}$. The displacement vector $\vec U_i$ is split into a strain component $\bar \epsilon \vec R_i$ a rotational component $\bar \omega \vec R_i$ and a dynamic component $\vec P_i$ obeying periodic boundary conditions $\vec U_i = \bar \epsilon \vec R_i + \bar \omega \vec R_i+ \vec P_i= \bar a \vec R_i+ \vec P_i$.

$$(\vec U_j . \vec R_{ij}- \vec U_i . \vec R_{ij})^2 = (\vec R_{ij}^T\bar a \vec R_{ij}+ \vec R_{ij}^T \vec P_{ij})^2$$

$$= (\vec R_{ij}^T\bar a \vec R_{ij})^2 + 2 \vec R_{ij}^T\bar a \vec R_{ij}\vec R_{ij}^T \vec P_{ij}$$

$$+ (\vec R_{ij}^T \vec P_{ij})^2$$ (76)

$$(\vec U_j - \vec U_i )^2 = \vec R_{ij}^T\bar a^T\bar a \vec R_{ij}$$

$$2 \vec R_{ij}^T\bar a^T\vec P_{ij} + \vec P_{ij}^T \vec P_{ij}$$ (77)

Equations (80) and (81) contain terms quadratic in $\bar a$, which will be considered as elastic energy contributions. The terms quadratic in $\vec P$ will contribute to the lattice dynamics. In addition there are also terms linear in $\bar a$ and $\vec P$. Thus the phonon Hamiltonian (78) can be separated into the elastic Energy $E_{el}$, the Einstein single ion oscillation term $H_{\rm E}$, the bilinear interaction term $H_{\rm int}$ and the mixing term $H_{mix}$

$$H_{\rm ph} = E_{el} + H_{mix}+ H_{\rm E}+H_{\rm int}$$ (78)