Crystal Field Phonon Interaction

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

**Figure 18:** Illustration of the transversal and longitudinal BvK springs
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The index

now denotes all atomic positions in a crystal, the momentum vector is $\mathbf p_i=a_0 \mathbf w_i$ and the displacement vector is $\hat \mathbf U_i$ . Fig. 18 shows the Born-van-Karman model definition of longitudinal springs $c_L$

and transversal springs $c_T$

used in the following derivation.

	$\displaystyle \hat H_{\rm phon}$	$\textstyle =$	$\displaystyle \sum_{i} \frac{a_0^2\hat \mathbf w_i^2}{2 m_i} + \frac{1}{2}\sum_... ...vert^2} (\hat \mathbf U_j . \mathbf R_{ij}-\hat \mathbf U_i . \mathbf R_{ij})^2$
			$\displaystyle + \frac{c_T(ij)}{2} (\hat \mathbf U_j - \hat \mathbf U_i )^2$	(76)

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 $\mathbf R_{ij}=\mathbf R_{j}-\mathbf R_{i}$ . The displacement vector $\hat \mathbf U_i$ is split into a strain component $\bar \epsilon \mathbf R_i$ a rotational component $\bar \omega \mathbf R_i$ and a dynamic component $\hat \mathbf P_i$ (displacement operator) obeying periodic boundary conditions $\hat \mathbf U_i = \bar \epsilon \mathbf R_i + \bar \omega \mathbf R_i+\hat \mathbf P_i= \bar a \mathbf R_i+\hat \mathbf P_i$ .

$\displaystyle (\hat \mathbf U_j . \mathbf R_{ij}-\hat \mathbf U_i . \mathbf R_{ij})^2$	$\textstyle =$	$\displaystyle (\mathbf R_{ij}^T\bar a \mathbf R_{ij}+ \mathbf R_{ij}^T \mathbf P_{ij})^2$
	$\textstyle =$	$\displaystyle (\mathbf R_{ij}^T\bar a \mathbf R_{ij})^2 + 2 \mathbf R_{ij}^T\bar a \mathbf R_{ij}\mathbf R_{ij}^T \hat \mathbf P_{ij}$
		$\displaystyle + (\mathbf R_{ij}^T \hat \mathbf P_{ij})^2$	(77)

	$\displaystyle (\hat \mathbf U_j - \hat \mathbf U_i )^2$	$\textstyle =$	$\displaystyle \mathbf R_{ij}^T\bar a^T\bar a \mathbf R_{ij}$
			$\displaystyle 2 \mathbf R_{ij}^T\bar a^T\hat \mathbf P_{ij} +\hat \mathbf P_{ij}^T \hat \mathbf P_{ij}$	(78)

Equations (81) and (82) contain terms quadratic in $\bar a$ , which will be considered as elastic energy contributions. The terms quadratic in $\hat \mathbf P$ will contribute to the lattice dynamics. In addition there are also terms linear in $\bar a$ and $\hat \mathbf P$ . Thus the phonon Hamiltonian (79) 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 $H_{mix}$

$\displaystyle \hat H_{\rm phon}$

$\textstyle =$

$\displaystyle E_{el} +\hat H_{mix}+\hat H_{\rm E}+\hat H_{\rm int}$

(79)