Bravais lattice - the simplest case

As mentioned in the previous section each non-equivalent lattice site results into a Lorentzian line in the energy domain or a exponential function in the time domain, respectively. The relatively complicated mathematical formalism of rate equations becomes much more concise in the case of a Bravais lattice. The definition of the Bravais lattice, i.e. a lattice with only one non-equivalent lattice site in the primitive cell, makes it possible to define only one jump frequency 1/$\tau$ and the jump matrix A from equation reduces into a scalar. The line width $\Gamma$ which is inserted into the frequency dependent refractive index (equation ) is in the case of N different jump vectors $l_{n}$

$$\Gamma (k_{R})=\frac {2\hbar}{\tau} \bigg{(} 1-\frac{1}{N} \sum _{n=1}^{N} e^{i k_{R}l _{n}} \bigg{)}$$ (73)

Especially the b.c.c. structure, which is a Bravais structure, plays an important role in this work for the section about the Fe/MgO(001). There are eight nearest neighbours (NN) in this structure. Assuming an NN jump-diffusion mechanism eight jump vectors $l_{n}$ can be defined (the origin of the used system is the atom in the center of the cube in figure ):

$$\begin{array}{lllrlrlllrlrll} l_{1}&=&a(&\frac{1}{2},&\frac{1}{2}... ...{1}{2}&)&\; & l_{8}=a(&-\frac{1}{2},&-\frac{1}{2},&-\frac{1}{2}&) & \end{array}$$ (74)

Two jump vectors are shown in figure . Inserting this set into equation the following simple formula for the line width $\Gamma$ can be calculated

$$\Gamma (k_{R}) = \frac {2\hbar}{\tau} \bigg{[} 1- \frac{1}{8}\bigg{(} e^{ik_{R3}\f... ...\frac{-a}{2}}4\cos(ik_{R1}\frac{a}{2}) \cos(ik_{R2}\frac{a}{2})\bigg{)}\bigg{]}$$

$$= \frac {2\hbar}{\tau} \bigg{(} 1- \cos (k_{R1} \frac{a}{2}) \cos (k_{R2} \frac{a}{2}) \cos (k_{R3} \frac{a}{2}) \bigg{)}$$ (75)

where $k_{Ri}$ is the $i$-th component of the reflected wave vector and $a$ is the lattice constant of iron ($a$=2.86 $\AA$) [44].

Figure: Simulated angular dependence of the line width for a nearest neighbour jump-diffusion mechanism sketched in the right part of the figure. The angle $\Phi$ is the angle between the reflected wave vector k$_{R}$ and the (010) axis of the b.c.c.structure.