Scattering rate

A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material.

The interaction picture
Define the unperturbed Hamiltonian by $$H_0$$, the time dependent perturbing Hamiltonian by $$H_1$$ and total Hamiltonian by $$H$$.

The eigenstates of the unperturbed Hamiltonian are assumed to be
 * $$ H=H_0+H_1\ $$
 * $$ H_0 |k\rang = E(k)|k\rang $$

In the interaction picture, the state ket is defined by
 * $$ |k(t)\rang _I= e^{iH_0 t /\hbar} |k(t)\rang_S= \sum_{k'} c_{k'}(t) |k'\rang $$

By a Schrödinger equation, we see
 * $$ i\hbar \frac{\partial}{\partial t} |k(t)\rang_I=H_{1I}|k(t)\rang_I $$

which is a Schrödinger-like equation with the total $$H$$ replaced by $$H_{1I}$$.

Solving the differential equation, we can find the coefficient of n-state.
 * $$ c_{k'}(t) =\delta_{k,k'} - \frac{i}{\hbar} \int_0^t dt' \;\lang k'|H_1(t')|k\rang \, e^{-i(E_k - E_{k'})t'/\hbar} $$

where, the zeroth-order term and first-order term are
 * $$c_{k'}^{(0)}=\delta_{k,k'}$$
 * $$c_{k'}^{(1)}=- \frac{i}{\hbar} \int_0^t dt' \;\lang k'|H_1(t')|k\rang \, e^{-i(E_k - E_{k'})t'/\hbar} $$

The transition rate
The probability of finding $$|k'\rang$$ is found by evaluating $$|c_{k'}(t)|^2$$.

In case of constant perturbation,$$c_{k'}^{(1)}$$ is calculated by
 * $$c_{k'}^{(1)}=\frac{\lang\ k'|H_1|k\rang }{E_{k'}-E_k}(1-e^{i(E_{k'} - E_k)t/\hbar})$$


 * $$|c_{k'}(t)|^2= |\lang\ k'|H_1|k\rang |^2\frac {sin ^2(\frac {E_{k'}-E_k} {2 \hbar}t)} { ( \frac {E_{k'}

-E_k} {2 \hbar} ) ^2 }\frac {1}{\hbar^2} $$

Using the equation which is
 * $$\lim_{\alpha \rightarrow \infty} \frac{1}{\pi} \frac{sin^2(\alpha x)}{\alpha x^2}= \delta(x)$$

The transition rate of an electron from the initial state $$k$$ to final state $$k'$$ is given by


 * $$P(k,k')=\frac {2 \pi} {\hbar} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) $$

where $$E_k$$ and $$E_{k'}$$ are the energies of the initial and final states including the perturbation state and ensures the $$\delta$$-function indicate energy conservation.

The scattering rate
The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by


 * $$w(k)=\sum_{k'}P(k,k')=\frac {2 \pi} {\hbar} \sum_{k'} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k)$$

The integral form is
 * $$w(k)=\frac {2 \pi} {\hbar} \frac {L^3} {(2 \pi)^3} \int d^3k' |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k)$$