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 ${\displaystyle H_{0}}$, the time dependent perturbing Hamiltonian by ${\displaystyle H_{1}}$ and total Hamiltonian by ${\displaystyle H}$.

The eigenstates of the unperturbed Hamiltonian are assumed to be

${\displaystyle H=H_{0}+H_{1}\ }$

${\displaystyle H_{0}|k\rangle =E(k)|k\rangle }$

In the interaction picture, the state ket is defined by

${\displaystyle |k(t)\rangle _{I}=e^{iH_{0}t/\hbar }|k(t)\rangle _{S}=\sum _{k'}c_{k'}(t)|k'\rangle }$

By a Schrödinger equation, we see

${\displaystyle i\hbar {\frac {\partial }{\partial t}}|k(t)\rangle _{I}=H_{1I}|k(t)\rangle _{I}}$

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

Solving the differential equation, we can find the coefficient of n-state.

${\displaystyle c_{k'}(t)=\delta _{k,k'}-{\frac {i}{\hbar }}\int _{0}^{t}dt'\;\langle k'|H_{1}(t')|k\rangle \,e^{-i(E_{k}-E_{k'})t'/\hbar }}$

where, the zeroth-order term and first-order term are

${\displaystyle c_{k'}^{(0)}=\delta _{k,k'}}$

${\displaystyle c_{k'}^{(1)}=-{\frac {i}{\hbar }}\int _{0}^{t}dt'\;\langle k'|H_{1}(t')|k\rangle \,e^{-i(E_{k}-E_{k'})t'/\hbar }}$

The transition rate

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

In case of constant perturbation,${\displaystyle c_{k'}^{(1)}}$ is calculated by

${\displaystyle c_{k'}^{(1)}={\frac {\langle \ k'|H_{1}|k\rangle }{E_{k'}-E_{k}}}(1-e^{i(E_{k'}-E_{k})t/\hbar })}$

${\displaystyle |c_{k'}(t)|^{2}=|\langle \ k'|H_{1}|k\rangle |^{2}{\frac {sin^{2}({\frac {E_{k'}-E_{k}}{2\hbar }}t)}{({\frac {E_{k'}-E_{k}}{2\hbar }})^{2}}}{\frac {1}{\hbar ^{2}}}}$

Using the equation which is

${\displaystyle \lim _{\alpha \rightarrow \infty }{\frac {1}{\pi }}{\frac {sin^{2}(\alpha x)}{\alpha x^{2}}}=\delta (x)}$

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

${\displaystyle P(k,k')={\frac {2\pi }{\hbar }}|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})}$

where ${\displaystyle E_{k}}$ and ${\displaystyle E_{k'}}$ are the energies of the initial and final states including the perturbation state and ensures the ${\displaystyle \delta }$-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

${\displaystyle w(k)=\sum _{k'}P(k,k')={\frac {2\pi }{\hbar }}\sum _{k'}|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})}$

The integral form is

${\displaystyle w(k)={\frac {2\pi }{\hbar }}{\frac {L^{3}}{(2\pi )^{3}}}\int d^{3}k'|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})}$

References

• C. Hamaguchi (2001). Basic Semiconductor Physics. Springer. pp. 196–253.
• J.J. Sakurai. Modern Quantum Mechanics. Addison Wesley Longman. pp. 316–319.