title: Scattering rate
text: # 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}\$
      $H_{0}|k\rangle =E(k)|k\rangle$
  
  In the interaction picture, the state ket is defined by
  
      $|k(t)\rangle _{I}=e^{iH_{0}t/\hbar }|k(t)\rangle _{S}=\sum _{k'}c_{k'}(t)|k'\rangle$
  
  By a Schrödinger equation, we see
  
      $i\hbar {\frac {\partial }{\partial t}}|k(t)\rangle _{I}=H_{1I}|k(t)\rangle _{I}$
  
  which is a Schrödinger-like equation with the total $H$ replaced by $H_{1I}$.
  
  Solving the differential equation, we can find the coefficient of n-state.
  
      $c_{k'}(t)=\delta _{k,k'}-{\frac {i}{\hbar }}\int _{0}^{t}dt'\;\langle k'|H_{1}(t')|k\rangle \,e^{-i(E_{k}-E_{k'})t'/\hbar }$
  
  where, the zeroth-order term and first-order term are
  
      $c_{k'}^{(0)}=\delta _{k,k'}$
      $c_{k'}^{(1)}=-{\frac {i}{\hbar }}\int _{0}^{t}dt'\;\langle k'|H_{1}(t')|k\rangle \,e^{-i(E_{k}-E_{k'})t'/\hbar }$
  
  ## The transition rate
  
  The probability of finding $|k'\rangle$ is found by evaluating $|c_{k'}(t)|^{2}$.
  
  In case of constant perturbation,$c_{k'}^{(1)}$ is calculated by
  
      $c_{k'}^{(1)}={\frac {\langle \ k'|H_{1}|k\rangle }{E_{k'}-E_{k}}}(1-e^{i(E_{k'}-E_{k})t/\hbar })$
      $|c_{k'}(t)|^{2}=|\langle \ k'|H_{1}|k\rangle |^{2}{\frac {sin^{2}({\frac {E_{k'}-E_{k}}{2\hbar }}t)}{({\frac {E_{k'}-E_{k}}{2\hbar }})^{2}}}{\frac {1}{\hbar ^{2}}}$
  
  Using the equation which is
  
      $\lim _{\alpha \rightarrow \infty }{\frac {1}{\pi }}{\frac {sin^{2}(\alpha x)}{\alpha x^{2}}}=\delta (x)$
  
  The transition rate of an electron from the initial state $k$ to final state $k'$ is given by
  
      $P(k,k')={\frac {2\pi }{\hbar }}|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})$
  
  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
  
      $w(k)=\sum _{k'}P(k,k')={\frac {2\pi }{\hbar }}\sum _{k'}|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})$
  
  The integral form is
  
      $w(k)={\frac {2\pi }{\hbar }}{\frac {L^{3}}{(2\pi )^{3}}}\int d^{3}k'|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})$
  