In physics, the scattering rate describes the rate at which a beam of particles is scattered while passing through a material. It represents the probability per unit time that a particle will be deflected from its original trajectory by an interaction, such as with impurities or phonons in a crystal lattice. The scattering rate, often denoted by w {\displaystyle w} or Γ {\displaystyle \Gamma } , is a crucial concept in solid-state physics and condensed matter physics, as it determines various material properties, including electrical conductivity and thermal conductivity.
The interaction picture Define the unperturbed Hamiltonian by H 0 {\displaystyle H_{0}} , the time dependent perturbing Hamiltonian by H 1 {\displaystyle H_{1}} and total Hamiltonian by H {\displaystyle H} . The eigenstates of the unperturbed Hamiltonian are assumed to be
H = H 0 + H 1 {\displaystyle H=H_{0}+H_{1}\ }
H 0 | k ⟩ = E ( k ) | k ⟩ {\displaystyle H_{0}|k\rangle =E(k)|k\rangle }
In the interaction picture, the state ket is defined by
| k ( t ) ⟩ I = e i H 0 t / ℏ | k ( t ) ⟩ S = ∑ k ′ c k ′ ( t ) | k ′ ⟩ {\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
i ℏ ∂ ∂ t | k ( t ) ⟩ I = H 1 I | k ( t ) ⟩ I {\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 H {\displaystyle H} replaced by H 1 I {\displaystyle H_{1I}} . Solving the differential equation, we can find the coefficient of n-state.
c k ′ ( t ) = δ k , k ′ − i ℏ ∫ 0 t d t ′ ⟨ k ′ | H 1 ( t ′ ) | k ⟩ e − i ( E k − E k ′ ) t ′ / ℏ {\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
c k ′ ( 0 ) = δ k , k ′ {\displaystyle c_{k'}^{(0)}=\delta _{k,k'}}
c k ′ ( 1 ) = − i ℏ ∫ 0 t d t ′ ⟨ k ′ | H 1 ( t ′ ) | k ⟩ e − i ( E k − E k ′ ) t ′ / ℏ {\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 | k ′ ⟩ {\displaystyle |k'\rangle } is found by evaluating | c k ′ ( t ) | 2 {\displaystyle |c_{k'}(t)|^{2}} . In case of constant perturbation, c k ′ ( 1 ) {\displaystyle c_{k'}^{(1)}} is calculated by
c k ′ ( 1 ) = ⟨ k ′ | H 1 | k ⟩ E k ′ − E k ( 1 − e i ( E k ′ − E k ) t / ℏ ) {\displaystyle 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 = | ⟨ k ′ | H 1 | k ⟩ | 2 s i n 2 ( E k ′ − E k 2 ℏ t ) ( E k ′ − E k 2 ℏ ) 2 1 ℏ 2 {\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
lim α → ∞ 1 π s i n 2 ( α x ) α x 2 = δ ( x ) {\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 k {\displaystyle k} to final state k ′ {\displaystyle k'} is given by
P ( k , k ′ ) = 2 π ℏ | ⟨ k ′ | H 1 | k ⟩ | 2 δ ( E k ′ − E k ) {\displaystyle P(k,k')={\frac {2\pi }{\hbar }}|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})}
where E k {\displaystyle E_{k}} and E k ′ {\displaystyle E_{k'}} are the energies of the initial and final states including the perturbation state and ensures the δ {\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
w ( k ) = ∑ k ′ P ( k , k ′ ) = 2 π ℏ ∑ k ′ | ⟨ k ′ | H 1 | k ⟩ | 2 δ ( E k ′ − E k ) {\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
w ( k ) = 2 π ℏ L 3 ( 2 π ) 3 ∫ d 3 k ′ | ⟨ k ′ | H 1 | k ⟩ | 2 δ ( E k ′ − E k ) {\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.