Scattering rate explained

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 (\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'} - (i)/(\hbar) \int ^{t} _{0} 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 ^{( 0 )} _{k'} = \delta _{k,k'} c ^{( 1 )} _{k'} =- (i)/(\hbar) \int ^{t} _{0} 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 ^{( 1 )} _{k'} is calculated by c ^{( 1 )} _{k'} = (\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 (sin ^{2 ( (E _{k'} - E _{k})/(2 \hbar)} t))/(( (E _{k'} - E _{k})/(2 \hbar) ) ^{2})} (1)/(\hbar ^{2})

Using the equation which is

\lim _{\alpha} (1)/(\pi) (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 byP(k,k') = (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') = (2 \pi)/(\hbar) \sum _{k'} |\lang k'|H _{1|k\rang} | ^{2} \delta(E _{k'} - E _{k)}

The integral form is

w(k) = (2 \pi)/(\hbar) (L ^{3})/(( 2 \pi) ^{3}) \int d ^{3k'} |\lang k'|H _{1|k\rang} | ^{2} \delta(E _{k'} - E _{k)}

References

• Book: C. Hamaguchi . Basic Semiconductor Physics . Springer . 2001 . 196–253.
• Book: J.J. Sakurai . . Addison Wesley Longman . 316–319.