Engineering: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 [math]\displaystyle{ H_0 }[/math], the time dependent perturbing Hamiltonian by [math]\displaystyle{ H_1 }[/math] and total Hamiltonian by [math]\displaystyle{ H }[/math].

The eigenstates of the unperturbed Hamiltonian are assumed to be

[math]\displaystyle{ H=H_0+H_1\ }[/math]

[math]\displaystyle{ H_0 |k\rang = E(k)|k\rang }[/math]

In the interaction picture, the state ket is defined by

[math]\displaystyle{ |k(t)\rang _I= e^{iH_0 t /\hbar} |k(t)\rang_S= \sum_{k'} c_{k'}(t) |k'\rang }[/math]

By a Schrödinger equation, we see

[math]\displaystyle{ i\hbar \frac{\partial}{\partial t} |k(t)\rang_I=H_{1I}|k(t)\rang_I }[/math]

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

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

[math]\displaystyle{ 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} }[/math]

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

[math]\displaystyle{ c_{k'}^{(0)}=\delta_{k,k'} }[/math]

[math]\displaystyle{ 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} }[/math]

The transition rate

The probability of finding [math]\displaystyle{ |k'\rang }[/math] is found by evaluating [math]\displaystyle{ |c_{k'}(t)|^2 }[/math].

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

[math]\displaystyle{ c_{k'}^{(1)}=\frac{\lang\ k'|H_1|k\rang }{E_{k'}-E_k}(1-e^{i(E_{k'} - E_k)t/\hbar}) }[/math]

[math]\displaystyle{ |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} }[/math]

Using the equation which is

[math]\displaystyle{ \lim_{\alpha \rightarrow \infty} \frac{1}{\pi} \frac{sin^2(\alpha x)}{\alpha x^2}= \delta(x) }[/math]

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

[math]\displaystyle{ P(k,k')=\frac {2 \pi} {\hbar} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) }[/math]

where [math]\displaystyle{ E_k }[/math] and [math]\displaystyle{ E_{k'} }[/math] are the energies of the initial and final states including the perturbation state and ensures the [math]\displaystyle{ \delta }[/math]-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

[math]\displaystyle{ 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) }[/math]

The integral form is

[math]\displaystyle{ 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) }[/math]

References

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