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 $H_0$, the time dependent perturbing Hamiltonian by $H_1$ and total Hamiltonian by $H$.

The eigenstates of the unperturbed Hamiltonian are assumed to be

$H=H_0+H_1$

$H_0|k⟩=E(k)|k⟩$

In the interaction picture, the state ket is defined by

$|k(t){⟩}_I=e^{i{H}_{0}t/\hslash }|k(t){⟩}_S=\sum _{k^{\prime }}{c}_{k^{\prime }}(t)|k^{\prime }⟩$

By a Schrödinger equation, we see

$i\hslash \frac{\partial }{\partial t}|k(t){⟩}_I=H_{1I}|k(t){⟩}_I$

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.

$c_{k^{\prime }}(t)={\delta }_{k,k^{\prime }}-\frac{i}{\hslash }\underset{0}{\overset{t}{\int }}d{t}^{\prime }⟨k^{\prime }|H_1(t^{\prime })|k⟩e^{-i(E_k-E_{k^{\prime }})t^{\prime }/\hslash }$

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

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

$c_{k^{\prime }}^{(1)}=-\frac{i}{\hslash }\underset{0}{\overset{t}{\int }}d{t}^{\prime }⟨k^{\prime }|H_1(t^{\prime })|k⟩e^{-i(E_k-E_{k^{\prime }})t^{\prime }/\hslash }$

The transition rate

The probability of finding $|k^{\prime }⟩$ is found by evaluating $|c_{k^{\prime }}(t){|}^2$.

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

$c_{k^{\prime }}^{(1)}=\frac{⟨k^{\prime }|H_1|k⟩}{E_{k^{\prime }}-E_k}(1-e^{i(E_{k^{\prime }}-E_k)t/\hslash })$

$|c_{k^{\prime }}(t){|}^2=|⟨k^{\prime }|H_1|k⟩{|}^2\frac{si{n}^2(\frac{E_{k^{\prime }}-E_k}{2\hslash }t)}{(\frac{E_{k^{\prime }}-E_k}{2\hslash }{)}^2}\frac{1}{{\hslash }^2}$

Using the equation which is

$\underset{\alpha \to \mathrm{\infty }}{\lim }\frac{1}{\pi }\frac{si{n}^2(\alpha x)}{\alpha x^2}=\delta (x)$

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

$P(k,k^{\prime })=\frac{2\pi }{\hslash }|⟨k^{\prime }|H_1|k⟩{|}^2\delta (E_{k^{\prime }}-E_k)$

where $E_k$ and $E_{k^{\prime }}$ 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

$w(k)=\sum _{k^{\prime }}P(k,k^{\prime })=\frac{2\pi }{\hslash }\sum _{k^{\prime }}|⟨k^{\prime }|H_1|k⟩{|}^2\delta (E_{k^{\prime }}-E_k)$

The integral form is

$w(k)=\frac{2\pi }{\hslash }\frac{L^3}{(2\pi {)}^3}\int d^3{k}^{\prime }|⟨k^{\prime }|H_1|k⟩{|}^2\delta (E_{k^{\prime }}-E_k)$

References

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