The Born's approximation

The Born's approximation is well known to give a quick solution for the Diffusion equation (or PDE : Partial Differential Equation) in the scattering formalism.

1. The Born series

Let's rewriteFrom the general solution of the equation of diffusion:

ψ_k(r) = e^ik.r -1/4π∫e^ik|r-r'|/|r-r'| U(r')ψ_k(r') dr'    (1)

We can see in the above equation that the term ψ_k(r') relating to the wave function is iterative. Let's then rewrite the same equation by expressing ψ_k(r') again in the second term of the second member in the equality. We have then:
ψ_k(r) = e^ik.r -1/4π∫e^ik|r-r'|/|r-r'| U(r') (e^ik.r' -1/4π∫e^ik|r'-r"|/|r'-r"| U(r") ψ_k(r")) dr'dr" = e^ik.r -1/4π∫e^ik|r-r'|/|r-r'| U(r') (e^ik.r') + (-1/4Ï€)²âˆ«e^ik|r-r'|/|r-r'| U(r') ∫e^ik|r'-r"|/|r'-r"| U(r") ψ_k(r")) dr'dr" + ...     (2)
This is known as Born series.
Commonly, we use only the two first term, and then keep the expression (1).

2. The Born approximation

The main idea is make the expression of the ψ_k(r), which is an asymptotic scattering, converge. When r tends to ∞ we can write:
|r - r'| = ((r- r')²) ^1/2 = ((r² - 2 rr'+ r'²)^1/2 = r ( 1 - 2 ur'/r + (r'/r)²)^1/2    (3)
U is the unit vector of the vector r.
r>>r' ther (r'/r)² is neclected, then:
|r - r'|= r( 1 - 2 ur'/r)^1/2 = r ( 1 - 2 ur'/r)^1/2 = r(1 - ur'/r)    (4)
The Green's function : G_k(r)= (-1/4Ï€) e^{ik|r - r'|} /|r - r'| ca be written as:
G_k(|r - r'|)= (-1/4Ï€)e^{ikr(1 - ur'/r)} e^ikur /r(1 - ur'/r). Neglecting again r' beside r, we have then:

G_k(|r - r'|)= (-1/4π)e^ikr e ^{- ikur'} /r    (5)


The function (1) can be written as :
ψ_k(r) = e^ik.r -1/4π∫e^{ikr(1 - ur'/r)}/(r(1 - ur'/r)) U(r')ψ_k(r') dr' = e^ik.r -1/4πe^ikr∫e ^{- ikur')}/(r(1 - ur'/r)) U(r')ψ_k(r') dr'
ψ_k(r) = e^ik.r - (1/4πr) e^ikr ∫e ^{- ikur'} U(r')ψ_k(r') dr'    (6)

Now, take the first order of the function ψ_k(r'), that is ψ_k(r') = e^ikr'. It follows that:
ψ_k(r) = e^ik.r - (1/4πr) e^ikr ∫e ^{- ikur'} U(r')e^ikr' dr' = e^ik.r - (1/4πr) e^ikr ∫e ^{- ikur'+ikr'} U(r') dr'

ψ_k(r') = e^ik.r - (1/4πr) e^ikr ∫e ^{ir'(k -ku)} U(r') dr'    (7)

The quantity k - ku is called the momentum transfert. Let q - k - ku ,as shown in the figure below:

The relationship (7) becomes:

ψ_k(r') = e^ik.r - (1/4πr) e^ikr ∫e ^iqr' U(r') dr'    (8)

Letting: f(q) = - (1/4π) ∫e ^iqr' U(r') dr'    (9)
Then:

ψ_k(r') = e^ik.r + e^ikr/r f(q)     (10)

The related scattering cross section is:
σ((q) = |f(q) |² = | - (1/4Ï€) ∫e ^iqr' U(r') dr'|²
Or:

σ((q) =(1/16Ï€²) | ∫e ^iqr' U(r') dr'|²Â Â Â  (11)

Let's suppose that the equation (1) is the Schrödinger equation.
k² = 2mE/(h/2Ï€)², U(r') = 2mV(r')/(h/2Ï€)²
Where m is the mass of the particle, E is its kinetic energy, V is its potential energy, and h the Planck's constant

We consider that the potential is central. Then : V(r'→) = V(r)
The relationship (9) becomes:
f(q) = - m/2π(h/2π) ∫e ^iqr' V(r') dr'    (12)
Let's consider the solid angle : r'² dr'sinθdθ dφ around the vector q, then:
∫e ^iqr' V(r') dr'= 2π∫∫e ^{iqr'cos(θ)} V(r') r'² dr'sinθdθ

For r': the integrale is from 0 to ∞
For θ from 0 to π

∫e ^iqr' V(r') dr' = 2π∫∫e ^{iqr'cos(θ)} V(r') r'² dr'sinθdθ    (13)
∫e ^{iqr'cos(θ)} sinθdθ = ∫e ^{iqr'cos(θ)}d(-cosθ)
= (- 1/iqr') (e^{- iqr'} - e^iqr') = (1/iqr') (2i) (sin(qr')) = (2/qr') sin(qr')    (14)

Then:

f(q) = - m/2Ï€(h/2Ï€) ∫e ^iqr' V(r') dr'= - m/2Ï€(h/2Ï€) (2Ï€) 2 ∫ V(r') 'sin(qr') /(qr') r'² dr'

f(q)^Born= (-2m/(h/2Ï€)²) ∫ V(r') 'sin(qr')/(qr') r'² dr'    (15)