The Born Approximation

Integral scattering equation for stationary states

The eigenvalue equation

E>0,

can be put into the form

with  E=h2k2/(2m)

The solutions to this equation may be written in the form

where f0(r) is a solution of the homogeneous equation  and G(r) is a solution of 

Verify:

                            0         acts only on r

                d(r-r')

The solutions of  are These are two linearly independent solutions of a second order differential equation.

(To verify that these are the solutions use

)

For a stationary scattering state we might choose f0(r)=eikz and G(r)=G+(r).

If |r'|<<|r| then and

This yields an integral expression for the scattering amplitude fk(q,f).

The Born approximation

).

This procedure can be repeated and yields the Born expansion.  The Born approximation is the first term in the Born expansion.

The scattering amplitude in the Born approximation is given by

Here 

The scattering amplitude is proportional to the Fourier transform of the potential energy.

 is the differential scattering cross section in the Born approximation.

The Yukawa potential

Let   As a®0 the potential becomes the Coulomb potential.  Then the scattering amplitude in the Born approximation is given by

Choose the coordinate system such that Then

                                                                  

 

Often you want to know the scattering cross section as a function of the scattering angle and not as a function of the momentum transfer.  Let q' be the angle between ki and kf. Then

p4.gif (1098 bytes)

Then 

The Coulomb potential

Let a®0, V0=Z1Z2e2. Then

This is Rutherford’s formula.  The Born approximation for the Yukawa potential gives Rutherford’s formula as a®0.  The total cross section skB®¥.  The total cross section is infinite, because the Coulomb potential has infinite range.

When is the Born approximation a good approximation?

In the integral we are replacing the exact solution fk(r') by the asymptotic incident wave eikz'.  Therefore fk(r') and eikz' should not be too different inside the range of the potential, i.e. in the region where U(r) contributes appreciably to the integral.

We therefore need that .  In the high k limit this inequality is easily satisfied because the integrand oscillates rapidly.  We can also satisfy the condition if the scattering potential is weak.

Problem:

Consider an electron of energy E0 and velocity v0 in the z-direction incident on an ionized Helium atom He+, with just one electron in its ground state.  Compute the differential cross section ds/dW for the incident electron to scatter into the solid angle dW about the spherical angles (q,f).  Explain your assumptions and approximations.
Solution:

For fast electrons we can use the Born approximation.  We then have

with q=k'-k, k the incident wave vector and a unit vector pointing in the direction (q,f).  (Note the switch in notation.)

                                                                                           

[

Take the limit as l®0.]

where is the atomic scattering form factor.

 

If we assume that F(q)=1, i.e. that the incoming electron sees a screened nucleus of charge 1, we obtain the Rutherford scattering cross section.