**
**

#### Integral scattering equation for stationary states

The eigenvalue equation

*E>0*,

can be put into the form

with
*E*=h^{2}*k*^{2}/(2*m*)

The solutions to this equation may be written in the form

where *f*_{0}(**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 *f*_{0}(**r**)=e^{ikz}
and *G(***r**)=G^{+}(**r**).

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

This yields an integral expression for the scattering amplitude *f*_{k}(q,f).

).

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 **k**_{i} and **k**_{f}. Then

Then

#### The Coulomb potential

Let *a*®0, V_{0}=Z_{1}Z_{2}e^{2}.
Then

This is Rutherford’s formula. The Born approximation for the Yukawa potential
gives Rutherford’s formula as *a*®0.
The total cross
section *s*_{k}^{B}®¥.
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 *f*_{k}(**r**')
by the asymptotic incident wave e^{ikz'}. Therefore *f*_{k}(**r**')
and e^{ikz'} 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.

Consider an electron of energy *E*_{0} and velocity **v**_{0}
in the z-direction incident on an ionized Helium atom *He*^{+}, with just one
electron in its ground state. Compute the differential cross section d*s*/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.