130_notes.dvi

Clearly there is more information available from scattering than whether a particle scatters or not.
For example,Rutherford discovered that atomic nucleusby seeing that high energy alpha
particles sometimes backscatter from a foil containing atoms. Theatomic model of the time did not
allow this since the positive charge was spread over a large volume. Wemeasure the probability to
scatter into different directions. This will also happen in the case of the BB and the billiard ball.
The polar angle of scattering will depend on the “impact parameter”of the incoming BB. We can
measure the scattering into some small solid angledΩ. The part of the cross sectionσthat scatters
into that solid angle can be called thedifferential cross sectiondσdΩ. The integral over solid angle
will give us back the total cross section.
∫
dσ
dΩ

dΩ =σ

The idea of cross sections and incident fluxes translates well to thequantum mechanics we are using.
If theincoming beam is a plane wave, that is a beam of particles of definite momentum or wave
number, we can describe it simply in terms of the number or particles per unit area per second, the
incident flux. Thescattered particle is also a plane wavegoing in the direction defined bydΩ.
What is left is the interaction between the target particle and the beam particle which causes the
transition from the initial plane wave state to the final plane wave state.

We have already studied one approximation method for scattering called a partial wave analysis (See
section 15.6). It is good for scattering potentials of limited range and for low energy scattering.
It divides the incoming plane wave in to partial waves with definite angular momentum. The high
angular momentum components of the wave will not scatter (much)because they are at large distance
from the scattering potential where that potential is very small. We may then deal with just the
first few terms (or even just theℓ= 0 term) in the expansion. We showed that the incoming partial
wave and the outgoing wave can differ only by a phase shift for elasticscattering. If we calculate
this phase shiftδℓ, we can then determine the differential scattering cross section.

Let’s review some of the equations. A plane wave can be decomposedinto a sum of spherical waves
with definite angular momenta which goes to a simple sum of incoming andoutgoing spherical waves
at larger.

eikz=eikrcosθ=

∑∞

ℓ=0

√

4 π(2ℓ+ 1)iℓjℓ(kr)Yℓ 0 →−

∑∞

ℓ=0

√

4 π(2ℓ+ 1)iℓ

1

2 ikr

(

e−i(kr−ℓπ/2)−ei(kr−ℓπ/2)

)

Yℓ 0

A potential causing elastic scattering will modify the phases of the outgoing spherical waves.

lim

r→∞

ψ=−

∑∞

ℓ=0

√

4 π(2ℓ+ 1)iℓ

1

2 ikr

(

e−i(kr−ℓπ/2)−e^2 iδℓ(k)ei(kr−ℓπ/2)

)

Yℓ 0

We can compute the differential cross section for elastic scattering.

dσ

dΩ

=

1

k^2

∣

∣

∣

∣

∣

∑

ℓ

(2ℓ+ 1)eiδℓ(k)sin(δℓ(k))Pℓ(cosθ)

∣

∣

∣

∣

∣

2

It is useful to write this in terms of the amplitudes of the scatteredwaves.

dσ

dΩ

= |f(θ,φ)|^2

f(θ,φ) =

1

k

∑

ℓ

(2ℓ+ 1)eiδℓ(k)sin(δℓ(k))Pℓ(cosθ) =

∑

ℓ

fℓ(θ,φ)