Cross section for scattering particle (mass,m; charge, Z1e)by a screened Coulomb potential (source,Z2e of infinite mass)

22

14

)(q

a

qF

In the limit a (no screening) this reducesto Rutherford’s scattering formula

2

42

2

)(1

4qF

m

d

d

Fermi’s Golden Rule

(see Gasiorowicz)

2222

22

2142

2

)cos1)(/2()/1(

161

4

paeZZ

m

d

d

using (1-cos)=sin2(/2)12

2

2222

221

)2/(sin4)/(

2

pa

eZmZ

d

d

2

222

221

)2/(sin4)2/(

Ema

eZZ

d

d

Using E=p2/2m (the classical expression) for the incoming particle

Breakdown of RutherfordScattering formula

When an incident particlegets close enough to the targetPb nucleus so that they interact

through the nuclear force (inaddition to the Coulomb forcethat acts when they are furtherapart) the Rutherford formula

no longer holds.

The point at which this breakdown occurs gives a

measure of the size of the nucleus.

R.M.Eisberg and C.E.PorterRev. Mod. Phys, 33, 190 (1961)

Radial probability distributions for a particle in a Coulomb potential(hydrogenic atom). Note the probability vanishes at r=0.

40Ca

12C

16O

Charge density distributionfor lead deduced from

electron scattering