1. Schrödinger’s equation
For “free” particles (unbounded in the “continuum”)
the solutions to
Schrödinger’s equation
with no potential
Sorry!…the V at left
is a volume appearing
for normalization V

2. q pf q = ki - kf =(pi-pf )/ħ pi q momentum transfer the momentum
given up (lost)
by the scattered
particle pf
q = ki - kf =(pi-pf )/ħ pi

3. Example
Cross section for scattering a particle of mass m and charge Z1e
by a Coulomb potential (source Z2 of comparatively ~infinite mass)
heavy nucleus or one locked
within a solid crystal.
At least partially screened
by the fraction of the electron
cloud not penetrated
(outside the atom appears neutral!)
screened
Coulomb potential
q1q2

4. I’ll let the direction
of q fix the z-axis
qr = qrcos

5. (0 – 1)

6. Cross section for scattering particle (mass,m; charge, Z1e)
by a screened Coulomb potential (source,Z2e of infinite mass)
In the limit a (no screening) this reduces
to Rutherford’s scattering formula

7. For an infinitely massive target:pi pf
For an infinitely massive target: pi pf pf pf
pi-pf ħ
pi-pf ħ pi =q pi q=
with

8. (see Gasiorowicz or favorite text)
Fermi’s Golden Rule
(see Gasiorowicz or favorite text)
using (1-cos)=sin2(/2) 1 2
Using E=p2/2m (the classical expression) for the incoming particle

9. Breakdown of Rutherford
Scattering formula
When an incident  particle
gets close enough to the target
Pb nucleus so that they interact
through the nuclear force (in
addition to the Coulomb force
that acts when they are further
apart) 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.Porter
Rev. Mod. Phys, 33, 190 (1961)

10. If that begins to provide some ideas of the size
of a nucleus…what’s it like inside?
How are the nucleons
(protons and neutrons) arranged?
How tightly packed in?
If we use the radial QM probability distributions
of electrons in a Coulomb potential as a guide:

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

12. Electron scattering
12C
40Ca
16O

15. Best fit for electron
scattering off gold.

17. Charge density distribution
for lead deduced from
electron scattering