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

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

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

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

(0 – 1)

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

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

(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

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)

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:

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

Electron scattering 12C 40Ca 16O

Best fit for electron scattering off gold.

Charge density distribution for lead deduced from electron scattering