Particle propertiescharacteristics specifically their interactions are often interpreted

Particle properties/characteristics specifically their interactions are often interpreted in terms of CROSS SECTIONS.

For elastically scattered projectiles: Ef , pf Ei , pi EN , p. N The recoiling particles are identical to the incoming particles but are in different quantum states The initial conditions may be precisely The simple 2 -body kinematics of scattering knowable fixes the energy of particles scattered through . only classically!

Nuclear Reactions Besides his famous scattering of particles off gold and lead foil, Rutherford observed the transmutation: or, if you prefer Whenever energetic particles (from a nuclear reactor or an accelerator) irradiate matter there is the possibility of a nuclear reaction

Classification of Nuclear Reactions • inelastic scattering individual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy • pickup reactions incident projectile collects additional nucleons from the target 16 O + d 158 O + 31 H (d, 3 H) 8 Ca + 32 He 40 Ca + 20 41 20 (3 He, ) • stripping reactions incident projectile leaves one or more nucleons behind in the target 90 91 (d, p) 40 Zr + d 40 Zr + p 3 He, d) 23 3 24 ( 11 Na + 2 He 12 Mg + d

[ 20 10 Ne]* Predicting a final outcome is much like rolling dice…the process is random!

V 0 0 1 x=0 2 3 x=a

continuity at x=0 requires solve for 2 A = C+ D

The cross section is defined by the ratio rate particles are scattered out of beam rate of particles focused onto target material/unit area a “counting” experiment notice it yields a measure, in units of area number of scattered particles/sec incident particles/(unit area sec) target site density × beamspot × target thickness how tightly focused or intense the beam is number of nuclear targets With a detector fixed to record data from a particular location , we measure the “differential” cross section: d /d.

Incident mono-energetic beam v Dt scattered particles A d N = number density in beam (particles per unit volume) N number of scattering centers in target intercepted by beamspot Solid angle d represents detector counting the d. N particles per unit time that scatter through into d FLUX = # of particles crossing through unit cross section per sec = Nv Dt A / Dt A = Nv Notice: q. Nv we call current, I, measured in Coulombs. d. N N F d. W d. N = s( )N F d W d. N = N F d s

d. N = FN s( )d W N F d s(q) the “differential” cross section R R R

the differential solid angle d for integration is sin d R Rsin d Rd Rsin Rd

Symmetry arguments allow us to immediately integrate out and consider rings defined by alone R Rsin d R R R Integrated over all solid angles Nscattered = N F ds TOTAL

The scattering rate d. Nscattered = N F ds TOTAL per unit time Particles IN (per unit time) = F Area(of beam spot) Particles scattered OUT (per unit time) = F Ns TOTAL

Scattering Probability (to a specific “final state” momentum pf ) Depends on “how much alike” the final and initial states are. assumed merely to be perturbed as it passes (quickly!) through the scattering potential The overlap of these wavefunctions is expressed by the “Matrix element” Potential perturbs the initial momentum state into a state best described as a linear (series) combination of possible final states… each weighted by the probability of that final state

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

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