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

 E i, p i E f, p f E N, p N The simple 2-body kinematics of scattering fixes the energy of particles scattered through . For elastically scattered projectiles: The recoiling particles are identical to the incoming particles but are in different quantum states The initial conditions may be precisely knowable 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 pickup reactions incident projectile collects additional nucleons from the target O + d  O + H (d, 3 H) Ca + He  Ca +  ( 3 He,  ) inelastic scattering individual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy Na + He  Mg + d Zr + d  Zr + p (d,p) ( 3 He,d) stripping reactions incident projectile leaves one or more nucleons behind in the target

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

V0V0 x = 0 x = a 123 0

continuity at x=0 requires solve for 2A = 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 number of scattered particles/sec incident particles/(unit area  sec)  target site density × beamspot × target thickness a “counting” experiment notice it yields a measure, in units of area With a detector fixed to record data from a particular location ,  we measure the “differential” cross section: d  /d . how tightly focused or intense the beam is number of nuclear targets

v   t d d  Incident mono-energetic beam scattered particles A 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 dN particles per unit time that scatter through  into d  FLUX = # of particles crossing through unit cross section per sec = Nv  t A /  t A = Nv Notice: qNv we call current, I, measured in Coulombs. dN N F d  dN =  N F d  dN = N F d 

 dN = F N  d   N F d  (q) the “differential” cross section R R R R R 

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

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

dN scattered = N F d  TOTAL The scattering rate per unit time Particles IN (per unit time) = F  A rea(of beam spot) Particles scattered OUT (per unit time) = F  N  TOTAL

Scattering Probability (to a specific “final state” momentum p f ) 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 q pipi pfpf q = k i  k f =(p i -p f )/ħ momentum transfer the momentum given up (lost) by the scattered particle