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]*

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 how tightly focused or intense the beam is density 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 t 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 t A / t 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 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

Earth Moon

Earth Moon

for some sense of spacing consider the ratio orbital diameters central body diameter ~ 10 s for moons/planets ~100 s for planets orbiting sun In a solid • interatomic spacing: 1 -5 Å (1 -5 10 -10 m) • nuclear radii: 1. 5 -5 Å (1. 5 -5 10 -15 m) the ratio orbital diameters ~ 66, 666 for atomic electron central body diameter orbitals to their own nucleus Carbon Oxygen Aluminum Iron Copper Lead 6 C 8 O 13 Al 26 Fe 29 Cu 82 Pb

A solid sheet of lead offers how much of a (cross sectional) physical target (and how much empty space) to a subatomic projectile? 207 Pb 82 w Number density, n: number of individual atoms (or scattering centers!) per unit volume n= NA / A where NA = Avogadro’s Number A = atomic weight (g) = density (g/cc) n= (11. 3 g/cc)(6. 02 1023/mole)/(207. 2 g/mole) = 3. 28 1022/cm 3

207 Pb 82 w For a thin enough layer n (Volume) (atomic cross section) = n (surface area w)(pr 2) as a fraction of the target’s area: = n (w)p(5 10 -13 cm)2 For 1 mm sheet of lead: 1 cm sheet of lead: 0. 00257 0. 0257

Actually a projectile “sees” nw nuclei per unit area but Znw electrons per unit area!

that general description of cross section let’s augmented with the specific example of Coulomb scattering

q 1 q 2 BOTH target and projectile will move in response to the forces between them. Recoil of target q 1 But here we are interested only in the scattered projectile

impact parameter, b

b q 2 d A beam of N incident particles strike a (thin foil) target. The beam spot (cross section of the beam) illuminates n scattering centers. If d. N counts the average number of particles scattered between and d using d. N/N = n d d = 2 b db becomes:

b q 2 d and so

b q 2 d

What about the ENERGY LOST in the collision? • the recoiling target carries energy • some of the projectile’s energy was surrendered • if the target is heavy • the recoil is small • the energy loss is insignificant Reminder: 1/ (3672 Z)

mvf mv 0 mvf ( (mv) = - mv 0 recoil momentum of target ) • large impact parameter b For small scattering ( ) and/or • large projectile speed v 0 vf vo

mvf /2 mv 0 Together with: Recognizing that all charges are simple multiples of the fundamental unit of the electron charge e, we write q 1 = Z 1 e q 2 = Z 2 e p

Recalling that kinetic energy K = ½mv 2 = (mv)2/(2 m) the transmitted kinetic energy (the energy lost in collision to the target) K = ( p)2/(2 mtarget)

Z 2≡Atomic Number, the number of protons (or electrons) q 2=Z 2 e q 1=Z 1 e

For nuclear collisions: mtarget 2 Z 2 mproton For collisions with atomic electrons: mtarget melectron q 1 = e

For nuclear collisions: mtarget 2 Z 2 mproton For collisions with atomic electrons: mtarget melectron q 1 = e Z 2 times as many of these occur!

mproton = 0. 000 000 001 6748 kg melectron = 0. 000 000 0009 kg The energy loss due to collisions with electrons is GREATER by a factor of

Notice this simple approximation shows that Why are a-particles “more ionizing” than b-particles?

energy loss speed

the probability that a particle, entering a target volume with energy E “collides” within and loses an amount of energy between E' and E' + d. E' P (E, E' ) d. E' dx ( 2 pb db ) ( dx NA Z/A ) Or P (E, E' ) d. E' dx = P 1 / (E')2

-d. E/dx = (4 p. Noz 2 e 4/mev 2)(Z/A)[ln{2 mev 2/I(1 -b 2)}-b 2] I = mean excitation (ionization) potential of atoms in target ~ Z 10 Ge. V Felix Bloch 103 Hans Bethe Range of d. E/dx for proton through various materials d. E/dx ~ 1/b 2 102 H 2 gas target 101 Pb target Logarithmic rise 100 101 102 E (Me. V) 104 105 106

bg Muon momentum [Ge. V/c] Particle Data Group, R. M. Barnett et al. , Phys. Rev. D 54 (1996) 1; Eur. Phys. J. C 3 (1998)

D. R. Nygren, J. N. Marx, Physics Today 31 (1978) 46 d. E/dx(ke. V/cm) m p a p d e Momentum [Ge. V/c]