PHY492: Nuclear & Particle Physics. Lecture 3 Homework 1 Nuclear Phenomenology

Transcription

1 PHY49: Nuclear & Particle Physics Lecture 3 Homework 1 Nuclear Phenomenology

2 Measuring cross sections in thin targets beam particles/s n beam m T = ρts mass of target n moles = m T A n nuclei = n moles A 0 n nuclei S # of target moles atomic mass A # target nuclei # target nuclei per unit area scattered particles detected per second n detected = n beam n nuclei S dω θ,φ ( )dω de tector January 17, 007 Carl Bromberg - Prof. of Physics

3 Rutherford scattering of 10 MeV α particles on Lead foil, 0.1 cm thick Problem 1.4 What is the counting rate at 90 in the lab, 100 cm away, in a counter 1 cm if there are 10 6 α particles/s hitting the target? dn = N 0 N S = N 0 A 0 ρt A dn = 0.5 s -1 ( dω θ,φ )dω ( dω θ,φ )dω What about at 5? dn = 34,000 s -1 dω θ ( ) = Z 4E sin 4 ( θ ) S = area of target is not given? ρ = mass/cm 3 ρt = mass/cm ρt A = moles/cm Z Z e 4E = Z Z e c 4E c = Z Z 4E A 0 ρt A = atoms/cm ( ) ( ) = Z Z 4E α c 197 MeVifm 137 January 17, 007 Carl Bromberg - Prof. of Physics 3

4 Problem 1.1: Rutherford scattering total cross section Differential Cross section in D&F Eq. 1.3 sin θ b = b E ( Z Z e ) dω θ ( ) = Z σ TOT 4E 1 sin 4 θ Scattering at angles larger than θ b ( θ > θ b ) = 8π Z 4E Total Cross section σ total = dω dω θ dω = sinθdθdφ January 17, 007 Carl Bromberg - Prof. of Physics 4 = 4π Z 4E = 4π Z 4E 1 sin θ b 1 dx x 3 sin θ b E Z Z e Area of disk radius b 1 = 4π Z 4E b = πb ( ) 4b E ( Z Z e )

5 Problem 1.3: N.R. two body scattering ς = m beam m target Target stationary (laboratory frame) Net zero momentum (center of mass frame) January 17, 007 Carl Bromberg - Prof. of Physics 5

6 Problem 1.5: Isotropic Center of Mass scattering Angle transformation cosθ cosθ Lab = CM + ς ( 1+ ς cosθ CM + ς ) 1 No backward scattering Cross section transformation dω Lab = ( ) 3 1+ ς cosθ + ς CM dω CM 1+ ς cosθ CM Lab scattering nearly isotropic January 17, 007 Carl Bromberg - Prof. of Physics 6

7 Problem 1.7: Relativistic electron Electron mass m = MeV/c Given electron momentum pc = mc Total energy E = p c + m c 4 = mc Velocity β = pc E = 1 Gamma γ = ( 1 β ) 1 = Kinetic energy T = E mc = ( 1)mc January 17, 007 Carl Bromberg - Prof. of Physics 7

8 Problem 1.9: Ultra-relativistic limit β 1 Ultra-relativistic limit tanθ Lab = γ CM β sinθ ( CM β cosθ CM + β ) sinθ CM γ CM CM cosθ CM + 1 ( ) Maximum at θ CM = π tanθ Lab = 1 γ CM 1 θ Lab = tan 1 = tan 1.1 γ CM = tan 1.01 ( ) = 5.7 ; γ CM = 10 ( ) = 0.57 ; γ CM = 100 January 17, 007 Carl Bromberg - Prof. of Physics 8

9 Problem 1.11: Scattering rates into a detector Alpha T=8 MeV, flux = 10 4 s -1, 0.1 mm Gold foil, ρ = 19.3 g/cm 3 Δθ =.05 rad, θ = 90 = π Δθ rad = 1.57 rad (note: dθ = Δθ if << 1 is satisfied!) θ Δθ =.05 rad, θ = 5 = π Δθ rad = rad ( dθ Δθ because << 1 not satisfied!) θ at 90 o dω θ ( ) = Z 4E 1 sin 4 θ =.0 b/sr; rate=3.5/s at 5 o ( dω θ ) = 1,000/s is greater than beam rate! but dθ is not correct! January 17, 007 Carl Bromberg - Prof. of Physics 9

10 Nuclear Numerology Nuclei are specified by three numbers A, Z, and N, and are labeled by the element X, in the periodic table with that Z : Z = # of protons (atomic number) N = # of neutrons (determines the isotope) A = Z + N (atomic weight, specifies the isotope) X = Mnemonic for element in periodic table For example, Calcium 40 : A Z X or A X Z or A X Z Z is not specified except by looking in a Periodic Table of Elements 40 Ca Z = 0, and therefore N = A-Z = 0 40 Ca 0 Neutron deficit Stable isotopes Neutron excess 34 Ca A Ca, A = 40, 4, 43, 44, 46, Ca Alice s Restaurant For binding energies January 17, 007 Carl Bromberg - Prof. of Physics 10

11 Half Lives ( s 1 s 10 7 s 100 s Stable January 17, 007 Carl Bromberg - Prof. of Physics 11

12 Decay modes ( spontaneous fission e capture, β + α Z p drip β stable n drip N January 17, 007 Carl Bromberg - Prof. of Physics 1

13 Topics to be covered size and shape mass and binding energy charge distribution angular momentum (spin) symmetries (parity) magnetic moments radioactivity energy levels reactions Physics of nuclei Nuclear size Binding energy per nucleon is < 1% of the nucleon mass. The protons and neutrons in the nucleus retain their particle properties. Assume nuclei are spherical and have a constant density (not compressed). V r 3 V A Nuclear radius r A 1 3 r = ( m) A 1 3 January 17, 007 Carl Bromberg - Prof. of Physics 13