PHYS 476/676 Nuclear and Particle PhysicsMean life and half lifeThe passage of energetic particles through matterCross-sectionsTypes of ScatteringDifferential cross-sectionsReaction RatesCharged particle cross-sections: Rutherford scatteringCharged particlesParticle at rest and related energy lossesStopping powerEnergy losses for extreme values of impact parametersSlide Number 13Particle Range RSlide Number 15Multiple scattering of charged particlesSlide Number 17Dominant interactions of energetic photonsSlide Number 19The relative penetrating power of energetic particlesPHYS 476/676 Nuclear and Particle PhysicsStudy of elementary particles through their interaction with matterWednesday, October 01, 2008 2Mean life and half life• A lot of particles are unstable• Some exist for shorter than a pico second.• W± and Z bosons have only a transient existence (you observe they were present based on the their interaction products)• Mean life is the mean time particle exists in isolation before int undergoes radioactive decay• It is usually labeled with τ• Particle has constant probability to decay 1/τ per unit timeProbability for particle to survive from t = 0 to t, P(t) = P(0) e-t/τ• The probability that the particle decays between times t, t + dt is clearly P(t) x (dt/τ), so the mean life is: • Half life T1/2 is the time at which there is a 50% probability that particle has decayed:• P(T1/2 ) = e-T/τ = ½• T1/2 = τln2 = 0.693 τ• Decay rate is: 1/τR∞0tP (t)(dt/τ )=R∞0te−t/τdt/τ = τWednesday, October 01, 2008 3The passage of energetic particles through matter• Energetic particles like α, n, p+, e-, photons, fission fragments deposit energy while passing through matter.• We learn about nuclear physics processes via detection and characterization of these energy depositions• Ionization is the most common way to lose energy in the material• Energy deposition in matter affects living organisms as well– Positive effects: destruction of malignant tissue in cancer treatment– Negative effects: cell DNA damage as a result of radiation exposure (workers in nuclear plants…)• The discussion will be limited to particles with energy up to 10 MeVWednesday, October 01, 2008 4Cross-sections• Interaction cross section is used to express the likelihood of interaction between particles.• Neutron cross-section – only interact via short range strong force (neutral particles); the same case for photons.• Picture: neutron approaching from distance to a nucleus at rest.• Cross-sectional area of the nucleus πa2 and probability of passing through area δA is δA/πa2• Neutron is QM wave-packet• Interaction with nucleus takes place: scattering, induced fission, radiative capture…• Probability of interaction = σtot / πa2• σtot is called total cross-section. Dimensions of area.• Depends on effective exposed area of nucleus, but also on neutron energy and interaction between nucleus and neutron.• It is QM in nature and σtot can be much larger than geometrical cross-section of the nucleus.aIncident neutronsWednesday, October 01, 2008 5Types of Scattering• Interaction is typically described in one of two coordinate systems: laboratory and center-of-mass. σtot is the same in both.• Typically, several interaction types can occur: reaction channels and they are all assigned a certain probability:– Elastic scattering– Inelastic scattering– Radiative capture• The total sum of all interaction probabilities has to add up to one. • Partial cross-section for reaction channel i is defined as σI = pi σtot• σtot = Σi σI• Photon cross-section calculated in similar manner (neutral particles).• However, photon interacts with atomic electrons, so the target is the whole atom instead of neutron.Wednesday, October 01, 2008 6Differential cross-sections• Differential cross-section represents interaction probability distribution as a function of angle for a specific reaction channel.• For neutron to be scattered in a small solid angle dΩ= sinθ dθ dφ, the probability is pe (θ, φ) dΩ• So we define elastic differential cross-section •• Measured in the lab with respect to fixed target.• Results different in the CM reference frame; kinematic transformation is straightforward.pe(θ, φ)dΩ =1σe(dσdΩ)dΩdσedΩR(dσedΩ)dΩ = σezyxIncident beamθφTargetWednesday, October 01, 2008 7Reaction Rates• Let’s have a broad collimated beam of mono-energetic neutrons.• ρn is the number density of neutrons in the beam• v is the neutron velocity.• Neutron flux is ρn v• ρn v dt ⋅πa2 is the number of neutrons passing through a circle of radius a, centered on nucleus• The probability of interaction between t and t+dt for nucleus in ground state is ρn v σtot dt.• The reaction rate is:• reaction rate = flux x cross-section• In the case of a single neutron, moving with velocity v through nuclei of number density ρnuc (reversed look: neutron at rest and nuclei are a beam), the reaction rate is ρn v σtot . Then, • the mean time τbefore interaction is τ= (ρn v σtot )-1, • The mean free path l = vτ= 1/(ρn σtot )• * τ<< 15 min (mean free life of neutron)Wednesday, October 01, 2008 8Charged particle cross-sections: Rutherford scattering• Charged particles influenced by Coulomb force• When a charged particle like proton is passing fixed target (nucleus Ze) deflection through an angle θ is approximately:•• where p is momentum, v velocity and b impact parameter.• For scattering between θ and θ –dθ:• Impact parameter from b to b + db corresponds to the effective area 2 πb db, so the elastic cross-section is:• The differential scattering cross-section for small angles is therefore:• This is a small angle limit (θ ~ sin θ) of Rutherford scattering formula.θ =(Ze24π²0)2bpvdθ =dbb2(2Ze24π²0pv)dσe=2πbdb =2π(2Ze24π²0pv)dθθ3dσedΩ=12πsinθdσedθ=(2Ze24π²0pv)1θ4Wednesday, October 01, 2008 9Charged particles• Protons and α-particles traveling through gas.• For E , 10 MeV, excitation and ionization of gas atoms and molecules are dominant mechanisms for energy loss• ‘Fast’ particle: charge ze, velocity v, energy E, passing a particle of charge z’e, mass mR , initially at rest.• Particle is moving along x-axis, while mR is at (0,b,0) where b is called impact parameter.• Equation of