2 Heavy Particle Collisions Charged particles interact in matter.Ionization and excitation of atomsNuclear interactions rareElectrons can lose most of their energy in a single collision with an atomic electron.Heavier charged particles lose a small fraction of their energy to atomic electrons with each collision.
3 Energy Transfer Assume an elastic collision. One dimension Moving particle M, VInitial energy E = ½ MV2Electron mass mOutgoing velocities Vf, vfThis gives a maximum energy transfer Qmax.
4 Relativistic Energy Transfer At high energy relativistic effects must be included.This reduces for heavy particles at low g, gm/M << 1.Typical ProblemCalculate the maximum energy a 100-MeV pion can transfer to an electron.mp = MeV = 280 meproblem is relativisticg = (K + mp)/ mp = 2.4Qmax = 5.88 MeV
5 Protons in Silicon The MIDAS detector measured proton energy loss. 125 MeV protonsThin wafer of siliconMIDAS detector 2001
6 Linear Energy Transfer Charged particles experience multiple interactions passing through matter.Integral of individual collisionsProbability P(Q) of energy transfer QRate of loss is the stopping power or LET or dE/dx.Use probability of a collision m (cm-1)
7 Energy Loss The stopping power can be derived semiclassically. Heavy particle Ze, VImpact parameter bCalculate the impulse and energy to the electron.
8 Impact Parameter Assume a uniform density of electrons. n per unit volumeThickness dxConsider an impact parameter range b to b+dbIntegrate over range of bEquivalent to range of QFind the number of electrons.Now find the energy loss per unit distance.
9 Stopping PowerThe impact parameter is related to characteristic frequencies.Compare the maximum b to the orbital frequency f.b/V < 1/fbmax = V/fCompare the minimum b to the de Broglie wavelength.bmin = h / mVThe classical Bohr stopping power is
10 Bethe FormulaA complete treatment of stopping power requires relativistic quantum mechanics.Include speed bMaterial dependent excitation energy
11 Silicon Stopping Power Protons and pions behave differently in matterDifferent massEnergy dependentMIDAS detector 2001
12 Range Range is the distance a particle travels before coming to rest. Stopping power represents energy per distance.Range based on energyUse Bethe formula with term that only depends on speedNumerically integratedUsed for mass comparisons
13 Alpha Penetration Typical Problem Part of the radon decay chain includes a 7.69 MeV alpha particle. What is the range of this particle in soft tissue?Use proton mass and charge equal to 1.Ma = 4, Z2 = 4Equivalent energy for an alpha is ¼ that of a proton.Use proton at 1.92 MeVApproximate tissue as water and find proton range from a table.2 MeV, Rp = g cm-2Units are density adjusted.Ra = cmAlpha can’t penetrate the skin.
14 Electron Interactions Electrons share the same interactions as protons.Coulomb interactions with atomic electronsLow mass changes resultElectrons also have stopping radiation: bremsstrahlungPositrons at low energy can annihilate.eeegeeZ
15 Beta CollisionsThere are a key differences between betas and heavy ions in matter.A large fractional energy changeIndistinguishable b- from e in quantum collisionBethe formula is modified for betas.
16 Radiative StoppingThe energy loss due to bremsstrahlung is based classical electromagnetism.High energyHigh absorber massThere is an approximate relation.-(dE/dx)/r in waterMeV cm2 g-1Energy col rad1 keV10 keV100 keV1 MeV10 MeV100 MeV1 GeV
17 Beta Range The range of betas in matter depends on the total dE/dx. Energy dependentMaterial dependentLike other measures, it is often scaled to the density.
18 Photon Interactions High energy photons interact with electrons. Photoelectric effectCompton effectThey also indirectly interact with nuclei.Pair production
19 Photoelectric Effect A photon can eject an electron from an atom. Photon is absorbedMinimum energy needed for interaction.Cross section decreases at high energygeZ
20 Compton EffectPhotons scattering from atomic electrons are described by the Compton effect.Conservation of energy and momentumg’gqfe
21 Compton Energy The frequency shift is independent of energy. The energy of the photon depends on the angle.Max at 180°Recoil angle for electron related to photon energy transferSmall q cot largeRecoil near 90°
22 Compton Cross SectionDifferential cross section can be derived quantum mechanically.Klein-NishinaScattering of photon on one electronUnits m2 sr-1Integrate to get cross section per electronMultiply by electron densityUnits m-1
23 Pair ProductionPhotons above twice the electron rest mass energy can create a electron positron pair.Minimum l = ÅThe nucleus is involved for momentum conservation.Probability increases with ZThis is a dominant effect at high energy.ge+e-Z
24 Total Photon Cross Section Photon cross sections are the sum of all effects.Photoelectric t, Compton sincoh, pair kCarbonLeadJ. H. Hubbell (1980)