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Transcript
Particle Interactions Heavy Particle Collisions • Charged particles interact in matter. – Ionization and excitation of atoms – Nuclear interactions rare • Electrons 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. Energy Transfer • Assume an elastic collision. – One dimension – Moving particle M, V – Initial energy E = ½ MV2 – Electron mass m – Outgoing velocities Vf, vf • This gives a maximum energy transfer Qmax. MV MV f mv f 1 2 MV 2 12 MV f2 12 mv2f Vf M m V M m Qmax 12 MV 2 12 MV f2 4mM 4mE 2 1 ( MV ) ( M m) 2 2 M Relativistic Energy Transfer • At high energy relativistic effects must be included. Qmax 2g 2 mV 2 1 2gm / M m2 / M 2 • This reduces for heavy particles at low g, gm/M << 1. Qmax 2g 2 mV 2 2g 2 2 mc2 Qmax 2(g 2 1)mc2 Typical Problem • Calculate the maximum energy a 100-MeV pion can transfer to an electron. • mp = 139.6 MeV = 280 me – problem is relativistic • g = (K + mp)/ mp = 2.4 • Qmax = 5.88 MeV Protons in Silicon • The MIDAS detector measured proton energy loss. – 125 MeV protons – Thin wafer of silicon MIDAS detector 2001 Linear Energy Transfer • Charged particles experience multiple interactions passing through matter. – Integral of individual collisions – Probability P(Q) of energy transfer Q Q Qmax Qmin QP (Q)dQ • Rate of loss is the stopping power or LET or dE/dx. – Use probability of a collision m (cm-1) Qmax dE mQ m QP (Q)dQ Qmin dx Energy Loss • The stopping power can be derived semiclassically. – Heavy particle Ze, V – Impact parameter b • Calculate the impulse and energy to the electron. kZe2 b p Fy dt 2 dt r r y Ze b p kZe 2 dt 2 2 3/ 2 (b V t ) 2 V b 2kZe2 p Vb r Fy kZe2 F 2 r Fx m,e x p 2 2k 2 Z 2 e 4 Q 2m mV 2b 2 Impact Parameter • Assume a uniform density of electrons. – n per unit volume – Thickness dx • Consider an impact parameter range b to b+db – Integrate over range of b – Equivalent to range of Q • Find the number of electrons. 2pnb(db)dx • Now find the energy loss per unit distance. dE 4pnk 2 Z 2e 4 db 2pn Qbdb b dx mV 2 dE 4pnk 2 Z 2e 4 bmax ln 2 dx mV bmin Stopping Power • The impact parameter is related to characteristic frequencies. • Compare the maximum b to the orbital frequency f. – b/V < 1/f – bmax = V/f • Compare the minimum b to the de Broglie wavelength. – bmin = h / mV • The classical Bohr stopping power is dE 4pnk 2 Z 2e 4 mV 2 ln 2 dx mV hf Bethe Formula • A complete treatment of stopping power requires relativistic quantum mechanics. – Include speed b – Material dependent excitation energy dE 4pnk 2 Z 2e 4 dx mc2 2 2mc2 2 2 ln 2 I ( 1 ) Silicon Stopping Power • Protons and pions behave differently in matter – Different mass – Energy dependent MIDAS detector 2001 Range • Range is the distance a particle travels before coming to rest. • Stopping power represents energy per distance. – Range based on energy • Use Bethe formula with term that only depends on speed – Numerically integrated – Used for mass comparisons R( K ) K 0 1 dE dE dx 1 R( K ) 2 Z 1 R( ) 2 Z 0 K 0 dE G( ) Mg ( )d M 2 f ( ) G( ) Z 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 = 4 M Ra ( ) 2 R p ( ) R p ( ) Z • Equivalent energy for an alpha is ¼ that of a proton. – Use proton at 1.92 MeV • Approximate tissue as water and find proton range from a table. – 2 MeV, Rp = 0.007 g cm-2 • Units are density adjusted. – Ra = 0.007 cm • Alpha can’t penetrate the skin. Electron Interactions • Electrons share the same interactions as protons. – Coulomb interactions with atomic electrons – Low mass changes result • Electrons also have stopping radiation: bremsstrahlung e e e g e e • Positrons at low energy can annihilate. Z Beta Collisions • There are a key differences between betas and heavy ions in matter. – A large fractional energy change – Indistinguishable from e in quantum collision • Bethe formula is modified for betas. 4pnk 2 e 4 dE 2 2 dx col mc 2mc2 2 ln F ( ) 2 I 1 2 F ( ) 2 F ( ) ln 2 2 2 1 ( 2 1 ) ln 2 8 14 10 4 23 24 2 ( 2) 2 ( 2)3 K mc2 Radiative Stopping • The energy loss due to bremsstrahlung is based classical electromagnetism. – High energy – High absorber mass 4nk 2 Z ( Z 1)e 4 dE 137mc2 dx rad 1 ln 2 3 • There is an approximate relation. KZ dE dE dx 700 dx rad col -(dE/dx)/r in water Energy 1 keV 10 keV 100 keV 1 MeV 10 MeV 100 MeV 1 GeV MeV cm2 g-1 col rad 126 0 23.2 0 4.2 0 1.87 0.017 2.00 0.183 2.20 2.40 2.40 26.3 Beta Range • The range of betas in matter depends on the total dE/dx. – Energy dependent – Material dependent • Like other measures, it is often scaled to the density. Photon Interactions • High energy photons interact with electrons. – Photoelectric effect – Compton effect • They also indirectly interact with nuclei. – Pair production Photoelectric Effect • A photon can eject an electron from an atom. – Photon is absorbed – Minimum energy needed for interaction. – Cross section decreases at high energy g e Z K e h Compton Effect • Photons scattering from atomic electrons are described by the Compton effect. – Conservation of energy and momentum g’ g f e h mc2 h E h h cos P cos f c c h sin P sin f c h h 1 h (1 cos ) 2 mc 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 transfer – Small cot large – Recoil near 90° h (1 cos ) mc K cot h (1 cos ) mc2 / h 1 cos h 1 2 tan f 2 mc Compton Cross Section • Differential cross section can be derived quantum mechanically. – Klein-Nishina – Scattering of photon on one electron – Units m2 sr-1 • Integrate to get cross section per electron – Multiply by electron density – Units m-1 d k 2e 4 d 2m 2 c 4 2pn 2 2 sin d sin d d Pair Production • Photons above twice the electron rest mass energy can create a electron positron pair. – Minimum = 0.012 Å • The nucleus is involved for momentum conservation. – Probability increases with Z • This is a dominant effect at high energy. h 2mc2 K K g e e Z Total Photon Cross Section • Photon cross sections are the sum of all effects. – Photoelectric , Compton incoh, pair k Carbon Lead J. H. Hubbell (1980)