Interaction of Heavy Charged Particles with Matter BAEN-625 Advances in Food Engineering Heavy charged particles y Charged particles other than the electron and positron Energy-loss mechanisms y A heavy particle traversing matter loses energy primarily thru the ionization and excitation of atoms ◦ Except in low velocities, it loses a negligible amount of energy in nuclear collisions y The moving particle exerts electromagnetic forces on atomic electrons and impart energy on them y The transferred energy may be sufficient to knock an electron out of an atom and thus ionize it y Or it may leave the atom in an excited state Heavy charged particle y Can transfer only small fraction of its energy in a single electronic collision y Its deflection in the collision is negligible y Thus it travels an almost straight path thru matter, y It loses energy continuously in small amounts thru collisions with atomic electrons Maximum Energy Transfer in a Single Collision y Assume ◦ the particle is moving rapidly compared to the electron ◦ Energy transferred is large compared with the BE (binding energy) of the electron in the atom ◦ The electron is free and at rest Maximum Energy Transfer in a Single Collision er in a ransfTum EnergMaximy Single Collision y Since energe conserdey and momentum arv 1 1 1 MV 2=MV 2 + mv 2 2 2 1 2 1 MV= MV1 + 1 mv y or Solving fV1: E=MV2/2 M()− m V V = Initial KE 1 M+ m 1 1 4mME Q =MV 2 −MV 2 = max 2 2 1 ()M+ m2 onIncident Particle is an Electr y Its mass is the same as that of the struck particle,M = m 4mME 4MME Q = = max M()+ m2 MM()+ 2 ME4 2 Q = =E max 4M 2 y ed in a singlere energ,eryEntir can be transf d-ball-type collisionbilliar Energy Transfer in a Single Collision if Incident Particle is an Electron er in a Single ransfTEnergum Maximy Collision- essionRelativistic Expr y T is small as long as elativistic is nonron An electr mc,est energywith the red compar2 = 0.511MeV 2γ2mV 2 Q = max 1m+ 2γ M /+ 2 m / 2 M γ=1 / 1 −β 2 β =V/ c Qmax in Proton Collision with Electron Proton Kinetic Qmax Maximum % Energy, E [MeV] Energy Transfer [MeV] 100Qmax/E 0.1 0.00022 0.22 1 0.0022 0.22 100 0.0219 0.22 100 0.229 0.23 1000 3.33 0.33 10000 136 1.4 100000 1060 10.6 1000000 53800 53.8 10000000 921000 92.1 Elastic or Inelastic y Equations shown before for Qmax are kinematic in nature y They follow from simultaneous conservation of momentum and KE y The assumption made to calculate energy loss was that the struck electron was not bound y Thus the collision being elastic y Charged-particle energy losses to atomic electrons are, in fact, inelastic Elastic Collision Elastic collision y Both momentum and kinetic energy are conserved y This implies that there is no dissipative force acting during the collision and that all of the kinetic energy of the objects before the collision is still in the form of kinetic energy afterward y For macroscopic objects which come into contact in a collision, there is always some dissipation and they are never perfectly elastic y In atomic or nuclear scattering, the collisions are typically elastic because the repulsive Coulomb force keeps the particles out of contact with each other. Examples of Elastic Collision y For a head-on collision with a stationary object of equal mass, the projectile will come to rest and the target will move off with equal velocity, like a head-on shot with the cue ball on a pool table. y This may be generalized to say that for a head-on elastic collision of equal masses, the velocities will always exchange. Examples of Elastic Collision y In a head-on elastic collision where the projectile is much more massive than the target, the velocity of the target particle after the collision will be about twice that of the projectile and the projectile velocity will be essentially unchanged. Examples of Elastic Collision y In a head-on elastic collision between a small projectile and a much more massive target, the projectile will bounce back with essentially the same speed and the massive target will be given a very small velocity. Inelastic Collision y Perfectly elastic collisions are those in which no kinetic energy is lost in the collision. y Macroscopic collisions are generally inelastic and do not conserve kinetic energy, though of course the total energy is conserved. y The extreme inelastic collision is one in which the colliding objects stick together after the collision Inelastic Collision Single-Collision Energy-Loss Spectra y Details about charged-particle penetration are embodied in the spectra of single-collision energy losses to atomic electrons y The collisions by which charged particles transfer energy to matter are inelastic y KE is lost in overcoming the BE of the struck electrons Single-Collision Energy-Loss Spectra In liquid water y The ordinate gives the probability density 0.06 ) 50-eV electrons W(Q) -1 y W(Q)dQ is the 0.04 5-MeV protons probability that a W(Q) (eV 150-eV electrons given collision will 0.00 result in an energy 0 50 100 Energy Loss Q (eV) loss between Q and Q + dQ Single-Collision Energy-Loss Spectra y For fast particles (speed > orbital In liquid water speed) ◦ Similarities in the region from 10- 70eV 0.06 y For slow charged particles 50-eV electrons ) ◦ The energy-loss spectra differ -1 from one another ◦ The time of interaction is longer 0.04 1-MeV protons than for fast particles W(Q) (eV ◦ The BE is more important 150-eV electrons ◦ Energy losses are closer to Qmax ◦ Slow particle excites atoms instead of ionizing them 0.00 0 50 100 y A minimum energy Qmin >0 is Energy Loss Q (eV) required for excitation or ionization of an atom Stopping Power y The average linear rate of energy loss of a heavy particle in a medium [MeV/cm] y Also referred as linear energy transfer (LET) of the particle Stopping Powers y Can be calculated from energy-loss spectra y For a given type of charged particle at a given energy, the SP is given by ◦ The probability μ per unit distance of travel that an electronic collision occurs ◦ The average energy loss per collision, Qmax Qmax Qavg =QW() Q dQ ∫Q min dE Qmax − =μQavg =QW μ () Q dQ dx ∫Qmin [1/cm] [MeV] [MeV/cm] Stopping Power-Semi Classical Calculation V Y ze Coulomb force zek 2 r F = 0 r 2 b Fy θ X Fx m -e Representation of the sudden Collision of a heavy charged Particle with an electron, Located at the origin XY -Semi Classical rewoStopping P Calculation y on is the The total momentum impared to the electrt collision is: ∞ ∞ ∞ 2 cosθ p F= dty = Fcosθ = dt0 k ze dt ∫− ∞ − ∫ ∞ −∫ ∞ r 2 0 ( timet = particle the heavycross Y - axis) cosθ = b/r ∞ cosθ ∞ b ∞ dt dt = 2 dt= 2 b ∫− ∞ r 2 ∫0 r 3 0b2() ∫ + 2 V 2 t 2 / 3 ∞ ⎡ t ⎤ 2 = 22b⎢ 2 2 2 1⎥ / 2= b⎣ () b+ V t ⎦0 Vb k2 ze2 p = 0 Vb p2 k2 z2 4e Q= = 0 2m mV2 b 2 Stopping Power-Semi Classical Calculation y In traversing a distance dx in a medium having a uniform density of n electrons per unit volume y The heavy particle encounters 2πnb db dx electrons at impact parameters between b and b + db y The energy lost to these electrons per unit distance traveled is 2πnQb db y The total linear rate energy loss is: 2 2 4 Q b dE max k4π z0 e nmax db −n =2π Qbd = b 2 = dx ∫Qmin mV ∫bmin b 2 2 4 dE k4π z0 e nbmax − = 2 ln dx mV bmin s r (Bethe’ewoRelativistic Stopping P Equation) y ons along The linear rate of energ loss to atomic electry vy charged paricle in a medium is the tthe path of a hea ysical quantity that determines the dose that the basic ph ers in the mediumparicle delivt 2 2 4 2 2 dE k4π z0 e⎡ n 2mc β 2 ⎤ − = 2 2 ⎢ln 2 − β ⎥ dx mc β ⎣ (I 1− β ) ⎦ 9 2− 2 8k0= . 99 × Nm 10 C atomic z = number ofparticle;magnitude the e = heavy of charge;electron number n = ofunit volumelectrons e in perthe medium electron rest mass; c speedm = of = light; β= V/c = speed of the particle relative to c excitation I = mean energy of the medium Stopping Power y Depends only on the charge ze and velocity β of the heavy particle y The relevant properties of the medium are its mean excitation energy I and the electronic density n y m is the mass of the target atomic electrons y Units: MeV/cm, mass stopping power-[MeV cm2/g] general,erwoStopping P y charged pary mediumvy ticle in any heaor anF dE5 . 09× −31z 10 2 n − =[ ( ) lnFIβ − ], MeV/cm dx β 2 eV 1.02× 6β 10 2 F(β )= ln − β 2 1− β 2 Mass Stopping Power y Useful quantity because it express the rate of energy loss of the charged particle per g/cm2 of the medium traversed y In gas –dE/dx depends on pressure, but –dE/ρdx does not y MSP does not differ greatly for materials with similar atomic composition (primarily light elements) 2 y For 10 MeV protons the MSP of H2O is 45.9 MeV cm /g and for 2 2 C14O10 44.2 cm /g, however for Pb(Z=82) the MSP is 17.5 cm /g y Heavy elements are less efficient on a g/cm2 basis for slowing down heavy charged particles (many of their electrons are too tightly bound in the inner shells to participate effectively in the absorption of energy) Mean Excitation Energies y SP equationom Can be calculated fr y It is the material parameter describing the of the target system to absorb ability energy y essions:Empirical expr 19 .⎧ 0eV , =Z 1 ⎪ 11I.