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dE/dx. Let’s next turn our attention to how charged particles lose energy in matter To start with we’ll consider only heavy charged particles like muons, pions, protons, alphas, heavy ions, … Effectively all charged particles except electrons
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dE/dx • Let’s next turn our attention to how charged particles lose energy in matter • To start with we’ll consider only heavy charged particles like muons, pions, protons, alphas, heavy ions, … • Effectively all charged particles except electrons • The mean energy loss of a charged particle through matter is described by the Bethe-Bloch equation
dE/dx • You’ll see
dE/dx • In particle physics, we call dE/dx the energy loss • In radiation and other branches of physics, dE/dx is called the stopping power or linear energy transfer (LET) anddE/rdx is called the mass stopping power
dE/dx • Assume • Electrons are free and initially at rest • Dp is small so the trajectory of the heavy particle is unaffected • Recoiling electron does not move appreciably • We’ll calculate the impulse (change in momentum) of the electron and use this to give the energy lost by the heavy charged particle
dE/dx • Note • We only consider collisions with atomic electrons and can neglect collisions with atomic nuclei because • Except for ions on high Z targets at low energy
dE/dx • bmin (short distance collisions) • In an elastic collision between a heavy particle and an electron
Classical dE/dx • bmax (long distance collisions) • We can invoke the adiabatic principle of QM • There will be no change if the interaction time is longer than the orbital period
dE/dx • Substituting we have • This is very close to Bohr’s 1915 result • Actually Bohr calculated the energy transfer to a harmonically bound electron and found
dE/dx • Our approximation is not too bad
dE/dx • Notes on the essential ingredients • Energy loss depends only on the velocity of the particle, not its mass • At low velocity, dE/dx decreases as 1/b2 • Reaches a minimum at b=0.96 or bg=3 • At high velocities, dE/dx increases as lng2 • Called the relativistic rise • Energy loss depends on the square of the charge of the incident particle • Energy loss depends on Z of the material
Bethe-Bloch dE/dx b=p/E g=E/m
Quantum Effects • Real energy transfers are discrete • QM energy > classical energy but the transfer occurs in a few collisions • Bethe calculated probabilities that the energy transferred would cause excitation or ionization • bmin must be consistent with the uncertainty principle • One needs to use the larger of • Bethe also included spin effects
Density Effect • So far we calculated the energy loss to one electron of one atom and then performed an incoherent sum • For large g, bmax > atomic dimensions • The atoms in between will be affected by the fields and these atoms themselves can produce perturbing fields at bmax • Atoms along the field will become polarized thus shielding electrons at bmax from the full electric field of the incident particle • Density effect = induced polarization will be greater in denser mediums
Density Effect • Calculation • Fermi (1940) was first • Sternheimer Phys Rev 88 (1952) 851 gives additional gory details • Jackson contains a calculation as well • The net effect is to reduce the logarithm by a factor of g • Instead of a relativistic rise we observe a less rapid rise called the Fermi plateau • And the remaining slow rise is due to large energy transfers to a few electrons
Density Effect • The density effect is usually estimated using Sternheimer’s parameterization • See tables on next slide
Bethe-Bloch Equation • K=0.307075 MeVcm2/g • I = mean excitation energy • Tmax is the maximum kinetic energy that can be imparted to a free electron • Accurate to about 1% for pion momenta between 40 MeV/c and 6 GeV/c • At lower energies additional corrections such as the shell correction must be made
Bethe-Bloch dE/dx b=p/E g=E/m
Tmax • Tmax is the maximum energy that can be imparted to electrons • Note it is in the logarithm and is also responsible for part of the dE/dx increase as the energy increases • Tmax is given by • Sometimes a low energy approximation is used
Tmax • Alpha particles from 252Cf fission
Mean Excitation Potential I • Approximately • I/Z = 12 eV for Z < 13 • I/Z = 10 eV for Z > 13 • Constants exist for most elements and should be used if more accuracy is needed
Other Effects • Shell effect • At low energies (when v~orbital velocity of bound electrons), the atomic binding energy must be accounted for • At velocities comparable to shell velocities, the dE/dx loss is reduced • Shell corrections go as -C/Z where C=f(b) • Relatively small effect (1%) at bg=0.3 but it can be as large as 10% in the range 1-100 MeV for protons • Bremsstrahlung • For heavy charged particles, this is important only at high energies (several hundred GeV muons in iron)
Low Energy • One large effect at low energies is that the incident particle will capture an electron for some of the time thus neutralizing itself • Thus the ionization losses will decrease • Energy losses from elastic scattering with nuclei also become important (and may dominate for heavy ions)
Low Energy dE/rdx~(z/b)2Z
dE/dx Values • For low energies (< 1000 MeV) tables of stopping power are available from NIST • http://physics.nist.gov/PhysRefData • For high energies, one can use dE/dxmin as a good estimate • http://pdg.lbl.gov
dE/dx Values • How much energy does a cosmic ray muon (E>1 GeV) deposit in a plastic scintillator 1 cm thick?
