1 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

2 dE/dx  You’ll see

3 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) and dE/  dx is called the mass stopping power

4 dE/dx  Assume Electrons are free and initially at rest  p 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

5 dE/dx

6

7  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

8 dE/dx  b min (short distance collisions) In an elastic collision between a heavy particle and an electron

9 Classical dE/dx  b max (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

10 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

11 dE/dx  Our approximation is not too bad

12 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/  2 Reaches a minimum at  =0.96 or  =3 At high velocities, dE/dx increases as ln  2  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

13 Bethe-Bloch dE/dx  =p/E  =E/m

14 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  b min must be consistent with the uncertainty principle One needs to use the larger of  Bethe also included spin effects

15 Density Effect  So far we calculated the energy loss to one electron of one atom and then performed an incoherent sum  For large , b max > atomic dimensions The atoms in between will be affected by the fields and these atoms themselves can produce perturbing fields at b max Atoms along the field will become polarized thus shielding electrons at b max from the full electric field of the incident particle  Density effect = induced polarization will be greater in denser mediums

16 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   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

17 Density Effect

18 Density Effect  The density effect is usually estimated using Sternheimer’s parameterization  See tables on next slide

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20 Bethe-Bloch Equation K= MeVcm 2 /g I = mean excitation energy T max 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

21 Bethe-Bloch dE/dx  =p/E  =E/m

22 T max  T max 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  T max is given by  Sometimes a low energy approximation is used

23 T max  Alpha particles from 252 Cf fission

24 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

25 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(  ) Relatively small effect (1%) at  =0.3 but it can be as large as 10% in the range MeV for protons  Bremsstrahlung For heavy charged particles, this is important only at high energies (several hundred GeV muons in iron)

26 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)

27 Low Energy dE/  dx~(z/  ) 2 Z

28 dE/dx Values  For low energies (< 1000 MeV) tables of stopping power are available from NIST  For high energies, one can use dE/dx min as a good estimate

29 dE/dx Values  How much energy does a cosmic ray muon (E>1 GeV) deposit in a plastic scintillator 1 cm thick?