Explanation of minimum ionizing radiation

[bcrowell]

Can anyone provide, or point me to, an explanation of minimum ionizing radiation? My understanding is that in the limit of ultrarelativistic velocities, a charged particle stopping in matter has a dE/dx that approaches some limit. Is this correct, and if so, why does it happen? Is it basically because an electron in the stopping material sees a pulse in the E and B fields, and the duration of the pulse is just $\Delta x/c$? But doesn't the intensity of the fields depend on how close you are to c? (I'm imagining a Lorentz-transformed Coulomb field.) Am I right in thinking that dE/dx would depend only on the properties of the stopping material and on the particle's charge? I'd imagine it wouldn't depend on the particle's mass, since the particle is ultrarelativistic, so its momentum is basically E/c.

Thanks in advance!

-Ben

 

[Vanadium 50]

Ionization depends on a particle's charge and velocity.

Slow particles have many chances to interact with an atom, so their ionization probability goes up. Fast particles have a lot of energy, so for the same fractional energy loss, more energy is lost. In between these two extremes lies a minimum.

 

[Bob S]

The best discussions of the Bethe Bloch energy loss equation can be found at

1) The GEANT4 CERN website:See Chapter 9 on page 157 in

http://geant4.web.cern.ch/geant4/UserDocumentation/UsersGuides/PhysicsReferenceManual/fo/PhysicsReferenceManual.pdf

2) The LBL Particle Data group website:

See Fig 27.1 and Eqn 27.3 in

http://pdg.lbl.gov/2009/reviews/rpp2009-rev-passage-particles-matter.pdf

The Bethe Bloch equation relates to "continuous energy loss" of incident charged particles (excluding electrons) interacting with atomic electrons in the stopping material.

There are two other types of interactions that are related to nuclear collisions (much smaller cross sections)

1) radiative; pair production and bremsstrahlung. Scales as 1/ (incident particle mass)² for specific incident energies.

2) inelastic nuclear interactions. Cross section ≈nuclear size.

See this page for radiation lengths, nuclear interaction lengths, etc in various materials.

http://pdg.lbl.gov/2009/reviews/rpp2009-rev-atomic-nuclear-prop.pdf

Bob S

 

[bcrowell]

Thanks, Vanadium 50 and Bob S, for the helpful posts!

So let's see if I have this right. I'm looking at the graph on p. 4 of the LBL particle group doc.

At very low energies, the velocity of the particle is comparable to the velocity of the atomic electrons. At higher energies, the atomic electrons basically feel an impulsive force from the fields of the particle. The time duration of this impulse falls off with velocity, leading to a reduction in the amount of energy transferred. The dE/dx keeps dropping with E for this reason, but at high enough energies, there is a countervailing effect that causes dE/dx to start rising again; is this because kinematically, the maximum energy transfer to a given electron is proportional to $\gamma$ in the limit of large $\gamma$? At some point in here, these two trends cancel one another, and we reach a minimum. The region around this minimum is quite flat for a long time, so people refer to anything in this range as minimum-ionizing.

At even higher energies (TeV), radiative effects like Bremstrahlung take over, and dE/dx starts rising again dramatically.

 

[Bob S]

A good derivation (the best I have seen) of the Bethe Bloch ionization energy loss equation is given in a book by David Ritson "Techniques of High Energy Physics" (Interscience 1961) pgs 4-24. Coulomb scattering off of nuclei is derived, then Coulomb scattering off of atomic electrons to derive the probability and energy distribution of the recoil electrons, which of course is the ionization energy loss. The log argument in dE/dx is ln[θ_max/θ_min], where θ_max and θ_min are the max and min deflection angles of the incident particle. For a momentum transfer p, the energy loss is p²/2m.

The final ionization energy loss equation is

dE/dx = -2π(N_ez²e⁴)/(β²mc²) [Ln(β²γ²mc²T_max) - Ln(I²)]

Ritson also discusses delta rays, density effect, ionization loss fluctuations, and range fluctuations.

The small dE/dx dependence on the sign of the charge of the incident particle is called the Barkas effect, which is based on analysis of curved positive and negative pion tracks in emulsions placed in the 184" cyclotron at Berkeley. Negative pions had a slightly longer range and correspondingly lower dE/dx. See Barkas Birnbaum Smith Phys Rev 101 778 (1956)

Bob S