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Electromagnetic interactions. Energy loss due to collisions An important fact: electron mass = 511 keV /c2, proton mass = 940 MeV/c2, so it is much easier to give an electron a "kick" than a nucleus, i.e. will be dominated by interactions with the electrons. Other types of e.m. interaction,

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electromagnetic interactions
Physics 70010 Modern LabElectromagnetic interactions
  • Energy loss due to collisions
    • An important fact: electron mass = 511 keV /c2, proton mass = 940 MeV/c2, so it is much easier to give an electron a "kick" than a nucleus, i.e. will be dominated by interactions with the electrons.
  • Other types of e.m. interaction,
    • bremsstrahlung and creation of electron-positron pairs by high-energy photons are sensitive to the electric field strength, so the interaction with the nucleus dominates.
  • Cerenkov/Transition radiation
    • A third category of interactions is sensitive to bulk properties of the matter, like dielectric constant. These interactions give rise to Cherenkov and transition radiation
Physics 70010 Modern Lab

Taking into account quantum-mechanical effects and using first-order

perturbation theory the Bethe-Bloch equation is obtained:

Tmax is the maximum energy transfer to a single electron:


Tmax is often approximated by 2me22. re is the classical

electron radius (re = e2 / mec2 = 2.82 x10-13 cm)

(radius of a classical distribution of the electron charge

with electrostatic self-energy equal to the electron mass).

I is the mean ionization energy.

NB: for high momentum particles

Substituting this and also e2 / mec2 for re gives eq. (2.19) of Fernow

Hans Albrecht Bethe

Felix Bloch

Physics 70010 Modern Lab
  • is the "density correction“:

It arises from the screening of remote electrons by close electrons, which results in a reduction of energy loss for higher energies (transverse electric field grows with !). The effect is largest in dense matter, i.e. in solids and liquids.

  • C is the "shell correction" :
  • Only important for low energies where the particle velocity has the same order of magnitude as the "velocity" of the atomic electrons.
  • For improved accuracy more correction factors need to be added, but the
  • particle data group claims that the accuracy in the form shown above for
  • energy loss of pions in copper for energies between 6 MeV and 6 GeV
  • about 1 %, with C set to 0.
  • Note that the Bethe-Bloch equation provides only the mean of the
  • "stopping power", but no information on fluctuations in it
Physics 70010 Modern Lab

dE/dx for pions as computed with Bethe-Bloch equation

dE/dx divided

by density 




slope due to 1/v2

high :



of 

due to





relativistic rise

due to ln 

  • about proportional to ne,

as ne = na Z = NA Z / A, -> ne ≈ NA / 2

From PDG, Summer 2002

Physics 70010 Modern Lab

Some phenomena not taken into account in the formula are :

  • Bremsstrahlung: photons produced predominantly in the electric field of the

nucleus. This is an important effect for light projectiles, i.e. in particular for

electrons and positrons

  • Generation of Cherenkov or transition radiation. Cherenkov radiation occurs when

charged particles move through a medium with a velocity larger than the velocity

of light in that medium. Transition radiation is generated when a highly relativistic

particle passes the boundary of two media with different dielectric constants. The

energy loss is small compared to the energy loss due to exciation and ionization

  • For electrons and positrons the Moller resp. Babha cross sections should be

used in the calculation of dE/dx, this leads to small corrections. Fernow

quotes, for  -> 1, Tmax set to 2me22 and without density and shell corrections:


Heavy particles:

range of stopping particles
Physics 70010 Modern LabRange of stopping particles

For thick enough material particles will be stopped, the range can

be calculated from (M = mass projectile, Z1 = charge projectile):

The Bethe-Bloch equation with Tmax approximated by 2me22 can be

written as:

f(v) can be replaced by g(E/M), as :

-> The dependency of R  Z12/M on E is

approximately material and projectile

independent( (dE/dx)/ is ~ material


Two different projectiles

with same energy:

Physics 70010 Modern Lab

Most of the energy

deposited at end of track

Fraction of particles


100 %

Sir William

Henry Bragg

Sir William

Lawrence Bragg





range R

Depth x in material

fluctuations in energy loss
Physics 70010 Modern LabFluctuations in energy loss
  • The energy transfer for each collision is determined by a probability distribution.
  • The collision process itself is also a process determined by a probability distribution.
  • The number of collisions per unit length of material is determined by a Gaussian distribution
  • the energy loss distribution usually is referred to as a "Landau" distribution. This is a distribution with a long tail for high values of the energy loss. The tail is caused by collisions with a high energy transfer.

Lev Davidovich Landau

Physics 70010 Modern Lab

From PDG, Summer 2002