1. Physics 70010 Modern Lab1 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, –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

2. Physics 70010 Modern Lab2 Taking into account quantum-mechanical effects and using first-order perturbation theory the Bethe-Bloch equation is obtained: T max is the maximum energy transfer to a single electron:, T max is often approximated by 2m e  2  2. r e is the classical electron radius (r e = e 2 / m e c 2 = 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 e 2 / m e c 2 for r e gives eq. (2.19) of Fernow Hans Albrecht Bethe Felix Bloch

3. Physics 70010 Modern Lab3  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

4. Physics 70010 Modern Lab4 dE/dx divided by density  (approximately material independent) dE/dx for pions as computed with Bethe-Bloch equation  about proportional to n e, as n e = n a Z = N A  Z / A, -> n e ≈ N A  / 2 slope due to 1/v 2 relativistic rise due to ln  high  : dE/dx independent of  due to density effect, "Fermi plateau" From PDG, Summer 2002

5. Physics 70010 Modern Lab5 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, T max set to 2m e  2  2 and without density and shell corrections: Electrons: Heavy particles:

6. Physics 70010 Modern Lab6 For thick enough material particles will be stopped, the range can be calculated from (M = mass projectile, Z 1 = charge projectile): The Bethe-Bloch equation with T max approximated by 2m e  2  2 can be written as: f(v) can be replaced by g(E/M), as : -> The dependency of R  Z 1 2 /M on E is approximately material and projectile independent( (dE/dx)/  is ~ material independent) Two different projectiles with same energy: Range of stopping particles

7. Physics 70010 Modern Lab7 Depth x in material Averange range R Fraction of particles surviving dE/dx Bragg curve Sir William Henry Bragg Sir William Lawrence Bragg 100 % Most of the energy deposited at end of track

8. Physics 70010 Modern Lab8 Lev Davidovich Landau Fluctuations 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.

9. Physics 70010 Modern Lab9 From PDG, Summer 2002