Heavy charged particle passage through matter Peter H. Hansen University of Copenhagen
Content Bohrs argument The Bethe-Bloch formula The Landau distribution Penetration range Biological effects
Bohrs argument The momentum transferred to an atomic electron by the passage of a heavy charged particle is in the impulse approximation: P = Fdt = e E dt = e since the longitudinal components cancel. E dx v, Use Gauss law on a long cylinder with the impact parameter, b as radius and the projectile line-of-flight as axis: E 2πbdx = 4πze,
Bohrs argument cont d and hence P = 2ze2 bv E = 2z2 e 4 m e v 2 b 2. The energy lost to all electrons located between b and b + db in a thickness dx is E(b) = 4πz2 e 4 m e v 2 ρ e valid between some b min and b max. db b dx,
Bohrs argument cont d Integrating over impact parameters, we get de dx = 4πz2 e 4 m e v 2 ρ e ln b max. b min The maximal transferable energy in a head-on collision is 2γ 2 m e v 2, whereby 2z 2 e 4 m e v 2 b 2 min = 2γ 2 m e v 2 b min = ze 2 γm e v 2.
Bohrs argument cont d The condition for b max is more tricky. The argument is that the interaction time must be short compared with the orbital period of the bound electron: leading finally to b γv τ, de dx = 4πz2 e 4 m e v 2 ρ e ln γ2 m e v 3 τ ze 2.
The Bethe-Bloch Formula The correct quantum mechanical calculation leads to where de cm2 = (0.1535 MeV dx g )ρ Z A β 2 [ ( 2me c 2 γ 2 β 2 ) W max ln 2β 2 δ 2 C ]. Z W max = I 2 2m e c 2 β 2 γ 2 1 + 2 m 2m e c 2 β 2 γ 2 e M 1 + β 2 γ 2 + m2 e M 2 is the maximal transferable energy in a knock-on collision with an atomic electron. z 2
The mean excitation potential The quantity I in the logarithmic term is essentially the average orbital frequancy from Bohrs formula time Plancks constant. It can be parametrized as I Z = 12 + 7 ev f or Z < 13 Z I Z = 9.76 + 58.8Z 1.19 ev f or Z 13 It is tabulated for various substances by eg http://pdg.lbl.gov.
The small corrections The density effect, δ arises from the fact that the electric field of the projectile tends to polarise the atoms along its path. This effectively shields the electrons from the field far away from the projectile and thus reduces the relativistic rise of the logarithmic term at very high momenta. See R.M.Sternheimer, Phys.Rev. 88 (1952) 851 for a parametrization. The shell correction, C is important only at low velocities where the orbital electron can not be considered as stationary wrt the incident projectile. See W.R.Leo, Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag for a parametrization.
The grand picture Stopping power [MeV cm 2 /g] 100 10 1 Lindhard- Scharff Nuclear losses µ Anderson- Ziegler Bethe-Bloch µ + on Cu Radiative effects reach 1% [GeV/c] Muon momentum Radiative Radiative losses Without δ 0.001 0.01 0.1 1 10 100 1000 10 4 10 5 10 6 βγ 0.1 1 10 100 1 10 100 1 10 100 [MeV/c] Minimum ionization E µc [TeV/c]
The Landau distribution For thin absorbers (eg gas sampling detectors), the number of ionization clusters of liberated electrons is not large enough for the Central Limit Theorem to hold. Because of the possibility of a large energy transfer in a single head-on collision, the probability distribution per sample for de/dx is very asymmetric with a long high-energy tail: This is the Landau distribution (see eg Kleinknecht), given by a complex integral (see also the GEANT manual).
de/dx sampling A way to deal with the Landau tails is to obtain many measurement samples and take a truncated mean of those. An excellent example is the de/dx in the ALEPH TPC. Here, each point is a truncated mean over at least 150 measurements on a single track with the lowest 8 and the highest 40 excluded. This results in an approximately Gaussian estimator with a width around 0.07 mip s. Note that, since the Bethe-Bloch formula depends only on beta, the de/dx of a particle with a given momentum can be used for particle identification in a wide range of momenta.
ALEPHs de/dx measurement
Penetration range The distance travelled by a charged particle before it stops is not an exact function of kinetic energy. This is because head-on collisions with electrons and nuclei are of statistical nature, causing a spread in the range called straggling. For not too high kinetic energies, the average range is approximately R = 0.004[g/cm 2 ] E 1.75 kin (E in MeV) for protons. This is more or less what we expect from the β 2 behaviour of de dx at low energies.
The Bragg peak Because of the β 2 behaviour, a proton or ion reaches its maximum ionization power just before it stops. This is called the Bragg peak. This make protons or light ions much better for cancer-therapy than the X-rays and electron guns we have in Denmark, since protons can deliver most of their energy at the Bragg peak to the sick tissue without destroying the healthy tissue in front. (anti-protons would be even better).
Equivalent dose The biological damage effects in average human tissue, however, vary with the kind of radiation. For example, 1 MeV neutrons causes 20 times more damage than electrons at the same dose. The equivalent dose is weighted by such factors. The unit is Sievert with the same dimension as Gray (J/kg). The biological damages are furthermore different (by up to a factor 20) in different organs of the body (the least in the skin and bone surfaces). The chemical or enzymatic restoration of damaged cells can not keep up with a sudden high dose. A sudden dose greater than 2-3 Sv is life-threatening.
Typical doses Ordinary people get most radiation from inhalation of Radon from the underground (typically 1.5 msv/yr). Stewardesses get more from cosmic radiation. Physicists working at nuclear labs in general get negligible doses. However, a dosimeter monitoring how much you get is advisable, if you work with radiation on a daily basis. Most radioactive sources do not produce radiation with energy high enough to penetrate the skin. Therefore it is most important to avoid inhaling or eating radioactive substances, which brings the radiation directly to the cells. This may also be why you should not smoke.
Shielding A high Z material (e.g. Pb) is best for shielding against γ-rays and positrons. For electrons low-z polystyrene or lucite work best in order to avoid bremsstrahlung (or a sandwich of low-z/high-z shield). For neutrons use paraffin, water or other hydrogeneous material. For high energy charged hadrons use the cheapest material (because you need a lot) with the highest density (i.e. Fe).