Bethe—Bloch equation

The final nuclear detector makes possible a separation of isobars based upon the principle that the range and rate of energy loss for particles of a given energy is atomic number dependent. Ions such as 14C and 14N have ranges in solids or gases that differ by over 20 percent at energies of about 14 MeV. The basis for this separation is the Bethe-Bloch equation [26,27], which can be simplified to read ... 

The mean rate of energy loss, i.e., the stopping power is given by the Bethe-Bloch equation... 

The Bethe-Bloch equation reproduces the experimental data in the energy region from a few MeV to a few GeV with the precision of about 1%. At lower energies atomic shell effects, at higher energies radiation losses come into play. 

Stopping power vs. relative momentum, py = p/Mc, for muons in copper. The solid curve indicates the total stopping power, the dash-dotted and dashed lines the Bethe-Bloch equation with and without density effect correction. The vertical bands separate the validity regions of various approximations indicated in the figure. The dotted line denoted with p. indicates the Barkas effect. In the Bethe-Bloch region the stopping power scales with the particle mass and Z/A of the medium... 

One can deduce via integration of the Bethe-Bloch equation the R range of particles losing energy in matter via ionization only. As the stopping power scales with momentum, R/M will... 

Stopping power due to ionization according to the Bethe-Bloch equation vs. relative momentum py = pIMc of heavy charged particles in liquid hydrogen, gaseous helium, carbon, aluminum, iron, tin and lead. Momentum scales are shown below the plot for muons, pions and protons and an energy scale for protons above... 

At low energies electron shell effects should also be taken into account by using the C/Z shell correction, see (Groom etal. 2001) for details. The Bethe-Bloch equation (O Eq. (8.1)) is based on first-order Born approximation. Higher order effects, important at low energies only, can be included by adding a term z L2(P ) in the square brackets of Eq. (8.1) where L2(P ) is an empirical function of particle speed. 

Stopping power due to ionization, according to the Bethe-Bloch equation, vs. kinetic energy of heavy charged particles in copper... 

The Bethe-Bloch equation has to be modified for this case, and it will be valid both for electrons and positrons. The maximal energy transfer will become Tnax = TJ2 and the Bethe-Bloch formula becomes (Leo 1987)... 

As it stands, ALbiocB. equation (16), when added to the Bethe formula, extends its range of validity into the classical regime. Alternatively one may introduee an inverse-Bloch correetion [10,11]... 

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