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Fast charged particles moving through matter interact with the electrons of atoms in the material. The interaction excites or ionizes the atoms. This leads to an energy loss of the traveling particle. The Bethe formula describes〔H. Bethe und J. Ashkin in "Experimental Nuclear Physics, ed. E. Segré, J. Wiley, New York, 1953, p. 253〕 the mean energy loss per distance travelled of swift charged particles (protons, alpha particles, atomic ions, but not electrons〔For electrons, the energy loss is slightly different due to their small mass and their indistinguishability, and since they suffer much larger losses by Bremsstrahlung〕) traversing matter (or alternatively the stopping power of the material). The non-relativistic version was found by Hans Bethe in 1930; the relativistic version (shown below) was found by him in 1932.〔Sigmund, Peter ''Particle Penetration and Radiation Effects. Springer Series in Solid State Sciences, 151.'' Berlin Heidelberg: Springer-Verlag. ISBN 3-540-31713-9 (2006)〕 The most probable energy loss differs from the mean energy loss and is described by the Landau-Vavilov distribution.〔H. Bichsel, Rev. Mod. Phys. 60, 663 (1988)〕 The Bethe formula is sometimes called "Bethe-Bloch formula", but this is misleading (see below). ==The formula== For a particle with speed ''v'', charge ''z'' (in multiples of the electron charge), and energy ''E'', traveling a distance ''x'' into a target of electron number density ''n'' and mean excitation potential ''I'', the relativistic version of the formula reads, in SI units:〔 where ''c'' is the speed of light and ''ε''0 the vacuum permittivity, , ''e'' and ''me'' the electron charge and rest mass respectively. Here, the electron density of the material can be calculated by : where ''ρ'' is the density of the material, ''Z'' its atomic number, ''A'' its relative atomic mass, ''NA'' the Avogadro number and ''Mu'' the Molar mass constant. In the figure to the right, the small circles are experimental results obtained from measurements of various authors, while the red curve is Bethe's formula.〔(【引用サイトリンク】url=http://www.exphys.jku.at/Stopping/ )〕 Evidently, Bethe's theory agrees very well with experiment at high energy. The agreement is even better when corrections are applied (see below). For low energies, i.e., for small velocities of the particle ''β'' << 1, the Bethe formula reduces to This can be seen by first replacing ''βc'' by ''v'' in eq. (1) and then neglecting ''β''2 because of its small size. At low energy, the energy loss according to the Bethe formula therefore decreases approximately as ''v''−2 with increasing energy. It reaches a minimum for approximately ''E'' = 3''Mc''2, where ''M'' is the mass of the particle (for protons, this would be about at 3000 MeV). For highly relativistic cases ''β'' ≈ 1, the energy loss increases again, logarithmically due to the transversal component of the electric field. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bethe formula」の詳細全文を読む スポンサード リンク
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