26. passage of particles through matter - Particle Data Group

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28 26. Passage of particles through matter 0.10 0.08 1 TeV muons on 3 m Fe Median 987 GeV/c dN/dp [1/(GeV/c)] 0.06 0.04 0.02 Mean 977 GeV/c FWHM 9 GeV/c 0.00 950 960 970 980 990 1000 Final momentum p [GeV/c] Figure 26.22: The momentum distribution of 1 TeV/c muons after traversing 3 m of iron as calculated withthe MARS14 Monte Carlo code [64] by S.I. Striganov [1]. 26.7. Čerenkov and transition radiation [5,69,70] A charged particle radiates if its velocity is greater than the local phase velocity of light (Čerenkov radiation) or if it crosses suddenly from one medium to another with different optical properties (transition radiation). Neither process is important for energy loss, but both are used in high-energy physics detectors. Čerenkov Radiation. The half-angle θ c of the Čerenkov cone for a particle with velocity βc in a medium with index of refraction n is θ c = arccos(1/nβ) ≈ √ 2(1 − 1/nβ) for small θ c , e.g. in gases. (26.36) The threshold velocity β t is 1/n, andγ t =1/(1−βt 2)1/2 . Therefore, β t γ t =1/(2δ +δ 2 ) 1/2 , where δ = n − 1. Values of δ for various commonly used gases are given as a function of pressure and wavelength in Ref. 71. For values at atmospheric pressure, see Table 6.1. Data for other commonly used materials are given in Ref. 72. The number of photons produced per unit path length of a particle with charge ze and per unit energy interval of the photons is or, equivalently, d 2 N dEdx = αz2 ~c sin2 θ c = α2 z 2 ( ) 1 r e m e c 2 1 − β 2 n 2 (E) ≈ 370 sin 2 θ c (E) eV −1 cm −1 (z =1), (26.37) d 2 ( ) N dxdλ = 2παz2 1 λ 2 1 − β 2 n 2 (λ) . (26.38) June 18, 2002 13:57
26. Passage of particles through matter 29 The index of refraction is a function of photon energy E, as is the sensitivity of the transducer used to detect the light. For practical use, Eq. (26.37) must be multiplied by the the transducer response function and integrated over the region for which βn(E)>1. Further details are given in the discussion of Čerenkov detectors in the Detectors section (Sec. 27 of this Review). Transition radiation. The energy radiated when a particle with charge ze crosses the boundary between vacuum and a medium with plasma frequency ω p is I = αz 2 γ~ω p /3 , (26.39) where √ ~ω p = 4πN e re 3 m e c 2 /α √ = 4πN e a 3 ∞ 2 × 13.6 eV. (26.40) Here N e is the electron density in the medium, r e is the classical electron radius, and a ∞ is the Bohr radius. For styrene and similar materials, √ 4πN e a 3 ∞ ≈ 0.8, so that ~ω p ≈ 20 eV. The typical emission angle is 1/γ. The radiation spectrum is logarithmically divergent at low energies and decreases rapidly for ~ω/γ~ω p > 1. About half the energy is emitted in the range 0.1 ≤ ~ω/γ~ω p ≤ 1. For a particle with γ =10 3 , the radiated photons are in the soft x-ray range 2 to 20 keV. The γ dependence of the emitted energy thus comes from the hardening of the spectrum rather than from an increased quantum yield. For a typical radiated photon energy of γ~ω p /4, the quantum yield is N γ ≈ 1 αz 2 γ~ω / p γ~ωp 2 3 4 ≈ 2 3 αz2 ≈ 0.5% × z 2 . (26.41) More precisely, the number of photons with energy ~ω >~ω 0 is given by [5] [ ( N γ (~ω >~ω 0 )= αz2 ln γ~ω ) ] 2 p − 1 + π2 , (26.42) π ~ω 0 12 within corrections of order (~ω 0 /γ~ω p ) 2 . The number of photons above a fixed energy ~ω 0 ≪ γ~ω p thus grows as (lnγ) 2 , but the number above a fixed fraction of γ~ω p (as in the example above) is constant. For example, for ~ω >γ~ω p /10, N γ =2.519 αz 2 /π =0.59% × z 2 . The yield can be increased by using a stack of plastic foils with gaps between. However, interference can be important, and the soft x rays are readily absorbed in the foils. The first problem can be overcome by choosing thicknesses and spacings large compared to the “formation length” D = γc/ω p , which in practical situations is tens of µm. Other practical problems are discussed in Sec. 27. References: 1. D.E. Groom, N.V. Mokhov, and S.I. Striganov, “Muon stopping-power and range tables: 10 MeV–100 TeV” Atomic Data and Nuclear Data Tables 78, 183–356 (2001). June 18, 2002 13:57
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