The Interaction of Radiation with Matter

Abstract

This document is part of Part 1 'Principles and Methods' of Subvolume B 'Detectors for Particles and Radiation' of Volume 21 'Elementary Particles' of Landolt-Börnstein - Group I 'Elementary Particles, Nuclei and Atoms'. It contains the Chapter '2 The Interaction of Radiation with Matter' with the content:

2 The Interaction of Radiation with Matter

2.1 Introduction

2.1.1 General concepts

2.1.2 Types of collisions

2.1.3 Observable effects of radiations

2.1.4 Stopping power dE/dx

2.2 Historical background

2.3 Description of the most frequent interactions of single fast charged particles

2.3.1 Narrow beams and straggling of heavy charged particles

2.3.2 Narrow beams of low energy electrons

2.3.3 Relation between track length and energy loss

2.3.4 Nuclear interactions

2.4 Photon interactions

2.4.1 Gases

2.4.2 Solids

2.4.3 Data for DOS

2.5 Interaction of heavy charged particles with matter

2.5.1 Inelastic scattering, excitation and ionization of atoms or condensed state matter

2.5.1.1 Bethe-Fano (B-F) method

2.5.1.2 Relativistic extension of B-F method

2.5.1.3 Fermi-virtual-photon (FVP) cross section

2.5.2 Integral quantities: moments

2.5.2.1 Comparison of moments: Si, Ne, P10

2.6 Electron collisions and bremsstrahlung

2.6.1 Electronic collisions

2.6.2 Bremsstrahlung BMS

2.7 Energy losses along tracks: multiple collisions and spectra

2.7.1 Monte Carlo method

2.7.2 Analytic methods

2.7.2.1 Convolutions

2.7.2.2 Laplace transforms

2.7.3 Analytic methods for thick absorbers

2.8 Evaluations and properties of straggling functions

2.8.1 Very thin absorbers

2.8.2 Thin absorbers

2.8.3 Thick absorbers

2.8.4 Comparisons of spectra

2.9 Energy deposition

2.9.1 Ionization

2.9.2 Delta rays

2.9.3 Auger electrons and photons

2.9.4 Cherenkov radiation

2.9.5 Transition radiation

2.9.6 Ion beams

2.10 Particle ID

2.11 Discussion and recommendations

2.12 Appendix

Appendices

2.12 Appendix

A Stopping power and track length

A.1 Stopping power M ₁

Bethe [18] derived the method described in Sects. 2.5.1.1 and 2.5.1.2. for atoms and calculated the first moment M ₁(v), Eq. (2.25), by using sum rules and defining the logarithmic mean excitation energyI. The nonrelativistic result is [53]

$$\frac{dE} {dx} \equiv {M}_{1}(v) = \frac{kN} {{\beta }^{2}} Z\ln \frac{{E}_{M} \cdot 2m{c}^{2}{\beta }^{2}} {{I}^{2}}$$

(2.41)

with I given by

$$Z\ln I = \int \nolimits \nolimits f(E,0)\ln E\ dE$$

(2.42)

It must be kept in mind that dE / dx is an average of the energy losses per unit segment length for single particles, Δ / x, as shown in Fig. 2.1. From Fig. 2.22-2.25 it is evident that dE / dx has a limited use for thin absorbers.

The use of the sum rules introduces an error which can be corrected with the shell corrections [53]. Other effects are treated with the Barkas and Bloch corrections etc. [101, 102, 103, 104]. The density effect, Cherenkov radiation etc. are implicit in Eq. (2.21), Sect. 2.9.4 [19].

In practice there will be little need to calculate M ₁(v) for the uses described in this book. Tables for protons and alpha particles for many absorbers are available [20, 65], also for muons [9].

A concept often used in particle physics is “minimum ionizing particles”. This usually refers to the minimum value of M ₁(β γ). For Si this minimum is at β γ ∼ 3. 5, but for Σ _t = M ₀ it is at β γ ∼ 16, and for Δ _p / x it is at β γ ∼ 4. 5 (Table 2.1), for gases near β γ ∼ 3. 5, Table 2.2.

Stopping power for electrons and positrons need a different analytic expression than heavy ions. Equations and tables are given in ICRU-37 [51], also see Sect. 2.6.

A.2 Total collision cross section M ₀

Analytic calculations for M ₀ with the Bethe-Fano equations given in Sects. 2.5.1.1 and 2.5.1.2 have been made [27, 52, 105, 106, 107]. Because of the complexity of the optical data, no simple expression similar to Eq. (2.41) is available. The analytic calculations differ considerably from the FVP calculations, Sect. 2.5.2.1. Calculations of M ₀ (also called inverse mean free path) have been made by several groups [20] (ORNL, Barcelona) with computer-analytic methods.

A.3 Restricted energy loss

The concept of restricted energy loss was introduced in the consideration of biological effects of radiations. Conceptually it is an energy deposition rather than an energy loss and therefore must be defined for a limited volume around a particle track. The customary definition, Eq. (27.9) in [23]

$${M}_{1}(v) \approx \frac{k} {{\beta }^{2}}{z}^{2}Z\ \frac{1} {2}\ln \frac{{E}_{cut} \cdot 2m{c}^{2}{\beta }^{2}{\gamma }^{2}} {{I}^{2}}$$

(2.43)

is valid for a cylindrical volume around the track with a radius given by the practical range of electrons with energy E _cut , see Table 2.6, Sect. 2.9.2. Because of the small number of energetic delta rays, Fig. 2.15, a MC calculation should be used for practical applications [54].

A.4 Ranges

Mean track lengths t (also called path lengths) for particles with initial kinetic energy T _u and final mean energy T _l can be calculated with the “CSDA approximation”

$$t ={ \int \nolimits \nolimits }_{{T}_{l}}^{{T}_{u} } \frac{dT'} {dT'/dx}$$

(2.44)

where dT / dx represents M ₁(v) of Eq. (2.41). Customarily if T _l = 0, t is called the particle range. Corrections for multiple scattering must be considered [98, 108], and nuclear interactions will reduce the particle fluence, see Sect. 2.3.4. Range data for muons can be found in [9], for pions, protons and alpha particles in [53, 65].

If T _l is finite, there will be an energy spectrum [77, 109] which does not correlate well with the track-straggling, see Fig. 2.4, Sect. 2.3.3.

Fig. 2.4

The dashed line gives the pdf of energy losses Δ in Ne for a segment length x = 9 cm for 500,000 particles as described in Fig. 2.1, Δ = (sc/10) keV. The peak value is at about 8.5 keV, the FWHM is 4.6 keV. The solid line gives the pdf of the segment length x = sc mm of particles which lost Δ ≤ 9 keV, as described in Fig. 2.3. The peak is at 86 mm. The two functions have opposite asymmetry, and they have no correlation. It will be difficult to compare the functions by using “stopping power”. It may be possible to measure these spectra with a GEM TPC [4, 23]. The quotient of the peaks is 1 keV/cm, much less than M ₁ = 1. 6 keV/cm.

B Rutherford type cross sections

B.1 Rutherford cross section and moments

The Rutherford cross section is given by Eq. (2.8). In order to calculate the moments, Eq. (2.25), integration limits must be defined. The upper limit must be E _M given in Eq. (2.2), a lower limit E _m , defined e.g. in Appendix B.4, is used.

$${M}_{0}(v) = \frac{k'} {{\beta }^{2}}\left [( \frac{1} {{E}_{m}} - \frac{1} {{E}_{M}}) - \frac{{\beta }^{2}} {{E}_{M}}\ln \frac{{E}_{M}} {{E}_{m}} \right ] \sim \frac{k'} {{\beta }^{2}} \frac{1} {{E}_{m}}$$

(2.45)

$${M}_{1}(v) = \frac{k'} {{\beta }^{2}}\left [\ln \frac{{E}_{M}} {{E}_{m}} - \frac{{\beta }^{2}} {{E}_{M}}({E}_{M} - {E}_{m})\right ] \sim \frac{k'} {{\beta }^{2}}\left [\ln \frac{{E}_{M}} {{E}_{m}} - {\beta }^{2}\right ]\ =\ \frac{k'} {{\beta }^{2}}\left [\ln \frac{{E}_{M}^{2}} {{I}^{2}} - {\beta }^{2}\right ]\ $$

(2.46)

From Figs. 2.9-2.11 it is clear that no plausible values of E _m can be defined from knowledge about atomic structure.

$${M}_{2}(v) = \frac{k'} {{\beta }^{2}}\left [({E}_{M} - {E}_{m}) - \frac{{\beta }^{2}} {2{E}_{M}}({E}_{M}^{2} - {E}_{ m}^{2})\right ] \sim k' \cdot 2m{c}^{2}\ \frac{1 - {\beta }^{2}/2} {1 - {\beta }^{2}}$$

(2.47)

where k′ = kN _A Z / A = 0. 1535 Z / A MeV cm²/g for an absorber Z, A.

The Landau [2] method to deal with E _m is given in Sect. B.4.

B.2 Rutherford cross section with shell structure

Inokuti [52] suggested an approximation by modifying the Rutherford cross section for each electron shell s with ν _s electrons as follows:

$${\sigma }_{\mathrm{s}}(E) = {\sigma }_{R}(E) \cdot {\nu }_{s} \cdot (1\ +\ \frac{4U} {3E})$$

(2.48)

to be used for E ≫ J _s where J _s is the ionization energy of shell s and with a parameter U similar to the average kinetic energy of the electrons in the shell. An application of this approach is described in Sect. III.B of [33]. This structure appears in Fig. 2.9 at least for the K-shell. It is likely that the use of this approximation would improve the FVP method shown in Fig. 2.8 and Eq. (2.24), see ref. [34].

B.3 Modified Rutherford cross section

In the “ALICE Technical Design Report of the Time Projection Chamber” [62], a modified Rutherford cross section was used, “AliRoot MC”

$${\sigma }_{\mathrm{A}}(E;v) = \frac{\kappa (v)} {{\beta }^{2}{E}^{2.2}}$$

(2.49)

with the restriction E _I < E < E _u, where E _I is the ionization energy of the gas and E _u is chosen arbitrarily to be 10 keV. κ(v) is a parameter chosen to give the value Σ_t(β γ = 3. 6) = 15 collisions/cm as suggested in [62, Fig. 7.1]. This cross section is shown by the dotted line in Fig. 2.11 and κ(v) / E ^{2. 2} appears to be a reasonable approximation to the FVP σ(E; v). Note that the Rutherford cross section differs by its slope. The exact choice of E _u is not critical as the reader may find by calculating Σ_t;A with Eq. (2.45). Further details are described in [54, 110].

To get a better approximation for Σ_t(v) for the scheme proposed in [62] the “Bethe-Bloch curve” was used to define the function Σ_t;A(v). This function was obtained from measurements of particle tracks in P10 (mainly by Lehraus et al., [111]). It is approximated by

$$f(\beta \gamma )\ =\ \frac{{P}_{1}} {{\beta }^{{P}_{4}}} \ \{{P}_{2} - {\beta }^{{P}_{4} } -\ln [{P}_{3} + \frac{1} {{(\beta \gamma )}^{{P}_{5}}} ]\}$$

(2.50)

and is shown as L[P10] in Fig. 2.16. Clearly this function also differs substantially from Σ_t;F(v). In addition it differs conceptually from Σ_t(v). The reason for the difference is that Eq. (2.50) was determined from experimental measurements of truncated mean values < C > for tracks. It also differs from the Bethe-Fano function M ₁(β γ) = dE / dx(β γ) shown in Fig. 2.16.

Monte Carlo calculations of straggling functions with the model of ref. [62] are compared to the analytic calculation with FVP σ(E) in Fig. 2.27. The differences seen are explained by the differences in the σ(E), Fig. 2.11. Predominant is the excess of σ(E) over σ _A(E) for 30 < E(eV ) < 300 resulting in the shift to greater Δ of the FVP function.

Fig. 2.27

Straggling functions for particles with β γ = 3. 6 traversing Ne segments with x = 7. 5 mm. Solid line: analytic FVP calculation, Σ_t = 13. 3 collisions/cm. × Monte Carlo calculation with ALICE TPC algorithm, with Σ_t = 15/cm. Even though the Σ_t is somewhat larger, the MC calculation has a smaller Δ _p . A Monte Carlo calculation with FVP σ(E) gives the same function as the analytic calculation [54]. The K-shell energy losses above Δ = 1. 3 keV do not appear in the MC calculation.

We see that the AliRoot [62] Monte Carlo calculation does not produce accurate straggling functions. Its attractiveness is that the functions used are analytic.

A modified approach which is just as simple as the current method but will produce more accurate straggling functions consists of

• replacing in Fig. 2.16 the function L[P10] by Σ_t[Ne]

• replacing in Fig. 2.14 the function given by the dotted line (from Eq. (2.49) by the FVP functions. A single tabulated function might be sufficient [54, 112].

B.4 Landau-Vavilov-Fano Laplace transform method

The straggling functions calculated by Landau and Vavilov [2, 3] were based on two approximations: the use of the Rutherford spectrum, Eq. (2.8), for the collision cross section and the requirement that the stopping power, Eq. (2.46), (written as α, Eq. 4 in [2]) give the Bethe stopping power, Eq. (2.41). In order to achieve this, E _m was chosen as ³⁴

$${E}_{m} = \frac{{I}^{2}} {{E}_{M}}$$

(2.51)

The value for M ₀ then is $M_0^\prime = \frac{k^\prime}{\beta^2} \cdot \frac{I^2}{E_M}$. It is larger than the Bethe-Fano value.

Example: Ar, β γ = 3. 6 β ² = 0. 93, I = 190 eV, ρ = 0. 0016 g/cm³, k′ / β ² = 120 eV/cm. Then E _m = 0. 0027 eV, M ₀ ′ = 44000 col/cm and M ₁ ′ = 2. 5 keV/cm. Clearly M ₀ ′ is much greater than given in Table 2.2, while M ₁ is quite close. This is plausible from a look at Fig. 2.10. Since the convolution method, Eq. (2.32), is equivalent to the Laplace transform method [6], the relative variance of the Poisson distribution, m _c ^{− 1 / 2}, as well as that of the convolutions, Fig. 2.18, will be much smaller for the Rutherford cross section, and the Landau functions are narrower, as seen in Fig. 2.24.

Functions calculated with modifications taking atomic binding into account [19, 113, 114, 115] are too wide, Fig. 2.14 in [33]. The calculation of the correction term D in [115] ³⁵ is incorrect. The correct term is given by Fano (p. 42 in [19]), [33]. If account is taken of the limit on E in Eq. (2.34) (E must be smaller than Δ in the calculation of δ ₂ [33]) the Vavilov-Fano functions are fair approximations for thin absorbers, Figs. 11-13 in ref. [33].

As far as calculation time is concerned it is similar for the convolution calculation. Because the mean energy loss ξ is used as a parameter this method has the advantage that it will be quite accurate for thicker segments. Because of the change in particle speed v, the upper limit of validity of all thin layer methods is reached if the mean energy loss exceeds 5 or 10%, see Fig. 2 of [47].