on MUON STOPPING POWER AND RANGE TABLES 10 MeV–100 TeV

Abstract The mean stopping power for high-energy muons in matter can be described by 〈− dE / dx 〉= a ( E )+ b ( E ) E , where a ( E ) is the electronic stopping power and b ( E ) is the energy-scaled contribution from radiative processes—bremsstrahlung, pair production, and photonuclear interactions. a ( E ) and b ( E ) are both slowly varying functions of the muon energy E where radiative effects are important. Tables of these stopping power contributions and continuous-slowing-down-approximation ranges (which neglect multiple scattering and range straggling) are given for a selection of elements, compounds, mixtures, and biological materials for incident kinetic energies in the range of 10 MeV to 100 TeV. Tables of the contributions to b ( E ) are given for the same materials.


Introduction
The mean stopping power for high-energy muons (or other heavy charged particles 1 ) in a material can be described by [1] − where E is the total energy, a(E) is the electronic stopping power, and b(E) is due to radiative processes-bremsstrahlung, pair production, and photonuclear interactions: The notation is convenient because a(E) and b(E) are slowly varying functions of E at the high energies where radiative contributions are important. b(E)E is less than 1% of a(E) for E < ∼ 100 GeV for most materials.
The "continuous-slowing-down-approximation" (CSDA) range is obtained from the integral where E 0 is sufficiently small that the result is insensitive to its exact value. At very high energies, where a and b are (essentially) constant, where E µc = a/b is the muon critical energy. The muon critical energy can be defined more precisely as the energy at which electronic losses and radiative losses are equal, in analogy to one of the ways of defining the critical energy for electrons. It is obtained by finding E µc such that a (E µc ) = E µc b (E µc ) .
The CSDA range is of limited usefulness, particularly at higher energies, because of the effect of fluctuations. (Fluctuations in radiative losses are discussed briefly in Section 4.6.) For example, the cosmic ray muon intensity falls very rapidly with energy, so that the flux observed deep underground is quite different from that to be expected from Eq. (3). We nonetheless calculate the CSDA range given by Eq. (3) as an indicator of actual muon range.
The important and well-studied subjects of stopping power fluctuations and range straggling in electronic energy loss [2,3] are not treated, even though they are much more serious for muons than for heavier particles: The fractional range straggling ( variance(range)/range) scales as 1/M for particles with the same velocity, and hence is three times larger for a 100 MeV muon than for a 900 MeV proton. In copper the fractional straggling varies from 4% at 10 MeV, through a minimum of 2.8% at 300 MeV, then rising through 5.7% at 10 GeV. Above ∼ 100 GeV straggling due to fluctuations in bremsstrahlung losses begins to dominate.
Multiple scattering is also neglected, but with more justification. One measure of multiple scattering is provided by the "detour factor" [3], the ratio of the average penetration depth to the average path length for a stopping particle. The detour factor is 0.98 in the worst case (uranium at our lowest energy). This ratio increases rapidly toward unity as the energy is increased or if the atomic weight of the absorber is decreased. Table 1: Definitions of most of the variables used in this report. The electronic charge e and the kinematic variables β = v/c and γ = 1/ √ 1 − β 2 have their usual definitions. Constants are from CODATA Recommended Values of the Fundamental Physical Constants: 1998 [7].
Parenthetical numbers after the values give the 1-standard deviation uncertainties in the last digits. In Sect. 4 the convention c = 1 is used.

Symbol
Definition Units or Value Charge of incident particle Z Atomic number of medium A Atomic mass of medium g mol −1 (Occasionally: atomic mass number) dimensionless K/A 4πN A r 2 e m e c 2 /A 0.307 075 MeV g −1 cm 2 for A = 1 g mol −1 I Mean excitation energy eV (Nota bene! ) δ Density effect correction to electronic energy loss hω p Plasma energy 4πN e r 3 e m e c 2 /α 28.816 ρ Z/A eV for ρ in g cm −3 N e Electron density w j Fraction by weight of the jth element in a compound or mixture ( w i = 1) n j number of the jth kind of atoms in a compound or mixture E µc Muon critical energy GeV ν Fractional energy transfer in an incident particle interaction ε νE, the energy transfer in a single interaction Tables of muon energy loss from a 1985 CERN internal report by Lohmann, Kopp, and Voss [4] have become the de facto world standard. This careful work serves as the benchmark for the present effort. Later theoretical work enables us to improve the calculations for low-Z elements (2 ≤ Z ≤ 10) and to make minor improvements elsewhere.
It is our intention to make this report sufficiently self-contained that the interested user can replicate our calculations, even though this results in our giving often-tedious detail. The necessary constants for electronic loss calculations and tables of b(E) for elements, for the mean radiative loss calculations, are also available as ASCII files at http://pdg.lbl.gov/AtomicNuclearProperties. These tables are more extensive than the subset of data actually presented in this paper.
There is one serious dilemma: We believe that the density effect corrections via the careful parameterizations of Sternheimer et al. [5] are more dependable than those calculated via their general algorithm [6]. But, as will be discussed in Section 3.2, better values for mean excitation energies are now available for a variety of materials. The changes are sometimes as great as 10%. Over much of our energy region of interest (for βγ > ∼ 1000) however, the density effect has "replaced" the mean excitation energy by the plasma energy, so that improvements in the mean excitation energy have no effect on the stopping power. We therefore continue to use sometimes-obsolete LBNL-44742 excitation values. How this affects our results will be discussed in Section 3.2. On the other hand, corrections to the densities used by Sternheimer et al. [5] are easily accommodated if the changes are small; this is done in several cases.
We present tables of stopping power and mean range for muons from kinetic energy T = 10 MeV to 100 TeV for most elements and a variety compounds and mixtures. Tables of b(E) are given for the same materials. In the case of elemental gases, tables are also given for the liquid state.
The symbols and constants used in this report are explained in Table 1.

Overview
The behavior of stopping power (= −dE/dx ) in copper over twelve decades of muon kinetic energy is shown by the solid curves in Fig. 1. Data below the breaks in the curves are from ICRU 49 [3], while data above the breaks are from our present calculations. Approximate boundaries between regions described by different theories or phenomenologies are indicated by the shaded vertical bands. While our main interest is at higher energies, some understanding of the behavior at lower energies is useful, in particular for starting range integrals. For β < α, below the first grey band in Fig. 1, the muon velocity is small compared with that of the valence electrons in the absorber. Following the work of Fermi and Teller [9], Lindhard and collaborators have constructed a successful semi-phenomenological model to describe ionizing energy losses in this regime, approximating the electronic structure of solids by a Fermi electron gas distribution [10]. The stopping power is found to be proportional to the projectile velocity. This region is marked by the dashed curve with the dotted extension in Fig. 1. However, below β ≈ 0.001-0.01 nonionizing energy losses via nuclear recoil become increasingly significant [3], finally dominating energy loss at very low energies.
Above β ≈ 0.05 or 0.1 (the second grey band in Fig. 1) one may make the opposite approximation, neglecting electronic motion within atoms. There is no satisfactory theory for the intermediate region, α < β < 0.1 (but see Ref. 11). There is, however, a rich experimental literature, which Andersen and Ziegler have used to construct phenomenological fits bridging the regions in which there is adequate theoretical understanding [12]. This is the interval between the grey bands shown in Fig. 1.
Electronic (ionization + excitation) losses in the high-velocity region are well described by Bethe's theory based on a first-order Born approximation [13], to which are added a number of corrections for the low-energy region and to account for the polarization of the medium at high energies. The curve falls to a broad minimum whose position for solid absorbers decreases from βγ = 3.5 to 3.0 as Z goes from 7 to 100. The mean electronic loss at the minimum value as a function of Z is shown in Fig. 2. The rise in Fig. 1 with further increases of the projectile energy (labeled "without δ" in Fig. 1) is less marked when the polarization effects are taken into account (dash-dotted curve).
Electronic losses at very high energies are somewhat modified by bremsstrahlung from the atomic electrons [14] (see Section 3.5) and other effects, such as form factor corrections [15]. These are of decreased importance because radiative energy losses begin to be significant above a few hundreds of GeV for even intermediate-Z absorbers. For example, radiative losses in copper dominate above E µc = 315 GeV. The radiative contribution, and hence the entire energy loss rate, increases nearly linearly with energy above 1 TeV or so.  . Solid curves indicate the total stopping power. Data below the break at T ≈ 0.5 MeV are scaled by the appropriate mass ratios from the π − and p tables in ICRU 49 [3], and data at higher energies are from the present calculations. Vertical bands indicate boundaries between different theoretical approximations or dominant physical processes. The short dotted lines labeled "µ − " illustrate the "Barkas effect" [8]. "Nuclear losses" indicates non-ionizing nuclear recoil energy losses, which are negligible here. LBNL-44742

Electronic energy losses of high-energy heavy particles
The physics formulae needed to describe the average electronic energy losses of a high-energy (β αZ) massive ( = electron) charged particle as it passes through matter have been reviewed elsewhere [3,[16][17][18][19][20][21][22][23][24]. ICRU 49 is particularly useful, although it is limited to protons and alpha particles (except for a short π − table) and to proton energies less than 10 GeV, corresponding to E < 1.1 GeV for muons. In this energy region nuclear recoil contributes negligibly to energy loss, and radiative losses, which typically become important above tens of GeV, and radiative losses can be added as an independent contribution.
For the moment, we leave open the possibilities that the charge is ze and that the particle might be something other than a muon. We briefly review the subject here in order to emphasize high-energy behavior.

Major contributions
The electronic stopping power 2 is calculated by summing the contributions of all possible inelastic scatterings. These are normally from lower to higher (bound or unbound) electronic energy states, so the particle loses a small amount of energy in each scattering. The kinetic energy of the scattered electron is Q.
In his derivation of the stopping power, Bethe [25] introduced the concept of "generalized oscillator strength" which is closely related to the inelastic-scattering form-factor [20].
1. Low-Q region. Here the reciprocal of the 3-momentum transfer (roughly an impact parameter) is large compared with atomic dimensions. The scattered electrons have kinetic energies up to some cutoff Q 1 , typically 0.01-0.1 eV [17]. For this region, Bethe approximated the generalized oscillator strength by the dipole oscillator strength f (ε) which is the generalized oscillator strength f (ε, q) for zero momentum transfer q (ε is the energy loss in a single collision). f (ε) is closely related to the optical absorption coefficient. He derived the following contribution to S: Here ln I = f (ε) ln ε dε. The denominator I 2 /2m e β 2 c 2 in the first (logarithmic) term is the effective lower cutoff on the integral over dQ/Q. This term comes from "longitudinal excitations" (the ordinary Coulomb potential), and the next two terms from transverse excitations. The low-Q region is associated with large impact parameters and hence with long distances. Polarization of the medium can seriously reduce this contribution, particularly at high energies where the transverse extension of the incident particle's electric field becomes substantial. The correction is usually made by subtracting a density-effect term δ, inside the square brackets of Eq. (6). This important correction is discussed in Sect. 3.4. 2. Intermediate-and high-Q regions. In an intermediate region atomic excitation energies are not necessarily small compared with Q, but transverse excitations can be neglected. At higher energies Q can be equated to the energy given to the electron, neglecting its binding energy. When the integration of the energy-weighted cross sections is carried out from Q 1 to some upper limit Q upper , one obtains Here Q max is the kinematic maximum possible electron recoil kinetic energy, given by Q upper is normally equal to Q max (as will be the case after the conclusion of this section), and cannot exceed Q max . The more general form given in Eq. (7) is useful in considering restricted energy loss, which is of relevance in considering the energy actually deposited in a thin absorber. At high energies (such that Q upper /Q max 1) the first term in the square brackets dominates. If Q upper is restricted to some maximum value, e.g. 0.5 MeV, then S high is essentially constant for Q max > Q upper . If Q upper = Q max the high-Q region stopping power rises with energy as ln Q max . In other words, the increase of S high with energy is associated with the production of high-energy recoil electrons, or δ-rays. A very small projectile mass dependence of the electronic stopping power is introduced by Q max , which otherwise depends only on projectile velocity. In Fano's discussion the low-energy approximation Q max ≈ 2m e c 2 β 2 γ 2 = 2m e p 2 /M 2 is implicit. Accordingly, Eq. (7) is more closely related to Rossi's form (see his Eqns. 2.3.6 and 2.5.4). This low-energy approximation is made in many papers of the Bevatron era, but is in error by a factor of two for a muon with T = 10.8 GeV. Note that Q max → E at very high energies.

Mean excitation energy
"The determination of the mean excitation energy is the principal non-trivial task in the evaluation of the Bethe stopping-power formula" [26]. Recommended values have varied substantially with time. Estimates based on experimental stopping-power measurements for protons, deuterons, and alpha particles and on oscillator-strength distributions and dielectric-response functions were given in ICRU 37 [27]. These were retained in ICRU 49 [3], where a useful comparison with other results is given, and they are used in the EGS4 [28] electron/photon transport code. These values (scaled by 1/Z) are shown in Fig. 3. The error estimates are from Table 2 in Ref. 26. As can be seen, I/Z 10 ± 1 eV for elements heavier than sulphur.
The figure also shows Bichsel's more recent determination of I for selected heavy elements [29]. He estimates uncertainties from 1.5% to 5%; the 5% errors are shown. The change from the ICRU 37 values is less than 7% in all of the 19 cases except for samarium (7.5%), tungsten (7.5%), bismuth (9.3%) and thorium (9.5%). In addition, the mean excitation energy for liquid water has been more recently determined to be 79.7 ± 0.05 eV [31], significantly higher than the ICRU 37 value, 75.0 eV. This reference also gives mean excitation energies for a variety of biological materials of interest here. In addition, Leung has described further corrections to stopping power theory due to relativistic effects of the target electrons [32]. Such effects could increase the stopping power by as much as 2% for high-Z targets. Bichsel has observed that this would be equivalent to lowering the mean excitation energy values for high-Z materials by as much as 10%.
We are strongly motivated to use the ICRU 37 mean excitation energies because of the availability of density effect parameters based on these values [5], yet in many cases the more recent values are superior and should be used. To investigate the consequences of errors or changes in the mean excitation energies, we ran a version of our code in which I was increased by 10% and no other changes were made. In the T µ = 10-100 MeV region, the stopping power increased by somewhat over 1% for carbon and iron. For lead it decreased by 2.6%-1.4% over this energy range. Since we did not modify the density-effect parameters, in particular C (see Eq. (12)), there was a residual ≈ 0.4% at high energies. The density-effect correction essentially replaces I by the plasma frequencyhω p for p/M > ∼ 1000, so the stopping power is completely insensitive to I for T > ∼ 100 GeV, or for the lower half of our stopping power tables. The range integral always has contributions from lower-energy parts, but these also become increasingly insignificant as the energy increases. We therefore feel justified in using the older data, for which dependable density-effect parameters are available.

Low-energy corrections
The distant-collision contribution to the stopping power given by Eq. (6) was obtained by Bethe [25] with the approximation that the velocities of atomic electrons are small compared with that of the projectile. More precisely, Bethe's approximation was to replace the generalized oscillator strength by the dipole oscillator strength f (ε) in obtaining this result. This leads to correction terms [16] which are different for each atomic shell. The "shell correction" for the jth shell is represented by −2C j /Z, so that an additional term −C/Z = − C j /Z appears in the square brackets of Eq. (6). Other ways to calculate the shell correction are discussed in Ref. 3. Unfortunately, the algorithms are not easy for the non-expert to implement.
The shell correction is not important at the energies of interest in this report. For example, the stopping power correction is 0.3% for a 10 MeV muon in iron, and 3% in uranium. It falls rapidly with increasing energy. But even at intermediate energies it plays a role in "starting" the range integral. To investigate its importance, and to compare our results with the proton stopping power and range-energy tables in ICRU 49 [3], we have used the simple but long-obsolete analytic LBNL-44742  [29]. The dotted curve is from the approximate formula of Barkas [30].
approximation for the shell correction introduced by Barkas [30]: 3 The accuracy of our results is addressed in Section 3.7.
In early Bevatron experiments Barkas et al. [8] found that negative pions had a somewhat greater range than positive pions with the same (small) initial energy. This was attributed to a departure from first-order Born theory [33], and is normally included by adding a term zL 1 (β) to the stopping-power formula. The effect has been measured for a number of negative/positive particle pairs, most recently for antiprotons/protons at the CERN LEAR facility [34].
It is illustrated by the µ − stopping-power segment shown in Fig. 1.
Bethe's stopping power theory is based on a first-order Born approximation. To obtain Bloch's result, a term z 2 L 2 (β) is added if results accurate at low energies are desired.
These corrections are discussed in detail in ICRU 49, and are mentioned here for completeness. They are not significant at the energies of concern in this report.
3 Explicitly, where η = βγ and I is in eV. This form is reasonably good only for η > 0.13 (T = 7.9 MeV for a proton, 0.89 MeV for a muon).

Density effect
As the particle energy increases its electric field flattens and extends, so that the distant-collision part of dE/dx (Eq. (6)) increases as ln βγ. However, real media become polarized, limiting this extension and effectively truncating part of this logarithmic rise. This "density effect" has been extensively treated over the years; see Refs. 5, and 6, and 28, and references therein. The approach is to subtract a density-effect correction, δ, from the distant-collision contribution, resulting in the δ/2 term in Eq. (15). At very high energies, wherehω p is the plasma energy defined in Table 1. As can be seen from Eq. (15), the effect of Eq. (10) is to replace I withhω p and to eliminate the explicit β 2 γ 2 dependence in the first (log) term in the square brackets. The remaining rise of the electronic stopping power comes from Q max , given by Eq. (8). The effect of the density correction is shown in Fig. 1. At some low energy (related to x 0 below) the density effect is insignificant, and above some high energy (see x 1 below) it is well described by the asymptotic form given in Eq. (10). Conductors require special treatment at the low-energy end. Sternheimer has proposed the parameterization [35] where x = log 10 (p/M ) = log 10 βγ. C is obtained by equating the high-energy case of Eq. (11) with the limit of Eq. (10), so that C = 2 ln (I/hω p ) + 1 .
The other parameters (a, k, x 0 , x 1 , and δ 0 ) are adjusted to give a best fit to the results of detailed numerical calculations for a logarithmically spaced grid of energy values. Note that C is the negative of the C used in earlier publications. A variety of different parameter sets are available. In some cases these result from a different fitting procedure having been used with the same model, and although the parameters look different the resulting δ is not sensibly different. For elements, the PEGS4 data [28] use the values from Ref. 36.
In a series of papers by Sternheimer, Seltzer, and Berger, the density-effect parameter tables are extended to nearly 300 elements, compounds, and mixtures. The chemical composition of the materials is given in Ref. 26. 4 The agreement with more detailed calculations or results obtained with other parameter sets is usually at the 0.5% level [37]; however, see Table IV in Ref. 38. We use the tables given in Ref. 5 for most of the present calculations. 5 The densities used in these tables are occasionally in error, or, in the case of some polymers with variable density, out of the usual range. In this and other cases we use Eq. (A.8) [6] to adjust the parameters; these are marked by footnotes in the tables in Section 5.
There remains the problem of obtaining the density-effect parameters if they have not been tabulated for the material of interest. This issue is of particular importance here in the case of cryogenic liquids such as N 2 , but is also of interest when dealing with a compound or mixture not tabulated by Sternheimer, Berger, and Seltzer [5]. The algorithm proposed by Sternheimer and Peierls [6] is discussed in Appendix A.
To some degree, both the adjustment of the parameters for a different density and the Sternheimer-Peierls algorithm can be checked by using those cases in the tables where parameters are given for different densities of the same material. When the "compact carbon" parameters are adjusted to the two other tabulated carbon densities, the difference in stopping power and range with those obtained directly is less than 0.2%. Calculation of parameters for a cryogenic liquid using the Sternheimer-Peierls algorithm can be checked for hydrogen and water. This method was used to calculate parameters for liquid hydrogen at bubble chamber density (0.060 g/cm 3 ), using the excitation energy for the liquid; at worst, at minimum ionization, −dE/dx was low by 2.5%, while the range was high by 1.1%. Deviations were smaller elsewhere. When the algorithm was used to calculate parameters for water using the excitation energy for steam, the result was 1% higher at minimum ionization than that obtained directly with the water parameters. Only a slight improvement was obtained by using the excitation energy given for water.
Hydrogen is always a worst case. Sternheimer, Berger, and Seltzer [5] tabulate parameters for both hydrogen gas and liquid hydrogen under bubble chamber conditions, so we have made calculations in both directions via the Sternheimer-Peierls algorithm and by scaling densities via Eq. (A.8). We conclude that the stopping power results in this report obtained with parameters scaled to different densities are accurate to within the overall 0.5% agreement level estimated by Seltzer and Berger [37], and that the parameters calculated for cryogenic liquids (except hydrogen) using the Sternheimer-Peierls algorithm could produce stopping power errors of slightly over 1% at minimum ionization, and less elsewhere.

Other high-energy corrections
Bremsstrahlung from atomic electrons in the case of incident muons was considered in a 1997 paper by Kelner, Kokoulin, and Petrukhin [14]. There are four lowest-order diagrams: Photon emission by the muon before and after photon exchange with the electron, and emission by the electron before and after photon exchange. The former diagrams result in losses nearly proportional to E, and are described by Eq. (19). The latter are properly part of electronic losses, and produce an additional term in the stopping power. To leading powers in logarithms, this loss is given by: As Kelner et al. observe, this addition is important at high energies, amounting to 2% of the electronic loss at 100 GeV and 4% at 1 TeV. It is included in our calculations. An additional spin-correction term, (1/4)(Q max /E) 2 , is included in the square brackets Eq. (7) if the incident particle is a muon (point-like and spin 1/2) [17]. Its contribution to the stopping power asymptotically approaches 0.038 (Z/A) MeV g −1 cm 2 , reaching 90% of that value at 200 GeV in most materials. In iron its fractional contribution reaches a maximum of 0.75% at 670 GeV. Although this contribution is well within uncertainties in the total stopping power, its inclusion avoids a systematic bias.
At energies above a few hundred GeV, the maximum 4-momentum transfer to the electron can exceed 1 GeV/c, so that, in the case of incident pions, protons, and other hadrons, cross sections are modified by the extended charge distributions of the projectiles. One might expect this "soft" LBNL-44742 cutoff to Q max to reduce the electronic stopping power. This problem has been investigated by J. D. Jackson [15], who concluded that corrections to dE/dx become important only at energies where radiative losses dominate. At lower energies the stopping power is almost unchanged, since its average, dominated by losses due to many soft collisions, is insensitive to the rare hard collisions. For muons the spin correction replaces this form-factor correction.
Jackson and McCarthy [40] have pointed out that the Barkas correction calculated by Fermi (but see their Ref. 20) persists at high energies; hence, a term should be added to the close-collision part of Eq. (15): This correction, which is ±0.00176 MeV g −1 cm 2 for z = ±1, Z/A = 1/2 and β = 1, produces range differences of a few parts per thousand between positive and negative muons near minimum ionization. At higher energies sign-indifferent radiative effects dominate. We neglect this correction.

Bethe-Bloch equation
We summarize this discussion with the Bethe-Bloch equation for muons in the form used in this paper: The final term, for bremsstrahlung from atomic electrons, is given by Eq. (13). Except for the very small projectile mass dependence introduced by Q max , this expression depends only on the projectile velocity. This means that a value of the stopping power for a particle with mass M 1 and kinetic energy T 1 is the same as the stopping power for a particle with mass M 2 at T 2 = (M 2 /M 1 )T 1 . Similarly, R/M is a function of T /M (or E/M , or p/M ).

Comparison with other ionizing energy loss calculations
Comparisons with the ICRU 49 proton tables have been made by running our code with the proton mass. A summary of the stopping power comparisons is given in Table 2, and of the CSDA range comparisons in Table 3. In general the agreement is regarded as adequate, but is worse at high atomic number and low energy. The simple shell correction given by Eq. (9) has been used, and under these conditions somewhat overcorrects. ICRU 49 concludes that the "differences between tabulated and experimental stopping powers are mostly smaller than 1% and hardly ever greater than 2%," and in the case of compounds and mixtures "the uncertainties are approximately three times as large as in the case of elements" [3].
Our muon tables start at T = 10 MeV, corresponding to a proton energy of about 100 MeV, so that only 100 MeV and above is relevant in the proton comparisons. For uranium the stopping power at 100 MeV is low by 0.8% and the range high by 1.9%. Without the shell correction the stopping power for this case is high by 1.7% and the range low by 2.5%. We make the shell correction only for elements. We conclude that in a worst-case scenario, PuCl 4 (which we do not tabulate) at 10 MeV, our results could be in error by nearly 3%. For lower-Z materials the agreement is consistent with ICRU 49. In any case the agreement improves rapidly with increasing energy.
Lohmann et al. [4] list muon electronic losses separately for hydrogen, iron, and uranium. Since they do not consider the contributions of bremsstrahlung from atomic electrons (Eq. (13)), we have made comparisons with this correction "turned off." Under these conditions, our results disagree LBNL-44742

Radiative losses
The radiative contribution to muon stopping power is conveniently written as is a slowly-varying function of energy which is asymptotically constant. As indicated earlier, it is usually written as a sum of contributions from bremsstrahlung, direct pair production, and photonuclear interactions: Here we describe the calculation of these contributions. Note that the convention c = 1 is used in all the formulae in this section.
In this section we specialize to M = M µ , although the results apply to any massive spin-1/2 pointlike particle. To a very rough approximation, the bremsstrahlung contribution scales as 1/M 2 , and the pair-production part as 1/M . The results below probably apply fairly well to charged pion radiative energy losses, although to the best of our knowledge radiative losses by spin-0 particles has not been treated.

Bremsstrahlung
The cross section for electron bremsstrahlung was obtained by Bethe and Heitler [41]. In the case of muons, it is necessary to take into account nuclear screening, which was first done consistently by Petrukhin and Shestakov [42]. A simple approximation for medium and heavy nuclei (Z > 10) was derived. Lohmann, Kopp, and Voss [4] also used this approximation, but for Z < 10 they set the nuclear screening correction equal to zero. As a result, their bremsstrahlung contribution for low-Z nuclei is overestimated by about 10%.
The CCFR collaboration [43] revised the Petrukhin and Shestakov [42] results, pointing out that Ref. 42 overestimates the nuclear screening correction by about 10%. Kelner et al. [44] later observed that the CCFR conclusion probably resulted from an incorrect treatment of the Bethe-Heitler formula. Their new calculations were in good agreement with the old ones by Petrukhin and Shestakov for medium and heavy nuclei, but in addition they proposed an approximation for light nuclei. An independent analysis was performed by the Bugaev group (see, e.g., Ref. 45). The results of Petrukhin and Shestakov and the Bugaev group for bremsstrahlung on screened nuclei agree to within a few percent.
All of the results mentioned above were derived in the Born approximation. It was recently shown [45] that the non-Born corrections in the region of low and high momentum transfers have the same order of magnitude but opposite signs. As a consequence, they nearly compensate each other.

LBNL-44742
The differential cross section for muon bremsstrahlung from a (screened) nucleus as given in Ref. 44 is used for the present paper: Here ν is the fraction of the muon's energy transferred to the photon, and where D n = 1. , and the nuclear screening correction ∆ n is given by The Thomas-Fermi potential for atomic electrons is assumed. A more precise calculation of the radiation logarithm using the Hartree-Fock model is described in Ref. 46, and the results agree with the Thomas-Fermi results within about 1% at high energies (total screening); the agreement is better at low energies. Since there is not yet a Hartree-Fock result for screening in the case of bremsstrahlung from atomic electrons, we prefer to use the form factors based on the Thomas-Fermi potential throughout.
To account for bremsstrahlung losses on atomic electrons, Z 2 in Eq. (16) is usually replaced with Z(Z + 1) (e.g., see Ref. 4). A much better approximation for the contribution from electrons, taking into account electronic binding and recoil, is given by [14]: In this case where B = 1429 for all elements but hydrogen, where B = 446, and δ = M 2 ν/2E(1 − ν), as above.
The cross section for bremsstrahlung as a function of fractional energy transfer ν is shown in Fig. 4. Although pair production dominates the radiative contributions to the stopping power, bremsstrahlung dominates at high ν.
The average energy loss −dE/dx due to bremsstrahlung is calculated by integrating the sum of these cross sections, as in Eq. (21) below.

Direct e + e − pair production
The cross section for direct e + e − pair production in a Coulomb field was first calculated by Racah [47]. Atomic screening was later taken into account by Kelner and Kotov [48]. With their approach, the average energy loss is obtained through a three-fold numerical integration. With the simple parameterization of the screening functions proposed by Kokoulin and Petrukhin [49], one obtains a double differential cross section for e + e − production. This formula is widely used in muon transport calculations (for example, see Ref. 4). Based on this work, a (rather complicated) analytic form for the energy spectrum of pairs created in screened muon-nucleus collisions was LBNL-44742 derived by Nikishov [50]. The explicit formula is given in Appendix B. The average energy loss for pair production is calculated by numerical integration: The same expression as for the nucleus is usually used to calculate the pair production contribution from atomic electrons, with Z 2 replaced with Z (e.g., Ref. 4). A more precise approach has recently been developed by Kelner [51], who proposed a simple parameterization of the energy loss based on a rigorous QED calculation. This formula for the electronic contribution to pair production energy loss by muons is valid to within 5% of the more laborious numerical result for E > 5 GeV, and is used for the present calculations: Here g = 4.4 × 10 −5 for hydrogen and g = 1.95 × 10 −5 for other materials. The cross section for direct pair production as a function of fractional energy transfer ν is shown in Fig. 4.

Photonuclear interactions
Several approaches have been developed to calculate the muon photonuclear cross section. The most widely used is that of Bezrukov and Bugaev [52]: Here ε is the energy loss of the muon and σ γN (ε) is the photoabsorption cross section defined below. Other values are given by x = 0.00282 A 1/3 σ γN (ε), m 2 1 = 0.54 GeV 2 , and m 2 2 = 1.8 GeV 2 . This cross section gives results consistent with other calculations to within 30% [4].
Recent measurements of photonuclear interaction of muon in rock performed by the MACRO collaboration [53] agree quite well with Monte Carlo simulations based on the Bezrukov and Bugaev cross section.
The total cross section σ γN (ε) for the photon-nucleon interaction appears as a normalization parameter in Ref. 52, which proposes a simple parameterization: This approximation is good enough only for muon energy loss ε > 5 GeV. For smaller ε, we use the experimental data given by Armstrong et al. [54]. The energy loss contribution is calculated by numerical integration of the differential cross section given by Eq. (23). The use here of a more precise photo-absorption cross section for ε < 5 than was used in the original model [52] does not change the photonuclear part of −dE/dx appreciably. The cross section for photonuclear interactions as a function of fractional energy transfer ν is shown in Fig. 4.   (13)). The subtraction loses significance at 10 GeV, where the radiative contribution is small.

Comparison with other works on muon radiative losses
Selected b values from our present calculations and according to Lohmann et al. [4] are listed in Table 4 and plotted in Fig. 5  For Z > 10 the results are nearly identical. For smaller atomic number, and at low energies, two effects are responsible for the differences: 1. In the nuclear part of bremsstrahlung, nuclear screening has only a weak energy dependence, and produces about a 4% reduction for hydrogen and a 10% reduction for carbon. This is apparent in our lower values of b tot for carbon and water as compared with Lohmann et al. Muon energy (GeV)  The circles for standard rock are from Gaisser and Stanev [55].

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replaced by Z(Z+0) in the low-energy limit, so that for hydrogen our contributions to these processes for 1-10 GeV are smaller by nearly a factor of two. Similarly, in the low-energy limit our bremsstrahlung and pair production contributions for carbon are 6/7 of Lohmann et al.'s values. The CERN RD 34 collaboration has measured the energy loss spectrum of 150 GeV muons in iron [56].
The energy deposition was measured in prototype hadron calorimeter modules for the ATLAS detector. Most probable electronic loss was subtracted, as was background from photonuclear interactions (which in this case is only about 7% of the total cross section). The remaining sensitivity was to energy loss by pair production (dominant at the smallest energy transfers, 0.01 < ν < 0.03), knock-on electrons (δ rays, included in the high-energy tail of the electronic loss and dominating for 0.03 < ν < 0.12), and bremsstrahlung (dominant for ν > 0.12). The RD 34 experimental results shown in Fig. 6 are expressed as the fractional deviation from the present calculations, where the Kelner et al. [14] model (basically that of Petrukhin and Shestakov) is used to obtain the bremsstrahlung component important at large ν. The fractional deviations from our results are also shown for the CCFR collaboration's revision of the Petrukhin and Shestakov cross section [43] and Rozental's formula [57]. The present calculation describes the data reasonably well, while the others are evidently less successful.   [56], shown as fractional deviations from the present calculation, including bremsstrahlung via Kelner et al. [14].
Heavy error bars indicate statistical errors only, while the light bars include systematic errors combined in quadrature. Deviations of alternate models from our calculations are shown by the solid line (Rozental's formula [57]) and the dashed line (the CCFR collaboration's revision of the Petrukhin and Shestakov cross section [43]) . LBNL-44742

Muon critical energy
Equation (5) defines the muon critical energy E µc as the energy for which electronic and radiative losses are equal. E µc for the chemical elements is shown in Fig. 7. The equality of electronic and radiative losses comes at a higher energy for gases than for solids and liquids because of the smaller density-effect correction for gases. Empirical functions have been fitted to these data for gases and for solids/liquids, in both cases excluding hydrogen from the fits. Since E µc depends upon ionization potentials and density-effect parameters as well as Z, the fits cannot be exact. Potassium, rubidium, and cesium are 3.6%, 3.2% and 3.4% high, respectively, while beryllium is 3.8% low. Most of the other solids and liquids fall within 2.5% of the fitted function. Among gases the worst fit is for neon (1.9% high).

Fluctuations in radiative energy loss
The radiative cross sections at several energies are shown in Fig. 4. The bremsstrahlung cross section varies roughly as 1/ν over most of the range (where ν is the fraction of the muon's energy transferred in a collision), while for pair production the distribution varies as ν −3 to ν −2 (see also Ref. 58). "Hard" losses are therefore more probable in bremsstrahlung, and in fact energy losses due to pair production may very nearly be treated as continuous. The photonuclear cross section has almost the same shape as the bremsstrahlung cross section at high ν, but it is about an order of magnitude lower.
An example illustrating the fluctuations is shown in Fig. 8. The distribution of final muon momenta was obtained by following 10 5 1 TeV/c muons through 3 m (2360 g/cm 2 ) of iron, using the MARS14 Monte Carlo code [59]. Our result is in nearly exact agreement with results obtained earlier with TRAMU [58]. The most probable loss is 8 GeV, or 3.4 MeV g −1 cm 2 . Our tables list a stopping power in iron as 9.82 MeV g −1 cm 2 for a 1 TeV muon, so that the mean loss should be 23 GeV, for a final energy (≈ momentum × c) of 977 GeV, far below the peak. This is also the mean calculated from the simulated output.  [59]. The comparative rarity of very low final momenta follows from the approach of the cross sections to zero as ν → 0.

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The full width at half maximum is 9 GeV/c, or 0.9%. The median is 988 GeV/c. 10% of the muons lost more than 3.2% of their energy and 2.6% lost more than 10% of their energy. Three out of the 10 5 candidates stopped in the iron, presumably because of very inelastic photonuclear interactions.
The classic case of the propagation of very high-energy muons in "standard rock" is discussed in Ref. 60 and references therein. Of special interest is Figure 3 of that paper, showing the survival probabilities as a function of depth for muons from 1 TeV to 10 6 TeV. The effects of the "radiative tail" are enormous; at 10 6 TeV only about 15% of the muons reach the CSDA range. At 10 TeV about half of them do, giving some indication of the usefulness of our CSDA ranges.
Treatment of radiative loss fluctuations in local energy deposit, range, or direction is beyond the scope of this paper. It is usually handled by Monte Carlo methods [58,60,61], although moment expansions are sometimes used when likelihoods need to be assigned to individual events. Electronic-loss straggling of high energy muons is described using a modified Vavilov distribution in Ref. 62.

Tabulated data
The contents of the main tables are shown in Tables 5-9. In this section we discuss the rationale in the selection of the elements, simple compounds, polymers, and biological materials for which tables of muon range-energy loss and radiative loss parameters are provided.
All "common" elements were selected. This included all elements with Z ≤ 38 (strontium), most elements through Z = 58 (cerium), and the more common heavy elements through Z = 94 (plutonium). Dysprosium (Z = 66) was included to avoid a large gap in Z between cerium and tantalum (Z = 73). The list is given in Table 5.
Cryogenic liquid forms of most of the elemental room-temperature gases (radon is the exception) are fairly common in laboratories, and hence were included as well. Because of differences in the density effect corrections, ionization energy loss differ between liquid and gaseous forms. Radiative losses are not affected by density, but the muon critical energy, where the electronic and radiative losses are the same, can be quite different. We have excluded gaseous bromine, (boiling point LBNL-44742 58.8 • C, although it is tabulated by Sternheimer et al. [5]. For carbon, two forms with different densities appear. In all, 74 range/energy-loss tables are given for 63 elements We should not overemphasize these differences: Related materials have similar stopping-power properties when these are listed in MeV cm 2 /g (as we do) rather than in MeV/cm. Liquid and gaseous xenon are not dissimilar in spite of a density ratio of 540. Plutonium is more than twice as dense as bismuth, but their stopping powers differ by only 5% at minimum ionization. The stopping powers of hydrocarbons are quite similar, as are those of many polymers and biological materials.
Atomic weights are given to the available significance. This varies with element, since the isotopic composition of samples from different sources varies. In general the atomic weights of elements with only one isotope are known to great precision [63].
The same "commonness" criterion was applied to the selection of the simple compounds listed in Table 6, with some qualifications: We limited ourselves to the compounds listed by Sternheimer et al. [5], which meant that certain common compounds such as NaCl were not available. Common inorganic scintillators (BaF 2 , BGO, CsI, LiF, LiI, NaI) are present. Materials such as trichloroethylene are included because of their role in important physics experiments. The list contains perhaps more hydrocarbons than necessary, in part to show the change of stopping power behavior as the H/C ratio changes (note the difference between acetylene and ethane). Liquid water and steam are both listed. Dry ice was included with some difficulty.
Polymers are listed in Table 7. Their energy loss behavior is quite similar except in the case of Teflon, which contains no hydrogen. "Thin film" polymers (Mylar, Kapton) were omitted. Polymers used for plastic scintillators (acrylic, polystyrene, polyvinyltolulene) are included. In some cases the name, like acrylic or polycarbonate, describes a family of polymers. The chemistry given is typical, and no great variation is to be expected except perhaps for "Bakelite," which is not very well characterized. Where space permits, the formula is given in such a way as to convey as much structural information as possible.
Mixtures of interest are given in Table 8. Muon energy loss in air is of great current interest, given atmospheric neutrino observations. Photographic emulsion is of more historic interest. Except for dry air (and, by definition, standard rock) none of the materials is particularly well characterized. The somewhat arbitrary concrete recipe is taken from The Reactor Handbook [64], and may be found, along with the other compositions, in Ref. 26.
For at least two generations, the depth of underground muon experiments has been reduced to depth in "standard rock." This is by definition the overburden of the Cayuga Rock Salt Mine near Ithaca, New York, where K. Greisen and collaborators made seminal observations of muons at substantial depths [1]. Ref. 1 says only "Most of the ground consists of shales of various types, with average density 2.65 g/cm 2 and average atomic number 11." Menon and Murthy later extended the definition: Z 2 /A = 5.5, Z/A = 0.5, and and ρ = 2.65 g/cm 2 [65]. It was thus not-quite-sodium. Lohmann et al. [4] further assumed the mean excitation energy and density effect parameters were those of calcium carbonate, with no adjustments for the slight density difference. We use their definition for this most important material.
Sternheimer et al. [5] list 14 biological materials and "phantoms," mixtures which have nearly identical responses to radiation as the biological materials they replace. Omitted materials can be approximated by those on the list: Brain (ICRP), lung (ICRP), skin (ICRP), testes (ICRP), soft tissue (ICRU 4-component), and striated muscle (ICRU) are quite similar to soft tissue (ICRP), as are several included materials such as eye lens (ICRP) and skeletal muscle (ICRP). Compact bone (ICRU) is similar to cortical bone (ICRP). Table 5: Index of tables for selected chemical elements. Physical states are indicated by "G" for gas, "D" for diatomic gas, "L" for liquid, and "S" for solid. Gases are evaluated at one atmosphere and 20 • C. The corresponding cryogenic liquids are evaluated at their boiling points at one atmosphere, and carbon is evaluated at several typical densities. Atomic weights are given to their experimental significance. Except where noted, densities are as given by Sternheimer, Berger, and Seltzer [5].

Element
Symbol  LBNL-44742 Table 6: Index of tables for selected simple compounds. Physical states are indicated by "G" for gas, "D" for diatomic gas, "L" for liquid, and "S" for solid. Gases are evaluated at one atmosphere and 20 • C. Except where noted, densities are those given by Sternheimer, Berger, and Seltzer [5]. Composition not explained may be found in Seltzer and Berger [26]  LBNL-44742 Table 7: Index of tables for selected high polymers. Except where noted, densities are those given by Sternheimer, Berger, and Seltzer [5]; actual densities of polymers will vary. Composition not explained may be found in Seltzer and Berger [26]  Notes: 1. Density effect parameters adjusted to this density using Eq. (A.8). Table 8: Index of tables for selected mixtures. Physical states are indicated by "G" for gas and "S" for solid. Gases are evaluated at one atmosphere and 20 • C. Densities are those given by Sternheimer, Berger, and Seltzer [5]. Composition may be found in Seltzer and Berger [26]  LBNL-44742 Table 9: Index of tables for selected biological materials. Physical states are indicated by "L" for liquid and "S" for solid. Densities are those given by Sternheimer, Berger, and Seltzer [5]. Composition may be found in Seltzer and Berger [26]  Appendix A. Stopping power and density-effect parameters for compounds and mixtures For most of the materials for which tables are given in this report, the relevant effective excitation energy and density-effect parameters have already been tabulated, and should be used. This section concerns how to deal with an untabulated compound or mixture.
It is usual to think of a compound or mixture as made up of thin layers of the pure elements (or, better, constituent compounds for which tabulated data are available) in the right proportion (Bragg additivity [66]) . Let n j be the number of the jth kind of atom in a compound (it need not be an integer for a mixture), and w j its weight fraction: When the Bethe-Bloch equation is inserted and the radiative terms added, the Z-dependent terms can be sorted out to find that the mixture or compound is equivalent to a single material with There are pitfalls in actually using Eq. (A.3) to calculate I . Since the electrons in a compound are more tightly bound than in the constituent elements, the effective I j are in general higher than those of the constituent elements. Exceptions are provided by diatomic gases and by metals in metallic alloys or compounds. Berger and Seltzer discuss ways to extend the Bragg additivity rule in lieu of a detailed calculation [26]: (a) For a select list of materials (carbon and some common gases), they propose alternate mean excitation energies in their Table 5.1 (or Table 6 in Ref. 26).
(b) For other elements, the excitation energies are multiplied by 1.13 before calculation of the mean (the "13% rule"). Although it is is not said, it would seem sensible to apply this rule in the case of a tightly-bound material such as CsI, and not apply it in the case of a metallic compound like Nb 3 Sn.
(c) Both are superseded by experimental numbers when available, as in the case of SiO 2 .
Bragg additivity has little meaning in calculating the density-effect correction. If the material of interest is not available in the tables of Ref. 5, then the algorithm given by Sternheimer and Peierls [6] should be used. Their recipe is given more succinctly in the EGS4 manual [28], and the following algorithm is adopted from that version for the parameterization of Section 3.4.
(a) I is obtained as described above (b) k is always taken as 3.00 (c) C = 2 ln(I/hω p ) + 1, with the plasma energyhω p obtained using the expression from Table 1 (e) In the case of gases, A.5

A.6
We have used this algorithm to calculate the density-effect coefficients for cryogenic liquids, which are not tabulated by Sternheimer, Berger, and Seltzer [5]. In this case, I for the gas was used.
One problem remains: Given the density-effect parameters, either from the literature [5] or from the algorithm given above, how does one modify them for the same material at a different density? This problem occurs for gases at different pressures, or for solid and liquids at different densities than those tabulated. In an early paper [6], Sternheimer noted that under quite general conditions δ r (p) = δ p √ r , A.7 where r = ρ/ρ 0 , the ratio of desired to tabulated densities, and the subscript r indicates the quantity evaluated at the desired density. This implies [6] that C r = C − ln r x 0r = x 0 − 1 2 log 10 r x 1r = x 1 − 1 2 log 10 r .

A.8
It is easily shown by matching different regions in Eq. (11) that the parameters a and k are unchanged by the transformation. This method was used to correct the parameters for several materials for which we prefer different densities, for example the density of fused silica. Liquid hydrogen is tabulated for the "bubble chamber density," for which we scaled the parameters to the density of liquid hydrogen at its boiling point at a pressure of one atmosphere.
Both algorithms were checked by calculating −dE/dx several ways, taking advantage of the fact that Ref. 5 lists both gaseous and liquid helium, three densities of carbon, and both steam and liquid water. The comparisons are discussed at the end of Sect. 3.4.

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Explanation of Tables 4   TABLE I: Muon energy loss rate and CSDA range for selected chemical elements.
The contents of Table I and other information are given in Table 5. The header defines the element and state, and gives the parameters used to calculate the electronic stopping power: Z: Atomic number. A: Atomic weight. ρ: Density. Gas density is evaluated at 20 • C. I: Mean excitation energy. a-δ 0 : Sternheimer et al. density effect parameters, as introduced in Sect. 3.4 and Eq. (11).
The body of the table presents ionizing energy loss (Eq. (15)) and the components of radiative loss rate (see Eqs. (1) and (2), as well as CSDA range obtained by integrating the total (Eq. (3)), as a function of the muon's initial kinetic energy T . The spacing of the independent variable is fairly uniform on a logarithmic scale. The corresponding momentum p is also given. The table is interrupted to show the points at which minimum ionization and muon critical energy (Eq. (5)) occur. A radiative loss field is left blank if the stopping power contribution is less than 0.0001.     Table 9. The format is identical with TABLE I, except that Z/A (Eq. (A.3))) replaces Z and A.  Table I. Interpolation in log E is used to generate the b values shown in TABLE I and for the ionizing loss calculations. The contents are listed in Table 5.  Table 6.  Table 7.  Table 8.  Table 9.