Corrections to the Higher Moments of the Relativistic Ion Energy-Loss Distribution Beyond the Born Approximation

Home , Felix Bloch

Corrections to the higher moments of the relativistic ion energy-loss distribution beyond the Born approximation. I. Z dependence of Mott’s corrections

O. Voskresenskaya

Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980 Russia

Abstract Using the Mott exact cross section for moderate relativistic energies, we calculated the corrections to the ﬁrst-order Born higher moments of the energy-loss distribution of charged particles in a wide range of the particle charge numbers. arXiv:1711.11572v4 [hep-ph] 12 Jun 2020

1 1 Introduction

Stopping power (the average energy loss of a particle The MCs are only available in a regime for which per unit path length measured e.g. in MeV/cm) is a separation into “distant collisions”, in which electron- necessary ingredient for many parts of basic science binding effects are included, and on “close collisions”, as well as for medical and technological applications for which the binding energy of the electrons can be [1, 2, 3]. The stopping power of a material is de- neglected (namely, at large velocities compared to scribed by the Bethe formula (the so-called “Bethe’s atomic velocities), is fair [14]1. They were first cal- stopping power formula”) [4, 5, 6]. A non-relativistic culated exactly by numerical integration of the Mott version of this formula was found by Hans Bethe in exact cross section in [16]. However, as the expres- [4]. Its relativistic version was obtained by him in sion for the MCs in (2) is extremely inconvenient [5]. for practical application, obtaining convenient and The relativistic version of this formula for average accurate MCs expressions becomes significant. ionization energy loss of charged particles reads In [17], devoted to the analysis of the contribution of close collisions to the average energy-loss of ¯ dE 2 relativistic ions in matter, it has been shown that − = 2ζ ln(Em/I) − β , dx the exact MC expression can be represented in the ζ = 0.1535(Z/β)2Z0/A, form of quite rapidly converging series of quantities 2 2 −1 Em = 2mc β (1 − β) , (1) bilinear in the Mott partial amplitudes, which can be easily computed. A comparison of this exact result where x denotes the distance traveled by a particle; with the approximate result of [18] was also consid- ered there. In this paper, we apply the exact MC Em is the maximum transferrable energy to an electron in a collision with the particle of velocity βc; m expression from [17] for computing the most com- is the electron mass; Z0 and A refer to the atomic mon characteristics of the energy-loss distributions number and the weight of the absorber. In the Bethe for moderate relativistic energies in a wide range of theory, the material is completely described by a sin- particle charge number (5 ≤ Z ≤ 95). gle number, the mean excitation potential I of the These characteristics (for instance, the ELS) are absorber. Felix Bloch has shown [7] that the mean of interest not only because of their fundamental as- excitation potential of the atoms is approximately pects, but also due to their impact on many appli- given by I=(10 eV ) Z0. If this approximation is in- cations in physics and other sciences. So, accurate troduced into formula (1) one obtains an expression information on the ELS is important in the applica- which is often called the “Bethe–Bloch formula” [8]. tion of the Rutherford backscattering spectrometry, The first-order Born approximation upon which nuclear reaction analysis, and scanning transmission the Bethe formula is based is inadequate to the task ion microscopy. As the ELS is one of the main fac- of accurately describing the energy-loss rate of highly tors limiting the depth resolution, understanding the charged particles (ions). The importance of a higher- shape of energy-loss distribution is important in the application of particle identification. In view of these order-correction term ΦM obtained by Nevill Mott [9] to the relativistic Bethe formula [4, 5] that reads and other applications, the study of ELS in matter become an interesting topic in recent years. dE¯ Nevertheless, most researchers have concentrated − = 2ζ ln(E /I) − β2 + Φ /2 , dx m M on stopping power, and only few investigators have 1 studied the ELS extensively [19–28]. Even less re- Φ = ∆ dE/dx¯ , M 2 M close sults devoted to the study of higher moments of the dE¯ Z dσ dσ energy-loss rate. However, it was recently shown ∆ = n Z0 M − B εdε , (2) M dx 0 dε dε that the higher moments of distributions of con- close served quantities measuring deviations from a Gaus- sian have a sensitivity to CP fluctuations [29]. The was pointed in [10]. In the above equations, ΦM article [30] reports the first measurement of higher denotes the so-called “Mott corrections” (MCs), n0 moments of the distributions from Au+Au collisions is the number of atoms of the substance per unit to search for signatures of the CP. volume, σB and σM signify the first-order Born The present communication is organized as and the Mott exact (calculated with the aid of the follows: We start (in Sec. 2) from a consideration Mott phase shifts) cross sections, respectively. The of an exact expression for the Mott phase-shift importance of σM for the scattering of heavy ions off formula. Using this expression, we get (in Sec. 3) electrons with respect to the energy-loss straggling a representation for the central moments (µn,M ) (“ELS”) of charged particles has been demonstrated in [11, 12], and the well known relativistic Bohr 1The idea of applying the Mott cross section to close colli- result [13] is exceeded by up to a factor of 3. sions was suggested in [15].

2 of the energy loss distribution in the form of fast 3 Computation of the central converging series. We also compare the computation moments of the energy-loss results for µn,M (Sec. 3) and for the normalized central moments ρk,M (Sec. 4) with the Born re- distribution sults. Finally (in Sec. 5), we summarize our findings. Remind now a few basic definitions. We define the n- th moment of a real-valued continuous function f(y) about a value y¯ as 2 Evaluation of the Mott exact Z ∞ n µn = (y − y¯) f(y) dy. phase-shift formula −∞ The moments about its mean are called “central moments”, and they describe the shape of the func- In the higher moments of the particle energy-loss dis- tion f(y). The first moment (µ ) is the mean. The tribution that determine its shape [31], the dominant 1 second central moment (µ = Ω¯ 2) is the variance.2 contribution is made by close collisions. The descrip- 2 Its positive square root is the standard deviation, tion of the lattеr in the framework of the Born ap- and it describes the width Ω¯ of the distribution. The proximation is valid only under condition Zα/β 1, third (µ ) and fourth (µ ) central moments are used where α ≈ 1/137 is the fine structure constant, and 3 4 to determine the normalized central moments, which β = v/c is the relative ion velocity in the labora- define the parameters of asymmetry and peakedness tory frame that is expressed in units of the speed of of the distribution, respectively. light c. In calculations of the characteristics of the Applied to our problem, the central moments of energy-loss distribution of heavy relativistic ions, we energy-loss distribution can be represented in the should expect significant deviations from the results Mott approximation as follows: in the Born approximation. π Z When we restrict consideration by moderate rel- dσM µ = 2πn Z0δx [δε(ϑ)]n sin ϑdϑ . (4) ativistic energies of incident particles such that one n,M 0 dΩ can neglect the effects of their electromagnetic struc- 0 ture, we can use in our calculations the expression Here, δx is the thickness of the layer of matter tra- for the scattering cross section in terms of the Mott versed by the ion. phase shifts [9]. Using the expression

As shown in [17], the Mott exact phase-shift for- m m mula can be represented as (2l + 1)xPl (x) = (l + 1 − m)Pl+1(x) m −(l + m)Pl−1(x) (5) 2 2 2 2 dσM ~ ξ |F (ϑ)| + |G(ϑ)| [32] and the orthogonality relation = 2 , (3) dΩ 2p sin (ϑ/2) 1 Z 2 (l + m)! ∞ P m(x)P n(x)dx = 1 δ , (6) X l1 l2 l1l2 F (ϑ) = FlPl(x),Fl = lCl − (l + 1)Cl+1, 2l1 + 1 (l1 − m)! −1 l=0 it is not difficult to obtain from (3) and (4) the fol- Γ(ρ − iν) l iπ(1−ρl) Cl = e , lowing representations for the quantities µn,M (n = Γ(ρl + 1 + iν) 2, 3, 4) in the form of rapidly converging series: p 2 2 ρl = l − (Zα) , ν = Zα/β , ∞ 2 G(ϑ) = − cos(ϑ/2) F 0(ϑ), x = cos(ϑ) , X h 2 Fl Fl−1 µ2,M = a (l + 1) ξ − p p 2l + 1 2l + 3 p = mcβ/ 1 − β2, ξ = ν 1 − β2, l=0 lF (l + 2)F 2i + l − l+1 , (7) 2l + 1 2l + 3 where Pl(x) denotes the Legendre polynomial of or- 2 ∞ ap X 1 h 2 ˜ 2 ˜ 2i der l with x = cos ϑ, ϑ is the scattering angle of µ3,M = ξ Fl + l(l + 1) Gl , m (2l + 1) the electron scattered by an ion in the center-of- l=0 mass system, F 0(ϑ) ≡ dF (ϑ)/dϑ, and Γ is the Euler l l + 1 F˜ = F − F − F , gamma function. l l 2l − 1 l−1 2l + 3 l+1 Below we apply the above dσ /dΩ representa- l − 1 l + 2 M G˜ = F − F − F , (8) tion to calculate the relative Mott corrections to l l 2l − 1 l−1 2l + 3 l+1 the first-order Born central moments and normal- 2The average square fluctuation in the energy loss, Ω¯ 2 = ized central moments of the relativistic ion energy- h(δε−hδεi)2i, where δε denotes the energy loss, is the so-called loss distribution. “straggling parameter”.

3 2 2 ∞ ˜ ˜ 2 p X h Fl Fl+1 Let us notice that the deviation from a simple µ = a (l + 1) ξ2 − 4,M m 2l + 1 2l + 3 Born approximation in determining the ELS can be l=0 very signiﬁcant. For instance, the observed in [24] lG˜ (l + 2)G˜ 2i + l − l+1 , energy straggling is about a factor of 5 lower than 2l + 1 2l + 3 the ﬁrst-order Born result. The ELS computed in our paper is generally larger than that in the Born a = 2πn Z0∆x( cβ)2/(1 − β2). (9) 0 ~ approximation by a factor of ∼ 2 − 3 for heavy ions. It is easily to verify that the terms of these series It is consistent with the results of the ELS measure- decrease as l−2n+1 (n = 2, 4) when l → ∞. ments for relativistic heavy ions from [12]. The re- In order to assess the applicability of the compu- sults obtained in our paper indicate the importance tation results in the Born approximation to describ- of the MCs in determining the central moments of ing the heavy ion close collisions with the atoms of the heavy ion energy-loss distributions at moderate matter that reads relativistic energies.

2 2 2 dσ 1 − β sin (ϑ/2) Table 1. The relative corrections δn(µ) to the B = ν2 ~ , (10) dΩ 2p sin4(ϑ/2) ﬁrst-order Born central moments µn,B (n = 2, 4): 2 Z dependence of δn(µ) for β = 0.75. µ = πn Z0δx ν2 ~ n,B 0 m Z δ2(µ) δ3(µ) δ4(µ) 2p2 n 1 β2 × − , (11) 10.0000 0.0484 0.0468 0.0424 m n − 1 n 20.0000 0.1172 0.1185 0.1112 the latter can be compared with the results of com- 30.0000 0.2106 0.2221 0.2145 putation using the formulae (7), (8), and (9). 40.0000 0.3330 0.3661 0.3629 For this purpose, we introduce the quantities 50.0000 0.4891 0.5607 0.5696 60.0000 0.6832 0.8171 0.8508 µn,M − µn,B 70.0000 0.9192 1.1505 1.2263 δn ≡ δn(µ) = , n = 2, 4, (12) µn,B 80.0000 1.1988 1.5727 1.7190 90.0000 1.5186 2.0942 2.3507 which characterize the accuracy of the Born approximation in computing the higher moments of the heavy ion energy-loss distribution. Thus, the use of the Born approximation in the Figure 1 and Table 1 show the computation re- analysis of close collisions of heavy ions with the sults of (12). They demonstrate the dependence of atoms of an irradiated material is at best possible δn(Z, β) on the ion charge numbers. only for the purpose of qualitative evaluation of the quantities under consideration.

4 Calculation of the normalized k-th central moments

Deﬁne now the normalized k-th central moment ρk ¯ k k/2 as the k-th central moment divided by Ω ≡ µ2 . In doing so, the following relations for the most important normalized 3rd and 4th central moments of the energy-loss distribution are valid: µ µ ρ = 3,M , ρ = 4,M . (13) 3,M 3/2 4,M 4/2 µ2,M ) µ2,M ) The normalized third central moment, the skewness, describes the stretching of the distribution to the right (+) or the left (−). The fourth central mo- Figure 1: Relative corrections δ to the Born central ment, the kurtosis, describes the peakedness of the n distribution (+, greater than normal, and −, less moments µn,B (n = 2, 4) of the energy-loss distribution of incident charged particles as a function of than normal). projectile charge number Z for β = 0.75. The relative Mott corrections to the normalized ﬁrst-order Born central moments ρk,B (k = 3, 4) can be expressed as

It is seen from them that the values of δn(µ) can ρk,M − ρk,B δk ≡ δk(ρ) = , k = 3, 4. (14) reach about two hundred percent for Z ∼ 90. ρk,B

4 The calculation results for the values of δk(ρ) form of quite rapidly converging series whose (k = 3, 4) are given in Figure 2 and Table 2. They terms are bilinear in the Mott partial ampli- demonstrate the dependence of δk(ρ) on the ion tudes, an algorithm is proposed to compute the charge number Z. These results show that the abso- most important central moments µn,M (n = lute values of the relative corrections δ3 to the nor- 2, 4) of the average energy-loss rates of rela- malized Born central moment ρ3,B of the relativis- tivistic charged particles in the Mott approxi- tic ion energy-loss rate do reach approximately 20%, mation. and the |δ | values increase to 50% for the heavy ions 4 • This algorithm reduces computing the µ to with Z ∼ 90. n,M a summing the fast converging (as l−2n+1 at l → ∞, l ∈ N0) inﬁnite series (7), (8), (9) and can be simply implemented using the numerical summation methods of converging series for a given level of precision. • Using the latter result, the parameters of the energy-loss distributions were calculated for charged particles of moderate relativistic energies in the Born and Mott approximations for a wide range 5 ≤ Z ≤ 95 of the particle charge numbers.

• The relative Mott corrections δn and δk to the Born values of the central moments µn,B (n = 2, 4) and the normalized central moments ρk,B (k = 3, 4) of the particle average energy- Figure 2: Relative corrections δk to the normalized Born loss distributions in an irradiated material were central moments ρk,B (k = 3, 4) of the energy-loss distri- also computed. bution of incident charged particles as a function of the particle charge number Z for β = 0.75. • It is shown that the relative Mott corrections δn(µ) to the Born central moments µn,B of the distributions can reach about 200% for heavy relativistic ions. This made it possible to ex- Table 2. The relative corrections δk(ρ) to the plain the results of experiment [12]. normalized ﬁrst-order Born central moments ρk: Z dependence of δk(ρ) for β = 0.75. • A conclusion is drawn on the applicability of the Born approximation to the consideration

Z −δ3(ρ) −δ4(ρ) of higher moments of the relativistic heavy ion 10.0000 0.0249 0.0516 energy-loss distributions at best only for the 20.0000 0.0529 0.1098 purpose of their qualitative evaluation. 30.0000 0.0825 0.1713 • The obtained results also mean that the ion 40.0000 0.1123 0.2330 energy-loss distribution, which was computed 50.0000 0.1411 0.2920 with the use of Mott’s corrections, are always 60.0000 0.1676 0.3468 less asymmetric and more normal than the dis- 70.0000 0.1912 0.3956 tributions computed in the Born approxima- 80.0000 0.2109 0.4376 tion. 90.0000 0.2259 0.4718 • The choice of an eﬃcient method for numerical calculations of the heavy ion energy-loss The decrease of the Mott normalized central mo- distributions and their parameters taking ac- ments (ρk,M ) in comparison with the Born one (ρk,B) count of the Mott corrections as well as com- for both light and heavy ions means that the ion parison of the computational results with the energy-loss rates, which were calculated with the use available experimental data will be the subject of Mott’s corrections, are always less asymmetric and of a further research. more “normal” (more close to Gaussian) than those in the Born approximation. Acknowledgments 5 Summary and outlook This work was supported by a grant from the Rus- • Based on the representation of the Mott cor- sian Foundation for Basic Research (project nos. 17- rections ∆M to the Bethe-Bloch formula in the 01-00661-а).

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