Antinucleus–nucleus cross sections

Production of anti-nuclei, especially anti-\(^4{\rm He}\), has been observed in nucleus-nucleus and proton-proton collisions by the RHIC and LHC experiments. Contemporary and future experimental studies of anti-nucleus production require a knowledge of anti-nucleus interaction cross sections with matter which are needed to estimate various experimental corrections, especially those due to particle losses which reduce the detected rate. Because only a few measurements of these cross sections exist, they were calculated using the Glauber approach [FG66][Fra68][DK85] and the Monte Carlo averaging method proposed in [AMZS84][SYuSZ89].

Two main considerations are used in the calculations: a parameterization of the amplitude of antinucleon-nucleon elastic scattering in the impact parameter representation and a parameterization of one-particle nuclear densities for various nuclei. The Gaussian form from [FG66][DK85] was used for the amplitude and for the nuclear density the Woods-Saxon distribution for intermediate and heavy nuclei and the Gaussian form for light nuclei was used, with parameters from the paper [WBB09]. Details of the calculations are presented in [eal11].

Resulting calculations agree rather well with experimental data on anti-proton interactions with light and heavy target nuclei (\(\chi^2/NoF\) = 258/112) which corresponds to an accuracy of \(\sim\)8% [eal11]. Nearly all available experimental data were analyzed to get this result. The predicted antideuteron-nucleus cross sections are in agreement with the corresponding experimental data [eal72].

Direct application of the Glauber approach in software packages like is ineffective due to the large number of numerical integrations required. To overcome this limitation, a parameterization of calculations [Gri09a][Gri09b] was used, with expressions for the total and inelastic cross sections as proposed above in the discussion of the Glauber-Gribov extension. Fitting the calculated Glauber cross sections yields the effective nuclear radii presented in the expressions for \(\bar pA\), \(\bar dA\), \(\bar tA\) and \(\bar \alpha A\) interactions:

\[R^{eff}_A=a\ A^b\ + \ c/A^{1/3}.\]

The quantities \(a\), \(b\) and \(c\) are given in [eal11].

As a result of these studies, the toolkit can now simulate anti-nucleus interactions with matter for projectiles with momenta between 100 MeV/c and 1 TeV/c per anti-nucleon.

Alternative nucleus-nucleus cross sections

The total reaction cross section has been studied both theoretically and experimentally and several empirical parameterizations of it have been developed. In Geant4 the total reaction cross sections are calculated using four such parameterizations: the Sihver[STS+93], Kox[eal87], Shen[SWF+89] and Tripathi[TCW97] formulae. Each of these is discussed in order below.

Sihver Formula

Of the four parameterizations, the Sihver formula has the simplest form:

\[\sigma_{R} = \pi r^{2}_{0} \left[A^{1/3}_{p} + A^{1/3}_{t} - b_{0} [A^{-1/3}_{p} + A^{-1/3}_{t}] \right]^{2}\]

where A\(_{p}\) and A\(_{t}\) are the mass numbers of the projectile and target nuclei, and

\[\begin{split}b_{0} &= 1.581-0.876(A^{-1/3}_{p} + A^{-1/3}_{t}) , \\ r_{0} &= 1.36 \mbox{ fm}.\end{split}\]

It consists of a nuclear geometrical term \((A^{1/3}_p + A^{1/3}_t)\) and an overlap or transparency parameter (\(b_0\)) for nucleons in the nucleus. The cross section is independent of energy and can be used for incident energies greater than 100 MeV/nucleon.

Kox and Shen Formulae

Both the Kox and Shen formulae are based on the strong absorption model. They express the total reaction cross section in terms of the interaction radius \(R\), the nucleus-nucleus interaction barrier \(B\), and the center-of-mass energy of the colliding system \(E_{CM}\):

\[\sigma_{R} = \pi R^{2} \left[1-\frac{B}{E_{CM}} \right].\]

Kox formula: Here \(B\) is the Coulomb barrier (\(B_c\)) of the projectile-target system and is given by

\[B_{c}=\frac{Z_{t}Z_{p}e^{2}}{r_{C} \left(A^{1/3}_{t}+A^{1/3}_{p} \right)},\]

where \(r_{C}\) = 1.3 fm, \(e\) is the electron charge and \(Z_t\), \(Z_p\) are the atomic numbers of the target and projectile nuclei. \(R\) is the interaction radius \(R_{int}\) which in the Kox formula is divided into volume and surface terms:

\[R_{int}=R_{vol}+R_{surf} .\]

\(R_{vol}\) and \(R_{surf}\) correspond to the energy-independent and energy-dependent components of the reactions, respectively. Collisions which have relatively small impact parameters are independent of both energy and mass number. These core collisions are parameterized by \(R_{vol}\). Therefore \(R_{vol}\) can depend only on the volume of the projectile and target nuclei:

\[R_{vol}=r_{0}\left( A^{1/3}_{t}+A^{1/3}_{p} \right) .\]

The second term of the interaction radius is a nuclear surface contribution and is parameterized by

\[R_{surf}=r_{0} \left[a\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}-c\right]+D.\]

The first term in brackets is the mass asymmetry which is related to the volume overlap of the projectile and target. The second term \(c\) is an energy-dependent parameter which takes into account increasing surface transparency as the projectile energy increases. \(D\) is the neutron-excess which becomes important in collisions of heavy or neutron-rich targets. It is given by

\[D=\frac{5(A_{t}-Z_{t})Z_{p}}{A_{p}A_{r}}.\]

The surface component (\(R_{surf}\)) of the interaction radius is actually not part of the simple framework of the strong absorption model, but a better reproduction of experimental results is possible when it is used.

The parameters \(r_0\), \(a\) and \(c\) are obtained using a \(\chi^{2}\) minimizing procedure with the experimental data. In this procedure the parameters \(r_{0}\) and \(a\) were fixed while \(c\) was allowed to vary only with the beam energy per nucleon. The best \(\chi^{2}\) fit is provided by \(r_{0}\) = 1.1 fm and \(a = 1.85\) with the corresponding values of \(c\) listed in Table III and shown in Fig. 12 of Ref. [eal87] as a function of beam energy per nucleon. This reference presents the values of \(c\) only in chart and figure form, which is not suitable for Monte Carlo calculations. Therefore a simple analytical function is used to calculate \(c\) in Geant4. The function is:

\[\begin{split}c &= -\frac{10}{x^{5}}+2.0 \mbox{ } \rm{for} \mbox{ } x \ge 1.5 \\ c &= \left(-\frac{10}{1.5^{5}}+2.0\right) \times \left (\frac{x}{1.5}\right)^{3} \mbox{ } \rm{for} \mbox{ } x < 1.5 ,\end{split}\]
\[x=\log(KE) ,\]

where \(KE\) is the projectile kinetic energy in units of MeV/nucleon in the laboratory system.

Shen formula: as mentioned earlier, this formula is also based on the strong absorption model, therefore it has a structure similar to the Kox formula:

\[\sigma_{R} = 10\pi R^{2} \left[1-\frac{B}{E_{CM}} \right].\]

However, different parameterized forms for \(R\) and \(B\) are applied. The interaction radius \(R\) is given by

\[R = r_0 \left[A^{1/3}_{t}+A^{1/3}_{p}+1.85\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}-C'(KE) \right] +\alpha\frac{5(A_{t}-Z_{t})Z_{p}}{A_{p}A_{r}}+\beta E^{-1/3}_{CM}\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}\]

where \(\alpha\), \(\beta\) and \(r_0\) are

\[\begin{split}\alpha &= 1 \mbox{ fm} \\ \beta &= 0.176 \mbox{ MeV}^{1/3} \cdot \mbox{fm} \\ r_0 &= 1.1 \mbox{fm}.\end{split}\]

In Ref. [SWF+89] as well, no functional form for \(C'(KE)\) is given. Hence the same simple analytical function is used by Geant4 to derive \(c\) values.

The second term \(B\) is called the nuclear-nuclear interaction barrier in the Shen formula and is given by

\[B=\frac{1.44Z_{t}Z_{p}}{r}-b\frac{R_{t}R_{p}}{R_{t}+R_{p}} (\mbox{MeV})\]

where \(r\), \(b\), \(R_t\) and \(R_p\) are given by

\[\begin{split}r &= R_{t}+R_{p}+3.2\mbox{ fm} \\ b &= 1\mbox{ MeV}\cdot \mbox{fm}^{-1} \\ R_{i} &= 1.12A^{1/3}_{i} -0.94A^{-1/3}_{i} ~ (i=t,p)\end{split}\]

The difference between the Kox and Shen formulae appears at energies below 30 MeV/nucleon. In this region the Shen formula shows better agreement with the experimental data in most cases.

Tripathi formula

Because the Tripathi formula is also based on the strong absorption model its form is similar to the Kox and Shen formulae:

(218)\[\sigma_{R} = \pi r_0^2 (A^{1/3}_{p}+A^{1/3}_{t}+\delta_{E})^{2} \left[1-\frac{B}{E_{CM}} \right],\]

where \(r_0\) = 1.1 fm. In the Tripathi formula \(B\) and \(R\) are given by

\[\begin{split}B &= \frac{1.44Z_{t}Z_{p}}{R}\\ R &= r_{p}+r_{t}+\frac{1.2(A^{1/3}_{p}+A^{1/3}_{t})}{E^{1/3}_{CM}}\end{split}\]

where \(r_i\) is the equivalent sphere radius and is related to the \(r_{rms,i}\) radius by

\[r_{i}=1.29r_{rms,i} ~ (i=p,t) .\]

\(\delta_{E}\) represents the energy-dependent term of the reaction cross section which is due mainly to transparency and Pauli blocking effects. It is given by

\[\delta_{E}=1.85S+(0.16S/E^{1/3}_{CM})-C_{KE}+[0.91(A_{t}-2Z_{t})Z_{p}/(A_{p}A_{t})],\]

where \(S\) is the mass asymmetry term given by

\[S=\frac{A^{1/3}_{p}A^{1/3}_{t}}{A^{1/3}_{p}+A^{1/3}_{t}}.\]

This is related to the volume overlap of the colliding system. The last term accounts for the isotope dependence of the reaction cross section and corresponds to the \(D\) term in the Kox formula and the second term of \(R\) in the Shen formula.

The term \(C_{KE}\) corresponds to \(c\) in Kox and \(C'(KE)\) in Shen and is given by

\[C_{E}=D_{Pauli}[1-\exp(-KE/40)]-0.292\exp(-KE/792)\times\cos(0.229KE^{0.453}) .\]

Here D\(_{Pauli}\) is related to the density dependence of the colliding system, scaled with respect to the density of the 12C+12C colliding system:

\[D_{Pauli} = 1.75 \frac{\rho_{A_p}+\rho_{A_t}}{\rho_{A_{\boldmath C}}+\rho_{A_{\boldmath C}}} .\]

The nuclear density is calculated in the hard sphere model. \(D_{Pauli}\,\) simulates the modifications of the reaction cross sections caused by Pauli blocking and is being introduced to the Tripathi formula for the first time. The modification of the reaction cross section due to Pauli blocking plays an important role at energies above 100 MeV/nucleon. Different forms of \(D_{Pauli}\,\) are used in the Tripathi formula for alpha-nucleus and lithium-nucleus collisions. For alpha-nucleus collisions,

\[D_{Pauli}=2.77 - (8.0\times 10^{-3} A_t) + (1.8\times 10^{-5}A^{2}_t) - 0.8/\{1+\exp[(250-KE)/75]\}\]

For lithium-nucleus collisions,

\[D_{Pauli}=D_{Pauli}/3.\]

Note that the Tripathi formula is not fully implemented in Geant4 and can only be used for projectile energies less than 1 GeV/nucleon.

Representative Cross Sections

Representative cross section results from the Sihver, Kox, Shen and Tripathi formulae in Geant4 are displayed in Table 38 and compared to the experimental measurements of Ref. [eal87].

Tripathi Formula for “light” Systems

For nuclear-nuclear interactions in which the projectile and/or target are light, Tripathi et al. [TCaJWW99] propose an alternative algorithm for determining the interaction cross section (implemented in the new class G4TripathiLightCrossSection). For such systems, Eq.(218) becomes:

\[\sigma _R = \pi r_0^2 [ A_p^{1/3} + A_t^{1/3} + \delta _E ]^2 \left(1 - R_C \frac{B}{E_{CM}} \right)X_m\]

\(R_C\) is a Coulomb multiplier, which is added since for light systems Eq.(218) overestimates the interaction distance, causing \(B\) (in Eq.(218)) to be underestimated. Values for \(R_C\) are given in Table 40.

\[X_m = 1 - X_1 \exp \left( { - \frac{E}{{X_1 S_L }}} \right)\]

where:

\[X_1 = 2.83 - \left( {3.1 \times 10^{ - 2} } \right)A_T + \left( {1.7 \times 10^{ - 4} } \right)A_T^2\]

except for neutron interactions with 4He, for which \(X_1\) is better approximated to 5.2, and the function \(S_L\) is given by:

\[S_L = 1.2 + 1.6\left[ {1 - \exp \left( { - \frac{E}{{15}}} \right)} \right]\]

For light nuclear-nuclear collisions, a slightly more general expression for \(C_E\) is used:

\[C_E = D\left[ {1 - \exp \left( { - \frac{E}{{T_1 }}} \right)} \right] - 0.292\exp \left( { - \frac{E}{{792}}} \right) \cdot \cos \left( {0.229E^{0.453} } \right)\]

\(D\) and \(T_1\) are dependent on the interaction, and are defined in tableTable 41.

Table 38 Representative total reaction cross sections

Proj.

Target

Elab

Exp. Results

Sihver

Kox

Shen

Tripathi

[MeV/n]

[mb]

\(^{12}\)C

\(^{12}\)C

30

1316\(\pm\)40

—

1295.04

1316.07

1269.24

83

965\(\pm\)30

—

957.183

969.107

989.96

200

864\(\pm\)45

868.571

885.502

893.854

864.56

300

858\(\pm\)60

868.571

871.088

878.293

857.414

870\(^1\) | 939\(\pm\)50

868.571

852.649

857.683

939.41

2100\(^1\) | 888\(\pm\)49

868.571

846.337

850.186

936.205

\(^{27}\)Al

30

1748\(\pm\)85

—

1801.4

1777.75

1701.03

83

1397\(\pm\)40

—

1407.64

1386.82

1405.61

200

1270\(\pm\)70

1224.95

1323.46

1301.54

1264.26

300

1220\(\pm\)85

1224.95

1306.54

1283.95

1257.62

\(^{89}\)Y

30

2724\(\pm\)300

—

2898.61

2725.23

2567.68

83

2124\(\pm\)140

—

2478.61

2344.26

2346.54

200

1885\(\pm\)120

2156.47

2391.26

2263.77

2206.01

300

1885\(\pm\)150

2156.47

2374.17

2247.55

2207.01

\(^{16}\)O

\(^{27}\)Al

30

1724\(\pm\)80

—

1965.85

1935.2

1872.23

\(^{89}\)Y

30

2707\(\pm\)330

—

3148.27

2957.06

2802.48

\(^{20}\)Ne

\(^{27}\)Al

30

2113\(\pm\)100

—

2097.86

2059.4

2016.32

100

1446\(\pm\)120

1473.87

1684.01

1658.31

1667.17

300

1328\(\pm\)120

1473.87

1611.88

1586.17

1559.16

\(^{108}\)Ag

300

2407\(\pm\)200\(^2\)

2730.69

3095.18

2939.86

2893.12
1. Data measured by Jaros et al. [eal78]
2. Natural silver was used in this measurement.
Table 40 Coulomb multiplier for light systems [TCaJWW99].

System

\(R_C\)

p + d

13.5

p + \(^3\)He

21

p + \(^4\)He

27

p + Li

2.2

d + d

13.5

d + \(^4\)He

13.5

d + C

6.0

\(^4\)He + Ta

0.6

\(^4\)He + Au

0.6
Table 41 Parameters D and T1 for light systems [TCaJWW99].

System

T1 [MeV]

D

G [MeV] (\(^4\)He + X only)

p + X

23

\(1.85 + \frac{{0.16}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\)

(Not applicable)

n + X

18

\(1.85 + \frac{{0.16}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\)

(Not applicable)

d + X

23

\(1.65 + \frac{{0.1}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\)

(Not applicable)

\(^3\)He + X)`

40

1.55

(Not applicable)

\(^4\)He + \(^4\)He

40

\(2.77 - 8.0 \times 10^{ - 3} A_T \) \(+ 1.8 \times 10^{ - 5} A_T^2 \) \(- \frac{{0.8}}{{1 + \exp \left( {\frac{{250 - E}}{G}} \right)}} \)

300

\(^4\)He + Be

25

(as for \(^4\)He + \(^4\)He)

300

\(^4\)He + N

40

(as for \(^4\)He + \(^4\)He)

500

\(^4\)He + Al

25

(as for \(^4\)He + \(^4\)He)

300

\(^4\)He + Fe

40

(as for \(^4\)He + \(^4\)He)

300

\(^4\)He + X (general)

40

(as for \(^4\)He + \(^4\)He)

75

AMZS84

V.V. Uzhinsky A.M. Zadorozhnyi and S.Yu. Shmakov. Soviet Journal of Nuclear Physics, 39:729, 1984. Russian original: Yadernaya Fizika 39 (1984) 1155.

DK85(1,2)

O.D. Dalkarov and V.A. Karmanov. Scattering of low-energy antiprotons from nuclei. Nuclear Physics A, 445(4):579–604, dec 1985. URL: https://doi.org/10.1016/0375-9474(85)90561-5, doi:10.1016/0375-9474(85)90561-5.

eal78

Jaros et al. Phys. Rev. C, 18:2273, 1978.

eal87(1,2,3,4)

S. Kox et al. Trends of total reaction cross sections for heavy ion collisions in the intermediate energy range. Phys. Rev. C, 35:1678–1691, 1987.

eal72

S.P. Denisov et al. Nuclear Physics B, 31:253, 1972.

eal11(1,2,3)

V. Uzhinsky et al. Physics Letters B, 705:235, 2011.

Fra68

V. Franco. Phys. Rev., 175:1376, 1968.

FG66(1,2)

V. Franco and R. J. Glauber. High-energy deuteron cross sections. Physical Review, 142(4):1195–1214, feb 1966. URL: https://doi.org/10.1103/PhysRev.142.1195, doi:10.1103/physrev.142.1195.

Gri09a

V. M. Grichine. A simplified glauber model for hadron–nucleus cross sections. The European Physical Journal C, 62(2):399–404, may 2009. URL: https://doi.org/10.1140/epjc/s10052-009-1033-z, doi:10.1140/epjc/s10052-009-1033-z.

Gri09b

V.M. Grichine. A simple model for integral hadron–nucleus and nucleus–nucleus cross-sections. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 267(14):2460–2462, jul 2009. URL: https://doi.org/10.1016/j.nimb.2009.05.020, doi:10.1016/j.nimb.2009.05.020.

SYuSZ89

V.V. Uzhinskii S.Yu. Shmakov and A.M. Zadorozhny. Computer Physics Communications, 54:125, 1989.

SWF+89(1,2)

W.-Q. Shen, B. Wang, J. Feng, W.-L. Zhan, Y.-T. Zhu, and E.-P. Feng. Total reaction cross section for heavy-ion collisions and its relation to the neutron excess degree of freedom. Nuclear Physics A, 491:130–146, 1989.

STS+93

L. Sihver, C. H. Tsao, R. Silberberg, T. Kanai, and A. F. Barghouty. Total reaction and partial cross section calculations in proton-nucleus (Zt≤26) and nucleus-nucleus reactions (Zp and Zt≤26). Physical Review C, 47(3):1225–1236, mar 1993. URL: https://doi.org/10.1103%2Fphysrevc.47.1225, doi:10.1103/physrevc.47.1225.

TCaJWW99(1,2,3,4,5)

R K Tripathi, F A Cucinotta, and and J W Wilson. Universal parameterization of absorption cross-sections - light systems. NASA Technical Paper, 1999.

TCW97

R. K. Tripathi, F. A. Cucinotta, and J. W. Wilson. Universal parameterization of absorption cross sections. Technical Report TP-3621, NASA Technical Paper, 1997.

WBB09

M. Rybczynski W. Broniowski and P. Bozek. Computer Physics Communications, 180:69, 2009.

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