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 [DK85, Fra68, FG66] 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 [DK85, FG66] 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 Shen[SWF+89] parameterization offers an alternative way to estimate the nucleus-nucleus total cross sections. The Shen approach is based on the strong absorption model. It expresses 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 \(E_{CM}\):

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

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

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

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.

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.

WBB09

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

Bibliography