
\chapter{Total Reaction Cross Section in Nucleus-nucleus Reactions}
\noindent
The transportation of heavy ions in matter is a subject of much interest in 
several fields of science. An important input for simulations of this process 
is the total reaction cross section, which is defined as the total 
($\sigma_{T}$) minus the elastic ($\sigma_{el}$) cross section for 
nucleus-nucleus reactions:
\begin{eqnarray*}
\sigma_{R} = \sigma_{T} - \sigma_{el} .
\end{eqnarray*}
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\cite{nnc.Sihver93}, 
Kox\cite{nnc.Kox87}, Shen\cite{nnc.Shen89} and Tripathi\cite{nnc.Tripathi97} 
formulae.  Each of these is discussed in order below.

\section{Sihver Formula}
\noindent
Of the four parameterizations, the Sihver formula has the simplest form:
\begin{equation}
\sigma_{R} = \pi r^{2}_{0}[A^{1/3}_{p} + A^{1/3}_{t} - b_{0} [A^{-1/3}_{p} + A^{-1/3}_{t}] ]^{2}
\end{equation}
where A$_{p}$ and A$_{t}$ are the mass numbers of the projectile and target 
nuclei, and
\begin{eqnarray*}
b_{0}=1.581-0.876(A^{-1/3}_{p} + A^{-1/3}_{t}) ,  
\end{eqnarray*}
\begin{eqnarray*}
r_{0}=1.36fm.
\end{eqnarray*}
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. 

\section{Kox and Shen Formulae}
\noindent
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 $E_{CM}$:
\begin{equation}
\sigma_{R} = \pi R^{2}[1-\frac{B}{E_{CM}}].
\end{equation}

\noindent
{\bf Kox formula:} Here $B$ is the Coulomb barrier ($B_c$) of the 
projectile-target system and is given by
\begin{eqnarray*}
B_{c}=\frac{Z_{t}Z_{p}e^{2}}{r_{C}(A^{1/3}_{t}+A^{1/3}_{p})},
\end{eqnarray*}
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:
\begin{eqnarray*}
R_{int}=R_{vol}+R_{surf} .
\end{eqnarray*}
$R_{vol}$ 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:
\begin{eqnarray*}
R_{vol}=r_{0}(A^{1/3}_{t}+A^{1/3}_{p}) .
\end{eqnarray*}

The second term of the interaction radius is a nuclear surface contribution and
is parameterized by
\begin{eqnarray*}
R_{surf}=r_{0}[a\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}-c]+D.
\end{eqnarray*}

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 
\begin{eqnarray*}
D=\frac{5(A_{t}-Z_{t})Z_{p}}{A_{p}A_{r}}.
\end{eqnarray*}
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.~\cite{nnc.Kox87} 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{eqnarray*}
c=-\frac{10}{x^{5}}+2.0 \mbox{   } \rm{for} \mbox{   } x \ge 1.5
\end{eqnarray*}
\begin{eqnarray*}
c=(-\frac{10}{1.5^{5}}+2.0)\times(\frac{x}{1.5})^{3} \mbox{   } \rm{for} \mbox{   } x < 1.5 ,
\end{eqnarray*}
\begin{eqnarray*}
x=log(KE) ,
\end{eqnarray*}
where $KE$ is the projectile kinetic energy in units of MeV/nucleon in the 
laboratory system. \\

\noindent
{\bf 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:
\begin{equation}
\sigma_{R} = 10\pi R^{2}[1-\frac{B}{E_{CM}}].
\end{equation}
However, different parameterized forms for $R$ and $B$ are applied.  The 
interaction radius $R$ is given by 
\begin{eqnarray*}
R=r_{0}[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)] \\
+\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}}
\end{eqnarray*}
where $\alpha$, $\beta$ and $r_0$ are
\begin{eqnarray*}
\alpha = 1 fm
\end{eqnarray*}
\begin{eqnarray*}
\beta = 0.176MeV^{1/3} \cdot fm
\end{eqnarray*}
\begin{eqnarray*}
r_{0}= 1.1 fm
\end{eqnarray*}
In Ref.~\cite{nnc.Shen89} 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
\begin{eqnarray*}
B=\frac{1.44Z_{t}Z_{p}}{r}-b\frac{R_{t}R_{p}}{R_{t}+R_{p}} (MeV)
\end{eqnarray*}
where $r$, $b$, $R_t$ and $R_p$ are given by
\begin{eqnarray*}
r=R_{t}+R_{p}+3.2fm
\end{eqnarray*}
\begin{eqnarray*}
b=1MeV\cdot fm^{-1}	
\end{eqnarray*}
\begin{eqnarray*}
R_{i}=1.12A^{1/3}_{i} -0.94A^{-1/3}_{i} ~ (i=t,p)
\end{eqnarray*}
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.

\section{Tripathi formula}
\noindent 
Because the Tripathi formula is also based on the strong absorption model its 
form is similar to the Kox and Shen formulae:
\begin{equation}
\sigma_{R} = \pi r_0^2 (A^{1/3}_{p}+A^{1/3}_{t}+\delta_{E})^{2}[1-\frac{B}{E_{CM}}],
\label{eqn15.4}
\end{equation}
where $r_0$ = 1.1 fm. In the Tripathi formula $B$ and $R$ are given by
\begin{eqnarray*}
B=\frac{1.44Z_{t}Z_{p}}{R}
\end{eqnarray*}
\begin{eqnarray*}
R=r_{p}+r_{t}+\frac{1.2(A^{1/3}_{p}+A^{1/3}_{t})}{E^{1/3}_{CM}}
\end{eqnarray*}
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 
\begin{eqnarray*}
\delta_{E}=1.85S+(0.16S/E^{1/3}_{CM})-C_{KE}+[0.91(A_{t}-2Z_{t})Z_{p}/(A_{p}A_{t})],
\end{eqnarray*}
where $S$ is the mass asymmetry term given by
\begin{eqnarray*}
S=\frac{A^{1/3}_{p}A^{1/3}_{t}}{A^{1/3}_{p}+A^{1/3}_{t}}.
\end{eqnarray*}
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 
\begin{eqnarray*}
C_{E}=D_{Pauli}[1-\exp(-KE/40)]-0.292\exp(-KE/792)\times\cos(0.229KE^{0.453}) .\,\\
\end{eqnarray*}
Here D$_{Pauli}$ is related to the density dependence of the colliding system, 
scaled with respect to the density of the $^{12}$C+$^{12}$C colliding 
system: \\
\begin{eqnarray*}
D_{Pauli} = 1.75 \frac{\rho_{A_p}+\rho_{A_t}}{\rho_{A_{\boldmath C}}+\rho_{A_{\boldmath C}}} .
\end{eqnarray*}
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. 
\noindent
For alpha-nucleus collisions,
\begin{eqnarray*}
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]\}\,
\end{eqnarray*}
\noindent
For lithium-nucleus collisions,
\begin{eqnarray*}
D_{Pauli}=D_{Pauli}/3.
\end{eqnarray*}
Note that the Tripathi formula is not fully implemented in Geant4 and can only 
be used for projectile energies less than 1 GeV/nucleon.


\section{Representative Cross Sections}


\noindent
Representative cross section results from the Sihver, Kox, Shen and Tripathi 
formulae in Geant4 are displayed in Table I and compared to the experimental 
measurements of Ref.~\cite{nnc.Kox87}.

\section{Tripathi Formula for "light" Systems}
\label{TripathiLight}
For nuclear-nuclear interactions in which the projectile and/or target are 
light, Tripathi {\normalsize\it{et al}} \cite{RefTripathiLight} propose an 
alternative algorithm for determining the interaction cross section 
(implemented in the new class G4TripathiLightCrossSection).  For such systems, 
Eq.\ref{eqn15.4} becomes:

\begin{equation}
 \sigma _R  = \pi r_0^2 [ A_p^{1/3} + A_t^{1/3} + \delta _E ]^2 
(1 - R_C \frac{B}{E_{CM}})X_m 
\label{eqn15.6}
\end{equation}

\noindent $R_C$ is a Coulomb multiplier, which is added since for light 
systems Eq. \ref{eqn15.4} overestimates the interaction distance, 
causing $B$ (in Eq. \ref{eqn15.4}) to be underestimated.  Values for $R_C$ are 
given in Table \ref{tab15.1}.

\begin{equation}
X_m  = 1 - X_1 \exp \left( { - \frac{E}{{X_1 S_L }}} \right)
\label{eqn15.7}
\end{equation}

\noindent where:

\begin{equation}
 X_1  = 2.83 - \left( {3.1 \times 10^{ - 2} } \right)A_T  + \left( {1.7 \times 10^{ - 4} } \right)A_T^2 
\label{eqn15.8}
\end{equation}

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

\begin{equation}
S_L  = 1.2 + 1.6\left[ {1 - \exp \left( { - \frac{E}{{15}}} \right)} \right]
\label{eqn15.9}
\end{equation}

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

\begin{equation}
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)
\label{eqn15.10}
\end{equation}

\noindent $D$ and $T_1$ are dependent on the interaction, and are defined 
in table \ref{tab15.2}.

\begin{table}
\begin{center}
\caption{Representative total reaction cross sections}
\label{nn-x-section-tb}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Proj.&Target&Elab&Exp. Results&Sihver&Kox&Shen&Tripathi\\
&&[MeV/n]&[mb]&&&&\\
\hline
&&&&&&&\\
$^{12}$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\\
\hline
\end{tabular}
\end{center}
1. Data measured by Jaros et al. \cite{nnc.Jaros78} \\ 
2. Natural silver was used in this measurement.
\end{table}

\begin{table}
\begin{center}
\caption{Coulomb multiplier for light systems \cite{RefTripathiLight}.}
\label{tab15.1}       % Give a unique label
\begin{tabular}{cc}
\hline\noalign{\smallskip}
System & \(R_C\) \\
\noalign{\smallskip}\hline\noalign{\smallskip}
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 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\vspace*{2cm}  % with the correct table height
\end{table}

\begin{table}
\caption{Parameters D and T1 for light systems \cite{RefTripathiLight}.}
\label{tab15.2}       % Give a unique label
% For LaTeX tables use
\begin{tabular}{cccc}
\hline\noalign{\smallskip}
System & T1 [MeV] & D & G [MeV] \\
& & & ($^4$He + X only) \\
\noalign{\smallskip}\hline\noalign{\smallskip}
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 &
\(
\begin{array}{l}
 D = 2.77 - 8.0 \times 10^{ - 3} A_T  \\ 
  + 1.8 \times 10^{ - 5} A_T^2  \\ 
  - \frac{{0.8}}{{1 + \exp \left( {\frac{{250 - E}}{G}} \right)}} \\ 
 \end{array}
\) &
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 \\

\noalign{\smallskip}\hline
\end{tabular}
% Or use
\vspace*{5cm}  % with the correct table height
\end{table}

\section{Status of this document}
\noindent
25.11.03  created by Tatsumi Koi\\
28.11.03  grammar check and re-wording by D.H. Wright\\
18.06.04  light system section added by Peter Truscott \\

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{nnc.Sihver93} 
 L. Sihver et al., Phys. Rev. C47, 1225 (1993).

\bibitem{nnc.Kox87} 
Kox et al. Phys. Rev. C35, 1678 (1987).

\bibitem{nnc.Shen89} 
Shen et al. Nucl. Phys. A491, 130 (1989).

\bibitem{nnc.Tripathi97} 
Tripathi et al, NASA Technical Paper 3621 (1997).

\bibitem{nnc.Jaros78} 
Jaros et al, Phys. Rev. C 18 2273 (1978).

\bibitem{RefTripathiLight}
% Format for Journal Reference
R K Tripathi, F A Cucinotta, and J W Wilson, "Universal parameterization of absorption cross-sections - Light systems," NASA Technical Paper TP-1999-209726, 1999.

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item 
 L. Sihver et al., Phys. Rev. C47, 1225 (1993).

\item 
Kox et al. Phys. Rev. C35, 1678 (1987).

\item 
Shen et al. Nucl. Phys. A491, 130 (1989).

\item 
Tripathi et al, NASA Technical Paper 3621 (1997).

\item 
Jaros et al, Phys. Rev. C 18 2273 (1978).

\item
% Format for Journal Reference
R K Tripathi, F A Cucinotta, and J W Wilson, "Universal parameterization of absorption cross-sections - Light systems," NASA Technical Paper TP-1999-209726, 1999.

\end{enumerate}

\end{htmlonly}

