[1211] | 1 | |
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| 2 | \chapter{Total Reaction Cross Section in Nucleus-nucleus Reactions} |
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| 3 | \noindent |
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| 4 | The transportation of heavy ions in matter is a subject of much interest in |
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| 5 | several fields of science. An important input for simulations of this process |
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| 6 | is the total reaction cross section, which is defined as the total |
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| 7 | ($\sigma_{T}$) minus the elastic ($\sigma_{el}$) cross section for |
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| 8 | nucleus-nucleus reactions: |
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| 9 | \begin{eqnarray*} |
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| 10 | \sigma_{R} = \sigma_{T} - \sigma_{el} . |
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| 11 | \end{eqnarray*} |
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| 12 | The total reaction cross section has been studied both theoretically and |
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| 13 | experimentally and several empirical parameterizations of it have been |
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| 14 | developed. In Geant4 the total reaction cross sections are calculated using |
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| 15 | four such parameterizations: the Sihver\cite{nnc.Sihver93}, |
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| 16 | Kox\cite{nnc.Kox87}, Shen\cite{nnc.Shen89} and Tripathi\cite{nnc.Tripathi97} |
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| 17 | formulae. Each of these is discussed in order below. |
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| 18 | |
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| 19 | \section{Sihver Formula} |
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| 20 | \noindent |
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| 21 | Of the four parameterizations, the Sihver formula has the simplest form: |
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| 22 | \begin{equation} |
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| 23 | \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} |
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| 24 | \end{equation} |
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| 25 | where A$_{p}$ and A$_{t}$ are the mass numbers of the projectile and target |
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| 26 | nuclei, and |
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| 27 | \begin{eqnarray*} |
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| 28 | b_{0}=1.581-0.876(A^{-1/3}_{p} + A^{-1/3}_{t}) , |
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| 29 | \end{eqnarray*} |
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| 30 | \begin{eqnarray*} |
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| 31 | r_{0}=1.36fm. |
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| 32 | \end{eqnarray*} |
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| 33 | It consists of a nuclear geometrical term $(A^{1/3}_p + A^{1/3}_t)$ and an |
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| 34 | overlap or transparency parameter ($b_0$) for nucleons in the nucleus. The |
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| 35 | cross section is independent of energy and can be used for incident energies |
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| 36 | greater than 100 MeV/nucleon. |
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| 37 | |
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| 38 | \section{Kox and Shen Formulae} |
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| 39 | \noindent |
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| 40 | Both the Kox and Shen formulae are based on the strong absorption model. They |
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| 41 | express the total reaction cross section in terms of the interaction radius |
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| 42 | $R$, the nucleus-nucleus interaction barrier $B$, and the center-of-mass energy |
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| 43 | of the colliding system $E_{CM}$: |
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| 44 | \begin{equation} |
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| 45 | \sigma_{R} = \pi R^{2}[1-\frac{B}{E_{CM}}]. |
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| 46 | \end{equation} |
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| 47 | |
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| 48 | \noindent |
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| 49 | {\bf Kox formula:} Here $B$ is the Coulomb barrier ($B_c$) of the |
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| 50 | projectile-target system and is given by |
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| 51 | \begin{eqnarray*} |
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| 52 | B_{c}=\frac{Z_{t}Z_{p}e^{2}}{r_{C}(A^{1/3}_{t}+A^{1/3}_{p})}, |
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| 53 | \end{eqnarray*} |
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| 54 | where $r_{C}$ = 1.3 fm, $e$ is the electron charge and $Z_t$, $Z_p$ are the |
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| 55 | atomic numbers of the target and projectile nuclei. $R$ is the interaction |
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| 56 | radius $R_{int}$ which in the Kox formula is divided into volume and surface |
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| 57 | terms: |
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| 58 | \begin{eqnarray*} |
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| 59 | R_{int}=R_{vol}+R_{surf} . |
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| 60 | \end{eqnarray*} |
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| 61 | $R_{vol}$ and $R_{surf}$ correspond to the energy-independent and |
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| 62 | energy-dependent components of the reactions, respectively. Collisions which |
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| 63 | have relatively small impact parameters are independent of both energy and mass |
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| 64 | number. These core collisions are parameterized by $R_{vol}$. Therefore |
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| 65 | $R_{vol}$ can depend only on the volume of the projectile and target nuclei: |
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| 66 | \begin{eqnarray*} |
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| 67 | R_{vol}=r_{0}(A^{1/3}_{t}+A^{1/3}_{p}) . |
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| 68 | \end{eqnarray*} |
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| 69 | |
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| 70 | The second term of the interaction radius is a nuclear surface contribution and |
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| 71 | is parameterized by |
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| 72 | \begin{eqnarray*} |
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| 73 | R_{surf}=r_{0}[a\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}-c]+D. |
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| 74 | \end{eqnarray*} |
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| 75 | |
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| 76 | The first term in brackets is the mass asymmetry which is related to the volume |
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| 77 | overlap of the projectile and target. The second term $c$ is an |
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| 78 | energy-dependent parameter which takes into account increasing surface |
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| 79 | transparency as the projectile energy increases. $D$ is the neutron-excess |
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| 80 | which becomes important in collisions of heavy or neutron-rich targets. It is |
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| 81 | given by |
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| 82 | \begin{eqnarray*} |
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| 83 | D=\frac{5(A_{t}-Z_{t})Z_{p}}{A_{p}A_{r}}. |
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| 84 | \end{eqnarray*} |
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| 85 | The surface component ($R_{surf}$) of the interaction radius is actually not |
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| 86 | part of the simple framework of the strong absorption model, but a better |
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| 87 | reproduction of experimental results is possible when it is used. |
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| 88 | |
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| 89 | The parameters $r_0$, $a$ and $c$ are obtained using a $\chi^{2}$ minimizing |
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| 90 | procedure with the experimental data. In this procedure the parameters $r_{0}$ |
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| 91 | and $a$ were fixed while $c$ was allowed to vary only with the beam energy per |
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| 92 | nucleon. The best $\chi^{2}$ fit is provided by $r_{0}$ = 1.1 fm and |
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| 93 | $a = 1.85$ with the corresponding values of $c$ listed in Table III and shown |
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| 94 | in Fig.~12 of Ref.~\cite{nnc.Kox87} as a function of beam energy per nucleon. |
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| 95 | This reference presents the values of $c$ only in chart and figure form, which |
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| 96 | is not suitable for Monte Carlo calculations. Therefore a simple analytical |
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| 97 | function is used to calculate $c$ in Geant4. The function is: |
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| 98 | \begin{eqnarray*} |
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| 99 | c=-\frac{10}{x^{5}}+2.0 \mbox{ } \rm{for} \mbox{ } x \ge 1.5 |
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| 100 | \end{eqnarray*} |
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| 101 | \begin{eqnarray*} |
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| 102 | c=(-\frac{10}{1.5^{5}}+2.0)\times(\frac{x}{1.5})^{3} \mbox{ } \rm{for} \mbox{ } x < 1.5 , |
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| 103 | \end{eqnarray*} |
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| 104 | \begin{eqnarray*} |
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| 105 | x=log(KE) , |
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| 106 | \end{eqnarray*} |
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| 107 | where $KE$ is the projectile kinetic energy in units of MeV/nucleon in the |
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| 108 | laboratory system. \\ |
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| 109 | |
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| 110 | \noindent |
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| 111 | {\bf Shen formula:} as mentioned earlier, this formula is also based on the |
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| 112 | strong absorption model, therefore it has a structure similar to the Kox |
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| 113 | formula: |
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| 114 | \begin{equation} |
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| 115 | \sigma_{R} = 10\pi R^{2}[1-\frac{B}{E_{CM}}]. |
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| 116 | \end{equation} |
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| 117 | However, different parameterized forms for $R$ and $B$ are applied. The |
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| 118 | interaction radius $R$ is given by |
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| 119 | \begin{eqnarray*} |
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| 120 | 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)] \\ |
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| 121 | +\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}} |
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| 122 | \end{eqnarray*} |
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| 123 | where $\alpha$, $\beta$ and $r_0$ are |
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| 124 | \begin{eqnarray*} |
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| 125 | \alpha = 1 fm |
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| 126 | \end{eqnarray*} |
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| 127 | \begin{eqnarray*} |
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| 128 | \beta = 0.176MeV^{1/3} \cdot fm |
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| 129 | \end{eqnarray*} |
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| 130 | \begin{eqnarray*} |
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| 131 | r_{0}= 1.1 fm |
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| 132 | \end{eqnarray*} |
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| 133 | In Ref.~\cite{nnc.Shen89} as well, no functional form for $C'(KE)$ is given. |
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| 134 | Hence the same simple analytical function is used by Geant4 to derive $c$ |
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| 135 | values. |
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| 136 | |
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| 137 | The second term $B$ is called the nuclear-nuclear interaction barrier in the |
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| 138 | Shen formula and is given by |
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| 139 | \begin{eqnarray*} |
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| 140 | B=\frac{1.44Z_{t}Z_{p}}{r}-b\frac{R_{t}R_{p}}{R_{t}+R_{p}} (MeV) |
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| 141 | \end{eqnarray*} |
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| 142 | where $r$, $b$, $R_t$ and $R_p$ are given by |
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| 143 | \begin{eqnarray*} |
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| 144 | r=R_{t}+R_{p}+3.2fm |
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| 145 | \end{eqnarray*} |
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| 146 | \begin{eqnarray*} |
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| 147 | b=1MeV\cdot fm^{-1} |
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| 148 | \end{eqnarray*} |
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| 149 | \begin{eqnarray*} |
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| 150 | R_{i}=1.12A^{1/3}_{i} -0.94A^{-1/3}_{i} ~ (i=t,p) |
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| 151 | \end{eqnarray*} |
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| 152 | The difference between the Kox and Shen formulae appears at energies below |
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| 153 | 30 MeV/nucleon. In this region the Shen formula shows better agreement with the |
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| 154 | experimental data in most cases. |
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| 155 | |
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| 156 | \section{Tripathi formula} |
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| 157 | \noindent |
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| 158 | Because the Tripathi formula is also based on the strong absorption model its |
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| 159 | form is similar to the Kox and Shen formulae: |
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| 160 | \begin{equation} |
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| 161 | \sigma_{R} = \pi r_0^2 (A^{1/3}_{p}+A^{1/3}_{t}+\delta_{E})^{2}[1-\frac{B}{E_{CM}}], |
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| 162 | \label{eqn15.4} |
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| 163 | \end{equation} |
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| 164 | where $r_0$ = 1.1 fm. In the Tripathi formula $B$ and $R$ are given by |
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| 165 | \begin{eqnarray*} |
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| 166 | B=\frac{1.44Z_{t}Z_{p}}{R} |
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| 167 | \end{eqnarray*} |
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| 168 | \begin{eqnarray*} |
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| 169 | R=r_{p}+r_{t}+\frac{1.2(A^{1/3}_{p}+A^{1/3}_{t})}{E^{1/3}_{CM}} |
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| 170 | \end{eqnarray*} |
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| 171 | where $r_i$ is the equivalent sphere radius and is related to the $r_{rms,i}$ |
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| 172 | radius by |
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| 173 | |
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| 174 | \[ |
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| 175 | r_{i}=1.29r_{rms,i} ~ (i=p,t) . |
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| 176 | \] |
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| 177 | |
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| 178 | $\delta_{E}$ represents the energy-dependent term of the reaction cross section |
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| 179 | which is due mainly to transparency and Pauli blocking effects. It is given by |
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| 180 | \begin{eqnarray*} |
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| 181 | \delta_{E}=1.85S+(0.16S/E^{1/3}_{CM})-C_{KE}+[0.91(A_{t}-2Z_{t})Z_{p}/(A_{p}A_{t})], |
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| 182 | \end{eqnarray*} |
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| 183 | where $S$ is the mass asymmetry term given by |
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| 184 | \begin{eqnarray*} |
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| 185 | S=\frac{A^{1/3}_{p}A^{1/3}_{t}}{A^{1/3}_{p}+A^{1/3}_{t}}. |
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| 186 | \end{eqnarray*} |
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| 187 | This is related to the volume overlap of the colliding system. The last term |
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| 188 | accounts for the isotope dependence of the reaction cross section and |
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| 189 | corresponds to the $D$ term in the Kox formula and the second term of $R$ in |
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| 190 | the Shen formula. |
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| 191 | |
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| 192 | The term $C_{KE}$ corresponds to $c$ in Kox and $C'(KE)$ in Shen and is given |
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| 193 | by |
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| 194 | \begin{eqnarray*} |
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| 195 | C_{E}=D_{Pauli}[1-\exp(-KE/40)]-0.292\exp(-KE/792)\times\cos(0.229KE^{0.453}) .\,\\ |
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| 196 | \end{eqnarray*} |
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| 197 | Here D$_{Pauli}$ is related to the density dependence of the colliding system, |
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| 198 | scaled with respect to the density of the $^{12}$C+$^{12}$C colliding |
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| 199 | system: \\ |
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| 200 | \begin{eqnarray*} |
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| 201 | D_{Pauli} = 1.75 \frac{\rho_{A_p}+\rho_{A_t}}{\rho_{A_{\boldmath C}}+\rho_{A_{\boldmath C}}} . |
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| 202 | \end{eqnarray*} |
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| 203 | The nuclear density is calculated in the hard sphere model. $D_{Pauli}\,$ |
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| 204 | simulates the modifications of the reaction cross sections caused by Pauli |
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| 205 | blocking and is being introduced to the Tripathi formula for the first time. |
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| 206 | The modification of the reaction cross section due to Pauli blocking plays an |
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| 207 | important role at energies above 100 MeV/nucleon. Different forms of |
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| 208 | $D_{Pauli}\,$ are used in the Tripathi formula for alpha-nucleus and |
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| 209 | lithium-nucleus collisions. |
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| 210 | \noindent |
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| 211 | For alpha-nucleus collisions, |
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| 212 | \begin{eqnarray*} |
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| 213 | D_{Pauli}=2.77 - (8.0\times 10^{-3} A_t) + (1.8\times 10^{-5}A^{2}_t) \\ |
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| 214 | - 0.8/\{1+\exp[(250-KE)/75]\}\, |
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| 215 | \end{eqnarray*} |
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| 216 | \noindent |
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| 217 | For lithium-nucleus collisions, |
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| 218 | \begin{eqnarray*} |
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| 219 | D_{Pauli}=D_{Pauli}/3. |
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| 220 | \end{eqnarray*} |
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| 221 | Note that the Tripathi formula is not fully implemented in Geant4 and can only |
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| 222 | be used for projectile energies less than 1 GeV/nucleon. |
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| 223 | |
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| 224 | |
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| 225 | \section{Representative Cross Sections} |
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| 226 | |
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| 227 | |
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| 228 | \noindent |
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| 229 | Representative cross section results from the Sihver, Kox, Shen and Tripathi |
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| 230 | formulae in Geant4 are displayed in Table I and compared to the experimental |
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| 231 | measurements of Ref.~\cite{nnc.Kox87}. |
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| 232 | |
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| 233 | \section{Tripathi Formula for "light" Systems} |
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| 234 | \label{TripathiLight} |
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| 235 | For nuclear-nuclear interactions in which the projectile and/or target are |
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| 236 | light, Tripathi {\normalsize\it{et al}} \cite{RefTripathiLight} propose an |
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| 237 | alternative algorithm for determining the interaction cross section |
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| 238 | (implemented in the new class G4TripathiLightCrossSection). For such systems, |
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| 239 | Eq.\ref{eqn15.4} becomes: |
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| 240 | |
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| 241 | \begin{equation} |
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| 242 | \sigma _R = \pi r_0^2 [ A_p^{1/3} + A_t^{1/3} + \delta _E ]^2 |
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| 243 | (1 - R_C \frac{B}{E_{CM}})X_m |
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| 244 | \label{eqn15.6} |
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| 245 | \end{equation} |
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| 246 | |
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| 247 | \noindent $R_C$ is a Coulomb multiplier, which is added since for light |
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| 248 | systems Eq. \ref{eqn15.4} overestimates the interaction distance, |
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| 249 | causing $B$ (in Eq. \ref{eqn15.4}) to be underestimated. Values for $R_C$ are |
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| 250 | given in Table \ref{tab15.1}. |
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| 251 | |
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| 252 | \begin{equation} |
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| 253 | X_m = 1 - X_1 \exp \left( { - \frac{E}{{X_1 S_L }}} \right) |
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| 254 | \label{eqn15.7} |
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| 255 | \end{equation} |
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| 256 | |
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| 257 | \noindent where: |
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| 258 | |
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| 259 | \begin{equation} |
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| 260 | X_1 = 2.83 - \left( {3.1 \times 10^{ - 2} } \right)A_T + \left( {1.7 \times 10^{ - 4} } \right)A_T^2 |
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| 261 | \label{eqn15.8} |
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| 262 | \end{equation} |
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| 263 | |
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| 264 | \noindent except for neutron interactions with $^4$He, for which $X_1$ is |
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| 265 | better approximated to 5.2, and the function $S_L$ is given by: |
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| 266 | |
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| 267 | \begin{equation} |
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| 268 | S_L = 1.2 + 1.6\left[ {1 - \exp \left( { - \frac{E}{{15}}} \right)} \right] |
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| 269 | \label{eqn15.9} |
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| 270 | \end{equation} |
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| 271 | |
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| 272 | \noindent For light nuclear-nuclear collisions, a slightly more general |
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| 273 | expression for $C_E$ is used: |
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| 274 | |
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| 275 | \begin{equation} |
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| 276 | 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) |
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| 277 | \label{eqn15.10} |
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| 278 | \end{equation} |
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| 279 | |
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| 280 | \noindent $D$ and $T_1$ are dependent on the interaction, and are defined |
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| 281 | in table \ref{tab15.2}. |
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| 282 | |
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| 283 | \begin{table} |
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| 284 | \begin{center} |
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| 285 | \caption{Representative total reaction cross sections} |
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| 286 | \label{nn-x-section-tb} |
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| 287 | \begin{tabular}{|c|c|c|c|c|c|c|c|} |
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| 288 | \hline |
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| 289 | Proj.&Target&Elab&Exp. Results&Sihver&Kox&Shen&Tripathi\\ |
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| 290 | &&[MeV/n]&[mb]&&&&\\ |
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| 291 | \hline |
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| 292 | &&&&&&&\\ |
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| 293 | $^{12}$C&$^{12}$C&30&1316$\pm$40&---&1295.04&1316.07&1269.24\\ |
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| 294 | &&83&965$\pm$30&---&957.183&969.107&989.96\\ |
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| 295 | &&200&864$\pm$45&868.571&885.502&893.854&864.56\\ |
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| 296 | &&300&858$\pm$60&868.571&871.088&878.293&857.414\\ |
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| 297 | &&870$^1$&939$\pm$50&868.571&852.649&857.683&939.41\\ |
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| 298 | &&2100$^1$&888$\pm$49&868.571&846.337&850.186&936.205\\ |
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| 299 | &$^{27}$Al&30&1748$\pm$85&---&1801.4&1777.75&1701.03\\ |
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| 300 | &&83&1397$\pm$40&---&1407.64&1386.82&1405.61\\ |
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| 301 | &&200&1270$\pm$70&1224.95&1323.46&1301.54&1264.26\\ |
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| 302 | &&300&1220$\pm$85&1224.95&1306.54&1283.95&1257.62\\ |
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| 303 | &$^{89}$Y&30&2724$\pm$300&---&2898.61&2725.23&2567.68\\ |
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| 304 | &&83&2124$\pm$140&---&2478.61&2344.26&2346.54\\ |
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| 305 | &&200&1885$\pm$120&2156.47&2391.26&2263.77&2206.01\\ |
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| 306 | &&300&1885$\pm$150&2156.47&2374.17&2247.55&2207.01\\ |
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| 307 | &&&&&&&\\ |
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| 308 | $^{16}$O&$^{27}$Al&30&1724$\pm$80&---&1965.85&1935.2&1872.23\\ |
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| 309 | &$^{89}$Y&30&2707$\pm$330&---&3148.27&2957.06&2802.48\\ |
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| 310 | &&&&&&&\\ |
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| 311 | $^{20}$Ne&$^{27}$Al&30&2113$\pm$100&---&2097.86&2059.4&2016.32\\ |
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| 312 | &&100&1446$\pm$120&1473.87&1684.01&1658.31&1667.17\\ |
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| 313 | &&300&1328$\pm$120&1473.87&1611.88&1586.17&1559.16\\ |
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| 314 | &$^{108}$Ag&300&2407$\pm$200$^2$&2730.69&3095.18&2939.86&2893.12\\ |
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| 315 | \hline |
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| 316 | \end{tabular} |
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| 317 | \end{center} |
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| 318 | 1. Data measured by Jaros et al. \cite{nnc.Jaros78} \\ |
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| 319 | 2. Natural silver was used in this measurement. |
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| 320 | \end{table} |
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| 321 | |
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| 322 | \begin{table} |
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| 323 | \begin{center} |
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| 324 | \caption{Coulomb multiplier for light systems \cite{RefTripathiLight}.} |
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| 325 | \label{tab15.1} % Give a unique label |
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| 326 | \begin{tabular}{cc} |
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| 327 | \hline\noalign{\smallskip} |
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| 328 | System & \(R_C\) \\ |
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| 329 | \noalign{\smallskip}\hline\noalign{\smallskip} |
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| 330 | p + d & 13.5 \\ |
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| 331 | p + $^3$He & 21 \\ |
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| 332 | p + $^4$He & 27 \\ |
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| 333 | p + Li & 2.2 \\ |
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| 334 | d + d & 13.5 \\ |
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| 335 | d + $^4$He & 13.5 \\ |
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| 336 | d + C & 6.0 \\ |
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| 337 | $^4$He + Ta & 0.6 \\ |
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| 338 | $^4$He + Au & 0.6 \\ |
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| 339 | \noalign{\smallskip}\hline |
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| 340 | \end{tabular} |
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| 341 | \end{center} |
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| 342 | \vspace*{2cm} % with the correct table height |
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| 343 | \end{table} |
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| 344 | |
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| 345 | \begin{table} |
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| 346 | \caption{Parameters D and T1 for light systems \cite{RefTripathiLight}.} |
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| 347 | \label{tab15.2} % Give a unique label |
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| 348 | % For LaTeX tables use |
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| 349 | \begin{tabular}{cccc} |
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| 350 | \hline\noalign{\smallskip} |
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| 351 | System & T1 [MeV] & D & G [MeV] \\ |
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| 352 | & & & ($^4$He + X only) \\ |
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| 353 | \noalign{\smallskip}\hline\noalign{\smallskip} |
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| 354 | p + X & 23 & |
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| 355 | \(1.85 + \frac{{0.16}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\) & |
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| 356 | (Not applicable) \\ |
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| 357 | |
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| 358 | n + X & 18 & |
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| 359 | \(1.85 + \frac{{0.16}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\) & |
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| 360 | (Not applicable) \\ |
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| 361 | |
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| 362 | d + X & 23 & |
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| 363 | \(1.65 + \frac{{0.1}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\) & |
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| 364 | (Not applicable) \\ |
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| 365 | |
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| 366 | $^3$He + X & 40 & 1.55 & (Not applicable) \\ |
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| 367 | |
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| 368 | $^4$He + $^4$He & 40 & |
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| 369 | \( |
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| 370 | \begin{array}{l} |
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| 371 | D = 2.77 - 8.0 \times 10^{ - 3} A_T \\ |
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| 372 | + 1.8 \times 10^{ - 5} A_T^2 \\ |
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| 373 | - \frac{{0.8}}{{1 + \exp \left( {\frac{{250 - E}}{G}} \right)}} \\ |
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| 374 | \end{array} |
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| 375 | \) & |
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| 376 | 300 \\ |
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| 377 | |
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| 378 | $^4$He + Be & 25 & (as for $^4$He + $^4$He) & 300 \\ |
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| 379 | $^4$He + N & 40 & (as for $^4$He + $^4$He) & 500 \\ |
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| 380 | $^4$He + Al & 25 & (as for $^4$He + $^4$He) & 300 \\ |
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| 381 | $^4$He + Fe & 40 & (as for $^4$He + $^4$He) & 300 \\ |
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| 382 | $^4$He + X (general) & 40 & (as for $^4$He + $^4$He) & 75 \\ |
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| 383 | |
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| 384 | \noalign{\smallskip}\hline |
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| 385 | \end{tabular} |
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| 386 | % Or use |
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| 387 | \vspace*{5cm} % with the correct table height |
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| 388 | \end{table} |
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| 389 | |
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| 390 | \section{Status of this document} |
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| 391 | \noindent |
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| 392 | 25.11.03 created by Tatsumi Koi\\ |
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| 393 | 28.11.03 grammar check and re-wording by D.H. Wright\\ |
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| 394 | 18.06.04 light system section added by Peter Truscott \\ |
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| 395 | |
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| 396 | \begin{latexonly} |
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| 397 | |
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| 398 | \begin{thebibliography}{99} |
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| 399 | |
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| 400 | \bibitem{nnc.Sihver93} |
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| 401 | L. Sihver et al., Phys. Rev. C47, 1225 (1993). |
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| 402 | |
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| 403 | \bibitem{nnc.Kox87} |
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| 404 | Kox et al. Phys. Rev. C35, 1678 (1987). |
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| 405 | |
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| 406 | \bibitem{nnc.Shen89} |
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| 407 | Shen et al. Nucl. Phys. A491, 130 (1989). |
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| 408 | |
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| 409 | \bibitem{nnc.Tripathi97} |
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| 410 | Tripathi et al, NASA Technical Paper 3621 (1997). |
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| 411 | |
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| 412 | \bibitem{nnc.Jaros78} |
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| 413 | Jaros et al, Phys. Rev. C 18 2273 (1978). |
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| 414 | |
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| 415 | \bibitem{RefTripathiLight} |
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| 416 | % Format for Journal Reference |
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| 417 | 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. |
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| 418 | |
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| 419 | \end{thebibliography} |
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| 420 | |
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| 421 | \end{latexonly} |
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| 422 | |
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| 423 | \begin{htmlonly} |
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| 424 | |
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| 425 | \section{Bibliography} |
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| 426 | |
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| 427 | \begin{enumerate} |
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| 428 | \item |
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| 429 | L. Sihver et al., Phys. Rev. C47, 1225 (1993). |
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| 430 | |
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| 431 | \item |
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| 432 | Kox et al. Phys. Rev. C35, 1678 (1987). |
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| 433 | |
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| 434 | \item |
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| 435 | Shen et al. Nucl. Phys. A491, 130 (1989). |
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| 436 | |
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| 437 | \item |
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| 438 | Tripathi et al, NASA Technical Paper 3621 (1997). |
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| 439 | |
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| 440 | \item |
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| 441 | Jaros et al, Phys. Rev. C 18 2273 (1978). |
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| 442 | |
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| 443 | \item |
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| 444 | % Format for Journal Reference |
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| 445 | 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. |
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| 446 | |
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| 447 | \end{enumerate} |
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| 448 | |
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| 449 | \end{htmlonly} |
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| 450 | |
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