1 | \section{Model description.} |
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2 | |
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3 | \hspace{1.0em} The Weisskopf treatment is an application of the |
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4 | detailed balance principle that relates the probabilities to go from |
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5 | a state $i$ to another $d$ and viceversa through the density of states |
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6 | in the two systems: |
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7 | %% |
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8 | \begin{equation} |
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9 | \label{evap:1} |
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10 | P_{i \rightarrow d} \rho(i) = P_{d \rightarrow i} \rho(d) |
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11 | \end{equation} |
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12 | %% |
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13 | where $P_{d \rightarrow i}$ is the probability per unit of time of a |
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14 | nucleus $d$ captures a particle $j$ and form a compound nucleus $i$ |
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15 | which is proportional to the compound nucleus cross section |
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16 | $\sigma_{\mathrm{inv}}$. Thus, the probability that a parent nucleus |
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17 | $i$ with an excitation energy $E^*$ emits a particle $j$ in its ground |
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18 | state with kinetic energy $\varepsilon$ is |
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19 | %% |
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20 | \begin{equation} |
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21 | \label{evap:2} |
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22 | P_j(\varepsilon) \mathrm{d}\varepsilon = g_j |
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23 | \sigma_{\mathrm{inv}}(\varepsilon) \frac{\rho_d(E_{\mathrm{max}} - |
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24 | \varepsilon)}{\rho_i(E^*)} \varepsilon \mathrm{d}\varepsilon |
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25 | \end{equation} |
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26 | %% |
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27 | where $\rho_i(E^*)$ is the level density of the evaporating nucleus, |
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28 | $\rho_d(E_{\mathrm{max}} - \varepsilon)$ that of the daugther |
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29 | (residual) nucleus after emission of a fragment $j$ and |
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30 | $E_{\mathrm{max}}$ is the maximum energy that can be carried by the |
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31 | ejectile. With the spin $s_j$ and the mass $m_j$ of the emitted |
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32 | particle, $g_j$ is expressed as $g_j = ( 2 s_j + 1 ) m_j / \pi^2 |
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33 | \hbar^2$. |
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34 | |
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35 | This formula must be implemented with a suitable form for the level |
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36 | density and inverse reaction cross section. We have followed, like |
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37 | many other implementations, the original work of Dostrovsky \textit{et al.} |
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38 | \cite{evap.Dostrovsky59} (which represents the first Monte Carlo |
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39 | code for the evaporation process) with slight modifications. The |
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40 | advantage of the Dostrovsky model is that it leds to a simple |
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41 | expression for equation \ref{evap:2} that can be analytically |
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42 | integrated and used for Monte Carlo sampling. |
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43 | |
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44 | |
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45 | \subsection{Cross sections for inverse reactions.} |
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46 | |
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47 | \hspace{1.0em} The cross section for inverse reaction is expressed by |
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48 | means of empirical equation \cite{evap.Dostrovsky59} |
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49 | %% |
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50 | \begin{equation} |
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51 | \label{evap:3} |
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52 | \sigma_{\mathrm{inv}}(\varepsilon) = \sigma_g \alpha \left( 1 + |
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53 | \frac{\beta}{\varepsilon} \right) |
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54 | \end{equation} |
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55 | %% |
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56 | where $\sigma_g = \pi R^2$ is the geometric cross section. |
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57 | |
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58 | In the case of neutrons, $\alpha = 0.76+2.2A^{-\frac{1}{3}}$ and |
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59 | $\beta = (2.12 A^{-\frac{2}{3}} - 0.050)/\alpha$ MeV. This equation |
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60 | gives a good agreement to those calculated from continuum theory |
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61 | \cite{evap.Blatt52} for intermediate nuclei down to $\varepsilon \sim 0.05$ |
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62 | MeV. For lower energies $\sigma_{\mathrm{inv},n}(\varepsilon)$ tends |
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63 | toward infinity, but this causes no difficulty because only the |
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64 | product $\sigma_{\mathrm{inv},n}(\varepsilon)\varepsilon$ enters in |
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65 | equation \ref{evap:2}. It should be noted, that the inverse cross |
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66 | section needed in \ref{evap:2} is that between a neutron of kinetic |
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67 | energy $\varepsilon$ and a nucleus in an excited state. |
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68 | |
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69 | For charged particles (p, d, t, $^3$He and $\alpha$), $\alpha = |
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70 | (1+c_j)$ and $\beta = -V_j$, where $c_j$ is a set of parameters |
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71 | calculated by Shapiro \cite{evap.Shapiro53} in order to provide a good fit |
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72 | to the continuum theory \cite{evap.Blatt52} cross sections and $V_j$ is the |
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73 | Coulomb barrier. |
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74 | |
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75 | \subsection{Coulomb barriers.} |
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76 | |
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77 | \hspace{1.0em} Coulomb repulsion, as calculated from elementary |
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78 | electrostatics are not directly applicable to the computation of |
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79 | reaction barriers but must be corrected in several ways. The first |
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80 | correction is for the quantum mechanical phenomenoon of barrier |
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81 | penetration. The proper quantum mechanical expressions for barrier |
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82 | penetration are far too complex to be used if one wishes to retain |
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83 | equation \ref{evap:2} in an integrable form. This can be approximately |
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84 | taken into account by multiplying the electrostatic Coulomb barrier by |
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85 | a coefficient $k_j$ designed to reproduce the barrier penetration |
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86 | approximately whose values are tabulated \cite{evap.Shapiro53}. |
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87 | %% |
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88 | \begin{equation} |
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89 | \label{evap:4} |
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90 | V_j = k_j \frac{Z_j Z_d e^2}{R_c} |
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91 | \end{equation} |
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92 | %% |
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93 | The second correction is for the separation of the centers of the |
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94 | nuclei at contact, $R_c$. We have computed this separation as $R_c = |
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95 | R_j + R_d$ where $R_{j,d} = r_c A_{j,d}^{1/3}$ and $r_c$ is |
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96 | given \cite{evap.Iljinov94} by |
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97 | %% |
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98 | \begin{equation} |
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99 | \label{evap:5} |
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100 | r_c = 2.173 \frac{1 + 0.006103 Z_j Z_d}{1 + 0.009443 Z_j Z_d} |
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101 | \end{equation} |
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102 | %% |
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103 | |
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104 | |
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105 | \subsection{Level densities.} |
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106 | |
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107 | \hspace{1.0em} The simplest and most widely used level density based |
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108 | on the Fermi gas model are those of Weisskopf \cite{evap.Weisskopf37} for |
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109 | a completely degenerate Fermi gas. We use this approach with the |
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110 | corrections for nucleon pairing proposed by Hurwitz and Bethe |
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111 | \cite{evap.Hurwitz51} which takes into account the displacements of the |
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112 | ground state: |
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113 | %% |
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114 | \begin{equation} |
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115 | \label{evap:6} |
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116 | \rho(E) = C \exp{\left( 2 \sqrt{a(E-\delta)} \right)} |
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117 | \end{equation} |
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118 | %% |
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119 | where $C$ is considered as constant and does not need to be specified |
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120 | since only ratios of level densities enter in equation \ref{evap:2}. |
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121 | $\delta$ is the pairing energy correction of the daughter nucleus |
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122 | evaluated by Cook \textit{et al.} \cite{evap.Cook67} and Gilbert and |
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123 | Cameron \cite{evap.Gilbert65} for those values not evaluated by Cook |
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124 | \textit{et al.}. The level density parameter is calculated according |
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125 | to: |
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126 | %% |
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127 | \begin{equation} |
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128 | \label{evap:7} |
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129 | a(E,A,Z) = \tilde{a}(A) \left \{ 1 + \frac{\delta}{E} [1 - \exp(-\gamma |
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130 | E)] \right\} |
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131 | \end{equation} |
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132 | %% |
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133 | and the parameters calculated by Iljinov \textit{et al.} |
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134 | \cite{evap.Iljinov92} and shell corrections of Truran, Cameron and Hilf |
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135 | \cite{evap.Truran70}. |
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136 | |
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137 | \subsection{Maximum energy available for evaporation.} |
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138 | |
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139 | \hspace{1.0em} The maximum energy avilable for the evaporation process |
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140 | (\textit{i.e.} the maximum kinetic energy of the outgoing fragment) is |
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141 | usually computed like $E^* - \delta - Q_j$ where is the separation |
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142 | energy of the fragment $j$: $Q_j = M_i - M_d - M_j$ and $M_i$, $M_d$ |
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143 | and $M_j$ are the nclear masses of the compound, residual and |
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144 | evporated nuclei respectively. However, that expression does not |
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145 | consider the recoil energy of the residual nucleus. In order to take |
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146 | into account the recoil energy we use the expression |
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147 | %% |
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148 | \begin{equation} |
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149 | \label{evap:8} |
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150 | \varepsilon_j^{\mathrm{max}} = \frac{(M_i + E^* - \delta)^2 + M_j^2 |
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151 | - M_d^2}{2 (M_i + E^* - \delta)} - M_j |
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152 | \end{equation} |
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153 | |
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154 | \subsection{Total decay width.} |
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155 | |
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156 | \hspace{1.0em} The total decay width for evaporation of a fragment $j$ |
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157 | can be obtained by integrating equation \ref{evap:2} over kinetic |
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158 | energy |
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159 | %% |
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160 | \begin{equation} |
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161 | \label{evap:9} |
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162 | \Gamma_j = \hbar \int_{V_j}^{\varepsilon_j^{\mathrm{max}}} |
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163 | P(\varepsilon_j) \mathrm{d}\varepsilon_j |
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164 | \end{equation} |
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165 | %% |
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166 | This integration can be performed analiticaly if we use equation |
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167 | \ref{evap:6} for level densities and equation \ref{evap:3} for inverse |
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168 | reaction cross section. Thus, the total width is given by |
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169 | %% |
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170 | \begin{eqnarray} |
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171 | \label{eq:10} |
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172 | \Gamma_j = \frac{g_j m_j R_d^2}{2 \pi \hbar^2} \frac{\alpha}{a_d^2} |
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173 | & \times & |
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174 | \Biggl \lgroup \biggl \{ \left(\beta a_d - \frac{3}{2}\right) + a_d |
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175 | (\varepsilon_j^{\mathrm{max}} - V_j) \biggr \} |
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176 | \exp{\left\{-\sqrt{a_i(E^*-\delta_i)}\right\}} + \nonumber \\ |
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177 | & & \biggl \{ (2\beta a_d - 3) \sqrt{a_d |
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178 | (\varepsilon_j^{\mathrm{max}} - V_j)} + 2 a_d |
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179 | (\varepsilon_j^{\mathrm{max}} - V_j) \biggr \} \times \nonumber \\ |
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180 | & & \exp{ |
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181 | \left\{ |
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182 | 2 \left[ |
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183 | \sqrt{a_d(\varepsilon_j^{\mathrm{max}} - V_j)} |
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184 | - |
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185 | \sqrt{a_i(E^* - \delta_i)} |
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186 | \right] |
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187 | \right\} |
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188 | } |
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189 | \Biggr \rgroup |
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190 | \end{eqnarray} |
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191 | %% |
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192 | where $a_d = a(A_d,Z_d,\varepsilon_j^{\mathrm{max}})$ and $a_i = |
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193 | a(A_i,Z_i,E^*)$. |
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194 | |
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195 | \section{GEM Model} |
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196 | |
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197 | \hspace{1.0em} As an alternative model we have implemented the |
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198 | generalized evaporation model (GEM) by Furihata |
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199 | \cite{evap.Furihata00}. This model considers emission of fragments heavier |
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200 | than $\alpha$ particles and uses a more accurate level density |
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201 | function for total decay width instead of the approximation used by |
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202 | Dostrovsky. We use the same set of parameters but for heavy ejectiles |
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203 | the parameters determined by Matsuse \textit{et al.} \cite{evap.Matsuse82} |
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204 | are used. |
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205 | |
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206 | Based on the Fermi gas model, the level density function is expressed |
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207 | as |
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208 | %% |
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209 | \begin{equation} |
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210 | \label{evap:11} |
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211 | \rho(E) = |
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212 | \left\{ \begin{array}{ll} |
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213 | \frac{\sqrt{\pi}}{12} |
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214 | \frac{e^{2\sqrt{a(E-\delta)}}}{a^{1/4}(E-\delta)^{5/4}} & |
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215 | \rm{for} \quad E \geq E_x \\ |
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216 | \frac{1}{T} e^{(E-E_0)/T} & \rm{for} \quad E < E_x |
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217 | \end{array} \right. |
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218 | \end{equation} |
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219 | %% |
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220 | where $E_x = U_x + \delta$ and $U_x = 150/M_d + 2.5$ ($M_d$ is the |
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221 | mass of the daughter nucleus). Nuclear temperature $T$ is given as |
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222 | $1/T = \sqrt{a/U_x} - 1.5U_x$, and $E_0$ is defined as $E_0 = E_x - |
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223 | T(\log{T} - \log{a}/4 - (5/4)\log{U_x} + 2 \sqrt{aU_x})$. |
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224 | |
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225 | By substituting equation \ref{evap:11} into equation \ref{evap:2} and |
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226 | integrating over kinetic energy can be obtained the following |
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227 | expression |
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228 | %% |
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229 | \begin{equation} |
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230 | \label{evap:12} |
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231 | \Gamma_j = \frac{ \sqrt{\pi} g_j \pi R_d^2 \alpha}{12 \rho(E^*)} \times |
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232 | \left\{ \begin{array}{ll} |
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233 | \{I_1(t,t) + (\beta+V)I_0(t)\} & |
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234 | \rm{for} \quad \varepsilon_j^{\mathrm{max}} - V_j < E_x \\ |
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235 | \{I_1(t,t_x)+I_3(s,s_x)e^s+ & \\ |
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236 | (\beta+V)(I_0(t_x)+I_2(s,s_x)e^s)\} & |
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237 | \rm{for} \quad \varepsilon_j^{\mathrm{max}} - V_j \geq E_x . |
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238 | \end{array} \right. |
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239 | \end{equation} |
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240 | %% |
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241 | $I_0(t)$, $I_1(t,t_x)$, $I_2(s,s_x)$, and $I_3(s,s_x)$ are expressed |
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242 | as: |
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243 | \begin{eqnarray} |
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244 | I_0(t) & = & e^{-E_0/T} (e^t -1) \\ |
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245 | I_1(t,t_x) & = & e^{-E_0/T} T \{(t - t_x + 1)e^{t_x} - t -1 \} \\ |
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246 | I_2(s,s_x) & = & 2\sqrt{2} \biggl \{ s^{-3/2} + 1.5 s^{-5/2} + 3.75 |
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247 | s^{-7/2} - \nonumber \\ |
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248 | & & (s_x^{-3/2} + 1.5 s_x^{-5/2} + 3.75 s_x^{-7/2}) \biggr \} \\ |
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249 | I_3(s,s_x) & = & \frac{1}{2\sqrt{2}} \Biggl [ 2 s^{-1/2} + 4 |
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250 | s^{-3/2} + 13.5 s^{-5/2} + 60.0 s^{-7/2} + \nonumber \\ |
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251 | & & 325.125 s^{-9/2} - |
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252 | \biggl \{ |
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253 | (s^2 - s_x^2) s_x^{-3/2} + |
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254 | (1.5s^2 + 0.5s_x^2) s_x^{-5/2} + \nonumber \\ |
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255 | & & |
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256 | (3.75s^2 + 0.25s_x^2) s_x^{-7/2} + |
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257 | (12.875s^2 + 0.625s_x^2) s_x^{-9/2} + \nonumber \\ |
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258 | & & |
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259 | (59.0625s^2 + 0.9375s_x^2) s_x^{-11/2} + \nonumber \\ |
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260 | & & |
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261 | (324.8s^2 + 3.28s_x^2) s_x^{-13/2} + |
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262 | \biggr \} |
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263 | \Biggr ] |
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264 | \end{eqnarray} |
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265 | %% |
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266 | where $t = (\varepsilon_j^{\mathrm{max}}-V_j)/T$, $t_x = E_x/T$, $s = 2 |
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267 | \sqrt{a(\varepsilon_j^{\mathrm{max}}-V_j -\delta_j)}$ and $s_x = 2 |
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268 | \sqrt{a(E_x - \delta)}$. |
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269 | |
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270 | Besides light fragments, 60 nuclides up to $^{28}$Mg are considered, |
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271 | not only in their ground states but also in their exited states, are |
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272 | considered. The excited state is assumed to survive if its lifetime |
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273 | $T_{1/2}$ is longer than the decay time, \textit{i. e.}, |
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274 | $T_{1/2}/\ln{2} > \hbar/\Gamma_j^*$, where $\Gamma_j^*$ is the |
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275 | emission width of the resonance calculated in the same manner as for |
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276 | ground state particle emission. The total emission width of an |
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277 | ejectile $j$ is summed over its ground state and all its excited |
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278 | states which satisfy the above condition. |
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279 | |
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280 | %%% Local Variables: |
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281 | %%% mode: latex |
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282 | %%% TeX-master: "EvaporationModel" |
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283 | %%% End: |
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