[1211] | 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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