1 | |
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2 | \section{The Total Probability for Photon Evaporation} |
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3 | \hspace{1.0em}As the first approximation we |
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4 | assume that |
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5 | dipole $E1$--transitions is the main source of $\gamma$--quanta from |
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6 | highly--excited nuclei \cite{evap.Iljinov92}. |
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7 | The probability to evaporate $\gamma$ |
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8 | in the energy interval $(\epsilon_{\gamma}, \epsilon_{\gamma}+d\epsilon_{\gamma})$ |
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9 | per unit of time is given |
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10 | \begin{equation} |
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11 | \label{SPE7}W_{\gamma}(\epsilon_{\gamma}) = |
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12 | \frac{1}{\pi^2 (\hbar c)^3}\sigma_{\gamma}(\epsilon_{\gamma}) |
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13 | \frac{\rho(E^{*}-\epsilon_{\gamma})} |
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14 | {\rho(E^{*})}\epsilon^2_{\gamma}, |
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15 | \end{equation} |
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16 | where $\sigma_{\gamma}(\epsilon_{\gamma})$ is the inverse (absorption of $\gamma$) |
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17 | reaction cross section, |
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18 | $\rho$ is a nucleus level density is defined by Eq. ($\ref{evap:6}$). |
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19 | |
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20 | |
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21 | The photoabsorption reaction cross section is given by the expression |
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22 | \begin{equation} |
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23 | \label{SPE8} |
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24 | \sigma _{\gamma}(\epsilon_{\gamma}) = |
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25 | \frac{\sigma_0 \epsilon^2_{\gamma} \Gamma^2_{R}} |
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26 | {(\epsilon^2_{\gamma} - E_{GDP}^2)^2 + \Gamma^2_R\epsilon^2_{\gamma}}, |
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27 | \end{equation} |
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28 | where $\sigma_0=2.5A$ mb, $\Gamma_R=0.3E_{GDP}$ and $E_{GDP}= 40.3 A^{-1/5}$ MeV are |
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29 | empirical parameters of the giant dipole resonance \cite{evap.Iljinov92}. |
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30 | The total radiation probability is |
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31 | \begin{equation} |
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32 | \label{SPE9}W_{\gamma} =\frac{3}{\pi^2 (\hbar c)^3}\int_{0}^{E^{*}} |
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33 | \sigma_{\gamma}(\epsilon_{\gamma}) |
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34 | \frac{\rho(E^{*}-\epsilon_{\gamma})} |
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35 | {\rho(E^{*})}\epsilon^2_{\gamma}d\epsilon_{\gamma}. |
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36 | \end{equation} |
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37 | The integration is performed numericaly. |
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38 | |
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39 | \subsection{Energy of evaporated photon} |
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40 | \hspace{1.0em}The energy of $\gamma$-quantum is sampled |
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41 | according to the Eq. $(\ref{SPE7})$ |
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42 | distribution. |
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43 | |
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44 | \section{Discrete photon evaporation} |
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45 | |
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46 | \hspace{1.0em} The last step of evaporation cascade consists of evaporation of |
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47 | photons with discrete energies. The competition between photons and |
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48 | fragments as well as giant resonance photons is neglected at this step. |
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49 | We consider the discrete E1, M1 and E2 photon transitions |
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50 | from tabulated isotopes. |
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51 | There are large number of isotopes \cite{evap.ENSDF} with the experimentally |
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52 | measured exited level energies, spins, parities and relative transitions |
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53 | probabilities. This information is |
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54 | implemented in the code. |
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55 | \section{Internal conversion electron emission} |
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56 | |
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57 | \hspace{1.0em} An important conpetitive channel to photon emission is internal |
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58 | conversion. To take this into account, the photon evaporation data-base was |
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59 | entended to include internal conversion coeffficients. |
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60 | |
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61 | The above constitute the first six columns of data in the photon evaporation |
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62 | files. The new version of the data base adds eleven new columns corresponding |
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63 | to: |
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64 | |
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65 | \begin{enumerate} |
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66 | \setcounter{enumi}{6} |
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67 | \item ratio of internal conversion to gamma-ray emmission probability |
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68 | \item - 17. internal conversion coefficients for shells K, L1, L2, L3, |
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69 | M1, M2, M3, M4, M5 and N+ respectively. These coefficients are normalised |
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70 | to 1.0 |
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71 | \end{enumerate} |
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72 | |
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73 | The calculation of the Internal Conversion Coefficients (ICCs) is |
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74 | done by a cubic spline interpolation of tabulalted data for the |
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75 | corresponding transition energy. These ICC tables, which we shall |
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76 | label Band \cite{spe.band}, R\"{o}sel \cite{spe.rosel} and Hager-Seltzer |
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77 | \cite{spe.hagsel}, are widely used |
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78 | and were provided in electronic format by staff at LBNL. The reliability |
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79 | of these tabulated data has been reviewed in Ref. \cite{spe.rys}. From |
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80 | tests carried out on these data we find that the ICCs calculated from all |
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81 | three tables are comparable within a 10\% uncertainty, which is better than |
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82 | what experimetal measurements are reported to be able to achive. |
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83 | |
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84 | The range in atomic number covered by these tables is Band: $1 <= Z <= 80$; |
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85 | R\"{o}sel: $30 <= Z <= 104$ and Hager-Seltzer: $3, 6, 10, 14 <= Z <= 103$. |
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86 | For simplicity and taking into account the completeness of the tables, |
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87 | we have used the Band table for $Z <= 80$ and R\"{o}sel for $81 <= Z <= 98$. |
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88 | |
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89 | The Band table provides a higher resolution of the ICC curves used in |
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90 | the interpolation and covers ten multipolarities for all elements up |
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91 | to $Z=80$, but it only includes ICCs for shells up to M5. In order to |
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92 | calculate the ICC of the N+ shell, the ICCs of all available M shells |
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93 | are added together and the total divided by 3. This is the scheme adopted |
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94 | in the LBNL ICC calculation code when using the Band table. The R\"{o}sel |
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95 | table includes ICCs for all shells in every atom and for $Z>80$ the N+ |
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96 | shell ICC is calculated by adding together the ICCs of all shells above M5. |
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97 | In this table only eight multipolarities have ICCs calculated for. |
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98 | |
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99 | |
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100 | \subsection{Multipolarity} |
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101 | The ENSDF data provides information on the multipolarity of the transition. |
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102 | The ICCs included in the photon evaporation data base refer to the |
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103 | multipolarity indicated in the ENSDF file for that transition. Only one |
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104 | type of mixed mulltipolarity is considered (M1+E2) and whenever the mixing |
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105 | ratio is provided in the ENSDF file, it is used to calculate the ICCs |
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106 | corresponding to the mixed multipolarity according the the formula: |
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107 | |
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108 | \begin{tabbing} |
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109 | \= xxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \kill |
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110 | \> \> - fraction in $M1 = 1/(1+\delta^{2})$ \\ |
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111 | \> \> - fraction in $E2 = \delta^{2}/(1+\delta^{2})$ \\ |
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112 | \> \\ |
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113 | \> where $\delta$ is the mixing ratio.\\ |
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114 | \end{tabbing} |
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115 | |
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116 | \subsection{Binding energy} |
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117 | For the production of an internal conversion electron, the energy of the |
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118 | transition must be at least the binding energy of the shell the electron |
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119 | is being released from. The binding energy corresponding to the various |
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120 | shells in all isotopes used in the ICC calculation has been taken from |
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121 | the Geant4 file G4AtomicShells.hh. |
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122 | |
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123 | \subsection{Isotopes} |
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124 | The list of isotopes included in the photon evaporation data base has been |
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125 | extended from $A<=240$ to $A<=250$. The highest atomic number included is |
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126 | $Z=98$ (this ensures that Americium sources can now be simulated). |
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127 | |
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128 | %%% Local Variables: |
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129 | %%% mode: latex |
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130 | %%% TeX-master: t |
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131 | %%% End: |
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