1 | \chapter[Cross-sections in Photonuclear/Electronuclear Reactions]{Cross-sections in Photonuclear and Electronuclear Reactions} |
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2 | \section{Approximation of Photonuclear Cross Sections.} |
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3 | |
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4 | The photonuclear cross sections parameterized in the |
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5 | {\tt G4PhotoNuclearCrossSection} class cover all incident photon energies from |
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6 | the hadron production threshold upward. The parameterization is subdivided |
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7 | into five energy regions, each corresponding to the physical process that |
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8 | dominates it. |
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9 | |
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10 | \begin{itemize} |
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11 | |
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12 | \item The Giant Dipole Resonance (GDR) region, depending on the nucleus, |
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13 | extends from 10 Mev up to 30 MeV. It usually consists of one large |
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14 | peak, though for some nuclei several peaks appear. |
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15 | |
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16 | \item The ``quasi-deuteron'' region extends from around 30 MeV up to the |
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17 | pion threshold and is characterized by small cross sections and a broad, |
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18 | low peak. |
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19 | |
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20 | \item The $\Delta$ region is characterized by the dominant peak in the |
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21 | cross section which extends from the pion threshold to 450 MeV. |
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22 | |
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23 | \item The Roper resonance region extends from roughly 450 MeV to 1.2 GeV. |
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24 | The cross section in this region is not strictly identified with the |
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25 | real Roper resonance because other processes also occur in this region. |
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26 | |
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27 | \item The Reggeon-Pomeron region extends upward from 1.2 GeV. |
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28 | |
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29 | \end{itemize} |
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30 | |
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31 | \noindent |
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32 | In the GEANT4 photonuclear data base there are about 50 nuclei for which the |
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33 | photonuclear absorption cross sections have been measured in the above |
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34 | energy ranges. For low energies this number could be enlarged, because for |
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35 | heavy nuclei the neutron photoproduction cross section is close to the total |
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36 | photo-absorption cross section. Currently, however, 14 nuclei are used in |
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37 | the parameterization: $^1$H, $^2$H, $^4$He, $^6$Li, $^7$Li, $^9$Be, |
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38 | $^{12}$C, $^{16}$O, $^{27}$Al, $^{40}$Ca, Cu, Sn, Pb, and U. The resulting |
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39 | cross section is a function of $A$ and $e = log(E_\gamma)$, where $E_\gamma$ |
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40 | is the energy of the incident photon. This function is the sum of the |
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41 | components which parameterize each energy region. \\ |
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42 | |
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43 | \noindent |
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44 | The cross section in the GDR region can be described as the sum of two |
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45 | peaks, |
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46 | \begin{equation} |
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47 | GDR(e) = th(e,b_1,s_1)\cdot exp(c_1-p_1\cdot e) + |
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48 | th(e,b_2,s_2)\cdot exp(c_2-p_2\cdot e) . |
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49 | \end{equation} |
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50 | The exponential parameterizes the falling edge of the resonance which |
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51 | behaves like a power law in $E_\gamma$. This behavior is expected from |
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52 | the CHIPS model, which includes the nonrelativistic phase space of nucleons |
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53 | to explain evaporation. The function |
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54 | \begin{equation} |
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55 | th(e,b,s) = \frac{1}{1+exp(\frac{b-e}{s})} , |
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56 | \end{equation} |
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57 | describes the rising edge of the resonance. It is the |
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58 | nuclear-barrier-reflection function and behaves like a threshold, cutting off |
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59 | the exponential. The exponential powers $p_1$ and $p_2$ are |
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60 | |
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61 | \begin{eqnarray*} |
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62 | p_1 = 1, p_2 = 2 \mbox{\hspace*{1mm} for \hspace*{7mm} $A < 4$ }\\ |
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63 | p_1 = 2, p_2 = 4 \mbox{\hspace*{1mm} for \hspace*{1mm} $4 \le A < 8$ }\\ |
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64 | p_1 = 3, p_2 = 6 \mbox{\hspace*{1mm} for $8 \le A < 12$} \\ |
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65 | p_1 = 4, p_2 = 8 \mbox{\hspace*{1mm} for \hspace*{6mm} $A \ge 12$} . |
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66 | \end{eqnarray*} |
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67 | |
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68 | \noindent |
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69 | The $A$-dependent parameters $b_i$, $c_i$ and $s_i$ were found for each of |
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70 | the 14 nuclei listed above and interpolated for other nuclei. \\ |
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71 | |
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72 | \noindent |
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73 | The $\Delta$ isobar region was parameterized as |
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74 | \begin{equation} |
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75 | \Delta (e,d,f,g,r,q)=\frac{d\cdot th(e,f,g)}{1+r\cdot (e-q)^2}, |
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76 | \label{Isobar} |
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77 | \end{equation} |
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78 | where $d$ is an overall normalization factor. $q$ can be interpreted as the |
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79 | energy of the $\Delta$ isobar and $r$ can be interpreted as the inverse of |
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80 | the $\Delta$ width. Once again $th$ is the threshold function. The |
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81 | $A$-dependence of these parameters is as follows: |
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82 | |
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83 | \begin{itemize} |
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84 | \item $d=0.41\cdot A$ (for $^1$H it is 0.55, for $^2$H it is 0.88), |
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85 | which means that the $\Delta$ yield is proportional |
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86 | to $A$; |
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87 | |
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88 | \item $f=5.13-.00075\cdot A$. $exp(f)$ shows how the pion threshold depends |
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89 | on $A$. It is clear that the threshold becomes 140 MeV only for uranium; |
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90 | for lighter nuclei it is higher. |
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91 | |
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92 | \item $g = 0.09$ for $A \ge 7$ and 0.04 for $A < 7$; |
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93 | |
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94 | \item $q=5.84-\frac{.09}{1+.003\cdot A^2}$, which means that the ``mass'' |
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95 | of the $\Delta$ isobar moves to lower energies; |
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96 | |
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97 | \item $r=11.9 - 1.24\cdot log(A)$. $r$ is 18.0 for $^1$H. |
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98 | The inverse width becomes smaller with $A$, hence the width increases. |
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99 | |
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100 | \end{itemize} |
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101 | The $A$-dependence of the $f$, $q$ and $r$ parameters is due to the |
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102 | $\Delta+N\rightarrow N+N$ reaction, which can take place in the nuclear |
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103 | medium below the pion threshold. \\ |
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104 | |
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105 | \noindent |
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106 | The quasi-deuteron contribution was parameterized with the same form as the |
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107 | $\Delta$ contribution but without the threshold function: |
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108 | \begin{equation} |
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109 | QD(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}. |
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110 | \label{QuasiD} |
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111 | \end{equation} |
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112 | For $^1$H and $^2$H the quasi-deuteron contribution is almost zero. For |
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113 | these nuclei the third baryonic resonance was used instead, so the |
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114 | parameters for these two nuclei are quite different, but trivial. |
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115 | The parameter values are given below. |
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116 | |
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117 | \begin{itemize} |
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118 | |
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119 | \item $v = \frac {exp(-1.7+a\cdot 0.84)}{1+exp(7\cdot (2.38-a))}$, where |
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120 | $a=log(A)$. This shows that the $A$-dependence in the quasi-deuteron |
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121 | region is stronger than $A^{0.84}$. It is clear from the denominator that |
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122 | this contribution is very small for light nuclei (up to $^6$Li or $^7$Li). |
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123 | For $^1$H it is 0.078 and for $^2$H it is 0.08, so the delta contribution |
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124 | does not appear to be growing. Its relative contribution disappears with |
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125 | $A$. |
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126 | |
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127 | \item $u = 3.7$ and $w = 0.4$. The experimental information is not |
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128 | sufficient to determine an $A$-dependence for these parameters. For both |
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129 | $^1$H and $^2$H $u = 6.93$ and $w = 90$, which may indicate contributions |
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130 | from the $\Delta$(1600) and $\Delta$(1620). |
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131 | |
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132 | \end{itemize} |
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133 | |
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134 | \noindent |
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135 | The transition Roper contribution was parameterized using the same form |
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136 | as the quasi-deuteron contribution: |
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137 | \begin{equation} |
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138 | Tr(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}. |
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139 | \label{Transition} |
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140 | \end{equation} |
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141 | Using $a=log(A)$, the values of the parameters are |
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142 | |
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143 | \begin{itemize} |
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144 | |
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145 | \item $v = exp(-2.+a\cdot 0.84)$. For $^1$H it is 0.22 and for $^2$H |
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146 | it is 0.34. |
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147 | |
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148 | \item $u = 6.46+a\cdot 0.061$ (for $^1$H and for $^2$H it is 6.57), so the |
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149 | ``mass'' of the Roper moves higher with $A$. |
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150 | |
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151 | \item $w = 0.1+a\cdot 1.65$. For $^1$H it is 20.0 and for $^2$H it is 15.0). |
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152 | \end{itemize} |
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153 | |
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154 | |
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155 | \noindent |
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156 | The Regge-Pomeron contribution was parametrized as follows: |
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157 | \begin{equation} |
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158 | RP(e,h)=h\cdot th(7.,0.2)\cdot (0.0116\cdot exp(e\cdot 0.16)+0.4\cdot exp(-e\cdot 0.2)), |
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159 | \label{Regge} |
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160 | \end{equation} |
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161 | where $h=A\cdot exp(-a\cdot (0.885+0.0048\cdot a))$ and, again, |
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162 | $a = log(A)$. The first exponential in Eq.~\ref{Regge} describes the Pomeron |
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163 | contribution while the second describes the Regge contribution. |
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164 | |
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165 | %The result of the approximation is shown in Fig.~\ref{photonuc} for 6 |
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166 | % of the 14 nuclei. |
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167 | %\begin{figure} |
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168 | % \resizebox{1.00\textwidth}{!} |
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169 | %{ |
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170 | %% hpw @@@@@ \includegraphics{photonuclear.eps} |
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171 | %} |
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172 | %\caption{Photoabsorbtion cross sections for 6 basic nuclei.} |
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173 | %\label{photonuc} |
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174 | %\end{figure} |
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175 | |
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176 | |
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177 | \section{Electronuclear Cross Sections and Reactions} |
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178 | |
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179 | Electronuclear reactions are so closely connected with photonuclear reactions |
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180 | that they are sometimes called ``photonuclear'' because the one-photon |
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181 | exchange mechanism dominates in electronuclear reactions. In this sense |
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182 | electrons can be replaced by a flux of equivalent photons. This is not |
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183 | completely true, because at high energies the Vector Dominance Model (VDM) or |
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184 | diffractive mechanisms are possible, but these types of reactions are beyond |
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185 | the scope of this discussion. |
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186 | |
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187 | \subsection[Common Notation for Electronuclear Reactions]{Common Notation for Different Approaches to Electronuclear Reactions} |
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188 | \label{threeApproaches} |
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189 | |
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190 | The Equivalent Photon Approximation (EPA) was proposed by |
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191 | E. Fermi \cite{Fermi} and developed by C. Weizsacker and E. Williams |
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192 | \cite{WeiWi} and by L. Landau and E. Lifshitz \cite{LanLif}. The |
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193 | covariant form of the EPA method was developed in Refs. \cite{Pomer} and |
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194 | \cite{Grib}. When using this method it is necessary to take into account |
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195 | that real photons are always transversely polarized while virtual photons |
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196 | may be longitudinally polarized. In general the differential cross section |
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197 | of the electronuclear interaction can be written as |
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198 | \begin{equation} |
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199 | \frac{d^2\sigma}{dydQ^2}=\frac{\alpha}{\pi Q^2}(S_{TL}\cdot(\sigma_T |
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200 | +\sigma_L)-S_L\cdot\sigma_L), |
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201 | \label{elNuc} |
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202 | \end{equation} |
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203 | where |
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204 | \begin{equation} |
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205 | S_{TL}=y\frac{1-y+\frac{y^2}{2}+\frac{Q^2}{4E^2} |
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206 | -\frac{m^2_e}{Q^2}(y^2+\frac{Q^2}{E^2})}{y^2+\frac{Q^2}{E^2}}, |
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207 | \label{STL} |
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208 | \end{equation} |
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209 | \begin{equation} |
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210 | S_L=\frac{y}{2}(1-\frac{2m_e^2}{Q^2}). |
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211 | \label{SL} |
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212 | \end{equation} |
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213 | The differential cross section of the electronuclear scattering can be |
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214 | rewritten as |
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215 | \begin{equation} |
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216 | \frac{d^2\sigma_{eA}}{dydQ^2}=\frac{\alpha y}{\pi Q^2}\left(\frac{(1-\frac{y}{2})^2} |
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217 | {y^2+\frac{Q^2}{E^2}}+\frac{1}{4}-\frac{m^2_e}{Q^2}\right)\sigma_{\gamma^*A}, |
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218 | \label{difBase} |
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219 | \end{equation} |
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220 | where $\sigma_{\gamma^*A}=\sigma_{\gamma A}(\nu)$ for small $Q^2$ and |
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221 | must be approximated as a function of $\epsilon$, $\nu$, and $Q^2$ for |
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222 | large $Q^2$. Interactions of longitudinal photons are included in the |
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223 | effective $\sigma_{\gamma^*A}$ cross section through the $\epsilon$ factor, |
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224 | but in the present GEANT4 method, the cross section of virtual photons is |
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225 | considered to be $\epsilon$-independent. The electronuclear problem, with |
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226 | respect to the interaction of virtual photons with nuclei, can thus be split |
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227 | in two. At small $Q^2$ it is possible to use the $\sigma_\gamma(\nu)$ cross |
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228 | section. In the $Q^2>>m^2_e$ region it is necessary to calculate the effective |
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229 | $\sigma_{\gamma^*}(\epsilon,\nu,Q^2)$ cross section. \\ |
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230 | |
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231 | \noindent |
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232 | Following the EPA notation, the differential cross section of electronuclear |
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233 | scattering can be related to the number of equivalent photons |
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234 | $dn=\frac{d\sigma}{\sigma_{\gamma^*}}$. For $y<<1$ and $Q^2<4m^2_e$ the |
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235 | canonical method \cite{encs.eqPhotons} leads to the simple result |
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236 | \begin{equation} |
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237 | \frac{ydn(y)}{dy}=-\frac{2\alpha}{\pi}ln(y). |
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238 | \label{neq} |
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239 | \end{equation} |
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240 | In \cite{Budnev} the integration over $Q^2$ for $\nu^2>>Q^2_{max}\simeq m^2_e$ |
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241 | leads to |
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242 | \begin{equation} |
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243 | \frac{ydn(y)}{dy}=-\frac{\alpha}{\pi}\left( |
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244 | \frac{1+(1-y)^2}{2}ln(\frac{y^2}{1-y})+(1-y)\right). |
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245 | \label{lowQ2EP} |
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246 | \end{equation} |
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247 | In the $y<<1$ limit this formula converges to Eq.(\ref{neq}). But the |
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248 | correspondence with Eq.(\ref{neq}) can be made more explicit if the exact |
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249 | integral |
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250 | \begin{equation} |
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251 | \frac{ydn(y)}{dy}=\frac{\alpha}{\pi}\left( |
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252 | \frac{1+(1-y)^2}{2}l_1-(1-y)l_2-\frac{(2-y)^2}{4}l_3\right), |
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253 | \label{diff} |
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254 | \end{equation} |
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255 | where $l_1=ln\left(\frac{Q^2_{max}}{Q^2_{min}}\right)$, |
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256 | $l_2=1-\frac{Q^2_{max}}{Q^2_{min}}$, |
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257 | $l_3=ln\left(\frac{y^2+Q^2_{max}/E^2}{y^2+Q^2_{min}/E^2}\right)$, |
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258 | $Q^2_{min}=\frac{m_e^2y^2}{1-y}$, |
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259 | is calculated for |
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260 | \begin{equation} |
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261 | Q^2_{max(m_e)}=\frac{4m^2_e}{1-y}. |
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262 | \label{Q2me} |
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263 | \end{equation} |
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264 | The factor $(1-y)$ is used arbitrarily to keep $Q^2_{max(m_e)}>Q^2_{min}$, |
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265 | which can be considered as a boundary between the low and high $Q^2$ |
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266 | regions. The full transverse photon flux can be calculated as an integral |
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267 | of Eq.(\ref{diff}) with the maximum possible upper limit |
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268 | \begin{equation} |
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269 | Q^2_{max(max)}=4E^2(1-y). |
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270 | \label{Q2max} |
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271 | \end{equation} |
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272 | The full transverse photon flux can be approximated by |
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273 | \begin{equation} |
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274 | \frac{ydn(y)}{dy}=-\frac{2\alpha}{\pi}\left( |
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275 | \frac{(2-y)^2+y^2}{2}ln(\gamma)-1\right), |
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276 | \label{neqHQ} |
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277 | \end{equation} |
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278 | where $\gamma=\frac{E}{m_e}$. It must be pointed out that neither this |
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279 | approximation nor Eq.(\ref{diff}) works at $y\simeq 1$; at this point |
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280 | $Q^2_{max(max)}$ becomes smaller than $Q^2_{min}$. The formal limit of the |
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281 | method is $y<1-\frac{1}{2\gamma}$. \\ |
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282 | \begin{figure}[tbp] |
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283 | \resizebox{0.95\textwidth}{!} |
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284 | { |
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285 | \includegraphics{hadronic/theory_driven/ChiralInvariantPhaseSpace/Fig12.eps} |
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286 | } |
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287 | \caption{Relative contribution of equivalent photons with small $Q^2$ |
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288 | to the total ``photon flux'' for (a) $1~GeV$ electrons and (b) $10~GeV$ |
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289 | electrons. In figures (c) and (d) the equivalent photon distribution |
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290 | $dn(\nu,Q^2)$ is multiplied by the photonuclear cross section |
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291 | $\sigma_{\gamma^*}(K,Q^2)$ and integrated over $Q^2$ in two regions: |
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292 | the dashed lines are integrals over the low-$Q^2$ equivalent |
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293 | photons (under the dashed line in the first two figures), and the |
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294 | solid lines are integrals over the high-$Q^2$ equivalent photons (above |
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295 | the dashed lines in the first two figures).} |
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296 | \label{nSigma} |
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297 | \end{figure} |
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298 | |
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299 | \noindent |
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300 | In Fig.~\ref{nSigma}(a,b) the energy distribution for the equivalent photons |
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301 | is shown. The low-$Q^2$ photon flux with the upper limit defined by |
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302 | Eq.(\ref{Q2me})) is compared with the full photon flux. The |
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303 | low-$Q^2$ photon flux is calculated using Eq.(\ref{neq}) (dashed lines) and |
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304 | using Eq.(\ref{diff}) (dotted lines). The full photon |
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305 | flux is calculated using Eq.(\ref{neqHQ}) (the solid lines) and using |
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306 | Eq.(\ref{diff}) with the upper limit defined by Eq.(\ref{Q2max}) (dash-dotted |
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307 | lines, which differ from the solid lines only at $\nu\approx E_e$). The |
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308 | conclusion is that in order to calculate either the number of low-$Q^2$ |
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309 | equivalent photons or the total number of equivalent photons one can use the |
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310 | simple approximations given by Eq.(\ref{neq}) and Eq.(\ref{neqHQ}), |
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311 | respectively, instead of using Eq.(\ref{diff}), which cannot be integrated |
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312 | over $y$ analytically. Comparing the low-$Q^2$ photon flux and the total |
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313 | photon flux it is possible to show that the low-$Q^2$ photon flux is about |
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314 | half of the the total. From the interaction point of view the decrease of |
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315 | $\sigma_{\gamma*}$ with increasing $Q^2$ must be taken into account. The |
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316 | cross section reduction for the virtual photons with large $Q^2$ is governed |
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317 | by two factors. First, the cross section drops with $Q^2$ as the squared |
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318 | dipole nucleonic form-factor |
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319 | \begin{equation} |
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320 | G^2_D(Q^2)\approx\left( 1+\frac{Q^2}{(843~MeV)^2}\right)^{-2}. |
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321 | \label{G2} |
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322 | \end{equation} |
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323 | Second, all the thresholds of the $\gamma A$ reactions are shifted to higher |
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324 | $\nu$ by a factor $\frac{Q^2}{2M}$, which is the difference between the $K$ |
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325 | and $\nu$ values. Following the method proposed in \cite{Brasse} |
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326 | the $\sigma_{\gamma^*}$ at large $Q^2$ can be approximated as |
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327 | \begin{equation} |
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328 | \sigma_{\gamma*}=(1-x)\sigma_\gamma(K)G^2_D(Q^2)e^{b(\epsilon,K)\cdot |
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329 | r+c(\epsilon,K)\cdot r^3}, |
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330 | \label{abc} |
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331 | \end{equation} |
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332 | where $r=\frac{1}{2}ln(\frac{Q^2+\nu^2}{K^2})$. The $\epsilon$-dependence of |
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333 | the $a(\epsilon,K)$ and $b(\epsilon,K)$ functions is weak, so for simplicity |
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334 | the $b(K)$ and $c(K)$ functions are averaged over $\epsilon$. They can be |
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335 | approximated as |
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336 | \begin{equation} |
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337 | b(K)\approx\left(\frac{K}{185~MeV}\right)^{0.85}, |
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338 | \label{bk} |
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339 | \end{equation} |
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340 | and |
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341 | \begin{equation} |
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342 | c(K)\approx-\left(\frac{K}{1390~MeV}\right)^{3}. |
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343 | \label{ck} |
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344 | \end{equation} |
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345 | |
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346 | \noindent |
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347 | The result of the integration of the photon flux multiplied by the |
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348 | cross section approximated by Eq.(\ref{abc}) is shown in |
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349 | Fig.~\ref{nSigma}(c,d). The integrated cross sections are shown |
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350 | separately for the low-$Q^2$ region ($Q^2<Q^2_{max(m_e)}$, dashed |
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351 | lines) and for the high-$Q^2$ region ($Q^2>Q^2_{max(m_e)}$, solid |
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352 | lines). These functions must be integrated over $ln(\nu)$, so it is |
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353 | clear that because of the Giant Dipole Resonance contribution, the |
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354 | low-$Q^2$ part covers more than half the total $eA\rightarrow hadrons$ |
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355 | cross section. But at $\nu>200~MeV$, where the hadron multiplicity |
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356 | increases, the large $Q^2$ part dominates. In this sense, for a better |
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357 | simulation of the production of hadrons by electrons, it is necessary to |
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358 | simulate the high-$Q^2$ part as well as the low-$Q^2$ part. \\ |
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359 | |
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360 | \noindent |
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361 | Taking into account the contribution of high-$Q^2$ photons it is possible to |
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362 | use Eq.(\ref{neqHQ}) with the over-estimated |
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363 | $\sigma_{\gamma^*A}=\sigma_{\gamma A}(\nu)$ cross section. The slightly |
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364 | over-estimated electronuclear cross section is |
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365 | \begin{equation} |
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366 | \sigma^*_{eA}=(2ln(\gamma)-1)\cdot J_1-\frac{ln(\gamma)}{E_e} |
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367 | \left( 2J_2-\frac{J_3}{E_e} \right). |
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368 | \label{eleNucHQ} |
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369 | \end{equation} |
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370 | where |
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371 | \begin{equation} |
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372 | J_1(E_e)=\frac{\alpha}{\pi}\int^{E_e}\sigma_{\gamma A}(\nu)dln(\nu) |
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373 | \label{J1} |
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374 | \end{equation} |
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375 | \begin{equation} |
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376 | J_2(E_e)=\frac{\alpha}{\pi}\int^{E_e}\nu\sigma_{\gamma A}(\nu)dln(\nu), |
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377 | \label{J2} |
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378 | \end{equation} |
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379 | and |
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380 | \begin{equation} |
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381 | J_3(E_e)=\frac{\alpha}{\pi}\int^{E_e}\nu^2\sigma_{\gamma A}(\nu )dln(\nu). |
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382 | \label{J3} |
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383 | \end{equation} |
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384 | The equivalent photon energy $\nu=yE$ can be obtained for a particular |
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385 | random number $R$ from the equation |
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386 | \begin{equation} |
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387 | R=\frac{(2ln(\gamma)-1)J_1(\nu)-\frac{ln(\gamma)}{E_e}(2J_2(\nu)-\frac{J_3(\nu)}{E_e})} |
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388 | {(2ln(\gamma)-1)J_1(E_e)-\frac{ln(\gamma)}{E_e}(2J_2(E_e)-\frac{J_3(E_e)}{E_e})}. |
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389 | \label{RnuHH} |
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390 | \end{equation} |
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391 | Eq.(\ref{diff}) is too complicated for the randomization of $Q^2$ but |
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392 | there is an easily randomized formula which approximates Eq.(\ref{diff}) |
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393 | above the hadronic threshold ($E>10~MeV$). It reads |
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394 | \begin{equation} |
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395 | \frac{\pi}{\alpha D(y)}\int^{Q^2}_{Q^2_{min}}\frac{ydn(y,Q^2)}{dydQ^2}dQ^2=-L(y,Q^2)-U(y), |
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396 | \label{RQ2HH} |
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397 | \end{equation} |
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398 | where |
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399 | \begin{equation} |
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400 | D(y)=1-y+\frac{y^2}{2}, |
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401 | \label{RQ2D} |
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402 | \end{equation} |
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403 | \begin{equation} |
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404 | L(y,Q^2)=ln\left( F(y)+(e^{P(y)}-1+\frac{Q^2}{Q^2_{min}})^{-1} \right), |
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405 | \label{RQ2L} |
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406 | \end{equation} |
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407 | and |
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408 | \begin{equation} |
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409 | U(y)=P(y)\cdot\left( 1-\frac{Q^2_{min}}{Q^2_{max}}\right), |
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410 | \label{RQ2U} |
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411 | \end{equation} |
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412 | with |
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413 | \begin{equation} |
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414 | F(y)=\frac{(2-y)(2-2y)}{y^2}\cdot\frac{Q^2_{min}}{Q^2_{max}} |
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415 | \label{RQ2F} |
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416 | \end{equation} |
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417 | and |
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418 | \begin{equation} |
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419 | P(y)=\frac{1-y}{D(y)}. |
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420 | \label{RQ2P} |
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421 | \end{equation} |
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422 | The $Q^2$ value can then be calculated as |
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423 | \begin{equation} |
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424 | \frac{Q^2}{Q^2_{min}}=1-e^{P(y)}+\left(e^{R\cdot |
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425 | L(y,Q^2_{max})-(1-R)\cdot U(y)}-F(y) \right)^{-1}, |
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426 | \label{Q2sol} |
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427 | \end{equation} |
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428 | where $R$ is a random number. In Fig.~\ref{Q2dep}, Eq.(\ref{diff}) (solid |
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429 | curve) is compared to Eq.(\ref{RQ2HH}) (dashed curve). Because the two |
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430 | curves are almost indistinguishable in the figure, this can be used as an |
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431 | illustration of the $Q^2$ spectrum of virtual photons, which is the derivative |
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432 | of these curves. An alternative approach is to use Eq.(\ref{diff}) for the |
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433 | randomization with a three dimensional table $\frac{ydn}{dy}(Q^2,y,E_e)$. |
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434 | \begin{figure}[tbp] |
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435 | \resizebox{0.95\textwidth}{!} |
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436 | { |
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437 | \includegraphics{hadronic/theory_driven/ChiralInvariantPhaseSpace/Fig13.eps} |
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438 | } |
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439 | \caption{Integrals of $Q^2$ spectra of virtual photons for three |
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440 | energies $10~MeV$, $100~MeV$, and $1~GeV$ at $y=0.001$, $y=0.5$, and $y=0.95$. |
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441 | The solid line corresponds to Eq.(\protect\ref{diff}) and the dashed |
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442 | line (which almost everywhere coincides with the solid line) |
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443 | corresponds to Eq.(\protect\ref{diff}).} |
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444 | \label{Q2dep} |
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445 | \end{figure} |
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446 | |
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447 | \noindent |
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448 | After the $\nu$ and $Q^2$ values have been found, the value of |
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449 | $\sigma_{\gamma^*A}(\nu,Q^2)$ is calculated using Eq.(\ref{abc}). |
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450 | If $R\cdot\sigma_{\gamma A}(\nu)>\sigma_{\gamma^*A}(\nu,Q^2)$, no |
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451 | interaction occurs and the electron keeps going. This ``do nothing'' |
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452 | process has low probability and cannot shadow other processes. |
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453 | |
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454 | |
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455 | \section {Status of this document} |
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456 | created by ? \\ |
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457 | 20.05.02 re-written by D.H. Wright \\ |
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458 | 01.12.02 expanded section on electronuclear cross sections - H.P. Wellisch \\ |
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459 | |
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460 | |
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461 | \begin{latexonly} |
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462 | |
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463 | \begin{thebibliography}{99} |
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464 | |
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465 | \bibitem{Fermi} E. Fermi, Z. Physik {\textbf{29}}, 315 (1924). |
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466 | |
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467 | \bibitem{WeiWi} K. F. von Weizsacker, Z. Physik {\textbf{88}}, 612 (1934), |
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468 | E. J. Williams, Phys. Rev. {\textbf{45}}, 729 (1934). |
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469 | |
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470 | \bibitem{LanLif} L. D. Landau and E. M. Lifshitz, |
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471 | Soc. Phys. {\textbf{6}}, 244 (1934). |
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472 | |
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473 | \bibitem{Pomer} I. Ya. Pomeranchuk and I. M. Shmushkevich, |
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474 | Nucl. Phys. {\textbf{23}}, 1295 (1961). |
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475 | |
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476 | \bibitem{Grib} V. N. Gribov {\textit {et~al.}}, ZhETF {\textbf{41}}, 1834 (1961). |
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477 | |
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478 | \bibitem{encs.eqPhotons} L. D. Landau, E. M. Lifshitz, ``Course of |
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479 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
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480 | Pergamon Press, p. 351, The method of equivalent photons. |
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481 | |
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482 | \bibitem{Budnev} V. M. Budnev {\textit {et~al.}}, Phys. Rep. {\textbf{15}}, 181 |
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483 | (1975). |
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484 | |
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485 | \bibitem{Brasse} F. W. Brasse {\textit {et~al.}}, Nucl. Phys. B {\textbf{110}}, 413 |
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486 | (1976). |
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487 | |
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488 | \end{thebibliography} |
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489 | |
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490 | \end{latexonly} |
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491 | |
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492 | \begin{htmlonly} |
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493 | |
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494 | \section{Bibliography} |
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495 | |
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496 | \begin{enumerate} |
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497 | \item E. Fermi, Z. Physik {\textbf{29}}, 315 (1924). |
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498 | |
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499 | \item K. F. von Weizsacker, Z. Physik {\textbf{88}}, 612 (1934), |
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500 | E. J. Williams, Phys. Rev. {\textbf{45}}, 729 (1934). |
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501 | |
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502 | \item L. D. Landau and E. M. Lifshitz, |
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503 | Soc. Phys. {\textbf{6}}, 244 (1934). |
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504 | |
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505 | \item I. Ya. Pomeranchuk and I. M. Shmushkevich, |
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506 | Nucl. Phys. {\textbf{23}}, 1295 (1961). |
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507 | |
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508 | \item V.N. Gribov {\textit {et~al.}}, ZhETF {\textbf{41}}, 1834 (1961). |
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509 | |
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510 | \item L.D. Landau, E. M. Lifshitz, ``Course of |
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511 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
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512 | Pergamon Press, p. 351, The method of equivalent photons. |
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513 | |
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514 | \item V.M. Budnev {\textit {et~al.}}, Phys. Rep. {\textbf{15}}, 181 |
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515 | (1975). |
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516 | |
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517 | \item F.W. Brasse {\textit {et~al.}}, Nucl. Phys. B {\textbf{110}}, 413 |
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518 | (1976). |
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519 | |
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520 | \end{enumerate} |
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521 | |
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522 | \end{htmlonly} |
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523 | |
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524 | |
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525 | |
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526 | |
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