[1208] | 1 | \chapter{Cross-sections in Photonuclear and Electronuclear Reactions} |
---|
| 2 | \section{Approximation of Photonuclear Cross Sections.} |
---|
| 3 | |
---|
| 4 | The photonuclear cross sections parameterized in the |
---|
| 5 | {\tt G4PhotoNuclearCrossSection} class cover all incident photon energies from |
---|
| 6 | the hadron production threshold upward. The parameterization is subdivided |
---|
| 7 | into five energy regions, each corresponding to the physical process that |
---|
| 8 | dominates it. |
---|
| 9 | |
---|
| 10 | \begin{itemize} |
---|
| 11 | |
---|
| 12 | \item The Giant Dipole Resonance (GDR) region, depending on the nucleus, |
---|
| 13 | extends from 10 Mev up to 30 MeV. It usually consists of one large |
---|
| 14 | peak, though for some nuclei several peaks appear. |
---|
| 15 | |
---|
| 16 | \item The ``quasi-deuteron'' region extends from around 30 MeV up to the |
---|
| 17 | pion threshold and is characterized by small cross sections and a broad, |
---|
| 18 | low peak. |
---|
| 19 | |
---|
| 20 | \item The $\Delta$ region is characterized by the dominant peak in the |
---|
| 21 | cross section which extends from the pion threshold to 450 MeV. |
---|
| 22 | |
---|
| 23 | \item The Roper resonance region extends from roughly 450 MeV to 1.2 GeV. |
---|
| 24 | The cross section in this region is not strictly identified with the |
---|
| 25 | real Roper resonance because other processes also occur in this region. |
---|
| 26 | |
---|
| 27 | \item The Reggeon-Pomeron region extends upward from 1.2 GeV. |
---|
| 28 | |
---|
| 29 | \end{itemize} |
---|
| 30 | |
---|
| 31 | \noindent |
---|
| 32 | In the GEANT4 photonuclear data base there are about 50 nuclei for which the |
---|
| 33 | photonuclear absorption cross sections have been measured in the above |
---|
| 34 | energy ranges. For low energies this number could be enlarged, because for |
---|
| 35 | heavy nuclei the neutron photoproduction cross section is close to the total |
---|
| 36 | photo-absorption cross section. Currently, however, 14 nuclei are used in |
---|
| 37 | the parameterization: $^1$H, $^2$H, $^4$He, $^6$Li, $^7$Li, $^9$Be, |
---|
| 38 | $^{12}$C, $^{16}$O, $^{27}$Al, $^{40}$Ca, Cu, Sn, Pb, and U. The resulting |
---|
| 39 | cross section is a function of $A$ and $e = log(E_\gamma)$, where $E_\gamma$ |
---|
| 40 | is the energy of the incident photon. This function is the sum of the |
---|
| 41 | components which parameterize each energy region. \\ |
---|
| 42 | |
---|
| 43 | \noindent |
---|
| 44 | The cross section in the GDR region can be described as the sum of two |
---|
| 45 | peaks, |
---|
| 46 | \begin{equation} |
---|
| 47 | GDR(e) = th(e,b_1,s_1)\cdot exp(c_1-p_1\cdot e) + |
---|
| 48 | th(e,b_2,s_2)\cdot exp(c_2-p_2\cdot e) . |
---|
| 49 | \end{equation} |
---|
| 50 | The exponential parameterizes the falling edge of the resonance which |
---|
| 51 | behaves like a power law in $E_\gamma$. This behavior is expected from |
---|
| 52 | the CHIPS model, which includes the nonrelativistic phase space of nucleons |
---|
| 53 | to explain evaporation. The function |
---|
| 54 | \begin{equation} |
---|
| 55 | th(e,b,s) = \frac{1}{1+exp(\frac{b-e}{s})} , |
---|
| 56 | \end{equation} |
---|
| 57 | describes the rising edge of the resonance. It is the |
---|
| 58 | nuclear-barrier-reflection function and behaves like a threshold, cutting off |
---|
| 59 | the exponential. The exponential powers $p_1$ and $p_2$ are |
---|
| 60 | |
---|
| 61 | \begin{eqnarray*} |
---|
| 62 | p_1 = 1, p_2 = 2 \mbox{\hspace*{1mm} for \hspace*{7mm} $A < 4$ }\\ |
---|
| 63 | p_1 = 2, p_2 = 4 \mbox{\hspace*{1mm} for \hspace*{1mm} $4 \le A < 8$ }\\ |
---|
| 64 | p_1 = 3, p_2 = 6 \mbox{\hspace*{1mm} for $8 \le A < 12$} \\ |
---|
| 65 | p_1 = 4, p_2 = 8 \mbox{\hspace*{1mm} for \hspace*{6mm} $A \ge 12$} . |
---|
| 66 | \end{eqnarray*} |
---|
| 67 | |
---|
| 68 | \noindent |
---|
| 69 | The $A$-dependent parameters $b_i$, $c_i$ and $s_i$ were found for each of |
---|
| 70 | the 14 nuclei listed above and interpolated for other nuclei. \\ |
---|
| 71 | |
---|
| 72 | \noindent |
---|
| 73 | The $\Delta$ isobar region was parameterized as |
---|
| 74 | \begin{equation} |
---|
| 75 | \Delta (e,d,f,g,r,q)=\frac{d\cdot th(e,f,g)}{1+r\cdot (e-q)^2}, |
---|
| 76 | \label{Isobar} |
---|
| 77 | \end{equation} |
---|
| 78 | where $d$ is an overall normalization factor. $q$ can be interpreted as the |
---|
| 79 | energy of the $\Delta$ isobar and $r$ can be interpreted as the inverse of |
---|
| 80 | the $\Delta$ width. Once again $th$ is the threshold function. The |
---|
| 81 | $A$-dependence of these parameters is as follows: |
---|
| 82 | |
---|
| 83 | \begin{itemize} |
---|
| 84 | \item $d=0.41\cdot A$ (for $^1$H it is 0.55, for $^2$H it is 0.88), |
---|
| 85 | which means that the $\Delta$ yield is proportional |
---|
| 86 | to $A$; |
---|
| 87 | |
---|
| 88 | \item $f=5.13-.00075\cdot A$. $exp(f)$ shows how the pion threshold depends |
---|
| 89 | on $A$. It is clear that the threshold becomes 140 MeV only for uranium; |
---|
| 90 | for lighter nuclei it is higher. |
---|
| 91 | |
---|
| 92 | \item $g = 0.09$ for $A \ge 7$ and 0.04 for $A < 7$; |
---|
| 93 | |
---|
| 94 | \item $q=5.84-\frac{.09}{1+.003\cdot A^2}$, which means that the ``mass'' |
---|
| 95 | of the $\Delta$ isobar moves to lower energies; |
---|
| 96 | |
---|
| 97 | \item $r=11.9 - 1.24\cdot log(A)$. $r$ is 18.0 for $^1$H. |
---|
| 98 | The inverse width becomes smaller with $A$, hence the width increases. |
---|
| 99 | |
---|
| 100 | \end{itemize} |
---|
| 101 | The $A$-dependence of the $f$, $q$ and $r$ parameters is due to the |
---|
| 102 | $\Delta+N\rightarrow N+N$ reaction, which can take place in the nuclear |
---|
| 103 | medium below the pion threshold. \\ |
---|
| 104 | |
---|
| 105 | \noindent |
---|
| 106 | The quasi-deuteron contribution was parameterized with the same form as the |
---|
| 107 | $\Delta$ contribution but without the threshold function: |
---|
| 108 | \begin{equation} |
---|
| 109 | QD(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}. |
---|
| 110 | \label{QuasiD} |
---|
| 111 | \end{equation} |
---|
| 112 | For $^1$H and $^2$H the quasi-deuteron contribution is almost zero. For |
---|
| 113 | these nuclei the third baryonic resonance was used instead, so the |
---|
| 114 | parameters for these two nuclei are quite different, but trivial. |
---|
| 115 | The parameter values are given below. |
---|
| 116 | |
---|
| 117 | \begin{itemize} |
---|
| 118 | |
---|
| 119 | \item $v = \frac {exp(-1.7+a\cdot 0.84)}{1+exp(7\cdot (2.38-a))}$, where |
---|
| 120 | $a=log(A)$. This shows that the $A$-dependence in the quasi-deuteron |
---|
| 121 | region is stronger than $A^{0.84}$. It is clear from the denominator that |
---|
| 122 | this contribution is very small for light nuclei (up to $^6$Li or $^7$Li). |
---|
| 123 | For $^1$H it is 0.078 and for $^2$H it is 0.08, so the delta contribution |
---|
| 124 | does not appear to be growing. Its relative contribution disappears with |
---|
| 125 | $A$. |
---|
| 126 | |
---|
| 127 | \item $u = 3.7$ and $w = 0.4$. The experimental information is not |
---|
| 128 | sufficient to determine an $A$-dependence for these parameters. For both |
---|
| 129 | $^1$H and $^2$H $u = 6.93$ and $w = 90$, which may indicate contributions |
---|
| 130 | from the $\Delta$(1600) and $\Delta$(1620). |
---|
| 131 | |
---|
| 132 | \end{itemize} |
---|
| 133 | |
---|
| 134 | \noindent |
---|
| 135 | The transition Roper contribution was parameterized using the same form |
---|
| 136 | as the quasi-deuteron contribution: |
---|
| 137 | \begin{equation} |
---|
| 138 | Tr(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}. |
---|
| 139 | \label{Transition} |
---|
| 140 | \end{equation} |
---|
| 141 | Using $a=log(A)$, the values of the parameters are |
---|
| 142 | |
---|
| 143 | \begin{itemize} |
---|
| 144 | |
---|
| 145 | \item $v = exp(-2.+a\cdot 0.84)$. For $^1$H it is 0.22 and for $^2$H |
---|
| 146 | it is 0.34. |
---|
| 147 | |
---|
| 148 | \item $u = 6.46+a\cdot 0.061$ (for $^1$H and for $^2$H it is 6.57), so the |
---|
| 149 | ``mass'' of the Roper moves higher with $A$. |
---|
| 150 | |
---|
| 151 | \item $w = 0.1+a\cdot 1.65$. For $^1$H it is 20.0 and for $^2$H it is 15.0). |
---|
| 152 | \end{itemize} |
---|
| 153 | |
---|
| 154 | |
---|
| 155 | \noindent |
---|
| 156 | The Regge-Pomeron contribution was parametrized as follows: |
---|
| 157 | \begin{equation} |
---|
| 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)), |
---|
| 159 | \label{Regge} |
---|
| 160 | \end{equation} |
---|
| 161 | where $h=A\cdot exp(-a\cdot (0.885+0.0048\cdot a))$ and, again, |
---|
| 162 | $a = log(A)$. The first exponential in Eq.~\ref{Regge} describes the Pomeron |
---|
| 163 | contribution while the second describes the Regge contribution. |
---|
| 164 | |
---|
| 165 | %The result of the approximation is shown in Fig.~\ref{photonuc} for 6 |
---|
| 166 | % of the 14 nuclei. |
---|
| 167 | %\begin{figure} |
---|
| 168 | % \resizebox{1.00\textwidth}{!} |
---|
| 169 | %{ |
---|
| 170 | %% hpw @@@@@ \includegraphics{photonuclear.eps} |
---|
| 171 | %} |
---|
| 172 | %\caption{Photoabsorbtion cross sections for 6 basic nuclei.} |
---|
| 173 | %\label{photonuc} |
---|
| 174 | %\end{figure} |
---|
| 175 | |
---|
| 176 | |
---|
| 177 | \section{Electronuclear Cross Sections and Reactions} |
---|
| 178 | |
---|
| 179 | Electronuclear reactions are so closely connected with photonuclear reactions |
---|
| 180 | that they are sometimes called ``photonuclear'' because the one-photon |
---|
| 181 | exchange mechanism dominates in electronuclear reactions. In this sense |
---|
| 182 | electrons can be replaced by a flux of equivalent photons. This is not |
---|
| 183 | completely true, because at high energies the Vector Dominance Model (VDM) or |
---|
| 184 | diffractive mechanisms are possible, but these types of reactions are beyond |
---|
| 185 | the scope of this discussion. |
---|
| 186 | |
---|
| 187 | \subsection{Common Notation for Different Approaches to Electronuclear |
---|
| 188 | Reactions} |
---|
| 189 | \label{threeApproaches} |
---|
| 190 | |
---|
| 191 | The Equivalent Photon Approximation (EPA) was proposed by |
---|
| 192 | E. Fermi \cite{Fermi} and developed by C. Weizsacker and E. Williams |
---|
| 193 | \cite{WeiWi} and by L. Landau and E. Lifshitz \cite{LanLif}. The |
---|
| 194 | covariant form of the EPA method was developed in Refs. \cite{Pomer} and |
---|
| 195 | \cite{Grib}. When using this method it is necessary to take into account |
---|
| 196 | that real photons are always transversely polarized while virtual photons |
---|
| 197 | may be longitudinally polarized. In general the differential cross section |
---|
| 198 | of the electronuclear interaction can be written as |
---|
| 199 | \begin{equation} |
---|
| 200 | \frac{d^2\sigma}{dydQ^2}=\frac{\alpha}{\pi Q^2}(S_{TL}\cdot(\sigma_T |
---|
| 201 | +\sigma_L)-S_L\cdot\sigma_L), |
---|
| 202 | \label{elNuc} |
---|
| 203 | \end{equation} |
---|
| 204 | where |
---|
| 205 | \begin{equation} |
---|
| 206 | S_{TL}=y\frac{1-y+\frac{y^2}{2}+\frac{Q^2}{4E^2} |
---|
| 207 | -\frac{m^2_e}{Q^2}(y^2+\frac{Q^2}{E^2})}{y^2+\frac{Q^2}{E^2}}, |
---|
| 208 | \label{STL} |
---|
| 209 | \end{equation} |
---|
| 210 | \begin{equation} |
---|
| 211 | S_L=\frac{y}{2}(1-\frac{2m_e^2}{Q^2}). |
---|
| 212 | \label{SL} |
---|
| 213 | \end{equation} |
---|
| 214 | The differential cross section of the electronuclear scattering can be |
---|
| 215 | rewritten as |
---|
| 216 | \begin{equation} |
---|
| 217 | \frac{d^2\sigma_{eA}}{dydQ^2}=\frac{\alpha y}{\pi Q^2}\left(\frac{(1-\frac{y}{2})^2} |
---|
| 218 | {y^2+\frac{Q^2}{E^2}}+\frac{1}{4}-\frac{m^2_e}{Q^2}\right)\sigma_{\gamma^*A}, |
---|
| 219 | \label{difBase} |
---|
| 220 | \end{equation} |
---|
| 221 | where $\sigma_{\gamma^*A}=\sigma_{\gamma A}(\nu)$ for small $Q^2$ and |
---|
| 222 | must be approximated as a function of $\epsilon$, $\nu$, and $Q^2$ for |
---|
| 223 | large $Q^2$. Interactions of longitudinal photons are included in the |
---|
| 224 | effective $\sigma_{\gamma^*A}$ cross section through the $\epsilon$ factor, |
---|
| 225 | but in the present GEANT4 method, the cross section of virtual photons is |
---|
| 226 | considered to be $\epsilon$-independent. The electronuclear problem, with |
---|
| 227 | respect to the interaction of virtual photons with nuclei, can thus be split |
---|
| 228 | in two. At small $Q^2$ it is possible to use the $\sigma_\gamma(\nu)$ cross |
---|
| 229 | section. In the $Q^2>>m^2_e$ region it is necessary to calculate the effective |
---|
| 230 | $\sigma_{\gamma^*}(\epsilon,\nu,Q^2)$ cross section. \\ |
---|
| 231 | |
---|
| 232 | \noindent |
---|
| 233 | Following the EPA notation, the differential cross section of electronuclear |
---|
| 234 | scattering can be related to the number of equivalent photons |
---|
| 235 | $dn=\frac{d\sigma}{\sigma_{\gamma^*}}$. For $y<<1$ and $Q^2<4m^2_e$ the |
---|
| 236 | canonical method \cite{encs.eqPhotons} leads to the simple result |
---|
| 237 | \begin{equation} |
---|
| 238 | \frac{ydn(y)}{dy}=-\frac{2\alpha}{\pi}ln(y). |
---|
| 239 | \label{neq} |
---|
| 240 | \end{equation} |
---|
| 241 | In \cite{Budnev} the integration over $Q^2$ for $\nu^2>>Q^2_{max}\simeq m^2_e$ |
---|
| 242 | leads to |
---|
| 243 | \begin{equation} |
---|
| 244 | \frac{ydn(y)}{dy}=-\frac{\alpha}{\pi}\left( |
---|
| 245 | \frac{1+(1-y)^2}{2}ln(\frac{y^2}{1-y})+(1-y)\right). |
---|
| 246 | \label{lowQ2EP} |
---|
| 247 | \end{equation} |
---|
| 248 | In the $y<<1$ limit this formula converges to Eq.(\ref{neq}). But the |
---|
| 249 | correspondence with Eq.(\ref{neq}) can be made more explicit if the exact |
---|
| 250 | integral |
---|
| 251 | \begin{equation} |
---|
| 252 | \frac{ydn(y)}{dy}=\frac{\alpha}{\pi}\left( |
---|
| 253 | \frac{1+(1-y)^2}{2}l_1-(1-y)l_2-\frac{(2-y)^2}{4}l_3\right), |
---|
| 254 | \label{diff} |
---|
| 255 | \end{equation} |
---|
| 256 | where $l_1=ln\left(\frac{Q^2_{max}}{Q^2_{min}}\right)$, |
---|
| 257 | $l_2=1-\frac{Q^2_{max}}{Q^2_{min}}$, |
---|
| 258 | $l_3=ln\left(\frac{y^2+Q^2_{max}/E^2}{y^2+Q^2_{min}/E^2}\right)$, |
---|
| 259 | $Q^2_{min}=\frac{m_e^2y^2}{1-y}$, |
---|
| 260 | is calculated for |
---|
| 261 | \begin{equation} |
---|
| 262 | Q^2_{max(m_e)}=\frac{4m^2_e}{1-y}. |
---|
| 263 | \label{Q2me} |
---|
| 264 | \end{equation} |
---|
| 265 | The factor $(1-y)$ is used arbitrarily to keep $Q^2_{max(m_e)}>Q^2_{min}$, |
---|
| 266 | which can be considered as a boundary between the low and high $Q^2$ |
---|
| 267 | regions. The full transverse photon flux can be calculated as an integral |
---|
| 268 | of Eq.(\ref{diff}) with the maximum possible upper limit |
---|
| 269 | \begin{equation} |
---|
| 270 | Q^2_{max(max)}=4E^2(1-y). |
---|
| 271 | \label{Q2max} |
---|
| 272 | \end{equation} |
---|
| 273 | The full transverse photon flux can be approximated by |
---|
| 274 | \begin{equation} |
---|
| 275 | \frac{ydn(y)}{dy}=-\frac{2\alpha}{\pi}\left( |
---|
| 276 | \frac{(2-y)^2+y^2}{2}ln(\gamma)-1\right), |
---|
| 277 | \label{neqHQ} |
---|
| 278 | \end{equation} |
---|
| 279 | where $\gamma=\frac{E}{m_e}$. It must be pointed out that neither this |
---|
| 280 | approximation nor Eq.(\ref{diff}) works at $y\simeq 1$; at this point |
---|
| 281 | $Q^2_{max(max)}$ becomes smaller than $Q^2_{min}$. The formal limit of the |
---|
| 282 | method is $y<1-\frac{1}{2\gamma}$. \\ |
---|
| 283 | \begin{figure}[tbp] |
---|
| 284 | \resizebox{0.95\textwidth}{!} |
---|
| 285 | { |
---|
| 286 | \includegraphics{hadronic/theory_driven/ChiralInvariantPhaseSpace/Fig12.eps} |
---|
| 287 | } |
---|
| 288 | \caption{Relative contribution of equivalent photons with small $Q^2$ |
---|
| 289 | to the total ``photon flux'' for (a) $1~GeV$ electrons and (b) $10~GeV$ |
---|
| 290 | electrons. In figures (c) and (d) the equivalent photon distribution |
---|
| 291 | $dn(\nu,Q^2)$ is multiplied by the photonuclear cross section |
---|
| 292 | $\sigma_{\gamma^*}(K,Q^2)$ and integrated over $Q^2$ in two regions: |
---|
| 293 | the dashed lines are integrals over the low-$Q^2$ equivalent |
---|
| 294 | photons (under the dashed line in the first two figures), and the |
---|
| 295 | solid lines are integrals over the high-$Q^2$ equivalent photons (above |
---|
| 296 | the dashed lines in the first two figures).} |
---|
| 297 | \label{nSigma} |
---|
| 298 | \end{figure} |
---|
| 299 | |
---|
| 300 | \noindent |
---|
| 301 | In Fig.~\ref{nSigma}(a,b) the energy distribution for the equivalent photons |
---|
| 302 | is shown. The low-$Q^2$ photon flux with the upper limit defined by |
---|
| 303 | Eq.(\ref{Q2me})) is compared with the full photon flux. The |
---|
| 304 | low-$Q^2$ photon flux is calculated using Eq.(\ref{neq}) (dashed lines) and |
---|
| 305 | using Eq.(\ref{diff}) (dotted lines). The full photon |
---|
| 306 | flux is calculated using Eq.(\ref{neqHQ}) (the solid lines) and using |
---|
| 307 | Eq.(\ref{diff}) with the upper limit defined by Eq.(\ref{Q2max}) (dash-dotted |
---|
| 308 | lines, which differ from the solid lines only at $\nu\approx E_e$). The |
---|
| 309 | conclusion is that in order to calculate either the number of low-$Q^2$ |
---|
| 310 | equivalent photons or the total number of equivalent photons one can use the |
---|
| 311 | simple approximations given by Eq.(\ref{neq}) and Eq.(\ref{neqHQ}), |
---|
| 312 | respectively, instead of using Eq.(\ref{diff}), which cannot be integrated |
---|
| 313 | over $y$ analytically. Comparing the low-$Q^2$ photon flux and the total |
---|
| 314 | photon flux it is possible to show that the low-$Q^2$ photon flux is about |
---|
| 315 | half of the the total. From the interaction point of view the decrease of |
---|
| 316 | $\sigma_{\gamma*}$ with increasing $Q^2$ must be taken into account. The |
---|
| 317 | cross section reduction for the virtual photons with large $Q^2$ is governed |
---|
| 318 | by two factors. First, the cross section drops with $Q^2$ as the squared |
---|
| 319 | dipole nucleonic form-factor |
---|
| 320 | \begin{equation} |
---|
| 321 | G^2_D(Q^2)\approx\left( 1+\frac{Q^2}{(843~MeV)^2}\right)^{-2}. |
---|
| 322 | \label{G2} |
---|
| 323 | \end{equation} |
---|
| 324 | Second, all the thresholds of the $\gamma A$ reactions are shifted to higher |
---|
| 325 | $\nu$ by a factor $\frac{Q^2}{2M}$, which is the difference between the $K$ |
---|
| 326 | and $\nu$ values. Following the method proposed in \cite{Brasse} |
---|
| 327 | the $\sigma_{\gamma^*}$ at large $Q^2$ can be approximated as |
---|
| 328 | \begin{equation} |
---|
| 329 | \sigma_{\gamma*}=(1-x)\sigma_\gamma(K)G^2_D(Q^2)e^{b(\epsilon,K)\cdot |
---|
| 330 | r+c(\epsilon,K)\cdot r^3}, |
---|
| 331 | \label{abc} |
---|
| 332 | \end{equation} |
---|
| 333 | where $r=\frac{1}{2}ln(\frac{Q^2+\nu^2}{K^2})$. The $\epsilon$-dependence of |
---|
| 334 | the $a(\epsilon,K)$ and $b(\epsilon,K)$ functions is weak, so for simplicity |
---|
| 335 | the $b(K)$ and $c(K)$ functions are averaged over $\epsilon$. They can be |
---|
| 336 | approximated as |
---|
| 337 | \begin{equation} |
---|
| 338 | b(K)\approx\left(\frac{K}{185~MeV}\right)^{0.85}, |
---|
| 339 | \label{bk} |
---|
| 340 | \end{equation} |
---|
| 341 | and |
---|
| 342 | \begin{equation} |
---|
| 343 | c(K)\approx-\left(\frac{K}{1390~MeV}\right)^{3}. |
---|
| 344 | \label{ck} |
---|
| 345 | \end{equation} |
---|
| 346 | |
---|
| 347 | \noindent |
---|
| 348 | The result of the integration of the photon flux multiplied by the |
---|
| 349 | cross section approximated by Eq.(\ref{abc}) is shown in |
---|
| 350 | Fig.~\ref{nSigma}(c,d). The integrated cross sections are shown |
---|
| 351 | separately for the low-$Q^2$ region ($Q^2<Q^2_{max(m_e)}$, dashed |
---|
| 352 | lines) and for the high-$Q^2$ region ($Q^2>Q^2_{max(m_e)}$, solid |
---|
| 353 | lines). These functions must be integrated over $ln(\nu)$, so it is |
---|
| 354 | clear that because of the Giant Dipole Resonance contribution, the |
---|
| 355 | low-$Q^2$ part covers more than half the total $eA\rightarrow hadrons$ |
---|
| 356 | cross section. But at $\nu>200~MeV$, where the hadron multiplicity |
---|
| 357 | increases, the large $Q^2$ part dominates. In this sense, for a better |
---|
| 358 | simulation of the production of hadrons by electrons, it is necessary to |
---|
| 359 | simulate the high-$Q^2$ part as well as the low-$Q^2$ part. \\ |
---|
| 360 | |
---|
| 361 | \noindent |
---|
| 362 | Taking into account the contribution of high-$Q^2$ photons it is possible to |
---|
| 363 | use Eq.(\ref{neqHQ}) with the over-estimated |
---|
| 364 | $\sigma_{\gamma^*A}=\sigma_{\gamma A}(\nu)$ cross section. The slightly |
---|
| 365 | over-estimated electronuclear cross section is |
---|
| 366 | \begin{equation} |
---|
| 367 | \sigma^*_{eA}=(2ln(\gamma)-1)\cdot J_1-\frac{ln(\gamma)}{E_e} |
---|
| 368 | \left( 2J_2-\frac{J_3}{E_e} \right). |
---|
| 369 | \label{eleNucHQ} |
---|
| 370 | \end{equation} |
---|
| 371 | where |
---|
| 372 | \begin{equation} |
---|
| 373 | J_1(E_e)=\frac{\alpha}{\pi}\int^{E_e}\sigma_{\gamma A}(\nu)dln(\nu) |
---|
| 374 | \label{J1} |
---|
| 375 | \end{equation} |
---|
| 376 | \begin{equation} |
---|
| 377 | J_2(E_e)=\frac{\alpha}{\pi}\int^{E_e}\nu\sigma_{\gamma A}(\nu)dln(\nu), |
---|
| 378 | \label{J2} |
---|
| 379 | \end{equation} |
---|
| 380 | and |
---|
| 381 | \begin{equation} |
---|
| 382 | J_3(E_e)=\frac{\alpha}{\pi}\int^{E_e}\nu^2\sigma_{\gamma A}(\nu )dln(\nu). |
---|
| 383 | \label{J3} |
---|
| 384 | \end{equation} |
---|
| 385 | The equivalent photon energy $\nu=yE$ can be obtained for a particular |
---|
| 386 | random number $R$ from the equation |
---|
| 387 | \begin{equation} |
---|
| 388 | R=\frac{(2ln(\gamma)-1)J_1(\nu)-\frac{ln(\gamma)}{E_e}(2J_2(\nu)-\frac{J_3(\nu)}{E_e})} |
---|
| 389 | {(2ln(\gamma)-1)J_1(E_e)-\frac{ln(\gamma)}{E_e}(2J_2(E_e)-\frac{J_3(E_e)}{E_e})}. |
---|
| 390 | \label{RnuHH} |
---|
| 391 | \end{equation} |
---|
| 392 | Eq.(\ref{diff}) is too complicated for the randomization of $Q^2$ but |
---|
| 393 | there is an easily randomized formula which approximates Eq.(\ref{diff}) |
---|
| 394 | above the hadronic threshold ($E>10~MeV$). It reads |
---|
| 395 | \begin{equation} |
---|
| 396 | \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), |
---|
| 397 | \label{RQ2HH} |
---|
| 398 | \end{equation} |
---|
| 399 | where |
---|
| 400 | \begin{equation} |
---|
| 401 | D(y)=1-y+\frac{y^2}{2}, |
---|
| 402 | \label{RQ2D} |
---|
| 403 | \end{equation} |
---|
| 404 | \begin{equation} |
---|
| 405 | L(y,Q^2)=ln\left( F(y)+(e^{P(y)}-1+\frac{Q^2}{Q^2_{min}})^{-1} \right), |
---|
| 406 | \label{RQ2L} |
---|
| 407 | \end{equation} |
---|
| 408 | and |
---|
| 409 | \begin{equation} |
---|
| 410 | U(y)=P(y)\cdot\left( 1-\frac{Q^2_{min}}{Q^2_{max}}\right), |
---|
| 411 | \label{RQ2U} |
---|
| 412 | \end{equation} |
---|
| 413 | with |
---|
| 414 | \begin{equation} |
---|
| 415 | F(y)=\frac{(2-y)(2-2y)}{y^2}\cdot\frac{Q^2_{min}}{Q^2_{max}} |
---|
| 416 | \label{RQ2F} |
---|
| 417 | \end{equation} |
---|
| 418 | and |
---|
| 419 | \begin{equation} |
---|
| 420 | P(y)=\frac{1-y}{D(y)}. |
---|
| 421 | \label{RQ2P} |
---|
| 422 | \end{equation} |
---|
| 423 | The $Q^2$ value can then be calculated as |
---|
| 424 | \begin{equation} |
---|
| 425 | \frac{Q^2}{Q^2_{min}}=1-e^{P(y)}+\left(e^{R\cdot |
---|
| 426 | L(y,Q^2_{max})-(1-R)\cdot U(y)}-F(y) \right)^{-1}, |
---|
| 427 | \label{Q2sol} |
---|
| 428 | \end{equation} |
---|
| 429 | where $R$ is a random number. In Fig.~\ref{Q2dep}, Eq.(\ref{diff}) (solid |
---|
| 430 | curve) is compared to Eq.(\ref{RQ2HH}) (dashed curve). Because the two |
---|
| 431 | curves are almost indistinguishable in the figure, this can be used as an |
---|
| 432 | illustration of the $Q^2$ spectrum of virtual photons, which is the derivative |
---|
| 433 | of these curves. An alternative approach is to use Eq.(\ref{diff}) for the |
---|
| 434 | randomization with a three dimensional table $\frac{ydn}{dy}(Q^2,y,E_e)$. |
---|
| 435 | \begin{figure}[tbp] |
---|
| 436 | \resizebox{0.95\textwidth}{!} |
---|
| 437 | { |
---|
| 438 | \includegraphics{hadronic/theory_driven/ChiralInvariantPhaseSpace/Fig13.eps} |
---|
| 439 | } |
---|
| 440 | \caption{Integrals of $Q^2$ spectra of virtual photons for three |
---|
| 441 | energies $10~MeV$, $100~MeV$, and $1~GeV$ at $y=0.001$, $y=0.5$, and $y=0.95$. |
---|
| 442 | The solid line corresponds to Eq.(\protect\ref{diff}) and the dashed |
---|
| 443 | line (which almost everywhere coincides with the solid line) |
---|
| 444 | corresponds to Eq.(\protect\ref{diff}).} |
---|
| 445 | \label{Q2dep} |
---|
| 446 | \end{figure} |
---|
| 447 | |
---|
| 448 | \noindent |
---|
| 449 | After the $\nu$ and $Q^2$ values have been found, the value of |
---|
| 450 | $\sigma_{\gamma^*A}(\nu,Q^2)$ is calculated using Eq.(\ref{abc}). |
---|
| 451 | If $R\cdot\sigma_{\gamma A}(\nu)>\sigma_{\gamma^*A}(\nu,Q^2)$, no |
---|
| 452 | interaction occurs and the electron keeps going. This ``do nothing'' |
---|
| 453 | process has low probability and cannot shadow other processes. |
---|
| 454 | |
---|
| 455 | |
---|
| 456 | \section {Status of this document} |
---|
| 457 | created by ? \\ |
---|
| 458 | 20.05.02 re-written by D.H. Wright \\ |
---|
| 459 | 01.12.02 expanded section on electronuclear cross sections - H.P. Wellisch \\ |
---|
| 460 | |
---|
| 461 | |
---|
| 462 | \begin{latexonly} |
---|
| 463 | |
---|
| 464 | \begin{thebibliography}{99} |
---|
| 465 | |
---|
| 466 | \bibitem{Fermi} E. Fermi, Z. Physik {\textbf{29}}, 315 (1924). |
---|
| 467 | |
---|
| 468 | \bibitem{WeiWi} K. F. von Weizsacker, Z. Physik {\textbf{88}}, 612 (1934), |
---|
| 469 | E. J. Williams, Phys. Rev. {\textbf{45}}, 729 (1934). |
---|
| 470 | |
---|
| 471 | \bibitem{LanLif} L. D. Landau and E. M. Lifshitz, |
---|
| 472 | Soc. Phys. {\textbf{6}}, 244 (1934). |
---|
| 473 | |
---|
| 474 | \bibitem{Pomer} I. Ya. Pomeranchuk and I. M. Shmushkevich, |
---|
| 475 | Nucl. Phys. {\textbf{23}}, 1295 (1961). |
---|
| 476 | |
---|
| 477 | \bibitem{Grib} V. N. Gribov {\textit {et~al.}}, ZhETF {\textbf{41}}, 1834 (1961). |
---|
| 478 | |
---|
| 479 | \bibitem{encs.eqPhotons} L. D. Landau, E. M. Lifshitz, ``Course of |
---|
| 480 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
---|
| 481 | Pergamon Press, p. 351, The method of equivalent photons. |
---|
| 482 | |
---|
| 483 | \bibitem{Budnev} V. M. Budnev {\textit {et~al.}}, Phys. Rep. {\textbf{15}}, 181 |
---|
| 484 | (1975). |
---|
| 485 | |
---|
| 486 | \bibitem{Brasse} F. W. Brasse {\textit {et~al.}}, Nucl. Phys. B {\textbf{110}}, 413 |
---|
| 487 | (1976). |
---|
| 488 | |
---|
| 489 | \end{thebibliography} |
---|
| 490 | |
---|
| 491 | \end{latexonly} |
---|
| 492 | |
---|
| 493 | \begin{htmlonly} |
---|
| 494 | |
---|
| 495 | \section{Bibliography} |
---|
| 496 | |
---|
| 497 | \begin{enumerate} |
---|
| 498 | \item E. Fermi, Z. Physik {\textbf{29}}, 315 (1924). |
---|
| 499 | |
---|
| 500 | \item K. F. von Weizsacker, Z. Physik {\textbf{88}}, 612 (1934), |
---|
| 501 | E. J. Williams, Phys. Rev. {\textbf{45}}, 729 (1934). |
---|
| 502 | |
---|
| 503 | \item L. D. Landau and E. M. Lifshitz, |
---|
| 504 | Soc. Phys. {\textbf{6}}, 244 (1934). |
---|
| 505 | |
---|
| 506 | \item I. Ya. Pomeranchuk and I. M. Shmushkevich, |
---|
| 507 | Nucl. Phys. {\textbf{23}}, 1295 (1961). |
---|
| 508 | |
---|
| 509 | \item V.N. Gribov {\textit {et~al.}}, ZhETF {\textbf{41}}, 1834 (1961). |
---|
| 510 | |
---|
| 511 | \item L.D. Landau, E. M. Lifshitz, ``Course of |
---|
| 512 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
---|
| 513 | Pergamon Press, p. 351, The method of equivalent photons. |
---|
| 514 | |
---|
| 515 | \item V.M. Budnev {\textit {et~al.}}, Phys. Rep. {\textbf{15}}, 181 |
---|
| 516 | (1975). |
---|
| 517 | |
---|
| 518 | \item F.W. Brasse {\textit {et~al.}}, Nucl. Phys. B {\textbf{110}}, 413 |
---|
| 519 | (1976). |
---|
| 520 | |
---|
| 521 | \end{enumerate} |
---|
| 522 | |
---|
| 523 | \end{htmlonly} |
---|
| 524 | |
---|
| 525 | |
---|
| 526 | |
---|
| 527 | |
---|