source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/ChiralInvariantPhaseSpace/PhotoElectroNucCrsSctn.tex @ 1211

\chapter[Cross-sections in Photonuclear/Electronuclear Reactions]{Cross-sections in Photonuclear and Electronuclear Reactions}
\section{Approximation of Photonuclear Cross Sections.}

The photonuclear cross sections parameterized in the 
{\tt G4PhotoNuclearCrossSection} class cover all incident photon energies from
the hadron production threshold upward.  The parameterization is subdivided 
into five energy regions, each corresponding to the physical process that 
dominates it.

\begin{itemize}

\item The Giant Dipole Resonance (GDR) region, depending on the nucleus,
      extends from 10 Mev up to 30 MeV.  It usually consists of one large
      peak, though for some nuclei several peaks appear.
 
\item The ``quasi-deuteron'' region extends from around 30 MeV up to the
      pion threshold and is characterized by small cross sections and a broad,
      low peak.

\item The $\Delta$ region is characterized by the dominant peak in the 
      cross section which extends from the pion threshold to 450 MeV.

\item The Roper resonance region extends from roughly 450 MeV to 1.2 GeV.
      The cross section in this region is not strictly identified with the
      real Roper resonance because other processes also occur in this region.  

\item The Reggeon-Pomeron region extends upward from 1.2 GeV. 

\end{itemize}

\noindent 
In the GEANT4 photonuclear data base there are about 50 nuclei for which the 
photonuclear absorption cross sections have been measured in the above 
energy ranges.  For low energies this number could be enlarged, because for 
heavy nuclei the neutron photoproduction cross section is close to the total 
photo-absorption cross section.  Currently, however, 14 nuclei are used in 
the parameterization: $^1$H, $^2$H, $^4$He, $^6$Li, $^7$Li, $^9$Be, 
$^{12}$C, $^{16}$O, $^{27}$Al, $^{40}$Ca, Cu, Sn, Pb, and U.  The resulting 
cross section is a function of $A$ and $e = log(E_\gamma)$, where $E_\gamma$ 
is the energy of the incident photon.  This function is the sum of the 
components which parameterize each energy region. \\

\noindent
The cross section in the GDR region can be described as the sum of two 
peaks,
\begin{equation}
GDR(e) = th(e,b_1,s_1)\cdot exp(c_1-p_1\cdot e) + 
         th(e,b_2,s_2)\cdot exp(c_2-p_2\cdot e) .
\end{equation}
The exponential parameterizes the falling edge of the resonance which 
behaves like a power law in $E_\gamma$.  This behavior is expected from  
the CHIPS model, which includes the nonrelativistic phase space of nucleons 
to explain evaporation.  The function
\begin{equation}
th(e,b,s) = \frac{1}{1+exp(\frac{b-e}{s})} ,
\end{equation}
describes the rising edge of the resonance.  It is the
nuclear-barrier-reflection function and behaves like a threshold, cutting off
the exponential.  The exponential powers $p_1$ and $p_2$ are

\begin{eqnarray*}
 p_1 = 1, p_2 = 2 \mbox{\hspace*{1mm} for \hspace*{7mm} $A < 4$ }\\ 
 p_1 = 2, p_2 = 4 \mbox{\hspace*{1mm} for \hspace*{1mm} $4 \le A < 8$ }\\
 p_1 = 3, p_2 = 6 \mbox{\hspace*{1mm} for $8 \le A < 12$} \\
 p_1 = 4, p_2 = 8 \mbox{\hspace*{1mm} for \hspace*{6mm} $A \ge 12$} .
\end{eqnarray*}
 
\noindent
The $A$-dependent parameters $b_i$, $c_i$ and $s_i$ were found for each of
the 14 nuclei listed above and interpolated for other nuclei. \\

\noindent
The $\Delta$ isobar region was parameterized as
\begin{equation}
\Delta (e,d,f,g,r,q)=\frac{d\cdot th(e,f,g)}{1+r\cdot (e-q)^2},
\label{Isobar}
\end{equation}
where $d$ is an overall normalization factor.  $q$ can be interpreted as the 
energy of the $\Delta$ isobar and $r$ can be interpreted as the inverse of 
the $\Delta$ width.  Once again $th$ is the threshold function.  The
$A$-dependence of these parameters is as follows:

\begin{itemize}
\item  $d=0.41\cdot A$ (for $^1$H it is 0.55, for $^2$H it is 0.88),
which means that the $\Delta$ yield is proportional
to $A$; 

\item  $f=5.13-.00075\cdot A$.  $exp(f)$ shows how the pion threshold depends
on $A$.  It is clear that the threshold becomes 140 MeV only for uranium; 
for lighter nuclei it is higher. 

\item  $g = 0.09$ for $A \ge 7$ and 0.04 for $A < 7$;

\item  $q=5.84-\frac{.09}{1+.003\cdot A^2}$, which means that the ``mass'' 
of the $\Delta$ isobar moves to lower energies;

\item $r=11.9 - 1.24\cdot log(A)$.  $r$ is 18.0 for $^1$H.
The inverse width becomes smaller with $A$, hence the width increases.

\end{itemize}
The $A$-dependence of the $f$, $q$ and $r$ parameters is due to the 
$\Delta+N\rightarrow N+N$ reaction, which can take place in the nuclear 
medium below the pion threshold. \\

\noindent
The quasi-deuteron contribution was parameterized with the same form as the 
$\Delta$ contribution but without the threshold function:
\begin{equation}
QD(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}.
\label{QuasiD}
\end{equation}
For $^1$H and $^2$H the quasi-deuteron contribution is almost zero.  For 
these nuclei the third baryonic resonance was used instead, so the 
parameters for these two nuclei are quite different, but trivial.
The parameter values are given below.

\begin{itemize}

\item  $v = \frac {exp(-1.7+a\cdot 0.84)}{1+exp(7\cdot (2.38-a))}$, where 
$a=log(A)$.  This shows that the $A$-dependence in the quasi-deuteron 
region is stronger than $A^{0.84}$.  It is clear from the denominator that 
this contribution is very small for light nuclei (up to $^6$Li or $^7$Li). 
For $^1$H it is 0.078 and for $^2$H it is 0.08, so the delta contribution 
does not appear to be growing.  Its relative contribution disappears with 
$A$.

\item  $u = 3.7$ and $w = 0.4$.  The experimental information is not 
sufficient to determine an $A$-dependence for these parameters.  For both 
$^1$H and $^2$H $u = 6.93$ and $w = 90$, which may indicate contributions
from the $\Delta$(1600) and $\Delta$(1620).

\end{itemize}

\noindent
The transition Roper contribution was parameterized using the same form
as the quasi-deuteron contribution:
\begin{equation}
Tr(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}.
\label{Transition}
\end{equation}
Using $a=log(A)$, the values of the parameters are 

\begin{itemize}

\item  $v = exp(-2.+a\cdot 0.84)$.  For $^1$H it is 0.22 and for $^2$H 
it is 0.34.

\item $u = 6.46+a\cdot 0.061$ (for $^1$H and for $^2$H it is 6.57), so the 
``mass'' of the Roper moves higher with $A$.

\item $w = 0.1+a\cdot 1.65$.  For $^1$H it is 20.0 and for $^2$H it is 15.0).
\end{itemize}


\noindent
The Regge-Pomeron contribution was parametrized as follows:
\begin{equation}
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)),
\label{Regge}
\end{equation}
where $h=A\cdot exp(-a\cdot (0.885+0.0048\cdot a))$ and, again,
$a = log(A)$.  The first exponential in Eq.~\ref{Regge} describes the Pomeron 
contribution while the second describes the Regge contribution.

%The result of the approximation is shown in Fig.~\ref{photonuc} for 6 
% of the 14 nuclei.
%\begin{figure}
%  \resizebox{1.00\textwidth}{!}
%{
%% hpw @@@@@   \includegraphics{photonuclear.eps}
%}
%\caption{Photoabsorbtion cross sections for 6 basic nuclei.}
%\label{photonuc}
%\end{figure}


\section{Electronuclear Cross Sections and Reactions}

Electronuclear reactions are so closely connected with photonuclear reactions 
that they are sometimes called ``photonuclear'' because the one-photon 
exchange mechanism dominates in electronuclear reactions.  In this sense 
electrons can be replaced by a flux of equivalent photons.  This is not 
completely true, because at high energies the Vector Dominance Model (VDM) or 
diffractive mechanisms are possible, but these types of reactions are beyond 
the scope of this discussion.

\subsection[Common Notation for Electronuclear Reactions]{Common Notation for Different Approaches to Electronuclear Reactions}
\label{threeApproaches}

The Equivalent Photon Approximation (EPA) was proposed by
E. Fermi \cite{Fermi} and developed by C. Weizsacker and E. Williams
\cite{WeiWi} and by L. Landau and E. Lifshitz \cite{LanLif}. The
covariant form of the EPA method was developed in Refs. \cite{Pomer} and 
\cite{Grib}.  When using this method it is necessary to take into account 
that real photons are always transversely polarized while virtual photons 
may be longitudinally polarized.  In general the differential cross section
of the electronuclear interaction can be written as
\begin{equation}
\frac{d^2\sigma}{dydQ^2}=\frac{\alpha}{\pi Q^2}(S_{TL}\cdot(\sigma_T
+\sigma_L)-S_L\cdot\sigma_L),
\label{elNuc}
\end{equation}
where
\begin{equation}
S_{TL}=y\frac{1-y+\frac{y^2}{2}+\frac{Q^2}{4E^2}
-\frac{m^2_e}{Q^2}(y^2+\frac{Q^2}{E^2})}{y^2+\frac{Q^2}{E^2}},
\label{STL}
\end{equation}
\begin{equation}
S_L=\frac{y}{2}(1-\frac{2m_e^2}{Q^2}).
\label{SL}
\end{equation}
The differential cross section of the electronuclear scattering can be
rewritten as
\begin{equation}
\frac{d^2\sigma_{eA}}{dydQ^2}=\frac{\alpha y}{\pi Q^2}\left(\frac{(1-\frac{y}{2})^2}
{y^2+\frac{Q^2}{E^2}}+\frac{1}{4}-\frac{m^2_e}{Q^2}\right)\sigma_{\gamma^*A},
\label{difBase}
\end{equation}
where $\sigma_{\gamma^*A}=\sigma_{\gamma A}(\nu)$ for small $Q^2$ and
must be approximated as a function of $\epsilon$, $\nu$, and $Q^2$ for
large $Q^2$.  Interactions of longitudinal photons are included in the 
effective $\sigma_{\gamma^*A}$ cross section through the $\epsilon$ factor, 
but in the present GEANT4 method, the cross section of virtual photons is 
considered to be $\epsilon$-independent.  The electronuclear problem, with
respect to the interaction of virtual photons with nuclei, can thus be split 
in two.  At small $Q^2$ it is possible to use the $\sigma_\gamma(\nu)$ cross 
section.  In the $Q^2>>m^2_e$ region it is necessary to calculate the effective
$\sigma_{\gamma^*}(\epsilon,\nu,Q^2)$ cross section. \\

\noindent 
Following the EPA notation, the differential cross section of electronuclear 
scattering can be related to the number of equivalent photons 
$dn=\frac{d\sigma}{\sigma_{\gamma^*}}$.  For $y<<1$ and $Q^2<4m^2_e$ the 
canonical method \cite{encs.eqPhotons} leads to the simple result
\begin{equation}
\frac{ydn(y)}{dy}=-\frac{2\alpha}{\pi}ln(y).
\label{neq}
\end{equation}
In \cite{Budnev} the integration over $Q^2$ for $\nu^2>>Q^2_{max}\simeq m^2_e$
leads to
\begin{equation}
\frac{ydn(y)}{dy}=-\frac{\alpha}{\pi}\left(
\frac{1+(1-y)^2}{2}ln(\frac{y^2}{1-y})+(1-y)\right).
\label{lowQ2EP}
\end{equation}
In the $y<<1$ limit this formula converges to Eq.(\ref{neq}).  But the
correspondence with Eq.(\ref{neq}) can be made more explicit if the exact 
integral
\begin{equation}
\frac{ydn(y)}{dy}=\frac{\alpha}{\pi}\left(
\frac{1+(1-y)^2}{2}l_1-(1-y)l_2-\frac{(2-y)^2}{4}l_3\right),
\label{diff}
\end{equation}
where $l_1=ln\left(\frac{Q^2_{max}}{Q^2_{min}}\right)$,
$l_2=1-\frac{Q^2_{max}}{Q^2_{min}}$, 
$l_3=ln\left(\frac{y^2+Q^2_{max}/E^2}{y^2+Q^2_{min}/E^2}\right)$,
$Q^2_{min}=\frac{m_e^2y^2}{1-y}$, 
is calculated for
\begin{equation}
Q^2_{max(m_e)}=\frac{4m^2_e}{1-y}.
\label{Q2me}
\end{equation}
The factor $(1-y)$ is used arbitrarily to keep $Q^2_{max(m_e)}>Q^2_{min}$, 
which can be considered as a boundary between the low and high $Q^2$ 
regions.  The full transverse photon flux can be calculated as an integral 
of Eq.(\ref{diff}) with the maximum possible upper limit
\begin{equation}
Q^2_{max(max)}=4E^2(1-y).
\label{Q2max}
\end{equation}
The full transverse photon flux can be approximated by
\begin{equation}
\frac{ydn(y)}{dy}=-\frac{2\alpha}{\pi}\left(
\frac{(2-y)^2+y^2}{2}ln(\gamma)-1\right),
\label{neqHQ}
\end{equation}
where $\gamma=\frac{E}{m_e}$.  It must be pointed out that neither this
approximation nor Eq.(\ref{diff}) works at $y\simeq 1$;  at this point 
$Q^2_{max(max)}$ becomes smaller than $Q^2_{min}$.  The formal limit of the 
method is $y<1-\frac{1}{2\gamma}$. \\
\begin{figure}[tbp]
\resizebox{0.95\textwidth}{!}
{
   \includegraphics{hadronic/theory_driven/ChiralInvariantPhaseSpace/Fig12.eps}
}
\caption{Relative contribution of equivalent photons with small $Q^2$
to the total ``photon flux'' for (a) $1~GeV$ electrons and (b) $10~GeV$
electrons.  In figures (c) and (d) the equivalent photon distribution 
$dn(\nu,Q^2)$ is multiplied by the photonuclear cross section
$\sigma_{\gamma^*}(K,Q^2)$ and integrated over $Q^2$ in two regions:
the dashed lines are integrals over the low-$Q^2$ equivalent
photons (under the dashed line in the first two figures), and the
solid lines are integrals over the high-$Q^2$ equivalent photons (above
the dashed lines in the first two figures).}
\label{nSigma}
\end{figure}

\noindent 
In Fig.~\ref{nSigma}(a,b) the energy distribution for the equivalent photons
is shown.  The low-$Q^2$ photon flux with the upper limit defined by
Eq.(\ref{Q2me})) is compared with the full photon flux.  The
low-$Q^2$ photon flux is calculated using Eq.(\ref{neq}) (dashed lines) and 
using Eq.(\ref{diff}) (dotted lines).  The full photon
flux is calculated using Eq.(\ref{neqHQ}) (the solid lines) and using
Eq.(\ref{diff}) with the upper limit defined by Eq.(\ref{Q2max}) (dash-dotted 
lines, which differ from the solid lines only at $\nu\approx E_e$).  The 
conclusion is that in order to calculate either the number of low-$Q^2$ 
equivalent photons or the total number of equivalent photons one can use the 
simple approximations given by Eq.(\ref{neq}) and Eq.(\ref{neqHQ}), 
respectively, instead of using Eq.(\ref{diff}), which cannot be integrated 
over $y$ analytically.  Comparing the low-$Q^2$ photon flux and the total 
photon flux it is possible to show that the low-$Q^2$ photon flux is about 
half of the the total.  From the interaction point of view the decrease of 
$\sigma_{\gamma*}$ with increasing $Q^2$ must be taken into account.  The 
cross section reduction for the virtual photons with large $Q^2$ is governed 
by two factors.  First, the cross section drops with $Q^2$ as the squared 
dipole nucleonic form-factor
\begin{equation}
G^2_D(Q^2)\approx\left( 1+\frac{Q^2}{(843~MeV)^2}\right)^{-2}.
\label{G2}
\end{equation}
Second, all the thresholds of the $\gamma A$ reactions are shifted to higher 
$\nu$ by a factor $\frac{Q^2}{2M}$, which is the difference between the $K$ 
and $\nu$ values.  Following the method proposed in \cite{Brasse}
the $\sigma_{\gamma^*}$ at large $Q^2$ can be approximated as
\begin{equation}
\sigma_{\gamma*}=(1-x)\sigma_\gamma(K)G^2_D(Q^2)e^{b(\epsilon,K)\cdot
r+c(\epsilon,K)\cdot r^3},
\label{abc}
\end{equation}
where $r=\frac{1}{2}ln(\frac{Q^2+\nu^2}{K^2})$.  The $\epsilon$-dependence of 
the $a(\epsilon,K)$ and $b(\epsilon,K)$ functions is weak, so for simplicity 
the $b(K)$ and $c(K)$ functions are averaged over $\epsilon$.  They can be 
approximated as
\begin{equation}
b(K)\approx\left(\frac{K}{185~MeV}\right)^{0.85},
\label{bk}
\end{equation}
and
\begin{equation}
c(K)\approx-\left(\frac{K}{1390~MeV}\right)^{3}.
\label{ck}
\end{equation}

\noindent 
The result of the integration of the photon flux multiplied by the
cross section approximated by Eq.(\ref{abc}) is shown in
Fig.~\ref{nSigma}(c,d).  The integrated cross sections are shown
separately for the low-$Q^2$ region ($Q^2<Q^2_{max(m_e)}$, dashed
lines) and for the high-$Q^2$ region ($Q^2>Q^2_{max(m_e)}$, solid
lines).  These functions must be integrated over $ln(\nu)$, so it is
clear that because of the Giant Dipole Resonance contribution, the 
low-$Q^2$ part covers more than half the total $eA\rightarrow hadrons$ 
cross section.  But at $\nu>200~MeV$, where the hadron multiplicity
increases, the large $Q^2$ part dominates.  In this sense, for a better 
simulation of the production of hadrons by electrons, it is necessary to 
simulate the high-$Q^2$ part as well as the low-$Q^2$ part. \\

\noindent 
Taking into account the contribution of high-$Q^2$ photons it is possible to 
use Eq.(\ref{neqHQ}) with the over-estimated
$\sigma_{\gamma^*A}=\sigma_{\gamma A}(\nu)$ cross section.  The slightly 
over-estimated electronuclear cross section is
\begin{equation}
\sigma^*_{eA}=(2ln(\gamma)-1)\cdot J_1-\frac{ln(\gamma)}{E_e}
\left( 2J_2-\frac{J_3}{E_e} \right).
\label{eleNucHQ}
\end{equation}
where
\begin{equation}
J_1(E_e)=\frac{\alpha}{\pi}\int^{E_e}\sigma_{\gamma A}(\nu)dln(\nu)
\label{J1}
\end{equation}
\begin{equation}
J_2(E_e)=\frac{\alpha}{\pi}\int^{E_e}\nu\sigma_{\gamma A}(\nu)dln(\nu),
\label{J2}
\end{equation}
and
\begin{equation}
J_3(E_e)=\frac{\alpha}{\pi}\int^{E_e}\nu^2\sigma_{\gamma A}(\nu )dln(\nu).
\label{J3}
\end{equation}
The equivalent photon energy $\nu=yE$ can be obtained for a particular 
random number $R$ from the equation 
\begin{equation}
R=\frac{(2ln(\gamma)-1)J_1(\nu)-\frac{ln(\gamma)}{E_e}(2J_2(\nu)-\frac{J_3(\nu)}{E_e})}
{(2ln(\gamma)-1)J_1(E_e)-\frac{ln(\gamma)}{E_e}(2J_2(E_e)-\frac{J_3(E_e)}{E_e})}.
\label{RnuHH}
\end{equation}
Eq.(\ref{diff}) is too complicated for the randomization of $Q^2$ but
there is an easily randomized formula which approximates Eq.(\ref{diff})
above the hadronic threshold ($E>10~MeV$).  It reads
\begin{equation}
\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),
\label{RQ2HH}
\end{equation}
where
\begin{equation}
D(y)=1-y+\frac{y^2}{2},
\label{RQ2D}
\end{equation}
\begin{equation}
L(y,Q^2)=ln\left( F(y)+(e^{P(y)}-1+\frac{Q^2}{Q^2_{min}})^{-1} \right),
\label{RQ2L}
\end{equation}
and
\begin{equation}
U(y)=P(y)\cdot\left( 1-\frac{Q^2_{min}}{Q^2_{max}}\right),
\label{RQ2U}
\end{equation}
with
\begin{equation}
F(y)=\frac{(2-y)(2-2y)}{y^2}\cdot\frac{Q^2_{min}}{Q^2_{max}}
\label{RQ2F}
\end{equation}
and
\begin{equation}
P(y)=\frac{1-y}{D(y)}.
\label{RQ2P}
\end{equation}
The $Q^2$ value can then be calculated as
\begin{equation}
\frac{Q^2}{Q^2_{min}}=1-e^{P(y)}+\left(e^{R\cdot
L(y,Q^2_{max})-(1-R)\cdot U(y)}-F(y) \right)^{-1},
\label{Q2sol}
\end{equation}
where $R$ is a random number.  In Fig.~\ref{Q2dep}, Eq.(\ref{diff}) (solid 
curve) is compared to Eq.(\ref{RQ2HH}) (dashed curve).  Because the two 
curves are almost indistinguishable in the figure, this can be used as an 
illustration of the $Q^2$ spectrum of virtual photons, which is the derivative 
of these curves.  An alternative approach is to use Eq.(\ref{diff}) for the 
randomization with a three dimensional table $\frac{ydn}{dy}(Q^2,y,E_e)$.
\begin{figure}[tbp]
\resizebox{0.95\textwidth}{!}
{
   \includegraphics{hadronic/theory_driven/ChiralInvariantPhaseSpace/Fig13.eps}
}
\caption{Integrals of $Q^2$ spectra of virtual photons for three
energies $10~MeV$, $100~MeV$, and $1~GeV$ at $y=0.001$, $y=0.5$, and $y=0.95$. 
The solid line corresponds to Eq.(\protect\ref{diff}) and the dashed
line (which almost everywhere coincides with the solid line)
corresponds to Eq.(\protect\ref{diff}).}
\label{Q2dep}
\end{figure}

\noindent 
After the $\nu$ and $Q^2$ values have been found, the value of
$\sigma_{\gamma^*A}(\nu,Q^2)$ is calculated using Eq.(\ref{abc}).
If $R\cdot\sigma_{\gamma A}(\nu)>\sigma_{\gamma^*A}(\nu,Q^2)$, no
interaction occurs and the electron keeps going. This ``do nothing''
process has low probability and cannot shadow other processes.


\section {Status of this document}
           created by ?              \\
 20.05.02  re-written by D.H. Wright \\
 01.12.02  expanded section on electronuclear cross sections - H.P. Wellisch \\


\begin{latexonly}

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\bibitem{WeiWi} K. F. von Weizsacker, Z. Physik {\textbf{88}}, 612 (1934),
E. J. Williams, Phys. Rev. {\textbf{45}}, 729 (1934).

\bibitem{LanLif} L. D. Landau and E. M. Lifshitz,
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\bibitem{Pomer} I. Ya. Pomeranchuk and I. M. Shmushkevich,
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\bibitem{Grib} V. N. Gribov {\textit {et~al.}}, ZhETF {\textbf{41}}, 1834 (1961).

\bibitem{encs.eqPhotons}  L. D. Landau, E. M. Lifshitz, ``Course of
Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'',
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\bibitem{Budnev} V. M. Budnev {\textit {et~al.}}, Phys. Rep. {\textbf{15}}, 181
(1975).

\bibitem{Brasse} F. W. Brasse {\textit {et~al.}}, Nucl. Phys. B {\textbf{110}}, 413
(1976).

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item E. Fermi, Z. Physik {\textbf{29}}, 315 (1924).

\item K. F. von Weizsacker, Z. Physik {\textbf{88}}, 612 (1934),
E. J. Williams, Phys. Rev. {\textbf{45}}, 729 (1934).

\item L. D. Landau and E. M. Lifshitz,
Soc. Phys. {\textbf{6}}, 244 (1934).

\item I. Ya. Pomeranchuk and I. M. Shmushkevich,
Nucl. Phys. {\textbf{23}}, 1295 (1961).

\item V.N. Gribov {\textit {et~al.}}, ZhETF {\textbf{41}}, 1834 (1961).

\item L.D. Landau, E. M. Lifshitz, ``Course of
Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'',
Pergamon Press, p. 351, The method of equivalent photons.

\item V.M. Budnev {\textit {et~al.}}, Phys. Rep. {\textbf{15}}, 181
(1975).

\item F.W. Brasse {\textit {et~al.}}, Nucl. Phys. B {\textbf{110}}, 413
(1976).

\end{enumerate}

\end{htmlonly}