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\begin{document}

\title[ELMS NuFact02 paper]{Calculations of Energy Loss and Multiple Scattering (ELMS) in Molecular Hydrogen}

\author{W W M Allison}   

\address{Physics Dept., Denys Wilkinson Building, Keble Rd., Oxford OX1 3RH, UK}

\begin{abstract}
To show that the principle of Ionisation Cooling  will work for muon beams we must be able to simulate energy loss and scattering in %%@
media reliably.  
We have three choices: we can use traditional calculations with their uncertainties; we can make measurements (MUSCAT); or, we can %%@
calculate the phenomena more carefully, looking afresh at the phenomena from first principles. In this paper we report on work %%@
following this third approach. We derive the double differential cross section for a collision with transverse momentum transfer $P_t$ %%@
and longitudinal momentum transfer $P_\ell$ from a knowledge of the UV and X-ray photoabsorption cross section of the medium, together %%@
with the known kinematics and dynamics of the scattering of point charges with screening. Distributions in energy loss and scattering %%@
may then be found by Monte Carlo techniques which take into account both correlations in scattering and energy loss, and the true %%@
effects of non-gaussian tails in distributions of interest. Preliminary results are reported for molecular hydrogen. Further work is %%@
in progress.
\end{abstract}


\section{Introduction}
Traditional methods of calculating energy loss and multiple scattering date from days when data on media were poor, computers were %%@
rare and the priority was on quick back-of-the-envelope results using simple parameters, $eg$ radiation length and mean ionisation %%@
potential. Today, given that the underlying physics is understood and that good atomic and molecular data are available, reliable %%@
calculations can be made. The ELMS (Energy Loss and Multiple Scattering) program is intended to do that. 

\section{The double differential cross section}

Collisions of the incident muon with the medium comprise:
\begin{enumerate}
\item Coulomb collisions with nuclei,
\item Coulomb collisions with constituent electrons
\item collisions with atoms as a whole. 
\end{enumerate}
Some years ago we described calculations of energy loss due to processes (ii) and (iii), depending only on the photoabsorption of the %%@
medium, but including $\delta$-rays, ionisation, excitation and Cherenkov Radiation in a unified way [Allison \& Cobb, 1980]. From the %%@
same starting point we have derived the cross section for transfer ($\hbar$\bf{k}$_t$, $\hbar k_\ell$) \rm to the muon:
\[ \frac{\rmd ^3\sigma}{\rmd ^2\bf {k}\mit _t\rmd k\mit _\ell} = 
\frac{\alpha\varepsilon_ 2}{\pi^2\beta N(k_\ell^2+k_t^2)}
\left[ \frac{\beta^4k_\ell^2k_t^2}{\left(k_\ell^2(1-\varepsilon _1\beta^2)+k_t^2\right)^2+\left(\varepsilon %%@
_2\beta^2k_\ell^2\right)^2}-
\frac{1}{\left|\varepsilon\right|^2}\right]\]  
where $N$ is the atomic density, $\varepsilon _1$ and $\varepsilon _2$ are the real and imaginary parts of the dielectric response %%@
function $\varepsilon(k,\omega)$. Following Allison \& Cobb $\varepsilon _2(k,\omega)$ is given in terms of the photoabsorption cross %%@
section for real photons $\sigma (\omega)$ by 
\[\varepsilon _2(k,\omega) = \frac{Nc}{\omega}\left[\sigma(\omega)\times H\left(\omega -\frac{\hbar k^2}{2m}\right) + \int^\omega _0 %%@
\sigma (\omega^\prime)\rmd \omega^\prime \times \delta \left(\omega -\frac{\hbar k^2}{2m}\right)   \right]\]
where $m$ is the electron mass. $\varepsilon _1$ is given in terms of $\varepsilon _2$ by causality (Kramers-Kronig Relations  %%@
[Jackson 1999]).
The first term describes atomic collisions as a whole, the second collisions with constituent electrons and the two are coupled  by %%@
the Thomas-Reiche-Kuhn Sum Rule. 

The cross section (ii) for hard collisions of an incident muon on a target electron is described Dirac Scattering [Perkins 1987] which %%@
involves a small spin-dependent modification to Rutherford Scattering. There is some rewriting to include the kinematics of %%@
relativistic recoil. 

For scattering by nuclei (i) we have the same $1/Q^4$ constituent scattering cross section, with form factors given by atomic %%@
screening and nuclear elastic scattering, both well known for hydrogen. 

\begin{figure}
\begin{center}
\includegraphics{xsection.eps}
\caption{The collision probability for\label{`firstfig`} 180 MeV/c muon incident on molecular hydrogen at liquid density. The %%@
left-right log scale is $10^{-8}\rightarrow 10^{9}$ eV in energy transfer. The front-back log scale is $10^1\rightarrow 10^{9}$ eV/c %%@
in transverse momentum transfer, $P_t$. The vertical log scale covers more than 8 decades. The lines of nuclear and electron %%@
constituent scattering are clearly separated. The latter joins on to the band of $P_t$-independent atomic collisions. (Note that some %%@
of the finer structures are plotting artefacts.)}
\end{center}
\end{figure}

The contribution of each process may be seen qualitatively in fig. \ref{`firstfig`}. Collisions with constituent electrons give %%@
correlated contributions to scattering and energy loss, while nuclear and atomic collisions give rise predominantly to scattering and %%@
energy loss respectively. 

\section{The ELMS program}

Modern data on the photoabsorption spectrum of molecular and atomic hydrogen are available [Berk2002]. These have been modified %%@
marginally to take account of some broadening and shifting of discrete transitions at liquid densities. Small effects on electron %%@
constituent scattering to include electron binding and fermi momentum have been taken into account. The different effect of nuclear %%@
screening by an electron with a Thomas Fermi distribution instead of the ground state atomic wavefunction has also been considered. %%@
These effects are small but they indicate the extent of systematic uncertainties.

From these cross sections the ELMS program generates by Monte Carlo a database of energy-loss and momentum transfer pairs in thin %%@
absorbers for a range of incident momenta. 'Thin' means that the cross section and the path length in the absorber are constant (~1mm, %%@
or much less at low momenta). From this database a second ELMS code generates single particle histories through finite absorbers, %%@
allowing for both changes of direction and momentum.
  
\begin{center}
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
\multicolumn{2}{|c|}{Multiple scattering}& \multicolumn{5}{c|}{Energy loss over density} \\ \hline
\multicolumn{2}{|c|}{$\theta$ (proj.) mrad.}& \multicolumn{5}{c|}{MeV g$^{-1}$ cm$^{2}$} \\ \hline
RMS  & RMS98\%  & Mean & RMS & Median & Most probable & Mean, atomic H \\ \hline
 7.55 & 6.94 &  4.64 & 0.65 & 4.46 & 4.25 & 4.32 \\ \hline
\end{tabular}
\caption{Multiple scattering and specific energy loss of 180 MeV/c muons in 10cm Molecular Hydrogen with $\rho = 0.0708$ g cm$^{-3}$. %%@
The last column is Atomic Hydrogen for comparison.} 
\end{table}
\end{center}

\section{Preliminary results}

Tracks are input to the code as 6-D phase space variables on the front side of an absorber; these variables are then generated at %%@
absorber exit. Table 1 shows some findings for an H$_2$ absorber of thickness 10cm, 0.708g cm$^{-2}$. Preliminary results of a %%@
detailed study of possible systematic errors suggest that errors on scattering are of order 1\% and on energy loss of order 2\%. We %%@
note that the extreme case of ignoring molecular binding altogether gives a 7\% shift in energy loss.
However the shape, tails and correlations of the distributions are also important. For example, the expected positive %%@
energy-scattering correlation is found: the mean $|P_t|$ of tracks in the smallest energy-loss decile is 1.55 MeV/c, while in the %%@
highest decile it is 1.77 MeV/c. A fuller report will be published latter. 
  
\References 

\item[] Allison W W M and Cobb J H 1980 {\it Ann. Rev. Nucl. Part. Sci.} {\bf 30}, 253
\item[] Perkins D H 1987 {\it Introduction to High Energy Physics}, 3rd edn., Addison Wesley
\item[] Berkowitz J 2002 {\it Atomic and Molecular Photoabsorption} Academic Press
\item[] Jackson J D 1999 {\it Classical Electrodynamics}, 3rd edn., Wiley

\endrefs

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