\section[Transition radiation]{Transition radiation}

\subsection{The Relationship of Transition Radiation to X-ray Cherenkov
Radiation}
X-ray transition radiation (XTR ) occurs when a relativistic charged
particle passes from one medium to another of a different dielectric
permittivity.  In order to describe this process it is useful to begin
with an explanation of X-ray Cherenkov radiation, which is closely related.

The mean number of X-ray Cherenkov radiation (XCR) photons of frequency 
$\omega$ emitted into an angle $\theta$ per unit distance along a particle
trajectory is ~\cite{griCR}
%
\begin{equation}
\label{Nxcr}
\frac{d^3 \bar{N}_{xcr}}{\hbar d\omega\,dx\,d\theta^2}=
\frac{\alpha}{\pi\hbar c}\frac{\omega}{c}\theta^2
\textrm{Im}\left\{Z\right\}.
\end{equation}
%
Here the quantity $Z$ is introduced as the {\em complex formation zone} of
XCR in the medium:
%
\begin{equation}
\label{Zj}
Z=\frac{L}{1-i\displaystyle\frac{L}{l}},\quad L=\frac{c}{\omega}
\left[\gamma^{-2}+\displaystyle\frac{\omega^2_p}{\omega^2}+\theta^2\right]^{-1}, 
\quad \gamma^{-2}=1-\beta^2.
\end{equation}
%
with $l$ and $\omega_p$ the photon absorption length and the plasma 
frequency, respectively, in the medium.  For the case of a transparent 
medium, $l \rightarrow \infty$ and the complex formation zone reduces to 
the {\em coherence length} $L$ of XCR.  The coherence length roughly 
corresponds to that part of the trajectory in which an XCR photon can be 
created. 

Introducing a complex quantity $Z$ with its imaginary part proportional to 
the absorption cross-section ($\sim l^{-1}$) is required in order to account
for absorption in the medium.  Usually, 
$\omega_p^2/\omega^2 \gg c/\omega l$.  Then it can be seen from Eqs. 
\ref{Nxcr} and \ref{Zj} that the number of emitted XCR photons is 
considerably suppressed and disappears in the limit of a transparent 
medium.  This is caused by the destructive interference between the photons 
emitted from different parts of the particle trajectory.

The destructive interference of X-ray Cherenkov radiation is removed if 
the particle crosses a boundary between two media with different 
dielectric permittivities, $\epsilon$, where 
%
\begin{equation}
\label{eps}
\epsilon=1-\frac{\omega^2_p}{\omega^2}+
i\frac{c}{\omega l}.
\end{equation}
%
Here the standard high-frequency approximation for the dielectric 
permittivity has been used.  This is valid for energy transfers larger than 
the $K$-shell excitation potential.

If layers of media are alternated with spacings of order $L$, the X-ray 
radiation yield from a trajectory of unit length can be increased by roughly 
$l/L$ times.  The radiation produced in this case is called X-ray transition 
radiation (XTR). 

\subsection{Calculating the X-ray Transition Radiation Yield}

Using the methods developed in Ref. \cite{gri01} one can derive the relation
describing the mean number of XTR photons generated per unit photon 
frequency and $\theta^2$ {\em inside} the radiator for a general XTR 
radiator consisting of $n$ different absorbing media with fluctuating 
thicknesses: 
%
\begin{eqnarray}
%\begin{equation}
\label{Nin}
&&\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2}=
\frac{\alpha}{\pi\hbar c^2}\omega\theta^2 
\textrm{Re}\left\{\sum_{i=1}^{n-1}(Z_{i}-Z_{i+1})^2+
\right. \\ 
&+&\left.
2\sum_{k=1}^{n-1}\,\sum_{i=1}^{k-1}(Z_{i}-Z_{i+1})\left[\prod_{j=i+1}^{k}F_{j}\right](Z_{k}-Z_{k+1})
\right\},\,F_j=\exp\left[-\frac{t_j}{2Z_j}\right]. \nonumber
%\end{equation}
\end{eqnarray}
%
In the case of gamma distributed gap thicknesses (foam or fiber radiators) the values 
$F_j$, ($j=1,2$) can be estimated as:
%
\begin{equation}
\label{Hj}
F_j = \int_0^{\infty}dt_j\,
\left(\frac{\nu_j}{\bar{t}_j}\right)^{\nu_j}
\frac{t_j^{\nu_j - 1}}{\Gamma(\nu_j)}
\exp\left[-\frac{\nu_j t_j}{\bar{t}_j}-\,i\frac{t_j}{2Z_j}\right]= \left[1 + 
\displaystyle i\frac{\bar{t}_j}{2Z_j\nu_j}\right]^{-\nu_j},
\end{equation}
%
where $Z_j$ is the complex formation zone of XTR 
(similar to relation \ref{Zj} for XCR) in the $j$-th medium 
\cite{gri01,g4xtr}.  $\Gamma$ is the Euler gamma function, $\bar{t}_j$ is 
the mean thickness of the $j$-th medium in the radiator and $\nu_j > 0$ is 
the parameter roughly describing the relative fluctuations of $t_j$.  In 
fact, the relative fluctuation is $\delta t_j/\bar{t}_j\sim 1/\sqrt{\nu_j}$. 

In the particular case of $n$ foils of the first medium ($Z_1, F_1$) 
interspersed with gas gaps of the second medium ($Z_2, F_2$), one obtains:
%
\begin{equation}
\label{Nn1}
\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2} = 
\frac{2\alpha}{\pi\hbar c^2}\omega\theta^2 
\textrm{Re}\left\{\langle R^{(n)}\rangle\right\},\quad F = F_1 F_2,
\end{equation}
%
\begin{equation}
\label{Rn}
\langle R^{(n)}\rangle=(Z_1-Z_2)^2\left\{n\frac{(1-F_1)(1-F_2)}{1-F}+
\frac{(1-F_1)^2F_2[1-F^n]}{(1-F)^2}\right\}.
\end{equation} 
%
Here $\langle R^{(n)}\rangle$ is the stack factor reflecting the radiator geometry. 
The integration of (\ref{Nn1}) with respect to $\theta^2$ can be simplified for the
case of a regular radiator ($\nu_{1,2}\rightarrow\infty$), transparent in terms
of XTR generation media, and $n\gg 1$~\cite{gar71}. The frequency spectrum of
emitted XTR photons is given by:
%
\begin{eqnarray}
\label{Nntr}
&&\frac{d \bar{N}_{in}}{\hbar d\omega}=
\int_{0}^{\sim 10\gamma^{-2}}d\theta^2\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2}=
\frac{4\alpha n}{\pi\hbar\omega}(C_1+C_2)^2 \nonumber \\
&&\cdot\sum_{k=k_{min}}^{k_{max}}
\frac{(k-C_{min})}{(k-C_1)^2(k+C_2)^2}
%\cdot \nonumber \\
\sin^2\left[\frac{\pi t_1}{t_1+t_2}(k+C_2)\right],\nonumber \\
\end{eqnarray}
%
%
\[
C_{1,2}=\frac{t_{1,2}(\omega^2_1-\omega^2_2)}{4\pi c\omega},\quad 
C_{min}=\frac{1}{4\pi c}\left[\frac{\omega(t_1+t_2)}{\gamma^2}+
\frac{t_1\omega^2_1+t_2\omega^2_2}{\omega}\right].
\]
The sum in (\ref{Nntr}) is defined by terms with $k\geq k_{min}$ corresponding
to the region of $\theta\geq 0$. Therefore $k_{min}$ should be the nearest
to $C_{min}$ integer $k_{min}\ge C_{min}$. The value of $k_{max}$ is defined by the
maximum emission angle $\theta^2_{max}\sim 10\gamma^{-2}$. It can be evaluated as the
integer part of
\[
C_{max}=C_{min}+\frac{\omega(t_1+t_2)}{4\pi c}\frac{10}{\gamma^2}, \quad 
k_{max}-k_{min}\sim10^2\div 10^3\gg 1.
\]
Numerically, however, only a few tens of terms contribute substantially to the
sum, that is, one can choose $k_{max}\sim k_{min}+20$. Equation (\ref{Nntr})
corresponds to the spectrum of the total number of photons emitted inside a
regular transparent radiator. Therefore the mean interaction length, $\lambda_{XTR}$,
of the XTR process in this kind of radiator can be introduced as:
\[
\lambda_{XTR}=n(t_1+t_2)\left[\int_{\hbar\omega_{min}}^{\hbar\omega_{max}}
\hbar d\omega\frac{d \bar{N}_{in}}{\hbar d\omega}\right]^{-1},
\]
where $\hbar\omega_{min}\sim 1$ keV, and $\hbar\omega_{max}\sim 100$ keV for the
majority of high energy physics experiments. Its value is constant along 
the particle trajectory in the approximation of a transparent regular radiator. 
The spectrum of the total number of XTR photons {\em after} regular transparent 
radiator is defined by (\ref{Nntr}) with:
\[
n\rightarrow n_{eff}=\sum_{k=0}^{n-1}\exp[-k(\sigma_1t_1+\sigma_2t_2)]=
\frac{1-\exp[-n(\sigma_1t_1+\sigma_2t_2)]}
{1-\exp[-(\sigma_1t_1+\sigma_2t_2)]},
\]
where $\sigma_1$ and $\sigma_2$ are the photo-absorption cross-sections corresponding 
to the photon frequency $\omega$ in the first and the second medium, respectively. 
With this correction taken into account the XTR absorption in the 
radiator (\ref{Nntr}) corresponds to the results of \cite{fab75}. In the more 
general case of the flux of XTR photons {\em after} a radiator, the XTR absorption 
can be taken into account with a calculation based on the stack factor derived 
in \cite{gar74}:
%
\begin{eqnarray}
\label{Rflux}
\langle R^{(n)}_{flux}\rangle&=& (L_1-L_2)^2\left\{
\frac{1-Q^n}{1-Q}\frac{(1 + Q_1)(1 + F) - 2F_1 - 2 Q_1 F_2}{2(1-F)}\right.\nonumber \\
&+&\left.\frac{(1 - F_1 )(Q_1 - F_1)F_2 (Q^n -F^n)}{(1 - F)(Q - F)}
\right\},
\end{eqnarray}
%
%
\[
Q = Q_1\cdot Q_2, \quad Q_j=\exp\left[-t_j/l_j\right]=\exp\left[-\sigma_j t_j\right],\quad j=1,2.
\]
Both XTR energy loss (\ref{Rn}) and flux (\ref{Rflux}) models can be implemented 
as a discrete electromagnetic process (see below). 


\subsection{Simulating X-ray Transition Radiation Production}

A typical XTR radiator consits of many ($\sim 100$) boundaries between different 
materials.  To improve the tracking performance in such a volume one can introduce 
an artificial material \cite{g4xtr}, which is the geometrical mixture of foil and 
gas contents.  Here is an example:
\begin{verbatim}
  // In DetectorConstruction of an application 
  // Preparation of mixed radiator material
  foilGasRatio  = fRadThickness/(fRadThickness+fGasGap);
  foilDensity  = 1.39*g/cm3;     // Mylar      
  gasDensity   = 1.2928*mg/cm3 ; // Air  
  totDensity   = foilDensity*foilGasRatio + 
                 gasDensity*(1.0-foilGasRatio);
  fractionFoil =  foilDensity*foilGasRatio/totDensity; 
  fractionGas  =  gasDensity*(1.0-foilGasRatio)/totDensity;      
  G4Material* radiatorMat = new G4Material("radiatorMat", 
                                            totDensity, 
                                            ncomponents = 2 );
  radiatorMat->AddMaterial( Mylar, fractionFoil );
  radiatorMat->AddMaterial( Air,   fractionGas  );
  G4cout << *(G4Material::GetMaterialTable()) << G4endl;  
  // materials of the TR radiator
  fRadiatorMat = radiatorMat;   // artificial for geometry  
  fFoilMat     = Mylar;  
  fGasMat      = Air;  
\end{verbatim}

This artificial material will be assigned to the logical volume in which
XTR will be generated:

\begin{verbatim}
  solidRadiator = new G4Box("Radiator",
                             1.1*AbsorberRadius , 
                             1.1*AbsorberRadius, 
                             0.5*radThick        );                          
  logicRadiator = new G4LogicalVolume( solidRadiator,   
                                       fRadiatorMat,  // !!!      
                                      "Radiator");                                                
  physiRadiator = new G4PVPlacement(0,
                                     G4ThreeVector(0,0,zRad),           
                                     "Radiator", logicRadiator,         
                                     physiWorld, false, 0       );      
\end{verbatim}

XTR photons generated by a relativistic charged particle intersecting a 
radiator with $2n$ interfaces between different media can be simulated by 
using the following algorithm.  First the total number of XTR photons is
estimated using a Poisson distribution about the mean number of photons 
given by the following expression: 
%
%\begin{equation}
%\label{Nn2}
\[
\bar{N}^{(n)}=\int_{\omega_1}^{\omega_2}d\omega
\int_{0}^{\theta_{max}^2}d\theta^2
\frac{d^2 \bar{N}^{(n)}}{d\omega\,d\theta^2}=
%\nonumber\\&=&
\frac{2\alpha}{\pi c^2}\int_{\omega_1}^{\omega_2}\omega d\omega
\int_{0}^{\theta_{max}^2}\theta^2 d\theta^2 
\textrm{Re}\left\{\langle R^{(n)}\rangle\right\}.
\]
%\end{equation}
%
Here $\theta_{max}^2\sim 10\gamma^{-2}$, $\hbar\omega_1\sim 1$~keV, 
$\hbar\omega_2\sim 100$~keV, and $\langle R^{(n)}\rangle$ correspond to the 
geometry of the experiment.  For events in which the number of XTR 
photons is not equal to zero, the energy and angle of each XTR quantum 
is sampled from the integral distributions obtained by the numerical 
integration of expression (\ref{Nn1}).  For example, the integral 
energy spectrum of emitted XTR photons, $\bar{N}^{(n)}_{>\omega}$, is defined 
from the following integral distribution:
%
%\begin{equation}
%\label{Nomega}
\[
\bar{N}^{(n)}_{>\omega}=\frac{2\alpha}{\pi c^2}
\int_{\omega}^{\omega_2}\omega d\omega
\int_{0}^{\theta_{max}^2}\theta^2 d\theta^2 
\textrm{Re}\left\{\langle R^{(n)}\rangle\right\}.
\]
%\end{equation}
%
In { \sc Geant4} XTR generation {\em inside} or {\em after} radiators is 
described as a discrete electromagnetic process. It is convenient for the 
description of tracks in magnetic fields and can be used for the cases when 
the radiating charge experiences a scattering inside the radiator. The base 
class {\tt G4VXTRenergyLoss} is responsible for the creation of tables with 
integral energy and angular distributions of XTR photons.  It also contains
the {\tt PostDoIt} function providing XTR photon generation and motion
(if fExitFlux=true) through a XTR radiator to its boundary.  Particular models 
like {\tt G4RegularXTRadiator} implement the pure virtual function 
{\tt GetStackFactor}, which calculates the response of the XTR radiator 
reflecting its geometry. Included below are some comments for the declaration 
of XTR in a user application.

In the physics list one should pass to the XTR process additional
details of the XTR radiator involved:

\begin{verbatim}
// In PhysicsList of an application
else if (particleName == "e-")  // Construct processes for electron with XTR 
{
   pmanager->AddProcess(new G4MultipleScattering, -1, 1,1 );
   pmanager->AddProcess(new G4eBremsstrahlung(),  -1,-1,1 ); 
   pmanager->AddProcess(new Em10StepCut(),        -1,-1,1 );
// in regular radiators:             
   pmanager->AddDiscreteProcess(                
   new G4RegularXTRadiator        // XTR dEdx in general regular radiator
// new G4XTRRegularRadModel        - XTR flux after general regular radiator
// new G4TransparentRegXTRadiator  - XTR dEdx in transparent 
//                                   regular radiator
// new G4XTRTransparentRegRadModel - XTR flux after transparent 
//                                   regular radiator
                         (pDet->GetLogicalRadiator(), // XTR radiator

                          pDet->GetFoilMaterial(), // real foil
                          pDet->GetGasMaterial(),  // real gas
                          pDet->GetFoilThick(),    // real geometry
                          pDet->GetGasThick(),
                          pDet->GetFoilNumber(),   
                          "RegularXTRadiator"));
// or for foam/fiber radiators:
   pmanager->AddDiscreteProcess(                
   new G4GammaXTRadiator    - XTR dEdx in general foam/fiber radiator
// new G4XTRGammaRadModel   - XTR flux after general foam/fiber radiator
                          ( pDet->GetLogicalRadiator(),
                            1000.,
                            100.,
                            pDet->GetFoilMaterial(),
                            pDet->GetGasMaterial(),
                            pDet->GetFoilThick(),
                            pDet->GetGasThick(),
                            pDet->GetFoilNumber(),
                            "GammaXTRadiator"));
} 
\end{verbatim}
Here for the foam/fiber radiators the values 1000 and 100 are the $\nu$ parameters 
(which can be varied) of the Gamma distribution for the foil and gas gaps, 
respectively. Classes G4TransparentRegXTRadiator and G4XTRTransparentRegRadModel 
correspond (\ref{Nntr}) to $n$ and $n_{eff}$, respectively.
 
\subsection{Status of this document}
18.11.05 modified by V.Grichine \\
29.11.02 re-written by D.H. Wright \\
29.05.02 created by V.Grichine \\

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{griCR}  V.M. Grichine, 
{\em Nucl. Instr. and Meth.}, {\bf A482} (2002) 629. 

\bibitem{gri01} V.M. Grichine, {\em Physics Letters}, {\bf B525} (2002) 225-239 

\bibitem{gar71} G.M. Garibyan,
{\em Sov. Phys. JETP} {\bf 32} (1971) 23.

\bibitem{fab75} C.W. Fabian and W. Struczinski
{\em Physics Letters}, {\bf B57 } (1975) 483.

\bibitem{gar74} G.M. Garibian, L.A. Gevorgian, and C. Yang,
{\em Sov. Phys.- JETP, 39 (1975) 265.}

\bibitem{g4xtr} J. Apostolakis, S. Giani, V. Grichine et al.,
{\em Comput. Phys. Commun.} {\bf 132} (2000) 241. 

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item  V.M. Grichine, 
{\em Nucl. Instr. and Meth.}, {\bf A482} (2002) 629. 

\item V.M. Grichine, {\em Physics Letters}, {\bf B525} (2002) 225-239 

\item G.M. Garibyan,
{\em Sov. Phys. JETP} {\bf 32} (1971) 23.

\item C.W. Fabian and W. Struczinski
{\em Physics Letters}, {\bf B57 } (1975) 483.

\item G.M. Garibian, L.A. Gevorgian, and C. Yang,
{\em Sov. Phys.- JETP, 39 (1975) 265.}

\item J. Apostolakis, S. Giani, V. Grichine et al.,
{\em Comput. Phys. Commun.} {\bf 132} (2000) 241. 

\end{enumerate}

\end{htmlonly}


