Transition radiation

The Relationship of Transition Radiation to X-ray Čerenkov 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 Čerenkov radiation, which is closely related.

The mean number of X-ray Čerenkov radiation (XCR) photons of frequency \(\omega\) emitted into an angle \(\theta\) per unit distance along a particle trajectory is [Gri02b]:

(204)\[\frac{d^3 \bar{N}_{xcr}}{\hbar d\omega\,dx\,d\theta^2}= \frac{\alpha}{\pi\hbar c}\frac{\omega}{c}\theta^2 \textrm{Im}\left\{Z\right\}.\]

Here the quantity \(Z\) is introduced as the complex formation zone of XCR in the medium:

(205)\[Z=\frac{L}{1-i\displaystyle\frac{L}{l}},\quad L=\frac{c}{\omega} \left[\gamma^{-2}+\displaystyle\frac{\omega^2_p}{\omega^2}+\theta^2\right]^{-1}, \quad \gamma^{-2}=1-\beta^2.\]

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 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. (204) and (205) 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 Čerenkov radiation is removed if the particle crosses a boundary between two media with different dielectric permittivities, \(\epsilon\), where

\[\epsilon=1-\frac{\omega^2_p}{\omega^2}+ i\frac{c}{\omega l}.\]

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).

Calculating the X-ray Transition Radiation Yield

Using the methods developed in Ref.[Gri02b] one can derive the relation describing the mean number of XTR photons generated per unit photon frequency and \(\theta^2\) inside the radiator for a general XTR radiator consisting of \(n\) different absorbing media with fluctuating thicknesses:

\[\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2}= \frac{\alpha}{\pi\hbar c^2}\omega\theta^2 \textrm{Re}\left\{\sum_{i=1}^{n-1}(Z_{i}-Z_{i+1})^2+ 2\sum_{k=1}^{n-1}\,\sum_{i=1}^{k-1}(Z_{i}-Z_{i+1})\left[\prod_{j=i+1}^{k}F_{j}\right](Z_{k}-Z_{k+1}) \right\},\]

\[F_j=\exp\left[-\frac{t_j}{2Z_j}\right].\]

In the case of gamma distributed gap thicknesses (foam or fiber radiators) the values \(F_j\), (\(j=1,2\)) can be estimated as:

\[F_j = \int_0^{\infty}dt_j\, \left(\frac{\nu_j}{\bar{t}_j}\right)^{\nu_j} \frac{t_j^{\nu_j - 1}}{\Gamma(\nu_j)} \exp\left[-\frac{\nu_j t_j}{\bar{t}_j}-\,i\frac{t_j}{2Z_j}\right]= \left[1 + \displaystyle i\frac{\bar{t}_j}{2Z_j\nu_j}\right]^{-\nu_j},\]

where \(Z_j\) is the complex formation zone of XTR (similar to relation (205) for XCR) in the \(j\)-th medium [eal00, Gri02a]. \(\Gamma\) 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:

(206)\[\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2} = \frac{2\alpha}{\pi\hbar c^2}\omega\theta^2 \textrm{Re}\left\{\langle R^{(n)}\rangle\right\},\quad F = F_1 F_2,\]

(207)\[\langle R^{(n)}\rangle=(Z_1-Z_2)^2\left\{n\frac{(1-F_1)(1-F_2)}{1-F}+ \frac{(1-F_1)^2F_2[1-F^n]}{(1-F)^2}\right\}.\]

Here \(\langle R^{(n)}\rangle\) is the stack factor reflecting the radiator geometry. The integration of ((206)) 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 \(n\gg 1\) [Gar71]. The frequency spectrum of emitted XTR photons is given by:

(208)\[\begin{split}\frac{d \bar{N}_{in}}{\hbar d\omega}= \int_{0}^{\sim 10\gamma^{-2}}d\theta^2\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2}= \frac{4\alpha n}{\pi\hbar\omega}(C_1+C_2)^2 \cdot\sum_{k=k_{min}}^{k_{max}} \frac{(k-C_{min})}{(k-C_1)^2(k+C_2)^2} %\cdot \nonumber \\ \sin^2\left[\frac{\pi t_1}{t_1+t_2}(k+C_2)\right],\end{split}\]

\[C_{1,2}=\frac{t_{1,2}(\omega^2_1-\omega^2_2)}{4\pi c\omega},\quad C_{min}=\frac{1}{4\pi c}\left[\frac{\omega(t_1+t_2)}{\gamma^2}+ \frac{t_1\omega^2_1+t_2\omega^2_2}{\omega}\right].\]

The sum in (208) 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

\[C_{max}=C_{min}+\frac{\omega(t_1+t_2)}{4\pi c}\frac{10}{\gamma^2}, \quad k_{max}-k_{min}\sim10^2 - 10^3\gg 1.\]

Numerically, however, only a few tens of terms contribute substantially to the sum, that is, one can choose \(k_{max}\sim k_{min}+20\). Eq.(208) 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 after regular transparent radiator is defined by (208) 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 ((208)) corresponds to the results of [FS75]. In the more general case of the flux of XTR photons after a radiator, the XTR absorption can be taken into account with a calculation based on the stack factor derived in [GMGY75]:

(209)\[\langle R^{(n)}_{flux}\rangle = (L_1-L_2)^2\left\{ \frac{1-Q^n}{1-Q}\frac{(1 + Q_1)(1 + F) - 2F_1 - 2 Q_1 F_2}{2(1-F)} \frac{(1 - F_1 )(Q_1 - F_1)F_2 (Q^n -F^n)}{(1 - F)(Q - F)} \right\},\]

\[Q = Q_1\cdot Q_2, \quad Q_j=\exp\left[-t_j/l_j\right]=\exp\left[-\sigma_j t_j\right],\quad j=1,2.\]

Both XTR energy loss (207) and flux (209) models can be implemented as a discrete electromagnetic process (see below).

Simulating X-ray Transition Radiation Production

A typical XTR radiator consists of many (\(\sim 100\)) boundaries between different materials. To improve the tracking performance in such a volume one can introduce an artificial material [eal00], which is the geometrical mixture of foil and gas contents. Here is an example:

// 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;

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

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       );

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:

\[\bar{N}^{(n)}=\int_{\omega_1}^{\omega_2}d\omega \int_{0}^{\theta_{max}^2}d\theta^2 \frac{d^2 \bar{N}^{(n)}}{d\omega\,d\theta^2}= \frac{2\alpha}{\pi c^2}\int_{\omega_1}^{\omega_2}\omega d\omega \int_{0}^{\theta_{max}^2}\theta^2 d\theta^2 \textrm{Re}\left\{\langle R^{(n)}\rangle\right\}.\]

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 (206). For example, the integral energy spectrum of emitted XTR photons, \(\bar{N}^{(n)}_{>\omega}\), is defined from the following integral distribution:

\[\bar{N}^{(n)}_{>\omega}=\frac{2\alpha}{\pi c^2} \int_{\omega}^{\omega_2}\omega d\omega \int_{0}^{\theta_{max}^2}\theta^2 d\theta^2 \textrm{Re}\left\{\langle R^{(n)}\rangle\right\}.\]

In Geant4 XTR generation inside or 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 G4VXTRenergyLoss is responsible for the creation of tables with integral energy and angular distributions of XTR photons. It also contains the PostDoIt function providing XTR photon generation and motion (if fExitFlux=true) through a XTR radiator to its boundary. Particular models like G4RegularXTRadiator implement the pure virtual function 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:

// 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"));
}

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 (208) to \(n\) and \(n_{eff}\), respectively.

Bibliography

eal00(1,2)

J. Apostolakis et al. Comput. Phys. Commun., 132():241, 2000.

FS75

C.W. Fabian and W. Struczinski. Physics Letters, B57():483, 1975.

GMGY75

G.M. Garibian, L.A. Gevorgian and C. Yang. Sov. Phys.- JETP, 39():265, 1975.

Gar71

G.M. Garibyan. Sov. Phys. JETP, 32():23, 1971.

Gri02a

V.M. Grichine. Generation of x-ray transition radiation inside complex radiators. Physics Letters B, 525(3-4):225–239, jan 2002. URL: https://doi.org/10.1016/S0370-2693(01)01443-5.

Gri02b(1,2)

V.M. Grichine. On the energy-angle distribution of cherenkov radiation in an absorbing medium. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 482(3):629–633, apr 2002. URL: https://doi.org/10.1016/S0168-9002(01)01927-1.