Considerations for design

 A practical transition radiation detector needs to take into account all the physics from the preceding sections. In most cases the length of the detector, L_det , is fixed either as the length along the beamline in a fixed target experiment or as the radial space in a colliding beam detector.

With a fixed detector length, optimisation is required for.

• The composition of the foils and the gas between them in the radiator.
• The foil thickness l₁ , to get the correct wavelength distribution of the transition radiation.
• The distance between the foils l₂ , which needs to be large enough to avoid formation zone effects as derived in (3.16).
• The number of foils, N_foil, in front of a detecting gas.
• The gas composition and the thickness of the gas l_gas .

The choice of xenon as the detecting gas is obvious from its short absorption length as seen in fig. 3.7. With a gas thickness of around 0.5 cm there is large absorption up to 10 keV. In the figure is also seen that 4 keV photons have an absorption length of just 300 $\mu$ m in a typical radiator foil giving only a small window for detecting transition radiation with $\omega_{gas}^{}$ $\simeq$ 7 keV.

For the foil material where low absorption is important a low Z material is the best. Since lithium and beryllium are reactive materials the best practical choices are carbohydrates like polyethylene or polypropylene. The gas between the foils in the radiator has to have as low electron density as possible to achieve a low plasma frequency and thereby limit the saturation effect (3.25). Helium is this aspect is the best choice with $\gamma$'_max = 26000 while for air $\gamma$'_max = 9600 for a 7 keV photon energy.

With $\omega_{gas}^{}$ and the radiator composition fixed the optimal foil thickness can be calculated from (3.31) and table 3.1 giving l_1optimal $\simeq$ 20 $\mu$ m. The actual thickness of the foil has then to take into account a typical crossing angle between the particles and the foil.

 

$$\omega_{P}^{} \rho$$ Material

polypropylene 20.87 0.935

$$\cdot$$ helium 0.27

$$\cdot$$ air 0.73

Kapton 24.5 1.39

$$\cdot$$ xenon 1.42

$$\cdot$$ Xe/CF₄/CO₂ 1.70

To perform well for emission angles above $\theta$ > 0.2 mrad (fig. 3.2) and for X-rays above 4 keV, (3.16) gives a minimum spacing of the foils of l_2minimal $\simeq$ 1.3 mm with air as the medium between the foils. For helium the minimum spacing is 2.2 mm.

The effective number of foils can now be calculated from (3.9) and fig. 3.7. The effective number of foils is below 35 for 5 keV X-rays. Hence not much is gained by having N_foil > 35 .

Adding up the different lengths gives

 

$$\approx$$ (68)

with only a small gain by increasing this length. At the same time the detector is not very efficient with an average number of photons

 

$$\gamma \gamma_{\text{single boundary}} \approx$$ (69)

created for an electron passing. Or rephrased: only every second electron passing the detector will emit a single transition radiation photon. The factor four comes from the optimised thickness of the foil for positive interference.

The solution to the problem is to place multiple copies of the radiator and detection gas behind each other. This also opens up the possibility to combine the particle identification with tracking.

To measure the energy deposited as ionisation in the xenon gas there are 2 methods called the Q and the N method.

In the Q method, one measures with ADC's or FADC's the total energy deposited in the gas which is a sum of the transition radiation in the radiator and ${\frac{{d}E}{{d}x}}$ from the particle which also passes through the detecting gas. The method is sensitive to background from the Landau tail of ${\frac{{d}E}{{d}x}}$ . Since the Q method is mainly sensitive to the X-rays with the highest energies, the number of foils can be enlarged in each layer because the absorption of X-rays in the radiator is only important for the soft part of the spectrum.

The N method instead, counts the number of clusters in the gas. A cluster is the volume of the gas where the ionisation from the electron knocked out by the X-ray photon is deposited. It typically has a diameter of 1 mm. This method is not very sensitive to ${\frac{{d}E}{{d}x}}$ since the main part of the ionisation is spatially spread out. The background mainly comes from $\delta$ -electrons which are electrons kicked out of the atomic shells by the primary particle with an energy high enough to make ionisation themselves.

While the Q method is expensive to implement for a multilayered detector with separate ADC's for each layer of detecting gas, the N-method can be incorporated in a multilayered detector in an elegant way. A threshold on the deposited energy in a single layer can give a simple yes/no answer if there was a cluster in a given layer. The probability for multiple clusters in a single layer is low and only a few clusters will be lost by just having a binary output. With a fixed probability for a cluster in each layer, particles passing all the layers will have a binomial distribution in the number of layers with a cluster registered.

The N method has to distinguish between two different binomial distributions from pions and electrons respectively for particle identification. The distributions have different average values and for binomial distributions there are no suffering from long Landau tails as with the Q method.

Since the total length of the detector is fixed and the length of each layer given by (3.39) also the number of layers is fixed. However, for the N method what counts is the mean number of clusters in all layers given by

 

< n_cluster >   = Np_layer (70)

where N is the number of layers and p_layer the probability for a cluster in each layer. This can also be expressed in terms of the number of detected clusters per unit length

 

$${\frac{{d}N_{\text{clusters detected}}}{{d}L}}$$ (71)

While the lower limit on l₂ , defined by the formation zone effect, is not very stringent the thickness of the Xe absorption layers are fixed and hence

 

$${\frac{L_{det}}{{N_{foil}}(l_1 + l_2 ) + l_{gas}}}$$ (72)

reaches a saturation when N_foill₂ is reduced to the point where N_foil(l₁ + l₂) = l_gas . Detailed simulations using the model described in section 5.6 shows an optimal performance for N_foil $\simeq$ 20 and l₂ $\simeq$ 0.3 mm. The maximal performance achievable is A_TR $\simeq$ 0.15 cluster/cm [35].