Simulation of the RICH-1 prototype

The RICH-1 prototype configurations are simulated to allow detailed comparisons of expected performance with data. The simulation is also used in the design of the prototypes themselves, to calculate the optimal position of the detector plane from the mirror. The program simulates photon emission in the radiators and assigns each a wavelength, Cherenkov angle and point of emission. Photons are then ray-traced to the HPD detectors and assigned to a hit. The program incorporates the following physical effects :

• Beam trajectory and composition : For each event, an input beam direction and position is defined according to the acceptance of the trigger scintillators. The input particle is defined to be either a $\pi^{-}_{}$, e^- or K^- in ratios of the real beam composition. The PS beam delivers particles with a given, fixed momentum to within $\pm$1%. The spread in momentum is also included in the simulation.
• Photon Generation and Chromatic Aberration : Photons emitted in the radiator are assigned an energy distribution ranging from 1.5 eV to 7.3 eV according to the relation :
  $${{dN_{\gamma}}\over{dE}}$$ = 370 sin²$$\theta_{\mathrm C}^{}$$L
  where $\theta_{\rm c}^{}$ is the Cherenkov angle, L is the radiator length in cm [4]. The dispersion of the refractive index is parameterised in the simulation by the Lorentz-Lorenz equation [6] :

  $${\frac{1+2cf(E)}{1-cf(E)}}$$ (1)

  where :
  

  $${\frac{4\pi a\rho N_A}{2M}} {\frac{\rho}{M}}$$ (2)

  $${\frac{F_1}{G_1^2-E^2}} {\frac{F_2}{G_2^2-E^2}}$$ (3)

  
  
  and $\rho$ is the density of the medium, M is molecular weight, E is the photon energy and F and G are Sellmeir coefficients. The latter are derived quantities, examples of which are tabulated in reference [7]. In the simulation, air is approximated by nitrogen, aerogel by quartz [4] (scaled by their relative densities) and C₄F₁₀ by C₂F₆ [8] scaled to fit measured data [9]. Figure 8 shows the simulated behaviour of the refractive indices for aerogel and C₄F₁₀ with respect to the wavelength of the emitted photon. Finally, the Cherenkov angle is generated with a uniform azimuthal angular distribution.

  Figure 8: Parameterisations of the refractive index of aerogel and C₄F₁₀ versus photon wavelength, as used by the simulation program described in the text. Datapoints shown are from [9].

  

• Emission Point & Beam Trajectory Uncertainty : In the LHCb RICH-1 counter, the mirror is split and tilted to direct the ring images out of the detector acceptance [1]. This introduces a photon emission point error. When reconstructing the Cherenkov angle, it is necessary to assume an emission point for the photons, which is taken to be the mid-point of the radiator.

  The beam trajectory uncertainty is assumed to be either determined by the trigger scintillator acceptance in the absence of the silicon telescope, or that determined by the silicon telescope pixel size and the nominal (0, 0, - 1) simulated beam direction is smeared accordingly.

• Ray tracing : Each photon is individually traced through the prototype geometry. The photon trajectory is modified by refraction at the gas/aerogel boundary, reflection from the mirror surface, absorption by the walls of the chamber or Rayleigh scattering in the aerogel. The chromatic dependence of the mirror reflection and transmission properties of the aerogel are used as measured, and are shown in Figures 3(b) and 3(d).
• Simulation of the HPD : The quantum efficiency of the HPD photocathode, as a function of photon wavelength, is supplied by the manufacturer and shown in Figure 3(c). From these data the simulation determines whether a photoelectron is produced. The simulation incorporates an 18% probability of a back-scattered electron being released from the silicon [10]. If a back-scattered electron is released, its flight path is calculated, assuming a uniform accelarating field of the HPD, and its new hit position recorded. Random electronic noise is also added to agree with observations in data.
• Pressure Variation of the Gas Radiator : The C₄F₁₀ gas recirculation system controls the pressure to approximately $\pm$20 mbar which leads to a periodic variation in the measured Cherenkov angle of approximately 0.7 mrad. This variation can be reduced substantially by deactivating the circulation system. This effectively decreases the rate of change of pressure, and allows measurements to be made over longer periods of time with greater stability.

An event display program is used to visualise the results of simulation and data events. An example of simulated events in RICH Configuration 2 is shown in Figure 9.

Figure 9: Event display showing the superposition of simulated events as in Configuration 2. The x and y axes are in units of mm.

Contributions to the Cherenkov angle resolution for RICH Configuration 2 with a C₄F₁₀ gas radiator are shown in Figure 10. Assuming an emission point for the photons to be the mid-point of the radiator induces a systematic contribution to the Cherenkov angle resolution shown in Figure 10(a). The effect of chromatic aberration is to induce a spread in the reconstructed values of $\theta_{\rm c}^{}$, and this contribution is shown in Figure 10(b). If in the reconstruction it is assumed that the beam particle follows the nominal trajectory, this results in an additional contribution to the Cherenkov angle resolution shown in Figure 10(c). Finally, the finite pixel size ( 2 x 2 mm²) affects the reconstructed Cherenkov angle, as indicated in Figure 10(d). Here the precise nature of the modulation of the $\theta_{\rm c}^{}$ distribution depends critically on the relative orientations of the Cherenkov ring relative to the line of the pixels in the HPD hexagonal structure. The effects of this contribution are simulated, however the precise alignment of the photon detectors relative to the ring remains a potential source of discrepancy between data and simulation. Results are averaged over compatible HPDs, in different orientations, in order to minimise this uncertainty.

Figure 10: Plots showing examples of reconstructed C₄F₁₀ Cherenkov angles when different contributions to the error are isolated in the simulation: (a) Emission point uncertainty only; (b) Chromatic aberration of the photons; (c) the particle position and direction uncertainty and; (d) the error due to the finite pixel size of 2 x 2mm².

The magnification of the incident position of photons arriving at the photocathode is measured to be 1.06 from laboratory studies using a scanning LED, and is included in the simulation. Finally, background signals from electronic noise are included in the simulation by smearing the expected photoelectron signal with a Gaussian whose sigma is determined by the pedestal width observed in data.