Interactions in flight.

The current version of SNOMAN recognises three possibilities for a Cerenkov photon being propagated from a vertex: either it will reach the next boundary with no interactions, or it will be Rayleigh scattered, or it will be absorbed, either by a wavelength shifter (wls) which will then re-emit the photon or all other materials which simply absorb the photon. This is determined by the routine TKMAIN, which calls GE_NEXT to calculate the distance to the next boundary, and PHINTL, which calculates the distance to the next interaction, and what that interaction will be. If the interaction length is longer than the distance to the next boundary, then the track is propagated to that boundary, and the code sets up a boundary vertex. If the interaction is nearer than the boundary, then the code propagates the track to the indicated point of the interaction, and sets up an interaction vertex with the interaction code set to that indicated by the previous call to PHINTL.

The subroutine ABSINT finds and returns the probability per unit track length of a Cerenkov photon being absorbed and whether it was absorbed by a wls. The values of the absorbtion probability is held in two arrays, one corresponding to media, the other correspoding to wls, tabulated at every 10nm. This tabulation is done during the initialisation in INPHI. Data is taken from the MEDA and MWLS titles banks and linearly interpolated/extrapolated through the wavelengths of interest. The relevant part of the MEDA bank has the form:

 1.0      #. scaling factor
 7.0      #. Number of data points
 254.    313.   366.   406.    436.    548.    578.      8*0.
 15.2e-4 4.1e-4 1.4e-4 1.1e-4  1.3e-4  5.8e-4  9.2e-4    8*0.

where the first number is a scale factor, and is applied (during initialisation) as a multiplier to the probability of an event. The second number is the number of valid data points, whilst the two rows of numbers are vectors containing the wavelength and probability of an absorbtion per unit track length. A similar structure exists in MWLS. The data in MWLS is still in a state of flux as certain quantities have yet to be measured. The data for heavy and light water is taken from Boivin et al [2] whilst the data describing the `standard' acrylic absorptivity is taken from the acceptance criteria document. `Good' and `Bad' acrylic describe some of the better and worse acceptable acrylic measurements, whilst 4-inch is data from measurements made on the 4-inch belly plate material.

There is no mechanism to ``switch off'' absorption, and, although this may effectively be done by setting the absorption probabilities to very small numbers, this is not recommended. This is because it is possible to get into a infinite loop due to a Cerenkov photon undergoing total internal reflection at a boundary in such a way that the photon next strikes the same boundary, but in a different place. The symmetry of the reflection from a spherical surface means that the second boundary interaction will also undergo total internal reflection, and so on. If the code cannot identify the material the code is attempting to propagate the Cerenkov photon through, it announces this fact and sets probability to an arbitrarily high number.

In the event of the photon being absorbed by something other than a wls, the vertex type is set to that of a sink vertex, which is then processed by the sink processor. If absorbed by a wls, the quantum efficiency is checked. If the photon is re-emitted then the track of the photon is changed to correspond to isotropic emission, its energy is accordly adjusted to correspond to the emission spectrum, the event time is increased by a time based on the decay time of the wls. Otherwise it becomes a sink vertex. Possible expansions to this code is the addition of polarization change, anisotropic emission, non-gaussian emission curves, and the use the scale factor in the MWLS to describe the order of the scaling wrt to concentration.

The subroutine RAYINT calculates and returns the probability of a Rayleigh scatter per unit path length, $\alpha$, for a Cerenkov photon, given the frequency of the photon and the medium it is in. Rayleigh's derivation of the formula [4]

$$\alpha = \frac{2k_{\omega}^{4}}{3\pi N}\left\vert n-1\right\vert^{2}$$ (13.7)

where $N$ is the number density of scattering particles, $n$ is the refractive index and $k_{\omega}$ is the wavenumber of the particle, is only valid in the limit $\left\vert n-1\right\vert \ll 1$, and is thus not applicable to the materials encountered in the SNO detector. Instead the Einstein-Smoluchowski formula (see [5]) must be used which defines $\alpha$ as

$$\alpha=\frac{1}{6\pi}\left(\frac{\omega}{c}\right)^{4} \left... ...silon_{r}-1)(\epsilon_{r}+2)}{3}\right\vert^{2}k_{B}T\beta_{T}$$ (13.8)

where $\omega$ is the angular frequency of the photon, $\epsilon_{r}$ is the relative permittivity, $k_{B}$ is Boltzmann's constant, $T$ the temperature and $\beta_{T}$ is the isothermal compressibility of the medium.

In the current version of the code, Rayleigh scattering is only implemented in the light water and heavy water regions of the detector, together with the acrylic (though it should be noted that the Rayleigh scattering is a strong function of how the acrylic was made, and these parameters are not yet fully understood. It should also be noted that the work of Boivin et al [2] suggests that the experimental scattering coefficient may be 2-3 times greater than the theory would predict.

If Rayleigh scattering is indicated, then the vertex processor will call the subroutine VXRAYL. The matrix element describing the Rayleigh scattering contains a term of the form $\vec{p}_1.\vec{p}_2$, where $\vec{p}_1$ and $\vec{p}_2$ are the initial and final state polarisations respectively. This produces a bias in the preferred directions of the Cerenkov photon, and since the direction and polarisation and direction vectors of any real photon must be perpendicular, also produces a constraint on the direction of the final state photon. This is implemented in the code, the code using the RAYSCATTER routine to perform the transform from initial photon coordinates to detector coordinates.

Anisotropic Rayleigh scattering is a small effect compared with the so-called isotropic scattering discussed in the code and is not considered (see [3] for more details). Mie scattering (scattering from individual particles) is also not implemented as the characteristics of the scattered distribution, and the probability of scattering, will be strongly dependant on parameters not known at this point (principally the size and density of particulate matter in the water).