Reflector Geometry

In principle, the reflectors are Winston cones [2], although they have been truncated somewhat to conform to the engineering requirements of the SNO detector. In practise, as the surface of the Winston cone does not have a tractable analytic form, they are modelled as a section of an elliptical torus. The parameters of the torus are determined by fitting an ellipse to the profile of the reflector, while the vertical sections are defined by the measured radii at the top and bottom of the reflector.

The presence of 18 strips of `omega' (q.v.), known as `petals', further complicates the model. They are modelled by assuming that the edges of the petals lie along the shape of the reflector described above, but that the petals are only bent in one direction, so that they are actually a section of an elliptical cylinder, defined by these edges. Thus the procedure for determining whether the reflector has been hit depends on solving the intersection of a line with a torus, as discussed above, giving a `proposed' solution $(x_p,y_p)$. This `proposed solution' can then be used to determine which petal was hit, and from this the equation of the cylinder can be deduced as follows. The number, $n$, of the petal is defined by the proposed solution $(x_p,y_p)$ by extracting the angle $\theta = \tan^{-1} y_p/x_p$; $n$ is then the integer part of $\theta/\alpha$, where $\alpha = 2\pi/18$ is the angular width of one of the eighteen petals. The equation of the petal in the x-y plane then becomes:

$$y = mx + c$$ (13.12)

where

$$m = - \frac{\cos(n+\frac{1}{2})\alpha}{\sin(n+\frac{1}{2})\alpha}$$ (13.13)

$$c = r\frac{\cos\frac{\alpha}{2}}{\sin(n+\frac{1}{2})\alpha}$$ (13.14)

However, $r$ is not a constant, but a function of $z$:

$$r - d = r' = a\sqrt{1 - \frac{z^2}{b^2}}$$ (13.15)

which will give the equation of a cylinder of elliptical cross-section. Solving for the intersection of a line with this elliptical cylinder will give a solution of the form:

$$t = \frac{-\Delta_2 \pm \sqrt{\Delta_2^2 - \Delta_1 \Delta_3}}{\Delta_1}$$ (13.16)

where

$$\Delta_1 = (v-mu)^2 + \left(\frac{eaw}{b}\right)^2$$ (13.17)

$$\Delta_2 = (y_0 - mx_0 - ed)(v-mu) + \left(\frac{ea}{b}\right)^2wz_0$$ (13.18)

$$\Delta_3 = (y_0 - mx_0 - ed)^2 - \left(\frac{ea}{b}\right)^2(b^2 - z_0^2)$$ (13.19)

$$e = \frac{\cos(\frac{\alpha}{2})}{\sin(n+\frac{1}{2})\alpha}$$ (13.20)

Using the fact that $\nabla F$ is the normal to the surface gives:

$$\underline{n} = \underline{i}(-m)$$

$$+ \underline{j}$$

$$+ \underline{k}\frac{eaz_0}{b\sqrt{b^2 - z^2}}$$ (13.21)

where $\underline{n}$ is not normalised.