## Geometry FactorsSince the intrinsic detection efficiency of the alpha-particle detector in a typical alpha-spectrometry
chamber is nearly 100 %, the overall efficiency of the counting system for a thin alpha-emitting source
is essentially the same as the geometry factor,
defined as the mean solid angle subtended by the detector window at the source, divided by In the equations below we’ll use spherical coordinates ( ), and Cartesian coordinates (r, θ, z), which are related to each other as follows:x, y, zVisualize the ) as horizontal, with the positive φ = π / 2-axis, or polar axis (z), rising upward.φ = 0## Solid Angle Subtended by a Disk at a Point on its AxisThe solid angle at
a point not on the surface is
generally given as an integral, which may be a surface integral or a line integral (around the boundary).
In spherical coordinates the solid angle subtended by D at the origin is given by the surface integral:DThe active window of an alpha-spec detector is typically a flat disk.
Let -plane at a fixed height xy above it, as shown belowhIn this case the surface integral yields the following equation for the solid angle subtended by the disk at the origin: where at the rim of the disk. The cosine of this angle is
given byφSo, we get: Although this expression is theoretically exact, in actual calculations it can produce large round-off
errors if using
an equation that is less susceptible to the effects of rounding. For example,Ωwhere 0 ≤ .
The following version should suffice as long as either Φ < π/2 or
h > 0 and
h = 0 (and neither R_{D} > 0 nor h is huge).R_{D}## Solid Angle Subtended by an EllipseSuppose now that , semi-minor axis a, eccentricity b, and maximum polar angle e, but it is still
centered on the polar axis at a height Φ above the horizontal plane.
(Note that htan .)
Now the solid angle subtended by Φ = a / h at the origin can be shown to beDwhere
sin but which in other respects is
compatible with the one used by
Abramowitz and Stegun. The definition that appears
in Numerical Recipes (see below), uses a different sign convention for α.nIf you can calculate this elliptic integral accurately, you can calculate is not too small. If it is too small, the fact that it is obtained as
the difference between Ω2π and another number that is almost equal to 2π implies that
round-off error can be relatively large.The value of
(Φ ≈ 0) or a / h ≈ 0
(e ≈ 1). When b / a ≈ 0, the solid angle
is approximated well by the quotient of the area of Φ ≈ 0 and D.
h^{2}Note: I’m sure there are better approximations.Suppose instead that from below, the value of the elliptic integral increases without bound.
A useful approximation in this case is:e → 1where is too small, the preceding approximation should be easier to calculate.Φ## Solid Angle Subtended by a Disk at a Point Not on Its AxisNext suppose at a
point D in the P-plane at a distance xy from the origin.rThe solid angle subtended by the disk is the same as the solid angle
subtended by the image of the disk obtained by a 3-D perspective transformation, where the view
plane is oriented so that the image is an ellipse centered on the orthogonal line that passes through the center
of projection P.
The trick then is to find the parameters of that ellipse The following equations provide the
necessary values; however, rounding error can be an issue in some circumstances.PIf we define then we get a substantial simplification, as shown below. When you use this equation for 2π.I use Carlson’s method for evaluating the elliptic integral, as presented in Numerical Recipes
[Press et al., 1992, 2007].
For extreme values of the arguments that make where and where
k = r R_{D} / L^{2} denotes the Gauss hypergeometric function:_{2}F_{1}In the equation above for Π(, multiplied by
n, k)4, is subtracted from h / L2π.
The first term of the series, where , can be
subtracted from j = 02π with good accuracy, leaving
the higher-order terms to be subtracted. If is not large (because e is not too far from the axis),
those remaining terms are relatively small and do not cause large rounding errors.
(If P, all the higher-order terms are zero, because e = 0.)n = k = 0## The Geometry Factor for an Extended SourceUsing this approach you obtain a function of r from the origin, assuming fixed values for P
and R_{D}. The geometry factor for an extended source is defined to be the mean solid angle
averaged over all points of the source, which is calculated by another integral.hGiven the function (see above) can be calculated using a good technique
for numerical integration, such as Gaussian quadrature or Simpson’s Rule.R_{S}If is very small and h or R_{S} > R_{D}, because the solid angle R_{S} ≈ R_{D} is almost a step function in the vicinity of Ω(r), dropping steeply from r = R_{D} to Ω ≈ 2π as Ω ≈ 0 increases.
In this case you can use Simpson’s Rule, which is robust and amenable to brute-force processing; or if r is small enough, you can use the fact that:hfor
R_{S} > R_{D}For 2π.
where
,
, and
For ,
these alternative equations are probably not great for actual calculations. And when the solid angle
is small, you still need to do some work to avoid large rounding errors.
(2) It is easier to calculate the solid angle subtended by a polygon than the solid angle subtended by a disk.
Approximating the disk by a regular polygon with the same area is a good practical option in the lab.
(3) Monte Carlo simulation is another popular approach, although it is inherently inexact.
On the other hand it can provide explicit uncertainty estimates.
(4) After my coworker described his use of my calculations at a radiochemistry conference
in October 2014, I knew I needed to complete my work on evaluating the combined standard uncertainty of
the geometry factor. I have done that now but I need to write it up clearly. (5) James Clerk Maxwell discussed the
solid angle subtended by an ellipse in A Treatise on Electricity and Magnetism, Chapter XIV. He mentioned a
solution using the elliptic integral of the third kind but preferred a different solution in terms of an
infinite series of spherical harmonics.r ≠ R_{D}
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