Geometry Factors

It is generally expected that the detec­tion and count­ing effi­ciency for alpha-particles emitted from a thin source in a typical alpha-spectrometry cham­ber should be es­sen­tially the same as the geom­etry fac­tor, defined as the mean solid angle sub­tended by the de­tec­tor win­dow at the source, divided by 4π. So, if you know the di­men­sions of the de­tec­tor and the source, and their rela­tive posi­tions and orien­ta­tions, in prin­ciple you can pre­dict the effi­ciency with­out a calibration.

In the equations below we’ll use spheri­cal co­ordi­nates (ρ, θ, φ), cylin­dri­cal co­ordi­nates (r, θ, z), and Cartesian co­ordi­nates (x, y, z), which are re­lated to each other as follows:

Visualize the xy-plane (φ = π / 2) as horizontal, with the positive z-axis, or polar axis (φ = 0), rising upward.

The solid angle Ω sub­tended by a sur­face D 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:

The active window of an alpha-spec de­tec­tor is typi­cally a flat disk. Let R_D denote the radius of the disk and assume the disk is centered on the polar axis, par­al­lel to the xy-plane at a fixed height h above it, as shown below

In this case the sur­face inte­gral yields the fol­low­ing equa­tion for Ω:

where Φ is the polar angle φ at the rim of the disk. The co­sine of this angle is given by

So, we get:

Although this expression is theo­reti­cally exact, in actual cal­cu­la­tions it can pro­duce large round-off errors if h ≫ R_D. In these situa­tions it is better to apply a few trig iden­tities and com­pute Ω using an equa­tion that is less sus­ceptible to the ef­fects of round­ing. For example,

where tan Φ = R_D / h. The fol­low­ing ver­sion should suf­fice as long as either h > 0 or R_D > 0.

Suppose now that D has an el­lip­ti­cal shape wtih semi-major axis a, semi-minor axis b, ec­cen­tric­ity e, and maxi­mum polar angle Φ, but it is still centered on the polar axis at a height h above the hori­zontal plane. (Note that tan Φ = a / h.) Now the solid angle sub­tended by D at the origin can be shown to be

where Π(n, k) denotes a com­plete Legendre el­lip­tic inte­gral of the third kind (with the same sign con­ven­tion for n used by Abramowitz and Stegun). So, if you can cal­cu­late this el­lip­tic inte­gral accu­rately, you can cal­cu­late Ω accurately, at least when the value of Ω is not too small. If it is too small, the fact that it is ob­tained as the dif­fer­ence be­tween 2π and another number that is almost equal to 2π im­plies that round-off error can be rela­tively large.

The value of Ω will be small if either Φ ≈ 0 (a / h ≈ 0) or e ≈ 1 (b / a ≈ 0). When Φ ≈ 0, the solid angle is ap­proxi­mated well by the quo­tient of the area of D and h². Note: I’m sure there are better approxi­mations.

Suppose instead that e ≈ 1. As e → 1, the value of the el­lip­tic inte­gral in­creases with­out bound. A use­ful ap­proxi­ma­tion in this case is:

where E(k, φ) de­notes a Legendre el­lip­tic inte­gral of the sec­ond kind. The el­lip­tic inte­gral here de­pends on Φ but not on e. If Φ is too small, the pre­ceding ap­proxi­ma­tion should be easier to calculate.

Next suppose D is again a disk but now con­sider the solid angle sub­tended by D at a point P in the xy-plane at a distance r from the origin. The solid angle sub­tended by the disk D at P 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.

If we define

then we get a sub­stan­tial sim­pli­fica­tion, as shown below.

When you use this equa­tion for Ω, there is little rea­son to worry about round-off error until you cal­cu­late the el­lip­tic inte­gral and per­form the final sub­trac­tion from 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, you can still ex­pect large rela­tive errors due to rounding.

Using this approach you obtain a func­tion Ω(r) for the solid angle de­fined in terms of the dis­tance r of P from the origin, assuming fixed values for R_D and h. 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. Given the func­tion Ω(r), the re­quired inte­gral for a disk-shaped source of non­zero radius R_S can be cal­cu­lated using a good tech­nique for nu­meri­cal integra­tion, such as Gauss­ian quad­ra­ture or Simpson’s Rule.

If h is not too small, Gaussian quad­ra­ture is likely to give ex­cellent results; but be care­ful with this tech­nique when h is very small and R_S > R_D or R_S ≈ R_D, be­cause the solid angle Ω(r) is almost a step func­tion in the vicinity of r = R_D, drop­ping steeply from Ω ≈ 2π to Ω ≈ 0 as r in­creases. In this case you can use Simpson’s Rule, which is more ro­bust and is ame­nable to brute-force proc­ess­ing; or if h is small enough, you can use the fact that:

For R_S < R_D, the limit is just 2π.

To be completed: (1) There are other (equiv­a­lent) equa­tions for Ω(r) that can be de­rived directly from the sur­face inte­gral; how­ever, my ver­sions of these are not simpler then those shown above. They are de­fined by cases, de­pend­ing on whether r < R_D, r = R_D, or r > R_D. For r ≠ R_D but r ≈ R_D, the alter­na­tive equa­tions are prob­ably not great for actual cal­cu­la­tions. (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.

Feedback: Got better ideas for do­ing these cal­cula­tions? Already know the sub­ject back­ward and for­ward, in­side and out? Or do you know the pub­lished ref­er­ence for the best (accu­rate, sim­ple, and el­egant) solu­tion cur­rently avail­able? Then en­lighten me by click­ing on the “Feedback” link below and send­ing me a note by e-mail.

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20 Mar 2010 Keith