Geometry Factors

Since 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 . 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. Or, if you meas­ure the effi­ciency for a source in one geom­etry, you can apply a cor­rec­tion fac­tor to esti­mate the effi­ciency for a dif­ferent geometry.

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:

geometry-coords.png

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

Solid Angle Subtended by a Disk at a Point on its Axis

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:

equation

The active window of an alpha-​spec de­tec­tor is typi­cally a flat disk. Let RD 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

figure
Figure 1 ‒ Solid angle subtended at a point on the axis (the origin)

In this case the sur­face inte­gral yields the fol­low­ing equa­tion for the solid angle sub­tended by the disk at the origin:

double integral

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

cosPhi.png

So, we get:

Omega1.png

Although this expression is theo­reti­cally exact, in actual cal­cu­la­tions it can pro­duce large round-off errors if h RD. 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,

Omega2.png

where tan Φ = RD / h and we use the fact that 0 ≤ Φ < π/2. The fol­low­ing ver­sion should suf­fice as long as either h > 0 or h = 0 and RD > 0 (and neither h nor RD is huge).

Omega3.png

Solid Angle Subtended by an Ellipse

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

Omega4.png

where Π(n, k) denotes a com­plete Legendre el­lip­tic inte­gral of the third kind, which is de­fined by the fol­low­ing equation.

Pi-def.png

Note: There are several vari­ants of the def­ini­tion of the func­tion Π. I favor the one used by Wolfram, which uses k instead of sin α 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 n.

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 and another number that is almost equal to 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.

Omega5.png

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

Omega6.png

where E(k, φ) denotes an ellip­tic inte­gral of the second kind. If Φ is too small, the pre­ceding ap­proxi­ma­tion should be easier to calculate.

Solid Angle Subtended by a Disk at a Point Not on Its Axis

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.

figure
Figure 2 ‒ Solid angle subtended at a point P off the axis

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.

Omega7.png

If we define

Omega-L.png

then we get a sub­stan­tial sim­pli­fica­tion, as shown below.

Omega-r-pre.png
Omega-r-simple.png

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 .

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 Ω very small, you can ex­pect large rela­tive errors due to round­ing in the final sub­trac­tion. The round­ing error can be re­duced in some cases by the use of a series for the ellip­tic integral. For example,

Pi-series.png

where n = e2 = r2 / L2 and k = rRD / L2 and where 2F1 denotes the Gauss hyper­geomet­ric function:

2F1-series.png

In the equation above for Ω, the value of Π(n, k), multi­plied by 4h / L, is sub­tracted from . The first term of the series, where j = 0, can be sub­tracted from with good accuracy, leav­ing the higher-​order terms to be sub­tracted. If e is not large (because P is not too far from the axis), those re­main­ing terms are rela­tively small and do not cause large round­ing errors. (If e = 0, all the higher-​order terms are zero, because n = k = 0.)

The Geometry Factor for an Extended Source

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 RD 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.

geometry-figure3-color.png
Figure 3 ‒ Mean solid angle for a uniform extended source

Given the func­tion Ω(r), the re­quired inte­gral for a disk-​shaped source of non­zero radius RS (see above) can be cal­cu­lated using a good tech­nique for nu­meri­cal integra­tion, such as Gauss­ian quad­ra­ture or Simpson’s Rule.

Omega-avg

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 RS > RD or RSRD, be­cause the solid angle Ω(r) is almost a step func­tion in the vicinity of r = RD, drop­ping steeply from Ω ≈ 2π to Ω ≈ 0 as r in­creases. In this case you can use Simpson’s Rule, which is ro­bust and ame­nable to brute-force proc­ess­ing; or if h is small enough, you can use the fact that:

Omega-lim.pngfor RS > RD

For RS < RD, the limit is just .


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, shown below, are less simple than those shown above.

Omega3cases.png

where K(k) denotes the com­plete ellip­tic inte­gral of the first kind and where

Omega3cases-C.png ,    Omega3cases-k.png ,   and    Omega3cases-n.png

For rRD but rRD, 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.

Updates: In 2017 I discovered the work of John T. Conway, who pub­lished a nice paper on this sub­ject in 2006. He pro­vided a closed-​form expres­sion for the aver­age geom­etry factor in terms of elliptic inte­grals and also an inte­gral that seems to give highly accurate results with­out much work. I have not explored all the cited refer­ences, but I recom­mend the paper anyway: “Generalizations of Ruby’s for­mula for the geo­metric effi­ciency of a parallel-​disk source and detector system,” Nuclear Instruments & Methods in Physics Research. A 562 (2006) 146–153.