Theoretical background

The concept of the effective solid angle (Ω) for calculation of semiconductor detector full energy peak efficiencies (ε_p)

In any gamma-spectrometric measurement with semiconductor detectors, the question of converting the number of counts (collected in a full energy peak) into the activity of the sample/source cannot be avoided. There are, in principle, three approaches to this issue:

Relative, where one tries to imitate as good as possible the source by a standard (or vice versa), while keeping the same counting conditions for the two. Paid enough care, the result is, in general, so accurate that cannot be surpassed by other methods. However, we all know what "enough care" means in practice. Combined with the inflexibility in respect with varying source&container parameters (shape, dimensions, material composition), this represents raison d'etre of the other approaches, as follows.
Absolute calculations (Monte Carlo methods) yield full energy peak efficiency (ε_p) for a given counting arrangement. It is essentially statistical treatment of the events which photons undergo - from being emitted by a source atom until the interaction with the detector active body - including the treatment of the so produced electrons, positrons and other subsequent energy carriers. This approach is beautifully exact, on condition that we consider sufficiently large number of incident photons, and that we know the details about

source, detector and intercepting layers' geometrical&compositional data,
the corresponding energy and angle dependent cross sections for various photon interactions, and
parameters characterizing electron/positron behaviour in all materials involved.

At present, inherent statistical uncertainty of Monte Carlo methods, unsatisfactory manufacturers' detector specifications and relatively poor knowledge of (some of) the above physical parameters are the limiting factors for its applicability. However, with speeding up of computers, and with more accurate detector specifications and cross section data (the determination of which is, on its turn, related to more careful and sophisticated measurements), it is reasonable to assume that the time works for absolute methods, and that in future they might be the dominant ones.

Semiempirical models, trying to conciliate the previous two. Semiempirical models commonly consist of two parts:

experimental (producing one kind or another of reference efficiency characteristic of the detector) and
relative-to-this calculation of ε_p.

Inflexibility of the relative method is avoided in this way, as well as the demand for some physical parameters needed in Monte Carlo calculations. Numerous variations exist within this approach, with emphases either to experimental or to computational part. Most of them simplify (or oversimplify) the physical model behind, i.e. the treatment of gamma-attenuation, geometry and detector response. It can be shown that only simultaneous differential treatment of these three factors is essentially justified. This fact is transformed into the concept of the effective solid angle (Ω), a calculated value incorporating the three components, and closely related to the detection efficiency.

Given a gamma-source (S) and a semiconductor detector (D), the effective solid angle is defined as:

(1)

with V_S = source volume, S_D = detector surface exposed to the source ("visible" by the source) and

(2)

Here T is point varying over V_S, P point varying over S_D, and n_u the external unit vector normal to infinitesimal area dσ at S_D. Eq. (1) is thus a five fold integral. Factor F_eff accounts for gamma-attenuation of the photon following the direction TP out of the detector active zone, while F_eff describes the probability of an energy degradable photon interaction with the detector material (i.e. coherent scattering excluded), initiating the detector response. The two factors include therefore geometrical and compositional parameters of the materials traversed by the photon.

With ε_p being proportional to Ω, the detection efficiency is found as:

(3)

where index "ref" denotes reference counting geometry to which the actual one is relative.

So as to apply this method the following should be known:

reference efficiency curve, usually obtained by counting calibrated source(s) at a reference geometry and covering gamma-energies (E_γ) in the region of interest (e.g. 50–3000 keV); considerable effort should be put in this phase to reach accurate ε_p vs. E_γ function, but it pays off in further exploitation;
geometrical and compositional data about the source, detector and all intercepting layers (for the latter e.g. source container and holder, detector end cap and housing, dead layers, etc.);
gamma-attenuation coefficients for all materials involved.

For a cylindrical source coaxially positioned with the detector, and with radius smaller than that of the detector (r₀ < R₀). Eq. (1) than gives

(4)

In the above, five fold integral is reduced to four fold due to axial symmetry. Disk and point sources are included in Eq. (4) (for L = 0, and L = 0, r₀ = 0, respectively).

For sources with radii larger than that of the detector (r₀ > R₀) we have:

(5)

with

Marinelli geometry can be described by:

(6)

When treating well detector we obtain:

(7)

In this case, points P and T have the following coordinates, respectively for the three terms:

[T (r,0, d + l), P (R cosφ, R sinφ, 0)],

[T (r,0, d + l), P (R_l cosφ, R_l sinφ, h)],

[T (r, 0, l), P (R cosφ, R sinφ, H_l)]

The above described effective solid angles and corresponding detector efficiencies represent theoretical basis for ANGLE software calculations. These account for the majority of counting situations in γ-spectrometry practice.

References

ANGLE v2.1 — New version of the computer code for semiconductor detector gamma-efficiency calculations, S. Jovanovic, A. Dlabac and N. Mihaljevic, Nuclear Instruments and Methods in Physics Research Section A, Volume 622, Issue 2, October 2010, Pages 385-391 Full article
Testing efficiency transfer codes for equivalence, T. Vidmar, N. Çelik, N. Cornejo Díaz, A. Dlabac, I.O.B. Ewa, J.A. Carrazana González, M. Hult, S. Jovanović, M.C. Lépy, N. Mihaljević, O. Sima, F. Tzika, M. Jurado Vargas, T. Vasilopoulou and G. Vidmar, Applied Radiation and Isotopes, Volume 68, Issue 2, February 2010, Pages 355-359 Full article
Reliability of two calculation codes for efficiency calibrations of HPGe detectors, K. Abbas, F. Simonelli, F. D’Alberti, M. Forte and M. F. Stroosnijder, Applied Radiation and Isotopes, Volume 56, Issue 5, May 2002, Pages 703-709 Full article
Methods and software for predicting germanium detector absolute full-energy peak efficiencies, K. R. Jackman and S. R. Biegalski, Journal of Radioanalytical and Nuclear Chemistry, Volume 279, Number 1/January, 2009, Pages 355-360 Full article
Calculation of the absolute peak efficiency of gamma-ray detectors for different counting geometries, L. Moens, J. De Donder, Lin Xi-lei, F. De Corte, A. De Wispelaere, A. Simonits and J. Hoste, Nuclear Instruments and Methods in Physics Research, Volume 187, Issues 2-3, 15 August 1981, Pages 451-472 Full article
Calculation of the peak efficiency of high-purity germanium detectors, L. Moens and J. Hoste, The International Journal of Applied Radiation and Isotopes, Volume 34, Issue 8, August 1983, Pages 1085-1095 Full article
ANGLE: A PC-code for semiconductor detector efficiency calculations, S. Jovanović, A. Dlabač, N. Mihaljević and P. Vukotić, Journal of Radioanalytical and Nuclear Chemistry, Volume 218, Number 1/April, 1997, Pages 13-20 Full article
On the applicability of the effective solid angle concept in activity determination of large cylindrical sources, P. Vukotić, N. Mihaljević, S. Jovanović, S. Dapčević, and F. Boreli, Journal of Radioanalytical and Nuclear Chemistry, Volume 218, Number 1/April, 1997, Pages 21-26 Full article
"EXTSANGLE" — An extension of the efficiency conversion program "SOLANG" to sources with a diameter larger than that of the Ge-detector, N. Mihaljević, S. Jovanović, F. De Corte, B. Smodiš, R. Jaćimović, G. Medin, A. De Wispelaere, P. Vukotić and P. Stegnar, Journal of Radioanalytical and Nuclear Chemistry, Volume 169, Number 1/March, 1993, Pages 209-218 Full article
Introduction of Marinelli effective solid angles for correcting the calibration of NaI(Tl) field gamma-ray spectrometry in TL/OSL dating, F. De Corte, S. M. Hossain, S. Jovanović, A. Dlabač, A. De Wispelaere, D. Vandenberghe and P. Van den Haute, Journal of Radioanalytical and Nuclear Chemistry, Volume 257, Number 3/September, 2003, Pages 551-555 Full article
Contribution of ²¹⁰Pb bremsstrahlung to the background of lead shielded gamma spectrometers, D. Mrđa, I. Bikit, M. Vesković and S. Forkapić, Nuclear Instruments and Methods in Physics Research Section A, Volume 572, Issue 2, 11 March 2007, Pages 739-744 Full article
Production of X-rays by cosmic-ray muons in heavily shielded gamma-ray spectrometers, I. Bikit, D. Mrda, I. Anicin, M. Veskovic, J. Slivka, M. Krmar, N. Todorovic and S. Forkapic, Nuclear Instruments and Methods in Physics Research Section A, Volume 606, Issue 3, 21 July 2009, Pages 495-500 Full article

Theoretical background

The concept of the effective solid angle (Ω) for calculation of semiconductor detector full energy peak efficiencies (εp)

References

The concept of the effective solid angle (Ω) for calculation of semiconductor detector full energy peak efficiencies (ε_p)