During the measurement of physical quantities, researchers try to reduce uncertainty to the extent possible. The measurement error in radiometry depends on the number of registered counts. Generally, the higher the number of counts is, the better the survey precision. The registered number of counts depends on sample activity, the probability of the γ quanta emission, named emission intensity, and the photon registration efficiency, named the absolute full energy peak efficiency (AFEPE) [1], which depends on photon energy. Obviously, the samplespecific activity is somehow determined, and in most cases, we cannot influence this activity. Additionally, we do not have an influence on the emission intensity, which is invariant and inherently bounded with a given radionuclide. Our control of AFEPE depends on choosing better sample geometry and measurement geometry [2]. Thus, to improve the survey accuracy, it is necessary to choose the best available geometries. Therefore, in practice, we need an effective tool that allows for choosing the best measurement condition before starting the measurement [3, 4].
For purposes of this report, the sample geometry indicates a particular sample to be measured characterised by its shape, density, atomic composition, homogeneity, and granulation. The measurement geometry indicates a sample with a previously defined sample geometry precisely located in relation to the detector, shielding, and absorbers, if any, in use. The integrated absolute full energy peak efficiency (IAFEPE) is the definite integral (Riemann integral) of the function eff(Eγ) of photon energy that approximates AFEPE values. Simultaneously, eff(Eγ,i) represents the value of the AFEPE for a particular photon’s energy. Photon energy Eγ,i is the energy of a particular γ quanta emitted from radionuclide i.
1. Riemann integral
2. IAFEPE
As was said previously, a better measurement geometry among those considered will result in a higher count rate. This means obtaining a larger net peak area for a particular FEAP in the case of a singular radionuclide emitting γ quanta with only one energy or the largest sum of all net peak areas in the other case.
Let k label the measurement geometry. Next, we consider only two sample geometries: “1” and “2”. For the photons emitted from a sample with geometry k, the measure of correctness of the chosen measurement geometry is the sum of all net peak areas (see Eq. 3):
$$\sum _{i=1}^{n}{N}_{i}^{k}={N}_{1}^{k}+{N}_{2}^{k}+\dots +{N}_{n}^{k}$$
Equation 3
where the subscript i – iterates the ith FEAP, n  number FEAPs in the γray spectrogram, the superscript k = 1, 2  labels the measurement geometry, and Nik [counts] is a net peak area of the ith FEAP. For the simplification of future divagation, we suppose that any singular radionuclide emits γ quanta with only one energy. However, this is not necessary.
It is possible to elaborate a particular net peak area based solely on the fundamental expression for the AFEPE represented by Eq. 4:
$$eff\left({E}_{\gamma ,i}\right)=\frac{{{N}_{ }}_{i}^{k}}{t\bullet {{P}_{\gamma ,i}}_{ }\bullet {A}_{i}}$$
Equation 4
where eff(Eγ,i) is the value of AFEPE(Eγ,i) for γ quanta with energy Eγ,I [keV], realised from sample contains nuclei i, Nik [counts] is the net peak area of the FEAP for γ quanta with energy Eγ,i that are realised from nucleus i from the sample measured in measurement geometry k, t [s] is the measurement time, Pγ,i is the probability of the particular γ quanta emission from nuclei i being measured, and Ai [Bq] is the activity of sample with nuclei i. For the simplification of future descriptions, Nik is called the net peak area. Notably, eff(Eγ,i) and Pγ,I are both dimensionless quantities. Additionally, AFEPE(Eγ,i) is a numerical value, while eff(Eγ) is a continuous function fitted to AFEPE(Eγ,i) values.
Let us introduce the quantum normalised net peak area, which is expressed in the following formula:
$${N}_{i}^{{\prime }k}=\frac{{N}_{i}^{k}}{{P}_{\gamma ,i}\cdot {A}_{i}}$$
Equation 5
The quantum \({N}_{i}^{{\prime }k}\) has its own physical meaning. It is a net peak area for a radionuclide that has both unitary activity and intensity.
The region of interest (ROI) contains counts represented by points on the Cartesian coordinate system distributed around a particular photon energy Eγ,i. Mainly, it is governed by Gaussian statistics [8], were the fundamental parameters are full width at half maximum height (FWHM) of the peak and σ  standard deviation. FWHM(Eγ) is a continuous function describing the distribution of FWHM against photon energy Eγ, while FWHM(Eγ,i) is the value of this function for a particular photon energy Eγ,i. The FWHM(Eγ) is usually obtained during a spectrometer energy calibration process called the shape calibration or simply spectrometer resolution. The FWHM(Eγ) is determined for a particular detector. Finally, we approximate a Gaussian function (see Fig. 1) by a rectangle with base ∆Ei expressed by FWHM(Eγ,i) [keV] (see Fig. 1 and Eq. 6) and height equal to eff(Eγ,i).
Obviously, the closer to ± ∞ integration limits are the approximation more precise is. In γray spectrometry, we generally operate with the limits range of ± σ or ± 2∙σ. For infinite integration limits, the integral of the Gaussian function is equal to the area of the proposed approximating rectangle. When the integration limits are ± σ, the field below the Gaussian curve covers 68% of the rectangular area, and in the case of ± 2σ, it covers an area of 95%. Thus, the appropriate rectangle has a physical meaning in γray spectrometry. The upper boundary of the definite integral of the Gaussian function is the idealization of counting photons with energy Eγ,i.
Presently, we link the photon energy with FWHM, i.e., a parameter that characterises the Gaussian distribution of the counts. Thus, we define the value of ∆Eγ,i (see Eq. 6), which is used in Fig. 1:
$${\varDelta E}_{\gamma ,i}=\left({E}_{i}+\frac{FWHM\left({E}_{\gamma ,i}\right)}{2}\right)\left({E}_{i}\frac{FWHM\left({E}_{\gamma ,i}\right) }{2}\right)$$
Equation 6
Thus, a value of net peak area affected photons with energy Eγ,i can be expressed by the following formula:
$${N}_{i}^{k}\cong P{ }_{\gamma ,i}\cdot {A}_{i}\cdot {eff}^{k}\left({E}_{\gamma ,i}\right){\bullet \varDelta E}_{\gamma ,i}\bullet t$$
Equation 7
or using the normalised net peak area (see Eq. 5):
$${{N}^{{\prime }}}_{i}^{k}\cong eff{ }^{k}\left({E}_{\gamma ,i}\right)\bullet {\varDelta E}_{\gamma ,i}\cdot t$$
Equation 8
It is unnecessary to explain how a peak area is estimated, and it is advisable to quote Gordon Gilmore—an icon of practical γray spectrometry—who said that ”the measurement of the peak area should require no more than a simple summation of the number of counts in each of those channels that we consider to be part of the peak” … [8].
It should be noted that for a particular radionuclide, irrespective of the measurement geometry, its emission intensity remains unchanged, and the measurement time must be the same for both geometries taken into consideration.
Let the symbol ℛ denote the effectiveness of the measurement process and the relation between the efficiency of the two analysed measurement geometries. It can be expressed by the following formula:
$$\mathcal{R}=\frac{\sum _{i}^{n}{{N}^{{\prime }}}_{i}^{1}}{\sum _{i}^{n}{{N}^{{\prime }}}_{i}^{2}}\cong \frac{\sum _{i}^{n}eff{ }^{1}\left({E}_{\gamma ,i}\right){\bullet \varDelta E}_{\gamma ,i}}{\sum _{i}^{n}eff{ }^{2}\left({E}_{\gamma ,i}\right)\bullet {\varDelta E}_{\gamma ,i}}$$
Equation 9
The measurement time is drawn before the summation operator and is shortened in the numerator and denominator. Obviously, ℛ can be greater or smaller than 1. In the first case, the measurement geometry “1” is more effective than “2”. One can see the symmetry between Eq. 1 and the newly introduced Eq. 8. It is worth emphasising that
$${eff}_{i}^{k}\left({E}_{\gamma ,i}\right)\bullet {\varDelta E}_{\gamma ,i}\equiv f\left({\xi }_{i}\right)\bullet {{\Delta }}_{i}$$
Equation 10
plays the same role as the value of a function f(ξ), whereas
$$\left({E}_{\gamma ,i}+\frac{FWHM\left({E}_{\gamma ,i}\right)}{2}\right)\le {\xi }_{i}\le \left({E}_{\gamma ,i}\frac{FWHM\left({E}_{\gamma ,i}\right)}{2}\right)$$
Equation 11
is the argument of this function (see Eq. 2). Thus, with respect to all previous divagations and Riemann integral definitions, we propose transforming Eq. 9 to the new form:
$$\mathcal{R}=\frac{{\int }_{{E}_{min}}^{{E}_{max}}{eff}^{1}\left({E}_{\gamma }\right){dE}_{\gamma }}{{\int }_{{E}_{min}}^{{E}_{max}}{eff}^{2}\left({E}_{\gamma }\right){dE}_{\gamma }}$$
Equation 12
where \({\int }_{{E}_{min}}^{{E}_{max}}{eff}^{1}\left({E}_{\gamma }\right){dE}_{\gamma }\) is the defined integral in the whole energy diapason \({E}_{\gamma }ϵ\)<\({E}_{min}\), \({E}_{max}\)> of the \({eff}^{1}\)(\({E}_{\gamma }\)) function. That is, for the sample measured in measurement geometry “1”, it is necessary to emphasise that we transform discrete values of photon registration efficiency into their continuous fitting function.
Thus, if one would like to complete a quantitative analysis of radionuclides that are in low activity and/or short halflife time (T1/2), she or he can at first check the value of IAFEPE expressed in arbitrary units [a.u.] (see Eq. 13):
$$IAFEPE={\int }_{{E}_{min}}^{{E}_{max}}{eff}^{ }\left({E}_{\gamma }\right){dE}_{\gamma }$$
Equation 13
3. Quantities derived from IAFEPE
IAFEPE itself is sufficient to assess the effective photon registration efficiency if the measured items are weightless samples or real samples but kept in different locations concerning a particular detector. When dealing with samples kept in different containers or contaminated areas with different sizes, it is necessary to operate with quanta directly related to IAFEPE and commonly with sample weight or surface area. The mass integrated absolute full energy peak efficiency (MIAFEPE) parameter combines the possibility of the sample being neutron activated or possessing contaminated liquid, solid or food and being measured by means of γray spectrometry. MIAFEPE is a product of sample mass and IAFEPE consolidation (see Eq. 14):
$$MIAFEPE = m\bullet IAFEPE$$
Equation 14
where m [g] is a sample mass.
Superficial integrated absolute full energy peak efficiency (SIAFEPE), which is a product of IAFEPE and the area of the contaminated surface, is a quantum that is useful for radiation monitoring. SIAFEPE is expressed by Eq. 15:
$$SIAFEPE=S \bullet IAFEPE$$
Equation 15
where S [m2] is a surface of contaminated area.
Part II: Numerical simulation  IAFEPE for a sample with changeable number or radionuclides inside To better understand the importance of the IAFEPE in case a measured sample retains very different numbers of radionuclides, a computer program that easily generates a graphical representation of considered cases was arranged. By using a random number, the algorithm generates the γray spectrum for a virtual sample that consists of a changeable number of γ quanta emitters with different strengths (Ai) and emission intensities (Pγ,i). The distance between two particular FEAPs is also changeable (however, it is not necessary) and expressed in FWHM. The energy efficiency calibration function and shape calibration are adopted from a real NaI detector. The divagation is performed for the NaI detector because its resolution, i.e., wide FEAPs, is best suited to graphical visualisation of the theoretical considerations cited above. The calculation is performed for noninterfering FEAPs only. However, this is not necessary. The type and manufacturer of the detector is not important here.
Next, the algorithm detects FEAPs, fits Gaussian functions and normalises them in accordance with Eq. 5. Using normalised Gaussian functions, the corresponding rectangles are calculated in such a way that their widths cover the whole energy range (however, this is not necessary). Summing of all rectangle areas leads to an IAFEPE value calculation. Thus, this is a practical proof of the IAFEPE applicability. The following sequence of figures presents the algorithm steps for different numbers of FEAPs.
Part III: Practical application of IAFEPE and derivate quantities
1. Laboratory practice
In laboratory practice, a Marinelli beaker is frequently used. It has a cup shape that allows a radioactive substance to cover a detector from almost all directions (see Fig. 3). In the above figure, two Marinelli beakers with different volumes are presented. For these sample geometries, the values of IAFEPE and MIAFEPE are calculated for water and presented in Table 1.
Table 1

Marinelli beaker with water

Table 1
The comparison of two Marinelli beakers with water. IAFEPE and MIAFEPE were obtained based on numerical characterisation of the particular HPGe detector. It should be noted that changing the detector implies obtaining the different function(s) for γ quanta registration efficiency.
size

small

large

mass [g]

520

2,300

IAFEPE [a.u.]

96.91

47.41

MIAFEPE [g∙a.u.]

≈ 50,000

≈ 109,000

Analyses of IAFEPE and MIAFEPE provide us with following information: the efficiency of photon registration emitted from the small container is twice that of the large one, but the mass of radioactive medium (water) compensates for the decreased efficiency and allows us to obtain the same radiometric survey precision (measurement error) during shorter measurement times by a factor of two. The large container possesses four times as much radioactive substance considering its volume. The efficiency is calculated for each Marinelli beaker based on particular detector numerical characteristics and LabSOCS™ [10], which is commercially available Monte Carlo based software for a numerically characterised detector. The type of detector is not important here.
2. Neutronics: stack of neutron activation foils
Let us consider a stack of 11 activation foils. Each has the same diameter (18 mm) and height (1 mm) but is made with different metals (see Fig. 4). The stack of foil is frequently activated at JET tokamak [11]. After activation with neutrons from deuterium plasma, a sample is placed on the centre of an HPGe detector endcap, and its activity is determined. The efficiency is calculated as previously described. After that, IAFEPE is the subject of subsequent calculations. The results are presented in Fig. 4:
It is seen that the sample density and nuclear charge (Z) essentially influences the photon registration efficiency. Thus, the photons released from the lightest sample with small Z made from aluminium are registered with the highest efficiency, unlike in the case of gold.
The energy for the maximum AFEPE increases mainly with sample density and Z. The maximum AFEPE decreases with increasing sample density and Z. This means that the photon registration conditions (efficiency) deteriorate with increasing metal density and Z. Qualitatively, the areas under the efficiency curve (i.e., IAFEPE) decrease with increasing sample density and Z. Quantitatively, this is proven using IAFEPE (see Fig. 4). The maxima AFEPE of sources with extreme values of density (aluminium and gold) changes by a factor of 3, while their mass changes by a factor of ~ 7. Simultaneously, the sample's IAFEPE is decreased. Parallel with the above two processes, a shift of photon energy that produces maxima of AFEPE is observed.
The most important conclusion coming directly from the above considerations is that changing the proportion between elements composing a mixed activation source can influence the efficiency of photon registration [12].
3. Radiation monitoring  surface contamination
Let us consider a flat contaminated area and detector that monitor radiation derived from the Earth’s surface [13]. The AFEPE is calculated for a row of circular flat discs with increasing diameter. These discs are homogeneously contaminated with radionuclides. All the calculations are made with the ISOCS™ [14]– the Monte Carlo based commercially available software for a numerically characterised detector. For the mentioned row of discs, the IAFEPE and SIAFEPE are calculated, and the results are presented in Fig. 5:
Analysis shows that for the contaminated surface area, there exists a threshold diameter (see Fig. 5b) above which the detector efficiency does not change, while the approximation function describing the SIAFEPE has a limit. The above presented result was obtained for a strictly defined LaBr3 detector with numerical characteristics. The type and manufacturer of the above detector are not important here.