Introduction

In γ-ray spectroscopy, one of the central problems is to determine the intensities of γ-rays observed with a high-resolution detector such as high purity germanium (HPGe). Using a spectrometer based on an HPGe detector results in γ-ray spectrum that contains sharp peaks and a continuum that arises mainly from Compton-scattering events in the detector. These sharp peaks due to complete absorption of the energies of the incoming γ-rays are called full energy peaks (FEP) [1]. In a spectrum, the areas of the peaks characterize the intensities or strengths of the sources can be determined with high precision and their energies can be used to identify the sources. Here we will concentrate on determination of intensities or emission rates from measured peak area. By definition the observed peak area in a unit time R γ, from a steady source of activity A is

$$ R_{\gamma } = AP_{\gamma } \varepsilon (E_{\gamma } )K, $$
(1)

where P γ is the absolute emission probability, the ε(E γ) is the detector absolute FEP efficiency (or briefly absolute efficiency) and K contains all correction factors for losses during the acquisition of the spectrum [1]. This includes dead time, absorption in the target and coincidence summing. The detector absolute efficiency is a smooth function of the γ-ray energy. It may include the γ-ray absorption by materials in between the active volume of the detector and the target or source. We have already studied the functional form of the absolute efficiency in the past [2, 3] and its behavior over a wide energy range in HPGe detectors [4–6].

We have received many requests from users of the Hypermet PC program [7–11] that the usage of orthogonal polynomials is too complicated and they would prefer to use a simple polynomial. Here we have to note that the program, which is freely available (http://www.iki.kfki.hu/nuclear/other/index_en.shtml), comes with an efficiency calculator that provides all necessary information of the fitted function. Nevertheless, inspired by the request for a simple polynomial, I decided to present the derivation of an efficiency fit with a simple polynomial. Furthermore, in this paper I would like to concentrate on the explicit calculation of the uncertainties of functions or efficiency functions fitted to the data using linear parameter models [2]. The results can be usefully applied in comparator experiments of prompt gamma activation analysis (PGAA) [12], the determination of cross sections in comparator experiments, and in correction for true coincidences in γ-ray spectroscopy.

In the first part of the article the formalism is introduced through the derivation of the uncertainty of a fitting polynomial using a special notation of uncertainty calculus as defined in the monograph of Jánossy [13]. The results are shown to be generally applicable to any linear form, for example in the fitting of angular distributions.

Calculation of uncertainty of relative detector efficiency

The relative γ-ray efficiency function of a γ-ray detector can be obtained from a least squares fit to relative efficiencies calculated from the data measured on well-established calibration sources. To obtain this we can use the relative emission probabilities and any value for the activity A of the source (example A = 1) in Eq. (1). The form we use in the fit is a polynomial function to these data on a log–log scale.

First, I consider a general polynomial fit to any measured data. To avoid ill conditioned equations, the independent scale is transformed to the (−1, 1) range. Furthermore, it is assumed that the parameters a i are independent for different index values, i, and that they exhibit Gaussian distributions. The form of a polynomial is

$$ P(x) = \sum\limits_{i = 0}^{n} {a_{i} x^{i} } , $$
(2)

where x is the independent variable and a i -s are the coefficients or parameters of the polynomial. The weighted least-squares equation that determines the coefficients is

$$ \chi^{2} = \sum\limits_{k = 1}^{m} {\frac{{\left( {y_{k} - \sum\nolimits_{i = 0}^{n} {a_{i} x_{k}^{i} } } \right)^{2} }}{{\sigma_{k}^{2} }}} , $$
(3)

where y k is the k th measured value with σ k uncertainty. It is generally required that the number of data points are more than the number of parameters i.e. m > n + 1. The solution is obtained from minimization of Eq. (3)

$$ \frac{{\partial \chi^{2} }}{{\partial a_{j} }} = \frac{\partial }{{\partial a_{j} }}\sum\limits_{k = 1}^{m} {\frac{{\left( {y_{k} - \sum\nolimits_{i = 0}^{n} {a_{i} x_{k}^{i} } } \right)^{2} }}{{\sigma_{k}^{2} }}} = 0. $$
(4)

The result of the calculation is

$$ \frac{{\partial \chi^{2} }}{{\partial a_{j} }} = - 2\sum\limits_{k = 1}^{m} {\frac{{\left( {y_{k} - \sum\nolimits_{i = 0}^{n} {a_{i} x_{k}^{i} } } \right)x_{k}^{j} }}{{\sigma_{k}^{2} }}} = - 2\underbrace {{\sum\limits_{k = 1}^{m} {\frac{{y_{k} x_{k}^{j} }}{{\sigma_{k}^{2} }}} }}_{{B_{j} }} \,+\, 2\sum\limits_{i = 0}^{n} {a_{i} \underbrace {{\sum\limits_{k = 1}^{m} {\frac{{x_{k}^{i} x_{k}^{j} }}{{\sigma_{k}^{2} }}} }}_{{V_{i,j} }}}\, = 0. $$
(5)

In vector form this can be written as

$$ \underline{\underline{V}} \cdot \underline{a} = \underline{B} . $$
(6)

The solution can be obtained by inversion of the matrix \( \underline{\underline{V}} \cdot \,\,\,\underline{a} = \underline{\underline{V}}^{ - 1} \cdot \underline{B} . \)

Before we derive the uncertainty of the polynomial, let us introduce handy notations (following Jánossy’s formalism [13]) for the calculation of the mean value of a statistical variable z having a distribution density f(z). We will note the mean or expected value of z with the bracket 〈 〉 operator

$$ \left\langle z \right\rangle = \int\limits_{ - \infty }^{\infty } {zf(z){\text{d}}z.} $$
(7)

We also introduce the operator δ as the difference of a statistical variable from its mean value [13].

$$ \delta \,z = z - \left\langle z \right\rangle = z - \int\limits_{ - \infty }^{\infty } {z\;f(z){\text{d}}z} . $$
(8)

Using these notations the mean square error \( \sigma_{z}^{2} \)of variable z is

$$ \sigma_{z}^{2} = \left\langle {(\delta \,z)^{2} } \right\rangle = \left\langle {(z - \left\langle z \right\rangle )(z - \left\langle z \right\rangle )} \right\rangle = \left\langle {z^{2} } \right\rangle - \left\langle z \right\rangle^{2} $$
(9)

Now we can calculate the variance–covariance for the elements of vector a using the solution of Eq. (6)

$$ \left\langle {\delta a_{i} \delta a_{k} } \right\rangle = \left\langle {\delta \sum\limits_{t} {V_{i,t}^{ - 1} B_{t} } \delta \sum\limits_{q} {V_{k,q}^{ - 1} B_{q} } } \right\rangle $$
(10)

Replacing the expression of B from Eq. (5) in Eq. (10) and assuming \( \delta x_{i} \equiv 0 \) or negligible we can write

$$ \left\langle {\delta a_{i} \delta a_{k} } \right\rangle = \left\langle {\sum\limits_{t} {V_{i,t}^{ - 1} \delta \sum\limits_{s = 1}^{m} {\frac{{y_{s} x_{s}^{t} }}{{\sigma_{s}^{2} }}} } \sum\limits_{q} {V_{k,q}^{ - 1} \delta \sum\limits_{h = 1}^{m} {\frac{{y_{h} x_{h}^{q} }}{{\sigma_{h}^{2} }}} } } \right\rangle $$
(11)

After rearrangement Eq. (11) can be written as

$$ \left\langle {\delta a_{i} \delta a_{k} } \right\rangle = \left\langle {\sum\limits_{t,q} {V_{i,t}^{ - 1} } V_{k,q}^{ - 1} \sum\limits_{s,h} {\frac{{\delta y_{s} x_{s}^{t} }}{{\sigma_{s}^{2} }}\frac{{\delta y_{h} x_{h}^{q} }}{{\sigma_{h}^{2} }}} } \right\rangle = \sum\limits_{t,q} {V_{i,t}^{ - 1} } V_{k,q}^{ - 1} \sum\limits_{s,h} {\frac{{x_{s}^{t} }}{{\sigma_{s}^{2} }}\frac{{x_{h}^{q} }}{{\sigma_{h}^{2} }}} \left\langle {\delta y_{s} \delta y_{h} } \right\rangle $$
(12)

Since the measured values are independent, the expression \( \left\langle {\delta y_{s} \delta y_{h} } \right\rangle \) is diagonal. If we perform the sum for index h then

$$ \left\langle {\delta a_{i} \delta a_{k} } \right\rangle = \sum\limits_{t,q} {V_{i,t}^{ - 1} } V_{k,q}^{ - 1} \sum\limits_{s} {\frac{{x_{s}^{t} }}{{\sigma_{s}^{2} }}\frac{{x_{s}^{q} }}{{\sigma_{s}^{2} }}} \sigma_{s}^{2} $$
(13)

The sum for index s is just matrix V t,q . By summing for index t we obtain

$$ \left\langle {\delta a_{i} \delta a_{k} } \right\rangle = \sum\limits_{q} {\delta_{i,q} V_{k,q}^{ - 1} } $$
(14)

where δ i,q is the Kronecker delta or the unity matrix. The final result, after performing the sum for index q is

$$ \left\langle {\delta a_{i} \delta a_{k} } \right\rangle = V_{i,k}^{ - 1} $$
(15)

From Eq. (15) we can see that inverse matrix \( \underline{\underline{V}}^{ - 1} \) is the variance–covariance matrix of the problem [14]. Its diagonal elements are the square of the uncertainties of the elements of the parameter vector a, which can be noted for each a i as \( \sigma_{ai}^{2} \).

The mean square error or uncertainty of the polynomial P(x) can be derived the same way

$$ \delta P(x) = \delta \left( {\sum\limits_{i = 0}^{n} {a_{i} x^{i} } } \right) = \sum\limits_{i = 1}^{n} {\delta a_{i} x^{i} } = \sum\limits_{i = 1}^{n} {\left( {a_{i} - \left\langle {a_{i} } \right\rangle } \right)x^{i} } . $$
(16)

The average of mean square of δP is

$$ \left\langle {\delta P(x)^{2} } \right\rangle = \left\langle {\delta \left( {\sum\limits_{i = 0}^{n} {a_{i} x^{i} } } \right)\delta \left( {\sum\limits_{k = 0}^{n} {a_{k} x^{k} } } \right)} \right\rangle = \sum\limits_{i,k} {\left\langle {\delta a_{i} \delta a_{k} } \right\rangle } x^{i} x^{k} . $$
(17)

If we substitute Eq. (15) in Eq. (17) we obtain

$$ \sigma_{P(x)}^{2} = \left\langle {\delta P(x)^{2} } \right\rangle = \sum\limits_{i,k} {V_{i,k}^{ - 1} x^{i} x^{k} } $$
(18)

In vector and matrix notation Eq. (18) is

$$ \sigma_{P(x)}^{2} = \underline{x} \underline{\underline{V}}^{ - 1} \underline{x} $$
(19)

where x = (1, x, x 2,…, x n). Finally, we have arrived at the general expression for the mean square error of a polynomial obtained from a fit to a set of data.

Here I note that substituting any linearly independent indexed functions of x in the place of the powers of x, we obtain a very general linear form for which the Eqs. (2–6 and 10–19) are the same. To be more specific one can apply this to an angular distribution fit by replacing the x i with the Legendre-polynomials LP i (x) (i.e. \( x^{i} \mapsto LP_{i} (x) \)).

Calculations the uncertainty of functions of relative efficiency

Returning to our special case, we have to replace x and y variables with the corresponding quantities of our efficiency fit; for an efficiency fit on a log–log scale x = ln(E), y = ln(ε). The uncertainty of ln(ε) is

$$ \sigma_{\ln (\varepsilon )} = \sqrt {\left\langle {\left( {\delta y} \right)^{2} } \right\rangle } ; $$
(20a)
$$ \delta y \approx \frac{{{\text{d}}y}}{{{\text{d}}\varepsilon }}\delta \varepsilon = \frac{\delta \varepsilon }{\varepsilon }; $$
(20b)
$$ \sigma_{\varepsilon } = \varepsilon \sigma_{\ln (\varepsilon )} = \varepsilon \sqrt {\underline{x} \underline{\underline{V}}^{ - 1} \underline{x} } . $$
(20c)

We can now calculate the uncertainty of a function F of ln(ε)

$$ \delta F(\ln (\varepsilon )) \approx \frac{{{\text{d}}F(\ln (\varepsilon ))}}{{{\text{d}}\ln (\varepsilon )}}\delta \ln (\varepsilon ) = \frac{{{\text{d}}F(\ln (\varepsilon ))}}{{{\text{d}}\ln (\varepsilon )}}\sigma_{\ln (\varepsilon )} . $$
(21)

A commonly occurring question is the uncertainty of the ratio of efficiencies at different energies, where using the \( \varepsilon = \exp (\ln (\varepsilon )) \) identity,

$$ \delta \left( {\frac{{\varepsilon (E_{1} )}}{{\varepsilon (E_{2} )}}} \right) = \frac{{\partial \exp (\ln (\varepsilon (E_{1} )))}}{{\varepsilon (E_{2} )\partial \ln (\varepsilon (E_{1} ))}}\frac{{\delta \varepsilon (E_{1} )}}{{\varepsilon (E_{1} )}} + \frac{{\partial \exp^{ - 1} (\ln (\varepsilon (E_{2} )))\varepsilon (E_{1} )}}{{\partial \ln (\varepsilon (E_{2} ))}}\frac{{\delta \varepsilon (E_{2} )}}{{\varepsilon (E_{2} )}}, $$
(22)

which after simplification becomes

$$ \delta \left( {\frac{{\varepsilon (E_{1} )}}{{\varepsilon (E_{2} )}}} \right) = \frac{{\delta \varepsilon (E_{1} )}}{{\varepsilon (E_{2} )}} - \frac{{\delta \varepsilon (E_{2} )\varepsilon (E_{1} )}}{{\varepsilon (E_{2} )^{2} }} = \frac{1}{{\varepsilon (E_{2} )^{2} }}\left( {\delta \varepsilon (E_{1} )\varepsilon (E_{2} ) - \delta \varepsilon (E_{2} )\varepsilon (E_{1} )} \right). $$
(23)

By calculating the expected value of the square of Eq. (23) we can derive the uncertainty

$$ \begin{gathered} \left\langle {\left[ {\delta \left( {\frac{{\varepsilon (E_{1} )}}{{\varepsilon (E_{2} )}}} \right)} \right]^{2} } \right\rangle = \frac{{\left( {\left\langle {\left[ {\delta \varepsilon (E_{1} )} \right]^{2} } \right\rangle \varepsilon (E_{2} )^{2} - 2\left\langle {\delta \varepsilon (E_{1} )\delta \varepsilon (E_{2} )} \right\rangle \varepsilon (E_{1} )\varepsilon (E_{2} ) + \left\langle {\left[ {\delta \varepsilon (E_{2} )} \right]^{2} } \right\rangle \varepsilon (E_{1} )^{2} } \right)}}{{\varepsilon (E_{2} )^{4} }}. \hfill \\ \hfill \\ \end{gathered} $$
(24)

In Eq. (24) we have to calculate the correlation term. It can be obtained in polynomial form using Eqs. (20a ) and (16) as

$$ \left\langle {\delta \varepsilon (E_{1} )\delta \varepsilon (E_{2} )} \right\rangle = \left\langle {\delta \ln (\varepsilon (E_{1} ))\delta \ln (\varepsilon (E_{2} ))} \right\rangle \varepsilon (E_{1} )\varepsilon (E_{2} ) = \left\langle {\sum\limits_{i = 1}^{n} {\delta a_{i} x_{1}^{i} } \sum\limits_{k = 1}^{n} {\delta a_{k} x_{2}^{k} } } \right\rangle \varepsilon (E_{1} )\varepsilon (E_{2} ). $$
(25)

After rearrangement

$$ \left\langle {\delta \varepsilon (E_{1} )\delta \varepsilon (E_{2} )} \right\rangle = \varepsilon (E_{1} )\varepsilon (E_{2} )\sum\limits_{i,k}^{n} {\left\langle {\delta a_{i} \delta a_{k} } \right\rangle x_{1}^{i} } x_{2}^{k} . $$
(26)

Using Eq. (15) to replace the bra-ket expression we have

$$ \left\langle {\delta \varepsilon (E_{1} )\delta \varepsilon (E_{2} )} \right\rangle = \varepsilon (E_{1} )\varepsilon (E_{2} )\underline{x}_{1} \underline{\underline{V}}^{ - 1} \underline{x}_{2} . $$
(27)

Substituting this result in Eq. (24) and using Eq. (20c) we obtain

$$ \sigma_{{\frac{{\varepsilon (E_{1} )}}{{\varepsilon (E_{2} )}}}}^{2} = \left( {\sigma_{{\ln (\varepsilon_{1} )}}^{2} - 2\underline{x}_{1} \underline{\underline{V}}^{ - 1} \underline{x}_{2} + \sigma_{{\ln (\varepsilon_{2} )}}^{2} } \right)\frac{{\varepsilon (E_{1} )^{2} }}{{\varepsilon (E_{2} )^{2} }}, $$
(28)

where x = (1, ln(E), ln(E)2, …, ln(E)n), the indices 1, 2 refer to the energies, and \( \underline{\underline{V}} \) is the matrix of the log–log fit. The nominator is fully symmetric in E 1 and E 2. If we set E 1 = E 2 then the uncertainty of the efficiency ratio becomes zero as one would expect it. We have to note that the result of Eq. 28 is not simply

$$ \left\langle {\delta \varepsilon (E_{1} )\delta \varepsilon (E_{2} )} \right\rangle \ne \varepsilon (E_{1} )\varepsilon (E_{2} )\sigma_{{\ln (\varepsilon_{1} )}} \sigma_{{\ln (\varepsilon_{2} )}} , $$
(29)

although it would also yield zero if E 1 = E 2, but for E 1 > E 2 it would yield a negative value for Eq. (24). Setting the correlation value in Eq. (28) to zero we obtain the uncorrelated calculation, which overestimates the uncertainty near the E 1 = E 2 values.

To show this we present a complete calculation for an efficiency fit and the calculation of the efficiency ratio function for different energies of E 1 and fixed E 2 = 500 keV value, showing both the correlated and uncorrelated uncertainty calculations in Fig. 1, and the same for E 2 = 5,000 keV in Fig. 2.

Fig. 1
figure 1

Calculation of the efficiency ratio function at different energies of E 1 and fixed a fix value of E 2 = 500 keV. The efficiency ratio is marked with squares (lilac in color), the relative uncertainty with correlation with diamonds (blue in color), and the uncorrelated uncertainty with triangles (red in color). (Color figure online)

Full size image
Fig. 2
figure 2

Calculation of the efficiency ratio function at different energies of E 1 and fixed a fix value of E 2 = 5,000 keV. The efficiency ratio is marked with squares (lilac in color), the relative uncertainty with correlation with diamonds (blue in color), and the uncorrelated uncertainty with triangles (red in color). (Color figure online)

Full size image

For the wide energy-range efficiency calculation, common for PGAA work [5] we used 60Co, 133Ba, 152Eu, 207Bi and 226Ra calibration sources and for the high energies the 14N(n, γ)15N in-beam source. The calculation was made by the Hypermet-PC efficiency routine as described in Ref. [5]. As we can see from the figures, the relative uncertainty of the ratio drops to a very low value near the fixed energy point and becomes exactly zero at the fixed point. This seems to be more dramatic for the 5,000 keV case. It is thus important the selection of the comparisons energy to minimize the overall uncertainty for the energy range of interest.

We note that this correlation appears only if the same efficiency function is used in the comparison. The current result in Eq. (28) is the correct representation of the correlation which was explicitly not shown in Ref. [12]. As I have already noted vector x in Eq. (28) can be a set of any linearly independent polynomials such as a set of orthogonal polynomials that was discussed in Refs. [5, 12]. In the case of efficiency functions arising from two different experiments, they are uncorrelated and then the correlation term in Eq. (28) should be set to zero. This and other uncertainty considerations for PGAA can be found in Ref. [15].

Another important case is needed for the true coincidence summing of cascading γ-rays. In this case the observed sum-peak area is proportional to the product of efficiencies at the two different energies. Thus, let us see what the uncertainty is in this case (again using a log–log fit)

$$ \delta \left( {\varepsilon (E_{1} )\varepsilon (E_{2} )} \right) = \left( {\delta \varepsilon (E_{1} )\varepsilon (E_{2} ) + \delta \varepsilon (E_{2} )\varepsilon (E_{1} )} \right). $$
(30)

The average of the square of Eq. (30), using Eqs. (27), (20a), (20b) and (20c), is

$$ \left\langle {\delta \left( {\varepsilon (E_{1} )\varepsilon (E_{2} )} \right)^{2} } \right\rangle = \left( {\sigma_{{\ln (\varepsilon_{1} )}}^{2} + 2\underline{x}_{1} \underline{\underline{V}}^{ - 1} \underline{x}_{2} + \sigma_{{\ln (\varepsilon_{2} )}}^{2} } \right)\varepsilon (E_{1} )^{2} \varepsilon (E_{2} )^{2} . $$
(31)

It clearly differs in the sign of the correlation term from Eq. (28), therefore at E 1 = E 2 instead of a minimum, Eq. (31) will have a maximum as is shown in Figs. 3 and 4.

Fig. 3
figure 3

Calculation of the efficiency product function at different energies of E 1 and a fixed value of E 2 = 500 keV. The efficiency product is marked with square (lilac in color), the relative uncertainty with correlation with diamonds (blue in color) and its uncorrelated uncertainty with triangle (red in color). (Color figure online)

Full size image
Fig. 4
figure 4

Calculation of the efficiency product function at different energies of E 1 and a fixed value of E 2 = 5,000 keV. The efficiency product is marked with square (lilac in color), the relative uncertainty with correlation with diamonds (blue in color) and its uncorrelated uncertainty with triangle (red in color). (Color figure online)

Full size image

Studying the figures we can conclude that a lower energy fixed point gives a smaller uncertainty in the case of efficiency products.

Comparator experiments and true coincidence summing corrections

In our laboratory the PGAA with internal comparator method is quite often used [15, 16]. In this method the sample nuclei are excited by a beam of cold neutrons. The radiative neutron capture or (n, γ) process excites the sample nuclei up to their neutron binding energy which is usually released with cascading, so called prompt γ-radiation in a very short time. The γ-radiation is observed with a Compton suppressed HPGe detector spectrometer and the digitized energy spectrum is recorded in a computer [17, 18]. The energy of a peak in the spectrum is characteristic for the emitting nucleus, while its area A X,γ is proportional to the amount of that nucleus in the sample. For a thin homogeneous sample, the following equation can be used to calculate the peak area

$$ A_{X,\gamma } = n_{X} \sigma_{X,\gamma } \varphi \;\varepsilon (E_{X,\gamma } )K(E_{\gamma } )t $$
(32)

where n X is the number of atoms of element X, σ X,γ is the partial γ-ray cross section, φ is the neutron flux, ε is the FEP efficiency, K is the correction factor for dead-time and absorptions of neutrons and γ-rays in the sample and t is the measuring time. The correction K(E γ ) can be factorized for its components. The symbol σ X,γ is a composite nuclear constant \( \sigma_{X,\gamma } = \sigma_{{X,{\text{th}}}} P_{X,\gamma } \theta_{X} \), where σ X,th is the thermal capture cross section, P X,γ is the γ-ray emission probability and θ X is the isotopic abundance.

A similar equation can be written for any other element in the sample, which we will call the comparator and denote with index C. Calculating the area ratio for element X and C we obtain

$$ \frac{{A_{X,\gamma } }}{{A_{C,\gamma } }} = \frac{{n_{X} \sigma_{X,\gamma } \varepsilon (E_{X,\gamma } )f(E_{X,\gamma } )}}{{n_{C} \sigma_{C,\gamma } \varepsilon (E_{C,\gamma } )f(E_{C,\gamma } )}}. $$
(33)

As we can see the flux, the time and the corrections except the one for the γ-attenuation f(E γ ) are cancelled out.

By rearrangement of Eq. (33) the elemental ratios can be calculated provided that the partial γ-ray cross sections are known.

$$ \frac{{n_{X} }}{{n_{C} }} = \frac{{A_{X,\gamma } \sigma_{C,\gamma } \varepsilon (E_{C,\gamma } )f(E_{C,\gamma } )}}{{A_{C,\gamma } \sigma_{X,\gamma } \varepsilon (E_{X,\gamma } )f(E_{X,\gamma } )}}. $$
(34)

Of course the efficiency ratios must also be measured with a well calibrated spectrometer and the correlation of efficiency values at different energies play a major role in the uncertainty calculation of the concentration ratio. This is the basic equation for nondestructive PGAA analysis of homogeneous samples to determine the atomic number concentration ratios in the samples or mass ratios if Eq. (34) is multiplied by the elemental mass ratios as it was shown in Ref. [15]. The applications of this formula are summarized in Refs. [19, 20].

In routine PGAA analysis, the direct application of formula (34) is rare, because PGAA is multi elemental method and capable for analysis of major and minor components and in some cases trace elements [15]. In addition, more than one peak of an element is used in the analysis of masses relative to the observed total mass or atomic number ratios relative to the observed total number of atoms. For some of the elements like oxygen the PGAA is not sensitive. In this case the masses of oxides are calculated, which can be done for geological samples with good accuracy [15].

An example for the usage of Eq. (34) can be given for a chemistry problem in which the question is the stoichiometry of a compound. In case of iron the oxidation state can be 2 or 3 which causes many problems if the compound is not stable. To improve the reliability of library data (the partial γ-ray cross sections), we decided to perform independent experiments on iron compounds that have not been measured earlier. We found that out of five compounds only the Fe2(SO4)3 had reliable stoichiometric composition by checking the atomic number ratios of iron and the comparator of sulfur, which in this case should be 2/3. In order to have the most precise value the Eq. (34) was used and compared to the uncorrelated efficiecy uncertainty calculation and to the full analysis (where many γ-rays were taken into the account for each element in the analysis) that is used in the PGAA.

The correlation of efficiency significantly lowered the uncertainty of the atomic number ratios for most intense peaks of Fe (352 keV) and S (841 keV). The result is presented with the graph of efficiency ratio function and its correlated and uncorrelated uncertainties in Fig. 5 and in Table 1 with the result of the three methods mentioned above.

Fig. 5
figure 5

The efficiency ratio \( \left( {\frac{\varepsilon (841)}{\varepsilon (352)}} \right) \) function (continuous blue line, left axis), the correlated uncertainty (long dashed red line, right axis), and the uncorrelated uncertainty (short dashed black line, right axis). Thick vertical line shows the place where the function values to be read. It is clearly visible that the correlated uncertainty is much smaller than the uncorrelated, despite the fact that at the pivot point no strong iron γ-ray could be found for the comparison. (Color figure online)

Full size image
Table 1 Comparison of stoichiometry and its uncertainty of iron and sulfur in ferric sulfate using three different calculation methods for the ratio and its uncertainty
Full size table

To make the result clearer we summarize the atomic number ratios in Table 1 with their uncertainties.

There is a clear agreement between the experiment and the theory with all of the evaluation methods, but Eq. (34) using correlated uncertainty provides the most strengthened result.

Another rearrangement of Eq. (33) will provide the elemental partial γ-ray cross sections if the elemental ratios are known

$$ \sigma_{X,\gamma } = \sigma_{C,\gamma } \frac{{n_{C} A_{X,\gamma } \varepsilon (E_{C,\gamma } )f(E_{C,\gamma } )}}{{n_{X} A_{C,\gamma } \varepsilon (E_{X,\gamma } )f(E_{X,\gamma } )}}. $$
(35)

The highest precision for the elemental partial γ-ray cross section can be obtained using stoichiometric compounds [21, 22], and were so used to determine libraries of such cross sections for analytical analysis in our laboratory [23, 24]. Application of the correlation of the efficiency ratio will lower the uncertainty to the value that can be calculated from the counting statistics as it was presented in the determination of the partial γ-ray cross section of the 1,885 keV γ ray of nitrogen in comparison to the primary standard of hydrogen in Ref. [12].

Instead of using elemental partial γ-ray cross sections, we can use Eq. (35) for internal calibration of unknown partial cross sections in enriched isotopic samples as well, if the cross section of any single comparator γ-ray is known. In this case n C and n X are identical and by definition θ is 1 [25]. Using the definition of partial γ-ray cross section, the total thermal neutron capture cross section σ th can be calculated if the emission probability P γ is known. The many ways for the determination of σ th based on Eq. (35) is summarized in Ref. [26]. All of these measurements involve the ratio of detector efficiencies.

Finally, we present the effect of efficiency correlations on the uncertainty budget for true coincidence summing. This phenomenon occurs due to the finite time resolution of detectors. When the decay constants of cascading γ-rays that hit the detector are shorter then the resolution time of the detector the deposited energies are summed [27]. The observed area of the summed peak can lie on top of the FEP of the crossover γ-ray, necessitating an additive correction, but also represents a loss from the FEPs of the summing γ-rays, which then need subtractive corrections. It is proportional to the product of the FEP efficiencies of the involved γ-rays; furthermore, it depends on the angular correlation of the γ-rays.

$$ A_{\gamma 1,\gamma 2} = n\sigma_{\gamma 1} P_{\gamma 2} \varphi \;\varepsilon (E_{\gamma 1} )\varepsilon (E_{\gamma 2} )K_{1,2} W(\theta )t $$
(36)

In Eq. (36) K 1,2 is not simply the product of K 1 and K 2 because the two γ-rays arriving practically at the same time, thus they behave with respect to the dead time like one γ-ray. Their absorption in the target must be calculated as the product of absorptions and finally the neutron absorption has to be counted only once. The angular correlation can be approximated with θ = 0, but more rigorously it is to be integrated for the detector solid angle around θ = 0. When calculating the uncertainty of true coincidence summing from formula (36), Eq. (31) should be used instead of the uncorrelated uncertainty calculation of efficiency product.

Equation (36) is not used in the PGAA practice. In fact, we try to avoid the close detection distance where it should be applied by setting the detector to sample distance sufficiently large. However, it must be applied in true coincidence correction of γ-ray intensities in close geometry decay studies. For this case Eq. (36) is

$$ A_{\gamma 1,\gamma 2} = AP_{\gamma 1} P_{\gamma 2} \;\varepsilon (E_{\gamma 1} )\varepsilon (E_{\gamma 2} )K_{1,2} W(\theta )t $$
(37)

Summary

Formalism was introduced for the derivation of the uncertainty of fitting polynomials. The results were shown to be generally applicable to any linear form, for example, in the fitting of angular distributions. The results were further generalized to calculate uncertainties of general functions defined on a log–log scale. This generalized uncertainty calculation formalism was applied to the ratio and the product of γ-ray detector efficiencies at different energies. The correlation strongly decreases the uncertainty of efficiency ratios at nearby energy values while increasing the uncertainty for their product.

Applications of the results were presented for comparator experiments.