**2. Efficiency transfer in gamma-ray spectrometry**

Because more and more nuclear installations reach the end of their life, the dismantling and decommissioning processes of them became a key topic in the nuclear industry. The radio‐ logical characterization of the systems, structures, equipment, components and the environ‐ ment represents a basic phase in the decommissioning process because allow the definition of the decommissioning strategy. This task is very important because it provides the basis for the correct classification of various types of waste, which in turn affects the decommissioning solution and the associated costs. The measurement method should be reliable and efficient. In addition, it should be flexible, able to provide proper results for the diversity of samples assessed with different compositions and densities, different shapes and possibly non-uniform activity distribution. The appropriate efficiency calibration in the deprived conditions is a challenging task.

missioning" is defined in the TRS 267 [1] as actions taken at the end of a facility useful life to retire the facility from service in a manner that provides adequate protection for the health and safety of the workers, public and environment. It is a complex process because it involves many operations such as detailed survey, decontamination and dismantling of power plant equip‐ ment and facilities, buildings and structure demolition, and managing the resulting waste and other materials that need to be taken into consideration, due to their effects on health and safety of the operating personnel, public and the environment. The decommissioning activities have expanded in the last years all over the world because many nuclear installations have been exhausting their lifetime. Careful planning and management are essential to ensure that decommissioning is fulfilled in a safe and cost-effective manner. A right evaluation of the radioactivity is very important affecting directly the starting point of the decommissioning

The characterization of the radioactive inventory in decommissioning wastes is described in TRS 267 [1] as a front-end task required to define the operational decommissioning plan and estimate costs and radiological risks associated with the plan. Once the decommissioning process is under way, regulatory, safety and waste disposal considerations require that the radioactive waste should be monitored and characterized. The objective of this characteriza‐ tion is to ensure that the waste will be handled and disposed of in a safe and economic manner. The methods and equipment used to characterize the radioactive waste resulted from decom‐ missioning vary considerably, depending upon the type and complexity of the facility and the

The work detailed in this chapter explores the specific gamma-ray spectrometry phenomena in different work conditions, relating the analysis, development and implementation in the radioactive waste management of specific investigation methods for gamma-ray spectrometry measurements that will produce reasonable measurement uncertainties [2] with lower cost and relatively short duration of data acquisition. The applicability and functionality of gammaray spectrometry methods to radiological characterization and free release of radioactive waste materials are presented, using experimental methods that are mostly combined with theoret‐

Because more and more nuclear installations reach the end of their life, the dismantling and decommissioning processes of them became a key topic in the nuclear industry. The radio‐ logical characterization of the systems, structures, equipment, components and the environ‐ ment represents a basic phase in the decommissioning process because allow the definition of the decommissioning strategy. This task is very important because it provides the basis for the correct classification of various types of waste, which in turn affects the decommissioning solution and the associated costs. The measurement method should be reliable and efficient. In addition, it should be flexible, able to provide proper results for the diversity of samples assessed with different compositions and densities, different shapes and possibly non-uniform

process. This can be the reason for unwanted delays between stages.

ical and simulation procedures using Monte Carlo computer codes.

**2. Efficiency transfer in gamma-ray spectrometry**

radionuclide mix from the plant.

116 Nuclear Material Performance

The experimental calibration of germanium detectors used in gamma-ray spectrometry [3] is difficult to achieve particularly for the geometry measurement that cannot be estimated as a point source. Therefore, when different samples with varies composition and densities are measured in various geometries, a large number of standards need to be measured to carry out the detection efficiency according to a specific sample matrix and geometry. The situation is more difficult in the case when the samples are measured with high efficiency detectors and the sample is placed close to detector, because in this case most radionuclides will give rise to important coincidence summing effects [4]. Consequently, the detection efficiency for a given energy depends not only on energy and experimental setup, but also on the radionuclide. The knowledge of the detection efficiency, which varies strongly with the source to detector distance, due to the geometry and absorption factors, is essential for operating these systems. Therefore, a comprehensive experimental calibration would require the measurement of a big number of standards, one for each geometry and matrix of interest, containing certified activities for each radionuclide that is present in the real samples. A better solution for determining the detection efficiency is the application of specific methods of calculation. In the gamma-ray spectrometry field, except the simulations performed for the calculation of the detection efficiencies, Monte Carlo simulation codes can also be used to evaluate the transfer factors [5]. Based on its relative sensitivity to the uncertainties of the detector parameters and of the calculation model, the method of efficiency transfer [6, 7], based on Monte Carlo simulation or on semi-empirical methods [6, 8, 9], is more and more relevant to evaluate the efficiency whenever direct experimental calibration is not accessible [10].

The applicability of the ETNA (Efficiency Transfer for Nuclide Activity) software to compute the efficiency transfer factors for various counting geometries used in routine laboratory measurements was examined. Thus, ETNA results were compared with experimental results (corrected for coincidence summing effects) [10, 11]. The detection efficiencies were calculated for NaI(Tl) and HPGe detectors. An approach using the experimental efficiency measured with point sources combined with theoretical procedures was applied for obtaining the peak efficiency *ε*(*E*) for disk sources measured with NaI(Tl) detector. Using the detection efficiencies for a reference point source geometry located at 10 cm distance from the high purity germa‐ nium (HPGe) detector, the applicability of the efficiency transfer method was checked once more.

#### **2.1. The efficiency calibration of the detectors using experimental measurements**

For the efficiency evaluation of the disk sources, the measurements were made with an Ortec gamma-ray spectrometry system consisting of a ScintiPack Photomultiplier Base with Preamplifier and High Voltage Supply type 296 and a DigiDART Digital Portable Multichannel Analyzer and lead collimator. The NaI(Tl) detector specifications are as follows: the diameter of the end cap of the detector is 3 × 3 inches, the crystal diameter is 8 cm, and the energy resolution is 70.62 keV at 1332 keV (60Co). The recommended operating bias is +1000 V.

The second Ortec gamma-ray spectrometry system used in this study for the examination of the applicability of the efficiency transfer method consisted of a high purity germanium detector, model GMX50P4, transplantable in Pop Top technology, with dimensions: 6.46 cm diameter, 7.5 cm length, 0.05 cm beryllium absorber layer and a Digital Portable Multichannel Analyzer type DigiDART. The main performance specifications of the HPGe warranted by the producer are presented in **Table 1**.


**Table 1.** HPGe performance specifications.

In the first step, certified standard point sources were used to evaluate the detector experi‐ mental efficiencies as a function of gamma-ray energies [3], for the NaI(Tl) detector.

The sources were measured in horizontal plane, at radial distances *r* = 0, 1, 2, 3 and 4 cm from the detector axis and at 0.8 cm from the face of the NaI(Tl) detector. A lead collimator was used in the measurements. Five sets of data were obtained for all the important gamma line involved in the study. From the graphic representation (**Figure 1**), it can be seen that the experimental efficiencies *ε*(*E*) for the NaI(Tl) detector do not present a smooth variation with the energy *E*.

In the case of the HPGe detector, the experimental detection efficiencies were evaluated for the detector-point sources distances of 2, 5, 10, 15 and 20 cm. The counting dead time of the measurements was in general controlled to be less than 7% and consequently corrected during the counting. The amplifier time constant was fixed to 12 *μs*. Cylindrical sources (**Table 2**) were also measured at 0, 1, 2 cm from the face of the HPGe detector. The counting dead time for these sources was less than 3%.


**Table 2.** Cylindrical sources.

134Cs and 137Cs radioactive sources were used to test the applicability of ETNA software for volume sources. Water and soil matrix have been chosen because they are most common in gamma-ray spectrometry laboratory.

The directly measured efficiency calibration curves obtained for the HPGe detector are represented in **Figure 1**. Was observed that the experimental efficiencies values *ε*(*E*) for the HPGe detector do not present a smooth variation with the energy, *E*.

**Figure 1.** The experimental values of the detection efficiencies for point sources measured with the NaI(Tl) and HPGe detectors.

The uncertainties (1*σ*) of the experimental efficiencies values were estimated using ISO standard [12] using uncertainties of the activities from the certificates and the uncertainties of the counting results. The values were up to 11% for *r=* 0, 1, 2, 3 and 4 cm distances from the NaI(Tl) symmetry axis and up to 3% for *h* = 2, 5 and 10 cm and up to 8% for *h* = 15 and 20 cm source-to-detector distances in the case of the HPGe detector.

#### **2.2. Coincidence summing corrections**

The second Ortec gamma-ray spectrometry system used in this study for the examination of the applicability of the efficiency transfer method consisted of a high purity germanium detector, model GMX50P4, transplantable in Pop Top technology, with dimensions: 6.46 cm diameter, 7.5 cm length, 0.05 cm beryllium absorber layer and a Digital Portable Multichannel Analyzer type DigiDART. The main performance specifications of the HPGe warranted by the

Resolution (FWHM) at 1.33 MeV, 60Co **2.2 keV** Peak-to-Compton ratio, 60Co 58:1 Relative efficiency at 1.33 MeV, 60Co 50% Peak shape (FWTM/FWHM), 60Co 2.0 Resolution (FWHM) at 5.9 keV, 55Fe 800 eV Recommended operating bias, negative 3300 V

In the first step, certified standard point sources were used to evaluate the detector experi‐

The sources were measured in horizontal plane, at radial distances *r* = 0, 1, 2, 3 and 4 cm from the detector axis and at 0.8 cm from the face of the NaI(Tl) detector. A lead collimator was used in the measurements. Five sets of data were obtained for all the important gamma line involved in the study. From the graphic representation (**Figure 1**), it can be seen that the experimental efficiencies *ε*(*E*) for the NaI(Tl) detector do not present a smooth variation with the energy

In the case of the HPGe detector, the experimental detection efficiencies were evaluated for the detector-point sources distances of 2, 5, 10, 15 and 20 cm. The counting dead time of the measurements was in general controlled to be less than 7% and consequently corrected during the counting. The amplifier time constant was fixed to 12 *μs*. Cylindrical sources (**Table 2**) were also measured at 0, 1, 2 cm from the face of the HPGe detector. The counting dead time for

134Cs and 137Cs radioactive sources were used to test the applicability of ETNA software for volume sources. Water and soil matrix have been chosen because they are most common in

**) Λ (Bq) u (%) (1σ)**

mental efficiencies as a function of gamma-ray energies [3], for the NaI(Tl) detector.

producer are presented in **Table 1**.

118 Nuclear Material Performance

**Table 1.** HPGe performance specifications.

these sources was less than 3%.

gamma-ray spectrometry laboratory.

**Table 2.** Cylindrical sources.

**Radionuclide H (cm) D (cm)** *ρ* **(g/cm3**

134Cs 3.2 7.4 1.0 1916 2.5 137Cs 3.3 7.4 1.4 1190 3.5

*E*.

The origin of the efficiency data deviation from smooth curves as a function of energy is due to the presence of significant coincidence summing effects in the case of 152Eu, 134Cs and 60Co sources [10, 11]. To remove the effects of coincidence summing, specific corrections were evaluated and applied to experimental efficiencies for the purpose to obtain useful efficiency curve. To evaluate the coincidence summing effects, it represents a difficult task, mainly when the nuclides present complex decay schemes. To obtain the correct efficiencies values for the 152Eu energy lines, the peak and total efficiencies are required for the energies of supplementary photons emitted by 152Eu nuclide. For instance, in the case of the peak with energy *E* = 121.78 keV, only coincidence losses are feasible. The process is produced when any photon from the 71 photons list is emitted instantaneously with the photon with *E* = 121.78 keV and together interacts with the detector. These photons span an energy range from χ-ray to *E* = 1647 keV, and therefore, the total efficiency for the energy in this range is needed. For the peak with *E* = 1408.01 keV, 21 moistures of various photons are possible and can contribute to sum peak effects. To evaluate the coincidence summing effects for the *E* = 1408.01 keV, the peak efficiency for all these photons is required.

A method available to correct these effects is the Monte Carlo method. A dedicated software called GESPECOR [6] has been applied, in order to evaluate the coincidence summing corrections. This is user-friendly Monte Carlo software useful for the computation of the efficiency [3], of matrix effects [13] and of coincidence summing corrections [4] in gamma-ray spectrometry with HPGe detectors.

Because GESPECOR is dedicated to germanium detectors, the code cannot be directly applied for NaI(Tl) detector. Therefore, in the case of NaI(Tl) detector, the coincidence summing correction factors have been evaluated using an iterative procedure. Both the decay scheme data evaluated by GESPECOR and the experimental values of the peak and of the total efficiencies for the point source measurements were needed. The procedure followed in the first iteration is represented in **Figure 2**.

**Figure 2.** Procedure for coincidence summing correction factors evaluation

Taking into account that the ratio between the peak and total efficiency is a smooth function of energy, a first estimate of the total efficiency as a function of energy was obtained (**Fig‐ ure 3**). This was possible even if only few directly measured total efficiency data were available.

**Figure 3.** The experimental ratio of total to peak efficiency versus the energy for the NaI(Tl) detector.

#### Dedicated Monte Carlo Procedures Applied in Gamma-ray Spectrometry Used in Decommissioning of Nuclear Facilities http://dx.doi.org/10.5772/62937 121

The obtained values of the peak and total efficiencies were used to calculate the coincidence summing correction factors in the first iteration. The correction factors were subsequently used to obtain improved values of the peak efficiencies for the NaI(Tl) detector (second iteration). The peak and total efficiencies values resulted in the second iteration obtained using the similar procedure followed in the first interation procedure used to evaluate the coincidence summing correction factors in the second iteration. Was observed that it was not necessary to proceed in a higher order iterations. The final values obtained for the peak efficiencies of the NaI(Tl) detector were achieved from the measured values of the peak efficiencies and the coincidence summing correction factors in the second iteration. In **Figure 4** are represented the detection efficiencies for point sources measured in horizontal plane at 0.8 cm from the end face of the detector, corrected for the effects of coincidence summing for NaI(Tl) detector, only for *r* = 0, 2, 4 cm.

A method available to correct these effects is the Monte Carlo method. A dedicated software called GESPECOR [6] has been applied, in order to evaluate the coincidence summing corrections. This is user-friendly Monte Carlo software useful for the computation of the efficiency [3], of matrix effects [13] and of coincidence summing corrections [4] in gamma-ray

Because GESPECOR is dedicated to germanium detectors, the code cannot be directly applied for NaI(Tl) detector. Therefore, in the case of NaI(Tl) detector, the coincidence summing correction factors have been evaluated using an iterative procedure. Both the decay scheme data evaluated by GESPECOR and the experimental values of the peak and of the total efficiencies for the point source measurements were needed. The procedure followed in the

Taking into account that the ratio between the peak and total efficiency is a smooth function of energy, a first estimate of the total efficiency as a function of energy was obtained (**Fig‐ ure 3**). This was possible even if only few directly measured total efficiency data were available.

**Figure 3.** The experimental ratio of total to peak efficiency versus the energy for the NaI(Tl) detector.

spectrometry with HPGe detectors.

120 Nuclear Material Performance

first iteration is represented in **Figure 2**.

**Figure 2.** Procedure for coincidence summing correction factors evaluation

**Figure 4.** The detection efficiency corrected for the effects of coincidence summing for NaI(Tl) detector.

In the case of HPGe detector, the coincidence summing effects are presented in the case of 152Eu, 134Cs and 60Co sources. Specific coincidence summing corrections were applied to the experimental efficiencies in order to obtain a generally useful efficiency curves for the HPGe detector. The values included in the HPGe detector manufacturer's data were used in com‐ putation as input detector data. The computed correction factors were subsequently used to obtain improved values of the peak efficiencies. In the case of cylindrical sources, the correc‐ tions were necessary only for 134Cs gel matrix. The detection efficiency curves (corrected for the effects of coincidence summing) in function of energy for the five source-to-HPGe detector distances obtained for the peak efficiency for point sources are represented in **Figure 5** only for *h* = 2, 10, 20 cm.

**Figure 5.** The detection efficiency corrected for the effects of coincidence summing for HPGe detector.

The efficiencies for the point sources obtained in this way for the reference measurement geometry (10 cm source-to-HPGe detector distance) could be used to evaluate the efficiency for other measurement geometries by the efficiency transfer method.

The uncertainties (1 *σ*) of the corrected efficiencies for point sources were up to 3% for NaI(Tl) detector and up to 3.5% for *h* = 2, 5 and 10 cm and 8.5% for *h* = 15 and 20 cm for HPGe detector. The uncertainties of the corrected efficiencies for cylindrical sources include additional uncertainties of the matrix effects; the resulting values were up to 8% (1*σ*).

#### **2.3. The efficiency transfer**

The gamma-ray spectrometry method is a relative method, necessitating standard radioactive sources for the efficiency calibration. When the standard source and the sample are the same, the next relation is applied for the computation of the activity of the sample:

#### Dedicated Monte Carlo Procedures Applied in Gamma-ray Spectrometry Used in Decommissioning of Nuclear Facilities http://dx.doi.org/10.5772/62937 123

$$
\Lambda\_{\rho} \left( E \right) = \Lambda\_{\epsilon} \left( E \right) \frac{R\_{\rho} \left( E \right)}{R\_{\epsilon} \left( E \right)} \tag{1}
$$

where *Λp*(*E*) and *Rp*(*E*) are the activity and the count rate for the sample, and *Λe*(*E*) and *Re*(*E*) are the activity and the count rate for the standard source corresponding to the peak with energy *E*.

In practice, it is difficult to use standard sources for all samples geometries. Accordingly, for this purpose, the efficiency transfer method can be used. Starting from the experimental efficiency for a reference point source, the efficiency for the sample can be evaluated using a mathematical model or simulation software.

Formally, the efficiency transfer method is based on the next equation [5]:

$$
\varepsilon\_{\text{(calc)}} = T \left( \frac{calc}{r \text{ef}} \right) \varepsilon\_{\text{(ref)}} \tag{2}
$$

where *<sup>T</sup>* ( *calc ref* ) is the transfer factor.

obtain improved values of the peak efficiencies. In the case of cylindrical sources, the correc‐ tions were necessary only for 134Cs gel matrix. The detection efficiency curves (corrected for the effects of coincidence summing) in function of energy for the five source-to-HPGe detector distances obtained for the peak efficiency for point sources are represented in **Figure 5** only

**Figure 5.** The detection efficiency corrected for the effects of coincidence summing for HPGe detector.

for other measurement geometries by the efficiency transfer method.

uncertainties of the matrix effects; the resulting values were up to 8% (1*σ*).

the next relation is applied for the computation of the activity of the sample:

**2.3. The efficiency transfer**

The efficiencies for the point sources obtained in this way for the reference measurement geometry (10 cm source-to-HPGe detector distance) could be used to evaluate the efficiency

The uncertainties (1 *σ*) of the corrected efficiencies for point sources were up to 3% for NaI(Tl) detector and up to 3.5% for *h* = 2, 5 and 10 cm and 8.5% for *h* = 15 and 20 cm for HPGe detector. The uncertainties of the corrected efficiencies for cylindrical sources include additional

The gamma-ray spectrometry method is a relative method, necessitating standard radioactive sources for the efficiency calibration. When the standard source and the sample are the same,

for *h* = 2, 10, 20 cm.

122 Nuclear Material Performance

The transfer factors can be calculated with the Monte Carlo [6] method or with more simplified procedures [14, 15] using the relationship between the simulated efficiency for measurement geometry and the efficiency for the reference geometry. The benefit of this method is that the results are less affected by the uncertainties of detector parameters, which represent a more important uncertainty source in the direct simulation of the efficiencies. Undoubtedly, an improper value will be considered for the detector radius, and this will strongly affect the values of the efficiencies calculated by simulation or evaluation by analytical computation, while the transfer factor is slightly sensitive to this incorrect value, because a wrong value will simultaneously affect both the efficiency calculated for the reference geometry and that for the geometry of interest and their ratio will be practically unchanged. The sensitivity of the efficiency to details of the input data and to the computation model was clearly revealed by Vidmar in 2008 [16]. Clearly, the efficiency transfer method offers better results in the case when the measurements of interest are similar to the reference measurements.

The National Laboratory Henri Becquerel (LNHB) from Saclay, France, developed in the early 2000s a software named ETNA (Efficiency Transfer for Nuclide Activity), dedicated for the calculation of the detector efficiency under measurement conditions different from those of calibration, and for the correction of the coincidence summing effects. The application of the ETNA software makes possible to greatly increase the accuracy of the results of quantitative analysis by gamma-ray spectrometry and avoid time-consuming measurement sequences [15].

The ETNA software was applied for the evaluation of the efficiencies for various geometries based on the efficiencies values for the reference measurement geometry.

#### *2.3.1. Computation of the efficiency for disk sources for the NaI(Tl) detector*

ETNA software was used in this section to achieve the efficiency transfer from a point source geometry to disk sources geometries for the NaI(Tl) detector. Due to lack of calibration certificates for standard disk sources, the efficiency calibration was calculated based on point sources measured with the NaI(Tl) detector.

Assuming symmetry of the cylindrical detector, for the determination of the experimental efficiencies in view of surface contamination measurement, the next relations were used:

$$N = \int\_{\left(\mathcal{S}\right)} \varepsilon\left(r\right) \Lambda\_{\mathcal{S}}\left(r, \varphi\right) d\mathcal{S} \tag{3}$$

$$\mathcal{L} \approx N = \int\_0^R \int\_0^{2\pi} \varepsilon \left( r \right) \Lambda\_S \left( r, \varphi \right) r dr d\varphi \tag{4}$$

$$\approx N = \int\_0^R \varepsilon\left(r\right) r dr \int\_0^{2\pi} \Lambda\_S\left(r, \varphi\right) d\varphi \tag{5}$$

$$\approx N = 2\pi \int\_0^R \varepsilon\left(r\right) \overline{\Lambda\_s(r)} r dr \tag{6}$$

where *N* is the peak count rate; *ε*(*r*) is the efficiency of a point source situated at distance *r* from the symmetry axis of the detector; *Λ<sup>S</sup>* (*r*, *ϕ*) is the surface activity of the source in the point of (*r*, *ϕ*) coordinates; *Λ<sup>S</sup>* ¯ (*r*)= <sup>1</sup> <sup>2</sup>*<sup>π</sup> ∫* 0 2*π Λ<sup>S</sup>* (*r*, *ϕ*)*dϕ*.

Considering that the surface activity is uniformly distributed, result that:

$$N = 2\pi \,\Lambda\_S \int\_0^R \varepsilon(r) r dr \tag{7}$$

$$N = \Lambda\_s I\left(R\right) = \pi R^2 \varepsilon \Lambda\_s \tag{8}$$

$$\text{for } \varepsilon = \frac{1}{\pi R^2} I(R) \tag{9}$$

$$\text{where: } \quad I(R) = 2\pi \int\_0^R \varepsilon(r) r dr \tag{10}$$

Then, the following result is obtained:

#### Dedicated Monte Carlo Procedures Applied in Gamma-ray Spectrometry Used in Decommissioning of Nuclear Facilities http://dx.doi.org/10.5772/62937 125

$$
\varepsilon = \frac{2}{R^2} \int\_0^R \varepsilon(r) r dr \tag{11}
$$

Using Eq. (11), the detection efficiency for the disk sources with radius of *r* = 1, 2, 3 and 4 cm for the NaI(Tl) detector was calculated. The values of the efficiencies *ε*(*r*) were taken from the measured efficiency curves, corrected for coincidence summing effects.

Using the ETNA software, the detection efficiencies for disk sources were calculated for the same type of detector and the same measuring geometry. The reference measurement is represented by a point source located at *r* = 0 cm (on the symmetry axis of the detector); thus, the reference efficiency is *ε*0. In **Figure 6**, the efficiency calibration curves, evaluated with the ETNA software, are represented for the disk sources with radius of *r* = 1, 2, 3 and 4 cm; and a comparison between the results obtained applying Eq. (11) and those evaluated using ETNA software for efficiencies detection for a disk source with *r* = 4 cm.

**Figure 6.** The efficiency calibration curves and ETNA vs. analytical calculation for the NaI(Tl) detector.

The values obtained with the analytical procedure are in accordance with the values resulted using ETNA software. The differences can arrive from the method used or from the input data. Should be mentioned that ETNA software calculates efficiencies for geometries in with the center of the source is placed only on the detector axis; different radial distances than that cannot be included.

#### *2.3.2. The ETNA computation for the HPGe detector*

*2.3.1. Computation of the efficiency for disk sources for the NaI(Tl) detector*

» =

» =

¯ (*r*)= <sup>1</sup> <sup>2</sup>*<sup>π</sup> ∫* 0 2*π*

Then, the following result is obtained:

sources measured with the NaI(Tl) detector.

124 Nuclear Material Performance

of (*r*, *ϕ*) coordinates; *Λ<sup>S</sup>*

ETNA software was used in this section to achieve the efficiency transfer from a point source geometry to disk sources geometries for the NaI(Tl) detector. Due to lack of calibration certificates for standard disk sources, the efficiency calibration was calculated based on point

Assuming symmetry of the cylindrical detector, for the determination of the experimental efficiencies in view of surface contamination measurement, the next relations were used:

> j

> > jj

> > > j j

ò (3)

ò ò (4)

ò ò (5)

ò *<sup>S</sup>* (6)

*r rdr* ò (7)

<sup>=</sup> (9)

ò (10)

(8)

( ) ( ) ( ) Λ , *<sup>S</sup> <sup>S</sup> N r r dS* <sup>=</sup> e

e

e

*Λ<sup>S</sup>* (*r*, *ϕ*)*dϕ*.

<sup>0</sup> 2 () *<sup>R</sup>*

( ) <sup>2</sup> *N IR R* = = Λ Λ *S S* p e

 e

2 <sup>1</sup> or ( ) *I R R*

> *I R r rdr* = p e

p

( ) ( ) <sup>0</sup> were : 2 *<sup>R</sup>*

Considering that the surface activity is uniformly distributed, result that:

*N <sup>S</sup>* = Lp

e

( ) ( ) <sup>2</sup> 0 0 Λ , *<sup>R</sup> N r r rdrd <sup>S</sup>* p

( ) ( ) <sup>2</sup> 0 0 Λ , *<sup>R</sup> N r rdr r d <sup>S</sup>* p

( ) <sup>0</sup> 2 () *<sup>R</sup>* »= L *N r r rdr* p e

where *N* is the peak count rate; *ε*(*r*) is the efficiency of a point source situated at distance *r* from the symmetry axis of the detector; *Λ<sup>S</sup>* (*r*, *ϕ*) is the surface activity of the source in the point

> The transfer of the efficiency from the reference point source geometry, *h* = 10 cm, to other point source geometries (distances from the detector end cap equal to 2, 5, 15 and 20 cm) and the computation of the efficiency for cylindrical samples with different matrices was done using ETNA software for the HPGe detector.

> Using the fitted efficiency data for the reference measurement as input, the description of the reference source, the description of the source for which the efficiencies are required, the

source-detector distance, the detector-absorber distance, ETNA software was applied for the computation of the efficiency for the other measurement geometries for the HPGe detector. The experimental efficiency curves were compared with the prediction of the ETNA software. Excepting the case of the smallest distance from the source to the detector, the discrepancies between ETNA and the experimental results were generally below 3%. In the case of the measurement at 2 cm distance from the detector, the discrepancies were higher being sensitive to the detector geometrical data. This is because the detector specifications established by the manufacturer of the detector were used without any optimization. Furthermore, the uncer‐ tainty of the distance between the crystal and the end cap (the manufacturer value) has a contribution in the uncertainty of the transfer factor, because of the change in the solid angle [10].

The HPGe detector efficiency transfer method has also been used for the efficiencies evaluation for the specific cylindrical sources. For this purpose, the matrix was considered water equiv‐ alent for the matrix with *ρ* = 1.0 g/cm3 and soil composition for the matrix with *ρ* = 1.4 g/cm3 .

The default attenuation coefficients foreseen by ETNA code were used for the matrices involved in the study.

In the case of the soil matrix, containing 137Cs, the ratio between the ETNA software values and the experimental values of the efficiency was 1.038 for *h* = 0 cm and 0.966 for *h* = 2 cm. The higher discrepancies in the case of 134Cs results (gel matrix) in comparison with the results for 137Cs (soil matrix) can be attributed to the uncertainty of coincidence summing effects and of the matrix effects.
