Corresponding author: Nikita S. Rykov ( rykov.nikita@gmail.com ) Academic editor: Georgy Tikhomirov
© 2019 Nikita S. Rykov, Gennady M. Bezhunov, Vyacheslav M. Gorbachev.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Rykov NS, Bezhunov GM, Gorbachev VM (2019) Use of mathematical modeling to extend the scope of application for the procedure of measuring the mass of ^{235}U in solid radioactive waste. Nuclear Energy and Technology 5(2): 163169. https://doi.org/10.3897/nucet.5.36478

The known dependence of absolute efficiency on energy and space for particular measurement conditions is used to determine the mass (activity) of ^{235}U in solid radioactive waste by gammaspectrometric method. The ISOCS system makes it possible to avoid laborious and timeconsuming calibration measurements using standard samples to obtain the absolute efficiency curve due to using the socalled characterized detector having a file with a set of efficiencies for various measurement geometries.
In many cases, the establishment of standard samples with parameters covering the ^{235}U mass measurement range in the variation intervals of influencing factors, including density, nonuniformity, isotopic composition, geometry, etc., is very expensive and, most often, not feasible. With regard for this, a computational and experimental approach is used based on results obtained by Monte Carlo method using the MCNP code with variation of the key influencing parameters in a broad range.
Calculations were performed for detectorrecorded spectra of gamma quanta from casks containing waste differing in the density of the cask content (the density was calculated with regard for the uranium contained in waste) – from 0.016 to 1 g/cm^{3}, in the mass of uranium in waste – from 0.64 g to 2 kg, and in the matrix material – graphite, cellulose, quartz, cellulose with 20 % of iron dust.
Applicability boundaries have been defined for the developed procedure to measure uraniumcontaining waste in terms of the material matrix (~ 2.2 %) and its density (~ 10 %) and the contribution of the uranium mass uncertainty in the cask (5 % for nonporous matrices, 10 % for porous matrices) to the obtained result has been estimated.
Nondestructive analysis of nuclear materials, solid radioactive waste, uranium mass, gammaspectrometry, ISOCS system, absolute efficiency curve, Monte Carlo method, MCNP code, measurement procedure, measurement procedure range
Inspection of casks with uraniumcontaining solid radioactive waste (SRW) has the purpose to determine the mass (activity) of ^{235}U in the casks. A highresolution detection unit and the In Situ Object Counting Systems (
The procedure used to develop the MP (OST 95 103532007, OST 95 102892005, OST 95 105712002) suggests that there is no notable quantity of sizeable solid inclusions (of several mm and more) from a material with a high Z in the SRW matrix and that the requirement of the distribution uniformity of both the SRW density and the ^{235}U mass (activity) in the volume is fulfilled. In reality, inclusions are however present and their impact needs to be assessed. The software part of the ISOCS system includes the Ratio Tester, a subprogram for indirect assessment of this factor. To reduce the effects of the ^{235}U mass (activity) irregular distribution in the SRW matrix on the measurement results, the cask (a 120 liter drum for the purpose hereof) is rotated in the process of measurement.
The known dependence of absolute efficiency on energy and space for particular measurement conditions needs to be used to determine the ^{235}U mass (activity) in SRW by gammaspectrometric method. The use of the ISOCS makes it possible to avoid laborious and timeconsuming calibration measurements based on standard samples to obtain the absolute efficiency curve due to using the socalled characterized detector including a file with a set of efficiencies for various measurement geometries.
In many cases, the establishment of a set of certified objects (CO) with parameters covering the ^{235}U mass measurement range in the variation intervals of influencing factors, including density, nonuniformity, isotopic composition, geometry, etc., is very expensive and, most often, not feasible. For this reason, use of computational methods is evolving worldwide for the experiment modeling (
Singletype casks with model matrices of various densities (from 0.016 g/cm^{3} to 0.708 g/cm^{3}) were used for measurements. For the ^{235}U mass range of 0.25 through 1.8 g (20 through 140 kBq), the measurements were based directly on models with COs. In the SRW ^{235}U mass range of 1.8 to 100 g (140 ÷ 8000 kBq), studies were performed computationally due to lack of representative standard samples.
The ISOCS system used for the studies had the following components:
The measurement geometry implemented by the ISOCS system’s software tools is shown in Fig.
Gamma peaks in a range of 140 to 220 keV, which include the base ^{235}U gamma peaks, are used for the analysis using the Geometry Master program. An intensive and readily detectable gamma peak of 185.7 keV is used for the quantitative analysis of ^{235}U.
Sources from an SSGS set and/or uranium samples are used for the energy calibration of the gammaspectrometer. Following an analysis of the uranium gamma spectrum, the Ratio Tester is used to determine, by analyzing the ^{235}U peaks of 143.8, 163.3, and 185.7 keV, the degree of the ^{235}U distribution homogeneity in the measured SRW cask.
For measurements, the SRW cask is placed on a platform equipped with a rotation device for the cask uniform rotation. The SRW cask weight was measured using a CAS floor scale of the DBII150 type with a weighing range of 10 to 150 kg. The matrix density was determined from the weight measurements with the drum filled up.
A mathematical model of the cask detector system was built for the MCNP code. It was assumed for easier calculations that there were no external radiation diffusers (walls). And the model included both the cask as such, which contained various materials simulating the potential waste matrix, and the detector’s lead collimator similar to that used in measurements.
A crosssection of the selected HPGe detector model is presented in Fig.
The design of the detector in the selected model reflects its characteristics known from the Canberra descriptions. Spectra of the ^{152}Eu source from a set of standard spectrometric gamma sources (SSGS) were measured to determine more accurately the geometrical parameters of the crystal and to confirm the validity of the detector model built. Two series of measurements were conducted:
The experiment conditions were modeled in the MCNP code based on the built detector model. In the event of the ^{152}Eu measurement, the area of the 121.8, 344.3 and 778.9 keV peaks was measured as the ratio of the recorded peak area per one emitted quantum multiplied by the total number of gamma quanta emitted by the source for the measurement time. These peaks were selected due to having a high intensity (over 10 %). The lines of 121.8 and 344.2 keV were chosen since this energy range includes the base gamma lines representative of ^{235}U. The same lines were used to determine the crystal area. The 778.8 keV line was used for the model verification in a more highenergy interval and to determine the crystal thickness.
The calculated and measured spectra from an uranium CO with a mass of 134.6 g and a ^{235}U enrichment of 89.2 % (at a distance of 33 cm from the detector to the sample) for the three base gamma peaks of the ^{235}U isotope with the energies 143.8, 163.3, and 185.7 keV were compared to confirm the accuracy of the model built.
To select the germanium crystal diameter and height, their values were varied in a range of 60 to 65 mm (diameter) and 30 to 35 mm (thickness). Table
Peak areas obtained as the result of the calculations and the experiment
Energy, keV  S_{m}  δS_{m}, %  S_{c}  δS_{c}, %  (1 – S_{c} / S_{m})⸱100 

^{152}Eu, center  
121.8  16155  0.83  16335  0.85  –1.11 
344.2  6458  1.29  6524  1.28  –1.02 
778.8  1422  2.83  1384  2.82  2.67 
^{152}Eum, shift  
121.8  11635  0.985  10789  0.79  7.27 
344.2  4481  1.55  4555  1.23  –1.65 
778.8  1000  3.44  1018  2.82  –1.80 
Standard uranium sample  
143.8  29945  0.63  29850  0.62  0.32 
163.3  16188  0.88  16092  0.88  0.59 
185.7  195800  0.23  197341  0.23  –0.79 
The value δS is taken from the results of processing a spectrum (actually acquired or modeled) by the Gamma Spectra Acquisition and Analysis program of the Genie2000 package (
A variant was used to build the calculation model for spectra from wastecontaining casks with variation of influencing factors which makes it possible to optmize the calculation as to the rate of acquiring the required statistics for the recorded pulses in the spectra obtained.
To expand the calculation statistics and to simulate the cask rotation effect, the detector was assumed to be a ring around the cask (Fig.
The cask was simulated as a cylinder with a 1 mm thick steel wall in which different waste types with a content of different NM quantities were modeled. Both radially and axially, the sample had the form of the real cask used in the experiment.
The number of the selected gamma quanta events was 2⸱10^{7} to 4⸱10^{7}, and the number of recorded pulses in the total absorption peak with the energy 185.7 keV was ~ 10^{5}. The number of the events selected varied depending on the density of the matrix of interest and the uranium mass in the cask. The yields of gamma quanta for ^{235}U were taken from (
Fig.
To make the analysis more convenient, data are presented in absolute units (the number of pulses in the energy channel per 1 g of ^{235}U per second). It can be seen from the available data that the base gamma lines in the energy range of interest coincide in terms of energy and area. Slightly excessive calculated gamma line intensities, as compared with the measured ones, are explained by the radiation selfshielding in uranyl nitrate solutions. After this effect was taken into account with an allowance introduced for the selfshielding in solutions in the measured spectra, the divergence of the 185.7 keV peak recording intensity in the calculation and in the experiment did not exceed 10 % for all considered densities of the caskcontained waste.
Therefore, the program used and the selected model make it possible to model correctly the gamma radiation spectra measurement, using a highresolution detector, on real complex wastesimulating objects for a reasonable computational time.
Detectorrecorded gamma quanta spectra were calculated from a cask containing waste differing in
The influence the variation of the factors above have on the determined intensity of the peak with the energy 185.7 keV and the measurement procedure application area were determined. The results are presented in the diagrams and tables below. The ^{235}U peak intensity (pulse/s/g) was calculated by summing up the pulses in the peak region with the background component deduced. Since the code yields the relative value of pulse/event, a coefficient was introduced that was equal to the yield of quanta of the considered line per 1 g of ^{235}U per second divided by the share of the gamma quanta with the energy 185.7 keV in the initially selected spectrum (depends on the isotopic composition of uranium and tabulated values of the yields of gamma quanta lines for the isotopes taken into account) and by the coefficient that allows for the ratio of the ring detector efficiency in the calculation to the efficiency of the real detector in the experiment. This makes it possible to simplify the analysis of the influence of factors and to compare the calculation with the experiment.
Fig.
It can be seen from Fig.
Count intensity in the 185.7 keV peak with the background deduction for different cellulose densities in the cask (experiment: 0.313 g of ^{235}U, 4.81 g of uranium; calculation: 1 g of ^{235}U, 20 g of uranium)
Waste material density, g/cm^{3}  Count intensity in 185.7 keV peak  

Calculation  Experiment  Difference (%)  
0.016  10.67  9.72  0.95 (9.77) 
0.2  7.219  –  – 
0.471  4.46  4.056  0.404 (9.96) 
0.708  3.279  3.107  0.172 (5.54) 
1  2.511  –  – 
The residual bias (RB) of the calculation and experiment agreement, which is equal to ~ 10 %, needs to be accepted for all density ranges within the calculation range during the MP development.
The matrix of waste may have different elemental compositions containing, among other things, segregated fractions like rags, gloves, concrete, debris, etc. The presence of metal chips, grit or moisture is not excluded as well. In most cases, the elemental composition of the matrix depends on elements with the atomic number up to that of calcium but, occasionally, there are admixtures of elements in a range from Z to iron.
Calculations were performed for waste in the form of cellulose (С_{6}Н_{10}О_{5}), quartz (SiO_{2}), graphite, and a mixture of 80 % of cellulose and 20 % of iron to assess the influence of the matrix type (elemental composition). The density for all waste types was 0.708 g/cm^{3} with which the effect from the difference in the elemental composition has the greatest possible range presented in measurements. The results are given in Table
Count intensities in the 185.7 keV peak with the background deduction for different waste matrices and different uranium masses in the cask (the waste material density is 0.708 g/cm^{3})
Uranium mass, g  Matrix composition  

Cellulose С_{6}Н_{10}О_{5}  Graphite С  Quartz SiO_{2}  80% of cellulose + 20% of iron  
Count intensity in the 185.7 keV peak  
1  3.367  –  –  3.301 
500  3.22  3.292  3.272  3.15 
Difference from cellulose, %  
500  –  –0.072 (–2.2)  –0.052 (–1.6)  0.07 (2.2) 
It can be seen from the table that the chemical composition of the matrix in the range of the atomic numbers of elements in a range from carbon to iron influences slightly the intensity of the count in the 185.7 keV peak (~ 2 % for the tested matrixes as compared with the cellulose matrix). This is explained by the fact that the mass attenuation factor for the 185.7 keV line depends weakly on the atomic number of the element in the considered range of matrices and the attenuation of gamma radiation will be defined practically by the density of the material contained in the cask. Based on the data obtained, the value of the RB from the matrix type was assumed to be equal to ~ 2.2 %.
At the same time, an increase in the density of the matrix with uranium homogeneously distributed throughout the cask volume will lead to a further attenuation of the line due to the high atomic number of uranium and, accordingly, to a comparatively greater mass attenuation factor. It can be seen from Table
Computational modeling makes it possible to investigate any range of uranium masses in the cask, while measurement capabilities are often greatly limited by small dimensions of uranium samples with a “zero” mass or by large dimensions of samples with a small uranium density to exclude intensive selfabsorption of gamma radiation in the local region of the sample as such. Calculations were performed for the uranium masses in the cask in a range of 0.64 g to 2 kg, with the statistical calculation error practically not depending on the uranium mass in the sample due to the calculation peculiarities when the statistics of the counts in the peak is defined just by the share of the considered line yield relative to all lines in the initial spectrum during modeling.
Table
Calculated intensities of the count in the 185.7 keV peak with the background deduction for different waste densities and different uranium masses in the cask
Uranium mass in cask, g  Waste material density, g/cm^{3}  
0.016  0.471  0.708  
Intensity of count in 185.7 keV peak  
0.64  –  –  3.28 
1  10.81  4.47  3.279 
5  –  4.466  – 
10  10.8  –  – 
20  –  4.464  – 
50  –  4.449  – 
100  10.59  4.46  3.252 
500  9.79  4.233  3.139 
1000  –  4.017  – 
2000  –  3.63  – 
The table data show that the selfabsorption of the 185.7 keV line in uranium, the uranium mass range in the cask being up to 100 g, may be neglected if there are no localized regions with a large density of uranium (pieces or nonporous lumps). And the larger the density of the matrix is as such, the smaller the relative influence of absorption in uranium is. With a uranium mass of 500 g, a downward bias is observed in the intensity of the 185.7 keV peak, normalized per 1 g of ^{235}U, in the amount of ~ 5 % for nonporous matrices and ~ 10 % for a porous matrix (0.016 g/cm^{3}). With a uranium mass of 1 kg, the downward bias reaches 11 % for nonporous matrices and ~ 20 % for porous matrices. This effect needs to be taken into account during measurements, specifically for waste with lowenriched uranium, when the total uranium mass in the cask is much larger than the measured mass of ^{235}U. At the same time, where the mass of ^{235}U obtained by measurements does not exceed several grams, the uranium enrichment may be neglected and the potential uncertainty from the unknown uranium enrichment (total uranium mass uncertainty) can be introduced to within the confidential boundaries of the relative RB.
The method proposed to calculate uraniumcontaining waste models makes it possible to estimate the contribution of errors from variation of influencing factors in a broad range with any required step which cannot be done traditionally using a CO.
An algorithm for building an efficient calculation model has been demonstrated. The adequacy of the model and the feasibility of describing correctly real waste types, while obtaining calculated gamma spectra that agree well with real measurements, have been proved by comparing the calculation results with the data of measuring uraniumcontaining samples. An example has been shown of optimizing the computational model by representing a detector in the form of a ring to simulate the sample rotation about its axis.
Based on the calculation results, with regard for direct measurement data, the contribution of the uncertainty of influencing factors to the error has been estimated for particular measured objects. The use of computational methods has made it possible to extend the measurement range of the ^{235}U mass in SWR (1.8 to 100 g) in the developed MP.
Boundaries have been defined for the developed procedure to measure uraniumcontaining waste from the material matrix (~ 2.2 %) and its density (~ 10 %), and the contribution of the uncertainty of the uranium mass in the cask to the obtained result has been estimated (5 % for nonporous matrices, 10 % for porous matrices).