Issue 
Radioprotection
Volume 57, Number 1, JanuaryMarch 2022



Page(s)  71  77  
DOI  https://doi.org/10.1051/radiopro/2021038  
Published online  20 January 2022 
Article
Gammaray shielding calculations using empirical formulas for buildup factor: application to a practical case
To Professor J.J. Fletcher
^{1}
Direction Générale de la Radioprotection et de la Sûreté Nucléaire, Ministère de l’Énergie et des Ressources Hydrauliques,
B.P. 1172,
Libreville, Gabon
^{2}
Laboratory of atomic, molecular and Nuclear Physics, Department of Physics, Faculty of Science, University of Yaounde I,
P.O. Box 812,
Yaounde, Cameroon
^{3}
Département de Physique, Faculté des Sciences, Université des Sciences et Techniques de Masuku,
B.P. 943,
Franceville, Gabon
^{4}
Institut de Cancérologie de Libreville,
B.P. 23902,
Libreville, Gabon
^{*} Corresponding author: p.ondomeye.dgrsn@gmail.com
Received:
7
December
2021
Accepted:
31
December
2021
Exposure of people to external ionizing radiation is controlled by time, distance and shielding. In the latter case, the radiation dose is reduced if shielding attenuates the radiation. The present study was aimed at comparing gammaray shielding calculation methods, the wellknown trialanderror method and a direct calculation method that make use of buildup factor’s empirical fitting formulas. The latter method was then applied to a practical case in gamma industrial radiography. The results showed that both methods provide comparable results. For high energies and highdensity materials, it has been observed that all the buildup factor forms studied provide similar values. Contrariwise, it has also been observed that the linear form of the buildup factor seemed to underestimate shielding thickness for low gamma energies (lower than ≈ 1 MeV). Furthermore, the application of the method using empirical buildup factor functions to a practical case in gamma industrial radiography showed that the calculations are in agreement with the measurements carried out on the field. This demonstrates that this method is efficient and can still be used even though more sophisticated and accurate methods are now available.
Key words: shielding / gammaray / buildup factor / air kerma / ambient dose equivalent
© SFRP, 2022
1 Introduction
Exposure of people to external ionizing radiation is controlled by three parameters: time, distance and shielding. The radiation dose received by individuals is proportional to the time they spent in the radiation field. The radiation dose is as well reduced substantially by increasing the distance from the source of radiation since it generally follows an inverse square law. Finally, the radiation dose is reduced if shielding attenuates the radiation (Podgorsk, 2005). Shielding design consists of three steps: (1) selecting a upper limit value for the effective dose in the occupied area (in general 0.5 μSv/h for a public area and 7.5 μSv/h for a supervised area); (2) estimating the radiation field in the occupied area considering only the unshielded radiation source; (3) obtaining the attenuation factor that is necessary to reduce the dose value from effective dose in (2) to the effective dose in (1). In a broad beam geometry, the measured dose rate will be greater than that described in a narrow beam geometry because scattered photons will also be detected. Such conditions usually apply to the shields required for protection from gammaray sources (Turner, 2012). Consideration of the buildup factor of gammarays is therefore essential for shielding calculations. The commonly used methods for computing accurate gammaray buildup factors are the iterative method, the invariant embedding method, the Monte Carlo method and the geometric progression (GP) method (Sharaf, 2015). Most of these methods are not only complicated but also time consuming. The GP fitting method is as good as the other method mentioned above and is relatively simple to apply. The only problem the persons interested in applying this method is that the fitting parameters, provided by the American Nuclear Society and the American National Standards Institutes, are not access free. Another alternative is the use of the trialanderror method, which has been widely used so far and proven efficient in practice (Cember, 2009; Turner, 2012). The issue with this method is that it is difficult to apply for radiation sources emitting more than two gammaray energies.
This work, in its first part, compares the results obtained by the trialanderror methods and those obtained by the use of empirical formulas to fit gammaray buildup factors. In the second part of the study, the latter method is applied to a practical case in gamma industrial radiography.
2 Materials and methods
2.1 Material
The features of the gamma radioactive source (Fig. 1) used during the evaluation of the shielded enclosure where radiography work is to be performed are presented in Table 1. The features of the survey meters used during this evaluation are given in Table 2.
Fig. 1
The gamma radioactive source used during the evaluation. Here the source is either within a fixed box on the transport vehicle (a) or within a simple transportation box (b). Are also shown the survey meters used to measure ambient dose equivalent rates (c) and the accessories (d) – remote control, ejection cable and guide tube – for exposing the sealed gamma source. 
Gamma radioactive source used during the evaluation of the shielded enclosure.
Survey meters used during the evaluation of the shielded enclosure.
3 Methods
Gammaray shielding calculations using empirical formulas for buildup factor are carried out. Depending on the case, either of the formulas below were used: (1) (2) where and are, respectively, the air kerma rate and the ambient dose equivalent rate calculated at distance d from the gamma radioactive source considered as punctual (the dimensions of the source are negligible compared to the distance d), (H^{*}(10)/K_{a})_{i} is the air kerma, K_{a}, to ambient dose equivalent, H^{*}(10), conversion coefficient. The conversion coefficient corresponding to gamma energy E_{i} were determined from the values given in ISO 40373 (ISO, 1999) standard for monoenergetic photon radiation using linear interpolation where needed; f_{i} is the gamma emission probability for energy E_{i}. Values of f_{i} and E_{i} for the radionuclides used in this work were taken from reference (IAEA, 2007). Particularly, those for Ir192 are given in Table 3. A is the activity of the gamma source, B_{i} is the buildup factor for energy E_{i}, μ_{i} is the linear attenuation coefficient for energy E_{i} and the relevant material, is the massenergy absorption coefficient in air for energy E_{i}, x is the shielding material thickness; μ_{i} and were determined from values provided by the National Institute of Standards and Technologies (NIST). The forms of buildup factor used in the study are as follows:

Linear

Quadratic

Berger

Taylor
In general, the goal is to determine the shielding thickness x given a known dose rate at a known distance d from the gamma la source. Equation (1) can be solved either by writing a computer program or by using an Excel sheet in which x is gradually increased, from the initial value x = 0 cm (unshielded source), until the desired dose rate value is reached. In the present work, the use of the second option (Excel sheet) was preferred.
Sometimes, one’s maybe interested in finding the distance d associated with either a controlled, a supervised or a public area whose corresponding dose rate values are set by the regulatory body.
The first part of the work consisted of calculating shielding thickness from some selected situations that has been dealt with in the literature (Cember, 2009; Turner, 2012) and then comparing the results obtained. These situations are listed below:

Situation 1
Determining the lead thickness needed if the air kerma rate at a distance of 1 m from a 1Ci ^{137}Cs point source is not to exceed 25 μGy/h.

Situation 2
Calculating the thickness of a lead shield needed to reduce the air kerma rate 1 m from a 10Ci point source of ^{42}K to 21.9 μGy/h (2.5 mR/h).

Situation 3
Determining the water depth needed if the air kerma rate at a point 6 m directly above a 10Ci point source of ^{24}Na is not to exceed 175.2 μGy/h (20 mR/h).

Situation 4
Finding the lead thickness needed to attenuate the air kerma rate from a 1Ci point source of ^{24}Na to 87.6 μGy/h (10 mR/h).
In the second part of the work, field measurements were carried out on the site where the shielded enclosure is located (Fig. 2).
The shielded enclosure has a length of 10.60 m, a width of 4.60 m and a height of 3 m. It is made of demountable ordinary concrete walls of 30 cm thick (Fig. 3). Ambient dose equivalent rates measurements were taken as illustrated in Figure 3. A 37Ci ^{192}Ir source and a collimator of attenuation 1/250th were used during the evaluation. The beam was directed towards the floor since the operator made the choice to carry out radiography work only in this configuration.
Gamma energies and the corresponding emission probabilities used in the present study for Ir192 (IAEA, 2007).
Fig. 2
Image of the shielded enclosure subject to testing. 
Fig. 3
Configuration of the shielded enclosure. Dose equivalent rates were measured at point C, about 4.6 m from a 37Ci ^{192}Ir source. 
4 Results and discussion
4.1 First part
The results obtained for the four situations mentioned previously are presented in Tables 4–7. The first observation is that values provided in this study, for all the buildup factor forms, are consistent with those provided in references (Cember, 2009; Turner, 2012) which were determined by the use of the trialanderror method. It is also observed, as shown in the results for situation 1, that the linear form of the buildup factor seems to underestimate shielding thickness for low gamma energies (lower than ≈ 1 MeV). This form of the buildup factor seems to provide the larger shielding thickness for high energies (greater that ≈ 1 MeV) and lowdensity materials (e.g., results for situation 3). For high energies and high density materials, all the buildup factor forms studied in the present work provide similar values (e.g., results for situations 2 and 4).
Comparison of the results for situation 1.
Comparison of the results for situation 2.
Comparison of the results for situation 3.
Comparison of the results for situation 4.
4.2 Second part
In this section, the results for the calculations are compared with measurements carried out in the field at a distance of about 4.6 m from the source, outside the shielded enclosure whose walls have a thickness x = 30 cm. The source was exposed with the use of a collimator of attenuation 1/250th, the beam being directed towards the floor. Results are given for the various forms of buildup factor used (Tab. 8).
These results show that the calculations are in agreement with the measurements carried out around the shielded enclosure. Indeed, the calculated ambient dose equivalent rates are in the range [0.53 μSv/h, 0.76 μSv/h] of the measurements performed, apart from the value calculated using the Taylor form of the buildup factor. This value is just above the measurement interval and, as such, may constitute the most conservative case from which radiation protection measures should be taken (e.g., delineation of areas).
Based on calculations for the different forms of buildup factor used in the present work, delineation of areas was carried out (Tab. 9) for the case of a 40Ci ^{192}Ir source as the shielded enclosure was approved by the regulatory body for a maximum activity not greater than this value. As expected, Taylor’s form of buildup factor provided the largest restricted areas. Based on Taylor’s formula of buildup factor, zoning should be carried out using values given in Table 9. For radiation protection reasons, these values were rounded up as shown in Figure 3. For these same reasons, the inside of the shielded enclosure is considered a controlled area.
The delineation of area was undertaken taking into account the use of a collimator of attenuation 1/250th and the fact that the beam was directed towards the floor. This configuration, as shown in the present study, reduces the zoning distance around the radiation source and may limit the skyshine effect. If, for any reason, the beam is directed to any wall, the zoning provided in this study will no longer be valid. Values in Table 9 for exposure of the source without collimator shall be used. However, the environment around the shielded enclosure does not allow this (Fig. 2). This means that additional shielding will be needed. Adding lead shielding on the interior surface of the shielded enclosure may be an appropriate solution.
Comparison of measurements results with those computed from the various forms of buildup factor.
Delineation of areas around the shielded enclosure for a ^{192}Ir source of 40 Ci maximum activity.
5 Conclusion
This work was aimed at comparing gammaray shielding calculation methods, the wellknown trialanderror method and a direct calculation method that make use of buildup factor’s empirical fitting formulas. The latter method was then applied to a practical case in gamma industrial radiography. It has been shown that both methods provide comparable results. Furthermore, the application of the method using empirical buildup factor functions to a practical case in gamma industrial radiography showed that the calculations are in agreement with the measurements carried out in the shielded enclosure. This demonstrates that this method is efficient and can still be used even though more sophisticated and accurate methods are now available.
Based on the field measurement configuration, a delineation of areas was carried out which would no longer be valid if, for any reason, the configuration is to be changed.
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding
This work has received operational support from the General Directorate of Radiation Protection and Nuclear Safety, Ministry of Energy and Hydraulic Resources, Gabon.
Ethical approval
Ethical approval was not required.
Informed consent
Informed consent was not required.
Authors contributions
P. Ondo Meye: Conceptualization, methodology, data acquisition and analysis, manuscript drafting. C. Chaley: Data acquisition and analysis, reviewing. S. Y. Loemba Mouandza: Methodology, data analysis, reviewing. B. C. Mabika Ndjembidouma: Data analysis, reviewing. G. H. BenBolie: Supervision, reviewing.
Acknowledgements
The authors are grateful to the anonymous reviewers for helping improve the work through their invaluable suggestions.
References
 Cember H. 2009. External radiation safety. Introduction to Health Physics (pp. 513–574). New York, USA: Mc Graw Hill Medical. [Google Scholar]
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Cite this article as: Ondo Meye P, Chaley C, Loemba Mouandza SY, Mabika Ndjembidouma BC, BenBolie GH. 2022. Gammaray shielding calculations using empirical formulas for buildup factor: application to a practical case
To Professor J.J. Fletcher. Radioprotection 57(1): 71–77
All Tables
Gamma energies and the corresponding emission probabilities used in the present study for Ir192 (IAEA, 2007).
Comparison of measurements results with those computed from the various forms of buildup factor.
Delineation of areas around the shielded enclosure for a ^{192}Ir source of 40 Ci maximum activity.
All Figures
Fig. 1
The gamma radioactive source used during the evaluation. Here the source is either within a fixed box on the transport vehicle (a) or within a simple transportation box (b). Are also shown the survey meters used to measure ambient dose equivalent rates (c) and the accessories (d) – remote control, ejection cable and guide tube – for exposing the sealed gamma source. 

In the text 
Fig. 2
Image of the shielded enclosure subject to testing. 

In the text 
Fig. 3
Configuration of the shielded enclosure. Dose equivalent rates were measured at point C, about 4.6 m from a 37Ci ^{192}Ir source. 

In the text 
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