Calculation of photon shielding properties for some neutron shielding materials

NUCLEAR ENERGY SCIENCE AND ENGINEERING

Calculation of photon shielding properties for some neutron shielding materials

A. M. El-Khayatt

Nuclear Science and Techniques

Vol.28, No.5

Article number 69

Published in print 01 May 2017

Available online 31 Mar 2017

DOI：10.1007/s41365-017-0222-y

36800

The objective of the present study is to calculate photon shielding parameters for seven polyethylene-based neutron shielding materials. The parameters include the effective atomic number (Z_eff), the effective electron density (N_eff) for photon interaction and photon energy absorption, and gamma-ray kerma coefficient (k_γ). The calculations of Z_eff are presented as a single-valued and are energy-dependent. While Z_eff values were calculated via simplistic power-law method, the energy-dependent Z_eff for photon interaction (Z_PI-eff) and photon energy absorption (Z_PEA-eff) are obtained via the direct method for energy ranges of 1 keV–100 GeV and 1 keV–20 MeV, respectively. The k_γcoefficients are calculated by summing the contributions of the major partial photon interactions for energy range of 1 keV–100 MeV. In most cases, data are presented relative to pure polyethylene to allow direct comparison over a range of energy. The results show that combination of polyethylene with other elements such as lithium and aluminum leads to neutron shielding material with more ability to absorb neutron and γ-rays. Also, the kerma coefficient first increases with Z of the additive element at low photon energies, and then converges with pure polyethylene at energies greater than 100 keV.

Neutron shielding materialsEffective atomic numberKerma coefficientγ-rays.

1 Introduction

Energetic radiations are of serious concern in nuclear powers and accelerator facilities, medical or industrial X-ray machines, and radioisotope production projects. So the radiation shielding is still an attractive topic for research, it aims to preserving both human safety and structural material which may be compromised from radiation exposure. Pure polyethylene (PE), with its high hydrogen concentration, is often used as a moderator to slow fast neutrons to thermal region. The combination of PE and other materials such as boron, lithium or silicon, etc., makes it an effective neutron shielding material for different purposes. For example borated PE is a useful for neutron shielding in areas of low and intermediate neutron fluxes. Adding lithium to PE can be a shielding for both γ-rays and neutrons. Mixing the PE and silicon leads to neutron shielding with high resistance to fire. Finally, γ-ray shielding parameters are important for the neutron shielding materials that are used in the presence of γ-rays or may be exposed to γ-rays from the (n, γ) reactions.

Effective atomic number (Z_eff) and effective electron density (N_eff) are useful parameters for describing radiation interaction with composite matter, in terms of equivalent elements. Many authors reported on γ-ray attenuation parameters for ‎concretes [1-3], glasses [4-7], lunar soil samples [8], building materials [9], alloys [10], low Z-materials [11, 12] and other materials [13-14]. However, plenty of them are usually restricted to a candidate γ-ray shielding material or to low-Z materials, which are more important in medical physics. Moreover, the γ-ray shielding properties seem to be limited to photon attenuation properties only, such as linear ‎attenuation coefficient (μ), mass attenuation coefficients (μ/ρ), and effective atomic numbers. In any case, kerma, K, subject had scarcely treated [15].

The kerma, an acronym for “kinetic energy released in materials”, has replaced the traditional exposure as the shielding design parameter [16]. In fact, gamma heating is the local energy deposition from γ-ray interactions. Often this can be done by estimating the kerma. To relate the radiation passing through a unit volume of a material of interest (fluence, Φ) to the energy release, K, in the material; the phrase “fluence-to-kerma factors” has been introduced by International Commission on Radiological Units and Measurements, ICRU [17] and widely used [18-20]. However, instead of using kerma factor, the kerma coefficient (k_γ, kerma per fluence) will be used through this paper, as the word coefficient implies a physical dimension whereas the word factor does not [21]. Kerma coefficient is of interest for biomedical applications, because it is used to convert photon fluence to kerma (absorbed dose). Also kerma coefficient, which is the key response function for nuclear heating, is important in many nuclear applications, particularly for fission and fusion power reactors [18, 20, 22].

In this work, we aimed at calculating photon shielding energy dependent parameters for seven polyethylene-based neutron shielding materials. The parameters include Z_eff and N_eff for both photon interaction and photon energy absorption for the energy regions of 1 keV–100 GeV and 1 keV–20 MeV, respectively. The k_γ was calculated for the energy region of 1 keV–100 MeV. For many cases, the results are presented relative to pure PE to allow direct comparison over a range of energy. Finally, we are also discussing the single-valued Z_eff and N_eff generated by the simple power law method, whereby the elemental constituents of a material are summed (weighted according to their atomic percentage or to their relative electron fraction) and raised to a power.

2. Computational method and theoretical basis

2.1. The single-valued effective atomic number

A simple way to evaluate Z_eff is the use of a simple power law:

Z_{eff} = {(\sum_{i}^{} a i^{e} Z_{i}^{m})}^{1 / m},

(1)

where Z_i is atomic number of the i^th element and a_i^e its fractional electronic content. In fact, the researchers consider different values for the exponent m, such as m was 2.94, 3.1, 3.4 and 3.5 by Mayneord [23], Hine [24], Tsaï and Cho [25], and Sellakumar et al. [26]. In addition, the corresponding electron density N_eff is given by

N_{eff} = (\sum_{i}^{} a i^{e} (Z_{i} / A_{i})),

(2)

where A_i is the atomic mass and N_A is the Avogadro’s constant.

Also, the Z_eff can be given by atomic percentage of each element, α_i^at, which is defined by the mass percentage w_i and A_i:

α_{i}^{at} = \frac{(w_{i} / A_{i})}{\sum_{i} (w_{i} / A_{i})} .

(3)

Literature review showed two more definitions: Eq.(4) by Puumalainen et al. [27] and Eq. (5) by Manninen et al. [28].

Z_{eff} = {(\frac{\sum_{i} a_{i}^{at} Z_{i}^{3}}{\sum_{i} a_{i}^{at} Z_{i}})}^{1 / 2},

(4)

Z_{eff} = \sum_{i} a_{i}^{at} Z_{i} .

(5)

It can easily be shown that a_i^e and α_i^at are very nearly equal.

Finally, Murty suggested a more simple expression for calculating the Z_eff [29],

Z_{eff} {(\sum_{i} w_{i} Z_{i}^{3.1})}^{1 / 3.1}

(6)

On the other hand, the average electron density is given by

< N > = N_{A} \frac{< A >}{< Z >},

(7)

where <A> and <Z> is the mean atomic mass and the mean atomic number, respectively:

< A > = \sum_{i} f_{i} A_{i},

(8)

< Z > = \sum_{i} f_{i} Z_{i},

(9)

where A is the atomic mass, Z is the atomic number, and f_i is the molar fraction of the i^th element (normalized so that ∑f_i =1).

This simple approach, however, is approximately valid at low energies where photoelectric absorption is dominating and is overly simplistic for many applications.

2.3. Calculation of energy-dependent Z_eff and N_eff

Energy-dependent Z_eff has been described elsewhere [30] in a more rigorous fashion, here only the final equations are given.

Z_{PI-eff} = \frac{{\sum_{i} f_{i} A_{i} (\frac{μ}{ρ})}_{i}}{{\sum_{i} f_{i} \frac{A_{i}}{Z_{i}} (\frac{μ}{ρ})}_{i}},

(10)

where (μ_en/ρ) is the mass attenuation coefficient. NXcom program has been used to calculate the mass attenuation coefficients for the energy range of 1 keV–100 GeV [31]. The effective electron density, N_PI-eff, is given by (in electron per gram):

N_{PI-eff} = N_{A} \frac{Z_{PI-eff}}{< A >} .

(11)

The effective atomic number (Z_PEA-eff) and the effective electron density (N_PEA-eff) for photon energy absorption can be obtained from Eqs. (10) and (11), respectively. Mass energy-absorption coefficients are calculated for the studied object at photon energies using the photon data from Hubbell and Seltzer [32].

2.2. Computation of photon kerma coefficients.

The kerma, K, is the quotient of dE_tr by dm, where dE_tr is the mean sum of the initial energies of all charged particles liberated by indirectly ionizing radiations such as neutrons and photons in a mass dm of a material [33], thus

K = d E_{t r} / d m

(12)

For a fluence, Φ, of uncharged particles of energy E, K in a specified material is given by

K = Φ E (μ_{t r} / ρ) = Ψ (μ_{t r} / ρ)

(13)

where (μ_tr/ρ) is the mass energy-transfer coefficient of the material for these particles, and Ψ is the energy fluence. The kerma per fluence, K /Φ, is termed the kerma coefficient, k, for uncharged particles of energy E in a specified material, thus

k = K / Φ = E (μ_{t r} / ρ) .

(14)

For low Z-materials (e.g., air, water and soft tissue) only a small part is expended in bremsstrahlung (collisions with atomic nuclei). Therefore, mass energy transfer coefficient (μ_tr/ρ) and mass energy mass energy-absorption coefficient (μ_en/ρ) are almost equal. Thereby, photon kerma coefficients (in Gy·cm²/photon) of such materials can be obtained by Eq.(15):

k_{γ} = k_{D} E_{γ} \sum_{i} w_{i} {[μ_{en} (E_{γ}) / ρ]}_{i}

(15)

where k_D= 1.602×10⁻¹⁰ Gy·g/MeV is the energy conversion coefficient.

On the other hand, when the bremsstrahlung production is not negligible, the major contributing reactions for gamma kerma coefficients (in Gy·cm²/photon) in the energy range of fusion systems are the photoelectric (Pe), Compton scattering (C), pair production reactions (pp), and coherent (no energy loss) [18]:

k_{γ} = k_{D} \sum_{i} w_{i} [σ_{Pe}^{i} E_{γ} + σ_{pp}^{i} (E_{γ} - 1.022) + σ_{C}^{i} (E_{γ} - E_{γ}^{i})]

(16)

where σ_peⁱ, σ_ppⁱ and σ_Cⁱ are photoelectric, pair production and Compton absorption cross-sections for element i, respectively. It is important to recognize that above 5 MeV, one in principle should include photonuclear absorption cross section σ_ph.n. However, this process is not readily amenable to systematic calculation and tabulation. Hence, σ_ph.n is ignored in current compilations, even though at its giant resonance peak between 5 and 40 MeV it can contribute between 2% (high-Z elements) and 6% (low-Z elements) to the total cross section σ_tot [34].

Implicit in the use of Eq. (16) is the assumption that all the gamma photon energy in the photoelectric process is deposited locally and in pair production, 1.022 MeV (energy of electron-positron masses) of the photon energy is not available for local deposition. In the Compton scattering reaction, the photon only deposits a fraction of its energy locally because the scattered photon carries the rest away.

3. Results and Discussion

The chemical composition of neutron shielding materials studied in the present work is given in Table 1 [35], for combinations of PE and other materials such as boron, lithium or silicon. The single-valued calculated energy dependent Z_eff and N_eff for all samples are listed in Table 2.

Elemental composition (wt%) of neutron shielding materials.

Sample No.	H	Li	B	C	O	Na	Mg	Al	Si	S	Cl	Ca	Ti	Mn	Fe	Zn	Sr
238	2.76	―	25.35	20.08	24.15	―	―	―	26.93	―	―	0.04	0.02	―	0.41	0.26	―
207HD	6.01	―	0.86	18.02	48.42	0.17	0.02	24.9	0.04	0.01	―	1.36	―	―	0.01	―	0.06
210	8.54	―	30	59.25	0.76	―	―	0.04	1.01	―	―	―	―	0.02	0.38	―	―
2015	8.6	7.5	0	57.76	26.13	0.001	―	―	―	―	0.004	0.001	―	―	―	―	―
261	8.94	―	10	36.65	44.41	―	―	―	―	―	―	―	―	―	―	―	―
201	11.6	―	5	61.2	22.2	―	―	―	―	―	―	―	―	―	―	―	―
213	14.4	―	―	85.6	―	―	―	―	―	―	―	―	―	―	―	―	―

The mean atomic mass, mean atomic number, average electron density and single-valued Z_eff and N_eff of neutron shielding materials.

Sample No.	<A>	<N>	Z _eff						N _eff * (×10²³ e·g⁻¹)	< N > (×10²³ e·g⁻¹)
			Ref.[29]	Ref.[20]	Ref.[24]	Ref.[25]	Ref.[26]	Ref.[28]
238	10.83	5.45	10.37	10.15	10.30	10.52	10.67	9.25	3.12	3.03
207HD	8.66	4.54	10.01	9.66	9.77	9.93	10.06	9.00	3.32	3.16
213	7.23	3.87	8.82	8.70	8.62	6.95	6.96	6.66	3.41	3.22
261	6.40	3.46	6.85	6.64	6.67	6.70	6.75	6.39	3.48	3.25
215	6.23	3.34	6.41	6.22	6.26	6.29	6.34	5.97	3.46	3.23
210	6.15	3.26	6.53	6.21	6.34	6.59	6.73	5.62	3.42	3.19
201	5.42	3.01	6.33	6.09	6.12	6.15	6.20	5.82	3.61	3.34

*Ref. [26]

Examples of such calculations are graphed in Fig. 1. The single-valued effective atomic numbers are only a rough approximation at low energies, and should be treated with some caution. At medium photon energies (about 0.5–3.5 MeV) where Compton scattering process is the most probable, one can notice that the Z_eff and N_eff are approximately equal to their mean values <Z> and <N>, respectively. Therefore the mean values can represent Z_eff and N_eff at medium photon energies only.

Fig. 1

The single-valued calculated energy dependent Z_PI-eff (a) and N_PI-eff (b) for Sample-238, and the mean atomic number.

3.1. The effective atomic number

The calculated values of energy dependent Z_PI-eff, N_PI-eff, Z_PEA-eff and N_PEA-eff for PE and six neutron shielding materials, and the ratios of (Z_PI-eff)_PE^m, (N_PI-eff)_PE^m, (Z_PEA-eff)_PE^m and (N_PEA-eff)_PE^m, are shown in Fig.2, as a function of photon energy. The energy dependence of Z_PI-eff (for total photon interaction) for all the neutron shielding materials have almost the same behavior. One can see that the curves of Z_eff are arranged in descending order according to their mean atomic number <Z>. Fig.2(a),shows three energy regions, where Z_eff is almost constant “plateau” for a given material, while the transition regions are characterized by a rapid variation of Z_eff. This energy behavior of Z_eff reflects relative importance of the three γ-ray processes: photoelectric, Compton and pair production. Since the plateau region is formed when only one photon interaction is dominant, whereas the transition one includes more than one interaction which making significant contributions to the total interaction. Therefore, we have three plateaus corresponding to the dominance regions of the three γ-ray interactions, photoelectric absorption (E < 0.01 MeV), Compton scattering (0.1 < E < 6 MeV), and pair production (E > 200 MeV). In addition two transition regions (0.01 < E < 0.1 MeV), and (6 < E < 200 MeV) are observed.

Fig. 2

Z_PI-eff (a), N_PI-eff (b), Z_PEA-eff (c) and N_PEA-eff (d) of the neutron shielding materials, and their ratios to as function of photon energy.

At a given photon energy, the interaction is proportional to Z ⁿ where n is about 4 for the photoelectric absorption, 1 for the Compton scattering, and 2 for pair production. The Z⁴ dependence of the photoelectric absorption cross section leads to heavy weight to the element with the highest atomic number and the maximum value of Z_eff is found in the region of photoelectric plateau. At intermediate energies, Compton scattering is the main interaction process, Z_eff is close to the mean atomic number of the material, as the Compton scattering cross section of a an element is proportional to Z. The minimum value of Z_eff is recorded in this intermediate plateau. Above 200 MeV, the pair production plateau is observed, where Z_eff is almost constant and its mean value is smaller than that obtained for photoelectric plateau. This is due to the fact that the pair production cross section is proportional to Z², giving less weight to the higher-Z elements than the photoelectric absorption cross section. The same arguments also hold for Z_PEA-eff as shown in Fig. 2(c).

To compare the Z_eff values of a given PE-based shielding material with that of polyethylene, Figs.2(e–h) present the ratios of Z_eff of the materials for photon interaction and photon energy absorption to those of PE as a function of photon energy. Figs. 2(e) and 2(g) show that the materials containing higher-Z elements, such as Samples 238 and 207HD, have Z_eff values systematically greater than that of PE over the full energy range studied. Such neutron shielding materials have a competitive advantage in attenuating gamma-rays.

3.2. The effective electron density

Figs. 2(b) and 2(d) show the energy dependence of N_eff for total photon interaction and photon energy absorption, respectively. Figs. 2(f) and 2(h) present the N_eff ratios with respect to PE. The behavior is similar to that of Z_eff, because the two parameters are closely related through the Eq. (7). However, in contrast to Z_eff, the obtained curves for N_eff are not arranged by their average electron density <N> (Table 1). Thus, we can conclude that the average electron density of the material does not give us any correct indication of real effective electron density value. Finally, it is obvious that the different single-valued effective atomic numbers, which are estimated using simple power law, are approximately valid at low energies where photoelectric absorption is dominating as shown in Fig. 1.

3.3. Kerma coefficients

The energy dependence of γ-ray kerma coefficient, which depends on the photon interaction cross sections, is given in Fig. 3. Surprisingly, there is no plateau, and only rapid variation of k_γ with energy is seen. The kerma coefficient curves of the studied materials have dips in the energy range of around 0.3 to 0.09 MeV, and the dip position shifts to higher energy with increasing concentration of high Z-elements. We have also seen that the K-, and L-absorption edges in graphs of 210, 207HD, and 238 samples due to the presence of high-Z elements such Zn and Sr (Table 3). Fig.3 also shows that the k_γ curves tend to converge after departure of the photoelectric absorption dominance region (E > 0.1 MeV), where the other photon interaction processes give less weight to the higher-Z additive elements than that given by photoelectric process.

Photon energies (in keV) of absorption edges above 1 keV.

Elements	Z	L3	L2	L1	K
Na	11	―	―	―	1.07
Mg	12	―	―	―	1.31
Al	13	―	―	―	1.56
Si	14	―	―	―	1.84
S	16	―	―	―	2.47
Ca	20	―	―	―	4.04
Ti	22	―	―	―	4.97
Mn	25	―	―	―	6.54
Fe	26	―	―	―	7.11
Zn	30	1.02	1.04	1.19	9.66
Sr	38	1.94	2.01	2.22	16.10

Fig. 3

Gamma-ray kerma coefficients for neutron shielding materials and polyethylene as a function of photon energy.

4. Conclusions

1. Combination of polyethylene with other elements such as lithium and aluminum leads to neutron shielding material with more ability to absorb neutron capture γ–rays, therfore these materials are good sheilding for neutron and gamma-ray.

2. Variations of Z_eff, and N_eff with photon energy showed three basic energy plataeus and two transition regions which are characterized by a rapid variation of Z_eff and N_eff. Both plateous and transition regions were attributed to dominance of one and more photon interaction processes, respectively.

3. The three energy plateous are approximately E < 0.01 MeV, 0.1 MeV < E < 6 MeV and E > 200 MeV.

4. The single-valued effective atomic numbers are calculated by different 6-expressions and the results are approximately valid at low energies Therefore should be treated with some caution.

5. The energy dependence of γ-ray kerma coefficients showed rapid variations with energy and they have dips in the energy range of around 0.3 to 0.9 MeV.

6. All curves of γ-ray kerma coefficients converge at energies greater than 0.1 MeV.

References

Akkurt , A. M. El-Khayatt,

Effective atomic number and electron density of marble concrete

. J. Radioanal. Nucl. Chem. 259(1), 633-638 (2013). doi: 10.1016/j.anucene.2013.04.021

Calculation of photon shielding properties for some neutron shielding materials

1 Introduction

2. Computational method and theoretical basis

2.1. The single-valued effective atomic number

2.3. Calculation of energy-dependent Zeff and Neff

2.2. Computation of photon kerma coefficients.

3. Results and Discussion

3.1. The effective atomic number

3.2. The effective electron density

3.3. Kerma coefficients

4. Conclusions

2.3. Calculation of energy-dependent Z_eff and N_eff