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 makes it an effective neutron shielding material for different purposes. For example, borated PE is 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 [1,2,3], glasses [4,5,6,7], 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 [18,19,20]. However, instead of using kerma factor, the kerma coefficient (k γ, kerma per fluence) will be used throughout 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:

$$ Z_{eff} = \left( {\sum\limits_{i} {a_{i}^{e} Z_{i}^{m} } } \right)^{1/m} , $$
(1)

where Z i is atomic number of the ith element, and a e i 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

$$ N_{\text{eff}} = N_{\text{A}} \left( {\sum\limits_{i} {a_{i}^{\text{e}} (Z_{i} A_{i} )} } \right), $$
(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 at i , which is defined by the mass percentage w i and A i :

$$ \alpha_{i}^{\text{at}} = \frac{{(w_{i} /A_{i} )}}{{\sum\limits_{i} {(w_{i} /A_{i} )} }} .$$
(3)

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

$$ Z_{\text{eff}} = \left( {\frac{{\sum\limits_{i} {\alpha_{i}^{\text{at}} Z_{i}^{3} } }}{{\sum\nolimits_{i} {\alpha_{i}^{\text{at}} Z_{i} } }}} \right)^{1/2} , $$
(4)
$$ Z_{eff} = \sum\limits_{i} {\alpha_{i}^{at} Z_{i} } . $$
(5)

It can easily be shown that a e i and α at i are very nearly equal.

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

$$ Z_{\text{eff}} = \left( {\sum\limits_{i} {w_{i} Z_{i}^{3.1} } } \right)^{1/3.1} $$
(6)

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

$$ \langle N\rangle = N_{\text{A}} \frac{\langle A\rangle }{\langle Z\rangle }, $$
(7)

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

$$ \langle A\rangle = \sum\limits_{i} {f_{i} A_{i} ,} $$
(8)
$$ \langle Z\rangle = \sum\limits_{i} {f_{i} Z_{i} ,} $$
(9)

where A is the atomic mass, Z is the atomic number, and f i is the molar fraction of the ith 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.

2.2 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.

$$ Z_{\text{PI - eff}} = \frac{{\sum\limits_{i} {f_{i} A_{i} \left( {\frac{\mu }{\rho }} \right)_{i} } }}{{\sum\limits_{i} {f_{i} \frac{{A_{i} }}{{Z_{i} }}\left( {\frac{\mu }{\rho }} \right)_{i} } }}, $$
(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):

$$ N_{\text{PI - eff}} = N_{\text{A}} \frac{{N_{\text{PI - eff}} }}{\langle A\rangle }. $$
(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].

2.3 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

$$ K = {\text{d}}E_{\text{tr}} /{\text{d}}m, $$
(12)

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

$$ K = \varPhi E(\mu_{\text{tr}} /\rho ) \, = \varPsi (\mu_{\text{tr}} /\rho ), $$
(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

$$ k = K/\varPhi = E(\mu_{\text{tr}} /\rho ). $$
(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):

$$ k_{\gamma } = k_{\text{D}} E_{\gamma } \sum\limits_{i} {w_{i} [\mu_{\text{en}} (E_{\gamma } )/\rho ]_{i} }, $$
(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]:

$$ k_{\gamma } = k_{\text{D}} \sum\limits_{i} {w_{i} \left[ {\sigma_{\text{Pe}}^{i} E_{\gamma } + \sigma_{\text{pp}}^{i} (E_{\gamma } - 1.022) + \sigma_{\text{C}}^{i} \left( {E_{\gamma } - E_{{_{\gamma } }}^{i} } \right)} \right]}, $$
(16)

where σ ipe , σ ipp , and σ iC 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.

Table 1 Elemental composition (wt%) of neutron shielding materials
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Table 2 The mean atomic mass, mean atomic number, average electron density and single-valued Z eff and N eff of neutron shielding materials
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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
figure 1

The single-valued calculated energy-dependent Z PI-eff (a) and N PI-eff (b) for Sample 238 and the mean atomic number

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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) mPE , (N PI-eff) mPE , (Z PEA-eff) mPE , and (N PEA-eff) mPE 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 has 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>. Figure 2a 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. The plateau region is formed when only one photon interaction is dominant, whereas the transition one includes more than one interaction which makes 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
figure 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

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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 Z 4 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, and Z eff is close to the mean atomic number of the material, as the Compton scattering cross section of 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 Z 2, 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. 2c.

To compare the Z eff values of a given PE-based shielding material with that of polyethylene, Fig. 2e–h presents 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. Figure 2e, g shows 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

Figure 2b, d shows the energy dependence of N eff for total photon interaction and photon energy absorption, respectively. Figure 2f, h presents 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 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 shown 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 0.3–0.09 MeV, and the dip position shifts to higher energy with increasing concentration of high-Z elements. We have also seen the K- and L-absorption edges in graphs of 210, 207HD, and 238 samples due to the presence of high-Z elements such as Zn and Sr (Table 3). Figure 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.

Fig. 3
figure 3

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

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Table 3 Photon energies (in keV) of absorption edges above 1 keV
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4 Conclusions

  1. 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; therefore, these materials are good shielding for neutron and gamma ray.

  2. 2.

    Variations of Z eff and N eff with photon energy showed three basic energy plateaus and two transition regions which are characterized by a rapid variation of Z eff and N eff. Both plateaus and transition regions were attributed to the dominance of one and more photon interaction processes, respectively.

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 6.

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