Introduction

Radiation has both particle and wave properties/features. Although these radiations are commonly applied in various fields of studies including physics, chemistry, material science, medical researches, industry, and agriculture, they are harmful to human life and the environment [1,2,3]. On the other hand, environmental remediation becomes a rising fields of investigation to eliminate the possible risks. There are two different techniques to manage and control the environmental pollution: First, using proper shielding materials method, and second, employing visible-light nanostructures strategy. As regards to the former method, scientists began to search for protective materials to diminish the hazardous effects of radiations on environment and human life [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. Some frequently utilized shielding materials such as lead-based composites are proper shields against gamma photons due to their high atomic number, high material density, and rigidity [8, 12, 16]. In addition to the remarkable shielding performance of the lead-based composites, they have some disadvantageous such as massive weight, toxic nature, and production of secondary radiations convincing researchers to find an alternative shielding materials. In order to choose appropriate material, various aspects should be considered. Availability, rigidity, low cost, good physical, and mechanical properties are some features of convenient shielding materials. With this in mind, concrete is widely used as shielding material. Unfortunately, high emission of greenhouse gases and destruction of natural resources open a new window into materials’ sciences. To evaluate the second method, some different visible-light nanostructures were suggested as substituent resources to eliminate environmental pollution [25,26,27,28,29]. These shielding composites like silver tungstate, nanostructures are activated by the sunlight and protect the environment. This method is known as echo-friendly technique. Zinatloo-Ajabshir et al. [29] investigated the copper effect on improving the electrochemical storage of hydrogen in CeO2 nanostructure for the first time. The nanostructured oxides were prepared by adding different weight percentages of copper inside CeO2. The outcomes reveal that adding copper into CeO2 will enhance energy storage, particularly hydrogen storage. Hence, this method could be beneficial as an environmentally friendly system to eliminate the possible risks of environment pollution. In this regard, polymer-based composites were attracted by the investigators [6, 7, 17, 21, 22]. More interestingly, polymer composites contain a large number of low Z materials like carbon, hydrogen, and oxygen; thereby, they have excellent shielding performance against neutrons. Accordingly, investigation of the neutron shielding quality of the selected vinyl ester composites is our main concern in this work. The excellent anti-corrosion and high chemical resistance are other advantageous of polymeric resources. Among different polymeric composites, vinyl ester composites are attracted considerable interest by researchers due to their unique characteristics such as chemical resistance, great strength, and low cost. Studies show that the applications of polymeric composites are found mainly in non-structural purposes due to their flammability, mechanical features. Numerous studies focus on enhancing the fire retardancy of the polymeric materials by adding various materials such as silicon- based materials, natural fibers such as hemp, flax, or bamboo as reinforcement to vinyl ester composites. Findings show that the recent technique was successful in inhibiting and controlling fire in polymers [30,31,32,33,34,35,36,37,38].

In spite of the fact that polymeric composites have perfect neutron shielding proficiency, few studies focus on the neutron aspects. Since neutrons are uncharged particles with high Relative Biological Effectiveness (RBE) which can be so harmful to human life and the environment [1], it requires further analysis by investigators. Therefore, the behavior of two colemanite- and barite-containing resources in terms of neutron–gamma photon shielding competence is studied in this research.

Having greater insight into the mentioned issue in terms of neutron attenuation and considering the importance of this subject, the current paper aims to apply an artificial radioisotope source, namely 252Cf neutron source, to evaluate neutron–gamma photon attenuation features of two colemanite- and barite-containing resources. The isotope 252Cf decays through alpha emission (~ 97%) into 248Cm and by spontaneous fission (~ 3%) into different fission products. As well as the 248Cm can decay in the form of alpha and spontaneous fission with the probability of 91.61% and 8.39%, respectively. This source emits 3.87 neutrons per fission on average with a specific activity of 0.536 mCi/g and average energy of 2.14 MeV. This isotope can be used in neutron radiography, reactor start-up sources, brachytherapy, etc. [39,40,41].

The 252Cf neutron source can also emit gamma photon in wide energy range up to 15 MeV that cannot be ignored in simulations and calculations [12, 13]. Thus, it is necessary to have some information about gamma photon that is emitted from 252Cf neutron source in terms of shielding quality. For this purpose, the Doppler Effect (DE) [12, 13, 40] is used in this paper to obtain the gamma photon spectrum which is produced from 252Cf neutron source for selected vinyl ester composites.

The present work aims to study the neutron–gamma photon shielding competence of two colemanite and barite vinyl ester series. The vinyl ester composites are irradiated by the 252Cf neutron source. The 252Cf radioisotope source is capsulated by the polyethylene cylinder and is simulated by the MCNPX Monte Carlo Simulation Code (MCSC). Two significant techniques, namely Watt Fission Distribution (WFD) and Doppler Effect (DE), are used to obtain the neutron and gamma photon spectra for two colemanite- and barite-containing resources. The Fast Neutron Removal Cross Section (FNRCS), equivalent absorbed dose rate, Half Value Layer (HVL), and Mean Free Path (MFP) are also estimated and outcomes are discussed in detail. Based on simulation results, Linear Attenuation Coefficient (LAC) and Effective Atomic Number (Zeff) versus incoming gamma photon are investigated. Additionally, the accumulations of gamma photon within the colemanite and barite series are derived by newly developed Phy-X: PSD software. It is worth recommending that the outcomes of this research would be useful for future applications of similar vinyl ester composites in terms of shielding purposes. Also, the present study can be a pioneer for future studies that are subjected to evaluate the shielding quality against neutron by 252Cf neutron source.

Materials and methods

The present work investigates the shielding strength of two colemanite- and barite-containing resources that are irradiated by 252Cf radio isotopic source against neutron and gamma photon. The codes, chemical compositions (mol%), and densities of the colemanite and barite vinyl ester series are enlisted in Tables 1, 2.

Table 1 Chemical compositions (mol%) and densities of colemanite vinyl ester series
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Table 2 Chemical compositions (mol%) and densities of barite vinyl ester series
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This paper utilizes a MCNPX Monte Carlo Simulation Code (MCSC), an analytical method, and Phy-X:PSD software to test the neutron–gamma photon shielding proficiency of the colemanite and barite vinyl ester composites. The simulations are classified into two different parts. first: neutron section and second: gamma photon section. In the neutron section, the Watt Fission Distribution (WFD) is used to derive the neutron spectrum; then, by using a definite card equivalent absorbed dose rate is obtained. Fast Neutron Removal Cross Section (FNRCS) and other neutron attenuation parameters are extracted by the analytical method. In the gamma photon section, Doppler Effect (DE) was exploited to obtain the gamma photon spectrum in the presence of the vinyl ester series. Other gamma photon attenuation characteristics can be extracted by the MCNPX simulation and Beer Lambert’s Law (BLL). To continue, in order to gain the equivalent atomic number, the accumulation of the gamma photon in different vinyl ester composites Phy-X: PSD software is employed.

MCNPX 2.6 Monte Carlo Simulation Code (MCSC)

A major requirement in radiation protection and nuclear investigations is choosing a reliable and exact method which is able to determine the radiation shielding characteristics of various shielding materials. Therefore, MCNPX Monte Carlo Simulation Code (MCSC) that is a general-purpose Monte Carlo N-Particle transport code developed by Los Alamos National Laboratory (LANL) is used to extract the neutron–gamma photon attenuation parameters [41]. The 252Cf neutron source which was surrounded by the cylindrical polyethylene, lead, and barite as first, second, and third layers, respectively, is simulated in this study. All the setup is cited in a stainless steel container with the dimensions of 50*50*50 cm3. The cubic sample with 2 cm thick is cited 10 cm away from the 252Cf source on top of the setup. The outside of the geometry was considered a void that means the code does not track the particles in that region. The simulation and calculations were done via a PC equipped with an Intel Core i5-4210U 1.70 GHz CPU, 6 GB of RAM. The 3-D schematic of this simulation setup that will be displayed in the visualization editor of MCNPX entitled MCNPX Visual Editor Version X_22S is shown in Fig. 1.

Fig. 1
figure 1

3-D schematic view of the simulation by visual editor

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Neutron particle attenuation parameters

Since the 252Cf source can emit a large amount of gamma photons, two separated programs are run to cover both neutron particle and gamma photon’s behavior. Thus, using Watt Fission Distribution (WFD) in the neutron input file [12, 13] and determining two fixed parameters of a = 1.180000 and b = 1.03419, the neutron spectrum is obtained [41]. The F5 ring tally card is used to calculate the neutron spectrum in the presence of the colemanite and barite shielding materials. To estimate the equivalent absorbed dose rate that has a significant role in terms of the neutron shielding application, F4 and DF4 (Dose Function) were applied, and then, the equivalent absorbed dose rate was extracted for different colemanite- and barite-containing resources. The DF4 card can be defined as: DF4 IU FAC IC INT.

Where IU, FAC, IC, and INT are the control units, normalization factor for dose, standard dose function, and energy interpolation, respectively.

Fast Neutron Removal Cross Section (FNRCS) that is an important factor to determine the quality of shielding materials can be derived using effective removal cross section by Eqs. 1 and 2, respectively:

$$ {\text{FNRCS}} = \mathop \sum \limits_{i} \rho_{i} \left( {\frac{{{\text{FNRCS}}}}{\rho }} \right)_{i} $$
(1)
$$ \rho_{i} = \omega_{i} \rho $$
(2)

where ωi and ρi are the weight fraction and partial density of the ith constituent element, respectively [42]. The value of \(\frac{{{\text{FNRCS}}}}{\rho }\) for each element is followed as Eq. 3 and Fig. 2 [43]:

$$ \frac{{{\text{FNRCS}}}}{\rho } = \left\{ {\begin{array}{*{20}l} {0.190 Z^{ - 0.743} } \hfill & {Z \le 8} \hfill \\ {0.125 Z^{ - 0.565} } \hfill & {Z > 8} \hfill \\ \end{array} } \right\} $$
(3)
Fig. 2
figure 2

\(\frac{{{\text{FNRCS}}}}{\rho }\) as a function of atomic weight [44]

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Half Value Layer (HVL) that plays an essential role to assess the shielding capacity of the materials is derived from \({\text{FNRCS}}({\text{cm}}^{ - 1}\)) and Eqs. 4, 5 [12, 42]:

$$ {\text{HVL}}\,\left( {{\text{cm}}} \right) = \frac{{{\text{Ln}}\left( 2 \right)}}{{{\text{FNRCS}}\,\left( {{\text{cm}}^{ - 1} } \right)}} $$
(4)
$${\text{MFP}}\,{\mkern 1mu} ({\text{cm)}} = \frac{1}{{{\text{FNRCS}}\,{\mkern 1mu} \left( {{\text{cm}}^{{ - 1}} } \right)}}$$
(5)

Gamma photon attenuation characteristics

In the second program, the gamma photon spectrum associated with the 252Cf spontaneous fission source was estimated based on the MCNPX Monte Carlo simulation Code (MCSC). Gamma photons that are produced from 252Cf neutron source are created through two ways: the prompt gamma rays associated with spontaneous fission and also from the resulting gamma rays which are produced as fission products and the second one gamma photon associated with alpha decay which contribute less than 0.1 percent of the total gamma emission and can be ignored [45]. Because of the high energy and velocity of the fission fragments, the Doppler Effect (DE) was used to calculate the gamma spectrum emitted from spontaneous fission through the equation below:

$$ E_{\gamma } = E_{0} \left( {1 \mp \frac{V}{C}} \right) $$
(6)

where E0 is the initial energy of the emitted gamma photon from the spontaneous fission at zero velocity and V is the velocity of the fission product. The fission fragments can be separated into heavy and light products which heavy fragments have the average velocity of 1.04*107 m/s, by energy difference (ΔE = \(E_{\gamma } - E_{0}\)) about 7% of the primary energy E0, and light fragments with the average velocity of 1.37*107 m/s, (ΔE = \(E_{\gamma } - E_{0} )\) would be about 9% of the primary energy E0. The products probabilities versus mass number are discussed elsewhere [12, 13]:

Using Doppler Effect (DE), the gamma photon energies were obtained, and applying Eq. 7, probabilities were extracted and given to the MCNPX code as an input file.

$$ {\text{SCf}}_{\gamma } \left( {E_{\gamma } } \right) = \left\{ {\begin{array}{*{20}l} {375E^{2} e^{{ - \frac{E}{0.109}}} + 0.468 e^{{ - \frac{E}{1.457}}} } \hfill & {E \le 1.5\;{\text{MeV}}} \hfill \\ {e^{{ - \frac{E}{0.851}}} } \hfill & {E > 1.5\;{\text{MeV}}} \hfill \\ \end{array} } \right\} $$
(7)

where \({\text{SCf}}_{\gamma } \left( {E_{\gamma } } \right)\) and E are the gamma photon spectrum and the incident gamma photon energy of spontaneous fission from 252Cf neutron source [46].

Hence, the gamma photon spectrum that is extracted from Doppler Effect is illustrated graphically in this research, and consequently, other related attenuation parameters are evaluated using Eqs. 8–11. Employing Beer Lambert’s Law (BLL), the Linear Attenuation Coefficient (LAC) is obtained through Eq. 8:

$$ I = I_{0} e^{{ - {\text{LAC}} * x}} $$
(8)

where I and I0 are attenuated and un-attenuated rays and x is the thickness of the sample. Based on the LAC, the Half Value Layer (HVL) values are given through the following equation [12]:

$$ {\text{HVL}}\,\left( {{\text{cm}}} \right) = \frac{{{\text{Ln}}\,\left( 2 \right)}}{{{\text{LAC}}\,\left( {{\text{cm}}^{ - 1} } \right)}} $$
(9)

The probabilities of the gamma photon transmission over a wide energy range are found by Transmission Factor (TF%) and also Radiation Protection Efficiency (RPE%). These parameters are determined by Eqs. 10, 11 [10]:

$$ {\text{TF}}\,\left( \% \right) = \frac{I}{{I_{0} }} \times 100 $$
(10)
$$ {\text{RPE}}\,\left( \% \right) = \left( {1 - \frac{I}{{I_{0} }}} \right) \times 100 = 100 - {\text{TF}}\% $$
(11)

Another important parameter so-called effective atomic number (Zeff) is calculated through Eq. 12:

$$ Z_{{{\text{eff}}}} = \frac{{\mathop \sum \nolimits_{i} f_{i} A_{i} \left( {\frac{{{\text{LAC}}}}{\rho }} \right)_{i} }}{{\mathop \sum \nolimits_{j} f_{j} \frac{{A_{j} }}{{Z_{j} }}\left( {\frac{{{\text{LAC}}}}{\rho }} \right)_{j} }} $$
(12)

where fi and Ai are fractional abundance and atomic weight, respectively [12, 13].

Newly developed Phy-X: PSD software

Having knowledge about gamma photon buildup factors like Energy Absorption Buildup Factor (EABF) and Exposure Buildup Factor (EBF) of various composites, glass, and also concrete is an essential issue for researchers and scientists to determine which materials can better block ionizing radiations. Thence, to assess buildup factors of colemanite and barite vinyl ester series the Phy-X: PSD software is used in this research. This software can be used to compute parameters relevant to radiation dosimetry in wide energy ranges. The compositions of the selected materials can be entered into the Phy-X: PSD code by either mol% or w% [47]. This software is available for any researcher at https://phy-x.net/.

Results and discussion

In this section, neutron and gamma photon attenuation characteristics of two candidate vinyl esters are estimated and the obtained results are discussed in detail. The 108 histories were run to achieve relative errors of less than 0.1% in all sections.

Colemanite and barite vinyl ester’s behavior against neutron

The neutron spectrum resulting from Watt Fission Distribution (WFD) is shown graphically in Fig. 3 for colemanite- and barite-containing resources. A continuous distribution is reported for the neutron flux from 0 to 12 MeV in Fig. 3. The graph’s behavior is similar to a fission reactor with the most probable energy of 0.7 MeV and average energy of 2.140 MeV [12, 13, 48]. Furthermore, the colemanite series flux is larger than the barite series in a wide energy range which convinces us colemanite series have excellent neutron stopping properties in comparison with the barite series.

Fig. 3
figure 3

Neutron flux for colemanite- and barite-containing resources

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Figure 4 represents the Fast Neutron Removal Cross Section (FNRCS) values of the colemanite- and barite-containing resources against 252Cf neutron source. As it is seen, chosen vinyl ester series have different trends. That means an increasing rate of colemanite in vinyl ester results in a gradual decrease in the FNRCS values, while the increasing rate of the barite in vinyl ester accounts for enhancement of the shielding ability against neutron. Thus, a rapid rise is observed for the barite series. This behavior may be due to adding more weight fraction of oxygen and hydrogen to the barite vinyl ester composite in comparison with the colemanite vinyl ester composite. For convenience sake, the Fast Neutron removal Cross Section for colemanite- and barite-containing resources is labeled as (FNRCS)V-C0%, (FNRCS)V-C20%, (FNRCS)V-C50%, (FNRCS)V-B0%, (FNRCS)V-B20%, and (FNRCS)V-B50%. Therefore, (FNRCS)V-C0%, (FNRCS)V-C20%, and (FNRCS)V-C50% equaling 9.905, 9.889, and 9.572 cm−1, while (FNRCS)V-B0%, (FNRCS)V-B20%, and (FNRCS)V-B50% lie in a wide range between 4.722 and 7.117 cm−1 which shows that even colemanite series possess lower density (according to Tables 1, 2), they own the highest FNRCS values and will have the best performance against neutron in comparison with the barite series. This fact convinces us that increasing the weight fraction of low Z elements causes an improvement in the neutron shielding ability and there is no relationship between FNRCS values and density of the composites that are in agreement with the previous researches [10, 13, 49].

Fig. 4
figure 4

Variation in the FNRCS versus colemanite- and barite-containing resources

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The equivalent absorbed dose rate of the selected colemanite and barite vinyl ester composites is estimated based on MCNPX simulation, and the findings are depicted in Fig. 5. As expected, colemanite series have the highest equivalent absorbed dose rate and can improve the neutron shielding proficiency while the barite series that possess a lower equivalent absorbed dose rate own the worst neutron shielding capability. Moreover, the increasing rate of colemanite in V-C series has a negative effect on neutron shielding proficiency, while the increasing rate of barite in the V-B series will enhance the neutron shielding performance. In addition, the FNRCS and equivalent absorbed dose rate of the colemanite and barite series have similar trends that mean the series with the highest dose absorption own the best shielding qualities against neutron.

Fig. 5
figure 5

Equivalent absorbed dose rate versus colemanite- and barite-containing resources

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The equivalent absorbed dose rate versus density for investigated shielding materials is calculated and shown in Fig. 6a, b. The equivalent absorbed dose rate related to colemanite vinyl ester series experiences a sharp reduction from V-C0% (with the density of 1.13 g.cm−3) to V-C50% (with the density of 1.56 g.cm−3), while a shrill increase is observed for barite vinyl ester series from V-B0% (with the density of 1.13 g.cm−1) to V-B50% (with the density of 2.3 g.cm−1). The findings show that the fast neutron removal cross section and neutron equivalent absorbed dose rate are strongly affected by the weight fractions of low Z materials. In other words, increasing the weight fractions of low Z materials will raise the values of FNRCS and equivalent absorbed dose rate.

Fig. 6
figure 6

Equivalent absorbed dose rate versus density for a colemanite- and b barite-containing resources

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The required thickness to reduce the initial intensity to half of it is achieved by the Half Value Layer (HVL) in Fig. 7. It can be beholden from this figure that V-B0% owns the highest HVL value of 0.1467598 cm, and consequently, it is the worst shielding materials among the rest of the colemanite and barite series. On the other hand, colemanite-containing resources possess the lower HVL and as a result have the best shielding performance against neutron. In addition, increasing rate of barite in the V-B series can better modify the neutron shielding capacity of the barite vinyl ester and leads to a sharp reduction in HVL values from V-B0% to V-B50%. Thus, a descending order of (HVL)V-B0% > (HVL)V-B20% > (HVL)V-B50% > (HVL)V-C50% > (HVL)V-C20% > (HVL)V-C0% is monitored for barite- and colemanite-containing vinyl ester.

Fig. 7
figure 7

Variation in the HVL versus colemanite- and barite-containing resources

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Another significant factor that provides the shielding nature of different composites is the Mean Free Path (MFP) that evaluates the distance traveling by the neutron particles between two subsequent collisions. As predicted, in Fig. 8. V-B50% that possesses the highest FNRCS owns the lowest MFP value of 0.1405111 cm. That means neutron particles can undergo a short distance between two following collisions in V-B50% in comparison with the other vinyl ester series. In order to clarify the outcomes, some numerical results are represented in this section. Accordingly, the MFP for the colemanite series varies between 0.1009591 and 0.1044714 cm, while this variation for barite series lies in a wide range of 0.2117893 to 0.1405111 cm.

Fig. 8
figure 8

Variation in the MFP versus colemanite- and barite-containing resources

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Colemanite and barite vinyl ester’s behavior against gamma photon

Using Doppler Effect (DE), the gamma photon spectra for two vinyl ester composites are demonstrated in Fig. 9. Although the graph behavior for the two types of vinyl ester samples is similar and close, the V-B series have the higher gamma flux in a wide energy range which proves that V-B series own a better shielding ability against gamma photon. In order to clarify the V-B shielding strength against gamma photon, other attenuation parameters are also estimated.

Fig. 9
figure 9

Gamma photon spectra for V-C0% and V-B0% vinyl ester composites

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As 252Cf neutron source can also emit gamma photon around 0 to 15 MeV, the Linear Attenuation Coefficient (LAC) of the candidate vinyl ester composites that is the most important parameter for determining shielding capacity is evaluated and depicted graphically in Fig. 10a, b. The LAC possesses maximum values at the low energy region, where the Photo-Electric interaction (PE) is predominant. This is because PE interaction is proportional to E−3.5 in this region [50]. Thereafter, in the energy range between 0.1 and 4 MeV, where Compton scattering (CS) is prevalent, the LAC was found to decrease moderately with gamma photon energies due to the variation in the CS interaction with E−1 in this range [10]. In some MeV’s, the LAC increases very slowly with the incident gamma energy due to the dominance of Pair Production (PP) interaction where the PP interaction varied with log E [50]. Having a better understanding of the shielding properties of colemanite- and barite-containing resources, the LAC is shown in a small energy range from 0 to 0.5 MeV. As it is seen, a sudden jump is discovered in low energy ranges of E = 0.088 MeV for barite series that can be due to the K-edge absorption of Pb element that is inserted into the barite vinyl ester [48]. From Fig. 10, it can be also figured out that increasing rate of colemanite and barite boosts the LAC value, following the pattern V-C0% < V-C20% < V-C50% < V-B0% < V-B20% < V-B50% is reported for the selected vinyl ester composites. Furthermore, the LAC variations for the colemanite vinyl ester series are close to each other and a small difference is realized, while these variations for the barite vinyl ester series occur in a wide range. Moreover, barite series have better shielding performance against gamma photon in comparison with colemanite series because of the higher density of the barite series and the strong relationship between LAC and density of the shielding materials.

Fig. 10
figure 10

Variation in the LAC versus incoming gamma photon energy ranging from a 0 to 15 MeV, b 0 to 0.5 MeV

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The effectiveness of the selected vinyl ester composites against gamma photon was qualified via Half Value Layer (HVL) in Fig. 11. Small HVL values are required for more effective shielding material. The acquired results exhibit a similar trend for both the colemanite and barite series. Furthermore, HVL values rise by increasing the gamma photon energy. In low energy regions, the HVL values are close to each other, and then, a progressive rise is reported for HVL values due to the Photo-Electric mechanism (PE). Thereafter, Compton Scattering (CS) and then Pair Production (PP) mechanisms become dominant in different energy ranges which may cause an increase in HVL values. Moreover, the accretion rate of the colemanite and barite in the vinyl ester will lessen the HVL values and consequently will enhance the shielding performance of the vinyl ester composites. From the obtained results, it can be also found that V-B50% possesses the thinnest HVL value and owns the best gamma photon shielding proficiency [50, 51].

Fig. 11
figure 11

HVL as a function of initial energy up to 15 MeV

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Figure 12a, b displays the fluctuation of the effective atomic number (Zeff) with respect to incident gamma photon energy up to 15 MeV. Although both colemanite and barite vinyl ester undergo Photo-Electric (PE), Compton Scattering (CS), and Pair Production (PP) mechanisms in different energy regions, barite series record jumps in low energy range. On account of the dominance of the photoelectric process in the low energy region, a sudden increase appears in the barite curve which may be due to the K-edge absorption of the Pb element (E = 0.088 MeV) that is inserted into the barite series. The lowest Zeff values were achieved at 1.5 MeV and varied between 8.09 and 11.58 from V-B0 to V-B50 and lie in a range of 3.69 to 5.25 from V-C0 to V-C50. As expected, this behavior (sharp rise and fall) is not seen for V-B0% that has 0 mol% Pb element. After that, Compton Scattering (CS) overcomes and Zeff has a moderate reduction by increasing the gamma photon energy. Above several MeV, the values of Zeff were found to increase again by increasing the gamma photon energy due to the Pair Production (PP) dominance. Figure 12 depicts that barite vinyl ester series have the greatest Zeff because of the larger abundance of Pb element [48, 52]. In conclusion, the sample with higher Zeff will encompass better shielding capabilities, while the sample with the lower Zeff value owns the lower shielding proficiencies. Similar trends are observed in the previous study done by other researchers [10, 50, 52, 53].

Fig. 12
figure 12

Variation in the Zeff versus energy ranging from a 0 to 15 MeV and b from 0 to 1 MeV for coelmanite and barite series

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To describe the shielding ability of two vinyl ester series, Transmission Factor (TF) and Radiation Protection Efficiency (RPE) of the selected vinyl ester series are derived based on MCNPX simulation via dividing final intensity by the initial intensity \(\left( {\frac{I}{{I_{0} }}} \right)\). Thus, the TF% and RPE% versus starting gamma photon energy are depicted in Figs. 13, 14. As expected, in the lower energy range a sharp rise is recorded for TF, while a gradual increase is seen in the higher energy range. This is due to the reduction in the wavelength and as a result decreasing the number of gamma photon interactions inside the selected composites. This fact causes an increase in the penetration power of incoming gamma photons. TF and RPE act opposite to each other. Thus, a different trend is recorded for RPE [16]. The lower TF% occurs for V-B50 which has the higher RPE%. Figures 13, 14 also emphasize that the sample coded as V-B50% which contains more amount of Pb element (0.008 mol%) exhibits better gamma photon performance.

Fig. 13
figure 13

Variation in the TF% versus incoming energy ranging from 0 to 15 MeV

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Fig. 14
figure 14

Variation in the RPE% versus incoming energy ranging from 0 to 15 MeV

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The equivalent atomic number (Zeq) of the selected vinyl ester series is presented in Fig. 15 via Phy-X: PSD software for wide energy ranging from 0.015 to 15 MeV. As anticipated, the equivalent atomic number’s behavior composed of various elements is similar to atomic numbers of an element for studied shielding materials, which characterizes the properties of these materials in terms of an equivalent element. The Zeq values undergo sharp rise, gradual reduction, and stable trends in low, intermediate, and high energy regions, respectively. This trend may be due to the atomic number (Z) dependency of cross section for different gamma photon mechanisms. That means, the Zeq is proportional to Z−1, Z, and Z2 in low, intermediate, and high energy ranges that are in agreement with the previous findings [54]. Hence, a descending order of (Zeq) V-B50% > (Zeq) V-B20% > (Zeq) V-B0% > (Zeq) V-C50% > (Zeq) V-C20% > (Zeq) V-C0% is obtained for Zeq curves related to vinyl ester series. It can be also figure out that the V-B50% owns the largest amount of high Z materials and possesses the highest equivalent atomic number.

Fig. 15
figure 15

Variation in the equivalent atomic number (Zeq) versus starting gamma photon energy for colemanite- and barite-containing resources

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The variation in the Exposure Buildup Factor (EBF) versus incoming energy at penetration depths of 1, 10, 20, 30, and 40 mfp is illustrated in Figs. 16a, b, 17a, b using newly developed Phy-X: PSD software [47]. In low energy region, EBF values are small that may be account for the dominance of the Photo Electric (PE) process in this range where all gamma photons are removed from the shielding materials. After that, by increasing the incoming gamma photon energy the Compton Scattering (CS) overtakes the Photo Electric (PE) mechanism, and as a result, it multiples the Compton Scattering (CS) process. Given this fact, the EBF increases up to Epe and reaches maximum. Then, in high energy region where the Pair Production (PP) process is dominant, the EBF value achieves a minimum level [50]. Moreover, the EBF value for both the colemanite and barite series rises by increasing the penetration depth from 1 to 40 mfp that is due to the increase in the number of scattered photons. The obtained results also reveal that increment of the penetration depth up to 40 mfp leads to a sharper peak at Epe that is because of the buildup of secondary gamma generated by electron–positron annihilation due to multiple scattering processes. According to the information that is mentioned above, by increasing the penetration depth up to 40 mfp, the thickness of the interaction rises, which may increase the scattering process.

Fig. 16
figure 16

Variation in the EBF versus starting gamma photon energy for V-C0% ranging from a 0.015 to 15 MeV and b 0.015 to 1 MeV, at 1, 10, 20, 30, and 40 mfp

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Fig. 17
figure 17

Variation in the EBF versus starting gamma photon energy for V-B0% ranging from a 0.015 to 15 MeV and b 0.015 to 1 MeV at 1, 10, 20, 30, and 40 mfp

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In addition, from Fig. 18 we conclude that in the low and intermediate energy regions, the EBF values for 10 mfp vary inversely with Zeq while in the high energy region (higher than 5 MeV) the EBF values of the present vinyl ester composites become directly proportional to Zeq that is accounting for the dominance of the Pair Production (PP) process in this range. Thus, V-B50% that possesses the highest Zeq values own the higher EBF in high energy regions which may be due to the highest amount of Pb element.

Fig. 18
figure 18

Variation in the EBF and Zeq versus energy for colemanite and barite resources

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The accumulation of the gamma photon that could be extracted with the help of the Energy Absorption Buildup Factor (EABF) is demonstrated graphically in Fig. 19a, b for the largest penetration depth of 40 mfp. The EABF/EBF ratio equals 1 for the low energy range which can be discussed that the estimated values of EABF equal to EBF. Afterward, colemanite and barite series act as the following pattern.

Fig. 19
figure 19

Accumulation of the gamma photons versus incoming energy ranging from a 0.015 to 15 MeV and b 0.015 to 1 MeV for V-C20% and V-B20% resources at 40 mfp

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The ratio of EABF/EBF reaches about 1.7 at Epe for colemanite series which is referring to the accumulation of gamma photon inside the colemanite series due to the dominance of Compton Scattering (CS), while the EABF/EBF ratio of the barite series has a sharp reduction to about 0 at Epe that is because of the K-edges absorption of Pb element in barite series. Above Epe, the EABF/EBF ratio drops down to lower than 1 for the colemanite series which may be due to the prevalence of the Pair Production (PP) process, while barite series undergo a rapid rise and gradual fall. All these observations can be justified in the following way: the case of EBF > EABF (EABF/EBF < 1) refers to that many gamma photons will be absorbed in the air as EBF is based on the energy absorption response of air, while the case of EBF < EABF (EABF/EBF > 1) refers to that many gamma photons that will be absorbed inside the selected colemanite and barite series because EABF relates to those gamma photons absorbed in the attenuating shielding materials.

Conclusions

The current investigation focuses on the determination of the neutron–gamma photon attenuation characteristics of two colemanite and barite vinyl ester composites based on MCNPX simulation, analytical calculations, and newly developed Phy-X: PSD software. New methods, namely Watt Fission Distribution (WFD) and Doppler Effect (DE), were applied to derive the neutron and gamma photon spectra of two colemanite and barite vinyl ester series which can be pioneer for future researches. On the other hand, the neutron shielding capacity of different polymeric materials so-called FNRCS is another new aspect of this project which can be so beneficial for future investigations. In addition, some related parameters such as HVL, MFP, Zeff, TF, RPE, and EBF that provide an important information about neutron–gamma shielding strength of colemanite and barite vinyl ester composites are evaluated. From the reported outcomes one can be concluded that:

  • The WFD shows that the colemanite series flux is larger than the barite series which proves that colemanite series have excellent neutron shielding properties in comparison with the barite series.

  • Colemanite-containing resources have the best shielding performance against neutron, and less volume is required to diminish the initial intensity of half of it.

  • The neutron can track the shortest distance between two subsequent collisions in V-C0% in comparison with the rest of the vinyl ester composites.

  • The V-B series have the largest gamma flux in comparison with the V-C series in a wide energy range.

  • Barite series have the highest LAC values and also a sharp peak is observed due to the K-edge absorption of the Pb element in barite series.

  • The V-B50 owns the lowest TF%, which it possesses the highest RPE%.

  • Before 5 MeV the EBF values for 10 mfp vary inversely with Zeq, while after 5 MeV, they are proportional with each other.

  • Among the studied vinyl ester composites, V-B0% owns the highest HVL value of 0.1467598 cm.

  • V-C50% has the best gamma photon shielding quality in comparison with the rest of the composites.