1 Introduction

The active use of radioactive sources in many areas of life has made essential the production of shielding materials to protect against the deleterious impacts of radiation. In this sense, researchers have carried out studies on many shielding materials such as concretes [1, 2], glasses [3, 4], and alloys [5]. Among the these material types, glasses have gained a major attention from researchers due to its physical and chemical properties such as ease of production, optical transparency, high corrosion resistance and eco-friendly [6]. Kavaz et al. [7,8,9] examined the physical, structural and gamma ray protection features of lithium borate glasses obtained by adding various minerals using different codes such as MCNPX and WinXCOM. Sayyed et al. [6, 10, 11] have reported many studies on the optical, structural and multiple nuclear radiation shielding features of various heavy metal-doped glass systems using WinXCOM software, GEANT4 and MCNPX simulation codes. Previously, Shams et al. surveyed the radiation shielding and mechanical properties of the TeO2–ZnO–NiO, PbO–Na2B4O7–CaO–Al2O3–SiO2 and Al2O3–Na2O–B2O3–Bi2O3 glass systems [12,13,14]. In all studies, it is seen that the use of heavy metal-containing oxides in glass synthesis increases density and nuclear radiation shielding ability of the material.

Amorphous alloys, i.e., metallic glasses, have been researched and developed about half a century ago. By adding new metals, many important properties of these glasses have been investigated for utilization in structural and functional applications. Previously, they were frangible and produced in micron thicknesses, which limited the application areas [15]. However, the studies led by a special working group in Japan [16,17,18,19] have overcome the problems in size and during production. Bulk metallic glasses own unique properties such as high toughness [20] and strength, good corrosion resistance [21], high elastic tensile, low internal friction and elastic modulus, good machinability, tensile strength up to 3000 MPa [22]. In addition, their superior performance against high temperature and pressure allows the use of these amorphous alloys in many industrial applications (industry, sports equipment, biomedicine and consumer electronics) [23].

Given the fact that the production of new amorphous alloy systems and many of their properties can be modified depending on the metals used, metallic glasses are extremely attractive materials for technological applications and scientific research. In recent years, Inoue’s works [17, 19, 24,25,26] have shown that the production of multicomponent bulk metallic glasses (BMG) which possess high-glass forming ability (GFA) could be produced. Many amorphous alloys such as ZrTiCuNiBe, TiNiCuSn, CuZrTiNi, NdFeCoAl, LaAlNi, FeCoNiZrNbB, FeAlGaPCB, PrCuNiAl, and PdNiCuP of 5–10-cm thickness were reported to be developed [15]. In the present century, BMGs are now regarded as the most sought materials in structural engineering due to their high mechanical properties [27]. Ion irradiation method is utilized for exploring the microstructure and mechanical properties of BMGs [28]. Information about their hardness, free volume and atomic structure can be obtained by methods such as X-ray diffraction and positron annihilation. Moreover, ion irradiation might also support to manufacture and use novel BMG for structural nuclear material under radiation medium [29].

In addition to the unique features mentioned in the above paragraphs BMGs with high density and both glass and alloy properties appear to be worth investigating for radiation protection studies. Due to the lack of studies on the shielding parameters of metallic glasses in the literature, the radiation shielding competences of NiPdP, CaMgZn, CuHfAl, ZrAlNiCu, PdNiP, and TiZrBeFe metallic glasses were surveyed in the current work. The main shielding parameter which is the mass attenuation coefficient (μ/ρ, cm2/g) was found from the chemical compositions of six metallic glasses. From that parameter, effective atomic number (Zeff), electron density (Nel), and half value layer (HVL) values of the BMGs were also calculated. Exposure buildup factor (EBF) values, another important gamma interaction parameter, were found for 1–40 mfp at 0.015–15 MeV photon energies by the Geometric Progression (GP)-fitting approach. Lastly, fast neutron removal cross-sections (ΣR), projected range (PR) and mass stopping power (MSP) for alpha (He+2) and proton (H1) particles have been found for the particle shielding effectiveness of selected metallic glasses. It can be emphasized that the outcomes obtained from the current study will beneficial for future enforcements of different types of metallic glass in the sense of radiation shielding purposes.

2 Materials and methods

As gamma photons pass through a material of a thickness (x), the intensity of the photons is attenuated, and this attenuation is defined by Beer–Lambert’s law [30]:

$$ \mu = \ln (I_{0} /I)/x, $$
(1)

where the I and I0 denote the incident and transmitted photon intensities and x refers the mass thickness of the irradiated material. \(\mu\) refers the linear attenuation coefficient (cm−1) of the sample and depends on the composition of the samples, the energy of the incident photon, the density and thickness of the sample. Another vital parameter which is the mass attenuation coefficient (\(\mu_{\rho }\), cm2/g) is an indication of the ability of the material to attenuate against gamma radiation. By utilizing the weight fraction (wi) of the ith component of a metallic glass sample, the \(\mu_{\rho }\) can be calculated according to the formula [31]:

$$ \mu_{\rho } = \sum\limits_{i} {w_{i} (\mu /\rho )_{i} } . $$
(2)

The detection of \(\mu_{\rho }\) values of the metallic glasses was acquired at 0.02–20 MeV gamma ray energies with using WinXCOM program [32].

Mean Free Path (MFP) is described as the average length of the path that a photon experiences successive interaction and MFP is given with next relation:

$$ {\text{MFP }} = \, ({1}/\mu ). $$
(3)

Half value layer (HVL) is an advantageous variable for designing and choosing any photon reduction material because it demonstrates the thickness of the material necessary to attenuate the radiation level to 50% of its initial value and HVL is founded with the next equation [33]:

$$ {\text{HVL}} = (\ln \, 2/\mu ). $$
(4)

The effective atomic number (Zeff) is a handy variable for explaining the attenuation of gamma radiation in multi-element medium and ensures information about the target material. Effective atomic number (Zeff) and effective electron density (Nel) of the metallic glasses were calculated according to the following relations:

$$ Z_{{{\text{eff}}}} = \frac{{\sum\nolimits_{i} {f_{i} A_{i} (\mu_{\rho } )_{i} } }}{{\sum\nolimits_{j} {f_{j} \frac{{A_{j} }}{{Z_{j} }}(\mu_{\rho } )_{j} } }} $$
(5)
$$ N_{{{\text{el}}}} = N_{A} \frac{{Z_{{{\text{ef}}f}} }}{\langle A\rangle }{\text{(electrons/g)}}{.} $$
(6)

To use a single atomic number in complex materials, an effective atomic number (Zeff) is needed. By dividing Zeff by the average atomic number, Nel is obtained, which is the number of electrons per unit mass.

Exposure buildup factor (EBF) depends on the energy absorption response of air and EBF is considered equivalent to the measured value of the dose absorbed in air by an unperturbed detector. Basically, an equivalent atomic number (Zeq) must be attributed to each material for the photon energies studied, when buildup factors are calculated for the chosen substance. Since the buildup factor chiefly arises from incoherent scattering, the Zeq is computed employing the ratio of incoherent scattering to total attenuation with incoherent scattering [34]. Geometric Progression (GP) fitting coefficients were determined using an interpolation formula, with the calculation of equivalent atom numbers. GP fitting parameters are obtained from the standard reference database ANSI/ANS-6.4.3. Then, the GP fitting parameters are used exposure buildup factors from the GP fitting formula that has been repeatedly mentioned in previous publications [35,36,37,38]

The effective removal cross-section (\(\sum R\)) is the probability of one neutron undergoing a specific reaction passing across the BMG medium. To compute the sum of \(\sum R\) of investigated samples, the next relation is used for each sample, respectively [1]:

$$ \sum {R = } \sum {W_{i} } \left( {\sum {R/\rho } } \right)_{i} $$
(7)

where Wi and \(\sum {R/\rho }\) refer the partial density(g/cm3) and the mass removal cross-section (cm2/g) of the ith element.

The mass stopping power (MSP) establishes how accomplished a material is in reducing the kinetic energy of the charged particles and projected range (PR) states the average range for stopping the charged particles. The very frequently employed method for finding stopping power and projected range is the computer software SRIM (Stopping and Range of Ions in Matter) developed by Ziegler et al. [39] SRIM-2008 version was utilized in that work [40].

3 Results and discussion

This study is related to extensive investigation of nuclear radiation shielding parameters of multi-element metallic glasses whose chemical formulas, codes and densities are given in Table 1. Figure 1 and Table 2 present the change in \(\mu_{\rho }\) values depending on photon energy. As indicated in Fig. 1, it was found that with the increment in gamma ray energy at the beginning, \(\mu_{\rho }\) declined rapidly and gradually decreased between 0.06 and 0.2 MeV. Initially, this notable reduction in \(\mu_{\rho }\) is related to photoelectric absorption, which is probably the main interaction between low energy photons and all metallic glasses. In all metallic glasses, \(\mu_{\rho }\) is almost constant beyond 0.02 MeV. Compton scattering and pair production are the main processes at medium and high-energy levels. In addition, \(\mu_{\rho }\) values have a sudden increase near 0.07 MeV for MG3 sample. Due to the fact that the possibility of photoelectric absorption occurs, the atomic number varies to Z4−5. Therefore, in the MG3 sample containing Hf, a small peak on the K X-ray absorption edge of Hf (65.33 keV) was seen. Furthermore, the chemical composition of the samples moved the \(\mu_{\rho }\) values and, therefore, all samples indicate comparable protection properties. MG3 and MG4 own the highest \(\mu_{\rho }\) values, whereas the lowest values of \(\mu_{\rho }\) pertain to MG2.

Table 1 Sample codes, name and density of the bulk metallic glasses
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Fig. 1
figure 1

The changes of mass attenuation coefficient of the BMGs with photon energy

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Table 2 The mass attenuation coefficient (\(\mu_{\rho }\), cm2/g) of the metallic glasses
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The HVL outcomes of the samples under examination at 0.02–20 MeV photon energies are plotted in Fig. 2. The results showed a relation between the density of and the attenuation ability of the samples. It is seen that low-density MG6 and MG2 samples have high HVL values. Enhancing the density cause lower HVL, which indicates that the high density influences assertively the shielding capacity of the glasses. Furthermore, as photon energy increases, HVL values of the samples begin to differ significantly. While low-energy photons up to 0.1 MeV need shorter distance to lose their energy, the high-energy photons mislay their energy at a longer thickness. While MG3 sample was more successful in stopping photons, MG6 showed the worst performance. In general, it can be concluded that BMGs are capable of protecting high-energy photons.

Fig. 2
figure 2

HVL values of the metallic glasses

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The Zeff curve as a function of the gamma ray energy for metallic glasses is demonstrated in Fig. 3. It is obvious that MG6 has the lowest Zeff due to elements with low atomic number compared to other samples. MG3 sample significantly outscore from other samples with respect to Zeff. The Zeff values of the studied samples are in the range of 21–46. The Nel values obtained using Zeff values are given in Fig. 4. Here, it is noteworthy that Nel values change in a different trend from the Zeff curve with photon energy. The metallic glasses with lower atomic weight own higher Nel values. MG5 and MG6 have the smallest Nel values, whereas MG2 and MG3 have the highest number of electrons per unit mass.

Fig. 3
figure 3

Variations of Zeff versus gamma energy for metallic glasses

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Fig. 4
figure 4

Variations of Nel versus gamma energy for glasses

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The calculated values of equivalent atomic number (Zeq) for metallic glasses are presented at Table 3. The GP fitting parameters of the EBF of MG2 sample is listed in Table 4 for photon energies of 0.15–15 MeV as an example. Figure 5a–f shows the change of EBF of the samples versus the photon energy at various penetration depths. With some exceptions generally, the EBF has smaller values at low and high energies whereas the higher values of EBF are seen for intermediate energy region. In the low-energy zone, it also showed sudden and high peaks around 0.02 MeV for MG1 and 0.06 MeV for MG3. In the low-energy zone, it showed sudden and high peaks around 0.02 MeV for MG1 and 0.06 MeV for MG3. That could be clarified depending on the partial photon interactions in the energy region. The photoelectric absorption is predominant at low-energy region and gamma photons are entirely absorbed and this could give rise a very small lifetime to photons which is insufficient for them to buildup in the medium. However, as photoelectric cross-section is proportional with Z4−5, in the case of samples including elements with high atomic numbers, abrupt increments occur near the absorption edge of the element. Therefore, the peaks were seen near the absorption edge of Zr in MG1 (17.99 keV) and Hf (65.35 keV) in MG3. There are sharp peaks at 24.35 keV on the absorption edge of the Pd element contained in MG4 and MG5. Peak intensities were varied depending on the percentage of Pd in the content of the MG4 and MG5 metallic glasses. At medium energies, Compton scattering photons are not fully deducted, but their energy are declined and the probability of photons multiplying in the environment increases. When the increases in the absorption edges are ignored, EBF values are high at middle energies. However, in general the EBF values possess maximum value of around 150 in this region. This is also quite good value for a shielding material. MG3 takes the lower EBF values among the other BMGs at medium energies. Electron–positron pair production happens at 1.022 MeV. Therefore, the sudden increments are observed for EBF values of the MG1, MG4 and MG5 at 1 MeV. As the pair production cross-section is proportional to Z2 at high energies, EBF values increased in metallic glasses with higher Zeq values. As the penetration depth rises, secondary photon formation increases and EBF values start to increase faster.

Table 3 Equivalent atomic numbers of the metallic glasses for the energy range 0.015–15 MeV
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Table 4 The EBF GP fitting coefficients (b, c, a, Xk and d) of MG2 sample
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Fig. 5
figure 5

a–f EBF at 0.015–15 MeV and 1–40 mfp for the metallic glasses

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The variation in EBF values of the BMGs depending on the penetration depth was found for selected fixed energies, 0.015, 1.5 and 15 MeV (Fig. 6a–c). It was obtained that at 0.015 MeV, the EBF values are about 1 and are almost constant with increasing photon energy except MG6. The changes in EBF with the penetration depth at 0.015 and 15 MeV are influenced by the elemental composition of the metallic glasses. At 1.50 MeV, At 1.5 MeV, it is seen that the variation of EBF values increases due to the Zeq values of the samples and became biggest at the large penetration depths. Since at high energies pair production process is the main interactions, EBF is raised with increasing Zeq at 15 MeV for 15–40 mfp. The electron–positron pair and the resulting radiation might leave the sample for low penetration depths. But, when the penetration depth is big, the secondary gamma photons contribute to photon buildup and, therefore, MG3 own the largest EBF values for 15 MeV (Fig. 6c).

Fig. 6
figure 6

a–c The EBF for the glasses up to 40 mfp at 0.015, 1.5, and 15 MeV

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The ΣR outcomes of the studied BMGs are obtained and are demonstrated in Fig. 7 and listed in Table 5. It is clear from Fig. 7 that the ΣR values are significantly different from each other and changed between 0.065 cm−1 for MG6 (ρ = 2.69 g/cm3) and 0.166 cm−1 for MG3 (ρ = 10.12 g/cm3). Its higher density than the other metallic glasses clarifies the largest \(\sum R\) value for the MG3 sample.

Fig. 7
figure 7

Effective removal cross-sections of the metallic glasses

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Table 5 Macroscopic effective removal cross-sections of bulk metallic glasses \(\sum R\) (cm−1)
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The Stopping and Range of Ions in Materials (SRIM) codes based on Monte Carlo simulation was written to allow the calculation of ion deposition profiles in materials exposed to energetic ion beam [40]. SRIM Monte Carlo code was employed to research proton and alpha mass stopping power (MSP), further proton and alpha projected range (PR) values of selected metallic glasses. The alpha and proton MSP values of metallic glasses against kinetic energy (KE) are presented in Fig. 8a, b. Initially with increasing kinetic energy, MSP values reach maximum. Then, the MSP values decrease as the kinetic energy further increase. As the elements with high atomic number increases in the metallic glasses, MSP values decrease and MG3 sample owns the lowest MSP values of both alpha and proton in the selected kinetic energy range. This result is also related to the high density of MG3 glass. Furthermore, Fig. 9a, b illustrates the variation of projected range (PR), which is another important proton and alpha protection properties, with kinetic energy. For the best alpha and proton protective materials, the lowest PR is required. Therefore, MG3 sample has the lowest PR values. As a result, it can be said that the MG3 is the perfect shield compared to the other samples to be utilized and developed for alpha, proton, gamma and neutron shielding practices.

Fig. 8
figure 8

Alpha and proton mass stopping power (MSP) as a function of kinetic energy for metallic glass samples

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Fig. 9
figure 9

Alpha and proton projected range (PR) as a function of kinetic energy for metallic glass samples

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4 Conclusion

The nuclear shielding calculations of six bulk metallic glasses were acquired in this work. The \(\mu_{\rho }\) values of the samples under study were found for 0.020–20 MeV photon energies using WinXCOM program. Besides, the HVL thickness of the metallic glasses was estimated using the linear attenuation coefficients. The outcomes exhibited that MG3 and MG6 samples own the largest and smallest HVL among the selected samples. The highest Zeff values were found to belong to samples containing heavy elements in the composition. In addition, it was concluded that the EBF values of MG3, MG4 and MG5 were lower than those of other samples. The outcomes show that as the Zeq values of the metallic glasses increased, the EBF values decreased and the elements Pd, Hf, and Zr, in the samples prevented the buildup of the photons at medium energies. On the other hand, due to the largest \(\sum R\) value of MG3, we can say that it possesses better neutron shielding efficiencies than the others. In addition, it was found that MG3 sample owns the lowest PR and the MSP values among the all of the samples. Taking into account all these, it can be concluded from this study that metallic glasses have superior shielding performance for gamma, neutral and charged particle radiations. It is seen clearly that selected metallic glasses are promising candidates for nuclear shielding applications as compared with some important shield materials.