TeO₂-PbO₂-Bi₂O₃-B₂O₃ glasses were prepared via. melt-quenching technique. The TeO₂, PbO₂, Bi₂O₃, and B₂O₃ oxides were weighed, and a batch of 15 g was prepared. The oxides were then ground together in an agate mortar for 30 min. The mixture was transferred to an alumina crucible. The batch was melted at 1000°C for 20 min. The molten mixture was quickly poured onto a pre-heated brass plate. To prevent cracking due to thermal stress, the glass discs were annealed for 2 h, then allowed to cool slowly. The photo of the samples is shown in Fig. 1.

The density was measured using the Archimedes method as [23,24,25]:

The chemical composition and density are listed in Table 1. The physical parameters are calculated using the standard formulae described in previous works and are provided in Table S1 of the supplementary information [26,27,28].

The elastic moduli were calculated using the Makishima-Mackenzie model [29,30,31]. The calculations involve the total atomic packing density (V_t) and the dissociation energy (G_t) as two input parameters:

In these equations, x_i is the mole fraction, and V_i is the packing density factor of the i-th oxide component.

The elastic moduli are determined as [26,27,28]:

Young’s modulus (E):

Bulk modulus (B):

Shear modulus (G):

Longitudinal modulus (L):

The absorption spectra were recorded using a PerkinElmer Lambda 19 UV-Vis spectrophotometer from 190–1000 nm. The E_g has been estimated using Tauc’s plot method. This method relates the absorption coefficient (α) and photon energy (hν) as [32,33]:

The quantity α h ν 1 / 2 is plotted against hν. The linear portion of this curve is then extrapolated to the x-axis where absorption is zero. The intercept provides the value of E_g. The E_g was utilized to calculate additional optical parameters and is shown in Table S2 of the supplementary information.

Optical electronegativity (χ*):

Linear dielectric susceptibility (χ⁽¹⁾):

Third-order nonlinear optical susceptibility (χ³):

Nonlinear refractive index (n₂^optical):

The linear attenuation coefficient (LAC) is obtained as: (13) I t = I 0 e − L A C . t I₀ and I_t are the incident and transmitted intensity of photons through a material of thickness t.

The MAC can be obtained as:

The HVL and MFP are obtained as:

The Phy-X software enables the accurate calculation of the MAC directly from a material’s chemical composition across various energy levels [34]. It automates the derivation of secondary shielding metrics, including the LAC, HVL and MFP. By replacing time-consuming manual calculations, Phy-X/PSD provides an efficient and reliable platform for the theoretical screening, comparison, and design of novel shielding products prior to physical fabrication. The obtained data is presented in Tables S3–S6 of the supplementary information.

The ρ increases from 4.759 (PbBi10) to 5.561 g cm⁻³ (PbBi16) as shown in Fig. 2. The lighter B₂O₃ (69.62 g mol⁻¹) is being progressively replaced by the much heavier HMOs, PbO₂ (239.2 g mol⁻¹) and Bi₂O₃ (465.9 g mol⁻¹) [35,36]. This also results in an overall rise in the mass of the glass matrix sharply with doping. The average molar mass (M_m) of the glass composition rises from 153.211 to 187.165 g mol⁻¹. The V_m increases from 32.194 to 33.657 cm³ mol⁻¹ (Fig. 2). This implies that the rate of volume expansion exceeds the rate of mass accumulation [35,36]. The addition of Pb⁴⁺ and Bi³⁺ ions expands the glass network, increasing the free volume (Fig. 2). The concentration of dopant metal ions (N) increases from 1.871 to 2.863 × 10²¹ ions cm⁻³. As the mole percentage of PbO₂ and Bi₂O₃ increases, the N naturally rises. The separation between boron atoms (d_B-B) decreases from 3.767 to 3.559 × 10⁻⁸ cm. As the content of B₂O₃ is reduced from 50 to 38 mol%, the boron structural units rearrange into smaller domains as shown in Fig. 3. This results in a slight reduction in the average inter-atomic distance between boron centers. The r_i decreases from 8.116 to 7.042 × 10⁻⁸ cm. The polaron radius (r_p) decreases from 3.270 to 2.838 × 10⁻⁸ cm. The ions come closer with increasing metal-ion number density, as shown in Fig. 3. This enhances the localisation of charge carriers, as indicated by the shrinking polaron radius [37]. The field strength (F) increases from 1.870 to 2.484 × 10¹⁵ cm². The reduction in r_i leads to a stronger local electric field. It influences the polarization of the surrounding oxygen ions. The OPD decreases from 80.761 to 75.468 as shown in Fig. 4. A lower OPD signifies a loosely packed structure. The addition of HMOs breaks the continuous B-O-B linkages. It results in the formation of NBOs and the disruption of the rigid network. The OMV increases from 12.382 to 13.251 cm³ mol⁻¹ as shown in Fig. 4. It confirms the increase in volume available per mole of oxygen. The elastic moduli decrease with the addition of PbO₂ and Bi₂O₃, as shown in Fig. 5. The mechanical stiffness reduces as the formation of NBOs disrupts the network. It is due to the replacement of stronger B-O bonds by weaker Pb-O and Bi-O bonds. The decrease in OPD also creates voids. It makes the material more susceptible to deformation under stress.

The Tauc’s plot is shown in Fig. 6. The E_g decreases from 2.969 eV to 2.813 eV as shown in Fig. 7. It is a direct consequence of structural depolymerization. The formation of NBOs creates localized defect states at the top of the valence band. The electrons in NBOs are less tightly bound than those in bridging oxygens. The energy required for an electronic transition is reduced. The n increases from 2.405 to 2.449 as shown in Fig. 7. This is due to the high concentration of HMOs. Pb⁴⁺ and Bi³⁺ are highly polarizable cations with larger ionic radii. It increases the electron density and polarizability resulting in a stronger interaction with incident light. The dielectric constant (ε) increases from 5.786 to 5.999. The optical dielectric constant (ε_opt) increases from 4.786 to 4.999. Since the dielectric behavior follows the refractive index trend. The R_m and α_m increase from 19.789 to 21.035 cm³ mol⁻¹ and 7.849 to 8.343 × 10⁻²⁴ cm³. The increase in R_m and α_m confirms the electronic softening of the glass. The electron clouds surrounding the oxygen and HMOs become more deformable due to formation of NBOs. The electronic polarizability (α_e) increases from 8.726 to 8.871 × 10²³. It is due to the specific contribution of electronic displacements to the total polarizability, further supporting the increase in the refractive index. The reflection loss (R_L) and transmission (T) increases from 0.170 to 0.177 and decrease from 0.709 to 0.699, respectively, as shown in Fig. 8. The reflection at the air-glass interface becomes more significant with a rise in refractive index. It leads to higher reflection losses and a reduction in optical transmission. The Metallization (M) decreases from 0.385 to 0.375. A less than 1 value of M indicates that the glasses are insulators. The decrease in M indicates a shift towards semi-conducting behavior [35,36]. The decrease in E_g increases the metallic character. The χ* decreases from 0.798 to 0.756. This parameter quantifies the ability of the anion matrix to hold valence electrons. It indicates that the electrons are less tightly held in the high-Pb/Bi glasses compared to the base glass. It is consistent with the formation of loosely bound NBOs. The χ⁽¹⁾ increases from 0.381 to 0.398. This linear optical parameter is directly derived from the refractive index and confirms the material’s increasing linear response to an optical field. The χ³ decreases from 1.751 to 1.548 × 10⁻¹⁵ esu. The n₂^optical decreases from 2.742 to 2.382 × 10⁻¹⁴ esu. It is due to the structural constraints imposed by the higher field strength (F) and reduced inter-nuclear distances, which may dampen the anharmonic motion of the electrons required for non-linear effects [37].

The MAC of the TeO₂–Bi₂O₃-PbO₂-B₂O₃ glass systems was computed utilizing Phy-X software. The energy interval (in MeV) employed in this investigation was 0.015 ≤ E ≤ 15. Fig. 9 depicts the MAC pattern of the TeO₂–Bi₂O₃-PbO₂-B₂O₃ glass systems. The MAC for all samples diminished as energy rose. The discontinuities in MAC at 0.04 and 0.1 MeV are due to the K-shell energy of heavy metals like Te, Pb, and Bi. The principles in radiation physics indicate that photons interact with a glass through three principal mechanisms. The initial phenomenon is termed the photoelectric effect (PE), illustrated in Fig. 9 for energy less than 0.8 MeV. It exerts significant effects on low-energy photons. For PbBi10 at 0.015 MeV, the MAC (in cm²/g) was 59.94, but diminished to 3.93 at 0.06 MeV. Identical behavior was noted in the PbBi12, PbBi14 and PbBi16 samples within this energy range. The second phenomenon, termed Compton scattering (CS), exhibits a cross section that is largely independent of elemental composition. As illustrated in Fig. 9, all samples within the range of 0.8 to 4 MeV exhibited roughly comparable MAC values. As an illustration, for PbBi10 and PbBi12 at 2 MeV, the MAC was 0.0439 and 0.0441 cm²/g, respectively. Pair production (PP) is the most crucial component for high photon energy region. The probability of occurrence of this phenomenon is proportional to Z². The MAC at the last few energies grew gradually up to 15 MeV, the maximum energy used in this investigation, as shown in Fig. 9. The findings of the attenuation graph showed that MAC rose when the concentrations of PbO₂ and Bi₂O₃ varied between 10 and 16 mol%. The sample containing the highest concentrations of PbO₂ and Bi₂O₃ (PbBi16) exhibited the greatest attenuation factor, while conversely, the sample with the lowest concentrations showed the least.

The LAC value, as seen in Fig. 10, was variable with energy and typically diminished sharply as the energy rose from 15 keV to 6 MeV. The glass reduces the intensity of low energy photons, preventing most from penetrating it. With increasing energy, the photons traversed the sample easily, indicating that the LAC was comparatively low for high-energy photons. As an illustration, for PbBi10 at 0.15 and 8 MeV, the LAC values were 4.99 and 0.171 cm⁻¹, respectively. The high-density material offers superior shielding compared to low-density glass. The PbBi16 sample possesses the greatest LAC values across all energy levels. PbBi10 exhibited the lowest density and the lowest LAC. Consequently, PbBi16 offers superior shielding owing to enhanced contacts compared to the PbBi10, PbBi12 and PbBi14 samples.

Two further parameters commonly employed to describe photon attenuation are the HVL and MFP. A reduced HVL and MFP are advantageous as they require a smaller space to attenuate an equivalent quantity of photons. The HVL of the TeO₂-Bi₂O₃-PbO₂-B₂O₃ glass systems is plotted in Fig. 11. It illustrates that PbBi16 exhibited the lowest HVL and optimal attenuation, attributable to the elevated weight fractions of Bi₂O₃ and PbO₂, as well as its comparatively high density. Conversely, the PbBi10, which had the lowest density and the least quantity of Bi₂O₃ and PbO₂, showed the worst attenuation. The HVL decreased with a rise in energy, leading to better attenuation against gamma rays. Additionally, Fig. 11 shows that the HVL increased with increasing energy. This implies that low-energy photons can be adequately shielded by a thin layer, but high-energy photons require a thicker layer. The HVL of the PbBi10–PbBi16 glasses at 0.06 MeV was 0.0371, 0.0341, 0.0315 and 0.0293 cm, whereas at 10 MeV, the values increased to 3.94, 3.62, 3.35 and 3.12 cm, respectively.

The MFP of the TeO₂-Bi₂O₃-PbO₂-B₂O₃ glasses is represented graphically in Fig. 12. The MFP for all glasses exhibited a quick increase up to 0.08 MeV with rising energy, followed by a slowing in the rate of growth. The reduction in MFP was attributed to the increased PbO₂ and Bi₂O₃ levels. A comparison of the MFP values of PbBi10 and PbBi16 revealed that PbBi10 needed a greater MFP than PbBi16. The MFP ratio between PbBi10 and PbBi16 was 1.41 at 0.1 MeV, 1.34 at 0.3 MeV, 1.23 at 0.6 MeV and 1.18 at 2 MeV. This signifies that the MFP ratio between PbBi10 and PbBi16 was comparatively elevated at lower energy, indicating that PbO₂ and Bi₂O₃ influenced the MFP, resulting in enhanced shielding efficiency at lower energies.

The comparative analysis was performed against several previously reported glass matrices to determine the practical standing of the synthesized glass system among existing radiation shields. The chemical compositions and heavy-metal contents of these reference materials differ. The evaluation of the current glasses against glasses modified with WO₃, TeO₂, GeO₂, MoO₃, or SrO allows for a direct assessment of the competitiveness and suitability for advanced industrial and medical applications. Fig. 13a compares the MAC of the TeO₂-Bi₂O₃-PbO₂-B₂O₃ glasses with those reported for WO₃-MgO-B₂O₃-TeO₂-GeO₂ glasses [38]. PbBi16 and PbBi14 have higher MAC than all the WO₃-MgO-B₂O₃-TeO₂-GeO₂ glasses, since it contains relatively high amounts of Bi₂O₃ and PbO₂. PbBi12 has comparable MAC with 15WO₃-35MgO-20B₂O₃-20TeO₂-10GeO₂ glass. Fig. 13b compares the MAC of the TeO₂-Bi₂O₃-PbO₂-B₂O₃ glasses with those reported for SrO-B₂O₃-TeO₂ with different modifiers [39]. All the TeO₂-Bi₂O₃-PbO₂-B₂O₃ glasses have higher MAC than SrO-B₂O₃-TeO₂ glasses with V₂O₅, MnO₂, MoO₃ and TiO₂. Fig. 13c compares the MAC of the TeO₂–Bi₂O₃-PbO₂-B₂O₃ glasses with those reported for zinc boro tellurite glasses modified with PbO, Bi₂O₃ and MoO₃ [40]. PbBi16 has comparable MAC with 10ZnO-35B₂O₃-35TeO₂-20Bi₂O₃. The zinc boro tellurite glass modified with MoO₃ has lower MAC than all the TeO₂-Bi₂O₃-PbO₂-B₂O₃ glasses, while the zinc boro tellurite glass modified with PbO has comparable MAC with PbBi12.