Mean Free Path Research Articles

Intermediate Diffusive-Ballistic Electron Conduction Around Mesoscopic Defects in Graphene.

Nondiffusive effects in charge transport become relevant as device sizes and features become comparable to the electronic mean free path. As a model system, we investigated the electric transport around mesoscopic defects in graphene with scanning tunneling potentiometry. Diffusive and ballistic contributions to the scattering dipole are probed by simultaneously resolving the nanoscale topography of pits in the graphene layer and measuring the local electrochemical potential in the surrounding area. We find evidence of transport in the intermediate regime between the diffusive and ballistic limits such that the magnitude of the electrochemical potential around the defects is substantially underestimated by diffusive models. Our experiments and modeling are supported by lattice-Boltzmann simulations, which highlight the importance of the ratio between defect size and mean free path in the intermediate transport regime. The magnitude of the scattering dipole depends on the shape of the pits in both the ballistic and diffusive transport modes. Remarkably, ballistic contributions to the electron transport are found at feature sizes larger than the mean free path and rapidly increase at lower sizes, having a noticeable impact already on mesoscopic length scales.

On the Diffusion Approximation of the Stationary Radiative Transfer Equation with Absorption and Emission

Abstract In this paper, we study the distribution of temperature of a body due to the transfer of radiation. Specifically, the boundary value problem for the stationary radiative transfer equation is considered. In all the analysis, we assume the so-called local thermal equilibrium (LTE), i.e., there is a well-defined temperature of the body at each point. We consider the limit in which the mean free path of the photons is much smaller than the characteristic length of the domain. In this case, we can approximate the solution by means of the so-called diffusion approximation. The analysis of this paper is restricted to the case in which the absorption coefficient is independent of the frequency $$ \nu $$ ν (the so-called Grey approximation). We ignore also scattering effects. Under these assumptions, we show that the density of radiative energy u, which is proportional to the fourth power of the temperature, solves in the limit an elliptic equation. The boundary values for that limit equation can be determined uniquely analyzing a suitable boundary layer problem. The method developed here allows to prove all the results using maximum principle arguments for a class of non-local elliptic equations.

Far-from-equilibrium attractors with Full Relativistic Boltzmann approach in 3+1D: moments of distribution function and anisotropic flows vn

We employ the Full Relativistic Boltzmann Transport approach for a conformal system in 3+1D to study the universal behaviour in moments of the distribution function and anisotropic flows. We investigate different transverse system sizes R and interaction strength η/s and identify universality classes based upon the interplay between R and the mean free path; we show that each of this classes can be identified by a particular value of the opacity γ^, which has been previously introduced in literature. Our results highlight that, at early times, the inverse Reynolds number and momentum moments of the distribution function display universal behaviour, converging to a 1D attractor driven by longitudinal expansion. This indicates that systems of different sizes and interaction strengths tend to approach equilibrium in a similar manner. We provide a detailed analysis of how the onset of transverse flow affects these moments at later times. Moreover, we investigate the system size and η/s dependence for the harmonic flows v2,v3,v4 and their response functions, along with the impact of the η/s and the system transverse size on the dissipation of initial azimuthal correlations in momentum space. Finally, we introduce the normalised elliptic flow v2/v2,eq, showing the emergence of attractor behaviour in the regime of large opacity. These results offer new insights into how different systems evolve towards equilibrium and the role that system size and interaction play in this process.

Three-dimensional diamond planar spiral detectors

Diamond’s superior carrier transport properties and unparalleled radiation tolerance make it an ideal material for alpha/neutron detection. High performing diamond detectors are already commercially available. However, even high quality single crystal diamond can degrade after high doses of radiation, resulting in a reduction in carrier mean free path. It is well known that reducing the carrier collection distance, by decreasing the detector electrode spacing, makes radiation detectors more tolerant of mean free path reduction, and therefore more resilient to high radiation doses. One approach for thin device fabrication involves using thin diamond substrates, which can be fragile. In this work, a thin detector has been fabricated using a thick, highly resilient 300 μm diamond substrate by utilising a 3D network of laser-written nano-carbon network electrodes. An optimised femto-second laser write process, utilising specialised optical arrangements, is used to realise planar configured diamond detectors, comprising two Ti/Pt/Au spiral electrodes, connected to internal spiral nano-carbon network ’wall’ electrodes, which extend 20 μm below the surface and have a 50 μm separation. It was found that introducing the nano-carbon network electrodes greatly improved the detector resolution and Charge Collection Efficiency. With close to 100% charge collection efficiency and ns rise times demonstrated, achieving “thin” detector performance, in “thick”, structural substrates.

Evaluation of the contribution of the mean free path in Co to the magnetoresistance in Co/Cu multilayers

Evaluation of the contribution of the mean free path in Co to the magnetoresistance in Co/Cu multilayers

High-Stability WSe2 Homojunction Photodetectors via Asymmetric Schottky and PIN Architectures

High-stability photovoltaic devices are crucial for low-power or passive applications in fields such as renewable energy, wearable electronics, and deep-space exploration. However, achieving stable and controllable doping in two-dimensional (2D) materials remains challenging, hindering the optimization of photovoltaic performance. Here, we fabricate three high-performance, self-driven photodetectors based on layered WSe2 with varying doping concentrations. By leveraging asymmetric Schottky barriers and introducing a defect-free, high-bandgap intrinsic region with a long mean free path, we construct a positive–intrinsic–negative (PIN) vertical homojunction that significantly enhances the photogenerated voltage, photon absorption, and carrier transport efficiency. The resulting PIN junction exhibits a photogenerated voltage of up to 0.58 V, a responsivity of 0.35 A/W, and an external quantum efficiency of 83.9%. Moreover, it maintains a reverse saturation current as low as 0.2 nA at 430 K. These results provide a promising route toward the development of high-responsivity, high-stability van der Waals devices and highlight the potential for 2D material-based technologies to operate reliably under extreme conditions.

On the Invalidity of the Extended Navier-Stokes Equations to Compute Rarefied Gas Flows in a Cylinder Array

On the Invalidity of the Extended Navier-Stokes Equations to Compute Rarefied Gas Flows in a Cylinder Array

Preparation and characterization of rubber-lead nanocomposite material as a protective shield against cobalt gamma radiation

The current study investigates the potential of styrene-butadiene rubber (SBR) nanocomposite reinforced with nano-lead (N-Pb) as a protective shield against gamma radiation emitted from a Cobalt-60 (Co-60) source. The influence of varying N-Pb concentrations (50–300 parts per hundred parts of rubber, pphr) on the structural, morphological, and radiation-shielding characteristics was investigated. The nanocomposite was characterized using several analytical techniques, including X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FTIR), and scanning electron microscopy (SEM). The results demonstrate that increasing the N-Pb concentration leads to enhanced crystallinity, improved homogeneity, and significantly enhanced gamma radiation shielding capabilities of the material. Notably, the nanocomposite exhibited a substantial decrease in gamma count rate (31.9% reduction with 300 pphr N-Pb), mean free path (81% decrease), half-value layer (77% decrease), and tenth-value layer (87% decrease) as the N-Pb content increased. These findings highlight the promising potential of SBR-N-Pb nanocomposites as a cost-effective and efficient shielding material for various gamma radiation applications.

Evaluation of Radiation Shielding Effectiveness of Some Metallic Alloys Used in the Nuclear Facilities

Abstract Several nuclear shielding parameters were evaluated for surgical stainless steel grades 304, 304 L, 316, and 316 L. The effective atomic number Zeff, mean free path (MFP), effective electron density Neff, half-value layer (HVL), and effective conductivity (Ceff) of the investigated alloys were evaluated via the mass attenuation coefficient (μ/ρ). The mass attenuation (μ/ρ) coefficients were computed for gamma-ray photons in the energy range from 15 keV to 15 MeV using Phy-X/PSD program. Fast neutron attenuation was analyzed by computing the removal cross section (ΣR, cm−1) using partial density method. The obtained results by Phy-X/PSD program were validated using NIST XCOM and Monte Carlo (MCNP4C) code. The stopping power of these alloys against electron/proton/α-particles was evaluated using the ESTAR and SRIM Monte Carlo code, considering total stopping power and projected range. Furthermore, the transmitted neutron fraction at different neutron energies was calculated using Neutron Calculatro-V2 code. These calculations were performed for thermal neutrons (25.4 eV) and fast neutrons (with energies of 4 and 4.5 MeV). The obtained results showed that 316 L alloy possesses good protection performance against gamma photons, charged particles, fast and thermal neutrons compared with other investigated alloys. Comparison of the calculated values revealed good agreement between Phy-X/PSD, NIST XCOM, and MCNP4C. This work should be informative for the potential uses of these materials in the nuclear industry to build effective radiation shielding.

Radiation shielding properties of xPbF2−20Na2O−10CaO−(70−x) B2O3 glass system

The influence of PbF2 incorporation on the physical and radiation shielding properties of six glass samples, labeled BPbF0 to BPbF25, was investigated. These samples have a composition of xPbF2−20Na2O−10CaO−(70−x) B2O3 where (0≤x≤25 in 5 mol.% increments). The physical parameters, including density and molar volume, were calculated. As PbF2 content increased, the density rose significantly from 2.476 g/cm³ to 4.077 g/cm³, attributed to the substitution of low molecular weight B2O3 with high molecular weight PbF2. Similarly, the molar volume increased from 26.95 cm³/mol to 27.138 cm³/mol, indicating the formation of non-bridging oxygen atoms and the expansion of the glass network. X-ray diffraction (XRD) analysis confirmed the amorphous nature of the samples, as no crystalline peaks were observed. Gamma radiation shielding parameters were theoretically evaluated using Phys-X/PSD software. The mass attenuation coefficient (MAC) and linear attenuation coefficient (LAC) were found to increase with higher PbF2 content at a given energy. Both coefficients decreased linearly between 0.015 and 0.08 MeV with increasing energy, remained nearly constant in the range of 1 to 3 MeV due to Compton scattering, and increased gradually beyond 3 MeV due to pair production. The half-value layer (HVL) and mean free path (MFP) decreased as PbF2 content increased, reflecting improved shielding efficiency. Furthermore, the effective atomic number (Zeff) and effective neutron density (Neff) also increased with higher PbF2 content. Compared to commercial glasses like RS-360 and various concretes, the prepared glass system demonstrated superior gamma-ray shielding properties. These findings suggest that the developed glass system is highly effective for gamma radiation shielding applications.

RADIATION AND FAST NEUTRON SHIELDING PROPERTIES OF NILO ALLOY

Nowadays, alloy materials have become increasingly important in radiation shielding due to their high density, which is crucial for effective radiation attenuation. The goal of this study was to evaluate several radiation shielding parameters, including the mass attenuation coefficient (MAC), half value layer (HVL), tenth value layer (TVL), and mean free path (MFP) for Nilo alloys 36, 42, and 48. These evaluations were conducted across a photon energy range of 1 keV to 100 GeV using the WinXCom software. Additionally, buildup factors (EABF and EBF) were simulated using the geometric progression (G-P) fitting method for energies ranging from 0.015 to 15 MeV, at depths corresponding to 1-40 mean free paths (mfp). Furthermore, fast neutron removal cross sections (ΣR) were determined using the partial density method for energies between 2 and 12 MeV. The results showed that Nilo alloy 48 exhibited the highest MAC and ΣR values, while its HVL, TVL, and MFP were the lowest among the samples. When comparing HVL, TVL, and MFP values of the Nilo alloys to hematite serpentine, it was found that all the Nilo alloy samples had lower values than hematite serpentine. These findings suggest that Nilo alloy 48 offers excellent radiation and fast neutron shielding, making it a promising candidate for the development of lead-free shielding materials.

Enhancing Radiation Shielding Efficiency of Nigella sativa Eumelanin Polymer Through Heavy Metals Doping.

Gamma radiation shielding is necessary for many applications; nevertheless, lead creates environmental risks. Eumelanin, a natural polymer, is a viable alternative, although its effectiveness is limited to lower gamma-ray energy. This research looks at how doping the herbal eumelanin polymer (Nigella sativa) with heavy metals including iron (Fe), copper (Cu), and zinc (Zn) affects its gamma radiation shielding characteristics. The inclusion of these metals considerably increases the linear attenuation coefficient (μ) and mass attenuation coefficient (μm) of eumelanin, especially at lower photon energies where the photoelectric effect is prominent. The μ value of pure eumelanin is 0.193 cm-1 at 59.5 keV. It goes up to 0.309 cm-1, 0.420 cm-1, and 0.393 cm-1 when Fe, Cu, and Zn are added, in that order. Similarly, the mass attenuation coefficients increase from 0.153 cm2/g for pure eumelanin to 0.230, 0.316, and 0.302 cm2/g for the Fe-, Cu-, and Zn-doped samples. At intermediate and higher energies (661.7 keV-to-1332.5 keV), where Compton scattering is the main interaction, differences in attenuation coefficients between samples are not as noticeable, which means that metal additions have less of an effect. The mean free path (MFP) and radiation protection efficiency (RPE) also show these behaviors. For example, at 59.5 keV the MFP drops from 5.172 cm for pure eumelanin to 3.244 cm for Mel-Fe, 2.385 cm for Mel-Cu, and 2.540 cm for Mel-Zn. RPE values also go up a lot at low energies. For example, at 59.5 keV Cu-doped eumelanin has the highest RPE of 34.251%, while pure eumelanin only has an RPE of 17.581%. However, at higher energies the RPE values for all samples converge, suggesting a more consistent performance. These findings suggest that doping eumelanin with Fe, Cu, and Zn is particularly effective for enhancing gamma-ray shielding at low energies, with copper (Cu) providing the most significant improvement overall, making these composites suitable for applications requiring enhanced radiation protection at lower gamma-ray energies.

Gamma-radiation insulating performance of AlON-hardened Na2O–Bi2O3–SiO2–BaO–Fe2O3–ZrO2 glasses

Aside high radiation cross-section, high mechanical strength is an essential quality for durable and effective glass shields. Many emerging glass shields are brittle with low strength parameters; consequently, limiting their longstanding applications. In this study, the use of AlON (aluminium oxynitride) to increase the hardness of a Zr-based glass system and the consequent effects on the glass density and gamma shielding capacity were investigated. AlON was produced from a combination of AlN and Al2O3 powders through the solid-phase reaction process at 1750 °C. The melt-and-quench process was then used to make the Zr-based (Na2O–Bi2O3–SiO2–BaO–Fe2O3–ZrO2) glass. The glass was homogeneously mixed with varying quantities (0 (GZr8), 4 (GZr8Al4), and 8% (GZr8Al8) by weight) of AlON powder. Using the FLUKA Monte Carlo code, the gamma photon interaction parameters of the AlON-doped glasses were obtained. The density of the glasses increased from 2.90 to 3.11 g/cm3 as the AlON mass proportion increased from 0 to 8%. For GZr8, GZr8Al4, and GZr8Al8, the mass attenuation coefficient had values in the range 0.0316–38.9421 cm2/g, 0.0315–38.8504 cm2/g, and 0.0311–37.0391 cm2/g, respectively. The range of the half-value layer and mean free path for 0.015–15 MeV photons is about 0.01–7.54 cm and 0.01–10.87 cm for GZr8, 0.01–7.19 cm and 0.01–10.38 cm for GZr8Al4, and 0.01–7.14 cm and 0.01–10.31 cm for GZr8Al8. The introduction of AlON into the glass matrix quenched photon buildup factors and enhanced the photon shielding ability of the GZr8 glass system. GZr8Al8 can displace many existing shielding materials, including glasses, concrete, and rocks, based on the analysis of the obtained results. Aside high gamma shielding efficiency, the mechanical strength and Pb-free nature are other attractive features that give the AlON-doped glasses an edge over many existing gamma shielding materials. The present glass system is useful for durable gamma-ray shielding of small-scale gamma sources gamma sources applied in medicine and radiation research.

Spin polarised quantised transport via one-dimensional nanowire-graphene contacts

Graphene spintronics offers a promising route to achieve low power 2D electronics for next generation classical and quantum computation. As device length scales are reduced to the limit of the electron mean free path, the transport mechanism crosses over to the ballistic regime. However, ballistic transport has yet to be shown in a graphene spintronic device, a necessary step towards realising ballistic spintronics. Here, we report ballistic injection of spin polarised carriers via one-dimensional contacts between magnetic nanowires and a high mobility graphene channel. The nanowire-graphene interface defines an effective constriction that confines charge carriers over a length scale smaller than that of their mean free path. This is evidenced by the observation of quantised conductance through the contacts with no applied magnetic field and a transition into the quantum Hall regime with increasing field strength. These effects occur in the absence of any constriction in the graphene itself and occur across several devices with transmission probability in the range T = 0.08 − 0.30.

Controlling the Radiation Shielding and Fluorescence Features of Polyvinyl Alcohol/Polyvinyl Pyrrolidone/Polyethylene Glycol by Incorporation with Erbium Doped Cobalt Ferrite Nanofillers

Our research described in this paper investigated the potential of polyvinyl alcohol (PVA), polyvinyl pyrrolidone (PVP), and polyethylene glycol (PEG) as a matrix for blended polymer nanocomposites, given their unique characteristics including biodegradability, mechanical strength, and water solubility. The incorporation of nanofillers, such as CoFe2-xErxO4 nanoparticles, significantly improved the performance of these composites, especially in optical and radiation attenuation applications. Key advancements include enhanced gamma-ray and neutron shielding, achieved through the high mass attenuation coefficients of the ferrite-based fillers. This research focused on the synthesizing of PVA/PVP/PEG blended polymer composites with CoFe2-xErxO4 nanoparticles via a casting method, with a detailed analysis of their structural, optical, and radiation-shielding properties. The findings aim to contribute to the development of sustainable, high-performance materials for advanced technological applications. The structure and crystallite sizes of the fillers were examined using the X-ray diffraction technique. The impact of the fillers on the structures and morphologies of the blends were also studied. According to CIE 1931 chromaticity diagrams samples, under an excitation wavelength of 380 nm showed a range of blue-violet hues, whereas samples under an excitation wavelength of 434 nm showed a range of violet-cyan hues, depending on the concentration of Er in the fillers. An increase in linear attenuation coefficient (LAC) at all photon energy levels and in mass attenuation coefficient (MAC) at lower energy levels was observed when the host blend was doped with CoFe2-xErxO4. Inclusion of CoFe2-xErxO4 into the host blend decreased both the half value layers (HVL) and tenth value layers (TVL), with this reduction becoming increasingly significant as the concentration of Er in the ferrite fillers was increased. Importantly, the incorporation of CoFe2-xErxO4 positively influenced the mean free path (MFP). In the low photon energy spectrum, the effective nuclear number (Z eff) values for the doped blends rose with an increasing amount of Er present in the ferrite fillers. The Z eff values displayed minimal variation in the middle to higher photon energy ranges with changes in Er content in the filler materials. A higher concentration of Er in the co-ferrite samples was associated with increased fast neutron removal cross-section (FNRCS) values. The filler’s Er content rose, while the exposure buildup factor (EBF), and energy absorption buildup factor (EABF) values changed slightly.

Polycarbonate Nanocomposite Films for Application in Electromagnetic Shielding

Polycarbonate nanocomposites (PC NCs) containing zinc oxide (ZnO), zinc sulfide (ZnS), and cerium oxide (CeO2) nanoparticles (NPs) can be effective alternatives to the γ ray shielding materials which are based on toxic lead. PC NCs shielding materials, free from lead, can be used for the shielding of γ rays in a wide range of energy by varying the type of NPs. The mass attenuation coefficient (MAC) and linear attenuation coefficient (LAC) were calculated for the PC/ZnO, PC/ZnS, and PC/CeO2 NCs together with the pure PC polymer using the WinXCom program. The attenuation coefficients were then used to compute the half value layer (HVL) and tenth value layer (TVL). The effective atomic number (Zeff), effective electron density (Neff), mean free path (MFP), atomic cross section (ACS), and electronic cross section (ECS) of the studied samples were computed as well. All of these shielding parameters were calculated for the incident energy range of the γ rays between 0.015 and 15 MeV, for the PC/ZnO, PC/ZnS, and PC/CeO2 NCs together with the pure PC polymer. The results indicated that the PC/CeO2 polymer NC had the best γ ray shielding properties when compared with PC/ZnS and PC/ZnO polymer NCs. Therefore, we concluded that the PC/CeO2 polymer NC can be a suitable candidate for application in radiation shielding.

Entropy driven inductive response of topological insulators

3D topological insulators are characterized by an insulating bulk and extended surface states exhibiting a helical spin texture. In this work, we investigate the hyperfine interaction between the spin-charge coupled transport of electrons and the nuclear spins in these surface states. Previous work has predicted that in the quantum spin Hall insulator phase, work can be extracted from a bath of polarized nuclear spins as a resource. We employ nonequilibrium Green’s function analysis to show that a similar effect exists on the surface of a 3D topological insulator, albeit rescaled by the ratio between electronic mean free path and device length. The induced current due to thermal relaxation of polarized nuclear spins has an inductive nature. We emphasize the inductive response by rewriting the current-voltage relation in harmonic response as a lumped element model containing two parallel resistors and an inductor. In a low-frequency analysis, a universal inductance value emerges that is only dependent on the device’s aspect ratio. This scaling offers a means of miniaturizing inductive circuit elements. An efficiency estimate follows from comparing the spin-flip induced current to the Ohmic contribution. The inductive effect is most prominent in topological insulators which have a large number of spinful nuclei per coherent segment, of which the volume is given by the mean free path length, Fermi wavelength and penetration depth of the surface state.

Two-dimensional particle-in-cell simulation to investigate electron heating mechanism in intermediate-pressure RF hollow cathode discharges

Abstract While electron heating in a low-pressure DC hollow discharge occurs by the electron pendulum effect in the hole, the heating mechanism of RF hollow discharges at high collisionality is not fully elucidated yet. Phase-resolved analysis with a two-dimensional particle-in-cell simulation enables the investigation of how the electron heating and density distribution changes by the electrode shape, bias voltage, secondary electron emission (SEE), and gas pressure in the RF hollow cathode discharge. Multiple density peak occurs depending on the energy relaxation mean free path of electrons, and anomalous electron heating occurs between the density peaks. Spatio-temporal analysis of electron heating at various locations is provided for DC bias and SEE variations.

Evaluation of radiation shielding characteristics of B4C, Fe2O3, Al2O3, and their composites

Abstract The growing use of nuclear radiation and the harmful effects of radiation exposure underscore the need for continued research into materials for radiation shielding. In the current study, shielding features of B4C, Fe2O3, Al2O3, and their composites (xB4C/yFe2O3/zAl2O3, x:y:z = 1:1:1, 3:4:3, and 3:3:4 wt%) were evaluated. The samples were analyzed using scanning electron microscopy (SEM) and x-ray diffraction (XRD). The gamma shielding parameters for the samples were determined using the Phy-X/PSD program. The results indicate that the sample S2 (Fe2O3) has a higher capacity to absorb γ-rays than the other analyzed samples. This is also supported by the notably lower mean free path and half-value layer values for S2 compared to the other samples. The experimental values of MAC for bulk and nano-samples at energy 0.662 MeV emitted from 137Cs isotope have been compared. Additionally, sample S1 (B4C) demonstrates significantly higher fast neutron attenuation than the other samples with a removal attenuation coefficient of 0.095 cm−1. Moreover, Thermal neutron scattering cross-sections of prepared samples were calculated for neutron energy of 0.025 eV with a neutron flux of 108 n/cm2/s. Therefore, the study found that thickness 1 mm of S1 sample prevented 67% of the incident neutrons, while thickness 2 mm was able to prevent almost all neutrons, which indicates that S1 sample is better compared to the other investigated samples. Meanwhile, the results indicate that the composites S4, S5, and S6 are highly efficient in shielding both gamma rays and neutrons.

Investigation of radiation shielding properties of CeO2 thin films prepared at different molarities

Abstract In this study, CeO2 thin films were produced using the spin coting method, which is one of the sol–gel methods, in six different molarities. X-ray diffraction (XRD) patterns revealed the characteristic peaks of the films, while Field Emission Scanning Electron Microscopy (FESEM) confirmed their homogeneous structure. Then, radiation shielding parameters like linear absorption coefficient (LAC), mass absorption coefficient (MAC), tent value layer (TVL), mean free path (MFP), and half value layer (HVL) were thoroughly examined. The results showed that increasing molarity had a significant effect on the thickness values of thin films and the absorption parameters were found to improve with increasing molarity. Both LAC and MAC values decrease as the energy level increases, but the increase in CeO2 molarity leads to a strong increase on these coefficients. The HVL value was also found to be 0.42 cm at the lowest energy of 14.957 keV and to be around 10 cm at the greatest energy of 59.543 keV (0.05 M). When the radiation energy applied to the material was raised from 14.957 keV to 59.543 keV, it was found that the MFP values of 0.05 M CeO2 thin films grew gradually from 0.61 cm to 14.51 cm. High energy radiation of 59.543 keV and a low density (0.05 M) medium resulted in peak TVL values of 33.423 cm, allowing the radiation to pass through the material with minimal interaction.