Hole Radius Research Articles

Analytical Model for Contaminant Transport in the CGCW and Aquifer Dual-Domain System Considering GMB Holes

Composite geomembrane cut-off walls (CGCW) have been widely used for the remediation of polluted sites, especially where the environmental conditions are complex. Accurate predictions of the GMB hole leakage and CGCW performance are essential for engineering design and cost control. This paper establishes empirical equations to predict the leakages through the CGCWs based on the numerical models. Additionally, an analytical solution for contaminant migration through the CGCW is proposed considering the effects of GMB holes. The accuracy of the established equations and analytical solution is verified by the numerical models. The key effects of the GMB thickness (TG), head loss (HG), cut-off wall hydraulic conductivity (kG), hole radius (rG) and shape on the leakage and CGCW performance are investigated. The results show that compared with other hole shapes, the leakage through the circular hole is lowest. This is mainly because the shape factor for the circular hole is 1.15–1.3 times lower than that for other shapes of holes with the same area. Additionally, the effects of the hole geometric properties and head loss on the CGCW performance can be more significant when the cut-off wall hydraulic coefficient is small. For example, the breakthrough time differences between the cases with rG = 0.005 m and 0.05 m are 0.8 and 5.0 years when kG = 10−10 and 10−9 m/s, respectively. This is because the impermeability of the CGCW is good when kG is small. This will weaken the impacts of the hole geometric properties on the leakage. The proposed empirical equations and analytical solution can provide effective suggestions for the design of the CGCW in different GMB hole cases.

Characteristics of stitching holes in plain-woven SiCf/SiC and its influence on tensile behavior

A certain number of stitching holes are randomly distributed in SiCf/SiC composites prepared by CVI method. The existence of these stitching holes disrupts the continuity of the internal fiber bundles and has a certain influence on the mechanical properties of the materials. In order to study the influence of stitching holes on the tensile behavior of plain-woven SiCf/SiC, this paper uses CT technology to scan the interior of plain-woven SiCf/SiC to obtain the morphology and distribution of the internal stitching holes. Then, cells containing stitching holes were modeled to study the influence of different characteristics of stitching holes on the tensile behavior of SiCf/SiC. Finally, the tensile behavior and damage process of the cell with stitching hole at the fracture location of the specimen and the cell with stitching hole near the fracture location were studied by modeling the specimen level cell with the thickness of the specimen. The conclusions are as follows: (1) A certain number of stitching holes were randomly distributed inside the specimen. The inclination angles of the stitching holes are mostly 70°–80°, and the radius of the stitching holes is mostly 0.25 mm–0.28 mm (2) The influence of stitching hole characteristics on the tensile behavior of plain-woven SiCf/SiC is as follows: Position > Hole radius > Inclination angle. Among the positional characteristics, the Type-Ⅱ model has the greatest influence on the weakening of the tensile behavior due to more pronounced stress concentration phenomenon. (3) The stitching hole at the edge of the specimen has the greatest influence on the weakening of the specimen strength and is accompanied by obvious stress concentration. In the section containing the stitching holes, the maximum local stresses in the fiber bundle and matrix are about 1.5 times that of the other two models, resulting in earlier damage initiation and faster accumulation.

Linear-acoustic effects of asymmetrical undercutting of toneholes of woodwind instruments.

The influence of the presence and magnitude of asymmetrical undercutting of the toneholes of woodwind instruments on the change in their effective radius and the corresponding shift of the eigenfrequencies of the air duct is considered. Within the framework of the electro-acoustic analogy, a tonehole with an asymmetrical undercutting along the axis of the main bore is presented as a cascade connection of two holes with different radii (undercutting and not undercutting parts of the hole), and with a sideway undercutting-in the form of a parallel connection. Formulas for numerical calculations are given that make it possible to determine the effective radius of such holes for the open state in the low-frequency approximation. Based on the obtained dependencies, using the transmission-matrix method, the eigenfrequencies of an air duct with one hole were calculated and compared with the results of computer modeling in the COMSOL Multiphysics 5.6 program. It is shown that increasing the degree and angle of undercutting leads to an increase in the effective radius, a displacement of the "center of gravity" of the sound hole along the main bore axis, and has effect on the shift of resonant frequency.

Optimization of highly sensitive three-layer photonic crystal fiber sensor based on plasmonic

In this work, a photonic crystal fiber (PCF) refractive index (RI) sensor has been designed and optimized. The RIs range covered by the sensor is from 1.38 to 1.41. The proposed optical sensor has three layers of air holes with 24, 12 and 6 holes in each layer. The geometry parameters of the proposed sensor (the radius of the air holes and the thickness of the plasmonic layer) have been optimized using the Nelder-Mead algorithm and the FEM numerical with the objective of achieving the highest sensitivity. To achieve an optimized structure with minimal sensitivity to fabrication errors, the rotation angle of the hole layers has been analyzed. The results indicate that, due to the specific geometry of the proposed structure, variations in the rotation angle and displacement of the air holes have no significant impact on the outcomes. The results also indicate that the sensor’s maximum amplitude sensitivity (AS), maximum wavelength sensitivity (WS), and figure of merit (FOM) are 10000 (RIU−1), 23000 (nm/RIU) and 131 (RIU−1) respectively. The optimized design provides high sensitivity, a wide diagnostic range for the detection of the analytes’ RIs, and the advantages of a gold plasmonic layer, ensuring high stability in biological environments. This combination results in enhanced performance of the sensor for various applications particularly in biosensing and medical fields. The designed structural geometry also eliminates the effects of tolerances in manufacturing processes which makes the proposed PCF device a very efficient sensor.

Quasinormal modes, thermodynamics and shadow of black holes in Hu–Sawicki f(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{f(R)}$$\\end{document} gravity theory

We derive novel black hole solutions in a modified gravity theory, namely the Hu–Sawicki model of f(R) gravity. After obtaining the black hole solution, we study the horizon radius of the black hole from the metric and then analyse the dependence of the model parameters on the horizon. We then use the 6th order WKB method to study the quasinormal modes of oscillations (QNMs) of the black hole perturbed by a scalar field. The dependence of the amplitude and damping part of the QNMs are analysed with respect to variations in model parameters and the error associated with the QNMs are also computed. After that we study some thermodynamic properties associated with the black hole such as its thermodynamic temperature as well as greybody factors. It is found that the black hole has the possibility of showcasing negative temperatures and is thermodynamically unstable for feasible values of model parameters. Then we analyse the geodesics and derive the photon sphere radius as well as the shadow radius of the black hole. The photon radius is independent of the model parameters while shadow radius showed fair amount of dependence on the model parameters. We tried to constrain the parameters with the help of Keck and VLTI observational data and obtained some bounds on m and c2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_{2}$$\\end{document} parameters.

Dual effect of string cloud and dark matter halos on particle motions, shadows and epicyclic oscillations around Schwarzschild black holes

In this work, we study the particle dynamics, shadows and quasi-periodic oscillations around the Schwarzschild black hole with three different dark matter halos and string cloud parameters. The motion of particles is considered with the string cloud parameter α and investigated for massive and massless particles. The photon orbits and inner stable circular orbits of these black holes are obtained from the effective potential. The black hole shadows are calculated using three different dark matter halos. The shadow radius of these black holes increases compared to a pure Schwarzschild black hole for different values of the string cloud parameter. Further, quasi-periodic oscillations are also discussed in the current analysis. Our study examines how the radial profiles of orbital and radial harmonic oscillation frequencies change with the string cloud parameter. This study examines the major properties of test particle's quasi-periodic oscillations near stable circular orbits in the black hole equatorial plane. Our findings indicate that observational tests for black holes influenced by dark matter halos and cloud strings are also feasible and viable.

Air temperature characteristics of cable-girder pin joint node of suspension bridge under pool fire

Under the action of high temperature, the mechanical properties of steel structure will be reduced. Cable-girder pin joint nodes (referred to as the node) of suspension bridges are composed of steel structures. Before studying the fire resistance of structures, it is necessary to accurately predict the air temperature characteristics under fire. In this paper, the FDS calculation model of the node is established for the pool fire. The air temperature characteristics of the node are clarified by numerical simulation and theoretical analysis. The effects of main factors such as fire location, wind speed, leakage hole radius and space height are studied. The unfavorable fire scenario and air heating mathematical model are obtained. The main conclusions are as follows: (1) As the fire source distance increases, the peak value of air temperature at the node gradually decreases, while the heating time steadily increases and eventually reaches a stable state. (2) The air temperature at the node initially increases and then gradually decreases as the wind speed increases. (3) Increasing the leakage hole radius from 0.02 m to 0.06 m results in a decrease of approximately 60 % in the air heating time at the node. (4) Increasing the space height from 0.25 m to 1.7 m leads to a maximum decrease of 72 % in air temperature at the node. (6) In the unfavorable fire scenario, the peak air temperature at the node can reach approximately 1200 °C, and the heating time ranges from 67 s to 149 s (7) A mathematical model of the air temperature rise curve, considering the influence of space height, is developed. This model can serve as a theoretical foundation and reference for the study of fire resistance of the node exposed to pool fires.

Asymmetric fluid flow in helical pipes inspired by shark intestines

Unlike human intestines, which are long, hollow tubes, the intestines of sharks and rays contain interior helical structures surrounding a cylindrical hole. One function of these structures may be to create asymmetric flow, favoring passage of fluid down the digestive tract, from anterior to posterior. Here, we design and 3D print biomimetic models of shark intestines, in both rigid and deformable materials. We use the rigid models to test which physical parameters of the interior helices (the pitch, the hole radius, the tilt angle, and the number of turns) yield the largest flow asymmetries. These asymmetries exceed those of traditional Tesla valves, structures specifically designed to create flow asymmetry without any moving parts. When we print the biomimetic models in elastomeric materials so that flow can couple to the structure's shape, flow asymmetry is significantly amplified; it is sevenfold larger in deformable structures than in rigid structures. Last, we 3D-print deformable versions of the intestine of a dogfish shark, based on a tomogram of a biological sample. This biomimic produces flow asymmetry comparable to traditional Tesla valves. The ability to influence the direction of a flow through a structure has applications in biological tissues and artificial devices across many scales, from large industrial pipelines to small microfluidic devices.

A quantitative analysis on the effect of crystal parameters on full energy peak efficiency of a coaxial high purity Germanium detector for the energy range 50 keV-2.5 MeV

High Purity Germanium (HPGe) detectors are essential instruments in gamma-ray spectrometry, offering high sensitivity and exceptional energy resolution. The full-energy-peak (FEP) efficiency is a critical parameter that influences the accuracy of activity concentration measurements of radionuclides. This study examines the FEP efficiency of a coaxial HPGe detector, focusing on variations in crystal length, crystal radius, and crystal hole dimensions. For varying crystal lengths, the efficiency values show negligible differences at low energy (50 keV) but significant increases at higher energies, indicating that longer crystal lengths enhance efficiency. Similarly, the efficiency increases with larger crystal radii across all energy levels suggesting substantial efficiency gains even at low energies. However, variations in crystal hole radius and depth, exhibit minimal impact on FEP efficiency across all tested energy levels. These findings highlight that optimizing crystal length and radius is more crucial for changing detector efficiency compared to modifying hole dimensions, providing valuable insights for optimizing Monte Carlo models and detector design.

Fermi model of a quantum black hole

We propose a quantum model of the Schwarzschild black hole as a quantum mechanics of a system of fermionic degrees of freedom. The system has a constant density of states and a Fermi energy that is inversely proportional to the size of the system. Assuming the equivalence principle, we show that the degeneracy pressure of the Fermi degrees of freedom is able to withstand the collapse of gravity if the radius of the system is given precisely by the horizon radius of the Schwarzschild black hole. In our model, the fermionic degrees of freedom at each energy level can be entangled in certain different ways, giving rise to a multitude of degenerate ground states of the system. The counting of these microstates reproduces precisely the Bekenstein-Hawking entropy. This simple Fermi model is universal and works also for the Reissner-Nordström charged black hole as well as black holes with a cosmological constant. From the properties of the Fermi variables, we propose that quantum gravity is characterized by a principle of where there can be no more than V/lP3 quantum states in any volume V. It implies a loss of spatial locality below the Planck length and suggests that any singularity predicted by general relativity is resolved and replaced by a quantum space in quantum gravity. In our model, a black hole spacetime is equipped with a uniform distribution of energy levels. This is another reason why black holes can be considered a simple harmonic oscillator of quantum gravity. Published by the American Physical Society 2024

Dynamic response mechanism of the hole subjected to the 3D point source disturbance in an isotropic formation

Accurate estimation of the effects of dynamic disturbances on stress concentration is crucial for the stability of rock engineering and the corresponding analytical approaches are needed. This study presents an analytical approach to calculate the relative stress distribution with a point source inside and outside a hole using the linear theory of elasticity. The Helmholtz potentials and Sommerfeld integral are employed to describe the displacement and stress components, and then formulate the equilibrium equations to solve the equivalent stress distribution around the hole. Numerical examples demonstrate the impact of model parameters on the equivalent stress, such as frequency, hole radius, source location, etc. It is found that sometimes high frequencies can make the equivalent stress greater far from the source than that close to the source. Additionally, when the ratio of the distance between the source and the hole axis to the hole radius exceeds ten, the equivalent stress distribution around the hole remains nearly constant. This approach can be used for the design and assessment of underground engineering structures' stability under dynamic disturbances.

Failure assessment of eccentric circular holes under compressive loading

AbstractThe present work aims to investigate the failure size effect on flattened disks containing an eccentric circular hole under mode I loading conditions. For this purpose, uniaxial compression tests are carried out on polymethyl methacrylate (PMMA) samples with holes. Depending on the hole radius and eccentricity, the energy release rate is either an increasing or decreasing function of the crack length, thus affecting the stability of crack propagation. Experimental results are interpreted and discussed through the coupled stress and energy criterion of Finite Fracture Mechanics. The approach lies on the assumption of a finite crack advance and it is implemented through the numerical estimation of the stress field and the Incremental Energy Release Rate functions. Finally, stability and crack speed propagation are discussed under the assumption of Linear Elastic Fracture Mechanics. Theoretical predictions reveal in agreement with experimental results thus demonstrating that the Coupled Criterion effectively captures the failure condition.

Quasinormal modes, Hawking radiation and absorption of the massless scalar field for Reissner–Nordström black hole surrounded by a cloud of strings in Rastall gravity

In this work, we investigate the quasinormal modes (QNMs), shadow radius, gray-body factor, Hawking radiation and absorption cross-section of the Reissner–Nordström black hole surrounded by a cloud of strings in Rastall gravity for a massless scalar field. We use the sixth-order Wenzel, Kramers and Brillouin (WKB) and unstable circular null geodesic methods to calculate QNMs. Moreover, the values obtained by the two methods are in very good agreement. The real part and absolute value of imaginary part of quasinormal frequencies decrease with the increase of [Formula: see text]. As [Formula: see text] increases, the real part of quasinormal frequencies increases while the absolute value of imaginary part first increases and then decreases. For [Formula: see text], they decrease with the increase of [Formula: see text]. While they first increase and then decrease with [Formula: see text] increasing at small value of [Formula: see text] when [Formula: see text]. Then using the unstable circular null geodesic method, we obtain the shadow radius of black hole. We find that shadow radius decreases first and then increases as the negative parameter [Formula: see text] increases. We also calculate the gray-body factor. According to it, Hawking radiation and absorption cross-section are calculated. We explore the influence of [Formula: see text], [Formula: see text] and [Formula: see text] on the energy emission rate. When other parameters are fixed, the black hole lives longer as [Formula: see text], positive parameter [Formula: see text] and [Formula: see text] increase. To obtain absorption cross-section, we adopt the partial wave method and sinc approximation method, and the results obtained by two methods have excellent consistency at mid-high frequency region. Further, we assess the effect of [Formula: see text], [Formula: see text] and [Formula: see text] on the absorption cross-section.

Investigate the optimum design of atomizing nozzles for coal dust suppression by using multifactor level response surface methodology

Mechanized coal extraction generates substantial dust, adversely affecting the physical and mental wellbeing of underground workers and degrading the operational environment. To address the limitations of traditional spray systems, which often suffer from poor atomization effects and limited ability to withstand wind interference, the innovative vortex atomizing nozzle has been developed specifically for dust suppression. This advanced solution was rigorously tested through detailed numerical simulations and comprehensive experimental investigations. These studies focused on examining the impact of critical parameters, including the velocity at the airflow director's exit, the angle of the exit hole, and its radius, on the efficacy of dust suppression. Findings highlighted the paramount importance of the airflow director's exit velocity in enhancing dust removal, with the exit hole's radius having the least impact. Optimal conditions for dust suppression were identified as an airflow director exit velocity of 20 m/s, an angle of 60°, and a radius of 0.6 mm. Field validations under these optimum parameters demonstrated unparalleled dust control efficiency, achieving a reduction in coal dust concentration exceeding 90 %. This apparatus offers a highly effective, user-friendly solution for comprehensive coal dust management across various mine locations.

Ultrahigh reflectivity photonic crystal membranes with optimal geometry

Photonic crystal (PhC) structures with subwavelength periods are widely used for diffractive optics, including high reflectivity membranes with nanoscale thickness. Here, we report on a design procedure for 2D PhC silicon nitride membrane mirrors providing optimal crystal geometry using simulation results obtained with rigorous coupled-wave analysis. The Downhill Simplex algorithm, a robust numerical approach to finding local extrema of a function of multiple variables, is used to optimize the period and hole radius of PhCs with both hexagonal and square lattices, as the membrane thickness is varied. Following these design principles, nanofabricated PhC membranes made from silicon nitride have been used as input couplers for an optical cavity, resulting in a maximum cavity finesse of 33 000, corresponding to a reflectivity of 0.999 82. The role played by the spot size of the cavity mode on the PhC was investigated, demonstrating the existence of an optimal spot size that agrees well with predictions. We find that, compared to the square lattice, the hexagonal lattice exhibits a spectrally wider reflective range, less sensitivity to fabrication tolerances, and higher reflectivity for membranes thinner than 200 nm, which may be advantageous in cavity optomechanical experiments. Finally, we find that all of the cavities that we have constructed exhibit well-resolved polarization mode splitting, which we expect is due primarily to a small amount of anisotropic stress in the silicon nitride and PhC asymmetry arising during fabrication.

Mechanical properties of C3N-BN hybrid nanosheets: Insights from molecular dynamics simulations

Molecular dynamics simulations have been utilized to explore the mechanical properties of the C3N-BN hybrid nanosheet across varying temperatures and strain rates, as well as in the presence of two different types of defects: circular holes and single vacancy atoms introduced into the hybrid structure. The findings revealed that the hybrid material shows intermediate mechanical properties compared to pure C3N and BN. The mechanical characterization of the hybrid suggests that variations in mechanical parameters remain relatively stable across different strain rates. However, temperature changes significantly affect the mechanical properties of the hybrid, with the highest failure stress recorded as 123.70 (GPa), failure strain of 21.57 %, and Young's Modulus of 747.25 (GPa) at 100 Kelvin. Expanding the hole radius decreases failure stress by approximately 76 %, 56 %, and 82 % when holes are exerted in the interface, BN, and CN sides of the hybrid, respectively, and reduces Young's modulus by about 65 %, 67 %, and 66 %. Introducing circular holes within the BN domain enhanced the hybrid structure's strength and resilience, demonstrating BN's superiority over C3N. Therefore, placing holes in the BN region is recommended for achieving superior mechanical properties in such structures. The introduction of single-atom vacancies revealed that carbon vacancies induced a far more significant mechanical strength reduction than vacancies of other atoms.

The thermal diffusion blocking effect of holes enclosing core fiber

We have developed a variety of optical fibers with holes enclosing the core to increase the service life of fiber optic devices under high-temperature and high-pressure environments by blocking the diffusion of dopants in the core with the help of air holes. Simulation results show that the fiber with uniformly distributed holes enclosing the core, the more the hole number, the smaller the core hole distance, and the larger the hole radius, the more significant the blocking effect on thermal diffusion. By comparing the refractive index distribution of three kinds of holes enclosing core fibers and single-mode fibers after prolonged high-temperature heating treatment, the thermal diffusion blocking effect of the holes enclosing core fiber is verified. Long-period fiber gratings fabricated by periodic refractive index modulation of holes enclosing core fibers can realize the measurement of various physical parameters. The thermal diffusion blocking effect of the holes enclosing core fiber enables the long-period fiber grating devices to work for a long time under high-temperature and high-pressure environments, showing the potential for practical applications in various industrial processes.

Prediction of Electric Load Neural Network Prediction Model for Big Data

In this study, the two-dimensional hexagonal Photonic crystal energy band structure was simulated using COMSOL Multiphysics field simulation software. The 2D hexagonal Photonic crystal structure was constructed by setting parameters such as periodic boundary conditions and air hole radius. Using the frequency domain solver of COMSOL software, the transmission and reflection spectra of the structure were calculated, and the energy band structure diagram was obtained. The effects of different parameters on the energy band structure were analyzed by adjusting the structure parameters. The results show that the fine tuning of the energy band structure of 2D hexagonal Photonic crystals can be realized by adjusting the radius of air holes and the periodic boundary conditions. This study provides a useful reference for further research on the properties and applications of 2D photonic crystals.

Study on the initiation criterion for double symmetric radial perforations emanating from circular hole under high water pressure considering T stress

Hydraulic fracturing is common technique to break the rock mass down under high water pressure. In this paper, the tensile failure of double symmetric radial perforations emanating from circular hole under far-field compression and high water pressure is studied. And the existing theoretical model does not consider the unequal water pressure applied on the surfaces of perforation and circular hole, it also neglects the effect of T stress, so the maximum tangential stress (MTS) criterion could be revised with considering T stress. Firstly, considering the unequal water pressure on the perforation surfaces and circular hole, the stress intensity factors (SIFs) are derived with conformal mapping method and compared with the semi-analytical solution. Secondly, the terms of T stress are derived by analogy to the open-crack under high water pressure. On this basis, the revised MTS criterion is obtained, and its validity is verified with model tests. Finally, the effects of the lateral pressure coefficient k, minimum horizontal principal stress σh(σh/σH remains constant), radius of circular hole a, hydraulic coefficient λ and radius of process zone rc on the initiation angle −θc, critical water pressure Pc and SIFs are analyzed respectively. It is found that −θc always increases firstly and then decreases with increasing the inclination angle α, whereas Pc firstly decreases and then increases as a/(c-a) varies from 0 to 1 and increases as α increases in the range of a/(c-a) = 1∼+∞. Then −θc increases with increasing σh and k, whereas Pc can intersect each other with increasing k and increases with increasing σh. As c-a remains constant, −θc increases firstly and then decreases with increasing the ratio a/c, whereas Pc keeps decreasing. Also −θc and Pc increase with decreasing λ. As rc increases, −θc decreases and Pc increases. As for SIFs, when α increases, KI decreases and KII increases firstly, then decreases. As α remains constant, KI increases firstly, then decreases with increasing k and KII keeps increasing. In the range of a/(c-a) = 0 ∼ 1, KI decreases with increasing a, and KII keeps increasing. Within the range of a/(c-a) = 1∼+∞, KI increases firstly, then decreases with increasing a, the increment of KII is not obvious. And KI decreases with decreasing λ.

Black hole interior quantization: a minimal uncertainty approach

In a previous work we studied the interior of the Schwarzschild black hole implementing an effective minimal length, by applying a modification to the Poisson brackets of the theory. In this work we perform a proper quantization of such a system. Specifically, we quantize the interior of the Schwarzschild black hole in two ways: once by using the standard quantum theory, and once by following a minimal uncertainty approach. Then, we compare the obtained results from the two approaches. We show that, as expected, the wave function in the standard approach diverges in the region where classical singularity is located and the expectation value of the Kretschmann scalar also blows up on this state in that region. On the other hand, by following a minimal uncertainty quantization approach, we obtain 5 new and important results as follows. (1) All the interior states remain well-defined and square-integrable. (2) The expectation value of the Kretschmann scalar on the states remains finite over the whole interior region, particularly where used to be the classical singularity, therefore signaling the resolution of the black hole singularity. (3) A new quantum number is found which plays a crucial role in determining the convergence of the norm of states, as well as the convergence and finiteness of the expectation value of the Kretschmann scalar. (4) A minimum for the radius of the (2-spheres in the) black holes is found (5) By demanding square-integrability of states in the whole interior region, an exact relation between the Barbero-Immirzi parameter and the minimal uncertainty scale is found.