Displacement Vector Research Articles

Investigating micro-cutting mechanism of glow discharge polymer based on material properties and removal behaviors analysis

Glow discharge polymer (GDP) is the unique artificial material for the target balls in Inertial Confinement Fusion tests, while its practical micromachining such as the fabrication of microstructure on material surfaces keeps challenging due to its unclear micro-cutting mechanism. Hence, this paper seeks to investigate the micro-cutting mechanism of GDP from the perspective of the material flow and cutting energy. To achieve it, the energy conservation equation of three cutting modes triggered by different ratios of uncut chip thickness to the tool cutting edge radius (RTS) was established based on cutting deformation behaviors. Meanwhile, the diamond cutting tests and the FEM simulation at different RTS were developed. The experimental observation of cutting forces and specific cutting forces verified the evolution of three cutting modes, including shearing, shearing-ploughing and ploughing in the micro-cutting of GDP with the decrease of RTS. Next, from the change of the node displacement vector observed from the simulated results, it can be seen that the real-time material flow behavior during micro-cutting of GDP varies obviously with the evolution of cutting modes. Besides, the fracture toughness Gc, and the energy dissipation of different cutting modes were analyzed. The proportion of the energy spent on material fracture (Gc=9.95 N/mm) is the largest one in shearing and shearing-ploughing modes, while in ploughing mode, the material plastic deformation consumed the most energy. The above results reveal the specific material properties and removal behaviors of GDP and contribute to optimizing the machining strategies for the practical micromachining of microstructures on material surfaces.

Fractional-order velocity-dilatation-rotation viscoelastic wave equation and numerical solution based on a constant-Q model

Viscoelastic wave equations based on the constant- Q (CQ) model can accurately describe the amplitude dissipation and phase distortion of waves in anelastic media. However, only three velocity or displacement components can be obtained directly by solving such equations. Starting from the time-domain second-order displacement viscoelastic wave equation, we derived the decoupled P- and S-wave displacement vector viscoelastic wave equation by using the polarization difference of P- and S-wave propagation in isotropic media. The equation can be transformed into the velocity-dilatation-rotation viscoelastic wave equation containing the first-order temporal derivative and fractional Laplacian operators, which can be solved directly by using the staggered-grid finite-difference and pseudospectral methods. We use the low-rank decomposition method to approximate the derived mixed space-wavenumber domain fractional Laplacian operators for modeling wave propagation in heterogeneous attenuating media. We also demonstrated the precision of our equation by comparing the numerical solutions with the analytical solutions. Furthermore, compared with the conventional velocity-stress viscoelastic wave equation, experimental results demonstrate that our equation can separate the pure P and S waves from the mixed wavefield during wavefield continuation. In addition, it can be separated into an equation containing predominantly an amplitude attenuation or a phase distortion term.

An alternative potential method for mixed steady‐state elastic oscillation problems

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

Co-rotational 3D beam element using quaternion algebra to account for large rotations: Formulation theory and static applications

This paper presents a co-rotational beam element based on quaternion algebra as a means of parameterizing large rotations. This co-rotational framework is based on a decomposition of beam kinematics into a rigid element frame, which follows the element and its pure deformation. Once the decomposition between rigid body motion and deformation is obtained, the principle of virtual work allows calculating the element response projected onto large displacements and rotations. This 3D co-rotational element lies within the framework of incremental formulations. The special feature of this formulation pertains to the decomposition of kinematics to extract the rigid body motion and pure deformations through use of quaternion algebra by solving an internal nonlinear kinematic equation. Thus, from the 2 quaternions defining global rotations and the 2 displacement vectors at the ends of the beam, this method is able to extract: 1 quaternion and 1 displacement vector parameterizing the rigid body motion, plus 2 other quaternions defining the internal rotations and beam elongation. The proposed formulation based on quaternion algebra therefore constitutes an alternative to the literature’s treatment of large-rotation kinematics using a quaternion algebra. The operators are also linearized in order to obtain the algorithmic stiffness operator. Eleven distinct static numerical applications are presented, along with comparisons from the literature in the aim of demonstrating the efficiency of this proposed element.

Lyra cosmologies with the dynamical system perspective

In this study, we revisit the homogeneous and isotropic, spatially curved model in the Lyra geometry framework. We apply the dynamical system approach to study the Universe evolution in model. The present cosmic dynamical system formulation is independent of the model parameter which yields the Universe expansion phases dominated by the stiff fluid, radiation, dark matter and dark energy. These expansion phases in the Lyra model are consistent with the General relativity model. In addition, the displacement vector field of the Lyra geometry model will give rise to the stiff matter dominated phase during early times of the Universe evolution. The cosmographic parameters, EoS parameter and statefinder diagnostic have been utilized in the dynamical system to illustrate the cosmic evolution of the Universe. The numerical solution technique is used to illustrate graphical behaviors of the cosmological parameters. On the basis of dynamical system and numerical analysis, we exhibit that the model explains accelerating universe expansion with transition from the decelerating phase and it will lead to the Λ cold dark matter (ΛCDM) model in the near future.

Observed soil arching-induced ground deformation and stress redistribution behind braced excavation

This paper presents observed arching-induced ground deformation and stress redistribution behind braced excavation using the top-down construction method. The soil properties around the excavation were determined by laboratory and field tests. The ground deformation, soil displacement vector, strain path, principal strain, maximum shear strain, lateral earth pressure, pore water pressure, and effective stress path are presented based on the measured data. The majority of soil behind the wall is under volumetric expansion, indicating consolidation, creep behavior, or a combination of both. Besides, two periods of increases in pore pressure are observed, due to stress transfer from the lower to the upper parts (i.e., soil arching effect). The deep inward movement of the wall and the nearby soil accounts for the distribution of lateral earth pressure acting on the wall. The soil located behind the area of maximum wall deformation and adjacent to the wall, as well as the soil below the excavation base intersected by the shear plane, is in an active stress state. The lateral earth pressure at 5 m from the left excavation wall showed minimal changes, due to the combined effects of soil arching from lateral excavation and shield tunneling.

A thin-walled cavity structure with double-layer tapered scatterer locally resonant metamaterial plates for extreme low-frequency attenuation

Locally resonant acoustic metamaterials (LRAMs) are effective spatial frequency filters due to their local resonance system. However, they own narrow stopbands, charge additional weight on the primary system, and operate only at the adjusted frequency range. In this paper, a novel dual-target LRAM is proposed based on coupling the cavity and convex mechanisms, utilizing the benefits of both sound-barrier and sound-absorbing types of acoustic metamaterials. A combination of the cavity and convex structures is presented and investigated for the first time, which exploits both the reflection and absorption theories simultaneously to improve the sound attenuation performance of acoustic metamaterials. By arranging two-layer finite periodic of 5×5 proposed convex unit cells along x and y directions and separating them by an air cavity, a supercell named hybrid locally resonant acoustic metamaterials (HLRAM) baffle is designed. The band structures, transmission spectrum, and displacement vector fields are calculated employing the finite element method (FEM). In addition, vibration modes at the edges of stopbands are computed and carefully analyzed to determine the formation mechanism, mechanics/dynamic response, and dispersion features of stopbands. Meanwhile, sensitivity analyses have been conducted on the model to investigate the influence of material and geometry parameters on dispersion characteristics. Equivalent spring-mass analytic models are used to direct the modifications of structure and the adjustment of structural and material parameters in order to achieve a wide stopband with a low opening frequency. Thereby, the effective stiffness and mass values influencing the starting and cutoff frequencies are adjusted in a desirable manner. The results show that the structure can generate a wide stopband that covers the frequency range of 64.9-805.6Hz, allowing the structure to block incoming acoustic waves with 35.42dB root mean square of noise attenuation (RMSNA) the given frequency range (0-1kHz). This study indicates that the HLRAM has a preponderance among conventional techniques as we can manipulate dispersion characteristics and the sound transmission loss (STL) based on the desired sound reduction level and shift stopbands to the intended frequency range. This can be achieved by altering cavity size, material, or geometry parameters.

The consolidation behavior of layered fractional viscoelastic soils considering groundwater

AbstractThis paper investigates the consolidation behavior of multi‐layered viscoelastic soils considering groundwater. First, the fractional Merchant viscoelastic model is introduced to describe the behavior of multi‐layered viscoelastic soils considering groundwater. Later, the governing equations are extended to a viscoelastic medium by virtue of the elastic‐viscoelastic corresponding principle in the Laplace–Hankel domain. According to the extended precise integration method, the soil layer is divided into a series of layer units. Then the relationship between general stress vector and general displacement vector on the top and bottom planes is established. Every two adjacent layer units are combined into one layer in each computational iteration. The solutions in the Laplace–Hankel domain are obtained by considering the boundary conditions, and numerical inversion is performed to obtain the solutions in the physical domain. The practicability of the present method is assessed by comparing the numerical results with those in the existing literature and done by ABAQUS. Finally, the effects of groundwater table, properties of the soils above groundwater table, load depth, viscoelastic parameters, and soil stratification are investigated.

Thermal contraction coordination behavior between unbound aggregate layer and asphalt mixture overlay based on the finite difference and discrete element coupling method

AbstractThe constraint action of the unbound aggregate layer underneath plays an important role in affecting the temperature strains in the top asphalt layer. The focus of the present paper is to investigate the interactive thermal contraction mechanisms between the asphalt mixture and granular base layers to offer a new perspective in promoting the understanding of the thermal cracking disease. In this paper, both experiments and simulations were conducted for the thermal contraction tests. A novel numerical modeling of virtual composite structures was proposed using wall‐zone coupling operation based on the finite difference and discrete element coupling method. The experimental composite structures containing three types of asphalt mixture overlays and five types of unbound aggregate base layers were developed for verification. The thermal contraction and restraint mechanism were revealed from both macro and microscopic scales. The proposed numerical modeling shows a 94% high accuracy. The particle displacement vector and contact force chain could explain the thermal contraction behavior of composite structures. Smaller particle motion displacement or stronger contact force chain result in a higher restraint strain of the asphalt overlay. The thermal contraction behavior can be coordinated through the compaction and loosening of unbound aggregates. The material parameters and cooling temperature differences in the asphalt overlay have slight effects on the constraint action of a base layer, while the gradation and mechanical parameters of the unbound aggregate layer show significant impaction. The parameters, cohesion C and friction angle φ, show a quadratic function with the restraint coefficient. This work has significant guidance on the selection of pavement structure and materials to improve the thermal cracking problem and lay the basis for the mechanical theoretical calculation to predict thermal cracking.

The equivalence principle for a plane gravitational wave in torsion-based and non-metricity-based teleparallel equivalents of general relativity

We study the energy–momentum characteristics of the plane “+”-polarized gravitational wave solution of general relativity in the teleparallel equivalent of general relativity (TEGR) and the symmetric teleparallel equivalent of general relativity (STEGR) using the previously constructed Noether currents. The current components describe energy–momentum locally measured by an observer if the displacement vector ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} is equal to the observer’s 4-velocity. To determine the non-dynamical connection in these theories, we use the unified “turning off” gravity principle. For a constructive analysis of the values of Noether currents and superpotentials in TEGR and STEGR, we use the concept of “gauges”. The gauge changing can affect the Noether current values. We study under what conditions the Noether current for the freely falling observer is zero. When they are established, the zero result can be interpreted as a correspondence to the equivalence principle, and it is a novelty for gravitational waves in the TEGR and STEGR. We highlight two important cases with positive and zero energy, which reproduce the results of previous works with a different approach for determining gravitational energy–momentum in the TEGR, and give their interpretation.

Justification of geodetic monitoring methodology of the retaining walls on the example of the embankment in Kremenchuk

Purpose. Development of a methodology for analyzing the results of geodetic measurements according to which it is possible to use the materials of past years for the needs of geodetic monitoring on the example of a retaining wall in the conditions of the city’s recreational area. Methodology. A technological scheme of geodetic monitoring of retaining walls has been developed with four main stages: analysis of initial data, design of geodetic monitoring, periodic observations, processing and analysis of geodetic monitoring results. The conditions of the recreational areas of the city determine the specifics of geodetic monitoring, limit the possibilities of choosing a scheme of the geodetic network and methods of measurements. In this regard, it is proposed to develop models of the development of deformation processes already at the first stage of geodetic monitoring, which will allow geodetic monitoring to be carried out with greater reliability in the future and avoid possible forecasting errors. Findings. The results of the analysis of geodetic measurements in the geodetic networks of Kremenchuk (coordinates and heights of wall signs) show the presence of horizontal and vertical displacements of the retaining wall. In the horizontal plane the retaining wall has shifted in the south-western direction, towards the Dnipro River. In the vertical plane, the retaining wall has subsided. The displacements of different parts of the retaining wall are uneven. The average annual rate of both horizontal and vertical displacements is equivalent and is approximately 1 mm/year. The values of absolute displacement vectors of wall signs in the horizontal plane exceed the accuracy of geodetic measurements and normative tolerances. Originality. Modeling of displacements of retaining walls in the conditions of recreational areas of the city is already underway, taking into account the analysis of the results of geodetic measurements of past years. Practical value. The data of the analysis of the results of geodetic measurements carried out in the geodetic densification networks of Kremenchuk indicate the presence of deformation processes and justify the need for their control through geodetic monitoring. The suggested models can be used as the comparative and combined analysis of future forecast changes based on previous and current results of measurements, which is a topic for another research.

A VARIATIONAL FORMULATION GOVERNED BY TWO BIPOTENTIALS FOR A FRICTIONLESS CONTACT MODEL

We consider a frictionless contact model whose constitutive law and contact condition are described by means of subdifferential inclusions. For this model, we deliver a variational formulation based on two bipotentials. Our formulation envisages the computation of a three-field unknown consisting of the displacement vector, the stress tensor and the normal stress on the contact zone, the contact being described by a generalized Winkler condition. Subsequently, we obtain existence and uniqueness results. Some properties of the solution are also discussed, focusing on the data dependence.

Structure and Charge Carrier Separation Promotion Effects of Antiphase Boundaries in Cesium Lead Bromide.

Defects in lead halide perovskites (LHPs) may have a significant impact on charge carrier separation, but the roles of the defects are not fully understood. Here, using aberration-corrected scanning transmission electron microscopy (STEM), different types of antiphase boundaries (APBs) are discovered in CsPbBr3 platelets. APBs with a displacement vector of 1/4[111] are characterized by double layers of CsBr layers at the (110) or (001) planes, while APBs at the (112) planes are formed through edge sharing of PbBr6 ̵octahedra. Significant lattice distortions are determined at (001) and (110) APBs on the basis of quantitative analyses of STEM images. Density functional theory calculations demonstrate that all three types of APBs can induce band offsets at their valence bands and conduction bands. The APBs are intended to promote the separation of photogenerated charge carriers in LHPs. These findings provide a crystal engineering technique for enhancing the optoelectronic properties of LHPs by controlling defects.

A computable quantification of multi-mode Gaussian coherence and complete monogamy relation

We propose a quantification CνGn of n-mode Gaussian coherence. The value of CνGn only depends on the covariance matrices and displacement vectors of continuous-variable states without any optimization procedures, and thus is easily calculated. For n=1, CνG1 is a proper Gaussian coherence measure of single-mode CV system. For n≥2, CνGn is invariant under any permutation of submodes, nonincreasing under any n-mode local incoherent Gaussian channels, vanishes at incoherent Gaussian states, and, satisfies the unification condition and the hierarchy condition that a multi-partite quantum correlation measure should obey. Thus CνGn is a multi-partite Gaussian correlation measure, which reveals, though the quantum coherence lives in single-partite systems, that the multi-mode coherence for continuous-variable systems can be regarded as a multipartite Gaussian correlation between modes, and such multi-partite Gaussian correlation is also a quantum resource. Moreover, we show that CνGn is completely monogamous as a multipartite Gaussian correlation measure. This means that the n-mode Gaussian coherence subjects to the principles of resource allocation. In addition, CνGn is an upper bound of the geometric-based single-mode Gaussian coherence measure CBu by the Bures distance at pure Gaussian states of mode ≤2 and can be used to detect coherence in any n-mode Gaussian states more efficiently.

The Effect of the Chemical Composition on Mechanical Properties of CMAS Diopside Glass Ceramics

Molecular dynamics simulations were performed on CaO-MgO-Al2O3-SiO2 (CMAS) diopside glass ceramics (GCs) to study the effect of nanocrystal on glass and the effect of chemical composition on mechanical properties. Under tensile loading, the GCs demonstrated that the strength lay between its glass and ceramic counterparts and maintained considerable ductility. Moreover, high Mg/Ca ion ratios are conductive to both the strength and ductility of GCs. In addition, Al ions should be avoided as far as possible since they would promote fracture. After analyzing the shear strain and displacement vector map for ion structures, we found that the presence of nanocrystal in glass changes the original deformation pattern and led to the deformation concentration surrounding the nanocrystal. A high Mg/Ca ion ratio would make the deformation more homogeneous, while a high Ca/Mg ion ratio would aggregate the deformation in the glass region near the nanocrystal. The existence of Al ions near the interface between glass and crystal would promote the formation of voids.

Environment geometry alters subiculum boundary vector cell receptive fields in adulthood and early development

Boundaries to movement form a specific class of landmark information used for navigation: Boundary Vector Cells (BVCs) are neurons which encode an animal’s location as a vector displacement from boundaries. Here we characterise the prevalence and spatial tuning of subiculum BVCs in adult and developing male rats, and investigate the relationship between BVC spatial firing and boundary geometry. BVC directional tunings align with environment walls in squares, but are uniformly distributed in circles, demonstrating that environmental geometry alters BVC receptive fields. Inserted barriers uncover both excitatory and inhibitory components to BVC receptive fields, demonstrating that inhibitory inputs contribute to BVC field formation. During post-natal development, subiculum BVCs mature slowly, contrasting with the earlier maturation of boundary-responsive cells in upstream Entorhinal Cortex. However, Subiculum and Entorhinal BVC receptive fields are altered by boundary geometry as early as tested, suggesting this is an inherent feature of the hippocampal representation of space.

Carrera Unified Formulation (CUF) for the composite plates and shells of revolution. Layer-wise models

Here, higher order layer-wise models of elastic composite multilayer plates and shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the thickness coordinates. Following the Unified Carrera Formulation (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order rectangular plate and cylindrical shell supported on the edges under sinusoidal loading, are considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric conical, spherical, elliptical and catenoidal shell fixed at the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress–strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order composite laminate shells.

Mechanical behavior of endodontically treated teeth: A three-dimensional finite element analysis using displacement vector.

This study aimed to analyze the effects of core materials, remaining tooth structures, and interfacial bonding on stress distribution in endodontically treated teeth using finite element analysis (FEA). Three-dimensional FEA was conducted using a reverse engineering technique based on maxillary premolars scanned by micro-computed tomography. Six models were generated with or without ferrules and with one of the following three abutment systems: metal core, resin core, or resin core with fiber posts. In each model, bonding and debonding were assumed in the dentin and surrounding structures: bonded and debonded models. The maximum principal stress values were recorded, and stress distribution of the entire restored teeth and dentin was generated. Furthermore, the distribution of the displacement vector of the debonded models was generated. In comparing the bonded and debonded models, the debonded models showed larger values for tensile stresses than those in bonded models for all abutment models. The models without ferrules rotated around the center of the abutment, whereas those with ferrules did not show remarkable displacement in the analysis. FEA assuming fracture of adhesive interface proved to be an effective method to clarify the significance of ferrules. It prevents stress concentration in dentin by reducing the rotation of the abutment, even when the adhesive fails.

Ultrahigh-speed imaging for high-impact concrete deformation analysis in pre- and post-cracking stages.

The evaluation of high-speed camera image sequence analysis results in concrete material testing under high-impact loading necessitates the consideration of the effect of the image quality on the measurement accuracy and thus on the potential of the geometric measurements derived from the image sequences. In this contribution, we evaluate the application potential of three ultrahigh-speed cameras with frame rates up to 10Mfps to analyze the deformation of concrete specimens before and after main crack formation in bending and compression tests. Specifically, we evaluate the Kirana 7M and Shimadzu HPV-X2 cameras with ISIS sensor architecture, and the Phantom TMX 7510 camera with BSI CMOS sensor technology. Three-point bending tests and split-Hopkinson pressure bar tests are performed on 160×40×40m m 3 cuboids and on 80mm long, 50mm diameter cylinders. Prior to main crack formation, the displacement vector field represents the specimen deformation, with higher values indicating the position where main cracks will initiate and propagate. Deformations of 80µm in 54µs for a bending test and of 154µm in 36.67µs for a compression test could be measured. The main cracks are then detected using displacement vector field discontinuity analysis techniques, and their evolution is followed to estimate the crack propagation velocity. Average velocities in bending tests between 603 and 854m/s have been determined over a time interval up to 40µs. An investigation of the camera sensor operation of the three optical devices is presented to assess their suitability for deformation analysis. Laboratory tests and real experimental results show that the quality of the propagation vector field, the crack detection, and the crack tip tracking are obviously affected by the image quality, but more significantly by the spatial and temporal resolution due to the small relative step deformations.

О корректности размера позитрония

It is noted that initially the minimum size of the hydrogen atom was determined by the semiclassical method, which is still widely used today. When calculating the minimum size of a positronium atom, the method tested on the hydrogen atom was not used. Instead, in the formula for the radius, the mass was replaced by the reduced mass of positronium. There would be nothing wrong with this if the above “hydrogen” method gave the same result. However, it is not. The difference between the formula for positronium and the “hydrogen” formula is the coefficient “2”, due to the fact that the positive charge is not in the center of mass. In the literature, the minimum size of a positronium atom (Bohr radius of positronium) was determined in accordance with the two-body problem method. In the particular case under consideration, the method of the two-body problem is redundant and can be successfully replaced by solving the problem of the motion of one of the bodies. This allows you to obtain more visual and obvious solutions without excessive formalism with the reduced mass and displacement vector. The doubly refined value of the positronium radius was determined using the semiclassical method. However, positronium is a quantum object. Therefore, a quantum mechanical consideration is also presented. The textbook solutions are taken as a basis. The solution to the problem of two particles is found in the form of an equivalent solution for one of the particles moving relative to a stationary center of mass, especially since there is a reliable quantum mechanical solution for the case of a spherically symmetric force field, which assumes a stationary center of force. The latter is especially important, since the mobility of the force center (electric charge) can lead to radiation and, consequently, a change in the energy balance. It has been established that the calculated minimum radius of positronium (the distance from its center of mass at which the probability of finding an electron/positron is maximum) is equal to four, and not two Bohr radii of hydrogen, as indicated in the literature. In this case, the semiclassical and quantum solutions do not contradict each other.