Hydrogenic Wave Functions Research Articles

Exciton binding and excitonic transition energies in wurtzite Zn1-xCdxO/ZnO quantum wells

Binding energy of the 1s exciton state in the Zn1-xCdxO/ZnO quantum wells, changing between 10 and 50 Å in width, has been calculated for x up to 0.2. The exciton wave function is represented in terms of hydrogenic wave functions and the effects of dielectric mismatch, exciton-phonon interactions and built-in electric field on the binding energy are considered. The calculated built-in electric field is around 19.6x MV cm−1 in a 20-Å Zn1-xCdxO/ZnO quantum well. The numerical results indicate that the exciton binding energy stays below its bulk value virtually for all x because of the built-in electric field dominating the quantum confinement effect. The binding energy is nearly 26 meV at maximum, barely above its bulk value, and about 13 meV at minimum. The contribution of the dielectric mismatch is up to 40% while that of the exciton-phonon interactions is negligible. The exciton binding energy increases up to 66 meV, well above the bulk value, in case the built-in electric field is neglected. A qualitative agreement between the calculated excitonic transition energies and the low temperature and low excitation intensity photoluminescence peak data is obtained although some ZnCdO material properties, such as low temperature band gap, are not available.

Influence of dense plasma on the energy levels and transition properties in highly charged ions

The studies of the influence of plasma environments on the level structures and transition properties for highly charged ions are presented. For the relativistic treatment, we implemented the multiconfiguration Dirac-Fock method incorporating the ion sphere (IS) model potential, in which the plasma screening is taken into account as a modified interaction potential between the electron and the nucleus. For the nonrelativistic treatment, analytical solutions of the Schrödinger equation with two types of the IS screened potential are proposed. The Ritz variation method is used with hydrogenic wave function as a trial wave function that contains two unknown variational parameters. Bound energies are derived from an energy equation, and the variational parameters are obtained from the minimisation condition of the expectation value of the energy. Numerical results for hydrogen-like ions in dense plasmas are presented as examples. A detailed analysis of the influence of relativistic effects on the energy levels and transition properties is also reported. Our results are compared with available results in the literature showing a good quantitative agreement.

MAPLE program for modelling hydrogen-like atoms in quantum mechanics with non-negative distribution function

The program is proposed for a realization of the symbolic algorithm based on the quantum mechanics with non-negative probability distribution function (QDF) and for calculations of energy levels for hydrogen-like atoms. The program is written up in the language MAPLE. In the framework of the algorithm an original Maple package for calculations of necessary functions, such as hydrogen wave functions, Sturmian functions and their Fourier-transforms, Clebsch-Gordan coefficients, etc. is proposed. Operators of observables are calculated on the basis of the QDF quantization rule. According to the Ritz method, eigenvalues of Ritz matrices represent spectral values of the quantity under investigation, i.e. energy. As an example, energy levels of hydrogen-like atoms are calculated and compared with experimental data retrieved from the NIST Atomic Spectra Database Levels Data. It turns out that this theory seems to be equivalent to the traditional quantum mechanics in regard to predictions of experimental values. However, the existence of a phase-space probabilistic quantum theory may be an important advance towards the explanation and interpretation of quantum mechanics.

Exciton binding energies in CdSe/MgSe quantum well structures

Binding energies of heavy- and light-hole excitons in CdSe/MgSe quantum wells have been determined in terms of quantum well width. Hydrogenic wave functions are assumed to represent exciton wave functions. Contribution of the polarization charges originating from the dielectric mismatch between the well and the barrier materials to the Coulomb interaction is calculated by means of image charge method. The effect of the exciton-phonon interaction on the exciton binding energies are included by means of an effective potential obtained through an exciton-bulk-optical-phonon Hamiltonian. The results indicate that both the modified Coulomb interaction due to the dielectric mismatch and the exciton-phonon interaction considerably change the exciton binding energy. The heavy-hole exciton binding energy in a 10-Å quantum well is seen to be enhanced as much as 87% compared to the value found when only the Coulomb interaction without dielectric mismatch is taken into account. The resulting binding energies of heavy-hole excitons range approximately between 57 and 41 meV for the quantum well width changing between 10 and 35 Å. Compared to the bulk CdSe exciton binding energy 15 meV, the exciton binding energies in CdSe/MgSe quantum wells are increased almost four times.

Role of nuclear charge change and nuclear recoil on shaking processes and their possible implication on physical processes

The probable role of the sudden nuclear charge change and nuclear recoil in the shaking processes during the neutron- or heavy-ion-induced nuclear reactions and weakly interacting massive particle–nucleus scattering has been investigated in the present work. Using hydrogenic wavefunctions, general analytical expressions of survival, shakeup/shakedown, and shakeoff probability have been derived for various subshells of hydrogen-like atomic systems. These expressions are employed to calculate the shaking, shakeup/shakedown, and shakeoff probabilities in some important cases of interest in the nuclear astrophysics and the dark matter search experiments. The results underline that the shaking processes are one of the probable channels of electronic transitions during the weakly interacting massive particle–nucleus scattering, which can be used to probe the dark matter in the sub-GeV regime. Further, it is found that the shaking processes initiating due to nuclear charge change and nuclear recoil during the nuclear reactions may influence the electronic configuration of the participating atomic systems and thus may affect the nuclear reaction measurements at astrophysically relevant energies.

Electronic wave function and binding effects in M-shell ionization of gold by protons

Abstract The measured M-X-ray production cross sections for protons, which are used in the particle induced X-ray emission (PIXE) technique, are systematically underestimated for low impact energies by the ECPSSR and ECUSAR theories. These theories, which are based on the plane wave Born approximation (PWBA) and use the screened hydrogenic wave functions, include corrections for the projectile Coulomb deflection and electron relativistic and binding effects. In the present paper, in order to interpret the observed disagreement at low impact energies, the systematic calculations of the M-shell ionization cross sections for gold were performed using the semiclassical (SCA) and the binary encounter (BEA) approximations in order to identify a role of the electronic wave function and electron binding effects. In these calculations the different wave functions, from nonrelativistic hydrogenic to selfconsistent Dirac-Hartree-Fock, were considered and the binding effect was treated within extreme separated- (SA) and united-atoms (UA) limits. The results are discussed in details and the observed discrepancies are attributed to inadequate description of the electron binding effect at the lowest impact energies for which the molecular approach is required.

Donor binding energies in a GaN/ZnGeN2 quantum well

Binding energies of a donor atom within a GaN/ZnGeN2 quantum well structure have been investigated. Hydrogenic type wave functions are assumed and the Schrödinger and Poisson's equations are solved self-consistently. The binding energies of the donor states 1s and 2p± associated with the first subband in the quantum well and the transition energies between them are represented as a function of the quantum well width, the donor position and the external magnetic field. The ground state 1s and the excited states 2p± have maximum binding energies as much as 64 and 11 meV, occurring in quantum wells of widths 10 and 18 Å, respectively. The binding energies are found to be an asymmetric function of the donor position with respect to the quantum well center because of the asymmetry in the band profile introduced by the built-in electric field in the structure. An external magnetic field up to 10 T is included into the calculations and it is seen that the excited states show a small Zeeman splitting, very close to the bulk GaN value, because of the heavy effective mass of electron.

Multiple ionization of Au by fast Sq+ ions of energy 0.4–3.8 MeV/amu

The measurements of the M- and N-shell ionization probabilities in gold bombarded by highly charged Sq+ (q=4–14) ions of energy 0.4–3.8MeV/amu are reported. The ionization probabilities were extracted from the measured L-X-ray spectra which were substantially modified by the multiple ionization effects. The observed L-X-ray line shifts and broadening allowed us (Banaś et al., 2002) to extract both the M- and N-shell ionization probabilities for the zero impact parameter. These ionization probabilities, which correspond to the moment of X-ray emission, were correlated with the ionization probabilities for the moment of ion impact ionization by using the vacancy rearrangement procedure developed here. In this way the ionization probabilities for the moment of ion impact multiple ionization were obtained. By comparing to the theoretical predictions of the semiclassical approximation (SCA) (Trautmann and Kauer, 1989) and the “geometrical model” (GM) (Sulik et al., 1987) based on the binary encounter approximation (BEA), the role of the electronic wave functions and the binding effect is discussed. Generally, the calculations using the hydrogenic wave functions reproduce the measured ionization probabilities.

Shaking process due to nuclear recoil during heavy ion-atom collisions

A theoretical study on the shaking process accompanying swift heavy ion-atom collisions is presented. Using the nonrelativistic hydrogenic wavefunctions, analytical expressions for the survival and shakeup/shakedown probability have been derived for the K-,L- and M-shell electrons of hydrogen-like atomic systems. The resulting expressions are used to calculate the shaking and shakeup/shakedown probability in the typical case of the recoiling nucleus of the multielectron (63Cu+) and hydrogen-like (63Cu28+) atomic systems, respectively. Interestingly, it is found that the requirement of the high colliding energy for the suddenness of the perturbation, can well fulfill the limiting condition of the necessary nuclear recoil for finite electron shaking during the ion-atom collisions. Further, the dependence of the shaking probability on atomic number has been depicted, and it has emerged that for a particular value of the recoil energy per nucleon, the shaking processes are more significant in the lower atomic number region.

Excitons in Orthorhombic and Tetragonal Hybrid Organic–Inorganic Perovskites

The small exciton binding energy Eb in hybrid organic–inorganic perovskites (HOIPs) enables their extraordinary photovoltaic performance. The measured Eb in MAPbI3 (MA = CH3NH3), the most studied HOIP, is 50% smaller in the orthorhombic phase than estimated from the high-frequency dielectric constant and drops an additional 25% in the tetragonal phase. Here we show that these puzzling exciton behaviors can be quantitatively explained by incomplete screening arising from polar optical phonons (POPs) and randomly oriented MA ions. Emission/absorption of POPs introduces a virtual radius of electron (hole), within which the screening due to POPs is ineffective. Randomly oriented dipoles of MA ions in the tetragonal phase mediate an extra long-range coupling between the electron and hole. The exciton Hamiltonian is accurately solved by using variational hydrogenic wave functions as the basis set. Our results consistently account for the observed exciton properties and reveal the impact of polar coupling on excitons in HOIPs.

13C ENDOR Spectroscopy of Lipoxygenase-Substrate Complexes Reveals the Structural Basis for C-H Activation by Tunneling.

In enzymatic C–H activation by hydrogen tunneling, reduced barrier width is important for efficient hydrogen wave function overlap during catalysis. For native enzymes displaying nonadiabatic tunneling, the dominant reactive hydrogen donor–acceptor distance (DAD) is typically ca. 2.7 Å, considerably shorter than normal van der Waals distances. Without a ground state substrate-bound structure for the prototypical nonadiabatic tunneling system, soybean lipoxygenase (SLO), it has remained unclear whether the requisite close tunneling distance occurs through an unusual ground state active site arrangement or by thermally sampling conformational substates. Herein, we introduce Mn2+ as a spin-probe surrogate for the SLO Fe ion; X-ray diffraction shows Mn-SLO is structurally faithful to the native enzyme. 13C ENDOR then reveals the locations of 13C10 and reactive 13C11 of linoleic acid relative to the metal; 1H ENDOR and molecular dynamics simulations of the fully solvated SLO model using ENDOR-derived restraints give additional metrical information. The resulting three-dimensional representation of the SLO active site ground state contains a reactive (a) conformer with hydrogen DAD of ∼3.1 Å, approximately van der Waals contact, plus an inactive (b) conformer with even longer DAD, establishing that stochastic conformational sampling is required to achieve reactive tunneling geometries. Tunneling-impaired SLO variants show increased DADs and variations in substrate positioning and rigidity, confirming previous kinetic and theoretical predictions of such behavior. Overall, this investigation highlights the (i) predictive power of nonadiabatic quantum treatments of proton-coupled electron transfer in SLO and (ii) sensitivity of ENDOR probes to test, detect, and corroborate kinetically predicted trends in active site reactivity and to reveal unexpected features of active site architecture.

Exciton binding energy in GaAsBiN spherical quantum dot heterostructures

The ground state exciton binding energies (EBE) of heavy hole excitons in GaAs1-x-yBixNy – GaAs spherical quantum dots (QD) are calculated using a variational approach under 1s hydrogenic wavefunctions within the framework of effective mass approximation. Both the nitrogen and the bismuth content in the material are found to affect the binding energy, in particular for larger nitrogen content and lower dot radii. Calculations also show that the ground state exciton binding energies of heavy holes increase more at smaller dot sizes as compared to that for the light hole excitons.

A Fortran program to calculate the matrix elements of the Coulomb interaction involving hydrogenic wave functions

The program MTRXCOUL [1] calculates the matrix elements of the Coulomb interaction between a charged particle and an atomic electron, ∫ψf∗(r)|R−r|−1ψi(r)dr. Bound-free transitions are considered, and non-relativistic hydrogenic wave functions are used. In this revised version a bug discovered in the F3Y CPC Program Library (PL) subprogram [2] is fixed. Furthermore, the COULCC CPC PL subprogram [3] applied for the calculations of the radial wave functions of the free states and the Bessel functions is replaced by the CPC PL subprogram DCOUL [4]. New version program summaryProgram Title: MTRXCOULProgram Files doi:http://dx.doi.org/10.17632/xyg9zrmzz2.1Licensing provisions: GNU GPL v3Programming language: Fortran 77Journal reference of previous version: Comput. Phys. Commun. 133 (2000) 119.Does the new version supersede the previous version?: YesReasons for the new version:1.In some applications MTRXCOUL led to unexpected results that were traced back to the erroneous execution of the subprogram F3Y [2]. For example, in some cases F3Y yielded completely different values for the inputs (l,m1,l,m2,l′,m′) and (l,m2,l,m1,l′,m′), while, for symmetry reason, one expects equal results. In the new version this error in F3Y was corrected.2.In MTRXCOUL the COULCC subprogram [3] is applied for the calculations of the radial wave functions of the free states RE,l(r) and the Bessel functions Jn(x). Since the publication of MTRXCOUL a relativistic version of the program, MTRDCOUL has also been developed and published [5]. In MTRDCOUL RE,l(r) and Jn(x) are calculated by the subprogram DCOUL written by Salvat et al. [4]. Since the latter program is suitable also for calculations of non-relativistic wave functions, to ensure consistency between MTRXCOUL and MTRDCOUL, in the revised program the COULCC [3] was replaced by DCOUL. Furthermore, in some applications DCOUL turned out to be more efficient than COULCC.Summary of revisions:1.In the line AAQQ0036 of the original F3Y program [2] the incorrect assignment A2=L1−M1−N2 was replaced by A2=L1−M1−N1. The corrected F3Y was tested by comparing its results with those obtained by a program written for the integral of the product of three spherical harmonics using the 369j program [6]. An agreement within 10−14 was found between the two calculations for all possible arguments of the function belonging to the values of li up to 10.2.In the RFINAL function of the revised MTRXCOUL the regular Coulomb function Fl(η,x) is calculated using the code DCOUL [4]. For E≠0Fl(η,x) is obtained using the SCOUL subroutine. For E=0RE,l(r) is expressed in terms of the Bessel function Jn(x). The latter is obtained calling the FCOUL subroutine of DCOUL in a separate program unit, BESSJ(N,X).Nature of problem: The theoretical description of the excitation and ionization of atoms by charged particle impact often requires the knowledge of the matrix elements of the Coulomb interaction. Considering that the program can easily be extended to the calculations of matrix elements between wave functions other than the hydrogenic ones, it may find a broad application including the treatment of the electron–electron correlation problems.Solution method: The algorithm is based on the multipole series expansion of the Coulomb potential.Additional comments including Restrictions and Unusual features: The matrix elements can be calculated with the following restrictions. The initial bound states are limited to 1s, 2s, 2p, 3s, 3p, 3d. The quantum number l in the final state has a maximum value of 10.AcknowledgmentsThis work was supported by the National Scientific Research Foundation (OTKA, Grant No. K109440). [1]L. Sarkadi, Comput. Phys. Commun. 133 (2000) 119.[2]A. Liberato de Brito, Comput. Phys. Commun. 25 (1982) 81.[3]I.J. Thompson and A.R. Barnett, Comput. Phys. Commun. 36 (1985) 363.[4]F. Salvat, J.M. Fernández-Varea and W. Williamson Jr., Comput. Phys. Commun. 90 (1995) 151.[5]L. Lugosi and L. Sarkadi, Comput. Phys. Commun. 141 (2001) 73.[6]L. Wei, Comput. Phys. Commun. 120 (1999) 222; Erratum: 182 (2011) 1199.AppendixTEST RUN OUTPUT [Display omitted]

Calculation of the matrix elements of the Coulomb interaction involving relativistic hydrogenic wave functions

The program MTRDCOUL [1] calculates the matrix elements of the Coulomb interaction between a charged particle and an atomic electron, ∫ψf∗(r)∣R−r∣−1ψi(r)dr. Bound-free transitions are considered, and relativistic hydrogenic wave functions are used. In this revised version a bug discovered in the F3Y CPC Program Library subprogram [2] is fixed. New version program summaryProgram Title: MTRDCOULProgram Files doi: “http://dx.doi.org/10.17632/4cmts2c49b.1 ”Licensing provisions: GNU GPL v3Programming language: Fortran 77Journal reference of previous version: Comput. Phys. Commun. 141 (2001) 73Does the new version supersede the previous version?: YesReasons for the new version: In some applications MTRDCOUL led to unexpected results that were traced back to the erroneous execution of the subprogram F3Y [2]. For example, in some cases F3Y yielded completely different values for the inputs (l,m1,l,m2,l′,m′) and (l,m2,l,m1,l′,m′), while, for symmetry reason, one expects equal results. In the new version this error in F3Y was corrected.Summary of revisions: In the line AAQQ0036 of the original F3Y program [2] the incorrect assignment A2=L1−M1−N2 was replaced by A2=L1−M1−N1. The corrected F3Y was tested by comparing its results with those obtained by a program written for the integral of the product of three spherical harmonics using the 369j program [3]. An agreement within 10−14 was found between the two calculations for all possible arguments of the function belonging to the values of li up to 10.Nature of problem: The theoretical description of the excitation and ionization of atoms by charged particle impact often requires the knowledge of the matrix elements of the Coulomb interaction. Considering that the program can easily be extended to the calculations of matrix elements between wave functions other than the hydrogenic ones, it may find a broad application including the treatment of the electron–electron correlation problems.Solution method: The algorithm is based on the multipole series expansion of the Coulomb potential.Additional comments including Restrictions and Unusual features: The matrix elements are calculated with the following restrictions. The initial bound states are limited to 1s1/2, 2s1/2, 2p1/2, 2p3/2, 3s1/2, 3p1/2, 3p3/2, 3d3/2, 3d5/2. The quantum number l in the final state has a maximum value of 10.AcknowledgmentsThis work was supported by the National Scientific Research Foundation (OTKA, Grant No. K109440). [1]L. Lugosi and L. Sarkadi, Comput. Phys. Commun. 141 (2001) 73.[2]A. Liberato de Brito, Comput. Phys. Commun. 25 (1982) 81.[3]L. Wei, Comput. Phys. Commun. 120 (1999) 222; Erratum: 182 (2011) 1199.AppendixTEST RUN OUTPUT [Display omitted]

Hydrogen atom wave function and eigen energy in the Rindler space

We study the hydrogen atom eigenstate energy and wave function in the Rindler space. The probability distribution is tilted because the electric field of the nucleus is no longer spherically symmetric. The hydrogen atom therefore cannot be treated exactly in the same way as what it is in an inertial frame. We also find that if the external force accelerates only the nucleus and then the nucleus accelerates its surrounding electrons through electromagnetic force, the electrons can tunnel through the local energy gap and split the hydrogen atom into an ion. This is similar to what one expects from the Stark effect. However, the critical acceleration is about 3×1022 m/s2. It is well beyond the gravitational acceleration on a regular star surface.

Shaking during Ion-Atom Collisions

The shaking (shakeup and shakeoff) processes accompanying ion–atom collisions are studied using nonrelativistic hydrogenic wave functions for the K-, Land M -shell electrons in the sudden approximation limit. The role of recoil amplitude in the shaking processes is discussed. It is found that the suddenness of collision between projectile and target nuclei plays a more definitive factor in the shaking of the respective atomic system than the recoil of nuclei.

Entropic properties ofD-dimensional Rydberg systems

The fundamental information-theoretic measures (the R\'enyi $R_{p}[\rho]$ and Tsallis $T_{p}[\rho]$ entropies, $p>0$) of the highly-excited (Rydberg) quantum states of the $D$-dimensional ($D>1$) hydrogenic systems, which include the Shannon entropy ($p \to 1$) and the disequilibrium ($p = 2$), are analytically determined by use of the strong asymptotics of the Laguerre orthogonal polynomials which control the wavefunctions of these states. We first realize that these quantities are derived from the entropic moments of the quantum-mechanical probability $\rho(\vec{r})$ densities associated to the Rydberg hydrogenic wavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$, which are closely connected to the $\mathfrak{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the ($n\to\infty$)-asymptotics of these norms in terms of the basic parameters of our system (the dimensionality $D$, the nuclear charge and the hyperquantum numbers $(n,l,\{\mu\}$) of the state) by use of recent techniques of approximation theory. Finally, these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states.

Excited-state positronium formation in positron–hydrogen collisions under weakly coupled plasmas

The effect of screening of weakly coupled plasma on positronium (Ps) formation in excited states in the scattering of a positron from the ground state of a hydrogen atom has been investigated using a distorted wave theory which includes screened dipole polarization potential. The effect of external plasma has been incorporated by using the Debye–Hückel screening model of the interacting charge particles. Variationally determined simple hydrogenic wave functions have been used to obtain the distorted wave scattering amplitude in a closed form. Effects of plasma screening on the differential and total cross sections have been studied in detail in the energy range 20–300 eV of incident positron. For the free atomic case, our results agree nicely with some of the most accurate results available in the literature. To the best of our knowledge, such a study on the differential and total cross sections for Ps formation in highly excited states in positron–hydrogen collisions under weakly coupled plasma is reported first time in the literature.

Dependence of heavy hole exciton binding energy and the strain distribution in GaAs1−xBix/GaAs finite spherical quantum dots on Bi content in the material

The ground state binding energy of heavy hole excitons confined in GaAs1−xBix/GaAs spherical quantum dots is calculated as a function of dot radius and the Bi content using a Variational method based on 1-s hydrogenic wave functions with effective mass approximation. The parameter shows strong dependence on the Bi mole fraction x, particularly at smaller values of the dot radii. The strain associated with the quantum dot is found to decrease exponentially with increase in dot radius and shows a linear increase with Bi composition.

Analytical evaluation of atomic form factors: Application to Rayleigh scattering

Atomic form factors are widely used for the characterization of targets and specimens, from crystallography to biology. By using recent mathematical results, here we derive an analytical expression for the atomic form factor within the independent particle model constructed from nonrelativistic screened hydrogenic wave functions. The range of validity of this analytical expression is checked by comparing the analytically obtained form factors with the ones obtained within the Hartee-Fock method. As an example, we apply our analytical expression for the atomic form factor to evaluate the differential cross section for Rayleigh scattering off neutral atoms.