Angular Quantum Number Research Articles

Tunable magnetic properties and magnetocaloric effect of TmGa by Ho substitution

The influence of Ho substitution for Tm atoms on the magnetic properties and magnetocaloric effect (MCE) of TmGa compound was systematically investigated according to magnetic measurements and neutron powder diffraction (NPD) experiments. The magnetic transitions of ${\mathrm{Tm}}_{1\ensuremath{-}x}{\mathrm{Ho}}_{x}\mathrm{Ga}$ compounds show different types by Ho substitution due to the variation of spin and orbital angular momentum quantum number and the complete magnetic diagram of ${\mathrm{Tm}}_{1\ensuremath{-}x}{\mathrm{Ho}}_{x}\mathrm{Ga}$ compounds was obtained. The spin reorientation (SR) transition of ${\mathrm{Tm}}_{0.1}{\mathrm{Ho}}_{0.9}\mathrm{Ga}$ compound was directly confirmed by variable-temperature NPD experiments. Results show that the magnetic moment orders along the $c$ axis at the temperatures between ${T}_{\mathrm{SR}}$ and ${T}_{\mathrm{C}}$ and it cants away from the $c$ axis towards the $ab$ plane upon cooling below ${T}_{\mathrm{SR}}$. Furthermore, Ho substitution plays a dominant role in MCE of ${\mathrm{Tm}}_{1\ensuremath{-}x}{\mathrm{Ho}}_{x}\mathrm{Ga}$ compounds. When $x=0.15$, the peak value of magnetic entropy change reaches the maximum value of 18.0 J/kg K under field change of 0--2 T. The refrigerant temperature span ($\ensuremath{\delta}{T}_{\mathrm{FWHM}}$) and refrigeration capacity of ${\mathrm{Tm}}_{0.85}{\mathrm{Ho}}_{0.15}\mathrm{Ga}$ compound show enhancement of 23.0 and 21.6%, correspondingly, compared with TmGa compound. The giant MCE of ${\mathrm{Tm}}_{0.85}{\mathrm{Ho}}_{0.15}\mathrm{Ga}$ compound results from the optimization of spin and total angular momentum quantum number by Ho substitution.

Monopole versus spherical harmonic superconductors: Topological repulsion, coexistence, and stability

The monopole harmonic superconductor (SC), proposed in doped Weyl semimetals as a pairing between the Fermi surfaces enclosing the Weyl points, is rather unusual, as it features the monopole charge inherited from the parent metallic phase. However, this state can compete with more conventional spherical harmonic pairings, such as an $s$-wave. We here demonstrate, within the framework of the weak coupling mean-field BCS theory, that the monopole and a conventional spherical harmonic SC quite generically coexist, while the repulsion can take place when the absolute value of the monopole charge matches the angular momentum quantum number of the spherical harmonic. As we show, this feature is a direct consequence of the topological nature of the monopole SC, and we dub it \emph{topological repulsion}. We illustrate the above principle with the example of the conventional $s-$ and $(p_x\pm ip_y)-$wave pairings competing with the monopole SC $Y_{-1,1,0}(\theta,\phi)$, which coexist in a finite region of the parameter space, and repel, respectively. Furthermore, the s-wave pairing is more stable both when the chemical potentials at the nodes are unequal, and in the presence of point-like charged impurities. Since the phase transition is discontinuous, close to the phase boundary, we predict that the Majorana surface modes at the interfaces between domains featuring the monopole and the trivial phases, such as an $s-$wave, will be the experimental signature of the monopole SC.

Interaction of the magnetic quadrupole moment of a non-relativistic particle with an electric field in the background of screw dislocations with a rotating frame

In this study, we considered a moving particle with a magnetic quadrupole moment in an elastic medium in the presence of a screw dislocation. We assumed a radial electric field in a rotating frame that leads a uniform effective magnetic field perpendicular to the plane of motion. We solved the Schrödinger equation to derive wave and energy eigenvalue functions by employing analytical methods for two interaction configurations: in the absence of potential and in the presence of a static scalar potential. Due to the topological defect in the medium, we observed a shift in the angular momentum quantum number which affects the energy eigenvalues and the wave function of the system.

Statistical and analytical approaches to finite-temperature magnetic properties of the compound SmFe12

To investigate the magnetic properties of $\mathrm{Sm}{\mathrm{Fe}}_{12}$, we construct an effective spin model, where magnetic moments, crystal-field (CF) parameters, and exchange fields at 0 K are determined by first-principles calculations. Finite-temperature magnetic properties are investigated by using this model. We further develop an analytical method with strong mixing of states with a different quantum number of angular momentum $J$ ($J$-mixing), which is caused by a strong exchange field acting on the spin component of $4f$ electrons. Comparing our analytical results with those calculated by Boltzmann statistics, we clarify that the previous analytical studies for Sm transition-metal compounds overestimate the $J$-mixing effects. The present method enables us to perform a quantitative analysis of the temperature dependence of magnetic anisotropy (MA) with high reliability. The analytical method with model approximations reveals that the $J$-mixing caused by the exchange field increases the spin angular momentum, which enhances the absolute value of the orbital angular momentum and MA constants via spin-orbit interaction. It is also clarified that these $J$-mixing effects remain even above room temperature. Magnetization of $\mathrm{Sm}{\mathrm{Fe}}_{12}$ shows a peculiar field dependence known as the first-order magnetization process (FOMP), where the magnetization shows an abrupt change at a certain magnetic field. The result of the analysis shows that the origin of FOMP is attributed to competitive MA constants between positive ${K}_{1}$ and negative ${K}_{2}$. The sign of ${K}_{1(2)}$ appears due to an increase in the CF potential denoted by the parameter ${A}_{2}^{0}\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}$ (${A}_{4}^{0}\ensuremath{\langle}{r}^{4}\ensuremath{\rangle}$) caused by hybridization between $3d$-electrons of Fe on the $8i$ ($8j$) site and $5d$ and $6p$ valence electrons on the Sm site. It is verified that the requirement for the appearance of FOMP is given as $\ensuremath{-}{K}_{2}<{K}_{1}<\ensuremath{-}6{K}_{2}$.

Solutions of Schrodinger equation and thermal properties of generalized trigonometric Poschl-Teller potential.

Analytical solutions of the Schrödinger equation for the generalized trigonometric Pöschl–Teller potential by using an appropriate approximation to the centrifugal term within the framework of the Functional Analysis Approach have been considered. Using the energy equation obtained, the partition function was calculated and other relevant thermodynamic properties. More so, we use the concept of the superstatistics to also evaluate the thermodynamics properties of the system. It is noted that the well-known normal statistics results are recovered in the absence of the deformation parameter and this is displayed graphically for the clarity of our results. We also obtain analytic forms for the energy eigenvalues and the bound state eigenfunction solutions are obtained in terms of the hypergeometric functions. The numerical energy spectra for different values of the principal and orbital quantum numbers are obtained. To show the accuracy of our results, we discuss some special cases by adjusting some potential parameters and also compute the numerical eigenvalue of the trigonometric Pöschl–Teller potential for comparison sake. However, it was found out that our results agree excellently with the results obtained via other methods

Carrier multiplication in semiconductor quantum dots

The creation of many electron–hole (e–h) pairs in semiconductors under the absorption of a single photon was named as carrier multiplication (CM) process. The probabilities of the creation of two and three e–h pairs under the absorption of a single photon in semiconductor quantum dots (QDs) were investigated in the second and in the third orders of the perturbation theory using as the perturbations the Hamiltonians of the electron–radiation and electron–electron Coulomb interactions. It was supposed that the photon in the initial state has a sufficient energy to create two or three e–h pairs with the electrons in the lowest conduction band and with the holes in the valence band on the lowest levels of the size quantization. Their envelope wave functions were determined as the solutions of the Schrödinger equation for the electron inside the impenetrable sphere with the radius R0 of the QD. They are expressed through the spherical Bessel functions. It was shown that the lowest energy level of the size quantization with orbital quantum number l=0 in the QDs with small radii R→0 has the wave function similar with the 3D Dirac delta function. In this approximation the probabilities of the CM were estimated.

Eigensolution techniques, expectation values and Fisher information of Wei potential function.

An approximate solution of the one-dimensional relativistic Klein-Gordon equation was obtained under the interaction of an improved expression for Wei potential energy function. The solution of the non-relativistic Schrödinger equation was obtained from the solution of the relativistic Klein-Gordon equation by certain mappings. We have calculated Fisher information for position space and momentum space via the computation of expectation values. The effects of some parameters of the Wei potential energy function on the Fisher information were fully examined graphically. We have also examined the effects of the quantum number n and the angular momentum quantum number ℓ on the expectation values and Fisher information respectively for some selected molecules. Our results revealed that the variation of most of the parameters of the Wei potential energy function against the Fisher information does not obey the Heisenberg uncertainty relation for Fisher information while that of the quantum number and angular momentum quantum number on Fisher information obeyed the relation.

The use of oscillator-strength sum rules for evaluating the angular-momentum dependence of the blackbody-radiation-induced broadening and shift of Rydberg-state energy levels

First evaluations of the blackbody-radiation (BBR)-induced shift and broadening of highly excited Rydberg states were based on the account of only the first term of the expansion for the Planck’s distribution in powers of the ratio of the BBR-photon energy to the environmental thermal energy. The corresponding asymptotic expression for the shift is proportional to a sum rule for the oscillator strengths, which is identical for all states, independently of their principal and angular-momentum quantum numbers. The broadening appeared proportional to the sum of the oscillator-strength first moments, which is dependent on only the principal quantum number, being independent of the state’s angular momentum. In this article, we demonstrate that the account of higher-order terms of the Planck’s distribution expansion provides corrections revealing the dependence of asymptotic shift and broadening on the Rydberg-state principal and angular-momentum quantum numbers. The strongest corrections are observed for states with low angular momenta, in agreement with earlier performed numerical calculations. Sum rules for higher moments of oscillator strengths are used for evaluating the corrections for the BBR-induced broadening and shift of Rydberg-state energy levels in neutral atoms and ions of arbitrary nuclear charge Z.

Greybody factor and quasinormal modes of Regular Black Holes

In this work, we investigate the quasinormal frequencies of a class of regular black hole solutions which generalize Bardeen and Hayward spacetimes. In particular, we analyze scalar, vector and gravitational perturbations of the black hole with the semianalytic WKB method. We analyze in detail the behaviour of the spectrum depending on the parameter p/q of the black hole, the quantum number of angular momentum and the s number. In addition, we compare our results with the classical solution valid for p = q = 1.

Radiative lifetimes of several doublet and quartet states of silicon boride

In this study, the potential energy curves were calculated for the X4Σ–, A4Π, B4Δ, C4Σ+, D4Σ–, E4Π, a2Π, b2Δ, c2Σ–, d2Σ+, and e2Π states of silicon boride. The transition dipole moments were computed for all dipole–allowed transitions between these states. The complete active space self-consistent field method was used for calculations, followed by the valence internally contracted multireference configuration interaction approach. The radiative lifetimes were approximately 1 – 10 μs, 0.1 – 1 μs, 1 – 10,000 ms, 1 – 1000 ms, and 1 μs for the A4Π, D4Σ–, B4Δ, C4Σ+, and E4Π states, respectively. Of these transitions between the six lowest–lying quartets, the spontaneous emissions from the A4Π – X4Σ–, D4Σ– – X4Σ–, E4Π – X4Σ–, and E4Π – A4Π systems were strong, whereas those from the B4Δ – A4Π, C4Σ+ – A4Π, and E4Π – B4Δ transitions were weak. The radiative lifetimes were approximately 0.1 – 1000 ms, 0.1 – 100 ms, 10 – 100 μs, and 1 – 10 μs for the b2Δ, c2Σ–, d2Σ+, and e2Π states, respectively. Of the transitions between the a2Π, b2Δ, c2Σ–, d2Σ+, and e2Π states, the spontaneous emissions from the e2Π – b2Δ and e2Π – c2Σ– systems were relatively strong, whereas those from the b2Δ – a2Π transition were relatively weak. The transition frequencies, Einstein A coefficients, and Franck–Condon factors of all spontaneous vibronic bands from these transitions were calculated. The results obtained in this study were compared with experimental and other theoretical values. The radiative–lifetime distribution varying with rotational angular quantum number J was calculated when J ≤ 50.5 for a particular vibrational level of these states.

Some aspects related to the transformation of the three body wave function built on the Gaussian basis

The three-body wave function built on the basis of the Gaussian function, calculated using the three-body Hamiltonian with the Pauli blocking operator is studied. As an example, the wave function of the ground state of the 9 Be was taken. Analytical expressions are presented for the overlap matrix elements of the basis function for both basic and alternative set of relative Jacobi coordinates. The correlation densities of the wave function are calculated and illustrated depending on the set of orbital quantum numbers.

Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

This work describes the fully analytical method for the calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper (Bagci and Hoggan in Phys Rev E 91(2):023303, 2015). The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.

On the geometric quantization of the ro-vibrational motion of homonuclear diatomic molecules

In the framework of geometric quantization we extend the harmonic oscillator rules to rotation and vibration of molecules. First, the geometric quantization of the rigid rotor is introduced. The novelty is that we found an explicit relation giving the impulse along the third axes, which takes discrete values and found that it depends not only on the principal quantum number, but also on the orbital and magnetic quantum numbers. Geometric quantization is further extended to the class of vibrational (damped) systems. The theory is sought as mathematical requirement, which can be traced in the analysis of integrable systems. It is shown that properties of the critical points of the momentum map can be the key to the integrability. A C⁎ isomorphism of the momentum map between the Lie algebra of physical observables and that of the harmonic oscillator induces the geometric quantization on former. We show how the method works, taking into account the Lennard-Jones' potential, which characterizes the interaction between molecules. A second example shows a physical system having the energy levels depending on the magnitude of the frequencies.

A phase-integral perspective on $$\alpha $$-decay

This paper applies the phase-integral method to the stationary theory of $$\alpha $$ -decay. The rigorous form of the connection formulas, and their one-directional nature that was not widely known in the physical literature, are applied. The condition for obtaining s-wave metastable states affects the stationary state at large distance from the nucleus, which is dominated by the cosine of the phase integral minus $${\pi \over 4}$$ . Accurate predictions for the lowest s-wave metastable state and mean life of the radioactive nucleus are obtained in the case of uranium. The final part of the paper describes the phase-integral algorithm for evaluating stationary states by means of a suitable choice of freely specifiable base function. Within this framework, original approximate formulas for the phase integrand with arbitrary values of the angular momentum quantum number are obtained.

Radiative lifetimes of the 11Σ–, 11Δ, 21Σ+, a3Π, 13Σ+, d3Δ, and e3Σ– states of carbon monosulfide

Numerous experimental and theoretical investigations studied the transition properties of the A1Π – X1Σ+ system of carbon monosulfide. However, only a few transition properties are currently available except for the A1Π state. Therefore, this study calculated the properties of the dipole-allowed transitions arising from the 11Σ–, 11Δ, 21Σ+, a3Π, 13Σ+, d3Δ, and e3Σ– states and the spin–forbidden transition from the a3Π to the X1Σ+ state. The radiative lifetimes were approximately 10 ns for the 21Σ+ state, 10 μs for the e3Σ– state, 10 – 100 μs for the d3Δ state, and 10 – 1000 μs for the 13Σ+ state. The emissions from the 21Σ+ – X1Σ+ system were strong. The radiative lifetimes were of the order of 0.1 – 1000 ms for the 11Δ state and 0.1 – 105 ms for the 11Σ– state. The lower the vibrational level of the 11Δ and 11Σ– states was, the longer the radiative lifetime became, suggesting that the spontaneous emissions arising from the lower vibrational levels of the two states were difficult to be measured through spectroscopy. The radiative lifetimes were of the order of 1 – 10 and 1 ms for the a3Π1 and a3Π0+ states, respectively; whereas those of the a3Π2 and a3Π0– states were extremely long. The radiative–lifetime distribution varying with the rotational angular quantum number J was investigated at J ≤ 70 for a particular vibrational level of the 11Σ–, 11Δ, 21Σ+, a3Π, 13Σ+, d3Δ, e3Σ–, a3Π1, and a3Π0+ states.

Effect of curvature on the eigenstates of magnetic skyrmions

Spectrum of spin eigenmodes localized on a ferromagnetic skyrmion pinned by a geometrical defect (bump) of magnetic films is studied theoretically. By means of direct numerical solution of the corresponding eigenvalue problem and finite element micromagnetic simulations we demonstrate, that the curvature can induce localized modes with higher azimuthal and radial quantum numbers, which are absent for planar skyrmions (for the same parameters). The eigenfrequencies of all modes, except the breathing and gyromodes decreases with increasing curvature. Due to the translational symmetry break, the zero translational mode of the skyrmion gains a finite frequency and forms the gyromode, which describes the uniform rotation of skyrmions around the equilibrium position. In order to treat the gyromotion analytically we developed a Thiele-like collective variable approach. We show that N\'{e}el skyrmions in curvilinear films experience a driving force originating from the gradient of the mean curvature. The gyrofrequency of the pinned skyrmion is proportional to the second derivative of the mean curvature at the point of equilibrium.

Ground-state phase diagram and excitation spectrum of a Bose-Einstein condensate with spin-orbital-angular-momentum coupling

We investigate the ground-state phase diagram and excitation spectrum of an interacting spinor Bose-Einstein condensate with spin-orbital-angular-momentum (SOAM) coupling realized in recent experiments by introducing atomic Raman transition with a pair of copropagating Laguerre-Gaussian laser beams that carry different orbital angular momenta (OAM) [Chen et al., Phys. Rev. Lett. 121, 113204 (2018) and Zhang et al., Phys. Rev. Lett. 122, 110402 (2019)]. Because of the ground-state degeneracy of the single-particle Hamiltonian at vanishing detuning, several angular-stripe phases, which are superposition of states with different angular quantum numbers, appear in the phase diagram. However, these phases normally exist at small detuning, which makes them hard to be probed in experiments. We show that for a large OAM difference of the laser beams, an asymmetric kind of angular-stripe phase can exist even at large detuning. The excitation spectra in different phases exhibit distinct features: In the angular-stripe phase there exist two gapless bands corresponding to the broken U(1) and rotational symmetries, while in the half-skyrmion phase the gapless band exhibits a roton-like structure. Our predictions of the angular-stripe phases and the low-energy excitations can be examined in recently realized BECs with SOAM coupling.

Radiative lifetimes and transition properties of boron monoxide cation

The transition dipole moments of dipole–allowed transitions between the following states were calculated in this study: X1Σ+, A1Π, B1Σ+, C1Δ, D1Σ–, and E1Π; a3Π, b3Σ+, c3Σ–, d3Π, and e3Δ, and those of the spin–forbidden transitions from the a3Π state to the X1Σ+ state of boron monoxide cation. The CASSCF method was used, followed by the valence icMRCI approach. The radiative lifetimes of the vibrational states were of the order of 1 – 10 μs for the A1Π state; 10 ns for the D1Σ–, e3Δ, and the first well of the B1Σ+ states; 1 – 1000 ns for the second well of the B1Σ+ state; 100 ns for the d3Π state; 0.1 – 1 μs for the b3Σ+ and E1Π states; 10 – 100 μs for the C1Δ state; and 10 – 10,000 μs for the c3Σ– state. The transition from the first well of the B1Σ+ state to the X1Σ+ state and that from the double well of the B1Σ+ state to the A1Π state were the strongest, followed by the D1Σ– – A1Π, E1Π – A1Π, d3Π – a3Π, and e3Δ – a3Π transitions. The transition from the second well of the B1Σ+ state to the X1Σ+ state was the weakest, followed by the C1Δ – A1Π, E1Π – B1Σ+, E1Π – C1Δ, E1Π – D1Σ–, c3Σ– – a3Π, and e3Δ – d3Π transitions. The variation in the radiative–lifetime distribution with the rotational angular quantum number J was evaluated. The radiative lifetimes of the vibrational states were of the order of 1 – 10 ms for the a3Π1 state and 10 ms for the a3Π0+ state. The a3Π1 – X1Σ+0+ and a3Π0+ – X1Σ+0+ transitions were weak. The transition properties reported herein may provide useful guidelines for related experimental and theoretical investigations in the near future.

Exact solutions of the rigid rotor in the electric field

AbstractWe first present a new constraint condition on the confluent Heun function HC(α, β, γ, δ, η;z) (β, γ ≥ 0, z ∈ [0, 1]) and then illustrate how to solve the rigid rotor in the electric field. We find its exact solutions unsolved previously through solving the Wronskian determinant. The present results compared with those by the perturbation methods are found to have a big difference for a large parameter a. We also present 2D and 3D probability density distributions by choosing different angular momentum quantum numbers l. We observe that the original eigenvalues with degeneracy (2 l + 1) are split into the (l + 1) state with approximate eigenvalues l(l + 1) for small a but large l.

The New “Noble Gas” Molecule: A Molecular Trip beyond Atoms

In a recent issue of Nature Chemistry, Jishan Wu and co-workers reported the hexacation species of a 3D conjugated molecular cage whose 50 π-electrons follow Hirsch’s rule of 3D aromaticity. We show how Hirsch’s rule connects hydrogenoid atoms, the (unprepared) decacation of the C60 fullerene, and the Wu group’s hexacation.