Magnetic Dipole Matrix Elements Research Articles

An ab initio spectroscopic model of the molecular oxygen atmospheric and infrared bands.

We present a unified variational treatment of the magnetic dipole matrix elements, Einstein coefficients and line strength for general open-shell diatomic molecules in the general purpose diatomic code Duo. Building on previous work in which similar expressions for the electric quadrupole transitions were developed, we also present a complete ab initio spectroscopic model for the infrared, electric dipole-forbidden, spectrum of the 16O2 molecule. The model covers seven states, namely the X 3Σ-g, a 1Δg, b 1Σ+g, I 1Πg, II 1Πg, I 3Πg and II 3Πg states, for which 7 potential energy, 6 electronic angular momentum, 7 spin-orbit, and 14 quadrupole moment curves are calculated using ic-MRCI theory and an aug-cc-pV5Z basis set. These curves are diabatised to remove avoided crossings between the excited Π states, and the resultant properties are used to produce a line list for higher-order transitions of astrophysical interest.

Relativistic many-body calculation of energies, transition rates, lifetimes, and multipole polarizabilities in Cs-like La iii

Excitation energies of the [Xe]$nd$ ($n=5\ensuremath{-}9$), [Xe]$ns$ ($n=6\ensuremath{-}10$), [Xe]$np$ ($n=6\ensuremath{-}9$), [Xe]$nf$ ($n=4\ensuremath{-}8$), and [Xe]$ng$ ($n=5\ensuremath{-}8$) states in La iii, where [Xe] = $1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{10}4{s}^{2}4{p}^{6}4{d}^{10}5{s}^{2}5{p}^{6}$, are evaluated. Electric dipole matrix elements for the allowed transitions between the low-lying [Xe]$nd$, [Xe]$ns$, [Xe]$np$, [Xe]$nf$, and [Xe]$ng$ states in the La iii ion are calculated using the high-precision relativistic all-order method where all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values are provided for a large number of electric dipole matrix elements, oscillator strengths, transition rates, and lifetimes. Scalar and tensor polarizabilities of the states listed above are evaluated. The uncertainties of the recommended values are estimated. Electric quadrupole and magnetic dipole matrix elements are calculated to determine lifetimes of the $5{d}_{5/2}$ and $6s$ metastable levels. The ground-state $E1$, $E2$, and $E3$ static polarizabilities are calculated. This work provides recommended values critically evaluated for their accuracy for a number of La iii atomic properties for use in planning and analysis of various experiments as well as theoretical modeling.

Simulating quantum magnets with symmetric top molecules

A correspondence is established between the electric dipole matrix elements of a polyatomic symmetric top molecule in a state with nonzero projection of the total angular momentum on the symmetry axis of the molecule and the magnetic dipole matrix elements of a magnetic dipole associated with an elemental spin F. It is shown that this correspondence makes it possible to perform quantum simulation of the single‐particle spectrum and the dipole‐dipole interactions of magnetic dipoles in a static external magnetic field with symmetric top molecules subject to a static external electric field . It is further shown that no such correspondence exists for 1Σ molecules in static fields, such as the alkali metal dimers. The effective spin angular momentum of the simulated magnetic dipole corresponds to the rotational angular momentum of the symmetric top molecule, and so quantum simulation of arbitrarily large integer spins is possible. Further, taking the molecule CH3F as an example, it is shown that the characteristic dipole‐dipole interaction energies of the simulated magnetic dipole are a factor of 620, 600, and 310 larger than for the highly magnetic atoms Chromium, Erbium, and Dysprosium, respectively. Several applications of the correspondence for many‐body physics are presented, including long‐range and anisotropic spin models with arbitrary integer spin S using symmetric top molecules in optical lattices, quantum simulation of molecular magnets, and spontaneous demagnetization of Bose‐Einstein condensates due to dipole‐dipole interactions. These results are expected to be relevant as cold symmetric top molecules reach quantum degeneracy through Stark deceleration and opto‐electrical cooling.

Critically evaluated theoretical atomic properties of Y iii

A systematic study of Y iii atomic properties is carried out using high-precision relativistic all-order method. Recommended values and estimates of their uncertainties are provided for a large number of electric-dipole reduced matrix elements, transition rates, and oscillator strengths for allowed transitions between $ns$, $n{p}_{j}$, $n{d}_{j}$, $n{f}_{j}$, and $n{g}_{j}$ levels with $n\ensuremath{\le}8$. The lifetimes of these levels are also evaluated. Electric-quadrupole and magnetic-dipole matrix elements are calculated to determine lifetimes of the $4{d}_{5/2}$ and $5s$ metastable levels. The ground-state $E1$, $E2$, and $E3$ static polarizabilities are calculated. This work provides recommended values critically evaluated for their accuracy for a number of Y iii atomic properties for use in theoretical modeling as well as planning and analysis of various experiments. We hope that the present study will stimulate further exploration of Y iii for various applications owing to its interesting structure of different low-lying metastable levels.

Relativistic many-body calculation of energies, oscillator strengths, transition rates, and lifetimes of Sc III ion

A systematic study of Sc iii atomic properties is carried out using a high-precision relativistic all-order method where all single, double, and partial-triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for the $ns$, $n{p}_{j}$, $n{d}_{j}$, $n{f}_{j}$, and $n{g}_{j}$ levels with $n\ensuremath{\leqslant}7$. Recommended values and estimates of their uncertainties are provided for a large number of electric-dipole transitions. Electric-quadrupole and magnetic-dipole matrix elements are evaluated to determine lifetimes of the $3{d}_{5/2}$ and $4s$ metastable levels. These calculations provide recommended values critically evaluated for their accuracy for a number of Sc iii atomic properties for use in theoretical modeling as well as planning and analysis of various experiments. We hope that the present study will stimulate further exploration of Sc iii for various applications owing to its interesting structure of different low-lying metastable levels.

Relativistic many-body calculations of energies, E2, and M1 transition rates of 4s 24p states in Ga-like ions

Ionization energies, fine-structure splitting of 4s 24p states, and magnetic-dipole and electric-quadrupole transition rates for the 4s 24p 2P 1/2–4s 24p 2P 3/2 transition are calculated for Ga-like ions with nuclear charge Z ranging from 31 to 100. First-, second-, third-order, and all-order Coulomb energies and first- and second-order Coulomb–Breit energies are calculated. Electric-quadrupole and magnetic-dipole matrix elements are evaluated to calculate the lifetime of the 4s 24p 2P 3/2 state. The matrix elements are calculated using both relativistic many-body perturbation theory, complete through third order, and the relativistic all-order method restricted to single and double excitations. These calculations provide a theoretical benchmark for comparison with experiment and theory.

Excitation energies, hyperfine constants,E1,E2, andM1transition rates, and lifetimes of6s2nlstates inTlIandPbII

Energies of np (n=6-9), ns (n=7-9), nd (n=6-8), and nf (n=5-6) states in Tl I and Pb II are obtained using relativistic many-body perturbation theory. Reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for the 72 possible electric-dipole transitions. Electric-quadrupole and magnetic-dipole matrix elements are evaluated to obtain np(3/2) - mp(1/2) (n,m=6,7) transition rates. Hyperfine constants A are evaluated for a number of states in 205Tl. First-, second-, third-, and all-order corrections to the energies and matrix elements and first- and second-order Breit corrections to energies are calculated. In our implementation of the all-order method, single and double excitations of Dirac-Fock wave functions are included to all orders in perturbation theory. These calculations provide a theoretical benchmark for comparison with experiment and theory.

New magnetic dipole phenomena in atomic nuclei

In recent years a number of nuclear phenomena was discovered that are accompanied by strong magnetic dipole (MI) transitions. Measurements of MZ matrix elements enabled nuclear physicists to deduce new structure information in a model-independent way, since the electromagnetic interaction is well understood. Due to the dominantly isovector character of the electromagnetic MI transition operator (see Appendix A), information on MI matrix elements helps to clarify various aspects of the nuclear isospin degree of freedom (see Appendix B). This is of particular importance as a prerequisite for making full use of newly established or planned radioactive ion beam (RIB) facilities probing nuclei at extreme values of isospin.

Coherent Superposition of Electric- and Magnetic-Dipole Spin-Flip Transitions in Zinc Blende Semiconductors

Spin resonance (SR) studies offer an important and unique opportunity for investigating coherent superposition of electric-and magnetic-dipole spin-flip transitions in semiconductors. We will focus on far-infrared studies of conduction-electron spin-flip transitions in narrow-gap semiconductors, where this effect is most clearly evident. Although the SR transition is normally electric-dipole-forbidden, it is well known that there exist mechanisms (e.g., “nonparabolicity” and inversion asymmetry) that relax these selection rules. In comparing the relative importance of these mechanisms, it will be shown that the combination of spin–orbit coupling and inversion asymmetry (as developed by E. I. Rashba and V. I. Sheka) is the dominant process allowing electric-dipole-induced spin-flip transitions. In this review, special attention will be given to the interference of the electric- and magnetic-dipole matrix elements, which provide a unique opportunity for determining the inversion-asymmetry parameter (including its sign) in zinc blende narrow-gap semiconductors. This effect can also serve as a basis for observing spin-based electromagnetically induced transparency—a phenomenon of considerable contemporary interest from both fundamental and applied viewpoints.

Microscopic theory of the second-order magnetoelectric susceptibility in the optical region: Case of boraciteCo3B7O13I

The microscopic theory of second-order magnetoelectric susceptibility in the optical region is presented for the case of optical transitions between states of Co 2 + ions in boracite Co 3 B 7 O 1 3 I. Using many particle wave functions of the Co 2 + ion in the crystal field of tetragonal symmetry D 2 d and accounting for spin-orbit and Zeeman interactions we obtain an expression for the second-order magnetoelectric susceptibility at optical frequencies, which takes into account the population of all Kramers doublets of the ground state and is valid in a wide temperature range. Both the splitting of Kramers doublets of the ground and excited states and the magnetic field dependence of electric and magnetic dipole matrix elements contribute to this susceptibility. The experimental data on dispersion, temperature dependences, and the absolute value of nonreciprocal birefringence in Co 3 B 7 O 1 3 I can be adequately described in the considered model.

Hyperfine structure, pressure shift and pressure broadening in the 648 nm transition of atomic bismuth by Faraday spectroscopy

The transition in atomic bismuth at 648 nm (6p3 4S32/-6p3 2D52/) has been studied by Faraday spectroscopy. Values for the upper level hyperfine constants have been deduced from the rotation spectrum as follows: A(2D52/)=2505 (1) MHz, B(2D52/)=38 (18) MHz. The ratio of the reduced electric quadrupole and magnetic dipole matrix elements was found to be -0.63 (3). Pressure broadening and shift of the transitions due to helium, neon, argon and krypton were also measured.

Measurement of the ratio of the squares of the reduced magnetic dipole matrix elements for the 876 nm and 648 nm transitions in atomic bismuth

Measurement of the ratio of the squares of the reduced magnetic dipole matrix elements for the 876 nm and 648 nm transitions in atomic bismuth

Far-infrared spin resonance in narrow-gap semiconductors

The author reviews far-infrared studies of spin resonance (SR) in narrow-gap semiconductors, with emphasis on SR of the conduction band. While the SR transition is normally electric-dipole-forbidden, it is well known that there exist mechanisms (e.g. 'non-parabolicity' or inversion asymmetry) which relax these selection rules. The author discusses these mechanisms and their relative importance. Although the author concentrates in this review on the Gamma 6-like conduction band of InSb, the author also reviews experimental results on other materials (e.g. HgSe, HgTe, Hg1-xMnxSe). The author discusses the anisotropy of the spin resonance and compares the g factors of free and donor-bound electrons. Furthermore, the author describes certain 'external' effects which influence the observation of SR in semiconductors, e.g. spin-spin exchange interaction in diluted magnetic semiconductors, plasma-shifted SR and uniaxial stress. Special attention is given to the interference of the electric dipole and magnetic dipole matrix elements, which provides a unique opportunity for the determination of the inversion asymmetry parameter (including its sign) in zincblende narrow-gap semiconductors.

Inversion asymmetry and hole magnetooptics in zinc-blende semiconductors.

We present a study of magnetooptical absorption by holes in zinc-blende semiconductors. We focus on transitions which, in the absence of the parity-violating Kane term in the Hamiltonian, are electric-dipole forbidden. These become allowed by virtue of mixing of states of different parity by the linear-k contribution to the effective Hamiltonian. The intensity of the resulting absorption is proportional to the square of the strength C of the Kane term. In addition to the electric-dipole transitions there are electric-quadrupole and magnetic-dipole transitions. Under certain conditions, in the Voigt geometrical arrangements, the matrix elements of the electric-dipole, the electric-quadrupole, and the magnetic-dipole moments give contributions to particular excitations and interference may result. The inversion-asymmetry contribution changes sign as the external magnetic field ${\mathrm{B}}_{0}$ is reversed while the electric-quadrupole and magnetic-dipole matrix elements change sign upon reversal of the incident photon momentum q. Thus, the interference may be made constructive or destructive by reversal of either ${\mathrm{B}}_{0}$ or q. We propose magnetooptical experiments which would allow the determination of both the magnitude and sign of C by measuring the change in the magnetooptic absorption upon reversal of ${\mathrm{B}}_{0}$. We give criteria for the existence of the interference and discuss application to InSb, GaSb, and HgTe.

Magnetic dipole transitions in deformed even-even nuclei

In the neutron-proton IBA model magnetic dipole matrix elements are derived for γ → γ and γ → g.s.b. transitions using a slightly broken neutron-proton ( F-spin) symmetry. The magnitude of the B(M1) values in a given nucleus as well as their variation with N and Z depends on the details of the symmetry breaking terms. A qualitative description of the experimental data is obtained.

Limits on the scaling of nucleon magnetic moments in nuclei

In view of the suggestion that nucleon magnetic moments inside nuclei may be modified due to a rescaling of the nucleon size, we investigate empirically how large such an effect can be. The method is based on a nearly model-independent scaling relation between the axial vector matrix element and the main part of the corresponding magnetic dipole matrix element supplemented by a small and well understood contribution from the one-pion exchange current. Taking the mass A = 3 and 12 systems as examples the upper limit, for such a change of the nucleon magnetic moment inside nuclei is found to be about 2%, considerably smaller than previous estimates in the literature.

Absorption and Faraday spectroscopy of the 876 nm line in Bi I

The transition 6p3 4S3/2-6p3 2D3/2 at 876 nm in atomic bismuth has been studied by laser spectroscopy. The absorption spectrum and the Faraday rotation spectrum have been measured over the frequency range occupied by the hyperfine structure, giving the following values for the interaction constants: A(2D3/2)=-1226.7(1.2) MHz, B(2D3/2)=-617(10) MHz. From the Faraday rotation spectrum the ratio of the reduced electric quadrupole and magnetic dipole matrix elements, (4S3/2//( omega /4 square root 3)*(-2er2C(2))//2D3/2)/(4S3/2// mu //2D3/2), was found to be 0.117(8). When the interaction constants are combined with data for the other levels of the 6p3 configuration, there is disagreement with the parameterisation of the fine and hyperfine structure given by Dembczynski et al. (1975, 1983).

The Crystal Field Scheme of PrCu6as Obtained by Inelastic Neutron Scattering

The inelastic neutron scattering was studied in both PrCu 6 and the reference system LaCu 6 in order to isolate the magnetic CF-transitions in orthorombic PrCu 6 . Ten CF-transitions could easily be resolved, leading to a CF-scheme with two singlets forming a nearly doubly degenerate groundstate (Δ≈0.02 men) without any coupling through a magnetic dipole matrix element (this doublet shows no quasielastic scattering). The first excited state is found to be 1.48 meV above that quasi-doublet ground state and the overall splitting comes out to be 8.95 meV. The static susceptibility calculated from this CF-scheme is in good agreement with the measured one, which is also presented in this paper.

Asymmetry measurement of the 2s1/2

When a beam of spin-polarized ${\mathrm{He}}^{+}$(2${\mathrm{s}}_{1/2}$) ions is quenched by an electric field E, the emitted radiation intensity contains a left-right-asymmetry term proportional to P\ensuremath{\cdot}k^\ifmmode\times\else\texttimes\fi{}E^, where P is the spin-polarization vector and k^ is the observation direction. The resulting asymmetry is proportional to the relativistic magnetic dipole matrix element 〈1${s}_{1/2}$,1/2\ensuremath{\Vert}${\mathrm{M}}_{1}$,0\ensuremath{\Vert}2s $_{1/2}$,1/2〉. The measured asymmetry (2.935\ifmmode\pm\else\textpm\fi{}0.337)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$ corresponds to the matrix element 〈1${s}_{1/2}$,1/2\ensuremath{\Vert}${\mathrm{M}}_{1}$,0\ensuremath{\Vert}2s $_{1/2}$,1/2〉 =-(0.2725\ifmmode\pm\else\textpm\fi{}0.0313)${\ensuremath{\alpha}}^{2}$e\ensuremath{\Elzxh}/mc , in agreement with the theoretical value -0.2794${\ensuremath{\alpha}}^{2}$e\ensuremath{\Elzxh}/mc. The measurement provides a direct test of the relativistic corrections to the magnetic dipole transition operator.

Interference of electric-dipole and magnetic-dipole interactions in conduction-electron-spin resonance in InSb.

The contributions of the electric-dipole and magnetic-dipole matrix elements to conduction-electron-spin resonance in zinc-blende crystals are investigated theoretically and experimentally. Using time-dependent perturbation analysis, we show that these contributions interfere at the resonance condition, resulting in an anomalous dependence of the resonance intensity on the sign of either the dc magnetic field B${\ensuremath{\rightarrow}}_{0}$ or the wave vector q\ensuremath{\rightarrow} of the photon. In the context of macroscopic dielectric response, it is shown that the above interference can be represented by time-reversal symmetry associated with the wave vector. The effects of electric-dipole--magnetic-dipole (EDMD) interference at spin resonance have been studied experimentally by far-infrared (FIR) magnetotransmission in a series of InSb samples with various orientations and electron concentrations. Experiments were performed in magnetic fields up to 60 kG in both Voigt and Faraday geometries, at FIR wavelengths 96.5, 118.8, 163, and 251.1 \ensuremath{\mu}m, in the temperature range between 2 and 35 K. Measurements of the absorption coefficient as a function of orientation of B${\ensuremath{\rightarrow}}_{0}$ in (100), (110), (111), and (112) planes, and as a function of the sign of B${\ensuremath{\rightarrow}}_{0}$ and q\ensuremath{\rightarrow}, are in excellent agreement with the predictions of the theory. These experiments provide an elegant method for determining the inversion-asymmetry parameter ${\ensuremath{\delta}}_{0}$, yielding a value of 56 a.u. (3.6\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}34}$ erg ${\mathrm{cm}}^{3}$). In addition to the EDMD interference, the FIR magnetotransmission spectra also conclusively demonstrate that the dominant mechanism allowing electric-dipole-excited spin resonance in InSb is inversion asymmetry.