Second-order Spin-orbit Interaction Research Articles

Robust thermal correlations induced by spin–orbit interactions

In this work, we demonstrate that the thermal nonclassical correlations can enhance and become more stable under the magnetic field and second-order spin–orbit interaction effect. Also, we show that the local quantum uncertainty measure can reveal purely quantum correlations beyond other quantifiers of correlations. Our analytical model describes a Heisenberg XYZ two-spin system with an inhomogeneous magnetic field and spin–orbit interaction. A special emphasis is devoted to local quantum uncertainty, concurrence, and Bell-nonlocality measures to investigate the correlations of the bipartite-system at the equilibrium temperature. Our results indicate that temperature, magnetic field, and magnetic field inhomogeneity with DMI and Kaplan–Shekhtman–Entin-Wohlman–Aharony (KSEA) exchanges all play a role in determining the degree of correlation between the quantum spins to some extent. The KSEA interaction greatly enhances the quantum correlation in the system whereas quantum quantifiers are more stable under the influence of the external magnetic field, especially when inhomogeneous and homogeneous magnetic field are near to zero. Furthermore, these findings suggest that the thermal LQU captures a stronger quantum correlation than the Bell-nonlocality and entanglement measures.

The gravitational two-body problem in the vicinity of the light ring: Insights from the black-hole–ring toy model

The physical properties of an axisymmetric black-hole–ring system are studied analytically within the framework of general relativity to second order in the dimensionless mass ratio μ≡m/M. In particular, we analyze the asymptotic behaviors of the binding energy and the total angular momentum of the two-body system in the vicinity of the light ring at R=3M, where the circular orbit becomes null. We find that both quantities diverge quadratically in μ(1−3M/R)−1 at the light ring. The reported divergent behavior of the physical quantities stems from the second-order spin–orbit interaction between the black hole and the orbiting object (the dragging of inertial frames by the orbiting ring). It is shown that this composed black-hole–ring toy model captures some of the essential features of the conservative dynamics of the (astrophysically more interesting) black-hole–particle system. In particular, we show that both systems share the same quadratic divergent behavior of the physical quantities near the light ring. Moreover, we prove that both systems are characterized by the same ratio E(2)(R→3M)J(2)(R→3M)=133, where E(2) and J(2) are the divergent second-order (self-interaction) expansion coefficients of the binding energies and the angular momenta of the systems, respectively.

Analytic toy model for the innermost stable circular orbit shift

A simple black-hole-ring system is proposed as a toy model for the two-body problem in general relativity. This toy-model yields the fractional shift $\Delta\Omega_{\text{isco}}/\Omega_{\text{isco}}={{29}\over{81\sqrt{2}}}\eta$ in the Schwarzschild ISCO (innermost stable circular orbit) frequency, where $\eta\equiv m/M_{\text{ir}}\ll 1$ is the dimensionless ratio between the mass of the particle and the irreducible mass of the black hole. Our model suggests that the second-order spin-orbit interaction between the black hole and the orbiting particle (the dragging of inertial frames) is the main element determining the observed value of the ISCO shift.

Anisotropy of the static dipole polarizability induced by the spin–orbit interaction: the S-state atoms N–Bi, Cr, Mo and Re

A systematic ab initio study is performed on the ground state static dipole polarizabilities of the 2 S +1 S, S >1/2, atoms from N to Bi, Cr, Mo, Mn, Tc and Re. The benchmark scalar-relativistic values of the scalar polarizability components are obtained using the coupled cluster method. The spin–orbit configuration interaction calculations are carried out for the anisotropic (tensor) polarizability components of these atoms (except Mn and Tc) that arise from the second-order spin–orbit interaction. The tensor polarizabilities are calculated for the first time and found to increase from 10 −5 (N) to 3.8 atomic units (Bi) approximately as the fourth power of the nuclear charge. The simple correlations and implication to magnetic trapping of cold atoms are discussed.

Molecules in high spin states III: The millimeter/submillimeter-wave spectrum of the MnCl radical (X 7Σ+)

The pure rotational spectrum of the MnCl radical (X (7)Sigma(+)) has been recorded in the range 141-535 GHz using millimeter-submillimeter direct absorption spectroscopy. This work is the first time the molecule has been studied with rotational resolution in its ground electronic state. MnCl was synthesized by the reaction of manganese vapor, produced in a Broida-type oven, with Cl(2). Transitions of both chlorine isotopomers were measured, as well as lines originating in several vibrationally excited states. The presence of several spin components and manganese hyperfine interactions resulted in quite complex spectra, consisting of multiple blended features. Because 42 rotational transitions were measured for Mn(35)Cl over a wide range of frequencies with high signal-to-noise, a very accurate set of rotational, fine structure, and hyperfine constants could be determined with the aid of spectral simulations. Spectroscopic constants were also determined for Mn(37)Cl and several vibrationally excited states. The values of the spin-rotation and spin-spin parameters were found to be relatively small (gamma=11.2658 MHz and lambda=1113.10 MHz for Mn(35)Cl); in the case of lambda, excited electronic states contributing to the second-order spin-orbit interaction may be canceling each other. The Fermi contact hyperfine term was found to be large in manganese chloride with b(F)(Mn(35)Cl)=397.71 MHz, a result of the manganese 4s character mixing into the 12sigma orbital. This orbital is spsigma hybridized, and contains some Mn 4psigma character, as well. Hence, it also contributes to the dipolar constant c, which is small and positive for this radical (c=32.35 MHz for Mn(35)Cl). The hyperfine parameters in MnCl are similar to those of MnH and MnF, suggesting that the bonding in these three molecules is comparable.

Elastic and Inelastic Collisions of Cold Spin-PolarizedC133sAtoms

We present theoretical calculations of thermally averaged elastic cross sections and spin-relaxation rate coefficients in the temperature range 0.1 to 300 mK for magnetically trapped, doubly polarized cesium atoms. The calculations reproduce the recent experimental results of Arndt et al. (1997) and Soding et al. (1998) for cases where the aSu ground state potential of Cs2 possesses a large scattering length (jA3j . 35 nm) and the magnitude of the second-order spin-orbit interaction is scaled to reproduce room temperature scattering data. [S0031-9007(98)06865-3]

Relation between the angular dependence of magnetic x-ray dichroism and anisotropic ground-state moments

We analyze the geometrical aspects of x-ray absorption spectroscopy as a function of the directions of the light polarization and the magnetization with respect to the crystalline axes of the material. Relations are derived for the angular dependence of the ground-state magnetic moments and the polarized core edge absorption signals in the presence of a second-order perturbation by an arbitrary interaction. Using the methods of the algebra of tensor operators the angular dependence can be written as a sum over different bipolar spherical harmonics. The example of second-order spin-orbit interaction in transition metals is worked out in detail. We show the importance of the magnetocrystalline anisotropy and describe a method to measure directly the anisotropy. A special choice of geometry, such as a transverse magnetic field, in combination with the sum rules for magnetic dichroism makes it possible to separate ground-state moments which cannot be distinguished in a collinear geometry. We also show that the anisotropy of the lattice can give strong variations in the observed spin to orbital magnetic moment ratio, which previously might have been ascribed to limitations in the application of the sum rules.

Estimating Bounds on Collisional Relaxation Rates of Spin-Polarized 87Rb Atoms at Ultracold Temperatures.

We present quantum scattering calculations for the collisional relaxation rate coefficient of spin-polarized 87Rb(f = 2,m = 2) atoms, which determines the loss rate of cold Rb atoms from a magnetic trap. Unlike the lighter alkali atoms, spin-polarized 87Rb atoms can undergo dipolar relaxation due to both the normal spin-spin dipole interaction and a second-order spin-orbit interaction with distant electronic states of the dimer. We present ab initio calculations for the second-order spin-orbit terms for both Rb2 and Cs2. The corrections lead to a reduction in the relaxation rate for 87Rb. Our primary concern is to analyze the sensitivity of the 87Rb trap loss to the uncertainties in the ground state molecular potentials. Since the scattering length for the a3Σ+u state is already known, the major uncertainties are associated with the X1Σ+g potential. After testing the effect of systematically modifying the short-range form of the molecular potentials over a reasonable range, and introducing our best estimate of the second-order spin-orbit interaction, we estimate that in the low temperature limit the rate coefficient for loss of Rb atoms from the f = 2,m = 2 state is between 0.4 × 10−15 cm3/s and 2.4 × 10−15 cm3/s (where this number counts two atoms lost per collision). In a pure condensate the rate coefficient would be reduced by 1/2.

Rotational, fine, and hyperfine analyses of the (0,0) band of the D 3Π–X 3Δ system of vanadium mononitride

The VN molecule has been produced in a molecular beam apparatus using a laser vaporization source and its D 3Π–X 3Δ(0,0) band has been studied by laser-induced fluorescence at low (∼0.1 cm−1) and sub-Doppler resolution (∼0.004 cm−1). Lifetimes of single rotational levels of the D 3Π0 component have been measured and interpreted. Rotational, fine, and hyperfine structures in six of the nine subbands possible for a 3Π ← 3Δ transition have been recorded. Both states exhibit a rapid transition from case (a) → case (b) coupling cases, manifested by reversals in the Landé patterns of the hyperfine structure. The data have been reduced to a set of 35 molecular constants using a modified case (aβ) effective Hamiltonian in which two additional magnetic hyperfine parameters are required for each state. The distortions in the hyperfine structure are due almost entirely to second-order spin–orbit interaction between states arising from the same configuration. Analysis of the derived parameters indicates that the X 3Δ state is well represented by the single electron configuration ...8σ2 3π4 9σ1 1δ1, in which the 9σ molecular orbital (MO) is a V 4s–4p hybrid (88% V 4s) and the 1δ MO is a pure V 3d orbital; the dominant configuration for the D 3Π state is ...8σ2 3π4 1δ1 4π1, in which the 4π MO is an antibonding orbital composed of at most 82% V 3dπ. The isoconfigurational a 1Δ and e 1Π states are calculated to lie 3390 and 2200 cm−1 above their respective high spin companions. The lambda doubling in the D 3Π0 component has been interpreted in terms of spin–orbit interactions with the B 3Σ− and d 1Σ+ states, both states arising from the ...8σ2 3π4 1δ2 configuration; the d 1Σ+ state is known [Simard, Masoni, and Hackett, J. Mol. Spectrosc. 136, 44 (1989)] to lie 102 cm−1 above D 3Π0, while the B 3Σ− state probably lies about 8000 cm−1 below.

M2,3 absorption spectroscopy of 3d transition-metal compounds

The author calculates M2,3 (3p) absorption spectra of the first-row transition-metal ions using a localized description for the 3dn to 3p53dn+1 excitation that includes 3p and 3d spin-orbit interaction and octahedral crystal-field symmetry. Each electronic or spin configuration has a distinctly different spectrum. The spectrum for a 3d9 initial state depends strongly on the second-order spin-orbit interaction. The calculated spectra provide a basis for the use of M2,3 absorption spectroscopy in materials science and biological science.

Point-group selection rules for ΔS=2 multiplicity change: Application to catalysis and organo-transition metal reaction

Electron spin–spin and second-order spin-orbit interaction operators are expanded as products of irreducible representations of symmetry point groups (Oh, Td, D5d, D6d, and C4v). From the transformation of the separated orbit and of the spin part, the selection rules for off-diagonal matrix elements may be deduced by taking direct products of the ‘‘initial’’ and ‘‘final’’ states. The special ΔMl selection rule for the orbital part of spin–spin interaction after expansion is also discussed. Emphasis is given to the ΔS=2 change connected by these operators. Possible examples of ΔS=2 change in d4, d5, and d6 configurations under the above mentioned point groups are given. As illustrations of the selection rules, the matrix elements for ΔS=2 and ΔMs=2 for these configurations are evaluated in the decoupled representation and given in terms of common parameters. The relevance of these multiplicity change to catalysis and reaction of organo-transition metal complexes is briefly alluded to.

Pair spectra studies of Cr3+ions in the (4A2×2T1) excited states of Cs3Cr2Cl9

A detailed optical absorption and Zeeman study of the spectrum of exchange coupled Cr3+ ions in Cs3Cr2Cl9 is presented, with particular reference to the singly-excited state where one ion of the pair is in the 4A2 ground state and the other is in the 2T1 optically excited state. The results are explained in terms of a new theoretical approach (K.R. Barry et al., ibid., vol.14, p.1281, 1981). Whereas the gross features of the spectrum may be understood without consideration of spin-dependent effects, it is shown that a small second-order spin-orbit interaction is needed to explain the fine details of the spectrum. In addition, an analysis of the vibronic structure associated with the pair transitions shows that the dominant spin-phonon coupling is to A1g modes which modulate the separation of the Cr3+ pair.

AB initio calculation of the spin dipole-dipole interaction in ground-state triplets

Ab initio results are presented for the spin dipole-dipole interaction in the ground-state linear triplet radicals NCN, CNN and CCO. The relative contributions from the first-order spin dipole-dipole interaction and the second-order spin-orbit interaction, to the observed spin coupling constant in these systems, are discussed.

The influence of second-order spin-orbit interaction in mixed configurations (d + s) N as compared to mutual magnetic interactions

The influence of second-order spin-orbit effects is investigated and compared to that of mutual magnetic interactions (spin-spin, spin-other orbit, orbit-orbit) in some well-analyzed spectra with mixed configurations (d + s) N (Os I, Nb II, Ru I, Fe I). In all spectra the influence of the magnetic interactions is found to be small in comparison to that of the effective interactions which represent second-order spin-orbit effects. In Os I the introduction of effective parameters gives rise to an important reduction of the mean error; in the other spectra their effect is non-negligible.

Optical-Absorption Spectrum of TetrahedralFe2+in CdIn2S4: Influence of a Weak Jahn-Teller Coupling

The optical absorption of ${\mathrm{Fe}}^{2+}$ ions substituted in the spinel Cd${\mathrm{In}}_{2}$${\mathrm{S}}_{4}$ has been studied between 6 and 300 \ifmmode^\circ\else\textdegree\fi{}K. For ${\mathrm{Fe}}^{2+}$ at tetrahedral sites, crystal field transitions have been observed at 2500 ${\mathrm{cm}}^{\ensuremath{-}1}$ ($^{5}E\ensuremath{\rightarrow}^{5}T_{2}$) and at 13 000 ${\mathrm{cm}}^{\ensuremath{-}1}$ ($^{5}E\ensuremath{\rightarrow}^{3}T_{1}$). At liquid-helium temperature the $^{5}E\ensuremath{\rightarrow}^{5}T_{2}$ transition shows a rich structure, attributed to the combined interaction of spin-orbit coupling and a Jahn-Teller active mode of vibration in the $^{5}T_{2}$ state. The frequencies and intensities of the stronger absorption lines of this $^{5}E\ensuremath{\rightarrow}^{5}T_{2}$ band are found to be in agreement with a numerical calculation of the energy-level scheme of the $^{5}T_{2}$ state, assuming a spin-orbit coupling constant of 70 ${\mathrm{cm}}^{\ensuremath{-}1}$ and the presence of a Jahn-Teller active mode of vibration of the iron-sulphur complex with a frequency of 300 ${\mathrm{cm}}^{\ensuremath{-}1}$ and a Jahn-Teller coupling energy of 210 ${\mathrm{cm}}^{\ensuremath{-}1}$. The results are compared with calculations of the moments of the spectral distribution and with measurements of the phonon frequencies. The $^{5}E$ ground state is split by second-order spin-orbit interaction. Measurements of the spectrum at different temperatures show that the two lowest sublevels of the $^{5}E$ state are separated by 18 ${\mathrm{cm}}^{\ensuremath{-}1}$.

Dynamical Jahn-Teller Effects for aV3+Ion in a MgO Crystal

A theory of the acoustic paramagnetic resonance APR for ${\mathrm{V}}^{3+}$ ions in MgO is developed to explain the results of Brabin-Smith and Rampton which shows the importance of the second-order spin-orbit interaction resulting from Jahn-Teller interactions, i.e., from the excited vibronic levels. Essentially, it is this interaction which is responsible for lifting the degeneracy of the spin-orbit ground states ${E}_{g}+{T}_{2g}$ of the ${\mathrm{V}}^{3+}$ ions in the cubic state. Straightforward APR experiments give the $g$ value within the excited triplet ${T}_{2g}$ and the magnitude of the zero-field separation ($D$) between the ${T}_{2g}$ and ${E}_{g}$ levels is deduced from the study of the dependence of the APR spectra on temperature. The magnitudes of the Jahn-Teller energy ${E}_{\mathrm{J}\mathrm{T}}$ and the frequency ${\ensuremath{\omega}}_{E}$ of the ${E}_{g}$ mode of vibration of the molecular cluster formed around a ${\mathrm{V}}^{3+}$ ion, which give the best fit with the experimentally observed parameters $g$ and $D$, are in good agreement with the estimate of ${E}_{\mathrm{J}\mathrm{T}}$ in an effective point-charge model and with the results on phonon sidebands in MgO crystals doped with different paramagnetic impurities, respectively. The strongly asymmetric nature of the line shape is explained by taking into account the presence of small random strain fields of tetragonal and orthorhombic nature about the cubic site. The experimentally determined hyperfine parameter $A$ is fitted in this theory, assuming the magnitude of the contact term $k$ to be 0.2.

On the theory of the spin-orbit interaction in the magnetoresistivity effects in ferromagnetic metals

Two physically different models, which have been used to explain magnetoresistance measurements in polycrystalline ferromagnetic metals, are extended to the case of single crystal ferromagnetic metals in an attempt to explain the observed anisotropy in the magnetoresistivity. It is shown in one of these models that the s-d scattering model, as developed by Smit, is inconsistent with the Döring equations unless it is assumed that the first-order spin-orbit interaction vanishes and only the second and higher orders remain. The second model which is considered, approaches the magnetoresistance effect from the viewpoint of the anisotropy in the velocity of the d-shell electrons; it is found that the anisotropy in the resistivity arises from the second-order spin-orbit interaction.