Order Spin-orbit Interactions Research Articles

N3LO spin-orbit interaction via the EFT of spinning gravitating objects

We present the derivation of the third subleading order (N3LO) spin-orbit interaction at the state of the art of post-Newtonian (PN) gravity via the EFT of spinning objects. The present sector contains the largest and most elaborate collection of Feynman graphs ever tackled to date in sectors with spin, and in all PN sectors up to third subleading order. Our computations are carried out via advanced multi-loop methods. Their most demanding aspect is the imperative transition to a generic dimension across the whole derivation, due to the emergence of dimensional-regularization poles across all loop orders as of the N3LO sectors. At this high order of sectors with spin, it is also critical to extend the formal procedure for the reduction of higher-order time derivatives of spin variables beyond linear order for the first time. This gives rise to a new unique contribution at the present sector. The full interaction potential in Lagrangian form and the general Hamiltonian are provided here for the first time. The consequent gravitational-wave (GW) gauge-invariant observables are also derived, including relations among the binding energy, angular momentum, and emitted frequency. Complete agreement is found between our results, and the binding energy of GW sources, and also with the extrapolated scattering angle in the scattering problem, derived via traditional GR. In contrast with the latter derivation, our framework is free-standing and generic, and has provided theory and results, which have been critical to establish the state of the art, and to push the precision frontier for the measurement of GWs.

Analytic Keplerian-type parametrization for general spinning compact binaries with leading order spin-orbit interactions

We derive a fully analytic Keplerian-type parametrization solution to conservative motion of spinning binary in ADM gauge. This solution is able to describe three dimensional motion of binaries of arbitrary eccentricity, mass ratio and initial configuration of spin angular momentum up to the leading order of post-Newtonian(PN) approximation and a linear order in spin. Based on our results waveforms can be quickly computed with high accuracy.

The remains of a spinning, hyperbolic encounter

We review a recently proposed approach to construct gravitational wave (GW) polarization states of unbound spinning compact binaries. Through this rather simple method, we are able to include corrections due to the dominant order spin-orbit interactions, in the quadrupolar approximation and in a semi-analytic way. We invoke the 1.5 post-Newtonian (PN) accurate quasi-Keplerian parametrization for the radial part of the dynamics and impose its temporal evolution in the PN accurate polarization states equations. Further, we compute 1PN accurate amplitude corrections for the polarization states of non-spinning compact binaries on hyperbolic orbits. As an interesting application, we perform comparisons with previously available results for both the GW signals in the case of non-spinning binaries and the theoretical prediction for the amplitude of the memory effect on the metric after the hyperbolic passage.

Spin-orbit TDDFT electronic structure of diplatinum(II,II) complexes.

[Pt2(μ-P2O5H2)4](4-) (Pt(pop)) and its perfluoroborated derivative [Pt2(μ-P2O5(BF2)2)4](4-) (Pt(pop-BF2)) are d(8)-d(8) complexes whose electronic excited states can drive reductions and oxidations of relatively inert substrates. We performed spin-orbit (SO) TDDFT calculations on these complexes that account for their absorption spectra across the entire UV-vis spectral region. The complexes exhibit both fluorescence and phosphorescence attributable, respectively, to singlet and triplet excited states of dσ*pσ origin. These features are energetically isolated from each other (∼7000 cm(-1) for (Pt(pop-BF2)) as well as from higher-lying states (5800 cm(-1)). The lowest (3)dσ*pσ state is split into three SO states by interactions with higher-lying singlet states with dπpσ and, to a lesser extent, pπpσ contributions. The spectroscopically allowed dσ*pσ SO state has ∼96% singlet character with small admixtures of higher triplets of partial dπpσ and pπpσ characters that also mix with (3)dσ*pσ, resulting in a second-order (1)dσ*pσ-(3)dσ*pσ SO interaction that facilitates intersystem crossing (ISC). All SO interactions involving the dσ*pσ states are weak because of large energy gaps to higher interacting states. The spectroscopically allowed dσ*pσ SO state is followed by a dense manifold of ligand-to-metal-metal charge transfer states, some with pπpσ (at lower energies) or dπpσ contributions (at higher energies). Spectroscopically active higher states are strongly spin-mixed. The electronic structure, state ordering, and relative energies are minimally perturbed when the calculation is performed at the optimized geometries of the (1)dσ*pσ and (3)dσ*pσ excited states (rather than the ground state). Results obtained for Pt(pop) are very similar, showing slightly smaller energy gaps and, possibly, an additional (1)dσ*pσ - (3)dσ*pσ second order SO interaction involving higher (1)dπpσ* states that could account in part for the much faster ISC. It also appears that (1)dσ*pσ → (3)dσ*pσ ISC requires a structural distortion that has a lower barrier for Pt(pop) than for the more rigid Pt(pop-BF2).

Gravitational waves from spinning compact binaries in hyperbolic orbits

Compact binaries in hyperbolic orbits are plausible gravitational wave (GW) sources for the upcoming and planned GW observatories. We develop an efficient prescription to compute post-Newtonian (PN) accurate ready-to-use GW polarization states for spinning compact binaries, influenced by the dominant order spin-orbit interactions, in hyperbolic orbits. This is achieved by invoking the 1.5PN accurate quasi-Keplerian parameterization for the radial sector of the orbital dynamics. We probe the influences of spins and gravitational radiation reaction on $h_+$ and $h_{\times}$ during the hyperbolic passage. It turns out that both polarization states exhibit the memory effect for GWs from spinning compact binaries in hyperbolic orbits. In contrast, only cross polarization state exhibits the memory effect for GWs from non-spinning compact binaries. Additionally, we compute 1PN accurate amplitude corrected GW polarization states for hyperbolic non-spinning compact binaries in a fully parametric manner and perform initial comparisons with the existing waveforms.

Analyzing Feshbach resonances: ALi6−Cs133case study

We provide a comprehensive comparison of a coupled channels calculation, the asymptotic bound state model (ABM), and the multichannel quantum defect theory (MQDT). Quantitative results for $^6$Li -$^{133}$Cs are presented and compared to previously measured $^6$Li -$^{133}$Cs Feshbach resonances (FRs) [M. Repp et al., Phys. Rev. A 87 010701(R) (2013)]. We demonstrate how the accuracy of the ABM can be stepwise improved by including magnetic dipole-dipole interactions and coupling to a non-dominant virtual state. We present a MQDT calculation, where magnetic dipole-dipole and second order spin-orbit interactions are included. A frame transformation formalism is introduced, which allows the assignment of measured FRs with only three parameters. All three models achieve a total rms error of < 1G on the observed FRs. We critically compare the different models in view of the accuracy for the description of FRs and the required input parameters for the calculations.

Canonical angles in a compact binary star system with spinning components: Approximative solution through next-to-leading-order spin-orbit interaction for circular orbits

This publication will deal with an explicit determination of the time evolution of the spin orientation axes and the evolution of the orbital phase in the case of circular orbits under next-to-leading order spin-orbit interactions. We modify the method of Schneider and Cui proposed in ["Theoreme \"uber Bewegungsintegrale und ihre Anwendungen in Bahntheorien", Verlag der Bayerischen Akademie der Wissenschaften, volume 212, 2005.] to iteratively remove oscillatory terms in the equations of motion for different masses that were not present in the case of equal masses. Our smallness parameter is chosen to be the difference of the symmetric mass ratio to the value 1/4. Before the first Lie transformation, the set of conserved quantities consists of the total angular momentum, the amplitudes of the orbital angular momentum and of the spins, $L, S_1,$ and $S_2$. In contrary, the magnitude of the total spin $S=|S_1+S_2|$ is not conserved and we wish to shift its non-conservation to higher orders of the smallness parameter. We perform the iterations explicitly to first order, while performing higher orders would mean no structural difference or harder mathematical difficulties. To apply this method, we develop a canonical system of spin variables reduced by the conservation law of total angular momentum, which is imposed on the phase space as a constraint. The result is an asymptotic series in $\epsilon$ that may be truncated appropriately considering the physical properties of the regarded system.

Hyperfine, rotational, and Zeeman structure of the lowest vibrational levels of theRb872(1)3Σg+state

We present the results of an experimental and theoretical study of the electronically excited $\tripletex$ state of $^{87}$Rb$_2$ molecules. The vibrational energies are measured for deeply bound states from the bottom up to $v'=15$ using laser spectroscopy of ultracold Rb$_2$ Feshbach molecules. The spectrum of each vibrational state is dominated by a 47\,GHz splitting into a $\cog$ and $\clg$ component caused mainly by a strong second order spin-orbit interaction. Our spectroscopy fully resolves the rotational, hyperfine, and Zeeman structure of the spectrum. We are able to describe to first order this structure using a simplified effective Hamiltonian.

Regular dynamics of canonical post-Newtonian Hamiltonian for spinning compact binaries with next-to-leading order spin-orbit interactions

This paper shows that a conservative canonical post-Newtonian Hamiltonian formulation of spinning compact binaries with a pure orbital part up to third post-Newtonian order and spin-orbit contributions at the next-to-leading post-Newtonian order is explicitly integrable and regular because there are 5 independent exact isolating integrals in the 10-dimensional phase space. With the help of symplectic integrators and the fast Lyapunov indicators of two nearby trajectories, numerical investigations also support the absence of chaos.

Motion and gravitational wave forms of eccentric compact binaries with orbital-angular-momentum-aligned spins under next-to-leading order in spin–orbit and leading order in spin(1)–spin(2) and spin-squared couplings

A quasi-Keplerian parameterization for the solutions of second post-Newtonian (PN) accurate equations of motion for spinning compact binaries is obtained including leading order spin–spin and next-to-leading order spin–orbit interactions. Rotational deformation of the compact objects is incorporated. For arbitrary mass ratios the spin orientations are taken to be parallel or anti-parallel to the orbital angular momentum vector. The emitted gravitational wave forms are given in analytic form up to 2PN point particle, 1.5PN spin–orbit and 1PN spin–spin contributions, whereby the spins are assumed to be of 0PN order.

Symplectic structure of post-Newtonian Hamiltonian for spinning compact binaries

The phase space of a Hamiltonian system is symplectic. However, the post-Newtonian Hamiltonian formulation of spinning compact binaries in existing publications does not have this property, when position, momentum and spin variables $[X, P, S_1, S_2]$ compose its phase space. This may give a convenient application of perturbation theory to the derivation of the post-Newtonian formulation, but also makes classic theories of a symplectic Hamiltonian system be a serious obstacle in application, especially in diagnosing integrability and nonintegrability from a dynamical system theory perspective. To completely understand the dynamical characteristic of the integrability or nonintegrability for the binary system, we construct a set of conjugate spin variables and reexpress the spin Hamiltonian part so as to make the complete Hamiltonian formulation symplectic. As a result, it is directly shown with the least number of independent isolating integrals that a conservative Hamiltonian compact binary system with both one spin and the pure orbital part to any post-Newtonian order is typically integrable and not chaotic. And conservative binary system consisting of two spins restricted to the leading order spin-orbit interaction and the pure orbital part at all post-Newtonian orders is also integrable, independently on the mass ratio. For all other various spinning cases, the onset of chaos is possible.

Stability of two-component alkali clusters formed on helium nanodroplets

The stability of two-component clusters consisting of light (Na or K) and heavy (Rb or Cs) alkali atoms formed on helium nanodroplets is studied by femtosecond laser ionization in combination with mass spectrometry. Characteristic stability patterns reflecting electron shell-closures are observed in dependence of the total number of atoms contained in the mixed clusters. Faster decay of the stability of mixed clusters compared to the pure light ones as a function of size indicates a destabilizing effect of heavy alkali atoms on light alkali clusters, presumably due to second order spin-orbit interaction.

Laboratory Detection of FeCO + ( X 4 Σ - ) by Millimeter/Submillimeter Velocity Modulation Spectroscopy

The millimeter/submillimeter spectrum of the molecular ion FeCO (X 4 S) has been recorded using velocity modulation spectroscopy. The molecular ion was created in an AC discharge of Fe(CO)5 and argon. Twentyseven rotational transitions, each consisting of four fine-structure components, were measured in the range 198– 418 GHz. The data were fit with a case b Hamiltonian, and rotational, spin-rotation, and spin-spin constants were determined. Because of the presence of higher order spin-orbit interactions, probably caused in part by a nearby 4 P excited state, numerous centrifugal distortion terms were needed for the spectral analysis. The value of g s, the third-order spin-rotation constant, was also remarkably large at 72.4 MHz. Rest frequencies for FeCO are now available for interstellar and circumstellar searches. This species may be present in molecular clouds, where CO is abundant and gas-phase iron should be in the form of Fe. Molecular ions such as FeCO could be the hidden carriers of metallic elements in such clouds. Subject headings: astrochemistry — ISM: molecules — line: identification — methods: laboratory — molecular data — stars: AGB and post-AGB

Deterministic nature of conservative post-Newtonian accurate dynamics of compact binaries with leading order spin-orbit interaction

We formally show that the conservative second post-Newtonian (PN) accurate dynamics of spinning compact binaries moving in eccentric orbits, when spin effects are restricted to the leading order spin-orbit interaction cannot be chaotic for the following two distinct cases: (i) the binary consists of compact objects of arbitrary mass, where only one of them is spinning with an arbitrary spin and (ii) the binary consists of equal-mass compact objects, having two arbitrary spins. We rest our arguments on the recent determination of PN accurate Keplerian-type parametric solutions to the above cases, indicating that the PN accurate dynamics is integrable in these two situations. We compare predictions of our case (i) with those from a numerical investigation of an equivalent scenario that observed chaos in the associated dynamics. We also present possible reasons for the discrepancies.

Post-Newtonian accurate parametric solution to the dynamics of spinning compact binaries in eccentric orbits: The leading order spin-orbit interaction

We derive Keplerian-type parametrization for the solution of post-Newtonian (PN) accurate conservative dynamics of spinning compact binaries moving in eccentric orbits. The PN accurate dynamics that we consider consists of the third post-Newtonian accurate conservative orbital dynamics influenced by the leading order spin effects, namely the leading order spin-orbit interactions. The orbital elements of the representation are explicitly given in terms of the conserved orbital energy, angular momentum and a quantity that characterizes the leading order spin-orbit interactions in Arnowitt, Deser, and Misner-type coordinates. Our parametric solution is applicable in the following two distinct cases: (i) the binary consists of equal mass compact objects, having two arbitrary spins, and (ii) the binary consists of compact objects of arbitrary mass, where only one of them is spinning with an arbitrary spin. As an application of our parametrization, we present gravitational wave polarizations, whose amplitudes are restricted to the leading quadrupolar order, suitable to describe gravitational radiation from spinning compact binaries moving in eccentric orbits. The present parametrization will be required to construct `ready to use' reference templates for gravitational waves from spinning compact binaries in inspiralling eccentric orbits. Our parametric solution for the post-Newtonian accurate conservative dynamics of spinning compact binaries clearly indicates, for the cases considered, the absence of chaos in these systems. Finally, we note that our parametrization provides the first step in deriving a fully second post-Newtonian accurate `timing formula', that may be useful for the radio observations of relativistic binary pulsars like J0737-3039.

Diyttrium: evidence for a 5∑ u− ground state from pulsed-field ionizationzero electron kinetic energy photoelectron spectroscopy and density functional calculations

Diyttrium has been studied by one-photon pulsed-field ionization-zero electron kinetic energy (PFI-ZEKE) photoelectron spectroscopy and density functional calculations. A spectrum consisting of 14 vibronic bands has been recorded at a resolution of ca. 0.5 cm −1 in the 39 880–40 840 cm −1 region. This spectrum is consistent with a X 4∑ g −-X 5∑ u − transition, in accord with the results of density functional calculations and earlier ab initio calculations. Both states suffer from extensive second — order spin-orbit interactions which produce a splitting of 210 cm −1 in the X 4∑ g − state and 68 cm −1 between the Ω = 2 and Ω = 0 spinorbit components of the X 5∑ u − state. These splittings serve to locate the perturbing 2∑ g + and 3∑ u states. The ionization potential, measured by the X 4∑ 1,2,g −-X 5∑ 0,u − (0,0) band, is determined to be 40 131(2) cm −1. The vibrational frequencies have been determined to be 197(2) cm −1 for the X 4∑ g − state and 185(2) cm −1 for the X 5∑ u − state. The results of density functional calculations using the Amsterdam density functional (ADF) and Gaussian92 codes are compared with experimental data and earlier ab initio results.

A study of the à 2B2–X̃ 2A1 band system of CaNH2 employing molecular beam optical Stark spectroscopy

The 000 à 2B2–X̃ 2A1 visible band system of a supersonic molecular beam sample of calcium amide, CaNH2, recorded in the presence of a variable static electric field has been studied using high resolution laser induced fluorescence spectroscopy. The determined spectroscopic parameters are (in cm−1), origin=15 464.366 9(8); A″=13.057 4(9); B″=0.300 47(5); C″=0.292 88(5); εbb″=0.001 1(3); εcc″=0.001 1(3); A′=11.448 7(2); B′=0.307 1(1); C′=0.299 2(1); εaa′=8.238(2); εbb′=0.005 2(6); εcc′=0.055 7(6); Δsk=−0.060 8(5). The rotational parameters are interpreted to give an approximate r0 structure. The large spin-rotation parameter for the à 2B2 state is discussed in terms of a second order spin–orbit interaction with other low-lying electronic states. Stark splitting of the PP11(1.5)+PQ12(0.5) branch feature was analyzed to produce a ground state permanent electric dipole moment, μ″, of 1.74(1) D. A comparison of μ″ with other monovalent calcium compounds is given.

Ligand field theory of metal sandwich complexes. Second order effects in the magnetic behaviour of dx configurations

Calculations have been made of second order contributions to the magnetic susceptibilities of transition metal sandwich complexes, using the pseudo-axial (C ∞ v) ligand field model. The temperature independent paramagnetism, due to second order Zeeman interactions between the ground state and the excited levels, has been evaluated for all d x systems, and for non-orbitally degenerate ground states the magnetic consequences of a zero field splitting, due to second order spin-orbit interactions, have been investigated.

The electronic spectra and magnetic circular dichroism of the uranyl ion

The MCD and absorption spectra of (Bu4N)UO2(NO3)3 have been measured over the range 18 000—33 000 cm-1 in a rigid polymer matrix at ∼10 K. A large number of sharp peaks are observed including structure in the high energy region previously referred to as the ‘structureless continuum’. It is shown that group theory places some rather strict limitations on the electronic states and the possible transitions between them, and we conclude that the most probable orbital ordering is e 1g < a 1g < a 2u < e 1u and that the low energy bands arise from transitions to the triplet states 3 E 1g and 3 E 2g . The higher energy band structure we assign as due to transitions to 3 E 1u . We show that second order spin-orbit interaction is necessary to account for the unsymmetric splitting of the triplets. Furthermore it is demonstrated that the symmetry D ∞h will not account for the intensity of absorption and MCD of the observed transitions and that the effective molecular symmetry is D 3.

Effect of the second order spin orbit and spin-spin interactions on the 4E (G) level of Mn 2+ in ZnS

An interpretation of the structure of the 6 A 1 → { 1 A 1, 4 E} zero phonon and phonon assisted transitions is given. Covalency effects show that the 6 A 1 → 4 A 1 and 6 A 1 → 4 E transitions are well separated; consequently none of the experimentally observed lines can be due to the 6 A 1 → 4 A 1 transition. The fine structure for the 4 E level is interpreted in terms of the spin-spin and spin-orbit interactions. The results obtained are consistent with experiments.