Electron Spin Angular Momentum Research Articles

Molecular-beam magnetic-resonance measurement of the spin-rotational interaction in RbKr

We have investigated the spin-rotation interaction in the van der Waals molecule RbKr using molecular-beam magnetic-resonance techniques. The bound $^{2}\ensuremath{\Sigma}$ molecules were produced in an expansion-cooled molecular beam and the interaction between the electronic spin angular momentum ($\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$) of the rubidium valence electron and the rotational angular momentum ($\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{N}}$) of the molecule was probed by magnetic resonance. A value for the strength of the interaction and a model of its dependence upon the internuclear separation were obtained. Results of numerical calculations of all bound and quasibound vibrational-rotational states of the RbKr ground-state potential are presented. These wave functions permit a detailed comparison of the results obtained by this technique and those obtained by M. Bouchiat et al. using optical-pumping techniques in a high-temperature gas cell. Excellent agreement is obtained when all of the relevant molecular states are considered.

On the Pressure Broadening of Molecular Multiplet Spectra

Gas-phase line shapes and band contours are calculated for microwave spectra involving fine structure levels due to electron spin and/or orbital angular momentum, or vibrational angular momentum. We use the impact approximation, and a semiclassical treatment of the molecular rotation. The line shapes are expressed in terms of a set of cross sections which depend on the transfer of angular momentum by collisions. Different results are found, depending on whether the spin and rotation are coupled together strongly [Case (a)] or weakly [Case (b)]. Good agreement is obtained with microwave measurements on NO [Case (a)] and O2 [Case (b)]. The results may be applied to study the angle-dependent shape of intermolecular forces.

Excitation cross section of some of the states of NE II, Ar II, and KR II by electron collision

The sudden perturbation method has been used to calculate the excitation cross section of some of the states of Ne II, Ar II, and Kr II by high energy electrons. In this method, the ground state of the noble gas atom, after a collision with a fast electron, is considered to lose one of its valence electrons in a time shorter than the relaxation time of the atom. The resulting configuration is then assumed to be a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|(p^{5}) ^{2}P_{j}\rangle</tex> state with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J = 3/2</tex> or 1/2, with the remaining five valence electrons having the same radial coordinates as in the original <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|(p^{6})^{1}S_{0}\rangle</tex> state of the neutral atom. The configuration <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|(p^{5})^{2}P_{j}\rangle</tex> is then expanded in terms of the ionic wave functions having the form of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|(p^{4})\bar{L}\bar{S}, nl', LSJ\rangle</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{L},\bar{S}</tex> designate the total orbital angular momentum and the total spin angular momentum of the core electrons, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l'</tex> is the orbital angular momentum of the excited running electron, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LSJ</tex> designates the final orbital, spin, and total angular momenta of the atom. The coefficients of the expansion are calculated in terms of the coefficients of fractional parentage and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3-j</tex> symbols. The radial wave functions for the neutral atoms and their excited ionic states for the involved radial integrals are obtained by using a computer program giving the self-consistent Hartree-Fock wave functions in the Slater approximation. Since no experimental measurements for the excitation cross sections with fast electrons are given, the results of the calculations are indirectly compared with the available work reported on the pulsed Ar II laser. Population of various excited ionic states is calculated assuming a uniform energy distribution of electrons having 1.8 times the ionization threshold of argon, and the results are compared with experimental data reported in the literature.

Theory of Indirect Exchange Interactions in Rare-Earth Metals

Liu has recently considered the effect of nonzero atomic orbital angular momentum on the exchange interactions between a localized magnetic moment and conduction electrons. By expanding a part of the band wave functions in spherical harmonics, he found that the leading term gave rise to the familiar scalar product interaction between conduction electron spin and total atomic angular momentum J. In the present paper, we derive Liu's result by a simpler calculation, and obtain the first correction term in a similar expansion. Our main motivation in doing the latter is to investigate the validity of the common use of the leading term only, doubt being raised, a priori, by the fact that the Fermi wave vector times the radial extent of the $4f$ function is not \ensuremath{\ll}1. We then investigate, in second-order perturbation theory, the magnetic interaction between atoms $n$ and $m$. The leading term is, of course, an isotropic Heisenberg-type interaction $A_{\mathrm{nm}}^{}{}_{}{}^{(0)}{\mathbf{J}}_{n}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{J}}_{m}$ corresponding to Liu's result. The first correction term gives rise to anisotropic interactions including pseudodipolar forces and an unconventional interaction which is quartic in the J's. An estimate for rare-earth metals, based on free conduction electrons and screened hydrogenic localized functions, suggests that the comparative importance of the correction term is very sensitive to the number of $4f$ electrons and also to the lattice structure. For Tb through Er, it appears that the leading term does dominate, the correction being roughly 10%; for Tm metal the correction term is \ensuremath{\sim}30%, whereas for a single pair of ${\mathrm{Tm}}^{3+}$ ions it is \ensuremath{\cong}80%. The leading term is also considered using the same type of one-electron functions. This gives roughly the right order of magnitude for the Curie temperature and the correct signs for first- and second-neighbor interactions; also the ratio of the latter interactions is in rough agreement with experiment.

Spin Density Matrices for Paramagnetic Molecules

The density of electron-spin angular momentum (erg-sec/cm3) at a point x, y, z in a paramagnetic molecule is ℏ Szρ(x, y, z), where Sz is the z component of the total electron spin angular momentum in units of ℏ. If the exact wave function for the molecule Ψ is built up from one-electron basis functions ψi, then the normalized spin density function ρ(x, y, z) can be expressed in terms of the ψi and the spin density matrix ρijρ(x,y,z)= ∑ ijρijψj*ψi.If the ψi are molecular orbitals, then ρij is the molecular orbital spin density matrix; if the ψi are atomic orbitals, then ρij is an atomic orbital spin density matrix. The equation of conservation of spin is Tr(Sρ)=1 where S is the non-orthogonality matrix, Sij=〈ψi|ψj〉. The elements of the spin density matrices are sensitive to overlap and electron correlation effects. This is illustrated by calculations of the spin density matrices of the ethylene positive ion radical, and of the allyl radical. The spin density matrix for the latter molecule is calculated using (a) the simple molecular orbital approximation, (b) the simple valence bond approximation, and (c) the complete π-electron interaction configurational treatment given by Chalvet and Daudel. It is shown that the general connection between σ-proton hyperfine splittings aN and the π-electron atomic orbital spin density matrix ρNN′, involves a (hyperfine interaction)-(exchange interaction) matrix QN′N″″N such that aN=Tr(QNρ).However, for practical calculations, observed hyperfine splittings may be used to estimate the diagonal elements of the spin density matrix, aN≈QρNN,where Q has the semiempirical value of —63 Mc or —23 gauss