Hexadecapole Interaction Research Articles

Irreducible spin precession theory applied to some topics in atomic and nuclear radio-frequency spectroscopy

Several problems from radio-frequency spectroscopy of atoms and nuclei are treated with irreducible spin precession theory. In the first part, effective field techniques are used to derive analytically single and multiple quantum double resonance lineshapes for atoms with a hyperfine structure in a high magnetic field. In the second part (as an extension to previous work), nuclear resonance signals are calculated for oriented nuclei subject to an electric hexadecapole interaction. Lineshapes of acoustically driven hexadecapole transitions are derived in closed form and compared to experiment. Further, multiple quantum NMR transitions within a hexadecapole shifted nuclear Zeeman structure are calculated, and some distinct features of hexadecapole effects on NMR lineshapes are pointed out. This last case is of current interest due to recent progress in NMR-line narrowing techniques. — In the Appendix, we give lineshape equations for single and double quantum NMR transitions on oriented (I=1)-nuclei subject to an electric quadrupole interaction; these equations are also being used in the atomic rf-spectroscopy calculations. The equations are exact to all orders of the interaction with the external fields.

Nuclear hexadecapole shielding factor for the holmium atom

The nuclear hexadecapole shielding factor ${R}_{H}$ for the $4f$ electrons of the holmium atom ($Z=67$) has been obtained. This calculation was carried out in connection with a recent atomic-beam experiment of Dankwort, Ferch, and Penselin, which indicates the presence of a nuclear hexadecapole interaction with the $4f$ electron, leading to a value of the nuclear hexadecapole moment $H({\mathrm{Ho}}^{165})=0.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}48}$ ${\mathrm{cm}}^{4}$. This result has to be corrected for shielding or antishielding effects of the holmium atomic core electrons on the $4f$ electron, which experiences the interaction with the nucleus. The present paper gives the procedure and the results of the calculation of ${R}_{H}$, which was found to be + 0.24 (shielding). Thus the value of the moment $H({\mathrm{Ho}}^{165})$ obtained by Dankwort et al. should be multiplied by the factor $\frac{1}{(1\ensuremath{-}{R}_{H})}=\frac{1}{0.76}$, giving a corrected value $H{({\mathrm{Ho}}^{165})}_{\mathrm{corr}}=1.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}48}$ ${\mathrm{cm}}^{4}$.

Hexadecapole interaction in the atomic ground state of165Ho

Using the atomic beam magnetic resonance method several hyperfine transitions in the ground state of165Ho were measured at magnetic fields near zero Oe. Including a hexadecapole interaction constantD a perfect fit of the seven hyperfine separations was possible giving the following results:A′=800.583645 (6)MHz,C′=−1.504 (37)kHzB′=−1668.00527 (33)MHz,D′=−0.137 (14)kHz. These interaction constants have been corrected for second order hyperfine interaction within the4I ground multiplet. The corrected constants are the following:A=800.583173 (36)MHz,C=−0.249 (140)kHzB=−1668.078 70 (330) MHz,D=−0.148 (16) kHz. Using a value for 〈r4f/−5〉Ho of Fraga a nuclear hexadecapole moment can be calculated:Π(165Ho)=0.89·10−48cm4. Because of severe uncertainties still present in the theory for calculating the electronic matrix elements this value can be only regarded as highly speculative.

Search for the Nuclear Electric Hexadecapole Moment ofIn115by Molecular-Beam Electric-Resonance Spectroscopy

This experiment with a molecular-beam electric-resonance spectrometer determined the hyperfine structure of indium fluoride, $^{115}\mathrm{In}$$^{19}\mathrm{F}$, to an accuracy of 1 part in ${10}^{6}$. The experiment was an attempt to observe the interaction of a nuclear electric hexadecapole moment of $^{115}\mathrm{In}$ that a simple order of magnitude calculation and a previous experiment showed might barely be observable. Although a hexadecapole interaction energy larger than 500 Hz could have been detected, no evidence of the interaction was found. To understand this result, the electric hexadecapole moment was calculated on the basis of a model in which the nuclear electric charge density was assumed to be uniform within a surface defined by $\mathcal{r}(\ensuremath{\theta}, \ensuremath{\varphi})={R}_{0}[1+{\ensuremath{\beta}}_{2}{P}_{2}(cos\ensuremath{\theta})+{\ensuremath{\beta}}_{4}{P}_{4}(cos\ensuremath{\theta})]$, where ${\ensuremath{\beta}}_{2}$ and ${\ensuremath{\beta}}_{4}$ are taken to have the same values as the corresponding parameters for the nuclear potential. The value of the hexadecapole interaction energy in $^{115}\mathrm{In}$$^{19}\mathrm{F}$ predicted by the model is 10 Hz, well below the limit of detectability with the spectrometer employed in this experiment.

Hyperfine Structure of Indium Fluoride

The radiofrequency spectrum of the indium fluoride molecule, 115In19F, has been measured with a high resolution molecular beam electric resonance spectrometer. We determined the hyperfine structure in the J=1 and the J=2 rotational states of several vibrational levels under conditions of very weak external electric and magnetic fields. The ∼700 MHz electric quadrupole interaction constant of the indium nucleus changes by 0.010(1) MHz between adjacent rotational states. We looked for, but did not find, an electric hexadecapole interaction of the indium nucleus; the upper limit for the (hexadecapole) interaction constant is 2 kHz.

Determination of hexadecapole transition moments by Coulomb excitation

For one even- A rotational nucleus ( 152Sm) the influence of the hexadecapole interaction on multiple Coulomb excitation has been calculated for a variety of bombarding conditions and different assumptions with regard to E4 transition moments. An experimental method is outlined by which the determination of E4 transition strengths is feasible. Extensive numerical results are presented which may be helpful in planning Coulomb excitation experiments.

Hyperfine Structure of Thallium Iodide and an Upper Limit for the Electric Hexadecapole Moment of the Iodine Nucleus

The hyperfine structure of thallium iodide has been studied in the J = 3 and J = 4 rotational states in an attempt to observe a nuclear electric hexadecapole effect. This study was carried out on a high-resolution molecular-beam electric resonance spectrometer at very weak electric and magnetic fields. The molecule is well described by the usual five hyperfine interaction constants which are (in kilohertz) Tl127205ITl127203IeqQ− 438 916.3 ± 0.5− 438 917.3 ± 1.0c13.05 ± 0.053.09 ± 0.1c234.65 ± 0.1534.36 ± 0.3c3− 2.48 ± 0.1− 2.59 ± 0.2c4− 6.67 ± 0.05− 6.57 ± 0.1. No evidence for a hexadecapole interaction constant as large as 500 Hz was found; this result is interpreted as setting an upper limit of ∼ 10−47 cm4 for the nuclear electric hexadecapole moment of 127I.

Theory of Phase Transitions in Solid Methanes. I. Electrostatic Multipole–Multipole Interaction

A systematic method of studying the electrostatic interaction between two multipoles is presented. It is based on the quantum theory of angular momentum. A system of two interacting multipoles which have tetrahedral symmetry is worked out in detail. New expressions for the octopole–octopole, octopole–hexadecapole, and hexadecapole–hexadecapole interaction potentials are found which will be applied to the study of the molecular dynamics in solid methanes in the articles to follow.

Quadrupole Antishielding Factors of Ions

Values of the quadrupole antishielding factor ${\ensuremath{\gamma}}_{\ensuremath{\infty}}$ have been obtained for the following ions: ${\mathrm{Y}}^{3+}$, ${\mathrm{In}}^{3+}$, ${\mathrm{Bi}}^{3+}$, and ${\mathrm{Am}}^{2+}$, using the method of direct solution of the inhomogeneous Schr\"odinger equation for the perturbed wave functions. The wave functions ${{v}_{1}}^{\ensuremath{'}} (nd\ensuremath{\rightarrow}s)$ and the resulting terms ${\ensuremath{\gamma}}_{\ensuremath{\infty}}(nd\ensuremath{\rightarrow}s)$ have been calculated for the case of ${\mathrm{Cu}}^{+}3d\ensuremath{\rightarrow}s$, ${\mathrm{Ag}}^{+}4d\ensuremath{\rightarrow}s$, and ${\mathrm{Bi}}^{3+}5d\ensuremath{\rightarrow}s$. We have also obtained the hexadecapole antishielding factor ${\ensuremath{\eta}}_{\ensuremath{\infty}}$ for the ${\mathrm{In}}^{3+}$ ion. Values of the lattice sums for $\frac{{\ensuremath{\partial}}^{4}V}{\ensuremath{\partial}{z}^{4}}$ pertaining to the nuclear hexadecapole interaction ($V=\mathrm{crystal}\mathrm{potential}$) have been determined for the following types of cubic lattice: simple cubic, NaCl structure, body-centered cubic, and face-centered cubic.

Nuclear Hexadecapole Interactions

The nuclear hexadecapole matrix elements for the static and the one-phonon nuclear interactions are developed and are evaluated for $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}3$ and $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}4$ nuclear transitions involving a spin-$\frac{9}{2}$ nucleus in a crystal with $\overline{4}3m$ symmetry. An expression for the saturation factor for a general interaction which gives rise to nuclear-spin transitions involving the change in the $z$ component of the spin by any amount $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}n$ is developed and is used to derive the angular variation of the one-phonon, $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}3$ and $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}4$ nuclear hexadecapole interactions. Finally a way to end the speculation about the observation of the hexadecapole interaction is presented.

CALCULATION OF THE STRUCTURE OF THE S1(0) AND S1(0)+S1(0) LINES IN THE INFRARED SPECTRUM OF SOLID HYDROGEN

At low ortho concentrations, the satellites appearing in the S1(0) infrared absorption feature of solid hydrogen arise from transitions in para molecules possessing one nearest-neighboring ortho molecule. The 15-fold degeneracy of the combination of the v = 1, J = 2 excited state of the para molecule and the v = 0, J = 1 ground state of the ortho molecule is partially lifted by the quadrupole–quadrupole interaction between the two molecules. The calculated structure of the S1(0) line is in good over-all agreement with the observed profile. The agreement is improved by taking into account the quadrupole–hexadecapole interaction. The best fit is obtained for a positive value of the hexadecapole moment of about 1.2 atomic units. The fine structure of the S1(0)+S1(0) line in the overtone region of pure parahydrogen, resulting from the splitting of the 25-fold degeneracy of the upper state by the combined quadrupole–quadrupole and quadrupole–hexadecapole interaction, is also calculated. The splitting and relative intensity of the two components of the resulting doublet structure are in close agreement with the experimental values. All the calculations are performed assuming a rigid close-packed hexagonal lattice structure. The discrepancy of about 20% between the theoretical values of the quadrupole moment of a hydrogen molecule in the v = 0 and v = 1 states and the empirical values deduced from the analysis of the S1(0) and S1(0)+S1(0) lines may be due to the neglect of the splittings arising from the coupling between the rotational motions of the molecules and the lattice vibrations.