Van Kranendonk Theory Research Articles

Vibron hopping and bond anharmonicity in hot dense hydrogen

The Raman-active vibron of dense hydrogen has been shown to exhibit unexpected changes as a function of pressure and temperature to above 100 GPa. To understand these results we have performed supercell-based calculations using Van Kranendonk theory taking into account the renormalization of the hopping parameter by the lattice vibrations. We find that the major temperature dependence at this level of theory comes from the differences in populations of rotational states. The theory provides a fair description of the experimental results up to 70 GPa. We examine in detail a number of assumptions made in the application of the Van Kranendonk model to hydrogen as a function of pressure and temperature. We also present results of hybrid path integral molecular dynamics calculations in the fluid state at a low pressure (7 GPa) near the melting temperature. An amorphous-solid model of the fluid predicts that the Raman vibron frequencies change little upon melting, in agreement with experiment. The Van Kranendonk theory with fixed rotational identities of the molecules tends to predict more peaks in the Raman spectrum than are observed experimentally.

Computations of Vibron Excitations and Raman Spectra of Solid Hydrogens

Numerical calculations for the vibronic states in the mixed ortho-para solid hydrogens are reviewed. They were performed within the supercell method on the basis of the Van Kranendonk theory and the Raman spectra were calculated and compared with experiment. Anderson localization is a feature of the results especially in the Raman scattering region of frequencies and especially when the ortho-para mixture is substantial. A simple approximate Hamiltonian model has been used and is surprisingly useful, leading to a suggestion that at high pressures the high frequency peak of the vibron spectrum of a mixed system is due to anisotropic Raman scattering from the ortho (para) hydrogen (deuterium) molecules, even under the assumption that the orbital angular momenta are disoriented.

Influence of order-disorder on the vibron excitations of H2 and D2 in ortho-para mixed crystals

We present a detailed experimental and theoretical study of the vibrons in ortho-para mixed crystals of solid hydrogen and deuterium at ambient and high pressure. Experimental results were obtained at ambient pressure and T=6–7 K (e.g., for hydrogen samples having ortho fractions of 19–62%) using high-resolution Fabry-Perot techniques, and at high pressure and T=77 K (e.g., hydrogen 50–50% ortho-para samples) using dispersive spectrographic techniques with diamond-anvil cells. The numerical calculations are based on the James and Van Kranendonk theory, and were performed by exactly diagonalizing the Hamiltonian for a large supercell of randomly placed molecular “species” on a crystalline lattice. Overall, excellent agreement between theory and experiment is obtained. The calculations show that disorder leads to Anderson localized vibrons for many of the pressures and concentrations studied experimentally and that a substantial portion of the Raman intensity is derived from these localized vibrons. We also calculate the species characteristics of the individual Raman peaks, the results of which suggest an explanation for the previously noted disagreement between experimental high-pressure results and the predictions of the van Kranendonk theory. Specifically, our analysis indicates that the higher frequency peak is associated with anisotropic scattering arising from partial alignment of J=1 angular momenta with respect to the crystallographic axes. Finally, our calculations show that the observed doublet structure in the lower frequency Raman peak for deuterium at low para (J=1) concentrations is well represented by added (para-molecule) diagonal terms in the van Kranendonk Hamiltonian that are plausibly associated with electric quadrupole-quadrupole interactions.

Density dependence of the intramolecular distance in solid H2: A. Spectroscopic determination.

We present an analysis of the measurements of the roton bands in p-${\mathrm{H}}_{2}$ at T=6 K. The quadrupole-quadrupole interaction gives the dominant anisotropic contribution up to at least 40 GPa and leads to a splitting of the J=0\ensuremath{\rightarrow}2 rotational transition into three equally spaced branches. In the framework of the Van Kranendonk theory, the effect of the anisotropic interaction can be accounted for and the remaining variation of the roton frequency is then related to the change of the bond length of the ${\mathrm{H}}_{2}$ molecule, ${\mathit{R}}_{\mathrm{intra}}$. The present determination of ${\mathit{R}}_{\mathrm{intrais}}$ semiquantitatively valid but points to an interesting variation, with a minimum around 30 GPa. By analogy with measurements done on ${\mathrm{I}}_{2}$ and ${\mathrm{Br}}_{2}$, and with ab initio calculations, this demonstrates the importance of charge-transfer interaction in dense ${\mathrm{H}}_{2}$, even at pressures which are far from the regime of molecular dissociation.

Nuclear spin-lattice relaxation of Co 59 in CoB

We report the result of the Co 59 nuclear spin-lattice relaxation time T 1 measurements in the diamagnetic monoboride CoB. The analysis of the data, in the 4.2–300 K temperature range, allows us to separate three contributions to the relaxation rate: first a Korringa process, ( T 1 K T) −1= 0.21 sec −1 K −1 (in good agreement with the temperature independent isotropic Knight shift) from which we deduced the Co 59 hyperfine constant A=6.2 ×10 −6eV, second an impurity contribution independent of temperature and third a quadrupolar term, T −1 1Q=3560 ( T θ D ) 2E( T θ D ) sec −1 , which is predominant at high temperature and well explained by the Van Kranendonk theory. It seems that it was the first time that such a quadrupolar effect was detected in a metallic compound. A remarkable coherency between Lundquist's three bands model and our experimental results has to be noted.

Collision-induced fundamental band of H 2 in H 2He and H 2Ne mixtures at different temperatures

The collision-induced fundamental infrared absorption band of hydrogen in binary mixtures H 2He and H 2Ne at 77, 195, 273, and 298 K has been studied with absorption path lengths of 27 and 105 cm for gas densities up to 530 amagat for several base densities of hydrogen. In each of these mixtures the enhancement absorption profiles show, in addition to the usual splitting of the Q branch into the main Q P and Q R components, a splitting of the S(1) line into the S P (1) and S R (1) components at all the experimental temperatures and a secondary splitting of the main Q P component into the Q P (3) and Q R (3) components at 273 and 298 K. The profiles of H 2He at 77 K also show a splitting of the S(0) line into S P (0) and S R (0). Integrated absorption coefficients were measured and binary and ternary absorption coefficients were derived. Van Kranendonk's theory of the ‘exponential-4’ model for the induced dipole moment was applied to the experimental binary absorption coefficients. The quadrupolar parts of these coefficients were calculated from the known molecular parameters and were then subtracted from the experimental values to obtain the overlap parts. The overlap parameters λ and ρ, giving respectively the magnitude and range of the overlap moment, were determined for each of the mixtures by obtaining the best fit of the calculated overlap part of the binary absorption coefficient as a function of temperature to the experimental values of the overlap parts. The values of λ, ρ, and μ(σ) (the overlap induced dipole moment at the Lennard-Jones intermolecular diameter σ) are as follows: Mixture λ ρ μ(σ) H 2He 5.6 × 10 −3 0.24 Å 2.92 × 10 −2 ea 0 H 2Ne 9.0 × 10 −3 0.29 Å 4.85 × 10 −2 ea 0

Investigation of the Rotary Lattice Mode in R2PtCl6 Compounds. II. From a Study of the 35Cl Nuclear Quadrupolar Spin–Lattice Relaxation

Measurements of the temperature and pressure variation of the 35Cl nuclear spin–lattice relaxation time in Rb2PtCl6 and Cs2PtCl6 are reported. The spin–lattice relaxation time is measured at atmospheric pressure for temperatures from 60 to 500 K and at four temperatures between 290 and 380 K for pressures to 5000 kg cm−2. Previously published data for K2PtCl6 are also included in the discussion. The Van Kranendonk theory of nuclear quadrupolar relaxation forms the basis of the analysis. The rotary lattice mode frequencies are deduced; they are of approximately the same magnitude and increase in the same sequence as the frequencies deduced from nuclear quadrupole resonance frequency measurements and from infrared and Raman data. An analysis of the pressure dependence of the spin–lattice relaxation time data yields order of magnitude pressure coefficients for the rotary mode frequencies. Finally, a thermodynamic analysis, which takes specific volume effects into account by incorporating both the temperature and pressure dependence of the data, is presented.

Structure of the Pressure-Induced Infrared Spectrum of Hydrogen in the First Overtone Region

The pressure-induced infrared spectrum of hydrogen in the first overtone region (1.1–1.25 μ) has been recorded at various temperatures and pressures. The spectrum observed at 24 K with a path length of 13.6 m and a gas density of 31 amagat is analyzed on the basis of the Van Kranendonk theory of quadrupole-induced absorption. The relative intensities of the 15 single and double transitions composing the band and the integrated intensity of the band are in satisfactory agreement with the theory.

Quadrupolar Nuclear Spin–Lattice Relaxation in R2MX6 Compounds

The Van Kranendonk theory of quadrupolar nuclear spin–lattice relaxation is adapted to the case of the X nuclei in R2MX6 compounds as observed in a pure nuclear quadrupole resonance experiment. The formula derived for the spin–lattice relaxation time is discussed with reference to experimental data for three R2PtCl6 compounds. The internal modes of the PtCl6 complex are shown to be relatively unimportant for spin–lattice relaxation of the chlorine nuclei. The experimental data yield average rotary mode frequencies of the PtCl6 complex in K2PtCl6, Rb2PtCl6, and Cs2PtCl6. The results are in good agreement with the theoretical predictions.

Rotating-Frame Nuclear-Double-Resonance Dynamics: Dipolar Fluctuation Spectrum in CaF2

A series of rotating-frame nuclear-double-resonance experiments is reported in which the NMR of the rare isotope $^{43}\mathrm{Ca}$ (natural abundance 0.13%) in Ca${\mathrm{F}}_{2}$ is detected via the observable resonance signal of the abundant $^{19}\mathrm{F}$ nuclei. The technique consists of (1) cooling the $^{19}\mathrm{F}$ dipolar-interaction energy reservoir by means of adiabatic demagnetization in the rotating frame, (2) coupling the rare $^{43}\mathrm{Ca}$ to this cooled reservoir by means of a rotating magnetic field at or near the $^{43}\mathrm{Ca}$ nuclear Larmor frequency, and (3) examining the final state of this reservoir by measuring the "dipolar" free-induction signal following a \textonequarter{}\ensuremath{\pi} rf pulse at the $^{19}\mathrm{F}$ Larmor frequency. With this basic format, results have been obtained and are reported here for (a) the relative heat capacities of the $^{43}\mathrm{Ca}$ rotating-frame Zeeman Hamiltonian and the $^{19}\mathrm{F}$ dipolar Hamiltonian terms, (b) the cross-relaxation time for the $^{43}\mathrm{Ca}$-$^{19}\mathrm{F}$ energy-transfer process as a function of rotating-frame effective field strength and orientation, (c) transient oscillations of the $^{19}\mathrm{F}$ dipolar spin temperature upon application of an rf pulse at the $^{43}\mathrm{Ca}$ resonance frequency, (d) the transverse (${T}_{2}$) relaxation process of the $^{43}\mathrm{Ca}$ nuclei, and (e) the spin-lattice relaxation time (${T}_{1}$) of the $^{43}\mathrm{Ca}$ nuclei. Theoretical expressions for results (a) and (b) are obtained using the well-known thermodynamic model to describe the state of the $^{19}\mathrm{F}$ and $^{43}\mathrm{Ca}$ spin systems. The $^{19}\mathrm{F}$-$^{43}\mathrm{Ca}$ dipolar coupling is treated as a perturbation to obtain a simple "Fermi Golden rule" expression for the cross-relaxation rate. This model, first applied to nuclear double resonance by Hartmann and Hahn, is combined here with the experimentally determined fluctuation spectrum of the $^{19}\mathrm{F}$-$^{43}\mathrm{Ca}$ coupling operator to yield calculated cross-relaxation rates in excellent agreement with the experimental results. The above-mentioned fluctuation spectrum is found to be a very nearly exponential function of the rotating-frame $^{43}\mathrm{Ca}$ Zeeman splitting over a range of cross-relaxation rates spanning nearly three orders of magnitude for two distinct orientations of the applied field relative to the crystal axes, and to be essentially independent of the orientation of the rotating-frame effective field. This rather surprising spectral form is crucial to the successful calculation of the results we find, and also leads to satisfactory agreement between theory and experiment for results (c) and (d). The transient oscillation phenomena (c) are the rotating-frame analog of the pulsed dc-field experiments carried out by Strombotne and Hahn. A density-matrix perturbation-expansion technique similar to that used by those authors is employed here to explain our results. The form of the $^{43}\mathrm{Ca}$ transverse decay process (d) is calculated by means of the Anderson-Weiss-model theory; the exponential decay time is closely approximated by a calculated result using the exponential fluctuation spectrum noted above, and the initial region of nonexponential behavior predicted by this model theory is clearly evident in our results. $^{43}\mathrm{Ca}$ ${T}_{1}$ values (e) were obtained at 300 and 355\ifmmode^\circ\else\textdegree\fi{}K with a variety of field orientations in the (100) plane of the crystal axes. The ${T}_{1}$ process is found to be isotropic and to have a temperature dependence suggesting that it is quadrupolar in origin. The temperature dependence of ${T}_{1}$ is in reasonable accord with the Van Kranendonk theory.

Nuclear-Quadrupole-Resonance Study of Lattice Dynamics inK2PtCl6

A pulsed magnetic-resonance spectrometer is used to measure pure NQR frequencies and relaxation times of the $^{35}\mathrm{Cl}$ and $^{37}\mathrm{Cl}$ nuclei in powdered samples of ${\mathrm{K}}_{2}$Pt${\mathrm{Cl}}_{6}$ over the temperature range $6lTl490\ifmmode^\circ\else\textdegree\fi{}$K. The ratio of frequencies is independent of temperature and equal to 1.2688\ifmmode\pm\else\textpm\fi{}0.001, in agreement with the ratio of quadrupole coupling constants as found for free Cl atoms. It is shown that a single low-frequency vibrational mode of frequency \ensuremath{\sim}38 ${\mathrm{cm}}^{\ensuremath{-}1}$ dominates the motional averaging of the electric field gradient at a Cl site. This mode is identified as the rotary lattice mode which corresponds to the threefold degenerate torsional oscillations of the ${[\mathrm{P}\mathrm{t}{\mathrm{Cl}}_{6}]}^{=}$ ion. A realistic model of the field gradient is proposed, and the data are discussed in terms of this model. The ${T}_{1}$ data span seven orders of magnitude from 100 \ensuremath{\mu}sec at 490\ifmmode^\circ\else\textdegree\fi{}K to 20 min at 6\ifmmode^\circ\else\textdegree\fi{}K. For $Tl320\ifmmode^\circ\else\textdegree\fi{}$K, $\frac{{T}_{1}(^{37}\mathrm{Cl})}{{T}_{1}(^{35}\mathrm{Cl})}\ensuremath{\simeq}1.6$; for $Tg320\ifmmode^\circ\else\textdegree\fi{}$K, ${T}_{1}(^{35}\mathrm{Cl})={T}_{1}(^{37}\mathrm{Cl})$. For $6lTl90\ifmmode^\circ\else\textdegree\fi{}$K, the temperature dependence of ${T}_{1}$ is consistent with the Van Kranendonk theory, assuming that the relaxation is dominated by a single optical mode of frequency \ensuremath{\sim}33 ${\mathrm{cm}}^{\ensuremath{-}1}$, that is, by the rotary lattice mode. For $90lTl320\ifmmode^\circ\else\textdegree\fi{}$K, the ${T}_{1}$ data increase more slowly with temperature than can be accounted for by the Van Kranendonk theory. This is thought to be because the amplitude of the torsional oscillations has become so large as to invalidate the assumptions of a harmonic theory. For $320lTl490\ifmmode^\circ\else\textdegree\fi{}$K the ${T}_{1}$ data suggest that the energy of the torsional oscillations has become sufficient for the onset of a hindered rotational motion. This would lead to a nonresonant relaxation process and a ${T}_{1}$ determined solely by the residence time of a ${[\mathrm{P}\mathrm{T}{\mathrm{Cl}}_{6}]}^{=}$ ion in one of its equilibrium orientations. An order-of-magnitude calculation gives support to such an interpretation. The ${T}_{2}$ data are dominated by a magnetic dipolar spin-spin interaction. This is verified experimentally by observing the magnetic field dependence of the spin-echo beats.

Nuclear relaxation time in mixed sodium salts

Synopsis Reduction of up to 50% was observed in nuclear magnetic resonance measurements of relaxation time T 1 of 23 Na in mixed, single phase sodium halide crystals. Van Kranendonk's theory of quadrupole nuclear relaxation mechanism, based on the point-charge model, has been slightly simplified at T/θ ≥ 1, and then applied to this problem. The calculations predict a much smaller reduction. Within the accuracy of the measurements, all relaxation transients show only a single relaxation time present which implies strong spin-spin coupling of all nuclei contributing to the signal, irrespective of the fact that some of its Zeeman levels have been shifted by static quadru-polar effects.

Angular Dependence of Nuclear Spin-Lattice Relaxation Time for Several Alkali Halide Crystals

The anisotropy of relaxation time for both 19F and 7Li has been examined in a pure single crystal of lithium fluoride, and is satisfactorily accounted for in terms of Bloembergen's spin-diffusion theory of relaxation. Confirmation is provided by the reduced anisotropy observed in two single crystals of lithium fluoride containing enhanced concentrations (50% and 90%) of 6Li. The lack of anisotropy of the 23Na relaxation time found in sodium chloride accords with the predictions of the theories of quadrupolar relaxation. A similar observation for the 79Br and 81Br relaxation times in potassium bromide is not inconsistent with van Kranendonk's theory and suggests that covalency does not make a strong contribution to the quadrupolar relaxation process.

Spin-Lattice Relaxation Time forI127in KI in the Temperature Range 2°K to 20°K

The spin-lattice relaxation time, ${T}_{1}$, for ${\mathrm{I}}^{127}$ in both single crystal and powdered KI has been measured in the temperature range 2\ifmmode^\circ\else\textdegree\fi{}K to 20\ifmmode^\circ\else\textdegree\fi{}K using a Pound-Watkins type nuclear magnetic resonance spectrometer. In the liquid hydrogen temperature range ${T}_{1}$ is approximately the same for both the single crystal and the powder specimens, varying from 20 sec at 20\ifmmode^\circ\else\textdegree\fi{}K to 70 sec at 14\ifmmode^\circ\else\textdegree\fi{}K. Both the temperature dependence and the magnitude of ${T}_{1}$ in this temperature range can be explained on the basis of a theory of nuclear quadrupolar spin-lattice relaxation proposed by Van Kranendonk. In the liquid helium temperature range (2\ifmmode^\circ\else\textdegree\fi{}K to 4\ifmmode^\circ\else\textdegree\fi{}K), ${T}_{1}$ is constant within experimental error. For the single crystal ${T}_{1}=6.9\ifmmode\times\else\texttimes\fi{}{10}^{3}$ sec and for the powder specimen ${T}_{1}=6.0\ifmmode\times\else\texttimes\fi{}{10}^{2}$ sec in this temperature range. These values of ${T}_{1}$ are less by orders of magnitude than is predicted using Van Kranendonk's theory. The ${\mathrm{I}}^{127}$ resonance (spin $I=\frac{5}{2}$ in units of $\ensuremath{\hbar}$) consisted of a narrow (less than 1 gauss half-width) central line with broad (>15 gauss) weak wings. This is attributed, according to Pound, to a shift in the nuclear magnetic energy levels brought about by the interaction of the nuclear quadrupole moment with electric field gradients present in the specimens.