James Keenan Model Research Articles

Rotational motions of CH4 molecules in weak orientational potentials of cubic symmetry

Nearly free rotational motions of CH4 molecules as substitutional impurities in argon, krypton, and xenon have been observed at low temperatures with inelastic neutron scattering. Besides energy transfer, the dependence of the scattered intensities on momentum transfer Q is used for the assignment of the experimentally observed lines to the various transitions of a spherical quantumrotor in an orientational potential of cubic symmetry. The measured intensities are in good agreement with theoretical predictions based on the extended James–Keenan model. Measurements with high energy resolution on solid CH4 in its antiferrorotational phase II were devoted to the determination of the Q dependence of the tunneling lines and the nearly free rotor lines. The results give the first direct experimental evidence for the value 3 for the ratio of the orientationally ordered molecules to those which are orientationally disordered in CH4-II.

Q dependence of rotational excitations of CH4 molecules in argon

Rotational excitations of CH4 molecules, matrix isolated in solid argon, are observed with inelastic neutron scattering (INS). In the argon matrix the orientational crystalline field Vc(Ω) leads to weakly hindered rotation. Experimental data are compared with the predictions of the ‘‘Extended James–Keenan model’’ for solid CH4 II. Deviations from complete orientational disorder are studied via the Q dependence of the scattered intensities. The transition energies for CH4 in Ar differ considerably from the corresponding transition energies in solid CH4 II due to the stronger crystalline field, while the transition matrix elements seem to be rather similar. The parameters β4=7.5 and β6=−25 of the expansion of the crystalline field into octahedral rotor functions are estimated from the experimental data and compared with predictions, based on different empirical potentials between nonbonding atoms.

Rotational excitation in phase II of solid A-CH4

Quantum rotational excitation energies in phase II of solid CH4 consisting of only A-spin species are calculated using the extended James-Keenan model. It is shown that the first excited rotational levels of the D2d and Oh molecules split into six levels at wavevector k=0. This result is compared with classical lattice dynamical calculations.

Theory of phase transitions in solid methanes. XIII. The differential neutron scattering cross section in phase II of solid CH4

Theoretical study of the differential neutron cross section in phase II of solid CH4 is carried out with special emphasis upon the dependence of the incident energy of the neutron and the scattering angle. Spin–rotational wave functions given by T. Yamamoto, Y. Kataoka, and K. Okada [J. Chem. Phys. 66, 2701 (1977)] are employed, which are obtained on the basis of the extended James–Keenan model. The interference effect of spin-dependent scattering by a tetrahedral four-proton system is explicitly taken into account according to Hama and Miyagi’s formalism [Prog. Theor. Phys. 50, 1142 (1973)]. The effect of inter- and intramolecular vibrations is included in terms of a Debye–Waller factor. The proton spin-dependent part of the cross section for an unpolarized neutron beam is calculated in the case of an equilibrium mixture of three nuclear spin species in a polycrystalline sample at around liquid helium temperature. The results are compared with available experimental data and the main features of observed spectra are reproduced by theory. Some predictions are made for future experimental work to deepen our understanding of solid methane.

Theory of phase transitions in solid methanes. XI. Infrared and Raman spectra of the ν3 and ν4 modes in phase II of solid CH4

The infrared and Raman spectra of the ν3 and ν4 modes of intramolecular vibration of methane in phase II have been theoretically studied on the basis of the extended James–Keenan model. Low-lying vibration–rotation states coupled through the first order Coriolis coupling are obtained. Two kinds of energy level schemes have been worked out, one for the molecules with an orientational order (D2d molecules) and the other for those without it (Oh molecules). The rotational structures (positions and intensities) of the ir spectra have been calculated, and a good agreement with experiment has been obtained, if the fundamental vibration frequencies of the ν3 and ν4 modes are put equal to 3012.1 and 1295.9 cm−1 for Oh molecules and 3007.3 and 1300.8 cm−1 for D2d molecules, respectively. Raman spectra are also treated and their rotational structures are predicted.

Theory of phase transitions in solid methanes. X. Centering around Phase II in solid CH4

Thermal, spectroscopic, and other properties of methane solids, especially those concerning Phase II of solid CH4 in nuclear spin species equilibration, are theoretically studied from a unified point of view, i.e., the extended James–Keenan model. It assumes a rigid lattice and treats the molecular motions with respect to the rotational degress of freedom in the crystal potential given by Yasuda [Prog. Theor. Phys. 45, 1361 (1971)]. Two adjustable parameters are introduced in order to adapt the assumed crystal potential to the actual situation in the solid state of CH4. Most of the calculations are carried out in the framework of the molecular field method in quantum statistical mechanics. The eight-sublattice antiferrorotational structure is assigned to Phase II. Thus we have two kinds of site Hamiltonians in this phase, the symmetry groups of which are the direct product groups ?hOh and ?dD2d. Basis functions are doubly symmetry adapted under each of these symmetry groups. Rotational functions are included up to J=8 (sometimes up to J=10). The accuracy of the calculations is tested and the errors in level spacings are estimated at a few percent. The level scheme obtained for Oh-site features hindered rotations, is independent of temperature, and applies also to all molecules in Phase I. The level scheme of D2d site bears the librational character in its lower energy part and has the lowest levels split through quantum tunneling (the tunneling levels). These level schemes are compared with the results of neutron inelastic scattering experiments and satisfactory agreements are obtained. The two-term crystalline field employed is justified through comparison with the result of the neutron diffraction experiment on Phase I of CD4. The transition between Phases I and II turns out to be of first order, and the reason for this is given. The nature of the transition is new, being neither the rotational melting proposed by Pauling nor the orientational order–disorder transition by Frenkel. Thermodynamic quantities are worked out, including the free energy, entropy, internal energy, specific heat, and the mean square of the proton spin angular momentum. Anomalous behaviors of the specific heat at low temperatures are studied in detail and compared with observation. The predicted structure of the tunneling levels is again and conclusively confirmed by experiment. The negative thermal expansion observed below about 10 K is nicely reproduced with an additional assumption on the response of the crystal potential upon varying the lattice spacing. The transition between Phases II and III observed at elevated pressure is qualitatively discussed with special reference to the role played by Oh molecules in Phase II, and a quantum nature of the transition below about 10 K is pointed out. The main predictions made in this report are as follows: (1) The tunneling levels have such temperature dependences below about 4 K that their level spacings at 0 K are about 10% larger than those at 4 K. Their effects on the Schottky anomaly in the specific heat and on the nuclear susceptibility at around 1 K are described in detail. (2) Apparently unusual quantum effects are predicted on the transition temperatures between Phases I and II. That of CD4 is the highest and those of CH4 and CT4 appear at about the same temperature. (3) Solid solutions of CH4 and Kr or Xe have double phase transitions in a certain CH4-rich region, the lowest temperature phase having no orientational order. If the conversion is not allowed, the lower transition does not occur. (4) Solid solutions of CH4 and CD4 have triple transitions in a certain CH4-rich region, the lowest temperature phase having the same structure as Phase II. (5) The tunneling levels of T species split into two levels, the upper one has the degeneracy six and the lower one the degeneracy three, and the separation is 0.01 K.

Short term proton spin conversion in solid CH4

Short term measurements were made of the static proton magnetic susceptibility of solid CH4 containing O2 impurities in the temperature range between 1.6 and 80 K. The samples were slowly crystallized and annealed above 80 K. A low level NMR Q-meter detector was used to measure the temperature dependence of the polarization of the hydrogen nuclei in methane relative to the polarization of fluorine nuclei in a thick teflon coil form surrounding the sample after thermal equilibration times of up to 2 h. The results can be consistently interpreted according to the James–Keenan model for Phase II of solid methane by assuming that, during a 2 h period, proton spin symmetry conversion essentially comes to equilibrium on the two disordered sublattices whose molecules are in states of nearly free rotation, but that conversion does not significantly occur within this time for molecules on the six remaining ordered sublattices.

A quantum statistical mechanical study of an extended James—Keenan model as applied to solid methane

A quantum statistical mechanical theory of an extended James—Keenan model is applied to solid CH 4. The calculations are carried out in the framework of the molecular field approximation in the truncated space spanned by rotational functions with J ⩽ 8. The transition between phases I and II is predicted to be of first order. A rotational energy level scheme is worked out that allows us to systematically understand various kinds of experimental facts including NMR, zero point entropy, far infrared absorption and neutron inelastic scattering.

A reexamination of the theory of phase transitions in crystalline heavy methane

Abstract The “analytic approach to the theory of phase transitions” of Weeks, Rice, and Kozak is applied to the problem of orientational transitions in solid heavy methane. Starting from the James-Keenan model of solid CD4, a generalized Hammerstein integral equation is derived and is carefully studied. On specializing to phases characterized by uniform orientational order, the Hammerstein equation is simplified; the possible bifurcation points of the resulting equation are found with the aid of Dolph's theorems and a linearized version of the equation. The more general case of nonuniformly ordered crystal structures is then treated, and the nonuniform order-disorder transition found by James and Keenan is shown to occur at a bifurcation point of the general Hammerstein equation. Results of the bifurcation analyses are compared with those of James and Keenan, and the implications of such comparisons for the “analytic theory” are discussed. In an appendix, the bifurcation points of the uniform order Hammerstein equation are treated from a rigorous mathematical standpoint, providing a formal justification for the more heuristic analyses of the text.

Quantum-Mechanical Theory of Orientational Ordering in Solid Methanes

The quantum-mechanical theory of the low temperature phase transitions of the solid methanes is discussed. It is shown how the standard techniques for treating the molecular field approximation can be applied to these problems. It is shown how internal and external symmetries should be combined so as to utilize the full symmetry of the problem. This analysis is quite general and applicable to all problems of orientational ordering. Results of detailed calculations for the James–Keenan model of CH4 and CD4 are presented and compared with experimental results. The calculations assume octupole interactions, include no crystal fields, and include all states up to J = 6. It is suggested that the inclusion of crystal field and possibly states of higher J is essential for describing the low temperature behavior.

Theory of Phase Transition in Solid Methanes. VI

The James-Keenan model which is extended by inclusion of a crystalline field of octa­ hedral symmetry is studied for the purpose of elucidating the mechanism of phase transitions and the nature of molecular rotation in the solid methane. All the calculations are made in the subspace J<4, J being the rotational quantum number. Four types of solid phase are examined, three of which are assumed to have the same sublattice structures and the same symmetry as in the disordered phase and the low and high temperature ordered phases of James and Keenan's classical theory, and the remainder has a layer structure. First, nuclear spin species A and T are treated separately. Then normal mixture of these nuclear spin species is examined with a further assumption that these nuclear spin species are randomly distributed in the lattice. A simplest possible form of octahedral field is assumed. The strength of the field is taken from the analysis made by Cabana, Savitsky and Hornig of the infrared absorption spectra by methane molecules dispersed in the matrix of Kr and Xe crystals. It is shown that thermodynamic properties of the extended James-Keenan model are similar to those of the original one and normal mixture exhibits the observed double transitions in solid methane when the magnitude of the effective octopole moment is chosen such that the observed upper transition temperature is reproduced. Nuclear spin species A is shown to be more stable than species T, which is in conformity with Wong, Noble, Bloom and Alexander's and Runolfsson, Mango and Borghini's experiments on the mean square value of the proton angular momentum at low temperatures. The effects of variation of. both the magnitude of the effective octopole moment and the strength of the crystalline field are also discussed.

Theory of Phase Transitions in Solid Methanes. II. Nuclear Spin Species A and T of Light Methane

The statistical mechanical properties of the James–Keenan model are studied with the purpose of elucidating the mechanism of phase transitions in the solid light methane. All the calculations are made on the basis of quantum statistical mechanics in the subspace with J ≦ 4, J being the rotational quantum number. The molecular field approximation method is employed. The low and high temperature-ordered phases are assumed to have the same sublattice structure as that which James and Keenan predicted in their classical theory. Under the provisional assumption of no interspecies conversion, nuclear spin species A and T are treated separately. The results of the numerical calculations indicate that the James–Keenan model is quite hopeful as a working model to understand the nature of the phase transitions in the whole family of the isotopic solid methanes.

Pair-Correlation Effects in a System of Coupled Rotators

Orientational transitions in systems of coupled molecular rotators have previously been treated by methods that take account of pair-correlation effects, but only in the case of molecules with axial symmetry coupled by forces that do not depend on the direction of the line of centers between molecules. The theory is here generalized to the case of direction-dependent couplings. The resulting integral equations are treated by expanding in powers of Tb/T and 1/z, where Tb is the branching temperature of the ordered solution under consideration, and z is the number of next neighbors to which a molecule is coupled. The zero-order solution of the problem is related to the solution in the internal-field approximation. The first-order correction for pair-correlation effects is determined by an integral equation. The theory is applied to the James—Keenan model of solid methane, for which the integral equation is easily solved. Inclusion of the first-order pair-correlation correction increases the estimated octopole moment of methane by some 10% over that derived by the internal field method, to 0.55×10—24 electron·cm3. The effect of pair correlations on the orientational distribution of the molecules is discussed on the basis of this theory. The calculations on solid methane are of illustrative value only, since the small moment of inertia of even CD4 makes quantum effects important, whereas the present theory is based on classical statistical mechanics.