Librational Motion in the α Phase of Solid N2 and CO

1971 ◽  
Vol 49 (23) ◽  
pp. 2898-2910 ◽  
Author(s):  
D. A. Goodings ◽  
M. Henkelman
Keyword(s):  
Lattice Spacing ◽  
Translational Motion ◽  
Harmonic Approximation ◽  
Simple Calculation ◽  
Angular Deviation ◽  
Α Phase ◽  
Librational Motion ◽  
Centers Of Mass ◽  
Measured Lattice Spacing ◽  
Solid N2

The librational motion in the α phase of solid N2 and CO has been studied in the approximation that the centers of mass of the molecules are fixed at the lattice sites (i.e. no translational motion). The potential between two molecules in the crystal was taken to be that derived by Kohin modified slightly to ensure that the molecules are in equilibrium at the measured lattice spacing at T = 0. This potential was expanded in the angle deviations of the molecules from their equilibrium orientations, and the librational frequencies at zero wave vector were calculated in the harmonic approximation. The results were found to be as much as 1.9 times higher than the frequencies measured by Raman scattering. This is attributed mainly to the failure of the harmonic approximation rather than to lack of knowledge of the parameters in the potential. This conclusion is supported by a simple calculation of the root mean square angular deviation of a molecular axis from its equilibrium orientation which for α-N2 is estimated to be 18° at T = 0. Finally, it is emphasized that the anisotropic attractive and repulsive dispersive terms and the quadrupole–quadrupole interaction make comparable contributions to the librational frequencies.

Librational motion in the α phase of solid nitrogen. I. Basic theory and results

1977 ◽  
Vol 55 (6) ◽  
pp. 554-572 ◽  
Author(s):  
Paul V. Dunmore
Keyword(s):  
Single Molecule ◽  
Translational Motion ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Intermolecular Potential ◽  
Α Phase ◽  
Solid Nitrogen ◽  
Librational Motion ◽  
The Mean

Librational motion in the α phase of solid nitrogen is studied on the basis of a model that ignores the translational motion of the molecular centres of mass. A general expansion in spherical harmonics is written down for an arbitrary two-body intermolecular potential, and methods are presented for obtaining the expansion coefficients for a particular potential model. The crystal Hamiltonian is written down and the mean-field approximation is briefly discussed. The eigenstates of the mean-field Hamiltonian are shown to correspond formally to the eigenstatesof the two-dimensional isotropic harmonic oscillator, and this correspondence is exploited to define boson creation and annihilation operators for excitations of a single molecule. The full crystal Hamiltonian is expressed in terms of these operators and the bilinear terms are diagonalized by an RPA treatment which is an extension of one given by Raich and Etters. Numerical results for the libron frequencies at the Γ and R points in the Brillouin zone are presented for the potential models proposed by Kohin and by Raich and Mills, the calculations having been performed with the intermolecular potential including terms as far as l = 6. The temperature dependence of the nuclear quadrupole resonance (NQR) frequency calculated with the Raich–Mills potential is shown to be in very good agreement with the measurements of Brookeman, McEnnan, and Scott. Finally the orientational probability density is presented as a function of temperature.

Librational motion in the α phase of solid nitrogen. III. Spin–Lattice relaxation

1977 ◽  
Vol 55 (21) ◽  
pp. 1848-1857 ◽  
Author(s):  
Paul V. Dunmore ◽  
D. A. Goodings
Keyword(s):  
Single Molecule ◽  
Lattice Relaxation ◽  
Spin Lattice Relaxation ◽  
Intermolecular Potential ◽  
Spin Lattice ◽  
Α Phase ◽  
Solid Nitrogen ◽  
Raman Process ◽  
Librational Motion ◽  
Quadrupolar Relaxation

The theory of spin–lattice relaxation of Van Kranendonk and Walker is adapted to the α phase of solid nitrogen. The quadrupolar relaxation of the 14N nucleus by the 'anharmonic Raman process' is calculated for the cubic anharmonic terms obtained from the theory presented in the first paper of this series. Different approximations to the Raich–Mills intermolecular potential are employed. When the expansion of the potential in spherical harmonics is truncated at l = 2, the calculated temperature dependence of T1 is in satisfactory agreement with the experimental measurements of DeReggi, Canepa, and Scott between 20 and 35 K. Below 15 K the calculated results for T1, are too long, probably due to the neglect of mixing between the librons and acoustic phonons. When the Raich–Mills potential includes the spherical harmonic terms with l = 4 and l = 6, the relaxation is found to increase by more than a factor of 40, resulting in T1, becoming much shorter than the experimental results over the high-temperature region. This arises mainly from single-molecule terms of three-fold symmetry in the potential, leading to the suspicion that these terms are unrealistically large.

Sodalite, Na4Si3Al3O12Cl: structure and ionic mobility at high temperatures by neutron diffraction

1996 ◽  
Vol 52 (4) ◽  
pp. 616-627 ◽  
Author(s):  
R. K. McMullan ◽  
S. Ghose ◽  
N. Haga ◽  
V. Schomaker
Keyword(s):  
Neutron Diffraction ◽  
Translational Motion ◽  
Site Occupancy ◽  
Ionic Mobility ◽  
Lattice Parameter ◽  
K Value ◽  
Bond Lengths ◽  
Librational Motion ◽  
Diffusion Paths ◽  
The Difference

Crystal data: Na4Si3Al3O12Cl, cubic, space group P43n, Z = 2, F(000) = 233.06 fm, μ n = 0.06 cm−1, lattice parameter ao  (Å) [T] (K) at eight temperatures: 8.882 (1) [295]; 8.902 (2) [500]; 8.912 (1) [600]; 8.923 (1) [700] 8.951 (2) [800]; 8.971(1) [900]; 8.988 (1) [1000]; 9.037 (1) [1200]. The crystal structure has been determined at six temperatures (295 ≤ T ≤ 1200 K) based on neutron diffraction data with (sin θ/λ) < 0.80 Å−1. Besides conventional parameters, the least-squares refinement model included thermal tensor parameters up to fourth-order for sodium and chlorine (295 ≤ T ≤ 1200 K) and up to third-order for oxygen (T ≥ 700 K), together with the 1:1 coupled site occupancy factors of sodium and chlorine (T = 1200 K). The indices-of-fit, wR(F 2), are in the range 0.015–0.028 with observation-to-parameter ratios from 7.0 to 8.6. Bond lengths and angles in the aluminosilicate framework have average e.s.d.'s less than 0.002 Å and 0.08°. Between 295 and 1200 K, the observed Si—O (Al—O) bond lengths differ by −0.015 Å (−0.012 Å); corrections for librating rigid SiO4 (AlO4) groups change the difference to +0.004 Å (+0.006 Å), compared with the 295 K value of 1.620 Å (1.741 Å). The unique Si—O—Al angle increases from 138.24° (295 K) to 146.87° (1200 K), while the Si and Al valence angles are virtually unchanged. Between 295 and 1200 K the [Na4Cl] clusters expand with increases in the Na—Cl bond lengths of 0.200 Å, with simultaneous increases in Na—O bond lengths of 0.145 Å and decreases in the shortest Na...O contact distances of 0.126 Å. The thermal expansion of sodalite is attributed to the increasing amplitudes of coupled translational motion of the Na+ ions and the librational motion of the [Al/ SiO4] tetrahedra, leading to the untwisting of the aluminosilicate framework. Maps of the probability density functions for Na+ and Cl− indicate ionic diffusion paths along (111) directions. There is a finite probability of finding the Na+ ion within the plane of the next-nearest O atoms, suggesting that Na+ jumps from an occupied to an unoccupied site in the next-nearest cage through the six-membered ring of [Al/SiO4] tetrahedra.

Librational motion in solid ?-N2

1972 ◽  
Vol 7 (5-6) ◽  
pp. 449-458 ◽  
Author(s):  
J. C. Raich ◽  
R. D. Etters
Keyword(s):  
Librational Motion ◽  
Solid N2

scholarly journals Numerical Simulation of Interaction Between Kr+ Ion and Rotating C60 Fullerene Towards for Nanoarchitectonics of Fullerene Materials

Crystals ◽  
2021 ◽  
Vol 11 (10) ◽  
pp. 1204
Author(s):  
Aleksandr V. Lun-Fu ◽  
Alexey M. Bubenchikov ◽  
Mikhail A. Bubenchikov ◽  
Vyacheslav A. Ovchinnikov
Keyword(s):  
Mathematical Model ◽  
Lattice Structure ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Translational Motion ◽  
C60 Fullerene ◽  
Special Role ◽  
Centrally Symmetric ◽  
Centers Of Mass ◽  
Angular Oscillations

Dynamics of charged fullerene in a surface layer of fullerite is studied under the influence of neutral or charged particles of the gas phase surrounding the fullerite material. The translational displacements of the nodes of the crystal lattice structure are determined by the equations of motion of the centers of mass of fullerenes. Central fullerene, which is described as a discrete set of sixty carbon atoms, plays a special role in the presented mathematical model. Angular oscillations and rotations of the central fullerene are described by the dynamic Euler equations. All other fullerenes have a centrally symmetric field of the potential of interaction with the surrounding atoms and molecules. In this regard, we use the hybrid discrete–continuous mathematical model with four potentials that describe the interactions between the surrounding fullerenes, smoothed fullerene and an atom, a pair of atoms, and electric charges. The results of a numerical study of influence of the Coulomb interaction on the rotational and translational motion of the C60 fullerene are presented.

Librational motion in the α phase of solid nitrogen. II. Neutron scattering

1977 ◽  
Vol 55 (6) ◽  
pp. 573-577 ◽  
Author(s):  
D. A. Goodings
Keyword(s):  
Neutron Scattering ◽  
Excellent Agreement ◽  
Structure Factor ◽  
Inelastic Neutron Scattering ◽  
Inelastic Neutron ◽  
Α Phase ◽  
Solid Nitrogen ◽  
Scattering Measurements ◽  
Librational Motion ◽  
The One

An expression is derived for the one-libron coherent inelastic structure factor for neutron scattering by libron modes in the cubic α phase of solid nitrogen. The results of calculations show that for certain modes at the Γ and R points in the first Brillouin zone and for the scattering vector Q along certain directions in reciprocal space the structure factor vanishes. The results are in excellent agreement with the recent inelastic neutron scattering measurements of Kjems and Dolling.

Lattice strain effects in the measurement of the Si lattice parameter by Laue-case double-crystal diffractometry

2004 ◽  
Vol 37 (5) ◽  
pp. 773-777 ◽  
Author(s):  
Giovanni Mana ◽  
Carlo Palmisano ◽  
Gianfranco Zosi
Keyword(s):  
Strain Gradient ◽  
Lattice Spacing ◽  
Lattice Strain ◽  
Numerical Solutions ◽  
Theoretical Framework ◽  
Lattice Parameter ◽  
X Rays ◽  
Double Crystal ◽  
Analytical And Numerical Solutions ◽  
Measured Lattice Spacing

The measurement of the (220) Bragg-plane spacing of Si by Laue-case double-crystal diffractometry with an uncertainty lower than 10−8requires a detailed theoretical framework that includes the study of lattice strain. In the present paper, the propagation of X-rays through a deformed crystal is re-examined and the influence of a constant strain gradient on the centre of the reflection domain is studied by means of Takagi's equations. Their analytical and numerical solutions indicate that the measured lattice spacing refers to the crystal entrance surface.

Microtomy of spherical Al2O3 powders

1983 ◽  
Vol 41 ◽  
pp. 74-75
Author(s):  
E.J. Jenkins ◽  
D.S. Tucker ◽  
J.J. Hren
Keyword(s):  
Thin Film ◽  
Size Range ◽  
Particle Aggregation ◽  
Ion Milling ◽  
Thin Sections ◽  
Α Phase ◽  
Convenient Means ◽  
Van Der Waals Attraction ◽  
Negative Feature ◽  
Film Substrate

The size range of mineral and ceramic particles of one to a few microns is awkward to prepare for examination by TEM. Electrons can be transmitted through smaller particles directly and larger particles can be thinned by crushing and dispersion onto a substrate or by embedding in a film followed by ion milling. Attempts at dispersion onto a thin film substrate often result in particle aggregation by van der Waals attraction. In the present work we studied 1-10 μm diameter Al2O3 spheres which were transformed from the amprphous state to the stable α phase.After the appropriate heat treatment, the spherical powders were embedded in as high a density as practicable in a hard EPON, and then microtomed into thin sections. There are several advantages to this method. Obviously, this is a rapid and convenient means to study the microstructure of serial slices. EDS, ELS, and diffraction studies are also considerably more informative. Furthermore, confidence in sampling reliability is considerably enhanced. The major negative feature is some distortion of the microstructure inherent to the microtoming operation; however, this appears to have been surprisingly small. The details of the method and some typical results follow.

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