Theory of crystal-field interactions in solid hydrogen. I. Single ortho impurities in solid parahydrogen

1979 ◽  
Vol 57 (7) ◽  
pp. 933-943 ◽  
Author(s):  
S. Luryi ◽  
J. Van Kranendonk
Keyword(s):  
Crystal Field ◽  
Pair Potential ◽  
Zero Point ◽  
Debye Model ◽  
Lattice Vibrations ◽  
Sound Waves ◽  
Quantum Crystal ◽  
Energy Effect ◽  
Out Of Plane ◽  
The Difference

The theory of the crystal-field splitting, Ve, of the J = 1 level of an ortho impurity in a parahydrogen matrix, due to the coupling of the rotational motion of the ortho molecule to the lattice vibrations, is developed taking due account of the quantum-crystal nature of the solid. Two contributions to Ve are identified, both arising from the anisotropy in the lattice vibrations. One contribution is due to the difference in the relative zero-point motion of the molecules in in-plane and out-of-plane pairs in the hcp lattice, and is parametrized in terms of the second moments of the pair distribution function. The other contribution is due to the local, quasi-state distortion of the lattice around the impurity induced by the terms in the coupling linear in the phonon variables. This self-energy effect is calculated using a generalized Debye model for the self-consistent harmonic phonons, in which the anisotropy of the velocity and of the polarization of the sound waves is parametrized in terms of the elastic constants of the crystal. The theory is compared with that of Raich and Kanney, which is shown to be based on unrealistic assumptions. The remaining uncertainties in the anisotropic pair potential and the phonon renormalization factors are discussed in connection with the available experimental data.

THEORY OF THE INTERACTION BETWEEN THE LATTICE VIBRATIONS AND THE ROTATIONAL MOTION IN SOLID HYDROGEN

1966 ◽  
Vol 44 (2) ◽  
pp. 313-335 ◽  
Author(s):  
J. Van Kranendonk ◽  
V. F. Sears
Keyword(s):  
Rotational Motion ◽  
Zero Point ◽  
Solid Hydrogen ◽  
The Self ◽  
Intermolecular Potential ◽  
Lattice Vibrations ◽  
Energy Effect ◽  
Self Energy ◽  
Blowing Up ◽  
Point Lattice

The effects of the interaction between the rotational motion of the molecules in solid hydrogen and the lattice vibrations, resulting from the anisotropic van der Waals forces, have been investigated theoretically. For the radial part of the anisotropic intermolecular potential an exp–6 model has been adopted. First, the effect of the lattice vibrations, and of the anistropic blowing up of the crystal by the zero-point lattice vibrations, is discussed. The effective anisotropic interaction resulting from averaging the instantaneous interaction over the lattice vibrations is calculated by assuming a Gaussian distribution for the modulation of the relative intermolecular separations by the lattice vibrations. Secondly, the displacement of the rotational levels due to the self-energy of the molecules in the lattice is calculated both classically and quantum mechanically, and the resulting shifts in the frequencies of the rotational transitions in solid hydrogen are given. Finally, the splitting of the rotational levels due to the anisotropy of the self-energy effect is calculated. The theory is applied to the calculation of the asymmetry of the S0(0) triplet in the rotational Raman spectrum of solid parahydrogen, and of the specific heat anomaly in solid hydrogen at low ortho-concentrations.

On the Use of Quantum Thermal Bath in Unimolecular Fragmentation Simulations

2019 ◽  
Author(s):  
Riccardo Spezia ◽  
Hichem Dammak
Keyword(s):  
Degrees Of Freedom ◽  
Molecular Simulations ◽  
Zero Point ◽  
Thermal Bath ◽  
Unimolecular Dissociation ◽  
Point Energy ◽  
Rotational Degrees Of Freedom ◽  
The Difference ◽  
Nuclear Effects

<div> <div> <div> <p>In the present work we have investigated the possibility of using the Quantum Thermal Bath (QTB) method in molecular simulations of unimolecular dissociation processes. Notably, QTB is aimed in introducing quantum nuclear effects with a com- putational time which is basically the same as in newtonian simulations. At this end we have considered the model fragmentation of CH4 for which an analytical function is present in the literature. Moreover, based on the same model a microcanonical algorithm which monitor zero-point energy of products, and eventually modifies tra- jectories, was recently proposed. We have thus compared classical and quantum rate constant with these different models. QTB seems to correctly reproduce some quantum features, in particular the difference between classical and quantum activation energies, making it a promising method to study unimolecular fragmentation of much complex systems with molecular simulations. The role of QTB thermostat on rotational degrees of freedom is also analyzed and discussed. </p> </div> </div> </div>

Microscopic Interpretation of Arrhenius Parameters

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Activation Energy ◽  
Transition State ◽  
Transition State Theory ◽  
Average Energy ◽  
Zero Point ◽  
State Theory ◽  
Exponential Factor ◽  
Quantum Tunnelling ◽  
Microscopic Interpretation ◽  
The Difference

This chapter reviews the microscopic interpretation of the pre-exponential factor and the activation energy in rate constant expressions of the Arrhenius form. The pre-exponential factor of apparent unimolecular reactions is, roughly, expected to be of the order of a vibrational frequency, whereas the pre-exponential factor of bimolecular reactions, roughly, is related to the number of collisions per unit time and per unit volume. The activation energy of an elementary reaction can be interpreted as the average energy of the molecules that react minus the average energy of the reactants. Specializing to conventional transition-state theory, the activation energy is related to the classical barrier height of the potential energy surface plus the difference in zero-point energies and average internal energies between the activated complex and the reactants. When quantum tunnelling is included in transition-state theory, the activation energy is reduced, compared to the interpretation given in conventional transition-state theory.

Theory Of 3d Transition Metal Defects In 3C SiC

1996 ◽  
Vol 442 ◽  
Author(s):  
Harald Overhof
Keyword(s):  
Electronic Structure ◽  
Transition Metal ◽  
Electronic Properties ◽  
Crystal Field ◽  
High Spin ◽  
Crystal Field Splitting ◽  
Large Crystal ◽  
Field Splitting ◽  
Vacancy Model ◽  
The Difference

AbstractThe electronic properties of 3d transition metal (TM) defects located on one of the four different tetrahedral positions in 3C SiC are shown to be quite site-dependent. We explain the differences for the 3d TMs on the two substitutional sites within the vacancy model: the difference of the electronic structure between the carbon vacancy VC and the silicon vacancy VSi is responsible for the differences of the 3d TMs. The electronic properties of 3d TMs on the two tetrahedral interstitial sites differ even more: the TMs surrounded tetrahedrally by four Si atoms experience a large crystal field splitting while the tetrahedral C environment does not give rise to a significant crystal field splitting at all. It is only in the latter case that high-spin configurations are predicted.

scholarly journals Structural elastic and thermal properties of SrxCd1−x O mixed compounds — a theoretical approach

2014 ◽  
Vol 32 (3) ◽  
pp. 350-357
Author(s):  
Purvee Bhardwaj
Keyword(s):  
Phase Transition ◽  
Vibrational Energy ◽  
Zero Point ◽  
Study Phase ◽  
Energy Effect ◽  
Phase Transition Pressure ◽  
Vegard’S Law ◽  
Extended Interaction ◽  
Experimental Values ◽  
Good Agreement

AbstractIn the present paper, the structural and mechanical properties of alkaline earth oxides mixed compound SrxCd1−x O (0 ≤ x ≤ 1) under high pressure have been reported. An extended interaction potential (EIP) model, including the zero point vibrational energy effect, has been developed for this study. Phase transition pressures are associated with a sudden collapse in volume. Phase transition pressure and associated volume collapses [ΔV (Pt)/V(0)] calculated from this approach are in good agreement with the experimental values for the parent compounds (x = 0 and x = 1). The results for the mixed crystal counterparts are also in fair agreement with experimental data generated from the application of Vegard’s law to the data for the parent compounds.

INFLUENCE OF CRYSTAL FIELD EXCITATIONS COUPLED TO LATTICE VIBRATIONS ON THERMODYNAMICAL PROPERTIES OF A TWO-LEVEL PARAMAGNET

2004 ◽  
Vol 18 (03) ◽  
pp. 337-348
Author(s):  
J. ULNER ◽  
J. PEISERT ◽  
K. DURCZEWSKI
Keyword(s):  
Green Function ◽  
Specific Heat ◽  
Crystal Field ◽  
Field Model ◽  
Lattice Vibrations ◽  
Coupling Constant ◽  
Phonon Field ◽  
Thermodynamical Properties ◽  
Weakly Coupled ◽  
Perturbative Method

A two-level crystal field model in the presence of the phonon field is considered. Spins are arranged on a lattice and weakly coupled to the lattice oscillations of small amplitudes. The anticommutator Green function perturbative method up to the second order in the coupling constant is applied to find thermodynamical quantities. The magnetic spectral function, describing the crystal field excitations, influences these quantities. The effect is studied numerically and illustrated in figures. In particular, the influence of the coupling on the Schottky specific heat is shown.

Deformation of Perforated Aluminium Plates under In-Plane Compressive Loading

2016 ◽  
Vol 710 ◽  
pp. 357-362
Author(s):  
Irene Scheperboer ◽  
Evangelos Efthymiou ◽  
Johan Maljaars
Keyword(s):  
Research Work ◽  
Compressive Loading ◽  
Local Buckling ◽  
Buckling Load ◽  
Structural Behaviour ◽  
Critical Buckling Load ◽  
Out Of Plane ◽  
The Difference ◽  
Ultimate Resistance ◽  
Multiple Holes

Aluminium plates containing a single hole or multiple holes in a row are recently becoming very popular among architects and consultant engineers in many constructional applications, due to their reduced weight, as well as facilitating ventilation and light penetration of the buildings. However, there are still uncertainties concerning their structural behaviour, preventing them from wider utilization. In the present paper, local buckling phenomenon of perforated aluminium plates has been studied using the finite element method. For the purposes of the research work, plates with simply supported edges in the out-of-plane direction and subjected to uniaxial compression are examined. In view of perforations, circular cut-outs and the total cut-out size has been varied between 5 and 40% of the total plate area. Moreover, different perforation patterns have been investigated, from a single, central cut-out to a more refined pattern consisting of up to 25 holes equally distributed over the plate. Regarding the material characteristics, several aluminium alloys are considered and compared to steel grade A36 on plates of different slenderness. For each case the critical (Euler) buckling load and the ultimate resistance has been determined.A study into the boundary conditions of the plate showed that the restrictions at the edges parallel to the load direction have a large influence on the critical buckling load. Restraining the top or bottom edge does not significantly influence the resistance of the plate.The results showed that the ultimate resistance of aluminium plates containing multiple holes occurs at considerably larger out-of-plane displacement as that of full plates. For very large total cut-out, a plate containing a central hole has a larger resistance than a plate with equal cut-out percentage but with multiple holes. The strength and deformation in the post-critical regime, i.e. the difference between the critical buckling load and the ultimate resistance, differs significantly for different number of holes and cut-out percentage.

Comparison of Failure Modes of Piping Systems With Wall Thinning Subjected to In-Plane, Out-of-Plane and Mixed Mode Bending Under Seismic Load: An Experimental Approach

2007 ◽  
Author(s):  
Izumi Nakamura ◽  
Akihito Otani ◽  
Masaki Shiratori
Keyword(s):  
Failure Modes ◽  
Seismic Load ◽  
Piping System ◽  
Shake Table ◽  
Shake Table Tests ◽  
Wall Thinning ◽  
Piping Systems ◽  
Out Of Plane ◽  
Plane Bending ◽  
The Difference

In order to investigate the influence of degradation on the dynamic behavior and failure modes of piping systems with local wall thinning, shake table tests using 3-D piping system models were conducted. About 50% full circumferential wall thinning at elbows was considered in the test. Three types of models were used in the shake table tests. The difference of the models was the applied bending direction to the thinned wall elbow. The bending direction considered in the tests was either of the in-plane bending, out-of-plane bending, or mixed bending of the in-plane and out-of-plane. These models were excited under the same input acceleration until failure occurred. Through these tests, the vibration characteristic and failure modes of piping models with wall thinning under seismic load were obtained. The test results showed that the out-of-plane bending is not significant for a sound elbow, but should be considered for a thinned wall elbow, because the life of piping models with wall thinning subjected to out-of-plane bending may reduce significantly.

Theoretical Modelling of Equations of States for Aluminum to Pressure up to 10 TPa and Temperature to 10 eV

2016 ◽  
Vol 1141 ◽  
pp. 121-124
Author(s):  
Nisarg K. Bhatt ◽  
R.H. Joshi ◽  
Brijmohan Y. Thakore ◽  
Ashvin R. Jani ◽  
P.R. Vyas
Keyword(s):  
Characteristic Temperature ◽  
Empirical Relation ◽  
Tight Binding ◽  
Extreme Temperature ◽  
Pair Potential ◽  
Debye Model ◽  
Energy Functional ◽  
Theoretical Modelling ◽  
Energy Curve ◽  
Fitting Procedure

Response of materials to extreme temperature (T) and/or Pressure (P) conditions can be studied through, e.g., equations of states (EOS), which link between macroscopic observations to the microscopic consequences. Accurate knowledge of EOS is therefore always desirable, and plethora of different forms of EOS is proposed in literature. Of which, the most frequently used Mie-Grüneisen EOS coupled either with Debye-Model or Einstein-model within the quasiharmonic approximation requires prior knowledge of Grüneisen parameter (γ), characteristic temperature (θ) and their volume variation. At the pair-potential level, the Einstein characteristic temperature (θE) is derived through second-order potential derivatives. Although, many potential-energy-functional (PEF) are proposed for metallic systems, all including many-body effects, however, in order to compute θE effective pair-potential has to be deduced from PEF. But this procedure relies on some unavoidable approximations and fitting procedure. In this context, we have used energy-functional based definition of effective θE and for its volume dependence. We have employed tight-binding second-moment approximation (TB-SMA) to deduce cold energy curve to deduce θE and γ. Further, anharmonicity associated in bonding is parameterized; and it is shown that the parameter describing it is related to the thermodynamic Grüneisen parameter. Thus, the present scheme also circumvents additional empirical relation for γ (V). The present proposal is then employed to deduce equations of states and related thermo-physical properties; taking aluminum as a prototype. Results so generated to pressures more than 10 TPa and temperatures as high as 110 kK are compared and discussed in light of other simulated and experimental findings.

On the cubic and hexagonal close-packed lattices

1955 ◽  
Vol 227 (1171) ◽  
pp. 447-465 ◽  
Keyword(s):  
Solid Helium ◽  
Zero Point ◽  
Lattice Energy ◽  
Close Packing ◽  
Inert Gases ◽  
Zero Point Energy ◽  
Point Energy ◽  
Hexagonal Close Packed ◽  
The Difference ◽  
Central Forces

The present paper was stimulated by the discovery by Dugdale & Simon (1953) of a polymorphic transition in solid helium. A discussion is given of the relative stability of the cubic and hexagonal close-packed lattices assuming central forces of the Mie—Lennard-Jones type. Taking static lattice energy alone into account the usual laws of force favour the hexagonal close-packed lattice, the difference in energy being about 0·01%. However, lattice dynamics indicates that the equivalent Debye Θ at the absolute zero is smaller for the cubic lattice, the difference being about 1%. Hence, ignoring zero-point energy, we should expect a transition to occur from hexagonal to cubic at an elevated temperature. The estimated temperature and energy of the transition are of the same order of magnitude as those observed experimentally in solid helium. An estimate is made of the effect of zero-point energy; the results can be applied with confidence to the heavier inert gases, but can only be considered as giving a qualitative indication for helium, since anharmonic effects are of great importance in this case. For the other inert gas solids it is concluded that the experimentally observed cubic close-packing at all temperatures must be due to non-central forces.

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