A method for determination of higher-order magnetic anisotropy constants — Importance of the cubic K3 and K4 for certain energy levels models

A method for determination of magnetocrystalline anisotropy constants of arbitrary order is proposed. The method is based on a least squares fitting of a phenomenological anisotropy energy for a given symmetry truncated at an arbitrary order term to a theoretical anisotropy energy computed exactly for a given energy level model. Several applications of the method to cubic systems are considered. The study reveals that the widely used expressions in the literature for the cubic constants K₁ and K₂ in terms of free energy for the three symmetry direction are of rather limitedvalidity only. The higher-order cubic constants K₃, K₄ and K₅ are determined besides the usual K₁ and K₂ in temperature range 0 to 300 K. The importance of the higher- order terms with respect to the first term in the cubic anisotropy energy is discussed. The results show that the cubic constants K₃ and K₄ cannot be neglected for most of the energy level models studied at certain temperatures.     

Director dynamics near the nematic-smectic-A phase transition

The Landau-de Gennes model for the free energy of a nematic liquid crystal near the phase transition to the smectic A-phase is used to determine the frequency dependence of the fluctuation corrections to the Frank elastic constants. It is shown that the interaction of the fluctuations of the smectic order parameter and the director results in corrections to all the Frank elastic constants. In the low-frequency limit (ω→0), the corrections to the Frank elastic constants K ₂₂ and K ₃₃ are the largest, and decrease to zero in the infinite-frequency limit. The correction to K ₁₁ is negative, and vanishes in both limits. The absolute value of the correction to K ₁₁ is the largest at frequencies in the megahertz range. It is shown that in oriented nematics the interaction of the smectic fluctuations and the director limits deviations of the director from the direction of preferred orientation, as a result of which relaxation of both inhomogeneous and homogeneous distortions of the director field can be observed. It is also shown that this gives rise to a frequency interval in the megahertz range in which shear waves begin to propagate in the nematic. The propagation speed of these waves is roughly a hundred times smaller than that of sound and strongly depends on the direction of propagation. Zh. éksp. Teor. Fiz. 114, 2022–2033 (December 1998)     

Structure and elastic properties of a nematic liquid crystal: A theoretical treatment and molecular dynamics simulation

The Frank elasticity constants which describe splay (K ₁), twist (K ₂), and bend (K ₃) distortion modes are investigated for 4-n-pentyl-4'-cyanobiphenyl (5CB) in the nematic liquid crystal. The calculations rest on statistical-mechanical approaches where the absolute values of K _i (i=1,2,3) are dependent on the direct correlation function (DCF) of the corresponding nematic state. The DCF was determined using the pair correlation function by solving the Ornstein-Zernike equation. The pair correlation function, in turn, was obtained from molecular dynamics (MD) trajectory. Three different approaches for calculations of the elasticity constants were employed based on different level of approximation about the orientational order and molecular correlations. The best agreement with experimental values of elasticity constants was obtained in a model where the full orientational distribution function was used. In addition we have investigated the approximation about spherical distribution of the intermolecular vectors in the nematic phase, often used in derivation of various mean-field theories and employed here for the construction of the DCF. We found that this assumption is not strictly valid, in particular a strong deviation from the isotropic distribution is observed for short intermolecular distances. Received 22 March 2000 and Received in final form 9 June 2000     

Influence of anisotropy constants of higher order on the magnetic reversal of uniaxial magnets

Within the framework of a simple model in which only homogeneous states of magnetization are considered, the influence of the constants of magnetic crystallographic anisotropy of higher order on the critical field of domain formation and the hysteresis loops of uniaxial magnets is investigated. It is shown that the critical field depends strongly on both the sign of K₁ and on the ratio of the magnitudes and signs of K₂/K₁ and K₃/K₁. On the basis of stability analysis for the free energy, hysteresis loops of uniaxial magnets as a function of K_i are calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 39–43, April, 1990.     

Light-field-induced spatially periodic freedericksz transition in a planar nematic cell

It is shown that a threshold spatially periodic reorientation of the director by a light field is possible in a planar nematic liquid crystal cell if the ratio of the Frank elastic constants, K ₂/K ₁, exceeds a critical value. The periodic director structure arising in the cell leads to a self-diffraction of the incident light wave. The dependences obtained for this phenomenon make it possible to determine the values of the elastic constants K ₁ and K ₂, the director reorientation threshold, and the period of the director structure from the experimental values of the self-diffraction angle.     

Surface elasticity and Fréedericksz’ threshold in a nematic cell weakly anchored

Summary The presence in the distortion free energy of theK ₁₃-term, depending on the first derivatives of the walls’ tilt angles, leads to some paradoxes. ButK ₁₃ can be shown to vanish in the linear elastic theory. Instead, the saddle splayK ₂₄-term is independent of the first derivatives of the anchoring angles. This fact allows us to obtain Fréedericksz’ threshold as a function of the involved elastic constants and of the anchoring strengths in the frame of the standard continuum theory. Work presented at the First USSR-Italy Bilateral Meeting on Liquid Crystals held in Portonovo, Ancona (Italy), September 30–October 2, 1987.     

Hydration of nitriles: an examination in terms of No Barrier Theory

Nitriles pose an interesting problem to the explanatory powers of organic chemistry because, despite the favorable overall thermodynamics of hydrolysis to the corresponding amides, the reactions are inherently slow. The rate determining step is hydration of the nitrile to give the imidic acid, which quickly tautomerizes to the amide. In terms of Marcus Theory, the intrinsic barriers for acid and base‐catalyzed hydration are higher for nitriles than for amides, which are themselves slow reactions. It is remarkable that hydration of a nitrile, for which the free energy change is close to zero except for polyhaloacetonitriles, is much slower than hydration of an amide, which is energetically unfavorable. This can be explained by No Barrier Theory in terms of the high energetic cost of the geometrical distortions in “one thing at a time” corner species. There are no experimental equilibrium constants for this initial hydration step, so we have determined them computationally. The free energy change for the initial hydration is small; it is the fast and energetically downhill second step, tautomerization to the amide, which makes the overall hydrolysis of nitrile to amide thermodynamically favorable. Very few of the pK_a values needed in the acid and base‐catalyzed mechanisms are known, so we used linear free energy relations and treat the parent pK_a values as adjustable parameters. This procedure leads to pK_a values in accord with expectation based on such data as are available and permits calculation of rate constants in satisfactory agreement with experiment. Copyright © 2013 John Wiley & Sons, Ltd.     

Inhomogeneous threshold director reorientation in a planar nematic flexoelectric cell with finite anchoring energy

The dependence of the threshold parameters and the period of the electric-field-induced spatially periodic reorientation of the director in a flexoelectric nematic liquid crystal (NLC) on the anchoring conditions at the surface of a planar NLC cell has been studied. The threshold electric field and the corresponding wave-number of the periodic structure of the director field have been numerically calculated for arbitrary values of the anchoring energy. In the case of strong anchoring, the corresponding analytical expressions are obtained in a single-constant approximation. A decrease in the azimuthal anchoring energy leads to an increase in the intervals of possible values of the flexoelectric parameter ν and the ratio K₂/K₁ of the Frank elastic constants. A decrease in the polar anchoring energy leads to narrowing of these intervals as compared to the case of infinitely strong anchoring at the NLC cell surface.     

The Millimeter-Wave Spectrum of 2,3-Dihydrofuran

The millimeter-wave spectrum of 2,3-dihydrofuran in the ground and five ring-puckering excited states has been measured in the frequency range 100–250 GHz. The ground and first ring-puckering excited states have been fitted to a two-state Hamiltonian including Coriolis coupling interaction. The determined energy difference of 18.684(7) cm⁻¹between these states and theaandbtype coupling parameters are consistent with the ring-puckering potential function and the previously observed dependence of the centrifugal distortion constants Δ_JK, Δ_K, and δ_K. A small ring-puckering dependence of the quartic centrifugal distortion constants Δ_Jand δ_Jhas been also observed. This dependence is well accounted for in terms of the ring-puckering potential function and the vibrational dependence of the rotational constants.     

Four constant potential function of amorphous selenium and evaluation of its thermodynamic and acoustic properties

Litov and Anderson after various considerations suggested a four constant potential function for a-Se as well as a-As₂S₃. Hence we also used a four constant potential function with the sole purpose of applying this potential function to obtain several acoustic, thermodynamic and other properties. We calculated several acoustic properties of a-Se like second order elastic constants (SOECs), their pressure derivatives, the longitudinal and transverse Grüneisen constant by two different methods, phonon frequencies, absorption band position through the use of Nath-Smith-Delaunay’s equation, and the thermodynamic properties like heat capacity, bulk modulus, thermal Grüneisen constant, the pressure derivative of the bulk modulus (dK _T/dP=C ₁), the pressure derivative ofC ₁ which is related to Anderson-Grüneisen parameter, pressure derivative of Grüneisen constant namelyγ′ _g which is related to second Grüneisen constant, characteristics of phonon frequencies, potential energy function through the use of fitted parameters and third order elastic constants. Finally we calculatedK _T at the reduced density ofρ/ρ ₀=1.1.K _T is obtained from the potential function with the fitted parameters. In all the above cases the calculated values are found to be in good agreement with experiment wherever available. In this connection it is important to point out that we eliminated ‘C’ a constant in the potential function using the equilibrium condition as was done by Litovet al in a-Se and Gerlichet al in the case of a-As₂S₃ as all amorphous substances are isotropic as mentioned by several authors. We contemplate to calculate several other properties for a-Se and a-As₂S₃ and present them at a later stage.     

Energieverlustmessungen mit 60 keV Elektronen an amorphem und polykristallinem Selen und Tellur und Bestimmung optischer Konstanten

The energy loss functions Im—(1/?) of amorphous and polycrystalline Selenium and Tellurium are determined from energy loss spectra of 60 keV electrons in the energy range up to 30 eV. The optical constants ?₁ and ?₂ are calculated from the energy loss function by Kramers Kronig analysis. The energy difference in the position of the volume plasma loss of amorphous and polycrystalline foils is compared with the calculated change in free electron plasma energy resulting from density change. Characteristic structure dependence of the optical constants are found to be similar for Selenium and Tellurium. They are discussed in terms of the results of band structure calculations.     

A study of π states of H2 + in the one centre approach

The statistical theory of the elastic constants of nematic liquid crystals is applied to detailed calculations based on well-defined approximations for the direct correlation function of Ornstein and Zernike and for the singlet orientational distribution. The model of hard spherocylinders with superimposed r ^-6 attraction modulated by a Maier-Saupe P ₂(cos ?_{1 2}) term, is used. The dependence on temperature, on the order parameter and on the length to breadth ratio, of the reduced elastic constants K* _i (i=1, 2, 3), is discussed in some detail and comparison with experiment is included.     

Perturbation theory of polar hard Gaussian overlap fluid mixtures

A perturbation theory of polar hard Gaussian overlap fluid mixture is discussed. Explicit analytic expressions for the second and third varial coefficients are given. Numerical results are estimated for the thermodynamic properties of quadrupolar hard Gaussian overlap fluid and fluid mixture. It is found that the excess free energy and internal energy depend on concentrationsc ₁,c ₂, molecular diameter ratioR, shape parameterK and the quadrupole momentsQ*₁,Q*₂.     

The structure of the distortion free-energy density in nematics: Second-order elasticity and surface terms

Summary By means of a phenomenological approach, we demonstrate that the mixed splay-bend elastic constantK ₁₃ in the free energy density of nematic liquid crystals must be considered zero, unless the bulk contributions of the squares of the distortion second-order derivatives are taken into account, together with the squares of the first-order derivatives times the second-order derivatives, and with the fourth powers of the first-order derivatives. Such contributions just reduce to one in the presence of—and close to—a threshold. Furthermore, the saddle-splayK ₂₄-term instead is shown always to play an essential role, as the bulk first-order elasticity, in determining the distortion free energy of nematics with weak anchoring subjected to spatial deformations. Finally, the new surfacelike elastic constants are shown to have a nilpotent character: thus they behave as well asK ₂₄ from the point of view of the variational calculus. Work presented at the second USSR-Italy Bilateral Meeting on Liquid Crystals held in Moscow, September 15–21, 1988.     

Anharmonic corrections for linear triatomic molecules subject to the Rennet-Teller effect

The effects of vibrational anharmonic terms and of the g_K -correction on the energy levels of a triatomic molecule in a degenerate electronic state are considered. The electronic wavefunctions are described using the approach first suggested in the original paper of Renner. Formulae for the anharmonic corrections in a number of different situations are derived. For an electronic Π state the corrections are given in the form where i runs over the various contributions and x ₁, x ₂, … depend on the anharmonic force constants. The functions F_i can be determined numerically (see equations (4.3) and 4.7)). For the case without spin-orbit interaction the F_i s are given explicitly to first order in ∈ in table 1. Further-more, the same results apply for levels with |K|=v ₂+1 even if the spinorbit interaction is not negligible. Explicit results for levels with |K|<v ₂ including spin-orbit interaction are given in tables 2 and 3. The cases with larger values for Λ (2, 3, …) are also considered. The energy level formulae for a Δ state differ from those derived earlier by Merer and Travis. In particular, the small separation between the vibronic Φ and Π levels with v ₂=1 is now found to be 4g_K . The additional terms that arise from end-over-end rotation of the molecule are discussed in § 6.     

Model and calculation of rate constants of nonradiative <Emphasis Type="Italic">T-S</Emphasis> intersystem conversion: Test by example of naphthalene and its chlorine derivatives

For in-plane spin states (s = z, y), the rate constants K _dg ^s of the nonradiative energy degradation T ₁ ⇝ S ₀ of the lowest triplet T ₁ state of naphthalene (NPH) and its dichloro-substituted derivatives at positions 1,4- and 2,3- of the molecule (1,4-NPH and 2,3-NPH) are calculated. A simple model is proposed for calculations that is based on the nonadiabatic approximation and uses all the out-of-plane vibrational modes of the molecule as promoting vibrational modes. As a result of calculations, the dependences of the rate constants K _dg and K _dg ^s on the positions of chlorine atoms in the molecule are obtained, which are consistent with the known data of magnetooptical measurements. The inversion of the ratio K _dg ^z : K _dg ^y in the 1,4-NPH and 2,3-NPH molecules is established.     

氢同位素氚水T2O(X1A1)的解析势能函数

获得T₂O(X¹A₁)解析势能函数的主要困难在于Born-Oppenheimer近似下T₂O(X¹A₁)与H₂O(X¹A₁)势能函数的不可区分性.然而,在Born-Oppenheimer近似下,分子势能函数实际上是键长、键角这些 关键词： 2O(X¹A₁)')" href="#">T₂O(X¹A₁) 同位素效应 解析势能函数     

The analytical potential energy function of flue gas SO2（X1A1）

The equilibrium structure of flue gas SO2 is optimized using the density functional theory （DFT）/B3P86 method and CC-PV5Z basis. The result shows that it has a bent （C2v, X1A1） ground state structure with an angle of 119.1184°. The vibronic frequencies and the force constants are also calculated. Based on the principles of atomic and molecular reaction statics （AMIIS）, the possible electronic states and reasonable dissociation limits for the ground state of SO2 molecule are determined. The potential functions of SO and 02 are fitted by the modified Murrell-Sorbie＋c6 （M-S＋c6） potential function and the fitted parameters, the force constants and the spectroscopic constants are obtained, which are all close to the experimental values. The analytic potential energy function of the SO2 （X1A1） molecule is derived using the many-body expansion theory. The contour liues are constructed, which show the static properties of SO2 （XIA1）, such as the equilibrium structure, the lowest energies, the most possible reaction channel, etc.     

Monomer counterrotations and tunneling splitting in CO dimer by data of millimeter wave spectroscopy

Analysis of the rotational spectrum of the molecular dimer (CO)₂ measured in the millimeter wave range has been performed and four new rotational states are revealed. Three of these states are characterized by almost free rotations of both monomers in the dimer. These states have approximately the same first term σ in the expansion of the rotational energy in powers of the rotational angular momentum J for various values of the momentum projections on the dimer axis (K=0, 1, 2) and various rotational constants B. The intrinsic rotational angular momenta of CO dimers, j₁=j₂=1, are determined from the σ value. In addition, a state with K=2 is found which corresponds to one of the known shape isomers of (CO)₂. The values of the tunneling splitting for each of the new states are determined. The results indicate that previous data on the suppressed tunneling are determined by the asymmetry of internal rotations in the CO monomers rather than by the _K value.     

Rotational analysis and line intensities of the HCO band at 296 K

This paper computes the rotational energy levels of the HCO transition, especially, the higher values of the rotational quantum numbers N_{Ka Kc} and K_a, with the rotational constants which are obtained via B3LYP method with 6-311G basis set, and the results show that the calculated frequencies using the computed vibration--rotation energy levels are in reasonable agreement with the data from the experiment. Meanwhile, the line intensities of HCO are first reported, the results are of significance for the studying HCO.