Physical meaning of nonholonomic strain measure

The paper overviews the main approaches to the introduction of strain measures. It has been concluded that the physical meaning of certain measures is insufficiently clear. Problems concerning the definition of the physical meaning of the nonholonomic asymmetric strain measure introduced in the framework of a multiscale model, which is based on the physical theory of plasticity, are discussed. This measure is calculated using the corotational integration of the asymmetric and frame-independent strain rate measure equal to the relative velocity gradient. The integration is carried in terms of the corotational coordinate system whose instantaneous motion is determined by averaging the spins of mesoscale elements. It is shown that if elastic distortions are neglected, the introduced mesoscale strain measure is equal to the sum (over all slip systems of the crystallite) of products of accumulated shear multiplied by the basis dyads of the slip systems. The averaging reveals that additional contribution to the value of the macroscale strain measure is attributed, along with elastic distortions, to corotational terms that appear due to different rotation rates of the introduced macroscale corotational frame and crystallite lattices. In view of the absence of analytical expressions for the meso- and macroscale spins, the physical meaning of the macroscale nonholonomic measure is defined in numerical experiments for several strain paths. Calculations have shown that contributions of the elastic and corotational components to the macroscale nonholonomic strain measure are negligible for strain paths of different complexity.     

On zero-field anisotropic static Kubo-Toyabe relaxation functions

For muons which have thermalized in the allowed stopping sites of a given crystal, zero-field static Kubo-Toyabe spin dynamics is expressed in terms of the crystal frame spherical harmonic coefficients of the local classical random magnetic field distributions associated with each stopping site. The resulting muon spin polarization involves nine observable relaxation functions which are the counter frame spherical tensor expansion coefficients of the second-rank dynamic motion tensor. These relaxation functions can be measured simultaneously using the skewed field technique for the same experimental conditions, that is, with a single apparatus and with a single crystal-counter frame orientation. The local field distributions of, in general, arbitrary symmetry are classical approximations to the magnetic field interactions between the spin of the muon and the spins of the nuclei associated with each site at which the muon has stopped. They are characterized by equating their moments with the quantal moments generated by the quantal magnetic fields. The observable consequences of the anisotropic second-order moments associated with the planar sites of hexagonal crystals are used as an illustration.     

Theoretical description of the methyl group relaxation in liquids

Spin-lattice and spin-spin relaxation times of protons and deuterons of methyl groups in liquids are calculated using a semiclassical form of the density operator formalism and symmetry adapted spherical tensors. The model of molecular motion assumes diffusion of the whole molecule containing the methyl group with some internal rotation. Special consideration is given to uniaxial internal rotation and anisotropic diffusion as well as to biaxial internal rotation and isotropic diffusion in interpreting the experimental results.     

Elastic moduli approximation of higher symmetry for the acoustical properties of an anisotropic material

The issue of how to define and determine an optimal acoustical fit to a set of anisotropic elastic constants is addressed. The optimal moduli are defined as those which minimize the mean-squared difference in the acoustical tensors between the given moduli and all possible moduli of a chosen higher material symmetry. The solution is shown to be identical to minimizing a Euclidean distance function, or equivalently, projecting the tensor of elastic stiffness onto the appropriate symmetry. This has implications for how to best select anisotropic constants to acoustically model complex materials.     

The Temperature Dependence of Nitroxide Spin–Label Interaction Parameters: a High-Field EPR Study of Intramolecular Motional Contributions

High-field W-band electron paramagnetic resonance (EPR) spectroscopy was utilized to study the temperature dependence of the magnetic interaction parameters (g-, hyperfine-, quadrupole tensors) of two types of doublet-state nitroxide spin probes in glass-forming ortho-terphenyl solution: a five-membered ring system of pyrroline type (model for the commonly used methane thiosulfonate spin label) and a six-membered ring system of piperidine type (model for the commonly used TOAC spin label). The analysis of the g- and hyperfine tensors in terms of their isotropic and anisotropic parts reveals at least two mechanisms of motion that are responsible for the temperature dependence of the interaction parameters. The first mechanism is attributed to the overall small-angle motion of the nitroxide molecule in the glassy matrix; it leads to an averaging of the anisotropies of the EPR parameters. The second mechanism originates in an intramolecular out-of-plane motion of oxygen in the nitroxide group. This type of motion is evidenced by comparing the experimental findings for the spin-interaction parameters with the results of density functional theory calculations. The harmonic oxygen out-of-plane vibrations result in a variation of both the isotropic and anisotropic parts of the g- and hyperfine tensors. In contrast, the quadrupole tensor is not influenced by this vibration mechanism in the temperature range under study (90–240 K). Consequences of the applicability of such typical nitroxide radicals for probing details of their protein environment and for studying librational dynamics in frozen solutions are discussed.     

Two-scale models of polycrystals: Evaluation of validity of Ilyushin’s isotropy postulate at large displacement gradients

This paper discusses multiscale models of inelastic deformation of single- and polycrystals, which are based on crystal plasticity theories, as applied to the verification and justification of Ilyushin’s isotropy postulate (in a special form) at large displacement gradients. Different approaches to motion decomposition on the macroscale into quasi-rigid (described by the motion of a corotational coordinate system) and strain-induced motion (a relatively moving coordinate system) are considered. The strain path is defined in terms of a moving coordinate system. Corresponding kinematic effects are defined in terms of a laboratory coordinate system. In this case, the loading process image is constructed and loading conditions are specified in terms of the moving coordinate system. Calculations are performed for two types of strain paths with different curvature by assuming two different hypotheses about quasi-rigid motion on the macroscale: (i) the spin of the moving coordinate system is equal to an averaged mesoscale spin, and (ii) the spin is equal to the macroscale vortex. It is shown that the isotropy postulate is more valid in the case of assuming the first hypothesis.     

E.P.R. evidence for the non-planarity of t-butyl radical

A discussion is given of the form of the scattering tensor of all possible normal modes in all possible electronically degenerate states of the important crystallographic point groups with doubly and triply degenerate representations. Arguments are put forward that asymmetric tensors do not contribute to the intensity of totally symmetric modes in electronically degenerate states. However, such tensors can contribute to the intensity of some of the other modes. The form of the scattering tensor is obtained from a study of corresponding transitions if the symmetry of the molecule is lowered so that degeneracies are removed. This lower symmetry situation is so chosen that it belongs to a group which is the subgroup of highest symmetry of the group which describes the symmetry of the molecule having degenerate electronic states.     

魔角旋转情形下固体的非均匀相互作用

各向异性非均匀相互作用的慢旋转MAS-NMR谱是一系列的旋转边带,这些边带的强度包含了该相互作用张量的全部信息,只要计算和分析少数几条边带就可以得到这些有用的结构信息,本文用不可约球谐张量算符表示非均匀相互作用,采用Bessel函数展开法分析计算非均匀相互作用产生的旋转边带,这种分析计算方法适用于各种非均匀相互作用。以化学位移和四极相互作用为例的实验与理论符合很好。 关键词：     

On the energy-momentum and spin tensors in the Riemann–Cartan space

General classical theories of material fields in an arbitrary Riemann–Cartan space are considered. For these theories, with the help of equations of balance, new non-trivially generalized, manifestly generally covariant expressions for canonical energy-momentum and spin tensors are constructed in the cases when a Lagrangian contains (a) an arbitrary set of tensorial material fields and their covariant derivatives up to the second order, as well as (b) the curvature tensor and (c) the torsion tensor with its covariant derivatives up to the second order. A non-trivial manifestly generally covariant generalization of the Belinfante symmetrization procedure, suitable for an arbitrary Riemann–Cartan space, is carried out. A covariant symmetrized energy-momentum tensor is constructed in a general form.     

The proton nuclear magnetic shielding tensors in biphenyl: experiment and theory

Line-narrowing multiple pulse techniques are applied to a spherical sample crystal of biphenyl. The 10 different proton shielding tensors in this compound are determined. The accuracy level for the tensor components is 0.3 ppm. The assignment of the measured tensors to the corresponding proton sites is given careful attention. Intermolecular shielding contributions are calculated by the induced magnetic point dipole model with empirical atom and bond susceptibilities (distant neighbours) and by a new quantum chemical method (near neighbours). Subtracting the intermolecular contributions from the (correctly assigned) measured shielding tensors leads to isolated-molecule shielding tensors for which there are symmetry relations. Compliance to these relations is the criterion for the correct assignment. The success of this program indicates that intermolecular proton shielding contributions can be calculated to better than 0.5 ppm. The isolated-molecule shielding tensors obtained from experiment and calculated intermolecular contributions are compared with isolated-molecule quantum chemical results. Expressed in the icosahedral tensor representation, the rms differences of the respective tensor components are below 0.5 ppm for all proton sites in biphenyl. In the isolated molecule, the least shielded direction of all protons is the perpendicular to the molecular plane. For the para proton, the intermediate principal direction is along the C-H bond. It is argued that these relations also hold for the protons in the isolated benzene molecule.     

含有五度旋转轴点群的超拉曼张量的计算

超拉曼散射涉及晶格振动, 因此它需要由动态张量来描述。根据坐标积与张量元在对称操作下的变换形式相同的原理, 通过C语言编程, 计算了属于不同不可约表示的动态张量, 并列出了含有五度旋转轴点群的三阶超拉曼张量。     

Faxén relations in solids-a generalized approach to particle motion in elasticity and viscoelasticity

A movable inclusion in an elastic material oscillates as a rigid body with six degrees of freedom. Displacement/rotation and force/moment tensors which express the motion of the inclusion in terms of the displacement and force at arbitrary exterior points are introduced. Using reciprocity arguments two general identities are derived relating these tensors. Applications of the identities to spherical particles provide several new results, including simple expressions for the force and moment on the particle due to plane wave excitation.     

Spin eigenfunctions and operators for the Dn groups

An automated procedure for determining symmetry-adapted spin eigenfunctions is developed for the D_n symmetry groups with n equivalent spins for arbitrary size of I and for n=3,…,6. These eigenfunctions are also eigenfunctions of the vector sum of the n equivalent nuclear spins. Generalized hyperfine spin operators are developed that have real matrix elements and that exploit the full symmetry. These spin operators can be combined with operators for other spins and molecular rotation using only real arithmetic.     

刚体定点转动的张量描述

通过引入转动张量来描述刚体的定点转动,避免了在用角位移描述刚体定点转动时所遇到的问题,即角位移在它是有限大小和无限小时属性发生了变化.验证了对于刚体定点无限小转动,可以分别采用角位移矢量和转动张量描述,两者是等价的.     

Unitary group decomposition of hamiltonian operators. II. SU (4) irreducible tensors,norms and their energy variation and symmetry breaking

SU(6) ? SU(4) tensor decomposition of effective interactions in the 2s-1d shell has been carried out to examine the relative importance of the various irreducible tensors in many-particle spaces. For this purpose norms of the irreducible tensors are evaluated in many-particle spaces. Variation of the expectation value of the square of the irreducible tensor parts with excitation energy has also been examined using the polynomial expansion method. A new measure of symmetry breaking that is theoretically more sound is derived which includes in its definition partial width as well as internal width. This is used to study SU(4) symmetry mixing in nuclei.     

Low-symmetry spin hamiltonian and crystal field tensors analysis: Fe(3+) in natrolite

Electron paramagnetic resonance study of a natural single crystal of natrolite was carried out at the frequency nu=36.772 GHz at room temperature. The angular dependence of the four symmetry-related spectra of Fe(3+) in the three crystallographic planes was fitted to a spin Hamiltonian (S=5/2) of symmetry C(i). The rank 4 crystal field tensors at tetrahedral sites were calculated using the point-charge model to determine the principal axes orientations of their cubic and trigonal components. The analysis of zero-field splitting tensors and comparison with crystal field ones suggests that Fe(3+) substitutes for Al(3+) with no significant distortion of the coordination tetrahedron in natrolite. Comparison of data for several natural and synthetic crystals reveals that the 4-rank zero-field splitting tensor invariants for Fe(3+) at the tetrahedral oxygen-coordinated sites are distinguishably smaller than those for Fe(3+) at octahedral sites. Such comparative analysis may help to determine the substitutional sites in other crystals.