Invariant tensors inSU (3)

The construction of independentSU (3) tensors out of octets of fields is considered by investigating numerically invariantSU (3) tensors. A method of obtaining independent sets of these to any rank is discussed and also independent sets are explicitly displayed up to fifth rank. It is shown that this approach allows us to obtain relations among the invariant tensors, and useful new identities involving thed _ijk andf _ijk tensors are exhibited.     

The momentum constraints of general relativity and spatial conformal isometries

Transverse-tracefree (TT-) tensors on (R ³,g _ab), withg _ab an asymptotically flat metric of fast decay at infinity, are studied. When the source tensor from which these TT tensors are constructed has fast fall-off at infinity, TT tensors allow a multipole-type expansion. Wheng _ab has no conformal Killing vectors (CKV's) it is proven that any finite but otherwise arbitrary set of moments can be realized by a suitable TT tensor. When CKV's exist there are obstructions — certain (combinations of) moments have to vanish — which we study.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P9376-PHY.Partially supported by Forbairt Grant SC/94/225.     

Canonical superenergy tensors in general relativity

We present the concept of superenergy tensors in the framework of general relativity (GR). These tensors were introduced constructively by the author years ago and they were obtained by a suitable averaging of the energy-momentum tensors or pseudotensors. Because in GR the Einstein canonical energy-momentum pseudotensor_E t _i ^k of the gravitational field and the canonical energy-momentum complex , matter and gravitation,are physically distinguished, we confine this paper to thecanonical superenergy tensor ^g S _i ^k of the gravitational field Γ _kl ⁱ and to the canonical total superenergy tensorS _i ^k = ^g S _i ^k + ^m S _i ^k of matter and gravitation only. These superenergy tensors can be obtained by the above-mentioned averaging of the pseudotensor_E t _i ^k and complex_E K _i ^k . We give the analytic forms of these two canonical superenergy tensors and show some of their possible applications in GR. The canonical superenergy tensor ^g S _i ^k of the gravitational field Γ _kl ⁱ can be used as asubstitute for the nonexisting energy-momentum tensor of this field.     

Spherical harmonics and cartesian tensor scalar product expansions of the distribution function

On the basis of the expansion of the distribution function in a sum of the spherical harmonics, the distribution functionf(v, r, t) is expanded in a series of scalar products of two Cartesian tensors term by term, i.e. The tensors and ^(l) (l=2, 3) are constructed in dependence on the spherical harmonic expansion coefficients (the tensors and ^(l) (l=0, 1) have been constructed by Jancel and Kahan [3]). On the basis of the knowledge of the analytic form off ₂ andf ₃ the equations forf ₁ f ₂ andf ₃ for the case of the Boltzmann's equation are determined.Technická 2, Praha 6, Czechoslovakia.     

Real Kasner and related complex “windmill” vacuum spacetime metrics

The Kasner family of vacuum solutions of Einstein's field equations admits a simply-transitiveH ₄, a four-parameter local homothetic group of motions which has an AbelianG ₃ subgroup. It is shown that a complex transformation of coordinates and constants exists which maps this family from the normal Kasner form into a form of vacuum metrics whose Weyl tensors are each Petrov type I and which were published in 1932 by Lewis. These metrics also admit a similarH ₄; however for one particular metric (for one parameter value) theH ₄ becomes aG ₄ and the resultant metric is one which was rediscovered by Petrov in 1962. These Lewis metrics are thus shown to be Kasner metrics over complex fields. Here they are calledwindmill metrics because of the rotating relationship between the coordinates and the Killing vector fields admitted. The principal null directions of thereal Kasner and the windmill metrics are discussed; the two families then provide illustrations of two degenerate classes of spacetime metrics whose Weyl tensors are of Petrov type I, as discussed elsewhere by Arianrhod and McIntosh. An extension of the windmill-type generation of metrics to some other families of metrics is also discussed.     

Proton ENDOR of VO(H2O)5 2+ in Mg(NH4)2(SO4)2)26H2O

ENDOR measurements at 25 K have been used to determine the hyperfine coupling tensors for all ten protons in the VO(H₂O)₅ ²⁺ ion in single crystals of Mg(NH₄)₂(SO₄)₂6H₂O. The traceless components of all the tensors are close to axial and their use in a point dipole treatment enables a very plausible geometrical model of the complex ion to be constructed. Six of the protons in the equatorial water molecules have substantial positive isotropic couplings and it is suggested that these reflect the direct admixture of hydrogen 1s components into the singly occupied orbital.     

Hydrogen bond studies

The deuteron magnetic resonance spectra from partially deuterated single crystals of NaHC₂O₄. H₂O have been studied. The quadrupole coupling tensors for stationary deuterons are determined at about 25°C. The quadrupole coupling constants and the assymmetry parameters η of the tensors for the two deuterons in the water molecule are, 235·2 ± 1·9 and 228·5 ± 1·6 kHz, and 0·09 ± 0·01 and 0·13 ± 0·01, respectively. The principal axes corresponding to the largest components both deviate by 0·9 ± 0·5° from their respective O-H directions. For one of the tensors this angular displacement is in the plane of the H₂O molecule and towards the other O-H direction. The displacement for the other tensor is out of the plane of the H₂O molecule.     

Iterative Lineshape Fitting of MAS NMR Spectra: A Tool to Investigate HomonuclearJCoupling in Isolated Spin Pairs

Possibilities and limitations of iterative lineshape fitting procedures of MAS NMR spectra of isolated homonuclear spin pairs, aiming at determination of magnitudes and orientations of the various interaction tensors, are explored. Requirements regarding experimental MAS NMR spectra as well as simulation and fitting procedures are discussed. Our examples chosen are the isolated³¹P spin pairs in solid Na₄P₂O₇· 10H₂O, (1), and Cd(NO₃)₂· 2PPh₃, (2). In both cases the two³¹P chemical shielding tensors in the molecular unit are related byC₂symmetry, and determination of the orientations of these two tensors in the molecular frame is possible. In addition, aspects of homonuclearJcoupling will be addressed. For 1, both magnitude and sign of²J_iso(³¹P,³¹P) (J_iso= −19.5 ± 2.5 Hz) are obtained; for 2, (J_iso= +139 ± 3 Hz) anisotropy ofJwith an orientation of theJ-coupling tensor collinear, or nearly collinear, with the dipolar coupling tensor can be excluded, while absence or presence of anisotropy ofJwith any other relative orientation of theJ-coupling tensor cannot be determined.     

On the electronic structure of N,N′-ethylenebis(acetylacetonatiminato)Co(II), Co(II)acacen

Single crystal E.P.R. and cobalt ENDOR measurements on N,N′-ethylenebis(acetylacetonatiminato)Co(II), Coacacen, diluted in Niacacen·1/2 H₂O are reported. Forbidden Δm _Co = 1,2 transitions in the E.P.R. spectra have been observed. The g-tensor (E.P.R.) and the cobalt hyperfine and quadrupole tensors (ENDOR) have been determined. The g- and cobalt hyperfine tensors are discussed. They support the |² A ₂,yz? groundstate proposed for four coordinated low-spin Schiff base complexes of cobalt(II). The measured quadrupole coupling is also compatible with a |² A ₂,yz? groundstate, if anisotropic contraction of the cobalt 3d orbitals is taken into consideration.     

Ab initio studies of dielectric,piezoelectric, and elastic properties of BaTiO 3 /SrTiO 3 ferroelectric superlattices

The phonon spectrum; crystal structure of the polar phase; spontaneous polarization; dielectric constant, piezoelectric, and elastic moduli tensors for free_standing and substrate_supported superlattices mBaTiO₃/nSrTiO₃ (with m = n = 1–4) were calculated within the density functional theory. The simulation of properties of the disordered Ba_0.5Sr_0.5TiO₃ solid solution using two special quasirandom SQS-4 structures and their comparison with the properties of the superlattices revealed a tendency of the BaTiO₃-SrTiO₃ system to superstructure ordering and showed that the superlattices are thermodynamically quite stable. The ground state of the free-standing superlattice corresponds to the monoclinic polar phase Cm, which transforms to the tetragonal polar phase P4mm under in-plane compressive strain of the superlattice and to the orthorhombic polar phase Amm2 under in-plane tensile strain. With a change in the in-plane lattice parameter, in the vicinity of boundaries between neighboring polar phases, some optical and acoustic modes soften and some components of the static dielectric constant, piezoelectric, and elastic moduli tensors diverge critically.     

Tensor polarization of 12C[2+ 1] in the 16O(13C,12C)17O reaction at 50 MeV

The ¹⁶O(¹³C,¹²C)¹⁷O reaction at 50 MeV has been investigated using the kinematical coincidence method. Polarization tensors t ₂₀ and t ₄₀ of ¹²C[2⁺ ₁] for the quantization axis taken along the direction of propagation have been measured by analyzing the energy spectrum of ¹²C[2⁺ ₁], modulated by the effect of γ ray emission. The deduced t ₄₀ values significantly deviate from zero, contrary to the prediction of the distorted-wave Born approximation theory based on one-step p shell neutron stripping without spin-dependent interactions. The phenomenological spin–orbit interaction necessary to reproduce the magnitude of measured t ₄₀ is found to be much larger than the folding model prediction. It is shown that the experimental polarization tensors as well as the cross sections can be reproduced by introducing multi-step processes involving excitations in ¹²C and ¹³C without introducing spin-dependent interactions. Received: 2 August 1999 / Revised version: 3 February 2000     

Do gravitational waves carry energy‐momentum and angular momentum?

We show that gravitational waves which possess a non‐vanishing Riemann tensor R_iklm ≠ 0 always carry energy‐momentum and angular momentum. Our proof uses canonical superenergy and supermomentum tensors for the gravitational field.     

Differential Line Broadening in the Presence of Quadrupolar–CSA Interference

The formalism for calculating the lineshape of a spin 1/2J-coupled to a high-spin nucleus undergoing quadrupolar and chemical shift anisotropy (CSA) relaxations is derived in the case where the tensors of both interactions are noncoincident and nonaxial. The expressions show that the CSA–quadrupolar interference term which is responsible for the asymmetry of lines involves a term depending on tensorial parameters. The effect of this term on the lineshapes is discussed with respect to three cases, namely coincident–axially symmetric, noncoincident–axially symmetric, and general noncoincident quadrupolar and CSA tensors. These cases are considered in the analysis of the lineshape of the¹H-decoupled spectra of the³¹P nucleusJ-coupled to the⁵⁹Co nucleus encountered in the tetrahedral cluster HFeCo₃(CO)₁₁PPh₂H.     

E.S.R. spectra of SiF3 radicals produced in a single crystal of SiF4

E.S.R. spectra of a γ-irradiated single crystal of SiF₄ were investigated. The spectra observed were attributed to SiF₃ radicals having ²⁸Si (I=0) and ²⁹Si (I=1/2) atoms. From the angular dependence of the spectral lines on rotation of the single crystal, hyperfine tensors were determined for three fluorine atoms and the ²⁹Si atom of the SiF₃ radical. The three fluorine atoms in the radical are equivalent, whereas the directions of their hyperfine tensors are different from one another owing to the pyramidal structure of the radical. In addition to the hyperfine analysis, the analysis of the superhyperfine structure due to neighbouring fluorine atoms gave information on the orientation of the radicals in the crystal and the mechanism of radical formation. The structure of the radical is discussed in comparison with that of the CF₃ radical.     

On explicit free field realization of current algebras

We construct the explicit free field representations of the current algebras so(2n)_k, so(2n+1)_k and sp(2n)_k for a generic positive integer n and an arbitrary level k. The corresponding energy–momentum tensors and screening currents of the first kind are also given in terms of free fields.     

Study of orientation characteristics of the 11B(5/2?, 4.445 MeV) nucleus in the 13C(d, αγ)11B reaction at Ed = 15.3 MeV

Double differential cross sections of the ¹³C(d, αγ)¹¹B(5/2⁻, 4.445 MeV) reaction were measured at E _d = 15.3 MeV. Angular dependences of the even components of density matrix spin tensors, magnetic sublevel populations, and components of multipole moment orientation tensors of the ¹¹B nucleus in this state were obtained. Experimental results were compared with theoretical data calculated under the assumption of different reaction mechanisms: deuteron pickup, heavy particle stripping, two-step mechanism of sequential cluster transfer, and compound nucleus formation.     

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field Q on a supermanifold M is called homological, if Q ² = 0. The operator of Lie derivative L _Q makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator L _Q and are represented by Q-invariant tensors made up of the homological vector field and a symmetric connection on M by means of the algebraic tensor operations and covariant differentiation.     

Einstein tensor and generalizations of Birkhoff's theorem

The Einstein tensors of metrics having a 3-parameter group of (global) isometries with 2-dimensional non-null orbitsG ₃(2,s/t) are studied in order to obtainalgebraic conditions guaranteeing an additional normal Killing vector. It is shown that Einstein spaces withG ₃(2,s/t) allow aG ₄. A critical review of some of the literature on Birkhoff's theorem and its generalizations is given.This work was started at the Department of Physics, Temple University, Philadelphia, Pa., and supported there by the Aerospace Research Laboratories of the Office of Aerospace Research, U.S.A.F.     

Anisotropic chemical shifts of 59Co in Co(NH3)6Cl

The chemical shift tensors of ⁵⁹Co in a single crystal of Co(NH₃)₆Cl₃ at the five nonequivalent cobalt sites were measured at room temperature. The principal axes of these tensors almost coincide with the principal directions of the electric field gradient tensors. Assuming that these directions are parallel to the symmetry axes about which the different Co(NH₃)₆³⁺ octahedra reorientate, possible reorientation axes of the octahedra at the five nonequivalent sites are proposed.