Relations between the electromagnetic form factors of the 0−- and 1−-mesons from the collinear groupsSU(3)⊗SU(3)⊗U(1) andSU(6) W

The relations between the electromagnetic matrix elements of the 18 low lying pseudoscalar and vector mesons predicted by the collinear groupsSU(3)?SU(3)?U(1) andSU(6) _W are derived. Using the hermiticity of the electromagnetic current operator, charge conjugation invariance and invariance under the full Lorentz group andSU(3)-symmetry separately, all electromagnetic matrix elements of these mesons are described by seven independent real form factors. After combining space-time with intrinsic properties of the particles this number is reduced to four by the minimal collinear groupSU(3) ?SU(3) ?U(1) and to three bySU(6) _W which involves more speculative assumptions. In the limit of low momentum transfer the predictions of both models become identical, depending on three real quantities. No disagreement with experiment has been found, as far as a comparison is possible.     

Predictions for the weak leptonic baryon form factors from the hybrid groupSU 3⊗SU 3⊗U 1

Relations between the form factors of the weak leptonic baryon interaction are obtained from invariance under the groupSU ₃?SU ₃?U ₁. The weak interaction operatorV _μ ?A _μ is assumed to behave like the corresponding components of representations35 ofSU ₆ with different parities. One gets the result that the well-known predictions of the staticSU ₆ theory, viz., pureF-coupling for the vector- and 3D+2F-coupling for the axial-vector part, are only valid in the limit of vanishing momentum transfer. In the same limit the resultC _A /C _V =?\(\tfrac{5}{3}\) is obtained if one further assumes thatV ₀ and\(\vec A\) belong to thesame representation35 ofSU ₆.     

Predictions for the electromagnetic formfactors of baryons from hybrid groups

The predictions of the hybrid groupSU ₃?SU ₃?U ₁ for the electromagnetic formfactors of baryons are studied in detail. Surprisingly, this small group leads already to strong restrictions. For example, it gives for the ratio of the magnetic formfactors of neutron and proton ?2/3 for all values of the momentum transfer variable. The results of the powerful (SU ₆) _W hybrid group which involves more speculative assumptions are also quoted. In addition, the theoretical foundation of the hybrid groups is discussed in the paper.     

Spectral Measures Associated to Rank two Lie Groups and Finite Subgroups of {GL(2,\mathbb{Z})}

Spectral measures for fundamental representations of the rank two Lie groups SU(3), Sp(2) and G₂ have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus \({\mathbb{T}^2}\) and are invariant under an action of the corresponding Weyl group, which is a subgroup of \({GL(2,\mathbb{Z})}\). Here we consider spectral measures invariant under an action of the other finite subgroups of \({GL(2,\mathbb{Z})}\). These spectral measures are all associated with fundamental representations of other rank two Lie groups, namely \({\mathbb{T}^2=U(1) \times U(1)}\), \({U(1) \times SU(2)}\), U(2), \({SU(2) \times SU(2)}\), SO(4) and PSU(3).     

TheSU(3) structure of the 3−/2 baryons

TheSU(3) structure of the 3^?/2 baryonsN ^*(1525),Ξ ^*(1820),Y ₁ ^* (1660),Y ₁ ^* (1715),Y ₀ ^* (1520) andY ₀ ^* (1700) is investigated on the hypothesis that they form a decupletoctet-singlet combination. Their branching ratios seem to be consistent with this picture. Mixing angles are determined, andΞ ^*(1820) is deduced to be mainly decuplet.     

The proton-antiproton annihilation at rest as predicted by theSU 6-covariantS-matrix formalism and byW-spin analysis

The proton-antiproton annihilation at rest into two mesons is calculated by two methods: (i) on the basis of theSU ₆ -covariantS-matrix formalism recently proposed byRothleitner andStech, and (ii) by assuming the additional invariance under the group (SU ₆) _W proposed byLipkin andMeshkov. In the first case the calculations give only few relations between transition amplitudes, whereas the larger symmetry group (SU ₆) _W actually determines all amplitudes up to a common factor. While agreement with experiment seems to be poor in some cases for the (SU ₆) _W , so far there are no experimental data available to test the relations of case (i).     

Optical and x-ray diffraction studies of the symmetry of distorted phases of the (NH<Subscript>4</Subscript>)<Subscript>3</Subscript>ZrF<Subscript>7</Subscript> crystal

(NH₄)₃ZrF₇ single crystals were grown, and polarization-optical and x-ray diffraction studies were performed on powders and crystalline plates of various cuts over a wide temperature range. Phase transitions are revealed at temperatures T _1↑ = 280 K, T _2↑ = 279.6 K, T _3↑ = 260–265 K, and T _4↑ = 238 K on heating and at T _1↓ = 280 K, T _2↓ = 269–270 K, T _3↓ = 246 K, and T _4↓ = 235 K on cooling. The sequence of changes in symmetry is established to be as follows: O _h ⁵ (Z = 4) ? D _2h ²⁵ (Z = 2) ? C _2h ³ (Z = 2) ? C _i ¹ (Z = 108) ? monoclinic²(Z = 216).     

Meson-baryon scattering with SU (3) invariant interactions

The static model invariant under SU₃ is discussed. The baryons and mesons are assigned according to the “eightfold way”, and the Low equation for the scattering matrix is derived. The scattering matric has been diagonalized for arbitrary mixing ofF- andD-type coupling and the crossing matrix has been calculated. To determine the mixing of the two couplings photoproduction cross sections have been calculated. From the comparison of theK ⁺ Λ andK ⁺ Σ0 production cross sections with experiment it follows that α=D/F=3.5. For this value of α the model predicts the 3/2 decublet resonance in very good agreement with the experimental situation.     

G2-Monopoles with Singularities (Examples)

G₂-Monopoles are solutions to gauge theoretical equations on G₂-manifolds. If the G₂-manifolds under consideration are compact, then any irreducible G₂-monopole must have singularities. It is then important to understand which kind of singularities G₂-monopoles can have. We give examples (in the noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building blocks in a gluing construction aimed at constructing non-Abelian ones.     

Raman scattering spectroscopy of inclusions of carbon in Al<Subscript>2</Subscript>O<Subscript>3</Subscript> films and its solid solutions with HfO<Subscript>2</Subscript>

We apply Raman scattering spectroscopy to study the nature of carbon inclusions in Al₂O₃ and (HfO₂) _x (Al₂O₃)_{1 ? x} films deposited using volatile complex compounds. Raman spectra of the films under investigation contain D and G vibrational modes, which indicate that carbon clusters of the sp ² configuration tend to form in the films. We estimate the size of clusters from the integrated intensity ratio I _D /I _G and find it to be in the range of 14–20 Å. The content of hydrogen in carbon clusters is calculated from the height of the photoluminescence pedestal and is found to vary from 14 to 30 at % depending on the regime of the film’s synthesis.     

Charge-exchange resonances and restoration of Wigner’s supersymmetry in heavy and superheavy nuclei

Various facets of the question of whether Wigner’s supersymmetry [SU(4) symmetry] may be restored in heavy and superheavy nuclei are analyzed on the basis of a comparison of the results of calculations with experimental data. The energy difference between the giant Gamow–Teller resonance and the analog resonance (the difference of E _G and E _A) according to calculations based on the theory of finite Fermi systems is presented for the case of 33 nuclei for which experimental data are available. The calculated difference ΔE _G–A of E _G and E _A tends to zero in heavier nuclei, showing evidence of the restoration of Wigner’s SU(4) symmetry. Also, the isotopic dependence of the Coulomb energy difference between neighboring isobaric nuclei is analyzed within the SU(4) approach for more than 400 nuclei in the mass-number range of A = 5–244. The restoration of Wigner’s SU(4) symmetry in heavy nuclei is confirmed. It is shown that the restoration of SU(4) symmetry is compatible with the possible existence of the stability island in the region of superheavy nuclei.     

Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II)

Let M be a six dimensional manifold, endowed with a cohomogeneity one action of G = SU₂ × SU₂, and \({M_{\rm reg} \subset M}\) its subset of regular points. We show that M _reg admits a smooth, 2-parameter family of G-invariant, non-isometric strict nearly Kähler structures and that a 1-parameter subfamily of such structures smoothly extends over a singular orbit of type S ³. This determines a new class of examples of nearly Kähler structures on T S ³.     

Zur Theorie der infinitesimalen Holonomiegruppe in der Allgemeinen Relativitätstheorie

Starting from the infinitesimal holonomy groupH _i of aV ⁴, (+++?) the spinholonomy group\(\tilde H_i \equiv \bar \sigma ^1 (H_i )\) defined by the covering isomorphism\(\sigma :G \to L_ + ^ \uparrow \) is introduced. In Einstein-spaces we may replace its real Lie-algebra by a complex one. With the complex calculus we may reproduce the results ofSchell, Goldberg andKerr with very much simplified proofs. A theorem on non-empty Einstein-spaces is given. In part 4 we prove a theorem on the connection between theH _i -behaviour of a vector (spinor) and its covariant derivative in aV ⁴. With its help we get in a simple manner the metiics of aV ⁴ with givenH _i and Dim (H _i ) <6; our results agree with those given byGoldberg andKerr, Cahen andDebever. Finally we make some new statements on imperfect holonomy groups.     

Comparison of theK ± p scattering data with the forward dispersion relations

The dispersion relations forK ^± p forward scattering were evaluated in order to test the compatibility with the experimental data and to obtain an estimation of the effective coupling constantG ^2/4π=(G _Λ ² +G _Σ ² )/4π.     

Die Winkelverteilung der Protonen der Reaktion B11 (d,p 0,1 B12 im Energiebereich von 0,7 ... 1,7 MeV

Angular distributions of the protons in this low-energy region have been measured by only few groups for obvious reasons: the energy of the emitted protons is very low. It will be shown that by using non-supported targets of B¹¹ with very little oxygen contamination the angular distributions can be obtained between approximately 5 and 165 degrees (cm.,p ₀-group) and between 20 and 165 degrees (cm.,p ₁) with high accuracy. The Butler-fits givel=1,r _B =5.2 fm forp ₀ andl=1,r _B =4.7 fm forp ₁. All angular distributions seem to exhibit strong participation of non-stripping mechanism.     

Zur Frage des Grundzustandes im Tb I-Spektrum

Analysis of the optical spectrum of Tb I confirms the conclusion reached byBender, Penselin andSchlüpmann from atomic beam magnetic resonance observations, that the lowest multiplet in the 4f ⁸5d6s ² configuration is⁸ G. The lowest level, however, is not⁸ G _15/2, but⁸ G _13/2, which is the true ground level of the Tb atom, provided the determination of the relative position of 4f ⁹6s ^{2 6} H _15/2 with respect to 4f ⁸5d6s ^{2 8} G _15/2 byBender c. s. is correct.     

<Emphasis Type="Italic">SU</Emphasis>(2|1) supersymmetric mechanics as a deformation of <Emphasis Type="Italic">N</Emphasis> = 4 mechanics

We give a brief review of SU(2|1) supersymmetric quantum mechanics based on the worldline realizations of the supergroup SU(2|1) in the appropriate N = 4, d = 1 superspaces. The corresponding SU(2|1) models are deformations of standard N = 4, d = 1 models by a mass parameter m.     

Superconducting properties of superlattices containing ferromagnetic layers

Using the microscopic theory formulated by de Gennes and extended by Takahashi and Tachiki, we calculate the transition temperatureT _c and the pair functionF for the superlattices consisting of superconducting and ferromagnetic layers. Superconducting layers. (s) and ferromagnetic layers (f) are modeled byV _s ≠0 andI _m,s =0 andV _f =0 andI _m,f ≠0, whereV _s .(V _f ) is the BCS coupling constant andI _m,s (I _m,f ) is the molecular field fors (f) layers.     

The structure of a vortex line and the lower critical field in superconducting alloys

The structure of an isolated vortex line, and the lower critical fieldH _c 1, is calculated by means of the generalized Ginzburg-Landau (GL) theory for arbitrary values of the GL-parameterk(≧1/√2) and the mean free pathl at temperaturesT in the vicinity ofT _c . The free energy functional including the corrections of order [1?(T/T _c )] to the GL-functional is derived exactly. The corresponding Euler-Lagrange equations determining the zero-order (GL) contributions and the corrections of order [1?(T/T _c )] to the order parameter,f(r), and the superfluid velocity,v(r), have been solved numerically. The shapes of the first-order corrections off(r), v(r), and the magnetic field,h(r) are found to depend markedly, for a given value ofκ, on a second parameter,α=0.882(ξ ₀ /l) (whereξ ₀ is theBCS-coherence-distance). The deviations from the GL-solutions become largest forh(r) at parameter valuesk≈ 1 andα ≈ 0(the deviation ofh(0) is about 6% atT=0.9T _c forκ=1 andα=0). The ratioH _c1/H _c (where the thermodynamic criticalH _c has the BCS-temperature-dependence) is found to increase slightly in the “clean” limit (α=0), and to decrease slightly in the “dirty” limit (α=∞) asT decreases (the variation ofH _c 1/H _c is always less than 3% for arbitrary values ofκ andα asT decreases fromT _c to 0.9T _c ).     

Quantenstatistik eines Gases mit verschiedener Bahn- und Spintemperatur

A system of particles with spin in a magnetic field may possess an orbital temperatureT _o different from the spin temperatureT _s (?0), if it is possible to neglect the energetic interaction between the orbital and the spin system. The calculation of the quantum statistical most probable distribution of identical independent particles on the orbital and spin energy levels yields the introduction of three Lagrange multipliers—according to the fact that the orbital and the spin energy and the number of particles are fixed—representing the orbital and spin temperature and a generalizedPlanck's “characteristic function”. Apart from the Boltzmann-approximation being valid in the case of small spin values forT _o ?T _e (T _e =customary degeneration temperature) and arbitraryT _s ?0, the distributions and the orbital and the spin energy depend onboth the temperaturesT _o andT _s coming from the principle of exclusion forFermi resp.Bose particles. The equations of state are discussed. There are four heat capacities, which possess characteristic peaks. In stead of the well-known temperature independence of the paramagnetism of degenerated conducting electrons one obtains χ～T _o /T _s . The behaviour of the Einstein-condensation of aBose gas is considered.