Crystal symmetry groups and their representations I lattice extensions of point groups in R2 and R3

We classify all point groups in R² and R³ according to their decomposition into m-fold semidirect products of cyclic groups. This product is characterized completely by means of m?1 matrices α⁽¹⁾,…, α^(m?1) with non-negative integer elements. For any arithmetic class [Pⁿ_i] in Rⁿ (n=2, 3) we define the set D(Pⁿ_i, T) of representative translations with the following properties: (a) The elements of D(Pⁿ_i, T) are in one-to-one correspondence with defining relations of Pⁿ_i; (b) D(Pⁿ_i, T) determines uniquely all equivalence classes of the factors of Pⁿ_i over T.We present: (i) a construction of all sets D(Pⁿ_i, T) in R² and R³; (ii) a construction of all algebraic classes of space groups and a derivation of all group operations which transform one algebraic class into another; (iii) a derivation of formulas expressing the multiplication table of any space group by means of the representative translations and matrices α⁽¹⁾,…, α^(m?1). The results (i), (ii) and (iii) are obtained without making use of the notion of non-primitive translations.     

The geometric SO(3) × D model

The geometric SO(3) × D model is formulated as a computationally viable submodel of the algebraic sp(3, R) model containing the Sp (1, R) model of Arickx et al. as a submodel. It is basically a scheme for constructing a space of states (like the space of Slater determinants) which can be employed either directly, for constrained variational quantal dynamics of the Hartree-Fock, RPA and TDHF type, or to construct a shell-model basis for the diagonalization of a microscopic hamiltonian. The results of HF calculations, restricted to SO(3) × D surfaces, are reported for the nuclei ⁴He, ⁸Be, ¹⁶O and ²⁰Ne.     

R-matrix analysis of the 12C(α, γ)16O reaction

The problems involved in extrapolating the ¹²C(α, γ)¹⁶O cross section to stellar energies are investigated. In the R-matrix formalism used, the extrapolated cross section is shown to be approximately independent of the interaction radius and the boundary condition number. Simultaneously good fits to recent ¹²C(α, γ)¹⁶O data and the ¹²C + α elastic scattering p-wave phase shift are shown.     

Poincaré subalgebras in so(4, 2) and sl(5, R)

Injective homomorphisms ε which map the Poincaré Lie algebra P into so(4, 2)=C are considered. Two mappings ε₁ and ε₂ are said to be Int-equivalent iff there exists an inner automorphism ρⁱⁿ_c> of C such that ε₁=ρⁱⁿ_cε₂. The set of all ε splits into four Int- equivalence classes. Mappings in different classes are equivalent with respect to non-inner automorphisms of C. A corresponding theorem holds for homomorphisms of P into sl (5, R). These algebraic properties give strong limitations for those integrable representations of P which can be found by restriction of integrable representations of so(4, 2) and of sl(5, R) to ε(P). Physical applications of the results are discussed.     

Total cross sections for 4,6He + 28Si reactions measured at 10–28 MeV/A

New results on the energy dependence of the total cross section (σ _R) for ⁶He scattering on ²⁸Si in the incident energy range 10–28 MeV/A are obtained. The α-particle-production cross sections for the ²⁸Si(⁶He, ⁴He)X channel are measured as well. The secondary beam of ⁶He with an intensity of up to 5×10⁴ particle/s was generated by bombarding a thick beryllium target with ～32-MeV/A ⁷Li ions. In the energy region below 17 MeV/A, σ _R increases sharply. The experimental dependences of the total cross sections are compared with the results of σ _R calculations using the double-folding potential within the optical model. The energy dependence of σ _R for ⁶He differs from that for the neighboring nuclei, which can be associated with the structural features of the former nucleus. The energy spectra of α particles produced in the ⁶He interactions with silicon indicate two mechanisms of their production: transfer reaction and ⁶He breakup in the field of the ²⁸Si nucleus.     

s- and p-wave neutron spectroscopy. Xe. Intermediate structure in 90Y and the generalized R-matrix theory

The resonance structure observed in the ⁸⁹Y(n, n)⁸⁹Y total cross section measurements in the range of 0.3 to 1.2 MeV incident energy was investigated using the generalized R-matrix theory of nuclear reactions and the doorway interpretation of intermediate structure. The energies and wave functions of the doorway resonances were calculated in a 2-particle and 3p-1h basis of the shell model. The model space and the parameters of the model calculation chosen were consistent with other shell model calculations in the mass-90 region. Several strong p-wave doorways with J^π = 0⁺, 1⁺, and 2⁺ were predicted by the model in the energy range studied. This is due to proximity of p-wave giant resonance. The escape widths Γ^↑ and the spreading widths Γ^↓ for these states were evaluated using the model wave functions and the R-matrix formalism. The calculated energy dependence of the total cross section shows that most of the predicted doorways are in general agreement with the observed anamolies with similar relative strength. More significantly, the underlying p-wave gross structure representing a grand average is of very similar shape in both theory and experiment. As expected in the mass 90-region, the s- and d-wave doorways contribute less significantly to the calculated resonance structure.     

Theoretical calculation of CH3Br/N2-broadening coefficients at various temperatures

The N₂-broadening coefficients of CH₃⁷⁹Br and CH₃⁸¹Br have been calculated for transitions belonging to the ^PP-, ^PQ-, ^PR-, ^RP-, ^RQ- and ^RR-branches of the ν₆ band near 10 μm, using a semiclassical impact theory. The intermolecular potential used, includes in addition to the overwhelming electrostatic interactions, induction and dispersion energy contributions, which are significant only for low J transitions with K approaching or equal to J. Comparisons have been performed with the extensive set of previous measurements at room temperature [4]. The theoretical results are in satisfactory agreement with the experimental data, and the J and K dependences are reasonably well reproduced. From calculations at 200, 230, 260 and 296 K, the temperature exponent of the collisional broadenings derived from a simple power law has been determined for each transition of each sub-branch and compared with a recent experimental evaluation [5]. Finally the N₂-broadening coefficients calculated at 296 K and their temperature exponents are given as supplementary materials of this paper for atmospheric applications and databases.     

Algebraic aspects of crystallography

Taking into account the fact that space groups are groups of transformations of Euclideann-dimensional space, non-equivalent systems of non-primitive translations are defined. They can be brought into one-to-one correspondence with the elements of the groupH ¹ (K, R ⁿ /Z ⁿ ) or with those of the groupH ¹ (K, Z ⁿ /kZ ⁿ )/H ¹ (K, Z ⁿ ). (K is a point group of orderk.) The consistency of these findings with the results of Part I is given by the isomorphisms $$H^2 (K,Z^n ) \cong H^1 (K,R^n /Z^n ) \cong H^1 (K,Z^n /kZ^n )/H^1 (K,Z^n ).$$ Theorems are proved giving the conditions for cohomology groupsH ^q (K, A) to be zero. These conditions are fulfilled in particular ifA=R ⁿ andK is a subgroup ofGL (n, R) that either is compact (thenq>0) or has a finite normal subgroup leaving no element ofR ⁿ invariant (thenq≧0). This implies that the affine, the Euclidean and the inhomogeneous Lorentz groups are the only extensions ofR ⁿ by the corresponding homogeneous groups. By way of illustration, the theory of this paper is applied to two 2-dimensional space groups.     

Elastic scattering of 56 MeV polarized deuterons on 4He

The differential cross section and the vector and tensor analyzing powers A_y, A_xx, A_yy and A_xz were measured for the d-⁴He elastic scattering at 56 MeV. The measurement of A_xz was performed using a deuteron beam polarized in the horizontal plane. An optical-model analysis of the experimental data was carried out. The magnitude of the tensor analyzing powers could not be reproduced without the tensor potential. By including the T_R type tensor potential, the optical-model calculations give a reasonable reproduction of the experimental data at θ_c.m. < 120°. The obtained T_R tensor potential was much stronger than that predicted by the folding model. The strength of the real T_R potential was roughly in accordance with that obtained from the optical-model analysis of d-⁴He elastic scattering at 20.2 MeV.     

Proton yields from quasielastic pion scattering on 27Al and 208Pb

A singles measurement of the quasielastic scattering of 255 MeV pions by ²⁷Al and ²⁰⁸Pb has been made. The measured values for the ratio R of π⁺ and π^? induced cross section, averaged over all measured angles of 2.2 ± 0.2 and 1.5 ± 0.1 for ²⁷Al and ²⁰⁸Pb respectively are substantially below the classical impulse approximation value of 11. A semiclassical nucleon charge exchange model is described for singles and coincidence measurements and is compared with the measured proton spectra. The comparison indicates that the singles proton spectra are dominated by events other than the quasielastic process whereas the coincidence data reported earlier and compared with the present calculations support the quasielastic interpretation. A strongly decreasing dependence of R on A for coincidence measurements may be interpreted as supporting the view that recoil nucleon charge exchange plays an important role in the quasielastic knockout process.     

π-N scattering in the operator droplet model

π-N scattering is treated within the framework of the operator droplet model. We have incorporated spin dependent effects, and we calculate the differential cross section, polarization and spin rotation parameters, R and A, at infinite energy. Considerable structures in R and A have been found in contrast with the prediction of the present favourite idea of S-channel helicity conservation for high-energy scattering. The original parametrization of the fit of Byers and Yang for π⁻p → π^On is used to calculate energy dependent amplitudes in elastic πN scattering. Without any new parameters, we have obtained a good fit for total cross sections, polarization and a consistent fit for the R- and A-parameters. Further possible developments are indicated.     

N2-broadening coefficients in the ν2 and ν4 bands of PH3

N₂-broadening coefficients are measured for 61 transitions of PH₃ in the ^QR branch of the ν₂ band and the ^PP, ^RP, ^SP, and ^PQ branches of the ν₄ band, using a tunable diode-laser spectrometer. The recorded lines with J values ranging from 1 to 16 and K from 0 to 11 are located between 1008 and 1106 cm⁻¹. The collisional widths are determined by fitting each spectral line with a Voigt profile, a Rautian profile, and a speed-dependent Rautian profile. The latter models provide larger broadening coefficients than the Voigt model. These coefficients have also been calculated on the basis of a semiclassical model of interacting linear molecules by considering an atom-atom Lennard-Jones potential in addition to the electrostatic contributions. The theoretical results are in good agreement with the experimental data and reproduce the J dependence of the broadenings, but their decrease at high J values is overestimated for the ^QR (J, K) transitions.     

A shock-tube measurement of the SiO(E1Σ+ − X1Σ+) transition moment

The sum of the squares of the electronic transition moments, Σ|R_e|², for the E¹Σ⁺ ?X¹Σ⁺ band system of SiO has been determined from absorption measurements conducted in the reflected-shock region of a shock tube. The test gas was produced by shock-heating a mixture of SiCl₄, N₂O and Ar, and the spectra were recorded photographically in the 150–230 nm wavelength range. The values of the Σ|R_e|² were determined by comparing the measured absorption spectra with those produced by a line-be-line synthetic spectrum calculation. The value of the Σ|R_e|² so deduced at an r-centroid value of 3.0 Bohr was 0.86±0.10 atomic units.     

Vibration-rotation matrix elements for diatomic molecules; vibration-rotation interaction functions Fv′v (m) for CO

General expressions for the fourth-order vibrational matrix elements are obtained for the transitions v → v′ (v ? v′ v + 4) using a sixth-power internuclear potential and a quartic dipole moment function. The results for the dipole moment coefficients M₀ to M₄ of CO and for some transition moments R^v′_v deduced from a calculation including successively second, third and fourth order perturbation theory are compared.Using the rotational potential function expanded through the cubic term as a perturbation, we have also obtained general expressions for the vibration-rotation matrix elements. The vibration-rotation interaction functions F^v′_v (m) are calculated for the transitions v → v′ (v = 0, 10, 20 and v ? v′ ? v + 4) of CO and the coefficients C, D, E and G of the quartic polynomials fitting these functions are deduced. Taking account of uncertainties in the matrix elements R⁰₀ to R⁴₀, an error estimate for the coefficients of F²₀(m) and F³₀(m) is given.     

Probing into the effect of Auger recombination mechanism on zero bias resistance-area product in In1−xGaxAs detector

D* (Detectivity), an important figure of merit for photodetectors, is limited by zero bias resistance-area product (R₀A). R₀A is determined by Auger recombination mechanism, depending on the composition, temperature, carrier concentration and other parameters of the photodetectors. To investigate R₀A of In_1−xGa_xAs infrared photodetectors, in this paper, theoretical analysis of Auger recombination mechanism was carried out in the room temperature, by taking CCCH, CHHL and CHHS into account. The calculated results show that there are significant influences on R₀A for various parameters in both p- and n-type regions of the devices. With carrier concentration around 10¹⁷ to 10¹⁸ cm⁻³, R₀A of 10⁸ Ω cm² (n-region) and 10⁶ Ω cm² (p-region) are obtained for x=0.47, when thickness and surface recombination velocity of the sample are 5 μm and 100 m/s, respectively.     

Magnetic charge can locally stabilize Kaluza-Klein bubbles

We construct a new 2-parameter family of static topological solitons in 5D minimal supergravity which are endowed with magnetic charge and mass. The solitons are asymptotically R⁴×S¹, where the radius of the S¹ has a lower bound Rs?Rmin. Setting up initial data on a Cauchy slice at a moment of time symmetry, we demonstrate that if Rs>Rmin these solitons correspond to a perturbatively stable “small” static bubble as well as an unstable “large” static bubble, whereas if Rs<Rmin there are no static bubbles. The energetics and thermodynamics of the magnetic black string are then discussed and it is shown that the locally stable bubble is the end point of a phase transition for an appropriate range of black string parameters.     

A study of N by means of the B(α, α)B reaction

The ¹¹B(α, α)¹¹B cross section has been measured as a function of energy and angle for bombarding energies from 4 to 8 MeV. Many resonances are observed. The data from 4 to 5 MeV were fitted using an R-matrix formalism with background phase shifts evolved from an optical-model potential. The three levels in ¹⁵N resulting from this analysis fit into the APCETS scheme for ¹⁵N.     

Is the threshold enhancement in γγ → ϱ0ϱ0 evidence for an isoscalar, JPC = 0++ exotic state?

The threshold enhancement in the cross section γγ → ?⁰?⁰, including the lower energy point reported by the CELLO group, and the observed angular distributions have been interpreted in terms of an isoscalar resonance with J^PC = 0⁺⁺, mass 1450–1550 MeV and width 100–150 MeV.     

Lie Superalgebras Based on 𝔤𝔩n Associated to the Adjoint Representation, and Invariant Geometric Structures Defined on Them

Finite-dimensional real and complex Lie superalgebras whose underlying Lie algebra is ????_n and whose odd module is ????_n itself under the adjoint representation are classified up to isomorphism. It is shown that for n≥3 there are one-parameter families of nonisomorphic such Lie superalgebras, plus another set of finitely many different isomorphism classes. For n=2 there are 10 different isomorphism classes over the real field, and 8 different over the complex numbers. For n=1 there are 2 different isomorphism classes over either ground field. Representatives on each isomorphism class are given, and their automorphism groups are determined. The question as to which representatives admit ?₂-graded, ad-invariant geometric structures (of orthogonal or symplectic type) is also addressed, and a precise list of which of such geometric structures can be defined on each isomorphism class is given. In particular, it is shown that ?₂-homogeneous, orthogonal, ad-invariant geometric structures must be odd. The case of ????₂ over the real field is further analyzed in order to determine for which of the equivalence classes that admit such a structure, that structure can be induced by an underlying Minkowski metric on the 4-dimensional (nongraded) ????₂.     

Helium- and argon-broadening coefficients of phosphine lines in the ν2 and ν4 bands

Pressure broadening of phosphine lines by helium and argon at room temperature has been experimentally investigated by high-resolution diode-laser spectroscopy. The broadening coefficients are measured for 38 transitions of PH₃ in the ^QR branch of the ν₂ band and in the ^PP and ^RP branches of the ν₄ band. The recorded lines with J values ranging from 3 to 14 and K from 0 to 10 are located between 1062 and 1094 cm⁻¹. The retrieval of the collisional widths is carried out by fitting each spectral line with a Voigt profile, a Rautian profile and a speed-dependent Rautian profile. The latter model provides larger broadening coefficients than the Voigt model. They are also calculated on the basis of a semiclassical model involving the atom-atom Lennard-Jones potential. The theoretical results are in reasonable agreement with the experimental data and reproduce the J and K dependencies of the broadenings.