On the multiplicities of irreducible representations of U(n) in some particular cases

Consider the tensor product of an arbitrary irreducible representation of U(n) of signature α with its contragredient. It is shown that the irreducible representation of signature (1, 0, 0,…, 0, -1) occurs in this product with multiplicity equal to the number of distinct components of signature α diminished by one.     

Canonical realizations of the lie algebras gl(n, R) and sl(n, R) I. Formulae and classification

The generators of the Lie algebra of the general linear group GL(n, R) and of the special linear group SL(n, R) are, recurrently, expressed through polynomials in the quantum canonical variables p_t and q_t. These realizations are skew-Hermitian, the Casimir operations are realized by constant multiples of identity element and, in dependence on the number of the canonical pairs used, they depend on d(d?1 for sl (n,R)), d = 2, …, n, free real parameters.     

Spinor-inverted solution to thirring model and its generalization to U(n) symmetry

Using the properties of massless free Fermi fields in (1-1) dimensions, it is shown that the spinor inverted form of Klaiber's operator solution to Thirring model is also a scale-invariant solution of the model. But unlike the former it admits a nonvanishing SU(n) current coupling in the generalization of the model to include U(n) symmetry. The value of this coupling constant is fixed and equals Dashen-Frishman number $\text{?4π}\text{(n + 1)}$. The general form of the 2m-point function is given and operates product expansions are exhibited in terms of composite local operators. Scale dimensions of all the bilinear and quadrilinear local operators with U(n) symmetry are computed and are found to depend on n. However, different parts of a composite local operator belonging to different irreducible U(n) representations have the same dimension.     

The matrix elements of irreducible representations of the group SO(n,1) in integral form

Using the method of induced representations, the matrix elements of unitary irreducible representations of the group SO(n,1) are found in integral form.     

Canonical Representations of Sp(1,n) Associated with Representations of Sp(1)

Canonical representations of Sp(1,n) associated with finite dimensional irreducible representations of Sp(1) are defined using vector-valued Berezin kernels. Their decomposition into irreducible representations is determined by decomposing the corresponding reproducing distributions in terms of positive definite trace spherical functions on Sp(1,n). The canonical representatons are also identified with the restriction to Sp(1,n) of certain maximal degenerate representations of SL(n+1,H). Received: 1 May 1998 / Accepted: 18 November 1998     

The multiphoton ionization spectrum of jet-cooled pyrimidine in the 3p Rydberg and B1 (π,n) states

Resonance-enhanced multiphoton ionization (REMPI) has been applied to study the n → 3p Rydberg transition of pyrimidine (jet-cooled sample and mass resolved spectrum). Only the one component, the 3p_z(B₂), appears in the (2 + 1) REMPI and the active vibrations are ν_6a = 622, ν₁ = 946, and ν_9a = 1116 cm⁻¹. The symmetry of the state was determined by polarization measurements (linear, circular polarization). The first (π^∗,n) ³B₁ triplet state appears as a one-photon resonance in the three-photon ionization process.     

Violation of the Mott-Ioffe-Regel limit: high-temperature resistivity of itinerant magnets Srn+1RunO3n+1 (n=2,3,∞) and CaRuO3

Sr_n+1Ru_nO_3n+1 represents a class of layered materials whose physical properties are a strong function of the number of Ru-O layers per unit cell, n. These materials along with the paramagnetic CaRuO₃ shares two major characteristics, namely, robust Fermi liquid behavior at low temperatures and anomalous transport behavior featured by linear temperature dependence of resistivity at high temperature. There is no crossover separating such two fundamentally different states. In this paper, we report results of our study that systematically addresses anisotropy and temperature dependence of resistivity as a function of n for the entire Sr_n+1Ru_nO_3n+1 series and CaRuO₃ and for a wide temperature range of It is found that the anomalous transport behavior correlates with magnetic susceptibility and that the scattering mechanism(s) driving the anomalous transport behavior becomes stronger with reducing dimensionality.     

Linear scaling of the interlayer relaxations of the vicinal Cu(p,p,p − 2) surfaces with the number of atom-rows in the terraces

We investigate the multilayer relaxation trends in the vicinal Cu(p,p,p − 2) surfaces employing the all-electron full-potential linearized augmented plane-wave method. Calculations are performed for the (3 3 1), (2 2 1), (5 5 3), (3 3 2), (7 7 5), and (4 4 3) surfaces, which have 3, 4, 5, 6, 7, and 8 atom-rows in the terrace, respectively. The following trends are identified: (i) The interlayer relaxations perpendicular to the surface scale almost linearly with the number of atom-rows in the terraces. (ii) The nearest-neighbor distances do not depend on the surface termination, but only on the local coordination. (iii) For Cu(p,p,p − 2) in which the topmost n surface layers have nearest-neighbor coordination smaller than the bulk Cu (calculated for the unrelaxed surfaces), the topmost (n − 1) interlayer spacings (d₁₂, … , d_n−1,n) contract compared with the unrelaxed spacing, while the nth interlayer spacing (d_n,n+1) expands. The next (n − 2) interlayer spacings (d_n+1,n+2, … , d_{2n−2,2n−1}) contract, while the interlayer spacing indicated by d_2n−1,2n expands. A similar rule was found for the relaxations parallel to the surfaces. These trends provide a better understanding of the atomic structure of vicinal Cu surfaces.     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

Electronic photodissociation spectra of the Arn-C4H2 (n=1-4) weakly bound cationic complexes

Electronic spectra of a series of weakly bound clusters consisting of argon (Ar_n, n=1-4) bound to the butadiyne cation, C₄H₂⁺, have been recorded in the visible range from 440 to 520 nm by photodissociation. The C₄H₂⁺ fragment signal was recorded as a function of the laser wavelength during excitation of the A←X electronic transition. The observed transitions were assigned to the band origin of the cationic complexes and to vibronic bands involving excitation of the ν₃ and ν₇ vibrational modes of the C₄H₂⁺ moiety, as well as combination bands of these modes. Comparison of the photodissociation spectra of the various clusters reveals a small blue shift, 25 cm⁻¹ of the band maxima relative to the corresponding transitions reported from gas phase spectra of the bare C₄H₂⁺ cation. The magnitude of the blue shift of each band increases with successive Ar solvation up to n=3. Furthermore, each band becomes increasingly broadened towards the red with the addition of Ar atoms due to an increasing number of unresolved transitions involving excited intermolecular modes.     

Density functional theory study of AunMn(n=1-8) clusters

Equilibrium geometries, relative stabilities, and magnetic properties of small Au_nMn (n=1-8) clusters have been investigated using density functional theory at the PW91P86 level. It is found that Mn atoms in the ground state Au_nMn isomers tend to occupy the most highly coordinated position and the lowest energy structure of Au_nMn clusters with even n is similar to that of pure Au_n+1 clusters, except for n=2. The substitution of Au atom in Au_n+1 cluster by a Mn atom improves the stability of the host clusters. Maximum peaks are observed for Au_nMn clusters at n=2, 4 on the size dependence of second-order energy differences and fragmentation energies, implying that the two clusters possess relatively higher stability. The HOMO-LUMO energy gaps of the ground state Au_nMn clusters show a pronounced odd-even oscillation with the number of Au atoms, and the energy gap of Au₂Mn cluster is the biggest among all the clusters. The magnetism calculations indicate that the total magnetic moment of Au_nMn cluster, which has a very large magnetic moment in comparison to the pure Au_n+1 cluster, is mainly localized on Mn atom.     

First-principles investigation of the surface structures of Cu(n, n−1, 0) (n=2, 3 and 4) stepped surfaces

Multilayer relaxation at high-index Cu(n, n−1, 0) (n=2, 3 and 4) stepped surfaces was determined by the first-principles pseudopotential plane wave (PPPW) method. For those surfaces that have n and n−1 atom-rows in terrace and subterrace, respectively, the topmost 2n−2 interlayer spacings contract, while the 2n−1st interlayer spacing expands. There is no similar rule found for the relaxations parallel to the surfaces. Compared with the bulk terminated structure, a thin compact layer, which consisted of the topmost 2n−1 atom layers and separated slightly from the underneath atom layers, makes the surface more flat after relaxation. The bond-lengths between the step edge (first layer) atom and its nearest-neighbors do not depend on the surface termination, but only on the local coordination.     

A systematic search for a realistic SU(n) tumbling gauge model

Within the framework of the dynamical symmetry breaking and the tumbling ideas, a systematic search was carried out in SU(n) groups with the ultimate aim of determining if a realistic and phenomenologically acceptable model exists which tumbles down to SU(3)_c ? U(1), or a suitable large group. To do so all the anomaly free and the asymptotically free fermion contents for any SU(n) were first determined. In order to have nontrivial tumbling the real and the pseudo-real representations have been eliminated, and the tumbling patterns of all the allowed complex ones in detail have been examined. No such realistic model has been found. These results combined with those of Srednicki's concerning the SO(4n + 2) and E₆ groups establish the fact that there cannot be any realistic tumbling gauge model within the context of the original tumbling hypotheses. Having thus established the need for a change of these hypotheses some suggestions and comment on various ways of remedying the problem are made.     

IRREDUCIBLE REPRESENTATIONS OF THE GRADED LIE ALGEBRA SU(m/n)

Using the definition of irreducible tensor operators of the Lie group SU(n), we ive a method of calculating the irreducible representations of the graded Lie algebra U(m/n). We also give some simple examples of irreducible representations of U(3/2) and SU(m/n).     

Investigations on field desorption products from silver surfaces by mass spectrometry

Field evaporation of silver and field desorption of silver surface compounds were investigated by analysing positive ions with a mass spectrometer. In particular, the well known adsorption states of oxygen, and further the interactions of H₂O, NH₃, H₂, CO and CH₄ were measured in the field ion mass spectrometer under steady state fields of > 0.1 V/Å with a sensitivity of < 0.1 ions s^?1 and at temperatures between 80 °K and 425 °K. Although oxygen is usually chemisorbed at Ag surfaces, no AgO⁺, AgO⁺₂ or other Ag-O compounds could be detected as positive ions, Ag⁺ and O₂⁺ are the only observed ions at best image fields in oxygen up to fields of field evaporation of Ag⁺(≈ 2.2 V/Å). Even after the actual adsorption of oxygen with zero-field (6 × 10⁵ Langmuir at 10^?3 Torr) at 323 °K and 473 °K and subsequent application of the desorption field at 210°K no silver-oxygen compounds were found in positive ionic form. Small quantities of AgO⁺ and AgO⁺₂ were only formed — besides Ag(H₂O)_x⁺ complexes — if atomic oxygen was supplied by the field induced dissociation of water.Gases which do not adsorb on silver under zero-field conditions (H₂, CO, CH₄, N₂) yield the ions Ag(H₂)_n, Ag(CO)_n⁺, n=1, 2; AgCH₄⁺, AgN₂⁺. The situation with H₂O and NH₃ is more complicated: Molecular ions [Ag(H₂O)_n]⁺·mH₂O, n=1,…, 4, m=1,…, 8 and [Ag(NH₃)_n]⁺·mNH₃, n=1, 2, m=1,…, 6 are found besides Ag⁺.From the temperature and field dependence conclusions are drawn about the mechanisms of evaporation and formation of ionic surface complexes. The activation energies of evaporation of Ag⁺ are found to depend on the square root of the field strength. In general, the generation of surface compounds can be described by field induced reactions rather than usual gas adsorption.     

Structural, electronic and magnetic properties of ConRh (n=1-8) clusters from density functional calculations

The geometries, stabilities, electronic and magnetic properties of Co_nRh (n=1-8) clusters have been investigated systematically within the framework of the generalized gradient approximation density-functional theory. The results indicate that the most stable structures of Co_nRh (n=1-8) clusters are all similar to those of corresponding Co_n+1 clusters. Maximum peaks of second-order energy difference are found at n=2, 4 and 7, indicating that these clusters possess relatively higher stability than their respective neighbors. The magnetism of the ground state of alloy clusters all displays ferromagnetic coupling except for Co₃Rh. In addition, the doped Rh atom exhibits an important influence on the magnetism of alloy clusters, e.g., compared with corresponding pure Co_n clusters, the local moment of Co atom is noticeably enhanced in Co_nRh alloy clusters at n=1, 2, 5, 6, 7 and 8, while reduced at n=3 and 4. Further analysis based on the average bond length, the charge transfer and the spin polarization has been made to clarify the different magnetic responses to Rh doping.     

Formation of Sim and SimCn clusters by C60 sputtering of Si

The secondary ion mass spectrum of silicon sputtered by high energy C₆₀⁺ ions in sputter equilibrium is found to be dominated by Si clusters and we report the relative yields of Si_m⁺ (1 ≤ m ≤ 15) and various Si_mC_n⁺ clusters (1 ≤ m ≤ 11 for n = 1; 1 ≤ m ≤ 6 for n = 2; 1 ≤ m ≤ 4 for n = 3). The yields of Si_m⁺ clusters up to Si₇⁺ are significant (between 0.1 and 0.6 of the Si⁺ yield) with even numbered clusters Si₄⁺ and Si₆⁺ having the highest probability of formation. The abundances of cluster ions between Si₈⁺ and Si₁₁⁺ are still significant (>1% relative to Si⁺) but drop by a factor of ∼100 between Si₁₁⁺ and Si₁₃⁺. The probability of formation of clusters Si₁₃⁺-Si₁₅⁺ is approximately constant at ∼5 × 10⁻⁴ relative to Si⁺ and rising a little for Si₁₅⁺, but clusters beyond Si₁₅ are not detected (Si_m≥16⁺/Si⁺ < 1 × 10⁻⁴). The probability of formation of Si_m⁺ and Si_mC_n⁺ clusters depends only very weakly on the C₆₀⁺ primary ion energy between 13.5 keV and 37.5 keV. The behaviour of Si_m⁺ and Si_mC_n⁺ cluster ions was also investigated for impacts onto a fresh Si surface to study the effects that saturation of the surface with C₆₀⁺ in reaching sputter equilibrium may have had on the measured abundances. By comparison, there are very minor amounts of pure Si_m⁺ clusters produced during C₆₀⁺ sputtering of silica (SiO₂) and various silicate minerals. The abundances for clusters heavier than Si₂⁺ are very small compared to the case where Si is the target.The data reported here suggest that Si_m⁺ and Si_mC_n⁺ cluster abundances may be consistent in a qualitative way with theoretical modelling by others which predicts each carbon atom to bind with 3-4 Si atoms in the sample. This experimental data may now be used to improve theoretical modelling.     

The bΣ(b0) → XΣ(X10,X21) and aΔ(a2) → X21 transitions of AsBr

Emission spectra of the b¹Σ⁺(b0⁺) → X³Σ⁻(X₁0⁺,X₂1) and a¹Δ(a2) → X₂1 transitions of AsBr have been measured in the near-infrared spectral region with a Fourier-transform spectrometer. The arsenic bromide radicals were generated in fast-flow systems by reaction of arsenic vapor (As_x) with bromine and were excited by microwave-discharged oxygen. The most prominent features in the spectrum are the Δv = +1,0,−1, and −2 band sequences of the b¹Σ⁺(b0⁺) → X³Σ⁻(X₁0⁺) transition in the range 11 700-12 700 cm⁻¹. With lower intensities, the Δv = 0 and −1 sequences of the b¹Σ⁺(b0⁺) → X³Σ⁻(X₂1) sub-system show up in the same range. Further to the red, between 6000 and 6700 cm⁻¹, the Δv = 0, +1, and −1 sequences of the hitherto unknown a¹Δ(a2) → X₂1 transition are observed. Analyses of medium- and high-resolution spectra have yielded improved molecular constants for the X₁0⁺, X₂1, and b0⁺ states and first values of the electronic energy and the vibrational constants of the a2 state.     

Planar nano-block structures Tin+1Al0.5Cn and Tin+1Cn (n = 1, and 2) from MAX phases: Structural,electronic properties and relative stability from first principles calculations

Structural, electronic properties and relative stability of quasi-two-dimensional (2D) free-standing planar nano-block (NBs) structures Ti_n+1Al_0.5C_n and Ti_n+1C_n (n = 1 and 2), which can be prepared using the recently developed procedure of exfoliation of corresponding NBs from MAX phases, were examined within first principles calculations in comparison with parent MAX phases Ti₃AlC₂ and Ti₂AlC. We found that in general Ti_n+1C_n and Ti_n+1Al_0.5C_n NBs retain the atomic geometries of the corresponding blocks of the MAX phases, but some structural distortions for the NBs occur owing to the lowering of the coordination number for atoms in the external Ti sheets of the nano-block structures. Our analysis based on their cohesive and formation energies reveals that the stability of the nano-block structures increases with index n (or, in other words, with a growth of the number of Ti–C bonds), the Al-containing NBs becoming more stable than the “pure” Ti–C NBs. Our data show that the magnetization of the simulated planar nano-block structures can be expected; so, for the Ti₃C₂ nano-block the most stable will be the spin configuration, where within each external Ti sheet the spins are coupled ferromagnetically together with antiferromagnetic ordering between opposite external titanium sheets of this nano-block.     

Constructions and properties of k out of n scalable secret image sharing

Recently, Wang et al. introduced a novel (2, n) scalable secret image sharing (SSIS) scheme, which can gradually reconstruct a secret image in a scalable manner in which the amount of secret information is proportional to the number of participants. However, Wang et al.’s scheme is only a simple 2-out-of-n case. In this paper, we consider (k, n)-SSIS schemes where a qualified set of participants consists of any k participants. We provide two approaches for a general construction for any k, 2 ? k ? n. For the special case k = 2, Approach 1 has the lesser shadow size than Wang et al.’s (2, n)-SSIS scheme, and Approach 2 is reduced to Wang et al.’s (2, n)-SSIS scheme. Although the authors claim that Wang et al.’s (2, n)-SSIS scheme can be easily extended to a general (k, n)-SISS scheme, actually the extension is not that easy as they claimed. For the completeness of describing the constructions and properties of a general (k, n)-SSIS scheme, both approaches are introduced in this paper.