Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

Structure and Vibrational Spectra of 1:1 Chloranilic Acid (CLA)—Tetramethylpyrazine (TMP) Complex

In searching for new molecular materials the crystals of 1:1 complexes of tetramethylpyrazine (TMP) with chloranilic acid (CLA) were grown. In the crystalline lattice of TMP.CLA infinite chains of hydrogen bonded components are formed. Very strong (d_(O·sN) = 2.692(2) {?}) hydrogen bonds without proton transfer are present. The components of the complex form stacks of parallel arrangement with the distance typical of van der Waals interaction. One can conclude that in packing, interactions via hydrogen bonds play a decisive role. In the IR spectrum one observes a broad, continuous absorption with well resolved trio typical of strong hydrogen bonds. A detailed analysis of IR and Raman spectra in the low frequency region related to deformation vibrations of CH₃ groups shows that strong interaction via hydrogen bonding only slightly affects the dynamics of these groups.     

Spectrum analysis of U-doped LaBr3 single crystals. Part 1: crystal-field analysis

Single crystals of U³⁺:LaBr₃ were grown by the Bridgman-Stockbarger technique. High-resolution polarized absorption spectra of the crystals were recorded at 4.2 K in the 4000-50,000 cm⁻¹ range. Sixty-four experimental crystal-field energy levels of the U³⁺ ion were fitted to a semiempirical Hamiltonian employing free-ion, one-electron crystal-field as well as two-particle correlation crystal-field (CCF) operators with an r.m.s. deviation of 28 cm⁻¹. The performed analysis of the spectra enabled the determination of crystal-field parameters and assignment of the observed 5f³→5f³ transitions. The effects of selected CCF operators on the splitting of some specific U³⁺ multiplets have been investigated and the obtained values of Hamiltonian parameters are discussed and compared with those reported in previous analyses.     

Matrix-isolation FT-IR and DFT theoretical studies of the intramolecular hydrogen bonding in Mannich bases

FT-IR Ar-matrix isolated spectra were studied for dichloro- (Cl₂-MB) and tetrachloroderivatives (Cl₄-MB) of the ortho Mannich base. The spectra were analyzed based on the DFT calculated frequencies and intensities and compared with those recorded in CCl₄ solution in the region of the ν(OH) and ν(OD) vibrations. The matrix-isolated spectra are characterized by narrower ν(OH) and ν(OD) bands with much better resolved fine structure than in solution. The fine structure originates from the anharmonic coupling with the low frequency modes as well as from Fermi resonance. The ν(OD) band shapes can be reproduced exclusively by assuming the Fermi resonance with overtones and summation of the frequencies of modes into which the bridge atoms are involved. The frequency isotopic ratio (ISR) is for both compounds 1.33 while the half-width ratios are equal to 1.82 and 1.94, for Cl₂-MB and Cl₄-MB, respectively.     

Vibrational spectra of the adduct of 1,8-bis(dimethylamino)naphthalene with dichloromaleic acid (DMAN x DCM)

The infra-red (IR), Raman (R) and inelastic incoherent neutron scattering (IINS) spectra, particularly in low frequency region, of the title ionic adduct were studied. It is shown that all low frequency vibrations (below 200 cm(-1)) of (CH3)2N groups of protonated 1,8-bis(dimethylamino)naphthalene (DMAN)--clearly observed in IINS spectra--are sensitive to the environment, i.e. to the type of counterion forming short contacts with C-H bonds of methyl groups. The internal frequencies were also calculated by ab initio method. The results are consistent with numerous observations of the counteranion effect on the geometry of the protonated DMAN. The conclusions are compared with structural and NMR studies reported recently for the 1,8-bis(dimethylamino)naphthalene with dichloromaleic acid (DMAN x DCM) adduct. The single crystal R polarized spectra taken over the frequency range 20-3200 cm(-1) were analyzed in detail. We have shown that a substantial difference in the IR spectrum of the dichloromaleic acid (DCM) anion in the DMAN adduct and in the potassium salt results from different geometries of OHO hydrogen bonds. In the case of potassium salt the chains of longer intermolecular hydrogen bonds are formed described by means of a double minimum potential.     

Unitary Geometric Algebra

The geometric significance of the imaginary unit in a complex geometric algebra has troubled the author for 40?years. In the unitary geometric algebra presented here, the imaginary i is a unit (pseudo) vector with square minus one which anti commutes with all of the real vectors. The resulting natural hermitian inner product and hermitian outer product induce a grading of the algebra into complex k-vectors. Basic orthogonality relationships are studied.     

IR spectra of [NHN]+ homoconjugated cations in 1-adamantanamine hemiperchlorate

The crystalline hemiperchlorate of primary 1-adamantanamine has been prepared. The occurence of strong [NHN]⁺ bridges has been found which show unusual temperature and isotope effects in IR spectrum. The region of the NH₂ vibrations has been analysed by comparing with that for pure amine and 1:1 salt.     

Spectrum of recoil nucleons in quasi-elastic neutrino-nucleus interactions

We have analyzed the consequences of introducing the local density approximation combined with an effective nuclear momentum-dependent potential into the CC quasi-elastic neutrino-nucleus scattering. We note that the distribution of recoil nucleons momenta becomes smooth for low momentum values and the sharp threshold is removed. Our results may be relevant for Sci-Fi detector analysis of K2K experiments. The total amount of observed recoil protons is reduced because some of them remain bound inside the nucleus. We compare theoretical predictions for a probability of such events with the results given by NUX + FLUKA MC simulations.Received: 23 July 2004, Revised: 8 November 2004, Published online: 23 December 2004J.T. Sobczyk: was supported by KBN grant 105/E-344/SPB/ICARUS/P-03/DZ211/2003-2005; C. Juszczak and J.A. Nowak were supported by LNGS-TARI P10/02     

Uniform illumination by diffractive shaping of independent light beams

This article presents a new optical set-up for illumination of a common area with a few independent light sources. The main aim is to create a uniform light distribution of a constant shape even if one of the light sources will stop working. In such a case, the shape of the light distribution should remain unchanged, however, the total light intensity will be lower. The novelty of the proposed solution is based on the usage of a small number of the independent light sources in opposite to a well known LED arrays approach. The beam shaping is made using Dammann gratings and specially designed blazed phase diffractive gratings. The article contains the theoretical analysis and the computer simulation verified by the experiment.