Paramagnetic resonance of Gd3+ in Sm (NO3)3. 6H2O single crystals

The paramagnetic resonance of Gd³⁺ in Sm(NO₃)₃.6H₂O single crystals, grown from solution, is studied at room temperature. A seven line spectrum for H//Z as well as for H//X corresponding to ΔM=±1 transitions are observed. The spin Hamiltonian analysis is presented.     

Paramagnetic resonance of Gd3+ in SmCl3·6H2O single crystals

The Paramagnetic Resonance of Gd³⁺ in SmCl₃·6H₂O single crystals, grown from solution, is studied at room temperature. A six line spectrum for H//Z and a seven line spectrum for H//X corresponding to ΔM=± 1 transitions are observed. Their angular variation in ZX plane from?=0° to?=90°, is studied and the spin-Hamiltonian analysis is presented. The probable amount of admixture of the next higher electronic state⁶P_7/2 with the ground state⁸S_7/2 is also estimated.     

Forbidden hyperfine transitions in electron spin resonance of Mn2+ in NaCl single crystal

Forbidden hyperfine transitions are observed in the electron spin resonance spectrum of divalent Mn⁵⁵ ion in NaCl single crystal for a particular associated pair. From the measurements of the M = + 1/2 → ?1/2, Δm = ± 1 transitions the parametersQ′ and Q″ of the nuclear electric quadrupole part of the spin-Hamiltonian H_o = Q′ [I_z ² ? 1/3 I (I + 1)] + Q″ (I_x ²?I_y ²) are found to be + 1.70 × 10^?4 cm.^?1 and +0.16 × 10^?4 cm.^?1 respectively.     

Isoliertheit und Stabilität von Flächen konstanter mittlerer Krümmung

In the Sobolev space H^m(B,?³), B the open unit disc in ?², we consider the set M_n of all conformally parametrized surfaces of constant mean curvature H with exactly n simple interior branch points (and no others). We denote by M*_n the set of all xεM_n with the following properties:

1. in every branch point the geometrical condition K_G¦x_Z¦≡O holds (K_G is the Gauss curvature and x_z is the complex gradient of the surface x).
2. the corresponding boundary value problem Δh+×_z{²(2H²-K_G)h=O,^hδB=O, is uniquely solvable.

We prove then, that the manifold M*=UM*_n is open and dense in the set of all surfaces of constant mean curvature H and that all x εM*_n are isolated and stable solutions of the Plateau problem corresponding to their boundary curves. In addition, the submanifold M*_n contains exactly all surfaces x for which the space of Jacobi fields is transversal (with exception of the 3-dimensional space of conformai directions) to the tangent space T_xM_n.     

The microwave spectrum of methyl cyanide andl-type doubling in CH3CN,CH3NC,CH3CCH and CF3CCH

The microwave spectrum of Methyl cyanide molecule has been obtained corresponding to the transitions J=0→1, J=1→2 and J=2→3 in the ground vibrational state as well as in the excited degenerate vibrational statesv ₈=1 andv ₈=2. The frequencies of the lines are measured by using a secondary standard spectrograph taking mostly the ammonia lines as standards. The value of the vibration-rotation interaction constanta ₈ has been found to be ?27·3±0·4 mc./sec. The spectrum obtained for the transitions J=1→2 and J=2→3 withv ₈=1 has been understood on the basis of Nielsen’s theory ofl-type doubling in symmetric top molecules. The assignments give a value of 0·94 for the coriolis interaction constant which is consistent with the value obtained from the infra-red spectral data. The earlier work on similar transitions involvingl-type doubling in the molecules CH₃NC, CH₃CCH and CF₃CCH has been reviewed and the inconsistencies in the values of some parameters like coriolis interaction constant ζ and the rotation constant D_JK obtained by earlier workers from their assignments are pointed out. A reassignment of the transitions and the involved parameters removes these inconsistencies and yields coriolis interaction constants which are in close agreement with those obtained from infra-red spectral data. The following are the different ζ values obtained:

$$\begin{gathered}\zeta _8 in CH_3 NC = 0 \cdot 93 \hfill \\\zeta _{10} in CH_3 CCH = 1 \cdot 00 \hfill \\\zeta _9 in CH_3 CCH = 0 \cdot 95 \hfill \\\zeta _{10} in CF_3 CCH = 0 \cdot 6 \hfill \\\end{gathered}$$     

Am×n矩阵的加边矩阵的奇异性问题

给定m×n阶矩阵A,我们给出了它的加边矩阵M=[A B C O] (1)为非奇的充分必要条件。其中O为r₁×r₂阶零矩阵。把M的逆矩阵记为分块形式M^-1=[A₁ B₂ C₃ O₄]其中C₁为n×m、C₂为n×r₁、C₃为r₂×m、C₄为r₂×r₁阶矩阵。在一定条件下,我们证明了其中的C₁为A的广义逆矩阵A⁺。     

Optical absorption of Co2+ doped NH4Cl: Phase transformation studies

The absorption spectrum of Co²⁺ doped NH₄Cl has been studied from the room temperature to the liquid nitrogen temperature. A sudden change in the spectrum is observed between 243° K and 233° K which is attributed to the phase transition in the crystal. From the observed spectrum it is suggested that Co²⁺ goes in interstitially as well as substitutionally. Both the types of centers exist at room temperature, but with decrease in temperature substitutional ions migrate to interstitial sites, the process being stimulated at the phase transformation point so that the 77° K spectrum seems to be mostly due to the interstitial centers. The 77° K spectrum is analyzed in the approximation of octahedral symmetry for interstitial ions and the band positions are fitted fairly well with B = 870 cm.^?1 Dq = 850 cm.^?1 and C = 4·4 B. A blue shift of about 100 cm.^?1 is observed for⁴T₁ (P) band at the phase transition which is attributed to the increase in Dq value with the anomalous lattice contraction at the phase transition. The decrease in the lattice parameter calculated from this blue shift is around 0·4% which is in good agreement with the results of X-ray measurements. Two possible models for the interstitial complex are examined and the one with fourfold chlorine coordination associated with two neutral water molecules at the first neighbour (NH₄)⁺ site lying along < 100> direction is suggested to be more probable.     

Studies in the hydrolysis of metal ions

The hydrolysis of cupric ion has been studied at various ionic strengths (0·01, 0·05, 0·1 and 0·5 M). The results are analyzed employing ‘core + links’ theory, log-log plot, normalization plot, and extrapolation method for obtaining the pure mononuclear curve. The stability constants of Cu₂(OH) ₂ ⁺⁺ , Cu₃(OH) ₄ ⁺⁺ , Cu(OH)⁺ and Cu(OH)₂ have been reported.     

The probable iodine molecular lasers in the violet and ultraviolet regions

Iodine when excited in the presence of argon shows three band systems in the regions 4420–4000 Å., 3,460–3015 Å. and 2785–2731 Å. all of which involve O _u ^Emphasis>+ (³π _u ) state at 15642 cm.^?1 as the common lower state. It is pointed out in this article that the transitions involving the first two band systems are suitable for laser action as these transitions are very strong and as their common lower state gets efficiently flushed out due to the crossing or touching of a O _u ^Emphasis>? repulsive state while a continued high frequency discharge through a mixture of iodine and argon would keep the upper state populated all the time. Tuning of the interferometer system is expected to make this laser action possible in a wide range of frequencies.     

Thermal expansion of ferroelectrics

Using the interference method the thermal expansion of lithium hydrazinium sulphate [Li(N₂H₅)SO₄] has been investigated in the temperature range ?170° C. to +220° C. The principal expansion coefficients along the crystallographic axes have been determined and at room temperature (25° C.) the values ofa _a,a _b anda _c are respectively 17·7, 13·5, and 43·0, ×10^?6 (°C.)^?1. The expansion coefficients are found to vary nonlinearly, and, in particular, along thec-axis the thermal expansion shows an anomalous behaviour between ?160°C. and ?60°C. and also in the neighbourhood of +130°C. These anomalies are explained as due to homomorphous transitions in which there is a reorientation of the ?NH ₊ ³ group about the N-N axis or a re-alignment of the ?NH₂ group.     

Oxidation of aldehydes by aqueous Ce (IV) in nitric acid

The kinetics of oxidation of acetaldehyde (at 20°, 31° and 40° C.), propionaldehyde (at 15°, 25° and 35° C.) and chloral hydrate (40°, 50° and 60° C.) by ceric nitrate in nitric acid medium [(H⁺) ≈ 0 · 5?1 · 5 M] were studied. The reactions were followed by determining rates of ceric disappearance for variations in [Ce⁴⁺], [RCHO], [H⁺], μ, temperature, etc. The stoichiometry, ΔCe⁴⁺ / Δ CH₃CHO ≈ 2 was established. The reactions were found to be 2nd Order—first order each with respect to [Ce⁴⁺] and [RCHO]. No complex formation between Ce⁴⁺ and aldehydes was observed. Neutral Ce (NO₃)₄ (H₂O)₂ and diol of the aldehyde were assumed the active species. The rate and thermodynamic data were calculated and discussed. The rates of oxidation were correlated with the structures of the aldehydes.     

Oxidation studies

Studies on oxidation of water by cobaltic ions in perchloric, nitric and sulfuric acid media in the temperature range 20–30° C. at constant ionic strength of 2·2 M were carried out. An overall order of 3/2 for cobaltic ion concentration was observed in perchloric, nitric acid media and at low [Co³⁺]<4×10^?3 M in sulfuric acid medium. At high [Co³⁺] in sulfuric acid medium an order 2 was observed. In all the three acid media an inverse dependence of the rates on [H⁺] was noticed. From the experimental results, an acid-independent reaction path was inferred in perchloric and nitric acid media. The effect of ionic strength, HSO ₄ ^? and initially added cobaltous ions as well as temperature was studied. Suitable reaction schemes were suggested to explain the experimental results and activation energy values for some of the rate parameters were evaluated. Kinetic studies on oxidation of tertiary amyl and tertiary butyl alcohols in sulfuric acid medium and that of the former in perchloric acid medium were carried out. The reaction rates were found to vary linearly with [Co³⁺] and [Alcohol]. From the effect of acidity and of [HSO ₄ ^? ] in H₂SO₄ medium on the reaction rates it was concluded that in HClO₄ medium CoOH²⁺ species were the active entities while in H₂SO₄ medium Co³⁺ as well as CoOH²⁺ species were the active entities. The influence of ionic strength, temperature, etc., on the raction rates were studied and activation energy values for some rate constants were evaluated.     

Laplace-Transformation nichtnegativer und vektorwertiger Maße

Calling a function f: R ₊ ^p →R with $$\sum\limits_{i = 1}^n { \sum\limits_{j = 1}^n {\alpha _i \alpha _j } f(t_i + t_j )} \geqslant 0 for all (\alpha _1 ,...,\alpha _n ) \varepsilon R^n ,$$ , (t₁,...,t_n∈(R ₊ ^p )ⁿ, n∈? positiv semidefinit, the Laplace-transformations of finite nonnegative measures on R ₊ ^p are charac terised as the continuous bounded positiv semidefinit functions. Let H be a real Hilbertspace. A σ-additive mapping \(M: \mathfrak{B}_ + ^p \to H\) is called an orthogonal measure iff 〈M(A), M(B)?=0 for A∩B=ø. Exactly those mappings Y: R ₊ ^p →H are Laplacetransformations of H-valued orthogonal measures, which are continuous and bounded and for which ?Y(s), Y(t)? is only a function of s + t. Using this result one obtains a representation theorem for continuous semi-grouphomomorphisms defined on R ₊ ^p with values in the “unit intervall” of the selfadjoint operators on H.     

Spectrum and continuum eigenfunctions of Schrödinger operators

We consider Schrödinger operators $\text{H = ?}\text{1}\text{2}\text{Δ + V}$ for a large class of potentials. V. We show that if H? = E? has a polynomially bounded solution ? then E is in the spectrum of H. This is accomplished by proving that the spectrum of H as an operator on L² is identical to its spectrum as an operator on the weighted L² space, L²_δ.     

Emission spectrum of bromine excited in the presence of argon—part I

Bromine was excited by an uncondensed transformer discharge in the presence of argon. The spectrum obtained was found to be different from what one gets without the presence of a foreign gas and consists of (1) a short discrete band system in the region 3150–2970 Å, (2) an extensive discrete band system in the region 2950–2670 Å, (3) a short and weak discrete system in the region 2660–2590 Å and a set of diffuse bands in the region 3340–3190 Å. The wavelengths and wave numbers of the band heads of the system 2950–2670 Å, as obtained from the measurements of the plates taken on the first order 21-ft. grating spectrograph, are given along with the vibrational analysis. This system is shown to be due to a transition from an upper electronic state at 51802 cm.^-1 with ω′_e = 150·5 cm.^-1 and ω′_e X′_e = 1·15 cm.^-1 to the well known³ II _α (O₂ ⁺) state at 15918 cm.^-1     

Potential functions for hydrogen bond interactions

A careful comparison of the distribution in the (R, θ)-plane of all NH … O hydrogen bonds with that for bonds between neutral NH and neutral C=O groups indicated that the latter has a larger mean R and a wider range of θ and that the distribution was also broader than for the average case. Therefore, the potential function developed earlier for an average NH … O hydrogen bond was modified to suit the peptide case. A three-parameter expression of the form \(V_{hb} = V_{min} + p_1 \Delta ^2 + q_1 e^{p_3 \Delta } \theta ^2 \) , with △ = R - R_min, was found to be satisfactory. By comparing the theoretically expected distribution in R and θ with observed data (although limited), the best values were found to bep ₁ = 25,p ₃ = ? 2 andq ₁ = 1 × 10^?3, with R_min = 2·95 Å and V_min = ? 4·5 kcal/mole. The procedure for obtaining a smooth transition from V_hb to the non-bonded potential V_nb for large R and θ is described, along with a flow chart useful for programming the formulae. Calculated values of ΔH, the enthalpy of formation of the hydrogen bond, using this function are in reasonable agreement with observation. When the atoms involved in the hydrogen bond occur in a five-membered ring as in the sequence a different formula for the potential function is needed, which is of the form V_hb = V_min +p ₁△² +q ₁ x ² wherex = θ ? 50° for θ ≥ 50°, withp ₁ = 15,q ₁ = 0·002, R_min = 2· Å and V_min = ? 2·5 kcal/mole.     

The convergence of the spectrum of a weakly connected domain

The paper deals with the convergence, as ε tends to zero, of the spectrum of the Neumann problem -Δυ_?=λ(?)υ_? in a «weakly connected» periodic domain Ω_? of ?³. The domain Ω_? is composed of a finite number of disjoint connected domains linked by thin bridges (curved plates or tubes). Under a few assumptions on the characteristic sizes of these bridges, we give an explicit asymptotic formula for the eigenvalues which tend to zero and we prove that the rest of the spectrum converges to the spectrum of an elliptic coupled system.     

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and ₂ ² (M) be the Sobolev space consisting of functions in L²(M) whose derivatives up to the order two are also in L²(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H ₂ ² (M),

Résolution des équations de Navier-Stokes dans des espaces anisotropes

In this paper we prove the global existence and uniqueness of the solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which H^δ in the i-th direction, δ₁+δ₂+δ₃=½,½< δ_i <½ and in a space which is L² in the first two directions and B_2,1^½ in the third direction, where H and B (respectively) denote the usual homogeneous Sobolev and Besov (respectively) spaces.