Schrödinger equation for a dirac bubble potential

The quantum-mechanical problem of a particle moving in a “Dirac bubble potential” U(r) = (λ/r_o)δ(r - r_o) is solved exactly for both bound and continuum states by making use of partial wave Green's functions G_l(r, r₀, k). Phase shifts are expressed in a compact form related to those for an impenetrable sphere.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

On the photoisomerisation of stilbene in n-alcohols

Photoisomerisation of t-stilbene in n-alcohols is discussed on the basis of recent experiments by Sundstro¨m and Gillbro. The reaction is proposed to consist of three steps: rotational relaxation on S₁^1rans →_k1S₁^p (p = perpendicular configuration) in a barrierless potential U₁(θ); a fast electronic transition S₁^p → S₀^p due to strong non-adiabatic interactions, and a rotational relaxation S₀^p →_k2S₀^cis(trans) in a barrierless potential U₀(θ); k₁ ≈ k₂. In the diffusion limit the photoexcited state lifetime, τ = [(k₁ + k₂)/2]⁻¹, is a linear function of the solvent viscosity in qualitative agreement with experiment.     

Temperature effects on the relaxation parameters of the 1-methylnaphthalene excimer

The excimer lifetime τ_D and the excimer fluorescence efficiency for 1-methylnaphthalene in ethanol have been determined between −30 and 60°C. Expressing the rate constant for excimer deactivation. k_D, in terms of radiative k_FD) and nonradiative (k_ID) processes as k_D = k_FD + k_ID it is found that k_FD is independent of tempeature and k_ID = 5 × 10⁶ + 3.2 × 10¹¹ exp (-ΔE/RT) sec⁻¹, where ΔE = 6.7 kcal mole⁻¹. The behaviour of k_FD and k_ID with temperature and the appearance of isoemissive points in a limited region of temperature are discussed.     

Prediction of ortho substituent effect in alkaline hydrolysis of phenyl esters of substituted benzoic acids in aqueous acetonitrile

The second-order rate constants k for the alkaline hydrolysis of phenyl esters of meta-, para- and ortho-substituted benzoic acids, X-C₆H₄CO₂C₆H₅, in aqueous 50.9% acetonitrile have been measured spectrophotometrically at 25°C. The log k values for meta and para derivatives correlated well with the Hammett σ_m,p substituent constants. The log k values for ortho-substituted phenyl benzoates showed good correlations with the Charton equation, containing the inductive, σ_I, resonance, σ^○ _R, and steric, E _s ^B, and Charton υ substituent constants. For ortho derivatives the predicted (log k _X)_calc values were calculated with equation (log k _ortho)_calc = (log k ^H _AN)_exp + 0.059 + 2.19σ_I + 0.304σ^○ _R + 2.79E _s ^B ? 0.0164ΔEσ_I — 0.0854ΔEσ^○ _R, where DE is the solvent electrophilicity, ΔE = E _AN — E _H20 = ?5.84 for aqueous 50.9% acetonitrile. The predicted (log k _X)_calc values for phenyl ortho-, meta- and para-substituted benzoates in aqueous 50.9% acetonitrile at 25°C precisely coincided with the experimental log k values determined in the present work. The substituent effects from the benzoyl moiety and aryl moiety were compared by correlating the log k values for the alkaline hydrolysis of phenyl esters of substituted benzoic acids, X-C₆H₄CO₂C₆H₅, in various media with the corresponding log k values for substituted phenyl benzoates, C₆H₅CO₂C₆H₄-X.     

A general formalism of deriving the pair potential of polymer chains for arbitrary interaction potentials between isolated chain segments at and close to the theta-point

A formalism has been worked out which allows to transform any non-punctiform segment-segment potential of isolated polymer segments ε of fairly short-ranged character into the pair-potentialU operating between linear polymer chains with a certain reference to the arguments as they have been originally put forward byFlory andKrigbaum. Although no restrictions are made in the derivation as to the repulsive or attractive contribution of the segment-segment potential ε because of some known general deficiencies of theFlory-Krigbaum treatment for exclusively repulsive interaction, the resulting equations are primarily intended to describe the thermodynamic situation at and close to the θ-point where repulsion and attraction—though working at different ranges of segment separation—cancel. As the equation derived is somewhat complicated two different approximate forms have been developed: The first one is based on aTaylor series expansion retaining terms up to the fourth order which allows to characterizeU by the second and the fourth moment of the pair segment-segment distribution function, β and γ (β being the so-called binary cluster integral of segment-segment interaction, which is considered to be zero for θ-conditions). In this caseU may be represented by an expression of the general form $$U/kT = A(1 - BR^2 )\exp \{ - bR^2 \} .$$ The second method is based on a separate integration over the repulsive and attractive ranges of ε giving the repulsive (U ₊) and the attractive (U _?) part ofU finally after some approximations leading to an equation of the general form $$U/kT = (U_ + + U_ - )/kT = A_1 \exp \{ - b_1 R^2 \} - A_2 \exp \{ - b_2 R^2 \} .$$ In both cases the knowledge of the exact form of ε is dispensable, only β and γ—or for the second case their repulsive (β₊ and γ₊) and attractive (β_? and γ_?) parts have to be known. It is shown that the approximations are in excellent accordance with the exact form so that they may be conveniently used to describe pair potentials of polymer chains and to analyze pair potentials of segment-segment interactions under the limitations and conditions indicated.     

Large isotope effect in the quenching of I(52P12) by benzene and benzene-d6

Spin—orbit relaxation of I(5²P_{$\text{1}\text{2}$})(ΔE = 0.94 eV) by benzene-d₆, has been studied at 297 K, using time-resolved atomic resonance fluorescence. A large isotope effect is observed, k_C6H6 = (4.6 ± 0.7) × 10^?13 cm³ molecule^?1 s^?1, and k_C6D6 = (9.9 ± 1.0) × 10^?15 cm³ molecule^?1 s^?1, despite evidence that formation of a bound collision complex may contribute to the quenching mechanism. The roles of resonant energy transfer channels, Franck—Condon factors and the density of final states, in the quenching process, are discussed.     

Synthesis and crystal structure of a new representative of the Rb2U2MoO10 uranomolybdate series

A new representative of the Rb₂U₂MoO₁₀ uranomolybdate series, was synthesized; its single crystals were grown and studied by X-ray diffraction. The atomic crystal structure was solved: monoclinic system, space group P2₁/c, a = 8.542(1) Å, b = 15.360(2) Å, c = 8.436(1) Å, b = 104.279(3)°, R ₁ = 0.064 for 1817 independent reflections with F > 4σ(F). The crystal structure is built of negatively charged infinite corrugated [U₂MoO₁₀] _2∞ ^δ? layers, which are linked through rubidium cations arranged between them. The compound was studied by IR spectroscopy; the absorption bands in the spectrum were assigned.     

Atomistic simulation on the site preference and mechanical properties of Th3Co4+xAl12−x and U3Co4+xAl12−x

An atomistic study is presented on the phase stability, site preference and lattice constants of the actinide intermetallic compounds Th₃Co_4+xAl_12−x and U₃Co_4+xAl_12−x. Calculations are based on a series of interatomic pair potentials related to the actinides and transition metals, which are obtained by a strict lattice inversion method. The lattice constants of Th₃Co_4+xAl_12−x and U₃Co_4+xAl_12−x are calculated for different values of x. The site preference of Co atoms at Al sites is also evaluated and the order is given as 6h, 4f, 2b and 12k for Th₃Co_4+xAl_12−x, and 6h, 4f, 12k and 2b for U₃Co_4+xAl_12−x. In addition, some simple mechanical properties such as the elastic constants and bulk modulus are investigated for the actinide compounds with complex structures.     

An accurate simplified data treatment for the initial adsorption kinetics in conditions of laminar convection in a slit: application to protein adsorption

We present the derivation of a simple approximation for the original expression of the adsorption rate [Langmuir 10 (1994) 3898] in conditions of laminar flow in a slit, to relate the measured initial kinetic constant k with the interfacial kinetic constant k_a and the transport-limited Lévêque constant k_Lev. The same method of derivation is applied here to get a simple approximation of the average kinetic constant 〈k〉 [Biomaterials 20 (1999) 1621]. For the local value, at distance x from the entrance of the slit, we propose k(x)/k_a=(u−1)(au−1)/(bu+1), where u=k(x)/k_Lev, a=0.452, b=−0.625, with a maximal error of 1% in comparison with the exact solution. For the average value over the length of the slit, we propose 〈k〉/k_a=(U−1)(AU−1)/(BU+1), where U=〈k〉/〈k_Lev〉, A=0.203, B=−0.273, with a maximal error of 0.03%. These approximations lead to an easy determination of the adsorption constant and diffusion coefficient D of the solute, as appropriate plots of experimental data provide k_a and D^2/3 as the intercepts of the curve with the ordinate and abscissa axes, respectively. It is pointed out that the linear approximation k⁻¹=k_a⁻¹+k_Lev⁻¹ would lead to the overestimation of both the diffusion coefficient and adsorption kinetic constant. As an example, the application to the analysis of experimental data for adsorption of α-chymotrypsin onto mica plates is provided.     

Pulse polarography: Part V. on the theory for the kinetic currents with an ece mechanism

We have derived an equation for the instantaneous limiting current in pulse polarography with an ECE mechanism. That equation has been derived for the expanding sphere electrode model (and also for the expanding plane electrode model) with the condition (k₁+k₂)>>1, k₁ and k₂ being the rate constants of the chemical reaction. We show that by an adequate choice of the time of application of the potential and time of the drop growth prior to potential application, it is possible — if the equilibrium constant for chemical reaction is known — to widen the interval of values for k₁ and k₂ that may be determined using this technique. At the same time, we also show that if the electrode sphericity is not taken into account, the values obtained for k₁ and k₂ are always lower than the real ones.     

A flash photolysis study of the elimination of dinitrogen from trans-(N2)2W(dppe)2; dppe = 1,2-bis-diphenylphosphinoethane

Flash photolysis of trans-(N2)2W(dppe)2 (1) at ?60, ?30, ?10°C, and room temperature indicates that loss of dinitrogen occurs stepwise via the following proposed intermediates. Photodissociation of 1 gives the transient A decaying with k1 ～ 4450 s?1 to the doubly coordinatively unsaturated species [W(dppe)2], B. Further reactions of B are dependent on the type of gas used to saturate the solutions. In N2-saturated media, B is efficiently reconverted into the starting complex 1 via (N2)W(dppe)2], C(N2), kN22 = 450 s?1, which in turn takes up a second molecule of N2, kN23 = 3.7 s?1. In CO-saturated solutions, trans-(CO)2W(dppe)2 is produced as the final product and the corresponding rate constants are kCO2 1500 s?1 for B → C(CO) and kCO3 = 1.14 s?1 for C(CO) → product. In Ar-saturated solvents, B is transformed, again in two steps; kAr2 = 1 s?1 and kAr3 = 0.1 s1?, to products of unknown structure.The different rate constants kN22, kCO2, kAr2 and kN23, kCO3 and kAr3, together the common activation energy of ca. 11 kcal/mol?1 for the three processes A → B, B → C(N2) and C(N2) → 1 suggest that the reactions of B and C occur by SN2-type displacement of coordinated solvent molecules by the incoming ligands.     

The molecular structure of trifluoroethene in the gas-phase as determined from electron diffraction and microwave data

The molecular structure of trifluoroethene was determined from electron diffraction data and the microwave rotational constants of the parent and deuterated molecule, corrected for zero-point vibrational motion. A GVFF adjusted to fit the vibrational frequencies was used for the correction. The molecule was found to be planar. Assuming equal geminal C1—F bond lengths, the following r_g distances and r_av angles are found: C1—F = 1.316 ± 0.011 Å, C2—F = 1.342 ± 0.024 Å, CC = 1.341 ± 0.012 Å, C—H = 1.100 ± 0.02 Å, ∠C—C—F1 = 123.1 ± 1.5°. ∠C—C—F2 = 124.0 ± 0.6°, ∠C—C—F3 = 120 ± 0.7° (Fl trans to F3) and ∠C—C—H = 124.0 ± 1.7°.The error limits include 3σ (σ = estimated standard deviation) and estimates of the systematic errors. The analysis suggests that all the C1—F distances are not equivalent, neither are the C2—C1—F angles, though the differences are not significant (10% level).     

Determination of the population,depopulation and the anisotropic spin lattice relaxation rates in the photoexcited triplet state of phenazine. A light modulation-EPR study

The photoexcited triplet state of phenazine in toluene glasses at 35 K is investigated by light modulation-EPR spectroscopy. From the transient EPR spectra and the kinetics in the three canonical orientations (p = x, y, z) the rate parameters are determined. Thus, the depopulation rate constants k_p, the anisotropic spin lattice relaxation rate constants W_p, and the ratios between the population constants A_p are calculated: k_x = (2.2 ± 0.3) × 10² s^?1, k_y = (0.21 ± 0.04) × 10² s^?1, k_z = (0.06 ± 0.03) × 10² s^?1, W_x = (8.6 ± 0.9) × 10³ s^?1, W_y = (11.0 ± 1.0) × 10³ s^?1, W_z = (14.0 ± 1.4) × 10³ s^?1, and A_x: _Ay:A_z ≈ 1:0.04:0.02. It is concluded therefore that the in-plane spin state |τ_x > is the active one.     

The relation between the high-temperature and room-temperature structure of CsCuCl3

The room-temperature structure of CsCuCl₃, spacegroup P6₁22, is described as the superposition of the high-temperature structure, spacegroup P6₃/mmc, and a softening vibration mode transforming as the degenerate irreducible representation τ5 (k = (0,0,2Π/3c)] of P6₃/mmc. It is argued that the presence of a third order invariant for k = (0,0,2Π/3c) in a power series expansion of the free energy in normal coordinates is the reason for a phase transition to occur at k = (0,0,2Π/3c). The connections with a cooperative Jahn-Teller effec: are indicated.     

Heat capacities of solid ortho—para mixtures of deuterium in the ordered state

The heat capacities of three ortho—para mixtures of solid deuterium in the rotationally ordered state as a function of temperature as well as the order—disorder transition temperature, T_c(x) for several compositions have been measured. The rotational heat capacities for all three mixtures, x (para or J = 1 mole fraction) = 0.69₉, 0.81₉ and 0.92₁ per mole of J = 1 species can be represented by a single function in terms of the reduced temperatures, T_c(x)/T, which to a good approximation is given by

, where Γ⁰_eff(1)/k = 1.0 deg is the effective electric quadrupolar coupling constant for the anisotropic interactions in pure para D₂ and T_c(1) = 4.05 degrees the order—disorder transition temperature. It is shown that this correlation follows from the assumption that both the libron energies and band widths of the libron modes scale with composition in a manner identical to the transition temperatures, namely

.     

Relationship between spontaneous generation of voltage and structural features in a single crystal of Pr0.6Ca0.4MnO3

Spontaneous generation of electrical voltage U was found for the perovskite-like single crystal of Pr_0.6Ca_0.4MnO₃, wherein the charge and orbital orderings occur at T _CO = 240 K and the antiferromagnetic one occurs at the Neel point T _N = 174 K. With decreasing the temperature, the U value increases first slowly (in the temperature range of 300 K-T _CO) and then more rapidly (T _CO-T _N). Starting from T _N, U increases exponentially and, at 85 K, reaches the value of 115 mV (in the ab plane) and 6.5 mV (along the c axis). The magnetic field has a different effect on the U value in different temperature ranges: in the temperature range of 85–130 K, it decreases U and, in the range of 130–240 K, increases U. The spontaneous voltage is thought to be caused by the existence of ferromagnetic and charge-orbital-ordered clusters of different topology in the sample. The latter clusters have own features and their structure are described by the non-centrosymmetric space group P2₁ nm.     

Rotation-vibration-translation energy transfer in laser excited Li2 (B1Πu)

Using the method of laser fluorescence, inelastic collisions with rare gas atoms of electronically excited ⁷Li₂ molecules in the υ = 2 and 4 levels were studied. Vibrational transitions ranging from Δ = +2 to ?4 were observed. The simultaneous rotational transitions were completely resolved, and detailed rate constants k_{Δυ, ΔJ} for specific collision- induced quantum jumps Δυ, ΔJ were determined. The effect of secondary rotational relaxation was eliminated by an extrapolation to zero pressure. By integration over ΔJ, rate constants k_Δυ, were found. They are, within the error limits, independent of the collision partner and on the initial υ (2 or 4) and depend rather weakly on Δυ. These findings are compared with theoretical results from various methods, generally based on a collinear collision model. The apparent disagreement in all respects suggests strongly the importance of rotational degrees of freedom in the collision. Experimental evidence for this is the large amount of V — R transfer observed, which about equals the V — T transfer. The mean cross sections σ(Δυ) for specific vibrational transitions Δυ range between 6 and 15 A², among the largest ever observed.     

Temperature dependence of the reaction rate constants for O(3P) atoms with C2H4, C3H6 and NO(M = N2O), determined by a modulation technique

Rate constants for the reaction of O(³P) atoms with C₃H₄, C₃H₆ and NO(M = N₂O) have been measured over the temperature range 300–392°K using a modulation-phase shift technique. The Arrhenius expressions obtained are:C₂H₄, k₂ = 3.37 × 10⁹ exp[?(1270 ± 200)/RT]liter mole^?1 sec^?1,C₃H₆, k₂ = 2.08 × 10⁹ exp[?(0 ± 300)/RT]liter mole^?1 sec^?1,NO(M = N₂O), k₁ = 9.6 × 10⁹ exp[(900 ± 200/RT]liter² mole^?2 sec^?1.These temperature dependencies of k₂ are in good agreement with recent flash photolysis-resonance flourescence measurements, although lower than previous literature values.     

Raman spectroscopy of powders: Effects of light absorption and scattering

New expressions are derived from Kubelka—Munk theory to describe the Raman intensities observed by back-scattering from powder samples. The equations relate the Raman intensity to the diffuse reflectance R_∞ of the sample for two cases: (i) for a series of samples having constant values of the scattering coefficient, s, but which vary in their values of the absorption coefficient, k; and (ii) for a series of samples having constant values of k, but differing in their values of s. The predicted intensity dependences are compared with the results of experiment.