Electron-vibration effects in Cs2SiF6 : Mn4+

We report a study of the ⁴ A _2g →² T _1g absorption band of Mn⁴⁺ in Cs₂SiF₆. The band shows several lines or groups of lines associated with transitions from the ⁴ A _2g ground state to the spin-orbit components (² T _1g )Γ₈ and (² T _1g )Γ₆ coupled to the three odd-parity vibrations v ₆(t _2u ), v ₄(t _1u ) and v ₃(t _1u ). The absorptions associated with the (² T _1g )Γ₈ electronic state have structure whereas those associated with the (² T _1g )Γ₆ do not. It is shown that the structure is a consequence of splitting of the Γ₈ × v vibronic multiplets by electron-vibration interaction. The intensity of the ⁴ A _2g →(² T _1g )Γ _i + v_j vibronic transitions are expressed in terms of a small number of parameters; two parameters for v(t _1u ) modes and three for v(t _2u ) modes. Plausible but not good fits to the low temperature Zeeman data and vibronic splitting patterns are obtained. The excitation spectrum of the Cs₂SiF₆ : Mn⁴⁺ in the region of the ⁴ A _2g →² E_g and ⁴ A _2g →² T _1g is recorded using a c.w. dye laser. This reveals numerous weaker lines involving combinational modes and even-parity modes v₅ (t _2g ), v ₂(e_g ) and v ₁(a _1g ). Several interesting electron-vibrational effects are observed. These are illustrated and discussed qualitatively.     

4 Ag ↔2 Eg Transition of Mn4+ in Cs2TiF6: MnF6 2-

We report and interpret the ⁴ A_2g ?² E_g absorption, emission, linear dichroism, MCD and MCE of the MnF₆ ^2- octahedra at sites of D_3d symmetry in single crystals of Cs₂TiF₆. The spectra show many sharp features but a simplified calculation using cubic basis functions successfully accounts for all of the major features and much of the detailed structure. The electronic levels are not split by the trigonal field in first order but trigonal distortion splits the three-fold degenerate v ₆(t_2u ) vibration of the octahedron into a two-fold degenerate v ₆(e_u ) and a non-degenerate v ₆(a_1u ) vibration and these two vibrational modes give rise to the features 213 cm^-1 and 246 cm^-1 from the zero-phonon line respectively. Similarly the v ₄(t_1u ) vibration is split into a v ₄(a_2u ) and v ₄(e_u ) vibration and give the features at 313 cm^-1 and 340 cm^-1 respectively.

Several transitions involving two or more quanta of vibration are clearly seen in the emission and MCE spectrum. At an energy shift of greater than 700 cm^-1 from the zero-phonon transition these features are found to consist of progressions in the v ₁(a_1g ) breathing mode and the v ₂(e_g ) two-fold degenerate Jahn-Teller active mode. This structure can again be explained, with one exception, in terms of a calculation which assumes cubic basis functions. The exception is the reversal in sign of the MCE at a frequency v ₆(e_u ) + v ₂(e_g ) compared with that at v ₆(e_u ) which is not fully understood.     

The electronic absorption and magnetic circular dichroism spectra of IrBr6 2- in several host crystals

The absorption and magnetic circular dichroism (M.C.D.) spectrum of the IrBr₆ ^2- ion at room and liquid helium temperature has been studied in the host crystals (NH₄)₂SnBr₆, K₂SnBr₆ and (C₂H₅NH₃)₂SnBr₆ in the region ～11 000–21 000 cm^-1. An interpretation of the spectrum is presented which differs significantly from those suggested previously. In order of increasing energy the allowed bands are assigned to the following ligand-to-metal charge-transfer transitions: E_g ″(² T _2g )→ U_u ′(² T _1u ) (13–14 000 cm^-1), E_g ″(² T _2g )→ E_u ″(² T _2u ) (16 800 cm^-1), and E_g ″(² T _2g )→ U_u ′(² T _2u ) (～ 18 300 cm^-1). Both our absorption and M.C.D. data strongly suggest a Jahn-Teller splitting of the U_u ′(² T _1u ) state but contradict a previous suggestion of such a splitting of the U_u ′(² T _2u ) state. Consideration of σ—π mixing in the t _1u (π + σ) molecular orbital suggests that the ～17 300 cm^-1 band arises from the orbitally-forbidden E_g ″(² T _2g )→ E_u ′(² T _1u ) transition. Bands in the 11 000–13 000 cm^-1 region are assigned to parity-forbidden charge-transfer transitions to states generated by the t _1g (π)→ t _2g excitation. The fine structure seen at liquid helium temperature in K₂SnBr₆ : Ir⁴⁺ both in the 14 500 cm^-1 band and overlying the E_g ″→ U_u ′(² T _2u ) band appears to be associated with parity-forbidden transitions.     

面心立方晶体中生成孪晶或六角密堆相的电子衍射分析

用指数变换矩阵将面心立方晶体的晶带轴[uvw]变换成孪晶的[u^tv^tw^t]及六角密堆结构的[u′v′w′],在略去这些指数的公共系数后,如u^(t2)+v^(t2)+w^(t2)>u²+v²+w²或u′,v′≠3n而u′+v′=3n,则孪晶或六角密堆结构与面心立方晶体有完全相重的电子衍射谱。这种标定的不唯一性是广泛存在的。 关键词：     

Franck-Condon Factors and r-Centroids for the (B [sbnd] X) and (C [sbnd] X) Bands of the 107Ag 109Ag Molecule

Arrays of Franck-Condon factors q(v′, v″) and r-centroids r(v′, v″) were computed using Morse potentials for C¹Π [sbnd] X¹Σ⁺ _g and B O⁺ _u [sbnd] X¹Σ⁺ _g bands of the ¹⁰⁷Ag ¹⁰⁹Ag molecule.     

(T1u+T1g)⊗(hg+τ1u)vibronic interaction and superconductivity in C 60 n− fullerides

The closeness of low-lying T_1u and T_1g levels of C ₆₀ ⁻ could enable their mixing under an odd parity vibration of (T_{1 u} + T_{1 g} ⊗ (h_g + τ_{1 u})type. In addition, the two levels are susceptible to Jahn-Teller interaction due to five-fold degenerate h_g vibrations. This complex problem of (T_1u+T_1g)⊗(h_g+τ_1u) vibronic interaction is transformed to a form similar to T_2g ⊗ (ε_g + τ_2g) vibronic problem of octahedral symmetry. The problem is analysed in an infinite coupling model and compared with the experimental spectroscopic results for the C ₆₀ ⁻ radical. The resulting parameters are used to calculate the pair-binding energy and superconducting transition temperature in C ₆₀ ⁿ⁻ fullerides. Vibronic mixing with the T_1g level is found to be responsible for maximising the pair-binding energy at the doping level n=3. It is also found to be an important source of T_c enhancement.     

Magnetic circular dichroism and absorption spectra of Cs2ZrCl6:Mo4+

Absorption and magnetic circular dichroism (M.C.D.) spectra of Mo⁴⁺ in Cs₂ZrCl₆ have been recorded under high resolution at liquid helium temperature over the range 20 000–38 000 cm^-1. All of the bands can be assigned to parity-allowed, ligand-to-metal charge-transfer transitions in both MoCl₆ ^2-(4d²) and an impurity ion which has been identified as MoOCl₅ ^2-(4d¹). The d¹ and d² systems are easily distinguished by the temperature dependence of their M.C.D. spectra, and detailed assignments are proposed for the latter. The absorption spectrum in the region 23 000–26 000 cm^-1 has recently been attributed to a d→d transition in the ion MoCl₆ ^-(4d¹) [1], but the M.C.D. spectrum proves this assignment to be incorrect, and we find no evidence for the presence of this singly-charged species. From the temperature dependence of the M.C.D. spectrum of MoCl₆ ^2-, we estimate the second-order spin-orbit splitting of the E and T ₂ components of the ³ T _1g ground state to be 27 ± 5 cm^-1 with E lowest. The electrostatic splitting of the ³ T _2u and ⁵ T _2u charge-transfer states arising from the t _1u (π + σ) →t _2g excitation is estimated to be close to 515 cm^-1.     

The analysis of nitrogen rotational and vibrational bands in a helium microhollow gas discharge

Emission spectroscopy is applied to measure the gas temperature T _g and the vibrational distribution of N₂ (C ³Π_u) and N₂ ⁺(B ²Σ_u ⁺) excited states from a helium microhollow gas discharge (MHGD) at atmospheric pressure. The rotational temperature T _rot of N₂ ⁺ is determined from relative intensity of the R‐branch lines of the N₂ ⁺(B ²Σ_u ⁺–X ²Σ_g ⁺) bands at 427.81 and 419.91 nm and the well‐known Boltzmann plot (BP). Using the same diagnostic technique, the rotationally resolved N₂(C ³Π_u–B ³Π_g) band at 380.49 nm is used to measure T _rot. Under our experimental conditions, T _g is equal to T _rot = 550–650 K for nitrogen molecules and shows a slight increase with the discharge current in the current range 3–10 mA. From the intensity ratio of two consecutive vibrational bands of the same sequence, the N₂(C ³Π_u) and N₂ ⁺(B ²Σ_u ⁺) vibrational temperature T _vib = 3,700–4,000 K is determined. It has been found that N₂ ⁺(B ²Σ_u ⁺) ions have non‐Boltzmann distribution in the helium MHGD, while N₂(C ³Π_u) molecules are populated according to the Boltzmann distribution. Following the Franck–Condon principle, the vibrational distribution of the ground state of N₂(X ¹Σ_g ⁺) molecules has been determined from the N₂(C ³Π_u) distribution using the inversion matrix of elements q _XC(ν ,ν ′).     

An invariant measure for the equationu tt −u xx +u 3=0

Numerical studies of the initial boundary-value problem of the semilinear wave equationu _tt –u _xx +u ³=0 subject to periodic boundary conditionsu(t, 0)=u(t, 2),u _t (t, 0)=u _t (t, 2) and initial conditionsu(0,x)=u ₀(x),u _t(0,x)=v ₀(x), whereu ₀(x) andv ₀(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial stateu ₀(x),v ₀(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above returning property.     

Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas

This paper studies an initial boundary value problem for a one-dimensional isentropic model system of compressible viscous gas with large external forces, represented by v _t –u _x =0,u _t +(av ^–) _x =(u _x /v) _x +f( ₀ ^x vdx,t), with (v(x, 0),u(x, 0))= (v ₀(x),u ₀(x)),u(0,t)=u(1,t)=0. Especially, the uniform boundedness of the solution in time is investigated. It is proved that for arbitrary large initial data and external forces, the problem uniquely has an uniformly bounded, global-in-time solution with also uniformly positive mass density, provided the adiabatic constant (>1) is suitably close to 1. The proof is based on L ²-energy estimates and a technique used in [9].     

A Construction of Blow up Solutions for Co-rotational Wave Maps

The existence of co-rotational finite time blow up solutions to the wave map problem from ${\mathbb{R}^{2+1} \to N}The existence of co-rotational finite time blow up solutions to the wave map problem from \mathbbR²⁺¹ ? N{\mathbb{R}^{2+1} \to N} , where N is a surface of revolution with metric d ρ ² + g(ρ)² dθ², g an entire function, is proven. These are of the form u(t,r)=Q(l(t)t)+R(t,r){u(t,r)=Q(\lambda(t)t)+\mathcal{R}(t,r)} , where Q is a time independent solution of the co-rotational wave map equation −u _tt  + u _rr  + r ⁻¹ u _r  = r ⁻² g(u)g′(u), λ(t) = t ^−1-ν, ν > 1/2 is arbitrary, and R{\mathcal{R}} is a term whose local energy goes to zero as t → 0.     

Experimental determination of the potential energy curves of the I(3/2u) and I(3/2g) states of Kr+ 2

The I(3/2u) and I(3/2g) states of Kr⁺ ₂ have been investigated by pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectroscopy following (2 + 1′) resonance-enhanced multiphoton excitation via the 0⁺ _g Rydberg state located below the Kr?([4p]⁵5p[1/2]₀) + Kr(¹S₀) dissociation limit of Kr₂. From the positions of a large number of vibrational bands in the spectra of the ⁸⁴Kr₂ and ⁸⁴Kr-⁸⁶Kr isotopomers, the adiabatic ionization potentials (IP(I(3/2u)) = 112672.4 ± 0.8cm^?1, IP(I(3/2g)) = 111 395.0 ± 1.4cm^?1), the dissociation energies (D ⁺ ₀(I(3/2u)) = 368.8 ± 2.0cm^?1, D ⁺ ₀(I(3/2g)) = 1646.2 ± 2.3cm^?1) and vibrational constants for both ionic states have been determined. Potential energy curves have been extracted which perfectly reproduce all experimental observations and are accurate over a wide range of energies and internuclear distances. The equilibrium internuclear distances (R ⁺ _e(I(3/2u)) = 4.11 ± 0.04 Å, R ⁺ _e(I(3/2g)) = 3.35 ± 0.10 Å) have been derived by comparing the intensity distribution in the PFI-ZEKE photoelectron spectra to calculated Franck-Condon factors. The dissociation energy of the I(3/2g) state and the equilibrium internuclear distance of the I(3/2u) state differ markedly from previously reported values.     

The magnetic circular dichroism spectrum of the 1 B 2u ←1 A 1g transition of benzene in the vapour phase

The magnetic circular dichroism (MCD) spectrum of the ¹ B _2u ←¹ A _1g transition of benzene has been measured in the vapour phase. Many of the bands due to transitions between single vibronic levels display A terms. It has been shown that the angular momentum arises by vibronic mixing both of the ¹ E _1u state with the ¹ B _1u state and of the ¹ E _2g states with the ¹ A _1g ground state by e _2g vibrations. The magnitudes and signs of the experimental and calculated ratios, A/D, for the A ₀ ⁰ vibronic origin are in excellent agreement. Two strong MCD progressions of opposite sign with B-term dispersion have been observed in regions of low absorption. These are identified with vibronic origins due to the v ₈ and v ₉ e _2g modes. By contrast the MCD spectrum of hexadeuterobenzene vapour has a much lower magneto-rotational strength and displays none of the striking features of the benzene MCD spectrum.     

The magnetic circular dichroism spectrum of the hexaquocobalt (II) ion in the infra-red region

The magnetic circular dichroism (MCD) and absorption spectra in the range 5500–10 250 cm^-1 of a single cubic crystal of [Co(H₂O)₆](BrO₃)₂ have been measured at various temperatures between 5 and 300 K. An analysis of the temperature variation of the zeroth absorption and MCD moments shows that the intensity of the E′ (⁴αT _1g ) → ⁴ T _2g transition is provided mainly through the two cobalt-oxygen t _1u vibrations. The signs and magnitudes of the overall b and c term are in reasonable accord with previously published calculations. The low energy side of the MCD band exhibits some zerophonon lines and a complex vibronic structure which appears to involve low energy lattice vibrations as well as the cobalt-oxygen skeletal vibrations.     

Electronic pair excitations of Mn2+ and Ni2+ in perovskite fluorides: involvement of the 3 T 1g a state of Ni2+

Crystals of KZnF₃ and KMgF₃ doped with Mn²⁺ and Ni²⁺ were used to study the spectroscopic properties of Mn²⁺-F^--Ni²⁺ pairs. Pair transitions to the doubly excited states ⁴ E_g ^a , ⁴ A _1g (Mn)³ T _1g ^a (Ni) and ⁴ E_g ^b (Mn)³ T _1g ^a (Ni) were observed. The participation of the spin-allowed ³ A _2g →³ T _1g ^a excitation on Ni²⁺ in the pair transition is explained by spin-orbit mixing between ³ T _1g ^a and ¹ E_g . The prominent electronic origins are assigned to the double spin-flip transitions ⁶ A _1g (Mn)³ A _2g (Ni) →⁴ E_gu(Mn)³ T _1g ^a (Γ₃)v(Ni) and ⁴ A _1g (Mn)³ T _1g ^a (Γ₃)v(Ni). The former lie at lower energy and are more intense than the corresponding ⁶ A _1g (Mn)³ A _2g (Ni) →⁴ E_gv(Mn)³ T _1g ^a (Γ₃)u(Ni) transitions involving two orbital jumps. The well-resolved vibronic structure is composed of three basic vibrations of ～ 150 cm^-1, ～ 294 cm^-1 and ～ 508 cm^-1 in the KZnF₃ host.     

E.S.R. evidence for the planarity of the t-butyl radical

Monte Carlo calculations are reported for the radial distribution function g ₂(r; λ) of a fluid in which the intermolecular pair potential is [u ^ref(r) + λu ^p(r)], u ^ref(r) being the Weeks-Chandler-Andersen (WCA) reference fluid, and [u ^ref(r) + u ^p(r)] being the Lennard-Jones (6, 12) fluid. The calculations are performed for λ values in the range 0 to 1, at the state condition ρσ³ = 0·80, kT/ε = 0·719. It is shown that at high densities the perturbation expansion of g ₂(r; λ = 1) about g ₂(r; λ = 0) is rapidly convergent, but that the corresponding expansion for y ₂(r; λ) = exp [βu(r; λ)] × g ₂(r; λ) is not. In addition Monte Carlo estimates of the individual terms that contribute to the first-order perturbation term, (?g ₂/?λ)_λ=0, are presented. It is shown that these terms are individually large, but that (?g ₂/?λ)_λ=0 is small because there is strong cancellation between the various terms. Consequently, the calculation of (?g ₂/?λ)_λ=0 is highly sensitive to the approximation used to evaluate the individual terms.     

Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation u _t =−(u u _xxx ) _x , modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u ₀(x)∈H ¹(ℝ), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L ¹-norm of the strong solution u(x,t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses analogies between the thin film equation and the porous medium equation. Received: 2 February 2001 / Accepted: 7 October 2001     

Vortex rings moving into turbulence

We report on an experimental study of turbulent vortex rings injected with velocity U _v0 into a grid-generated turbulent flow (with RMS streamwise velocity u _*) and followed relative to the mean flow. The initial Reynolds number of the vortices varies from 4500 to 11,500. The turbulence was characterised by an intensity I_t =u _*/U _v0, which varied over the range 0<I_t <0.03. A mathematical model based on a stochastic model of the vortex core is developed to explain and interpret the results. The vortex radius grows diffusively in time with the rate of increase of the square of the vortex radius increasing linearly with I_t . As the vortices grow, they slow down sufficiently rapidly in a manner that they penetrate a finite distance into the turbulence. The vortex velocity, averaged over many experiments, showed an initial t ^?1 decay, consistent with Maxworthy’s experiments. The analysis and experiments show that such vortices ultimately only move a finite distance from their point of generation and this distance varies inversely with I_t .     

Calculation of the quenching rate constants for electronically excited singlet molecular nitrogen

The quenching rate constants for the singlet states (a′)⁽¹⁾Σ _u ⁻ (v = 1−17), a ⁽¹⁾Π _g (v = 0−14), and w ⁽¹⁾Δ _u (v = 0−13) of molecular nitrogen colliding with an N₂ molecule are calculated using quantum-chemical approximations. It is shown for the first time that both the intramolecular and intermolecular processes of electronic excitation transfer are significant for these states. Calculated rate constants are in satisfactory agreement with experimental data.