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Volume 64, Issue 2, 15 January 1976

Effect of temperature and quencher concentration on vibrational relaxation in condensed media
View Description Hide DescriptionIn this paper, we are concerned with the temperature effect on vibrational relaxation and the vibrational energy transfer from the vibrationally excited donor to the acceptor. For the temperature effect, we present numerical results to show the temperature dependence of the rate constant of vibrational relaxation and to discuss the validity of the rate constants obtained from the use of the weak coupling approximation and the strong coupling approximation. It is shown that although the temperature effect is extremely large over the temperature range T=0 to T=ϑ_{ E }, the Einstein temperature of the medium, for the temperature range T=0 to T=0.3 ϑ_{ E }, the rate constant varies slowly with temperature. For the vibrational energy transfer, we derive the master equation to describe the time dependent behavior of the excited donor, and the expression for the rate constant of vibrational energy transfer. The master equation is solved to study the temporal behavior of the excited donor as a function of the acceptor concentration. Numerical results are presented to demonstrate the theoretical results.

Cross sections and rate constants for low‐temperature ^{4}He–H_{2} vibrational relaxation
View Description Hide DescriptionConverged coupled‐states integral cross sections were determined for the vibrational relaxation of the v=1 j=0 level of p‐H_{2} in collisions with ^{4}He. The collision energies ranged from 0.005 to 0.4 eV. The Gordon–Secrest (GS) potential was used with both a harmonic (HO) and rotating‐Morse oscillator (MO) description of the H_{2} molecule. Additional calculations incorporated modifications in the long‐range and spherically symmetric (V _{0}) parts of the purely repulsive GSsurface. Rotational coupling plays a major role in vibrational relaxation even at thermal energies. Although the relative importance of individual vibration–rotation transitions is a sensitive function of the choice of intramolecular potential, the HO and MO total relaxation cross sections are nearly identical. These total relaxation cross sections exhibit a power‐law dependence on the initial translational energy down to ∼0.1 eV above threshold; below which point a positive curvature appears for all the surfaces considered. Rate constants for vibrational relaxation were computed and compared with experiment for 60 °K <T<500 °K. The GS rates lie below the experimental values and exhibit a less pronounced curvature at low temperatures. Introducing the correct long‐range attraction into the GSV _{0} potential results in a modest increase in curvature. Agreement with experiment is obtained only with the use of the semiempiricalV _{0} potential of Shafer and Gordon.

On the significance of finite propagation speeds in multicomponent reacting systems
View Description Hide DescriptionA continuum model of diffusion in a reacting mixture is built around a constitutive relation for momentum exchange by frictional interaction between diffusing species. The resulting continuity and momentum equations incorporate a finite relaxation time for diffusion. Under appropriate conditions, linearization of these equations produces a coupled system of hyperbolic equations for small disturbances of a stationary state. Fick’s law can be recovered from the linear equations by assuming instantaneous relaxation of the flux, provided that the stationary state is uniform. Fick’s law is generally inconsistent with the momentum equation when the stationary state is nonuniform. The stability of uniform stationary solutions predicted by the parabolic system obtained when Fick’s law is used is compared with the stability predicted by the hyperbolic system. When the former predicts stability and has at least one pair of complex‐conjugate roots, the latter may predict that the stationary state is unstable. Thus, inclusion of relaxation in the model can lead to qualitatively different predictions of stability.

Spin–rotation measurements and magnetic shielding in CH_{2}CF_{2}
View Description Hide DescriptionRotational transitions 1_{01}→0_{00}, 2_{12}→1_{11}, 2_{11}→2_{12}, and 3_{21}→3_{22} of 1,1 difluoroethylene were studied using a molecular beammaser spectrometer. Emission transitions were observed with 6 kHz linewidth (FWHM) and hyperfine structure due to fluorine and hydrogen interactions was resolved. The diagonal elements of the fluorine spin–rotation tensor are M _{ a a }=−13.4±3.6 kHz, M _{ b b }=−10.94±0.68 kHz and M _{ c c }=−6.06±0.68 kHz. This data is used to determine paramagnetic contributions to the fluorine chemical shifttensor.Analysis of the data yielded line centers for the rotational transitions of 15 774 348±3 kHz for 1_{01}→0_{00}, 26 465 055±5 for 2_{12}→1_{11}, and 14 419 906±3 kHz for 3_{21}→3_{22}, and 15 250 615±3 kHz for 2_{11}→2_{12}. Results of semiempirical calculations of diamagnetic contributions to the shielding tenors are presented for CH_{2}CF_{2} and other molecules. Comparisons of total shielding tensors are made for related molecules.

Photoabsorption cross sections of two‐electron atoms by the coordinate rotation method: Application to H^{−} and several states of He
View Description Hide DescriptionThe coordinate rotation method, recently extended by us to treat photoabsorption processes, is used to obtain photoabsorption cross sections for several two‐electron atoms. The calculations are performed using standard configuration–interaction methods; the need for atomic continuum wavefunctions is completely avoided in this approach. We have computed the photodetachment cross section of H^{−} and photoionization cross sections for He in its ground and 2 ^{1} S states. In all cases, the computed cross sections agree well with results obtained by numerical integration and with available experimental data.

A Monte Carlo study of ion–water clusters
View Description Hide DescriptionThe procedures developed in this paper have been employed to calculate theoretical free energies of formation of ion–water clusters for comparison with experiment. Gibbs free energies were calculated for the gas phase reaction, ion(H_{2}O)_{ N−1}+H_{2}O_{(vapor)}=ion(H_{2}O)_{ N }, for the Li^{+}, Na^{+}, K^{+}, Cl^{−}, and F^{−} ions and for N=1–6. The standard state for all calculations was taken as 298 °K and 1 atm. The Monte Carlo method was used to evaluate the appropriate classical expressions of statistical mechanics by employing the intermolecular potential functions recently developed from a b i n i t i o Hartree–Fock calculations. Enthalpies, entropies, and structural information were also calculated. Agreement with experiment is sufficiently good to demonstrate the feasibility of this approach.

Self‐consistent field with pseudowavefunctions
View Description Hide DescriptionA computational method is given in which the energy of an atom is computed by using pseudowavefunctions only. The method centers on a model energy expression E _{ M } which is similar to the Hartree–Fock energy expression, but contains only pseudowavefunctions. A theorem is proved according to which the Hartree–Fock orbitals can be transformed by a linear transformation into a set of uniquely defined pseudowavefunctions which have the property that when substituted into E _{ M } this quantity will closely approximate the Hartree–Fock energyE _{ F }. The new method is then formulated by identifying the total energy of an atom with the minimum of E _{ M }. Application of the energy minimum principle leads to a set of equations for the pseudowavefunctions which are similar to but simpler than the Hartree–Fock equations. These equations contain pseudopotentials for which explicit expressions are derived. The possibility of replacing these pseudopotentials by simpler model potentials is discussed and the criteria for the selection of the model potential are outlined.

Modified ’’ideal three phase model’’ and the melting of alkali metals
View Description Hide DescriptionA model potential for real materials is proposed that consists of inverse‐power, 1/r ^{ n }, repulsion with a density dependent power, n (ρ), and K a c attraction, −γ^{3} exp(−γr). The function n (ρ) is determined semiempirically from compressibility factor data along the melting line. The model is applied to the alkali metals and is capable of reproducing the Kraut–Kennedy law and the maximum in the melting curve.

Semiclassical eigenvalues for nonseparable systems: Nonperturbative solution of the Hamilton–Jacobi equation in action‐angle variables
View Description Hide DescriptionIt is shown how the Hamilton–Jacobi equation for a multidimensional nonseparable system can be efficiently solved directly in action‐angle variables. This allows one to construct the total (classical) Hamiltonian as a function of the ’’good’’ action‐angle variables which are the complete set of constants of the motion of the system; requiring the action variables to be integers then provides the semiclassical eigenvalues. Numerical results are presented for a two‐dimensional potential well, and one sees that the semiclassical eigenvalues are in good agreement with the exact quantum mechanical values even for the case of large nonseparable coupling.

On the hyperfine splitting of the hydrogen atom in a spherical box
View Description Hide DescriptionThe hyperfine splitting of the hydrogenlike atom confined to a sphere of radius r _{0} is examined as a function of r _{0}. The wavefunction is made to vanish at r=r _{0} and is normalized to unity over the finite sphere. The effect of decreasing r _{0} (i.e., compressing the sphere isotropically) on the hyperfine splittingA of the ground state is nonlinear, with A increasing approximately as r ^{−1} _{0}. Comparison with atomic hydrogen trapped in α‐quartz is made.

On the ultraviolet spectrum of t r a n s‐1,3‐butadiene
View Description Hide DescriptionThe ultraviolet absorption spectra of t r a n s‐1,3‐butadiene, 1,1,4,4‐t r a n s‐1,3‐butadiene‐d _{4}, and t r a n s‐1,3‐butadiene‐d _{6} between 2300 and 1350 Å have been recorded and analyzed. Four Rydberg series were identified (δ=0.087, 0.21, 0.42, and 0.67). A transition‐by‐transition comparison of this analysis with those in the literature shows that several of the previously assigned transitions are misidentified. A continuum originating around 1450 Å is reported.

^{14}N nuclear quadrupole resonance and relaxation measurements of sodium nitrite
View Description Hide DescriptionMeasurements of the quadrupole coupling constant, asymmetry parameter, inverse linewidth parameter T*_{2}, spin–spin relaxation timeT _{2}, and spin–lattice relaxation timesT _{1s } and T _{1l } are reported for sodium nitrite in the temperature region from 77–467 °K. The relations between the spin–lattice relaxation times and the transition probabilities W _{+}, W _{Δ}, and W _{−} are derived for a spin one system, and the transition probabilities are calculated from the T _{1} data. The results are compared with other NaNO_{2} studies and support the work of Avagadro e t a l. in the temperature region near the ferroelectric phase transition. They also indicate the existence of short‐range order above this transition.

Collisional quenching of O(^{1} S) by rare gas atoms and collision‐induced emission by O(^{1} S)‐rare gas eximers
View Description Hide DescriptionThe quenching and collision‐induced emission of the O(^{1} S) atom by the rare gas atoms M=Xe, Kr, Ar, Ne, He has been investigated at 201 and 291 K. After pulsed photolytic production of O(^{1} S) the emission decay rate as well as the intensity was investigated as a function of added gas concentration. Second‐order O(^{1} S) collisional quenching coefficients, k _{exp}, and efficiency coefficients, k*_{exp}, for the collisionally induced eximer emission have been measured. The formation of the O(^{1} S)Xe eximer occurred with thermal equilibrium in the O(^{1} S)Xe state by steady‐state termolecular recombination, the O(^{1} S)Xe potential well depth being ∼0.06 eV. For Kr, Ar, Ne,and He the quenching and emission enhancement did not exhibit a temperature dependence, showing that thermal equilibrium and steady‐state recombination was applicable. The potential depths are small compared to k T, where T=201 K. The O(^{1} S) removal occurs practically only via O(^{1} S) + M (+ M ) O(^{1} S)M(+M), O(^{1} S) MO(^{1} S)M+h v, with A* being an averaged molecular radiation probability. Other potential processes, in particular bimolecular quenching is unimportant. The mechanism is determined quantitatively by k _{exp}=A⋅k*_{exp}= A*⋅ (k _{ f }/k _{ r }) =A*⋅K, where K is the O(^{1} S)M equilibrium constant,A the O(^{1} D←^{1} S) Einstein coefficient. From k _{exp}, k*_{exp} and K values for A* have been derived. Also upper limits of coefficients for the collisional removal of O(^{1} S)M have been obtained. The numerical results are given in the paper.

Mode coupling description of dynamics in dilute polymer solutions
View Description Hide DescriptionA calculation of the dynamic structure factor for a dilute polymer solution which explicitly takes into account the coupling to the viscous modes of the fluid is presented. The techniques of mode–mode coupling are used to extract this contribution from the Liouville equation for the relevant polymer variable. In the small wave vector regime it is shown that this contribution yields exactly the hydrodynamic contribution to the diffusion coefficient which appears in other theories when the interactions between the monomers via the solvent are calculated by using the Oseen tensor. For large wave vectors the long range correlations in the polymer chain give rise to a nonlocal memory kernel similar to that of the anomalous part of the mutual diffusion coefficient near the binary mixture critical point. The form of this wave vector dependence is examined for several models of the polymer chain.

EPR study of NO_{2} and NO_{3} produced in urea nitrate by uv irradiation, x‐irradiation, and constituent tritium atom decay
View Description Hide DescriptionNO_{2} and NO_{3} radicals have been produced by uv irradiation, x‐irradiation, and constituent tritium atom decay in single crystals of urea nitrate. Two types of NO_{3} and five types of NO_{2} were produced, and which of these were observed depended on the type of irradiation and the temperature. At 77 °K two types (two different orientations) of NO_{2} were formed from tritium decay while only one of these was produced by x‐irradiation. The various forms of NO_{2} were found to irreversibly interconvert from one to another as the temperature increased. Different mechanisms for defect formation have been proposed to account for the different types of radicals formed by x‐irradiation and tritium decay at room temperature and 77 °K. The NO_{3} species which resulted from any of the methods of irradiation at room temperature showed no ^{14}N hfs but did exhibit a significant interaction with five neighboring hydrogen bonded protons. Deuterium substitution caused the proton hfs to collapse into one line, thus simplifying the analysis. This NO_{3} was the same one reported formed by γ‐irradiation. A different type of NO_{3} without proton hfs was formed when either x‐irradiation or tritium substitution was used at 77 °K. These two types did not interconvert. From evaluation of the spin‐Hamiltonian parameters the electronic ground states for these NO_{3} radicals have been elucidated. The low temperature form is found to be a π radical and the high temperature form is a σ radical.

Structural phase transitions in solids with applied stresses and fields, and effect of isotopic impurities on the free energy
View Description Hide DescriptionSubject to the assumptions described in the text, we obtain the following expression for the transition temperature T _{ c } associated with a structural phase transition:T _{ c } = α (ΔU + J_{ i } P _{ i }Δa _{ i })/k _{ B }, α ≡ 2/ln(‖Φ‖/‖Φ′‖). Here ΔU is the change in energy, the P _{ i } are given by the pressure, stresses, and applied fields, the Δa _{ i } are given by the changes in volume, strains, and dipole moments, and ‖Φ‖ is the determinant of the force constant matrix. We also give a result which may be useful for solids containing isotopic impurities: At high temperatures and in the harmonic approximation, ΔG = −(3/2) k _{ B } T J_{ k }ln(m _{ k }′/m _{ k }), where ΔG is the change in Gibbs free energy when some of the particle masses are changed from m _{ k } to m _{ k }′.

Pattern formation in reacting and diffusing systems
View Description Hide DescriptionA mechanism is described, whereby stable sharply differentiated (dissipative) structures can evolve naturally within a mixture of reacting and diffusing substances. Our model has two reacting components, with one diffusion coefficient much smaller than the other. Unlike patterned states obtained by small amplitude analysis near uniform states, our structures have large amplitude and serve to divide the reactor into subregions, each corresponding to a distinct phase for the system. The evolution of the structured stationary state from an arbitrary initial distribution occurs in two stages. The first involves differentiation into subregions, and the second involves the migration of the boundaries of the subregions into a stable final configuration. A singular perturbation analysis and the theory of motion of wavefronts is used to deduce these qualitative properties.

A ^{3}Σ^{+} _{ u } molecules in the N_{2} afterglow
View Description Hide DescriptionExcitation of N_{2} by electrons with energy below the ionization threshold produces a strong afterglow in the B ^{3}Π_{ g }–A ^{3}Σ^{+} _{ u } system. The precursor is identified as the A ^{3}Σ^{+} _{ u } state in the v≳6 vibrational levels. The peak electron cross section of the A ^{3}Σ^{+} _{ u }–X ^{1}Σ^{+} _{ g } system is estimated at about three times the B ^{3}Π_{ g }–X ^{1}Σ^{+} _{ g } transition. Deactivation rates of the A ^{3}Σ^{+} _{ u } v≳6 levels by X ^{1}Σ^{+} _{ g } molecules vary over the 8E−12 cm^{3}sec^{−1}–5E−11 cm^{3}sec^{−1} range. There is some evidence that the higher levels may relax at rates comparable to that of electronic deactivation. Deactivation of B ^{3}Π_{ g } by X ^{1}Σ^{+} _{ g } molecules has a strong dependence on the vibrational level of the B ^{3}Π_{ g } state. The rate coefficients for this process vary between 1E−11 cm^{3}sec^{−1} and 1E‐10 cm^{3}sec^{−1}, with no measureable contribution by vibrational relaxation. The results suggest that production rates of the B ^{3}Π_{ g } state in the Lewis–Rayleigh (L–R) afterglow are much more uniformly distributed over the vibrational levels than has been previously assumed. We also suggest it is unlikely that significant amounts of energy pass through the ^{5}Σ^{+} _{ g } state in the L–R afterglow. About 25% of the energy of atomic nitrogen recombination enters the B ^{3}Π_{ g } state in the L–R afterglow, according to the present results. The factors controlling the A ^{3}Σ^{+} _{ u } v=0,1 population in the L–R afterglow appear to be much less well defined than has been suggested in previous literature.

Low temperature specific heat and phonon anomalies in transition metal compounds
View Description Hide DescriptionThe specific heat of the following pairs of transition metalcompounds (TMC’s) has been measured in the temperature range 1.8<T<100 K : TiN/TiC, NbC/ZrC, and TaC/HfC. Theoretical results for the lattice specific heat, obtained from phonon densities calculated on the basis of appropriate shell models agree well with the experimental data. Values for Debye temperatures and Sommerfeld coefficients are given and compared with previously published data. We show that from the difference in the lattice specific heat between the superconducting and the normal compound of each pair, it is possible to determine the average frequency of the lowest‐lying phonon anomalies occurring in the superconducting TMC. For TiN, NbC, and TaC, this frequency was determined to lie at 5.8, 4.8, and 3.1 THz, respectively. These values compare favorably with results from neutron scattering experiments. Thus, specific heatmeasurements can serve as a valuable tool to determine the mean frequency of the phonon anomalies in the superconducting compounds of this class of materials.

HF infrared chemiluminescence: Energy disposal and the role of the radical fragment in the abstraction of hydrogen from polyatomic molecules by F atoms
View Description Hide DescriptionHF infrared chemiluminescence has been utilized to study the energy disposal for the abstraction of hydrogen by fluorine atoms from polyatomic molecules which yield radical fragments with large stabilization energies. The prototype systems selected for study, methyl benzenes, phenol, and acetonitrile, are cases which yield resonance stabilized radicals as products. Comparison is made to the energy disposal from the reaction of F with the primary C–H bonds of aliphatic hydrocarbons, which have smaller radical stabilization energies. In general the radical stabilization energy, which is associated with major changes in geometry of the radical relative to the parent molecules, was not available to the HF product. The reactions of F + benzene and ethylene also were studied to provide reference data for different types of C–H bonds. The HF vibrational energy distributions have been interpreted using an extension of the information theory which previously has been applied to three body reactions. Vibrational surprisal analyses are developed and discussed for three models of the reference (prior) product distributions: (i) the polyatomic fragment product was treated as an atom, i.e., the three body case, (ii) the rotations of the radical fragment were added to the three body model, (iii) a complete model including all vibrational and rotational modes of the polyatomic radical fragment. For (iii) with the use of the full thermochemical exoergicity linear surprisal plots were found and these plots were used to assign relative populations to HF (v=0). The information‐theoretic parameters from the three reference models are compared for a series of F+HR reactions in which R increases in complexity from Cl to CH_{2}C_{6}H_{5}. For reactions with large product stabilization energies, calculations for (i) and (ii) were done with a reduced ’’effective available’’ energy corresponding to the assumption that the energy available to HF was less than the full exoergicity. Some insight is gained into the role of the R fragment in the energy disposal.