Volume 55, Issue 1, 01 July 1971

On the Motion of Aerosols in Nonuniform Gases. I
View Description Hide DescriptionThe use of a variational approach for solving the problem of the motion of aerosol particles in nonuniform gases, at all rarefactions, is proposed. In particular, explicit variational formulations are given for the frictional drag and the force in diffusiophoresis.

Electron Spin Resonance and Relaxation Studies of Internal Motion in CH_{2}CO_{2} ^{−} Radicals Trapped in γ‐Irradiated Sodium Acetate Crystals
View Description Hide DescriptionCH_{2}CO_{2} ^{−} radicals are stable in γ‐irradiated sodium acetate crystals. The ESR spectra of CH_{2}CO_{2} ^{−} and carbon‐13 labeled radicals, CH_{2} ^{13}CO_{2} ^{−} and ^{13}CH_{2}CO_{2} ^{−}, were studied in detail in the temperature range of 77°K to room temperature. It was found that the CH_{2} group executes an internal motion which is characterized by proton exchange in the oriented radicals. The motion is slowed on lowering the temperature, giving rise to an electron spin relaxation time of the order of microseconds at 77°K. The correlation time for the motion was found to be temperature dependent, varying as exp with an activation energy of the order of 3 kcal mole^{−1}.

Statistical Mechanics for Long Semiflexible Molecules: A Model for the Nematic Mesophase
View Description Hide DescriptionA model system for long semiflexible rods is introduced to investigate the nematic mesophase. The free energy is separated into two parts , where represents a fully aligned, hard system and a correction which in our case is evaluated by the counting technique of DiMarzio extended to the case of nonrigid rods. may be estimated by a variety of techniques. We find several interesting features: (1) for both rigid and flexible cases the anisotropic phase undergoes a second transition at still higher densities to what we take to be the solid phase; (2) for the flexible case sensible values of length‐to‐breadth ratio and flexibility give the isotropic–anisotropic transition at liquid densities and with in keeping with experiment and in marked contrast to all available results for strictly rigid systems; (3) for a given temperature there is a lower bound to the stiffness energy (associated with making a bend) required for the existence of an isotropic–anisotropic transition. In the present model this energy has the value .

Radial Expansion of Orbitals and the Near Degeneracy of the and Excited States of Octahedral Mn(II) Complexes
View Description Hide DescriptionThe relationship between the wavefunctions for mostly molecular orbitals and the energies of the nearly degenerate and excited states of octahedral Mn^{2+} complexes is investigated from two points of view. First, the expressions relating the energy splitting to the degree of covalency have been extended within the framework of the intermediate neglect of differential overlap (INDO) approximation to include the large but previously neglected two‐center Coulomb integrals and all ligand one‐center Coulomb and exchange integrals. The results resemble those previously obtained in their inability to account for the spectral assignment of the state below for the MnF_{6} ^{4−} complex in the salts KMnF_{3}, RbMnF_{3}, and MnF_{2}. Second, the assumption of identical metal‐ion repulsion integrals for the and molecular orbitals is discarded in a simple analysis of an all‐electron molecular SCF calculation for the MnF_{6} ^{4−} complex. While both the and orbitals contain a metal‐ion orbital that is expanded relative to that for the free Mn^{2+} ion, the slightly greater radial expansion for the orbital is shown to qualitatively account for the observed ordering of the and states.

Experimental Core Polarization of Transition Ions from Electron Paramagnetic Resonance Data
View Description Hide DescriptionTreatment of available electron paramagnetic resonance data of ^{2}D ions of the and transition series taking into account the transfer of unpaired electronic density to the ligand molecules leads to experimental values of the core polarization within 20% of the theoretical estimates. Experimental values are given for Nb^{3+}, Mo^{5+}, Ag^{2+}, VO^{2+}, and Cu^{2+}. Tentative values are also given for two ions: Pt^{3+} and Re^{6+}.

Microwave Spectrum of BF
View Description Hide DescriptionDirect rotational transitions have been observed in the ground vibrational state of ^{10}B^{19}F and ^{11}B^{19}F produced in the gas phase by a discharge in flowing BF_{3}. Measurements on the transitions yield , and . New molecular parameters are derived for ^{11}BF including , and . The electric dipole moment of BF is determined to be 0.5 ± 0.2 D.

Reaction Kinetics in Stochastic Models
View Description Hide DescriptionTwo solublemodels of chemical reaction kinetics are treated. It is found that the equilibrium approximation overestimates the true rate constants by a factor that is typically about 2 and may often be much larger. The mathematical mechanism by which a high activation energy is reflected in a long chemical relaxation time is studied and is found to be the same as the mechanism by which, in quantum mechanics,tunneling through a high potential barrier between two wells in resonance leads to a slight separation between the two lowest energy levels.

Resonance Offset Effects in Multiple‐Pulse NMR Experiments
View Description Hide DescriptionThe behavior of multiple‐pulse NMR line narrowing experiments in solids as a function of the frequency offset from resonance is examined experimentally and theoretically. Resolution in the four‐pulse, six‐pulse, and phase‐alternated tetrahedral angle experiments was found to increase on going off resonance. These results are explained using coherent averaging theory. The effective Hamiltonian which determines the width of the spectral line off resonance must be averaged over the motion caused by both the rf pulse Hamiltonian and the resonance offset Hamiltonian. It is assumed that the period of the motion caused by one of these two cyclic Hamiltonians is much shorter than the other so that the averages can be carried out sequentially rather than simultaneously. Two relaxation rates are evident in the four‐pulse experiment on solids, corresponding to relaxation parallel and perpendicular to one of the body diagonals, e.g., the (111) direction, in the coordinate frame appropriate to that experiment. For solids containing a single magnetic nuclear species the two rates appear to be equal on resonance, but off resonance relaxation along the (111) direction is slower than relaxation perpendicular to it. This behavior is explained qualitatively by the fact that average Hamiltonians which do not in general commute with do commute with this operator when they are further averaged over the motion caused by an offset from resonance. The effective heteronuclear dipole–dipole interaction Hamiltonian commutes with both on and off resonance, which accounts for the difference in the two relaxation rates independent of resonance offset observed in solids containing more than one magnetic nuclear species.

Collision Dynamics and the Statistical Theories of Chemical Reactions. II. Comparison of Reaction Probabilities
View Description Hide DescriptionThe reaction probability obtained as a function of total energy from transition‐state theory for classical collinear atom, diatomic‐molecule exchange reactions is compared with the results of trajectory calculations for the same potential surface. For a well‐defined barrier in the symmetric H, H_{2}reaction, there is excellent agreement up to rather high energies. Other reactions (e.g., H_{2}+Br⇄HBr+H) yield considerably larger deviations; these can be shown to result from a transmission coefficient that is less than unity, from a nonequilibrium distribution in the transition region, or from both effects.

Nonequilibrium Contributions to the Rate of Reaction. IV. Explicit Time‐Dependent Solutions
View Description Hide DescriptionThe explicit time dependence of the corrections to the equilibrium rate of reaction and the equilibrium rate of change of the temperature in one‐component monatomic systems is obtained by integrating the nonlinear Boltzmann equation with a moment method. The calculations are carried out for the model systems employed in the earlier studies. For sufficiently small values of the ratio of the elastic and reactive relaxation times, it was found that in a time the time‐dependent corrections increased rapidly from zero to an asymptotic value. For times greater than , the asymptotic values of the corrections increased or decreased slowly and approximately linearly with time. The maximum values of the corrections as a function of time (in instances where a maximum value was obtained) were compared with the corresponding result obtained employing the Chapman–Enskog solution of the Boltzmann equation. This qualitative comparison indicates that for the Chapman–Enskog result differs from the present result by 10%. For , the deviation between the two results is reduced to 1%–3%. In the limit as , the present results coincide exactly with the Chapman–Enskog result.

Ultraviolet Absorption Spectra of Transition Metal Atoms in Rare‐Gas Matrices
View Description Hide DescriptionAbsorption spectra from 2200 to 4000 Å have been obtained of chromium,manganese,iron,cobalt,copper,nickel,tin, and palladium atoms trapped in argon matrices at 4.2°K and of iron and copper in krypton and xenon matrices at 4.2 and 20°K. Observed transitions were found to correlate with gas phase transitions, under the assumption of a matrix and atomic configuration dependent shift of the atomic transitions. Energy shifts of the transitions were inversely proportional to matrix atom size. Configurations with an odd number of electrons were shifted less than those with an even number. Within a given configuration, the transitions at higher energies were shifted more than those at lower energies. A Lennard‐Jones potential was unsuccessful in generating the observed energy shifts caused by the interaction between the trapped atom and the matrix. Atom diffusion and resultant aggregation within the matrix both during the condensation of the solid from the gas phase and during warming was found to be a significant effect. Oscillator strengths, calculated from observed spectra, were from 10 to 100 times less than gas phase values.

Core‐Electron Binding Energies and Slater Atomic Shielding Constants
View Description Hide DescriptionThe Slater–Zener expression for the energy of an atom is generalized to include hypothetical states for which the Slater atomic shielding constants do not minimize the energy. The generalized energy expression is used with Slater's rules for atomic shielding constants to study the Koopmans' theoremionization potential for core electrons and its dependence on nuclear charge and the number of valence electrons on an atom. An electronic relaxation energy for x‐ray states is defined and is shown to be proportional to the number of electrons in shells outside the core hole, independent of nuclear charge, and quadratic in the change of shielding constant for each shell upon core‐hole formation. Finally, arguments are given to show that the core hole should be thought of as localized, even when there are several equivalent atomic sites over which it may be delocalized.

Improved Perturbation Theory for Inelastic Encounters
View Description Hide DescriptionAn improved perturbation theory for inelastic bimolecular collisions is proposed, which is based upon the suitable choice of an effective distortion potential derived from a second‐order adiabatic optical potential. Since the new theory involves simply a rescaling of the original effective interaction, essentially no additional labor is required to obtain the corrected transition probabilities, at least for the special case of a one‐dimensional problem. For example, an application to the prototype problem of vibrational relaxation, namely the collinear collision of an atom with a harmonic oscillator, reduces the error of the usual distorted wave approximation from 116% to 5%, averaging over all cases for which exact multistate results exist for comparison. Furthermore, a simple prediction concerning the accuracy of the distorted wave approximation for resonance transitions is borne out by exact results.

Hall Coefficient of Metal–Ammonia Solutions
View Description Hide DescriptionThe Hall coefficient and electrical conductivity of Li–NH_{3}, K–NH_{3}, and Ca–NH_{3} solutions were measured as a function of concentration and temperature. The temperature range extended from 203–243°K and the concentration extended from metal saturation down to ≈ 4 MPM. The Hall coefficient was temperature independent for all solutions studied, but upon dilution, increased to values above the free‐electron value at different electron concentrations for the monovalent and divalent solutions. The conductivity values ranged from 100–5000 Ω^{−1}·cm^{−1}, and was positive. Using an ionized impurity scattering model and an adjustable effective mass for the metal–ammonia solutions, a qualitative fit for the conductivity, Hall coefficient, and thermopower can be made in the region 4–8 MPM. The loss of degeneracy upon dilution seems to be responsible for anomalies in this region, and hence the metal–nonmetal transition can be narrowed down to the 1–4 MPM region.

Test of Combining Rules for Intermolecular Distances. Potential Function Constants from Second Virial Coefficients
View Description Hide DescriptionA new combining rule for equilibrium distances in intermolecular potential functions, , is tested by means of experimental second virial coefficients of gas mixtures. It is found to be superior to the Lorentz “arithmetic mean” rule.

Fusion Curve and Electrical Conductivity of Molten HgCl_{2} and HgI_{2} at Elevated Pressure
View Description Hide DescriptionThe fusion curves of HgCl_{2} and HgI_{2} were measured to a pressure of 22 kbar. The melting temperature can be represented as a function of pressure by , where is the melting temperature in degrees Centigrade at a pressure in kilobars; and are constants. Values for (deg), (deg kbar^{−1}), and (deg kbar^{−2}) are 276, 259; 19.8, 17.1, and − 0.26, − 0.41, respectively. The electrical conductivities of molten HgCl_{2} and HgI_{2} were measured to 805 and 620°C, respectively, at a pressure of 5.4 kbar. The electrical conductivity of HgCl_{2}(l) was also measured from 557 to 634°C at . At these elevated pressures varies exponentially with , i.e., . At 5.4 kbar, and 0.36 (ν·cm)^{−1}; and 2.53 ± 0.12 kcal/mole, respectively, for molten HgCl_{2} and HgI_{2}. The behavior of HgCl_{2}(l) and HgI_{2}(l) at this elevated pressure is now typical for a strong electrolyte. This is in contrast to their behavior at ordinary pressures where HgCl_{2} shows a maximum in its vs curve and HgI_{2} has a negative temperature coefficient of from the onset of melting. The conductivities of molten HgCl_{2} and HgI_{2} were also examined from 3 to 20 kbar at constant . At 600°K, for HgCl_{2} = 12.4; at 555°K this ratio for HgI_{2} is 1.66. Thus pressure increases the conductivity of these salts. This is attributed to a greater degree of ionic dissociation at elevated pressures.

Photoionization Cross Sections for the Ground State Configuration of Atomic Nitrogen and Oxygen
View Description Hide DescriptionPhotoionization cross sections for the ground state configuration of N and O were calculated for photoelectron energies up to 50 eV using the scaled Fermi–Thomas potential method of Stewart and Rotenberg. All outer‐ and inner‐shell transitions contributing to the cross section in this energy range were included. Results are in good agreement with experiment and with results obtained by more elaborate but numerically much slower methods of calculation such as Hartree–Fock–Slater and related methods.

Determination of Thermal Transport Properties from Thermal Transpiration Measurements
View Description Hide DescriptionThe magnitude of the thermomolecular pressure difference across capillary tubes has been measured for the gases argon, nitrogen, oxygen, carbon monoxide, and methane at mean temperatures of 162.3, 233.5, 292.0, and 340.5°K. From these data, the translational part of the thermal conductivity (Eucken factor) has been determined for each of the polyatomic gases. Estimates have been made for the rotational relaxation number and the total thermal conductivity (Eucken factor) for these gases using the thermal conductivity formulas of Mason and Monchick (in the full and linearized forms) and of Saxena and co‐workers. It is observed that: (a) These three sets of formulas are equivalent, except for a scaling of the rotational relaxation number; (b) the rotational relaxation number for these gases is an increasing function of temperature; and (c) the values of the Eucken factor and rotational relaxation number obtained here are in very good agreement with the data for these parameters gotten by other methods.

Interpretation of the ^{57}Fe Isomer Shift by Means of Atomic Hartree–Fock Calculations on a Number of Ionic States
View Description Hide DescriptionThe Hartree–Fock method is used to calculate the values of for a number of configurations of different Fe ions. An interpolation formula for as a function of the occupation number of valence electron orbitals is given. Experimental Mössbauer isomer shifts are correlated to estimated configurations which are compared to available MO calculations for a few cases.

Proton Spin Relaxation in the Liquid Crystal PAA
View Description Hide DescriptionThe proton spin–lattice relaxation time of the liquid crystalpara‐azoxyanisole (PAA) was studied at 30 MHz in the temperature range 125–445°K. The rotating frame experiment was performed at 15 and 30 MHz in a rotating field of 6 G over the temperature interval 125–425°K, and the rotating field strength dependence of was studied in the range 1.6–16 G at three different temperatures. These observations show that in the nematic phase three processes are contributing to the spin–lattice relaxation: a slow cooperative mode, , the translational diffusion, and the order parameter fluctuation as proposed by Pincus. In the isotropic phase the order parameter fluctuations contribute much less because of the small size of the order clusters. The frequency and temperature dependence of various mechanisms are discussed and it is shown that the time autocorrelation function of the collective order fluctuations decays nonexponentially in the nematic phase.