Volume 47, Issue 10, 15 November 1967

Heat Capacity and Entropy of Mn(NH_{4})_{2}(SO_{4})_{2}·6H_{2}O from 1° to 4°K in Fields up to 24 000 Oe Perpendicular to the b Axis; Direct Observation of Magnetic Imperfection
View Description Hide DescriptionThe magnetization behavior of a spherical single crystal of Mn(NH_{4})_{2}(SO_{4})_{2}·6H_{2}O has been investigated from 1° to 4°K in fields up to 24 000 Oe perpendicular to the crystal plane, along 15 (approximately) isentropic paths for entropies ranging from about 1.6 cal mole^{−1}·deg^{−1} to 3.5 cal mole^{−1}·deg^{−1}. The heat capacity of the specimen has been measured at 0, 4360, 6450, 12 900, and 17 760 Oe. The two sets of measurements have been combined to obtain the entropy as a function of field and temperature over the range covered. The heat capacity and entropy are in all cases close to the values predicted by a Brillouin function with S=5/2, g=2.000 with small but significant deviations due to departures from ``ideal'' paramagnetic behavior. The deviations are shown clearly in the behavior of (H/T) along isentropes, which is presented for 10 particular values of the entropy. They correspond qualitatively to that predicted by the crystalline field obtained from other sources.

Theoretical Studies of Two‐Photon Absorption Processes. II. Model Calculations
View Description Hide DescriptionTwo‐photon absorption cross sections are calculated for the hydrogen atom and the hydrogen molecule. Three methods of calculation are employed: (1) a second‐order time‐dependent perturbation expansion; (2) a ``mean‐energy'' approximation; (3) a sum‐rule equivalent of perturbation theory. The various methods are compared and evaluated in terms of their general utility in describing double‐photon absorption processes.

Thermal Diffusion in Multicomponent Gas Mixtures
View Description Hide DescriptionThe thermal diffusion factor has been studied by an effective four‐tube trennschaukel for a ternary and a quaternary system. The systems studied are, (1) H_{2}–CO_{2} with He added as the third component and (2) H_{2}–CO_{2} with the He–Ne mixture added as the third and the fourth components. A pronounced dip has been obtained in the thermal diffusion factor vs composition curves for both the ternary and the quaternary systems. Attempt has been made to interpret the data in terms of the recent theory for thermal diffusion in polynary gas mixtures.

Floating Spherical Gaussian Orbital Model of Molecular Structure. I. Computational Procedure. LiH as an Example
View Description Hide DescriptionThe Kimball—Neumark spherical Gaussian orbital model is extended to apply to the singlet ground states of the general molecule with localized orbitals. Formulas are presented for energy, electron density, dipole moment, and the forces on nuclei and the computational procedure is described. The model is applied to LiH and the results are discussed in detail.

Floating Spherical Gaussian Orbital Model of Molecular Structure. II. One‐ and Two‐Electron‐Pair Systems
View Description Hide DescriptionThe floating spherical Gaussian orbital (FSGO) model is applied to the He and Be atom isoelectronic sequences and to the molecules and ions: H_{2}, He_{2} ^{+ +}, HeH^{+}, H_{3} ^{+}, H_{4} ^{+ +}, HeH^{−}, LiH, and BeH^{+}. Bond lengths are predicted generally to within 5% of accurate values. The He–He repulsive interaction is also calculated.

Microwave Spectrum of Propylene Imine
View Description Hide DescriptionThe microwave spectrum of propylene imine has been investigated in the range 9–40 GHz. The rotational constants associated with two species of molecules were found to be 16 893.00, 6533.71, and 5761.26 MHz for the trans isomer and 16 665.66, 6558.64, and 5812.36 MHz for what is probably the cis isomer. The dipole‐moment components for the trans isomer were found to be μ_{ a } = 0.21±0.06, μ_{ b } = 0.81±0.03, and μ_{ c } = 1.36±0.02 D, and for the other species of molecule, μ_{ a } = 1.28±0.01, μ_{ b } = 0.95±0.08, and μ_{ c } = 0.78±0.08 D.

Electron Spin Resonance Investigation of Photochromic Sodalites
View Description Hide DescriptionSynthetic sodalites containing sulfur and showing considerable photochromic activity have been investigated by ESR. The center responsible for the color has been shown to be an electron trapped at a chlorine vacancy. The origin of the electron which is reversibly transferred during the processes of coloration and bleaching is believed to be the ion S_{2} ^{−}.

Semiclassical Theory of Inelastic Scattering
View Description Hide DescriptionThe Schiff approximation is applied to the problem of collisions of two asymmetric molecules. The treatment includes effects of energy interchange between translational and internal degrees of freedom, and it includes transitions to states not directly coupled by the intermolecular potential. The differential and partial scattering cross sections are obtained in terms of a phase‐shift matrix which is a generalization of the WKB phase shift. This matrix may be obtained simply as an integral of the matrix of the intermolecular potential over the classical trajectory. Semiclassical approximations similar to those used in the case of elastic scattering are made, and the angular distribution reduces to the previously obtained classical expression. The approximations used in the derivation are that l and k (the orbital angular momentum quantum number and the wavenumber) must be large and Δk/k must be small. The calculations are reasonably simple for scattering below about 30° where the scattering may be considered as a perturbation on a straight‐line trajectory.

Influence of Nonbonded Interactions on Molecular Geometry and Energy: Calculations for Hydrocarbons Based on Urey—Bradley Field
View Description Hide DescriptionA modified Urey—Bradley potential energy function comprised of quadratic terms for bond stretches, bond‐angle bends, and torsional displacements together with analytical expressions for pairwise nonbonded interactions was chosen to represent the force field for hydrocarbon molecules. Quadratic constants were taken from the spectroscopic U–B analyses of Schachtschneider and Snyder [Spectrochim. Acta 19, 117 (1963)], while the nonbonded functions adopted were those proposed by Bartell [J. Chem. Phys. 32, 827 (1960)]. Reference bond angles for the quadratic terms were taken to be 109.5° or 120° for tetrahedral or trigonal coordination, respectively. Reference single‐bond lengths and the torsional constant were adjusted to fit the experimental data for CH_{4} and C_{2}H_{6}. Double bonds and ring bonds in cyclopropyl compounds were considered to be rigid. The above selections served to establish a universal model force field for hydrocarbons with no remaining adjustable parameters. The potential energy functions for a variety of saturated hydrocarbons and several olefins and cyclopropyl derivatives were minimized with respect to independent structure parameters (i.e., bond stretches, bends, and internal rotations). Even though all C–H (and C–C) bonds were input to be identical to those in CH_{4} (and C_{2}H_{6}) except for nonbonded environment, the bond lengths and angles corresponding to the minimum potential energy exhibited an appreciable variation from molecule to molecule, as did also the strain energies of geometric and rotational isomers. Calculated trends in structures, isomerization energies, and barriers to rotation agreed quite well with experimentally observed trends, provided that experimental isomerization energies were corrected to 0°K and zero‐point energies were taken into account. Some novel features of the results and applications of the model for predicting deformations in strained systems are discussed. The present study differs from previous work in the area of ``molecular mechanics'' in the use of a more general force field, in allowing the strained molecules to relax in all degrees of freedom (except for unsaturated groups and cyclopropyl rings), in the selection of molecular systems, and in a detailed comparison with experiment.

Born Inelastic Differential Cross Sections in H_{2}
View Description Hide DescriptionFirst Born differential cross sections are calculated for inelastic electron scattering of 300‐eV electrons off H_{2} for excitations from the ground electronic—vibrational state into the vibrational levels of the B, B′, C, D, and D′ Rydberg states. The initial and final state electronic wavefunctions were approximated by Hartree—Fock functions. In order to facilitate the calculation, the molecular orbitals were expanded in a linear combination of Gaussian‐type atomic orbitals.
The variation of the electronic scattering amplitude was examined as a function of molecular orientation, internuclear separation, and scattering angle (or electron momentum transfer). It is shown that the use of the Franck—Condon factors to determine relative intensities for scattering into different vibrational levels yields errors of less than 20%, and hence these factors are a good approximation to the relative intensities.
The theoretical differential cross section for zero‐angle scattering is in good agreement with the most recent experimental data, and transition moments extracted from electron scattering experiments within the context of the Franck—Condon principle are in agreement with the theoretical values.

Systematic Approximations to the Partition Function for Crystalline Solids
View Description Hide DescriptionThe two‐dimensional harmonic system based upon a triangular lattice with nearest‐neighbor interactions only is used as a test case to compare several systematic approximation schemes for obtaining the partition function of a crystalline solid. Attention is focused on the high‐temperature limit where the free energy function F_{N} has the form F_{N}/Nk_{B}T = −2lnT ^{*}+D. T ^{*} is a reduced temperature. Comparisons are made on the results obtained for the additive constant D. The various schemes examined are (1) a modified cell‐cluster expansion, (2) a sequence of m×∞ strips, i.e., extended tunnel models, (3) extrapolation of the properties of m×m systems with fixed boundary conditions, and (4) extrapolation of the properties of m×m systems with periodic boundary conditions. The results demonstrate that a cell‐cluster expansion through sixth order (i.e., including all six‐particle figures) gives a better estimate of the free energy than a 10×∞ system, a 12×12 system under fixed boundary conditions, or an 11×11 system under periodic boundary conditions. The exact value of D to four significant figures was obtained by extrapolating the results of the m×m systems with periodic boundary conditions, and the result is 0.8256.

ESR and Optical Spectroscopy of CrO_{4} ^{3−} in Chlorospodiosite, Ca_{2}PO_{4}Cl
View Description Hide DescriptionStudies of the spectral properties of CrO_{4} ^{3−} ion in Ca_{2}PO_{4}Cl (chlorospodiosite) single crystals confirm the structural results established by single‐crystal x‐ray structure analyses. Detailed electron spin resonance measurements of a single crystal of Ca_{2}(PO_{4}, CrO_{4})Cl are described and the appropriate spin‐Hamiltonian parameters determined. It is shown that the unpaired electron (3d ^{1}) in its ground state occupies a d _{ z2} orbital, which is largely concentrated on the chromium atom. The visible and ultraviolet absorptionspectrum of a single crystal of Ca_{2}PO_{4}Cl containing CrO_{4} ^{3−} is described and shown to be similar to the spectrum of MnO_{4} ^{2−}, which had been interpreted on the basis of a Ballhausen and Liehr molecular‐orbital level scheme. The infrared absorption spectra show more splitting of the vibrational bands of the chromate (V) compound (Ca_{2}CrO_{4}Cl) than of the phosphate (Ca_{2}PO_{4}Cl), in agreement with the structural results which indicated a greater distortion of the CrO_{4} ^{3−} than that of the PO_{4} ^{3−} tetrahedra. The infrared spectra of spodiosites and related compounds with apatite structure are compared.

Rate of Attachment of Gaseous Electrons to Nitrogen Dioxide
View Description Hide DescriptionThe rates of attachment of gaseous thermal electrons (300°K) to NO_{2} in the presence of several inert gases have been measured. In the pressure range from 3 to 70 mm, the rate of attachment is independent of the pressure, but dependent on the nature of the inert gas. A mechanism is proposed that accounts for these observations.

Matrix Elements for Configuration d ^{4} in a Weak Octahedral Field Using Racah Methods
View Description Hide DescriptionEmploying Racah algebra, the general formula for calculating the matrix elements of a weak crystalline field for configuration d^{n} has been derived. Applying this formula the matrix elements for d ^{4} (including spin—orbit coupling interaction) have been calculated and tabulated.

Thermal Dependence of Fluorescence and Lifetimes of Sm^{2+} in Several Host Lattices
View Description Hide DescriptionFluorescence spectra and lifetimes of Sm^{2+} in CaF_{2} and SrF_{2} were studied from 15° to 300°K and in BaClF from 15° to 650°K. A sharp line appears above a wide background at ∼100°K and 14 115 cm^{−1} for CaF_{2}:Sm^{2+} and at 140°K and 14 350 cm^{−1} for SrF_{2}:Sm^{2+}. Lifetimes at 80°K are ∼2×10^{−6} and ∼10^{−3} secs, respectively, in agreement with previous workers' results. BaClF:Sm^{2+} presents several sharp lines corresponding to transitions from three metastable states ^{5} D _{0} (14 542 cm^{−1}), ^{5} D _{1} (15 878 cm^{−1}), and ^{5} D _{2} (17 820 cm^{−1}) to the ground‐state multiplets ^{7} F _{0} to ^{7} F _{4}. The most efficient states at 300°, 80°, and 15°K are, respectively, ^{5} D _{0}, ^{5} D _{1}, and ^{5} D _{2}; above 350°K, the ^{5} D _{1}fluorescence increases relatively to the ^{5} D _{0}fluorescence, contrary to what happens from 77° to 350°K. Several ^{5} D _{0} lines persist and remain sharp (∼4 Å at 650°K) even at high temperatures with no observable background. Under pulsed excitation the decays of the fluorescence lines are consistent with an optical excitation in the 4f ^{5}5d bands, which is transferred non‐radiatively to the ^{5} D _{2} state and then cascades to lower levels with rates slower with decreasing temperature and for the ^{5} D _{1}—^{5} D _{0}energy gap. At 300°K the ^{5} D _{1}fluorescence presents a double decay, but at higher temperatures the ^{5} D _{1} decay is purely exponential with the same time constant of ^{5} D _{0}. The continuous and pulsed fluorescence data furnish evidence of pumping by phonon annihilation in the ^{5} D _{1} state from the lower ^{5} D _{0} state at temperatures greater than 300°K.

Stark Effect in Gas‐Phase Electron Resonance. The Dipole Moments of ClO, BrO, SH, and ^{1}Δ SO
View Description Hide DescriptionWe have studied the effects of electric fields on the electron resonance spectra of a number of diatomic free radicals in the gas phase. The use of high‐frequency electric modulation has a number of advantages over conventional magnetic modulation, which are discussed. The theory of the Stark effect in molecules having Λ degeneracy or near degeneracy is developed and accounts quantitatively for our observations. We report determinations of the electric dipole moments of ClO, BrO, and SH in their electronic ground states, and SO in its ^{1}Δ excited state.

Electron Spin Resonance Studies of a Reaction Intermediate in Strontium Chloride
View Description Hide DescriptionThe reaction of strontium chloride powder or single crystal with fluorine at room temperature produces a defect, stable after removal of fluorine but located close to the reaction interface only, which has an electron spin resonance absorption. The ESRspectrum is consistent with a model of the defect as a chlorine atom displaced from an anion site towards a neighboring anion vacancy. The principal directions of the hyperfine tensor and gtensor are two twofold axes and a fourfold axis of the SrCl_{2} crystal. The tensor components indicate that the unpaired electron is localized on the Cl atom, and suggest that the atom is subject to a strong crystal field determined chiefly by two nearest‐neighbor cations which define a twofold axis of the crystal. The unpaired electron is in an orbital mainly of p character and aligned along a twofold axis which is probably the one perpendicular to the line of the cations. The single‐crystal spectrum, although having orientation‐dependent line positions, has line shapes and intensities resembling those of a powderspectrum. This suggests a range of relative displacements of Cl atom and neighboring cations along a ``reaction coordinate'' which is probably a fourfold axis of the crystal.

Correlation Effect in the Time‐Dependent Concentration Fluctuations of the Isobutyric Acid—Water System near its Critical Mixing Point
View Description Hide DescriptionMeasurements by the laser homodyne ``self‐beating'' technique were made of the quasielastic Rayleigh linewidth in a critical binary liquid mixture, isobutyric acid—water (T_{c} =26.12°C), at temperature distances 0.07_{1}°, 0.19_{6}°, 0.36°, 0.73°, and 1.16°C above the critical solution temperature over a range of scattering angles. Deviations from the Landau—Placzek theory on the K ^{2}(K=ks, k=2π/λ, s=2 sinΘ/2) dependence of the half‐width of the central (Rayleigh) component (T) have been observed. Experimental data are in good agreement with Fixman's modification which takes into account the effect of long‐range correlations in binary liquid mixtures and has the form T=aK ^{2}(1+K ^{2}/κ^{2}) with κ corresponding to the inverse correlation length of Ornstein and Zernike.

Rotatory Diffusion Constant of a Chain Molecule
View Description Hide DescriptionThe rotatory diffusion constant of a chain molecule of spherical configuration is evaluated by the method which the author developed recently for the intrinsic viscosity. The theory is simpler than that given by Riseman and Kirkwood and yields a more general result in terms of the moment of the segment distribution function. The general result reduces to a simple expression for a flexible chain in terms of the molecular constants based on a pearl‐necklace model. The relation between the intrinsic viscosity and the rotatory diffusion constant is proved to be the same as Riseman and Kirkwood's. This is due to the same flow mechanism underlying the viscosity and diffusion phenomena.

Quenching of Hg^{*}(^{3} P _{1}) Atoms by Hydrogen Molecules Investigated by Phase‐Space Theory
View Description Hide DescriptionPhase‐space theory is used to investigate product distribution and internal energy distribution among various products in the mercury‐sensitized decomposition of hydrogen molecules. Calculated quantum yields at room temperature for H atoms, HgH, and undissociated hydrogen molecules are 0.52, 0.16, and 0.58, respectively. The H‐atom yield is much lower than the possible maximum of 2.0 but agrees well with an experimental value of about 0.5. The majority of undissociated hydrogen molecules is produced in vibrationally excited states. In mercury‐sensitized decompositions, these vibrationally hot H_{2} molecules may undergo reactions which were, in the past, attributed solely to H‐atom reactions.