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Volume 32, Issue 1, 01 January 1960
32(1960); http://dx.doi.org/10.1063/1.1700900View Description Hide Description
A theory for a flame in a solid whose speed is controlled by diffusion of radicals and heat is applied to slowly propagating flashes observed in solid nitrogen deposited at 4°K from a microwavedischarge. Reasonable agreement is obtained with observed speeds.
Molecular Orbital Studies of Diatomic Molecules. I. Method of Computation for Single Configurations of Heteronuclear Systems32(1960); http://dx.doi.org/10.1063/1.1700944View Description Hide Description
This paper describes a method of molecular orbital calculation for heteronuclear diatomic molecules, in the approximation of a single spatial configuration which is a product of one‐electron functions. The method is applicable to systems in which the one‐electron orbitals are not required to be either orthogonal or filled in pairs. A practical procedure is presented for treating permutational symmetry and spin for an arbitrary spatialwave function, and using this procedure we consider simultaneously all the spin configurations associated with the chosen spatialwave function. The wave functions used are parametric expressions in spheroidal coordinates, and include the usual Slater orbitals as special cases. All necessary integrals are evaluated in a manner suitable for machine computation. A computer program to implement this method of calculation has been prepared, and with it an approximate energy level for a 6‐electron molecule can be obtained in about 8 min. Provision was made to program the computer to vary the parameters of the wave functions so as to find automatically the wave function of minimum energy of the parametric form chosen. Only a part of the calculation need be repeated after varying parameters, and the optimum retention of computed data for re‐use is discussed.
32(1960); http://dx.doi.org/10.1063/1.1700904View Description Hide Description
The purposes of this article are: (1) to call attention to the fact that bimolecular reactions (A+B→AB) in solids and liquids do not, in general, follow simple second‐order kinetics and, (2) to discuss some of the complications arising when deviations occur. In a previous paper a general expression for the rate of bimolecular reactions in liquids and solids was derived. The general reaction rate is second‐order in the concentrations but the ``rate constant'' is time dependent. That expression is simplified and presented here in terms of three physically significant parameters, D, the sum of the diffusion coefficients of species A and B, r 0, the A—B capture radius, and s, the ratio of the probability that a pair of particles A—B separated by a capture radius r 0 will react, to the probability that they will diffuse apart before reaction can occur. The dependence of the ``rate constants'' on these parameters and on the time is illustrated graphically. The physical significance of the limiting cases is discussed. A simple, accurate, physically meaningful approximation to the rather abstract general rate expression is presented. An experimental method of distinguishing bimolecular reactions when they do not follow simple second‐order kinetics is discussed. Reactions involving particles with long‐range forces are considered briefly.
32(1960); http://dx.doi.org/10.1063/1.1700910View Description Hide Description
This paper deals with heterogeneous chemical reactions within systems in which the relevant geometric dimensions are small compared with the mean free path of gas molecules. A typical example can be found in a surface reaction occurring at moderate pressures inside the pore of a catalyst with a radius of several angstroms.
Specific problems of this type have in the past been formulated, subject to the well‐known limitations of ordinary diffusion theory, in terms of differential equations for the species concentrations. This in turn supposes that the reaction pattern is dependent on local conditions. A more general and exact formulation given in the following shows that the concentration of a molecular species at a point is determined by the concentration distributions of all species stemming from the entire system. Thus one is led to a formulation in terms of integral equations. In an application of the theory the validity of the conventional differential equation approach is investigated and is found to yield satisfactory results only under certain limiting conditions.
32(1960); http://dx.doi.org/10.1063/1.1700919View Description Hide Description
The mean life τ2 and fraction I 2 of positrons which form 3 Spositronium have been measured for annihilation in 20 organic liquids—alcohols, glycols, acids, benzene derivatives, and others. By using the measured value of τ2, it is possible to calculate 〈σν〉Av where σ is the molecular cross section for annihilating 3 Spositronium on collision and ν is the velocity of the positronium atom. The results show that in general 〈σν〉Av is related to the ``size'' of the molecule although there are some consistent exceptions to this rule. There is also a rough correlation between τ2 and density, τ2 being shortest for the most dense liquids and longest for the least dense ones.
32(1960); http://dx.doi.org/10.1063/1.1700943View Description Hide Description
High‐resolution spectra of NH3, yielding 1800 lines between 3060–3580 cm−1, are presented. Analysis of the ν3 fundamental is complete through J′=9 and presents no unexpected features. Resolution of the K substructure in the ν1 fundamental shows that several perturbations are present. The most important of these, involving a Fermi resonance with the parallel component of 2ν4, and a Coriolis resonance with its perpendicular component, are clarified. Lines in both components of 2ν4 are identified through J′=6, and a strong Coriolisinteraction between the l=0 and l=2 states is observed and discussed.
32(1960); http://dx.doi.org/10.1063/1.1700945View Description Hide Description
Small‐angle x‐ray measurements on the system C7F16‐i‐C8H18 at temperatures near the consolute temperature show that a large degree of clustering takes place. Starting with pure i‐C8H18, which exhibits no scattering, the scattered intensity increases with increasing C7F16 concentration in a manner which indicates that the clusters are increasing in size. The maximum size is attained at a volume fraction of ∼0.50, at which point the clusters are made up of ∼140 C7F16 molecules. With a further increase in concentration, interaction peaks first appear in the intensity curves, indicating an approach to close packing, and then the slope of the curves begins to decrease, indicating that a phase reversal has taken place giving rise to smaller clusters of hydrocarbon in perfluorocarbon.
32(1960); http://dx.doi.org/10.1063/1.1700946View Description Hide Description
The isotropic hyperfine interactions of radicals or ions have been discussed from the viewpoint of the self‐consistent field (SCF) molecular orbital theory. In these phenomena, the most important electron configurations concerned are those arising due to the one electron excitations from the double occupied orbital to the vacant orbital together with the lowest energy configuration. The protonhyperfine splitting of the methyl radical was calculated nonempirically by using the SCF‐LCAO‐MOs in adopting the first order perturbation. The agreement with the observed value is quite good when one uses MOs that include the 3s AO of carbon atom, while it is not so satisfactory in the case of the MOs without the 3s AO, because of the improvement of the form of the totally symmetric antibonding vacant orbital which gives the main contribution to the phenomena.
32(1960); http://dx.doi.org/10.1063/1.1700947View Description Hide Description
The paramagnetic resonance absorption of crystalline NpF6 has been studied at 3 cm wavelength and at the boiling point of liquid He. The data were fitted to the spin Hamiltonian,. The parameters are isotropic within the errors of the measurements, with g = −0.604 and | A/hc | = 0.1100 cm−1. These results are interpreted in terms of the configuration f 1 perturbed by spin‐orbit coupling and an octahedral molecular potential. The relationships of the present results to optical data and measurements of magnetic susceptibility for the same substance are discussed. Hyperfine structure due to F is observed and is anisotropic.
32(1960); http://dx.doi.org/10.1063/1.1700948View Description Hide Description
The Ewald‐Born theory of double refraction is applied to wurtzite‐type compounds. Except for the small distortion and polarization, these compounds have identical near‐neighbor symmetry to zinc blende (cubic) structures, and hence the double refraction is caused primarily by the distant‐neighbor interactions. The effects of anisotropy of the Lorentz‐Lorenz force and of the Coulomb force are calculated. In the (ionic) approximation of zero cation polarizability the theory predicts a double refraction larger than the measured one by a factor ∼2 for all six compounds that have been measured. In the (covalent) approximation of zero effective ionic charge the theory underestimates the double refraction of AlN, but may be made to agree with the measured values of the other compounds by a suitable choice of atomic polarizabilities. In both approximations the theory predicts n 0<ne, in agreement with long‐wavelength observations. Anisotropicinteraction of polarized light with the solid as a whole, as in exciton transitions, seems to account for the inversion to n 0>ne near the absorption edge of CdS.
32(1960); http://dx.doi.org/10.1063/1.1700949View Description Hide Description
Exact protonmagnetic resonance eigenspectra of the spin Hamiltonian are calculated for a system containing one group of six identical protons and one group of two identical protons. The method of calculation considers each group of identical protons as a composite ``particle'' with fixed total spin, and does not require determination of the explicit form of the zero‐order eigenfunctions. The calculated spectra are compared with experimental high‐resolution spectra of propane at both 40 Mc and 60 Mc. Thevalues of p, the ratio of the spin coupling constant to the chemical shift, are found to be 0.415 and 0.277, respectively, at the two frequencies. The frequency‐independent spin coupling constant is 7.26 cps.
32(1960); http://dx.doi.org/10.1063/1.1700950View Description Hide Description
A theory of separation of ortho and para hydrogen, by adsorption at low temperatures, is developed by considering the energy levels of a three‐dimensional hindered rotator. The Schroedinger equation and boundary conditions are identical with those for prolate spheroidal wave functions and tables of the latter have been employed in calculating the energies as functions of the barrier height. Separation factors, at low surface coverage, have also been computed as a function of barrier height and these pass continuously from unity at zero barrier height over into the limiting separation factor for a two‐dimensional rotator (in the adsorbed state) which was used by Sandler in the approximate calculation of separation factors. The theory predicts that ortho hydrogen is more strongly adsorbed than para hydrogen at all barrier heights and that para deuterium is more strongly adsorbed than ortho deuterium. Moreover, the ortho‐para hydrogen separation factor is larger at all barrier heights than the para‐ortho separation factor for deuterium. These results agree with the experiments of Cunningham, Chapin, and Johnston. Calculated and observed separation factors for hydrogen and deuterium (at low surface coverages) are not in accurate quantitative agreement, however this may be due to the fact that experimental separation factors for both isotopes were not determined under comparable conditions. Calculations have been made first by assuming that rotation and (center of mass) vibration are separable. This model is then refined to take into account the interaction of rotation and vibration.
The separation of the hydrogen‐deuterium isotopes is also considered. Although the inclusion of interaction between vibration and rotation does not greatly change the ortho‐para separation factors, this effect has a marked influence on the calculated isotope separation factors. It is of considerable interest that isotope separation factors are strongly dependent on the ortho‐para composition of the isotope mixture, at least at low surface coverages, in the theory developed here.
32(1960); http://dx.doi.org/10.1063/1.1700951View Description Hide Description
A survey of the observed quadrupole coupling constants of the deuteron is presented. It is concluded on the basis of calculations that the field gradients at hydrogen nuclei will be positive in almost all molecules. The simple wave functions greatly underestimate the field gradients of the bonding electrons. A good part, but not all, of the discrepancy, previously noted by White, can be removed with the use of a contracted charge distribution and ionic character. The wave function of Ellison and Shull gives a very bad description of the field gradients at the deuterons in D2O but the experimental data indicate the direction for future improvement. The field gradients at deuterons remain attractive quantities for characterizing the charge density in bonds involving hydrogen atoms.
32(1960); http://dx.doi.org/10.1063/1.1700952View Description Hide Description
The burning mechanism of a solid propellant is described in terms of a pyrolysis law, a premixed laminar flame, and nothing more. A quantitatively accurate burning rate formula is derived from the theory. The predicted burning rate pressure dependence agrees with the experimental results for ammonium perchlorate propellants.
32(1960); http://dx.doi.org/10.1063/1.1700953View Description Hide Description
Previous studies of the growth of alumina crystals have demonstrated the validity of current crystal growththeory for a high‐temperature system involving a chemical reaction. The nucleation of alumina on a perfect alumina surface has been studied in the present report at very high supersaturations. The nucleation event is not consistent with the classical theory of nucleation, but rather is described by the Cahn‐Hilliard theory of nucleation based on their treatment of the thermodynamics of nonuniform systems. In addition, the recrystallization of polycrystalline deposits to give near perfect crystal was observed.
32(1960); http://dx.doi.org/10.1063/1.1700954View Description Hide Description
In this paper the cellmodel for the liquid state, as proposed by Lennard‐Jones and Devonshire, is recast in terms of quantum statistical mechanics. A method is developed by which the energy levels and wave functions in the liquid cell can be determined to any prescribed accuracy. Thus, in the present work, no additional approximations are introduced which are not inherent in the cellmodel. When a sufficient number of energy levels is obtained, the partition function and thus the thermodynamic properties are derived.
A method is devised by which from a knowledge of the wave function, the derivative of the corresponding energy with respect to the volume may be obtained. This derivative is required for the evaluation of the pressure.
The theory is applied to H2 and D2 at a density near that of the crystals at 0°K. The corresponding classical Lennard‐Jones and Devonshire thermodynamic functions are obtained and compared with the results of the present theory. The calculations involved were performed on a Bendix G—15D computer.
The quantized cell theory predicts that for hydrogen and deuterium experimentally detectable differences in energy,entropy, and specific heat must persist at temperatures as high as − 150°C.
32(1960); http://dx.doi.org/10.1063/1.1700880View Description Hide Description
The HD molecule has a small permanent electric dipole moment as a consequence of the asymmetry of nuclear vibration. The complete molecular Hamiltonian contains a term which is not invariant to inversion of electronic coordinates through the molecular center. This gives rise to an axial asymmetry of electronic charge. Treating the term as a perturbation, it is found that the 1Σ g + electronic ground state is perturbed only by states of 1Σ u + symmetry. Assuming that only the lowest such state (the B state) contributes, an upper bound on the permanent moment is established. For the ground vibrational state this is computed to be 8.89×10−4 Debye unit, directed in the sense H+D−.
32(1960); http://dx.doi.org/10.1063/1.1700881View Description Hide Description
A method is developed for solving second‐order perturbation problems without explicit knowledge of the higher molecular states. This method involves, in principle, a transformation of basis from the set of orthonormalized eigenfunctions of the unperturbed molecular Hamiltonian to an arbitrary set of functions not, in general, either normalized or mutually orthogonal, but of such symmetry that perturbation matrix elements connecting them to the ground state do not vanish. If the new basis is chosen expediently, convergence to second‐order eigenvalues may be attained with a small number of functions. The method is related to the conventional perturbation and variational approaches. As an illustration, the perturbed harmonic oscillator is treated.
32(1960); http://dx.doi.org/10.1063/1.1700882View Description Hide Description
The theory of the temperature dependence of pure quadrupole frequencies under constant volume conditions is modified when constant pressure conditions apply. For the case of molecular crystals the first and second derivatives of frequency with respect to temperature have been calculated assuming that (i) the quadrupole coupling constant is independent of volume, and (ii) the lattice frequencies vary linearly with temperature. These derivatives have been evaluated experimentally for the α phase of p‐dichlorobenzene, and comparisons made with the theory; good agreement with previous work on the pressure dependence of quadrupole frequencies is obtained.