Volume 43, Issue 6, 15 September 1965

Theory of Phenomenological Coefficients in Solid‐State Diffusion. I. General Expressions
View Description Hide DescriptionGeneral expressions for the phenomenological coefficients in terms of jump probabilities have been derived for isothermal solid‐state diffusion using linear response theory of the Kubo type. The assumptions required are: (a) the system evolves according to the master equation of a Markoff process, (b) a weak assumption about the general form of the perturbation to the transition probabilities in an applied field, (c) the detailed‐balance principle as applied to a Markoff process for a system at equilibrium. The Onsager relations are obeyed.
The advantages of using these results as a starting point for detailed calculations compared with conventional methods are briefly discussed and further developments noted.
An appendix discusses the application of the same method to thermal diffusion.

Inelastic Neutron Spectra and the Vibrational Modes of the Hydrogen Layer in Alkali and Alkaline‐Earth Hydroxides
View Description Hide DescriptionInelastic neutron scatteringspectra of pure and deuterated lithium hydroxides and of Ca(OH)_{2}, Ca(OD)_{2}, and Mg(OH)_{2} are presented. These spectra are explained by the interaction of neutrons with lattice vibrations in the scatterers. Special emphasis is given to the part of the spectra which is predominantly influenced by vibrations of the hydrogen layer atoms.

Structure and Partition Function of Liquid Water. I. Examination of the Model of Némethy and Scheraga
View Description Hide DescriptionThe Némethy and Scheraga model for the thermodynamic properties of liquid water is compared with the experimental results of Buijs and Choppin. It is shown that the two sets of data do not agree and an explanation for the discrepancy is suggested.

Structure and Partition Function of Liquid Water. II. Examination of the Results of Buijs and Choppin
View Description Hide DescriptionIt is shown that the experimental results of Buijs and Choppin are not consistent with any thermodynamic model of liquid water based on three species of molecules, each with a sharp level of positive energy.

Structure and Partition Function of Liquid Water. III. Development of the Partition Function for a Band Model of Water
View Description Hide DescriptionA partition function for liquid water is developed, based on the three types of water molecules postulated by Buijs and Choppin. The main feature of this approach is that the concept of discrete energy levels (corresponding to zero, one, and two hydrogen bonds) is abandoned and replaced by the idea of energy bands. The parameters of the partition function are evaluated from the thermodynamic data at 50°C, the results of Buijs and Choppin, and the assumption of dipole—dipole forces within the liquid. These parameters are then used to calculate the thermodynamic functions over the range 0° to 100°C. The calculated results are in very good agreement with the experimental observations, the agreement with C_{v} being generally better than 0.2%. The experimental results of Buijs and Choppin and our band model of water are therefore consistent with each other and also with thermodynamic data on water.

NMR Second Moment of Solid Cyclododecane
View Description Hide DescriptionThe NMR second moment for the rigid lattice of cyclododecane has been measured and is compared with theoretical values calculated for models which have been derived from an x‐ray analysis. The NMR data complement the other data such that the conformation of the molecule could be determined unambiguously.

Reinterpretation of the EPR Spectrum of N,N′‐Tetramethylbenzidine Cation
View Description Hide DescriptionThe EPR spectrum of N,N,N′,N′‐tetramethylbenzidine monocation has been completely reanalyzed and has been found to have the following hyperfine coupling constants in acetonitrile: a _{N}=4.86 G, a _{H methyl}=4.74 G, a _{H3,5,3′,5′}=1.63 G, and a _{H2,6,2′,6′}=0.79 G.

Isotopic Thermal‐Diffusion Factor for Xenon
View Description Hide DescriptionThe isotopic thermal‐diffusion factor α_{0} for xenon has been determined over a mean temperature range of 245° to 543°K. An artificial mixture of xenon isotopes containing approximately 18% ^{129}Xe and 15% ^{136}Xe was subjected to thermal diffusion in a 10‐tube swing separator; the isotopic thermal‐diffusion factor was obtained from the change of concentrations of these two isotopes only. A best‐fitting procedure yields potential parameters α=16 and ε/k=257°K for the exponential‐six interaction‐potential model, whereas the Lennard‐Jones (12:6) model fails to reproduce the experimental results. This exp‐6 potential was then used to calculate the values of diffusion,viscosity,thermal conductivity, and second virial coefficient, and a reasonably good agreement was obtained with r_{m} =4.41 Å.
An intercomparison of the observed and the theoretically calculated isotopic thermal‐diffusion factors for all the inert gases from T ^{*}=0.6 to 60 is also presented.

Charge Dependence in Ionic Solvation
View Description Hide DescriptionSystems consisting of a single electrically neutral or ionic species in interaction with a large number of solvent molecules are treated in general quantum‐statistical‐mechanical terms. Such systems are assumed to have Hamiltonians, for each electronic state, which exhibit a parametric dependence upon charge not exceeding quadratic. Changes in the Helmholtz free‐energy, energy, and entropy functions are determined formally for the process of ionic solvation due to Born. On the basis of a general argument that an electrically neutral species acquires a nonzero potential upon solvation, the changes in the cited thermodynamic functions are shown to be capable of taking on positive values for a range of values of the charge parameter. This behavior is in direct contrast with expectations from the Born expression for such changes. Semiempirical treatment of data available for solvation free energies of aqueous ions permits an estimate, of 1 V, to be made of this potential. A possible experimental test of this result is described.

Effects of Pressure on the Electrical Conductivity of Some Organic Charge‐Transfer Complexes
View Description Hide DescriptionThe effects of pressure and temperature on the electrical conductivity of some highly conductive charge‐transfer complexes such as dibenzo[c, d]phenothiazine—dichlorodicyano‐p‐benzoquinone (2:1), dibenzo‐[c, d]phenothiazine—dibromodicyano‐p‐benzoquinone (3:2), tetrathiotetracene—o‐chloranil (3:1), tetrathiotetracene—o‐bromanil (3:1) complexes are reported. The resistivity of these complexes under pressure follows the usual exponential law; 1/σ=ρ=ρ_{0} exp (E/kT). The measurements are carried out at a temperature range of 27°—120°C. The highest pressure applied is 40 kbar. The resistivity decreases with increasing pressure. However, this change is only of the order of 10 (from 1 to 30 000 bar). The activation energies also decrease with increasing applied pressure and the values in the tetrathiotetracene complexes become almost zero at about 35 kbar. The ρ_{0} obtained for these complexes increases with increasing pressure. The effect of pressure on the electrical properties of the organic solids is discussed.

Electric and Magnetic Properties of the Strontium Ferrates
View Description Hide DescriptionA series of strontium ferrate compositions ranging from SrFeO_{2.7} to SrFeO_{3.0} have been produced by equilibration at oxygen pressures ranging from 0.2 to 855 atm. The stoichiometric composition SrFeO_{3.0} exhibits a simple cubic perovskite structure (a=3.850 Å). Ceramic disks of this material are conductive (ρ≈10^{−3} Ω·cm) and their temperature dependence is characteristic of metallic conduction. Magnetic measurements indicate that SrFeO_{3.0} is antiferromagnetic below 130°K. With increasing oxygen deficiency, specimens show increased cell constants and eventually tetragonal distortion. Decrease in the Néel temperature and increased resistivity accompany these changes.

Correlation Functions in Nuclear Relaxation. I. Analysis of Random Motions from Field Dependence of Relaxation Times
View Description Hide DescriptionThe contributions to T _{1} in a fluid arising from spin—rotational interaction and chemical‐shielding anisotropy are discussed in terms of their simultaneous use to analyze the detailed nature of random rotations without the need of a microscopic theory. This possibility often exists because of the involvement of different correlation functions of the lattice variables and the experimental distinguishability of the latter mechanism through its magnetic‐field dependence. The method is illustrated by comparing the results to be expected for three different random processes: (1) rotational diffusion, (2) random walk among the m_{j} states of a rotational manifold of fixed J with no selection rules, (3) the same with a rigorous Δm_{j} =±1 selection rule. Possible favorable experimental cases are discussed.

Phase Shifts and the Quantum‐Mechanical Hamilton—Jacobi Equation
View Description Hide DescriptionA method of obtaining absolute phase shifts by integration of the quantum‐mechanical Hamilton—Jacobi equation is developed and applied to the example of particles interacting through a Lennard‐Jones potential. A new expression for the absolute phase shift is obtained in terms of an irregular solution of the Schrödinger equation.

Theory of Hypochromism
View Description Hide DescriptionA unified theory of complex refractive index in fluids is described. The theory shows that hypo‐ (or hyper‐) chromism is an inevitable consequence of significant anisotropic relative correlation. Formulas are quoted which contain, among other significant terms, the more general form of those given by Tinoco and others. Factors of dispersive refractive index and multiple‐scattering terms neglected in the previous treatments are shown to be important. Three alternative reasons why pronounced hypochromism may be accompanied by small frequency shifts in the dipole approximation are indicated: in general, hypochromism is accompanied by changes of band or line shapes as well as shifts of absorption maxima. A proof of a sum rule for integrated absorption in small isolated polymers is adumbrated: under less stringent conditions the sum rule appears to fail. The arguments are developed in much greater detail elsewhere.

Inelastic Electron Scattering from Atoms at Medium Energies. II. Polarization Effects
View Description Hide DescriptionThe effect of inelastic electron scattering on the elastic, total, and discrete inelastic differential scattering cross sections is investigated in the second Born approximation. It is shown that in this approximation the polarization corrections can be written as a function of the Coulomb phase shift and other auxiliary functions to within an additive constant in the Coulomb phase. This constant is chosen by imposing the physically reasonable condition that the polarization correction must tend to zero with increasing scattering angle. The results of the calculation of polarization effects on the elasticscattering amplitudes for Ar and U are given for an incident electron energy of 40 kV.

Nuclear Quadrupole Resonance and Bonding in Crystalline Ammonia
View Description Hide DescriptionNuclear quadrupole coupling constants (eQq) of ^{14}N have been determined as a function of temperature for each of the isotopic species NH_{3}, NH_{2}D, NHD_{2}, and ND_{3} in the crystalline state. The temperature dependences are explained, following a theory proposed by Bayer, in terms of an averaging of the electric field gradient (q) by torsional lattice motions. Frequencies and moments of inertia known from other spectroscopic studies of NH_{3} and ND_{3} were employed in the calculations, which also led to values for the quadrupole coupling constants without zero‐point vibrations. These values turned out to be the same (−3.47 Mc/sec) within 1% for NH_{3} and ND_{3}, showing that the different field gradients for the various isotopic species in the crystalline solid are well explained in terms of lattice motions.
Comparison of the vibrationless, or ``quasistatic'' values, with the coupling constants of the gaseous molecules (−4.08 Mc/sec) then gave the shift produced by the crystalline field, here called the quasistatic shift. This is large relative to the changes produced by isotopic substitution in the gaseous molecule. Calculations of the electric field and the field gradient produced at a given ^{14}N nucleus by surrounding molecules in the known crystal structure were made to explain the quasistatic shift. Molecules containing the nearest hydrogens were treated as a distribution of four point charges, and the others, to a total of about 100, were taken as point dipoles. Both models were consistent with a crystalline dipole moment evaluated from the gaseous value and the known molecular polarizability, taking into account the reaction field of the lattice. The quasistatic shift was orders of magnitude too large to be accounted for by the direct field gradient of surrounding neighbors; it is better described in terms of a redistribution of electrons in the nitrogen p orbitals by polarization of the molecule in the crystal field.

Cell‐Cluster Method for Hard Squares
View Description Hide DescriptionThe partition function of four hard squares at high densities is calculated, and all the cell‐cluster integrals associated with this partition function are evaluated. The use of higher cell‐cluster integrals in this formalism does not ensure monatonic approach to the true partition function.

Thermodynamic Properties of Thionyl Fluoride from 13°K to Its Boiling Point. Entropy from Molecular and Spectroscopic Data
View Description Hide DescriptionThe heat capacity of thionyl fluoride has been measured from 12° to 230°K for a sample of 99.976 mole% purity as determined by the melting‐point method. The heat of fusion was 1505.9 cal/mole at the solid—liquid—vapor equilibrium temperature of 143.25°K. The heat of vaporization at the normal boiling point of 228.84°K was 5091 cal/mole. The vapor pressure of the liquid to the normal boiling point has been measured, and the data are represented bywhich was obtained by minimizing the square of the residuals of the pressure.
The density of the liquid is represented by d(g/cc)=2.389–0.003398T.
The entropy of thionyl fluoride gas in the standard state at the normal boiling point calculated from the experimental data is 63.56 cal/(mole·°K). A spectroscopicentropy of 63.28 cal/(mole·°K) is calculated from the product of the moments of inertia, I_{A}I_{B}I_{C} =1.6562×10^{−114} g^{3}·cm^{6} from microwave data and the following frequency assignment: v _{1}(a′)=1336 cm^{−1}; v _{2}(a′)=808 cm^{−1}; v _{3}(a′)=530 cm^{−1}; v _{4}(a′)=378 cm^{−1}; v _{5}(a″)=748 cm^{−1}; and v _{6}(a″)=396 cm^{−1}. The discrepancy between the two entropies is accountable in terms of the experimental uncertainty and therefore, the molecular parameters and frequency assignment are confirmed.

Ferromagnets and Simple Fluids near the Critical Point: Some Thermodynamic Inequalities
View Description Hide DescriptionIt is often assumed that thermodynamic variables have a simple power‐law behavior near the liquid—vapor critical point of a simple fluid; for example, that the difference in density of saturated liquid and vapor decreases as (T_{c}—T)^{β} as the temperature T approaches its critical value T_{c}. Thermodynamic arguments are used to derive several inequalities relating exponents which describe the behavior of the isothermal compressibility, specific heat at constant volume, and various other quantities near the critical point. Many of the inequalities apply equally to the analogous problem of a ferromagnet near its Curie point. They are based on the usual ``stability'' or ``convexity'' conditions on the thermodynamic potentials together with other plausible, but less general, hypotheses.

Ion—Molecule Reactions in Hydrogen—Rare‐Gas Mixtures
View Description Hide DescriptionIon—molecule reactions in rare‐gas—hydrogen systems have been studied by high‐pressure mass spectrometry, using a specially designed ion source which employs the beta rays of tritium as ionizing medium. The rate constants for reactions of H_{3} ^{+} with rare gases and of rare‐gas ions with hydrogen have been measured. The results are in qualitative agreement with radiolysis experiments on hydrogen—deuterium exchange, and if a quantitative comparison is attempted, it is possible to estimate the rate constant of the chain‐propagation reaction. Experiments are also reported for N_{2}–H_{2} and O_{2}–H_{2} mixtures.