J. H. Van Vleck

The Jahn‐Teller Effect and Crystalline Stark Splitting for Clusters of the Form XY6

The present article develops the mathematical theory needed in connection with the preceding paper, and with a study of paramagnetic relaxation to appear later. Jahn and Teller have shown that the normal coordinate theory for a molecular cluster such as X·6H2O (X=Ti, V, Cr) is materially different from usual if the X ion is degenerate. Namely, it is impossible to eliminate linear terms in the atomic displacements simultaneously from all substates of the degenerate level. Consequently, stability is achieved only if the degeneracy is lifted, and so the water group distorts itself so that it is not cubically arranged. A calculation is made of the magnitude of the effect, which has an important bearing on magnetic behavior. In order to allow for the disturbing effect of distant atoms, the cluster X·6H2O is not treated as free, but rather as embedded in a trigonal field of force. The Stark splittings in such fields, quite apart from the Jahn‐Teller effect, are hence obtained as a corollary.

On the Isotope Corrections in Molecular Spectra

According to the simple theory of the rigid rotator and harmonic oscillator, the coefficients of J(J+1) and (v+½) in the energy should be inversely proportional to respectively the reduced mass and the square root thereof. Actually various small corrections, important only in compounds containing hydrogen or deuterium, make the mass dependence slightly different, and hence, at least in principle, explain why spectroscopic observations may yield slightly incorrect mass ratios for isotopes. The corrections are due to four causes (a) anharmonicity, (b) interaction between vibration and electronic motion even when perturbations between different electronic states are neglected, (c) L uncoupling or in other words perturbations between states differing by one unit in Λ, (d) interaction between states of equal Λ. The formulas for the various corrections are collected—some old, some new or more exact than previously. The relative magnitudes of the different effects are estimated. Incidentally, perturbations betwe...

A Theoretical Comparison of the Visual, Aural, and Meter Reception of Pulsed Signals in the Presence of Noise

The present paper examines theoretically the relative sensitivity of the detection of signal pulses in the presence of noise, (a) by observation of an oscilloscope, (b) by aural perception, in which one listens to the fundamental or a low harmonic of the pulse repetition frequency (PRF), (c) and by a meter. The metering scheme may be either aperiodic, where the rectified current is fed directly to a meter with a long time‐constant, or periodic, where the rectified current is sent through an audio‐filter tuned to the PRF, given a supplementary rectification, and then passed through the meter. The dependence of the sensitivities of the different methods on various relevant parameters is studied in some detail. These parameters include the width and the shape of the IF response, the pulse length, the PRF, and in aural or meter reception, the duration of the gate, the width of the audio‐filter, and/or the time constant of the meter. The descriptive survey of the results is given in Part I and the mathematical...

On the Role of Dipole‐Dipole Coupling in Dielectric Media

The mathematical treatment of dipole‐dipole coupling in my previous article on magnetism applies with but little modification to the electric case. Because of fluctuation effects, the use of the usual Lorentz local field E+4πP/3 and of the Clausius‐Mossotti formula cannot be justified except as a first approximation valid at low density. Consequently one need not accept the C‐M prediction that polar liquids should exhibit the electric analogue of ferro‐magnetism. A Gaussian approximation, or better still, a formula based on Onsagers field would never allow ferro‐magnetism. Consequently the hypothesis of hindered rotation, as in the theories of Fowler and Debye, may not be necessary to explain the absence of spontaneous polarization, as well as the non‐occurrence of much saturation curvature in strong applied fields. It is a weakness of their theory that this hypothesis cannot consistently be employed both for such purposes and to explain discontinuities in dielectric constants and specific heats at lower...

On the Theory of the Structure of CH4 and Related Molecules: Part III

Explicit calculations are made to examine whether the tetrahedral model of CH4 is really the most stable in the Hund‐Mulliken and Slater‐Pauling theories. The secular determinant involved in the H‐M procedure is too complicated (degree 8) to solve generally, but three methods of approximation are given applicable in three different limiting cases, in all of which the tetrahedral model proves to be that of least energy quite irrespective of repulsions between the H atoms. The fact that two of the methods do not assume the carbon s‐p separation to be small shows that s‐p hybridization is not a necessary condition for tetrahedral valence bonds. Calculations are also given which show that in the Heitler‐London‐Pauling‐Slater method the regular tetrahedron is superior to other models of somewhat lower symmetry. It is further calculated according to both the H‐M and H‐L‐P‐S schemes that in compounds of the form CH2X2, CHX3, and CH3X, the most stable models are tetrahedra of less symmetry than the regular tetrah...

Effect of the Anisotropic Exchange and the Crystalline Field on the Magnetic Susceptibility of Eu2O3

This paper invokes the combined action of the crystalline field and exchange in an attempt to explain theoretically why the susceptibility χ of cubic Eu2O3 is found by experimenters to be 30% higher than that for the free Eu3+ ion at temperatures below 50°K. The crystalline field is known approximately from optical studies, whereas the coefficient in the exchange coupling is treated as an empirical parameter. This parameter cannot be deduced from a comparison with Gd2O3 as we have shown elsewhere that there are contributions from anisotropic exchange which do not enter in the coupling of S‐state ions such as Gd3+, but which do not necessarily vanish for Eu3+–Eu3+ interaction. The crystalline field is found to account for only 42% of the observed enhancement of χ, whereas the isotropic exchange of the type present in Gd2O3 in fact reduces χ by a small amount, thus demonstrating the importance of the anisotropic exchange. The latter needs to be opposite in sign and five times larger than the conventional isotropic exchange in order to explain the observed χ at T = 0°K. Then the temperature dependence of χ calculated with the molecular field approximation agrees with experiment very well.

Note on the crystal field parameters of rare earth garnets

Abstract The results of various investigators on the crystal field parameters of garnets of the heavy rare earths are examined and compared, including discussion of their relation on the model which we used in the preceding paper on TmGaG. A number of inconsistencies and errors in the literature are noted. The empirical ratios of sixth to fourth order terms which have been reported are practically always larger, sometimes by about a factor 10, than the values obtained by a point charge calculation with Freeman-Watson radii for the 4 f electrons.

Note on the Zeeman Effect of Chrome Alum

It is shown that the Zeeman patterns observed by Spedding and Nutting are comprehensible if and only if the lines in question are intersystem combinations.

Note on the gyromagnetic ratio of Co++ and on the Jahn-teller effect in Fe++

Abstract By the method of pseudo-angular momentum is meant the utilization of the fact that in a triply degenerate level in a cubic field, the orbital angular momentum obeys the same commutation rules as for a p state, except for a proportionality factor. This method is used to calculate the gyromagnetic ratio of a Co++ ion in a cubic field. The value thus obtained is 1.65, in better agreement with Sucksmiths experimental measurements on CoSO4 and CoCl2 than is the spin-only value g = 2, even though these materials do not have cubic symmetry. The other application is to the Jahn- Teller effect in Fe++. An explanation is given why Low finds no evidence of this effect in his magnetic resonance measurements on Fe++, and at the same time this mechanism plays an important role in the Abragam-Pryce theory of the magnetic susceptibility of copper fluosilicate. At first sight, one would expect the Fe++ ion to show a large J-T effect, as its lowest level is triply degenerate even when spin-orbit interaction is included; however, the linear terms in the distortional coordinates have very small coefficients and so are ineffective.

Line-breadths and the theory of magnetism

The tradit ional start ing point for any discussion of line breadths is the simple collision theory of Lorentz (1). We can omit radiation damping, for it is un impor tan t in the microwave region. The Lorentz model is basically a harmonic oscillator subject to collisions of exceedingly short duration. At each such impact there is a sudden change in phase. I t is further assumed tha t there is no persistance o r ~ hang-over ~> in phase, so tha t after collision one value of the phase is as likely as another. (Actually, as shown by VAN VLEC~; and WEISSKOPF (2), the various phases should be weighted somewhat unevenly conforming to the Bol tzmann factor for the instantaneous v~lue of the energy in the applied field. The resulting corrections, however, introduce only terms in ]/(~+co0) rather than 1/(o~ O~o), and are of no importance near reso a ce. We shall omit them). The mathemat ica l ~nMysis for the Lo~entz model runs as follows. The differential equation for a harmonic oscillator subject to a periodic field