K W H Stevens

A one-dimensional barrier and time-dependent tunnelling

There has recently been a growth in interest in electron tunnelling through barriers, and in particular in the question of whether it is possible to talk of a tunnelling time and, if so, what is it? A brief account of a study of the signal velocity of wavepackets travelling into a rectangular barrier was given recently, and the present paper is an extended version of this theory. In particular it is shown that the signal velocity of a free electron is the same as its group velocity, which in turn is the same as that which a classical particle particle of the same energy would have. It is also shown that when the wavepacket enters a classically forbidden region the signal velocity continues to have a meaning, and indeed a pulse-like wavefunction travels very much like a light pulse in a dispersive and attenuating medium. The expression for the signal velocity is symmetrical with respect to the barrier height, so that pulses well below the barrier travel faster, though with greater attenuation, than pulses near the top of the barrier. This result agrees with conclusions reached by some other workers. It is also shown that an electron responds to an accelerating field in the barrier as would be expected for a classical particle. It is argued that the not uncommon statement that the non nor eigenstates of a free particle describe a beam of particles is incorrect for electrons, for such a solution can at most describe two electrons. The theory indicates that by using orthogonal wavepackets it is possible to construct many-electron wavefunctions which do describe beams of electrons, and then free electrons travel with the group (and signal) velocity.

An explanation of the electron paramagnetic transitions and low temperature entropy anomalies of hexammine nickel halides

A point charge electrostatic model is used to explain the sudden increase in electron paramagnetic resonance linewidth at critical temperatures Tc and the low temperature entropy anomalies exhibited by hexammine nickel halides. It is suggested that these results are due to hindered rotation of the H3 groups between different equilibria arrangements of H3 protons within the solids. Barriers within the crystals are computed by assigning charges β|e| to each proton site, and summing the β2e2/rij terms between protons i and j on the same and on neighbouring Ni(NH3)6 clusters. Low and high energy arrangements of protons within each cluster are obtained. The model explains the electron paramagnetic resonance results by predicting that the nickel ion experiences a change from trigonal to cubic symmetry, occurring when the nearest neighbour electrostatic interaction between protons on neighbouring clusters is thermally overcome. The entropy anomalies, corresponding to a depopulation of one member of a ground state split by rotation of individual H3 groups, give values for β|e|. Reasonably consistent values for the three halides (similar ¼|e|) lend support to the validity of the main predictions.

Intermediate valence-a view of the theoretical situation

The phenomenon of intermediate valence (IV) has recently received a good deal of experimental and theoretical attention, but in spite of all the effort the phenomenon is not well understood, particularly the low temperature observations. The theoretical situation is reviewed, and those areas where the understanding seems to be reasonably satisfactory and also those where it is less satisfactory are delineated. The topic is introduced through a simple description of IV. This is followed by a brief account of some of the experimental results from which it is seen that while there are some phenomena which can be understood in an elementary way, there are others which are quite puzzling. A more detailed study is then made of the type of Hamiltonian which it would seem appropriate to use, and this is followed by an account of the approximations which have been used reasonably successfully (thermodynamic-type treatments and high-temperature thermodynamic perturbation theory). Attention is focused on the problem of the ground- and low-lying energy levels.

The Jahn-Teller effect of octahedrally co-ordinated 3d4 ions

The chromous ion in magnesium oxide is studied and an interpretation is given of the acoustic paramagnetic resonance results of Marshall and Rampton. Because of the very strong orbit-lattice interaction it is important to take full account of the Jahn-Teller effect. The cluster approximation justified previously is used, and then are added the spin-orbit coupling, anharmonic forces, external magnetic field and random lattice strains. The resultant secular equations are usually solved numerically in terms of parameters to describe the strains and anharmonic couplings. It is postulated that the observed line shapes reflect random strain distributions and that peaks in the lines are correlated with turning points in the strain-resonance field curves. Parameters are chosen to fit the observations for a definite field orientation and it is shown that a good interpretation can then be given for the results for a variety of orientations. The energy-level diagram of the chromous ion is shown to be quite complicated, and it is pointed out that many more transitions than have yet been observed can be expected.

An effective Hamiltonian for the Jahn-Teller coupling of T1g and T2g ions

An investigation is made of the properties of paramagnetic ions which have orbital triplets interacting with lattice vibrations. The linear coupling between the orbital levels and local distortions which have Eg symmetry are eliminated by a sequence of transformations on the Hamiltonian. These lead to an equivalent Hamiltonian in which the Jahn-Teller energy appears explicitly and most of the other operators have modified forms. The Hamiltonian of the lattice is discussed in more detail than usual by the introduction of symmetry adapted coordinates and momenta. Explicit formulae are obtained for the Jahn-Teller energy in terms of elastic constants and coupling coefficients, and for the displacements of lattice points, which are a consequence of the Jahn-Teller effect. It is also shown how a continuum model of the lattice can be introduced, and used to derive a cosθ/R2 dependence for the distortion associated with the Jahn-Teller effect. The analysis throws a good deal of light on the reason why the cluster model works so well.

The theory of planar methyl rotation

For some time it has not been clear that the standard theory of methyl group rotation in a plane takes full account of the antisymmetry requirement imposed by protons. This paper examines the properties of a free and a hindered methyl rotator, primarily by algebraic and symmetry methods. It is found that the full theory is not as simple as that usually given, and that there are some differences in the results, which throw new light on methyl rotation.

Pair spectra studies of Cr3+ ions in the (4A2 × 2T1) excited states of Cs3Cr2Cl9

A detailed optical absorption and Zeeman study of the spectrum of exchange coupled Cr3+ ions in Cs3Cr2Cl9 is presented, with particular reference to the singly-excited state where one ion of the pair is in the 4A2 ground state and the other is in the 2T1 optically excited state. The results are explained in terms of a new theoretical approach (K.R. Barry et al., ibid., vol.14, p.1281, 1981). Whereas the gross features of the spectrum may be understood without consideration of spin-dependent effects, it is shown that a small second-order spin-orbit interaction is needed to explain the fine details of the spectrum. In addition, an analysis of the vibronic structure associated with the pair transitions shows that the dominant spin-phonon coupling is to A1g modes which modulate the separation of the Cr3+ pair.

Fluctuating valence in SmS

In the ground state of normal SmS, each rare-earth ion is in a (4f)6 7F0 state, with all the other electrons filling valence bands. The symmetry of the ground state is therefore known. From an examination of the observed properties it is reasoned that when the lattice collapses under hydrostatic pressure (preserving the cubic symmetry) the new ground state has the same symmetry as the ground state of the uncollapsed phase. Since the new ground state shows fluctuating valence, the problem is to incorporate this feature into its detailed description. It is shown that this can be done in a variety of ways. It is further shown that a band of low-lying states, each with J=0 at each site but with k not=0, can also be constructed, as well as higher bands with some sites having J=1. The picture which emerges is thought to be helpful in providing a theoretical framework within which to discuss the phenomena associated with fluctuating valence. Experimental tests of the model are suggested, from a theory which can be developed further if substantiated.

Jahn-Teller effects in orbital triplets. I. Second-order contributions for D(T2) ions in cubic crystals coupled to E-type lattice distortions

Transformation methods are used to calculate the second-order contributions to the effective hamiltonian of a D(T2) ion coupled to E-type displacements in a cubic crystal. It is found that it is only possible to relate the first- and second-order Jahn-Teller reduction factors when a cluster model is used. For other lattice models, the parameters are unrelated. Finally, the orbital contribution to the magnetic susceptibility of a crystal containing delta (T2) ions is calculated.

Exchange interactions in rare earth chalcogenides

The author obtains explicit formulae for the admixtures of the J=1 level of Sm2+ (4f)6 7F into the J=0 level by exchange coupling to an adjacent Eu2+ (4f)7 8S ion. The problem is formulated using states which are antisymmetric with respect to interchanges of electrons. An unperturbed Hamiltonian is defined which is symmetric with respect to interchanges, and the difference between it and the full Hamiltonian is treated by perturbation theory, taken to second order. It is found that the dominant processes are those in which, in the intermediate states, either two electrons, one from each 4f shell, have been excited into unoccupied orbitals, or two electrons from closed shells have been excited into the 4f shells, one electron to each rare earth ion. By the use of equivalent operator techniques, it is shown that the admixtures can be derived from an equivalent anisotropic exchange interaction, with the important feature that the spin operators which appear are defined in second quantized forms and are not identical with the commonly used spin operators. It is suggested that in phenomenological exchange spin Hamiltonians these new definitions should be used.