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We find a Lagrangian density from which the equations of motion for the Lang-muir solitons follow in the usual way. We show how this Lagrangian leads to the usual conservation laws. For the one-dimensional case we discuss how a consideration of these conservation laws can help us to understand some of the results obtained in numerical experiments on the behaviour of a strongly turbulent plasma. We point out that the situation in the three-dimensional case may be fundamentally different, and we discuss near-sonic perturbations and Karpman's treatment of these.

We discuss various aspects of the spontaneous generation of magnetic fields in a Langmuir plasma. We first of all show that the correct general expression for the ponderomotive force leads to the solenoidal current responsible for the magnetic-field generation. We derive the ponderomotive-force expression and also the magnetic-field generation equations from a two-time-scale two-fluid description. We also use a kinetic approach to derive the magnetic-field generation equations. We discuss the stability of monochromatic Langmuir waves and show that they are subject to both the ordinary modulational instability and to a magneto-modulational instability. We show that the coupled nonlinear equations describing the electric field strength amplitude, the plasma density, and the self-generated magnetic field can, under certain conditions, be reduced to a generalized cubic nonlinear Schrodinger equation. We finally show, by using a virial theorem, that the self-generated magnetic field does not stabilize the wave collapse.

For pt.I see ibid., vol.25, no.6, p.665 (1983). The authors review various methods for obtaining an expression for the ponderomotive force on a magnetized plasma; these methods are based on a fluid, kinetic, or a stress-tensor approach. They propose a new method which employs a fluid-dielectric model of the plasma; this method seems to be simple and generally applicable. They first of all calculate an expression for nonstationary waves in an isotropic plasma and find agreement with earlier results. They then derive an expression for the ponderomotive force in a magnetized plasma for the case of stationary waves and compare their result with earlier attempts to obtain such an expression.

Using the Zakharov equations in their Fourier-transformed form, we consider the development of the modulational instability (MI) both for the monochromatic and the finite-width cases. In the static approximation and considering a monochromatic Langmuir wave which is coupled only to a single pair of Stokes and anti-Stokes Langmuir perturbations, we show that the resultant set of equations is integrable and we discuss the analytical solution of these equations. We show how a finite-width driver will lead to a threshold for the MI. We compare our results with those obtained by other authors.

In this paper we discuss the turbulent state of an unmagnetized, completed ionized hydrogen plasma for the case when the electron temperature is much higher than the ion temperature so that the main modes to be considered are the Langmuir and ion-sound modes, while the transverse, electromagnetic mode can for most purposes be neglected. We give in the introduction a brief discussion of the quasi-linear theory of weak Langmuir turbulence as developed by Tsytovich and coworkers. We discuss why the peak of the resulting spectrum occurs at a finite wavenumber; the reason is that the removal of energy through electron collisions is faster than either thermalization of the ion-component or the transfer to yet lower wavenumbers through scattering by ions. In section 2 we sketch Zakharov's derivation of the basic equations which in the linear form describe the three modes mentioned earlier and which describe the nonlinear modulational instability (M.I.). In section 3 we give the linear analysis of the equations derived in section 2 and show that the decay instability of finite-amplitude Langmuir waves is fully covered by these equations. We also discuss the general dispersion relations for perturbations of such finite-amplitude waves and show that they lead to both the M.I. and the decay instability. We evaluate the growth rates for various kinds of perturbations. It is shown that all our results are valid only provided the energy density of the Langmuir modes W is less than the kinetic energy density nT e of the electrons. In section 4 we discuss under what circumtances in the linear approximation the M.I. dominates over other dissipative mechanisms and we show that it may well be the dominant mechanism for the case of strong turbulence, when W/nT e ≡ W > = m e /m i . Section 5 is devoted to a discussion of the one-dimensional variant of our basic equations. We give a brief discussion of various soliton-bearing equations such as the Kortweg-de Vries equation (KdV), the sine-Gordon equation (SG), and the non-linear Schrodinger equation (NLS).We discuss the possible connection between the number of polynomial conserved densities (p.c.d.) which lead to integrals of motion and the number of solitons in the strict sense of the term which can occur in a single exact solution of the equation. We also discuss the evolution of a system of Langmuir solitons in terms of their interactions with one another and with ion-sound. In section 6 we first of all show that in three-dimensions soliton-like structures are unstable against adiabatic collapse. We then discuss the dynamics of such collapsing “cavitons” and show that the theory of collapsing cavitons predicts that the turbulent energy should increase as the 2 3 power of the pumping rate. We discuss the fact that it may be difficult to avoid overcrowding of cavitons which would violate their independence. We finally briefly discuss the few relevant experiments and mention some ideas for future research.

The authors derive for the case of a magnetized plasma the analogues of the Zakharov equations for an isotropic plasma. They discuss the linearized dispersion relations following from these equations.

We derive a set of generalized Zakharov equations valid for both electrostatic and electromagnetic, that is, potential and non-potential, perturbations which include corrections due to higher electron non-linearities and allowing for a breakdown of slow-time scale quasi-neutrality and we show how these correction terms may halt the Langmuir collapse in two or three dimensions.

A brief discussion is given of possible approaches to developing a theory of turbulent states in plasmas in which there is a considerable energy content in various wave modes.

On considere l'effet d'une turbulence variant rapidement dans le temps dans un plasma de Langmuir magnetise sans collision sur le taux de croissance d'un mode de dechirure. On montre que dans des conditions typiques, le taux de croissance peut etre augmente

Synopsis An attempt is made to study in a simplified model of a binary substitutional alloy the influence of the differences in atomic radii of the constituent atoms and in the potential energies of the various possible pairs of atoms. The theory is developed for linear, simple cubic, and body centred cubic lattices, and it is found that the order-disorder transition in the three-dimensional lattices can be either first or second order depending on the differences between the various potential energies. The results are applied to the case of -brass. The agreement of theory with experiment is as good as can be expected from a theory involving necessarily a large number of simplifying assumptions. The effect of lattice vibrations on our results is considered briefly.

Summary The difficulty of the redundant zeros in the theory of the characteristic matrix, which has been pointed out by Ma 4) is studied in some detail. Another difficulty inherent to the theory in its present form is pointed out.

After a qualitative discussion of the advantages of the density matrix and of the different ways to introduce it (the statistical, quantum mechanical and operational methods of approach), § 2 deals with the general properties of the density matrix, including a discussion of pure cases and mixtures. A brief discussion is given of Green function techniques and of the relation between Green functions and correlation functions. A discussion of recent developments in the evaluation of partition functions concludes the first part of this article dealing with the theory of density matrix techniques. Sections 5 to 9 discuss applications. The first application is the quantum-chemical one to many-body systems in their ground state, that is, systems at absolute zero, and it is shown how the density matrix fits into the Hartree-Fock and Thomas-Fermi schemes. A brief discussion is given of the theory of diamagnetism. This is followed by a discussion of non-equilibrium processes and of Kubo's approach to transport theory. After that the polarization of beams of electrons or of photons is discussed and it is indicated how density matrix techniques can be used to treat scattering processes. Section 9 concludes this part of the paper by a brief account of density matrix theory applications to resonance and relaxation phenomena. Finally, the theory of measurement in quantum mechanics is considered.

After a brief introduction discussing the basis of molecular-field theories, we introduce a generalization of Anderson’s treatment of antiferromagnetism which enables us to take higher-order exchange interactions into account as well as anisotropy terms. We restrict ourselves to terms quadratic in the spins. This theory is developed in detail for face-centred cubic, body-centred cubic, simple cubic, and body-centred tetragonal magnetic lattices as well as for rutile structure antiferromagnetics. The general theory is developed in such a way that we can predict the possible preferred directions along which the spins in the antiferromagnetics will be alined. Until more information is available about the values of the various exchange parameters, it is not possible to predict theoretically which of the possible spin arrangements will, in fact, be the most stable one. On comparing experimental evidence on the spin arrangements in face-centred cubic and rutile structure antiferromagnetics we find, however, that almost all arrangements found experimentally are included in those predicted by the theory.

This paper is a sequel to an earlier one (ter Haar & Lines 1962 referred to as A) in which we applied a molecular-field treatment to anisotropic antiferromagnetics. In the present paper we apply spin-wave theory to investigate the influence of anisotropy of nearest-neighbour interactions and of the occurrence of next-nearest-neighbour interactions on the stability of the types of order found in A. After a brief introduction, face-centred cubic antiferromagnetics are considered in the second section. We find that there is no type of f.c.c. order which is stable for nearest-neighbour isotropic exchange interactions only. For the case of type 1 order with all spins along the direction of the unique cubic axis the order is stabilized by a small amount of anisotropy in the nearest-neighbour interaction. This is the only f.c.c. order which we found to be stable for nearest-neighbour interactions only. The influence of the more-remote-neighbour interactions is probably small for this case. For the case of type 1 order with all spins perpendicular to the unique cubic axis, we find that this type of order is only stable, provided interactions more remote than the nearest-neighbour ones occur. As far as type 2 order is concerned, the case where the preferred direction of order is in one of the ferromagnetically ordered planes turned out to be too complicated to be treated, but the case where the preferred direction is perpendicular to the ferromagnetic planes and the isotropic case can be treated. The orders in the latter cases are stable, provided the next-nearest-neighbour interactions are not too weak. If they are too weak, type 3 A order is the stable one. Type 3A order with the spins oriented along the unique cubic axis is stable, provided there is a small amount of isotropic antiferromagnetic next-nearest-neighbour interaction present. Type 3A order with spins perpendicular to the unique cubic axis is stable only if we include second and third nearest-neighbour interactions of sufficient magnitude. For most of these cases we have computed the spin-wave ground-state energy and the average value in this ground state of the total sublattice spin-component along the preferred direction; this value should be close to its maximum for the spin-wave treatment to be reliable. We observe that for all orders considered here there is a general rule: the order is not stable, if it is possible to single out a plane in the structure for which the average interactions between atoms within the plane and those outside is zero. In §2 we discuss the body-centred tetragonal lattice. We find that type 1 order is stable, provided the isotropic next-nearest-neighbour exchange interaction is larger than the nearestneighbour exchange interaction. If their ratio is less than 0.5 the so-called rutile type diagonal order—or type 2 order—is stable whenever its existence is predicted by the molecular-field theory. In the latter case one must introduce four sets of spin-waves rather than the two sets occurring for the other types of order considered in the present paper. In the last section we consider antiferromagnetic resonance. We find that the resonance frequency observed for MnO agrees rather better with the exchange interaction deduced from susceptibility measurements than with the value of this interaction deduced from mixed-salt para-magnetic-resonance measurements. For the case of MnF 2 we find a resonance wavelength of about 0.95 mm as against the experimental wavelength of 1.15 mm. We finally predict resonance frequencies of 15.0 and 19.1 cm-1 for (NH 4 ) 2 IrCl 6 and K 2 IrCl 6 if they should show type 1 order and of 10.6 and 13.5 cm-1, if the order should be type 3A.

The statistical aspect of the second law of thermodynamics is illustrated by a discussion of two simple models. In both cases the most probable behaviour, and the mean life and the average time of recurrence of different states of fluctuation are obtained and discussed.

Synopsis Kramers' variational method is used to derive the partition functions per spin for the cubic ferromagnetic structures. These partition functions are compared with the partition functions derived by the approximate methods of Bethe, Kirkwood and Zernike. The case of the two dimensional square ferromagnet with both nearest and next nearest neighbour interactions is also briefly discussed.

Synopsis It is shown how simple kinetic arguments can be used to derive approximate expressions for the thermal conductivity, electrical conductivity, thermoelectric power, Hall constant, and magnetoresistance of metals and semi-conductors, including in some cases the contribution from lattice currents.

We develop a theory for the attenuation of sound in liquid helium, valid at temperatures below about 1°K, by considering three-phonon processes. We also consider some higher-order contributions to the attenuation and consider the influence of finite lifetimes on the velocity of sound. We compare our results with experimental data and also discuss theories developed by other authors.

After a brief introduction to the problem and a discussion of the basic assumptions to be made, we derive the Bloch equation—relaxing to zero magnetization and containing one relaxation time only—through a coarse-graining in time. We also derive an integro-differential relaxation equation which is shown to lead to the Bloch equation in the case where there is no driving field.

We present a Green function calculation for the susceptibility of a Heisenberg antiferromagnet. In the random phase approximation the susceptibility at the Neel point has the molecular field value and the perpendicular susceptibility is independent of temperature. These results are modified in the generalised Callen approximation which we also consider. We include a brief mention of spin-flopping and give an expression for the temperature dependence of the critical field.

Synopsis We have applied the results of a previous paper to evaluate the Hall coefficient of bismuth at liquid helium temperatures, using a generalisation of Dingle's expression for the Fermi level as a function of the magnetic field strength and Shoenberg's parameters for the energy surfaces. The oscillatory part of the Hall coefficient is satisfactorily reproduced, but our theory predicts a linear behaviour of the non-oscillatory part, while a quadratic dependence is found experimentally. Including the contribution to the conductivity due to holes changes the field dependence of the steady part of the Hall coefficient from H to H1.3, and does not significantly alter the absolute magnitude or the oscillatory part.

Summary A brief account is given of the reasons why the many-body problem has recently become the subject of a rapidly expanding literature. After a survey of the main ideas of recent theories, it is shown how these ideas have played a role in giving us a better understanding of such topics as superconductivity, plasmas, and liquid helium. In the theory of the many-body problem the main branches of theoretical physics applied are Hamiltonian mechanics or quantum mechanics (to describe the motion of the constituent particles of the many-body system under consideration) and statistical mechanics (to describe the collective action of the large number of particles in the system). As these subjects are relatively advanced we have discussed in an appendix the main concepts of Hamiltonian mechanics and some of those of quantum statistics, including the definition and some of the properties of fermions and bosons.

We discuss the limitations of various expressions governing the determination of the limiting polarization of radio-waves and slightly extend earlier work, which was limited to the case where the wave-vector is parallel to the density gradient, to the special case where the magnitic field is parallel to the density gradient, but the wave-vector can be at an angle to both. We consider the problem especially in connection with the problem of pulsar emission.

The consideration of the statistical aspect of the second law of thermodynamics of a previous paper is extended. The influence of fluctuations in the wind-wood model of the Ehrenfests is investigated, and the relative occurrence, mean life and time of recurrence of such fluctuations are calculated. These are found to agree with the results obtained for the simpler models considered previously.

We consider that a relativistic formulation of thermodynamics or statistical mechanics has as its aim to provide a unique correspondence between the descriptions which different observers in uniform relative motion will give of a thermodynamic system. We show how this favours Ott's formulation of relativistic thermodynamics. We also show in how far one is led to the same conclusion by considering a relativistic formuation of statistical mechanics. In our discussions we restrict ourselves throughout to the special theory of relativity.

Synopsis It is suggested that the conductivity tensor may give useful information about the shape of the Fermi-surface.

It is shown how one can derive Bloch's induction equation from the equation of motion of the density matrix by coarse-graining in time. The relaxation of the magnetization is assumed to be due to a fluctuating magnetic field acting upon the spins. The derivation does not provide any new information but is sufficiently simple that it may be used as a lecture application of the density matrix.

It is shown that cosmic rays may have caused sufficient ionisation in a solar disc in aKant-von Weizsacker type of theory about the origin of the solar system to produce a conductivity high enough for magnetohydrodynamic effects to be important.

It is pointed out that the density matrices p HF used by quantum chemists and others in the discussion of the Hartree-Fock self-consistent field method differ from the density matrices p SM introduced in statistical physics, and that the idempotency condition for p HF has therefore a physical meaning which is different from the same condition for p SM . A suggestion is made for a consistent nomenclature.

The equations of motion of the Fourier components of the density are used to derive both the hydrodynamic equation of continuity and the hydrodynamic equation of motion. A brief discussion is given of the quantum mechanical case.

Makinson’s extension of Wilson’s treatment of the second-order effects in metals is used to derive an expression for the contribution of the lattice current to the thermo-electric power of metals at those temperatures where electron-phonon scattering predominates. It is found that in this temperature region one may expect the thermo-electric effect to show a sign opposite to the one which follows from the simple electron theory of metals. This is because the term due to the departure from equilibrium of the lattice distribution is larger than the usual term and is of opposite sign. If the temperature is greatly decreased or increased, the usual term predominates. The effect discussed may have a bearing on the behaviour of the thermo-electric power of the alkali metals, although it cannot explain this behaviour completely.

Synopsis The variational method developed by Kramers and Wannier is applied to the I s i n g model of a simple cubic ferromagnet and in this way equations are derived from which the approximate values of the partition function and thus of the configurational energy and specific heat can be obtained. These equations are solved both by series expansions and by numerical methods at temperatures above the Curie temperature.

We treat Dyson's ideal-spin-wave Hamiltonian by Green's function methods. In the second section we consider only noninteracting Dyson spin waves. In the third section we consider interacting spin waves, using the simplest possible decoupling assumption. Our results lead then to the same expressions for the magnetization and the spin-wave specific heats as Dyson in the limit as $S\ensuremath{\rightarrow}\ensuremath{\infty}$ up to terms of order ${T}^{4}$. The difference is of order $\frac{{T}^{4}}{{S}^{2}}$. If we introduce a higher-order decoupling, we obtain results identical with those of Oguchi, which in turn agree fairly closely with those of Dyson. We also obtain a spin-wave dispersion law which is a slight improvement of the one obtained by Brout and Englert, especially for small values of $S$. In three appendices we discuss, respectively, the decoupling process, the shifts in the spin-wave energies, and their "damping."

Synopsis A Lennard-Jones and an “exp-6” potential function for neon are investigated for the possible existence of a bound state of a neon molecule, by a method used by de Boer in the case of helium. It is concluded that there exists a bound state with an energy of about 5 × 10-5 atomic units ( 0.002 eV).

(1956). On the thermal conductivity and thermoelectric power of semiconductors. Advances in Physics: Vol. 5, No. 18, pp. 241-269.

A higher-order decoupling similar to the first-order Bogolyubov-Tyablikov decoupling is used to find various correlation functions in the paramagnetic region for a Heisenberg ferromagnet. Expressions are given for the susceptibility, the nearest-neighbor correlation, and autocorrelation at the critical point as well as at high temperatures.

From the experiments on the deviation of the cornea's surface from a sphere, information can be obtained about the anisotropy of the tangential tensions acting in the cornea.