Selected Scientific Papers of Sir Rudolf Peierls

The following sections are included:

• PREFACE

• CHRONOLOGY OF THE LIFE OF SIR RUDOLF ERNST PEIERLS

• CONTENTS

• Acknowledgements

It will be shown that one can derive from Bloch's calculations qualitatively correct conclusions about the galvano-magnetic effects: in particular, both signs are obtained for the Hall effect, which the Sommerfeld Theory had not been able to produce, and the order of magnitude of the changes in resistance is obtained…

The phenonmenon of the Hall effect is largely analogous to the deflection of cathode rays in a magnetic field, except that in some metals it produces a sign that is different from what is expected. An explanation of this paradox was impossible as long as the electrons were visualised as freely-moving in the metal, for then the analogy to cathode rays would be literally true…

The conditions for the existence of stationary states are established for a special type of potential functions, such as they exist in connexion with problems arising from the formation of molecules. Among other results, it is found that there always exist stationary solutions for a simple potential “well”, although this is not necessarily so in the presence of short-range repulsive forces.

The following sections are included:

• Introduction

• Harmonic forces

• Perturbation energy

• Statistics for the classical calculation

• The H-Theorem for classical mechanics

• The H-Theorem, quantum-mechanical treatment

• Discussion of the conditions for interference

• Classical formulation of thermal conduction

• Quantum-mechanical approach to thermal conduction

• Influence of lattice perturbations

• Comparison with experiment

• Summary

The following sections are included:

• Introduction

• General Method

• Electrical Conductivity

• Heat Conduction

• Summary

The following remarks are supplementary to two previous papers by the author on electric and thermal conductivity…

The electromagnetic field and its interaction with matter are described by a Schrödinger equation in the configuration space of the light quanta. The results are identical with those of Heisenberg and Pauli.

It is shown by considering possible methods of measurement that all the physical quantities occurring in wave mechanics can in general no longer be defined in the relativistic rannge. This is related to the well-known failure of the methods of wave mechanics in that range.

This is a report on an investigation by the author2 into the mechanism of the absorption of light by solids in the visible and ultra-violet spectrum…

The circumstances under which the free enemy in quantum statistics can be calculated without knowing the stationary states of the system are investigated. With the methods developed here, the diamagnetic susceptibility of free electrons, the effect of collisions on it, and the magnetic behaviour of bound electrons are studied. It is determined up to what field strengths the susceptibility remains independent of the field.

Calculations of diamagnetism are extended to the case of strong fields, and the resulting anomalies are compared with those observed in the case of bismuth.

A criterion by Kretschmann for the applicability of the theory is basically confirmed, although the formula for the bounds for the applicability have to be modified. The essential results of the theory concerning the temperature dependence of the conductivity are not affected, however. A second argument against the usual theory is refuted.

In the statistical treatment of mechanical systems, and in particular in the electron theory metals, a simplifying assumption is made which is shown not to be valid for metals at high temperatures. But just in this latter case other simplifications make a treatment of this problem possible, so that the usual results can still be justified in almost all cases, with the exception of the case of semi-conductors.

For many, physically very diverse but formally analogous, phenomena it is found experimentally that a transition takes place from an ordered state to a state of disorder at a certain temperature (Cf. the preceding talk by R.H.Fowler (Helv. Phys. Acta .7, Suppl. 2, 72-80, 1934). Typical examples for this are the melting points of solids and the Curie point of ferromagnetism…

THE view has recently been put forward¹, that a neutral particle of about electronic mass, and spin ½ℏ (where ℏ = h / 2π) exits, and that this 'neutrino' is emitted together with an electron in β-decay…

Although it seems very unlikely that neutrinos, after having been emitted in a nuclear process, give rise to any detectable ionisation¹, we would like to point out that it is not impossible in principle to decide experimentally whether they exist. One possible experiment would be to check…

The following sections are included:

• Introduction

• Wave Equation for H²

• Absorption of γ-Rays by diplons

• Capture of Neutrons by Protons

• Scattering of γ-Rays by diplons

• Disintegration of Diplons Under Electron Bombardment

• Limitations of the employed model

• Summary

The mass defects of the lightest nuclei, particularly the ratio between that of the diplon and the α-particle, make it very probable that the range of the interaction force between proton and neutron is very small, of the order of about 10^-13 cm…

The following sections are included:

• Introduction

• General remarks about statistical errors

• General formulæ

• Determination of decay constant: Discussion of Usual Methods

• Determination of decay constant: Rigorous treatment

• Most efficient lenght of series

• Nature effect: approximate treatment

• Nature effect: Rigorous treatment

• Determination of absorpton coefficient

• Summary

The following sections are included:

• Introduction

• Cubic face-centred lattice

• Approximation for s = 0

• Formation of Superlattice

• Energy content

• Discussion of Results

• Summary

The problem of a degenerate relativistic gas has recently become of importance for work on stellar structure. Some controversy has arisen as to whether there is an equation of state in the usual sense of the word, i.e. whether the pressure-density relation of such a gas enclosed in a certain volume would be independent of the shape of this volume. This might seem sufficiently obvious to make a proof unnecessary, but in view of the controversy it is perhaps worth while to give a proof…

The following sections are included:

• INTRODUCTION

• ASUMPTIONS

• LONG-SHAPED BODY IN LONGITUDINAL FIELD

• SOLUTION FOR AN ELLIPSOID

• COMPARISON WITH EXPERIMENT

• DISCUSSION OF FIELD EQUATIONS

• CALORIC EFFECTS

• SUMMARY

Statement of problem. In a recent paper, Fowler discusses the adsorption isotherm for a monatomic layer, assuming an interaction between neighbouring atoms in this layer. In this paper we do not intend to give a contribution to the physical problem, but merely to show how the statistical problem involved can be solved by a method due to Bethe…

Ising discussed the following model of a ferromagnetic body: Assume N elementary magnets of moment μ to be arranged in a regular lattice; each of them is supposed to have only two possible orientations, which we call positive and negative. Assume further that there is an interaction energy U for each pair of neighbouring magnets of opposite direction. Further, there is an external magnetic field of magnitude H such as to produce an additional energy of – μH (+ μH) for each magnet with positive (negative) direction…

It is well known that in regions in which the refractive index varies sufficiently slowly, Schrodinger's equation can be very simply treated by using its connexion with Hamilton-Jacobi's differential equation. It is also known that a similar approximation is possible in regions of slowly varying imaginary refractive index (total refiexion). For the latter case the method was developed in papers by Jeffreys (1924), Wentzel (1926), Brillouin (1926) and Kramers (1926). These papers discuss also the behaviour of the wave function in the neighbourhood of the limit between the regions of real and imaginary refractive index…

IT was found by Rollin and by Keesom and Keesom that liquid helium II shows an extraordinarily high heat conductivity, of the order of 190 cal./deg.cm.sec. We have measured the heat conductivity with the following method, which is essentially a combination of a vapour-pressure thermometer with a liquid-helium manometer. (The method was developed in discussion with Dr. J. D. Cockroft.) An evacuated…

Bohr has shown that in a collision between two nuclei of which at least one is heavy, an unstable compound nucleus will be temporarily formed and that the lifetime of such a compound nucleus, measured on a nuclear scale, is usually very large. For this reason these compound nuclei have fairly well-defined energy levels ("virtual" or "resonance" levels ), and the positions of these levels and the properties of the metastable states belonging to them determine the cross-sections for all nuclear reactions…

It is well known that the eigenfunction ψ₁ of the lowest state of any system has the property of making the integral…

Nuclear Reactions in the Continuous Energy Region IT is typical for nuclear reactions initiated by collisions or radiation that they may, to a large extent, be considered as taking place in two steps: the formation of a highly excited compound system and its subsequent disintegration or radiative transition to a less excited state.…

It is well known that a single neutron may cause a nuclear reaction chain of considerable magnitude, if it moves in a medium in which the number of secondary neutrons which are produced by neutron impact is, on the average, greater than the number of absorbed neutrons. From recent experiments it would appear that this condition might be satisfied in the case of uranium…

FEnmi's original theory of (β-decays made a definite prediction as to the energy distribution of the electrons emitted from a (β-active element. It was found that the experimental distribution curves did not agree in shape with this prediction in the sense that the number of electrons of low energy (relative to the upper limit of the spectrum) was considerably greater in the experimental than in the theoretical curves…

Calculations are made of the size of a dislocation and of the critical shear stress for its motion.

The possible construction of 'super-bombs' based on a nuclear chain reaction in uranium has been discussed a great deal and arguments have been brought forward which seemed to exclude this possibility. We wish here to point out and discuss a possibility which seems to have been overlooked in these earlier discussions…

The following sections are included:

• INTRODUCTION

• THE NOTION OF THE "COMPOUND NUCLEUS"

• THE LEVEL DISTRIBUTION AND THE STATE OF NUCLEAR MATTER

• DISINTEGRATION

• NUCLEAR REACTIONS: THE DISPERSION FORMULA

• THE CASE OF OVERLAPPING LEVELS

• COLLISIONS OF SLOW NEUTRONS WITH NUCLEI

• FAST NEUTRONS AND CHARGED PARTICLES

• NUCLEAR PHOTO-EFFECT

• NUCLEAR FISSION

• REFERENCES

The following sections are included:

• Separation Potential

• The Connexions in a Cascade Plant

• The Case of Many Fractions

• Efficiency of Various Methods of Separation
  • Thermal Diffusion
  • Evaporation at Low Pressure
  • Diffusion through a Porous Wall
  • Centrifuge
  • Velocity Selector

In a recent paper, Eddington raises an objection against the customary use of the Lorentz transformation in quantum mechanics, as for instance when applied to the theory of the hydrogen atom or the behaviour of a degenerate gas. This objection seems to us to be mainly based on a misunderstanding, and our purpose here is to show that the practice of theoretical physicists on this point is quite consistent . The issue is a little confused because Eddington's system of mechanics is in many important respects completely different from quantum mechanics, and although Eddington's objection is to an alleged illogical practice in quantum mechanics he occasionally makes use of concepts which have no place there…

After the detonation of an explosive in a medium, a wave across which there is a discontinuity of pressure, temperature and density travels through the medium. To determine its propagation the properties of the medium are required over a large range of temperature, although if the medium is air estimates show that the density is probably never much greater than ten times the normal value…

the following I discuss the properties , in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type…

A definition of Poisson brackets is given which is related to the action principle, but does not require the introduction of canonical variables. This permits the laws for forming both the commutators of canonical theory and the anticommutators of Fermi-Dirac particles to be stated in a manifestly covariant way. Examples of the use of this method are given. The last section discusses tentatively the extension to the case of equations which cannot be written in canonical form.

Assuming as an early stage in the expansion of the universe a homogeneous fluid of nuclear density and low temperature, it is shown that, for reasonable values of the constants, this will, on expansion, leave the matter in the form of droplets of the same properties as those found in the Mayer-Teller “polyneutron” theory. However, this model leads necessarily to an abundance curve in which the amount of heavy elements is at least comparable to that of the light elements, contrary to experience.

The conditions for absence of non-causal actions over large time intervals in non-local theories are expressed as conditions on the Fourier transform of the form function.

The paper investigates the possibility of introducing ‘non-local’ interactions, i.e. interactions represented by four-dimensional integral operations, in order to eliminate divergences in the quantum theory of interacting fields. In particular, a type of equation is discussed which preserves all the required invariance properties, including gauge invariance and macroscopic causality. It turns out that equations of this type still give divergent results. The origin of these divergences is discussed, and it is shown that if there is any way of formulating a finite theory it would have to be very different from the one investigated here.

Starting from the functional equations governing the Green function of a single nucleon moving in an external field with radiative corrections, as given by Schwinger, a formulation is developed which relates this Green function to that of a nucleon moving in an arbitrary external field, without radiative corrections. In the case of neutral scalar meson theory in which the recoil of the nucleon is neglected, the Green function is obtained in closed form. Mass and Green-function renormalizations are easily done completely, and the singularities of the solution investigated, proving to be an interesting illustration of the expected behaviour in more realistic cases.

The purpose of this note is to present a simple proof of the rule usually given for obtaining the spectrum of normal frequencies of a crystal. The rule is that the distribution of frequencies is the same as those of a hypothetical crystal satisfying the 'cyclic boundary condition'. We shall in the following refer to the hypothetical case as the 'mathematical crystal'…

A method for evaluating transition amplitudes for bound electrons in second order in the effects of the radiation field is outlined. An example of the type of problem concerned is the coherent scattering of γ-rays by the K electrons in heavy atoms. The static field in which the electron moves is taken into account exactly; no expansion is made in its effects. In the usual perturbation theory this is equivalent to summing matrix elements over intermediate states which are solutions of the wave equation including the static potential. In the method presented here, however, the sum over radial eigenstates for a particular angular momentum of intermediate state is replaced by quadratures of the products of known functions with the solution of a pair of coupled inhomogeneous differential equations.

THE one-particle propagator, which DYSON denotes by S_F1', is an important quantity in field theory, and a good knowledge of it, better than that obtainable from perturbation theory, would be useful. Several authors have shown that, considered as a function of the variable…

The fact that the total electron energy in a lattice of atoms may favour a reduction of symmetry takes a particularly extreme form in one dimension, where both the edge of the Fermi distribution and the position of the discontinuity produced by the symmetryreducing deformation consist of single points, so that there is no problem in making them coincide exactly…

Collective motions of a many-body system, such as translation, rotation and oscillation, may be incorporated approximately in a shell-model description by regarding the shell-model wave function as a trial function for a variational approach, and exploiting the fact that the expectation value of the energy is then independent of the location and orientation, and approximately independent of the size and shape of the potential well. If these degeneracies are removed in the usual way, one is led naturally to wave functions containing both shell-model and collective aspects and to approximate values for the translational, rotational and vibrational energies.

The model of Fermi interactions which involves an intermediate charged boson of spin one is formulated and found to be capable of reproducing all the results of the direct interaction theory with A and V coupling provided the mass of the intermediate particle is sufficiently large. The only apparent difficulty for the model is the predicted radiative decay of the muon.

The non-relativistic problem of scattering of a particle by a target possessing discrete excited states can be expressed in terms of ‘physical’ resonance states, i.e. solutions of the wave equation for complex energy in which in the asymptotic form of the wave function in each channel one of the two possible exponential terms (which for real energy represent the incoming and outgoing wave) vanishes. This representation is possible provided the interaction between the particles and the target vanishes exactly beyond a certain distance. If the interaction decreases exponentially a similar representation may in some cases still be obtained by analytic continuation; it contains also ‘redundant’ eigenstates in which the coefficient of one of the asymptotic waves tends to infinity. Possible generalizations of the method are discussed.

Historically, the first attempts to describe nuclei were based on the idea of a shell model, mainly because this description had been very successful in the case of the atom. However, the validity of such an approach seemed doubtful in view of the strength and the short range of the nuclear forces, as compared to the weaker long-range forces in the atom. These doubts were reinforced by the discovery of slow-neutron resonances, which clearly indicated the importance of many-body features which are neglected in the shell-model approach, and by the success of the compoundnucleus model of Niels Bohr, which in its extreme form idealizes the nucleus by treating it as a system of strongly interacting, and strongly corelated particles, like the molecules in a liquid drop…

I would like to talk very briefly about one possible change one may ring on the, in the first place, phenomenological nuclear force, which is not new. It was mentioned at Kingston a year ago, and also some papers along these lines have been published by Levinger's group, who have been working on this idea, and I think several people have discussed this. Now before I say what exactly the project is, I would like to make a remark, first of all, on the extent to which the idea of a nuclear-force or two-nucleoninteraction is really defined…

The translational (rotational) motion of a nucleus is described in terms of a linear combination of wave functions, each of which is generated from the Hartree-Fock wave function of the shell model, including deformation, by a linear (angular) displacement and a linear (angular) velocity. The angular velocity is represented in the same way as in the work of Thouless, and if this is based on a fully self-consistent treatment the moment of inertia found for rotation about a fixed axis is the same as that found by Thouless. The three-dimensional case is treated by the same method. The method can be regarded as an extension of that of Peierls and Yoccoz, arranged to give in the translational problem exactly the correct kinetic energy of the centre-of-mass motion.

The paper discusses the method proposed by Villars for describing the rotational states of nuclei. For simplicity only two-dimensional rotation is considered for the ground-state band of an even–even nucleus in which the last occupied sub-shell of the deformed potential is completely filled. It is shown how to avoid inconsistencies arising from the multi-valuedness of the angle variable which appears in the formalism. In the Villars method the moment of inertia appears in terms of a sum over the eigenstates of the ‘intrinsic’ Hamiltonian. Since these are not calculable exactly, one needs an approximation scheme. A possible scheme based on the Hartree–Fock approximation is used, though for the present this is approximated further by the use of harmonic oscillator wave functions. The case of ²⁸Si is treated by way of illustration. The detailed model is too unrealistic to expect physically significant results; it does suggest, however, that the method is suitable for practical evaluation. Further steps for improving the accuracy are discussed.

want here to make some comments on a very old problem, namely the apparent contradiction between the reversibility of the microscopic laws of physics and the evident irreversibility of statistical mechanics and thermodynamics. To avoid misunderstanding , it should be clear from the outset that these remarks have no connection with the recent doubts about time reversal invariance in weak interactions; or if there is any such connection I am not aware of it…

Elementary considerations support the result of Abraham, by which the momentum is, for given energy, inversely proportional to the refractive index, rather than proportional to it, as predicted by Minkowski. This conclusion disagrees, however, with the experiment of Jones & Richards.

For quantum systems with a symmetry it is often convenient to start from a simple soluble model lacking the symmetry, and restore the symmetry by projection. If this is done for all eigenstates of the model system, one obtains an overoomplete set of basis functions, which is not suitable for standard perturbation theory.

The paper develops a method by which one can do perturbation theory in this situation. The method is illustrated by application to the problem of the centre-of-mass motion of a nucleus, the symmetry being the translational and Galileo invariance.

The force on a moving ion in an electron gas to which an external electric field may be applied, is investigated in a semiclassical approximation, using a simple collision-time model for the mechanism which keeps the electron gas in equilibrium. It is found that in this simple model the screening of the ion has no effect on the force, so that the only correction to the force acting is that due to the ‘wind effect’, the momentum transfer due to the scattering of the electrons which are streaming past the ion.

A complete solution of the equations is given for a weakly charged ion, to first order in the charge.

It is shown that a perturbation expansion for projected states proposed earlier satisfies the Sohrödinger equation. Modified forms of the series are suggested, and criteria for their convergence are discussed. The method is applied to the ‘Lipkin model’ for a two-level manybody problem, which is exactly soluble. The second-order term in the perturbation series is shown to give a good approximation, and is compared with alternative methods. The convergence factor of the series is shown to be better than that of ordinary perturbation theory, which in most cases of this model is divergent.

The perturbation theory for projected states is shown to converge for a schematic one-dimensional model of . Ordinary perturbation theory diverges for the odd state. It converges for the even state, but very marginally for large interatomic distance.

The following sections are included:

• INTRODUCTION

• THE MOST NAIVE TRANSPORT PROBLEM

• ANGLE DEPENDENT SCATTERING; MANY COLLISION TIMES

• A STILL MORE GENERAL TRANSPORT COLLISION TIME

• HOW VALID IS THE BOLTZMANN EQUATION?

• THE SITUATION IS BETTER THAN IT SEEMS

• THERE ARE STILL RESERVATIONS

• PAULI'S REMARK AND OFF-DIAGONAL ELEMENTS

• SUMMARY OF LIMITS OF VALIDITY

• EXTENSION TO MANY-BODY PROBLEM

• NEGLECT OF CORRELATIONS

• CONSERVATION LAWS

• LIMIT ON COLLISION RATE

• OMISSION OF OFF-DIAGONAL TERMS: RIGOROUS DERIVATIONS

• ILLUSTRATIONS: LATTICE THERMAL CONDUCTIVITY

• CONTENT OF BOLTZMANN EQUATION

• EXPONENTIAL BEHAVIOUR

• THE USE OF MODERN METHODS

• CONCLUSIONS

• REFERENCES

Considerations of momentum conservation, applied to the simple semi-classical model studied in a previous paper, lead to a relation between the force on the ion and the resistance change of the metal caused by the ion. The result includes the direct force of the field on the ion, any screening correction, the ‘electron wind’, and the force due to the carrier density modulation.

It is shown that neither Minkowski's result, according to which the ratio of momentum to energy for a light wave in a medium of refractive index n is n/c, nor that of Abraham, who found 1/nc, is correct. For a broad wave in a uniform medium, the correct answer is given by (2.12) with . For weak refraction it is approximately equal to the average of the Abraham and Minkowski results. Abraham's formula gives correctly the part of the momentum which resides in the electromagnetic field, but not the mechanical momentum of the medium which travels with the light pulse. Minkowski's formula gives the pseudo-momentum, a quantity of physical interest. The momentum change upon reflexion or transmission usually involves also acoustic transients, these are discussed for some simple cases.

The paper shows, for a simple model, how the wave functions belonging to complex energy eigenvalues (resonance states) can be regarded as part of a complete set of states, containing also bound states and a continuum of (generally complex) wave functions. This forms the basis of an expansion, which is used to describe inelastic scattering in the plane-wave Born approximation, taking the same simple model for the target.

The Perey-Buck approximation is studied, and corrections to it are discussed, for the physically interesting case in which both the wavelength and the range of the non-locality are small compared to the distances over which the wavelength and the interactions change appreciably. In one dimension the Perey-Buck formula is the leading term of an expansion in the spirit of WKB. Further terms in the expansion are defined and terms up to second order given explicitly (sect. 2). This expansion is equivalent to that of Perey and Saxon, but it displays the orders of magnitude of the terms more clearly (sect. 3). For three-dimensional problems with central symmetry the angular integration can be performed in the non-local form, and the approximation of sect. 2 applied to the radial equation (sect. 4). The approximation is tested numerically on two simple models resembling the n-α and n-⁴⁰Ca interactions. In the latter case there is no significant difference between our approximation and that of Perey and Buck, both being satisfactory; in the former case agreement is achieved only by taking into account the second-order corrections (sect. 5).

Physicists tend to use models of various kinds to aid their understanding of complicated physical situations. The models differ by the degree of simplification or exaggeration they involve, according to the purpose for which they are used. It is suggested that one might distinguish seven different types. Examples of each type are discussed, together with the examples of the kind of confusion that can result if the nature of the model used is misunderstood.

The paper investigates a model of a liquid with some dispersion and some nonlinearity in Lagrange variables. A wavepacket of sound of small amplitude, and of a wavelength long compared with the characteristic wavelength of the dispersion law, will in general split in two, unless the oscillatory amplitude is accompanied by a non-oscillatory motion of an amplitude proportional to the square of the amplitude of the oscillatory part. The total momentum carried is then ambiguous, and it may not be possible to define the total momentum of such a pulse.

The following sections are included:

• Introduction

• Definition of pseudomomentum

• Pseudomomentum of flow

• Immersed object

• Electromagnetic field

• Polarizable medium

• Light pulse

• Momentum of sound

• Effect of dispersion and nonlinearity

• Momentum of light

• References

The paper analyses the measurement process on the basis of the interpretation of the wave function, or more generally the density matrix, as describing our knowledge of the system observed. There has to be a division between the system, as described, and the observer, but usually there is wide flexibility in placing this division within the chain of correlations involved in the measurement. This chain must, in particular, involve three steps: (1) a correlation between the quantity to be observed and the apparatus, (2) and uncontrolled interaction which leads to the loss of information required by the uncertainty principle, and (3) the recognition of the result by the observer.

The shape of extended objects in classically forbidden regions is shown to undergo expansion analogous to Lorentz contraction of a relativistic body of finite velocities. The problem of two interacting Dirac particles moving in one dimension is solved explicitly and the results are generalized to soliton solutions of field theories. An estimate of the effect on tunneling rates is also given, including solitons in (CH)_z.

In a stimulating article (Physics World August p33) the late John Bell professed dissatisfaction with the foundations of quantum mechanics as usually presented, particularly in connection with the so-called "collapse of the wavefunction" as a result of a measurement. He agreed that for all practical purposes the use of quantum mechanics by qualified practitioners leads to well defined answers which, where they can be checked, agree with experiment…

There are three ways in which an expected symmetry may not hold exactly: first, if the basic laws do not respect the symmetry exactly; second, if the initial or boundary conditions do not obey the symmetry; and third, by a spontaneous breakdown, which can happen in two ways, here called SBS1 and SBS2, which will be defined and illustrated. Similar to them is the symmetry-breaking approximation, in which a symmetric system is approximated by an asymmetric formalism to make it easier to handle certain correlations.

The following sections are included:

• Complete Bibliography for Sir Rudolf Peierls