Perspectives in Theoretical Physics. The Collected Papers of E. M. Lifshitz

731 Pages·1992·42.682 MB·English

by  J. B. Sykes, D. ter Haar and L. P. Pitaevskii (Auth.)

Ε. Μ. LIFSHITZ
Perspectives in
Theoretical Physics
THE COLLECTED PAPERS OF
Ε. M. LIFSHITZ
Edited by
L. P. PITAEVSKM
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First edition 1992
Library of Congress Cataloging-in-Publication Data
Lifshits, Ε. M. (Evgenii Mikhailovich)
[Selections. English. 1991 ]
Perspectives in theoretical physics: the collected papers
of Ε. M. Lifshitz/edited by L. P. Pitaevskii
p. cm.
1. Physics. I. Pitaevskii, L. P. (Lev Petrovich) II. Title.
QC71.L532513 1991 90-46224
British Library Cataloguing in Publication Data
Lifshitz, Ε. M. (Evgeni Mikhailovich)
Perspectives in theoretical physics: the collected papers
of Ε. M. Lifshitz.
1. Theoretical physics
I. Title II. Pitaevski, L. P.
530.1
ISBN 0-08-036364-4
Printed in Great Britain by BPCC Wheatons Ltd., Exeter
Introduction
Evgenii Mikhailovich Lifshitz was born on 21 February 1915 in Kharkov,
which at that time was the capital of the Ukraine. His father was a gastro-
enterologist and a professor at the Institute of Medicine. His younger brother
was Ilya Mikhailovich Lifshitz, the theoretical solid state physicist-like his
brother an Academician and a Lenin Prize winner. Ε. M. Lifshitz studied at
the Chemical College in Kharkov from 1929 to 1931 and after that from 1931
to 1933 in the Physics and Mechanics Faculty of the Kharkov Mechanics and
Machine Building Institute. From 1933 to 1938 he worked, first as a graduate
student under L. D. Landau until his Ph.D. examination in 1934, and then as
a senior research assistant in the Ukrainian Physicotechnical Institute. From
February to May 1938 he worked in Moscow at the Ail-Union Leather
Institute, returning to Kharkov to the Chemical Technology Institute, where
he stayed from September 1938 to June 1939. This period of rapid changes of
places of work coincided with Landau's move from Kharkov to Moscow,
Landau's unjust arrest, and his fortunate release-due to Kapitza's inter­
vention with the authorities-a year later. These difficulties also explain the
gap of three months when Lifshitz lived in the Crimea with his first wife,
Elena Konstantinovna Berezovskaya.
In 1939 Lifshitz's D.Sc. thesis was accepted by the Leningrad State
University and apart from a period during the war, when the whole Institute
of Physical Problems was evacuated to Kazan, he spent the rest of his life
from September 1939 in Moscow at the Institute of Physical Problems of the
USSR Academy of Sciences. Until 1956, when he switched all his activities to
his work on the Journal of Experimental and Theoretical Physics (JETP) and
the Course of Theoretical Physics, he taught at several institutes: Kharkov
University, Kharkov Mechanics and Machine Building Institute, Kharkov
Chemical Technology Institute, Moscow State University, and the Moscow
Pedagogical Institute.
Lifshitz received many honours. In 1945 he received the Order of the Red
Star for his work for the army, and in 1954 the Order of the Red Banner of
Labour and also a State Prize; in 1958 the Lomonosov Prize of the USSR
Academy of Sciences, and in 1962, jointly with Landau, the Lenin Prize. In
1966 he was elected a Corresponding Member of the USSR Academy of
Sciences, he was awarded the Landau Prize of the USSR Academy of
Sciences in 1974 and elected an Academician in 1979; in 1983 he was elected a
1
2 Perspectives in Theoretical Physics
Foreign Member of the Royal Society, and in 1985 he received an honorary
doctorate from Budapest University.
Lifshitz had a son from his first marriage who is a pathological anatomist,
and Lifshitz is survived by his second wife, Zinaida Ivanovna. He died on
29 October 1985 after a heart operation.
When one comes to Lifshitz's scientific career, one must distinguish three
important parts: his work - originally with Landau-on the classical Course
of Theoretical Physics, his work as editor of JETP, and his scientific papers.
JETP is for Soviet physics what the Physical Review is for the USA - the main
physics journal. Lifshitz for about 30 years dealt with all aspects of all the
papers submitted to JETP, and its excellence as a primary physics journal is
largely due to Lifshitz's work.
Lifshitz will probably be mostly remembered for having been the second
part of'Landau and Lifshitz', the masterly 10-volume textbook of theoretical
physics. A large part of Lifshitz's scientific output can be found in the Course
of Theoretical Physics. After Landau's accident in January 1962, from which
he did not recover, Lifshitz alone was responsible not only for preparing new
editions of the volumes which had already appeared, but also for
writing-with new collaborators, all from the Landau school-the volumes
which were still needed to complete the set. Many of Lifshitz's papers
provided sections for the various volumes of the Course of Theoretical
Physics.
Many generations of physicists-not only in the USSR-have used
'Landau and Lifshitz' in their daily (creative and educational) activity. The
complete course, which was completed in 1979, consists of 10 volumes:
Mechanics, Theory of Fields, Quantum Mechanics, Quantum Electrodynamics,
Statistical Physics (Classical and Quantum Statistics and Theory of the Con­
densed State), Fluid Mechanics, Theory of Elasticity, Electrodynamics of
Continuous Media, and Physical Kinetics. The idea for this course was
conceived in the early 1930s when Landau was in Kharkov, and the early
chapters were based on lecture notes.
The guiding idea of the course consists of arriving by the shortest path to
the solution of specific problems without getting bogged down in arguments
and the laying of foundations. It is, however, not just a handbook of
mathematical methods. The whole exposition is based on physical concepts,
either general ones-such as conservation laws-or model ones-such as a
collisionless plasma or a perfect gas.
The course has been internationally acclaimed: it has been translated in its
entirety into English, German, French, Japanese, Italian, and Hungarian,
while parts have been translated into Spanish, Portuguese, Serbo-Croatian,
Rumanian, Polish, Bulgarian, Chinese, Vietnamese, Greek, Hindi, and
Slovak.
In the light of Lifshitz's activities for JETP and the Course of Theoretical
Physical it is not surprising that a feature of Lifshitz's work as a scientist is the
Introduction 3
relatively small number of papers he published. The present volume contains
1 7 , 23713 2 , 3 41, 4 1
only 48 items, and of those six are review articles. ' '
Most of Lifshitz's papers deal with specific problems in theoretical physics
rather than with general theories. This is a consequence of the fact that in
working on the Course of Theoretical Physics he came upon as yet unsolved
points which he felt should be incorporated in the volume he was working on.
This has as a consequence that most papers are quite extensive, and one finds
21
only two short notes among the papers. The first one is a brief paper
3
showing that the chemical potential of liquid He can, over a fairly wide
2
temperature range, be represented as a power series in T \ this paper was a
precursor of the Fermi liquid theory devised by Landau several years later.
We quote an interesting-prophetic-sentence from this paper: Theoret­
ically, liquid helium-3 should have a specific heat proportional to the
temperature as it is probably a quantum liquid with a "Fermi-type" energy
39
spectrum like that of the "electron liquid" in a metal.' The second brief note
reports briefly more extensive work given in detail in several other papers.
172732
Of the review papers, three ' ' deal with the behaviour of liquid helium
3 134
at low temperatures, and especially with its superfluid properties. Two '
deal with forces between condensed bodies, a topic on which he published
41
extensively (see below), while the last review paper dealt with cosmological
problems to which he devoted so much time towards the end of his life.
1
A number of his earlier papers were written with Landau. His first paper
dealt with pair production during a collision of two particles, a topic he
2
returned to in his second paper which he wrote by himself. In these papers,
which appeared only a few years after the discovery of the positron, he
discussed electron-positron pair formation in heavy-particle collisions. In the
first paper Landau and he study the ultra-relativistic limit, that is, the case
where the relative velocity of the two colliding bodies is close to the velocity
of light. In this case one must use second-order perturbation theory, as first-
order perturbation theory gives an effect which in the ultra-relativistic limit is
much smaller than the second-order effect. We can then neglect the difference
between the particle trajectories and straight lines; that is, neglect the
interactions between the two colliding particles, and the effect is due merely
to the superposed fields of these particles. In this case the pair-production
cross-section varies as the cube of the logarithm of the energy of the colliding
particles. In the follow-up of this paper Lifshitz considered the case where the
relative velocity of the two colliding particles is small as compared to the
velocity of light. In this case one must distinguish several regions. The first
region is the one where the velocity is so small that the inverse time of
2
collision is much less than mc /ft, where m is the electron mass and c the speed
of light; in that case the pair-production cross-section is small and decreases
The numbers refer to the order of papers in this volume.
4 Perspectives in Theoretical Physics
exponentially with decreasing velocity. That region is therefore of no further
interest. For larger velocities one should distinguish the case where the
motion of the colliding particles can be treated classically and the case where
we need quantum theory to describe it. It turns out, however, that the pair-
production cross-section is the same in these two cases. For these cases
Lifshitz first of all repeated the calculations from the earlier paper; that is, he
calculated the second-order cross-section, in order to be able to compare it
with the first-order effect, where the wavefunctions of the colliding particles
are perturbed by their interaction. The results of the calculations of both the
first- and the second-order cross-sections show (1) that the first-order cross-
section is now proportional to the cube of the logarithm of the relative
velocity of the two colliding particles, but (2) that in this case the second-
order cross-section is proportional to the tenth power of that velocity, and (3)
that the first-order effect dominates over the second-order one as soon as the
relative velocity falls below 0.3c.
Lifshitz returned a few years later to problems in nuclear physics and
7-9
discussed nuclear reactions involving deuterons, such as the disinteg­
ration of the deuteron.
In 1936 Lifshitz wrote two papers on the photoelectromotive force in
semiconductors. The first one was with Landau and the second one by
4
Lifshitz alone. In the first paper they calculate the electromotive force arising
in a circuit which includes a semiconductor when one of its contacts is
illuminated, both when there are no holes and when there are holes present.
5
In the second paper the same calculations are repeated for the case when
there is an external magnetic field present.
Possibly the most important one of the papers Lifshitz published with
3
Landau as coauthor is the one which contains the so-called Landau-Lifshitz
equation of motion for the magnetisation in ferromagnets. This paper dis­
cusses also the domain structure of ferromagnets and the problem of ferro­
magnetic resonance.
It is well known that a ferromagnetic crystal above the Curie point will
consist of domains, each of which is practically magnetised to saturation, but
in such a way that the total magnetic moment of the crystal is zero. The shape
of the domains-which was the concern of the discussion by Landau and
Lifshitz-is determined by the requirement of minimum energy of the system
as a whole. If the ferromagnetic crystal has one axis of easy magnetisation,
there are three sources of energy:
1. the magnetic energy, partly inside and partly outside the crystal;
2. the anisotropy energy, deriving from the fact that not all magnetic
moments in the crystal will be along the axis of easy magnetisation;
and
3. the exchange energy, deriving from the fact that not all magnetic moments
will be parallel to one another.
Introduction 5
Writing down expressions for these three energies Landau and Lifshitz and
find (1) the width of the Bloch wall, (2) the width of the domains, and (3) the
shape of the closure domains. They also show that the domains have a layer
rather than a needle structure.
They next considered what would happen to the magnetisation if
an external magnetic field is applied to the ferromagnet. By considering
not only the precession of the magnetisation due to this field, but also
possible damping forces due to spin-spin interactions which will tend to
align the magnetisation with the external field, they find the Landau-Lifshitz
equation of motion, which can be used to find the frequency of ferromag­
netic resonance. Landau and Lifshitz, in fact, discuss, the case of a har­
monically varying magnetic field without, however, discussing resonance
phenomena.
6
In 1937 Lifshitz published a paper on the properties of an electron gas in a
magnetic field. In that paper he introduced the so-called drift approximation
and also derived an expression for the collision integral in a magnetised
plasma.
1011
During the war Lifshitz published two important papers ' on the theory
of second-order phase transitions, which supplement Landau's classical work
on the subject. There have been radical changes in this- theory and in the
theory of critical phenomena in the years which followed the publication of
these papers. The elucidation of the decisive role played by fluctuations in the
immediate vicinity of the critical point, the discovery of similarity properties,
and the possibility of calculating critical indices, have made the theory of
these phase transitions quite different from the Landau theory. However, the
principal points have not only been preserved, but have acquired greater
importance. They relate to the connection between the nature of the trans­
ition and the change in symmetry of the system. The geometrical symmetry
approach has been retained, and this is the reason why Lifshitz's work in this
field has not lost its value. Before briefly discussing this, we want to stress one
other fact. Lifshitz's paper is based on the use of group theory, and one should
realise that a thorough knowledge of group theory in those days was a
rarity, and showed a more than usually profound mastery of mathematical
techniques.
In his original papers Landau had indicated only one reason for a
transition ceasing to be a second-order one-the presence of cubic terms in
the expansion of the thermodynamic potential. In that case the transition
becomes first-order. However, Lifshitz pointed out that, in fact, there is only a
limited number of possible second-order phase transitions. This is related to
the fact that a second-order phase transition is possible only between phases
possessing different symmetries where the elements of the symmetry group of
the second phase form a subgroup of the symmetry group of the first phase.
The number of possible second-order phase transitions is thus limited by the
6 Perspectives in Theoretical Physics
total number of space groups. Lifshitz in his first paper enumerates all
possible transitions for the case of Bravais lattices. The following general
theorems illustrate the kind of transitions which are possible:
1. A second-order phase transition can occur for any change in structure
which halves the number of symmetry transformations; such a change may
occur either by a doubling of the unit cell for a given crystal class or by a
halving of the number of rotations and reflections for a given unit cell.
2. Second-order phase transitions cannot occur for changes in structure
which reduce to one-third the number of symmetry transformations, as
this would imply the presence of cubic terms in the thermodynamic
potential.
In the second paper Lifshitz considers second-order phase transitions in
substitutional alloys. He shows that Landau's thermodynamic theory im­
poses restrictions which are not always realised in order-disorder theories
such as those by Bethe and Peierls. He poses a question similar to the one in
his first paper: For what types of order in alloys can a Curie point exist; that
is, can a second-order phase transition take place. He considers body-
centred-cubic, face-centred-cubic, and hexagonal-close-packed lattices, as
superstructures are known to occur in those lattices. Using the same tech­
nique as in his first paper he then lists all possible transitions.
In Landau's theory of the superfluidity of helium II he devotes a short
section to the propagation of sound in helium II. He finds that there are two
sound velocities, but does not comment any further on this. A few years later
12
in 1944 Lifshitz discussed sound in helium II in much more detail. In this
paper he also coins the terms first and second sound. He discusses in some
details how these two kinds of sound can be generated, and points out that it
is very difficult to produce second sound by the usual means of producing
sound, that is by a vibrating body. He then suggests that it may be generated
by having a surface with a temperature which fluctuates periodically in time,
and points out that in second sound the temperature vibrations are much
stronger than the pressure vibrations. These predictions were verified by
Peshkov's experiments after the war.
2 228
Apart from some papers on miscellaneous topics, including two ' on
hydrodynamical subjects with Landau, the remainder of Lifshitz's scientific
ceuvre was devoted to two topics: the forces between solid objects and
cosmological problems. In 1948 Casimir, in a paper with Polder the aim of
which was to account for anomalous experimental results for forces between
colloid particles, had pointed out that retardation effects would modify the
forces between molecules and between solid bodies, increasing the index of
the attractive London-van der Waals forces by unity. This idea was taken up
by Lifshitz-together with Dzyaloshinskii, Pitaevskii, Abrikosova, and
2 3 , 2 5 , 2 6 , 3 14 , 3 3 , 3
Deryagin-in a number of papers. Lifshitz's idea was that the
interaction between the bodies is transmitted through a fluctuating electro-
Introduction 7
magnetic field. Because of thermodynamic fluctuations this field is always
present in the interior of a material medium and also extends beyond its
boundaries. Of course, at very low temperatures the fluctuations are of a
purely quantum nature. The theory was also used to consider properties of
thin liquid films on the surface of a solid body, as well as forces between
bodies separated by a dielectric.
1535 48
The last group of papers ' " deals with various cosmological prob­
lems. With Belinskii, Khalatnikov, and various other collaborators he con­
sidered mostly the general problem of singularities in cosmological solutions.
Another topic he investigated was the question of the gravitational stability
of the isotropic model, as this model seems to give an adequate description of
the present-day state of the universe, considered on a large scale. He found
that the situation is quite satisfactory for an expanding universe, but not so
satisfactory for a contracting universe. His studies were all based on the
Einstein equations in their classical form without a cosmological term.
D. TER HAAR