Table Of ContentINTERATOMIC
POTENTIALS
IAN M. TORRENS
Centre d'Études Nucléaires de Saclay
France
ACADEMIC PRESS
New York and London 1972
Observe how system into system runs
ALEXANDER POPE
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PREFACE
The subject of the interaction between atoms is one which crosses the frontiers
of many different fields of physical research. Wherever a problem is treated on
an atomic scale there is a need for some knowledge of the forces which exist
between the atoms. It is these forces which decide much of what we can observe
in natural phenomena and a familiarity with them enables us to manipulate
the raw materials we find in nature to suit our various needs. Interatomic
forces are not simple because the atom itself is not an elementary particle.
It is a composite body consisting of nucleus and orbital electrons which must
be treated quantum mechanically. Consequently, any theory of interatomic
forces or potentials must deal with the complicated problem of many-body
interactions.
A large volume of literature exists in a number of different fields concern
ing the interaction between atoms, both at high energies where they penetrate
each other until the nuclear repulsion predominates and at low energies or
large interatomic separations. The scope of this literature is very wide, and
few extensive reviews are available to aid the researcher new to the subject of
atomic forces. To an increasing extent different fields of theoretical study call
for at least an empirical knowledge of the interatomic potential and its varia
tion with distance. In particular, with the modern large-capacity high-speed
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xix Preface
computer the treatment of physical problems numerically on an atomistic
scale has become quite feasible and suitable interaction potentials between
atoms are required. Even without the aid of the computer, simple analytical
or statistical theories often call for an interatomic potential based on theoret
ical considerations or experimental data.
From the personal experience of the author, it is very difficult,when
confronted with the atomic interaction problem at an early stage in a research
career, to obtain a suitable interatomic potential from previous work. An
elaborate literature search is usually required, and it is very easy to miss
important points and to decide upon what are in some cases unsuitable forms
of potential for treating the problem under consideration. Essentially, this
lack of a comprehensive treatise on the subject of interatomic potentials was
instrumental in the decision to write the present monograph. I have no illu
sions as to its role as a comprehensive work. For example, inelastic or charge-
exchange atomic interactions are mentioned only very briefly, and the book
might reasonably be accused of being slightly biased (for low-energy poten
tials) toward the metallic form of solid. Its purpose is rather to serve as an
elementary introduction to the subject for those new to the field and an easy
reference book for more experienced researchers. To this end I have included
a considerable number of references to original articles, reviews, and books
on different aspects of the subject.
Throughout the book the aim is to sacrifice involved mathematical
derivations on the altar of more qualitative physical considerations. It is of
course unrealistic to exclude mathematics completely, as the large number of
equations bears witness, but such as there are should present little difficulty.
An undergraduate course in mathematics and elementary quantum mechanics
should suffice to understand any of the physical principles presented here.
I apologize to the reader if it occasionally seems that I have made an equation
appear from thin air. In such cases several pages of mathematics would have
been required to justify its existence, and the original reference, listed at the
end of the chapter, is available for consultation by those interested in probing
more deeply into a particular topic.
Consistency of notation always presents a problem when gathering
material from widely differing sources. One generally finds that similarly
defined quantities often differ by a few physical constants, electronic charges,
or powers of atomic number! In such cases I have tried to point out any
differences in notation between this book and earlier original papers relevant
to the particular topic.
ACKNOWLEDGMENTS
It is a great pleasure for me to acknowledge the assistance of a number of
friends and colleagues during the preparation of this manuscript. Dr. Yves
Adda and other members of the Département d'Etudes Métallurgiques have
been sources of encouragement throughout the work. I am grateful to
Professor David Lazarus, Dr. Yves Quéré, Dr. Maurice Gerl, Dr. Mark
Robinson, Dr. Dirck Van Vliet, Professor Norman March, and Dr. Daniel
Schiff for making valuable comments on the whole or part of the manuscript.
The assistance of Messrs. Michel Bendazzoli and Pierre Truchot in the pre
paration and photography of a number of the figures has been very much
appreciated. In addition, my thanks are due to the Department de Calcul
Electronique at Saclay for the use of the computers and peripheral devices
in connection with some of the tables and figures. Finally, I wish to thank all
those authors who very kindly gave me permission to reproduce their figures
and in many cases sent me photographs or lent me original drawings.
xiii
INTRODUCTION
The atomic theory of matter is relatively ancient, predating the actual
experimental evidence for the existence of the atom by over two millenia.
Following the hypotheses of earlier Greek philosophers, Lucretius, writing
in the first century B.C., gave a speculative but quite lucid account of the
atomic composition of matter. In this theory the interlocking between different
shapes of atoms gave rise to the physical properties of materials. This idea of
the physical form of the atom determining its interaction with others persisted
through the period of post-Renaissance science, and it was not until the
eighteenth century that the concept of an interatomic force independent of the
form of the atom became current. Newton tried to explain various chemical
and physical phenomena by postulating a force which was attractive at short
distances but repulsive at larger separations. Although erroneous, this was
the introduction of the interaction of atoms at a distance without the necessity
that they touch each other. At about the same time, a remarkably clear
sighted account of the force between atoms was put forward by Boscovich.
His model suggested a repulsive force at small distance followed by several
1
2 Introduction
attraction-repulsion oscillations of the force distance curve. The zero-force
points were those at which atoms combined to form stable arrangements.
Considering the amount of experimental information available at that time,
this was no minor achievement. Boscovich's potential shows some remarkable
correspondence with very recent forms of interaction presented at various
stages of this book.
Modern theories of atomic forces began with the discovery at the turn of
the century of the composite nature of the atom and of the nature of the
interaction between its electrically charged components. The force between
atoms was revealed to be many-body in nature, made up of nucleus-nucleus,
electron-nucleus and electron-electron interactions. With the development of
quantum mechanics it became theoretically possible to write down a rigorous
expression for the interaction of two atoms. However, the solution of the
relevant equations was shown to be extremely complicated except for the
simplest systems and quite substantial approximations were necessary.
Although with present-day computing systems quite large advances have
been made in solving the many-body problem, it is more usual to represent the
interatomic force by an approximative expression. This is achieved either by
approximating the theory in some way or by adjusting parameters to experi
mental data, or by a combination of both methods. The purpose of this book is
to discuss the physical principles behind a range of such atomic interactions
and to show how they may be applied to some atomic problems.
CHAPTER I
THE NATURE OF INTERATOMIC FORCES
1.1 ATOMIC INTERACTIONS IN PHYSICAL PHENOMENA
Almost all physical phenomena, discounting the world inside the atomic
nucleus, may be attributed directly or indirectly to the forces between atoms.
Basic concepts such as temperature and pressure, the strength of a solid or
the viscosity of a liquid, as well as our own physical form and that of this
book, are intimately related to the forces between atoms.
It has long been known that an atom, made up of nucleus and orbital
electrons, is largely empty space, like the solar system. Therefore, the idea of
an atom as a basic entity is rather nebulous. But such are the forces binding
the electrons to the nucleus that except for quite high energy collisions the
atom may be thought of as an entity about as penetrable, from the point of
view of another atom, as a solid hard rubber ball. This is, of course, an over
simplification. In order to estimate the force acting between atoms in an
ensemble or in a collision process, it is often necessary to take into account
the elementary components of the atom. Thus, in the rubber ball analogy, we
3
4 I The Nature of Interatomic Forces
would have to explain why the hardness varies with the energy of a collision
and why under certain circumstances two rubber balls attract each other
when they are not touching.
With the normal concept of an atom made up of a central nucleus and
orbital electrons, let us consider the forces intervening when it interacts with
another atom. We discount any subnuclear phenomena since the forces
holding together the nuclear components—neutrons, protons, and mesons—
are of a completely different nature and orders of magnitude stronger than any
interatomic forces. Thus the nucleus is effectively a solid body of diameter
~10"12 cm with a positive charge depending on the number of protons
present. If there were no orbital electrons the force between two nuclei would
be Coulombic, of the form
F^^Z.Z^jr2 (1.1)
where Z and Z are the numbers of protons contained in the nucleus.
x 2
The nucleus -b- orbital electrons constitute an atom of diameter ~ 10" 8 cm,
the effect of the electrons being to neutralize the charge of the nucleus. Con
sequently, if atoms were considered as entities there should be no forces
between them at distances greater than that at which they touch. However, at
separations comparable with the atomic diameter, the Coulomb interactions
of the outer electrons of atom 1 with those of atom 2 and with the nucleus
of atom 2 (and vice versa) have a significant effect. The two atoms no longer
view each other as entities and the interaction becomes a rather complicated
many-body problem. The same problem exists in an even more acute form
when the electron shells of two atoms interpenetrate during an energetic
collision.
To treat this many-body problem accurately for all types of atomic inter
action would defy the most powerful computing systems available now or in
the foreseeable future. Fortunately, a number of approximations and averag
ing procedures under different physical situations enable us to obtain
analytical expressions or numerical values for the interatomic forces which
may be applied with varying degrees of confidence to atomic problems. The
greater part of this book will be devoted to the various approximations which
render the many-body problem of atomic interactions less intractable.
For most purposes the force between two atoms is expressed in terms of
their potential energy of interaction, or interatomic potential. It depends to a
first approximation on the separation r between the atoms; the relation be
tween the force F(r) and the potential V(r) is
F(r)=-(d/dr)[V(r)] (1.2)
Strictly speaking, the potential may also depend on the relative positions of
