Table Of ContentLong-Lived
States
•
In
Collisions
Author
Slobodan D. Bosanac
The Rugjer Boskovic Institute
Zagreb, Croatia
Yugoslavia
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Bosanac, Slobodan D.
Long-lived states in collisions.
Bibliography: p.
Includes index.
1. Collisions (Nuclear physics) I. Title.
QC794.6.C6B67 1988 539.7’54 87-17845
ISBN 0-8493-6871-5
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To my mother
PREFACE
The idea of writing a book about the long-lived states in collisions was brewing for some
time, but starting it came quite suddenly. It was a coincidence of two events. One was that
new ideas developed in understanding of such states, and the other was a discussion with
a friend, Prof. Nenad Trinajstic, about some aspects of these states, after which he suggested
that I write a book about all these. It was a challenge that I accepted, not realizing what
was awaiting me.
First, I had to convince myself that such a book is really needed. The importance of long-
lived states in collisions cannot be disputed. In fact, some of the most interesting processes
have such states as their intermediary. For example, large numbers of long-lived states
appear in collision of stable elementary particles, e.g., protons, electrons, positrons, etc.,
and for some specific reasons they are also called elementary particles. Some of them live
for such a long time that they even leave a clearly visible trace in, say, a bubble chamber.
An alpha-decaying nucleus is also an example of a long-lived state, but it is treated as a
decay process rather than as an intermediary state in collision. Less obvious but a very
important example of the long-lived states is connected with the time irreversible processes
in systems with large number degrees of freedom, such as a glass of water. Usually such
systems are very successfully treated by the statistical theories, but these theories do not
answer the basic question: What causes the time irreversibility of the processes in such
systems, when it is known that the equations of motion for atoms and molecules are time-
reversible? Again, if one looks in details, what seems to be the time irreversible process is
in fact manifestation of the properties of the long-lived states which are formed in collisions
of large clusters of particles.
These scattered examples only show that one finds long-lived states on different micro-
scopic levels. It is not clear, however, that in all these cases the long-lived states may have
some common properties. One common feature in those examples is that the systems are
quantum, i.e., their properties must be described by the quantum theory. The classical theory
is seldom good and only for the qualitative understanding of these systems. However, in
the quantum theory of scattering there is a phenomenon called resonance which, it is argued,
represents a long-lived state and therefore, in principle, all the above-mentioned long-lived
states can be described by these resonances. There are numerous books which describe
resonances, and therefore adding this one to an already big pile would just be repeating
what is already known about them, or at best it would be applying this knowledge to atom
and molecule collisions. Whether this is sufficiently interesting to justify writing a book is
questionable. However, there is a small problem with the long-lived states which haunted
me for long time, but I thought it could be solved in terms of resonances, although it was
not obvious to me how. Imagine that we collide a particle with a large cluster of particles
(say an atom with a large molecule). If the forces between these two species are attractive
from their infinite until a relatively small separation (which is the case with atoms and
molecules) then on impact with the cluster the incoming particle may be able to transfer
most of its kinetic energy into vibrations of the cluster, and stay bound to it. This process
can be very nicely observed in the classical simulation of collision, and indeed it was
described on many occasions for the case of atom-molecule systems. It was further observed
that the original particle stays bound to the cluster for a long time, until part of the excess
energy is not released by the dissociation process, i.e., by the fragmentation of the cluster.
It was also observed that formation of the intermediary long-lived state is not very sensitive
on the variation of the incident energy of the particle. In other words, a long-lived state is
formed even for a relatively large variation of the collision energy. Yet it is known that
resonances have a unique property. The longer they live the narrower the interval of the
collision energy within which they are observed. Therefore it appears that the long-lived
states, which we described, have properties that are completely opposite to those of reso-
nances. Of course, it is not legitimate to compare these two kinds of long-lived states because
the former is a result of the classical simulation while the latter is the quantum phenomenon.
However, one can always make a quantum system behave more and more classically, by
taking smaller and smaller value of the Planck's constant h. In such a case it is known that
the number of resonances, per unit interval of energy, increases and therefore in the end
one obtains the classical effect of the long-lived states as the interference effect of many
overlapping resonances. Such an explanation cannot be excluded, but it did not look right
to me. This was further substantiated by the fact that in the same limit the width of resonances
also descreases. Since one cannot make beautiful theories by having only the feeling that
something is not right, the idea that there may be other types of long-lived states, besides
resonances, was waiting until the proof was available that this is indeed the case. Further
research showed that resonances are not long-lived states in a true sense, which, at least to
me, was quite intriguing. At this point I thought that the results were interesting enough so
that writing a book on the long-lived states would have some justification.
The plan for the book was that it would be useful to scientists other than atom and
molecular physicists, and so I tried to make it as general as possible, however, with the
emphasis on the atom and molecule collisions. There are some specific features of these
collisions which must be taken into account, but they are not of such character that the
theory is less general. On the contrary, by considering these features one learns a great deal
about the collision theory for the systems with hard core potentials and about the quantum
theory in the limit of "very small h". In addition, throughout the book we will use the
method of the complex angular momentum analysis of the scattering amplitude. In its original
form, when this method was introduced in the potential scattering theory, it was applied to
systems for which the implicit assumption was that they do not have hard core. Therefore,
original theory cannot be used for description of atom and molecule collisions, but with
suitable modification it becomes a powerful tool.
The book contains essentially two parts. A review of the classical, quantum, and semi-
classical theories of collisions are given in the first part, while their applications to the atom
and molecule collisions are given in the second part. The idea in the first part was to give
an overview of essential parts in the general theory of collisions which are relevant to the
study of the long-lived states. Originally this part was planned to be brief, but in the course
of writing it became obvious that very little is known about the topic. When it was finished
I realized that this "would-be introduction" had taken almost half of the book. It contains
what I believe to be the major points in the theory of collisions, with extensive discussions
of the drawbacks in the classical and quantum theory. As an illustration of these drawbacks
I have discussed simple examples. For some, these examples may appear too trivial to be
mentioned, but I hope that they serve the purpose. On the other hand, if they tum out to
be failures, I hope that they will make the interested reader think about the better ones. It
would be disastrous if these examples led the reader to the wrong conclusions.
In this part of the book I have also included a chapter on decay. These processes are not
the subject of the collision theory, but very often it is assumed that decays are "half" of
an appropriate scattering problem. Intuitively this is an attractive model, especially if a
decaying system lives for a long time before it splits into fragments. The model is founded
on the belief that the long-lived states "forget" their origin and therefore, no matter how
they are formed, they will always fragment in one way. However, such a belief is not
realistic, as discussed in this chapter. Furthermore, it is shown that there are very few truly
decaying processes. Many of them appear to be decays, however, they have their history,
even when long-lived states are involved.
Analysis of the long-lived states in atom and molecule collisions is given in the second
part of the book. Since there is a large amount of information about simple systems, and
as one goes to more complex systems the knowledge about them becomes poorer, the chapters
likewise shrink towards the end of the book. Complex systems are difficult to analyze
because the relevant equations of motion are difficult to solve. Also there are some fun-
damental problems, which are not related to our inability of solving the equations of motion,
but are connected with the properties of long-lived states in complex systems. The roots of
these problems are mentioned in the first part of the book, and they are traced to the immensely
complicated analytic structure of the scattering amplitude. Therefore as one goes from simple
systems to more complex ones, the methods of analysis change from the quantum to entirely
classical theory of scattering. One reason is that for complex systems it is much easier to
solve the initial value problem (which is the classical theory) than the boundary value problem
(which is the quantum theory). The other reason is that in the quantum theory of scattering
one makes expansion of the wave function in the asymptotic states, which is poorly con-
vergent. In the classical theory this problem does not exist, and therefore it is much easier
to apply this theory than the quantum. However, the meaning of the results obtained from
the classical theory of collisions is questionable, as is partly discussed in the chapter on the
semiclassical theory of scattering.
Overall this book is the theoretician's view of the long-lived states in collisions. I did not
discuss the experimenal side of this problem or, in particular, the methods how to measure
the time delay. This does not mean that I am totally ignorant of the experimental side of
scientific research, and for this I am greatly thankful to my friends, Prof. U. Buck and Prof.
J. P. Toennies, both from the MPI fiir Stromungsforschung in Gottingen. They have taught
me the necessary rudiments about the experiments in atomic and molecular collisions, which
every theoretician should know. I am also thankful to them for many discussions, and long
collaboration in this field. As a theoretician I should be thankful to many colleagues from
different fields of science, from whom I have learned that the basis of the most important
laws in nature is simplicity. From them I have also learned that the beauty of nature lies in
a variety of details, which are nothing else but reflections of a few basic laws. I am thankful
to Prof. J. N. Murrell from the University of Sussex, who taught me how to be a scientist
and to my father Prof. T. Bosanac, with whom I had endless discussions about different
problems in science, although very often it was he who did the talking. I would be doing
a great injustice if I only thanked those who directly or indirectly contributed to the scientific
content of the book. Without the patience and understanding of my wife, Ana, this book
would never have seen the daylight. I am grateful to her for this and also for many lonely
months when she was waiting for me to return from visits to my colleagues. Our children,
Jana and Iva, were too young to understand why their father was very often too busy with
science to be able to play with them. I only hope that one day they will forgive me for this.
In the end, I would like to thank my friend Prof. H. Kroto from the University of Sussex,
for using free time in his immensely busy schedule to critically review the manuscript and
Dr. Z. Bacic from the Institute of Physics in Zagreb, for making valuable comments about
the book. Also, thanks go to my secretary, Mrs. B. Spoljar, who painstakingly typed my
handwriting which sometimes was even difficult for me to read.
THE AUTHOR
Prof. Slobodan Danko Bosanac, Ph.D., is Science Advisor in the Department of Physical
Chemistry at the Rugjer Boskovic Institute in Zagreb, Croatia, Yugoslavia.
Prof. Bosanac received his higher education at the University of Zagreb where he graduated
from the Department of Theoretical Physics in the field of high energy physics. After
graduation, he obtained the SRC grant from the British government to do research at the
University of Sussex under the supervision of Prof. J. N. Murrell. During the period between
1969 and 1972 he worked on the problems of bound states of three atom systems using the
hyperspherical coordinates. The theory was applied to a particular problem of He trimer.
He also worked on the reactive collision problems in connection with vibrational predis-
sociation of carbon dioxide molecules.
In 1972 he obtained a Ph.D. from the University of Sussex and obtained a 2-year SRC
postdoctoral grant to work with Dr. G. G. Balint-Kurti at the University of Bristol, England.
During this period, he worked on the rotational inelastic problems of simple systems. He
returned to Zagreb in 1974 where he continued research at the Rugjer Boskovic Institute
until 1975 when he was in the Military Service for 1 year.
From 1974 to 1976 he became interested in the theoretical problems of long-lived states
in collisions. For this, he generalized the method of complex angular momenta to molecular
collisions and developed perturbation theory for the poles of S-matrix. In 1977 he obtained
the Lady Davis Fellowship to do research at the Hebrew University of Jerusalem with Prof.
R. B. Gerber. During this year, he formulated the semiclassical perturbation theory of
collisions, which was necessary for proper treatment of molecular collision problems. Also,
the technique of complex angular momentum poles was applied to the inversion problem of
scattering.
After his return to Zagreb, Prof. Bosanac investigated various inelastic collision processes
in order to understand the mechanism of molecular energy transfer and rules of formation
of long-lived states. Models of rotational and vibrational energy transfer were formulated
which explained its basic features. He initiated several international projects with the groups
at the Max-Planck-Institute for Fluid Dynamics in Gottingen, the University of Sussex, and
the University of Florida in Gainesville. He made numerous extended visits to Universities
in Europe and the U.S. where he was involved in various projects in the field of molecular
energy transfer.
Prof. Bosanac has published over 50 scientific papers related to molecular energy transfer
and general topics in mathematical physics.
TABLE OF CONTENTS
Chapter I
Classical Theory of Collisions ......................................................... .
I. General Aspects .................................................................. .
II. Limitations of Classical Theory .................................................. 13
Chapter 2
Quantum Theory of Scattering ........................................................ 19
I. General Aspects .................................................................. 19
II. Resonances and Long-Lived States .............................................. 31
Chapter 3
Semiclassical Theory of Scattering .................................................... 41
Chapter 4
Decays .................................................................................. 57
Chapter 5
Two-Atom Systems ..................................................................... 69
I. General Theory .................................................................. 69
A. Transformation of Scattering Amplitude .................................. 69
B. Properties of Regge Poles and Residues .................................. 79
C. Perturbation Theory of Poles and Residues ............................... 83
II. Elastic Collisions ................................................................ 90
A. Properties of Regge Poles and Residues .................................. 90
B. Time-Delay Analysis of Scattering Amplitude ............................ 97
C. Analysis of Cross-Sections .............................................. 103
D. Historical Overview ..................................................... 109
III. Excitation of Electrons - Two Channel Problem .............................. Ill
Chapter 6
Atom-Molecule Systems ............................................................... 123
I. Scattering on Spherical Potential. ............................................... 123
A. Vibrationally Inelastic Collisions ........................................ 123
B. Inelastic Total Cross-Sections ........................................... 134
C. Classical Long-Lived States ............................................. 135
II. Scattering on Nonspherical Potential. ........................................... 141
A. Scattering without Energy Transfer. ..................................... 141
B. Two-Dimensional Scattering ............................................. 145
C. Rotational Energy Transfer .............................................. 149
Appendix A ............................................................................ 157
Appendix B ............................................................................ 159
Appendix C ............................................................................ 161
Appendix D ............................................................................ 163
References .............................................................................. 167
Index ................................................................................... 169
1
Chapter 1
CLASSICAL THEORY OF COLLISIONS
I. GENERAL ASPECTS
Newton's equations of motion form the basis of the classical theory of collision. They
are second order nonlinear differential equations in time, and their solution represents the
time evolution of the coordinates of all particles. Classical theory is an initial value problem
in one variable (time), which means that the evolution of systems of particles is uniquely
determined if at some instant of time all their coordinates and velocities are known. The set
of all coordinates can be written as a column vector U and will be called a trajectory of the
0
system of particles. The time derivative of such a trajectory is U, and thus, Newton's
equations are the equations for this trajectory.
Newton's equations can be formulated in various ways. One of them is on the basis of
the principle of least action, the Hamilton principle, which says that the integral of a function
S, called the Lagrange function, between two points in time has its extreme value only along
the classical trajectory. Explicitly, if we write the deviation from the classical trajectory U
as the column vector E(t), then the integral
I(E) = f'' S(U + E, u + E) dt (1)
,,
is sensitive to second order in E(t), i.e.,
(2)
If E is zero at the end points, i.e., E(t1) = E(t2) = 0, it follows, from this principle, that
the Lagrange function S satisfies the set of equations
as d (as) (3)
~ = di a til
where U1 is the ith element of U. The Lagrange function has the form
S = T(U) - V(U) (4)
where T is the kinetic energy and V(U) is the potential. In Cartesian coordinates the kinetic
energy for a system of n + 1 particles has the form
T = -21 n1'L"+~J1' l m I (x0 2I + y0 2I + z0 ~) (5)
where ml is the mass of the ith particle and x, y, and Z1 are the respective three-dimensional
(3D) coordinates. The potential is independent of velocities U and time, a necessary re-
quirement for conservative systems, i.e., those which conserve energy. It is also a function
of the relative separations of atoms and molecules.
The set of Equations 3 describes the motion of particles relative to an arbitrary coordinate
system, the laboratory coordinate system (L). It is convenient, however, to work in the
