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The background for the details of multichannel scattering calculations has now been established. We consider methods based on the integral-equation formulation of chapter 6. These momentum-space methods have proved accurate at all energies in a sufficient variety of situations to justify the belief that they can be generally applied. In some situations sufficient accuracy is achieved without resorting to the full power of the integral-equation solution. The methods used in these situations are distorted-wave methods. Their relationship to the full solutions will be examined in a simplified illustrative case. A brief outline will be given of alternative methods based on a coordinate-space formulation of the multichannel problem.
There are two characteristic difficulties of multichannel many-fermion problems. The first is that computational methods can of course directly address only a finite number of channels whereas the physical problem has an infinite number of discrete channels and the ionisation continuum. The second is that the electrons are identical so that the formulation in terms of one-electron states must be explicitly antisymmetric in the position (or momentum) and spin coordinates.
We first show how to set up the problem within the framework of formal scattering theory using antisymmetric products of one-electron states. The problem is then formulated in terms of the calculation of reduced T-matrix elements relating the absolute values of initial and final momenta in different angular-momentum states. This depends on a knowledge of the corresponding potential matrix elements, whose calculation we treat in detail. We then show how the target continuum is accounted for in the scattering formalism.
We have considered the measurement of observables in electron–atom collisions and the description of the structure of the target and residual atomic states. We are now in a position to develop the formal theory of the reaction mechanism. Our understanding of potential scattering serves as a useful example of the concepts involved.
Reactions are understood in terms of channels. A channel is a quantum state of the projectile–target system when the projectile and target are so far apart that they do not interact. It is specified by the incident energy and spin projection of the projectile and the quantum state of the N-electron target, which may be bound or ionised.
The reaction mechanism is studied by considering targets whose description is simple and, at least from the spectroscopic point of view, believable within an accuracy appropriate to the scattering experiment. Hydrogen is the obvious example, although experiments are difficult because of the need to make the atomic target by dissociating molecules. Sodium is a target for which a large quantity of experimental data is available and whose structure can be quite well described for the lower-energy states. When the reaction mechanism is sufficiently understood the reaction may be used as a probe for the structure of more-complicated target or residual systems.
Formulation of the problem
Scattering theory concerns a collision of two bodies, that may change the state of one or both of the bodies. In our application one body (the projectile) is an electron, whose internal state is specified by its spin-projection quantum number v.
The problem of N bound electrons interacting under the Coulomb attraction of a single nucleus is the basis of the extensive field of atomic-spectroscopy. For many years experimental information about the bound eigenstates of an atom or ion was obtained mainly from the photons emitted after random excitations by collisions in a gas. Energy-level differences are measured very accurately. We also have experimental data for the transition rates (oscillator strengths) of the photons from many transitions. Photon spectroscopy has the advantage that the photon interacts relatively weakly with the atom so that the emission mechanism is described very accurately by first-order perturbation theory. One disadvantage is that the accessibility of states to observation is restricted by the dipole selection rule.
Photon spectroscopy associates two numbers with the pair of states involved in a transition, the energy-level difference and the transition rate. The correlated emission directions of photons in successive transitions are determined trivially by the dipole selection rule. In most cases it is impossible to solve the many-body problem accurately enough to reproduce spectroscopic data within experimental error and we are left wondering how good our theoretical methods really are.
Because our description of differential cross sections for momentum transfer in a reaction initiated by an electron beam depends on our ability to describe both the structure and the reaction mechanism, scattering provides much more information about bound states. This is even more true of ionisation. The information is less accurate than from photon spectroscopy and is obtained only after a thorough understanding of reactions, the subject of this book, is achieved. The understanding of structure and reactions is of course achieved iteratively.
Quantitative studies of the scattering of electrons by atoms began in 1921 with Ramsauer's measurements of total collision cross sections. Ramsauer (1921) with his single-collision beam technique showed that electron–atom collision cross sections for noble gas targets pass through maxima and minima as the electron energy is varied, and can have very low minima at low electron energies. The marked transparency of rare gases over a small energy range to low energy (~1 eV) electrons was also noted by Townsend and Bailey (1922) in swarm experiments. This result was in total disagreement with the classical theory of scattering, which predicts a monotonic increase in the total collision cross section with decreasing energy. The Ramsauer–Townsend effect provided a powerful impetus to the development of quantum collision theory.
Although the history of electron impact cross-section measurements is quite long, the instrumentation and the experimental techniques used have continued to evolve, and have improved significantly in recent years. Part of the motivation for this progress has been the need for electron collision data in such fields as laser physics and development, astrophysics, plasma devices, upper atmospheric processes and radiation physics. The development of electron–atom collision studies has also been strongly motivated by the need of data for testing and developing suitable theories of the scattering and collision processes, and for providing a tool for obtaining detailed information on the structure of the target atoms and molecules and final collision products.
From the very beginning of our involvement in the investigation of light induced polarization of angular momenta of molecules we were fascinated by the variety of information about the properties of molecules which they bear. At the same time the description and interpretation of these phenomena appeared to us to be extremely complicated and unclear. In fact, at times it seemed as if our computers understood the problem better than we did.
This book is an attempt to clarify the processes during the course of which polarized (ordered) angular momenta distribution is created in an ensemble of molecules in the gas phase by the effect of light. We discuss the effect of static external magnetic and electric fields on the angular momenta distribution. In particular, we wish to emphasize the ‘geometric’ meaning and interpretation of the phenomena. This may, we believe, be a further step in attempts to simplify the theoretical description, thus making it more accessible to a wide range of users, both physicists and chemists.
The fundamental basis for optical polarization (alignment, orientation) of angular momenta is the law of conservation of angular momenta in photon–molecule interaction. In this book we examine a variety of macroscopic manifestations of spatial anisotropy of angular momenta, such as angular distribution and polarization of emitted light, including changes under non-linear absorption, and the influence of collisions and external fields. Quantum angular momentum theory, in particular that which is based on irreducible tensorial set representation, presents a well-developed approach that is widely used in subatomic, atomic and molecular physics.
The advancement of knowledge of electron–atom collisions depends on an iterative interaction of experiment and theory. Experimentalists need an understanding of theory at the level that will enable them to design experiments that contribute to the overall understanding of the subject. They must also be able to distinguish critically between approximations. Theorists need to know what is likely to be experimentally possible and how to assess the accuracy of experimental techniques and the assumptions behind them. We have aimed to give this understanding to students who have completed a program of undergraduate laboratory, mechanics, electromagnetic theory and quantum mechanics courses.
Furthermore we have attempted to give experimentalists sufficient detail to enable them to set up a significant experiment. With the development of position-sensitive detectors, high-resolution analysers and monochromators, fast-pulse techniques, tuneable high-resolution lasers, and sources of polarised electrons and atoms, experimental techniques have made enormous advances in recent years. They have become sophisticated and flexible allowing complete measurements to be made. Therefore particular emphasis is given to experiments in which the kinematics is completely determined. When more than one particle is emitted in the collision process, such measurements involve coincidence techniques. These are discussed in detail for electron–electron and electron–photon detection in the final state. The production of polarised beams of electrons and atoms is also discussed, since such beams are needed for studying spin-dependent scattering parameters. Overall our aim is to give a sufficient understanding of these techniques to enable the motivated reader to design and set up suitable experiments.
The detailed study of the motion of electrons in the field of a nucleus has been made possible by quite recent developments in experimental and calculational techniques. Historically it is one of the newest of sciences. Yet conceptually and logically it is very close to the earliest beginnings of physics. Its fascination lies in the fact that it is possible to probe deeper into the dynamics of this system than of any other because there are no serious difficulties in the observation of sufficiently-resolved quantum states or in the understanding of the elementary two-body interaction.
The utility of the study is twofold. First the understanding of the collisions of electrons with single-nucleus electronic systems is essential to the understanding of many astrophysical and terrestrial systems, among the latter being the upper atmosphere, lasers and plasmas. Perhaps more important is its use for developing and sharpening experimental and calculational techniques which do not require much further development for the study of the electronic properties of multinucleus systems in the fields of molecular chemistry and biology and of condensed-matter physics.
For many years after Galileo's discovery of the basic kinematic law of conservation of momentum, and his understanding of the interconversion of kinetic and potential energy in some simple terrestrial systems, there was only one system in which the dynamical details were understood. This was the gravitational two-body system, whose understanding depended on Newton's discovery of the 1/r law governing the potential energy. By understanding the dynamics we mean keeping track of all the relevant energy and momentum changes in the system and being able to predict them accurately.