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.

Introduction

When the construction details of a lens are known, it is possible to avoid all theory and to determine its aberrations numerically by using Snell's law and geometry. This process is called ray tracing. The procedure is straightforward. Select a ray coming from a chosen object point. Follow its path as it traverses the lens, and find its intersection point with the image plane. Do this for several other rays coming from the same object point. Ideally all the intersection points should coincide and form a perfect image point, but this is hardly ever the case. Usually the intersection points fall in slightly different spots, and the information obtained about their spread must be used to predict how well the lens will perform when it is put to its intended use.

Ray tracing consists of two steps applied alternately, translation and refraction. Given a ray specified by a point and a direction, (1) its intersection point with the next lens surface must be determined, and (2) this point found, Snell's law must be used to find the direction of the ray after refraction. These two steps are repeated until the image plane is reached. We used this process in chapter 10, but there we restricted ourselves to the small angle approximation. This led to simple linear relations. Now we have to use the exact equations, which are unpleasantly complicated.

Prior to the introduction of mechanical desk calculators in the 1930s and 1940s, logarithms were used to carry out these calculations.

Entrance and exit pupil

Fig. 11.1(a) shows a lens followed by an aperture (diaphragm, stop) P′Q′ in the image space. How can we aim a ray in the object space so that it will pass through the diaphragm? To answer this question we image the diaphragm backwards through the lens. This image is labeled PQ in the figure. If a ray in the object space intersects the PQ-plane in a point between P and Q, it has to pass through the image that the lens forms of this intersection point, which will be found between P′ and Q′. So the ray will pass through the diaphragm. If, on the other hand, a ray intersects the PQ-plane outside PQ, it will eventually pass through a point outside P′Q′, i.e. it will not pass through the diaphragm. So any ray passing through the actual aperture P′Q′ must pass through its backwards image PQ in the object space. This backwards image is called the entrance pupil.

When the stop is placed in the object space a similar result holds, in reverse. With the arrangement shown in fig. 11.1(b) we cannot produce just any ray in the image space; only those rays that pass through the image P″Q″ of diaphragm P′Q′ can be realized. The effective aperture found by imaging the actual stop in the image space is called the exit pupil.

In most lenses the stop that determines which rays will reach the image space is buried inside the lens, as shown in fig. 11.1(c).

Basic properties

Thin lenses are made up of elements that are so thin, separated by air gaps that are so small, that all the axial distances may, for all practical purposes, be set equal to zero. Doublets and triplets used as telescope objectives are common examples. For lenses of this nature the effect on the aberrations of introducing the thicknesses is small, so that only small changes in the surface curvatures are needed when, in the end, the finite thicknesses must be accounted for. We will only consider thin lenses used in air.

The rays through the axial point of the lens are undeviated as they emerge into the image space. This point is therefore imaged free from spherical aberration, and, because Abbe's sine rule is clearly satisfied, free from coma as well. If the pupil coincides with the lens, the rays through the center of the lens are the chief rays. As they move on undeviated, there can be no distortion. There is, however, a great deal of astigmatism.

To evaluate the astigmatism, we consider, in the spirit of chapter 23, rays close to one of the chief rays. In the calculation of the four-by-four matrix for these rays the thicknesses are set to zero, so the translation matrices reduce to unit matrices and can be left out of the product. First consider a single thin element. The field angle, i.e. the angle between the axis and the chief ray outside of the lens, is ψ.