There are, in general, many possible transitions of electrons in atoms. In some processes of practical interest more than one electron may undergo a transition. Such multiple electron transitions are the topic of the next and subsequent chapters. In this chapter the simpler topic of single electron transitions is considered, where the activity of a single electron in an atom is the focus of attention. Even this relatively simple case may be impossible to fully understand if the electron of interest is influenced by other electrons in the atom. So in this chapter the interdependency of electrons in the system is ignored. That is, the electrons are treated independently. Typically, such an independent electron is regarded as beginning in an initial state characterized by some effective nuclear charge ZT and a set of quantum numbers n, l, m, s, ms from which all possible properties (e.g., energy, shape, magnetic properties, etc.) may be determined. Interaction with something else, (usually a particle of charge Z and velocity, v), may cause a transition to a different final state of the atom.

The simplest transition occurs in interaction of atomic hydrogen with a structureless projectile. There are various ways to evaluate the transition probability for such a system. Exact calculations usually require use of a computer. Approximate calculations may be done more easily. Calculations for many electron systems are often done approximately using single electron transition probabilities.

Correlation

There is a significant difference between complex and merely large. This difference is related to the the notion of correlation which defines the rules of interdependency in large systems. The relevant question is: how may one make complicated things from simple ones? Biological systems are complex because the atomic and molecular subsystems are correlated. From the point of view of atomic physics correlation in condensed matter, chemistry and biology is determined at least in part by electron correlation in chemical bonds and the complex interdependent structures of electronic densities. Understanding correlation in this broad sense is a major challenge common to most of science and much of technology. This is sometimes referred to as the many body problem. In a general sense correlation is a conceptual bridge from properties of individuals to properties of groups or families.

The concept of correlation arises in many different contexts. ‘Individual’ may mean an individual electron, an individual molecule or in principle an individual person, musical note or ingredient in a recipe. In this book individual refers to electron for the most part. In this case the interaction between individuals is well known, namely l/r12. However, that does not mean that electron correlation is well understood in general. Although much has been done to investigate correlation in various areas of physics, chemistry, statistics, biology and materials science, in many cases little is well understood except in the limit of weak correlation.

In previous chapters interactions with structureless point charge projectiles have been considered. There are many interactions, however, which involve at least two atomic centers with one or more electrons on each center. In such cases the projectile is not well localized and there is a need to integrate over the non localized electron cloud density of the projectile. Evaluation of cross sections and transition rates for such processes requires a method for dealing with at least four interacting bodies. If multiple electron transitions occur on any of the atomic centers, then some form of even higher order many body theory is required. In general such a many body description is difficult.

In this chapter the probability amplitude for a transition of a target electron caused by a charged projectile carrying an electron is formulated. This probability amplitude may be used for transitions of multiple target electrons if the correlation interaction between the target electrons is neglected. Unless the projectile is simply considered as an effective point projectile with a charge Zeff, the interaction between the target and the projectile electrons may not be ignored. Since this interaction is between electrons on two different atomic centers, the effects of this interaction have been referred to as two center correlation effects (Cf. section 6.2.4).

Introduction

In this chapter interactions of photons with atoms are considered. Here the emphasis is on systems interacting with weak electromagnetic fields so that a single atomic electron interacts with a single photon. Initially interactions with a single electron are considered. In this case the photon tends to probe in a comparatively delicate way the details of the atomic wavefunction (e.g. effects of static correlation in multi-electron atoms). Later two electron transitions are considered. Because these two electron transitions are often negligible in the absence of electron correlation the two electron transitions are usually a direct probe of the dynamics of electron correlation.

In previous chapters the impact parameter (or particle) picture has been used wherever possible in order to recover the product form for the transition probability in the limit of zero correlation. However, here the likelihood of interacting with more than a single photon is quite small since the electromagnetic field of a photon, even for strong laser fields, is almost always small compared with the electric field provided by the target nucleus. Consequently, this independent electron limit is not often useful. Also, photon wavepackets are usually much larger in size than an atom. Consequently the wave picture is used where the electric and magnetic fields of the photon are considered to be plane waves. Transformation to the particle picture may be done using the usual Fourier transform from the scattering amplitude to the probability amplitude (Cf. section 3.3.3).

Introduction

Ophthalmoscopes are optical instruments designed for the visual inspection of the internal structure of the eye, but most commonly the retina. However, because the amount of light reflected from the subject's eye is very low, the observer will only see an image if the subject's retina is well illuminated and thus an auxiliary illuminating system is an essential component of an ophthalmoscope. Thus ophthalmoscopes consist of two main components: a viewing system and an illuminating system.

There are a number of different designs for the viewing system, and these can be divided into two groups, as follows:

1. (a) Direct ophthalmoscopes. These are the simpler of the two types. They are discussed in detail in the next section, Section 29.1.

2. (b) Indirect ophthalmoscopes. The viewing system is more complicated than for direct ophthalmoscopes, having extra lenses between the subject's and observer's eyes. The extra complexity allows independent control over field-of-view and magnification. Indirect ophthalmoscopes are discussed in detail in Sections 29.2 and 29.3.

In this chapter, we will concentrate on the viewing system and only make a brief mention of the illumination system of the direct ophthalmoscope.

The magnification in direct ophthalmoscopy is often quoted as 15. In contrast, the magnification of indirect ophthalmoscopy is quoted as being much lower, usually in the region of about (–)3. However, once we look at the construction of indirect ophthalmoscopes, we will see that there is some potential ambiguity in the way that the magnification of an indirect ophthalmoscope is defined, but for the moment we will leave that problem aside and begin by looking at the properties of the direct ophthalmoscope.

Introduction

A wide variety of visual optical instruments are designed for binocular viewing, but probably the most well known are binoculars. Binocular viewing seems to offer a number of advantages over monocular viewing. Monocular viewing usually requires one eye to be closed, which may often lead to fatigue or discomfort, especially over extended viewing periods, although it may be possible for some people to keep both eyes open when viewing monocularly. This is possible if the image of the other eye is suppressed. In comparison, binocular viewing uses both eyes and hence is probably far less fatiguing. Binocular instruments also have the potential for stereoscopic viewing. Thus superficially, binocular viewing seems to be superior to monocular viewing. However a badly designed, badly manufactured or damaged binocular system can lead to significant problems in binocular viewing. Therefore we should be aware of the need for design and constructional tolerances for binocular instruments.

Stereoscopic and non-stereoscopic constructions

Not all binocular systems provide a stereoscopic image. The production of stereoscopic images requires that the two optical axes at the eyepieces be separated in object space. Let us look at several constructions and see how this requirement may be achieved.

Non-stereoscopic systems

Figure 37.1 shows a possible schematic construction of a binocular instrument, in which the two optical axes are identical in object space and hence the eyes will not see a stereoscopic image. In practical systems, the beam-splitting and deviations are often done with prisms rather than with mirrors as depicted in the diagram.

Introduction

A number of visual optical instruments are designed to measure the angles between two distant objects or the distance of an object. The determination of the angle between two distant objects is frequently done in surveying. Using simple rules of trigonometry, the location of any point can be determined if the direction of two other points and the distances between any two pairs of points are known. The determination of the elevation of celestial objects combined with astronomical tables and an accurate clock can be used to determine a position on the earth's surface, for surveying and navigation.

In many applications, these instruments have been superseded by other quicker or more accurate means. For example distance measurement can now be done with laser rangefinders and satellites are used for routine navigation. However, visual optical instruments are still used in some applications and are still worthy of some attention.

Angle measuring instruments

The theodolite

The theodolite is in essence a telescope of medium magnification with an eyepiece containing an alignment graticule. It is mounted such that it can rotate through horizontal and vertical circles, so that its horizontal and vertical direction of pointing can be measured. The theodolite is fitted with a levelling bubble so that the horizontal scale or table can be accurately made horizontal. If the vertical scale is zero for this horizontal scale then the elevation of an object can be absolutely read from the vertical scale.

Introduction

Optics is the study of light whereas visual optics is the study of the optical properties of the eye and sight. Ancient civilizations such as those of Greece were familiar with some of the properties of light, for example the laws of reflection. However, the Greeks misunderstood the nature of sight and the optical principles of the eye. They believed that light was emitted by the eye and only produced a visual response when the emitted rays struck an object. Many centuries passed before it was realized that light passes from the object to the eye and not from the eye to the object.

We will see later in this book, when we come to look at the optics of the eye, that the ability to sense the visual word around us is limited by the optical properties of the eye and its defects. For example before the advent of optical instruments, the smallest creature that could be seen with the unaided eye was about 0.05 mm in length and the mountains of the moon were unknown. Of particular frustration must have been the deterioration of eyesight with age. For example, as we age, the closest point of clear sight recedes, making it more and more difficult to do some things that we enjoy or need to do, such as reading and fine craft work. The discovery or invention of optical instruments enabled these restrictions to be overcome and allowed mankind to discover a world that was much more complex than ever envisaged, from the discovery of micro flora and fauna to countless galaxies far out in space.