A dictionary definition of acceleration is an increase in speed from which one understands that a charged-particle accelerator would increase the speed of charged particles – as indeed it does. However, today's accelerators work at ultra-relativistic energies and it is not so much the particle's speed that increases as its mass. For example, between 1 MeV and 1 GeV an electron gains speed modestly from approximately 95% of the speed of light to what is virtually the full value, but its mass leaps forward from approximately three times its rest value to around 2000 times. This anomaly led Ginzton, Hansen and Kennedy to propose the names mass aggrandiser or ponderator, but neither became fashionable. More strictly one should speak of a momentum aggrandiser, but since this is sure to be as unfashionable as the others, we are left with the simple name accelerator. The accelerator family is, however, very large, so the authors will concentrate on synchrotrons and storage rings with only brief references to linear accelerators and many of the early circular machines.

Although universities often include some lectures on accelerators in their physics courses, there are very few courses which can claim to be principally about accelerators. The machines and the expertise in this field are mainly in national and international laboratories. Since these laboratories have a more mission-orientated approach than universities, relatively few books have been written and the accelerator community has relied heavily on a ‘learning-by-working apprenticeship’ for newcomers and on personal contacts and conferences for the dissemination of knowledge.

CHROMATIC EFFECTS

Chromatic effects are caused by the momentum dependence of the focal properties of lattice elements. The momentum of a particle is closely analogous to the frequency of light in classical optics and it is for this reason that the name chromaticity has been adopted (Gk. chroma colour). Figure 6.1 illustrates the effect of the momentum dependence of the focal length of a lens, first for a beam in a dispersion-free region, which is the normal case in classical optics and secondly for a beam with finite dispersion, which is the normal case in the arcs of an accelerator. The differential bending in dipoles gives rise to the dispersion function D(s), which is also strictly a chromatic effect, but since the dispersion function is well understood and readily calculated by optics programs, it is not usually included under the title of chromaticity.

Optical systems can be made achromatic by combining lenses made from different glasses, whose refractive indices behave differently with wavelength, but the focal strength of a magnetic lens always varies with momentum according to the same law, so that accelerators have no equivalent of a ‘different glass’. Instead, the accelerator designer uses sextupoles, set in regions with finite dispersion. This can be rather complex and far from ideal. The complexity arises because errors that occur in a dispersion-free region, as shown in Figure 6.1(a), cannot be corrected locally.

From the time that Newton first proposed that there was a universal force of gravity inversely proportional to the square of the distance between two point masses, there have been recurrent investigations of how far that rule was correct, and many different alternative forms have been suggested. The other assumption that Newton made, that the force of gravity did not depend on the chemical composition of bodies, has also been questioned from time to time; Newton himself carried out the first experimental test of what has become known as the weak principle of equivalence. It has often been suggested that some apparently anomalous behaviour in celestial mechanics should be ascribed to a failure of the inverse square law; indeed Clairaut developed the first analytical theory of the motion of the Moon because of discrepancies between Newton's theory and observation that might have been due to an inverse-cube component of the force. As with all subsequent studies before general relativity, careful analysis showed that the effects were consistent with the inverse square law. General relativity predicts a small deviation from the inverse square law close to very massive bodies, a deviation that has been confirmed by careful observation.

The motions of celestial bodies about each other are, with very minor exceptions, unable to reveal any departure from the weak principle of equivalence; if such departures are to be detected, they must be sought in laboratory experiments or geophysical observations.

Three hundred years after Newton published Philosophiae Naturalis Principia Mathematica the subject of gravitation is as lively a subject for theoretical and experimental study as ever it has been (Hawking & Israel, 1987). Theorists endeavour to relate gravity to quantum mechanics and to develop theories that will unify the description of gravity with that of all other physical forces. Experimenters have looked for gravitational radiation, for anomalies in the motion of the Moon that would correspond to a failure of the gravitational weak principle of equivalence, for deviations from the inverse square law and for various other effects that would be inconsistent with general relativity. The cosmological implications of general relativity continue to be elaborated and various ways of using space vehicles to test notions of gravitation have been proposed. In particular, the last three decades have seen a considerable effort devoted to applying modern techniques of measurement and detection of small forces to experiments on gravitation that can be done within an ordinary physics laboratory, and it is those that are the subject of this book.

Our scope is indeed quite restricted. It is concerned with experiments where the conditions are under the experimenter's control, in contrast to observation, where they are not. It is concerned with experiments that can be done within a more or less ordinary-sized room, that is to say, the distances between attracting body and attracted body do not usually exceed a few metres and may often be much less, while the masses of gravitating bodies are of the order of kilograms or much less.

Introduction

Thermal noise is unavoidable and sets the fundamental limit to the detectability of the response of an oscillator to any gravitational effect, but it is not the only disturbance to which an oscillator may be subject. Other forces may act on the mass of a torsion pendulum if it is subject to electric or magnetic or extraneous gravitational fields. The point of support of a torsion pendulum or other mechanical oscillator may be disturbed by ground motion. Ground motion is predominantly translational and so might be thought not to affect a torsion pendulum to a first approximation. However, all practical oscillators have parasitic modes of oscillation besides the dominant one, and although in linear theory normal modes are independent, in real non-linear systems modes are coupled. Thus, even if in theory seismic ground motion had no component of rotation about a vertical axis, none the less there would be some coupling between the primary rotational mode of a torsion pendulum and its oscillations in a vertical plane. In practice, therefore, any disturbance of a mechanical oscillator may masquerade as a response to a gravitational signal.

External sources of noise can be avoided with proper design of experiments. In this chapter we shall discuss both the sources of external disturbance and also the ways in which oscillators of different design respond.

Ground disturbance

Sources of ground noise

We begin with a discussion of seismic motions that move the point of support of a pendulum.

Introduction

The essence of the principle of equivalence goes back to Galileo and Newton who asserted that the weight of a body, the force acting on it in a gravitational field, was proportional to its mass, the quantity of matter in it, irrespective of its constitution. This is usually known as the weak principle of equivalence and is the cornerstone of Newtonian gravitational theory and the necessary condition for many other theories of gravitation including the theory of general relativity. In recent times, however, it was found that the weak principle of equivalence was not sufficient to support all theories and the principle has been extended as (1) Einstein's principle of equivalence and (2) the strong principle of equivalence.

Following a brief discussion of the principle of equivalence, this chapter is devoted to an account of the principal experimental studies of the weak principle of equivalence.

Einstein's principle of equivalence

Gravitation is one of the three fundamental interactions in nature and a question at the heart of the understanding of gravitation is whether or how other fundamental physical forces change in the presence of a gravitational force.

Einstein answered this fundamental question with the assertion that in a non-spinning laboratory falling freely in a gravitational field, the non-gravitational laws of physics do not change. That means that the other two fundamental interactions of physics – the electro-weak force and the strong force between nucleons – all couple in the same way with a gravitational interaction, namely: in a freely falling laboratory, the non-gravitational laws of physics are Lorentz invariant as in the theory of special relativity.

Introduction

In tests of the weak principle of equivalence, exact calculations of the attractions of masses are not necessary, but they are essential in experiments to test the inverse square law and to measure the gravitational constant. In fact, the calculation of the gravitational attraction of laboratory masses is usually not at all simple, because the dimensions of the masses are comparable with the separations between them, so that neither the test mass nor the attracting mass can be treated as a point object. In the following sections we discuss the gravitational attractions of laboratory masses with various common geometrical shapes. We present the results in terms of the gravitational efficiency, that is, the ratio of the gravitational attraction of a laboratory mass at a certain separation to that of a point mass with the same mass and separation. Furthermore, the precision demanded in measurements of separations of masses, the most difficult measurements in the determination of G and the test of gravitational law, depends on the geometry of the masses. These effects can have a strong influence on the conduct and final results of an experiment and it is essential to discuss in detail the calculation of potentials and attractions before going on to describe experiments.

Masses of three forms are often used in the laboratory: spheres, cylinders and rectangular prisms. The formula for the gravitational attraction of a sphere is well known and simple, but in practice it is not possible to manufacture an ideal sphere, the practical problem is usually how the real precision of manufacture affects the results; cylinders and prisms can be made very precisely but calculating the attraction is difficult.

Introduction

Although the weak principle of equivalence has been verified for ordinary macroscopic matter to very high precision, two questions remain open:

1. Is the principle valid for antimatter? Although indirect evidence from virtual antimatter in nuclei and short-lived antiparticles suggests that antimatter may have normal gravitational properties, no direct tests of the validity of the weak principle of equivalence for antimatter have been made.

2. Is the principle valid for microparticles? As the test bodies in macroscopic experiments are formed of neutrons, protons and electrons bound in nuclei, there is no doubt about the validity of the weak principle of equivalence for bound particles. However, the possibility of the principle of equivalence being violated for free particles should be studied.

Two main features characterize laboratory tests of the weak principle of equivalence for free elementary particles, both the consequence of their small masses. (1) When forces on substantial masses of bulk material are compared, a null experiment based on comparing different test bodies of two kinds of material can be devised. That is not possible for microscopic particles, and the gravitational accelerations have to be measured directly and subsequently compared with the acceleration of ordinary bulk matter to obtain the Eötvös coefficient. (2) The gravitational forces are very weak, even in the field of the Earth (which is the strongest attractive field), and so the accuracy of any experiment is very poor compared with Eötvös-type experiments using bulk masses.

We have not dealt in this book with all possible experiments on gravitation that have been or could be carried out in the laboratory, whether on the ground or in a space vehicle, but have concentrated on those on which most work has been done and from which most results have been obtained. That is because we have been concerned more with questions of experimental design and technique rather than with the bearing of the results on theories of gravitation. Something was said of that in the Introduction and we simply call attention again to recent reviews such as those of Cook (1987b), Will (1987) and others in the Newton Tercentary review of Hawking & Israel (1987). We have restricted our accounts to the weak principle of equivalence, the inverse square law and the measurement of the constant of gravitation partly because in numbers of results they dominate the subject, but more importantly because, having been so frequently and thoroughly studied, it seems that all the significant issues of experimental method and design are brought out when they are considered.

It was observed in the conclusion of the last chapter on the constant of gravitation, that the definition and calculation of the entire attraction upon a detector such as a torsion balance is no simple matter, and that applies equally to experiments on the inverse square law, as may be shown by the details of the calculations that were necessary in the experiments of Chen et al., (1984).

We turn now to describe some important general features of quantum field theory so that one can obtain a better overall view of the theory. This will be done by continuing to use the scalar theory as an illustration. We shall first investigate the structure of the interacting propagator — the two-point Green's function. This structure will tell us how to construct single-particle states for an interacting theory. The single-particle state construction will then be extended to multiparticle states, and the relationship of n-point Green's functions to scattering amplitudes will be obtained. This relationship is called the “reduction formula”. The interacting propagator can also describe unstable particles, and this will be explained including an outline of how such particles are produced and detected and how very short-lived particles appear as resonances in scattering amplitudes. The effective action will then be introduced, and it will be shown to be the generating functional of connected amplitudes. As a by-product, the cluster decomposition theorem will be described and the fact that the power of Planck's constant ħ counts the number of closed loops in a graph will be explained. Finally, the Legendre transform of the effective action will be examined. It is the generating function of single-particle irreducible, connected graphs. Its restriction to constant fields defines the effective potential which is a useful instrument for describing spontaneously broken symmetry.