The natural line broadening resulting from electromagnetic or acoustic radiative processes, the only causes of broadening discussed so far, by no means exhausts the mechanisms available and indeed is usually so minor an effect as to be of small practical importance. We may distinguish broadening due to different behaviour on the part of different members of an ensemble from broadening exhibited by each member on its own. In the first class are Doppler broadening and broadening due to variations in environment, to which may be added the effects of slight differences between superficially similar systems (e.g. different isotopes). In the second class, in addition to radiative processes, must be counted anything, especially collision with other atoms, which interrupts or distorts the wavetrain emitted by a single system so as to widen its spread of Fourier components. There is a very large literature on these effects, whose detailed analysis is both taxing and controversial. No attempt will be made here to go beyond an elementary discussion and illustration of some of the leading ideas, with examples of how the line-width may be reduced or its effects mitigated for the purpose of high-precision measurements of the central frequency. We start with the second class of processes, and for our purpose the two-level system provides an adequate model, with ammonia as a practical realization.

The absorption spectrum of ammonia at a rather low pressure, 1.2 torr, in the wavelength range from 1.1 to 1.5 cm is shown in fig. 18.8, each line resulting from transitions between pairs of levels in different rotational states of the molecule, as defined by the pairs of quantum numbers above each line.

We begin with some examples of one-dimensional vibration in a nonparabolic potential chosen to permit complete analytical solution of Schrödinger's time-independent equation. These examples are of isochronous vibrators which classically have a frequency independent of amplitude, and which might be expected therefore to have energy levels equally spaced at a separation of ħω0. This expectation turns out to be very nearly right and inspires a certain confidence in the semi-classical procedure developed by (among others) Bohr, Wilson and Sommerfeld. We therefore apply this procedure to some non-isochronous systems and find once more rather good agreement with the results of exact quantum mechanics. Periodic systems in fact can often be treated semi-classically with adequate accuracy, and significant economy of effort in comparison with strict quantum-mechanical analysis. This approach pays handsomely when we turn in the next chapter to the quantization of electron cyclotron orbits which, as already discussed in chapter 8, are closely related to harmonic oscillators. Conduction electrons in semi-conductors, and still more in metals, have their behaviour modified by the lattice through which they move, and a complete quantal treatment has never been achieved. It is clear, however, from approximate calculations, often of great complexity, that the semiclassical method describes most of the interesting physical processes correctly and very simply. In chapter 15 we shall describe in outline some of the effects which can be treated quite well enough for most purposes without even writing down Schrödinger's equation.

Linear systems whose parameters are independent of time possess, as has been abundantly illustrated already, well-defined normal modes from which their motion can be synthesized by superposition; and the response to an applied force, varying with time, can be written in terms of the response to each separate Fourier component of the force. The same is not true of non-linear systems, since superposition is no longer a valid procedure for synthesizing the response. Every anharmonic system responds differently to a given form of time-dependent force, and even when the response has been found in any special case it will not scale up unchanged in response to an amplification of the force. Thus the response to a sinusoidal force is in general non-sinusoidal, the waveform changing with the amplitude of the force. There are very few general statements that can be made about the character of the response. One cannot even assert that the oscillations of the vibrator will have the same fundamental frequency as the applied force – it may respond at a subharmonic frequency, i.e. an integral submultiple of that of the force, or the response may be asynchronous to the point of randomness. Even when order prevails, with regular vibration at the fundamental or subharmonic frequency, changing the amplitude or frequency of the applied force to an infinitesimal degree may have the effect of throwing the response into an entirely new pattern.

A resonant system acted upon by an oscillatory force presents a straightforward enough problem if it is passive and linear, especially if the force is applied by some prime mover that is uninfluenced by the response it excites. Such problems are the subject matter of chapter 6, while nonlinear passive systems, as discussed in chapter 9, are more complicated to handle. If the prime mover is influenced by the response, additional complexities enter, and this chapter treats of some of these. As is to be expected, linear systems present the least difficulty, and we shall begin with the behaviour of two coupled resonant systems, each of which may be thought of as driving the other and being perturbed by the reaction of the other back on it. Examples have already appeared earlier, as for instance the coupled pendulums discussed in chapter 2, and the coupled resonant lines in chapter 7. In both cases we noted a very general characteristic of such systems, that even if they are tuned to the same frequency before being coupled, they do not vibrate at this frequency when coupled, but have resonances which move progressively further from the original frequency as the coupling is strengthened. It would perhaps be logical, having considered this problem, to proceed to coupled, passive, non-linear vibrators; but these are so difficult that we shall leave them alone. It is not quite so hard to derive useful results for the behaviour of self-maintained oscillators when perturbed either by the injection of a steady signal or by being coupled to a similar oscillator.

As is appropriate in an exposition of the quantum mechanics of vibrators, we started with a quantal calculation and have finished with one; but we have never strayed far from the classical models. We began, indeed, in chapter 13 the task of establishing limits to the validity of classical reasoning, and the very last example of this chapter has been used to demonstrate, somewhat perversely it might be thought, that the method of quantum mechanics can occasionally be applied to systems which physicists and engineers would instinctively regard as classical. So long as the discussions give insight into physical processes and reveal the strengths and weaknesses of the analytical tools available, no apology is needed and no defence is offered except the evidence of the book itself. Even the rather simple systems which provide material for the whole of the volume demonstrate clearly the increased power available to those who can handle both classical and quantal reasoning. The reader who wishes to apply his skill to complex vibratory processes will find his tasks eased if he can use whichever seems advantageous at each stage, and be confident that his understanding of simple processes will enable him to recognize the dangers inherent in both approaches – the danger of carrying classical reasoning too far into the quantum domain, and the danger of forcing over-simplifications on physical systems to make them amenable to the unforgiving methodology of quantum mechanics.

Ten years ago, when I started writing on the physics of vibration, I had in mind a single volume. Four years ago I was reconciled to the need for two, and now must confess that the complexity of the subject has made it advisable to get the second of the projected three parts into print without waiting for the third to accompany it between the same covers. It is still my hope to do justice to the vibrations of extended systems, but the difficulties are considerable and not made easier by the vigour with which some of the central topics are being pursued at present.

Of all the encouragement I have enjoyed I particularly wish to record with the warmest thanks the help of Dr Edmund Crouch and Dr John Hannay who long ago, as research students, derived for me some solutions of Schrödinger's equation which provided a stout anchor for my thoughts: Dr Bob Butcher who has firmly guided me in my brief excursions into structural chemistry and molecular spectra: and Dr Andrew Phillips who devoted more time than he could have been expected to spare to a critical reading of much of the typescript. If the state of that typescript as delivered to the printers did not achieve even a modest standard of tidiness, the fault is entirely the consequence of copious afterthoughts on my part, and in no way to be blamed on Mrs Janet Thulborn whose patient and faithful typing deserved better respect, and has certainly earned my gratitude.

In combining Parts 1 and 2 into a single volume only minor changes have been made, apart from the correction of errors. I have not attempted to bring the treatment up to date even for such rapidly expanding fields as the study of chaos in non-linear systems, but have been content to add a small number of references. Where an argument could be corrected, clarified or extended in the space of a few sentences, these are signposted in the margin and placed at the end of the chapter. The system of marginal cross-references has met with critical approval and I have taken the opportunity of adding to their number.

My hopes of ever completing a further volume on the vibrations of extended systems have by now grown faint. There are too many exciting new ideas that are not yet ready for the type of exposition that suited the present work. Fortunately the development, by both classical and quantal methods, of the physics of simple vibrators produces a reasonably selfcontained argument, and I am grateful, as always, to Cambridge University Press for making possible this synthesis of concepts which in modern physics should be regarded always as complementary, never as antagonistic.

The writing of this book has occupied several years, and now that it is finished it is time to ask what sort of a book it is. For all its length, it turns out to be only the first volume out of two, any hope of covering the topic in a single volume having vanished as the project developed. It must be obvious, therefore, that it is not a textbook in the sense of an adjunct to a conventional course of lectures (there are not even any questions at the ends of the chapters). On the other hand, it is certainly not an advanced treatise, for many of the more difficult topics are treated at a much lower level than is to be found in the specialist works devoted to them. Moreover, I can claim no professional skill in most of them, and this is both a confession and an advertisement. For by writing about them in a way that illuminates for me the essential physical thought underlying what is often a very complicated calculation, I hope to have provided a treatment that will enlighten others in the same unlearned state. The field is very wide, ranging from applied mathematics (non-linear vibration and stability theory) to electrical engineering (oscillators), and taking in masers, nuclear magnetic resonance, neutron scattering and many other matters on the way. Nobody could hope to make himself a master of all these, and few advanced treatises dealing with one topic think fit to mention the analogies with others.

The original maser of Gordon, Zeiger and Townes, driven by a focussed stream of ammonia molecules in their antisymmetrical state, provides a conveniently explicit example on which to base a discussion of the principles underlying coherent excitation of a vibrator by stimulated emission. It was shown in chapter 18 how a quadrupole electrostatic lens served to separate symmetric from antisymmetric states, and we shall assume that separation is perfect; it is easy to extend the argument to include a proportion of molecules in the symmetric state. In addition we shall ignore any complications arising from the multitude of rotational states leading to the fine structure shown in fig. 18.8, and shall assume that only one line contributes, for example the strong 3,3 line at 23.9 GHz. Since the microwave cavity resonator, if it is to be excited by the molecules, must normally be very closely tuned to their natural frequency this assumption is realistic.

The simplest intuitive approach to the maser is by way of Einstein's treatment of radiation in terms of stimulated and spontaneous processes.† Excited molecules passing through the resonator, when it is already in an oscillatory condition, are stimulated by the field; if the resonator frequency is well matched to the molecular levels they may make a transition down to the ground state and on leaving the resonator have 2Δ0 less energy than when they entered.