In this chapter we shall describe the main features in the procedure for measuring the set of relative integrated intensities ρhkl of the crystal. We shall assume that the difTractometer has been correctly positioned and alined with respect to the incident beam. The methods of making these adjustments are fully described by Furnas (1957).

Choice of crystal size and shape

To minimize errors arising from the effects of absorption, extinction and simultaneous reflexions, the crystal must be as small as possible. The ultimate limit to the specimen size is determined by counting statistics (see §10.7), and, because of the greater intensity of primary X–ray beams compared with monochromatic neutron beams, the crystal size tends to be much larger in neutron diffraction. In the early period of neutron diffraction the linear dimensions of the crystal were measured in centimetres, but now, with the availability of high–flux reactors and with the use of automatic methods to speed up the collection of data, the size of crystal is down to 1 or 2 mm.

On p. 240 we gave the criterion μR ∼ 1 as a rough indication of the optimum size of the crystal, where μ is the linear absorption coefficient and R the average crystal radius. From Table XIX, this corresponds to an average radius for X–ray work of about 0.1 mm. The diffracted intensity decreases with the size of the unit cell, so that larger specimens are used when the cell size is large and the absorption is relatively low.

Introduction

Before describing the measurement of the coherent Bragg reflexions, we shall discuss the various sources of background scattering occurring with the reflexions. The background scattering introduces both systematic and random errors in the determination of the diffracted intensity. Systematic errors arise from those components of the background scattering which have a non–linear dependence on scattering angle in the neighbourhood of the Bragg peak: the contribution of the background to the peak cannot then be estimated by simply extrapolating background measurements taken on either side of the peak. The random or statistical error associated with the presence of the background is discussed later in Chapter 10: we shall note there the importance of reducing the background as much as possible in order to improve the statistics of counting, especially in measuring weak reflexions.

Thus the conditions under which the reflexions are measured must be chosen with two points in mind: first, we must obtain a good estimate of the background under the Bragg peak, in order to make a valid background subtraction; secondly, the background must be as small as possible, so as to enhance the signal–to–background ratio.

We have referred already (see p. 7) to the two principal methods of measuring a set of Bragg reflexions. The first is the inclination method, which is related to the photographic Weissenberg technique. The second, the normal–beam equatorial method, has no counterpart in photographic work. In this chapter we shall describe the diffraction geometry associated with these two methods and derive formulae for the setting angles of the crystal and detector, both in a general form and in various simplified forms for special settings and particular crystal symmetries. We shall then compare the two geometries and show how the special settings which are used in either method are related to one another. Finally, we shall discuss the problem of measuring several reflexions at the same time with a diffractometer.

It is necessary to describe first the geometrical requirements for setting the crystal and detector and for measuring the reflexion. Bragg's law imposes certain geometrical conditions on the positions of the crystal and the detector, and these conditions must be satisfied before the measurement begins. Once the crystal and detector are correctly set, the reflexion is measured by counting the number of diffracted X–ray quanta or slow neutrons received by the detector as the crystal rotates uniformly through the Bragg reflecting region. These geometrical considerations are best described in terms of the reciprocal lattice and the Ewald sphere of reflexion.

Introduction

There is no one ideal X–ray or neutron diffractometer which is equally well suited to the many different investigations which have to be carried out. The principal factors in the performance of a diffractometer are:

1. Accuracy of intensity measurement

2. Speed of operation

3. Number of accessible reflexions

4. Amount of manual intervention required

5. Accessibility of specimen

6. Availability of a computer

7. Versatility

8. Reliability

9. Cost

These varying requirements frequently conflict with one another and the choice of instrument for a given application will depend on the relative importance attached to them. We shall compare later the performance of two specific instruments under the headings listed above.

As we have seen in Chapter 2, diffractometers can be constructed according to two different geometrical arrangements, leading to a division into equatorial and inclination instruments. All diffractometers suitable for the collection of three–dimensional intensity data must have a number of shafts capable of being set independently: in the two arrangements the rotational degrees of freedom are allocated differently between crystal and detector shafts.

The general design of the diffractometer is dictated primarily by the type of geometrical arrangement, and by the method adopted in setting the shafts. The collimator, the goniometer head which supports the crystal, alinement aids such as viewing telescopes and even the detector itself can be thought of as exchangeable attachments which can be selected in accordance with the particular investigation in hand, and are of secondary importance in influencing the design of the instrument.

Quantitative neutron diffraction studies were not possible until 1945 with the advent of the nuclear reactor as a powerful source of neutrons. A high–flux reactor, such as the Harwell Dido or Pluto research reactors, has a central flux of about 1014 slow neutrons/cm2/sec. These neutrons move in all directions and only a proportion of about 1 in 105 travel in the right direction down the collimator; of these collimated neutrons, in turn, a fraction of between 10−3 and 10−2 has the right wavelength to be reflected by the monochromator. The collimated flux of monochromatic neutrons striking the sample is, therefore, 106 to 107/cm2/sec. This compares with a flux exceeding 1010 quanta/cm2/sec at the sample in the X–ray case. To compensate for this disparity in the incident flux and for the smaller cross–section for scattering of neutrons as compared with X–rays (Bacon, 1962), larger samples are used in neutron diffraction and the time required to count the diffracted neutrons is usually made longer.

In this chapter we shall describe briefly the collimators and monochromators used for the production of the primary neutron beam striking the sample. Only those points are discussed which relate to single crystal diffractometry: other neutron techniques are covered in Chapter 4 of G. E. Bacon's Neutron Diffraction (1962) and in Chapter 3 of Thermal Neutron Scattering (1965), edited by P. A. Egelstaff. In §7.3 on monochromators we include a discussion of the resolution of neutron reflexions, as this is closely connected with the properties of monochromators.

Introduction

All radiation detectors used in diffractometry function in a basically similar fashion: the detection of an individual incident X–ray quantum or neutron results in the collection of a certain quantity of electrical charge at the input terminal of the detecting circuitry. No matter what the actual detector may be, this resultant charge may be dealt with in one of two ways. Either the charge which corresponds to the arrival of a given number of pulses per unit time is integrated and the resultant current is taken as a measure of the incident intensity, or the individual pulses of charge are counted, if necessary after amplification and shaping.

The first method, that of current measurement, is used only with X–ray ionization chambers which are rarely employed today. All other X–ray and neutron detectors are ‘counters’ which produce discrete pulses. These detectors have the important feature that the result of an intensity measurement is given directly in digital form as the number of incident quanta or neutrons in a given time interval: such digital data are thus already in a suitable form for further processing.

It is easy to specify the requirements to be met by an ideal detector for use in diffraction studies.

Until recently, nearly all crystal–structure determinations with X–rays were carried out using photographic methods. Although the majority of crystal structures are still solved using photographic data, the past few years have witnessed profound changes in experimental techniques with the introduction of single crystal diffracton ters and with the application of automation procedures to their control. These changes, together with advances in crystallographic computing techniques, portend a sharp increase in both the quantity and quality of future crystallographic work.

In this chapter we shall sketch first the development of photographic and counter methods of measuring the set of X–ray intensities diffracted by a single crystal. This is followed by a survey of single crystal techniques in neutron diffraction and by a comparison of these techniques with X–ray methods. The chapter finishes with a discussion of automatic diffractometers and of the accuracy, speed and cost of counter methods. Our main aim in this chapter is to introduce briefly many points which are explained more fully later in the book.

X–ray techniques for measuring Bragg reflexions

The first diffraction pattern from a crystal, copper sulphate, was recorded on a photographic plate (Friedrich, Knipping & von Laue, 1912). Shortly afterwards, the ionization spectrometer (Bragg & Bragg, 1913) was developed and used both for the measurement of the wavelengths of X–ray spectra and for the determination of crystal structures.

The determination of a crystal structure normally proceeds in three distinct stages. The first is the measurement of the intensities of the Bragg reflexions and the calculation from them of amplitudes, reduced to a common scale and corrected for various geometrical and physical factors. These amplitudes are known as ‘observed structure amplitudes’ or ‘observed structure factors’. The second stage is the solution of the phase problem: the phases of the reflexions cannot be measured directly, and yet they must be derived in some way before the structure can be solved by Fourier methods. Because of uncertainties in the amplitudes and phases, this first structure is only approximately correct. The third stage in the structure determination consists of refining the approximate atomic positions so as to obtain the best possible agreement between the observed structure factors and the ‘calculated structure factors’, that is, those calculated from the approximate atomic positions of the successive stages of refinement.

This book describes counter methods of obtaining the set of observed structure factors of a single crystal, and so is concerned with the first stage only. The rapidly developing interest in automatic methods of collecting structure–factor data, and in measuring intensities to a high level of accuracy, have stimulated the development of counter methods. Photographic methods are still widely used in X–ray crystallography, but in neutron diffraction the single crystal diffractometer has remained the basic instrument for measuring neutron structure amplitudes since systematic studies began in the early 1950's.

Introduction

In this and the next chapter we shall study the methods which are used to ensure that primary beams of the desired wavelength and divergence fall upon the specimen. Since these methods are quite different for X–rays and neutrons we shall here depart from our usual procedure and consider X–rays in this chapter and neutrons in the next.

X–ray intensity measurements are generally made with radiation from an X–ray tube with a copper or molybdenum target. In many investigations it is sufficient to remove the Kβ component from the direct beam by simple filtration: this procedure is discussed in §6.4. In addition to the Kα–doublet the incident beam then still contains a considerable proportion of non–characteristic ‘white’ radiation, whose effect is suppressed by the use of balanced filters, also treated in §6.4.

With either method of filtration the direct beam from the X–ray target strikes the crystal. The beam divergence is determined entirely by the dimensions of the source collimator, the crystal and the focal spot; suitable dimensions of these are discussed in §§6.2 and 6.3.

The highest degree of monochromatization occurs when the primary beam is reflected from a crystal monochromator. In this case the divergence of the beam which falls on the specimen is in part determined by the mosaic spread of the monochromator. Monochromators are discussed in §6.5.

X–ray sources

It is not proposed to discuss the production and properties of X–rays in the present monograph, as these subjects have been fully covered elsewhere.