## André Authier

Print publication date: 2013

Print ISBN-13: 9780199659845

Published to Oxford Scholarship Online: September 2013

DOI: 10.1093/acprof:oso/9780199659845.001.0001

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# X‐Rays as a Branch of Optics

Chapter:
(p.213) 9 X‐Rays as a Branch of Optics
Source:
Early Days of X-ray Crystallography
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199659845.003.0009

# Abstract and Keywords

This chapter is devoted to optical properties of X-rays. In the first section the early measurements of optical properties of X-rays are discussed briefly (specular reflection, refraction, diffraction by a slit). In the ensuing sections the principle of the Ewald and Laue dynamical theories of X-ray diffraction is presented. The special optical properties of X-ray wavefields are described, such as their direction of propagation inside the crystal, anomalous absorption, and standing waves, as well as their application to the Kossel effect and the location of impurities at crystal surfaces. It is shown how the deviation from Bragg’s law, which is due to the effect of refraction, was measured experimentally, and the principle of the double-crystal spectrometer is given. The discovery of the Compton effect is also related, and its consequences as to the nature of X-rays are discussed.

[Laue's] discovery was primarily a contribution to optics.

Sir C. W. Raman (1937)

# 9.1 Optical properties of X‐rays

The title of this chapter is taken from A. H. Compton's Nobel lecture (1927), in which pointed out that ‘it has not always been recognized that X‐rays is a branch of optics’, but that ‘there is hardly a phenomenon in the realm of light whose parallel is not found in the realm of X‐rays’. All the properties characteristic of light have been found also to be characteristic of X‐rays:

1. 1. Specular reflection. Röntgen and his contemporaries tried in vain to observe specular reflection of X‐rays (Sections 4.3 and 4.4). This is not surprising, since the refractive index of matter for X‐rays is only very slightly less than 1, as shown by Darwin (1914a, Section 8.4.1) and Lorentz (1916). The existence of the refractive index for X‐rays was first detected by the observation of deviations from Bragg's law (Section 9.3). Total reflection can only be observed if the glancing angle is smaller than a critical angle, $ω = 2 δ o$, and the refractive index, n, is given by (8.12). A. H. Compton (1923a) was the first to observe the specular reflection of X‐rays from plate glass and silver mirrors. The work was presented at the meeting of the American Physical Society in Washington on 21/22 April 1922.1 A beam collimated by two slits, and of effective width 2 minutes of arc fell on the mirror, and the totally reflected beam was detected with an ionization spectrometer. For an X‐ray wavelength of 1.279 Å from a tungsten anticathode, Compton measured the critical angle to be 10 minutes of arc for crown glass and 22.5 minutes of arc for silver. The corresponding experimental values of δo (the deviation of the refractive index from 1) were 4.2 × 10−6 and 21.5 × 10−6, respectively, to be compared with the theoretical values 5.2 × 10−6 and 19.8 × 10−6, respectively. Specular reflection was also observed, one year later, by M. Siegbahn and O. Lundquist, from a silver mirror.2

2. 2. Refraction. For the same reasons, refraction of X‐rays was not observed by the early investigators (Röntgen 1895, Perrin 1896, Voller and Walter 1897, and others). Barkla (1916) made an attempt to observe the refraction by putting two crystals of potassium bromide one above the other, so as to form two refracting prisms end to end, and letting an X‐ray beam pass between the two crystals; but he observed no effect. The first successful observation was by Larsson et al. (1924) in Siegbahn's laboratory with a glass prism, using a photographic method to observe the deviation of the X‐ray beam. They used an X‐ray tube with an anticathode of mixed copper and iron, and separated the deviations due to Cu and Fe X‐radiation. The next observation was by B. Davis and C. M. Slack (1925) with a prism of copper and an ionization chamber and by the (p.214) same authors with a prism of aluminium inserted in the path of the X‐ray beam between the two calcite crystals of a double‐crystal spectrometer (Davis and Slack 1926). More sensitive versions of the latter experiment were developed much later, when highly perfect crystals became available, for instance by Okkerse (1963) and by Malgrange et al. (1968), and with the X‐ray interferometer developed by Bonse and Hart (1965).

3. 3. Diffraction by a slit. The experiments by Haga and Wind, and by Walter and Pohl, have been discussed in Section 5.9. Diffraction by a ruled reflection grating was first observed by Compton and Doan (1925), at a small glancing angle, within the critical angle, with Cu and Mo radiation.

4. 4. Diffuse scattering was observed by Röntgen himself and by the early investigators (Section 5.4). Diffuse scattering by crystals is superimposed on the Bragg peaks and is due to thermal vibrations and static disorder in the structure, but there is, in fact, no clear‐cut separation between the two.

5. 5. Polarization of X‐rays was discovered by Barkla (1905)—see Section 5.6.

6. 6. Emission and absorption spectra, see Section 10.6.

7. 7. Photoelectric effect. The emission of electrons following the absorption of X‐rays was first observed by Perrin (1897), Curie and Sagnac (1900), and by Dorn (1900), and was studied further by Innes (1907), see Section 5.10 and A. H. Compton (1926). The photoelectric effect is now used, for instance, in X‐ray photoelectron spectroscopy.

Nowadays, many optical systems involving X‐ray optics are used at synchrotron facilities for focusing or imaging purposes—curved mirrors, wave‐guides, Fresnel zone plates, and diffractive and refractive lenses; for a review, see, for instance, Authier 2003 or 〈http://www.x-ray-optics.com〉. Furthermore, when X‐rays are diffracted at the Bragg angle by perfect crystals, entirely different optical properties appear, and these are summarized in Section 9.7.

# 9.2 Ewald's dynamical theory of X‐ray diffraction, 1917

## 9.2.1 The dispersion surface

Ewald's dynamical theory of diffraction (Ewald 1917) was developed by him during World War I while he was stationed on the Russian front, where he was servicing a mobile medical X‐ray unit (Ewald 1962a). The theory is derived from his thesis, and it is similarly based on Maxwell's equations. Its name can be traced back to Maxwell's paper, A dynamical theory of the electromagnetic field,3 in which Maxwell explains that ‘it may be called a Dynamical Theory, because it assumes that in space there is matter in motion, by which the observed electromagnetic phenomena are produced’.

The most important result of Ewald's dispersion theory (Section 6.2) is that the optical field inside the crystal can be expanded as a sum of plane waves. The corresponding total electric field E is:

$Display mathematics$
(9.1)

where Eh and Kh are the amplitudes and the wave vectors of the diffracted waves propagating in the crystal, respectively, ν is their frequency and r a position vector. The vectors Kh = HP are vectors in reciprocal space, called Anregungsvektoren by Ewald, where H is a reciprocal (p.215)

Fig. 9.1 Left: Full line: Dispersion surface in the reciprocal lattice. O: origin of the reciprocal lattice; H: node of the reciprocal lattice. h: reciprocal lattice vector; P: a tiepoint; OP: incident wave; HP: reflected wave; dashed circle: trace of the Ewald sphere. Right: Enlarged view of the centre of the diagram on the left.

lattice node (Fig. 9.1). The summation is over all the reciprocal lattice vectors, h = OH, where O is the origin of the reciprocal lattice. Expansion (9.1) was called wavefield by Laue (1931a) and is sometimes called Ewald wave. Solid‐state physicists call it a Bloch wave, after Bloch (1928), who derived the theory of electrons in solids, and introduced a similar expression for the wave function of the electrons.

The most frequent case is that when there are only two nodes of the reciprocal lattice that lie on the Ewald sphere (the ‘two‐beam case’). Two waves only propagate inside the crystal, the reflected wave of wave vector Kh = HP, and the refracted wave, Ko = OP (Fig. 9.1, Left). The extremity, P, of the wave vectors was called Anregungspunkt, excitation point, by Ewald (1917), but he later called it ‘tiepoint’ to emphasize the connection between the two waves. Indeed, the two waves propagate together in the crystal; they interfere to generate standing waves, and they both undergo the same anomalous absorption properties.

The relation between the amplitudes of the waves are deduced from Maxwell's equations. It is, for a plane‐polarized wave:

$Display mathematics$
(9.2)

which expresses the self‐consistency of the problem. The summation is over all the reciprocal lattice nodes and χ′h is the h′ coefficient of the Fourier expansion of the polarizability of the medium. For the set of linear equations (9.2) to have a non‐trivial solution, its determinant must be equal to zero. The corresponding secular equation is the ‘dispersion equation’. It is the equation of the surface on which the tiepoint P must lie—the ‘dispersion surface’.

The only terms in expansion (9.1) that have a non‐negligible amplitude are those for which the resonance factors $1 / ( K h 2 − k 2 )$ are very large, namely those for which Kh is not very different from the wave number in a vacuum, k. They correspond to those reciprocal lattice points that are close to the Ewald sphere, O and H in the two‐beam case. Far from the Bragg (p.216) condition for any reflection, there is only one such term, and one wave only propagates inside the crystal. In the two‐beam case, the dispersion surface is composed of two sheets connecting the two spheres centred at O and H and of radii n/λ, where n is the refractive index, branch (1) and branch (2) (Fig. 9.1). All the properties of the diffracted waves are deduced from the analysis of the dispersion surface.

## 9.2.2 Wavefields excited in the crystal by the incident wave: reflection profiles

The next step is to introduce the boundary conditions and then to find out which are the waves actually excited in the crystal. The dispersion surface is the equivalent of the surface of indices in optics and one simply applies Huygens's construction. Two geometrical situations are to be distinguished: transmission, or Laue geometry, and reflection, or Bragg geometry.

• Transmission geometry: the normal, n, to the entrance surface of the crystal cuts across both branches of the dispersion surface (Fig. 9.2, Left). There are two tiepoints, P 1 and P 2, and two wavefields propagating inside the crystal. The boundary conditions are applied in the same way at the exit surface of the crystal to determine the reflected wave. The variations of its intensity with the glancing angle, θ, of the incident wave are shown in Fig. 9.2, Right.

• Reflection geometry: the normal, n, to the entrance surface intersects one branch only of the dispersion surface, at P′ and P″ (Fig. 9.3, Left). The wavefield corresponding to P″ would propagate towards the outside of the crystal and is not excited. The wavefield corresponding to P′ propagates towards the inside of the crystal and is the only one excited. If the normal to the entrance surface lies within the shaded region in the figure, the intersection points are imaginary and there is total reflection, as shown in Fig. 9.3, Right, representing the variations of the reflected intensity with glancing angle of the incident wave. It is the famous ‘top‐hat’ curve.

Twenty years after his main article, P. P. Ewald published a development of his theory for any kind of lattice and taking the full structure factor into account (Ewald 1937). Ewald's dynamical

Fig. 9.2 Transmission geometry. Left: Dispersion surface; P 1, P 2: tiepoints of the waves excited in the crystal by the incident wave; OP1: refracted wave; HP 1: reflected wave. Right: Reflection profile; horizontal axis: glancing angle of the incident beam.

(p.217)

Fig. 9.3 Reflection geometry. Left: Construction in reciprocal space. P′: tiepoint of the wavefield propagating inside the crystal; OP′: refracted wave; HP′: reflected wave. Right: Reflection profile; horizontal axis: glancing angle of the incident beam.

Fig. 9.4 Plane-wave Pendellösung. Top : As predicted by Ewald (1927). After Ewald (1927), with kind permission of Springer Science and Business Media. Bottom: As observed by Malgrange and Authier (1965) with a wedge-shaped silicon crystal and MoK α radiation.

theory was extended to the case of electrons by H. Bethe,4 soon after the discovery of electron diffraction (Bethe 1928).

## 9.2.3 Pendellösung

In the transmission geometry, the diffracted waves overlap as they propagate inside the crystal. The waves associated with the two branches of the dispersion surface interfere, and Ewald (1917) predicted that a periodic transfer of energy should occur between the waves diffracted in the incident and reflected directions, as shown in Fig. 9.4 reproduced from a later publication (p.218) (Ewald, 1927). He called this effect Pendellösung, after the German verb, pendeln, to oscillate. It took more than forty years until it was observed in real life, by Kato and Lang (1959) for spherical waves and by Malgrange and Authier (1965) in the plane wave case. It is now commonplace and is also observed in the reflection geometry. The early developments of Ewald's dynamical theory of diffraction have been recounted by him in Ewald (1962a, 1969b, 1979a) and in an Annex to Authier (2003).

# 9.3 Deviation from Bragg's law

It has been shown in Section 8.4.2 that the reflection peak is shifted in reflection geometry with respect to Bragg's angle by a small angle, Δθ, given by (8.17); see Figs. 8.8 and, particularly, 9.3. This shift was first observed by students of M. Siegbahn in Lund University, Sweden. Very careful measurements of the Bragg angle were needed for the determination of X‐ray wavelengths. The first to detect the deviation from Bragg's angle was W. Stenström (1919), who wrote about it in his thesis. He compared the values of sin θ/n = λ/2d with increasing values of n and found that they were not constant. The work was pursued in more detail by E. Hjalmar (1920, 1923), who observed discrepancies in the values of the wavelength of Cu 1 for several orders of n in gypsum, and by A. Larsson, who measured the Bragg angle for the reflection of copper 1 up to the 11th order (Siegbahn 1925). They used a very accurate X‐ray spectrograph developed by Siegbahn (1919a, b) for the measurement of X‐ray spectral lines. The deviation from Bragg's law was suspected by W. Duane and R. A. Patterson (1920) with measurements of tungsten L lines with high order reflections on calcite and an ionization spectrometer, but it lay within experimental errors. It was studied more fully in B. Davis's laboratory in Columbia University, New York, USA, with the double‐crystal spectrometer, first by Davis and Terrill (1922) with calcite and Mo 1, and then, at B. Davis's suggestion, with crystals ground so as to give an asymmetric reflection, which increases the deviation, by Hatley (1924) with calcite, also with Mo 1, and by Nardroff (1924) with pyrite and MoK β 1, Cu 1 and CuKβ.

It was Ewald (1920, 1925) who explained the shift by the effect of refraction and calculated it with his dynamical theory. Siegbahn (1925) checked the explanation with crystals of gypsum and calcite and agreed with Ewald's explanation.

# 9.4 The double‐crystal spectrometer

Double‐crystal settings using twice the same crystal with the same reflection are used to produce monochromatic radiation (Fig. 9.5). They were first used by Wulffand Uspenski (1913b) and Wagner and Glocker (1913) to confirm that a Laue spot corresponded to a given wavelength (Section 7.2). They were used for the measurement of absolute integrated intensities (Compton 1917b; W. L. Bragg et al. 1921a, b; Wagner and Kulenkampf 1922; Bearden 1927), and for the accurate measurement of X‐ray wavelengths (Compton 1931; Bearden 1931).

B. Davis (Fig. 9.6) was one of the main developers of the double‐crystal technique in the United States, at Columbia University in New York. He and W. M. Stemple were the first to record rocking curves and to show the value of the double‐crystal spectrometer as an instrument of high resolving power (Davis and Stemple 1921). The source of X‐rays was a Coolidge tube with a tungsten anticathode, and the two crystals were calcite crystals in the parallel setting (Fig. 9.5, Left). Davis and Stemple compared crystals that had various origins and degrees of (p.219)

Fig. 9.5 Double-crystal settings. Left: Parallel, (+n,−n) non-dispersive; if rays 1 and 2 satisfy Bragg's condition on the first crystal, both also satisfy Bragg's condition on the second crystal. Right: Antiparallel, (+n,+n) dispersive; if rays 1 and 2 both satisfy Bragg's condition on the first crystal, and one of them satisfies Bragg's condition on the second crystal, the other one does not. After Authier (2003).

perfection, and had cleaved or polished surfaces. The reflected intensity was measured with an ionization chamber when rocking the second crystal. The half‐widths and the maximum reflectivities depended on the degree of perfection of the crystals and on the state of the surfaces. With two freshly cleaved very clear Iceland spar crystals (not polished), they obtained a beautiful rocking curve, with a half‐width of 16 seconds of arc and a maximum reflectivity nearly 50%, which showed that, indeed, their crystals were highly perfect. With a less perfect crystal, the half‐width was 57 seconds of arc. They also made measurements with different (p.220) wavelengths and found that the wavelength dependence of the maximum reflectivity decreased with increasing perfection of the crystals. Allison (1932) and Tu (1932) with the double‐crystal spectrometer and highly perfect calcite crystals obtained very good agreement with the results of dynamical theory and found no trace of mosaicity. It was the first time that it was clearly proved that there could be crystals perfect enough to diffract according to the perfect‐crystal theory.

Wagner and Kulenkampf (1922) in Munich were the first to distinguish the two possible settings for the two crystals. Consider, in the ‘parallel’ setting (Fig. 9.5, Left), two rays, 1 and 2, with slightly different glancing angles on the reflecting planes, and which satisfy Bragg's condition on the first crystal for wavelengths, λ1 and λ2 respectively. Both satisfy Bragg's angle on the second crystal because their glancing angles on the second crystal are equal to their glancing angles on the first crystal. The setting is non‐dispersive. In the ‘anti‐parallel’ setting (Fig. 9.5, Right), on the contrary, if ray 1 satisfies Bragg's condition on the second crystal, ray 2 does not, because its glancing angle in the second crystal is different from its glancing angle on the first crystal. Only monochromatic rays of wavelength λ1 are reflected by the second crystal. The setting is dispersive. The two settings are denoted (+n,−n) and (+n,+n), respectively.

Ehrenberg and Mark (1927), and Ehrenberg and von Süsich (1927) in Berlin‐Dahlem, recorded very narrow rocking curves with diamond and calcite, respectively, with Mo radiation. An important progress in the theory of double‐crystal settings was made by Ehrenberg, Ewald and Mark (1928) who showed that the reflectivity observed when rocking the second crystal is the convolution of the intrinsic reflectivities of the two crystals. They measured the half‐widths of the rocking curves and compared them with the values calculated with Ewald's dynamical theory. They found a rather good agreement for the 111 reflections of zinc‐blende and diamond. Reasonable agreement between measured and calculated rocking curve widths was also observed by Davis and Purks (1929) for highly perfect crystals of calcite.

The optical and geometrical properties of the double‐crystal spectrometer were discussed in detail by M. Schwarzchild (1928) in B. Davis's laboratory. A detailed account of the theory is given in Compton and Allison's textbook.5 The double‐crystal settings are the archetypes of the multiple optical devices used for beam conditioning with synchrotron radiation (Authier 2003).

# 9.5 The Compton effect and the corpuscular nature of X‐rays, 1923

The discovery of the Compton effect by A. H. Compton (1923b) was a major event in the history of physics. It was the outcome of five years of work, both experimental and theoretical. Compton gave his own account of how he arrived at this result,6 and the historians of science have commented in detail the successive steps of his thought.7

After his thesis, Compton's interests shifted toward scattering phenomena. He tried to find classical explanations for anomalies unaccounted for by Thomson's scattering theory. For instance, the intensity of γ‐rays scattered by a plate should be the same on both sides of the plate, but, in fact, it is higher on the emergent side. To explain this effect, Compton suggested the existence of electrons of various sizes and shapes. The work was presented at the 1917 (p.221) December Meeting of the American Physical Society, while he was with Westinghouse, and published in 1919, but was not very convincing. Compton then went to Rutherford's laboratory in Cambridge, England, on a National Research Council Fellowship. There, he studied the progressive softening of γ‐rays with angle of scattering (Compton 1921), which had first been observed by Sadler and Mesham (1915). This effect was not new but Compton's experiments were quantitative and far more accurate than those of his predecessors.

In the ship, on his way back to the USA, Compton (Fig. 8.4) drew the plans of the experiments he intended to undertake at Washington University in St Louis, Missouri, where he had been appointed Professor of Physics.8 There, he carefully measured the increase in wavelength of the scattered X‐rays with an ionization spectrometer and a special X‐ray tube designed by him, and blown by him. The increase in wavelength is a maximum at a scattering angle of 90°. Compton's results indicated a kind of fluorescence whose wavelength was independent of the nature of the substance used as a scatterer, and depended on that of the incident rays, which he called ‘general’ fluorescence, as opposed to the usual ‘characteristic’ fluorescence. Furthermore, with C. F. Hagenow, he showed that this new fluorescence was polarized. He suggested that it was ‘emitted at the instant of liberation of the secondary cathode rays from the atoms’, by a mechanism distinct from that of the ‘true’ scattering without change of wavelength (Thomson scattering). These results were presented at the Meeting of the American Physical Society in Washington on 22 April 1921. In a short note published in March 1922, Compton suggested a mechanism involving a Doppler shift of the X‐rays radiated by the recoil electron.9

In a paper that constituted the third part of a report by a Committee on X‐ray spectra of the National Research Council,10 Compton summarized the discussion on the softening of secondary radiation in a very clear way.11 Part of the secondary rays is of the same nature as the primary radiation, but with an increased wavelength. It looks like scattered radiation, but the increase in wavelength cannot be explained by classical theory. It could be explained by a Doppler shift on the basis of a quantum effect. The scattering electron receives a momentum hν/c, where ν is the frequency of the primary radiation, and the conservation of energy demands that the electron shall recoil with a momentum hν′ /c, where ν′ is the frequency of the secondary radiation. But, Compton concluded, ‘it has not been possible to account on either basis [classical or quantum] for all the observed phenomena’. The paper was published in October 1922. Then, in a communication to the American Physical Society on 1 December 1922, Compton gave the explanation of the effect in terms of light quanta, whereby an X‐ray quantum transfers part of its momentum, q1, to an electron which acquires a momentum qe (Fig. 9.7):

$Display mathematics$
(9.3)

where q2 is the momentum of the scattered X‐ray quantum.

The change in wavelength is Δλ = (2h/mc) sin2 θ, where 2θ is the angle between the incident and the scattered X‐ray quanta (Fig. 9.7).12 The recoil electrons thus predicted by (p.222)

Fig. 9.7 Compton scattering. An incident X-ray quantum transfers part of its momentum q1 to the electron.

Fig. 9.8 Fifth Solvay Conference at Institut International de Physique Solvay, Brussels, 1927. Top row : A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. de Donder, E. Schrödinger, J. E. Verschaffelt, W. Pauli, W. Heisenberg, R. H. Fowler, L. Brillouin. Middle row: P. Debye, M. Knudsen, W. L. Bragg, H. A. Kramers, P. A. M. Dirac, A. H. Compton, L. de Broglie, M. Born, N. Bohr. Bottom row: I. Langmuir, M. Planck, M. Curie, H. A. Lorentz, A. Einstein, P. Langevin, Ch.-E. Guye, C. T. R. Wilson, O. W. Richardson. Photo Benjamin Couprie.

Compton were observed a few months later by C. T. R. Wilson13 who was to share the 1927 Nobel Prize for Physics with Compton, and by W. Bothe.14 Figure 9.8 shows A. H. Compton (p.223) and C. T. R. Wilson at the Fifth Solvay Conference in Brussels, in 1927, where the newly formulated quantum theory was discussed by the leading physicists of the time.

The idea of light quanta, although more than ten years old, was not very familiar to the physicists of Anglo‐Saxon countries. According to Carl H. Eckart (1902–1973), a graduate student who had a desk in the office of the Australian‐born physicist, G. E. M. Jauncey (1888–1947), next door to Compton's, there was, some time in the autumn of 1922, a discussion between Jauncey and Compton on a series of papers by Schrödinger on light quanta that may have triggered Compton's train of thought15—a view shared by J. Jenkin.16 In Europe, the idea of light quanta was quite familiar. P. Debye (1923), on the basis of Compton's report in Bull. Nat. Res. Council, proposed on 14 March 1923 a mechanism similar to Compton's.

Compton's discovery was the source of a lively controversy between him and W. Duane.17 Experiments in Duane's laboratory had not confirmed Compton's measurements, and Duane did not accept Compton's proposed mechanism with X‐ray quanta. The controversy lasted until, finally, Compton's experimental results were confirmed in Duane's laboratory.

# 9.6 Laue's dynamical theory of X‐ray diffraction, 1931

Laue admired Ewald's thesis (Fig. 9.9) and considered it one of the all‐time masterpieces in mathematical physics (Laue 1931a). He noted, however, that in it crystals were represented by

Fig. 9.9 Left: Paul P. Ewald in 1950, after Authier (2009). Right: Max von Laue in 1950. Photo Deutsches Museum Munich.

(p.224) a discrete distribution of single‐point dipoles, while the recent theories about the structure of the atom pointed to a continuous distribution of electronic charge, a view confirmed by the studies of W. L. Bragg et al. (1922) and James et al. (1928) on rock‐salt. This led him to reformulate Ewald's dynamical theory on an entirely different basis. He assumed a continuous distribution of the dielectric susceptibility of the medium for X‐rays and considered it to be proportional to the electron density. A theory of the diffraction of X‐rays by a medium with a continuous dielectric susceptibility had already been developed in Vienna by Lohr (1924), but it did not have any practical applicability.

Laue's theory is, in fact, simpler than Ewald's and is the more widely used of the two. It consists in looking for solutions of Maxwell's equations in a medium with a triply periodic dielectric susceptibility. The electric negative and positive charges are distributed in a continuous way throughout the whole volume of the crystal, and cancel out so as to ensure the neutrality of the crystal. The local electric charge and density of current may therefore be put equal to zero in Maxwell's equations.

Laue found it more convenient to represent the electromagnetic field through the electric displacement D because div D = 0. By elimination of the electric and magnetic fields in Maxwell's equations, one obtains the propagation equation:

$Display mathematics$
(9.4)

The dielectric susceptibility χ can be expanded in a Fourier series:

$Display mathematics$
(9.5)

where the coefficients χh are proportional to the structure factors Fh.

The electric displacement is therefore also triply periodic and can be expanded in a Fourier series analogous to (9.1),

$Display mathematics$
(9.6)

which expresses the wavefield propagating in the crystal.

By substitution of the expansions of χ and D in the propagation equation (9.4), one finds that the amplitudes Dh satisfy a set of equations similar to (9.2), from which the dispersion surface can be deduced in the same way.

In his 1931 article, Laue discussed the properties of the dispersion surface and derived expressions for the reflected intensity in both the transmission and reflection geometries, which are more convenient than Ewald's, but, at the time, he did not go further into the study of dynamical diffraction. A first extension of the theory included Laue's explanation of the contrast of Kossel lines (Laue 1941). A second edition of that book was published in 1945 with only small changes. The third edition, which included discussions of the properties of wavefields, such as anomalous absorption and propagation direction, appeared on the year of Laue's accidental death (Laue 1960). An account of Laue's theory covering the progresses that took place during the forty years that followed can be found in Authier (2003).

Laue's theory of a continuous distribution of dielectric susceptibility was later justified quantum‐mechanically by G. Moliere (1939). The correspondence between Ewald's and Laue's (p.225) dynamical theories was worked out by H. Wagenfeld (1968). The dynamical theory had been developed by Ewald and Laue for an incident plane wave. It was extended to the case of an incident spherical wave by N. Kato (1960) and to that of any type of wave by S. Takagi (1962).

# 9.7 Optical properties of wavefields

The notion of wavefield was introduced initially as a purely mathematical entity. For Ewald (1913a, 1917), expression (9.1) described the optical field in the crystal. For Laue (1931 a), equation (9.6) expressed the solution of the propagation equation (9.4). That notion did not appear in Darwin's theory. In fact, wavefields have a physical reality; wavefields belonging to different branches of the dispersion surface undergo different anomalous absorption (Borrmann 1941, 1950), propagate along different directions inside the crystal (Borrmann 1954; Borrmann et al. 1955), interfere to produce Pendellösung fringes (Kato and Lang 1959, Malgrange and Authier 1965), and the path of individual wavefields can even be isolated (Authier 1960, 1961). The first evidence of the physical existence of the wavefields came from Laue's interpretation of the contrast of Kossel lines, based on the standing waves generated by the interference of the waves which constitute a wavefield.

## 9.7.1Kossel lines

Kossel lines occur when the fluorescent radiation from one type of atom in a crystal is Bragg‐reflected by the lattice planes of that same crystal. One speaks then of lattice sources. The primary radiation maybe either electrons or X‐rays. These lines lie at the intersections of cones having as axes the normals to each family of lattice planes with the photographic plate. As an example, the lines due to the reflections on (1̄11), (111̄) and (11̄1) can be observed in Fig. 9.10, Left. These line are in general dark or light or have a double contrast, dark–light or light–dark.

Fig. 9.10 Left: Kossel lines in a copper crystal with Cu radiation. Photo H. Voges, reproduced in Laue (1960). Right: variation of the intensity of the wavefield |D|2 across the reflection domain, recalculated after Laue (1935).

(p.226) This effect was surmised by Mie (1923), Clark and Duane (1923) and by Kossel (1924), but was not explained by them. It was clearly observed for the first time in 1935, by Kossel18 and his co‐workers using electrons as the primary source (Kossel 1935; Kossel and Voges 1935; Kossel et al. 1935). The Kossel lines, which are analogous to the Kikuchi lines observed in electron diffraction, were also studied in detail by G. Borrmann for his PhD thesis, prepared under the supervision of Kossel in Danzig (now Gdansk); see Authier and Klapper (2007). Borrmann used X‐rays as the primary source, both in the reflection setting (Borrmann 1935, 1936) and in the transmission setting (Borrmann 1938), and, like Kossel and his co‐workers, a copper crystal as the source of the diffracted secondary radiation. As mentioned in Section 6.5, W. Friedrich and P. Knipping had chosen a copper salt for their first attempts at observing diffraction of X‐rays by a crystal, because they thought it would have something to do with fluorescence radiation, and copper was a suitable element for that.

Borrmann observed that, in the transmission geometry, the double contrast of the lines is inverse for thick crystals (Borrmann 1938). This was not explained at the time but was, in fact, the first indication of anomalous absorption (Schülke and Brümmer 1962).

It was Laue (1935) who explained the fine structure of the Kossel lines in the reflection geometry, using the properties of wavefields and the reciprocity theorem. The intensity of the wavefield excited by an incident plane wave is, after equation (9.6), in the two‐beam case:

$Display mathematics$
(9.7)
$Display mathematics$
(9.8)

where Ψ is the phase of Dh/Do. Fig. 9.10, Right, shows its variations across the reflection domain, in the reflection geometry. Laue (1935) argued that, according to the reciprocity theorem, the intensity distribution in space of the beams resulting from the reflection of the spherical waves emitted by the lattice sources should be identical. The fact that it was, indeed, what is observed was considered by Laue, at the time he was writing his 1960 book (1959), as the only direct evidence of the physical existence of the wavefields.

## 9.7.2Standing waves

Expression (9.8) of the wavefields shows that the interference of the waves Do and Dh generates a set of standing waves in the crystal (Laue 1941). The term cos 2π(h · r+ Ψ) indicates that the nodes lie on planes parallel to the lattice planes and that their periodicity is equal to 1 / h = dhkl/n where dhkl is the periodicity of the hkℓ family of lattice planes and n is the order of the reflection.

1. 1. In transmission geometry, phase Ψ is equal to π for wavefields associated with branch (1) of the dispersion surface and to 0 for wavefields associated with branch (2). The nodes of standing waves therefore lie on the lattice planes (planes of maximum electronic density) for wavefields associated with branch (1) of the dispersion surface (Fig. 9.1). These wavefields undergo anomalously low absorption. The antinodes (maxima of electric field) of the wavefields associated with branch (2) lie on the lattice planes. They undergo anomalously high absorption.

2. (p.227) 2. In reflection geometry, the phase Ψ varies from π to 0 across the total reflection domain. For an incidence on the low‐angle side of the reflection domain, the nodes of standing waves lie on the lattice planes. The absorption is anomalously low. As the incidence sweeps the reflection domain, the nodes are progressively shifted until they lie at mid‐distance between the reflecting planes for an incidence on the high‐angle side of the reflection domain. It is then the antinodes which lie on the reflecting planes, and the absorption is anomalously high.

This shift of the system of nodes and antinodes in the reflection geometry when one rocks the crystal through the reflection domain can be made use of to localize the position of atoms in the crystal. If the incident radiation excites the emission of secondary radiation, either fluorescent X‐rays or photoelectrons, by atoms of the crystal or impurity atoms, this emission will be maximum when the atom lies at an antinode of electric field. The position of these atoms can therefore be localized by detecting this secondary radiation with an appropriate detector synchronously with the recording of the intensity reflected by the crystal.19

## 9.7.3 Anomalous absorption

Anomalous absorption of X‐rays, which is one of the most remarkable properties of wavefields, was discovered by G. Borrmann20 (1941) and bears his name (Borrmann effect). G Borrmann and his students played a decisive role in the first revival of the dynamical theory. When Ewald submitted his habilitation thesis in 1917, Sommerfeld had found it a beautiful mathematical construction but predicted that it would never have any practical applications. These came more than twenty years later, with Borrmann's investigations.

The discovery of anomalous absorption came from the observation by Borrmann of the forward‐diffracted beams transmitted through good‐quality crystals of calcite and quartz of various thicknesses, but only the quartz results were published at the time (Hildebrandt 1995, 2002; Authier and Klapper 2007). His experimental set‐up was the same as that already used by Rutherford and Andrade (1914ft) to measure the wavelength of γ‐rays diffracted by a rock‐salt crystal: a point source and a very divergent beam—the wide‐angle method (see Section 7.10.3). The trace of the forward‐diffracted beam was expected to show a deficit of intensity against the back‐ground because of the intensity drawn out of the incident beam by the reflected beam. It was the contrary that was observed, which baffled Laue considerably. It could only mean an anomalously low absorption. Laue (1949) accounted for the effect by calculating the intensities of the reflected and forward‐diffracted beams taking absorption into account. Borrmann (1950, 1954) made very careful measurements of the anomalous absorption with calcite crystals and explained it by the relative positions of the nodes and antinodes of the standing wavefields with respect to the lattice planes (Section 9.7.2). Wavefields associated with branch (2) are, in practice, completely absorbed out in thick crystals.

(p.228) Anomalous absorption takes place in a similar way in reflection geometry and is exhibited by the reflection profiles, which become asymmetric. This effect was first observed by Renninger (1955).

## 9.7.4 Path of the wavefields: Borrmann triangle, or fan

A surprising result of Borrmann's 1950 article on anomalous absorption in calcite had been that the propagation of X‐rays in thick crystals was neither along the incident nor the reflected directions, but in between, along the lattice planes. This had already been guessed at earlier, in a very qualitative way, by Murdock (1934), who had observed ‘triple Laue spots’ in quartz crystals. Laue had at first not been convinced by Borrmann's observations. But, from Maxwell's theory of electromagnetism, it is known that the direction of propagation of the energy of an electromagnetic wave is along the Poynting vector, S = R e(EH*), where R e(E) is the real part of E and H* the complex conjugate of H. Laue (1952c) calculated the Poynting vector by means of the dynamical theory and showed that it is normal to the dispersion surface (Fig. 9.11, Left). A natural incident beam is divergent and should therefore excite tie points along the whole dispersion surface. It is therefore to be expected that there should be wavefields propagating inside the crystal along all the directions lying between the incident and reflected directions (Borrmann 1954, 1959). They fill out what is now called the Borrmann triangle, or fan (Fig. 9.11, Right). The absorption is minimum for waves propagating along the lattice planes (maximum anomalous absorption effect), and, for thick crystals, are the only ones observed, as Borrmann (1950) had shown. The path of wavefields in a calcite crystal was then studied carefully by Borrmann et al. (1955), confirming Laue's calculations. That calculation was later generalized by Kato (1958) to the n‐beam case, but Ewald (1958) pointed out that Laue's and Kato's calculations implied incident plane waves, which was not the case in practice. He then proposed a very simple physical proof by substituting wave bundles for plane waves and showing that their group velocity is along the normal to the dispersion surface.

Fig. 9.11 Propagation of wavefields in a crystal. Left: reciprocal space. so: normal to the crystal surface; P 1, P 2: tiepoints of the two waves excited in the crystal; S1, S2: Poynting vectors. Right: Borrmann fan in direct space. After Borrmann (1959).

(p.229)

Fig. 9.12 Experimental proof of the double refraction of X-rays in a silicon crystal. Left: experimental setup. Right: traces of the reflected and forward diffracted beams on the photographic plate. After Authier (2003).

The perturbations in the propagation of wavefields by crystal defects such as dislocations, stacking faults, twins, inclusions, etc. can be taken advantage of to visualize these defects by topographic imaging techniques (Lang 1958, Borrmann et al. 1958).

## 9.7.5Double refraction

An incident plane wave excites two wavefields inside the crystal in the transmission geometry, with tiepoints P 1 and P 2 (Fig. 9.11, Left), and with Poynting vectors S1, S2. The two wavefields therefore propagate along separate paths inside the crystal. In the general case of unpolarized radiation, there are in fact four wavefields, two for each direction of polarization. This is why Borrmann (1955) spoke of quadruple refraction (Vierfachbrechung) of X-rays. In practice, the paths corresponding to the two directions of polarization are so close to each other that it is hopeless to try to observe their separation. The separation of the paths of wavefields 1 and 2 is in principle also impossible to observe, since either the incident wave is a spherical wave and all the possible directions of propagation within the Borrmann fan are excited, or it is a plane wave and its lateral expansion is, by definition, infinite. The paths of the two wavefields then overlap and cannot be separated.

A way to go round this difficulty was found by isolating from the Borrmann fan a wave packet that is narrow in both direct and reciprocal space (Authier 1960). The paths of the two packets of wavefields, 1 and 2, can then be separated (Fig. 9.12, Left). The result is shown in Fig. 9.12, Right (Authier 1961). It provides the most direct experimental proof of the physical existence of the wavefields.

## Notes:

(1) Compton, A. H. (1922). Phys. Rev. 20, 84.

(2) Siegbahn, M. and Lundquist, O. (1923). Röntgenstrålarnas total reflexion. Fysisk Tidsskr. 21, 170–171.

(3) Maxwell, J. C. (1865). Phil. Trans. Roy. Soc. 155, 459–512.

(4) Hans Bethe, born 2 July 1906 in Strasbourg, then Germany, died 6 March 2005 in Ithaca, NY, USA, was a German-American nuclear physicist. He studied physics at the Goethe University in Frankfurt, and at the University of Munich where he obtained his PhD prepared under the supervision of A. Sommerfeld. He became Privatdozent at the University of Munich in 1930, and Assistant Professor at the University of Tübingen in 1933, but soon after emigrated, first to England, then to the USA. He was appointed Assistant Professor at Cornell University, Ithaca, NY, in 1935, and Professor in 1937. He was married to Ewald's daughter, Rose, in 1939. Bethe was awarded the 1967 Nobel Prize in Physics for ‘his contributions to the theory of nuclear reactions, especially his discoveries concerning the energy production in stars’.

(5) Compton, A. H. and Allison, S. K. (1935). X-rays in Theory and Experiment. Van Nostrand, New York.

(6) Compton, A. H. (1961). The scattering of X-rays as particles. Am. J. Phys. 29, 817–820.

(7) R. H. Stuewer (1975). The Compton Effect. Turning Point in Physics. Science History Publications. New York.

(8) R. S. Shankland (1973). Scientific Papers of Arthur Holly Compton. University of Chicago Press.

(9) Compton, A. H. (1922). The spectrum of secondary rays. Phys. Rev. 19, 267–268.

(10) The Committee consisted of W. Duane, A. H. Compton, B. Davis, A. W. Hull, and D. L. Webster.

(11) A. H. Compton (1922). Secondary radiations produced by X-rays. Bull. Nat. Res. Council. 4, number 20, 1–56.

(12) Cooper, M., Mijnarends, P., Shiotani, N., Sakai, N., and Bansil, A. (2004). X-ray Compton scattering. Oxford Series on Synchrotron Radiation. Oxford University Press.

(13) C. T. R. Wilson (1923). Investigations on X-rays and β‐rays by the cloud method. Proc. Roy. Soc. A. 104, 1–24.

(14) W. Bothe (1923). Über eine neue Sekundärstrahlung der Röntgenstrahlen. Z. Phys. 16, 319–320.

(15) Interview of Dr Carl Eckart by John L. Heilbron on 31 May 1962, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD, USA, 〈http://www.aip.org/history/ohilist/4586.html〉.

(16) J. Jenkin (2002). G. E. M. Jauncey and the Compton Effect. Phys. Perspect. 4, 320–332.

(17) S. K. Allison (1965). Arthur Holly Compton, 1892–1962. Bibliographical Memoirs of the National Academy of Science. Washington, DC, USA.

(19) See footnote 23 in Chapter 7.

(19) The X-ray Standing Wave Technique: Principles and Applications. J. Zegenhagen and A. Kazimirov, editors (2013). World Scientific, Singapore.

(20) Gerhard Borrmann, born 30 April 1908, died 12 April 2006, was a German physicist. He received his higher education at the Technische Universität München and the Technische Hochschule Danzig (now Gdansk, Poland) where he was awarded the title Diplom-Ingenieur in 1930, and where he obtained his PhD in 1936. He became then Kossel's assistant. In 1938, he was called by M. von Laue to the Kaiser-Wilhelm-Institut für Physikalische Chemie und Elektrochemie in Berlin-Dahlem (now Fritz-Haber-Institut der Max-Planck-Gesellschaft), where he turned to the study of reflection by perfect crystals. When Laue was appointed Director of the Fritz-Haber-Institut, Borrmann became head of a department of his own (Kristalloptik der Röntgenstrahlen).