Dynamical Theory of X-Ray Diffraction

by Authier, André

• ISBN13: 9780198528920
• ISBN10: 0198528922
• Edition: Revised
• Format: Paperback
• Copyright: 2004-01-08
• Publisher: Oxford University Press

Summary

The dynamical theory of diffraction has witnessed exciting developments since the advent of synchrotron radiation. This book provides an up-to-date account of the theory of diffraction and its applications. The first part serves as an introduction to the subject, presenting early developments and the basic results. It is followed by a detailed development of the diffraction and propagation properties of x-rays in perfect crystals and by an extension of the theory to the case of slightly and highly deformed crystals. The last part gives three applications of the theory: X-ray optics for synchrotron radiation, locations of atoms at surfaces, and X-ray diffraction topography. The book is richly illustrated and contains a wide range of references to the literature. It will be a most useful reference work for graduate students, lecturers and researchers.

Author Biography

Andre Authier is Professor Emeritus at Universite Pierre et Marie Curie, Paris in France. He is co-editor of Acta Crystallographica, Section A and editor of International Tables of Crstallography, Volume D.

Table of Contents

I Background and basic results	1	(112)
1 Historical developments	3	(25)
1.1 Prologue	3	(1)
1.2 The discovery of X-ray diffraction	4	(1)
1.3 The geometrical theory of diffraction	5	(1)
1.4 Darwin's dynamical theory of diffraction	6	(2)
1.5 Extinction theories	8	(3)
1.6 Ewald's dynamical theory	11	(2)
1.7 Early confirmations of the dynamical theory	13	(1)
1.8 Laue's dynamical theory	14	(1)
1.9 Umweganregung and Aufhellung	14	(2)
1.10 The properties of wavefields	16	(9)
1.10.1 Anomalous absorption (the Borrmann effect)	16	(4)
1.10.2 Wavefield trajectories	20	(3)
1.10.3 Pendellösung	23	(2)
1.11 Diffraction by deformed crystals	25	(1)
1.12 Modern times	26	(2)
2 Properties of the electromagnetic held-propagation and scattering	28	(29)
2.1 Maxwell's equations	28	(1)
2.2 The electrodynamic potentials in vacuum	29	(2)
2.2.1 The vector and scalar potentials	29	(1)
2.2.2 The retarded potentials	30	(1)
2.3 The electrodynamic potentials in polarized media	31	(1)
2.4 Hertz vectors (polarization potentials)	31	(2)
2.5 Propagation of an electromagnetic wave in vacuum	33	(1)
2.6 Scattering of X-rays by an electron	33	(3)
2.7 Polarizability of matter for X-rays	36	(7)
2.7.1 Elementary dispersion theory	36	(1)
2.7.2 Fourier expansion of the polarizability	37	(4)
2.7.3 Index of refraction	41	(1)
2.7.4 Absorption	41	(2)
2.8 Ewald's dispersion theory	43	(6)
2.9 Propagation equation of an electromagnetic wave in materials in Laue's dynamical theory	49	(1)
2.9.1 Laue's basic assumption	49	(1)
2.9.2 Propagation equation	49	(1)
2.10 Specular reflection-Fresnel relations	50	(7)
3 Geometrical theory of X-ray diffraction	57	(11)
3.1 Classical scattering by an electron polarization	57	(1)
3.2 Amplitude diffracted by a periodic electron distribution	58	(3)
3.3 Intensity diffracted by a small crystal	61	(2)
3.4 Reflectivity	63	(2)
3.5 Integrated intensity	65	(2)
3.6 Mosaic crystals	67	(1)
4 Elementary dynamical theory	68	(45)
4.1 Limitations of the geometrical theory	68	(1)
4.2 Introduction of the dispersion surface	69	(2)
4.3 Analogy with the band theory of solids	71	(2)
4.4 Propagation equation	73	(1)
4.5 Fundamental equations of dynamical theory	74	(5)
4.6 Amplitude ratio of the refracted and reflected waves	79	(1)
4.7 Solutions of plane-wave dynamical theory	80	(8)
4.7.1 Boundary conditions	80	(1)
4.7.2 Departure from Bragg's angle of the incident wave	81	(1)
4.7.3 Transmission and reflection geometries	82	(3)
4.7.4 Deviation parameter	85	(1)
4.7.5 Determination of the tiepoints	85	(2)
4.7.6 Effective absorption coefficient	87	(1)
4.8 The diffracted waves in the transmission geometry	88	(11)
4.8.1 Double refraction	88	(1)
4.8.2 Boundary conditions for the amplitudes at the entrance surface	88	(1)
4.8.3 Intensities of the reflected and refracted waves	89	(1)
4.8.4 Anomalous absorption	90	(2)
4.8.5 Boundary conditions at the exit surface	92	(2)
4.8.6 Reflectivity	94	(2)
4.8.7 Pendellösung	96	(2)
4.8.8 Integrated intensity	98	(1)
4.9 The diffracted waves in the reflection geometry	99	(5)
4.9.1 Tiepoints	99	(1)
4.9.2 Thick crystals-total reflection	99	(3)
4.9.3 Thin crystals	102	(2)
4.10 Influence of the asymmetry on the position and width of the rocking curve and of the angular distribution of the reflected beam	104	(3)
4.11 Comparison with geometrical theory	107	(3)
4.12 Dynamical diffraction by quasicrystals	110	(3)
II Advanced dynamical theory	113	(240)
5 Properties of wavefields	115	(40)
5.1 Relations between the field vectors	115	(2)
5.2 Fundamental equations of the dynamical theory	117	(1)
5.3 The dispersion equation in the two-beam case	118	(3)
5.4 Poynting vector of the wavefields	121	(2)
5.5 Determination of the tiepoints-geometrical interpretation of the deviation parameter	123	(13)
5.5.1 Boundary condition for the wavevectors	123	(2)
5.5.2 Deviation from Bragg's angle of the middle of the reflection domain	125	(1)
5.5.3 Coordinates of the tiepoint	126	(2)
5.5.4 Deviation parameter, Pendellösung distance and Darwin width in the transmission geometry	128	(4)
5.5.5 Deviation parameter, extinction distance, penetration depth and Darwin width in the reflection geometry	132	(3)
5.5.6 Index of refraction for dynamical diffraction	135	(1)
5.6 The deviation parameter in absorbing crystals	136	(1)
5.7 Amplitude ratio of the refracted and reflected waves	136	(3)
5.7.1 Phase of the amplitude ratio in the transmission geometry	137	(1)
5.7.2 Phase of the amplitude ratio in the reflection geometry	138	(1)
5.8 Anomalous absorption	139	(9)
5.8.1 Effective absorption coefficient in the transmission geometry	139	(2)
5.8.2 Absorption coefficient in the propagation direction	141	(1)
5.8.3 Discussion of anomalous absorption-properties of the standing wavefield	142	(5)
5.8.4 Anomalous absorption in the reflection geometry-penetration depth	147	(1)
5.9 Dispersion surface when the Bragg angle is close to 2π/2	148	(7)
5.9.1 Deviation from Bragg's angle and Darwin width	148	(3)
5.9.2 Dispersion surface	151	(2)
5.9.3 Penetration depth	153	(1)
5.9.4 Applications	154	(1)
6 Intensities of plane waves in the transmission geometry	155	(18)
6.1 Boundary conditions for the amplitudes at the entrance surface	155	(2)
6.2 Amplitudes of the refracted and reflected waves	157	(4)
6.3 Boundary conditions for the wavevectors at the exit surface	161	(5)
6.3.1 Condition for the existence of two outgoing waves	161	(2)
6.3.2 Wavevectors of the outgoing waves (Laue-Laue geometry)	163	(2)
6.3.3 Laue-Bragg geometry	165	(1)
6.4 Rocking curves of the reflected and refracted beams	166	(5)
6.4.1 Boundary conditions for the amplitudes at the exit surface	166	(2)
6.4.2 Reflectivity	168	(1)
6.4.3 Properties of the rocking curves	169	(2)
6.5 Integrated intensity	171	(2)
7 Intensities of plane waves in the reflection geometry	173	(16)
7.1 Thick absorbing crystals	173	(8)
7.1.1 Reflectivity	173	(2)
7.1.2 Shape of the rocking curves	175	(6)
7.2 Standing waves	181	(4)
7.3 Thin crystals	185	(4)
7.3.1 Boundary conditions for the amplitudes	185	(1)
7.3.2 Reflectivity	186	(3)
8 Dynamical diffraction in highly asymmetric coplanar and non-coplanar geometries	189	(36)
8.1 Introduction	189	(1)
8.2 Diffraction at grazing incidence or grazing emergence	190	(2)
8.3 Deviation from Bragg's incidence of the middle of the reflection domain	192	(5)
8.3.1 Grazing incidence and Bragg geometry	192	(3)
8.3.2 Grazing incidence, Laue geometry	195	(1)
8.3.3 Grazing emergence	196	(1)
8.4 Variation of the Darwin width for a grazing incidence	197	(3)
8.5 Variation of the width of the diffracted beam for a grazing emergence	200	(1)
8.6 Equation of the dispersion surface	201	(5)
8.7 Relation with the traditional dynamical theory	206	(1)
8.8 Specularly and Bragg-reflected intensities	207	(6)
8.8.1 Boundary conditions for the amplitudes at the entrance surface	207	(3)
8.8.2 Specularly and Bragg-reflected intensities for a grazing incidence and the Bragg geometry (semi-infinite crystal)	210	(3)
8.8.3 Specularly and Bragg-reflected intensities for a grazing incidence and the Laue geometry	213	(1)
8.9 Grazing incidence diffraction (non-coplanar geometry)	213	(12)
8.9.1 Introduction	213	(3)
8.9.2 Three-dimensional representation of the dispersion surface	216	(1)
8.9.3 Tiepoints excited by the incident wave	216	(7)
8.9.4 Equation of the dispersion surface	223	(1)
8.9.5 Amplitudes of the waves	224	(1)
9 n-beam dynamical diffraction	225	(24)
9.1 Introduction	225	(1)
9.2 The general three-beam case	226	(10)
9.2.1 Renninger-scans	226	(1)
9.2.2 Fundamental equations of the dynamical theory	227	(6)
9.2.3 Solution in the general case	233	(2)
9.2.4 Energy flow	235	(1)
9.3 The three-beam coplanar case	236	(1)
9.4 Determination of phases using n-beam diffraction	236	(6)
9.5 The super-Borrmann effect	242	(7)
9.5.1 Experimental evidence	242	(1)
9.5.2 Solution of the 111, 111 case	243	(3)
9.5.3 Anomalous absorption coefficient	246	(3)
10 Spherical-wave dynamical theory: I. Kato's theory	249	(28)
10.1 Extension of the dynamical theory to any kind of incident wave	249	(1)
10.2 Fourier expansion of a spherical wave in plane waves	250	(5)
10.2.1 Principle of Kato's spherical-wave theory	250	(1)
10.2.2 The incident wave is a scalar wave	250	(3)
10.2.3 The incident wave is a vector wave	253	(2)
10.3 Direct integration in the transmission geometry	255	(5)
10.3.1 The geometrical conditions	255	(2)
10.3.2 Stationary phase method	257	(1)
10.3.3 Amplitude distribution on the exit surface-reflected wave	257	(3)
10.3.4 Amplitude distribution on the exit surface-refracted wave	260	(1)
10.4 Intensity distribution on the exit surface	260	(3)
10.5 Equal-intensity (Pendellösung) fringes	263	(1)
10.6 Integration by the stationary phase method	264	(4)
10.7 Integrated intensity	268	(1)
10.8 Influence of polarization	269	(1)
10.9 Bragg geometry	269	(5)
10.10 Diffraction of ultrashort pulses	274	(1)
Appendix: Geometrical interpretation of n/ square root of s(γh) + n2 in the transmission geometry	274	(3)
11 Spherical-wave dynamical theory: II. Takagi's theory	277	(27)
11.1 Introduction	277	(2)
11.2 Generalized fundamental equations	279	(6)
11.2.1 Modulated waves	279	(1)
11.2.2 Takagi's equations	280	(3)
11.2.3 Boundary conditions for the amplitudes at the entrance surface	283	(2)
11.3 Reduction of Takagi's equations in the plane-wave case	285	(1)
11.4 Absorbing crystals	286	(1)
11.5 Analytical resolution of Takagi's equations for perfect crystals	286	(1)
11.6 Analytical solution for a point source using the method of integral equations	287	(4)
11.6.1 Transmission geometry	288	(2)
11.6.2 Reflection geometry	290	(1)
11.7 Analytical resolution of Takagi's equations using the Riemann function	291	(4)
11.7.1 Hyperbolic nature of Takagi's equations	291	(1)
11.7.2 General expression of the reflected and refracted waves	292	(1)
11.7.3 Determination of the Riemann function	293	(2)
11.7.4 General solution of Takagi's equations	295	(1)
11.8 Analytical solution for an incident spherical wave using the method of Riemann functions	295	(4)
11.8.1 The incident wave is a point source located on the entrance surface	295	(1)
11.8.2 The incident wave is a point source located away from the entrance surface	296	(2)
11.8.3 Conservation of energy	298	(1)
Appendix: Hyperbolic partial differential equations	299	(5)
Characteristics	299	(2)
Adjoint differential expression	301	(3)
12 Ray tracing in perfect crystals	304	(49)
12.1 Ray tracing	304	(1)
12.2 The Structure of real waves	305	(1)
12.3 Wavepackets made of the superposition of separate plane waves	306	(2)
12.4 Wavepackets made of a continuous distribution of wavevectors	308	(2)
12.5 Group velocity and Poynting vector	310	(1)
12.6 Angular amplification	311	(6)
12.7 Intensity distribution along the base of the Borrmann triangle (transmission geometry)	317	(6)
12.8 Geometrical properties of wavefield trajectories within the Borrmann triangle	323	(1)
12.8.1 Wavefields propagating along the median, AE, of the Borrmann triangle	323	(1)
12.8.2 Properties of the trajectories of the two wavefields excited by a plane wave	323	(1)
12.9 Experimental proof of double refraction	324	(2)
12.10 Experimental observation of the separation of the wavefield paths	326	(9)
12.10.1 Experimental setup	326	(2)
12.10.2 Focalization of the various wavelengths	328	(1)
12.10.3 Separation of wavefield paths in the transmission case	329	(1)
12.10.4 Plane-wave Pendellösung	330	(2)
12.10.5 Application to the measurement of the index of refraction	332	(3)
12.11 Fresnel diffraction near the Bragg incidence	335	(4)
12.12 Ray tracing in finite crystals	339	(10)
12.12.1 Introduction	339	(2)
12.12.2 Bragg-Laue geometry-pseudo-plane waves	341	(2)
12.12.3 Bragg-Bragg geometry; multiple reflections of a pseudo-plane wave in thin crystals	343	(1)
12.12.4 Laue-Bragg geometry-Borrmann-Lehmann fringes	344	(5)
12.13 Coherence of extended, non-strictly monochromatic sources	349	(4)
III Extension of the dynamical theory to deformed crystals	353	(82)
13 X-ray (racing in slightly deformed crystals	355	(51)
13.1 X-ray propagation in deformed materials	355	(2)
13.1.1 The different degrees of deformation	355	(1)
13.1.2 Principle of ray theories for weak deformations	356	(1)
13.2 Effective misorientation	357	(6)
13.2.1 Local reciprocal lattice vector	357	(2)
13.2.2 Effective misorientation in direct space	359	(1)
13.2.3 Effective misorientation in reciprocal space	360	(2)
13.2.4 Strain gradient	362	(1)
13.3 Polarizability of a deformed crystal	363	(1)
13.4 The Eikonal approximation	363	(5)
13.4.1 Justification of the concept of local dispersion surface	363	(2)
13.4.2 Fermat's principle	365	(3)
13.5 Ray trajectories	368	(7)
13.5.1 Local dispersion surface	368	(1)
13.5.2 Local wavevectors	369	(1)
13.5.3 Differential equation of the wavefield trajectories	369	(6)
13.6 The case of a constant strain gradient	375	(11)
13.6.1 Equation of the ray trajectory with respect to the lattice planes	375	(2)
13.6.2 Ray trajectories in the transmission geometry	377	(2)
13.6.3 Pure bending	379	(3)
13.6.4 Temperature gradient	382	(1)
13.6.5 Ray trajectories in the reflection geometry	382	(4)
13.7 Diffracted intensities-plane-wave case	386	(9)
13.7.1 Zero absorption	386	(3)
13.7.2 Absorbing crystals (transmission geometry)	389	(1)
13.7.3 Expression of the diffracted intensities for a constant strain gradient	389	(2)
13.7.4 Discussion of the intensity distribution for a constant strain gradient	391	(4)
13.8 Diffracted intensities-spherical-wave case	395	(10)
13.8.1 Pendellösung in slightly deformed crystals	395	(2)
13.8.2 Phase of the refracted wave in a deformed crystal	397	(4)
13.8.3 Expression of the phase in terms of the coordinates in direct space	401	(2)
13.8.4 Shape of the Pendellösung fringes in a deformed crystal	403	(2)
13.9 Lameller model	405	(1)
14 Propagation of X-rays in highly deformed crystals	406	(29)
14.1 Introduction	406	(1)
14.2 Takagi's equations in a deformed crystal	406	(3)
14.3 Resolution of Takagi's equations in the deformed crystal case	409	(12)
14.3.1 Small deformations, limit of the validity of the Eikonal approximation	409	(1)
14.3.2 Analytical resolution of Takagi's equations	410	(5)
14.3.3 Numerical integration	415	(5)
14.3.4 Applications	420	(1)
14.4 Ray concept applied to highly distorted crystals	421	(5)
14.4.1 Generalization of the notion of wavefields, interbranch scattering	421	(2)
14.4.2 Example: X-ray propagation in a crystal with a concentration gradient (Keitel et al. 1999)	423	(3)
14.5 Statistical dynamical theories	426	(6)
14.5.1 Introduction	426	(2)
14.5.2 Principle of Kato's statistical dynamical theory	428	(3)
14.5.3 Experimental tests of the statistical dynamical theory	431	(1)
Appendix: Resolution of Takagi's equations in the case of a constant strain gradient using Laplace transforms (Katagawa and Kato 1974)	432	(3)
IV Applications	435	(137)
15 X-ray optics	437	(58)
15.1 X-ray sources	437	(8)
15.1.1 X-ray tubes	437	(2)
15.1.2 Synchrotron radiation	439	(6)
15.2 Flat monochromators	445	(11)
15.2.1 Introduction	445	(1)
15.2.2 Monochromator crystals	446	(3)
15.2.3 Multiple-reflection monochromators	449	(7)
15.3 Applications of multiple-crystal arrangements to beam conditioning	456	(17)
15.3.1 suppression of tails	456	(3)
15.3.2 Wavelength scanner	459	(1)
15.3.3 Production of beams with a very narrow angular spread	459	(2)
15.3.4 Harmonic suppression	461	(6)
15.3.5 Production of polarized radiation	467	(6)
15.4 Focusing optics	473	(10)
15.4.1 Introduction	473	(1)
15.4.2 Mirrors	474	(2)
15.4.3 Multilayers	476	(1)
15.4.4 Curved crystals	477	(1)
15.4.5 Fresnel zone plates	478	(2)
15.4.6 Bragg-Fresnel lenses	480	(1)
15.4.7 Refractive lenses	481	(1)
15.4.8 X-ray wave-guides	482	(1)
15.5 X-ray interferometers	483	(6)
15.5.1 Principle	483	(2)
15.5.2 Applications	485	(4)
15.6 Imaging with X-rays	489	(6)
15.6.1 Introduction	489	(1)
15.6.2 Phase contrast imaging	489	(6)
16 Location of atoms at surfaces and interfaces using X-ray standing waves	495	(18)
16.1 Principle	495	(3)
16.2 Theory	498	(4)
16.2.1 Fluorescence yield	498	(4)
16.2.2 Influence of thermal vibrations	502	(1)
16.3 Bulk crystals	502	(2)
16.3.1 Extinction effect	502	(1)
16.3.2 Determination of the polarity of heteropolar crystals	503	(1)
16.4 Solution to the surface registration problem	504	(3)
16.5 Thin films and buried interfaces	507	(3)
16.5.1 Simple model	507	(2)
16.5.2 Calculation of the standing pattern in an overlayer with the dynamical theory	509	(1)
16.6 Standing waves in deformed crystals	510	(1)
16.7 Standing waves due to specular reflection	511	(2)
17 X-ray diffraction topography	513	(59)
17.1 Introduction	513	(1)
17.2 Single-crystal reflection topography (Berg-Barrett technique)	514	(6)
17.2.1 Principle	514	(2)
17.2.2 Image formation	516	(2)
17.2.3 Penetration depth	518	(1)
17.2.4 Stereographic views	519	(1)
17.2.5 Applications	520	(1)
17.3 Single-crystal transmission topography	520	(44)
17.3.1 Early history	520	(3)
17.3.2 Principle of section topographs	523	(5)
17.3.3 Projection topographs	528	(10)
17.3.4 Dislocation images	538	(13)
17.3.5 Images of planar defects	551	(10)
17.3.6 Applications	561	(3)
17.4 Double- or multiple-crystal topography	564
17.4.1 Principle of double-crystal topography	564	(3)
17.4.2 Plane-wave crystal topography	567	(1)
17.4.3 Synchrotron double-crystal topography	568	(1)
17.4.4 Mapping of distortions and of lattice parameter variations	569	(1)
17.4.5 Equal-strain or equal-lattice parameter contours	569	(1)
17.4.6 Double-crystal setting for high spatial resolution topography	570
Appendices
Appendix 1 Useful formulae	572	(5)
Appendix 2 The early days of dynamical theory	577	(7)
References	584	(64)
List of symbols	648	(4)
Index of Names	652	(9)
Index of subjects	661