(International Tables) Reciprocal space

International Tables for Crystallography
Volume B: Reciprocal space

Second online edition (2010) ISBN: 978-1-4020-8205-4 doi: 10.1107/97809553602060000108 | 1 | 2 |

Edited by U. Shmueli

Preface to the third edition (p. xiv) | html | pdf |
U. Shmueli

General relationships and techniques
- 1.1. Reciprocal space in crystallography (pp. 2-9) | html | pdf | zxc chapter contents |
  U. Shmueli
  - 1.1.1. Introduction (p. 2) | html | pdf |
  - 1.1.2. Reciprocal lattice in crystallography (pp. 2-3) | html | pdf |
  - 1.1.3. Fundamental relationships (pp. 3-5) | html | pdf |
    - 1.1.3.1. Basis vectors (p. 3) | html | pdf |
    - 1.1.3.2. Volumes (p. 4) | html | pdf |
    - 1.1.3.3. Angular relationships (p. 4) | html | pdf |
    - 1.1.3.4. Matrices of metric tensors (pp. 4-5) | html | pdf |
  - 1.1.4. Tensor-algebraic formulation (pp. 5-7) | html | pdf |
    - 1.1.4.1. Conventions (p. 5) | html | pdf |
    - 1.1.4.2. Transformations (p. 5) | html | pdf |
    - 1.1.4.3. Scalar products (pp. 5-6) | html | pdf |
    - 1.1.4.4. Examples (pp. 6-7) | html | pdf |
  - 1.1.5. Transformations (pp. 7-8) | html | pdf |
    - 1.1.5.1. Transformations of coordinates (pp. 7-8) | html | pdf |
    - 1.1.5.2. Example (p. 8) | html | pdf |
  - 1.1.6. Some analytical aspects of the reciprocal space (pp. 8-9) | html | pdf |
    - 1.1.6.1. Continuous Fourier transform (p. 8) | html | pdf |
    - 1.1.6.2. Discrete Fourier transform (pp. 8-9) | html | pdf |
    - 1.1.6.3. Bloch's theorem (p. 9) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.1.4.1. Derivation of the general expression for the rotation operator (p. 7) | html | pdf |
- 1.2. The structure factor (pp. 10-23) | html | pdf | zxc chapter contents |
  P. Coppens
  - 1.2.1. Introduction (p. 10) | html | pdf |
  - 1.2.2. General scattering expression for X-rays (p. 10) | html | pdf |
  - 1.2.3. Scattering by a crystal: definition of a structure factor (p. 10) | html | pdf |
  - 1.2.4. The isolated-atom approximation in X-ray diffraction (pp. 10-11) | html | pdf |
  - 1.2.5. Scattering of thermal neutrons (p. 11) | html | pdf |
    - 1.2.5.1. Nuclear scattering (p. 11) | html | pdf |
    - 1.2.5.2. Magnetic scattering (p. 11) | html | pdf |
  - 1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism (pp. 11-12) | html | pdf |
  - 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion (pp. 12-15) | html | pdf |
    - 1.2.7.1. Direct-space description of aspherical atoms (pp. 12-14) | html | pdf |
    - 1.2.7.2. Reciprocal-space description of aspherical atoms (pp. 14-15) | html | pdf |
  - 1.2.8. Fourier transform of orbital products (pp. 15-17) | html | pdf |
    - 1.2.8.1. One-centre orbital products (pp. 15-17) | html | pdf |
    - 1.2.8.2. Two-centre orbital products (p. 17) | html | pdf |
  - 1.2.9. The atomic temperature factor (p. 17) | html | pdf |
  - 1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation (pp. 17-18) | html | pdf |
  - 1.2.11. Rigid-body analysis (pp. 18-20) | html | pdf |
  - 1.2.12. Treatment of anharmonicity (pp. 20-22) | html | pdf |
    - 1.2.12.1. The Gram–Charlier expansion (pp. 20-21) | html | pdf |
    - 1.2.12.2. The cumulant expansion (pp. 21-22) | html | pdf |
    - 1.2.12.3. The one-particle potential (OPP) model (p. 22) | html | pdf |
    - 1.2.12.4. Relative merits of the three expansions (p. 22) | html | pdf |
  - 1.2.13. The generalized structure factor (p. 22) | html | pdf |
  - 1.2.14. Conclusion (p. 22) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 1.2.7.1. Real spherical harmonic functions (x, y, z are direction cosines) (pp. 13-14) | html | pdf |
    - Table 1.2.7.2. Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981) (p. 15) | html | pdf |
    - Table 1.2.7.3. `Kubic harmonic' functions (pp. 16-17) | html | pdf |
    - Table 1.2.7.4. Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Su & Coppens, 1990) (p. 18) | html | pdf |
    - Table 1.2.8.1. Products of complex spherical harmonics as defined by equation (1.2.7.2a) (p. 19) | html | pdf |
    - Table 1.2.8.2. Products of real spherical harmonics as defined by equations (1.2.7.2b) and (1.2.7.2c) (p. 19) | html | pdf |
    - Table 1.2.8.3. Products of two real spherical harmonic functions $[y_{lmp}]$ in terms of the density functions $[d_{lmp}]$ defined by equation (1.2.7.3b) (p. 20) | html | pdf |
    - Table 1.2.11.1. The arrays $[G_{ijkl}]$ and $[H_{ijkl}]$ to be used in the observational equations $[U_{ij} = G_{ijkl} L_{kl} + H_{ijkl} S_{kl} + T_{ij}]$ [equation (1.2.11.9)] (p. 20) | html | pdf |
    - Table 1.2.12.1. Some Hermite polynomials (Johnson & Levy, 1974; Zucker & Schulz, 1982) (p. 21) | html | pdf |
- 1.3. Fourier transforms in crystallography: theory, algorithms and applications (pp. 24-113) | html | pdf | zxc chapter contents |
  G. Bricogne
  - 1.3.1. General introduction (p. 24) | html | pdf |
  - 1.3.2. The mathematical theory of the Fourier transformation (pp. 24-52) | html | pdf |
    - 1.3.2.1. Introduction (pp. 24-25) | html | pdf |
    - 1.3.2.2. Preliminary notions and notation (pp. 25-28) | html | pdf |
      - 1.3.2.2.1. Metric and topological notions in $[{\bb R}^{n}]$ (p. 25) | html | pdf |
      - 1.3.2.2.2. Functions over $[{\bb R}^{n}]$ (pp. 25-26) | html | pdf |
      - 1.3.2.2.3. Multi-index notation (p. 26) | html | pdf |
      - 1.3.2.2.4. Integration, $[L^{p}]$ spaces (pp. 26-27) | html | pdf |
      - 1.3.2.2.5. Tensor products. Fubini's theorem (p. 27) | html | pdf |
      - 1.3.2.2.6. Topology in function spaces  (pp. 27-28) | html | pdf |
        1.3.2.2.6.1. General topology  (pp. 27-28) | html | pdf |
        1.3.2.2.6.2. Topological vector spaces  (p. 28) | html | pdf |
    - 1.3.2.3. Elements of the theory of distributions (pp. 28-35) | html | pdf |
      - 1.3.2.3.1. Origins (pp. 28-29) | html | pdf |
      - 1.3.2.3.2. Rationale (p. 29) | html | pdf |
      - 1.3.2.3.3. Test-function spaces  (pp. 29-30) | html | pdf |
        1.3.2.3.3.1. Topology on $[{\scr E}(\Omega)]$   (p. 29) | html | pdf |
        1.3.2.3.3.2. Topology on $[{\scr D}_{k} (\Omega)]$   (p. 29) | html | pdf |
        1.3.2.3.3.3. Topology on $[{\scr D}(\Omega)]$   (pp. 29-30) | html | pdf |
        1.3.2.3.3.4. Topologies on $[{\scr E}^{(m)}, {\scr D}_{k}^{(m)},{\scr D}^{(m)}]$   (p. 30) | html | pdf |
      - 1.3.2.3.4. Definition of distributions (p. 30) | html | pdf |
      - 1.3.2.3.5. First examples of distributions (p. 30) | html | pdf |
      - 1.3.2.3.6. Distributions associated to locally integrable functions (p. 30) | html | pdf |
      - 1.3.2.3.7. Support of a distribution (p. 30) | html | pdf |
      - 1.3.2.3.8. Convergence of distributions (pp. 30-31) | html | pdf |
      - 1.3.2.3.9. Operations on distributions  (pp. 31-35) | html | pdf |
        1.3.2.3.9.1. Differentiation  (pp. 31-32) | html | pdf |
        1.3.2.3.9.2. Integration of distributions in dimension 1  (p. 32) | html | pdf |
        1.3.2.3.9.3. Multiplication of distributions by functions  (p. 32) | html | pdf |
        1.3.2.3.9.4. Division of distributions by functions  (pp. 32-33) | html | pdf |
        1.3.2.3.9.5. Transformation of coordinates  (p. 33) | html | pdf |
        1.3.2.3.9.6. Tensor product of distributions  (pp. 33-34) | html | pdf |
        1.3.2.3.9.7. Convolution of distributions  (pp. 34-35) | html | pdf |
    - 1.3.2.4. Fourier transforms of functions (pp. 35-39) | html | pdf |
      - 1.3.2.4.1. Introduction (p. 35) | html | pdf |
      - 1.3.2.4.2. Fourier transforms in $[L^{1}]$   (pp. 35-37) | html | pdf |
        1.3.2.4.2.1. Linearity  (p. 35) | html | pdf |
        1.3.2.4.2.2. Effect of affine coordinate transformations  (p. 35) | html | pdf |
        1.3.2.4.2.3. Conjugate symmetry  (p. 35) | html | pdf |
        1.3.2.4.2.4. Tensor product property  (p. 35) | html | pdf |
        1.3.2.4.2.5. Convolution property  (pp. 35-36) | html | pdf |
        1.3.2.4.2.6. Reciprocity property  (p. 36) | html | pdf |
        1.3.2.4.2.7. Riemann–Lebesgue lemma  (p. 36) | html | pdf |
        1.3.2.4.2.8. Differentiation  (p. 36) | html | pdf |
        1.3.2.4.2.9. Decrease at infinity  (p. 36) | html | pdf |
        1.3.2.4.2.10. The Paley–Wiener theorem  (pp. 36-37) | html | pdf |
      - 1.3.2.4.3. Fourier transforms in $[L^{2}]$   (p. 37) | html | pdf |
        1.3.2.4.3.1. Invariance of $[L^{2}]$   (p. 37) | html | pdf |
        1.3.2.4.3.2. Reciprocity  (p. 37) | html | pdf |
        1.3.2.4.3.3. Isometry  (p. 37) | html | pdf |
        1.3.2.4.3.4. Eigenspace decomposition of $[L^{2}]$   (p. 37) | html | pdf |
        1.3.2.4.3.5. The convolution theorem and the isometry property  (p. 37) | html | pdf |
      - 1.3.2.4.4. Fourier transforms in $[{\scr S}]$   (pp. 37-39) | html | pdf |
        1.3.2.4.4.1. Definition and properties of $[{\scr S}]$   (pp. 37-38) | html | pdf |
        1.3.2.4.4.2. Gaussian functions and Hermite functions  (pp. 38-39) | html | pdf |
        1.3.2.4.4.3. Heisenberg's inequality, Hardy's theorem  (p. 39) | html | pdf |
        1.3.2.4.4.4. Symmetry property  (p. 39) | html | pdf |
      - 1.3.2.4.5. Various writings of Fourier transforms (p. 39) | html | pdf |
      - 1.3.2.4.6. Tables of Fourier transforms (p. 39) | html | pdf |
    - 1.3.2.5. Fourier transforms of tempered distributions (pp. 39-42) | html | pdf |
      - 1.3.2.5.1. Introduction (p. 39) | html | pdf |
      - 1.3.2.5.2. $[{\scr S}]$ as a test-function space (p. 40) | html | pdf |
      - 1.3.2.5.3. Definition and examples of tempered distributions (p. 40) | html | pdf |
      - 1.3.2.5.4. Fourier transforms of tempered distributions (p. 40) | html | pdf |
      - 1.3.2.5.5. Transposition of basic properties (p. 40) | html | pdf |
      - 1.3.2.5.6. Transforms of δ-functions (pp. 40-41) | html | pdf |
      - 1.3.2.5.7. Reciprocity theorem (p. 41) | html | pdf |
      - 1.3.2.5.8. Multiplication and convolution (p. 41) | html | pdf |
      - 1.3.2.5.9. $[L^{2}]$ aspects, Sobolev spaces (pp. 41-42) | html | pdf |
    - 1.3.2.6. Periodic distributions and Fourier series (pp. 42-47) | html | pdf |
      - 1.3.2.6.1. Terminology (p. 42) | html | pdf |
      - 1.3.2.6.2. $[{\bb Z}^{n}]$ -periodic distributions in $[{\bb R}^{n}]$ (p. 42) | html | pdf |
      - 1.3.2.6.3. Identification with distributions over $[{\bb R}^{n}/{\bb Z}^{n}]$ (p. 42) | html | pdf |
      - 1.3.2.6.4. Fourier transforms of periodic distributions (pp. 42-43) | html | pdf |
      - 1.3.2.6.5. The case of nonstandard period lattices (pp. 43-44) | html | pdf |
      - 1.3.2.6.6. Duality between periodization and sampling (p. 44) | html | pdf |
      - 1.3.2.6.7. The Poisson summation formula (p. 44) | html | pdf |
      - 1.3.2.6.8. Convolution of Fourier series (p. 44) | html | pdf |
      - 1.3.2.6.9. Toeplitz forms, Szegö's theorem  (pp. 44-45) | html | pdf |
        1.3.2.6.9.1. Toeplitz forms  (pp. 44-45) | html | pdf |
        1.3.2.6.9.2. The Toeplitz–Carathéodory–Herglotz theorem  (p. 45) | html | pdf |
        1.3.2.6.9.3. Asymptotic distribution of eigenvalues of Toeplitz forms  (p. 45) | html | pdf |
        1.3.2.6.9.4. Consequences of Szegö's theorem  (p. 45) | html | pdf |
      - 1.3.2.6.10. Convergence of Fourier series  (pp. 45-47) | html | pdf |
        1.3.2.6.10.1. Classical $[L^{1}]$ theory  (p. 46) | html | pdf |
        1.3.2.6.10.2. Classical $[L^{2}]$ theory  (p. 47) | html | pdf |
        1.3.2.6.10.3. The viewpoint of distribution theory  (p. 47) | html | pdf |
    - 1.3.2.7. The discrete Fourier transformation (pp. 47-52) | html | pdf |
      - 1.3.2.7.1. Shannon's sampling theorem and interpolation formula (p. 47) | html | pdf |
      - 1.3.2.7.2. Duality between subdivision and decimation of period lattices  (pp. 48-49) | html | pdf |
        1.3.2.7.2.1. Geometric description of sublattices  (p. 48) | html | pdf |
        1.3.2.7.2.2. Sublattice relations for reciprocal lattices  (p. 48) | html | pdf |
        1.3.2.7.2.3. Relation between lattice distributions  (p. 48) | html | pdf |
        1.3.2.7.2.4. Relation between Fourier transforms  (pp. 48-49) | html | pdf |
        1.3.2.7.2.5. Sublattice relations in terms of periodic distributions  (p. 49) | html | pdf |
      - 1.3.2.7.3. Discretization of the Fourier transformation (pp. 49-51) | html | pdf |
      - 1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT) (p. 51) | html | pdf |
      - 1.3.2.7.5. Properties of the discrete Fourier transform (pp. 51-52) | html | pdf |
  - 1.3.3. Numerical computation of the discrete Fourier transform (pp. 52-62) | html | pdf |
    - 1.3.3.1. Introduction (p. 52) | html | pdf |
    - 1.3.3.2. One-dimensional algorithms (pp. 52-57) | html | pdf |
      - 1.3.3.2.1. The Cooley–Tukey algorithm (pp. 52-54) | html | pdf |
      - 1.3.3.2.2. The Good (or prime factor) algorithm  (pp. 54-55) | html | pdf |
        1.3.3.2.2.1. Ring structure on $[{\bb Z}/N{\bb Z}]$   (p. 54) | html | pdf |
        1.3.3.2.2.2. The Chinese remainder theorem  (p. 54) | html | pdf |
        1.3.3.2.2.3. The prime factor algorithm  (pp. 54-55) | html | pdf |
      - 1.3.3.2.3. The Rader algorithm  (pp. 55-56) | html | pdf |
        1.3.3.2.3.1. N an odd prime  (pp. 55-56) | html | pdf |
        1.3.3.2.3.2. N a power of an odd prime  (p. 56) | html | pdf |
        1.3.3.2.3.3. N a power of 2  (p. 56) | html | pdf |
      - 1.3.3.2.4. The Winograd algorithms (pp. 56-57) | html | pdf |
    - 1.3.3.3. Multidimensional algorithms (pp. 58-62) | html | pdf |
      - 1.3.3.3.1. The method of successive one-dimensional transforms (p. 58) | html | pdf |
      - 1.3.3.3.2. Multidimensional factorization  (pp. 58-61) | html | pdf |
        1.3.3.3.2.1. Multidimensional Cooley–Tukey factorization  (pp. 58-59) | html | pdf |
        1.3.3.3.2.2. Multidimensional prime factor algorithm  (pp. 59-60) | html | pdf |
        1.3.3.3.2.3. Nesting of Winograd small FFTs  (p. 60) | html | pdf |
        1.3.3.3.2.4. The Nussbaumer–Quandalle algorithm  (pp. 60-61) | html | pdf |
      - 1.3.3.3.3. Global algorithm design  (pp. 61-62) | html | pdf |
        1.3.3.3.3.1. From local pieces to global algorithms  (p. 61) | html | pdf |
        1.3.3.3.3.2. Computer architecture considerations  (p. 61) | html | pdf |
        1.3.3.3.3.3. The Johnson–Burrus family of algorithms  (p. 62) | html | pdf |
  - 1.3.4. Crystallographic applications of Fourier transforms (pp. 62-106) | html | pdf |
    - 1.3.4.1. Introduction (p. 62) | html | pdf |
    - 1.3.4.2. Crystallographic Fourier transform theory (pp. 62-76) | html | pdf |
      - 1.3.4.2.1. Crystal periodicity  (pp. 62-68) | html | pdf |
        1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors  (pp. 62-63) | html | pdf |
        1.3.4.2.1.2. Structure factors in terms of form factors  (p. 63) | html | pdf |
        1.3.4.2.1.3. Fourier series for the electron density and its summation  (pp. 63-64) | html | pdf |
        1.3.4.2.1.4. Friedel's law, anomalous scatterers  (p. 64) | html | pdf |
        1.3.4.2.1.5. Parseval's identity and other $[L^{2}]$ theorems  (p. 64) | html | pdf |
        1.3.4.2.1.6. Convolution, correlation and Patterson function  (pp. 64-65) | html | pdf |
        1.3.4.2.1.7. Sampling theorems, continuous transforms, interpolation  (pp. 65-66) | html | pdf |
        1.3.4.2.1.8. Sections and projections  (pp. 66-67) | html | pdf |
        1.3.4.2.1.9. Differential syntheses  (p. 67) | html | pdf |
        1.3.4.2.1.10. Toeplitz forms, determinantal inequalities and Szegö's theorem  (pp. 67-68) | html | pdf |
      - 1.3.4.2.2. Crystal symmetry  (pp. 68-76) | html | pdf |
        1.3.4.2.2.1. Crystallographic groups  (p. 68) | html | pdf |
        1.3.4.2.2.2. Groups and group actions  (pp. 68-70) | html | pdf |
        1.3.4.2.2.3. Classification of crystallographic groups  (pp. 70-71) | html | pdf |
        1.3.4.2.2.4. Crystallographic group action in real space  (pp. 71-72) | html | pdf |
        1.3.4.2.2.5. Crystallographic group action in reciprocal space  (pp. 72-73) | html | pdf |
        1.3.4.2.2.6. Structure-factor calculation  (pp. 73-74) | html | pdf |
        1.3.4.2.2.7. Electron-density calculations  (p. 74) | html | pdf |
        1.3.4.2.2.8. Parseval's theorem with crystallographic symmetry  (p. 74) | html | pdf |
        1.3.4.2.2.9. Convolution theorems with crystallographic symmetry  (p. 75) | html | pdf |
        1.3.4.2.2.10. Correlation and Patterson functions  (pp. 75-76) | html | pdf |
    - 1.3.4.3. Crystallographic discrete Fourier transform algorithms (pp. 76-91) | html | pdf |
      - 1.3.4.3.1. Historical introduction (pp. 76-77) | html | pdf |
      - 1.3.4.3.2. Defining relations and symmetry considerations (pp. 77-78) | html | pdf |
      - 1.3.4.3.3. Interaction between symmetry and decomposition (pp. 78-79) | html | pdf |
      - 1.3.4.3.4. Interaction between symmetry and factorization  (pp. 79-85) | html | pdf |
        1.3.4.3.4.1. Multidimensional Cooley–Tukey factorization  (pp. 79-82) | html | pdf |
        1.3.4.3.4.2. Multidimensional Good factorization  (p. 82) | html | pdf |
        1.3.4.3.4.3. Crystallographic extension of the Rader/Winograd factorization  (pp. 82-85) | html | pdf |
      - 1.3.4.3.5. Treatment of conjugate and parity-related symmetry properties  (pp. 85-89) | html | pdf |
        1.3.4.3.5.1. Hermitian-symmetric or real-valued transforms  (pp. 85-87) | html | pdf |
        1.3.4.3.5.2. Hermitian-antisymmetric or pure imaginary transforms  (p. 87) | html | pdf |
        1.3.4.3.5.3. Complex symmetric and antisymmetric transforms  (pp. 87-88) | html | pdf |
        1.3.4.3.5.4. Real symmetric transforms  (p. 88) | html | pdf |
        1.3.4.3.5.5. Real antisymmetric transforms  (p. 88) | html | pdf |
        1.3.4.3.5.6. Generalized multiplexing  (pp. 88-89) | html | pdf |
      - 1.3.4.3.6. Global crystallographic algorithms  (pp. 89-91) | html | pdf |
        1.3.4.3.6.1. Triclinic groups  (p. 89) | html | pdf |
        1.3.4.3.6.2. Monoclinic groups  (p. 89) | html | pdf |
        1.3.4.3.6.3. Orthorhombic groups  (pp. 89-90) | html | pdf |
        1.3.4.3.6.4. Trigonal, tetragonal and hexagonal groups  (p. 90) | html | pdf |
        1.3.4.3.6.5. Cubic groups  (p. 90) | html | pdf |
        1.3.4.3.6.6. Treatment of centred lattices  (p. 90) | html | pdf |
        1.3.4.3.6.7. Programming considerations  (pp. 90-91) | html | pdf |
    - 1.3.4.4. Basic crystallographic computations (pp. 91-100) | html | pdf |
      - 1.3.4.4.1. Introduction (p. 91) | html | pdf |
      - 1.3.4.4.2. Fourier synthesis of electron-density maps (p. 91) | html | pdf |
      - 1.3.4.4.3. Fourier analysis of modified electron-density maps  (pp. 91-93) | html | pdf |
        1.3.4.4.3.1. Squaring  (p. 91) | html | pdf |
        1.3.4.4.3.2. Other nonlinear operations  (pp. 91-92) | html | pdf |
        1.3.4.4.3.3. Solvent flattening  (p. 92) | html | pdf |
        1.3.4.4.3.4. Molecular averaging by noncrystallographic symmetries  (pp. 92-93) | html | pdf |
        1.3.4.4.3.5. Molecular-envelope transforms via Green's theorem  (p. 93) | html | pdf |
      - 1.3.4.4.4. Structure factors from model atomic parameters (p. 93) | html | pdf |
      - 1.3.4.4.5. Structure factors via model electron-density maps (pp. 93-94) | html | pdf |
      - 1.3.4.4.6. Derivatives for variational phasing techniques (pp. 94-95) | html | pdf |
      - 1.3.4.4.7. Derivatives for model refinement  (pp. 95-100) | html | pdf |
        1.3.4.4.7.1. The method of least squares  (pp. 95-96) | html | pdf |
        1.3.4.4.7.2. Booth's differential Fourier syntheses  (p. 96) | html | pdf |
        1.3.4.4.7.3. Booth's method of steepest descents  (p. 96) | html | pdf |
        1.3.4.4.7.4. Cochran's Fourier method  (pp. 96-97) | html | pdf |
        1.3.4.4.7.5. Cruickshank's modified Fourier method  (pp. 97-98) | html | pdf |
        1.3.4.4.7.6. Agarwal's FFT implementation of the Fourier method  (p. 98) | html | pdf |
        1.3.4.4.7.7. Lifchitz's reformulation  (p. 98) | html | pdf |
        1.3.4.4.7.8. A simplified derivation  (pp. 98-99) | html | pdf |
        1.3.4.4.7.9. Discussion of macromolecular refinement techniques  (p. 99) | html | pdf |
        1.3.4.4.7.10. Sampling considerations  (pp. 99-100) | html | pdf |
      - 1.3.4.4.8. Miscellaneous correlation functions (p. 100) | html | pdf |
    - 1.3.4.5. Related applications (pp. 100-106) | html | pdf |
      - 1.3.4.5.1. Helical diffraction  (pp. 100-102) | html | pdf |
        1.3.4.5.1.1. Circular harmonic expansions in polar coordinates  (p. 100) | html | pdf |
        1.3.4.5.1.2. The Fourier transform in polar coordinates  (p. 101) | html | pdf |
        1.3.4.5.1.3. The transform of an axially periodic fibre  (p. 101) | html | pdf |
        1.3.4.5.1.4. Helical symmetry and associated selection rules  (pp. 101-102) | html | pdf |
      - 1.3.4.5.2. Application to probability theory and direct methods  (pp. 102-106) | html | pdf |
        1.3.4.5.2.1. Analytical methods of probability theory  (pp. 102-104) | html | pdf |
        1.3.4.5.2.2. The statistical theory of phase determination  (pp. 104-106) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.3.3.1. A few global algorithms for computing a 400-point DFT (p. 61) | html | pdf |
- 1.4. Symmetry in reciprocal space (pp. 114-174) | html | pdf | zxc chapter contents |
  U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve
  - 1.4.1. Introduction (p. 114) | html | pdf |
    U. Shmueli
  - 1.4.2. Effects of symmetry on the Fourier image of the crystal (pp. 114-117) | html | pdf |
    U. Shmueli
    - 1.4.2.1. Point-group symmetry of the reciprocal lattice (pp. 114-115) | html | pdf |
    - 1.4.2.2. Relationship between structure factors at symmetry-related points of the reciprocal lattice (pp. 115-116) | html | pdf |
    - 1.4.2.3. Symmetry factors for space-group-specific Fourier summations (p. 116) | html | pdf |
    - 1.4.2.4. Symmetry factors for space-group-specific structure-factor formulae (pp. 116-117) | html | pdf |
  - 1.4.3. Structure-factor tables (pp. 117-119) | html | pdf |
    U. Shmueli
    - 1.4.3.1. Some general remarks (p. 117) | html | pdf |
    - 1.4.3.2. Preparation of the structure-factor tables (p. 117) | html | pdf |
    - 1.4.3.3. Symbolic representation of A and B (pp. 118-119) | html | pdf |
    - 1.4.3.4. Arrangement of the tables (p. 119) | html | pdf |
  - 1.4.4. Symmetry in reciprocal space: space-group tables (pp. 119-122) | html | pdf |
    U. Shmueli
    - 1.4.4.1. Introduction (p. 119) | html | pdf |
    - 1.4.4.2. Arrangement of the space-group tables (pp. 119-120) | html | pdf |
    - 1.4.4.3. Effect of direct-space transformations (p. 120) | html | pdf |
    - 1.4.4.4. Symmetry in Fourier space (pp. 120-121) | html | pdf |
    - 1.4.4.5. Relationships between direct and reciprocal Bravais lattices (pp. 121-122) | html | pdf |
  - Appendix 1.4.1. Comments on the preparation and usage of the tables (p. 122) | html | pdf |
    U. Shmueli
  - Appendix 1.4.2. Space-group symbols for numeric and symbolic computations (pp. 122-134) | html | pdf |
    - A1.4.2.1. Introduction (pp. 122-123) | html | pdf |
      U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve
    - A1.4.2.2. Explicit symbols (pp. 123-127) | html | pdf |
      U. Shmueli
    - A1.4.2.3. Hall symbols (pp. 127-134) | html | pdf |
      S. R. Hall and R. W. Grosse-Kunstleve
  - Appendix 1.4.3. Structure-factor tables (pp. 135-161) | html | pdf |
    U. Shmueli
  - Appendix 1.4.4. Crystallographic space groups in reciprocal space (pp. 162-173) | html | pdf |
    U. Shmueli
  - References | html | pdf |
  - Tables
    - Table 1.4.4.1. Correspondence between types of centring in direct and reciprocal lattices (p. 121) | html | pdf |
    - Table A1.4.2.1. Explicit symbols (pp. 124-126) | html | pdf |
    - Table A1.4.2.2. Lattice symbol L (p. 127) | html | pdf |
    - Table A1.4.2.3. Translation symbol T (p. 127) | html | pdf |
    - Table A1.4.2.4. Rotation matrices for principal axes (p. 128) | html | pdf |
    - Table A1.4.2.5. Rotation matrices for face-diagonal axes (p. 128) | html | pdf |
    - Table A1.4.2.6. Rotation matrix for the body-diagonal axis (p. 128) | html | pdf |
    - Table A1.4.2.7. Hall symbols (pp. 130-134) | html | pdf |
    - Table A1.4.3.1. Plane groups (p. 135) | html | pdf |
    - Table A1.4.3.2. Triclinic space groups (p. 135) | html | pdf |
    - Table A1.4.3.3. Monoclinic space groups (pp. 136-137) | html | pdf |
    - Table A1.4.3.4. Orthorhombic space groups (pp. 138-140) | html | pdf |
    - Table A1.4.3.5. Tetragonal space groups (pp. 141-149) | html | pdf |
    - Table A1.4.3.6. Trigonal and hexagonal space groups (pp. 150-155) | html | pdf |
    - Table A1.4.3.7. Cubic space groups (pp. 156-161) | html | pdf |
    - Table A1.4.4.1. Crystallographic space groups in reciprocal space (pp. 162-173) | html | pdf |
- 1.5. Crystallographic viewpoints in the classification of space-group representations (pp. 175-192) | html | pdf | zxc chapter contents |
  M. I. Aroyo and H. Wondratschek
  - 1.5.1. List of abbreviations and symbols (p. 175) | html | pdf |
  - 1.5.2. Introduction (pp. 175-176) | html | pdf |
  - 1.5.3. Basic concepts (pp. 176-179) | html | pdf |
    - 1.5.3.1. Representations of finite groups (p. 176) | html | pdf |
    - 1.5.3.2. Space groups (pp. 176-177) | html | pdf |
    - 1.5.3.3. Representations of the translation group $[{\cal T}]$ and the reciprocal lattice (pp. 177-178) | html | pdf |
    - 1.5.3.4. Irreducible representations of space groups and the reciprocal-space group (pp. 178-179) | html | pdf |
  - 1.5.4. Conventions in the classification of space-group irreps (pp. 179-181) | html | pdf |
    - 1.5.4.1. Fundamental regions (p. 179) | html | pdf |
    - 1.5.4.2. Minimal domains (pp. 179-180) | html | pdf |
    - 1.5.4.3. Wintgen positions (pp. 180-181) | html | pdf |
  - 1.5.5. Examples and discussion (pp. 181-191) | html | pdf |
    - 1.5.5.1. Guide to the figures (pp. 182-184) | html | pdf |
    - 1.5.5.2. Guide to the k-vector tables (pp. 184-188) | html | pdf |
    - 1.5.5.3. Figures and tables (pp. 188-189) | html | pdf |
    - 1.5.5.4. Discussion (pp. 189-191) | html | pdf |
      - 1.5.5.4.1. Representation domains and asymmetric units (pp. 189-190) | html | pdf |
      - 1.5.5.4.2. Splitting of k-vector types (p. 190) | html | pdf |
      - 1.5.5.4.3. k-vector types for non-holosymmetric space groups (p. 190) | html | pdf |
      - 1.5.5.4.4. Ranges of independent parameters (pp. 190-191) | html | pdf |
  - 1.5.6. Conclusions (p. 191) | html | pdf |
  - Appendix 1.5.1. Reciprocal-space groups $[({\cal G})^{*}]$ (p. 192) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.5.5.1. Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class $[m\overline{3}mI]$ (p. 183) | html | pdf |
    - Fig. 1.5.5.2. Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class $[m\overline{3}I]$ (p. 184) | html | pdf |
    - Fig. 1.5.5.3. Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : $[c/a \,\lt \, 1]$ (p. 185) | html | pdf |
    - Fig. 1.5.5.4. Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : $[c/a\,\gt \, 1]$ (p. 186) | html | pdf |
    - Fig. 1.5.5.5. Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : $[a^{-2} \,\lt \, b^{-2}+c^{-2}]$ , $[b^{-2} \,\lt \, c^{-2}+a^{-2}]$ and $[c^{-2} \,\lt \, a^{-2}+b^{-2}]$ (p. 187) | html | pdf |
    - Fig. 1.5.5.6. Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : $[c^{-2}\,\gt \, a^{-2}+b^{-2}]$ (p. 188) | html | pdf |
    - Fig. 1.5.5.7. Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : $[a^{-2}\,\gt \, b^{-2}+c^{-2}]$ (p. 189) | html | pdf |
  - Tables
    - Table 1.5.4.1. Conventional coefficients $[(k_{i})^{T}]$ of k expressed by the adjusted coefficients $[(k_{ai})]$ of IT A for the different Bravais types of lattices in direct space (p. 181) | html | pdf |
    - Table 1.5.4.2. Primitive coefficients $[(k_{pi})^{T}]$ of k from CDML expressed by the adjusted coefficients $[(k_{ai})]$ of IT A for the different Bravais types of lattices in direct space (p. 181) | html | pdf |
    - Table 1.5.5.1. List of k-vector types for arithmetic crystal class $[m\overline{3}mI]$ (p. 183) | html | pdf |
    - Table 1.5.5.2. List of k-vector types for arithmetic crystal class $[m\overline{3}I]$ (p. 184) | html | pdf |
    - Table 1.5.5.3. List of k-vector types for arithmetic crystal class : $[c/a \,\lt\, 1]$ (p. 185) | html | pdf |
    - Table 1.5.5.4. List of k-vector types for arithmetic crystal class : $[c/a \,\gt\, 1]$ (p. 186) | html | pdf |
    - Table 1.5.5.5. List of k-vector types for arithmetic crystal class : $[a^{-2} \,\lt\, b^{-2}+c^{-2}]$ , $[b^{-2} \,\lt\, c^{-2}+a^{-2}]$ and $[c^{-2} \,\lt\, a^{-2}+b^{-2}]$ (p. 187) | html | pdf |
    - Table 1.5.5.6. List of k-vector types for arithmetic crystal class : $[c^{-2}>a^{-2}+b^{-2}]$ (p. 188) | html | pdf |
    - Table 1.5.5.7. List of k-vector types for arithmetic crystal class : $[a^{-2}\,\gt \, b^{-2}+c^{-2}]$ (p. 189) | html | pdf |

Reciprocal space in crystal-structure determination
- 2.1. Statistical properties of the weighted reciprocal lattice (pp. 195-214) | html | pdf | zxc chapter contents |
  U. Shmueli and A. J. C. Wilson
  - 2.1.1. Introduction (p. 195) | html | pdf |
  - 2.1.2. The average intensity of general reflections (pp. 195-196) | html | pdf |
    - 2.1.2.1. Mathematical background (pp. 195-196) | html | pdf |
    - 2.1.2.2. Physical background (p. 196) | html | pdf |
    - 2.1.2.3. An approximation for organic compounds (p. 196) | html | pdf |
    - 2.1.2.4. Effect of centring (p. 196) | html | pdf |
  - 2.1.3. The average intensity of zones and rows (pp. 196-197) | html | pdf |
    - 2.1.3.1. Symmetry elements producing systematic absences (p. 196) | html | pdf |
    - 2.1.3.2. Symmetry elements not producing systematic absences (pp. 196-197) | html | pdf |
    - 2.1.3.3. More than one symmetry element (p. 197) | html | pdf |
  - 2.1.4. Probability density distributions – mathematical preliminaries (pp. 197-200) | html | pdf |
    - 2.1.4.1. Characteristic functions (pp. 197-198) | html | pdf |
    - 2.1.4.2. The cumulant-generating function (p. 198) | html | pdf |
    - 2.1.4.3. The central-limit theorem (pp. 198-199) | html | pdf |
    - 2.1.4.4. Conditions of validity (p. 199) | html | pdf |
    - 2.1.4.5. Non-independent variables (pp. 199-200) | html | pdf |
  - 2.1.5. Ideal probability density distributions (pp. 200-202) | html | pdf |
    - 2.1.5.1. Ideal acentric distributions (p. 200) | html | pdf |
    - 2.1.5.2. Ideal centric distributions (p. 200) | html | pdf |
    - 2.1.5.3. Effect of other symmetry elements on the ideal acentric and centric distributions (pp. 200-201) | html | pdf |
    - 2.1.5.4. Other ideal distributions (p. 201) | html | pdf |
    - 2.1.5.5. Relation to distributions of I (p. 201) | html | pdf |
    - 2.1.5.6. Cumulative distribution functions (pp. 201-202) | html | pdf |
  - 2.1.6. Distributions of sums, averages and ratios (pp. 202-203) | html | pdf |
    - 2.1.6.1. Distributions of sums and averages (p. 202) | html | pdf |
    - 2.1.6.2. Distribution of ratios (pp. 202-203) | html | pdf |
    - 2.1.6.3. Intensities scaled to the local average (p. 203) | html | pdf |
    - 2.1.6.4. The use of normal approximations (p. 203) | html | pdf |
  - 2.1.7. Non-ideal distributions: the correction-factor approach (pp. 203-207) | html | pdf |
    - 2.1.7.1. Introduction (pp. 203-204) | html | pdf |
    - 2.1.7.2. Mathematical background (p. 204) | html | pdf |
    - 2.1.7.3. Application to centric and acentric distributions (pp. 204-205) | html | pdf |
    - 2.1.7.4. Fourier versus Hermite approximations (pp. 205-207) | html | pdf |
  - 2.1.8. Non-ideal distributions: the Fourier method (pp. 207-212) | html | pdf |
    - 2.1.8.1. General representations of p.d.f.'s of by Fourier series (p. 208) | html | pdf |
    - 2.1.8.2. Fourier–Bessel series (pp. 208-209) | html | pdf |
    - 2.1.8.3. Simple examples (p. 209) | html | pdf |
    - 2.1.8.4. A more complicated example (pp. 209-211) | html | pdf |
    - 2.1.8.5. Atomic characteristic functions (pp. 211-212) | html | pdf |
    - 2.1.8.6. Other non-ideal Fourier p.d.f.'s (p. 212) | html | pdf |
    - 2.1.8.7. Comparison of the correction-factor and Fourier approaches (p. 212) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.1.7.1. Atomic heterogeneity and intensity statistics (p. 204) | html | pdf |
    - Fig. 2.1.8.1. The same recalculated histogram as in Fig. 2.1.7.1 along with the centric correction-factor p.d.f (p. 212) | html | pdf |
  - Tables
    - Table 2.1.3.1. Intensity-distribution effects of symmetry elements and centred lattices causing systematic absences (p. 196) | html | pdf |
    - Table 2.1.3.2. Intensity-distribution effects of symmetry elements not causing systematic absences (p. 197) | html | pdf |
    - Table 2.1.3.3. Average multiples for the 32 point groups (modified from Rogers, 1950) (p. 197) | html | pdf |
    - Table 2.1.5.1. Some properties of gamma and beta distributions (p. 201) | html | pdf |
    - Table 2.1.7.1. Some even absolute moments of the trigonometric structure factor (pp. 206-207) | html | pdf |
    - Table 2.1.7.2. Closed expressions for $[\gamma_{2k}]$ [equation (2.1.7.11)] for space groups of low symmetry (p. 207) | html | pdf |
    - Table 2.1.8.1. Atomic contributions to characteristic functions for (pp. 210-211) | html | pdf |
- 2.2. Direct methods (pp. 215-243) | html | pdf | zxc chapter contents |
  C. Giacovazzo
  - 2.2.1. List of symbols and abbreviations (p. 215) | html | pdf |
  - 2.2.2. Introduction (p. 215) | html | pdf |
  - 2.2.3. Origin specification (pp. 215-216) | html | pdf |
  - 2.2.4. Normalized structure factors (pp. 216-221) | html | pdf |
    - 2.2.4.1. Definition of normalized structure factor (pp. 216-218) | html | pdf |
    - 2.2.4.2. Definition of quasi-normalized structure factor (pp. 218-219) | html | pdf |
    - 2.2.4.3. The calculation of normalized structure factors (pp. 219-221) | html | pdf |
    - 2.2.4.4. Probability distributions of normalized structure factors (p. 221) | html | pdf |
  - 2.2.5. Phase-determining formulae (pp. 221-230) | html | pdf |
    - 2.2.5.1. Inequalities among structure factors (pp. 221-222) | html | pdf |
    - 2.2.5.2. Probabilistic phase relationships for structure invariants (pp. 222-223) | html | pdf |
    - 2.2.5.3. Triplet relationships (pp. 223-224) | html | pdf |
    - 2.2.5.4. Triplet relationships using structural information (pp. 224-225) | html | pdf |
    - 2.2.5.5. Quartet phase relationships (pp. 225-227) | html | pdf |
    - 2.2.5.6. Quintet phase relationships (p. 227) | html | pdf |
    - 2.2.5.7. Determinantal formulae (pp. 227-228) | html | pdf |
    - 2.2.5.8. Algebraic relationships for structure seminvariants (pp. 228-229) | html | pdf |
    - 2.2.5.9. Formulae estimating one-phase structure seminvariants of the first rank (p. 229) | html | pdf |
    - 2.2.5.10. Formulae estimating two-phase structure seminvariants of the first rank (pp. 229-230) | html | pdf |
  - 2.2.6. Direct methods in real and reciprocal space: Sayre's equation (pp. 230-231) | html | pdf |
  - 2.2.7. Scheme of procedure for phase determination: the small-molecule case (pp. 231-232) | html | pdf |
  - 2.2.8. Other multisolution methods applied to small molecules (pp. 232-234) | html | pdf |
  - 2.2.9. Some references to direct-methods packages: the small-molecule case (pp. 234-235) | html | pdf |
  - 2.2.10. Direct methods in macromolecular crystallography (pp. 235-239) | html | pdf |
    - 2.2.10.1. Introduction (p. 235) | html | pdf |
    - 2.2.10.2. Ab initio crystal structure solution of proteins (pp. 235-236) | html | pdf |
    - 2.2.10.3. Integration of direct methods with isomorphous replacement techniques (p. 236) | html | pdf |
    - 2.2.10.4. SIR–MIR case: one-step procedures (pp. 236-237) | html | pdf |
    - 2.2.10.5. SIR–MIR case: the two-step procedure. Finding the heavy-atom substructure by direct methods (p. 237) | html | pdf |
    - 2.2.10.6. SIR–MIR case: protein phasing by direct methods (p. 237) | html | pdf |
    - 2.2.10.7. Integration of anomalous-dispersion techniques with direct methods (pp. 237-238) | html | pdf |
    - 2.2.10.8. The SAD case: the one-step procedures (p. 238) | html | pdf |
    - 2.2.10.9. SAD–MAD case: the two-step procedures. Finding the anomalous-scatterer substructure by direct methods (pp. 238-239) | html | pdf |
    - 2.2.10.10. SAD–MAD case: protein phasing by direct methods (p. 239) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.2.4.1. Probability density functions for cs. and ncs. crystals (p. 221) | html | pdf |
    - Fig. 2.2.4.2. Cumulative distribution functions for cs. and ncs. crystals (p. 222) | html | pdf |
    - Fig. 2.2.5.1. Curves of (2.2.5.6) for some values of $[2\sigma_{3} \sigma_{2}^{-3/2} |E_{\bf h} E_{\bf k} E_{{\bf h} - {\bf k}}|]$ (p. 223) | html | pdf |
    - Fig. 2.2.5.2. Variance (in square radians) as a function of α (p. 223) | html | pdf |
    - Fig. 2.2.5.3. Distributions (2.2.5.18) (solid curve) and (2.2.5.20) (dashed curve) for the indicated values in three typical cases (p. 225) | html | pdf |
    - Fig. 2.2.7.1. The form of w as given by (2.2.7.2) (p. 232) | html | pdf |
  - Tables
    - Table 2.2.3.1. Allowed origin translations, seminvariant moduli and phases for centrosymmetric primitive space groups (p. 217) | html | pdf |
    - Table 2.2.3.2. Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric primitive space groups (pp. 218-219) | html | pdf |
    - Table 2.2.3.3. Allowed origin translations, seminvariant moduli and phases for centrosymmetric non-primitive space groups (p. 220) | html | pdf |
    - Table 2.2.3.4. Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric non-primitive space groups (pp. 220-221) | html | pdf |
    - Table 2.2.4.1. Moments of the distributions (2.2.4.4) and (2.2.4.5) (p. 222) | html | pdf |
    - Table 2.2.5.1. List of quartets symmetry equivalent to $[\Phi = \Phi_{1}]$ in the class mmm (p. 226) | html | pdf |
    - Table 2.2.8.1. Magic-integer sequences for small numbers of phases (n) together with the number of sets produced and the root-mean-square error in the phases (p. 233) | html | pdf |
- 2.3. Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing (pp. 244-281) | html | pdf | zxc chapter contents |
  L. Tong, M. G. Rossmann and E. Arnold
  - 2.3.1. Introduction (pp. 244-247) | html | pdf |
    - 2.3.1.1. Background (pp. 244-245) | html | pdf |
    - 2.3.1.2. Limits to the number of resolved vectors (p. 245) | html | pdf |
    - 2.3.1.3. Modifications: origin removal, sharpening etc. (pp. 245-246) | html | pdf |
    - 2.3.1.4. Homometric structures and the uniqueness of structure solutions; enantiomorphic solutions (pp. 246-247) | html | pdf |
    - 2.3.1.5. The Patterson synthesis of the second kind (p. 247) | html | pdf |
  - 2.3.2. Interpretation of Patterson maps (pp. 247-251) | html | pdf |
    - 2.3.2.1. Simple solutions in the triclinic cell. Selection of the origin (pp. 247-248) | html | pdf |
    - 2.3.2.2. Harker sections (p. 248) | html | pdf |
    - 2.3.2.3. Finding heavy atoms (p. 249) | html | pdf |
    - 2.3.2.4. Superposition methods. Image detection (pp. 249-250) | html | pdf |
    - 2.3.2.5. Systematic computerized Patterson vector-search procedures. Looking for rigid bodies (p. 251) | html | pdf |
  - 2.3.3. Isomorphous replacement difference Pattersons (pp. 251-255) | html | pdf |
    - 2.3.3.1. Introduction (p. 251) | html | pdf |
    - 2.3.3.2. Finding heavy atoms with centrosymmetric projections (pp. 251-252) | html | pdf |
    - 2.3.3.3. Finding heavy atoms with three-dimensional methods (p. 252) | html | pdf |
    - 2.3.3.4. Correlation functions (pp. 252-253) | html | pdf |
    - 2.3.3.5. Interpretation of isomorphous difference Pattersons (pp. 253-254) | html | pdf |
    - 2.3.3.6. Direct structure determination from difference Pattersons (pp. 254-255) | html | pdf |
    - 2.3.3.7. Isomorphism and size of the heavy-atom substitution (p. 255) | html | pdf |
  - 2.3.4. Anomalous dispersion (pp. 255-257) | html | pdf |
    - 2.3.4.1. Introduction (pp. 255-256) | html | pdf |
    - 2.3.4.2. The $[P_{\rm s}({\bf u})]$ function (p. 256) | html | pdf |
    - 2.3.4.3. The position of anomalous scatterers (pp. 256-257) | html | pdf |
    - 2.3.4.4. Computer programs for automated location of atomic positions from Patterson maps (p. 257) | html | pdf |
  - 2.3.5. Noncrystallographic symmetry (pp. 258-260) | html | pdf |
    - 2.3.5.1. Definitions (pp. 258-259) | html | pdf |
    - 2.3.5.2. Interpretation of Pattersons in the presence of noncrystallographic symmetry (pp. 259-260) | html | pdf |
  - 2.3.6. Rotation functions (pp. 260-269) | html | pdf |
    - 2.3.6.1. Introduction (pp. 260-261) | html | pdf |
    - 2.3.6.2. Matrix algebra (pp. 262-263) | html | pdf |
    - 2.3.6.3. Symmetry (pp. 263-264) | html | pdf |
    - 2.3.6.4. Sampling, background and interpretation (pp. 264-267) | html | pdf |
    - 2.3.6.5. The fast rotation function (p. 268) | html | pdf |
    - 2.3.6.6. Locked rotation functions (pp. 268-269) | html | pdf |
  - 2.3.7. Translation functions (pp. 269-272) | html | pdf |
    - 2.3.7.1. Introduction (p. 269) | html | pdf |
    - 2.3.7.2. Position of a noncrystallographic element relating two unknown structures (p. 270) | html | pdf |
    - 2.3.7.3. Position of a known molecular structure in an unknown unit cell (pp. 270-271) | html | pdf |
    - 2.3.7.4. Position of a noncrystallographic symmetry element in a poorly defined electron-density map (p. 271) | html | pdf |
    - 2.3.7.5. Locked translation function (pp. 271-272) | html | pdf |
    - 2.3.7.6. Computer programs for rotation and translation function calculations (p. 272) | html | pdf |
  - 2.3.8. Molecular replacement (pp. 272-275) | html | pdf |
    - 2.3.8.1. Using a known molecular fragment (pp. 272-273) | html | pdf |
    - 2.3.8.2. Using noncrystallographic symmetry for phase improvement (pp. 273-274) | html | pdf |
    - 2.3.8.3. Update on noncrystallographic averaging and density-modification methods (pp. 274-275) | html | pdf |
    - 2.3.8.4. Equivalence of real- and reciprocal-space molecular replacement (p. 275) | html | pdf |
  - 2.3.9. Conclusions (p. 275) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.3.1.1. Effect of `sharpening' Patterson coefficients (p. 246) | html | pdf |
    - Fig. 2.3.1.2. (c) The point Patterson of the two homometric structures in (a) and (b) (p. 246) | html | pdf |
    - Fig. 2.3.2.1. Origin selection in the interpretation of a Patterson of a one-dimensional centrosymmetric structure (p. 247) | html | pdf |
    - Fig. 2.3.2.2. The c-axis projection of cuprous chloride azomethane complex (C₂H₆Cl₂Cu₂N₂) (p. 247) | html | pdf |
    - Fig. 2.3.2.3. Atoms ABCD, arranged as a quadrilateral, generate a Patterson which is the sum of the images of the quadrilateral when each atom is placed on the origin in turn (p. 250) | html | pdf |
    - Fig. 2.3.3.1. Three different cases which can occur in the relation of the native, $[{\bf F}_{\rm N}]$ , and heavy-atom derivative, $[{\bf F}_{\rm NH}]$ , structure factors for centrosymmetric reflections (p. 251) | html | pdf |
    - Fig. 2.3.3.2. Vector triangle showing the relationship between $[{\bf F}_{\rm N}]$ , $[{\bf F}_{\rm NH}]$ and $[{\bf F}_{\rm H}]$ , where $[{\bf F}_{\rm NH} = {\bf F}_{\rm N} + {\bf F}_{\rm H}]$ (p. 252) | html | pdf |
    - Fig. 2.3.3.3. A Patterson with coefficients $[({\bf F}_{\rm NH1} - {\bf F}_{\rm NH2})^{2}]$ will be equivalent to a Patterson whose coefficients are $[(AB)^{2}]$ (p. 253) | html | pdf |
    - Fig. 2.3.3.4. The phase α of the native compound (structure factor $[{\bf F}_{\rm N}]$ ) is determined either as being equal to, or 180° out of phase with, the presumed heavy-atom contribution when only a single isomorphous compound is available (p. 254) | html | pdf |
    - Fig. 2.3.3.5. Let (a) be the original structure which contains three heavy atoms ABC in a noncentrosymmetric configuration (p. 254) | html | pdf |
    - Fig. 2.3.3.6. A plot of mean isomorphous differences as a function of resolution (p. 255) | html | pdf |
    - Fig. 2.3.4.1. (a) A model structure with an anomalous scatterer at A (p. 256) | html | pdf |
    - Fig. 2.3.4.2. Anomalous-dispersion effect for a molecule (p. 257) | html | pdf |
    - Fig. 2.3.5.1. The two-dimensional periodic design shows crystallographic twofold axes perpendicular to the page and local noncrystallographic rotation axes in the plane of the paper (p. 258) | html | pdf |
    - Fig. 2.3.5.2. The objects $[A_{1}B_{1}]$ and $[A_{2}B_{2}]$ are related by an improper rotation θ (p. 258) | html | pdf |
    - Fig. 2.3.5.3. The position of the twofold rotation axis which relates the two piglets is completely arbitrary (p. 259) | html | pdf |
    - Fig. 2.3.6.1. Shape of the interference function G for a spherical envelope of radius R at a distance H from the reciprocal-space origin (p. 261) | html | pdf |
    - Fig. 2.3.6.2. Relationships of the orthogonal axes $[X_{1}, X_{2}, X_{3}]$ to the crystallographic axes $[a_{1}, a_{2}, a_{3}]$ (p. 262) | html | pdf |
    - Fig. 2.3.6.3. Eulerian angles $[\theta_{1}, \theta_{2}, \theta_{3}]$ relating the rotated axes $[X'_{1}, X'_{2}, X'_{3}]$ to the original unrotated orthogonal axes $[X_{1}, X_{2}, X_{3}]$ (p. 262) | html | pdf |
    - Fig. 2.3.6.4. Variables ψ and φ are polar coordinates which specify a direction about which the axes may be rotated through an angle κ (p. 262) | html | pdf |
    - Fig. 2.3.6.5. Rotation space group diagram for the rotation function of a Pmmm Patterson function $[(P_{1})]$ against a Patterson function $[(P_{2})]$ (p. 264) | html | pdf |
    - Fig. 2.3.6.6. The locked rotation function, L, applied to the determination of the orientation of the common cold virus (p. 267) | html | pdf |
    - Fig. 2.3.7.1. Crosses represent atoms in a two-dimensional model structure (p. 270) | html | pdf |
    - Fig. 2.3.7.2. Vectors arising from the structure in Fig. 2.3.7.1 (p. 270) | html | pdf |
  - Tables
    - Table 2.3.1.1. Matrix representation of Patterson peaks (p. 245) | html | pdf |
    - Table 2.3.2.1. Coordinates of Patterson peaks for C₂H₆Cl₂Cu₂N₂ projection (p. 248) | html | pdf |
    - Table 2.3.2.2. Square matrix representation of vector interactions in a Patterson of a crystal with M crystallographic asymmetric units each containing N atoms (p. 248) | html | pdf |
    - Table 2.3.2.3. Position of Harker sections within a Patterson (p. 249) | html | pdf |
    - Table 2.3.5.1. Possible types of vector searches (p. 259) | html | pdf |
    - Table 2.3.5.2. Orientation of the glyceraldehyde-3-phosphate dehydrogenase molecular twofold axis in the orthorhombic cell (p. 259) | html | pdf |
    - Table 2.3.6.1. Different types of uses for the rotation function (p. 260) | html | pdf |
    - Table 2.3.6.2. Eulerian symmetry elements for all possible types of space-group rotations (p. 264) | html | pdf |
    - Table 2.3.6.3. Numbering of the rotation-function space groups (p. 264) | html | pdf |
    - Table 2.3.6.4. Rotation-function Eulerian space groups (pp. 265-266) | html | pdf |
    - Table 2.3.8.1. Molecular replacement: phase refinement as an iterative process (p. 273) | html | pdf |
- 2.4. Isomorphous replacement and anomalous scattering (pp. 282-296) | html | pdf | zxc chapter contents |
  M. Vijayan and S. Ramaseshan
  - 2.4.1. Introduction (p. 282) | html | pdf |
  - 2.4.2. Isomorphous replacement method (pp. 282-283) | html | pdf |
    - 2.4.2.1. Isomorphous replacement and isomorphous addition (pp. 282-283) | html | pdf |
    - 2.4.2.2. Single isomorphous replacement method (p. 283) | html | pdf |
    - 2.4.2.3. Multiple isomorphous replacement method (p. 283) | html | pdf |
  - 2.4.3. Anomalous-scattering method (pp. 284-287) | html | pdf |
    - 2.4.3.1. Dispersion correction (p. 284) | html | pdf |
    - 2.4.3.2. Violation of Friedel's law (pp. 284-285) | html | pdf |
    - 2.4.3.3. Friedel and Bijvoet pairs (p. 285) | html | pdf |
    - 2.4.3.4. Determination of absolute configuration (pp. 285-286) | html | pdf |
    - 2.4.3.5. Determination of phase angles (p. 286) | html | pdf |
    - 2.4.3.6. Anomalous scattering without phase change (p. 286) | html | pdf |
    - 2.4.3.7. Treatment of anomalous scattering in structure refinement (p. 287) | html | pdf |
  - 2.4.4. Isomorphous replacement and anomalous scattering in protein crystallography (pp. 287-293) | html | pdf |
    - 2.4.4.1. Protein heavy-atom derivatives (p. 287) | html | pdf |
    - 2.4.4.2. Determination of heavy-atom parameters (pp. 287-288) | html | pdf |
    - 2.4.4.3. Refinement of heavy-atom parameters (pp. 289-290) | html | pdf |
    - 2.4.4.4. Treatment of errors in phase evaluation: Blow and Crick formulation (pp. 290-291) | html | pdf |
    - 2.4.4.5. Use of anomalous scattering in phase evaluation (p. 291) | html | pdf |
    - 2.4.4.6. Estimation of r.m.s. error (pp. 291-292) | html | pdf |
    - 2.4.4.7. Suggested modifications to Blow and Crick formulation and the inclusion of phase information from other sources (pp. 292-293) | html | pdf |
    - 2.4.4.8. Fourier representation of anomalous scatterers (p. 293) | html | pdf |
  - 2.4.5. Anomalous scattering of neutrons and synchrotron radiation. The multiwavelength method (pp. 293-294) | html | pdf |
    - 2.4.5.1. Neutron anomalous scattering (pp. 293-294) | html | pdf |
    - 2.4.5.2. Anomalous scattering of synchrotron radiation (p. 294) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.4.2.1. Vector relationship between $[{\bf F}_{\rm N}]$ and $[{\bf F}_{\rm M}\ (\equiv {\bf F}_{\rm NH})]$ (p. 283) | html | pdf |
    - Fig. 2.4.2.2. Relationship between $[\alpha_{\rm N}]$ , $[\alpha_{\rm H}]$ and $[\varphi]$ (p. 283) | html | pdf |
    - Fig. 2.4.2.3. Harker construction when two heavy-atom derivatives are available (p. 284) | html | pdf |
    - Fig. 2.4.3.1. Variation of (a) and (b) as a function of atomic number for Cu $[K\alpha]$ and Mo $[K\alpha]$ radiations (p. 284) | html | pdf |
    - Fig. 2.4.3.2. Vector diagram illustrating the violation of Friedel's law when $[{\bf F}''_{\rm Q} \neq 0]$ (p. 285) | html | pdf |
    - Fig. 2.4.3.3. A composite view of the vector relationship between $[{\bf F}_{\rm N}(+)]$ and $[{\bf F}_{\rm N}(-)]$ (p. 285) | html | pdf |
    - Fig. 2.4.4.1. Distribution of intersections in the Harker construction under non-ideal conditions (p. 290) | html | pdf |
    - Fig. 2.4.4.2. Vector diagram indicating the calculated structure factor, $[{\bf D}_{{\rm H}i}(\alpha)]$ , of the ith heavy-atom derivative (p. 290) | html | pdf |
    - Fig. 2.4.4.3. The probability distribution of the protein phase angle (p. 290) | html | pdf |
    - Fig. 2.4.4.4. Harker construction using anomalous-scattering data from a single derivative (p. 291) | html | pdf |
  - Tables
    - Table 2.4.3.1. Phase angles of different components of the structure factor in space group P222 (p. 285) | html | pdf |
- 2.5. Electron diffraction and electron microscopy in structure determination (pp. 297-402) | html | pdf | zxc chapter contents |
  J. M. Cowley, J. C. H. Spence, M. Tanaka, B. K. Vainshtein, B. B. Zvyagin, P. A. Penczek and D. L. Dorset
  - 2.5.1. Foreword (pp. 297-299) | html | pdf |
    J. M. Cowley and J. C. H. Spence
  - 2.5.2. Electron diffraction and electron microscopy (pp. 299-307) | html | pdf |
    J. M. Cowley
    - 2.5.2.1. Introduction (p. 299) | html | pdf |
    - 2.5.2.2. The interactions of electrons with matter (pp. 299-301) | html | pdf |
    - 2.5.2.3. Recommended sign conventions (p. 301) | html | pdf |
    - 2.5.2.4. Scattering of electrons by crystals; approximations (pp. 301-302) | html | pdf |
    - 2.5.2.5. Kinematical diffraction formulae (pp. 302-303) | html | pdf |
    - 2.5.2.6. Imaging with electrons (pp. 303-304) | html | pdf |
    - 2.5.2.7. Imaging of very thin and weakly scattering objects (pp. 304-306) | html | pdf |
    - 2.5.2.8. Crystal structure imaging (p. 306) | html | pdf |
    - 2.5.2.9. Image resolution (pp. 306-307) | html | pdf |
    - 2.5.2.10. Electron diffraction in electron microscopes (p. 307) | html | pdf |
  - 2.5.3. Point-group and space-group determination by convergent-beam electron diffraction (pp. 307-356) | html | pdf |
    M. Tanaka
    - 2.5.3.1. Introduction (pp. 307-308) | html | pdf |
    - 2.5.3.2. Point-group determination (pp. 308-318) | html | pdf |
      - 2.5.3.2.1. Symmetry elements of a specimen and diffraction groups (pp. 308-309) | html | pdf |
      - 2.5.3.2.2. Identification of three-dimensional symmetry elements (pp. 309-310) | html | pdf |
      - 2.5.3.2.3. Identification of two-dimensional symmetry elements (p. 310) | html | pdf |
      - 2.5.3.2.4. Diffraction-group determination (pp. 310-311) | html | pdf |
      - 2.5.3.2.5. Point-group determination (p. 311) | html | pdf |
      - 2.5.3.2.6. Projection diffraction groups (pp. 311-312) | html | pdf |
      - 2.5.3.2.7. Symmetrical many-beam method (pp. 312-318) | html | pdf |
    - 2.5.3.3. Space-group determination (pp. 318-344) | html | pdf |
      - 2.5.3.3.1. Lattice-type determination (p. 318) | html | pdf |
      - 2.5.3.3.2. Identification of screw axes and glide planes (pp. 318-323) | html | pdf |
      - 2.5.3.3.3. Space-group determination (pp. 323-333) | html | pdf |
      - 2.5.3.3.4. Dynamical extinction in HOLZ reflections (pp. 333-334) | html | pdf |
      - 2.5.3.3.5. Symmetry elements observed by CBED (p. 334) | html | pdf |
      - 2.5.3.3.6. Examples of space-group determination (pp. 334-344) | html | pdf |
    - 2.5.3.4. Symmetry determination of incommensurate crystals (pp. 344-352) | html | pdf |
      - 2.5.3.4.1. General remarks (pp. 344-348) | html | pdf |
      - 2.5.3.4.2. Point-group determination (pp. 348-349) | html | pdf |
      - 2.5.3.4.3. Space-group determination (pp. 349-352) | html | pdf |
    - 2.5.3.5. Symmetry determination of quasicrystals (pp. 352-356) | html | pdf |
      - 2.5.3.5.1. Icosahedral quasicrystals (pp. 352-354) | html | pdf |
      - 2.5.3.5.2. Decagonal quasicrystals (pp. 354-356) | html | pdf |
  - 2.5.4. Electron-diffraction structure analysis (EDSA) (pp. 356-361) | html | pdf |
    B. K. Vainshtein and B. B. Zvyagin
    - 2.5.4.1. Introduction (p. 356) | html | pdf |
    - 2.5.4.2. The geometry of ED patterns (pp. 356-358) | html | pdf |
    - 2.5.4.3. Intensities of diffraction beams (pp. 358-359) | html | pdf |
    - 2.5.4.4. Structure analysis (pp. 359-361) | html | pdf |
  - 2.5.5. Image reconstruction (pp. 361-366) | html | pdf |
    B. K. Vainshtein
    - 2.5.5.1. Introduction (p. 361) | html | pdf |
    - 2.5.5.2. Thin weak phase objects at optimal defocus (pp. 361-362) | html | pdf |
    - 2.5.5.3. An account of absorption (pp. 362-363) | html | pdf |
    - 2.5.5.4. Thick crystals (p. 363) | html | pdf |
    - 2.5.5.5. Image enhancement (pp. 363-366) | html | pdf |
  - 2.5.6. Three-dimensional reconstruction (pp. 366-375) | html | pdf |
    B. K. Vainshtein and P. A. Penczek
    - 2.5.6.1. The object and its projection (pp. 366-367) | html | pdf |
    - 2.5.6.2. 3D reconstruction in the general case (pp. 367-368) | html | pdf |
    - 2.5.6.3. Discretization and interpolation (pp. 368-369) | html | pdf |
    - 2.5.6.4. The algebraic and iterative methods (pp. 369-371) | html | pdf |
    - 2.5.6.5. Filtered backprojection (pp. 371-372) | html | pdf |
    - 2.5.6.6. Direct Fourier inversion (pp. 372-375) | html | pdf |
    - 2.5.6.7. 3D reconstruction of symmetric objects (p. 375) | html | pdf |
  - 2.5.7. Single-particle reconstruction (pp. 375-388) | html | pdf |
    P. A. Penczek
    - 2.5.7.1. Formation of projection images in single-particle reconstruction (pp. 375-376) | html | pdf |
    - 2.5.7.2. Structure determination in single-particle reconstruction (pp. 376-377) | html | pdf |
    - 2.5.7.3. Electron microscopy and data digitization (pp. 377-378) | html | pdf |
    - 2.5.7.4. Assessment of the data quality and estimation of the image formation parameters (p. 378) | html | pdf |
    - 2.5.7.5. 2D data analysis – particle picking (p. 379) | html | pdf |
    - 2.5.7.6. 2D alignment of EM images (pp. 379-381) | html | pdf |
    - 2.5.7.7. Initial determination of 3D structure using tilt experiments (pp. 381-382) | html | pdf |
    - 2.5.7.8. Ab initio 3D structure determination using computational methods (pp. 382-383) | html | pdf |
    - 2.5.7.9. Refinement of a 3D structure (pp. 383-385) | html | pdf |
    - 2.5.7.10. Resolution estimation and analysis of errors in single-particle reconstruction (pp. 385-386) | html | pdf |
    - 2.5.7.11. Analysis of 3D cryo-EM maps (pp. 386-388) | html | pdf |
  - 2.5.8. Direct phase determination in electron crystallography (pp. 388-394) | html | pdf |
    D. L. Dorset
    - 2.5.8.1. Problems with `traditional' phasing techniques (pp. 388-389) | html | pdf |
    - 2.5.8.2. Direct phase determination from electron micrographs (p. 389) | html | pdf |
    - 2.5.8.3. Probabilistic estimate of phase invariant sums (pp. 389-391) | html | pdf |
    - 2.5.8.4. The tangent formula (pp. 391-392) | html | pdf |
    - 2.5.8.5. Density modification (p. 392) | html | pdf |
    - 2.5.8.6. Convolution techniques (p. 392) | html | pdf |
    - 2.5.8.7. Maximum entropy and likelihood (pp. 392-393) | html | pdf |
    - 2.5.8.8. Influence of multiple scattering on direct electron-crystallographic structure analysis (pp. 393-394) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.5.2.1. The variation with accelerating voltage of electrons of (a) the wavelength, λ and (b) the quantity $[\lambda [1 + (h^{2} / m_{0}^{2} c^{2} \lambda^{2})] = \lambda_{\rm c} / \beta]$ (p. 301) | html | pdf |
    - Fig. 2.5.2.2. Diagram representing the critical components of a conventional transmission electron microscope (TEM) and a scanning transmission electron microscope (STEM) (p. 304) | html | pdf |
    - Fig. 2.5.2.3. The functions $[\chi (U)]$ , the phase factor for the transfer function of a lens given by equation (2.5.2.33), and $[\sin \chi (U)]$ for the Scherzer optimum defocus condition (p. 305) | html | pdf |
    - Fig. 2.5.3.1. Four symmetry elements m′, i, 2′ and $[\bar 4]$ of an infinitely extended parallel-sided specimen (p. 309) | html | pdf |
    - Fig. 2.5.3.2. Illustration of symmetries appearing in dark-field patterns (DPs) (p. 310) | html | pdf |
    - Fig. 2.5.3.3. Illustration of symmetries appearing in dark-field patterns (DPs) and a pair of dark-field patterns (±DP) for the combinations of symmetry elements (p. 310) | html | pdf |
    - Fig. 2.5.3.4. Relation between diffraction groups and crystal point groups (p. 312) | html | pdf |
    - Fig. 2.5.3.5. CBED patterns of Si taken with the [111] incidence (p. 314) | html | pdf |
    - Fig. 2.5.3.6. CBED pattern of Si taken with the [100] incidence (p. 314) | html | pdf |
    - Fig. 2.5.3.7. Illustration of symmetries appearing in hexagonal six-beam, square four-beam and rectangular four-beam dark-field patterns expected for all the diffraction groups except for 1, 1_R, 2, 2_R and 21_R (pp. 315-317) | html | pdf |
    - Fig. 2.5.3.8. CBED patterns of FeS₂ taken with the [111] incidence (p. 320) | html | pdf |
    - Fig. 2.5.3.9. CBED patterns of V₃Si taken with the [110] incidence (p. 321) | html | pdf |
    - Fig. 2.5.3.10. Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a b-glide plane and a 2₁ screw axis (p. 322) | html | pdf |
    - Fig. 2.5.3.11. CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS₂ (p. 322) | html | pdf |
    - Fig. 2.5.3.12. CBED pattern of FeS₂ taken with the [110] electron-beam incidence (p. 322) | html | pdf |
    - Fig. 2.5.3.13. Illustration of dynamical extinction lines appearing in HOLZ reflections due to glide planes (p. 344) | html | pdf |
    - Fig. 2.5.3.14. HOLZ CBED pattern of FeS₂ (p. 344) | html | pdf |
    - Fig. 2.5.3.15. CBED patterns of rutile (p. 345) | html | pdf |
    - Fig. 2.5.3.16. CBED patterns of Sm₃Se₄ (p. 346) | html | pdf |
    - Fig. 2.5.3.17. The (3 + 1)-dimensional description of one-dimensionally modulated crystals (p. 347) | html | pdf |
    - Fig. 2.5.3.18. (a) Schematic diffraction pattern from a modulated crystal (p. 347) | html | pdf |
    - Fig. 2.5.3.19. CBED patterns of the incommensurate phase of Sr₂Nb₂O₇ taken at 60 kV (p. 349) | html | pdf |
    - Fig. 2.5.3.20. (a) Mirror symmetry of modulation waves ( $[_{\;1}^m ]$ ) τ₄ = 0 (p. 350) | html | pdf |
    - Fig. 2.5.3.21. (a) Umweganregung paths a, b and c to the 0011 forbidden reflection (p. 350) | html | pdf |
    - Fig. 2.5.3.22. Schematic diffraction pattern at the [001] incidence of Sr₂Nb₂O₇ (p. 351) | html | pdf |
    - Fig. 2.5.3.23. Diffraction pattern of Sr₂Nb₂O₇ taken with [001] incidence at 60 kV (p. 351) | html | pdf |
    - Fig. 2.5.3.24. Three pairs of ZOLZ [(a), (c) and (e)] and HOLZ [(b), (d) and (f)] CBED patterns taken at 60 kV from an area of Al₇₄Mn₂₀Si₆ (p. 352) | html | pdf |
    - Fig. 2.5.3.25. CBED patterns of Al₇₄Mn₂₀Si₆ taken with an electron incidence along the threefold axis (p. 353) | html | pdf |
    - Fig. 2.5.3.26. CBED patterns of metastable Al₇₀Ni₁₅Fe₁₅ taken from a 3 nm diameter area (p. 353) | html | pdf |
    - Fig. 2.5.3.27. CBED patterns of metastable Al₇₀Ni₂₀Fe₁₀ taken from a 3 nm diameter area (p. 355) | html | pdf |
    - Fig. 2.5.4.1. Ewald spheres in reciprocal space (p. 357) | html | pdf |
    - Fig. 2.5.4.2. Triclinic reciprocal lattice (p. 357) | html | pdf |
    - Fig. 2.5.4.3. Formation of ellipses on an electron-diffraction pattern from an oblique texture (p. 358) | html | pdf |
    - Fig. 2.5.5.1. The χ function and two components of the Scherzer phase function sin $[\chi (U)]$ and cos $[\chi (U)]$ (p. 362) | html | pdf |
    - Fig. 2.5.5.2. (a) Diagram of an optical diffractometer (p. 365) | html | pdf |
    - Fig. 2.5.6.1. A three-dimensional object $[\varphi_{3}]$ and its two-dimensional projection $[\varphi_{2}]$ (p. 367) | html | pdf |
    - Fig. 2.5.6.2. The projection sphere and projection $[\varphi _2 ( {{\bf x}_\boldtau } )]$ of $[\varphi_{3}({\bf r})]$ along τ onto the plane $[{\boldtau} \perp {\bf x}]$ (p. 367) | html | pdf |
    - Fig. 2.5.6.3. (a) Formation of a backprojection function; (b) projection synthesis (2.5.6.9) is a superposition of these functions (p. 368) | html | pdf |
    - Fig. 2.5.6.4. Discretization in two dimensions (d = 2) (p. 368) | html | pdf |
    - Fig. 2.5.6.5. Nonuniform distribution of projections (p. 371) | html | pdf |
    - Fig. 2.5.6.6. Kaiser–Bessel window function used in the gridding reconstruction algorithm GDFR (p. 373) | html | pdf |
    - Fig. 2.5.6.7. Partitions of the sampling Fourier space using Voronoi diagrams in direct inversion algorithms (p. 374) | html | pdf |
    - Fig. 2.5.6.8. (a) Plot of correlation coefficients (CCs) calculated between Fourier transforms of the reconstructed structure and the original phantom as a function of the magnitude of the spatial frequency using five reconstruction algorithms (p. 374) | html | pdf |
    - Fig. 2.5.7.1. Typical steps performed in a single-particle cryo-EM structure determination project (p. 377) | html | pdf |
    - Fig. 2.5.7.2. The geometrical constraints of the 2D alignment problem (p. 380) | html | pdf |
    - Fig. 2.5.7.3. Principle of random conical tilt reconstruction (p. 381) | html | pdf |
    - Fig. 2.5.7.4. Schematic of the 3D projection-alignment procedure (p. 384) | html | pdf |
    - Fig. 2.5.7.5. Schematic of 3D projection alignment with CTF correction performed on the level of 3D maps reconstructed from projection images sorted into groups that share similar defocus settings (p. 384) | html | pdf |
    - Fig. 2.5.7.6. A high-resolution cryo-EM map allows backbone tracing (p. 387) | html | pdf |
    - Fig. 2.5.7.7. Hrs (blue) embedded into the membrane (yellow) of an early endosome (p. 387) | html | pdf |
    - Fig. 2.5.8.1. Potential map for diketopiperazine (p. 390) | html | pdf |
  - Tables
    - Table 2.5.2.1. Standard crystallographic and alternative crystallographic sign conventions for electron diffraction (p. 302) | html | pdf |
    - Table 2.5.3.1. Two- and three-dimensional symmetry elements of an infinitely extended parallel-sided specimen (p. 309) | html | pdf |
    - Table 2.5.3.2. Symmetry elements of an infinitely extended parallel-sided specimen and diffraction groups (p. 309) | html | pdf |
    - Table 2.5.3.3. Symmetries of different patterns for diffraction and projection diffraction groups (p. 311) | html | pdf |
    - Table 2.5.3.4. Diffraction groups expected at various crystal orientations for 32 point groups (pp. 313-314) | html | pdf |
    - Table 2.5.3.5. Symmetries of hexagonal six-beam CBED patterns for diffraction groups (p. 318) | html | pdf |
    - Table 2.5.3.6. Symmetries of square four-beam CBED patterns for diffraction groups (p. 319) | html | pdf |
    - Table 2.5.3.7. Symmetries of rectangular four-beam CBED patterns for diffraction groups (p. 319) | html | pdf |
    - Table 2.5.3.8. Dynamical extinction rules for an infinitely extended parallel-sided specimen (p. 323) | html | pdf |
    - Table 2.5.3.9. Dynamical extinction lines appearing in ZOLZ reflections for all crystal space groups except Nos. 1 and 2 (pp. 324-332) | html | pdf |
    - Table 2.5.3.10. Space-group sets indistinguishable by dynamical extinction lines (p. 333) | html | pdf |
    - Table 2.5.3.11. Space-group sets distinguishable by coherent CBED (p. 333) | html | pdf |
    - Table 2.5.3.12. Dynamical extinction lines appearing in HOLZ reflections for crystal space groups that have mirror and glide planes (pp. 335-343) | html | pdf |
    - Table 2.5.3.13. Wavevectors, point- and space-group symbols and CBED symmetries of one-dimensionally modulated crystals (p. 349) | html | pdf |
    - Table 2.5.3.14. Diffraction groups and CBED symmetries for two icosahedral point groups (p. 353) | html | pdf |
    - Table 2.5.3.15. Pentagonal and decagonal point groups constructed by analogy with trigonal and hexagonal point groups (p. 354) | html | pdf |

Dual bases in crystallographic computing
- 3.1. Distances, angles, and their standard uncertainties (pp. 404-409) | html | pdf | zxc chapter contents |
  D. E. Sands
  - 3.1.1. Introduction (p. 404) | html | pdf |
  - 3.1.2. Scalar product (p. 404) | html | pdf |
  - 3.1.3. Length of a vector (p. 404) | html | pdf |
  - 3.1.4. Angle between two vectors (pp. 404-405) | html | pdf |
  - 3.1.5. Vector product (p. 405) | html | pdf |
  - 3.1.6. Permutation tensors (p. 405) | html | pdf |
  - 3.1.7. Components of vector product (p. 405) | html | pdf |
  - 3.1.8. Some vector relationships (pp. 405-406) | html | pdf |
    - 3.1.8.1. Triple vector product (p. 405) | html | pdf |
    - 3.1.8.2. Scalar product of vector products (p. 406) | html | pdf |
    - 3.1.8.3. Vector product of vector products (p. 406) | html | pdf |
  - 3.1.9. Planes (p. 406) | html | pdf |
  - 3.1.10. Variance–covariance matrices (pp. 406-408) | html | pdf |
  - 3.1.11. Mean values (p. 408) | html | pdf |
  - 3.1.12. Computation (pp. 408-409) | html | pdf |
  - References | html | pdf |
- 3.2. The least-squares plane (pp. 410-417) | html | pdf | zxc chapter contents |
  R. E. Marsh and V. Schomaker
  - 3.2.1. Introduction (p. 410) | html | pdf |
  - 3.2.2. Least-squares plane based on uncorrelated, isotropic weights (pp. 410-413) | html | pdf |
    - 3.2.2.1. Error propagation (pp. 411-412) | html | pdf |
    - 3.2.2.2. The standard uncertainty of the distance from an atom to the plane (pp. 412-413) | html | pdf |
  - 3.2.3. The proper least-squares plane, with Gaussian weights (pp. 413-416) | html | pdf |
    - 3.2.3.1. Formulation and solution of the general Gaussian plane (pp. 413-415) | html | pdf |
    - 3.2.3.2. Concluding remarks (pp. 415-416) | html | pdf |
  - Appendix 3.2.1. (pp. 416-417) | html | pdf |
  - References | html | pdf |
- 3.3. Molecular modelling and graphics (pp. 418-448) | html | pdf | zxc chapter contents |
  R. Diamond and L. M. D. Cranswick
  - 3.3.1. Graphics (pp. 418-434) | html | pdf |
    R. Diamond
    - 3.3.1.1. Coordinate systems, notation and standards (pp. 418-419) | html | pdf |
      - 3.3.1.1.1. Cartesian and crystallographic coordinates (p. 418) | html | pdf |
      - 3.3.1.1.2. Homogeneous coordinates (pp. 418-419) | html | pdf |
      - 3.3.1.1.3. Notation (p. 419) | html | pdf |
      - 3.3.1.1.4. Standards (p. 419) | html | pdf |
    - 3.3.1.2. Orthogonal (or rotation) matrices (pp. 419-426) | html | pdf |
      - 3.3.1.2.1. General form (pp. 419-422) | html | pdf |
      - 3.3.1.2.2. Measurement of rotations and strains from coordinates (pp. 422-425) | html | pdf |
      - 3.3.1.2.3. Orthogonalization of impure rotations (p. 425) | html | pdf |
      - 3.3.1.2.4. Eigenvalues and eigenvectors of orthogonal matrices (pp. 425-426) | html | pdf |
    - 3.3.1.3. Projection transformations and spaces (pp. 426-431) | html | pdf |
      - 3.3.1.3.1. Definitions (p. 426) | html | pdf |
      - 3.3.1.3.2. Translation (p. 426) | html | pdf |
      - 3.3.1.3.3. Rotation (p. 426) | html | pdf |
      - 3.3.1.3.4. Scale (p. 426) | html | pdf |
      - 3.3.1.3.5. Windowing and perspective (pp. 426-428) | html | pdf |
      - 3.3.1.3.6. Stereoviews (p. 428) | html | pdf |
      - 3.3.1.3.7. Viewports (pp. 428-429) | html | pdf |
      - 3.3.1.3.8. Compound transformations (pp. 429-430) | html | pdf |
      - 3.3.1.3.9. Inverse transformations (p. 430) | html | pdf |
      - 3.3.1.3.10. The three-axis joystick (pp. 430-431) | html | pdf |
      - 3.3.1.3.11. Other useful rotations (p. 431) | html | pdf |
      - 3.3.1.3.12. Symmetry (p. 431) | html | pdf |
    - 3.3.1.4. Modelling transformations (pp. 431-432) | html | pdf |
      - 3.3.1.4.1. Rotation about a bond (p. 431) | html | pdf |
      - 3.3.1.4.2. Stacked transformations (pp. 431-432) | html | pdf |
    - 3.3.1.5. Drawing techniques (pp. 432-434) | html | pdf |
      - 3.3.1.5.1. Types of hardware (pp. 432-433) | html | pdf |
      - 3.3.1.5.2. Optimization of line drawings (p. 433) | html | pdf |
      - 3.3.1.5.3. Representation of surfaces by lines (p. 433) | html | pdf |
      - 3.3.1.5.4. Representation of surfaces by dots (p. 433) | html | pdf |
      - 3.3.1.5.5. Representation of surfaces by shading (pp. 433-434) | html | pdf |
      - 3.3.1.5.6. Advanced hidden-line and hidden-surface algorithms (p. 434) | html | pdf |
  - 3.3.2. Molecular modelling, problems and approaches (pp. 434-438) | html | pdf |
    R. Diamond
    - 3.3.2.1. Connectivity (p. 435) | html | pdf |
      - 3.3.2.1.1. Connectivity tables (p. 435) | html | pdf |
      - 3.3.2.1.2. Implied connectivity (p. 435) | html | pdf |
    - 3.3.2.2. Modelling methods (pp. 435-438) | html | pdf |
      - 3.3.2.2.1. Methods based on conformational variables (pp. 435-437) | html | pdf |
      - 3.3.2.2.2. Methods based on positional coordinates (p. 437) | html | pdf |
      - 3.3.2.2.3. Approaches to the problem of multiple minima (pp. 437-438) | html | pdf |
  - 3.3.3. Implementations (pp. 438-442) | html | pdf |
    R. Diamond
    - 3.3.3.1. Systems for the display and modification of retrieved data (pp. 438-439) | html | pdf |
      - 3.3.3.1.1. ORTEP (p. 438) | html | pdf |
      - 3.3.3.1.2. Feldmann's system (pp. 438-439) | html | pdf |
      - 3.3.3.1.3. Lesk & Hardman software (p. 439) | html | pdf |
      - 3.3.3.1.4. GRAMPS (p. 439) | html | pdf |
      - 3.3.3.1.5. Takenaka & Sasada's system (p. 439) | html | pdf |
      - 3.3.3.1.6. MIDAS (p. 439) | html | pdf |
      - 3.3.3.1.7. Insight (p. 439) | html | pdf |
      - 3.3.3.1.8. PLUTO (p. 439) | html | pdf |
      - 3.3.3.1.9. MDKINO (p. 439) | html | pdf |
    - 3.3.3.2. Molecular-modelling systems based on electron density (pp. 439-442) | html | pdf |
      - 3.3.3.2.1. CHEMGRAF (pp. 439-440) | html | pdf |
      - 3.3.3.2.2. GRIP (p. 440) | html | pdf |
      - 3.3.3.2.3. Barry, Denson & North's systems (p. 440) | html | pdf |
      - 3.3.3.2.4. MMS-X (p. 440) | html | pdf |
      - 3.3.3.2.5. Texas A&M University system (p. 440) | html | pdf |
      - 3.3.3.2.6. Bilder (pp. 440-441) | html | pdf |
      - 3.3.3.2.7. Frodo (p. 441) | html | pdf |
      - 3.3.3.2.8. Guide (p. 441) | html | pdf |
      - 3.3.3.2.9. HYDRA (p. 441) | html | pdf |
      - 3.3.3.2.10. O (pp. 441-442) | html | pdf |
    - 3.3.3.3. Molecular-modelling systems based on other criteria (p. 442) | html | pdf |
      - 3.3.3.3.1. Molbuild , Rings, PRXBLD and MM2/MMP2 (p. 442) | html | pdf |
      - 3.3.3.3.2. Script (p. 442) | html | pdf |
      - 3.3.3.3.3. CHARMM (p. 442) | html | pdf |
      - 3.3.3.3.4. Commercial systems (p. 442) | html | pdf |
  - 3.3.4. Graphics software for the display of small and medium-sized molecules (pp. 442-445) | html | pdf |
    L. M. D. Cranswick
    - 3.3.4.1. Introduction (pp. 442-443) | html | pdf |
    - 3.3.4.2. Types of crystal structure display and functionality (pp. 443-445) | html | pdf |
      - 3.3.4.2.1. Ball and stick (p. 443) | html | pdf |
      - 3.3.4.2.2. Anisotropic displacement parameters (p. 443) | html | pdf |
      - 3.3.4.2.3. Mean-square displacement amplitude (p. 443) | html | pdf |
      - 3.3.4.2.4. Polyhedral display (pp. 443-444) | html | pdf |
      - 3.3.4.2.5. Cartesian coordinates (p. 445) | html | pdf |
      - 3.3.4.2.6. Comparing or overlaying crystal structures (p. 445) | html | pdf |
      - 3.3.4.2.7. Extended structures and topology analysis (p. 445) | html | pdf |
      - 3.3.4.2.8. Magnetic crystal structure display (p. 445) | html | pdf |
      - 3.3.4.2.9. Incommensurate crystal structures (p. 445) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 3.3.1.1. The relationship between display-space coordinates (X, Y, Z, W) and picture-space coordinates (x, y, z, w) derived from them by the window transformation, U (p. 427) | html | pdf |
    - Fig. 3.3.1.2. Schematic representation of a simple branched-chain molecule with a stationary root and two extremities (p. 432) | html | pdf |
  - Tables
    - Table 3.3.4.1. Functionality of software for crystal structure display (p. 443) | html | pdf |
    - Table 3.3.4.2. Availability of software for crystal structure display (p. 444) | html | pdf |
- 3.4. Accelerated convergence treatment of R⁻ⁿ lattice sums (pp. 449-457) | html | pdf | zxc chapter contents |
  D. E. Williams
  - 3.4.1. Introduction (p. 449) | html | pdf |
  - 3.4.2. Definition and behaviour of the direct-space sum (p. 449) | html | pdf |
  - 3.4.3. Preliminary description of the method (pp. 449-450) | html | pdf |
  - 3.4.4. Preliminary derivation to obtain a formula which accelerates the convergence of an R⁻ⁿ sum over lattice points X(d) (pp. 450-452) | html | pdf |
  - 3.4.5. Extension of the method to a composite lattice (pp. 452-453) | html | pdf |
  - 3.4.6. The case of n = 1 (Coulombic lattice energy) (pp. 453-454) | html | pdf |
  - 3.4.7. The cases of n = 2 and n = 3 (p. 454) | html | pdf |
  - 3.4.8. Derivation of the accelerated convergence formula via the Patterson function (p. 454) | html | pdf |
  - 3.4.9. Evaluation of the incomplete gamma function (pp. 454-455) | html | pdf |
  - 3.4.10. Summation over the asymmetric unit and elimination of intramolecular energy terms (p. 455) | html | pdf |
  - 3.4.11. Reference formulae for particular values of n (pp. 455-456) | html | pdf |
  - 3.4.12. Numerical illustrations (pp. 456-457) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 3.4.2.1. Untreated lattice-sum results for the Coulombic energy (n = 1) of sodium chloride (kJ mol⁻¹, Å); the lattice constant is taken as 5.628 Å (p. 449) | html | pdf |
    - Table 3.4.2.2. Untreated lattice-sum results for the dispersion energy (n = 6) of crystalline benzene (kJ mol⁻¹, Å) (p. 450) | html | pdf |
    - Table 3.4.12.1. Accelerated-convergence results for the Coulombic sum (n = 1) of sodium chloride (kJ mol⁻¹, Å): the direct sum plus the constant term (p. 456) | html | pdf |
    - Table 3.4.12.2. The reciprocal-lattice results (kJ mol⁻¹, Å) for the Coulombic sum (n = 1) of sodium chloride (p. 456) | html | pdf |
    - Table 3.4.12.3. Accelerated-convergence results for the dispersion sum (n = 6) of crystalline benzene (kJ mol⁻¹, Å); the figures shown are the direct-lattice sum plus the two constant terms (p. 456) | html | pdf |
    - Table 3.4.12.4. The reciprocal-lattice results (kJ mol⁻¹, Å) for the dispersion sum (n = 6) of crystalline benzene (p. 456) | html | pdf |
    - Table 3.4.12.5. Approximate time (s) required to evaluate the dispersion sum (n = 6) for crystalline benzene within 0.001 kJ mol⁻¹ truncation error (p. 457) | html | pdf |
- 3.5. Extensions of the Ewald method for Coulomb interactions in crystals (pp. 458-481) | html | pdf | zxc chapter contents |
  T. A. Darden
  - 3.5.1. Introduction (pp. 458-460) | html | pdf |
  - 3.5.2. Lattice sums of point charges (pp. 460-471) | html | pdf |
    - 3.5.2.1. Basic quantities (pp. 460-461) | html | pdf |
    - 3.5.2.2. Ewald sum: first derivation (pp. 461-463) | html | pdf |
    - 3.5.2.3. Ewald sum: more complete derivation (pp. 463-467) | html | pdf |
      - 3.5.2.3.1. The case of p > 3 (pp. 464-465) | html | pdf |
      - 3.5.2.3.2. The case p = 1 (pp. 465-467) | html | pdf |
    - 3.5.2.4. The surface term (p. 467) | html | pdf |
    - 3.5.2.5. The polarization response (pp. 467-470) | html | pdf |
    - 3.5.2.6. Calculating the pressure (pp. 470-471) | html | pdf |
  - 3.5.3. Generalization to Gaussian- and Hermite-based continuous charge distributions (pp. 471-474) | html | pdf |
    - 3.5.3.1. Lattice sums of interacting spherical Gaussians (pp. 471-473) | html | pdf |
    - 3.5.3.2. Higher angular momentum: Hermite Gaussians (pp. 473-474) | html | pdf |
  - 3.5.4. Computational efficiency (pp. 474-480) | html | pdf |
    - 3.5.4.1. The direct sum (pp. 474-475) | html | pdf |
    - 3.5.4.2. Reciprocal-sum speed-ups: the smooth particle mesh Ewald and fast Fourier Poisson methods (pp. 475-480) | html | pdf |
      - 3.5.4.2.1. The smooth PME approach  (pp. 475-478) | html | pdf |
        3.5.4.2.1.1. Properties of B-splines  (pp. 476-477) | html | pdf |
        3.5.4.2.1.2. B-spline approximation to trigonometric functions  (pp. 477-478) | html | pdf |
      - 3.5.4.2.2. The fast Fourier Poisson approach (pp. 478-480) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 3.5.4.1. Timing versus accuracy for smooth PME, FFP and regular Ewald sum reciprocal-space approaches (p. 479) | html | pdf |

Diffuse scattering and related topics
- 4.1. Thermal diffuse scattering of X-rays and neutrons (pp. 484-491) | html | pdf | zxc chapter contents |
  B. T. M. Willis
  - 4.1.1. Introduction (p. 484) | html | pdf |
  - 4.1.2. Dynamics of three-dimensional crystals (pp. 484-487) | html | pdf |
    - 4.1.2.1. Equations of motion (pp. 485-486) | html | pdf |
    - 4.1.2.2. Quantization of normal modes. Phonons (p. 486) | html | pdf |
    - 4.1.2.3. Einstein and Debye models (p. 486) | html | pdf |
    - 4.1.2.4. Molecular crystals (pp. 486-487) | html | pdf |
  - 4.1.3. Scattering of X-rays by thermal vibrations (pp. 487-488) | html | pdf |
  - 4.1.4. Scattering of neutrons by thermal vibrations (pp. 488-489) | html | pdf |
  - 4.1.5. Phonon dispersion relations (pp. 489-490) | html | pdf |
    - 4.1.5.1. Measurement with X-rays (pp. 489-490) | html | pdf |
    - 4.1.5.2. Measurement with neutrons (p. 490) | html | pdf |
    - 4.1.5.3. Interpretation of dispersion relations (p. 490) | html | pdf |
  - 4.1.6. Measurement of elastic constants (pp. 490-491) | html | pdf |
  - References | html | pdf |
- 4.2. Disorder diffuse scattering of X-rays and neutrons (pp. 492-539) | html | pdf | zxc chapter contents |
  F. Frey, H. Boysen and H. Jagodzinski
  - 4.2.1. Introduction (pp. 492-493) | html | pdf |
  - 4.2.2. Basic scattering theory (pp. 493-495) | html | pdf |
    - 4.2.2.1. General (pp. 493-494) | html | pdf |
    - 4.2.2.2. X-ray and neutron scattering (pp. 494-495) | html | pdf |
  - 4.2.3. Qualitative treatment of structural disorder (pp. 495-507) | html | pdf |
    - 4.2.3.1. Basic mathematics (pp. 495-496) | html | pdf |
    - 4.2.3.2. Fourier transforms (pp. 496-497) | html | pdf |
    - 4.2.3.3. General formulation of a disorder problem (p. 497) | html | pdf |
    - 4.2.3.4. General aspects of diffuse scattering (pp. 497-498) | html | pdf |
    - 4.2.3.5. Types of diffuse scattering (pp. 498-507) | html | pdf |
      - 4.2.3.5.1. Substitutional fluctuations, occupational disorder (pp. 498-499) | html | pdf |
      - 4.2.3.5.2. Displacement fluctuations, displacive disorder (p. 499) | html | pdf |
      - 4.2.3.5.3. Clusters and domains (pp. 499-502) | html | pdf |
      - 4.2.3.5.4. One-dimensional disorder (pp. 502-506) | html | pdf |
      - 4.2.3.5.5. Two-dimensional disorder (p. 506) | html | pdf |
      - 4.2.3.5.6. Three-dimensional disorder (pp. 506-507) | html | pdf |
    - 4.2.3.6. Symmetry (p. 507) | html | pdf |
  - 4.2.4. General guidelines for analysing a disorder problem (pp. 507-509) | html | pdf |
  - 4.2.5. Quantitative interpretation (pp. 509-526) | html | pdf |
    - 4.2.5.1. Introduction (p. 509) | html | pdf |
    - 4.2.5.2. Layered structures: one-dimensional disorder (pp. 509-514) | html | pdf |
      - 4.2.5.2.1. Stacking disorder in close-packed structures (pp. 512-514) | html | pdf |
    - 4.2.5.3. Chain-like structures: two-dimensional disorder (pp. 514-518) | html | pdf |
      - 4.2.5.3.1. Randomly distributed collinear chains (pp. 514-516) | html | pdf |
      - 4.2.5.3.2. Disorder within randomly distributed collinear chains (pp. 516-517) | html | pdf |
      - 4.2.5.3.3. Correlations between almost collinear chains (pp. 517-518) | html | pdf |
    - 4.2.5.4. Defects, short-range ordering, clustering: three-dimensional disorder (pp. 518-524) | html | pdf |
      - 4.2.5.4.1. General formulation (elastic diffuse scattering) (pp. 518-520) | html | pdf |
      - 4.2.5.4.2. Random distribution of defects (p. 520) | html | pdf |
      - 4.2.5.4.3. Short-range order in multicomponent systems (pp. 520-521) | html | pdf |
      - 4.2.5.4.4. Displacements: general remarks (pp. 521-522) | html | pdf |
      - 4.2.5.4.5. Distortions in binary systems (pp. 522-524) | html | pdf |
      - 4.2.5.4.6. Comparison with the cluster method (p. 524) | html | pdf |
    - 4.2.5.5. Molecular crystals: orientational disorder (pp. 524-526) | html | pdf |
  - 4.2.6. Disorder diffuse scattering from aperiodic crystals (pp. 526-528) | html | pdf |
    - 4.2.6.1. Incommensurately modulated structures (p. 527) | html | pdf |
    - 4.2.6.2. Composite structures (p. 527) | html | pdf |
    - 4.2.6.3. Quasicrystals (pp. 527-528) | html | pdf |
  - 4.2.7. Computer simulations and modelling (pp. 528-530) | html | pdf |
    - 4.2.7.1. Introduction (p. 528) | html | pdf |
    - 4.2.7.2. Simulation programs (p. 528) | html | pdf |
    - 4.2.7.3. Modelling procedures (pp. 528-530) | html | pdf |
      - 4.2.7.3.1. Molecular dynamics (p. 529) | html | pdf |
      - 4.2.7.3.2. Monte Carlo calculations (pp. 529-530) | html | pdf |
      - 4.2.7.3.3. General remarks (p. 530) | html | pdf |
  - 4.2.8. Experimental techniques and data evaluation (pp. 530-535) | html | pdf |
    - 4.2.8.1. Single-crystal techniques (pp. 530-534) | html | pdf |
    - 4.2.8.2. Powders and polycrystals (pp. 534-535) | html | pdf |
    - 4.2.8.3. Total diffraction pattern (p. 535) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.2.3.1. Model of the two-dimensional distribution of point defects, causing changes in the surroundings (p. 500) | html | pdf |
    - Fig. 4.2.3.2. One-dimensional Patterson functions of various point-defect distributions (p. 501) | html | pdf |
    - Fig. 4.2.3.3. Periodic array of domains consisting of two different atoms, represented by different heights (p. 502) | html | pdf |
    - Fig. 4.2.3.4. Influence of distortions at the boundary of domains and separation into two parts (p. 504) | html | pdf |
    - Fig. 4.2.3.5. Behaviour of p₁₁(m) and p₁₂(m) in mixed crystals (unmixing) (p. 505) | html | pdf |
    - Fig. 4.2.3.6. Decomposition of Fig. 4.2.3.5(a) into (a) a periodic and (b) a rapidly convergent part (p. 505) | html | pdf |
    - Fig. 4.2.3.7. The same distribution (cf. Fig. 4.2.3.5) in the case of superstructure formation (p. 506) | html | pdf |
    - Fig. 4.2.5.1. Construction of the correlation function in the method of overlapping clusters (p. 521) | html | pdf |
    - Fig. 4.2.8.1. Schematic sketch of a diffractometer setting (p. 533) | html | pdf |
    - Fig. 4.2.8.2. Line profiles in powder diffraction for diffuse peaks (full line), continuous streaks (dot–dash lines) and continuous planes (broken lines) (p. 535) | html | pdf |
- 4.3. Diffuse scattering in electron diffraction (pp. 540-546) | html | pdf | zxc chapter contents |
  J. M. Cowley and J. K. Gjønnes
  - 4.3.1. Introduction (pp. 540-541) | html | pdf |
  - 4.3.2. Inelastic scattering (pp. 541-542) | html | pdf |
  - 4.3.3. Kinematical and pseudo-kinematical scattering (p. 542) | html | pdf |
  - 4.3.4. Dynamical scattering: Bragg scattering effects (pp. 542-544) | html | pdf |
  - 4.3.5. Multislice calculations for diffraction and imaging (p. 544) | html | pdf |
  - 4.3.6. Qualitative interpretation of diffuse scattering of electrons (pp. 544-545) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.3.1.1. Comparison between the kinematical inelastic scattering (full line) and elastic scattering (broken) for electrons and X-rays (p. 540) | html | pdf |
    - Fig. 4.3.1.2. Electron-diffraction pattern from a disordered crystal of 17Nb₂O₅·48WO₃ (p. 541) | html | pdf |
- 4.4. Scattering from mesomorphic structures (pp. 547-566) | html | pdf | zxc chapter contents |
  P. S. Pershan
  - 4.4.1. Introduction (pp. 547-549) | html | pdf |
  - 4.4.2. The nematic phase (pp. 549-551) | html | pdf |
  - 4.4.3. Smectic-A and smectic-C phases (pp. 551-554) | html | pdf |
    - 4.4.3.1. Homogeneous smectic-A and smectic-C phases (pp. 551-553) | html | pdf |
    - 4.4.3.2. Modulated smectic-A and smectic-C phases (p. 553) | html | pdf |
    - 4.4.3.3. Surface effects (pp. 553-554) | html | pdf |
  - 4.4.4. Phases with in-plane order (pp. 554-561) | html | pdf |
    - 4.4.4.1. Hexatic phases in two dimensions (pp. 555-556) | html | pdf |
    - 4.4.4.2. Hexatic phases in three dimensions (pp. 556-558) | html | pdf |
      - 4.4.4.2.1. Hexatic-B (p. 556) | html | pdf |
      - 4.4.4.2.2. Smectic-F, smectic-I (pp. 556-558) | html | pdf |
    - 4.4.4.3. Crystalline phases with molecular rotation (pp. 558-560) | html | pdf |
      - 4.4.4.3.1. Crystal-B (pp. 558-560) | html | pdf |
      - 4.4.4.3.2. Crystal-G, crystal-J (p. 560) | html | pdf |
    - 4.4.4.4. Crystalline phases with herringbone packing (pp. 560-561) | html | pdf |
      - 4.4.4.4.1. Crystal-E (p. 560) | html | pdf |
      - 4.4.4.4.2. Crystal-H, crystal-K (pp. 560-561) | html | pdf |
  - 4.4.5. Discotic phases (p. 561) | html | pdf |
  - 4.4.6. Other phases (pp. 561-562) | html | pdf |
    - 4.4.6.1. Phases with intermediate molecular tilt: smectic-L, crystalline-M,N (p. 562) | html | pdf |
    - 4.4.6.2. Nematic to smectic-A phase transition (p. 562) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.4.1.1. Illustration of the progression of order throughout the sequence of mesomorphic phases that are based on `rod-like' molecules (p. 547) | html | pdf |
    - Fig. 4.4.1.2. Schematic illustration of the real-space molecular order and the scattering cross sections in reciprocal space for the: (a) nematic; (b) smectic-A; and (c), (d) smectic-C phases (p. 548) | html | pdf |
    - Fig. 4.4.1.3. Chemical formulae for some of the molecules that form thermotropic liquid crystals (p. 549) | html | pdf |
    - Fig. 4.4.3.1. (a) Schematic illustration of the necessary condition for coupling between order parameters when $[|{\bf q}_{2}|\,\lt\, 2|{\bf q}_{1}|]$ ; $[|{\bf q}| =]$ $[ (|{\bf q}_{2}|^{2} - |{\bf q}_{1}|^{2})^{1/2} = |{\bf q}_{1}| \sin (\alpha)]$ (p. 553) | html | pdf |
    - Fig. 4.4.3.2. Specular reflectivity of ∼8 keV X-rays from the air–liquid interface of the nematic liquid crystal 8OCB 0.05 K above the nematic to smectic-A transition temperature (p. 554) | html | pdf |
    - Fig. 4.4.4.1. (a) Schematic illustration of the geometry and (b) kinematics of X-ray scattering from a freely suspended smectic film (p. 555) | html | pdf |
    - Fig. 4.4.4.2. Typical $[Q_{L}]$ scans from the crystalline-B phases of (a) a free film of 7O.7 (p. 555) | html | pdf |
    - Fig. 4.4.4.3. Scattering intensities in reciprocal space from two-dimensional: (a) liquid; (b) crystal; (c) normal hexatic; and tilted hexatics (p. 556) | html | pdf |
    - Fig. 4.4.4.4. Scattering intensities in reciprocal space from three-dimensional tilted hexatic phases (p. 557) | html | pdf |
    - Fig. 4.4.4.5. The phase diagram for free films of 7O.7 as a function of thickness and temperature (p. 557) | html | pdf |
    - Fig. 4.4.4.6. Location of the Bragg peaks in one 60° section of reciprocal space for the three-dimensional crystalline-B phases observed in thick films of 7O.7 (p. 559) | html | pdf |
    - Fig. 4.4.4.7. (a) The `herringbone' stacking suggested for the crystalline-E phase in which molecular rotation is partially restricted (p. 560) | html | pdf |
    - Fig. 4.4.5.1. Schematic illustration of the molecular stacking for the discotic (a) $[D_{2}]$ and (b) $[D_{1}]$ phases (p. 561) | html | pdf |
  - Tables
    - Table 4.4.1.1. Some of the symmetry properties of the series of three-dimensional phases described in Fig. 4.4.1.1 (p. 547) | html | pdf |
    - Table 4.4.1.2. The symmetry properties of the two-dimensional hexatic and crystalline phases (p. 548) | html | pdf |
    - Table 4.4.2.1. Summary of critical exponents from X-ray scattering studies of the nematic to smectic-A phase transition (p. 551) | html | pdf |
- 4.5. Polymer crystallography (pp. 567-589) | html | pdf | zxc chapter contents |
  R. P. Millane and D. L. Dorset
  - 4.5.1. Overview (p. 567) | html | pdf |
    R. P. Millane and D. L. Dorset
  - 4.5.2. X-ray fibre diffraction analysis (pp. 568-583) | html | pdf |
    R. P. Millane
    - 4.5.2.1. Introduction (p. 568) | html | pdf |
    - 4.5.2.2. Fibre specimens (p. 568) | html | pdf |
    - 4.5.2.3. Diffraction by helical structures (pp. 568-570) | html | pdf |
      - 4.5.2.3.1. Helix symmetry (p. 569) | html | pdf |
      - 4.5.2.3.2. Diffraction by helical structures (pp. 569-570) | html | pdf |
      - 4.5.2.3.3. Approximate helix symmetry (p. 570) | html | pdf |
    - 4.5.2.4. Diffraction by fibres (pp. 570-574) | html | pdf |
      - 4.5.2.4.1. Noncrystalline fibres (p. 570) | html | pdf |
      - 4.5.2.4.2. Polycrystalline fibres (pp. 570-571) | html | pdf |
      - 4.5.2.4.3. Random copolymers (pp. 571-572) | html | pdf |
      - 4.5.2.4.4. Partially crystalline fibres (pp. 572-574) | html | pdf |
    - 4.5.2.5. Processing diffraction data (pp. 574-576) | html | pdf |
    - 4.5.2.6. Structure determination (pp. 576-583) | html | pdf |
      - 4.5.2.6.1. Overview (p. 576) | html | pdf |
      - 4.5.2.6.2. Helix symmetry, cell constants and space-group symmetry (p. 576) | html | pdf |
      - 4.5.2.6.3. Patterson functions (p. 577) | html | pdf |
      - 4.5.2.6.4. Molecular model building (pp. 577-579) | html | pdf |
      - 4.5.2.6.5. Difference Fourier synthesis (p. 579) | html | pdf |
      - 4.5.2.6.6. Multidimensional isomorphous replacement (pp. 579-581) | html | pdf |
      - 4.5.2.6.7. Other techniques (pp. 581-582) | html | pdf |
      - 4.5.2.6.8. Reliability (pp. 582-583) | html | pdf |
  - 4.5.3. Electron crystallography of polymers (pp. 583-586) | html | pdf |
    D. L. Dorset
    - 4.5.3.1. Is polymer electron crystallography possible? (p. 583) | html | pdf |
    - 4.5.3.2. Crystallization and data collection (pp. 583-584) | html | pdf |
    - 4.5.3.3. Crystal structure analysis (p. 584) | html | pdf |
    - 4.5.3.4. Examples of crystal structure analyses (pp. 584-586) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.5.2.1. Fibre diffraction geometry (p. 574) | html | pdf |
    - Fig. 4.5.2.2. Flow chart of the molecular-model-building approach to structure determination (Arnott, 1980) (p. 578) | html | pdf |
    - Fig. 4.5.3.1. Initial potential map for poly-γ-methyl-L-glutamate (plane group pg) found with phases generated by the Sayre equation (p. 585) | html | pdf |
    - Fig. 4.5.3.2. Crystal structures of linear polymers determined from three-dimensional data (p. 586) | html | pdf |
  - Tables
    - Table 4.5.3.1. Structure analysis of poly-γ-methyl-L-glutamate in the β form (p. 585) | html | pdf |
- 4.6. Reciprocal-space images of aperiodic crystals (pp. 590-624) | html | pdf | zxc chapter contents |
  W. Steurer and T. Haibach
  - 4.6.1. Introduction (pp. 590-591) | html | pdf |
  - 4.6.2. The n-dimensional description of aperiodic crystals (pp. 591-598) | html | pdf |
    - 4.6.2.1. Basic concepts (p. 591) | html | pdf |
    - 4.6.2.2. 1D incommensurately modulated structures (pp. 591-593) | html | pdf |
    - 4.6.2.3. 1D composite structures (pp. 593-594) | html | pdf |
    - 4.6.2.4. 1D quasiperiodic structures (pp. 594-597) | html | pdf |
    - 4.6.2.5. 1D structures with fractal atomic surfaces (pp. 597-598) | html | pdf |
  - 4.6.3. Reciprocal-space images (pp. 598-621) | html | pdf |
    - 4.6.3.1. Incommensurately modulated structures (IMSs) (pp. 598-602) | html | pdf |
      - 4.6.3.1.1. Indexing (p. 599) | html | pdf |
      - 4.6.3.1.2. Diffraction symmetry (pp. 599-600) | html | pdf |
      - 4.6.3.1.3. Structure factor (pp. 600-602) | html | pdf |
    - 4.6.3.2. Composite structures (CSs) (pp. 602-603) | html | pdf |
      - 4.6.3.2.1. Indexing (p. 603) | html | pdf |
      - 4.6.3.2.2. Diffraction symmetry (p. 603) | html | pdf |
      - 4.6.3.2.3. Structure factor (p. 603) | html | pdf |
    - 4.6.3.3. Quasiperiodic structures (QSs) (pp. 603-621) | html | pdf |
      - 4.6.3.3.1. 3D structures with 1D quasiperiodic order  (pp. 603-607) | html | pdf |
        4.6.3.3.1.1. Indexing  (pp. 603-604) | html | pdf |
        4.6.3.3.1.2. Diffraction symmetry  (p. 604) | html | pdf |
        4.6.3.3.1.3. Structure factor  (pp. 604-605) | html | pdf |
        4.6.3.3.1.4. Intensity statistics  (pp. 605-606) | html | pdf |
        4.6.3.3.1.5. Relationships between structure factors at symmetry-related points of the Fourier image  (pp. 606-607) | html | pdf |
      - 4.6.3.3.2. Decagonal phases  (pp. 607-613) | html | pdf |
        4.6.3.3.2.1. Indexing  (p. 610) | html | pdf |
        4.6.3.3.2.2. Diffraction symmetry  (pp. 610-611) | html | pdf |
        4.6.3.3.2.3. Structure factor  (pp. 611-612) | html | pdf |
        4.6.3.3.2.4. Intensity statistics  (pp. 612-613) | html | pdf |
        4.6.3.3.2.5. Relationships between structure factors at symmetry-related points of the Fourier image  (p. 613) | html | pdf |
      - 4.6.3.3.3. Icosahedral phases  (pp. 613-621) | html | pdf |
        4.6.3.3.3.1. Indexing  (pp. 616-618) | html | pdf |
        4.6.3.3.3.2. Diffraction symmetry  (p. 618) | html | pdf |
        4.6.3.3.3.3. Structure factor  (p. 619) | html | pdf |
        4.6.3.3.3.4. Intensity statistics  (p. 619) | html | pdf |
        4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image  (pp. 619-621) | html | pdf |
  - 4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals (pp. 621-623) | html | pdf |
    - 4.6.4.1. Data-collection strategies (pp. 621-623) | html | pdf |
    - 4.6.4.2. Commensurability versus incommensurability (p. 623) | html | pdf |
    - 4.6.4.3. Twinning and nanodomain structures (p. 623) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.6.2.1. The combination of a basic structure $[s({\bf r})]$ , with period a, and a sinusoidal modulation function , with amplitude A, period λ and $[t = {\bf q}\cdot {\bf r}]$ , gives a modulated structure (MS) $[s_{m}({\bf r})]$ (p. 591) | html | pdf |
    - Fig. 4.6.2.2. 2D embedding of the sinusoidally modulated structure illustrated in Fig. 4.6.2.1 (p. 592) | html | pdf |
    - Fig. 4.6.2.3. Schematic representation of the 2D reciprocal-space embedding of the 1D sinusoidally modulated structure depicted in Figs. 4.6.2.1 and 4.6.2.2 (p. 593) | html | pdf |
    - Fig. 4.6.2.4. 2D embedding of a 1D composite structure without mutual interaction of the subsystems (p. 593) | html | pdf |
    - Fig. 4.6.2.5. 2D embedding of a 1D composite structure with mutual interaction of the subsystems causing modulations (p. 594) | html | pdf |
    - Fig. 4.6.2.6. Schematic representation of the reciprocal space of the embedded 1D composite structure depicted in Fig. 4.6.2.4 (p. 594) | html | pdf |
    - Fig. 4.6.2.7. Schematic representation of the reciprocal space of the embedded 1D composite structure depicted in Fig. 4.6.2.5 (p. 594) | html | pdf |
    - Fig. 4.6.2.8. 2D embedding of the Fibonacci chain (p. 596) | html | pdf |
    - Fig. 4.6.2.9. Schematic representation of the reciprocal space of the embedded Fibonacci chain depicted in Fig. 4.6.2.8 (p. 596) | html | pdf |
    - Fig. 4.6.2.10. Reciprocal space of the embedded Fibonacci chain as a modulated structure (p. 597) | html | pdf |
    - Fig. 4.6.2.11. 2D direct-space embedding of the Fibonacci chain as a modulated structure (p. 598) | html | pdf |
    - Fig. 4.6.2.12. (a) Three steps in the development of the fractal atomic surface of the squared Fibonacci sequence starting from an initiator and a generator (p. 598) | html | pdf |
    - Fig. 4.6.3.1. Schematic diffraction patterns for IMSs with (a) 1D, (b) 2D and (c) 3D modulation (p. 599) | html | pdf |
    - Fig. 4.6.3.2. The relationships between the coordinates $[x_{1k}, x_{4k}, \bar{x}_{1}, \bar{x}_{4}]$ and the modulation function $[u_{1k}]$ in a special section of the $[(3 + d)\hbox{D}]$ space (p. 601) | html | pdf |
    - Fig. 4.6.3.3. Schematic diffraction patterns for 3D IMSs with (a) 1D harmonic and (b) rectangular density modulation (p. 602) | html | pdf |
    - Fig. 4.6.3.4. The relative structure-factor magnitudes of mth-order satellite reflections for a harmonic displacive modulation are proportional to the values of the mth-order Bessel function $[J_{m}(x)]$ (p. 602) | html | pdf |
    - Fig. 4.6.3.5. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component $[|{\bf H}^{\parallel}|]$ of the diffraction vector (p. 604) | html | pdf |
    - Fig. 4.6.3.6. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component $[|{\bf H}^{\perp}|]$ of the diffraction vector (p. 604) | html | pdf |
    - Fig. 4.6.3.7. The structure factors $[F({\bf H})]$ of the Fibonacci chain decorated with aluminium atoms (U_overall = 0.005 Å²) as a function of the parallel (above) and the perpendicular (below) component of the diffraction vector (p. 605) | html | pdf |
    - Fig. 4.6.3.8. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component $[|{\bf H}^{\parallel}|]$ of the diffraction vector (p. 606) | html | pdf |
    - Fig. 4.6.3.9. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component $[|{\bf H}^{\perp}|]$ of the diffraction vector (p. 606) | html | pdf |
    - Fig. 4.6.3.10. Part … LSLLSLSL … of a Fibonacci sequence $[s({\bf r})]$ before and after scaling by the factor τ (p. 607) | html | pdf |
    - Fig. 4.6.3.11. Scaling operations of the Fibonacci sequence (p. 607) | html | pdf |
    - Fig. 4.6.3.12. Reciprocal basis of the decagonal phase in the 5D description projected upon $[{\bf V}^{\parallel}]$ (above left) and $[{\bf V}^{\perp}]$ (above right) (p. 608) | html | pdf |
    - Fig. 4.6.3.13. A section of a Penrose tiling (thin lines) superposed by its τ-deflated tiling (above, thick lines) and by its $[\tau^{2}]$ -deflated tiling (below, thick lines) (p. 609) | html | pdf |
    - Fig. 4.6.3.14. Scaling symmetry of a pentagram superposed on the Penrose tiling (p. 609) | html | pdf |
    - Fig. 4.6.3.15. Atomic surface of the Penrose tiling in the 5D hypercubic description (p. 610) | html | pdf |
    - Fig. 4.6.3.16. Projection of the 4D hyper-rhombohedral unit cell of the Penrose tiling in the 4D description upon the perpendicular space (p. 610) | html | pdf |
    - Fig. 4.6.3.17. Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) (p. 611) | html | pdf |
    - Fig. 4.6.3.18. Enlarged section of Fig. 4.6.3.17 (p. 611) | html | pdf |
    - Fig. 4.6.3.19. The perpendicular-space diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) (p. 611) | html | pdf |
    - Fig. 4.6.3.20. Enlarged section of Fig. 4.6.3.19 showing the projected 4D reciprocal-lattice unit cell (p. 611) | html | pdf |
    - Fig. 4.6.3.21. Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\parallel}]$ (p. 612) | html | pdf |
    - Fig. 4.6.3.22. Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ (p. 613) | html | pdf |
    - Fig. 4.6.3.23. Radial distribution function of the structure-factor magnitudes $[|F({\bf H})|]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ (p. 613) | html | pdf |
    - Fig. 4.6.3.24. Pentagrammal relationships between scaling symmetry-related positive structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 614) | html | pdf |
    - Fig. 4.6.3.25. Pentagrammal relationships between scaling symmetry-related negative structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 614) | html | pdf |
    - Fig. 4.6.3.26. Pentagrammal relationships between scaling symmetry-related structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 614) | html | pdf |
    - Fig. 4.6.3.27. Pentagrammal relationships between scaling symmetry-related structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in perpendicular space (p. 614) | html | pdf |
    - Fig. 4.6.3.28. Perspective (a) parallel- and (b) perpendicular-space views of the reciprocal basis of the 3D Penrose tiling (p. 615) | html | pdf |
    - Fig. 4.6.3.29. Schematic representation of a rotation in 6D space (p. 615) | html | pdf |
    - Fig. 4.6.3.30. The two unit tiles of the 3D Penrose tiling: a prolate $[[\alpha_{\rm p} = \hbox{arc} \cos (5^{-1/2}) \simeq 63.44^{\circ}]]$ and an oblate $[(\alpha_{\rm o} = 180^{\circ} - \alpha_{\rm p})]$ rhombohedron with equal edge lengths $[a_{\rm r}]$ (p. 616) | html | pdf |
    - Fig. 4.6.3.31. Atomic surface of the 3D Penrose tiling in the 6D hypercubic description (p. 616) | html | pdf |
    - Fig. 4.6.3.32. Perspective parallel-space view of the two alternative reciprocal bases of the 3D Penrose tiling (p. 616) | html | pdf |
    - Fig. 4.6.3.33. Physical-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 617) | html | pdf |
    - Fig. 4.6.3.34. Perpendicular-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 618) | html | pdf |
    - Fig. 4.6.3.35. Radial distribution function of the structure factors $[F({\bf H})]$ of the 3D Penrose tiling (p. 619) | html | pdf |
    - Fig. 4.6.3.36. Parallel-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (p. 620) | html | pdf |
    - Fig. 4.6.3.37. Perpendicular-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (p. 620) | html | pdf |
    - Fig. 4.6.4.1. Simulated diffraction patterns of (a) the $[\simeq 52\;\hbox{\AA}]$ single-crystal approximant of decagonal Al–Co–Ni, (b) the fivefold twinned approximant, and (c) the decagonal phase itself (p. 621) | html | pdf |
    - Fig. 4.6.4.2. Zero-layer X-ray diffraction patterns of decaprismatic Al_73.5C_21.7Ni_4.8 crystals (p. 622) | html | pdf |
  - Tables
    - Table 4.6.2.1. Expansion of the Fibonacci sequence $[B_{n} = \sigma^{n}(\hbox{L})]$ by repeated action of the substitution rule σ: $[\hbox{S} \rightarrow \hbox{L}]$ , $[\hbox{L} \rightarrow \hbox{LS}]$ (p. 595) | html | pdf |
    - Table 4.6.3.1. 3D point groups of order k describing the diffraction symmetry and corresponding 5D decagonal space groups with reflection conditions (see Rabson et al., 1991) (p. 612) | html | pdf |
    - Table 4.6.3.2. 3D point groups of order k describing the diffraction symmetry and corresponding 6D decagonal space groups with reflection conditions (see Levitov & Rhyner, 1988; Rokhsar et al., 1988) (p. 618) | html | pdf |
    - Table 4.6.4.1. Intensity statistics of the Fibonacci chain for a total of 161 322 reflections with $[-1000 \leq h_{i} \leq 1000]$ and $[0 \leq \sin \theta/\lambda \leq 2\;\hbox{\AA}^{-1}]$ (p. 620) | html | pdf |

Dynamical theory and its applications
- 5.1. Dynamical theory of X-ray diffraction (pp. 626-646) | html | pdf | zxc chapter contents |
  A. Authier
  - 5.1.1. Introduction (p. 626) | html | pdf |
  - 5.1.2. Fundamentals of plane-wave dynamical theory (pp. 626-630) | html | pdf |
    - 5.1.2.1. Propagation equation (pp. 626-628) | html | pdf |
    - 5.1.2.2. Wavefields (p. 628) | html | pdf |
    - 5.1.2.3. Boundary conditions at the entrance surface (p. 628) | html | pdf |
    - 5.1.2.4. Fundamental equations of dynamical theory (pp. 628-629) | html | pdf |
    - 5.1.2.5. Dispersion surface (pp. 629-630) | html | pdf |
    - 5.1.2.6. Propagation direction (p. 630) | html | pdf |
  - 5.1.3. Solutions of plane-wave dynamical theory (pp. 630-633) | html | pdf |
    - 5.1.3.1. Departure from Bragg's law of the incident wave (p. 630) | html | pdf |
    - 5.1.3.2. Transmission and reflection geometries (p. 631) | html | pdf |
    - 5.1.3.3. Middle of the reflection domain (p. 631) | html | pdf |
    - 5.1.3.4. Deviation parameter (pp. 631-632) | html | pdf |
    - 5.1.3.5. Pendellösung and extinction distances (pp. 632-633) | html | pdf |
    - 5.1.3.6. Solution of the dynamical theory (p. 633) | html | pdf |
    - 5.1.3.7. Geometrical interpretation of the solution in the zero-absorption case (p. 633) | html | pdf |
      - 5.1.3.7.1. Transmission geometry (p. 633) | html | pdf |
      - 5.1.3.7.2. Reflection geometry (p. 633) | html | pdf |
  - 5.1.4. Standing waves (p. 633) | html | pdf |
  - 5.1.5. Anomalous absorption (pp. 633-634) | html | pdf |
  - 5.1.6. Intensities of plane waves in transmission geometry (pp. 634-638) | html | pdf |
    - 5.1.6.1. Absorption coefficient (p. 634) | html | pdf |
    - 5.1.6.2. Boundary conditions for the amplitudes at the entrance surface – intensities of the reflected and refracted waves (pp. 634-635) | html | pdf |
    - 5.1.6.3. Boundary conditions at the exit surface (p. 635) | html | pdf |
      - 5.1.6.3.1. Wavevectors (p. 635) | html | pdf |
      - 5.1.6.3.2. Amplitudes – Pendellösung (p. 635) | html | pdf |
    - 5.1.6.4. Reflecting power (pp. 635-636) | html | pdf |
    - 5.1.6.5. Integrated intensity (p. 637) | html | pdf |
      - 5.1.6.5.1. Non-absorbing crystals (p. 637) | html | pdf |
      - 5.1.6.5.2. Absorbing crystals (p. 637) | html | pdf |
    - 5.1.6.6. Thin crystals – comparison with geometrical theory (pp. 637-638) | html | pdf |
  - 5.1.7. Intensity of plane waves in reflection geometry (pp. 638-640) | html | pdf |
    - 5.1.7.1. Thick crystals (pp. 638-639) | html | pdf |
      - 5.1.7.1.1. Non-absorbing crystals (p. 638) | html | pdf |
      - 5.1.7.1.2. Absorbing crystals (pp. 638-639) | html | pdf |
    - 5.1.7.2. Thin crystals (pp. 639-640) | html | pdf |
      - 5.1.7.2.1. Non-absorbing crystals (pp. 639-640) | html | pdf |
      - 5.1.7.2.2. Absorbing crystals (p. 640) | html | pdf |
  - 5.1.8. Real waves (pp. 640-642) | html | pdf |
    - 5.1.8.1. Introduction (pp. 640-641) | html | pdf |
    - 5.1.8.2. Borrmann triangle (p. 641) | html | pdf |
    - 5.1.8.3. Spherical-wave Pendellösung (pp. 641-642) | html | pdf |
  - Appendix 5.1.1. Basic equations (pp. 642-644) | html | pdf |
    - A5.1.1.1. Dielectric susceptibility – classical derivation (pp. 642-643) | html | pdf |
    - A5.1.1.2. Maxwell's equations (p. 643) | html | pdf |
    - A5.1.1.3. Propagation equation (pp. 643-644) | html | pdf |
    - A5.1.1.4. Poynting vector (p. 644) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 5.1.2.1. Bragg reflection (p. 627) | html | pdf |
    - Fig. 5.1.2.2. Boundary condition for wavevectors at the entrance surface of the crystal (p. 628) | html | pdf |
    - Fig. 5.1.2.3. Intersection of the dispersion surface with the plane of incidence (p. 629) | html | pdf |
    - Fig. 5.1.2.4. Intersection of the dispersion surface with the plane of incidence shown in greater detail (p. 629) | html | pdf |
    - Fig. 5.1.2.5. Dispersion surface for the two states of polarization (p. 630) | html | pdf |
    - Fig. 5.1.3.1. Departure from Bragg's law of an incident wave (p. 630) | html | pdf |
    - Fig. 5.1.3.2. Transmission, or Laue, geometry (p. 631) | html | pdf |
    - Fig. 5.1.3.3. Reflection, or Bragg, geometry (p. 631) | html | pdf |
    - Fig. 5.1.3.4. Boundary conditions at the entrance surface for transmission geometry (p. 632) | html | pdf |
    - Fig. 5.1.3.5. Boundary conditions at the entrance surface for reflection geometry (p. 632) | html | pdf |
    - Fig. 5.1.6.1. Variation of the effective absorption with the deviation parameter in the transmission case for the 400 reflection of GaAs using Cu Kα radiation (p. 634) | html | pdf |
    - Fig. 5.1.6.2. Variation of the intensities of the reflected and refracted waves in an absorbing crystal for the 220 reflection of Si using Mo Kα radiation, t = 1 mm (μt = 1.42) (p. 635) | html | pdf |
    - Fig. 5.1.6.3. Boundary condition for the wavevectors at the exit surface (p. 636) | html | pdf |
    - Fig. 5.1.6.4. Decomposition of a wavefield into its two components when it reaches the exit surface (p. 636) | html | pdf |
    - Fig. 5.1.6.5. Cross sections of the incident, $[{\bf K}_{{\bf o}}^{(a)}]$ , refracted, $[{\bf K}_{{\bf o}}]$ , and reflected, $[{\bf K}_{{\bf h}}]$ , waves (p. 636) | html | pdf |
    - Fig. 5.1.6.6. Theoretical rocking curves in the transmission case for non-absorbing crystals and for various values of $[t/\Lambda_{\rm L}]$ (p. 637) | html | pdf |
    - Fig. 5.1.6.7. Variations with crystal thickness of the integrated intensity in the transmission case (no absorption) (arbitrary units) (p. 637) | html | pdf |
    - Fig. 5.1.7.1. Theoretical rocking curve in the reflection case for a non-absorbing thick crystal in terms of the deviation parameter (p. 638) | html | pdf |
    - Fig. 5.1.7.2. Theoretical rocking curve in the reflection case for a thick absorbing crystal (p. 639) | html | pdf |
    - Fig. 5.1.7.3. Bragg case: thick crystals (p. 639) | html | pdf |
    - Fig. 5.1.7.4. Bragg case: thin crystals (p. 639) | html | pdf |
    - Fig. 5.1.8.1. Borrmann triangle (p. 641) | html | pdf |
    - Fig. 5.1.8.2. Packet of wavefields of divergence Δα excited in the crystal by an incident wavepacket of angular width $[\delta (\Delta \theta)]$ (p. 641) | html | pdf |
    - Fig. 5.1.8.3. Intensity distribution along the base of the Borrmann triangle (p. 642) | html | pdf |
    - Fig. 5.1.8.4. Interference at the origin of the Pendellösung fringes in the case of an incident spherical wave (p. 642) | html | pdf |
    - Fig. 5.1.8.5. Spherical-wave Pendellösung fringes observed on a wedge-shaped crystal (p. 643) | html | pdf |
- 5.2. Dynamical theory of electron diffraction (pp. 647-653) | html | pdf | zxc chapter contents |
  A. F. Moodie, J. M. Cowley and P. Goodman
  - 5.2.1. Introduction (p. 647) | html | pdf |
  - 5.2.2. The defining equations (p. 647) | html | pdf |
  - 5.2.3. Forward scattering (pp. 647-648) | html | pdf |
  - 5.2.4. Evolution operator (p. 648) | html | pdf |
  - 5.2.5. Projection approximation – real-space solution (p. 648) | html | pdf |
  - 5.2.6. Semi-reciprocal space (pp. 648-649) | html | pdf |
  - 5.2.7. Two-beam approximation (p. 649) | html | pdf |
  - 5.2.8. Eigenvalue approach (pp. 649-650) | html | pdf |
  - 5.2.9. Translational invariance (p. 650) | html | pdf |
  - 5.2.10. Bloch-wave formulations (p. 650) | html | pdf |
  - 5.2.11. Dispersion surfaces (p. 650) | html | pdf |
  - 5.2.12. Multislice (p. 651) | html | pdf |
  - 5.2.13. Born series (pp. 651-652) | html | pdf |
  - 5.2.14. Approximations (p. 652) | html | pdf |
  - References | html | pdf |
- 5.3. Dynamical theory of neutron diffraction (pp. 654-664) | html | pdf | zxc chapter contents |
  M. Schlenker and J.-P. Guigay
  - 5.3.1. Introduction (p. 654) | html | pdf |
  - 5.3.2. Comparison between X-rays and neutrons with spin neglected (pp. 654-655) | html | pdf |
    - 5.3.2.1. The neutron and its interactions (p. 654) | html | pdf |
    - 5.3.2.2. Scattering lengths and refractive index (pp. 654-655) | html | pdf |
    - 5.3.2.3. Absorption (p. 655) | html | pdf |
    - 5.3.2.4. Differences between neutron and X-ray scattering (p. 655) | html | pdf |
    - 5.3.2.5. Translating X-ray dynamical theory into the neutron case (p. 655) | html | pdf |
  - 5.3.3. Neutron spin, and diffraction by perfect magnetic crystals (pp. 655-658) | html | pdf |
    - 5.3.3.1. Polarization of a neutron beam and the Larmor precession in a uniform magnetic field (pp. 655-656) | html | pdf |
    - 5.3.3.2. Magnetic scattering by a single ion having unpaired electrons (pp. 656-657) | html | pdf |
    - 5.3.3.3. Dynamical theory in the case of perfect ferromagnetic or collinear ferrimagnetic crystals (pp. 657-658) | html | pdf |
    - 5.3.3.4. The dynamical theory in the case of perfect collinear antiferromagnetic crystals (p. 658) | html | pdf |
    - 5.3.3.5. The flipping ratio (p. 658) | html | pdf |
  - 5.3.4. Extinction in neutron diffraction (nonmagnetic case) (pp. 658-659) | html | pdf |
  - 5.3.5. Effect of external fields on neutron scattering by perfect crystals (p. 659) | html | pdf |
  - 5.3.6. Experimental tests of the dynamical theory of neutron scattering (pp. 659-660) | html | pdf |
  - 5.3.7. Applications of the dynamical theory of neutron scattering (pp. 660-661) | html | pdf |
    - 5.3.7.1. Neutron optics (p. 660) | html | pdf |
    - 5.3.7.2. Measurement of scattering lengths by Pendellösung effects (p. 660) | html | pdf |
    - 5.3.7.3. Neutron interferometry (pp. 660-661) | html | pdf |
    - 5.3.7.4. Neutron diffraction topography and other imaging methods (p. 661) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 5.3.3.1. Schematic plot of the two-beam dispersion surface in the case of a purely magnetic reflection such that $[{\bf Q}_{{\bf h}} = {\bf Q}_{-{\bf h}} = {\bf Q}_{{\bf 0}}]$ and that the angle between $[{\bf Q}_{{\bf 0}}]$ and $[{\bf Q}_{{\bf h}}]$ is equal to $[\pi /4]$ (p. 657) | html | pdf |

International Tables for CrystallographyVolume B: Reciprocal space

Edited by U. Shmueli

Contents

International Tables for Crystallography
Volume B: Reciprocal space