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(Adapted from: James A. Kaduk IIT Adjunct Faculty, Chemistry). Lattices and Symmetry Scattering and Diffraction (Physics). Bragg’s Law. Fundamental statement of the way waves scatter off of planes of scatterers n l = 2 d sin q Rearranging, (1/2 d ) = (1/ n ) sin q / l
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(Adapted from: James A. Kaduk IIT Adjunct Faculty, Chemistry) Lattices and SymmetryScattering and Diffraction (Physics)
Bragg’s Law Fundamental statement of the way waves scatter off of planes of scatterers nl = 2dsinq Rearranging, (1/2d) = (1/n)sinq/l We can formulate this so that the n disappears, so that (1/2d) = sinq/l
Bragg’s Law V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 148 (2003)
Optical Diffraction PSSC Physics, Figure 18-A, p. 202-203 (1965)
Optical Diffraction PSSC Physics, Figure 18-B, p. 202-203 (1965)
Optical Diffraction D. Halliday and R. Resnick, Physics, p. 1124 (1962)
Scattering by One Electron Oscillation direction of the electron X-ray beam Electron Electric vector of The incident beam
Electromagnetic Waves E • E(t=0;z) = A cos2p(z/l) A A z
E Z new wave Consider a new wave displaced by a distance Z from the original wave:Z corresponds to a phase shift2π(Z/λ) = α z original wave
Scattering by Two Electrons p q s0 2 • Let the magnitudes of s0 and s = 1/λ • Diffracted beams 1 and 2 have the same magnitude, but differ in phase because of the path difference p + q r 1 s 2 1
Phase difference of wave 2 with respect to wave 1 • 2pr•(s-s0)/l = 2pr•S/ l where S = (s-s0)
The Atomic Scattering Factor • f(S) = ∫ r(r)exp(2pir•S)d3r= 2 ∫ r(r)cos(2pr•S)d3r
Atomic Scattering Factors V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 213 (2003)
Scattering from a Row of Atoms M. J. Buerger, X-ray Crystallography, Fig. 15, p. 33 (1942)
Scattering by a Plane of Atoms M. J. Buerger, X-ray Crystallography, Fig. 16, p. 34 (1942)
Scattering by a Unit Cell J. Drenth, Principles of Protein X-ray Crystallography, Fig. 4.12 p. 80 (1999)
Scattering by a Unit Cell • fj(S) = fjexp(2pirj•S) for each atom j in the unit cell • For the entire ensemble of atoms, • F(S) = ∑j=1nfjexp(2pirj•S)
The crystallographer’s world view Reality can be more complex!
Molecules pack together in a regular pattern to form a crystal. There are two aspects to this pattern: Periodicity Symmetry First consider the periodicity…
To describe the periodicity, we superimpose (mentally) on the crystal structure a lattice. A lattice is a regular array of geometrical points, each of which has the same environment (they are all equivalent).
A unit cell of a lattice (or crystal) is a volume which can describe the lattice using only translations. In 3 dimensions (for crystallographers), this volume is a parallelepiped. Such a volume can be defined by six numbers – the lengths of the three sides, and the angles between them – or three basis vectors.
Descriptions of the Unit Cell a, b, c, , , a, b, cx1a + x2b + x3c, 0 xn< 1lattice points = ha + kb + lc, hkl integersdomain of influence – Dirichlet domain, Voronoi domain, Wigner-Seitz cell, Brillouin zone
The Reduced Cell • 3 shortest non-coplanar translations • Main Conditions (shortest vectors) • Special Conditions (unique) • May not exhibit the true symmetry
Point Symmetry Elements • A point symmetry operation does not alter at least one point upon which it operates • Rotation axes • Mirror planes • Rotation-inversion axes (rotation-reflection) • Center Screw axes and glide planes are not point symmetry elements!
Symmetry Operations • A proper symmetry operation does not invert the handedness of a chiral object • Rotation • Screw axis • Translation • An improper symmetry operation inverts the handedness of a chiral object • Reflection • Inversion • Glide plane • Rotation-inversion
Not all combinations of symmetry elements are possible. In addition, some point symmetry elements are not possible if there is to be translational symmetry as well. There are only 32 crystallographic point groups consistent with periodicity in three dimensions.
