Crystals Statics. Structural Properties. Geometry of lattices. Aug 23, 2018
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1 Crystals Statics. Structural Properties. Geometry of lattices Aug 23, 2018
2 Crystals Why (among all condensed phases - liquids, gases) look at crystals? We can take advantage of the translational symmetry, as well as point group symmetries, greatly simplifying description, and getting insights into material properties Electronic, optical, thermal, transport, and other properties of solids are best expressed in crystals
3 A bit of history (before the discovery of X-ray diraction) Studies of symmetry (not particularly in crystals) go back many centuries (stacking of cannon balls, shapes of snowakes) René Just Haüy - French mineralogist, called Father of modern crystallography By cleaving crystals and studying angles between faces, in a rst scientic approach, Haüy surmised they consisted of identical repeating units (1784) Studies of crystal symmetry developed throughout 19th century, introducing point groups, Bravais lattices (1848), and space groups The 230 space groups: E. S. Fedorov (1891), A. Schoenies (1891), and W. Barlow (1894)
4 A bit of history (before the discovery of X-ray diraction) Lord Kelvin The molecular tactics of a crystal (1894)
5 A bit of history rst diraction of X-rays by crystal Wavelength of X-rays was roughly known at the time...lattice constants are ca. 10 times greater than the conjectured wavelengths of the X-rays. (Friedrich, Knipping and Laue, 1912) For the rst time, atomic dimensions of the crystal lattice were known
6 Crystal structure = Lattice + Basis If atoms at a certain place found lowest energy pattern, they will likely do same elsewhere This leads to the translational symmetry of the crystal, carrying structure into itself, T = n 1 a 1 + n 2 a 2 + n 3 a 3 n 1, n 2, n 3 - integers, r = r + T, and the vectors a 1, a 2, a 3 are said to generate the lattice Real crystal is a single or a few atoms (basis) propagated, repeated over Bravais lattice Crystal structure = Lattice + Basis Label basis atoms within the unit cell with integer index j Their positions are: r j = x j a 1 + y j a 2 + z j a 3 xj, y j, z j - fractional (crystal) coordinates of atoms (as opposed to Cartesian)
7 Bravais lattices Bravais (1848) - long before any knowledge about atoms showed that in three-dimensional space only 14 dierent lattices (point-systems), are possible These Bravais lattices are just sets of mathematical points, placed according to T = n 1 a 1 + n 2 a 2 + n 3 a 3, no material content yet By another denition, a Bravais lattice is an innite array of discrete points that appears exactly same, from whichever point the array is viewed (if a Maxwell demon were to see this array from any of the points, he would see exactly the same array from each point) In 1D, there is only 1 possible Bravais lattice, in 2D there are 5, and in 3D there are 14
8 Idealizations used Crystal structure = Lattice + Basis No surface, L (crystal dimension) a (interatomic distance) If N total is the total number of atoms in a crystal, surface area S N 2/3, while volume V N total total Number of atoms near the surface is relatively small when N total is large No thermal displacements (mean probabilities, not instant positions) No defects of structure
9 Primitive and conventional lattices Set of smallest a's dening cell with smallest volume and representing every lattice point are called primitive translation vectors Their choice is not unique, but the volume is invariant, smallest, containing 1 lattice point per primitive cell V = a 1 a 2 a 3 Check: choose a 2, a 3 and as the base of the parallelepiped, then its base area is A = a 2 ( a 3 sin () a 2 ˆ, a 3 ) = a 2 a 3, and the volume is V = a 1 A cos a 1 ˆ, A = a 1 A Nonprimitive lattice (with larger volume and containing more than 1 lattice point) representing the symmetry of the crystal is also often used
10 Dierent choices of primitive lattices Choice of the primitive lattice is not unique If a 1, a 2, a 3 and a 1, a 2, a 3 are two dierent primitive lattices, then a i = 3 k=1 α ika k where α ik are integers Conversely, a i = 3 k=1 β ika k with integer β ik Then, determinants det α ik and det β ik are reciprocals of each other and also integers, hence they are both either +1 or 1 We thus have det α ik = ±1 as a necessary and sucient condition for the lattice a 1, a 2, a 3 to be primitive
11 Primitive unit cell One choice for the points of the cell is r = x 1 a 1 + x 2 a 2 + x 3 a 3, with x i between 0 and 1. In this case, a 1, a 2, a 3 are the edges of the cell The disadvantage of this choice is that the cell is not showing the full symmetry of the Bravais lattice Another special choice is a so-called Wigner-Seitz unit cell - region of space around a lattice point that is closer to that point than to any other lattice point Has all the symmetries of the Bravais lattice Yet another choice to reect lattice symmetry is to use nonprimitive cell For disordered materials, Wigner-Seitz cell is generalized (same denition) to Voronoi polyhedra, a set of space-lling cells
12 Space groups Besides translation group T, a crystal can be invariant w.r.t. to a group of other symmetry operations (rotations, reections, inversions) that leave at least one point at the same place (point group) Group must satisfy certain rules (product of operations also member of the group, is associative, there is unique unit element I, and unique inverse element A 1 for each element A such that AA 1 = A 1 A = I) Only 2-, 3-, 4-, and 6-fold rotations are possible in periodic structures dividing (together with other elements) 14 3D Bravais lattices into 7 crystal systems Combined with translations, point group operations form 230 space groups (73 symmorphic, i.e. those where point group operations and translations are separable and 157 non-symmorphic, with screw rotations and glide reections)
13 Examples of crystal structures Face-centered cubic (fcc) structure Primitive lattice: a 1 = (1/2) ay + (1/2) az a 2 = (1/2) ax + (1/2) az a 3 = (1/2) ax + (1/2) ay Basis: B 1 = 0 (Cu) Other elements with fcc structure: Al, Ni, Sr, Rh, Pd, Ag, Ce, Tb, Ir, Pt, Au, Pb, Th, also inert gases: Ne, Ar, Kr, Xe Related is hexagonal close packed (hcp) structure - atomic planes stacked as ABABAB..., as opposed to ABCABCABC... in fcc Elements with hcp structure: Mg, Be, Sc, Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Gd, Tb, Dy, Ho, Er, Tm, Lu, Hf, Re, Os, Tl
14 Examples of crystal structures Body-centered cubic Primitive lattice: a 1 = (1/2) ax + (1/2) ay + (1/2) az a 2 = + (1/2) ax (1/2) ay + (1/2) az a 3 = + (1/2) ax + (1/2) ay (1/2) az Basis: B 1 = 0 (W) Other elements with bcc structure: Li (at room temp.), Na, K, V, Cr, Fe, Rb, Nb, Mo, Cs, Ba, Eu, Ta
15 Examples of crystal structure Sodium chloride Primitive lattice: a 1 = (1/2) ay + (1/2) az a 2 = (1/2) ax + (1/2) az a 3 = (1/2) ax + (1/2) ay Basis: B 1 = 0 B 2 = (1/2) a 1 + (1/2) a 2 + (1/2) a 3 (Na) (Cl)
16 Examples of crystal structures Diamond Primitive lattice: a 1 = (1/2) ay + (1/2) az a 2 = (1/2) ax + (1/2) az a 3 = (1/2) ax + (1/2) ay Basis: B1 = (1/8) a1 (1/8) a2 (1/8) a3 (C) B 2 = + (1/8) a 1 + (1/8) a 2 + (1/8) a 3 (C) Other elements with diamond structure: Si, Ge, α-sn (gray)
17 Examples of crystal structures Zincblende Primitive lattice: a 1 = (1/2) ay + (1/2) az a 2 = (1/2) ax + (1/2) az a 3 = (1/2) ax + (1/2) ay Basis: B 1 = 0 B 2 = + (1/4) a 1 + (1/4) a 2 + (1/4) a 3 (Zn) (S)
18 Examples of crystal structures Graphite Primitive lattice: a 1 = (1/2) ax ( 3/2 ) ay a 2 = (1/2) ax + ( 3/2 ) ay a 3 = cz Basis: B 1 = (1/4) a 3 B 2 = (3/4) a 3 B 3 = (1/3) a 1 + (2/3) a 2 + (1/4) a 3 B 4 = (2/3) a 1 + (1/3) a 2 + (3/4) a 3 (C) (C) (C) (C)
19 Examples of crystal structures Co 7 W 6 alloy in the temperature range K 13 atoms in the rhombohedral primitive cell (a 1 = a 2 = a 3, a 1, ˆa 2 = a 1, ˆa 3 = a 2, ˆa 3 90 ) Conventional hexagonal cell is 3 the primitive, with 39 atoms
20 Index system for crystal planes and directions Plane dened by 3 points - take points of intersection with crystal axes - 3 integers (if plane to axis, assume ) Take reciprocals, reduce to smallest integers with same ratio, these are indices labeling the plane Indices (hkl), e.g. (111), for intercept on the negative side of axis, use a minus sign above, e.g. (1 11) Use curly braces for symmetry equivalent planes, e.g. {111} The indices [uvw] dening a direction in a crystal are the set of smallest integers with the ratio of the components of the vector in this direction (in crystal coordinates) For example, the a 1 axis is in the [100] direction, and the a 2 is [0 10] direction All directions equivalent by symmetry are designated by the angular brackets, uvw In cubic crystals, the direction [hkl] is perpendicular to a plane (hkl) with same indices (prove it?)
21 Index system for crystal planes and directions Example: A plane intersecting the a 1, a 2, and a 3, axes at points 7, 7, and 3 (in units of the corresponding axis length), respectively Reciprocals are 1/7, 1/7, and 1/3 The smallest integers with same ratio as reciprocals are 3, 3, and 7 This set of planes is thus (337)
22 Direct imaging of atomic structure Transmission electron microscopy (TEM), scanning electron microscopy (SEM) Use wave properties of electrons and take advantage of their much shorter wavelength than visible light (e.g., 5 pm at 50 kev kinetic energy) to image samples with a much better spatial resolution than the light-optical microscope Scanning tunneling microscopy (STM) and atomic force microscopy (AFM) Produce information about electronic density (STM) or interaction force (AFM) at dierent points in space near the surface with atomic-level resolution
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