Post on 18-Jun-2020
Crystals Statics. Structural Properties. Geometryof lattices
Aug 23, 2018
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
A bit of history (before the discovery of X-ray di�raction)
▶ Studies of symmetry (not particularly in crystals) go back many
centuries (stacking of cannon balls, shapes of snow�akes)
▶ René Just Haüy - French mineralogist, called �Father of
modern crystallography�
▶ By cleaving crystals and studying angles between faces, in a
�rst scienti�c 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. Schoen�ies
(1891), and W. Barlow (1894)
A bit of history (before the discovery of X-ray di�raction)
▶ Lord Kelvin �The molecular tactics of a crystal� (1894)
A bit of history
▶ 1912 - �rst di�raction 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
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 = n1a1 + n2a2 + n3a3
▶ n1, n2, n3 - integers, r′ = r+T, and the vectors a1,a2,a3 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: rj = xja1 + yja2 + zja3▶ xj , yj , zj - fractional (crystal) coordinates of atoms (as
opposed to Cartesian)
Bravais lattices
▶ Bravais (1848) - long before any knowledge about atoms
showed that in three-dimensional space only 14 di�erent
lattices (point-systems), are possible
▶ These Bravais lattices are just sets of mathematical points,
placed according to T = n1a1 + n2a2 + n3a3, no �material
content� yet
▶ By another de�nition, a Bravais lattice is an in�nite 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
Idealizations used
▶ Crystal structure = Lattice + Basis
▶ No surface, L (crystal dimension) ≫ a (interatomic distance)
▶ If Ntotal is the total number of atoms in a crystal, surface area
S ∼ N2/3total
, while volume V ∼ Ntotal
▶ Number of atoms near the surface is relatively small when
Ntotal is large
▶ No thermal displacements (mean probabilities, not instant
positions)
▶ No defects of structure
Primitive and conventional lattices
▶ Set of �smallest� a's de�ning 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 = |a1 · a2 × a3|▶ Check: choose a2,a3 and as the base of the parallelepiped,
then its base area is A = a2a3 sin ( ˆa2,a3) = |a2 × a3|, andthe volume is V = a1A cos
(ˆa1,A
)= |a1 ·A|
▶ Nonprimitive lattice (with larger volume and containing more
than 1 lattice point) representing the symmetry of the crystal
is also often used
Di�erent choices of primitive lattices
▶ Choice of the primitive lattice is not unique
▶ If a1,a2,a3 and a′1,a′2,a
′3 are two di�erent primitive lattices,
then a′i =∑3
k=1 αikak where αik are integers
▶ Conversely, ai =∑3
k=1 βika′k with integer βik
▶ Then, determinants det |αik| and det |βik| are reciprocals ofeach other and also integers, hence they are both either +1 or
−1
▶ We thus have det |αik| = ±1 as a necessary and su�cient
condition for the lattice a′1,a′2,a
′3 to be primitive
Primitive unit cell
▶ One choice for the points of the cell is
r = x1a1 + x2a2 + x3a3, with xi between 0 and 1. In this
case, a1,a2,a3 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 re�ect lattice symmetry is to use
nonprimitive cell
▶ For disordered materials, Wigner-Seitz cell is generalized (same
de�nition) to Voronoi polyhedra, a set of space-�lling cells
Space groups
▶ Besides translation group T, a crystal can be invariant w.r.t.
to a group of other symmetry operations (rotations,
re�ections, 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−1A = 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 re�ections)
Examples of crystal structures▶ Face-centered cubic (fcc) structure
▶ Primitive lattice:
▶a1 = (1/2) aY + (1/2) aZa2 = (1/2) aX+ (1/2) aZa3 = (1/2) aX+ (1/2) aY
▶ Basis:▶ B1 = 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 - atomicplanes 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
Examples of crystal structures
▶ Body-centered cubic▶ Primitive lattice:
▶a1 = − (1/2) aX+ (1/2) aY + (1/2) aZa2 = +(1/2) aX− (1/2) aY + (1/2) aZa3 = +(1/2) aX+ (1/2) aY − (1/2) aZ
▶ Basis:▶ B1 = 0 (W)
▶ Other elements with bcc structure: Li (at room temp.), Na, K,V, Cr, Fe, Rb, Nb, Mo, Cs, Ba, Eu, Ta
Examples of crystal structure
▶ Sodium chloride▶ Primitive lattice:
▶a1 = (1/2) aY + (1/2) aZa2 = (1/2) aX+ (1/2) aZa3 = (1/2) aX+ (1/2) aY
▶ Basis:
▶ B1 = 0 (Na)B2 = (1/2)a1 + (1/2)a2 + (1/2)a3 (Cl)
Examples of crystal structures
▶ Diamond▶ Primitive lattice:
▶a1 = (1/2) aY + (1/2) aZa2 = (1/2) aX+ (1/2) aZa3 = (1/2) aX+ (1/2) aY
▶ Basis:
▶ B1 = − (1/8)a1 − (1/8)a2 − (1/8)a3 (C)B2 = +(1/8)a1 + (1/8)a2 + (1/8)a3 (C)
▶ Other elements with diamond structure: Si, Ge, α-Sn (gray)
Examples of crystal structures
▶ Zincblende▶ Primitive lattice:
▶a1 = (1/2) aY + (1/2) aZa2 = (1/2) aX+ (1/2) aZa3 = (1/2) aX+ (1/2) aY
▶ Basis:
▶ B1 = 0 (Zn)B2 = +(1/4)a1 + (1/4)a2 + (1/4)a3 (S)
Examples of crystal structures
▶ Graphite▶ Primitive lattice:
▶a1 = (1/2) aX−
(√3/2
)aY
a2 = (1/2) aX+(√
3/2)aY
a3 = cZ
▶ Basis:
▶
B1 = (1/4)a3 (C)B2 = (3/4)a3 (C)
B3 = (1/3)a1 + (2/3)a2 + (1/4)a3 (C)B4 = (2/3)a1 + (1/3)a2 + (3/4)a3 (C)
Examples of crystal structures
▶ Co7W6 alloy in the temperature range 500�1950 K
▶ 13 atoms in the rhombohedral primitive cell (a1 = a2 = a3,ˆa1,a2 = ˆa1,a3 = ˆa2,a3 ̸= 90◦)
▶ Conventional hexagonal cell is 3× the primitive, with 39 atoms
Index system for crystal planes and directions
▶ Plane de�ned 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. (11̄1)
▶ Use curly braces for symmetry equivalent planes, e.g. {111}▶ The indices [uvw] de�ning 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 a1 axis is in the [100] direction, and the −a2is [01̄0] 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?)
Index system for crystal planes and directions
▶ Example:▶ A plane intersecting the a1, a2, and a3, 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)
Direct imaging of atomic structure
▶ Transmission electron microscopy (TEM), scanning electronmicroscopy (SEM)
▶ Use wave properties of electrons and take advantage of theirmuch shorter wavelength than visible light (e.g., 5 pm at 50keV kinetic energy) to image samples with a much betterspatial resolution than the light-optical microscope
▶ Scanning tunneling microscopy (STM) and atomic forcemicroscopy (AFM)
▶ Produce information about electronic density (STM) orinteraction force (AFM) at di�erent points in space near thesurface with atomic-level resolution