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