Definitions of Basis, Lattice, and Crystals

 

Basis

A crystal is formed by the periodic repetition of a group of atoms in all directions. That group of atoms is called the basis.

 

  • The basis of a crystal can be one or more atoms.
  • An ideal crystal consists of infinite repetitions of the basis.

 

Lattice

The infinite array of mathematical points which (describes how the basis is repeated) forms a periodic spatial arrangement is called the lattice.

  • The lattice looks identical from whichever points you view the array.
  • Note that lattice is not a crystal.

 

  • In three dimensions, the lattice can be identified by three independent vectors \(\boldsymbol{a}_1\), \(\boldsymbol{a}_2\) and \(\boldsymbol{a}_3\).
  • The position of each point \(\boldsymbol{R}\) can be written as a linear combination of these vectors, \[\boldsymbol{R}=n_1 \boldsymbol{a}_1+n_2 \boldsymbol{a}_2+n_3 \boldsymbol{a}_3 \tag{1}\] where \(n_1\), \(n_2\), and \(n_3\) are integers.
  • The vectors that can generate or span the lattice are not unique. For example, see the following figure.
Different choices for vectors spanning the lattice

 

  • Recall that the volume of a parallelepiped with axes \(\boldsymbol{a}_1\) , \(\boldsymbol{a}_2\) and \(\boldsymbol{a}_3\) is given by \(|\boldsymbol{a}_1\cdot(\boldsymbol{a}_2\times \boldsymbol{a}_3)|\) .

 

Primitive Cells vs. Unit Cells

Primitive Cell

Of the vectors satisfying Eq. (1), those that form a parallelepiped with the smallest volume, are called primitive translation vectors, and the parallelepiped they form is known as the primitive cell.

 

Of the vectors satisfying Eq. (1), those that form a parallelepiped with the smallest volume, are called primitive translation vectors, and the parallelepiped they form is known as the primitive cell.

 

Unit Cell

To better demonstrate the symmetry of the entire lattice, sometimes non-primitive translation vectors are used to specify the lattice. In this case, the parallelepiped is called the unit cell.

  • The unit cell may or may not be identical to the primitive cell.
Primitive cell is shown in gray

 

Unit cell of a face-centered cubic structure

 

Primitive unit cell of a face-centered cubic structure. As drawn, the primitive translation vectors are \[\boldsymbol{a}_1=\frac{a}{2}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}})\quad \boldsymbol{a}_2=\frac{a}{2}(\hat{\boldsymbol{y}}+\hat{\boldsymbol{z}})\quad \boldsymbol{a}_3=\frac{a}{2}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{z}})\] 

Lattice Parameters and Packing Fraction

Lattice Parameters

Lattice Parameters
The length of axes of the unit cell (called lattice constants \(a\), \(b\), and \(c\)) and the angle between the axes (\(\alpha\), \(\beta\) and \(\gamma\)) specify the unit cell. The lattice constants and the three angles between them are termed lattice parameters. See the following figure.

 

Packing Fraction

  • We try to pack \(N\) hard spheres (cannot deform)
  • The total volume of the spheres is \[V_s=N \frac{4}{3}\pi R^3\]
  • The volume these spheres occupy \(V>V_S\) (there are spacing)\[{\rm Packing\ Fraction}=\frac{N \frac{4}{3}\pi R^3}{V}\]

 

Bravais lattices

  • Auguste Bravais, a French physicist, identified 14 distinct lattices in three dimensions.
  • There are 5 Bravais lattices in two dimensions (shown below)
  • In crystallography, all lattices are traditionally called Bravais lattices or translation lattices.

 

Five Bravais Lattices in Two Dimensions

Oblique Rectangular Centerd Rectangular
Square Hexgonal (Rhombic)  

 

 

Seven Crystal Systems

 

Isometric (or Cubic)

$a=b=c$

$\alpha=\beta=\gamma=90^\circ$

Tetragonal

$a=b\ne c$

$\alpha = \beta = \gamma = 90^\circ$

Orthorhombic

$a\ne b\ne c$

$\alpha = \beta = \gamma = 90^\circ$

Hexagonal

$a=b\ne c$

$\alpha = \beta = 90^\circ, \gamma = 120^\circ$

 

Triclinic

$a\ne b\ne c$

$\alpha \ne \beta \ne \gamma$

Monoclinic

$a\ne b\ne c$

$\alpha = \beta = 90^\circ \ne \gamma$

Rhombohedral (or Trigonal)

$a=b=c$

$\alpha = \beta = \gamma < 120^\circ, \ne 90^\circ$

 

Cubic Lattices

Primitive Body-Centered Face-Centered

 

Simple Cubic (SC) Crystal

  • Location of all lattice points \[\boldsymbol{R}=a(u\, \hat{\boldsymbol{x}}+v\, \hat{\boldsymbol{y}}+w\, \hat{\boldsymbol{z}})=a[u\ v\ w]\]
  • Polonium (Po) is the only metal that forms a simple cubic unit cell.
  • Each lattice point is shared by 8 neighboring units
  • average volume occupied by each lattice point\(=\)
    average volume occupied by each atom in SC\(=\Omega_{SC}=\frac{a^3}{8\times\frac{1}{8}}=a^3\)

 

Body-Centered Cubic (BCC) Crystal

  • Location of all lattice points \[\begin{aligned} \boldsymbol{R}&=a(u\, \hat{\boldsymbol{x}}+v\, \hat{\boldsymbol{y}}+w\, \hat{\boldsymbol{z}})=a[u\ v\ w]\\ \boldsymbol{R}&=a((u+0.5)\, \hat{\boldsymbol{x}}+(v+0.5)\, \hat{\boldsymbol{y}}+(w+0.5)\, \hat{\boldsymbol{z}})=a[u+0.5\ v+0.5\ w+0.5]\end{aligned}\]
  • average volume occupied by each lattice point\(=\)
    average volume occupied by each atom in BCC\(=\Omega_{BCC}=\frac{a^3}{8\times\frac{1}{8}+1}=\frac{a^3}{2}\)
  • Nearest neighbor distance = \(d=2r=a\frac{\sqrt{3}}{2}\)
  • \[\text{Packing fraction}=\frac{2\times\frac{4\pi}{3}\times(\frac{a\sqrt{3}}{4})^3}{a^3}=\frac{\pi\sqrt{3}}{8}\approx 0.68\]

The primitive translation vectors are: \[\boldsymbol{a}_1=\frac{a}{2}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}}-\hat{\boldsymbol{z}})\] \[\boldsymbol{a}_2=\frac{a}{2}(-\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}}+\hat{\boldsymbol{z}})\] \[\boldsymbol{a}_3=\frac{a}{2}(\hat{\boldsymbol{x}}-\hat{\boldsymbol{y}}+\hat{\boldsymbol{z}})\]

  • The primitive unit cell is a rhombohedron of edge \(a\frac{\sqrt{3}}{2}\)
  • The angle between adjacent edges is \(109^\circ 28’\)

  • Number of nearest neighbors = 8
  • Number of second nearest neighbors = 6
  • Second nearest neighbor distance = \(a\)

Some Elements with BCC structure

Barium Ba Chromium Cr
Cesium Cs \(\alpha-\)Iron \(\alpha-\)Fe
Potassium K Lithium Li
Molybdenum Mo Sodium Na
Niobium Nb Rubidium Rb
Tantalum Ta Titanium Ti
Vanadium V Tungsten W

 

Face-Centered Cubic (FCC) Crystal

 

  • Location of all lattice points \[\begin{aligned} \boldsymbol{R}&=a[u\ v\ w]\\ \boldsymbol{R}&=a[u+0.5\ v\ w]\\ \boldsymbol{R}&=a[u\ v+0.5\ w]\\ \boldsymbol{R}&=a[u\ v\ w+0.5]\end{aligned}\]
  • average volume occupied by each lattice point\(=\)
    average volume occupied by each atom in FCC\(=\Omega_{FCC}=\frac{a^3}{8\times\frac{1}{8}+6\times\frac{1}{2}}=\frac{a^3}{4}\)
  • Nearest neighbor distance = \(d=2r=\frac{a}{\sqrt{2}}\)
  • \[\text{Packing fraction}=\frac{4\times\frac{4\pi}{3}\times(\frac{a\sqrt{2}}{4})^3}{a^3}=\frac{\pi\sqrt{2}}{6}\approx 0.74\]

 

The primitive cell vs the unit cell 

The primitive translation vectors are: \[\boldsymbol{a}_1=\frac{a}{2}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}})\] \[\boldsymbol{a}_2=\frac{a}{2}(\hat{\boldsymbol{y}}+\hat{\boldsymbol{z}})\] \[\boldsymbol{a}_3=\frac{a}{2}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{z}})\]

 

  • The angle between adjacent edges is \(60^\circ\)

The reference atom is red.  Blue, green, and orange points are the 12 nearest neighbors of the reference atoms, and the purple points are its 6 next nearest neighbors atoms.

  • Packing fraction \(\frac{1}{6}\pi\sqrt{2}\approx 0.740\)
  • Some elements with fcc structure: Ar, Ag, Al, Au, Ca, Ce, \(\beta-\)Co, Cu, Ir, Kr, La, Ne, Ni, Pb, Pd, Pr, Pt, \(\delta-\)Pu, Rh, Sc, Sr, Th, Xe, Yb

 

Characteristics of Cubic structure

  simple cubic b.c.c. f.c.c.
volume of conventional cell \(a^3\) \(a^3\) \(a^3\)
no. of lattice points per cell 1 2 4
no. of nearest neighbors
(coordination number)
6 8 12
no. of 2nd nearest neighbors 12 6 6
nearest neighbor distance \(a\) \(\frac{\sqrt{3}}{2}a\approx 0.866 a\) \(\frac{a}{\sqrt{2}}\approx 0.707a\)
2nd nearest neighbor distance \(a\sqrt{2}\) \(a\) \(a\)
packing fraction \(\pi/6\approx 0.52\) \(\pi\sqrt{3}/8\approx 0.68\) \(\pi\sqrt{2}/6\approx 0.74\)

[111], [101], and [110] describe directions \(m\), \(t\), and \(n\), respectively.

 

Crystal Structure May Change with Temperature

  • If the temperature changes, some materials might undergo a phase change.
  • For example, at atmospheric pressure (105 Pa) and below 912 °C, pure iron (Fe) has a BCC structure, known as ⍺-iron. If we heat iron above  912 °C, its structure changes to an FCC structure, known as 𝛾-iron. Above 1394 °C, the structure changes back to BCC, known as 𝛿-iron. You may see the phase diagram of pure iron in the following figure. 

 

 

Hexagonal Close-Packed (HCP) Structure

 

 

  • 30 elements crystallize in hcp form.
  • HCP crystal has a hexagonal lattice and a multi-atom basis.
  • It can be viewed as two nested simple hexagonal Bravais lattice shifted by \(\boldsymbol{a}_1/3+ \boldsymbol{a}_2/3+\boldsymbol{a}_3/2\).
    where

\[\boldsymbol{a}_1=a\hat{\boldsymbol{x}},\ \boldsymbol{a}_2=\frac{a}{2}\hat{\boldsymbol{x}}+\frac{a\sqrt{3}}{2}\hat{\boldsymbol{y}},\quad \boldsymbol{a}_3=c\,\hat{\boldsymbol{z}}\]

    • average vol. occupied by each lattice point \[\Omega_{hex\ lattice}=\frac{\sqrt{3}}{2}a^2 c\]
    • average vol. occupied by each atom \[\Omega_{HCP}=\frac{\Omega_{hex\ lattice}}{2}\]

 

Comparing Close-Packed Structures

The left is the hcp structure, and the right is the fcc structure.
hcp fcc

 

 

  • In this figure, the left structure is hcp, and the right is fcc
  • volume fraction = 0.74
  • number of nearest neighbors (coordination number) is 12 for both hcp and fcc structures
  • Although a hexagonal close-packing of equal atoms is only obtained if \(c/a=\sqrt{8/3}\approx 1.63\), the term hcp is used for any structure described previously.

 

Elements with hcp structures

Element \large c/a Element \large c/a
Ideal 1.63    
Be 1.56 Cd 1.89
Ce 1.63 \(\alpha\)-Co 1.62
Dy 1.57 Er 1.57
Gd 1.59 He (2K) 1.63
Hf 1.58 Ho 1.57
La 1.62 Lu 1.59
Mg 1.62 Nd 1.61
Os 1.58 Pr 1.61
Re 1.62 Ru 1.59
Tb 1.58 Ti 1.59
Tl 1.60 Tm 1.57
Y 1.57 Zn 1.59

[adapted from Ashcroft, Mermin, Solid State Physics]

 

Diamond Crystal

  • It is the structure of carbon in a diamond crystal
  • It can be viewed as two interpenetrating fcc lattices displaced by \(\frac{a}{4}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}}+\hat{\boldsymbol{z}})\)
  • Or it can be imagined as the fcc lattice with two point basis \(\boldsymbol{0}\) and \(\frac{a}{4}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}}+\hat{\boldsymbol{z}})\).
  • Coordination number is 4
  • Packing fraction is \(\frac{\sqrt{3}}{16}\pi\approx 0.34\)

 

(a) Tetrahedral bond in a diamond structure (b) Diamond structure projected on a cube face. Fractions denote the height above the base in units of \(a\)

 

3D Diamond Structure

  • Elements with diamond structure: C (diamond), Si, Ge, \(\alpha\)-Sn (gray)
  • average volume occupied by each atom: \(\Omega_{DC}=\frac{\Omega_{FCC}}{2}=\frac{a^3}{8}\)

 

Sodium Chloride Structure

  • Na\(^{+}\) and Cl\(^-\) ions are placed on alternate points of a simple cubic structure
  • The lattice is fcc; the basis consists of Na\(^{+}\) and Cl\(^-\)

NaCl structure, Blue atoms represent Na atoms and Green ones represent Cl atoms. 

Some compounds with sodium chloride structure

LiF LiCl LiBr LiI    
NaF NaCl NaBr NaI    
RbF RbCl RbBr RbI    
CsF          
AgF AgCl AgBr      
MgO MgS MgSe      
CaO CaS CaSe CaTe    
SrO SrS SrSe SrTe    
BaO BaS BaSe BaTe    

[Adapted from Ashcroft, Mermin, Solid State Physics]

 

Cesium Chloride Structure

  • Cs\(^{+}\) and Cl\(^-\) ions are placed at \(\boldsymbol{0}\) and body center position \(\frac{a}{2}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}}+\hat{\boldsymbol{y}})\), respectively.
  • The lattice is simple cubic; the basis consists of Cs\(^{+}\) and Cl\(^-\)

 

CsCl structure. Blue spheres represent Cl atoms and the big red sphere represents the Cs atom which is much larger than the Cl atoms.

  • Some compounds with the cesium chloride structure:
    CsCl CsBr CsI
    TlCl TlBr TlI

 

Miller Indices

Miller Indices for Directions in Cubic Structure

  • It is often required to specify certain directions and planes in crystals. To this end, we use Miller indices.
  • \([hkl]\) represents the direction vector \(h\hat{\boldsymbol{x}}+k\hat{\boldsymbol{y}}+l\hat{\boldsymbol{z}}\) of a line passing through the origin.
  • These integers \(h\), \(k\) and \(l\) must be the smallest numbers that will give the desired direction. i.e. we write \([111]\) not \([222]\).
  • If a component is negative, it is conventionally specified by placing a bar over the corresponding index. For example, we write \([1\bar{1}1]\) instead of \([1 -1 1]\).
  • Coordinates in angle brackets such as \(\langle123\rangle\) denote a family of directions that are equivalent due to symmetry operations, such as [123], [132], [321], [\(\bar{1}23\)], [\(\bar{1}\bar{2}3\)], etc.

[111], [101], and [110] describe directions \(m\), \(t\), and \(n\), respectively.

 

 Miller Indices for Planes in Cubic Structure

  • A crystallographic plane is denoted by the Miller indices of the direction normal to the plane, but instead of brackets we use parenthesis, i.e. (\(hkl\))
$(100)$ $(110)$ $(111)$
  • “Coordinates in curly brackets or braces such as \(\{100\}\) denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.”
  • For a cubic system, a family \(\{hkl\}\) consists of all the planes given by the permutations of the number \(h\), \(k\), \(l\) and their negatives.
  • If the symmetry of the system is lower than that of cubic, not all the planes given by the permutations necessarily belong to a family. For example, in a rhombohedral system, we have \(\{100\}=\{(100),(\bar{1}00),(010),(0\bar{1}0),(001),(00\bar{1})\}\), but in an orthorhombic system \(\{100\}\) family has only two members \((100)\) and \((\bar{1}00)\).

 

Miller Indices for hcp

  • Miller indices contains 4 digits instead of 3 digits
  • \([hkil]\) means: \[\{\alpha (h \boldsymbol{a}_1+k\boldsymbol{a}_2+i\boldsymbol{a}_3+l \boldsymbol{c}): \alpha\in\mathbb{R}\}\] and \[h+k+i=0\]
  • Directions along axes \(\boldsymbol{a}_1\), \(\boldsymbol{a}_2\) and \(\boldsymbol{a}_3\) are of type \(\langle \bar{1} 2\bar{1} 0\rangle\).
  • \((hkil)\) is a plane the normal direction of which is \([hkil]\).
Determination of indices for a digonal axis if Type I – $[2\overline{11}0]$ Determination of indices for a digonal axis if Type II – $[10\overline{1}0]$

 

Further Reading

  • Ashcroft, N.W., Mermin, N.D., Solid State Physics, Harcourt College Publishers, 1976.
  • De Graef, M., McHenry, M.E., Structure of Materials, Cambridge University Press, 2007.
  • Kittel, C., Introduction to Solid State Physics, 8th ed., Wiley, 2004.