### Crystal

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

- 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.

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

- The unit cell may or may not be identical to the primitive cell.

*Unit cell of a face-centered cubic structure*

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

### Bravais lattices

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

*The five fundamental 2D Bravais lattices: (1) oblique, (2) rectangular, (3) centered rectangular, (4) hexagonal (rhombic), and (5) square*

### 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)
$a=b=c$ $\alpha = \beta = \gamma < 120^\circ, \ne 90^\circ$ |

Crystal System | Length | Angles |

Cubic | $a=b=c$ | $\alpha = \beta = \gamma = 90^\circ$ |

Trigonal | $a=b=c$ | $\alpha = \beta = \gamma < 120^\circ, \ne 90^\circ$ |

Hexagonal | $a=b\ne c$ | $\alpha = \beta = 90^\circ, \gamma = 120^\circ$ |

Tetragonal | $a=b\ne c$ | $\alpha = \beta = \gamma = 90^\circ$ |

Orthorhombic | $a\ne b\ne c$ | $\alpha = \beta = \gamma = 90^\circ$ |

Monoclinic | $a\ne b\ne c$ | $\alpha = \beta = 90^\circ \ne \gamma$ |

Troclinic | $a\ne b\ne c$ | $\alpha \ne \beta \ne \gamma$ |

### 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)\[{\rm Packing\ Fraction}=\frac{N \frac{4}{3}\pi R^3}{V}\]

### Cubic lattice

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 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 \(60\circ\)

- 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.*

### Phase Diagram of Pure Iron

### 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 \[\Omega_{HCP}=\frac{\Omega_{hex\ lattice}}{2}\]

### Close Packed Structures

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 in the previous slide.

### Elements with hcp structures

Element | \(c/a\) | Element | \(c/a\) |
---|---|---|---|

Ideal | |||

Be | 1.56 | Cd | 1.89 |

Ce | \(\alpha\)-Co | 1.62 | |

Dy | 1.57 | Er | 1.57 |

Gd | 1.59 | He (2K) | |

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) | (b) |

*(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 \(a\)*

- Elements with diamond structure: C (diamond), Si, Ge, \(\alpha\)-Sn (gre`y)
- 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
*

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
*

- Some compounds with the cesium chloride structure:

CsCl CsBr CsI TlCl TlBr TlI

### 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.