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**Crystal**, any solid material in which array of atoms of solid are arranged in a orderly and periodic pattern and whose surface regularity reflects its internal symmetry.

Crystalline Vs Amorphous solids :

**Crystalline** are those material which exhibits 3D long range periodicity of arrangement of atom or molecules or ions. Crystallinity can be seen in metals, many ceramics and some polymer.

**Amorphous** have no periodic packing of atoms or molecules or ions. Occures from rapid cooling.

CRYSTALLINE SOLIDS | AMORPHOUS SOLIDS |
---|---|

Sharp melting point | No particular melting point |

Anisotropic | Isotropic |

True solid | Pseudo solid |

Symmetrical | Unsymmetrical |

More rigid | Less rigid |

Long range order | Short range order |

Example: Potassium nitrate, copper | Example: Fibre Glass, PVC, Teflon etc. |

What is Lattice :

➧ A lattice is an infinite array of points

➧ Lattice is a translationally periodic arrangement of points

➧ Each lattice point is identical to every other point, i.e., the environment around one lattice site is identical to that around every other.

➧ Each point in the lattice is defined by \( R = n_{1}a_{1} + n_{2}a_{2} + n_{3}a_{3} \)

where \( n_{1} , n_{2} \textrm{ and } n_{3} \) is a set of three integers and \( a_{1} , a_{2} \textrm{ and } a_{3} \) are (known as the primitive vectors) three non coplanar vectors.

Unit Cell :

From Above figure, any parallelogram (and parallelopiped in 3D), connecting all lattice points is called a unit cell. There are infinite possibilities for unit cells.

In simple words **The smallest repeating unit of the crystal lattice is the unit cell, and it should refect crystal's symmetry. **

**Types of Unit cell : **

• **Primitive Unit cell :** Unit cell having one formula unit or one lattice point

• **Non-primitive Unit cell :** Unit Cell having more than one formula units or lattice points

Crystal Structure :

➧ Crystal structure can be constructed in many ways. Example : A Lattice and Motif (i.e., atom of group of atoms).

➧ There are many combinations of lattices and motifs possible that can result in the same crystal. There is no unique choice.
\[ \textrm{ Crystal Structure = Lattice + Motif } \]

Based on XRD technique, all crystalline materials are classified into 7-crystal system. And these 7-crystal system further divided into 14 Bravais lattices .

Crystal System | Geometry | Crystal Symmetry | Bravais Lattices |
---|---|---|---|

Cube | \( a = b = c \textrm{ ; } \alpha = \beta = \gamma = 90^\circ \) | 4 , 3-fold rotation axes | 3 → SC, BCC, FCC |

Tetragonal | \( a = b \neq c \textrm{ ; } \alpha = \beta = \gamma = 90^\circ \) | 1 , 4-fold rotation axes | 2 → ST, BCT |

Orthorhombic | \( a \neq b \neq c \textrm{ ; } \alpha = \beta = \gamma = 90^\circ \) | 3 , \( \perp \) 2-fold rotation axes | 4 → SO, BCO, FCO, ECO |

Rhombohedral | \( a = b = c \textrm{ ; } \alpha = \beta = \gamma \neq 90^\circ \) | 1 , 3-fold axis symmetry | 1 → SR |

Hexagonal | \( a = b \neq c \textrm{ ; } \alpha = \beta = 90^\circ \textrm{ ; } \gamma = 120^\circ \) | 1 , 6-fold rotation axes | 2 → SH |

Monoclinic | \( a \neq b \neq c \textrm{ ; } \alpha = \beta = 90^\circ \neq \gamma \) | 1 , 2-fold rotation axes | 2 → SM, ECM |

Triclinic | J\( a \neq b \neq c \textrm{ ; } \alpha \neq \beta \neq \gamma \neq 90^\circ \) | None | 1 → S.Triclinic |

Total Bravais Lattices : |
14 |

FCC (Face Centered Cubic) :

(1) 8 Atoms, At every corner contributing \( \frac{1}{8} \) => 8 Corner atoms \( \times \) \( \frac{1}{8} \) contribution = 1 atom

(2) 6 Atoms, At every face center contributing \( \frac{1}{2} \) => 6 Face centered atoms \( \times \) \( \frac{1}{2} \) contribution = 3 atom

Total number of atoms = 1+3 = 4 atoms.

(1) FCC Elements are generally strong and ductile

(2) Examples :- \( \gamma-Fe \), Al, Cu, Ni, Au, Ag, Pt.

BCC (Base Centered Cubic) :

(1) 8 Atoms, At every corner contributing \( \frac{1}{8} \) => 8 Corner atoms \( \times \) \( \frac{1}{8} \) contribution = 1 atom

(2) 1 Atoms, At every Body center contributing 1 => 1 atom

Total number of atoms = 1+1 = 2 atoms.

(1) BCC Elements are generally hard and brittle

(2) Examples :- \( \alpha-Fe \), \( \delta-Fe \), W, Cr, V, Mo, Ta.

HCP (Hexagonal Closest Packed) :

(1) 12 Atoms, At every corner contributing \( \frac{1}{6} \) => 12 Corner atoms \( \times \) \( \frac{1}{6} \) contribution = 2 atom

(2) 2 Atoms, At top and bottom face center contributing \( \frac{1}{2} \) => 2 Face centered atoms \( \times \) \( \frac{1}{2} \) contribution = 1 atom

(3) 3 Atoms, Inside HCP structure contributing 1 => 3 atom

Total number of atoms = 2 + 1 + 3 = 6 atoms or 2 Atom per Unit (1HCP has 3 Unit structures )

(1) HCP Elements are generally less ductile compare to FCC

(2) For regular HCP Element \( \frac{c}{a} = \sqrt{\frac{8}{3}} = 1.632 \).

(3) Examples :- Mg, Zn, Zr, Ti, Co.

Coordination Number :

CN is define as the total number of

SC | FCC | BCC | HCP | DC | |
---|---|---|---|---|---|

CN | 6 | 12 | 8 | 12 | 4 |

Atomic Packing Factor :

APF is define as the ratio of total avg. volume( Number of atoms \(\times \) total number of atom ) occupied by atoms to the volume of the unit cell.

SC | FCC | BCC | HCP | DC | |
---|---|---|---|---|---|

APF | 0.52 | 0.72 | 0.68 | 0.72 | 0.34 |

Volume Density or Theoratical Density or X-Ray Density :

Where ,

Z = Number of atoms in unit cell (atoms)

\(M_{w} = \) Molar Mass (g/mol)

\(N_{A} = \) Avogadro’s number = \( 6.023 \times 10^{23} \) \( \frac{atom}{mol} \)

V = Volume of unit cell = \( a^3 \) \( ( m^3 ) \)

Planer Density :

Planer Density is define as the Ratio of number of atoms who's centers are intersected by the plane to the area of the plane itself.

Plane | FCC | BCC |
---|---|---|

(100) | \[ \frac{2}{a^2} \] | \[ \frac{1}{a^2} \] |

(110) | \[ \frac{2}{\sqrt{2}a^2} \] | \[ \frac{2}{\sqrt{2}a^2} \textrm{ } (CPP) \] |

(111) | \[ \frac{2}{(\sqrt{3}/2)a^2} \textrm{ } (CPP) \] | \[ \frac{1/2}{(\sqrt{3}/2)a^2} \] |

For FCC : (111) Plane

For BCC : (110) Plane

Linear Density :

Linear density defined as the number of atoms whose centers are intersected by a line to the length of the line itself.

Direction | FCC | BCC |
---|---|---|

[100] | \[ \frac{1}{a} \] | \[ \frac{1}{a} \] |

[110] | \[ \frac{2}{\sqrt{2}a} \textrm{ } (CPD) \] | \[ \frac{1}{\sqrt{2}a} \] |

(111) | \[ \frac{1}{(\sqrt{3})a} \] | \[ \frac{2}{(\sqrt{3})a} \textrm{ } (CPD)\] |

For FCC : [110] Direction

For BCC : [111] Direction