Unit1 Crystal Structure Bravais lattice Materials science plays

Unit-1 Crystal Structure & Bravais lattice.

Materials science plays a key role in almost all aspects of modern life and in the technologies and equipment we rely upon as a matter of routine. Almost all the components have relied upon advances in materials science and the work of materials scientists!

Some of the materials involved and their application: Display. This relies upon the combination of a liquid crystal display and a touch screen for communication with the device. The touch screen is made from a conductive but transparent material, indium tin oxide, a ceramic conductor. Interconnects: Interconnects that provide the links between components are now made of copper, not aluminium, for higher speed and efficiency. ( Further may be by CNT, Nanowires… Wireless Microwave circuits need capacitors which are ceramic insulators whose structure and composition is carefully controlled to optimise the capacitance. Ics: At the heart of the i. Phone are a number of integrated circuits (ICs) built upon billions of individual transistors, all of which rely on precise control of the semiconductor material, silicon, to which has been added dopant atoms to change the silicon’s electronic properties. Adding just a few dopant atoms per million silicon atoms can change the conductivity many orders of magnitude! Battery: The battery is a modern Li-ion battery where the atomic structure of the electrodes is carefully controlled to enable the diffusion of the Li ions. Headphones. Most headphones use modern magnetic materials whose structure and composition has been developed to produce very strong permanent magnets. This is part of a transducer that turns electrical signals into sound.

Classification and Terminology Traditionally states of matter can be classed into 3 ‘classical’ groups: Gases Liquids Solids Metals Ceramics Groups of materials stressing for example their electrical and magnetic properties, so we can further explain about Semiconductors Superconductors Hard and Soft Magnetic Materials Polymers

In addition already studied about the solids that are crystalline (that have a crystal structure) and non-crystalline in nature. A crystalline solid is one in which the atoms are arranged in a periodic fashion. A non-crystalline material is nonperiodic it does not have long-range order but can have ‘short range order’ where the local arrangement of atoms (and the local bonding) is approximately the same as in a crystal.

Crystal structure: the mode in which atoms, ions, or molecules are geometrically arranged. A crystal structure refers to the unique and systematic arrangement of atoms or molecules that are in a crystalline solid. The atoms or molecules are in a specified pattern.

Importance of Atomic Structure What is common to all materials “they all are composed of atoms” So it is very important to understand the properties of material, and to improve those properties by adding or removing atoms, we need to know the material’s atomic structure.

The properties (whether mechanical, electrical, chemical etc) of all solid materials dependent upon are 1. the relative positions of the atoms in the solid and 2. their mutual interaction i. e. the nature of the bonding (whether e. g. covalent, ionic, metallic, van der Waals). carbon-carbon interactions lead to a very directional covalent bond called a sp 3 bond

As Richard Feynman said: ‘It would be very easy to make an analysis of any complicated chemical substance; all one would have to do would be to look at it and see where the atoms are…’ als ri e t a m c i g n m i o d t n a ta eir s r e g th d n u din …. t r ta stan ture s s r c ’ e t u d e r l t n s u o S by

We already know that A lattice is a regular array of points. Each point must have the same number of neighbors as every other point and the neighbors must always be found at the same distances and directions. All points are in the same environment.

Lattice ---An array of points repeating periodically. crystal lattice A regular arrangement of the essential particles of a crystal in a three dimensional space

Possibilities of unit cells

Crystal: Primitive The smallest three-dimensional portion of a complete space lattice, which when repeated over and again in different directions produces the complete space lattice. The size and shape of a unit cell is determined by the lengths of the edges of the unit cell (a, b and c) and by the angles.

Bravais Lattice Bravais in 1948, shows that there are 5 Bravais lattices in 2 D and 14 Bravais lattices in 3 D under the 7 crystal systems They are commonly known as Bravais lattices

1 D Lattices Construction of a 1 D lattice q Let us construct a 1 D lattice starting with two points These points are shown as ‘finite’ circles for better ‘visibility’! The point on the right has one to the left and hence by the requirement of identical surrounding the one of the left should have one more to the left By a similar argument there should be one more to the left and one to the right This would lead to an infinite number of points The infinity on the sides would often be left out from schematics In 1 D spherical space a lattice can be finite!

1 D Lattices Starting with a point the lattice translation vector (basis vector) can generate the lattice q In 1 D there is only one kind of lattice. q This lattice can be described by a single lattice parameter (a). q To obtain a 1 D crystal this lattice has to be decorated with a pattern. q The unit cell for this lattice is a line segment of length a.

2 D Lattices q 2 D lattices can be generated with two basis vectors q. There are five distinct 2 D lattices: 1 Square 2 Rectangle 3 Centered Rectangle 4 120 Rhombus 5 Parallelogram (general)

Five distinct 2 D lattices

2 D Lattices b a Two distances: a, b There are three lattice parameters which describe this lattice Two basis vectors generate the lattice One angle: = 90 in the current example

3 D Lattices q 3 D lattices can be generated with three basis vectors q 3 basis vectors generate a 3 D lattice q. The unit cell of a general 3 D lattice is described by 6 numbers 6 lattice parameters 3 distances (a, b, c) 3 angles ( , , )

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Bravais Lattice 7 Crystal Classes with 4 possible unit cell types Symmetry indicates that only 14 3 -D lattice types occur

q There are 14 distinct 3 D lattices which come under 7 Crystal Systems The BRAVAIS LATTICES (with shapes of unit cells as) : Cube Square Prism (Tetragonal) Rectangular Prism (Orthorhombic) 120 Rhombic Prism (Hexagonal) (a = b = c, = = = 90 ) (a = b c, = = 90 , = 120 ) Parallelepiped (Equilateral, Equiangular) (Trigonal) Parallelogram Prism (Monoclinic) Parallelepiped (general) (Triclinic) (a = b = c, = = 90 ) (a b c, )

Thanks

A description of the characteristics of 14 Bravais lattices of three dimensions along with the axial relationship for the class of crystal lattices, i. e. seven systems to which each belongs are summarized in Table 1

Based on pure symmetry considerations, there are only fourteen independent ways of arranging points in three-dimensional space,

Every type of unit cell is characterized by the number of lattice points (not the atoms) in it. The number of lattice points in unit cell can be calculated by appreciating the following: Contribution of lattice point at the corner = 1/8 th of the point Contribution of the lattice point at the face = 1/2 of the point Contribution of the lattice point at the centre = 1 of the point For example: The number of lattice points per unit cell for simple cubic (SC), body centered cubic (BCC) and face centered cubic (FCC) lattices are 1, 2 and 4, respectively.

Now, we shall discuss about the seven type of basic systems mentioned in Table 1.

(i) Cubic Crystal System: All those crystals which have three equal axes and are at right angles to each other and in which all the atoms are arranged in a regular cube are said to be cubic crystals (Fig. 3. 8). The most common examples of this system are cube and octahedron as shown in Fig. 3. 8(a) and (c). In a cubic crystal system, we have

Atomic Packing Factor (APF): This is defined as the ratio of total volume of atoms in a unit cell to the total volume of the unit cell. This is also called relative density of packing (RDP). Thus In a simple cubic cell, no. of atoms in all corners =(1/8 )x 8 = 1 Radius of an atom = r and volume of cubic cell = a 3 = (2 r )3 Therefore

(ii) Tetragonal Crystal System: This includes all those crystals, which have three axes at right angles to each other and two of these axes (say horizontal) are equal, while third (say vertical) is different (i. e. , either longer or shorter than the other two). The most common examples of this system of crystals are regular tetragonal and pyramids

(iii) Hexagonal Crystal System: All those crystals which have four axes falls under this system. Three of these axes (say horizontal) are equal and meet each other at an angle of 60° and the fourth axis (say vertical) is different, i. e. either longer or shorter than the other three axes.

(iv) Orthorhombic Crystal System The space lattice is simple. The crystal axes are perpendicular to one another but all the three axes are essentially of unequal lengths (Fig. 3. 12).

(v) Rhom-bohedral or Trigonal Crystal System Three axes are equal and are equally inclined to each other at an angle. Other 90° rhombohedral prisms and pyramids (Fig. 3. 13(b) and (c)) are the most common examples of this crystal system.

(vi) Monoclinic Crystal System: Two of the crystal axes are perpendicular to each other, but the third is obliquely inclined. The repetitive intervals are different along all the three axes. Monoclinic lattices may be simple or base-centered.

(vii) Triclinic Crystal System: All those crystals, which have three unequal axes and none of them is at right angles to the other two axes (Fig. 3. 15) are included in this crystal system. The repetitive intervals are different along all the three axes. All irregular crystals belong to this class.

CRYSTAL STRUCTURE FOR METALLIC STRUCTURE The most common 3 types of space lattice or unit cells with which most common metals crystallise, Body-centered cubic (BCC) Face-centered cubic (FCC) Hexagonal closed-packed (HCP)