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. Ics: At the heart of the iPhone 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! 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… 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. Wireless Microwave circuits need capacitors which are ceramic insulators whose structure and composition is carefully controlled to optimise the capacitance.

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

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 non-periodic 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.

What is common to all materials “they all are composed of atoms” 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.

of all solid materials are dependent upon (whether mechanical, electrical, chemical etc) of all solid materials are dependent upon 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). The properties carbon-carbon interactions lead to a very directional covalent bond called a sp3 bond

So let’s start understanding materials by understanding their atomic 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…’ So let’s start understanding materials by understanding their atomic structure….

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. A regular arrangement of the essential particles of a crystal in a three dimensional space crystal lattice

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 2D and 14 Bravais lattices in 3D under the 7 crystal systems They are commonly known as Bravais lattices

1D Lattices Construction of a 1D lattice Let us construct a 1D 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 1D spherical space a lattice can be finite!

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

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

Five distinct 2D lattices

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

3D lattices can be generated with three basis vectors 3 basis vectors generate a 3D lattice The unit cell of a general 3D lattice is described by 6 numbers  6 lattice parameters  3 distances (a, b, c)  3 angles (, , )

Not fit to scale

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

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

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 = a3 = (2r )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 the 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.

Hexagonal closed-packed (HCP) 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)