Basics of Materials Science and Engineering Short Notes Part-2

By Vineet Vijay|Updated : December 9th, 2019

Basics of Material Science & Engineering

1.Crystal Structures

Crystal structure is one of the most important aspects of materials science and engineering as many properties of materials depend on their crystal structures. The basic principles of many materials characterization techniques such as X-ray diffraction (XRD), Transmission electron microscopy (TEM) are based on crystallography. Therefore, understanding the basics of crystal structures is of paramount importance.

Atomic Arrangement

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Crystalline – periodic arrangement of atoms: definite repetitive pattern

Non-crystalline or Amorphous – random arrangement of atoms.

The periodicity of atoms in crystalline solids can be described by a network of points in space called lattice.

1.1 Space lattice:  A space lattice can be defined as a three dimensional array of points, each of which has identical surroundings. 

  • If the periodicity along a line is a, then position of any point along the line can be obtained by a simple translation, ru = ua.
  • Similarly ruv= ua + vb will repeat the point along a 2D plane, where u and v are integers.byjusexamprep1.2 Crystal Systems

The space lattice points in a crystal are occupied by atoms.

The position of any atom in the 3D lattice can be described by a vector ruvw= ua + vb + wc, where u, v and w are integers

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The three unit vectors, a, b, c can define a cell as shown by the shaded region in Fig. (a) This cell is known as unit cell (Fig. b) which when repeated in the three dimensions generates the crystal structure.

Other crystal systems with different arrangement of unit vectors and angles are shown below:

 

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1.3 Crystal Planes

1.3.1 Miller Indices

Planes in a crystal are described by notations called Miller Indices

Miller indices of a plane, indicated by h k l, are given by the reciprocal of the intercepts of the plane on the three axes.

The plane, which intersects X axis at 1 (one lattice parameter) and is parallel to Y and Z axes, has Miller indices

h = 1/1 = 1, k = 1/∞ = 0, l = 1/∞ = 0. It is written as (hkl) = (100).

Miller indices of some other planes in the cubic system are shown below:

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To find the Miller Indices of a plane, follow these steps:

  • Determine the intercepts of the plane along the crystal axes
  • Take the reciprocals
  • Clear fractions
  • Reduce to lowest terms and enclose in brackets ()
  • Ex: Intercepts on a, b, c : ¾, ½, ¼ (h k l) = (4/3, 2, 4) = (2 3 6)
  • Planes can also have negative intercept e.g. 1, -1/2, 1 h k l = 1 -2 1. This is denoted as ( 1 2 1 )

1.3.2 Family of Planes

Planes having similar indices are equivalent, e.g. faces of the cube (100), (010) and (001). This is termed as a family of planes and denoted as {100} which includes all the (100) combinations including negative indices.

1.3.3 Interplanar Spacing

The spacing between planes in a crystal is known as interplanar spacing and is denoted as dhkl. In the cubic system spacing between the (hkl) planes is given as

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         For example, dhkl of {111} planes d111= a/√3 

 

1.3.4 Crystal Directions

The directions in a crystal are given by specifying the coordinates (u, v, w) of a point on a vector (ruvw) passing through the origin. ruvw = ua + vb + wc. It is indicated as [uvw]. For example, the direction [110] lies on a vector r110 whose projection lengths on x and y axes are one unit (in terms of unit vectors a and b).

Directions of a form or family like [110], [101], [011] are written as <110>

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1.3.5 Coordination Number

Coordination number is the number of nearest neighbor to a particular atom in the crystal.

In the FCC lattice each atom is in contact with 12 neighbor atoms. FCC coordination number Z = 12

For example, the face centered atom in the front face is in contact with four corner atoms and four other face-centered atoms behind it (two sides, top and bottom) and is also touching four face-centered atoms of the unit cell in front of it as shown in given figure.

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The coordination number of BCC crystal is 8.

The body centered atom is in contact with all the eight corner atoms. Each corner atom is shared by eight unit cells and hence, each of these atoms is in touch with eight body centered atoms as shown in figure below:

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1.3.6 Atomic Packing Fraction (APF)

Atomic packing factor (APF) or packing efficiency indicates how closely atoms are packed in a unit cell and is given by the ratio of volume of atoms in the unit cell and volume of the unit cell.

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The table given below shows the Atomic Packing Fraction for different crystal systems:

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1.3.7 Planar Density

Planar density (PD) refers to density of atomic packing on a particular plane.

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 For example, there are 2 atoms (1/4 x 4 corner atoms +1/2 x 2 side atoms) in the {110} planes in the FCC lattice.

Planar density of {110} planes in the FCC crystal   

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 2. Lattice Defects

 Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atoms or molecules in most crystalline materials is not perfect. The regular patterns are interrupted by crystallographic defects.

2.1 Types of Lattice Defects

  1. Point defects
  2. Line defects
  3. Surface defects
  4. Volume defects

2.1.1 Point Defects:

Vacancy – An atom missing from regular lattice position.

Vacancies are present invariably in all materials.

Interstitialcy – An atom trapped in the interstitial point (a point intermediate between regular lattice points) is called an interstitialcy.

An impurity atom at the regular or interstitial position in

the lattice is another type of point defect.

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In ceramic materials point defects occur in pair to maintain the electroneutrality.

A cation-vacancy and a cation-interstitial pair is known as Frenkel defect.

A cation vacancy-anion vacancy pair is known as a Schottky defect.

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 2.1.2 Line Defects(Dislocations)

Dislocation is the region of localized lattice distortion which separates the slipped and unslipped portion of the crystal.

  • The upper region of the crystal over the slip plane has slipped relative to the bottom portion. The line (AD) between the slipped and unslipped portions is the dislocation.
  • The magnitude and direction of slip produced by dislocation (pink shaded) is the Burger vector, b, of the dislocation.

It is of two types: Edge and Screw Dislocations

Edge Dislocations: In one type of dislocations, the Burger vector is perpendicular to the dislocation line and the distortion produces an extra half-plane above the slip plane.

Screw Dislocations: The other type of dislocation is the screw dislocation where the Burger vector is parallel to the dislocation line (AD).

The trace of the atomic planes around the screw dislocation makes a spiral or helical path (pink shade) like a screw and hence, the name.

2.1.3 Surface Defects (Grain Boundary)

  • Most crystalline solids are an aggregate of several crystals. Such materials are called polycrystalline.
  • Each crystal is known as a grain. The boundary between the grains is the grain boundary
  • A grain boundary is a region of atomic disorder in the lattice only a few atomic diameter wide.
  • Grain boundaries act as obstacles to dislocation motion. Hence, presence of more grain boundaries (finer grain size) will increase the strength.

2.1.4 Volume Defects

These are basically of 3 types:

  • Porosity
  • Inclusions
  • Cracks

  These defects form during manufacturing processes for various reasons and are harmful to the material.

  • Examples of Volume defects:
    Casting blow holes, porosity – Gas entrapment during melting and pouring. Improper welding parameters/practice
  • Shrinkage cavity due to improper risering
  • Non-metallic inclusions – Slag, oxide particles or sand entrapment
  • Cracks – Uneven heating/cooling, thermal mismatch, constrained expansion/contraction all leading to stress development

              

Next - Basics of Material Science & Engineering Part 3

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