MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of A Learner’s Guide

UNIT CELLS (UC)  A unit cell (also sometimes causally referred to as a cell) is a representative unit of the structure.  which when translationally repeated (by the basis vector(s)) gives the whole structure.  The term unit should not be confused with ‘having one’ lattice point or motif (The term primitive or sometimes simple is reserved for that).  If the structure is a lattice, the unit cell will be unit of that (hence will have points* only).lattice  If the structure under considerations is a crystal, then the unit cell will also contain atoms (or ions or molecules etc.).  Note: Instead of full atoms (or other units) only a part of the entity may be present in the unit cell (a single unit cell)crystal  The dimension of the unit cell will match the dimension of the structure**:  If the lattice is 1D  the unit cell will be 1D, if the crystal is 3D  then the unit cell will be 3D, if the lattice is nD  the unit cell will be nD.nD Unit cell Crystal Lattice of a Will contain lattice points only Will contain entities which decorate the lattice * Strangely in crystallography often we even ‘split a point’ (and say that 1/8 th belongs to the UC). ** One can envisage other possibilities– e.g. a 2D motif may be repeated only along one direction (i.e. the crystal is 3D but the repeat direction is along 1D)

ADDITIONAL POINTS  A cell is a finite representation of the infinite lattice/crystal  A cell is a line segment (1D) or a parallelogram (2D) or a parallelopiped (3D) with lattice points at their corners  This is the convention  If the lattice points are only at the corners, the cell is primitive. Hence, a primitive unit cell is one, wherein the lattice points are only at the corners of the unit cell (or the ends of a line segment unit cell in 1D)  If there are lattice points in the cell other than the corners, the cell is non-primitive. Why Unit Cells? Instead of drawing the whole structure I can draw a representative part and specify the repetition pattern.     Consider an infinite pattern made of squares This can be thought of as a single square repeated in x and y directions This way the infinite information content of a crystal can be reduced to the information required to specify the contents of a unit cell (along with the lattice translation vectors).

In general the following types of unit cells can be defined:  Primitive unit cell  Non-primitive unit cells  Voronoi cells  Wigner-Seitz cells

1D

 Unit cell of a 1D lattice is a line segment of length = the lattice parameter  this is the PRIMITIVE UNIT CELL (i.e. has one lattice point per cell). Each of these lattice points contributes half a lattice point to the unit cell Contributions to the unit cell: Left point = 0.5, Middle point = 1, Right point = 0.5. Total = 2 Primitive UC Doubly Non- primitive UC Triply Non- primitive UC 1D

 Unit cell of a 1D crystal will contain Motifs in addition to lattice pointsMotifslattice  NOTE:  The only kind of motifs possible in 1D are line segments  Hence in ‘reality’ 1D crystals are not possible as Motifs typically have a finite dimension (however we shall call them 1D crystals and use them for illustration of concepts)  We could have 2D or 3D motifs repeated along 1D (hence periodicity and ‘crystallinity’ is only along 1D) Each of these atoms contributes ‘half-atom’ to the unit cell  Unit cell in 1D is described by 1 (one) lattice parameter: a Though the whole lattice point is shown only half belongs to the UC Though this is the correct unit cell Often unit cells will be drawn like this Correct unit cell

2D

 Unit cell in 2D is described by 3 lattice parameters: a, b,   b a  Special cases include: a = b;  = 90  or 120  2D  Unit Cell shapes in 2D  Lattice parameters  Square  (a = b,  = 90  )  Rectangle  (a, b,  = 90  )  120  Rhombus  (a = b,  = 120  )  Parallelogram (general)  (a, b,  )

Note: Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!! 2D Rectangular lattice 90  Note: these are the basis vectors (and included angle) for UC-1 above UC-1 Note: basis vectors (& included angle) will change based on the ‘unit cell’ chosen [which implies that lattice parameters will change as well !]  90 

IMPORTANT Symmetry (or the kind) of the Lattice or the crystal is not altered by our choice of unit cell!! You say this is obvious  I agree!

 When possible we chose a primitive unit cell  The factors governing the choice of unit cell are:  Symmetry of the Unit Cell  should be maximum (corresponding to lattice*)  Size of the Unit Cell  should be minimum  Convention  if above fails to resolve the issue we use some convention (We will see later - using an example- that convention is not without common sense!)Symmetry How to choose a unit cell? How does convention come into play in the choice of unit cell for Orthorhombic lattices? * The lattice may have higher symmetry than the crystal→ but in choosing the unit cell we focus on the lattice. E.g. if we decorate a square lattice with a triangle motif, we land up with a rectangle crystal. But we prefer to chose a square unit cell for the crystal as well.

This is nothing but a square lattice viewed at 45  ! Centred square lattice = Simple square lattice Continued…

 In this case the primitive (square) and the non-primitive square cell both have the same symmetry  But the primitive square cell is chosen as it has the smaller size  The primitive parallelogram cell is not chosen as it has a lower symmetry  The lattice has 4-fold symmetries as shown  The square cells also have 4-fold symmetry  The parallelogram cell does NOT have 4-fold symmetry (only 2-fold  lower symmetry) Note these are symmetries of the UC and not of the lattice!

Centred Rectangular Lattice Note that the distribution of symmetry elements has not changed (as compared to the Simple Rectangular Lattice) Lattice parameters: a, b,  = 90  Continued… Unit Cell of Lattice

Simple rectangular Crystal (Not a centred crystal) True Unit Cell of Crystal Now the UC of the crystal will have a motif Note that the UC has entities of the motif in parts! Part of the structure The centres of only the green circles are lattice points (of course equivalently the centres of only the maroon circles) Though the whole lattice point is shown only one fourth belongs to the UC Unit Cell the way it is usually shown Correct unit cell

Solved Example Choice of Lattice, Motif, UC, Symmetry Elements etc are illustrated in the example (try and understand those concepts with which you are familiar at this juncture and postpone the other concepts for a later discussion) Click here

3D

 In order to define translations in 3-d space, we need 3 non-coplanar vectors.  With the help of these three vectors, it is possible to construct a parallelepiped called a UNIT CELL.  Conventionally, the fundamental translation vector is taken from one lattice point to the next in the chosen direction. Cells- 3D

The symbol “  ” implies → “need not be equal to”. Some common names of unit cells are given here → alternate names are also used for these cells. As we have noted in many places, these are conventional unit cells chosen and alternate unit cells are also possible for the structure (for which this unit cell shape is chosen). Also, as we have noted elsewhere, these are unit cell shapes and not be confused with the definition of crystal system (i.e. these unit cell shapes do not define the crystal system). Shape Preferred unit cell for _____ crystal system Constraints on lattice parameters (distances) Constraints on lattice parameters (angles) CubeCubica = b = c  =  =  = 90  Square PrismTetragonal a = b  c  =  =  = 90  Rectangular PrismOrthorhombic a  b  c  =  =  = 90  120  Rhombic Prism Hexagonal a = b  c  =  = 90 ,  = 120  Parallopiped (Equilateral, Equiangular) Trigonal (rhombohedral)a = b = c  =  =   90  Paralleogramic PrismMonoclinic a  b  c  =  = 90    Parallopiped (general)Triclinic a  b  c          Unit Cell shapes in 3D:

Different kinds of CELLS Unit cell A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal. Primitive unit cell For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space. Wigner-Seitz cell A Wigner-Seitz cell is a particular kind of primitive cell, which has the same symmetry as the lattice.

Q & A How many ‘shapes’ of primitive unit cells are possible?  1D → one.  2D, 3D → Infinite (few examples in 2D given below). 1D 2D

Q & A

 The conventional unit cell chosen has lattice points at the ends of line segment/corners/vertices … Should a Unit Cell have Lattice Points only at the Corners? Conventional UCs 1D 2D Funda Check  But in principle any unit cell like the ones below (space filling) should work fine! (all the illustrated cells fill space!)

 We had earlier seen that conventional choice of unit cells can ‘cut into’ the lattice points (and hence into entities of motif) (as below).  Choices of some non-conventional cells (like the ones drawn before) can alleviate this problem of ‘cutting into’ lattice points.  The new unit cell may still (or may not as below) cut into parts of the motif. Problem: UC has entities of the motif in parts! New choice of “non- conventional” cell Corners of the unit cell still have to be lattice points By convention

Define a unit cell. Funda Check  An unit cell unit of the structure [is a line segment in 1D, a parallelogram in 2D and a parallelepiped in 3D], such that lattice points are at the ends of the line segment (1D) and at the vertices of the parallelogram (2D) or parallelepiped (3D); which when repeated by the translational symmetry vector(s) generates the whole structure (lattice or crystal).  A primitive unit cell has only one lattice point per cell, which are at: the ends of the line segment (1D) and at the vertices of the parallelogram (2D) or parallelepiped (3D). (A primitive unit cell need not have 1 atom per cell!!)

 Is a primitive unit cell with the symmetry of the lattice  Created by Voronoi tessellation of space  The region enclosed by the Wigner-Seitz cell is closer to a given lattice point than to any other lattice point Wigner-Seitz Cell

Square lattice Centred Rectangular lattice Wigner-Seitz cells

BCC Tetrakaidecahedron The Tetrakaidecahedron is a space filling solid, which is semi-regular.

FCC Rhombic Dodecahedron The rhombic dodecahedron has been considered as the least ‘photogenic’ solid! This is also a ‘semi-regular’ space filling solid. This is the coordination polyhedron for the BCC lattice.