MATERIALS SCIENCE & ENGINEERING

1. MATERIALS SCIENCE & ENGINEERING Part of A Learner’s Guide AN INTRODUCTORY E-BOOK Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email:anandh@iitk.ac.in, URL:home.iitk.ac.in/~anandh http://home.iitk.ac.in/~anandh/E-book.htm

2. UNIT CELLS (UC) • An 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) • 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) • 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 Will contain lattice points only Lattice Unit cell of a Will contain entities which decorate the lattice Crystal

3. Why Unit Cells? Instead of drawing the whole structure I can draw a representative part and specify the repetition pattern 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. • If there are lattice points in the cell other than the corners, the cell is non-primitive.

4. In general the following types of unit cells can be defined • Primitive unit cell • Non-primitive unit cells • Voronoi cells • Wigner-Seitz cells

5. 1D

6. 1D Contributions to the unit cell: Left point = 0.5, Middle point = 1, Right point = 0.5. Total = 2 • 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 Primitive UC Doubly Non-primitive UC Triply Non-primitive UC

7. Unit cell of a 1D crystal will contain Motifs in addition to lattice points • 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) Though the whole lattice point is shown only half belongs to the UC Each of these atoms contributes ‘half-atom’ to the unit cell Though this is the correct unit cell Often unit cells will be drawn like this • Unit cell in 1D is described by 1 (one) lattice parameter: a

8. 2D

9. 2D b  a • Unit cell in 2D is described by 3 lattice parameters: a, b,  • Special cases include: a = b;  = 90 or 120 • Unit Cell shapes in 2D  Lattice parametersSquare(a = b,  = 90) Rectangle(a, b,  = 90)  120 Rhombus (a = b,  = 120)  Parallelogram (general)  (a, b, )

10. 2D Rectangular lattice Note: Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!!

11. 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!

12. How to choose a unit cell? • 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!) How does convention come into play in the choice of unit cell for Orthorhombic lattices?

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

14. 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!

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

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

17. Click here 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) Solved Example

18. 3D

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

20. Unit Cell shapes in 3D: Cube  (a = b = c,  =  =  = 90) Square Prism  (a = b  c,  =  =  = 90) Rectangular Prism  (a  b  c,  =  =  = 90) 120 Rhombic Prism  (a = b  c,  =  = 90,  = 120) Parallopiped (Equilateral, Equiangular)  (a = b = c,  =  =   90) Paralleogramic Prism  (a  b  c,  =  = 90  ) Parallopiped (general)  (a  b  c,     ) Some common names of unit cells are given here → alternate names are also used for these cells

21. 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.

22. Q & A

23. Should a Unit Cell have Lattice Points only at the Corners? • The conventional unit cell chosen has lattice points at the corners/vertices … 1D Conventional UCs 2D

24. But in principle any unit cell like the ones below (space filling) should work fine!(all the illustrated UC fill space!)

25. 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 New choice of non-conventional cell Problem: UC has entities of the motif in parts!

26. Wigner-Seitz 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

27. Square lattice Centred Rectangular lattice Wigner-Seitz cells

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

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