Symmetry and Properties of Crystals (MSE638) Important Concepts of Crystallography

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1 Symmetry and Properties of Crystals (MSE638) Important Concepts of Crystallography Somnath Bhowmick Materials Science and Engineering, IIT Kanpur January 8, 2019

2 Crystallography Study of the crystals/patterns In crystallography symmetry can refer to symmetry of Crystal Motif Lattice Unit cell What we observe is a crystal. Individual motif (atom/molecule): observed in electron microscope. Analogy: Crystal Buliding, Motif Brick Lattice: infinite array of regularly spaced points. It is an abstract idea. Frequently used terms like lattice vibration is a misnomer! How to represent the infinity into something finite? Unit cell: a small space defined such that when repeated (by translation), it generates the infinite lattice 2 / 10

3 Crystal and motif Crystal: constituent atoms arranged in a periodic array Crystal: representative units (single atom/ group of atoms) repeated in space in regular intervals The representative unit is called the motif An asymmetric motif can exist in both right and left handed form Note that, a pair of left and right handed object are related by a reflection about a mirror plane, but they can not be superimposed by translation or rotation Such a pair are called enantiomorphous L is going to be our favorite motif!! 3 / 10

4 L More about Right and Left in a plane (2D) Y Y X (x,y) L L (x,y) X X (x,y) Y Change coordinate system such that +ve diection of X-axis reverses 2 nd figure can not coincide with 1 st, becuase no rotation of XY -coordinate system can make it coincide with XY Change coordinate system such that +ve diection of both axis reverse 3 rd figure coincide with the 1 st by 180 rotation x x: reflection across Y y ȳ: reflection across X (x, y) ( x, ȳ): 180 rotation about the origin Reversal of one coordinate axis generates an enantiomorphous figure Reversal of two coordinate axes generates a congruent figure 4 / 10

5 Building a crystal/pattern Take a motif & repeat it in regular intervals of length T (translation) T has direction and magnitude (vector), but no unique origin Symmetry operation act on all of the space. Thus, ideal pattern/crystal extends from to + Translation is a symmetry operation that any crystal must have (holds good till we discuss quasi-crystal) Any other symmetry operation that leaves the pattern unchanged? 5 / 10

6 Building a crystal/pattern All the patterns above have a 180 rotation symmetry. The second pattern has a horizontal and vertical mirror. Is there a mirror plane in third pattern also? If you look at individual motifs, then yes. Caution: symmetry operation act on all of the space, including other symmetry elements. A mirror in the third pattern creates another tilted translation vector which does not exist in the original pattern. 1 & 3: 180 rotation; 2: 180 rotation & reflection (horizontal and vertical mirror). Symmetry of the motif and crystal/pattern need not be the same. 6 / 10

7 Lattice and Motif Motif can be of any shape So, instead of array of motifs, let s work with an array of regularly spaced points (lattice points) Pattern/crystal Lattice + Motif Lattice is an array of fictitious points that manifests the translation periodicity of a crystal/pattern Number of crystals huge, but possible number of lattices limited Let s look at some examples. 7 / 10

8 Copper and NaCl crystal Both NaCl and copper crystal have face centered cubic lattice. Is there any implication in terms of symmetries? Let s go back to 2D. 8 / 10

9 Lattice and Motif Lattice has the highest symmetry mirror planes and 4-fold rotation. Symmetries of crystal symmetries of lattice. The pattern/crystal has the symmetry of motif, although we started with a lattice having higher symmetry. 9 / 10

10 Take home message Crystalline object at least posses translation and can have additional symmetries. Crystal Lattice + Motif. Symmetry of the lattice, motif and crystal need not be the same. Symmetries of crystal symmetries of lattice. The crystal is left only with the symmetry of the motif. Golden rule in crystallography: a symmetry operation act on all of the space, including the other symmetry elements. 10 / 10