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

The Next Level in Crystallography  From: 1  Lattice + Motif to 2  (Lattice + Motif* + Symmetry) to 3  (Space Group + Motif*) to 4  Space Group + Asymmetric Unit (+ Wyckoff positions) International Tables of Crystallography Ed.: Th. Hahn International Union of Crystallography, (2005) Advanced Reading

 Crystal = Lattice + Motif is a simple definition of a crystal.  But, this definition does not bring out the most important aspect of a crystal → the symmetry of a crystal.  In general a crystal will have more symmetry than just translational symmetry (except triclinic crystals which may have only translational symmetry. Note: some triclinic crystals may have inversion symmetry also).  Additional symmetries which a crystal may possess include: rotation, inversion, glide reflection etc.  Crystallography is the language used to describe crystals, which is one of succinctness*.  For large motifs there might be considerable ‘expenditure in parameters’ to describe the motif. Additional symmetry present in the structure may help us reduce this expenditure. Progress from simpler to complex definitions of crystals * terseness

Emphasis: Why go for concept of Space Group, Asymmetric Units and Wyckoff positions? (when life seemed nice and simple with Crystal = Lattice + Motif)!  Let us look at the labour (‘information’) reduction already achieved?  We reduced the description of an infinite crystal into the description of the contents of a unit cell along with the x,y,z translations.  For non-primitive cells we reduced the information further by taking the contents associated with just one lattice point (which is then repeated at each lattice point within the cell).  We can reduce the information further by using any symmetry available in the motif (and crystal) in addition to the translational symmetry already considered. Click here to see a 2D example to illustrate this aspect

 We will introduce a intermediate step before we understand crystals using the definition of a Crystal = Space Group + (Asymmetric Unit) + Wyckoff Positions.  This step is a definition of a crystal in terms of: Crystal = Lattice + Motif* + Symmetry ($).  It should be understood that this step is only for understating the concepts.  In the definition Crystal = Asymmetric Unit + Space Group (+Wyckoff Positions)  Asymmetric unit is a region of space and not just an entity like a motif.  One way to reach this definition is by considering a crystal to be: Crystal = Space Group + Motif* ($) (where motif* is an entity- e.g could be a Carbon atom) ($) If a student feels uncomfortable with these intermediate stops he can work with Crystal = Asymmetric Unit + Space group Some intermediate steps to make the transition ‘smoother’

Crystal = Lattice + Motif Method-1  In the example below the Al 12 W Frank-Kasper phase can be generated by placing an icosahedron of Al atoms (i.e. 12 atoms of Al) with a W atom at its centre (this combination of 12 Al atoms and one W atom is the motif) in each lattice point of a BCC lattice. = BCC + Motif Lattice Al 12 W crystal

2D Example Crystal = Lattice + Motif = + Lattice Motif Crystal This example is carried further here.

Crystal = Lattice + (Motif* + Symmetry) (this symmetry is not the point group symmetry of the structure) Method-2  Motif* → What the Point Group operates on to generate the structure (can be different from the Motif in “Lattice + Motif” picture  In the example below the Al 12 W Frank-Kasper phase can be generated by placing an icosahedron of Al atoms (i.e. 12 atoms of Al) with a W atom at its centre (this combination of 12 Al atoms and one W atom is the motif) in each lattice point of a BCC lattice.  The icosahedron itself can be generated starting with one Al atom and imposing symmetry  In the Lattice + Motif Description we have to give the position of each of the 12 Al atoms, while if we use the symmetry present in the motif (i.e. symmetry) then we need to consider only one Al atom. Hence we have reduced the number of parameters which we need to use. = BCC ++

Crystal = Space Group + Motif* Method-3  Motif* → What the space group operates on to generate the structure (can be different from the Motif in “Lattice + Motif” picture  In the example considered (next slide) we generate the diamond cubic structure starting with a Carbon atom.

Generated by the repeated action of the 4 1 screw Let us see how the Diamond Cubic structure can be generated using the Space Group (as below) and a Carbon atom as the Motif* Let us start with a single carbon atom at (½, ½, 0) and a 4 1 screw operator with axis passing through (x = ½, y = ¼) Starting Carbon atom The screw axis generates an infinite column of atoms along ‘c’ axis

Next let us introduce the diamond glide operator  to the 4 1 screw There are other ways of proceeding from here → the option chosen is to operate the diamond glide on 4 1 screw and not directly on atoms already generated The diamond glide (d) operator moves a copy of the screw (note the copy is 4 3 ) Note: symmetry operators act on entire space → including on other symmetry operators present Needless to say the screw operator will act on the diamond glide operator (not shown here)

The new screw further generates additional column of atoms and additional screw axes Additional screws generated

These screw axes further generate additional column of atoms and more screw axes → spreading their extent in 2D x-y plane to infinity Hence, 4 1 /d is enough to generate the entire DC structure!

Crystal = Space Group + (Asymmetric Unit) + Wyckoff Positions Method-4  This is the official language of crystallography.  Often the visualization of space group operation in 3D to give a structure is very difficult.  We will consider a 2D example to understand the concept.  The Asymmetric unit (AU) is a region of space, which repeated by the space group tiles entire space.  Species are assigned to the AU by Wyckoff positions (Wyckoff labels along with specification of variables (x,y,z) therein).  If the AU is tiled to fill space then automatically the ‘entities’ contained therein (like atoms etc.) are repeated to form the crystal.

= Crystal = Space Group + Asymmetric Unit 2D Example or Space group = p4gm + Continued… With symmetry elements overlaid Note that the asymmetric unit contains part of the entities of the motif and unit cell Asymmetric Unit: x  [0, 0.5], y  [0, 0.5], x + y  0.5 More details can be found here Wyckoff positions → 2a, 4c, 4c +

Let us see how the combination of p4gm, Asymmetric Unit (AU) and Wyckoff positions gives us the crystal Starting with the AU apply ‘g’ (glide reflection) Now apply the 4-fold rotation (at centre) The blue triangle is produced from the yellow triangle This gives us the contents of one unit cell → now translational symmetry (the “p” in the p4gm) can fill the entire 2D plane and hence give us the entire crystal in 2D The ‘entities’ (i.e. open and filled circles) are shown only for reference → The AU is a region of space (which of course contains a part of the contents of a unit cell) Note: Instead of working with AU and g (followed by 4), we could have worked with AU and m (followed by 4). The use as illustrated was to show the effect of the ‘g’ operator (in addition).