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. Symmetry Operators Translation  Rotation  Inversion  Mirror Roto-inversion  Roto-reflection  Glide reflection  Screw axis Point Groups, Space Groups Wyckoff Positions SYMMETRY Advanced Reading Elementary Crystallography M.J. Buerger John Wiley & Sons Inc., New York (1956) Crystallography and Crystal Chemistry F. Donald Bloss Holt Rinehart and Winston, Inc., New York (1971) Fantastic book with good illustrations http://pd.chem.ucl.ac.uk/pdnn/pdindex.htm From the ‘heart’ of crystallography

3. Pathway to understanding symmetry & crystal structures If this page makes you a little queasy- skip it! Individual (simple) Symmetry Operators A These are the ‘tools’ you would require for understanding lattices and crystal structures Compound & Combination of Symmetry Operators B How symmetry operators without translation combine Point Groups 7 Crystal Systems C 32 point groups in 3D Lattices have only 7 distinct point groups How symmetry operators with translation combine D Space Groups 14 Bravais Lattices Lattices have only 14 distinct space groups 230 space groups in 3D How to repeat? Part of the geometry which cannot come from symmetry (what to repeat!) Asymmetric Unit E Where to put your stuff (say atoms) to get a crystal from space groups F Wyckoff Positions

4. Why study symmetry? • Crystals are an important class of materials. • Crystals (and in fact quasicrystals) are defined based on symmetry.The symmetry being referred to in this context is geometrical symmetry. • Symmetry helps reduce the ‘infinite’ amount of information required to describe a crystal into a finite (preferably small) amount of information.* • Any property of a crystal will have at least the symmetry of the crystal  the Neumann Principle ( A property can have higher symmetry than the crystal). • One obvious manifestation of the symmetry inherent in a crystal, is the external shape of the crystal. • Symmetry (in conjunction with other elements) helps us define an infinite crystal in a succinct manner. (Infinite information is reduced to finite tangible information- will return to this soon). Note the facets KDP crystals grown from solution * More about how this ‘happens’ soon

5. Symmetry of What? • In crystallography (the language of describing crystals) when we talk of Symmetry; the natural question which arises is: Symmetry of What? • The symmetry under consideration could be of one the following entities: Lattice  Crystal  Motif  Unit cell(these are distinct and should not be confused with one another!) • When the symmetry is normally used, it is the symmetry of the crystal being referred to. Lattice Note the facets Motif Or the molecule Symmetry of the Crystal Unit Cell KDP crystals grown from solution Eumorphic crystal (equilibrium shape and growth shape of the crystal) The external shape of the crystal corresponds to the point group symmetry of the crystal

6. SYMMETRY • If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation. SYMMETRY OPERATOR • Given a general point a symmetry operator leaves a finite set of points in space • A symmetry operator closes space onto itself

7. Some of the concepts in this slide are ‘pretty’ advanced and can be learnt later via hyperlinks NOTES: • Presence of symmetry enables us to consider only a part of a object (or other entity  which could even be infinite) in conjunction with the symmetry operators (see coming slide for explanations) • All symmetry operators may not be required to understand/analyze/generate a structure ( but a few basic ones are) • The effects of many symmetry operators may be identical (especially in lower dimensions or when mirror symmetric objects are not involved) • Certain combination* of symmetry operators (without a translational component) can also leave a finite set of points$ and these are called the Point Groups We can have point groups in 1D, 2D, 3D or nD in general. • Certain combination* of symmetry operators involving translations can leave a periodic array of (finite set of) objects in space  the Space Groups We can have space groups in 1D, 2D, 3D or nD in general. * Only certain specific combinations are allowed which possess this property $ Given a general point

8. Why do I need to talk about symmetry and symmetry operators? • If the object, collection of objects, crystal etc. (which is under consideration) has some symmetry then the whole need not be described, but only a part can be described along with the symmetry operators. • For example consider a square (as below).  An infinite tiling of squares can be thought of as a single square repeated in x and y directions  Further one half of the square with a mirror plane (mirror line in 2D) can give the whole square.  Or a quarter of a square with two mirror planes or  a diagonal half of the quarter with three mirror planes. (note: mirror planes in 2D are lines)  Consider an infinite pattern made of squares m   m  This can be thought of as a single square repeated in x and y directions m • Else one could have considered a quarter of the object along with a 4-fold rotation operation (with symbol  and which rotates space by 90). Now imagine that the 1/8th triangle had a 1000 atoms in it- we will have to give the coordinates of a 7000 atoms less! Hence, we use the language of symmetry to be terse in our description (i.e. to supply minimum information)

9. Classification of Symmetry Operators Dimension of the Operator Takes an object to its mirror form or not Based on If the operator acts at a point or moves a point (i.e. outside a unit cell) If it plays a role in the shape of a crystal or not (Macroscopic/Microscopic)

10. Classification of Symmetry Operators Dimension of the Operator Based on Lower dimensional space

11. Classification of Symmetry Operators Takes an object to its mirror form or not Based on Translation Takes object to same form → Proper Type I Rotation Symmetries Roto-reflection Mirror Type II Inversion Roto-inversion Takes object to enantiomorphic form → improper (Mirror image form)

12. Classification of Symmetry Operators If it plays a role in the shape of a crystal or not (Macroscopic/Microscopic) Based on Rotation Mirror Influence the external shape of the crystal Macroscopic Inversion Symmetries Screw Axes Microscopic Glide Reflection Do not Influence the external shape of the crystal

13. Classification of Symmetry Operators If the operator acts at a point or moves a point Based on Points remain ‘localized’ and we land up with a finite number of points Rotation Roto-reflection Mirror Roto-inversion Does not move a point Inversion Symmetries (i.e. outside a unit cell) Translation Moves a point Screw Axes Points ‘move’ to create an infinite array Glide Reflection

14. Notation for representing left and right handed objects • To start with we use the notation as described below. (Occasionally deviating from this as well!). • Ultimately, we will turn to ‘International Tables of Crystallography’ symbols in b/w. +R R +L L

15. Translation • The translation symmetry operator (t) moves an point or an object by a displacement t or a distance t. • A periodic array of points or objects is said to posses translational symmetry. • Translational symmetry could be in 2D or 3D (or in general nD). • If we have translational symmetry in a pattern then instead of describing the entire pattern we can describe the ‘repeat unit’ and the translation vector(s). t   t         

16. Mirror and Inversion With note on left and right handed objects • The left hand of a human being cannot be superimposed on the right hand by mere translations and rotations • The left hand is related to the right hand by a mirror symmetry operation (m) (or the inversion as below). • The right hand is called the enantiomorphic form of the left hand • Another operator which takes objects to enantiomorphic forms is the inversion operator (i) (in the figure to the right below- between the two hands (in the mid-plane) at the centre is an inversion operator)` m Inversion operator Horizontal Mirror Vertical Mirror

17. Rotation Axis • Rotation axis rotates a general point (and hence entire space) around the axis by a certain angle • On repeated operation (rotation) the ‘starting’ point leaves a set of ‘identity-points*’ before coming into coincidence with itself. • As we are interested mainly with crystals, we are interested in those rotations axes which are compatible with translational symmetry → these are the (1), 2, 3, 4, 6 – fold axis. If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where: The rotations compatible with translational symmetry are  (1, 2, 3, 4, 6) Click here for proof  Crystals can only have 1, 2, 3, 4 or 6 fold symmetry * explained in an upcoming slide

18. Symbol for 2-fold axis =180 2-fold rotation axis n=2 Then the operation of the 2-fold leaves two points Symbol for 3-fold axis =120 3-fold rotation axis n=3

19. n=4 4-fold rotation axis =90 n=6 6-fold rotation axis =60

20. ‘Putting together’ two symmetry operators • Symmetry operators can be put together in two ways: (i) in combination, (ii) as a compound. • In a combination the ‘individuality’ of the symmetry operators is not lost. • In a compound two symmetry operators perform ‘as-if’ they are a single operator (their ‘individuality’ is not fully expressed). The component operations express themselves and should be compatible with lattice translation Combination How can symmetry operators be put together? Compound Two symmetry operations performed in sequence as a single event → identity of the individual operators lost

21. Compound Symmetry Operators • We have so far considered: Translation, mirror, inversion & rotation operators. These are simple symmetry operators. • Roto-inversion and Roto-reflection are compoundsymmetry operators which do not involve translation → both these take left handed objects to right handed forms For generating point groups (to be considered later) one of the two operators is sufficient and hence we will consider roto-inversion only in future considerations. • Glide reflection and Screw are compound symmetry operators involving translation → (Only) Glide reflection takes left handed objects to right handed form • It is important to note in these operations the ‘complete’ compound operator acts before leaving a identity-point (i.e. Roto-inversion is NOT rotation followed by a inversion). • In some cases these compound operators can be broken down into a combination of two operators.  In a combination (unlike a compound) the individual operators express themselves fully – i.e. the first operator acts first and then the second acts on the result of the first operation.

22. Compound symmetry operator Roto-inversion • A roto-inversion operator rotates a point/object and then inverts it (inversion operation) in one go. • A left handed object will be taken to its right handed form by the operation. • We will only consider 1,2,3,4,6 - fold rotations (crystallographic) as a part of the roto-inversion operation. Roto-inversion operations Compatible with translational symmetry

23. Compound symmetry operator Screw Axis • A screw (axis) operator rotates a point/object and then moves it a fraction of the repeat distance in one go. • The faction which the screw axes move is called the Pitch of the screw. • We will only consider (1, 2, 3, 4, 6) - fold rotations (crystallographic) as a part of the screw axes. • The screw axes to be considered are: 21  31, 32  41, 42, 43 61, 62, 63, 64, 65 • The normal and screw axis both give the same effect on the external symmetry of the crystal. • All identity points have the same enantiomorphic form (i.e. all objects created by the screw operator are all either left-handed or all are right-handed)

24. The 32 axis produces a rotation of 120 along with a translation of 2/3. • The set of points generated are: (0,0) (120,2/3) (240,4/31/3) (360,6/32)… • This is equivalent to a left handed screw (LHS) of pitch 1/3 Note: these are not fraction calculations! m

25. The 43 axis is a RHS with a pitch of 3/4 • The set of points generated are: (0,0) (90,3/4) (180,6/42/41/2) (270,9/41/4)… • The effect of 43 axis can be thought of as a LHS with a pitch of 1/4 Note: these are not fraction calculations!

26. The 42axis generates the following set of points:(0,0) (90,1/2) (180,2/21) (270,3/21/2) (360,4/22) • The grey arrowhead maps the (270,3/2) point to (270,1/2)→ to keep points within unit cell Note: these are not fraction calculations!

28. Compound symmetry operator Glide Reflection • A glide (reflection) operator move a point/object by a fraction of the repeat distance and reflects the object in one go. • Kinds of ‘Glides’ are considered in crystallography: Axial Glide (a, b, c) →  Diagonal Glide (n) →  Diamond Glide (d) →

29. Different type of glides

30. Complete set of symmetry operators 3 numbers each 3 numbers each

31. Point Groups and Space Groups • We have so far considered various types of symmetry operators- those with translation and those without (keeping our focus on those related to crystals). • The symmetry operators without translation (rotation, inversion, mirror, roto-inversion, roto-reflection) leave a finite number of identity-points and even those involving translation (glide and screw) leave a finite number of identity-points within the unit cell. • Symmetry operators which do not involve translation can combine with one another in certain specific ways so as to leave a finite number of identity-points (i.e. arbitrary combinations are not possible).  The number of such possible combinations (along with ‘single’ symmetry operators) is 32 and these are called the 32 Point Groups. One such combination is 4mm*  An example of a disallowed combination is 22 (with an included angle of (say) 15)*. • There are 7 distinct point group symmetries of lattices (14 Bravais Lattices) which correspond to the 7Crystal Systems. • When all symmetry elements are allowed to combine- including those with translation- then we end up with 230space groups. • There are 14 distinct space group symmetries of Lattices → the 14 Bravais Lattices Leave at least one point unmoved * Considered in upcoming slides We shall not formally derive the 32 point groups or the 230 space groups- interested readers may consult Elementary Crystallography by M.J. Buerger

32. Allowed combinations • As mentioned before only some combinations of symmetry operators are allowed. • 4mm is an allowed combination (as below) provided that the two mirrors are at 45 and the line of intersection of the mirror is the line of the 4-fold axis. When ever we write a symbol for a combination (according to the Hermann–Mauguin notation)- the symbol has a precise meaning w.r.t to the relative orientation of the component operators. • As shown below 2-fold axes with an included angle of 30 is an allowed combination → leading to point group 622 → Starting with just the two 2-fold axes- by repeated action of the two folds twelve 2-fold axes are created → which automatically implies that a 6-fold is perpendicular to the two 2-folds! Active 2-fold is in red 622 The 2-folds have been coloured differently to understand the origin of the 6-fold 622 4mm Right Handed Left Handed

33. Disallowed combinations • Most of the possible combinations of symmetry elements are actually disallowed! If we randomly chose two rotation axes and put them at some random angle- more likely than not that would be a disallowed combination (note that there are only 22 allowed combinations → along with the single operators (10 in number) we get the 32 point groups) • As shown below two 2-fold axis with an included angle of (say) 15 is a disallowed combination → this is because the presence of two 2-folds with an included angle of 15 implies the presence of a 12 fold perpendicular to the plane of the 2-folds → which is a disallowed rotational symmetry in crystallography. • Another example of a combination which is disallowed is (say) two 2-fold axes with an included angle of 7 (360 is not divisible by 7!). In this case: the action of one two fold on the other repeatedly, would lead to an infinite number of two folds on the plane and hence an infinite number of points (if we start with one point) (i.e. space would not close on itself!).

34. Minimum set of symmetry operators required to create point groups • Though there are a number of symmetry operators, we have already seen a redundancy in their combined action. E.g. 6 = 3/m. • This reaises the question: what is the minimum number of symmetry opertors required to create all possible combinations (i.e. the 32 point groups)? • The answer is we can create all the 32 point groups using Only Rotations and Roto-inversions! R  Rotation 32 Point Groups R  Roto-inversion

35. Point group symmetry of Lattices → 7 The 7 crystal systems Symmetries actingat a point 32 point groups R R Along with symmetrieshaving a translation G + S 230 space groups Space group symmetry of Lattices →14 The 14 Bravais lattices R + R → rotations compatible with translational symmetry (1, 2, 3, 4, 6)

36. The 32 Point Groups Highest symmetry class is in blue The possible combinations of crystallographic symmetry operators Point groups on Blue are Holohedral symmetry classes (highest symmetry for a crystal system)→ these 7 point groups are the only possible symmetries of lattices

37. Identity Points/Objects • If we start with a general point, then the operation of symmetry operator(s) will leave a (finite) set of points. These symmetrically related set of points are called identity points. • An extension of the concept of Identity points is to use identity objects which can show left or right handedness. (Some examples are shown below). • The number of identity points is the Order of the symmetry operator or the point group and is a ‘measure of the symmetry’ of the point group (/operator). Alternate diagram 4-fold leaves 4 identity points 4mm Right Handed 4mm Left Handed 4mm point group leaves 8 identity points: 4 left handed (orange circle) and 4 right handed (green circle) Right Handed Left Handed

38. Concept of Sub-group • A point group of higher symmetry may contain within it the operations of one of more point group(s) of lower symmetry. The lower symmetry point group(s) are called Subgroup(s) of the higher symmetry point group. • E.g.  the 4 point group contains the operations of 2 point group the 4mm point group contains the operations of the point groups 4, 2, m 4-fold contains 2-fold Click here to know more

39. How do we go from a space group to a crystal? Why space groups at all? Why not work with Lattice + Motif picture? Click here • The Space Groupgives us a distribution of symmetry elements in space.(Given this distribution some points in space have a higher symmetry than others.) • If the Asymmetric Unitis used as a tile, then this tile in conjunction with the space group can fill entire space. Like unit cell (as a tile) in conjunction with basis vectors can fill entire space. • Wyckoff Positions for atomic species distribute (put) the atomic entities with respect to the symmetry operators. Wyckoff positions specify Site Symmetry and Occupancy by entities (usually atomic species) Further values for variables in Wyckoff table (x,y,z) have to be specified. • Obviously there is noScale in ‘symmetry related stuff’→ scale has to be added in via the Lattice Parameters (Unit Cell Parameters → Lengths and Angles consistent with the space group). Making a Crystal Space Group+ Asymmetric Unit + Wyckoff Positions + Lattice Parameters Consistent with the crystal system Site symmetry, Values for variables & Occupancy Asymmetric Unit is that part of the crystal which cannot be generated using symmetry operators→ “Crystal  Symmetry = Asymmetric Unit”

40. Positioning a object with respect to the symmetry elements In this part we briefly consider the effect of positioning an object with respect to the distribution of symmetry elements • As seen in the example of 4mm point group- placing an object in special positions reduces the number of identity-points/’objects’ produced by the point group. General site  8 identity-points (4R, 4L) On mirror plane (m1)  4 identity-points On mirror plane (m2)  4 identity-points Site symmetry 4mm 1 identity-point Object has to have mirror symmetry (bilateral symmetry) Note: this is for a point group and not for a lattice  the black lines are not unit cells

41. Positioning a object with respect to the symmetry elements • So it is clear that merely specifying the symmetry operations (along with their distribution) is not sufficient to generate a pattern. We have to know where the entity (say atoms) are positioned with respect to the symmetry operators in the unit cell. • It is also clear that if an object is placed on a (or a combination of) symmetry operator(s), it has to be compatible with the symmetry operator(s). E.g.  a left handed object cannot be placed on a mirror plane or  a rectangular object cannot be placed on a 4 fold axis or a square can be placed on a 4mm site only if the diagonal of the square coincides with a mirror • This positioning also determines the number of such entities within the unit cell → called the Multiplicity.  Higher the site symmetry → lower will be the number of entities in the unit cell • Wyckoff had developed a notation to label the symmetry positions Each site (with a certain symmetry) is labeled with a alphabet. The labeling starts with the highest symmetry sites (‘a’ for highest symmetry, then ‘b’ …) Many different sites with differing symmetry may have the same multiplicity → but they will have different ‘Wyckoff’ labels.

42. Positioning of a motif w.r.t to the symmetry elements of a lattice  Wyckoff positions A 2D lattice with symmetry elements

43. Any site of lower symmetry should exclude site(s) of higher symmetry [e.g. (x,x) in site f cannot take values (0,0) or (½, ½)] 4mm f g e b d c a Number of Identi-points

44. Within a line or a region special points/lines of higher symmetry have to be excluded. f g e b d c a f Exclude thesepoints Exclude thesepoints d e Exclude thesepoints

45. Identify the 2D space group (plane group) of the crystal below. What is the Wyckoff symbol for each of the species? What is the Stoichiometry of the crystal? Solved Example A B C Unit cell

46. Looking at the space group table we can see that it matches with p4mm space group. First we overlay the symmetry operators Stoichiometry by inspection and by the Wyckoff symbols isA1 B1 C4 Asymmetric unit Can we have fractional numbers for Wyckoff occupancy? Slide 15

47. Effect of decoration of a lattice on the symmetry We briefly consider this aspect here- details can be found in the topics on Geometry of Crystals and Making Crystals. • An Infinite Lattice can be represented by a Unit Cell. • On decorating the lattice with objects the symmetry of the lattice may be: Maintained lowered • A special type of object (/collection of objects), which is repeated identically (in shape, orientation colour etc.) at each lattice point is called a Motif.

48. Consider a square (which could also function as a unit cell of a crystal if decorated with a motif) • The square shape (and also the collection of four points in the corners of a square) have some basic symmetries as shown below Complete set of symmetries i Square Which can be written as 4 m1 m2 i Symmetries 4-fold Which can be further abbreviated as 4mm mv = m1 md = m2 4 points at the vertices of a square

49. Effect of the decoration  a 2D example Two kinds of decoration are shown  (i) for an isolated object, (ii) an object which can be an unit cell. Redundant mirrors which need not be drawn 4mm Redundant inversion centre Can be a unit cell for a 2D crystal 4mm Decoration retaining the symmetry This is not a Motif as it is not repeated identically at each point Motif

50. m mm m Possible UCs of Crystals Motifs