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Constructing crystals in 1D, 2D & 3D Understanding them using the language of:  Lattices  Symmetry

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.

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Constructing crystals in 1D, 2D & 3D Understanding them using the language of:  Lattices  Symmetry

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  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 LET US MAKE SOME CRYSTALS Constructing crystals in 1D, 2D & 3D Understanding them using the language of: Lattices  Symmetry Additional consultations http://cst-www.nrl.navy.mil/lattice/index.html

  2. 1D

  3. Making a 1D Crystal • Some of the concepts are best illustrated in lower dimensions  hence we shall construct some 1D and 2D crystals before jumping into 3D • A strict 1D crystal = 1D lattice + 1D motif • The only kind of 1D motif is a line segment(s) (though in principle a collection of points can be included). Lattice + Motif = Crystal

  4. Other ways of making the same crystal • We had mentioned before that motifs need not sit on the lattice point- they are merely associated with a lattice point • Here is an example: Note: For illustration purposes we will often relax this strict requirement of a 1D motif We will put 2D motifs on 1D lattice to get many of the useful concepts across 1D lattice +2D Motif* It has been shown that 1D crystals cannot be stable!! *looks like 3D due to the shading!

  5. Each of these atoms contributes ‘half-atom’ to the unit cell

  6. Time to brush-up some symmetry concepts before going ahead Lattices have the highest symmetry (Which is allowed for it) Crystals based on the latticecan have lower symmetry In the coming slides we will understand this IMPORTANT point If any of the coming 7 slides make you a little uncomfortable – you can skip them (however, they might look difficult – but they are actually easy)

  7. As we had pointed out we can understand some of the concepts of crystallography better by ‘putting’ 2D motifs on a 1D lattice. These kinds of patterns are called Frieze groups and there are 7 types of them (based on symmetry). Progressive lowering of symmetry in an 1D lattice illustration using the frieze groups Consider a 1D lattice with lattice parameter ‘a’ Asymmetric Unit Unit cell a • Asymmetric Unitis that part of the structure (region of space), which in combination with the symmetries (Space Group) of the lattice/crystal gives the complete structure (either the lattice or the crystal) • (though typically the concept is used for crystals only) The concept of the Asymmetric Unit will become clear in the coming slides The unit cell is a line segment in 1D  shown with a finite ‘y-direction’ extent for clarity and for understating some of the crystals which are coming-up

  8. This 1D lattice has some symmetries apart from translation. The complete set is: • Translation (t) • Horizontal Mirror (mh) • Vertical Mirror at Lattice Points (mv1) • Vertical Mirror between Lattice Points (mv2) • Note: • The symmetry operators (t, mv1, mv2) are enough to generate the lattice • But, there are some redundant symmetry operators which develop due to their operation • In this example they are 2-fold axis or Inversion Centres (and for that matter mh) t mh mv1 mv2 mmm Or more concisely mh mmm The intersection points of the mirror planesgive rise to redundant inversion centres (i) Three mirror planes mv1 mv2 mirror

  9. Note of Redundant Symmetry Operators t mmm Three mirror planes Redundant inversion centres Redundant 2-fold axes • It is true that some basic set of symmetry operators (set-1) can generate the structure (lattice or crystal) • It is also true that some more symmetry operators can be identified which were not envisaged in the basic set  (called ‘redundant’) • But then, we could have started with different set of operators (set-2) and call some of the operators used in set-1 as redundant •  the lattice has some symmetries  which we call basic and which we call redundant is up to us! How do these symmetries create this lattice? Click here to see how symmetry operators generate the 1D lattice mirror

  10. Asymmetric Unit • We have already seen that Unit Cell is the least part of the structure which can be used to construct the structure using translations (only). • Asymmetric Unit is that part of the structure (usually a region of space), which in combination with the symmetries (Space Group) of the lattice/crystal gives the complete structure (either the lattice or the crystal) (though typically the concept is used for crystals only) • Simpler phrasing: It is the least part of the structure (region of space) which can be used to build the structure using the symmetry elements in the structure (Space Group) Asymmetric Unit + mv2 + mh Lattice point Which is theUnit Cell Unit cell If we had started with the asymmetric unit of a crystal then we would have obtained a crystal instead of a lattice + t Lattice

  11. Decoration of the lattice with a motif  may reduce the symmetry of the crystal t 1 mmm Decoration with a “sufficiently” symmetric motif does not reduce the symmetry of the lattice Instead of the double headed arrow we could have used a circle (most symmetrical object possible in 2D) t 2 mm Decoration with a motif which is a ‘single headed arrow’ will lead to the loss of 1 mirror plane mirror

  12. t 3 mg Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centrethe translational symmetry has been reduced to ‘2a’ t ii 4 2 inversion centres glide reflection mirror

  13. t 5 m 1 mirror plane t g 6 1 glide reflection translational symmetry of ‘2a’ t 7 No symmetry except translation glide reflection mirror

  14. 2D Video: Making 2D crystal using discs

  15. Making a 2D Crystal • Some aspects we have already seen in 1D  but 2D many more concepts can be clarified in 2D • 2D crystal = 2D lattice + 2D motif • As before we can relax this requirement and put 1D or 3D motifs! • We shall make various crystals starting with a 2D lattice and putting motifs and we shall analyze the crystal which has thus been created Continued

  16. Square Lattice Circle Motif + = Square Crystal Continued…

  17. Square Lattice Circle Motif + = Square Crystal Symmetry of the lattice and crystal identical  Square Crystal Including mirrors 4mm Continued…

  18. Important Note > Symmetry of the Motif Symmetry of the lattice Hence Symmetry of the lattice and Crystal identical (symmetry of the lattice is preserved)  Square Crystal Symmetry of the Motif • Any fold rotational axis allowed! (through the centre of the circle) • Mirror in any orientation passing through the centre allowed! • Centre of inversion at the centre of the circle

  19. What do the ‘adjectives’ like square mean in the context of the lattice, crystal etc? Funda Check • Let us consider the square lattice and square crystal as before. • In the case of the square lattice → the word square refers to the symmetry of the lattice(and not the geometry of the unit cell!). • In the case of the square crystal → the word square refers to the symmetry of the crystal(and not the geometry of the unit cell!)

  20. = Square Motif + Square Lattice Square Crystal Continued…

  21. Important Note = Symmetry of the Motif Symmetry of the lattice Hence Symmetry of the lattice and Crystal identical  Square Crystal 4mm Symmetry of the Motif • 4mm symmetry Continued…

  22. If the Symmetry of the Motif  Symmetry of the Lattice The Symmetry of the lattice and the Crystal are identical Important Rule i.e. Symmetry of the lattice is NOT lowered  but is preserved Common surviving symmetry determines the crystal system

  23. In a the above example we are assuming that the square is favourably orientedAnd that there are symmetry elements common to the lattice and the motif = Square Motif + Square Lattice Square Crystal Rotated 4

  24. How do we understand the crystal made out of rotated squares? Funda Check • Is the lattice square → YES(it has 4mmsymmetry) • Is the crystal square → YES(but it has 4symmetry → since it has at least a 4-fold rotation axis- we classify it under square crystal- we could have called it a square’ crystal or something else as well!) • Is the ‘preferred’ unit cell square → YES(it has square geometry) • Is the motif a square → YES(just so happens in this example- though rotated wrt to the lattice) Infinite other choices of unit cells are possible → click here to know more

  25. Square Lattice Triangle Motif + = Square Crystal Rectangle Crystal Symmetry of the lattice and crystal different  NOT a Square Crystal m Here the word square does not imply the shape in the usual sense Continued…

  26. Only one set of parallel mirrors left Symmetry of the structure m

  27. Important Note < Symmetry of the Motif Symmetry of the lattice The symmetry of the motif determines the symmetry of the crystal  it is lowered to match the symmetry of the motif (common symmetry elements between the lattice and motif  which survive) (i.e. the crystal structure has only the symmetry of the motif left: even though the lattice started of with a higher symmetry)  Rectangle Crystal (has no 4-folds but has mirror) Symmetry of the Motif Note that the word ‘Rectangle’ denotes the symmetry of the crystal and NOT the shape of the UC • Mirror • 3-fold Continued…

  28. If the Symmetry of the Motif < Symmetry of the Lattice The Symmetry of the lattice and the Crystal are NOT identical Important Rule i.e. Symmetry of the lattice is lowered  with only common symmetry elements

  29. How do we understand the crystal made out of triangles? Funda Check • Is the lattice square → YES(it has 4mmsymmetry) • Is the crystal square → NO(it has only msymmetry → hence it is a rectangle crystal) • Is the unit cell square → YES(it has square geometry) (we have already noted that other shapes of unit cells are also possible) • Is the motif a square → NO(it is a triangle!)

  30. = Triangle Motif + Square Lattice Parallelogram Crystal Rotated Crystal has No symmetry except translational symmetry as there are no symmetry elements common to the lattice and the motif (given its orientation)

  31. Some more twists

  32. Square Lattice Random shaped Motif + In Single Orientation = Square Crystal Parallelogram Crystal Symmetry of the lattice and crystal different  NOT Square Crystal No Symmetry Except translation

  33. Square Lattice Random shaped Object + Randomly oriented at each point = Square Crystal Amorphous Material(Glass) Symmetry of the lattice and crystal different  NOT even a Crystal No Symmetry

  34. Is there not some kind of order visible in the amorphous structure considered before? How can understand this structure then? Funda Check • YES, there is positional order but no orientational order. • If we ignore the orientational order (e.g. if the entities are rotating constantly- and the above picture is a time ‘snapshot’- then the time average of the motif is ‘like a circle’) • Hence, this structure can be considered to be a ‘crystal’ with positional order, but without orientational order! Click here to know more

  35. Summary of 2D Crystals Click here to see a summary of 2D lattices that these crystals are built on

  36. From the previous slides you must have seen that crystals have: CRYSTALS Orientational Order Positional Order Later on we shall discuss that motifs can be: MOTIFS Geometrical entities Physical Property In practice some of the strict conditions imposed might be relaxed and we might call a something a crystal even if • Orientational order is missing • There is only average orientational or positional order • Only the geometrical entity has been considered in the definition of the crystal and not the physical property

  37. 3D

  38. Making a 3D Crystal • A strict 3D crystal = 3D lattice + 3D motif • We have 14 3D Bravais lattices to chose from • As an intellectual exercise we can put 1D or 2D motifs in a 3D lattice as well(we could also try putting higher dimensional motifs like 4D motifs!!) • We will illustrate some examples to understand some of the basic concepts (most of which we have already explained in 1D and 2D)

  39. Sphere Motif Simple Cubic (SC) Lattice + Graded Shading to give 3D effect Simple Cubic Crystal Unit cell of the SC lattice = • If these spheres were ‘spherical atoms’ then the atoms would be touching each other • The kind of model shown is known as the ‘Ball and Stick Model’

  40. To know more about Close Packed Crystals click here Sphere Motif + Body Centred Cubic (BCC) Lattice Atom at (½, ½, ½) Body Centred Cubic Crystal Atom at (0, 0, 0) = Unit cell of the BCC lattice Space filling model Central atom is coloured differently for better visibility So when one usually talks about a BCC crystal what is meant is a BCC lattice decorated with a mono-atomic motif Note: BCC is a lattice and not a crystal

  41. Sphere Motif Face Centred Cubic (FCC) Lattice + Close Packed implies CLOSEST PACKED Cubic Close Packed Crystal(Sometimes casually called the FCC crystal) Point at (½, ½, 0) Point at (0, 0, 0) Unit cell of the FCC lattice = Space filling model So when one talks about a FCC crystal what is meant is a FCC lattice decorated with a mono-atomic motif Note: FCC is a lattice and not a crystal

  42. More views All atoms are identical- coloured differently for better visibility

  43. Two Ion Motif Face Centred Cubic (FCC) Lattice + NaCl Crystal = Cl Ion at (0, 0, 0) Na+ Ion at (½, 0, 0) Note: This is not a close packed crystal Has a packing fraction of 0.67

  44. Two Carbon atom Motif(0,0,0) & (¼, ¼, ¼) Face Centred Cubic (FCC) Lattice + Diamond Cubic Crystal = Tetrahedral bonding of C (sp3 hybridized) It requires a little thinking to convince yourself that the two atom motif actually sits at all lattice points! Note: This is not a close packed crystal There are no close packed directions in this crystal either!

  45. Two Ion Motif Face Centred Cubic (FCC) Lattice + NaCl Crystal Cl Ion at (0, 0, 0) = Na+ Ion at (½, 0, 0) The Na+ ions sit in the positions (but not inside) of the octahedral voids in an CCP crystal  click here to know more Solved Example Note: This is not a close packed crystal Has a packing fraction of 0.67

  46. NaCl crystal: further points Click here: Ordered Crystals This crystal can be considered as two interpenetrating FCC sublattices decorated with Na+ and Cl respectively Inter-penetration of just 2 UC are shown here

  47. More views Coordination around Na+ and Cl ions

  48. Now we present 3D analogues of the 2D cases considered before:those with only translational symmetry and those without any symmetry The blue outline is NO longer a Unit Cell!! Triclinic Crystal(having only translational symmetry) Amorphous Material (Glass) (having no symmetry what so ever)

  49. Making Some Molecular Crystals • We have seen that the symmetry (and positioning) of the motif plays an important role in the symmetry of the crystal. • Let us now consider some examples of Molecular Crystals to see practical examples of symmetry of the motif vis a vis the symmetry of the crystal.(click here to know more about molecular crystals → Molecular Crystals) • It is seen that there is no simple relationship between the symmetry of the molecule and the symmetry of the crystal structure. As noted before: Symmetry of the molecule may be retained in crystal packing (example of hexamethylenetetramine) or May be lowered (example of Benzene)

  50. Funda Check • From reading some of the material presented in the chapter one might get a feeling that there is no connection between ‘geometry’ and ‘symmetry’. I.e. a crystal made out of lattice with square geometry can have any (given set) of symmetries. • In ‘atomic’ systems (crystals made of atomic entities) we expect that these two aspects are connected (and not arbitrary). The hyperlink below explains this aspect. Click here → connection between geometry and symmetry

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