Lectures 1&2_7f2f27b4-d81d-4225-9e4a-52ebc9e0bf90 (1).pptx

CONDENSED MATTER PHYSICA

1. SEE625 Structural, Microstructural and Spectroscopic Characterization of Materials Ashish Garg Srinivas Yadavalli Sustainable Energy Engineering [email protected] [email protected] Contact:

2. Course Objectives For students from diverse backgrounds working on materials science questions How materials appear at different length scales? Micro/nano-structure Order at atomic length scale Defects How do we “characterize” structure and composition of materials? Imaging: Optical, SEM, TEM, AFM, STEM Diffraction: XRD, Electron Diffraction Spectroscopy: EDS, EELS, EPMA, SIMS Optical: UV-Vis, Ellipsometry Advanced Characterization Techniques Bond Characterization: XPS, Raman, FTIR Thermal Analysis: DSC, DTA, TGA

3. Course Details Assignments (Bi-weekly): 20% Project: 20% Study a characterization method in depth Five-page report and a 10 min presentation Mid-semester Exam: 30% End-semester Exam: 30% Grading Three hours per week: 3.30 pm-5.00 pm (Tuesday, Thursday)

4. Why Characterize Materials? Every atom, molecule, and their order has a story to tell! Austenite Slow Cooling Moderate Cooling Rapid Cooling Pearlite Bainite Martensite Vary the processing conditions of steel to control structure, properties (Steel At 800 ˚C) Control the structure at different length scales Very Hard, Brittle Softer, Ductile Moderate Hardness, Tough Structural Beams Automotive components Cutting Tools, Wear Resistant

5. Structure-Property Correlations Different arrangement of atoms leads to different properties Allotropes of Carbon Diamond Graphite Amorphous Carbon Hard and Transparent Variable Properties Opaque and Soft Crystalline Materials Amorphous Materials (Well-defined long range order) (No order) TEM Imaging to See Atoms

6. Detecting Defects in Crystals Local Deviations in Long Range Order are Called Grain Boundaries 10 nm Individual Grains are Single Crystalline Disorder at GBs GBs can render a complete different set of properties for a material GB density is again controlled by the way we make the material Same Material with Different Grain Sizes have Different Optical Properties (More Examples in Future Lectures) Ordering and Orientation of Atoms are Fundamental to Determining a Material’s Behavior

7. Detecting Phase Characteristics and Boundaries A high resolution TEM image of alternating GaAs and AlGaAs layers (Simone Montanari PhD thesis (2005)) http://www.doitpoms.ac.uk/tlplib/epitaxial-growth/printall.php TEM Image Epitaxial Layers Similar Crystal Structure and interatomic distances in GaAs and AlAs Orientation of Grains Different for the Two Phases Phases and order identified using Electron Diffraction, EELS (while visualizing under a TEM) Chapter 1: Crystal Structure and Bravais Lattices

8. Crystal Structure and Bravais Lattices Lattice, Basis (Motif) Unit Cell Primitive and Non-Primitive Cells Crystal Systems Directions, Planes, Miller Indices, Weiss zone law

9. Crystal structure = Bravais lattice + atomic basis Bravais lattice: an infinite array of discrete points that appears the same regardless of the point from which it is viewed Atomic basis: set of atoms associated with each lattice point Crystal Structure and Bravais Lattices No regular arrangement Regular arrangement with identical neighbourhood (single atom basis) Regular arrangement with identical neighbourhood (three atom basis)

10. A B No, A and B don ’ t have identical neighbourhood A B Yes Is it a Point Lattice? Is it a Point Lattice? Detecting Lattice Lattice is a regular arrangement of points in space with identical neighbourhood

11. Basis can consist of one molecule or more than one molecule depending on their relative orientation with respect to each other provided a group of them makes a repeatable pattern. Identifying Basis

12. Motif (A-B) Motif (2A-2B) A 2 B 2 is the repeatable unit AB is the repeatable unit The atomic units (atom basis) are hung along the underlying framework, the lattice. Examples:

13. Bravais lattice Atomic basis Examples:

14. The smallest repeating unit that, when translated through space, recreates the entire lattice. The unit cell is defined by its lattice vectors, which describe the edges of the cell. Unit Cell Bravais lattice: Infinite set of points that look identical from each point In 2D: Need 2 independent vectors: a 1 , a 2 . These are called the basis vectors. Lengths are called lattice parameters ( a,b ) Parallelogram created by basis vectors is called unit cell Different choices of basis vectors possible For defining a Bravais lattice, choose cell with: smallest dimensions, highest symmetry a 1 a 2 

15. Bravais Lattice Types 1) Oblique lattice a 1  a 2 ,  90 deg 2) Rectangular lattice a 1  a 2 ,  =90 deg  a 2 a 1 3) Square lattice a 1 = a 2 ,  =90 deg 4) Hexagonal lattice a 1 = a 2 ,  =120 deg  a a a  a  a 2 a 1 Lattice: A general term for a regular, repeating arrangement of points. Bravais Lattice: A specific type of lattice used in crystallography, characterized by its periodic arrangement and the smallest repeating unit that fills space.

16. 5 th lattice is not primitive Consider this set of points: Could be oblique lattice But there’s also another lattice with higher symmetry

17. What do we mean by symmetry? A symmetry operation is one that we can perform on a system that transforms it into something that is identical to the starting configuraiton Point symmetry – keep one point fixed Symmetry operations in 2-D 1) Rotation e.g., 2-fold rotation: rotate about an axis twice (by 180 deg) e.g., 3-fold rotation: rotate about an axis 3 times Only 1,2,3,4 and 6-fold rotation consistent with filling space

18. Symmetry operations in 2-D 2) Mirror plane x,y  - x,y (like looking at object in mirror) x,y -x,y 3) Center of symmetry (inversion) x,y  -x,-y x,y -x,-y This object doesn’t have inversion symmetry But does have mirror plane along 45 deg axis This object doesn’t have mirror symmetry for mirror plane along x or y axis Lay inverted object on top of itself – not the same

19. 1) Oblique lattice 2-fold rotation only 2) Rectangular lattice 2-fold rotation 2 mirror planes 3) Square lattice 2- and 4-fold rotation 4 mirror planes 4) Hexagonal lattice 2-, 3- and 6-fold rotation 4 mirror planes Point symmetry of 2D Bravais lattices (all have inversion symmetry)

20. If we describe it as oblique it only has 2-fold symmetry Go back to 5th lattice If we describe it as rectangular C it has 2-fold rotational symmetry and 2 mirror planes Using centered lattice let’s us show symmetry in system Every lattice in 2D is either: oblique, rectangular, square, hexagonal or rectangular C

21. Crystal Structure and Bravais Lattices Lattice, Basis (Motif) Unit Cell Primitive and Non-Primitive Cells Crystal Systems Directions, Planes, Miller Indices, Weiss zone law

22. Five Bravais Lattices in 2D Space 1) Oblique lattice 2-fold rotation only 2) Rectangular lattice 2-fold rotation 2 mirror planes 3) Square lattice 2- and 4-fold rotation 4 mirror planes 4) Hexagonal lattice 2-, 3- and 6-fold rotation 4 mirror planes

23. If we describe it as oblique it has only 2-fold symmetry 5th lattice… If we describe it as rectangular C it has 2-fold rotational symmetry and 2 mirror planes Using centered lattice let’s us show symmetry in system Every lattice in 2D is either: oblique, rectangular, square, hexagonal or rectangular C

24. Key definitions Lattice: An infinite array of discrete points that appears the same regardless of the point from which it is viewed. Just translate along any of the basis vectors (edges of corresponding unit cell), the pattern should repeat Unit Cell: The smallest repeating unit that, when translated through space, recreates the entire lattice. The unit cell is defined by its lattice vectors, which describe the edges of the cell. Primitive and Non-Primitive Cells: A primitive cell is the smallest repeating unit in a crystal lattice containing exactly one lattice point, while a non-primitive (conventional) cell is a larger unit chosen to reflect the full symmetry of the crystal, often containing multiple lattice points. a 1 a 2

25. Questions: Identify the rotational symmetry axes and mirror planes in the various the 2D Bravais lattices What is the 2D Bravais lattice and atom basis associated with the crystal structure (b)? Is it a primitive cell? Mention two examples each of primitive and non-primitive cells possible in this structure. Identify the Bravais lattice and atom basis for the arrangement (a) (a) (b) How many crystal systems are defined in 2D space?

26. 3-D Space Bravais lattice in 3-dimensions Reminder: infinite set of points that look identical from each point To describe the Bravais lattice, need three independent vectors: a 1 , a 2 and a 3 These are called the basis vectors Parallelopiped created by basis vectors is called unit cell Origins of unit cells at R = n 1 a 1 + n 2 a 2 + n 3 a 3 where n’s are all integers Unit cells fill up all space (no gaps) a 1 a 2 a 3

27. Lengths of basis vectors are called lattice parameters: a, b, c Different angles between basis vectors: , ,  Different arrangements of lengths and angles form 7 crystal systems | a 3 | = c | a 1 | = a | a 2 | = b    Cubic a = b = c  =  =  = 90 Hexagonal a = b ≠ c  =  = 90,  = 120 Tetragonal a = b ≠ c  =  =  = 90 Rhombohedral a = b = c  =  =  ≠ 90 (or trigonal) Orthorhombic a ≠ b ≠ c  =  =  = 90 Monoclinic a ≠ b ≠ c  =  = 90,  ≠ 90 Triclinic a ≠ b ≠ c  ≠  ≠  ≠ 90

28. Different crystal systems distinguished by unit cell shape and symmetry Symmetry operation – after operation performed, system is same as when it started e.g., rotation about an axis Tetragonal system a = b ≠ c  =  =  = 90 4-fold rotational axis Symmetric w.r.t. rotation by 90 deg 2-fold rotational axes Symmetric w.r.t. rotation by 180 deg Properties of material have same symmetry (or greater) than crystal structure

29. Make Bravais lattices from the crystal systems If there is only one lattice point in the unit cell, it is called primitive (P). In addition, for some crystal systems there can be more than one lattice point in the unit cell: a 1 a) Base-centered (C): lattice point at center of one set of faces (orthorhombic, monoclinic) Extra lattice point at ½ a 1 + ½ a 2 b) Body-centered (I) or inner centered: lattice point at center of unit cell (cubic, tetragonal, orthorhombic) Extra lattice point at ½ a 1 + ½ a 2 + ½ a 3 a 2 a 3 c) Face-centered (F): lattice point at center of each face of the unit cell (cubic, orthorhombic) Extra lattice points at: ½ a 1 + ½ a 2 ½ a 1 + ½ a 3 ½ a 2 + ½ a 3 Always possible to choose a set of basis vectors so that unit cell is primitive

30. Why use centering? Every lattice could be described in terms of 3 basis vectors (primitive lattice)  But this would obscure the symmetry e.g., centered rectangular lattice in 2d In 3D, consider FCC lattice Has lattice points on corners of cube and all faces Note: this is a Bravais lattice Looks identical from all the lattice points Has symmetry of cube Primitive basis vectors and unit cell shown in red Doesn’t have symmetry of cube (no 4-fold axes) What are primitive basis vectors for BCC lattice?

31. Not all centering possible for every lattice - Some are redundant (creates same lattice with different basis) E.g., base-centered tetragonal  rotated primitive tetragonal  Or symmetry changes to another lattice (e.g., 1-face-centered tetragonal)  lose 4-fold symmetry axis might qualify as a orthorhombic Bravais lattice? - Or isn’t lattice (consider 2-face centered orthorhombic) 

32. Centering only allowed for certain crystal systems 1) Cubic: P, I and F 2) Tetragonal: P and I

33. Centering only allowed for certain crystal systems 3) Orthorhombic: P, C, I and F 4) Monoclinic: P and C

34. Results is 14 Bravais lattices All crystals can be described in terms of one of these lattices and atomic basis http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Lecture1/Lec1.html#anchor2

35. Planes and Directions Essential for the completeness of crystal structures Millers Indices (in the names of William Hallowes Miller) Crystallographic Planes Identification of various faces seen on the crystal ( h,k,l ) for a plane or { h,k,l } for identical set of planes; h, k, l are integers Directions Atomic directions in the crystal [ u,v,w ] for a direction or < u,v,w > for identical set of directions u, v, w are integers

36. Determination of a Crystal Plane A crystallographic plane in a crystal satisfies following equation a/h, b/k, and c/l are the intercepts of the plane on x, y, and z axes. a,b,c are the unit cell lengths h, k, l are the integers called as Miller indices and the plane is represented as (h, k, l)   Unit Cell Parameters: 4A, 8A and 3A Reciprocal of fractional intercepts: 2, 4/3, 1 Convert to smallest set of integers: (6, 4, 3) Fractional intercepts: 2A/4A, 6A/8A, 3A/3A

37. Crystal Planes Interplanar angle is given by (cubic only) Interplanar spacing is given by (Cubic)

38. Denoted as [ u,v,w ] and family of directions in the form < uvw > Vector components of the direction resolved along each of the crystal axis reduced to smallest set of integers Crystal Directions [1 1 1] X Y Z [101] X Y Z [231] We want the vector to touch the surface of the cube: so divide the miller indices by the highest value index For [231], the line originating from origin should meet the point 2a/3 along x-axis, b along y-axis, c/3 along z-axis Plane with h,k,l miller indices is normal to the direction with h,k,l miller indices

39. Family of Directions and Planes [1 1 1] X Y Z [1 1 1] [1 1 1] X Y Z [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] belong to the family of directions <111> Directions and planes of the same family have their indices forming permutations You can similarly visualize for the family of <110> X Y Z ( 1 1 1 ) ( 1 1 1 ) ( 1 1 1) ( 1 1 1 ) ( 1 1 1) belong to the family of directions {111} You can similarly visualize for the family of {110}

40. a 1 a 3 a 2 c Hexagonal system Directions [ uvw ] [UV T W]   We need to define a new 4 indices system. Why? [100] [010] [110] With the 3-indices notation, the equivalent directions cannot be denoted appropriately. E.g., [100] [010] [110]   Question: Convert [111] to four-index notation and mark on the unit cell!

41. Planes ( hkl ) becomes ( hkil ) h+k =- i Hexagonal system: Crystallographic Planes a 1 a 3 a 2 c ( 1 00) ( 1 1 ) ( 1 10 ) ( 10 1 ) Question: Draw (1122) plane! {10 1 0} family of planes

42. Weiss Zone Law Condition for a direction [U V W] to lie on a plane (h k l) h.U + k.V + l.W = 0 U = k 1 l 2 − k 2 l 1 V = l 1 h 2 − l 2 h 1 W = h 1 k 2 − h 2 k 1 A direction (UVW) common to two planes (h 1 k 1 l 1 ) and (h 2 k 2 l 2 ), called as zone axis can be found as X Z Find the direction common to planes ( 1 1 1 ) and Y ( 1 1 1) Use Weiss Zone Law as well derive the solution!