1. CRYSTAL STRUCTURE&X-RAY DIFFRACTIONDr. Y. NARASIMHA MURTHY Ph.DSRI SAI BABA NATIONAL COLLEGE (Autonomous)ANANTAPUR-515001-A.P(INDIA)firstname.lastname@example.org
2. Classification of Matter
3. SolidsSolids are again classified in to twotypes Crystalline Non-Crystalline (Amorphous)
4. What is a Crystalline solid?A crystal or crystalline solid is a solidmaterial, whose constituent atoms,molecules, or ions are arranged in anorderly repeating pattern extending inall three spatial dimensions.So a crystal is characterized by regulararrangement of atoms or molecules
8. Amorphous Solid• Amorphous (Non-crystalline) Solid iscomposed of randomly orientated atoms ,ions, or molecules that do not formdefined patterns or lattice structures.• Amorphous materials have order only withina few atomic or molecular dimensions.
9. • Amorphous materials do not haveany long-range order, but they havevarying degrees of short-range order.• Examples to amorphous materialsinclude amorphous silicon, plastics,and glasses.• Amorphous silicon can be used insolar cells and thin film transistors.
11. What are the Crystal properties?o Crystals have sharp melting pointso They have long range positional ordero Crystals are anisotropic(Properties change depending on thedirection)o Crystals exhibit Bi-refringenceo Some crystals exhibit piezoelectric effect& Ferroelectric effect etc…also
12. What is Space lattice ?• An infinite array ofpoints in space,• Each point hasidenticalsurroundings to allothers.• Arrays arearranged exactlyin a periodicmanner.αabCB EDO Ayx
13. Translational Lattice Vectors – 2DA space lattice is a set ofpoints such that a translationfrom any point in the lattice bya vector;R = l a + m blocates an exactly equivalentpoint, i.e. a point with thesame environment as P . Thisis translational symmetry. Thevectors a, b are known aslattice vectors and (l,m) is apair of integers whose valuesdepend on the lattice point.
14. • For a three dimensional latticeR = la + mb +ncHere a, b and c are non co-planar vectors• The choice of lattice vectors is notunique. Thus one could equally well takethe vectors a, b and c as a lattice vectors.
15. Basis & Unit cell• A group of atoms or moleculesidentical in composition is called thebasisor• A group of atoms which describecrystal structure
16. Unit Cell• The smallest component of thecrystal (group of atoms, ions ormolecules), which when stackedtogether with pure translationalrepetition reproduces the wholecrystal.
18. 2D Unit Cell example -(NaCl)
19. Choice of origin is arbitrary - latticepoints need not be atoms - but unitcell size should always be the same.
20. This is also a unit cell -it doesn’t matter if you start from Na or Cl
21. This is NOT a unit cell even thoughthey are all the same - empty space isnot allowed!
22. In 2Dimensional space this is a unit cellbut in 3 dimensional space it is NOT
23. Now Crystal structure !!Crystal lattice + basis = Crystal structure• Crystal structure can be obtained byattaching atoms, groups of atoms ormolecules which are called basis (motif)to the lattice sides of the lattice point.
24. The unit cell and,consequently, theentire lattice, isuniquelydetermined by thesix latticeconstants: a, b, c,α, β and γ. Thesesix parameters arealso called as basiclattice parameters.
25. Primitive cell• The unit cell formed by the primitives a,band c is called primitive cell. A primitivecell will have only one lattice point. Ifthere are two are more lattice points it isnot considered as a primitive cell.• As most of the unit cells of various crystallattice contains two are more latticepoints, its not necessary that every unitcell is primitive.
26. Crystal systems• We know that a three dimensionalspace lattice is generated by repeatedtranslation of three non-coplanarvectors a, b, c. Based on the latticeparameters we can have 7 popularcrystal systems shown in the table
27. Table-1Crystal system Unit vector AnglesCubic a= b=c α =β =√=90Tetragonal a = b≠ c α =β =√=90Orthorhombic a ≠ b ≠ c α =β =√=90Monoclinic a ≠ b ≠ c α =β =90 ≠√Triclinic a ≠ b ≠ c α ≠ β ≠√ ≠90Trigonal a= b=c α =β =√≠90Hexagonal a= b ≠ c α =β=90√=120
28. Bravais lattices• In 1850, M. A. Bravais showed thatidentical points can be arrangedspatially to produce 14 types of regularpattern. These 14 space lattices areknown as ‘Bravais lattices’.
31. Coordination Number• Coordination Number (CN) : The Bravaislattice points closest to a given point arethe nearest neighbours.• Because the Bravais lattice is periodic, allpoints have the same number of nearestneighbours or coordination number. It is aproperty of the lattice.• A simple cubic has coordination number 6;a body-centered cubic lattice, 8; and a face-centered cubic lattice,12.
32. Atomic Packing Factor• Atomic Packing Factor (APF) isdefined as the volume of atomswithin the unit cell divided by thevolume of the unit cell.
33. Simple Cubic (SC)• Simple Cubic has one lattice point so itsprimitive cell.• In the unit cell on the left, the atoms at thecorners are cut because only a portion (inthis case 1/8) belongs to that cell. The rest ofthe atom belongs to neighboring cells.• Coordinatination number of simple cubic is 6.
35. Atomic Packing Factor of SC
36. Body Centered Cubic (BCC)• As shown, BCC has two latticepoints so BCC is a non-primitivecell.• BCC has eight nearest neighbors.Each atom is in contact with itsneighbors only along the body-diagonal directions.• Many metals (Fe, Li, Na.. etc),including the alkalis and severaltransition elements choose theBCC structure.
37. Atomic Packing Factor of BCC2 (0,433a)
38. Face Centered Cubic (FCC)• There are atoms at the corners of the unitcell and at the center of each face.• Face centered cubic has 4 atoms so itsnon primitive cell.• Many of common metals (Cu, Ni, Pb ..etc)crystallize in FCC structure.
39. Face Centered Cubic (FCC)
40. Atomic Packing Factor of FCCFCC0.74
41. HEXAGONAL SYSTEM A crystal system in which three equal coplanar axesintersect at an angle of 60, and a perpendicular tothe others, is of a different length.
42. TRICLINIC & MONOCLINIC CRYSTAL SYSTEMTRICLINIC & MONOCLINIC CRYSTAL SYSTEMTriclinic minerals are the least symmetrical. Theirthree axes are all different lengths and none of themare perpendicular to each other. These minerals arethe most difficult to recognize.Monoclinic (Simple)α = γ = 90o, ß ≠ 90oa ≠ b ≠cTriclinic (Simple)α ≠ ß ≠ γ ≠ 90oa ≠ b ≠ cMonoclinic (Base Centered)α = γ = 90o, ß ≠ 90oa ≠ b ≠ c,
43. ORTHORHOMBIC SYSTEMOrthorhombic (Simple)α = ß = γ = 90oa ≠ b ≠ cOrthorhombic (Base-centred)α = ß = γ = 90oa ≠ b ≠ cOrthorhombic (BC)α = ß = γ = 90oa ≠ b ≠ cOrthorhombic (FC)α = ß = γ = 90oa ≠ b ≠ c
44. TETRAGONAL SYSTEMTetragonal (P)α = ß = γ = 90oa = b ≠ cTetragonal (BC)α = ß = γ = 90oa = b ≠ c
45. Rhombohedral (R) or TrigonalRhombohedral (R) or Trigonal (S)a = b = c, α = ß = γ ≠ 90o
46. Crystal Directions• We choose one lattice point on the line as an origin, saythe point O. Choice of origin is completely arbitrary, sinceevery lattice point is identical.• Then we choose the lattice vector joining O to any point onthe line, say point T. This vector can be written as;R = la + mb + ncTo distinguish a lattice direction from a lattice point, thetriple is enclosed in square brackets [ ... ] is used. [l, m, n]• [l, m, n] is the smallest integer of the same relative ratios.
47. 210X = 1 , Y = ½ , Z = 0[1 ½ 0] [2 1 0]
48. Negative directions• When we write thedirection [n1n2n3]depend on the origin,negative directions canbe written as• R = l a + m b + n c• Direction must besmallest integers.
49. Examples of crystal directionsX = 1 , Y = 0 , Z = 0 ► [1 0 0]
50. Crystal Planes• Within a crystal lattice it is possible to identify setsof equally spaced parallel planes. These are calledlattice planes.• In the figure density of lattice points on each planeof a set is the same and all lattice points arecontained on each set of planes.baba
51. MILLER INDICES FORCRYSTALLOGRAPHIC PLANES• William HallowesMiller in 1839 was able togive each face a unique label of threesmall integers, the Miller Indices• Definition: Miller Indices are thereciprocals of the fractional intercepts(with fractions cleared) which the planemakes with the crystallographic x,y,z axesof the three nonparallel edges of the cubicunit cell.
52. Miller IndicesMiller Indices are a symbolic vector representation for theorientation of an atomic plane in a crystal lattice and aredefined as the reciprocals of the fractional intercepts whichthe plane makes with the crystallographic axes.To determine Miller indices of a plane, we use the followingsteps1) Determine the intercepts of the plane along eachof the three crystallographic directions2) Take the reciprocals of the intercepts3) If fractions result, multiply each by thedenominator of the smallest fraction
53. IMPORTANT HINTS:• When a plane is parallel to anyaxis,the intercept of the planeon that axis is infinity.So,theMiller index for that axis is Zero• A bar is put on the Miller indexwhen the intercept of a plane onany axis is negative• The normal drawn to a plane(h,k,l) gives the direction [h,k,l]
57. Example-4(1/2, 0, 0)(0,1,0)
58. Miller Indices
59. Spacing between planes in acubic crystal isl+k+ha=d 222hklWhere dhkl = inter-planar spacing between planes with Millerindices h, k and l.a = lattice constant (edge of the cube)h, k, l = Miller indices of cubic planes being considered.
60. X-Ray diffraction• X-ray crystallography, also called X-raydiffraction, is used to determine crystalstructures by interpreting the diffractionpatterns formed when X-rays are scatteredby the electrons of atoms in crystallinesolids. X-rays are sent through a crystal toreveal the pattern in which the moleculesand atoms contained within the crystal arearranged.
61. • This x-ray crystallography was developedby physicists William Lawrence Bragg andhis father William Henry Bragg. In 1912-1913, the younger Bragg developedBragg’s law, which connects the observedscattering with reflections from evenlyspaced planes within the crystal.