1. CRYSTAL STRUCTURE & X-RAY DIFFRACTION Dr. Y. NARASIMHA MURTHYPh.D SRI SAI BABA NATIONAL COLLEGE (Autonomous) ANANTAPUR-515001-A.P(INDIA)

2. Classification of Matter 3. Solids

Solids are again classified in to two types

Crystalline

Non-Crystalline (Amorphous)

4. What is a Crystalline solid?

A crystal or crystalline solid is a solid material, whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions.

So a crystal is characterized by regular arrangement of atoms or molecules

5. Examples !

Non-Metallic crystals:

Ice, Carbon, Diamond, Nacl, Kcl etc

Metallic Crystals:

Copper, Silver, Aluminium, Tungsten, Magnesiumetc

6. Crystalline Solid 7. Single crystal Single Crystalexample 8. Amorphous Solid

Amorphous (Non-crystalline) Solid iscomposedofrandomly orientatedatoms , ions, or moleculesthatdonotform defined patterns or lattice structures .

Amorphousmaterials have order only within a few atomic or molecular dimensions.

9.

Amorphousmaterials do not have any long-range order, but they have varying degrees of short-range order.

Examples to a morphousmaterials includea morphoussilicon, plastics, and glasses.

Amorphoussilicon can be used in solar cells and thin film transistors.

10. Non-crystalline 11. What are the Crystal properties?

Crystals havesharp melting points

They have long range positional order

Crystals are anisotropic

(Properties change depending on the

direction)

Crystals exhibit Bi-refringence

Some crystals exhibit piezoelectric effect

& Ferroelectric effect etcalso

12. What is Space lattice ?

An infinite array of points in space,

Each point has identical surroundings to all others.

Arrays are arranged exactly in a periodic manner.

a b C B E D O A y x 13. Translational Lattice Vectors 2 D A space lattice is a set of points such that a translation from any pointin the lattice by a vector; R = l a + m b locates an exactlyequivalentpoint,i.e.a point with the same environment asP. This is translational symmetry. The vectorsa ,bare known as lattice vectors and (l,m) is a pair of integers whose values depend on the lattice point . 14.

For a three dimensional lattice

R =la + mb +nc

Here a, b and c are non co-planar vectors

The choice of lattice vectors is not unique . Thusone could equally well take the vectors a, b and cas a lattice vectors.

15. Basis & Unit cell

A group of atoms or molecules identical in composition is called the basis

or

A group of atoms which describe crystal structure

16. Unit Cell

The smallest component of the crystal (group of atoms, ions or molecules), whichwhen stackedtogetherwith pure translational repetition reproduces the whole crystal .

17. a 2D-Crystal S S S S 18. 2DUnit Cellexample -(NaCl) 19. Choice of origin is arbitrary - lattice points need not be atoms -butunit cell size should always be the same . 20. This is also a unit cell -it doesnt matter if you start from Na or Cl 21. This isNOT a unit celleven though they are all the same - empty space is not allowed! 22. In 2Dimensional spacethis is a unit cell butin 3 dimensional space it isNOT 23. Now Crystal structure !!

Crystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif)to the lattice sides of the lattice point.

Crystal lattice + basis = Crystal structure 24.

The unit cell and, consequently, the entire lattice, isuniquely determined by the six lattice constants: a, b, c, , and. These six parameters are also called as basic lattice parameters.

25. Primitive cell

The unit cell formed by the primitives a,b and c is called primitive cell.A primitive cell will have only one lattice point.If there are two are more lattice points it is not considered as a primitive cell.

As most of the unit cells of various crystal lattice contains two are more lattice points, its not necessary that every unit cell is primitive.

26. 27. Crystal systems

We know that a three dimensional space lattice is generated by repeated translation of three non-coplanar vectors a, b, c.Based on the lattice parameters we can have 7 popular crystal systems shown in the table

28. Table-1 = =90 =120 a= b c Hexagonal= =90 a= b=c Trigonal 90a b c Triclinic = =90 a b c Monoclinic = ==90 a b c Orthorhombic = ==90 a = b c Tetragonal = ==90 a= b=c Cubic Angles Unit vector Crystal system 29. Bravais lattices

In 1850, M. A. Bravais showed that identical points can be arranged spatially to produce 14 types of regular pattern. These 14 space lattices are known as Bravais lattices.

30. 14 Bravais lattices C Base centred 7 P Simple Orthorhombic 6 I Body centred 5 P Simple Tetragonal 4 F Face centred 3 I Body centred 2 P Simple Cubic 1 Symbol Bravais lattices Crystal Type S.No 31. P Simple Hexgonal 14 P Simple Trigonal 13 P Simple Triclinic 12 C Base centred 11 P Simple Monoclinic 10 F Face centred 9 I Body centred 8 32. 33. Coordination Number

Coordination Number (CN) : The Bravais lattice points closest to a given point are the nearest neighbours .

Because the Bravais lattice is periodic, all points have the same number of nearest neighbours or coordination number. It is a property of the lattice .

A simple cubic has coordination number 6; a body-centered cubic lattice, 8; and a face-centered cubic lattice,12.

34. Atomic Packing Factor

Atomic Packing Factor (APF)is defined as the volume of atoms within the unit cell divided by the volume of the unit cell.

35. Simple Cubic (SC)

Simple Cubic has one lattice point so its primitive cell.

Inthe unit cell on the left, the atoms at the corners are cut because only a portion (in this case 1/8) belongs to that cell. The rest of the atom belongs to neighboring cells .

Coordinatination number of simple cubic is 6.

36. a b c 37. Atomic Packing Factor of SC 38. Body Centered Cubic (BCC)

As shown, BCC has two lattice points soBCC isanon-primitive cell.

BCC has eight nearest neighbors.Each atom is in contact with itsneighbors only alongthebody-diagonal directions.

Manymetals(Fe, Li, Na.. etc) , including the alkalis and several transition elements choose the BCC structure.

39. Atomic Packing Factor of BCC 2 (0,433a) 40. Face Centered Cubic (FCC)

There are atoms at the corners of the unit cell and at the center of each face.

Face centered cubic has 4 atoms so its non primitive cell.

Many of common metals (Cu, Ni, Pb ..etc) crystallize in FCC structure.

41. 42. Face Centered Cubic (FCC) 43. Atomic Packing Factor of FCC FCC 0 . 74 44. HEXAGONAL SYSTEM

A crystal system in which three equal coplanar axes intersect at an angle of 60, and a perpendicular to the others, is of a different length.

45. TRICLINIC &MONOCLINIC CRYSTAL SYSTEM

Triclinic minerals are the least symmetrical. Their three axes are all different lengths and none of them are perpendicular to each other. These minerals are the most difficult to recognize .

Monoclinic ( Simple ) == 90 o , 90 oa b c Triclinic ( Simple ) 90 o a b c Monoclinic (Base C entered )== 90 o , 90 oa b c, 46. ORTHORHOMBIC SYSTEM Orthorhombic ( Simple )= == 90 o a b c Orthorhombic (B ase-centred) = == 90 o a b c Orthorhombic (BC) = == 90 oa b c Orthorhombic (FC)= == 90 oa b c 47. TETRAGONAL SYSTEM Tetragonal (P)= == 90 o a = b c Tetragonal (BC)= == 90 oa = b c 48. Rhombohedral (R)o r Trigonal Rhombohedral (R)o r Trigonal(S) a = b = c,= = 90 o 49. Crystal Directions

We choose one lattice point on the line as an origin, say the point O. Choice of origin is completely arbitrary, since every lattice point is identical.

Then we choose the lattice vector joining O to any point on the line, say point T. This vector can be written as;

R = la + mb + nc

To distinguish a latticedirectionfrom a latticepoint , the triple is enclosed in square brackets [ ... ] is used. [l, m, n]

[l, m, n] is the smallest integer of thesame relative ratios .

50. 210 X = 1 , Y = , Z = 0 [1 0] [ 2 1 0] 51. Negative directions

When we write the direction [n 1 n 2 n 3 ] depend on the origin, negative directions can be written as

R = l a + m b + n c

Direction must be

smallest integers.

52. Examples of crystal directions X = 1 , Y = 0 , Z = 0 [1 0 0] 53. Crystal Planes

Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are calledlattice planes .

In the figure density oflattice points on each plane of a set is the sameand all lattice points are contained on each set ofplanes .

b a b a 54. MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES

William HallowesMiller in 1839 was able to give each face a unique label of three small integers, the Miller Indices

Definition:Miller Indices are the reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with the crystallographic x,y,z axes of the three nonparallel edges of the cubic unit cell.

55. Miller Indices

Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as thereciprocals of the fractional interceptswhich the plane makes with the crystallographic axes.

To determine Miller indices of a plane, we use thefollowing steps

1) Determine the intercepts of the plane along each

of the three crystallographic directions

2) Take the reciprocals of the intercepts

3) If fractions result, multiply each by the

denominator of the smallest fraction

56. IMPORTANT HINTS:

When a plane is parallel to any axis,the intercept of the plane on that axis is infinity.So,the Miller index for that axis is Zero

A bar is put on the Miller index when the intercept of a plane on any axis is negative

The normal drawn to a plane (h,k,l) gives the direction [h,k,l]

57. Example-1 (1,0,0) 58. Example-2 (1,0,0) (0,1,0) 59. Example-3 (1,0,0) (0,1,0) (0,0,1) 60. Example-4 (1/2, 0, 0) (0,1,0) 61. Miller Indices 62. Spacing between planes in a cubic crystal isWhere d hkl=inter-planar spacing between planes withMiller indices h, k and l. a=lattice constant (edge of the cube) h, k, l=Miller indices of cubic planes being considered. 63. X-Ray diffraction

X-ray crystallography, also called X-ray diffraction, is used to determine crystal structures by interpreting the diffraction patterns formed when X-rays are scattered by the electrons of atoms in crystalline solids. X-rays are sent through a crystal to reveal the pattern in which the molecules and atoms contained within the crystal are arranged.

64.

This x-ray crystallographywas developedby physicists William Lawrence Bragg and his father William Henry Bragg. In 1912-1913, the younger Bragg developed Braggs law, which connects the observed scattering with reflections from evenly spaced planes within the crystal.

65. X-Ray Diffraction Braggs Law : 2dsin = n 66. ThanQ Any Queries ?...