- 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
- 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 !
- Ice, Carbon, Diamond, Nacl, Kcl etc
- 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
- (Properties change depending on the
- 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
- 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
- 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;
- 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
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 ?...