CRYSTAL STRUCTURE & X-RAY DIFFRACTION : CRYSTAL STRUCTURE & X-RAY DIFFRACTION Dr. Y. NARASIMHA MURTHY Ph.D
SRI SAI BABA NATIONAL COLLEGE (Autonomous)
firstname.lastname@example.org Classification of Matter : Classification of Matter Solids : Solids Solids are again classified in to two types
Non-Crystalline (Amorphous) What is a Crystalline solid? : 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 Examples ! : Examples ! Non-Metallic crystals:
Ice, Carbon, Diamond, Nacl, Kcl etc…
Copper, Silver, Aluminium, Tungsten, Magnesium etc… Crystalline Solid : Crystalline Solid Single crystal : Single crystal Single Crystal example Amorphous Solid : Amorphous Solid Amorphous (Non-crystalline) Solid is composed of randomly orientated atoms , ions, or molecules that do not form defined patterns or lattice structures.
Amorphous materials have order only within a few atomic or molecular dimensions. Slide 9: Amorphous materials do not have any long-range order, but they have varying degrees of short-range order.
Examples to amorphous materials include amorphous silicon, plastics, and glasses.
Amorphous silicon can be used in solar cells and thin film transistors. Slide 10: Non-crystalline What are the Crystal properties? : What are the Crystal properties? Crystals have sharp melting points
They have long range positional order
Crystals are anisotropic
(Properties change depending on the
Crystals exhibit Bi-refringence
Some crystals exhibit piezoelectric effect
& Ferroelectric effect etc…also What is Space lattice ? : 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. Translational Lattice Vectors – 2D : Translational Lattice Vectors – 2D A space lattice is a set of points such that a translation from any point in the lattice by a vector;
R = l a + m b
locates an exactly equivalent point, i.e. a point with the same environment as P . This is translational symmetry. The vectors a, b are known as lattice vectors and (l,m) is a pair of integers whose values depend on the lattice point. Slide 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. Thus one could equally well take the vectors a, b and c as a lattice vectors. Basis & Unit cell : Basis & Unit cell A group of atoms or molecules identical in composition is called the basis
A group of atoms which describe crystal structure Unit Cell : Unit Cell The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal. Slide 17: a 2D-Crystal 2D Unit Cell example -(NaCl) : 2D Unit Cell example -(NaCl) Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same. : Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same. This is also a unit cell - it doesn’t matter if you start from Na or Cl : This is also a unit cell - it doesn’t matter if you start from Na or Cl This is NOT a unit cell even though they are all the same - empty space is not allowed! : This is NOT a unit cell even though they are all the same - empty space is not allowed! In 2Dimensional space this is a unit cellbut in 3 dimensional space it is NOT : In 2Dimensional space this is a unit cellbut in 3 dimensional space it is NOT Now Crystal structure !! : Now Crystal structure !! Crystal lattice + basis = 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. Slide 24: The unit cell and, consequently, the entire lattice, is uniquely determined by the six lattice constants: a, b, c, α, β and γ. These six parameters are also called as basic lattice parameters. Primitive cell : 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. Crystal systems : 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 Table-1 : Table-1 Bravais lattices : 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’. 14 Bravais lattices : 14 Bravais lattices Coordination Number : 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. Atomic Packing Factor : 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. Simple Cubic (SC) : Simple Cubic (SC) Simple Cubic has one lattice point so its primitive cell.
In the 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. Atomic Packing Factor of SC : Atomic Packing Factor of SC Body Centered Cubic (BCC) : Body Centered Cubic (BCC) As shown, BCC has two lattice points so BCC is a non-primitive cell.
BCC has eight nearest neighbors. Each atom is in contact with its neighbors only along the body-diagonal directions.
Many metals (Fe, Li, Na.. etc), including the alkalis and several transition elements choose the BCC structure. Atomic Packing Factor of BCC : Atomic Packing Factor of BCC Face Centered Cubic (FCC) : 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. Slide 42: Face Centered Cubic (FCC) Atomic Packing Factor of FCC : Atomic Packing Factor of FCC HEXAGONAL SYSTEM : 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. TRICLINIC & MONOCLINIC CRYSTAL SYSTEM : 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) a = g = 90o, ß ¹ 90o
a ¹ b ¹c Triclinic (Simple) a ¹ ß ¹ g ¹ 90
oa ¹ b ¹ c Monoclinic (Base Centered) a = g = 90o, ß ¹ 90o
a ¹ b ¹ c, ORTHORHOMBIC SYSTEM : ORTHORHOMBIC SYSTEM Orthorhombic (Simple) a = ß = g = 90o
a ¹ b ¹ c Orthorhombic (Base-centred)a = ß = g = 90o
a ¹ b ¹ c Orthorhombic (BC) a = ß = g = 90o
a ¹ b ¹ c Orthorhombic (FC) a = ß = g = 90o
a ¹ b ¹ c TETRAGONAL SYSTEM : TETRAGONAL SYSTEM Tetragonal (P) a = ß = g = 90o
a = b ¹ c Tetragonal (BC) a = ß = g = 90o
a = b ¹ c Rhombohedral (R) or Trigonal : Rhombohedral (R) or Trigonal Rhombohedral (R) or Trigonal (S) a = b = c, a = ß = g ¹ 90o Crystal Directions : 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 lattice direction from a lattice point, the triple is enclosed in square brackets [ ... ] is used. [l, m, n]
[l, m, n] is the smallest integer of the same relative ratios. Negative directions : Negative directions When we write the direction [n1n2n3] depend on the origin, negative directions can be written as
R = l a + m b + n c
Direction must be
smallest integers. Examples of crystal directions : Examples of crystal directions X = 1 , Y = 0 , Z = 0 ► [1 0 0] Crystal Planes : Crystal Planes Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are called lattice planes.
In the figure density of lattice points on each plane of a set is the same and all lattice points are contained on each set of planes. MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES : 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. Miller Indices : Miller Indices Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.
To determine Miller indices of a plane, we use the following 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 IMPORTANT HINTS: : 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] Example-1 : Example-1 Example-2 : Example-2 Example-3 : Example-3 Example-4 : Example-4 Miller Indices : Miller Indices Spacing between planes in a cubic crystal is : Spacing between planes in a cubic crystal is Where dhkl = inter-planar spacing between planes with Miller indices h, k and l.a = lattice constant (edge of the cube)h, k, l = Miller indices of cubic planes being considered. X-Ray diffraction : 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. Slide 64: This x-ray crystallography was developed by physicists William Lawrence Bragg and his father William Henry Bragg. In 1912-1913, the younger Bragg developed Bragg’s law, which connects the observed scattering with reflections from evenly spaced planes within the crystal. X-Ray DiffractionBragg’s Law : 2dsinΘ = nλ : X-Ray DiffractionBragg’s Law : 2dsinΘ = nλ Slide 66: ThanQ
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