CRYSTAL STRUCTURE & X-RAY DIFFRACTION

Classification of Matter

Solids Solids are again classified in to two types Crystalline Non-Crystalline (Amorphous)

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 ! Non-Metallic crystals: Ice, Carbon, Diamond, Nacl, Kcl etc… Copper, Silver, Aluminium, Tungsten, Magnesium etc…

Crystalline Solid

Single crystal Single Crystal example

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.

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.

Non-crystalline

What are the Crystal properties? Crystals have sharp 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 etc…also

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. α a b C B E D O A y x

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.

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 A group of atoms or molecules identical in composition is called the basis or A group of atoms which describe crystal structure

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.

S S S S a 2D-Crystal

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.

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!

In 2Dimensional space this is a unit cell but in 3 dimensional space it is NOT

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

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 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 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 Cubic a= b=c α =β =√=90 Tetragonal a = b≠ c Orthorhombic Crystal system Unit vector Angles Cubic a= b=c α =β =√=90 Tetragonal a = b≠ c Orthorhombic a ≠ b ≠ c Monoclinic α =β =90 ≠√ Triclinic α ≠ β ≠√ ≠90 Trigonal α =β =√≠90 Hexagonal a= b ≠ c α =β=90 √=120

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 S.No Crystal Type Bravais lattices Symbol 1 Cubic Simple P 2 Body centred I 3 Face centred F 4 Tetragonal 5 6 Orthorhombic 7 Base centred C

8 Body centred I 9 Face centred F 10 Monoclinic Simple P 11 Base centred C 12 Triclinic 13 Trigonal 14 Hexgonal

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 (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 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.

a b c

Atomic Packing Factor of SC

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 2 (0,433a)

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.

Face Centered Cubic (FCC)

Atomic Packing Factor of FCC 0.74

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 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. Triclinic (Simple) a ¹ ß ¹ g ¹ 90 oa ¹ b ¹ c Monoclinic (Simple) a = g = 90o, ß ¹ 90o a ¹ b ¹c Monoclinic (Base Centered) a = g = 90o, ß ¹ 90o a ¹ b ¹ c,

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 (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 (S) a = b = c, a = ß = g ¹ 90o

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.

210 X = 1 , Y = ½ , Z = 0 [1 ½ 0] [2 1 0]

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 X = 1 , Y = 0 , Z = 0 ► [1 0 0]

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. b a b a

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 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: 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 (1,0,0)

Example-2 (1,0,0) (0,1,0)

Example-3 (1,0,0) (0,1,0) (0,0,1)

Example-4 (1/2, 0, 0) (0,1,0)

Miller Indices

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 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.

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 Diffraction Bragg’s Law : 2dsinΘ = nλ

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