UNIT-1 CRYSTAL PHYSICS

Introduction In nature matter can exist in three different states namely, solids, liquids and gases. In case of solids, the atoms or molecules are attached to one another with strong forces of attraction. Thus, a solid material has a definite volume and shape. Further, the physical structure of solids and their properties are related to the arrangement of atoms or molecules within the solid.

Classification of solids Based on the atomic arrangement within the solid, they are classified into three categories, namely single crystalline solids, polycrystalline solids and amorphous solids. They are (i) Single crystalline solids (ii) Polycrystalline solids (iii) Amorphous solids

Single crystalline solids Single crystals are polyhedrons that have a distinct shape for each material and are bound by smooth shiny faces and straight edges. In crystalline solids, the atoms are arranged in a regular periodic pattern in three dimensions as shown in Fig.

The arrangement of atoms in specific relation to each other is called “order”. Single crystals possess both short range and long range order. Single crystalline solids are anisotropic substances. Examples are quartz, alum, diamond, rock- salt etc.

Poly crystalline solids In polycrystalline solids, the periodicity of structure gets interrupted at certain boundaries called grain boundaries. Polycrystalline solids consists of fine grains oriented in different directions as shown in Fig. Polycrystalline solids are isotropic substances. Majority of the natural solids have polycrystalline structure. Ex. Metals

Amorphous solids In amorphous solids, the periodicity of atoms is completely absent i.e., the atoms are oriented randomly as shown in Fig. These materials possess short-range order only. These are isotropic substances. Ex: Glass, Rubber and many polymers.

Lattice and Space Lattice A regular arrangement of points in one, two or three dimensions represents the lattice. The symmetry of crystal has been explained easily by assuming an array of points in space about which the atoms are located. This leads to the concept of space lattice.

Let us consider a two dimensional square array of atoms along X- and Y- directions as shown in Fig. The line joining the any two points is known as translational vector or basis vector. These vectors are also known as unit cell primitives.

Consider a lattice point as origin at an arbitrary point ‘O’ as shown in Fig. Let ‘P’ be a lattice point at a distance ‘r’ from the origin. The position vector of the point ‘P’ can be represented by translational vector as given below:

In two-dimensional space lattice, where n1 and n2 are integers and their values depend on the position of the lattice point under consideration w.r.t. the origin. Similarly in three-dimensional space lattice:

Basis A unit assembly of atoms identical in composition and orientation is known as basis. It is also known as pattern. When the basis is represented with correct periodicity in all directions, then it gives the actual structure of crystal.

Consider a two- dimensional crystal structure which contains two different atoms as shown in Fig. From this figure it is clear that the basis is identical in composition, arrangement and orientation.

Unit Cell The smallest possible geometrical figure which when repeated regularly in three-dimensional space gives the actual crystal structure is known as unit cell. The choice of unit cell is not unique but it can be constructed in a number of ways. Let us consider a two dimensional crystal with periodic array of atoms as shown in Fig. This figure shows the different ways of representing unit cell in two-dimensional crystal.

Lattice Parameters 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.

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

Crystal system Unit vector Angles Cubic a= b=c α =β =  =900 Tetragonal a = b≠ c α =β = =900 Orthorhombic a ≠ b ≠ c Monoclinic α =β =900 ≠  Triclinic α ≠ β ≠ ≠ 900 Trigonal α =β = ≠ 900 Hexagonal a= b ≠ c α =β=900, =1200

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 (APF) 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] a b c 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.

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.

2 (0,433a)

Face Centered Cubic [BCC] There are atoms at the corners of the unit cell and at the centre of each face. Face cantered cubic has 4 atoms so its non primitive cell. Many of common metals (Cu, Ni, Pb ..etc) crystallize in FCC structure.

NaCl Structure The NaCl crystal is a system of Na+ and Cl ions arranged alternately in a “cubic” pattern in space. The unit cell can be drawn with either the Na+ ions at the corners or with the Cl ions at the corners. A unit cell of NaCl lattice with Na+ ions at the corners is shown in following figure.

The Na+ ions are situated at the corners as well as at the centers of the faces of the cube i.e., Na+ ions lie on a FCC lattice. Similarly, Cl ions lie in FCC lattice, but their lattice being relatively displaced half the edge length of the unit cell along each axis. Thus NaCl crystal is formed by interpenetration of two FCC sub-lattices of Na+ and Cl ions. But, within the unit cell, there must be equal number of Na+ and Cl ions.

Diamond Structure In a diamond crystal, the carbon atoms are linked by directional covalent bonds. The entire diamond lattice is constructed of tetrahedral units as shown in Fig.

The diamond lattice is composed of two inter penetrating FCC sub-lattices along the body diagonal by cube edge. One sub-lattice (X, say) has its origin at the point (0,0,0) and the other sub-lattice (Y, say) has its origin quarter of the way along the body diagonal i.e., at the point ( , , ), where ‘a’ is the lattice constant. The position of atoms in the unit cell of the diamond structure projected on a cubic face as shown in figure. The points at 0 and ½ are on the FCC lattice; those at ¼ and ¾ are on a similar lattice displaced along the body diagonal by of its length.

Graphite Structure The structure of graphite consists of many flat layers of hexagons as shown in figure. These layers are known as graphene sheets.

Each graphene sheet is itself a giant molecule Each graphene sheet is itself a giant molecule. Carbon is in group IV of the periodic table and so it has four electrons in its outer shell. Each carbon atom in the layer is joined by strong covalent bonds to only three other carbon atoms at an angle 120o and a weaker bond with a fourth atom lying in the adjacent layer. This fourth bond makes an angle of 90o with the plane of the layer. The unit cells are right hexagonal prisms with edges 1.42 and 3.35 (see Figure).

The distance between the layers is 2 The distance between the layers is 2.36 times more than the C-C covalent bond between the atoms in one layer. Thus, the bonds between layers are weak. Since there are no covalent bonds between the layers, the layer can easily slide over each other making graphite soft and slippery and an excellent lubricant like oil. The fourth electron between the layers is delocalized. It is a free electron and these free electrons between the layers allow graphite to conduct electricity and heat.

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 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 Determine the intercepts of the plane along each of the three crystallographic directions Take the reciprocals of the intercepts If fractions result, multiply each by the denominator of the smallest fraction

Example -1

Example -2

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

Interplanar spacing Consider a rectangular coordinate system with origin at O Assume a plane ABC with Miller indices (h k l). Let this plane makes intercepts OA, OB and OC on X-, Y- and Z- axes, respectively. Let the normal to the plane from the origin intercept the plane at ‘N’. The distance (d) of the normal from the origin is called interplanar spacing i.e., ON = d.

X- ray diffraction – Bragg’s Law 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. Bragg’s principle: The X-rays reflected from various parallel planes of a crystal interfere constructively when the path difference is equal to integral multiple of wavelength of incident X-rays. Bragg’s Law : 2dsinΘ = nλ

Bragg’s principle: The X-rays reflected from various parallel planes of a crystal interfere constructively when the path difference is equal to integral multiple of wavelength of incident X-rays.

Powder Method The powder method is used to determine the value of the lattice parameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal. If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards.

A sample of some hundreds of crystals (i. e A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film. A very small amount of powdered material is sealed into a fine capillary tube made from glass that does not diffract x-rays. The specimen is placed in the Debye Scherrer camera and is accurately aligned to be in the centre of the camera. X-rays enter the camera through a collimator.

The powder diffracts the x-rays in accordance with Braggs law to produce cones of diffracted beams. These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs on the film.

The full opening angle (4θ) of diffracted cone is determined by measuring the distance (S) between two corresponding arcs on the powder photograph using the relation, Knowing all the possible values of ‘θ’ and considering only the first order reflections from all possible planes, we can calculate the interplanar spacing for various sets of parallel planes using the equation,