SOLID STATE PHYSICS Crystalline Solids: In crystalline solids, the basic building blocks (atoms, ions, molecules or group of atoms or ions) are arranged in 3-d in regular &periodic manner. These solids have well defined geometrical shape. They have definite melting point. Amorphous Solids: In amorphous solids, the basic building blocks are arranged randomly in 3-d. They do not have a well defiened melting point. Distinction between Crystalline & Amorphous Solids: Crystalline solid: The atoms or molecules of the solids are periodic in space. Amorphous solid: The atoms or molecules of the solids are not periodic in space. Crystalline solid: These solids possess definite external geometry. Amorphous solid: These solids do not possess definite shape & size. Crystalline solid: All the bonds between atoms of the crystalline solids are of equal strength. Amorphous solid: All the bonds between atoms of the crystalline solids are not of equal strength. Crystalline solid: Some crystalline solids are anisotropic i.e. the magnitude of physical properties(such as refractive index, conductivity etc.) are different along different directions of the crystal. Amorphous solid: Amorphous solids are isotropic. Crystalline solid: These are having sharp melting and boiling point(because all the bonds break at the same time). Amorphous solid: These do not posses sharp melting and boiling point(because weaker bonds will break first and stronger bonds will break next). [Example: As the temperature of glass is gradually raised, it softens and starts flowing without any sharp change from solid state to liquid state.] Crystalline solid: Example: Calcite, mica, Quartz, diamond etc.. Amorphous solid: Example: glass, plastic rubber. Lattice + Basis = Crystal Basis: The atoms, ions, molecules or group of atoms or ions, which are repeated in 3-d in a regular periodic manner to form the crystal are called basis of the crystal. This is the basic building block of the crystal. Lattice: The lattice is a collection of geometrical points in 3-d space arranged in a regular periodic manner. Unit Cell: A unit cell is the smallest geometric figure, which on repetition in 3-d space gives the actual crystal structure. Primitive Cell: It is the smallest unit cell. In primitive cells the lattice points are only at the corners. In a non-primitive cell there are lattice points inside the cell.
Primitive lattice vectors are denoted by , , . If we choose any lattice point as origin, then the position vector of any other lattice point is represented by ; where lattice points along the direction determined by are integers representing the number of , , . Direct translation vector expressed as the projections of the vector on the axes determined by ; where , , , the integers, are . Depending on the relative magnitudes and orientations of the primitive lattice vectors , and there are seven crystal systems and fourteen types of lattices in 3-d. The 14 types of lattices are called Bravais lattices. Seven crystal systems(& their corresponding Bravais lattice) are: Cubic(sc, bcc,fcc),Orthorhombic(bco,fco,eco),Tetragonal(st,bct),Trigonal(st),Hexagonal(sh),MOnoclinic(sm,ecm) & Triclinic (st). Primitive Lattice Vectors ( , , ): There can be infinite number of directions and planes in a crystal lattice. The directions & planes are specified in terms of , , called Primitive lattice vectors. Miller Indices: The various crystal planes are specified by a set of three numbers or indices called Miller Indices. Procedure for assignment of Miller Indices: For the procedure go through the following example. Example 1: In a cubic crystal, a given plane intercepts the axes at 2a, 3b & 3c. Find the Miller indices of the plane. Solution: Step 1: [Find the intercepts of planes with the crystallographic axis interms of lattice parameters a,b & c]. The intercepts, in terms of lattice parameters are 2,3,3. Step 2: [Take reciprocal of the intercepts] Reciprocal of the intercepts are 1/2 ,1/3 and 1/3. Step 3: [Convert the reciprocal to the nearest whole number ] The smallest whole number is 3,2,2. Step 4: [The whole numbers represent the Miller Indices of the plane & are enclosed within parentheses( ).] Hence the Miller Indices of the plane are (3 2 2 ). NOTE: If the intercept on a given axis is on the – ve side, the corresponding index number is represented by a bar over it.
For example: The miler indices & also parallel to ‘b’ axis. Inter planner Spacing (d): represents the plane intercepts the ‘c’ axis on the negative side The spacing between two adjacent (h k l) planes is given by For simple cubic , Hence . . Reciprocal Lattice: Reciprocal lattice is the arrays of points that are obtained at the end of normal drawn to a common origin whose length is proportional to the reciprocal of inter planar spacing. Reciprocal Lattice Vector: It is defined as the vector whose magnitude is equal to i.e. reciprocal to inter planar spacing. It’s direction is parallel to normal of plane. We know , , are primitive lattice vectors of direct lattice. Let vectors of reciprocal lattice. These can be expressed as ; Where ; , , are primitive lattice . is the volume of the primitive cell of direct lattice. A general reciprocal lattice vector is + + , where are integers. Brillowin Zone: The primitive cell of the reciprocal lattice is called the Brillowin zone of the crystal. Bragg’s Law: The crystal is imagined to be a set of parallel planes, each plane containing lattice points. Bragg gave a simple method of analyzing the scattering of X-rays for crystal planes. X-rays being diffracted by the crystal lattice if their wavelength approximately equal to the distance between two consecutive atomic planes of the crystal. A monochromatic X-ray beam of wavelength ‘ ’ is incident at an angle ‘ ’ to Bragg’s planes. Glancing angle (the angle which the incident ray makes with the plane of incidence or Bragg’s plane) Inter planar spacing=PQ. AP & CQ: Incident X-rays; PB & QD: Reflected rays; P & Q: Lattice Points; Draw PM⊥ CQ & PN ⊥ QD. Path difference between the two rays (APB & CQD)= MQ +QN= PQ sinθ + PQ sinθ =2d sin θ. For constructive interference, Path difference = nλ => 2d sin θ = n λ; where n=1,2,3….. This is the Bragg’s Law.
Mode of the fiber: As we have discussed in the earlier article about the light propagation through an optical fiber after total internal reflection. The path followed by the light in a fiber is called the mode of the fiber. The light can pass through one path ( that is mono mode or single mode fiber) or more than one path (that is called multi mode fiber). On the basis of this, the fiber is divided into two types: 1. Step index fiber 2. Graded index fiber Let us discuss the difference between the two: 1. Step index fiber is of two types viz; mono mode fiber and multi mode fiber. Graded index fiber is of only of one type that is multi mode fiber. 2. The refractive index of the core of the step index fiber is constant throughout the core. The refractive index of the core of the graded index fiber is maximum at the center of the core and then it decreases towards core-cladding interface. 3. Number of modes for step index fiber N = V2/2, where V is cut off frequency or normalized frequency or V- number Number of modes for graded index fiber is N = V2/ 4. 4. V number can be less that 2.405 or more that 2.405 for step index fiber V number is always more than 2.405 for graded index fiber. 1. In a cubic crystal, a given plane intercepts the axes at a, 2b & 3c. Find the Miller indices of the plane. Solution: : Step 1: The intercepts, in terms of lattice parameters are 1,2,3. Step 2: Reciprocal of the intercepts are 1 ,1/2 and 1/3. Step 3: The smallest whole number is 6,3,2. Step 4: Hence the Miller Indices of the plane are (6 3 2 ). 2. In a crystal, a given plane is parallel to the ‘a’ axis and ‘c’ axis and intercepts the ‘b’ axes at b. Find the Miller Indices. Solution: Step 1: The intercepts, in terms of lattice parameters a, b, c are . Step 2: Reciprocal of the intercepts are 0, 1, 0. Step 3: The smallest whole number is 0, 1, 0. Step 4: Hence the Miller Indices of the plane are (0 1 0 ). (Ans.) 3. In an ortho-rhombic crystal a lattice plane cuts intercepts of lengths axes. Deduce the Miller indices of the plane, where Solution: are primitive vectors of the units cell. Step 1: The intercepts, in terms of lattice parameters a, b, c are Step 2: Reciprocal of the intercepts are Step 3: The smallest whole number is along three . . . Step 4: Hence the Miller Indices of the plane are . (Ans.)