Lectures & Presentations

Crystal structure: introduction to lattice properties.

Lattice and crystals. 
A lattice is a three-dimensional periodic array of identical building blocks. The building blocks are atoms or groups of atoms. The crystals usually come with imperfection of structure and impurities. 

The periodicity of crystals is well established by the experimental studies of X-ray, neutron and electron diffraction patterns. 

A solid is a crystal if the positions of the atoms in it are exactly periodic. Here is a diagram that represents this property ideally. 

i. Distance between two nearest neighbors is ‘a’ along x-axis and ‘b’ along y-axis, where x–, and y– axes are not necessarily orthogonal.  

ii. A perfect crystal maintains the periodicity for -infinity < x < infinity and -infinity < y < infinity . The points A, B, C are equivalents. That means for an observer at A, the environment at A is exactly the same as it is for an observer at B or C.
This is expressed by saying crystals have translational symmetry. e.g. if the crystal is translated by a vector R — joining two atoms, the appearance of the crystal remains unchanged.

The atoms have no restrictions as to which location they preside over, as long as that position can be occupied by any  atom, it can be taken over by any other given atom, and all others would relent.  

Imperfection in crystals. 
There are no perfect crystals though, defined the above way. All crystals have some degree of imperfections. There are 3 basic examples of imperfections. 

i. Atoms near the surface have a different environment than atoms deep inside the crystal. 

ii. Due to thermal vibrations, equilibrium position of atoms are distorted, which depends on temperature T.

iii. Atoms always contain foreign elements known as impurities. 

The effect of imperfections can be neglected in very ideal crystals. Imperfections lead to interesting physical properties of crystals. E.g. Resistivity of metals is a result of thermal vibrations of atoms. — We will discuss this at a later time, in this course. 

When atoms are replaced by geometrical points, geometrical patterns depicting the periodicity of the crystals are obtained. They do not have any physical contents. Such geometrical patterns are known as "Lattice" or "crystal Lattice". 

Bravais and non-Bravais lattices. 
There are two classes of lattices, Bravais and non-Bravais lattices.

Bravais Lattice: In a Bravais lattice all lattice points are equivalent, hence all atoms of the crystal are of the same kind. 

Non-Bravais Lattice: In a non-Bravais lattice some of the lattice points are not equivalent. 

This is easily understood by the following diagram. 

Entropy, probability and equilibrium in thermodynamic systems.

The current lecture numbered lecture – I and II, is intended to be an introduction to the statistical mechanics paper of a Physics honors degree. It was delivered to the same class, on 22 November 2017.

Topics covered: 
i. Micro and macro state. 

ii. Entropy and thermodynamic probability and thermal equilibrium.  

Thermodynamic limit. 

Lets consider a physical system which is composed of N identical particles, in a volume of V. N is an extremely large number, typically in the order of 10^{23}.

Lets confine ourselves to the “thermodynamic limit”. i.e. N goes to infinity, V goes to infinity so that; n = N/V is fixed at a value chosen.
Important note: The ratio n is known as number density or particle number density — also concentration is sometimes used instead of density. One can distinguish them by referring to mass concentration vs number concentration. In a similar way one must distinguish number density from the not so unrelated parameter by the name mass-density. 

Extensive properties. 

In the thermodynamic limit, the “extensive properties” of the system such as energy E and entropy S are directly proportional to the size of the system, viz. N or V.

Intensive properties. 
Similarly the “intensive properties” such as temperature T, pressure P and chemical potential (mu) are independent of the size.  

Four-vectors and conservation laws in relativity

This lecture was delivered to the final year honors class of 3 year science degree students on 21 November 2017 as part of the Classical Dynamics paper.

In this lecture we will discuss some of the important tools of relativistic mechanics. We will discuss the idea of proper-time, 4-velocity, 4-acceleration, 4-momentum, 4-force and related conservation law of the 4-momentum.

A. Proper-time. 
The proper time is the time interval in the rest-frame of any event. The proper time is related to time-interval in other inertial frame by: tau = (1/gamma)t where gamma  > 1 always.

Gamma is the Lorentz factor or Lorentz boost factor directly related to the speed of an object in speed-of-light units, i.e. beta.

gamma = 1/sqrt{1-v^2/c^2}

Hence proper-time is the smallest possible time interval for an object in motion in among all possible inertial frames of reference and it occurs in the rest frame.

d(tau) < dt

Proper-time is necessary to define other basic quantities in theory of relativity if we are to preserve their basic meaning in terms of the non-relativistic mechanics definitions.

B. Four velocity. 
Four velocity of a particle is the rate of change of 4-displacement …

So, …  is the position vector — or space-time interval in the Minkowski  space — akin to the difference of two 3-dimensional vector in coordinate space, this time with 4 coordinates rather than 3.

The proper-time interval d(tau) is a Lorentz invariant i.e. when we move between arbitrary inertial frames of references given by the Lorentz factor beta or  gamma this interval retains its value — because it retains its form. Any variable which would retain its form under such transformation are said to be Lorentz invariant quantities.

Relativistic Doppler effect

Relativistic Doppler effect. 

There is an apparent shift in the observed frequency of any electromagnetic wave (light) when there is any relative motion between the source of light and the observer. This can be easily determined by using the 4-vector formulation of theory of relativity.

Lets discuss the details of this phenomena under two situations.

A. Source is at rest and observer is in motion. 
Lets us consider two inertial frames S and S’. S’ is moving wrt S, along the x-axis with speed v = (beta) c where the observer is at rest in S’ frame but the source is at rest in the  S frame.

Introduction to special theory of relativity.

Special Theory of Relativity:
Galilean Transformations,. Newtonian Relativity.

This was a lecture delivered to physics-elective class of a 3 year non-physics degree students on 10th April 2017. This is also a good exposition to honors students and anyone at an introductory level of the special theory of relativity, with requisite mathematical background. 

Let us consider an inertial frame of reference S. The space and time coordinates of any event occurring in frame S are given by x, y, z, t.

Now let us consider another frame of reference S’ which is inertial but moves wrt frame S at speed v, along +x direction.

The coordinates of the same event in the S’ frame are given as: x’, y’, z’, t’. The relationship among the coordinates of any event in two different frames of reference both of which are inertial frames, is known as Galilean Coordinate Transformation or Galilean Transformation.

If we assume that time passes by at the same rate in both S and S’ frames, the resulting laws satisfy Newtonian Relativity. We say time is an absolute quantity in an infinitude of equivalent inertial frames of references as the rate of time change is independent of the particular inertial frame of reference we have chosen. Consequently: t = t’.

The above equation is known as velocity addition rule in Newtonian Relativity. This is valid only for classical mechanics in the sense of speed of objects and speed of frame of reference, which are quite insignificant with respect to the speed-of-light value.

Velocity addition is nothing but a relation of velocities of objects in different frames among each other. So its exactly what we call “relative velocities” in elementary mechanics. Relative velocity, velocity addition and velocity transformation are the exact same thing. Read more about these here and here. The second link also expounds on what happens when speeds approach that of light.