Properties of Longitudinal Progressive Waves

Longitudinal wave motion refers to wave motion in which particles of medium vibrate along the direction of propagation of wave.

Properties:

1. All the particles have same Amplitude, Frequency and Time Period

2. There is a gradual Phase difference between successive particles

3. All the particles vibrating in Phase will be at a distance equal to nƛ. Here n=1,2,3etc. It means the   minimum distance between two particles vibrating in Phase is equal to wave length.

4. When the particle moves in same distance as that of wave, it is in a region of compression.

5. When the particle moves in opposite direction as that of wave it is in a region of Refraction.

6. When the particle is at mean position, it is a region of maximum Compression or Refraction.

7. When the particle is at extreme position, the medium around particles has its normal density, with compression on one side and rare fraction on other side.

Ionic Conductivity - Detailed Explanation

Thermal Radiation - Important points

The process of heat transfer from a body by virtue of its temperature with out involvement of intervening medium is called Radiation. The radiant energy is transported by electromagnetic waves because these waves can travel through vacuum.

The Radiation emitted by a body by virtue of its temperature is called Thermal Radiation. It is an inherent property of all bodies.

According to Prevost theory of heat exchanger, every body emits and absorbs radiant energy continuously as long as its temperature is above 0 K.

At low temperature, the emission rate is small while at higher temperatures it increases rapidly as 4th power of absolute temperature. 

At ordinary and moderate high temperature, mostly longer waves(infrared) are emitted but at very high temperatures shorter waves are also emitted.

Properties of Thermal Radiation:

i)  It travels through empty space with the velocity of light.

ii) It undergoes Reflection, Refraction and total internal reflection obeying the same law as light.

iii) It exhibits the phenomenon of interference, diffraction and polarisation.

iv) It exerts a small, but finite pressure on the surface on which it is incident. This is called as pressure of thermal radiation.

v)  It obeys inverse square law

Some important terms related to Thermal Radiation are  Spectral Energy Density, Total Energy Density, Emmisive Power & absorptive power

Spectral Energy Density

Spectral Energy Density for a particular wavelength is the energy per unit volume per unit range of wavelength.

Total Energy Density

Total energy density of thermal radiation at any point is the total radiant energy per unit volume around that point due to all wavelengths.

Emmissive Power

The emissive power of a body at a given temperature and for a given wavelength, is defined as the ratio radiant energy absorbed for a second by unit surface area of the body per unit wavelength range.

Absorptive power

The absorptive power of a body at a given temperature and for a given wavelength is defined as the ratio of radiant energy absorbed per second by unit surface area of the body to the total energy falling per second on the same area. 

Black body and its Radiation

What is a Black body?

Energy distribution in black body radiation

Laws of Black body Radiation

Kirchoff's Law

Stefan's-Boltzmann law

Wien's Law

Rayleigh-Jeans Law

Planck's Law

MILLER INDICES

Paulis Hypothesis of Beta Decay

Pauli introduced concept of third particle, a neutral particle which gets emitted in β⁻decay. This particle has the generic name of Neutrino.

The total energy is shared by 3 particles

The recoil nucleus
The Electron
The Neutrino

Because of its comparatively great mass, the recoil energy of Nucleus is very small and nearly all Kinetic Energy is shared between the Beta particle and the Neutrino.

In addition to laws of conservation of charge and energy, we must also apply the laws of conservation Linear and angular momentum to every nuclear process. Taking our reference system as the parent nucleus at rest, the vector sum of Linear Momenta of the recoil nucleus, the beta particle and neutrino must be zero.

To conserve angular momentum in β⁻decay, we note that parent and daughter nuclei are isobars; i.e. they have equal number of nucleons. Hence, the total change in nuclear angular momenta will be either zero or an integral multiple of ℏ.

The beta particle has an intrinsic spin angular momenta of 1/2ℏ.The vector sum of angular momenta of Neutrino and beta particle will be either zero or one in units of ℏ.

The present accepted theory, which is supported by experimental evidence shows that there are two types of neutrino or two components of Neutrino. It has been found that the axis of spin of neutrino is parallel to its direction of motion; one type spins according to the left hand rule with respect to its direction of motion as its axis, the other component spins according to right hand rule.

The first type is usually called neutrino represented by symbol 𝜈, the second type is called antineutrino.
 
The spin vector of neutrino points opposite to direction of its motion.

The spin vector of anti neutrino points in the direction of its motion.

Another way of saying this is that the helicity of neutrino is negative and that of anti neutrino is positive or one has right handed helicity and other has left handed helicity.
 

Nature of Orbits for a body projected from earth with different speeds

Minimum velocity required for an object to orbit around the Earth  is  Vs =√gR = 8KmSec⁻¹

Let Ve be the escape velocity  required for a body to escape Earth's gravitational field.

Let if  'V' be velocity with which a body is projected from Earth.

Then,

1. V < Vs → body falls to ground
2. V=Vs → body rotates round the Earth in circular orbit closer to surface of Earth.
3. Vs < V < Ve → body revolves in elliptical orbit
4. V=Ve → body just escapes from Gravitational Field
5. V>Ve → body moves in interstellar space with velocity equal to √❲V²-Ve²❳
6. V<Ve → Total Energy of body is Negative 
7. V=Ve → Total Energy of body is Zero.

Pair Production - Conversion of Radiation into Matter

The cloud chamber experiments revealed that a Photon can give up its energy to materialize as two electrons of opposite charge. Certainly the Photon must have an energy of atleast 2mₑc² in order to produce a pair.

No photon, regardless of its energy, can produce a pair in a perfect vacuum.

Pair Production is strictly an Electromagnetic Process. It seems to occur mostly in the intense electric field near the nucleus rather than inside the nucleus.

At higher energies or with heavy targets it is typically reasonable to ignore the energy transferred to target, so that nearly all energy from Photon goes to electron-positron pair.

Energy equation

h𝜈 → 2mₑc²+E1+E2

holds approximately.

 mₑc²→ rest energy of each electron

 E1, E2 → Kinetic Energies of particles

 The heavier the target, the more nearly the equation is satisfied. 

Pair Production can occur in the vicinity of an electron.

Pair Annihilation

Positron and Electron coalesce to produce atleast two photons

e⁺ + e⁻→2𝛾

Annihilation into three or more Photons is possible but less likely. Each extra photon tends to supress the rate of annihilation by a factor of order of magnitude of fine structure constant 1/137.

A Positron moves thru matter and forms ion pairs giving up energy in the process. There is about 2% chance that a Positron will hit an electron and annihilate.

But more likely output is that Positron will stop and become attracted to an electron. The atom formed by these two particles is called Positronium.

The Positron-Electron system drops into successively lower energy states, emitting (low energy) photons, until it arrives in ground state.

Properties of Positronium

The lowest Bohr orbit of Positronium is one for which n=1 and l=0, so that the lowest is an S-state.

The S state has fine structure due to the spins of particles; when the two spins are oppositely directed, the atom is in a ¹S state. When the two spins are parallel, it is in a ³S state, and has higher energy.

The triplet state is a meta stable state and has longer life time than singlet state.

The life time of singlet state was revealed by J.Pirenne, J A wheeler and is of order 10⁻¹º Sec.

The life time of Triplet state was revealed by Ore & Powell and is about 1.4x10⁻⁷ Sec.

Annihilation radiation emitted by combination of electron-Positron pair in ¹S state should consist of two gamma ray photons emitted simultaneously.

Radiation from ³S state of this system should consist of 3 𝛾 ray Photons emitted simultaneously.

The first experimental evidence for formation of Positronium was obtained by M. Deutsch, who observed time delay between emission of Positron from ²²Na and appearance of annihilation photon from substance in which Positrons are observed. Several gases N₂, O₂ etc are used as absorbers of Positrons. The time delay is due to formation of Positronium.

Properties of Pions

• Pions are Mesons
• There are 3 kinds of Pions: π⁺, π⁻, π⁰
• Either charged Pion possess a mass of 139.6 MeV and neutral Pion is 135.0 MeV.
• Pions have spin zero.
• P+P → π⁺ + n + P
• P+P → π⁰ + P + P
• P+n → π⁻ + P + P

• Charged Pions decay into Muons (Weak Process in Decay):

π⁺ → 𝜇⁺ + 𝜈

π⁻ → 𝜇⁻ + 𝜈

• The mean life is 2.6 x 10⁻⁸ Sec. 

• The neutral Pion decays in different way; process is 

π⁰ → 𝛾 + 𝛾 ; This decay is Electromagnetic in nature.

The presence of photons in final state leads us to expect the process is electromagnetic in nature.

The Photons from the decay always seem to come from the spot at which π⁰was produced in some bombardment process. The measurement of life time of such a short lived object is not easy but emulsion techniques provide enough spatial resolution so that in case of rare decay modes

 π⁰ → 𝛾 + 𝛾

π⁰ → 𝛾 + 𝛾 

it is barely possible to measure separation of electrons from place at which  π⁰ was produced.

•  The mean life of  π⁰ is about 0.89 x 10⁻¹⁶ Sec.