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.

 

CLASSIFICATION OF ELEMENTARY PARTICLES

How to calculate Electric Field from a Uniform Plane Sheet of Charge?

Assume that the sheet is infinite in extent and that the charge per unit area is 𝛔.

Considerations of symmetry lead us to believe that a field direction is every where Normal to Plane and if we have no field from any other charges in the world, the fields must be same in magnitude on each side.

Let us choose a Gaussian surface - a rectangular box that cuts thru the sheet as shown in figure below.

The field is Normal to these two faces. The two faces parallel to sheet will have equal areas say A.

As the electric field 'E' is parallel to area element dS;

∫E.dS = E∫dS = EA

The total flux from two faces is given by

∫E.dS1 + ∫E.dS2 = EA+EA

The total charge enclosed in the box is  𝛔A.

So according to Gauss Law, EA+EA = 𝛔A.

E=𝛔/2𝛆₀

What is Internal Conversion?

It is a process which enables an excited Nuclear state to come down to some lower state with out emission of a Gamma Photon. The energy ∆E involved in this Nuclear transition gets transferred directly to bound electron of atom. Such a electron gets knocked out of atom. Electrons like this are called conversion electrons and the process is called internal conversion.

It is interesting to note that wave mechanically, an atom electron spends part of its time inside a nucleus. This probability is highest for K-shell electrons which are closest to Nucleus. For such a case, Nucleus may de excite not by Ɣ- emission but by giving excitation energy ∆E directly to a K-shell electron.

Internal conversion is also possible for higher atomic levels L,M etc.

The kinetic energy of converted electron 'Ke' is Ke=∆E-Bₑ

  Bₑ - atomic binding energy of electron

∆E = Ei-Ef ; Nuclear Excitation energy

Usually continuous 𝜷-spectra are super imposed by discrete lines due to conversion.

It was wrongly believed that internal conversion process is like Photoelectric effect; a Ɣ-photon emitted by a nucleus is absorbed by orbital electron which is emitted as in photoelectric effect.

The simplest situation which disproves this is a transition between two states having spin equal to zero.

A 0→0 transition (∆I=0) is forbidden for all multipole orders and so Ɣ-emission by nucleus is completely forbidden.

However 0→0 transition is readily found to proceed by internal conversion. The experiment was performed on 0.7MeV level of ⁷²Ge. This is a 0→0 transition and it was found that conversion electrons can be detected, but there is a complete absence of Ɣ- ray emission.

In 1932, Taylor and Mott suggested that transition probability 'λ' from a Nuclear state 'a' to a Nuclear state 'b' is sum of two terms

λ=λₑ+λᵧ

λₑ & λᵧ are partial decay constants for conversion electron emission  and for gamma emission respectively.

Ratio between two decay constants is called conversion coefficeint and is measured as ratio between total number of conversion electrons emitted  (N) and total no. of gamma rays (N) emitted in same transition over the same time.

Conversion Coeff(α) = Nₑ/Nᵧ=  λₑ/λᵧ

 value of 'α' is found to depend on transition energy, multipole character of transition and atomic number Z.  

Born-Haber Cycle

PROPERTIES OF CRYSTAL STRUCTURES

The Seven Crystal Systems and Fourteen Bravais Lattices

ALL ABOUT HELIUM-I & HELIUM-II

Helium was the last gas to be liquefied on account of having lowest critical temperature -268 C of all known gases.

It is a colorless transparent very voltaile liquid and has the lowest boiling point at 4.2K at a pressure of 

1 atmosphere.

A peculiar property arising in the Helium system  is that solid cannot be obtained merely by lowering the temperature of the liquid.

Kamerlingh Onnes failed to solidify Helium despite the fact that he has reached a temperature of 0.84K.

Solid Helium was first obtained by "Keesom" who subjected liquid Helium to very high pressures. The solid is obtained at a pressure of 250 atm at 4.2K while it soldifies at only 23 atm at 1.1K.

Thus it is necessary to increase the pressure simultaneously while lowering the temperature of liquid.

Later investigations revealed that at high enough pressure, solid Helium could be obtained in equilibrium with the vapor at temperatures well above the critical point of gas. Thus at 5800 atm, solid Helium is obtained at a temperature as high as 42K. This is the curious property of Helium system that although liquid cannot exist above the critical temperature(5.2K), solid can exist if sufficiently great pressure is applied. Hence at high enough pressure, the melting pressure, the melting point exceeds the critical temperature and solid Helium melts to form gas.

The phase equilibria of Helium are represented diagrametically in Fig.

Phase diagram shows that it is entirely different from that of all other substances.

The fusion curve(or solid liquid phase line) and the saturated vapor pressure curve (or liquid vapor phase line) do not meet in a point, as in case of other substances and if we pursue the vapor pressure curve down to lower temperature it is found that the vapour and the liquid continue to be in equilibrium down to the absolute zero. Thus the 3 phases solid, liquid and vapor are never found to coexist or in otherwords, Helium has no triple point in conventional sense.

The helium will not solidify even at 0K if it is not subjected to pressures exceeding 25 atmospheres.

The SVP curve on the other hand, appears to proceed normally to the left towardsthe Origin(P=0, T=0) but to the right it terminates at critical point corresponding to a temperature of 5.2K and a pressure of 2.26 atmospheres.

The point A(2.19K) is known as the Lambda point of liquid Helium under its own pressure.

Phase Transition (Lambda Transition)

This transition divides the liquid state into two phases, Helium I and Helium II.

The fusion curve and SVP curve are joined by the Lambda line running between the points 'E'(T=1.75K, P=30atm) and A(T=2.19K and P=0.05atm) with Helium-I to its right and Helium-II to the left.

Thus, Helium is present in the liquid form on either side of Lambda line.

Kamerlingh Onnes, in course of his investigations found that liquid Helium shows an extremely interesting behavior if it is cooled below its boiling point (4.2K) to about 2.18K, he found that the density passes through an abrupt maximum at 2.19K decreasing slightly there after as shown in Fig.

The density first rise with fall of temperature from 4.2K up to 2.19K reaches a maximum value 0.1262 at 2.19K and then decreases with the decrease of temperature.

Thus below 2.19K, the liquid Helium which was contracting when cooled now begins to expand.

Specific Heat of liquid Helium at constant volume:

Cv increases upto 2.19K and at this temperature there is a sudden an abnormal increases in its value.

Beyond 2.19K the specific heat first decreases and then increases.

The Specific Heat temperature graph at 2.19K looks like the Greek letter Lambda and hence this temperature at which specific heat changes abruptly is called Lambda point.

Liquid Helium above 2.19K which behaves in a normal way is called liquid Helium-I.

Liquid Helium below 2.19K is called liquid Helium-II because of its abnormal properties.

No heat is evolved or absorbed during the transition from one form of Helium to another i.e. No latent heat is involved in He-I to  He-II and vice versa transition which suggests that 

a) The entropy is continuous across the curve i.e. The entropy of He-II is same as that of He-I.

b) There is no change of density during transitions i.e. density of both types of liquid is about same. 

while the viscosity of liquid increases with decrease in temperature, that of liquid He-I decreases showing He-I resembles a gas.

The viscosity of He-II is almost zero.

Peculiar Properties of Helium-II

A) Super Fluidity

At Lambda Point, the rate of flow increases abruptly and below it, the flow is found to be extraordinary large thus proving experimental evidence of a very low viscosity of liquid He-II.

The values of viscosity coefficient plotted as obtained by oscillating disc method.

There is a sharp discontinuity at Lambda point. The viscosity falls by a factor of about ten on passing through the lambda point.

Kapitza found that Ratio of Viscosity of HeliumII to HeliumI is approximately 10^-3.

Thus liquid He-II has practically zero viscosity and can flow rapidly with out resistance. This property is known as Super fluidity.

B) High Heat Conductivity

Helium-II is found to have an extraordinary high coefficient of thermal conductivity.
The heat transported per unit temperature gradient is several 10 times as great as that in copper at room temperature.
He-II is said to be about 800 times more conducting than copper.

It is found that heat flow in He-II is not proportional to temperature gradient.

Daunt and mendelsohn pointed out phenomenon of superconductivity in He-II.

C) Formation of films over solid surfaces

Liquid He-II can creep along solid surfaces in the form of high mobile film generally known as "Rollin" (also called as Rollinsimon) film.

The properties of film were investigated by Daunt and Mendelsohn.

If a tube containing Helium-II is placed in Helium-II bath

a) If liquid level inside tube is less, flow of liquid takes place from bath to tube.
b) Flow from tube to bath.
c)  Liquid inside the tube creeps out along surface of tube collects at its bottom in form of drops and falls into bath till tube becomes empty.

Thus liquid He-II seems to defy gravity by creeping out of containing vessel by coating the walls with a thin film of the liquid.

ALPHA DECAY PARADOX - BARRIER PENETRATION

Brief Description of Maxwell's Equations

The Bloch Theorem