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Sunday, September 17, 2017

How does Stern Gerlach Experiment proved spatial quantization and spin of electron

THE STERN-GERLACH EXPERIMENT



 Procedure implemented:


The silver atoms beam is produced by heating silver in a small electric oven. The beam is passed through an inhomogenous Magnetic Field.

Arrangement to produce inhomogenous Magnetic Field


We have one of pole pieces of the magnet flat with a cylindrical groove and the other in the form of a knife edge, parallel to groove.

The intensity of magnetic field increases as we go towards upper knife edge pole from center and it decreases as we go below towards lower pole.

A photographic plate is arranged to record the configuration of beam after its passage through the field.

The whole arrangement is placed in a vacuum. In absence of magnetic field, a trace of form of a narrow strip is obtained as shown in fig (a).

In presence of inhomogenous magnetic field the strip splits up into two components as shown in fig(b). 



The splitting of silver beam into two components in inhomogenous field verifies existence of electro spin and postulate of space quantization as shown below: 

Silver has an atomic number 47. According to Pauli's exclusion principle, all inner shells and sub shells are completely filled except outer most electron in 5S state.  Thus, it is a monovalent element.
The 5S electron is responsible for magnetic moment of atom.

When all silver atoms possessing a magnetic moment 'μᴊ' pass through inhomogenous magnetic field, they experience different amount of force in vertical direction depending on their orientation and alignment with magnetic field.

If magnetic moment  'μᴊ' can have all possible orientations then beam of Silver atoms consisting of Millions of atoms having all possible orientations will spread out into a broad continuous band on emerging from magnetic field. So a broad continuous patch should be observed on photographic plate.

Experimentally only two narrow strips are obtained on photographic plate. Therefore, predictions of classical physics are not correct in this case.

The two narrow strips show that 'μᴊ' cannot have all possible orientations, but only two possible orientations as shown in below figure



We know that 'μᴊ' is proportional to angular momentum 'J' and hence direction of 'J' relative to a well defined direction is given by

J = [√j(j+1)   ]* h/2Π

There are (2j+1) possible orientations of J. The Stern-Gerlach experiment shows that (2j+1)=2 or j=1/2.

Thus, J=(√3 /2)* (h/2Π)

It is known that angular momentum 'J' of Silver atoms entirely due to spin of its valence electrons. Thus, we conclude that the electron has a spin angular momentum

S = [√s(s+1)]*(h/2Π); where s=1/2.

Thus, Stern and Gerlach found that intial beam split into two distinct parts, corresponding to two opposite spin orientations in magnetic field that are permitted by space quantization.


Thursday, August 17, 2017

THE RAINBOW - EXPLANATION

Of all the optical phenomenon in every day life, the rainbow is loveliest.

Reflection of sun light by the rain drops is certainly an essential  element of an explanation but refraction-plays a role, too.

The following figure shows crucial path of light




The circle represents the cross section of a spherical rain drop. For the light ray, the sequence is Refraction, Reflection, and Refraction.

The angle (less than 90o) between the incident direction and the emergent direction is called as "Return Angle".

 A Ray from the sun strikes the spherical rain drop and some light is refracted into the water. Here we may ignore the portion that is reflected by the drops' surface. Next, the Ray proceeds to the far side of drop and is reflected there. Now we may ignore the portion that is refracted. Finally, the ray strikes the underside of drop and is refracted out into the air.

In precisely which direction does the emergent ray travel?

Two rules - i) Snell's Law for Refraction & ii) Equality of the angles of incidence and reflection

 suffice for answering that question. Once the initial point of contact between the rain drop and the ray from the sun has been specified. Also because the index of refraction depends on color, we must specify the color of light.

Lets start considering Red. Working out the complete path for many rays-i.e. for many different initial points of contact-reveals a surprising geometric property: The return angle for red light never exceeds 42.5 deg C, and most rays have a return angle 42 deg.



So return angle will decide the color of light reaching our vision of sight. Hence, different colors emerge from different sets of rain drops and produce a colored Rainbow.



Monday, August 14, 2017

What are Light Pipes

Sending light in a staright line is easy. Getting it around corners -with out losing intensity, is a different story.

Even when a light beam reflects from a good silvered mirror, not all of the incident light bounces off; some is absorbed by metal, as much as 20%. A sequence of reflections, in which 20% is lost at each bounce, will quickly reduce a powerful beam to a faint trace of its former itself.

A light pipe is made of glass, designed in such a fashion that there is no refraction of Light out of glass, and hence, there is no loss of intensity.




The Glass just "pipes" the light up and around the curve.

In commercial use, light pipes are drawn out as very long, fine fibers of glass. The diameter is about 0.1mm and field of applications itself is called fiber optics.


SPEED OF LIGHT IN VACUUM

Newton composed his "Opticks", which he published in 1704, he addressed the question of speed and wrote as follows:

[Light is propogated from luminous bodies in Time, and speeds about seven or eight minutes of an hour in passing from Sun to Earth.

This was observed first by Roemer, and then by others, by means of eclipses of satellites of Jupiter. For these eclipses, when the Earth is between the Sun and Jupiter, happen about seven or eight minutes sooner than they ought to do by the tables, and when the Earth is beyond the Sun they happen seven or eight minutes later than they ought  to do; The reason being, that the lights of satellites has farther to go in the latter case than in the former by diameter of earths orbit. ]

Newton is referring to Olaus Roemers calculations of 1676.

The Astronomers of seventeenth century had studied 'IO' (satellie of Jupiter) ever since Galileo discovered the Moons of Jupiter in 1610. They noted the eclipses and sought to predict their recurrence. But the eclipses -as

observed on Earth - did not recur perfectly periodical. If one used observations made when the Earth was nearest to Jupiter to predict when eclipses will terminate when the earth is farthest from Jupiter (about half a year

later), then the eclipses actually seem to terminate later than caluclated. The delay reasoned Roemer, is due to travel time for light to cross Earth's Orbit.

Newton gives the Time for Light to travel from Sun to Earth as 7 or 8 minutes. Let us average and take 15 minutes as the time for light to cross diameter of Earth's Orbit.

Already in Newton's day, the diameter of Earth's orbit was known to be 3 x 1011 meters.

Speed of Light in Vacuum = Diameter of Earth's Orbit / Apparent delay in eclipse termination
                                           = 3x10¹¹/900 Sec = 3x10⁸ m/sec


Saturday, July 8, 2017

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.

Tuesday, February 7, 2017

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.
 

Friday, February 3, 2017

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.

 mₑc²rest 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.


Sunday, January 29, 2017

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.  

Thursday, January 26, 2017

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.