Electric Current in Atoms - Bohr Magneton ; magnetic moment of electron in orbit
The revolution of electron in its orbit around nucleus resembles a magnetic dipole and the magnetic moment due to this orbital motion of electron is
𝜇ₑ₁ = - (e/2m) x angular momentum
angular momentum = mr²w
The minus sign indicates that dipole moment points in direction opposite to vector representing angular momentum.
The ratio of magnetic dipole moment of the electron due to its orbital motion and angular momentum of orbital motion is called "orbital gyro magnetic ratio" represented by '𝛾'.
𝛾 = (magnetic moment/angular momentum) = e/2m
The strength of magnetic dipole is given by
𝜇ₑ₁ = -𝜇B.l;
𝜇B - Bohr Magneton = (eh/4πm) = 9.27 x 10⁻²⁴ Amp.m²
Therefore, '𝜇B' represents magnetic moment of an elementary permanent magnetic dipole.
As we know that for a 'l' value there exists a quantum number 'ml' such that it takes +l to -l values hence for a d-electron for eg:
𝜇ₑ₁ = - (e/2m) x angular momentum
angular momentum = mr²w
The minus sign indicates that dipole moment points in direction opposite to vector representing angular momentum.
The ratio of magnetic dipole moment of the electron due to its orbital motion and angular momentum of orbital motion is called "orbital gyro magnetic ratio" represented by '𝛾'.
𝛾 = (magnetic moment/angular momentum) = e/2m
The strength of magnetic dipole is given by
𝜇ₑ₁ = -𝜇B.l;
𝜇B - Bohr Magneton = (eh/4πm) = 9.27 x 10⁻²⁴ Amp.m²
Therefore, '𝜇B' represents magnetic moment of an elementary permanent magnetic dipole.
As we know that for a 'l' value there exists a quantum number 'ml' such that it takes +l to -l values hence for a d-electron for eg:
corresponding possible magnetic moment along direction of field are 2𝜇B, 𝜇B, 0, -𝜇B, -2𝜇B
Therefore 𝜇ₑ₁ = -𝜇B.ml
Characteristics of electrical conduction in Metals
The general characteristics of electrical conduction in metals are summarized as follows:
1) The electrical current density in the steady state is proportional to electric field strength
(Ohm's law).
2) For pure specimens, the electric conductivity (σ) and the thermal conductivity (σ') vary with temperature as follows:
1) The electrical current density in the steady state is proportional to electric field strength
(Ohm's law).
2) For pure specimens, the electric conductivity (σ) and the thermal conductivity (σ') vary with temperature as follows:
σ∝T⁻¹ and σ' =const (for T > θD); θD is characteristic Debye temperature.
so that σ' / σT is independent of temperature (Weidmann-Franz law)
For T < θD;
σ∝T⁻⁵ and σ' = T⁻² where 'θD' is characteristic Debye Temperature.
The relation ρ∝T⁵ is known as Bloch-Gruneisen T⁵ law.
3) For metals that exhibit the phenomenon of superconductivity, their resistivity disappears at temperature above 0Kand below critical temperature for superconducting phase transition (critical temp=4.15K) for mercury.
4) For metals containing small amounts of impurities, the electrical resistivity(ρ) may be written as
ρ = ρ₀ + ρᴘ(T)
where 'ρ₀' is a constant that increases with increasing impurity content and ρ(T) is temperature dependent part of resistivity. This is known as Mattheissen's rule.
5) For most metals, the electrical resistivity decreases with increasing pressure.
6) The resistivity of alloys that exhibit order-disorder transitions shows pronounced minima corresponding to ordered phases.
Discovery of Artificial Disintegration
The artificial transmutation of one element into another is first accomplished by Rutherford in 1919.
The chamber 'c' was filled with a gas such as Nitrogen and Alpha particles from a radioactive source at 'A' were absorbed in the gas. A sheet of silver foil 'F', itself thick enough to absorb the alpha particles was placed over an opening in the side of chamber. A zinc sulphide screen 'S' was placed outside this opening and a microscope 'M' was used for observing any scintillatons - occuring on the screen 'S'. Scintillations were observed when the chamber was filled with Nitrogen, but when the Nitrogen was replaced by Oxygen or Carbondioxide no scintillations were observed.
Rutherford concluded that the scintillations were produced by high energy particles which were ejected from Nitrogen nuclei as a result of bombardment of these nuclei by alpha particles.
Magnetic deflection experiments indicated that these particles were Hydrogen nuclei or Protons.
Later experiments by Rutherford and Chadwick showed that these ejected Protons had Ranges upto 40cm in air.
Other light elements in the Range from Boron to Potassium were also disintegrated by bombardment with alpha particles.
The disintegration of Nuclei has also been studied with Wilson cloud chamber. One of the first of these investigations was that of Blackett, who photographed the tracks of alpha particles in a Wilson cloud chamber containing 90% Nitrogen and 10% Oxygen. The majority of tracks photographed were straight tracks typical of alpha particle tracks.
Many of the tracks were observed to be forked tracks, indicating that an inelastic collision had taken place between an alpha particle and a Nitrogen Nucleus.
Measurement of the tracks showed that momentum of system was conserved but that the sum of kinetic energies of particles after impact was less than kinetic energy of alpha particle before impact.
On the basis of theory of nucleus advanced by Bohr, the disintegration of Nitrogen by bombardment with alpha particles may be thought as consisting of two separate parts.
The first is the capture of the alpha particle by Nitrogen nucleus which resulted in the formation of a new compound nucleus.
The second is the breaking up of compound nucleus into two particles, one of which is a Proton.
These two processes can be represented by means of a nuclear reaction equation analogous to one representing a chemical reaction.
The nuclear reaction equation for this process is
₂He⁴ + ₇N¹⁴ ---------> (₉F¹⁸✷) -------> ₈O¹⁷ ⁺ ₁H¹ + Q
Q is energy evolved or absorbed during nuclear reaction
Q ---> -Ve ---> energy has been absorbed (endothermic)
Q ---> +Ve ---> energy has been evolved (exothermic)
Q ---> nuclear reaction energy or disintegration energy
If sum of masses of the final particles exceeds that of initial particles, 'Q' must be negative; the energy absorbed in such a nuclear reaction must have been obtained from kinetic energies of the particle.
If 'E1' is kinetic energy of alpha particle just before capture, 'E2' the kinetic energy of Proton, 'E3' the kinetic energy of product nucleus,
Q = E2+E3-E1
In those cases in which Q is positive the sum of kinetic energies of final particles will be greater than kinetic energy of incident alpha particle.
The chamber 'c' was filled with a gas such as Nitrogen and Alpha particles from a radioactive source at 'A' were absorbed in the gas. A sheet of silver foil 'F', itself thick enough to absorb the alpha particles was placed over an opening in the side of chamber. A zinc sulphide screen 'S' was placed outside this opening and a microscope 'M' was used for observing any scintillatons - occuring on the screen 'S'. Scintillations were observed when the chamber was filled with Nitrogen, but when the Nitrogen was replaced by Oxygen or Carbondioxide no scintillations were observed.
Rutherford concluded that the scintillations were produced by high energy particles which were ejected from Nitrogen nuclei as a result of bombardment of these nuclei by alpha particles.
Magnetic deflection experiments indicated that these particles were Hydrogen nuclei or Protons.
Later experiments by Rutherford and Chadwick showed that these ejected Protons had Ranges upto 40cm in air.
Other light elements in the Range from Boron to Potassium were also disintegrated by bombardment with alpha particles.
The disintegration of Nuclei has also been studied with Wilson cloud chamber. One of the first of these investigations was that of Blackett, who photographed the tracks of alpha particles in a Wilson cloud chamber containing 90% Nitrogen and 10% Oxygen. The majority of tracks photographed were straight tracks typical of alpha particle tracks.
Many of the tracks were observed to be forked tracks, indicating that an inelastic collision had taken place between an alpha particle and a Nitrogen Nucleus.
Measurement of the tracks showed that momentum of system was conserved but that the sum of kinetic energies of particles after impact was less than kinetic energy of alpha particle before impact.
On the basis of theory of nucleus advanced by Bohr, the disintegration of Nitrogen by bombardment with alpha particles may be thought as consisting of two separate parts.
The first is the capture of the alpha particle by Nitrogen nucleus which resulted in the formation of a new compound nucleus.
The second is the breaking up of compound nucleus into two particles, one of which is a Proton.
These two processes can be represented by means of a nuclear reaction equation analogous to one representing a chemical reaction.
The nuclear reaction equation for this process is
₂He⁴ + ₇N¹⁴ ---------> (₉F¹⁸✷) -------> ₈O¹⁷ ⁺ ₁H¹ + Q
Q is energy evolved or absorbed during nuclear reaction
Q ---> -Ve ---> energy has been absorbed (endothermic)
Q ---> +Ve ---> energy has been evolved (exothermic)
Q ---> nuclear reaction energy or disintegration energy
If sum of masses of the final particles exceeds that of initial particles, 'Q' must be negative; the energy absorbed in such a nuclear reaction must have been obtained from kinetic energies of the particle.
If 'E1' is kinetic energy of alpha particle just before capture, 'E2' the kinetic energy of Proton, 'E3' the kinetic energy of product nucleus,
Q = E2+E3-E1
In those cases in which Q is positive the sum of kinetic energies of final particles will be greater than kinetic energy of incident alpha particle.
Discovery of Meson
Yukawa predicted that it is due to the exchange of a massive particle between the nucleons leading to a short range force.
A result of much calculation is that the Range of a force is of same order of magnitude as compton wave length of exchanged particle. By analogy the nuclear force has a Range of about 1.4 x 10⁻¹³ cm.
A particle for which ℏ/mc = 1.4 x 10⁻¹³ cm will have its rest mass energy equal to 150 MeV or about 275 times the mass of electron.
The name Mesotron was given to this exchanged particle whose mass is intermediate between that of electron and Proton. The modern name is Meson.
In 1937, a particle believed to be of the type was discovered by "S H Neddermeyer" and "C D Anderson" and independently by "J C Street" and "E C Stevenson" in cloud chamber studies of cosmic rays.
Estimates of the mass of this Meson were made from measurements of curvature of its track in a magnetic field which yielded values for mass of Meson in neighbourhood of 200 electron masses. Both positive and negative particles were observed.
WB Fretter (1946) made some very careful measurements of masses of mu particles, using two cloud chambers, one above the other. They were expanded simultaneously when ever a penetrating particle passed through them. This was accomplished by placing the Geiger Counters above each chamber, the two sets of actuating the expansion mechanism whenever an ionizing particle passed through them as shown in below Fig.
The upper cloud chamber was placed in a magnetic induction of 5300 Gauss so that momentum of particle could be measured. The lower cloud chamber had a set of lead plates 0.5 inch thick and placed 1.5 inch apart so that Range in lead of particles could be measured. Out of 2100 tracks observed, 26 were suitable for measurement, their mass determination is yielded a value of 202Me.
The present accepted value is 207mₑ.
Later Occhialini and Powell and D M Perkins using a special nuclear emulsion photographic plates exposed at high altitudes, observed that some of Mesons stopped in photographic emulsions and produced so called stars - that is, nuclear disintegration with the emission of slow protons or alpha particles.
The photographs showed the curved track of heavy Meson which is named '𝚷' Meson; when captured by a nucleus in the emulsion, the resulting nuclear disintegration produces a star in which 3 charged particles are emitted.
The kinetic energy of muon emitted in the decay of a Pi Meson is always same and is equal to about 4 MeV.
𝚷⁺ ------------> 𝛍⁺ + 𝝂
𝚷⁻ ------------> 𝛍⁻ + 𝝂' ; 𝝂' is anti neutrino
A result of much calculation is that the Range of a force is of same order of magnitude as compton wave length of exchanged particle. By analogy the nuclear force has a Range of about 1.4 x 10⁻¹³ cm.
A particle for which ℏ/mc = 1.4 x 10⁻¹³ cm will have its rest mass energy equal to 150 MeV or about 275 times the mass of electron.
The name Mesotron was given to this exchanged particle whose mass is intermediate between that of electron and Proton. The modern name is Meson.
In 1937, a particle believed to be of the type was discovered by "S H Neddermeyer" and "C D Anderson" and independently by "J C Street" and "E C Stevenson" in cloud chamber studies of cosmic rays.
Estimates of the mass of this Meson were made from measurements of curvature of its track in a magnetic field which yielded values for mass of Meson in neighbourhood of 200 electron masses. Both positive and negative particles were observed.
WB Fretter (1946) made some very careful measurements of masses of mu particles, using two cloud chambers, one above the other. They were expanded simultaneously when ever a penetrating particle passed through them. This was accomplished by placing the Geiger Counters above each chamber, the two sets of actuating the expansion mechanism whenever an ionizing particle passed through them as shown in below Fig.
The upper cloud chamber was placed in a magnetic induction of 5300 Gauss so that momentum of particle could be measured. The lower cloud chamber had a set of lead plates 0.5 inch thick and placed 1.5 inch apart so that Range in lead of particles could be measured. Out of 2100 tracks observed, 26 were suitable for measurement, their mass determination is yielded a value of 202Me.
The present accepted value is 207mₑ.
Later Occhialini and Powell and D M Perkins using a special nuclear emulsion photographic plates exposed at high altitudes, observed that some of Mesons stopped in photographic emulsions and produced so called stars - that is, nuclear disintegration with the emission of slow protons or alpha particles.
The photographs showed the curved track of heavy Meson which is named '𝚷' Meson; when captured by a nucleus in the emulsion, the resulting nuclear disintegration produces a star in which 3 charged particles are emitted.
The kinetic energy of muon emitted in the decay of a Pi Meson is always same and is equal to about 4 MeV.
𝚷⁺ ------------> 𝛍⁺ + 𝝂
𝚷⁻ ------------> 𝛍⁻ + 𝝂' ; 𝝂' is anti neutrino
Properties of Stationary Waves
When two simple harmonic waves of same amplitude, frequency and time period travel in opposite directions in a straight line, the resultant wave obtained is called a stationary or a standing wave.
Properties of stationary waves:
1) In these waves, nodes and anti nodes are formed alternately.
Nodes are positions where particles are at their mean positions having maximum strain.
Anti nodes are positions where the particles vibrate with maximum amplitude having minimum strain.
2) The medium is split into segments and all particles of same segment vibrate in phase. The particles in one segment have a phase difference of '𝜫 ' with the particles in neighboring segment.
3) Condensations and rarefractions do not travel forward as in progressive wave but they appear and disappear alternately at same place.
4) As condensations and rarefractions do not travel forward there is no transfer of energy.
5) The distance between two adjacent nodes is 'ƛ/2' and also the distance between two adjacent antinodes is 'ƛ/4'. Between the two nodes there is anti node and vice versa.
6) The general appearance of wave can be represented by a sine curve but it reduces to straight line twice in each time period.
Properties of stationary waves:
1) In these waves, nodes and anti nodes are formed alternately.
Nodes are positions where particles are at their mean positions having maximum strain.
Anti nodes are positions where the particles vibrate with maximum amplitude having minimum strain.
2) The medium is split into segments and all particles of same segment vibrate in phase. The particles in one segment have a phase difference of '𝜫 ' with the particles in neighboring segment.
3) Condensations and rarefractions do not travel forward as in progressive wave but they appear and disappear alternately at same place.
4) As condensations and rarefractions do not travel forward there is no transfer of energy.
5) The distance between two adjacent nodes is 'ƛ/2' and also the distance between two adjacent antinodes is 'ƛ/4'. Between the two nodes there is anti node and vice versa.
6) The general appearance of wave can be represented by a sine curve but it reduces to straight line twice in each time period.
REFLECTION, REFRACTION AT PLANE SURFACES
LAWS OF REFLECTION
The angle of incidence is equal to angle of reflection.
The incident ray, Normal and Reflected ray ray all lie in one plane.
PROPERTIES OF IMAGE FORMED BY PLANE MIRROR
- The image formed by a plane mirror is "virtual", "erect" and laterally reversed.
- The size of image is equal to size of object.
- The image is as far behind the mirror as the source is in front of it.
- When the plane mirror is rotated through certain angle, the reflected ray turns through double the angle.
- When two plane mirrors are kept facing each other at an angle '𝛳 ' and an object is placed between them, multiple images of the object are formed as a result of multiple successive reflections.
a) If (360/𝛳) is "even", then no. of images is given by n = (360/𝛳)-1
b) If (360/𝛳) is "odd", then following two situations arise
i) If object lies symmetrically, then n = (360/𝛳)-1
ii) If object lies unsymmetrically, then n = 360/𝛳
c) When two plane mirrors are placed parallel to each other, then
n = (360/0) = ∞ (infinite no. of images)
Note:-
I) The point object for a mirror is a point from which the rays incident on mirror actually diverge or towards which the incident rays appear to converge.
II) An optical image is a point where rays of light either intersect or appear to do so.
REFRACTION OF LIGHT
The Refracted ray bends towards the Normal when the second medium is denser than first medium and vice versa.
The deviation 'D' suffered by refracted ray is given by D = i-r
2. Snells Law: For any media, the ratio of sine of angle of incidence to sine of angle of refraction is a constant for a light beam of particular wavelength.
sini/sinr = 𝜇2/𝜇1 = constant
Refractive index 𝜇 = velocity of light in vacuum / velocity of light in medium
The deviation 'D' suffered by refracted ray is given by D = i-r
LAWS OF REFRACTION
1. The Incident ray, the Refracted ray and the Normal to surface separating two media lie in one plane.2. Snells Law: For any media, the ratio of sine of angle of incidence to sine of angle of refraction is a constant for a light beam of particular wavelength.
sini/sinr = 𝜇2/𝜇1 = constant
Refractive index 𝜇 = velocity of light in vacuum / velocity of light in medium
Nature of orbits of satellites for different speeds
Let
'V' be velocity with which a body is projected from Earth.
Vs be minimum velocity of object to orbit around earth
Ve be escape velocity from surface of earth
then if,
i) V < Vs --- body falls to ground
ii) V = Vs --- body rotates round earth in circular orbit closer to surface of Earth
iii) Vs < V < Ve --- body revolves in elliptical orbit
iv) V = Ve ----------body just escapes from gravitational field
v) V > Ve --------- body moves in interstellar space with velocity equal to √V² -Ve²
vi) V<Ve ---------- Total energy of body is negative
vii) V =Ve ---------- Total energy of body is zero
'V' be velocity with which a body is projected from Earth.
Vs be minimum velocity of object to orbit around earth
Ve be escape velocity from surface of earth
then if,
i) V < Vs --- body falls to ground
ii) V = Vs --- body rotates round earth in circular orbit closer to surface of Earth
iii) Vs < V < Ve --- body revolves in elliptical orbit
iv) V = Ve ----------body just escapes from gravitational field
v) V > Ve --------- body moves in interstellar space with velocity equal to √V² -Ve²
vi) V<Ve ---------- Total energy of body is negative
vii) V =Ve ---------- Total energy of body is zero
Satellites - Important points to be noted
1. Orbital velocity of satellite is independent of mass of the satellite but depends on mass of planet and radius of orbit.
2. A satellite orbiting around a planet will have both Potential energy and Kinetic energy. Here Potential energy is negative and Kinetic energy is positive.
3. Total energy of satellite is negative.
4. With the increase of height of orbit from surface of planet, for a satellite
5. A satellite orbiting very close to surface of Earth is known as its surface satellite. Orbital velocity for such a satellite is V = √gR = 8 Km.S⁻¹.
6. Relative velocity of parking satellite with respective to Earth is zero.
7. Orbital linear velocity is about 3 Km.Sec⁻¹.
8. A satellite cannot be coast in a stable orbit in a plane not passing through the Earth's center.
9. If two satellite move around the Earth in its equitorial plane such that one moves from West to East and other from East to West and other from East to West, the time period of revolution of first satellite will be more compared to other.
10. If a rocket launched in equitorial plane from West to East, advantage is up to 0.47 Km.Sec⁻¹ in the launching speed.
11. If the Kinetic energy of an orbiting satellite is E, its Potential Energy will be -2E and total energy will be -E.
12. If a body is in a satellite which does not produce its own gravity, its true weight in that satellite W' is given by
W'/W =mg'/mg ; W' = W/(1+[h/R])²
W - Weight of body on Earth
h - Height of orbit of satellite
R - Radius of Earth
so true weight is lesser than its weight on Earth.
13. Apparent weight of a body in a satellite is zero and is independent of radius of orbit .
2. A satellite orbiting around a planet will have both Potential energy and Kinetic energy. Here Potential energy is negative and Kinetic energy is positive.
3. Total energy of satellite is negative.
4. With the increase of height of orbit from surface of planet, for a satellite
a) Potential energy increases (from more negative to less negative)
b) Kinetic energy decreases
c) Orbital velocity decreases
d) Total energy increases
e) Period of revolution increases
6. Relative velocity of parking satellite with respective to Earth is zero.
7. Orbital linear velocity is about 3 Km.Sec⁻¹.
8. A satellite cannot be coast in a stable orbit in a plane not passing through the Earth's center.
9. If two satellite move around the Earth in its equitorial plane such that one moves from West to East and other from East to West and other from East to West, the time period of revolution of first satellite will be more compared to other.
10. If a rocket launched in equitorial plane from West to East, advantage is up to 0.47 Km.Sec⁻¹ in the launching speed.
11. If the Kinetic energy of an orbiting satellite is E, its Potential Energy will be -2E and total energy will be -E.
12. If a body is in a satellite which does not produce its own gravity, its true weight in that satellite W' is given by
W'/W =mg'/mg ; W' = W/(1+[h/R])²
W - Weight of body on Earth
h - Height of orbit of satellite
R - Radius of Earth
so true weight is lesser than its weight on Earth.
13. Apparent weight of a body in a satellite is zero and is independent of radius of orbit .
FRICTION - Important Points to be remembered
1. The force which opposes the relative motion of two surfaces of bodies in contact, is called as "frictional force".
2. Friction is tangential force between the contact surfaces of two bodies.
3. Friction is due to Electromagnetic Forces between the surfaces in contact.
4. Friction is due to molecular interaction at the surfaces in contact.Friction is due to adhesive forces between molecules of two surfaces in contact.
5. Friction depends on nature of surfaces in contact and on the impurities present on these surfaces.
6. Normal Force: When two bodies are in contact or when one body is placed over another body, the contact force which either body exerts on other normal to contact surface is called Normal Force or Normal Reaction.
4. Friction is due to molecular interaction at the surfaces in contact.Friction is due to adhesive forces between molecules of two surfaces in contact.
5. Friction depends on nature of surfaces in contact and on the impurities present on these surfaces.
6. Normal Force: When two bodies are in contact or when one body is placed over another body, the contact force which either body exerts on other normal to contact surface is called Normal Force or Normal Reaction.
7. Friction is proportional to Normal Force.
8. Limiting Friction is least force necessary to set a body into motion.
9. Sliding Friction is the friction which comes into play when the surface of an object moves relative to the surface of another object.
8. Limiting Friction is least force necessary to set a body into motion.
9. Sliding Friction is the friction which comes into play when the surface of an object moves relative to the surface of another object.
10. Static friction is the friction which comes into play when surfaces of the objects are at rest relative to each other even there is an external force acted upon.
11. Static friction is a self adjusting force.
12. Kinetic Friction is not a self adjusting friction.
12. Kinetic Friction is not a self adjusting friction.
13. The substances which reduce friction are called as lubricants.
14. Generally coefficient of static friction is less than 1 but in some cases it may exceed 1.
15. Frictional force is a "Non-Conservative" force.
16. If a body of mass 'm' is on the floor of a lift which is moving with uniform acceleration 'a', Normal force on body or its apparent weight is
N = mg ±ma = m(g±a)
a) If the lift moves up, then N = m(g+a)
b) If the lift moves down, then N = m(g-a)
c) If the lift falls freely, then N=0
d) If the lift moves with uniform velocity, then a =0, and N=mg
17. When a person falls on a rough road, the frictional force exerted by road on him is along his direction of motion.
18. The angle made by resultant of Normal force and Limiting friction with Normal force is called angle of friction. The tangent of this angle gives coefficient of static friction.
19. The substances which reduce friction are called Lubricants.
14. Generally coefficient of static friction is less than 1 but in some cases it may exceed 1.
15. Frictional force is a "Non-Conservative" force.
16. If a body of mass 'm' is on the floor of a lift which is moving with uniform acceleration 'a', Normal force on body or its apparent weight is
N = mg ±ma = m(g±a)
a) If the lift moves up, then N = m(g+a)
b) If the lift moves down, then N = m(g-a)
c) If the lift falls freely, then N=0
d) If the lift moves with uniform velocity, then a =0, and N=mg
17. When a person falls on a rough road, the frictional force exerted by road on him is along his direction of motion.
18. The angle made by resultant of Normal force and Limiting friction with Normal force is called angle of friction. The tangent of this angle gives coefficient of static friction.
19. The substances which reduce friction are called Lubricants.
20. A good lubricant must be highly viscous and low volatile in nature.
21. The frictional force exerted by fluids is also called as "drag".
22. Frictional force on an object in a fluid depend on its speed with respect to fluid, on the shape of the object and on the nature of fluid.
23. Friction can produce heat.
Crystal Growth Methods - Brief Explanation
Various types of crystal growth methods are
Growth from Water Solution
Growth from Flux
Hydro Thermal Growth
Electrode Deposition
Gel Growth
This technique is used for soluble crystals like sugar, salt crystals for example NaCl, KCl, KBr are used. Their growth rates are very small. They have 5 mole percent solubility.
Nucleation is one such process. Liquid containing crystal solution solution having low viscosity is taken into a beaker. Crystal which has to be grown is taken in very small size which is called as seed crystal. We have to hang this crystal in the liquid in beaker. The molecules join crystal to form the crystal big in size.
Advantages of this technique
a) Growth is at temperature well below the melting point
b) High quality crystals can be obtained
c) Doping with suitable materials could be done
d) Solid solution can be grown easily
So when pressure is exerted on crystal then melting point of material decreases (of about 50000 pounds per sq. inch). So special devices such as autoclaves are used for this purpose.
conditions of growth in hydro thermal process:
i) The Temperature
ii) The Pressure
iii) The temperature difference between top and bottom ends of autoclave
Growth from Water Solution
Growth from Flux
Hydro Thermal Growth
Electrode Deposition
Gel Growth
Growth from Water Solution
Nucleation is one such process. Liquid containing crystal solution solution having low viscosity is taken into a beaker. Crystal which has to be grown is taken in very small size which is called as seed crystal. We have to hang this crystal in the liquid in beaker. The molecules join crystal to form the crystal big in size.
Growth from Flux
This method is used for crystals which are not dissoluble. This technique uses oxide crystals/metal crystals. Crystals like quartz having high melting point of 1400 oC which is attained at higher energy are grown using this technique. For this they are combined with other crystals called as flux whose benefit is to reduce the melting point of crystals to form.Advantages of this technique
a) Growth is at temperature well below the melting point
b) High quality crystals can be obtained
c) Doping with suitable materials could be done
d) Solid solution can be grown easily
Hydro thermal growth
This method is used for crystals whose melting point is very high. For instance, Al2O3 cannot be soluble in water. Normally Al2O3 dissolves in water at critical temperature of 353 ⁰C.So when pressure is exerted on crystal then melting point of material decreases (of about 50000 pounds per sq. inch). So special devices such as autoclaves are used for this purpose.
conditions of growth in hydro thermal process:
i) The Temperature
ii) The Pressure
iii) The temperature difference between top and bottom ends of autoclave
Charactersitics of electron in one dimensional periodic potential
CRYSTAL MOMENTUM
For free electron, the quantity ℏk represents true momentum of electron as follows:
E= (ℏk)²/2m = (1/2m)*(ℏ)²*(k)² = (1/2m)*(h/2𝝅)²*(2𝝅/ƛ)² = (1/2m)*(h/ƛ)² = P²/2m
Therefore, the dynamical behavior of free electron can be represented by true momentum.
But when we consider an electron in periodic potential, ℏk doesn't represent true momentum. ℏk does not represent true momentum. The energy doesn't vary with 'k' as in previous case.
The true instantaneous momentum of an electron in presence of lattice potential is not a constant of motion and cannot be calculated by quantum mechanical method we take average value.
So in order to describe the dynamical behavior of electron in periodic potential we introduce a new type of momentum called as Crystal Momentum.
When we deal with interactions of electrons with lattice, we use conservation of crystal momentum and not of true momentum.
The crystal momentum is perfectly well defined constant for a state of given energy.
VELOCITY
The quantum mechanical part describes that the velocity of electron in a one dimensional lattice will be equal to Group Velocity of waves representing the electron.
v = (dw/dk) ...................................................................................................(1)
where 'w' is angular frequency of debroglie waves.
Eqn(1) depends on actual E-K curve.
(dE/dk) = ℏ (dw/dk) ; v=(1/ℏ)(dE/dk) ..........................................................(2)
for free electrons, substituting E= ℏk, v=p/m
giving linear variation of 'v' with 'k'.
In band theory of solids, however, 'E' is not proportional to k².
The variation of 'E' with 'k' is as sown in fig:
using this type of variation of 'E' with 'k' as shown in fig below.
We observe that at bottom (k=0) of energy band, the velocity is zero and as the value of 'k' increases ('E' increases) the velocity increases reaching its maximum at k=kₒ, where kₒ corresponds to "point of inflection" on E-K curve. Beyond this point the velocity begins to decrease and finally assumes zero value at k=𝝅/a, which is top of band. These are entirely new features which do not appear at all in behavior of free electrons.
It is known for long time that an electron has well defined mass and when accelerated by an electric field, it obeys Newtonian Mechanics. What happens when electron is to be accelerated inside a crystal? How will it react to electric field.
The mass of an electron inside the crystal appears, in general, different from free electron mass and is usually referred to as "effective mass".
The velocity of an electron in one dimensional lattice is given by
v = (2𝝅/ℏ)(dE/dk)..................................................................................................(3)
a= dv/dt = (2𝝅/h)(d²E/dk²)*(dk/dt) .......................................................................(4)
so we have to find value of dk/dt.
Let an electron is subjected to influence of an electric field of strength 'E' for a time dt. If velocity of electron is v, the distance travelled in time dt=vdt
Therefore Work done, dE=(e𝜀)*v*dt
we know
v = (2𝝅/h)(dE/dk) ; therefore dE= (e𝜀)*(2𝝅/h)*(dE/dk)*dt
(dk/dt) = 2𝝅e𝜀/h .....................................................................................................(5)
substituting (5) in (4), we get
a = (2𝝅/h)² *(e𝜀)*(d²E/dk²) ...................................................................................(6)
For free particle, a= m(dv/dt) = eE;
a=e*E/m ..................................................................................................................(7)
comparing (6) & (7) , both forms are identical, we introduce a new mass known as effective mass given by
m* = (h/4𝝅) * (d²E/dk²)⁻¹ ......................................................................................(8)
For free electron,
m* = m
Physically speaking near the bottom of band the effective mass m* has a constant value which is positive because the quadratic eqn [E ∝k²] is satisfied near the bottom of band.
But as 'k' increases m* is no longer a constant, being now a function of k, because quadratic relation is no longer valid.
The degree of freedom of an electron is defined by a factor
For free electron, the quantity ℏk represents true momentum of electron as follows:
E= (ℏk)²/2m = (1/2m)*(ℏ)²*(k)² = (1/2m)*(h/2𝝅)²*(2𝝅/ƛ)² = (1/2m)*(h/ƛ)² = P²/2m
Therefore, the dynamical behavior of free electron can be represented by true momentum.
But when we consider an electron in periodic potential, ℏk doesn't represent true momentum. ℏk does not represent true momentum. The energy doesn't vary with 'k' as in previous case.
The true instantaneous momentum of an electron in presence of lattice potential is not a constant of motion and cannot be calculated by quantum mechanical method we take average value.
So in order to describe the dynamical behavior of electron in periodic potential we introduce a new type of momentum called as Crystal Momentum.
When we deal with interactions of electrons with lattice, we use conservation of crystal momentum and not of true momentum.
The crystal momentum is perfectly well defined constant for a state of given energy.
VELOCITY
The quantum mechanical part describes that the velocity of electron in a one dimensional lattice will be equal to Group Velocity of waves representing the electron.
v = (dw/dk) ...................................................................................................(1)
where 'w' is angular frequency of debroglie waves.
Eqn(1) depends on actual E-K curve.
(dE/dk) = ℏ (dw/dk) ; v=(1/ℏ)(dE/dk) ..........................................................(2)
for free electrons, substituting E= ℏk, v=p/m
giving linear variation of 'v' with 'k'.
In band theory of solids, however, 'E' is not proportional to k².
The variation of 'E' with 'k' is as sown in fig:
using this type of variation of 'E' with 'k' as shown in fig below.
We observe that at bottom (k=0) of energy band, the velocity is zero and as the value of 'k' increases ('E' increases) the velocity increases reaching its maximum at k=kₒ, where kₒ corresponds to "point of inflection" on E-K curve. Beyond this point the velocity begins to decrease and finally assumes zero value at k=𝝅/a, which is top of band. These are entirely new features which do not appear at all in behavior of free electrons.
EFFECTIVE MASS OF ELECTRON
It is known for long time that an electron has well defined mass and when accelerated by an electric field, it obeys Newtonian Mechanics. What happens when electron is to be accelerated inside a crystal? How will it react to electric field.
The mass of an electron inside the crystal appears, in general, different from free electron mass and is usually referred to as "effective mass".
The velocity of an electron in one dimensional lattice is given by
v = (2𝝅/ℏ)(dE/dk)..................................................................................................(3)
a= dv/dt = (2𝝅/h)(d²E/dk²)*(dk/dt) .......................................................................(4)
so we have to find value of dk/dt.
Let an electron is subjected to influence of an electric field of strength 'E' for a time dt. If velocity of electron is v, the distance travelled in time dt=vdt
Therefore Work done, dE=(e𝜀)*v*dt
we know
v = (2𝝅/h)(dE/dk) ; therefore dE= (e𝜀)*(2𝝅/h)*(dE/dk)*dt
(dk/dt) = 2𝝅e𝜀/h .....................................................................................................(5)
substituting (5) in (4), we get
a = (2𝝅/h)² *(e𝜀)*(d²E/dk²) ...................................................................................(6)
For free particle, a= m(dv/dt) = eE;
a=e*E/m ..................................................................................................................(7)
comparing (6) & (7) , both forms are identical, we introduce a new mass known as effective mass given by
m* = (h/4𝝅) * (d²E/dk²)⁻¹ ......................................................................................(8)
For free electron,
m* = m
- Effective mass can also be determined using "Cyclotron Resonance Experiment".
- Effective mass need not always be greater than 'm'. It can be smaller than 'm'.
- It can be negative.
Physically speaking near the bottom of band the effective mass m* has a constant value which is positive because the quadratic eqn [E ∝k²] is satisfied near the bottom of band.
But as 'k' increases m* is no longer a constant, being now a function of k, because quadratic relation is no longer valid.
The degree of freedom of an electron is defined by a factor
fk = (m/m*) = (m/ℏ2)*(d²E/dk²)
fk is measure of extent to which an electron in state 'k' is free.
fk is measure of extent to which an electron in state 'k' is free.
Principles of Special Theory of Light
1. Does the speed
of light depend on motion of source of light?
No, the motion
of light is not affected by motion of source of light.
2. Is photon a
particle?
The photon is
a particle of light, but it doesn’t possess all essential properties we ascribe
to a tiny ball i.e. photon doesn’t behave as a common sense particle but it has
got some peculiar properties.
3. When we follow
Albert Einstein in developing special theory of relatively, we are developing a
theory of space and time.
4. The principles of special theory of light.
Principle 1:
Colloquial
statement: If we are in unaccelerated vehicle, its motion has no effect
on the way things happen inside it.
Formal statement:
The laws of physics are the same in all unaccelerated reference frames.
Principle 2: The
motion of light is not affected by motion of source of light.
5. The special theory of relativity
Special:
The word special in name arises because we employ only unaccelerated reference
frames, not all reference frames that one can think of. In other words, we
special to the way things appear when observed from uniformly moving reference
frames.
Relativity:-The
word relativity comes from a phrase coined by Henri Poincare, an eminent French
physicist and mathematician.
In 1904, Poincare was invited to address the international
congress of arts and science, held in st Louis to commemorate the 100th
anniversary of Louisiana Purchase. Poincare spoke of a principle of relativity.
If you are in plane on its way from Chicago to phoenix,
another plane making the return flight, over wheat fields of Kansas. A farmer,
looking up, notes that you are flying south west at 500 miles/hr relative to
his wheat fields.
The pilot of return flight notes that the distance between
the two planes is decreasing at about 1000 miles/hr. So far as the pilot is
concerned, you are travelling at about 1000 miles/hr relative to his plane.
The essence is this: statements
about uniform motion relative to a specified reference frame wheat fields or
another air plane are meaningful.
A quantitative statement about uniform motion without
specification of a reference frame is not meaningful. Why? Because our
principle 1 says we cannot discern uniform motion without recourse to some
reference frame.
Take first the colloquial form of that principle if we are
in an unaccelerated vehicle, its motion has no effect on the way things happen
inside it. So by just doing experiments inside the vehicles, we have no way to
assign a velocity to the vehicle. Only if we look out of window and thereby use
wheat fields of Kansas as an outside reference frame. We can decide on velocity
(velocity to that outside reference frame).
Theory: It
appears because principles 1 & 2
are generations from observation and experiment.
6. THE CONSTANCY OF SPEED OF LIGHT
- Observes in all un accelerated reference frames measure the same speed for light ( in vacuum) from any given source.
- They all measure 3*10 8m/sec always for light in vacuum.
- This remarkable property is called “constancy of speed of light”.
Note:-Some
factors other than light may be observed differently in unaccelerated frames.
7. An “event” is anything
that happens at some definite locations at some definite time. Proto typical
examples are your birth, assassination of Abraham Lincoln etc. In contrast, a
forest fire that sweeps across 10000 acres in 5 days does not constitute an
“event” because the fire is spread out in space and time.
The adjective “definite” means “distinct” or “limited” for any one observing the happening.
8. THE RELATIVITY OF
SIMULTANEITY:
- Spatially separated events that are simultaneous in one frame are, in general, not simultaneous when viewed from other reference frame.
- Simultaneity is a relative concept, but not an absolute one.
- The concept of simultaneity between two events in different space points has an exact meaning only in relation to a given inertial system i.e. “Each frame of reference has its own particular time”.
- To measure the length of an object means to locate its end points simultaneously. As simultaneity depends on frame of reference, the length measurements will also depend on frame of reference.
- Thus, “The length i.e. Space is a relative concept, not an absolute one”.
- Thus there is no such thing as an absolute, global “now”.
Brewsters Law
When light is transferred from glass surface it may be polarized. It was explained by Brewster. According to Brewster the light reflected from a surface is completely polarized if the reflected beam and the beam refracted into material form a right angle. If the incident beam is polarized in plane of incidence there will be no reflection at all. Only if the incident beam is polarized normal to the plane of incidence it will be reflected.
Reason for above property
In the reflecting material, the light is polarized transversely and we know that it is motion of charges in material which generates emergent beam which we call as reflected beam.The source of this so called reflected light is not simply that the incident beam is reflected; our deeper understanding of this phenomenon tells us that the incident beam drives an oscillation of charges in the material, which in turn generates reflected beam.
From below fig, it is clear that only oscillations normal to paper can radiate in direction of reflection and consequently reflected beam will be polarized normal to plane of incidence. If the incident beam is polarized in plane of incidence there will be no reflected light.
This phenomenon is readily demonstrated by reflecting a linear polarized beam from a flat piece of glass. If glass is turned to different angles of incidence to polarized beam, sharp attenuation of reflected intensity is observed when angle of incidence passes through Brewsters angle. This attenuation is observed only if plane of polarization lies in plane of incidence. If plane of polarization is normal to plane of incidence, the usual reflected intensity is observed at all angles.
BLOCH THEOREM
Bloch assumed that electrons move in a perfect periodic potential. He considered one dimensional array of lattice. The potential of electron at positive ion site is zero and is maximum in between. So long any line passing through the centers of positive ions, the potential variation must be as shown in below figure.
So Bloch gave a condition which is
𝚿(x+Na)=𝚿(x) .............................................................................................................(1)
It is considered as boundary condition.
Consider Schrodinger wave equation for one dimensional lattice.
(d²𝚿(x)/dx²) + (2m/ħ²)*[E-V(x)]*𝚿(x) = 0 .................................................................(2)
The Schrodinger equation for an electron in the potential at x+a is
[d²𝚿(x+a)/d(x+a)²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0 ..............................................(3)
Because of periodicity,
[d/d(x+a)] = d/dx ; V(x+a) = V(x)
With this, eqn (3) reduces to
[d²𝚿(x+a)/dx²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0 ...................................................(4)
This is Schrodinger equation at x+a.
as 𝚿 at x+a is also obeying Schrodinger wave equation as 𝚿 at x there should exist a relation between 𝚿(x+a) & 𝚿(x).
Let 𝚿(x+a) = A𝚿(x)..................................................................................................(5)
𝚿(x+2a) = A²𝚿(x) [i.e. A𝚿(x+a) = A.A𝚿(x) = A²𝚿(x)]
𝚿(x+na) = Aⁿ𝚿(x)
from eqn(1), Aⁿ =1 [i.e by using bloch condition]
Aⁿ =exp(2πij) [i.e. exp(2πij) =1 for j=01,2............]
or
A=exp(2πij/n)
Therefore, 𝚿(x+a) = exp(2πij/n)*𝚿(x) --------------------------------------------------(6) [ from eqn 5]
𝚿(x) can be written in terms of other function Uk(x )
𝚿(x) = exp(ikx)*Uk(x) where k=(2πj/n) ..................................................................(7)
From eqns (6) & (7),
exp[ik(x+a)]*Uk(x+a) = exp(2πij/n)*exp(ikx)*Uk(x)
exp[ika]*Uk(x+a) = exp(2πij/n)*Uk(x)
noting that Ka = 2πj/n,
we can write that Uk(x+a) = Uk(x) ..........................................................................(8)
So Bloch gave a condition which is
𝚿(x+Na)=𝚿(x) .............................................................................................................(1)
It is considered as boundary condition.
Consider Schrodinger wave equation for one dimensional lattice.
(d²𝚿(x)/dx²) + (2m/ħ²)*[E-V(x)]*𝚿(x) = 0 .................................................................(2)
The Schrodinger equation for an electron in the potential at x+a is
[d²𝚿(x+a)/d(x+a)²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0 ..............................................(3)
Because of periodicity,
[d/d(x+a)] = d/dx ; V(x+a) = V(x)
With this, eqn (3) reduces to
[d²𝚿(x+a)/dx²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0 ...................................................(4)
This is Schrodinger equation at x+a.
as 𝚿 at x+a is also obeying Schrodinger wave equation as 𝚿 at x there should exist a relation between 𝚿(x+a) & 𝚿(x).
Let 𝚿(x+a) = A𝚿(x)..................................................................................................(5)
𝚿(x+2a) = A²𝚿(x) [i.e. A𝚿(x+a) = A.A𝚿(x) = A²𝚿(x)]
𝚿(x+na) = Aⁿ𝚿(x)
from eqn(1), Aⁿ =1 [i.e by using bloch condition]
Aⁿ =exp(2πij) [i.e. exp(2πij) =1 for j=01,2............]
or
A=exp(2πij/n)
Therefore, 𝚿(x+a) = exp(2πij/n)*𝚿(x) --------------------------------------------------(6) [ from eqn 5]
𝚿(x) can be written in terms of other function Uk(x )
𝚿(x) = exp(ikx)*Uk(x) where k=(2πj/n) ..................................................................(7)
From eqns (6) & (7),
exp[ik(x+a)]*Uk(x+a) = exp(2πij/n)*exp(ikx)*Uk(x)
exp[ika]*Uk(x+a) = exp(2πij/n)*Uk(x)
noting that Ka = 2πj/n,
we can write that Uk(x+a) = Uk(x) ..........................................................................(8)
Conclusion
Bloch Theorem is a mathematical theorem and it gives us the form of electron wave function in a periodic potential.
𝚿(x) = exp(ikx)*Uk(x) represents Plane Wave
Thus, electron in a one dimensional lattice behaves a a plane wave.It only gives Wave nature of electron.
GIBBS CANONICAL ENSEMBLE
(SYSTEM IN CONTACT WITH A HEAT RESERVOIR)
Canonical ensemble describes those systems that are not isolated but are in contact with a heat reservoir.
The system under consideration together with a heat reservoir forms a closed system and then system of interest is taken as sub system of this closed system.
Any sub system of an isolated system in statistical equilibrium can be represented by a canonical ensemble.
The probability density of a canonical ensemble depends both on Energy 'E' and temperature 'T'
It is given by
Partition function for canonical ensemble
The statistical energy of a system in a canonical ensemble is given by
Thermodynamical entropy 'S' of a system in a canonical ensemble is given by
The entropy at absolute zero in a canonical ensemble can be expressed as
gₒ is statistical weight of ground state
Canonical ensemble describes those systems that are not isolated but are in contact with a heat reservoir.
The system under consideration together with a heat reservoir forms a closed system and then system of interest is taken as sub system of this closed system.
Any sub system of an isolated system in statistical equilibrium can be represented by a canonical ensemble.
The probability density of a canonical ensemble depends both on Energy 'E' and temperature 'T'
It is given by
ρ(E,T) = Ae-E/𝜏
Partition function for canonical ensemble
z=(1/h³ⁿ.n!)∫exp(-E(q,p)/KTdΓ
The statistical energy of a system in a canonical ensemble is given by
𝜎 = logZ +(E/𝜏); 𝜎 = logZ +T[∂(log Z)/∂T]
Thermodynamical entropy 'S' of a system in a canonical ensemble is given by
S=K𝜎
S=KlogZ + (E/T); U=KT2[∂(log Z)/∂T]
The entropy at absolute zero in a canonical ensemble can be expressed as
S=K.log gₒ
gₒ is statistical weight of ground state
What is Chemical Potential?
For a chemical system, molar free energy is known as Chemical Potential.
A chemical substance that is free to move from one place to another place, will move spontaneously from a state of higher chemical potential to a state of lower chemical potential.
In the position of equilibrium, the chemical potential is constant through the entire system.
Let us consider a general heterogeneous system consisting of an independent components in several coexisting phases.
To start with, it is convenient to describe a given phase by its chemical composition, which is specified by the no. of mole 'Ni' of each species i, its volume V and its entropy 'S'.
If we consider internal energy (U)
'μi' is chemical potential of component 'i' in given phase.
We can also consider chemical potential 'μ' in terms of Helmoltz free energy 'F'.
The chemical potentials are thus the rate of change of free energy per mole, at constant volume and temperature.
μ can also be expressed as
A System in external field will be in equilibrium if the temperature and chemical potential of each component of the system is constant through out, i.e.
A chemical substance that is free to move from one place to another place, will move spontaneously from a state of higher chemical potential to a state of lower chemical potential.
In the position of equilibrium, the chemical potential is constant through the entire system.
Let us consider a general heterogeneous system consisting of an independent components in several coexisting phases.
To start with, it is convenient to describe a given phase by its chemical composition, which is specified by the no. of mole 'Ni' of each species i, its volume V and its entropy 'S'.
If we consider internal energy (U)
U=U(S,V,N₁,N₂,.....Nᵢ,.....Nn)
μi=❴∂U/∂Nᵢ❵S,V,Nj ; j= except 'i'
'μi' is chemical potential of component 'i' in given phase.
dU=TdS-PdV+Σμᵢi.dNᵢ for i=1...n
We can also consider chemical potential 'μ' in terms of Helmoltz free energy 'F'.
F = F(T,V,N₁,N₂......Nn)
μ1=❴∂F/∂N1❵T,V,N₂,....
μ2=❴∂F/∂N2❵T,V,N₁,N₃,....
The chemical potentials are thus the rate of change of free energy per mole, at constant volume and temperature.
μ can also be expressed as
μi=❴∂G/∂Nᵢ❵T,P,Nj
A System in external field will be in equilibrium if the temperature and chemical potential of each component of the system is constant through out, i.e.
dT₁=0 and dμᵢ=0
Subscribe to:
Posts (Atom)