Download Text Book: Fundamentals of Physics Authors: Halliday, Resnick

PHY102- GENERAL PHYSICS II
Text Book: Fundamentals of Physics
Authors: Halliday, Resnick & Walker
Edition: 8th Extended
Lecture Schedules
TOPICS:
Ch. 39 – More About Matter Waves
Ch. 40 – All About Atoms
Ch. 41– Conduction of Electricity in Solids.
Dr. O.k. oWOadE
Rm. 214 {Dept. of Physics and Engineering
Physics.}
Thermal Radiation
•The spectra of electromagnetic radiation (light) emitted by hot
objects.
•The total intensity of the emitted radiation is proportional to the
fourth power of the absolute temperature (T⁴).
•Since the temperature of all objects are above absolute zero, all
objects emit thermal radiations.
•At every temperature, this radiation is in the infrared red region
and so is not visible.
•Increased temperature causes the radiation to shift to slower
wavelength and the colour to change to yellow-orange.
•Although we see only a particular color, there is a continuous
spectrum (all wavelengths) at all temperatures.
•The curves shown do not correspond to any real radiating body,
they are for an idealized blackbody. A black body is an idealized
system that absorbs (and emits) all the radiation that is incident
on it. DIAGRAM
•More radiation is emitted at every wavelength and the waves length of
the maximum – intensity component ( max) becomes shorter. This
wavelength shift obey a relationship named Wein’s displacement law.
max T  2.99 *10 mK
3
max = wavelength of the radiation (in meters) of which maximum
intensity occurs.
T = temperature of the body (in Kelvin)
•It can be used to determine the wavelength of the maximum spectral
component if the temperature is known.
• Photons is quantized and exists in elementary amount (quanta).
•From classical idea that light is a sinusoidal waves with a wavelength,
,
frequency f, and speed c such that
c
f 

Photon in line with Einstein’s proposal.
E  hf
The quantum of light waves of the frequency f, has the energy E, h is the
plank constant .
h = 6.63 x 10¯³⁴Js = 4.14 x 10⁻¹⁵ eV.s
Einstein proposed that when light is absorbed or emitted by an object
(matter), the absorption or emission events occurs in the atoms of the
object.
The light is emitted and absorbed as photons. We assume that all the light
emitted by the lamp reaches (and this absorbed by) the sphere, the rate R at
which photons are absorbed by the sphere is equal to the rate the photons
are emitted by the lamp.
Pemit
rateofenergyemission
Remit 
R  Remit
ratepereimmittedphoton
Pemit Pemit 


hf
hc

E
The Photoelectric Effect
• If you direct a beam of light of short wavelength into a clean metal
surface, the light will cause electrons to leave that surface (light will
eject the electrons from the surface ).
• Photoelectric effect is used in many devices, including TV, camera,
camcorder, and night vision viewers.
•The potential difference V vary until it reaches a certain value
called the stopping potential Vstop at which point the reading of
meter A has just dropped to zero.
The kinetic energy K
of the most energetic electrons
max
K max  eVstop
Note that the photoelectric effect does not occur if the frequency
is below a certain cut off frequency fₒ or if the wavelength is
greater than the corresponding cut off wavelength 0  c
f
0
Work Function Q of a metal is the energy which must be supplied to
enable an electron to escape from its surface. It is expressed in
electron volts (eV).
To eject escape from the target, an electron must pick up a certain
minimum energy Q, where Q is a property of the target material
called its work function. If the energy hf, transferred to an electron
by a photon exceeds the work function of the material (if hf >Q), the
electron escape the target.
If the energy transferred does not exceed the work function (hf < Q),
the electron cannot escape.
hf  K max  Q
K max  hf  Q
eVstop  hf  Q
Q
h
Vstop  
f


e
e


Q  hf  h
0
0
•Einstein extended his concept of light quantum (photons) by
proposing that a quantum of light has linear momentum.
hf
h
P

c

Arthur Compton (1923) at Washington University in St. Louis carried
out an experiment that supported the view that both momentum and
energy are transformed via photons.
Using the conservation law of energy, suggest a collision between xray and initially stationary free electron in the target. The conservation
of energy then gives us
hf  hf ' K
•In which hf is the energy of the incident x-ray photon, hf´ is the
energy of the scattered x –ray photon, K is the kinetic energy of the
recoiling electron.
•For the electron’s kinetic energy. Here m is the electron’s mass and
ᵞ is the Lorentz factor.
K  mc2 (  1)
1
 
1 (
v 2
)
c
hf  hf '  mc 2 (  1)
h
h
 '  mc 2 (  1)


but p  h
, the magnitude of the amount of the momentum of the

incident photon
h
'
The magnitude of the recoiling electron’s momentum is p  mv
and
h h
 ' cos   mv cos  ......x  axis
 
h
0  ' sin   mv sin  ......... y  axis

but   ( ' )
, the compton shift of the scattered x –rays.
The five collision variable ( , ' , v, ,  )
, eliminate v and ϕ
which deal only with the recoiling electron.
h
 
(1  cos  )
mc
……compton shift
The particle is loosely bond electron, and thus one would
substitute the mass of an electron for m to evaluate the compton
wavelength for compton scattering from an electron.
Electron and matter waves
A beam of light is a waves, but it transfers energy and
momentum to matter at point via photons.
de Broglie suggested p= h/λ might apply not only in to photons
but also to electrons
Momentum p to a photon of light with wavelength λ.
λ = h/p (de Broglie wavelength). To assign a wavelength λ to a
particle with momentum of magnitude p.
de Broglie’s prediction of the existence of matter waves.
EXAMPLES……………………………
SCHRODINGER’S EQUATION
Waves function φ(x, y, z, t) is written in the form.
 ( x, y, z, t )   ( x, y, z )e
 iwt
where w = 2πf is the angular frequency of the matter waves.
Suppose that a matter waves reaches a particle detector that is small,
then probability that a particle will be detected in a specified time .
interval is proportional to |ψ|² where |ψ| is the absolute value of the
waves function at the location of the detector.
|ψ|² is called probability density. And How do we find the wave function?
Sound waves and waves on the strings are Mechanical waves, while light
waves are described by Maxwell’s equation. Matter waves are described
by Schrodinger’s equation which is reduces to:
d 2 8 2 m
E  U ( x)  0

2
2
dx
h
Schrodinger’s equation, one dimensional motion, this
equation describes a free particles i.e
The Schrodinger’s equation describes a free particles that
is, a moving particle on which no net force acts.
The particle total kinetic energy in this case is all kinetic
energies and thus E = 1/2mv²
d 2
8 2 m

2
dx
h2
 mv 2

 2



  0

d 2
p


2



2
dx
h 

note (λ = h/p) , 1/λ= p/h where λ is the de Broglie wavelength
of the moving particles
2π/λ as the angular wave number (k)
d 2
2

k
 0
2
dx
Schrondinger’s equation of free particle
The general solution
 ( x)  Ae  Be
ikx
ikx
Where A and B are arbitrary constants.
Using
 ( x, y, z, t )   ( x, y, z )e iwt
 ( x, t )   ( x)e iwt  ( Aeikx  Be ikx )
 ( x, t )  Aei ( kx  wt )  Be i ( kx  wt )
Heisenberg’s Uncertainty Principle
•Measured values cannot be assigned to the position and the
momentum p of a particle simultaneously with unlimited precision i.e
our inability to predict the position of a free particle.
x.p x   

y.p y   

z.p z   
Heisenberg’s uncertainty Principle
Note ħ = h/2π
Where ∆x and ∆pₓ represent the intrinsic uncertainty in the
measurements of x components of ṝ and ṕ with parallel meanings
for the y and z terms.
•The particle whose probability density is a free particle, that is no
force acts on its, and so its momentum ṕ must be constant.
Without making a point of it – that we can determine ṕ with the
absolute precision, in other words, we assumed that ∆pₓ = ∆py =
∆pz =0
•The assumption then requires ∆x →∞, ∆y →∞, ∆z →∞. With such
infinitely great uncertainties, the position of the particle is
completely unspecified.
Quantized Energies
•Potential energies of the electron as a function of its position along the xaxis of the idealized trap. DIAGRAM
An electron can be trapped in the U = 0.
•When the electron is in the central cylinder, its potential energy U (=-eV) is
zero because there the potential V is zero.
•If the electron could get outside this region, its potential energy would be
positive and of infinite magnitude because there V→∞ .
•The electric potential energy U(x) of an electron confined to the central
cylinder of the idealized trap U = 0 for 0 <x< L and U →∞ for x <0 and x >L.
•Since electron is non relativistic , this momentum magnitude P is related to
kinetic energy K by P = (2mk)⅟2 where m= mass of the electron.
For an electron moving within the central cylinder , where U = 0, the total
(mechanical) energy E is equal to the kinetic energy.
h
h
 
p
2mE
•For states or discrete standing waves pattern in which the string can
oscillate, are those for which the length L of the string is equal to an integer
number of half-wavelengths.
L
n
2
•
for n = 1, 2, 3, ….
Eq (1)
•Each value of n identifies a state of the oscillating string, using the
language of quantum physics with integer n , a quantum number .
•For each state of string permitted , the transverse displacement of the
string at any position x along the string.
nx
yn ( x)  A sin(
),
•
for n= 1, 2, 3, eq (2) in which the quantum
L
number n identifies the oscillation pattern and A depends on the time at
which you inspect the string.
put eq(2) and in (1) and make E
h2
2
En  (
)
n
8mL2
for n = 1, 2, 3, ……(4)
•The positive integer n here is the quantum number of the electron’s
quantum state in the trap. These are the lowest five energy levels
allowed in the electrons. DIAGRAM
•The quantum state with the lowest possible energy level E, allowed with
quantum number n = 1 is called the ground state of the electron.
•All the quantum states with greater energies (n=2 or greater) are called
excited states of the electron.
•The amount of energy that is required for the electron change of state is
E  Ehigh  Elow
•If a confined electron emits a photon, the energy hf of that photon must equal
to the energy difference ∆E between the initial energy level of the electron and
a lower level.
•The electron tends to de-excite, rather than remain in an excited state until it
reaches the ground state (n = 1).
•If the electron is to de-excite, it must lose just enough energy to jump to a
lower energy level.
•If it is to loose energy by emitting light, then the loss of energy must be
by emission of photon.
PROBABILITY OF DETECTION
The probability P(x) that an electron can be detected at position x within
the well
(Probability P(x) of detection in width dx centered on the position x) =
2

( x ) at position ˣ) x width dx
n
(probability density
or
P( x)   n2 ( x)dx
2
2
2 nx

(
x
)

A
sin
(
)
n
The probability density is
L
for n =1 , 2, 3….
•To find the probability that the electron can be detected in any finite
section of the well-say, between points x1 and point x2
(Probability of detection between x1 and x2) =
 n2 ( x )

x2
x1
x2
P(x)   A 2 Sin 2 (
x1
nx
)dx
L
Normalization
The product  2 ( x ) dx gives the probability that an electron in an infinite
n
well can be
detected in the interval of the x axis that lies between x and x + dx

2

 n ( x)dx  1
(normalization equation)

which is used to evaluate the amplitude of a wave function.
Quantum Dots
The highly developed techniques used to fabricate computer chips can be
used to construct, used to construct atom by atom, individual potential
energy wells that behave, in many respects, like artificial atoms.
•It has promising applications in electronic optics and computer
technology.
Two and Three-Dimensional Electron Traps
This is a two –dimensional trap with infinite potential walls.
DIAGRAM
 nx,ny 
A
n y
n
2
2
Sin ( x x)
Sin (
y)
Lx
L
Ly
L
2
L
and
Enx,ny
2
h2
h2
h 2 nx2 n y
2
2
(
)nx  (
)n y 
(  )
8mL2x
8mL2y
8m L2x L2y
For a rectangular box
DIAGRAM
Enx,ny,nz
h 2 nx n y nz

( 2  2  2)
8m Lx Ly Lz
nz is the third quantum number for fitting the matter waves into
width Lz
Bohr Model of the Hydrogen Atom
Balmer wavelength
for n= 3, 4, 5, and 6
1
1
 R( 2  2 ) where R is a constant

2
n
1
But in 1913, Bohr made the following assumptions
(1) The electron in an hydrogen atom orbits the nucleus in a circle
much like Earth orbits the sun.
(2) The magnitude of the angular momentum Ḹ of the electron in its
orbits is restricted to the values
L= n  for n = 1, 2, 3, …..
For orbital Radius in Quantized Bohr Model
| q1 || q2 |
F k
r2
coulomb force
q1 = charge –e of the electron and q2 = charge +e
of the nucleous (the proton)
F = ma …… radial axis
1 e2
v2

 m(  )
2
40 r
r
Angular momentum of a particle of mass m and speed v
rmv  n
v 
n
rm
replacing
L2 0 2
r
n
2
me

with h/2

for n = 1, 2, 3, ….
r  an 2
h 2 0
a
 5.291772 x10 11 m
2
me
 52.92 pm
Energy changes
hf  E  Ehigh  Elow
me4
1
1

(

)
2 2
2
2
 8e0 L c nhigh nlow
1
 R(
1
1

)
2
2
nlow nhigh
me4
R
=1.097373 x 107m-1 (Rydberg Constant)
2 3
8e0 h c
Properties of Atoms
(1) Atoms are stable – Essentially all the atoms that form our tangible
world have existed without change for billions of years.
Think about it what would the world be like if atoms continually changed
into other forms, perhaps every few weeks or every few years?
(2) Atoms combine with each other.
They stick to form stable molecules and stack up to form rigid solids.
Atoms exist only in discrete quantum states, each having a certain energy.
An atom can make a transition from one state to another by emitting light (to
jump to a lower energy state Elow ) or by absorbing light to jump up to a
higher energy level Ehigh ) .
Light is emitted or absorbed as a photon with energy
hf  Ehigh  Elow
The electron is trapped in an atom or is free, it has an intrisic spin
angular momentum Ŝ which is the basic characteristic of an electron,
like its mass and electronic charge.
Ŝ is quantized and depended on a spin quantum numbers s, which is
always ½ for electrons (and for protons and neutrons).
Depend on a spin magnetic quantum numbers ms with only value of
+1/2 or -1/2
Orbital angular momentum L of an electrons in an atom is quantized i.e
only certain values.
L
l (l  1)
X-Rays
X-rays was discovered in 1895 by Rontgen.
They are produced whenever cathode rays (e-s) are brought to rest
matter. DIAGRAM
X-ray is highly evacuated and contains an anode and a tungsten filament
connected to a cathode. The e-s are obtained from the filament by
thermionic emission and are accelerated to the anode by a p.d up to
100kV. The anode is a copper block inclined to the e- stream and having
a small target of tungsten, or another high-melting-point metal on which
e-s are focused by the cathode. The tube has a lead shield with a small
window to allow the passage of the x-ray beam. Very small % of the K.E.
of the e-s is converted into x-rays. The rest of the K.E becomes internal
energy of the anode which has to be kept cool by circulating oil or water
through channels in it.
The intensity of the x-ray beam increases when nos of e-s hitting the
target increases and this is controlled by the filament current.
The quality or penetrating power of the x-rays is determined by the speed
attained by the e-s and increases with the p.d across the tube.
(1) “Soft” x-rays only penetrate such objects as flesh.
(2) “hard” x-rays can penetrate much more solid
matter.
Properties
-They travel in straight line
-They readily penetrate matter, penetration is least containing elements of
high density and high atomic number. Lead is a much better absorber
than ordinary soda glass.
-They are not deflected by electric or magnetic field
-They eject e-s from matter by the photoelectric effect
-They ionize a gas, permitting it to conduct
-They cause certain substances to fluoresce
-They affect a photographic emulsion in a similar manner to light.
USES OF X-RAYS
(1)Medicine- x-rays can damage healthy cells of the human body
-Suspected bone fractures can be investigated since x-rays of a certain
hardness can penetrate flesh but not bone.
-In the treatment of cancer by radiotherapy, very hard x-rays are used to
destroy the cancer cells.
(2)Industry- Castings and Welded joints can be inspected for internal
imperfection using x-rays.
(3) X-ray crystallography –The study of crystal structure by x-rays is
now a powerful method of scientific research
2d sin Q  n 
(4) X-ray Spectrometer
The Bragg x-ray spectrometer was developed to measure (i) x-ray
wavelengths and (ii) Spacing of atoms in crystals
(5) X-ray powder photography- Instead of a single crystal, a
polycrystalline specimen or a crystalline powder is used.
X-ray Spectra and the quantum theory
(a)Continuous and line Spectra
The radiation from an x-ray tube can be analyzed with a spectrometer and
an intensity wavelength graph obtained to show the spectral distribution.
(i) A continuous spectrum which has a definite lower wavelength limit,
increases to a maximum and then decreases gradually in the layer
wavelengths.
(ii) A line spectrum consisting of groups or series of two or three peaks of
high-intensity radiation superimposed on the continuous spectrum. The
series are denoted by the letter K, L, M etc.
(b)Certain features of the continuous spectrum are readily explained by the
quantum theory. Thus the existence of a definite minimum wavelength can
be justified if we assume that this radiation consists of x-ray photons
produced by e-s which have given up all their K.E. in a single encounter with
a target atom.
If such an e- has mass m and speed v on striking the target, the energy hf of
the photon is given by
hf  1 / 2mv
2
For workdone on a charge e-
eV  1 / 2mv
2
hf  eV
V increases, wavelent minmum decreasesi e x-rays of higher frequency
and greater penetrating power are emitted
CONDUCTION OF ELECTRICITY IN SOLIDS
Crystalline Solids: Solids whose atoms are arranged in a repetitive
three-dimensional structure called a lattice.
•Solid such as wood, plastic, glass, and rubber whose atoms are not
arranged in such repetitive patterns are not considered.
•Three electrical properties that can be used to distinguish among
crystalline solids are resistivity , temperature, coefficient of resistivity
and number density of charge carriers . Solids can be broadly divided
into insulators (very high resistivity), metals (low resistivity), positive
and low coeff. Of resistivity, large carriers)and semiconductors (high
resistivity, negative and high coeff. Of resistivity and small carriers).
•An isolated atom can exist in only a discrete set of energy levels. As
atoms come together to form a solid, the levels of the individual atoms
merge to form the discrete energy bands of the solids. These energy
band are separated by energy gaps, each of which corresponds to a
range of energies that no electron may possess.
•Any energy band is made up of an enormous number of very closely
spaced levels. The pauli exclusion principle asserts that only one
electron may occupy each of these levels.
Insulators: The highest band containing electrons is completely filled
and is separated from the vacant band above it by as energy gap so
large that electrons can essentially never become thermally agitated
enough to jump across the gap.
Metals: The highest band that contains any electrons is only partially
filled. The energy of the highest filled level at a temperature of 0K is
called the Fermi energy E for the metal, e.g For Copper E=7.0 eV. The
electrons in the partially filled band are the conduction electrons and
their number is
(number of conduction electrons in sample) = (number of atoms in
sample) x (number of valence electrons per atom)
The number of atoms in a sample is given by
(number of atoms in sample) = sample mass (Msam)/Atomic
mass
= sample mass (Msam)/(Molar mass M)/Na
= (materials density)(sample volume V)/(Molar mass M)/Na
The number density n of the conduction electrons is
n = Number of conduction electrons in sample / sample volume V
The density of states function N(E) is the number of available energy levels
per unit volume of the sample and per unit energy interval and is given by
8 2m 3 / 2
N (E)  (
) E 1/ 2
3
h
density of states (m-3)-1
The occupancy probability P(E), the probability that a given available
state will be occupied by an electron, is
P( E ) 
1
e
( E  EF ) / KT
1
Occupancy probability
The density of occupied states No(E) is given by the product of the two
qualities
No(E) = N(E)P(E) ----Density of occupied states
Fermi energy for a metal can be found by integrating No(E) for T =0
from E = 0 to E = EF
2
3
0.121h 2 2 / 3
2/3 h
2/3
EF  (
) ( )n  (
)n
m
m
16 2
SEMICONDUCTORS
The band structure of semiconductor is like that of an insulator except
that the gap width Eg is much smaller in the semiconductor. For Silicon (a
semiconductor) at room temperature , thermal agitation raises a few
electrons to the conduction band, leaving an equal number of holes in the
valence band.
•The number of electrons in the conduction band of silicon can be increased
greatly by doping with small amounts of Phosphorus thus forming n-type
material.
•The number of holes in the valence band can be greatly increased by doping
with aluminum, thus forming P-type material.
The p-n Junction
A p-n junction is a single semiconducting crystal with one end doped to
form p-type material and the other end doped to form n-type material,
the two types meeting at a junction plane.
At thermal equilibrium: The majority carriers (electrons on the n side
and holes on the p side) diffuse across the junction plane, producing a
diffusion current Idiff.
The minority carriers (holes on the n side and electrons on the p side)
are swept across the junction plane, forming a drift current Idrift. These
two currents are equal in magnitude, making the net current zero.
A depletion zone, consisting largely of charged donor and acceptor
ions, forms across the junction plane.
A contact potential difference V0 develops across the depletion zone.
Applications of the p-n Junction
•When a p.d is applied across a p-n junction, the device conducts
electricity more readily for one polarity of the applied potential
difference than for the other. Thus, a p-n junction can serve as a
junction rectifier.
•When a p-n junction is forward biased, it can emit light, hence can
serve as a light-emitting diode (LED)
  c / f  hc / Eg
THANK YOU
THE END