Transcript n - Tarleton State University

CHAPTER 3
Planck’s Constant
3-1 Atoms and Radiation in Equilibrium
3-2 Thermal Radiation Spectrum
3-3 Quantization of Electromagnetic
Radiation
3-4 Atomic Spectra and the Bohr Model
Max Karl Ernst Ludwig Planck
(1858-1947)
We have no right to assume that any physical laws exist, or if they have
existed up until now, or that they will continue to exist in a similar
manner in the future.
An important scientific innovation rarely makes its way by gradually
winning over and converting its opponents. What does happen is that
the opponents gradually die out.
- Max Planck
Blackbody Radiation
Why is Black Body Radiation important?
When matter is heated, it emits
radiation.
A blackbody is a cavity with a
material that only emits thermal
radiation. Incoming radiation is
absorbed in the cavity.
Blackbody radiation is theoretically interesting because the
radiation properties of the blackbody are independent of the
particular material. Physicists can study the properties of intensity
versus wavelength at fixed temperatures.
Rayleigh-Jeans Formula
Lord Rayleigh used the classical
theories of electromagnetism and
thermodynamics to show that the
blackbody spectral distribution
should be:
u( )  8 kT  4
u ( )  8 kT  4
Integrating u ( ) from 0  , we find

 u( )d   0 when   0
0
It approaches the data at longer wavelengths, but it deviates badly at
short wavelengths. This problem for small wavelengths became
known as the ultraviolet catastrophe and was one of the
outstanding exceptions that classical physics could not explain.
Planck’s Radiation Law
Planck assumed that the radiation in the cavity was emitted (and
absorbed) by some sort of “oscillators.” He used Boltzman’s
statistical methods to arrive at the following formula that fit the
blackbody radiation data.
8 hc
u ( )  hc  kT
e
1
5
Planck’s radiation law
Planck made two modifications to the classical theory:
The oscillators (of electromagnetic origin) can only have certain
discrete energies, En = nhf, where n is an integer, n is the frequency,
and h is called Planck’s constant: h = 6.6261 × 10−34 J·s.
The oscillators can absorb or emit energy in discrete multiples of the
fundamental quantum of energy given by:
E  hf
Planck’s Radiation Law
Exercise 3-1: Derive Planck’s radiation law.
Application: The Big Bang theory
predicts black body radiation. This
radiation was discovered in 1965
by A. Penzias & R. Wilson. Cosmic
Background Explorer (COBE) and
Wilkinson Microwave Anisotropy
Probe (WMAP) detected this
radiation field at 2.725± 0.001 °K
This data supports the Big
Bang Theory
Stefan-Boltzmann Law
The total power radiated increases with the temperature:
R   T
4
This is known as the Stefan-Boltzmann law, with the constant σ
experimentally measured to be 5.6705 × 10−8 W / (m2 · K4).
The emissivity є (є = 1 for an idealized blackbody) is simply the
ratio of the emissive power of an object to that of an ideal
blackbody and is always less than 1.
Wien’s Displacement Law
The spectral intensity R(,T ) is the total power radiated per unit area
per unit wavelength at a given temperature.
Wien’s displacement law: The maximum of the spectrum shifts to
smaller wavelengths as the temperature is increased.
mT  constant=2.898 10 m.K
-3
Planck’s Radiation Law
Exercise 3-2 : Show that Stefan’s Boltzmann law, Wein’s
displacement law and Rayleigh-Jean’s law can be derived from
Planck’s law
Exercise 3-3: What is the average energy of an oscillator that has a
frequency given by hf=kT according to Planck’s calculations?
Exercise 3-4: How Hot is a Star? Measurement of the wavelength
at which spectral distribution R(λ) from the Sun is maximum is
found to be at 500nm, how hot is the surface of the Sun?
Exercise 3-5: How Big is a Star? Measurement of the wavelength at
which spectral distribution R(λ) from a certain star is maximum
indicates that the star’s surface temperature is 3000K. If the star is
also found to radiate 100 times the power Psun radiated by the Sun,
how big is the star? Take the Sun’s surface temperature as 5800 K.
What is a Photon?
Planck introduced the idea of a photon or quanta. A cavity emits
radiation by way of quanta. How does the radiation travel in space?
We think that radiation is a wave phenomenon however the energy
content is delivered to atoms in concentrated groups of waves
(quanta).
Properties of a Photon
If a photon is to be considered as a particle we must be able to
describe its mass, momentum, energy, statistics etc.
 Energy of a photon
-34
-15
E  hf ; h=6.626  10 J .s  4.136  10 eV .s
E  Energy, f  frequency

limiting value 1/2hf otherwise integral multiple of hf
Interaction of photons with matter
Complete absorption or partial absorption with the photon
adjusting its frequency to remain as particle

Intensity of photon
Intensity  number of photons
intensity has nothing to do with the energy of photons
Properties of Photons
Constant
h of photon
h defines the smallest quantum angular momentum of a particle
hf
hf
hf
 E  hf  mc  m  2 and p  2 c 
c
c
c
2
Exercise 6 Show that h has units of angular momentum
Mass and momentum of photon
photons move with velocity v=c
E  hf  mc 2 
m0
1 v c
2
2
c 2  m0 


1  v2 c2 
hf
0
2
c
Photons have no rest mass m0
Photon
is not a material particle since rest mass is zero, it is a wave
structure that behaves like a particle
Properties of Photons
Charge
of a photon
photons do not carry charge, however they can eject charge particles
from matter when they impinge on atoms
Photon
Statistics
Consider the radiation as a gas of photon. Photons move randomly like
molecules in a gas and have wide range of energies but same velocity
Statistics of photons is described by Bose-Einstein. We can talk about
intensity and temperature in the same way as density and temperature.
Photoelectric Effect
Methods of electron emission:
Thermionic emission: Applying
heat allows electrons to gain
enough energy to escape.
Secondary emission: The electron gains enough energy by transfer
from another high-speed particle that strikes the material from outside.
Field emission: A strong external electric field pulls the electron out of
the material.
Photoelectric effect: Incident light (electromagnetic radiation) shining
on the material transfers energy to the electrons, allowing them to
escape. We call the ejected electrons photoelectrons.
Photo-electric Effect
Experimental Setup
Photo-electric effect
observations
Electron
kinetic
energy
The kinetic energy of
the photoelectrons is
independent of the
light intensity.
The kinetic energy of
the photoelectrons, for
a given emitting
material, depends only
on the frequency of
the light.
Classically, the kinetic
energy of the
photoelectrons should
increase with the light
intensity and not
depend on the
frequency.
Photoelectric effect
observations
Electron
kinetic
energy
There was a threshold
frequency of the light,
below which no
photoelectrons were
ejected (related to the
work function f of the
emitter material).
The existence of a threshold frequency is completely inexplicable in
classical theory.
Photoelectric effect
observations
(number of
electrons)
When photoelectrons
are produced, their
number is proportional
to the intensity of light.
Also, the photoelectrons
are emitted almost
instantly following
illumination of the
photocathode,
independent of the
intensity of the light.
Classical theory predicted that, for
extremely low light intensities, a long
time would elapse before any one
electron could obtain sufficient
energy to escape. We observe,
however, that the photoelectrons are
ejected almost immediately.
Einstein’s Theory: Photons
Einstein suggested that the electro-magnetic radiation field is
quantized into particles called photons. Each photon has the energy
quantum:
E  hf
where f is the frequency of the light and h is Planck’s constant.
Alternatively,
E  h
where:
h  h 2
Einstein’s Theory
Conservation of energy yields:
1 2
hf  f  mv
2
where f is the work function of the metal (potential energy to be
overcome before an electron could escape).
In reality, the data were a bit more
complex. Because the electron’s energy
can be reduced by the emitter material,
consider fmax (not f):
1

eV0   mv 2   hf  f
2
max
Example – Photoelectric Effect
Exercise 3-6: An experiment shows that when electromagnetic
radiation of wavelength 270 nm falls on an aluminum surface,
photoelectrons are emitted. The most energetic of these are
stopped by a potential difference of 0.46 volts. Use this information
to calculate the work function of aluminum in electron volts.
Exercise 3-7: The threshold wavelength of potassium is 558 nm.
What is the work function for potassium? What is the stopping
potential when light of 400 nm is incident on potassium?
Exercise 3-8 Light of wavelength 400 nm and intensity 10-2 W/m2 is
incident on potassium. Estimate the time lag expected classically.
Atomic Spectra
Newton discovers the
dispersion of light
Invention of Spectroscopy
Spectra
Three Kinds of spectra
Solid, liquid or a dense gas
excited to emit a continuous
spectrum
Light passing through low density
gas excites atoms to produce
emission spectra
Light passing through cool low
density gas results in absorption
spectra
Line Spectra
Chemical elements were observed to produce unique
wavelengths of light when burned or excited in an electrical
discharge.
Balmer Series
In 1885, Johann Balmer found an
empirical formula for the
wavelength of the visible
hydrogen line spectra in nm:
2
n
n  364.6 2
nm
n 4
Rydberg-Ritz Formula
As more scientists
discovered emission lines
at infrared and ultraviolet
wavelengths, the Balmer
series equation was
extended to the Rydberg
equation:
1
mn
1 
 1
 R  2  2  for n>m, R=Rydberg constant
n 
m
For Hydrogen R=R H  1.096776  107 m 1
For Heavy atom R=R   1.097373  107 m 1
The Classical Atomic Model
Consider an atom as a planetary
system.
The Newton’s 2nd Law force of
attraction on the electron by the
nucleus is:
kZe2 mv 2
F 2 
r
r
where v is the tangential velocity of the
electron:
2
kZe
1
v2 
 mv 2  kze2 2r
mr
2
The total energy is then:
E  K V 
2
2
kZe kZe
kZe


2r
r
2r
2
This is negative, so
the system is bound,
which is good.
The Classical Atomic Model
Exercise 7: Show that in the classical model the frequency of
radiation for an accelerating electron is
f 
1
r
3/ 2
and the Energy is
1
E
r
The Planetary Model is Doomed
From classical E&M theory, an accelerated electric charge radiates
energy (electromagnetic radiation), which means the total energy
must decrease. So the radius r must decrease!!
Electron
crashes
into the
nucleus!?
Electron
does not
crash in
the Bohr
model
Physics had reached a turning point in 1900 with Planck’s
hypothesis of the quantum behavior of radiation, so a radical
solution would be considered possible.
The Bohr Model of the Hydrogen Atom
Bohr’s general assumptions:
n=2
n=1
1. Stationary states, in which orbiting
electrons do not radiate energy, exist in
atoms and have well-defined energies, E.
Transitions can occur between them,
yielding light of energy: Bohr frequency
condition
E = Ei − Ef = hf
2. Classical laws of physics do not apply
to transitions between stationary states,
but they do apply elsewhere.
3. The angular momentum of the nth state is:
where n is called the Principal Quantum
Number.
n=3
n
Angular
momentum is
quantized!
Consequences of the Bohr Model
The angular momentum is:
L  mvr  n
So the velocity is:
v  n / mr
2
2
kZe
mv
From; F  2 

r
r
2
So:
Solving for r:
2
12
 kZe 
v

mr


n
kZe

2 2
mr
mr
2
a0
2
n 2 a0
n2
rn 

2
mkZe
Z
a0 is called the Bohr radius.
Bohr Radius
The Bohr radius,
a0 
2
mke
2
 0.0529nm
is the radius of the unexcited hydrogen atom.
The “ground” state Hydrogen atom diameter is:
Energy of an electron
Exercise 3-9: Show that the energy of an electron in any atom at orbit n
is quantized and that it gives the ground state energy of Hydrogen
atom to be -13.6eV.
Z2
En   E0 2 n  1, 2,3,...
n
mk 2 e 4
where E0  
2 2
Rydberg-Ritz formula
Exercise 3-10: Derive the Ryderberg-Ritz formula
 1
E0 mke4
1 
 R 2  2  , R 

=Rydberg constant
3
 nf


n
hc 4 c
i


For Hydrogen R=R H  1.096776 107 m1
1
For Heavy atom R=R   1.097373 107 m1
Transitions in the Hydrogen Atom
The atom will remain in the excited state for a short time before
emitting a photon and returning to a lower stationary state. In
equilibrium, all hydrogen atoms exist in n = 1.
Successes and Failures of the
Bohr Model
Success:
Why should the nucleus of the atom be
kept fixed?
The electron and hydrogen nucleus
actually revolve about their mutual
center of mass.
Conservation of momentum require that
the momenta of nucleus and electron
equal in magnitude. The total kinetic
energy is then
p2
p2 M  m 2 p2
Ek 


p 
2 M 2m 2mM
2
mM
m
where  

M  m 1 m M
The electron mass is replaced
by its reduced mass:
Limitations of the
Bohr Model
The Bohr model was a great
step in the new quantum
theory, but it had its limitations.
Failures:
Works only for single-electron (“hydrogenic”) atoms.
Could not account for the intensities or the fine structure of
the spectral lines (for example, in magnetic fields).
Could not explain the binding of atoms into molecules.
Rydberg-Ritz Formula -Example
Exercise 3-11 Use the Rydberg-Ritz formula to calculate the first
line of Balmer, Lyman and Paschen series for the Hydrogen atom.
The Correspondence
Principle
When energy levels are
very close quantization
should have little effect
Bohr’s correspondence
principle is rather obvious:
In the limits where classical and
quantum theories should agree,
the quantum theory must reduce
the classical result.
The Correspondence Principle
Exercise 3-12: Show that in the limit of large quantum number the Bohr
frequency is the same as the classical frequency.
Fine Structure Constant



In Bohr’s theory we know that transitions can
occur for ∆n≥1, for small n values. If we allow this
for large n and calculate the classical and Bohr
frequencies as in previous exercise, we will find
that they do not agree.
To avoid this disagreement, A. Sommerfeld
introduced special relativity and elliptical orbits.
From Bohr orbit in hydrogen for n=1, we have
mvr  n
v
mr1

m

2
mke 2


ke 2
v ke 2 1.44eV .nm
1
 



c
c 197.3eV .nm 137
We will skip the
mathematical
treatment of A.
Sommerfeld work
α is called the fine structure constant
Fine Structure Constant

The fine structure constant can be understood in the following way.
For each circular orbit rn and energy En a set of n elliptical orbits
exist whose major axis are the same but they have different
eccentricities and thus different velocities and Energies.
Electron transitions depend on the eccentricities of the initial and
final orbits and on the major axes, thus resulting in splitting of
energy levels of n called fine-structure splitting.
Fine structure constant leads to the notion of electron spin
Example
Exercise 3-13: Show that the energy levels of oscillators in simple
harmonic motion are quantized.
Do it yourself exercise: solve the differential equation
2
dx
2


x

0
2
dt
Example
Exercise 3-14: Derive the Bohr quantum condition from WilsonSommerfeld quantization rule
X-Ray Production: Theory
An energetic electron
passing through matter will
radiate photons and lose kinetic
energy, called bremsstrahlung.
Since momentum is conserved,
the nucleus absorbs very little
energy, and it can be ignored.
The final energy of the electron is
determined from the conservation
of energy to be:
E f  Ei  hf
Ei
Ef
hn
X-Ray Production: Theory
If photons can transfer energy to electrons, can
part or all of the kinetic energy of electron be
converted into photons?
Ei
“The Inverse photoelectric effect”
Ef
This was discovered before the work of Planck
and Einstein
hn
Observation of X Rays
1895: Wilhelm Röntgen studied the
effects of cathode rays passing
through various materials. He
noticed that a phosphorescent
screen near the tube glowed during
some of these experiments. These
new rays were unaffected by
magnetic fields and penetrated
materials more than cathode rays.
He called them x rays and deduced
that they were produced by the
cathode rays bombarding the glass
walls of his vacuum tube.
Wilhelm Röntgen
X-Ray
Production:
Experiment
Current passing through a filament produces copious numbers of
electrons by thermionic emission. If one focuses these electrons by
a cathode structure into a beam and accelerates them by potential
differences of thousands of volts until they impinge on a metal
anode surface, they produce x rays by bremsstrahlung as they stop
in the anode material.
Electromagnetic theory Predicts X-Ray
Accelerated charges produce electromagnetic waves, when fast
moving electrons are brought to rest, they are certainly accelerated
1906: Barkla found that X-Ray show Polarization, this establishing
that X-Rays are waves.
X-Rays have wavelength range 0.1 nm – 100 nm
Even though classical theory predicts x-ray’s, the experimental data
is not explainable.
Two distinctive features
1.Some targets have enhanced
peaks. For example
Molybdenum shows two peaks
at specific wavelengths.
a) This is due to
rearrangement of electrons
of the target material after
bombardment.
b) X-ray ‘s have a continuous
spectrum
2.No matter what the target , the
threshold wavelength depends
on the accelerating potential
Inverse Photoelectric Effect
Since the work function of the target
is of the order of few eV, whereas the
accelerating potential is thousand of
eV
1

eV0   mv 2   hf  f
2
max
eVo  hf max 
From photoelectric effect
hc
min
Duane-Hunt rule
X-Ray Spectra
Bohr-Rutherford picture of the atom
can also be applied to heavy
elements
Shells have letter names:
K shell for n = 1
L shell for n = 2
The atom is most stable in its
ground state.
An electron from higher
shells will fill the inner-shell vacancy at lower energy.
When it occurs in a heavy atom, the radiation emitted is an x-ray.
It has the energy E (x ray) = Eu − Eℓ.
Atomic Number and Moseley
The x-rays have names:
L shell to K shell: Kα x-ray
M shell to K shell: Kβ x-ray
etc.
G.J. Moseley studied x-ray
emission in 1913.
Atomic number Z = number
of protons in the nucleus.
Moseley found a
relationship between the
frequencies of the
characteristic x-ray and Z.
f
12
 An ( Z  b)
Moseley found this relation
holds for the Kα x-ray with b=1 and
different An values (from quantum
mechanics):
Moseley’s Empirical Results
The K x-ray is produced from the
n = 2 to n = 1 transition.
In general, the K series of x-ray frequencies are:
Form the Bohr model with Z=Z-1
mk 2e4
1
1
2 1
2
f 
(Z  1)  2  2   cR (Z  1) 1  2 
3
4
1 n 
 n 
1 

where A  cR  1  2 
 n 
2
n
We use Z-1 instead of Z because one electron is already
present in the K-shell and so shields the other's from the
nucleus’ charge.
Moseley’s research clarified the importance of Z and the electron
shells for all the elements, not just for hydrogen.
Moseley’s Empirical Results
For the L series of x-ray wavelength the
frequencies are
 1 1 
f  cR  2  2  ( Z  7.4) 2
2 n 
Neodymium Z=60 and
Samarium Z=62
b) Promethium Z=61
c) All three elements
together
a)
Example X-Ray Spectra
Exercise 3-15: Calculate the wavelength of Kα line of molybdenum, Z=42,
and compare with the value λ=0.0721nm measured by Moseley.
Aguer (oh-zhay) Effect
In 1923 Pierre Auger discovered an alternative to X-ray emission.
The atom may eject a third electron from a higher-energy outer shell via
radiationless process called Auger effect
E3  E  E2  E1
KE  E  E3
EdN(e)/dE plot:
a) Auger spectrum of Cu
b) Al and Al2O3 together
note the energy shift in
the larges peak due to
adjustment in Al
electron shell energies
X-Ray Scattering
Max von Laue suggested that if x-rays were a form of electromagnetic
radiation, interference effects should be observed.
Crystals act as three-dimensional gratings, scattering the waves and
producing observable interference effects.
Bragg’s Law
William Lawrence Bragg
interpreted the x-ray
scattering as the reflection of
the incident x-ray beam from
a unique set of planes of
atoms within the crystal.
There are two conditions for
constructive interference of
the scattered x rays:
1) The angle of incidence must equal the
angle of reflection of the outgoing wave.
2) The difference in path lengths must be
an integral number of wavelengths.
Bragg’s Law: nλ = 2d sin θ (n = integer)
The Bragg Spectrometer
A Bragg spectrometer scatters x
rays from crystals. The intensity of
the diffracted beam is determined
as a function of scattering angle
by rotating the crystal and the
detector.
When a beam of x rays passes
through a powdered crystal, the
dots become a series of rings.
Examples
Exercise 3-16: What is the shortest wavelength present in a radiation if
the electrons are accelerated to 50,000 volts?
Exercise 3-16:The spacing of one set of crystals planes in common salt is
d=0.282nm. A monochromatic beam of X-rays produces a Bragg
maximum when its glancing angle is 7 degrees. Assuming that this is the
first order maximum (n=1), find the wavelength of the X-rays, what is the
minimum possible accelerating voltage Vo that produced the X-rays?