Download 3. The Experimental Basis of Quantum Theory

CHAPTER 3
Prelude to Quantum Theory
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Discovery of the X-Ray and the Electron
Determination of Electron Charge
Line Spectra
Quantization
Blackbody Radiation
Photoelectric Effect
X-Ray Production
Compton Effect
Pair Production and Annihilation
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
Prof. Rick Trebino, Georgia Tech, www.frog.gatech.edu
3.1: Discovery of the X-Ray
and the Electron
In the 1890s
scientists and
engineers were
familiar with
cathode rays.
These rays were
generated from
one of the metal
plates in an
evacuated tube
with a large electric
potential across it.
It was surmised that cathode rays had something to do with atoms.
It was known that cathode rays could penetrate matter and were
deflected by magnetic and electric fields.
Observation of X-Rays
In 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
(1845-1923)
Röntgen’s
X-Ray Tube
Röntgen used x-rays to make a shadowgram
of the bones of a hand on a phosphorescent
screen.
Thomson’s Cathode-Ray Experiment
In 1897, J.J. Thomson used an evacuated cathode-ray tube to
show that the cathode rays were negatively charged particles
(electrons) by deflecting them in electric and magnetic fields.
Thomson’s Experiment: e/m
Thomson’s method of measuring the
ratio of the electron’s charge to mass
was to send electrons through a
region containing an electric field and
then also with a magnetic field
perpendicular to it.
J. J. Thomson (1856-1940)
y
e

v0

B

ℓ





q
x
E
Calculation of e/m
An electron moving through the
electric field is accelerated by a
force:
Fy  ma y  eE
q << 1, so vx ≈ v0
vy
t  / v0
ayt
(eE/m)( /v0)


Electron deflection angle: tan(q ) 
v x v0
v0
unknown
To find v0, turn on the magnetic field, which deflects the electron
against the electric field force. Use:
F  eE  ev0  B  0
And adjust the magnetic field until the net force is zero.
E   v0  B
 v0  E / B
Charge-to-mass ratio:
 tan(q ) 
e E tan(q )

m
B2
eE
m ( E/B) 2
3.2: Determination
of Electron
Charge
Millikan’s 1911 oil-drop experiment
determined e, hence also m.
Robert Andrews Millikan
(1868 – 1953)
Using an electric field,
Millikan suspended
tiny charged oil drops.
Then he turned off the
field and watched
them free-fall.
Calculation of the oil drop charge
The electric field balanced gravity and suspended
a charged oil drop:
V
Fy  eE  e  mdrop g
d
 e  mdrop gd / V
Turning off the electric field, the drop mass, mdrop, could be determined
from Stokes’ relationship of the terminal velocity, vt, to the drop density,
, and the air viscosity, h :
2
vt  2 g  r / 9h
Solving for the drop radius:
And using:
r  3 h vt / 2 g 
mdrop  43  r 3   e  18 (h 3 v3t / 2 g  )1/2 d / V
Millikan experimental results
Because a drop could have 1, 2, 3, or more electrons on it, the data
looked like this:
Thousands of experiments showed that
there is a basic quantized electron charge:
e = 1.602 x 10-19 C
3.3: Line Spectra
Chemical elements were observed to produce unique wavelengths of
light when burned or excited in an electrical discharge.
Balmer Series
In 1885, a school teacher, Johann Balmer, found an empirical formula
for the wavelength of the visible hydrogen line spectra in nm:
4
 1 1 

 2 2
 364.56nm  2 k 
1
(where k = 3,4,5…)
Rydberg Equation
As more scientists discovered emission lines at infrared and ultraviolet
wavelengths, the Balmer series equation was extended to the Rydberg
equation:
1 
 1
 RH  2  2 

n k 
1
RH  1.096776 x 107 m-1
3.5: Blackbody radiation
Blackbody radiation is emitted from a hot body. It's anything but black!
The name comes from the assumption that the body absorbs at every
frequency and hence would look black at low temperature.
Blackbody radiation was (and still is) theoretically interesting because
the radiation properties
of the blackbody are
independent of the
particular material.
19th-century physicists
studied the spectral
intensity versus
wavelength at various
temperatures.
Wien’s Displacement Law
Wien’s
Displacement Law:
The maximum of the
spectrum shifts to
shorter wavelengths
as the temperature
is increased.
But what about the
precise shape of the
spectral intensity?
S(,T)
The spectral intensity S(,T) is the total power radiated per unit area
per unit wavelength from a blackbody at a given temperature.
Lord Rayleigh used the
classical theories of
electromagnetism and
thermodynamics to show
that the blackbody
spectral distribution
should be:
S ( , T ) X

Spectral Intensity
Rayleigh-Jeans Formula
2 ckT
4
Wavelength (nm)
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 Boltzmann’s
statistical methods.
But Planck made two modifications to the classical theory:
The oscillators (of electromagnetic origin) can only have certain discrete
energies, En = nhn, where n is an integer, n is the frequency, and h is a
constant: h = 6.6261 × 10−34 J·s—now called Planck’s constant.
The oscillators can absorb or emit energy
in discrete multiples of the fundamental
quantum of energy given by:
DE = hn
He obtained a formula that fit the blackbody radiation data perfectly:
S ( , T ) 
2 c 2 h

5
1
e
hc /  kT
1
Planck’s radiation law
3.6: 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.
Photo-electric Effect
Experimental Setup
Photo-electric effect:
Classical theory vs.
measurements
Classical theory predicted that the
electrons absorb energy from the
beam at a fixed rate. So, for
extremely low light intensities, a
long time would elapse before any
one electron could obtain sufficient
energy to escape.
But, in actual measurements, the
electrons are emitted almost
instantly following illumination of
the cathode, independent of the
intensity of the light.
Initial observations by
Heinrich Hertz 1887
Photo-electric effect:
Classical theory vs.
measurements
Electron
kinetic
energy
n0
Classical theory also
predicted that the kinetic
energy of the electrons
should increase with the
light intensity and not
depend on the light
frequency.
In fact, the kinetic energy
of the electrons depends
only on the frequency of
the light.
Also, the number of
electrons was
proportional to the light
intensity.
Photo-electric
effect:
classical
theory vs.
observations
Threshold
frequencies
Electron
kinetic
energy
n0
Experiments also
revealed a threshold
frequency of the light,
below which no electrons
were ejected.
The existence of a threshold frequency was completely inexplicable
in classical theory.
Einstein’s theory: Photons!
Flying in the face of a century of observations that light is a wave,
Einstein suggested that light is somehow quantized into particles we
now call photons. Each photon has energy:
E  hn
where n is the frequency of the light and h is Planck’s constant.
Alternatively, because n = w/2:
E w
where:
 h / 2
Einstein’s theory
Conservation of energy yields:
Electron kinetic energy (K) =
= Photon energy  Potential energy to be overcome
K  hn  f
where f is the work function
of the metal (potential energy
to be overcome before an
electron could escape).
This simple—but highly
unintuitive—idea explained all
the properties of the photoelectric effect!
Electron
kinetic
energy
n0
Other scientists’ reactions to Einstein’s
explanation of the photo-electric effect
The data were actually less clear than these slides have
implied.
Even though it proved his own theory, Planck was skeptical.
Millikan spent ten years trying to disprove it, but finally
grudgingly published data supporting it in 1916 and won the
Nobel Prize in 1923 for it (and his oil-drop experiment).
But Millikan still didn’t believe in the photon concept.
Einstein finally won the Nobel Prize for this work in 1921
(but wasn’t actually awarded it until 1922).
3.7: Inverse photo-electric effect:
Bremsstrahlung x-ray production
An energetic electron
passing through matter will
radiate photons and lose kinetic
energy, called Bremsstrahlung.
Unlike photons, electrons can lose
some of their energy.
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  hn
e
Ei
Ef
nucleus
hn
Bremsstrahlung X-ray
production:
experiment
Current in a filament produces electrons by thermionic emission.
Accelerated by thousands of volts, the electrons smash into the
anode, where they produce x-rays.
Their energy before and after:
Ei = eV0
and
Ef ~ 0
Inverse photoelectric effect
Neglect the work function because it’s
small compared to the electron energy,
which is large here. Conservation of
energy requires that the maximum
photon energy equal the electron kinetic
energy. This yields the Duane-Hunt
limit, first found experimentally. The
minimum photon wavelength depends
only on the accelerating voltage and is
the same for all targets.
eV0  hn max 
 min
hc
min
hc

eV0
Photons also have momentum!
Use our expression for the relativistic
energy to find the momentum of a photon,
which has no mass:
E  (mc )  p c
2
2 2
Alternatively:
When
radiation
pressure is
important:
2 2

E hn h
p 

c
c 
h 2
p
 k
2 
Comet tails (other forces are small)
Viking spacecraft (would've missed Mars by 15,000 km)
Stellar interiors (resists gravity)
3.8: Compton Effect
Ep , pp
Photons have energy and momentum:
E  hc / 
p h/
e
When a photon enters matter, it can
interact with one of the electrons. The
laws of conservation of energy and
momentum apply, as in any elastic
collision between two particles.
This yields the change in
wavelength of the scattered
photon, known as the
Compton effect:
Ee , pe
Ep’ , pp ’
h
D      
1  cos q 
mc
3.9: Pair Production and Annihilation
In 1932, C. D. Anderson observed a
positively charged electron (e+) in
cosmic radiation. This particle, called a
positron, had been predicted to exist
several years earlier by Paul Dirac.
A photon’s energy can be converted
entirely into an electron and a positron
in a process called pair production:
 e e


But it doesn’t happen in empty space.
Paul Dirac
(1902 - 1984)
Pair Production
in Empty Space
E  pc  hn
Energy conservation :
E
hn  E  E
+
The total energies for the electron and positron are:
Since me ≠ 0:
E
E2  p2c 2  me2c 4
E  p c
This yields a lower limit on the photon energy:
Momentum conservation in the
direction of propagation (×c) yields:
hn  p c  p c
hn  pc  pc cos(q )  p c cos(q )
This yields an upper limit on the photon energy:
hn  p c  p c
A contradiction! Pair production in empty space is impossible!
Pair Production
in Matter
In the presence of matter, the
nucleus absorbs some energy
and momentum, making the
process possible.
The photon energy required for
pair production in the presence
of matter is:
hn  E  E  Knucleus
hn  2me c 2  1.022 MeV
Pair Annihilation
A positron passing through matter
will likely annihilate with an
electron. The electron and positron
can form an atom-like configuration
first, called positronium.
Pair annihilation in empty space
produces two photons to conserve
momentum.
Annihilation near a nucleus can
result in a single photon.
hv
hv
Pair annihilation
yielding two photons
Conservation of momentum:
hv1 / c  hv2 / c  0
So the two photons will have the
same frequency:
v1  v2  v
Conservation of energy:
2me c 2  hv1  hv2  2hv
The two photons from positronium
annihilation will move in opposite
directions with an energy:
hv  me c 2  0.511 MeV
hv
hv
PositronEmission
Tomography
PET scan
of a normal
brain
Radioactivity
Radioactive decay is the process by
which an unstable atomic nucleus
loses energy by emitting particles
and/or radiation. The emission is
spontaneous—the nucleus decays
without collision with another particle.
Henri Becquerel
1852-1908
Radioactivity was first discovered in 1896 by
the French scientist Henri Becquerel while
working on phosphorescent materials. He
wrapped a photographic plate in black paper
and placed various phosphorescent salts on it.
All results were negative until he used uranium
salts. The result with these compounds was a
deep blackening of the plate.
Types of radioactivity
There are many types of radioactive
emissions. Becquerel, Ernst Rutherford,
and Marie and Pierre Curie (~1900)
were able to classify them into various
types.
They
included:
Marie Skłodowska Curie
1867-1934
a-rays:
Helium nuclei (2 protons and 2 neutrons)
b-rays:
Electrons (or positrons)
-rays:
-rays
N-rays
In 1903, Rene Blondlot, of
Nancy, France, discovered a new
ray that he called the N-ray.
N-rays had remarkable
properties and could only be
seen by dispersing them with an
aluminum prism and then by
observing weak luminescence in
the dark with dark-adapted
human eyes.
Rene Blondlot 1849–1930
Alas, N-rays didn’t actually exist.
After ~300 papers were published on N-rays, American scientist,
R.W. Wood, visited Blondlot’s lab and secretly removed the
aluminum prism, and Blondlot could still see them. N-rays are
probably the greatest case of self-delusion in the history of science.