Transcript Physics 124 : Particles and Waves

1) photoelectric effect
2) electron kicked to a higher energy
state (excited state)
3) scattering
4) pair production: electron and positron
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The momentum of a photon
 In special relativity there is an expression relating the energy E
of a particle to its momentum p and mass m:
E  p c m c
2
2 2
2 4
 For a photon, which has zero mass, this expression becomes:
E  pc
 Thus, the momentum of a photon is its energy divided by the
speed of light c. Using E=hf, and lf=c, we find that the
momentum of a single photon of wavelength l is
p
h
l
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The Compton Effect
 the Compton effect is the
scattering of a photon off
of an electron that’s
initially at rest
Arthur Compton
(1892-1962)
 if the photon has enough
energy (X-ray energies or
higher), the scattering
behaves like an elastic
collision between particles
 the energy and momentum
of the system is conserved
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The Compton Effect
 To calculate what happens, we use the same principles as for an elastic
collision, however, the fact that one particle is massless (the photon)
has some “strange” consequences.
 if the two particles were massive, we’d had the situation we studied before.
Note in particular:
 no deflection angle … since particle 2 is at rest, the problem reduces to a one
dimensional collision
 the velocity of particle 1 will change after the impact, and if m2>m1, particle 1 will
get scattered backwards
Classical
elastic
scattering
Compton
scattering
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The Compton Effect
 Since the photon is massless, it always moves at the speed of
light.
 the photon does loose momentum and energy during the collision
(giving it to the electron), consequently its wavelength increases
 the “reason” there is a deflection angle, is that otherwise it would
be impossible for the system to conserve both energy and linear
momentum
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The Compton Effect



Calculating the Compton effect:

the incident photon has frequency f, hence wavelength l=c/f

the photon is scattered into an angle q, and in the process its frequency changes to f’ (and
correspondingly l’=c/f’)

the electron is initially at rest, and afterwards gains a velocity v. The angle at which the electron
is scattered is q’
Conservation of energy:
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hf  hf   me v 2
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Conservation of momentum in the x direction
h
h
 cos( q )  me v cos( q ' )
l l'

v
Conservation of momentum in the y direction
0
l
q'
l'
h
sin(q )  me v sin(q ' )
l'
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The Compton Effect
The preceding is a rather messy set of
equations to solve … here is the key
result:
h
1  cosq 
l ' l 
mec
The quantity h/mec is called the
Compton wavelength of the electron,
and has a value of 2.43x10-12m.
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The Wave Nature of Matter
 In 1923 de Broglie suggested that if
light has both wave-like and particlelike properties, shouldn’t all matter?
 Specifically, he proposed that the
wavelength l of any particle is related
to its momentum p by
h
l
p
 For a matter particle, l is called the de
Broglie wavelength of the particle
Louis de Broglie
(1892-1987)
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Electron Diffraction
 The previous example illustrates how difficult it would be to reproduce a Youngtype double slit experiment to demonstrate electron diffraction
 However, in a typical crystal lattice the interatomic spacing between atoms in
the crystal is of order 10-10m, and scattering a beam of electrons off a pure
crystal produces an observable diffraction pattern
 this is what Davisson & Germer did in 1927 to confirm de Broglie’s hypothesis
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Electron Diffraction
 However, more recently people have been able to
duplicate a double slit experiment with electrons.
 The images below show a striking example of this,
where electrons are fired, one at a time, toward a
double slit
 the positions of the electrons that make it through
the slit and hit the screen are recorded
 with time, the characteristic double-slit diffraction
pattern appears
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Electron Diffraction
 What’s going on here? How can a single electron
“interfere” with itself? … it’s the wave-particle
duality again.
 an electron is a particle, but its dynamics (i.e. its motion) is governed by
a matter wave, or its so called wave function
 the amplitude-squared of the electron’s wave function is interpreted as the
probability of the electron being at that location
 places where the amplitude are high (low) indicate a high (low) probability of
finding the electron
 so when a single electron is sent at a double slit, its matter wave governs
how it moves through the slit and strikes the screen on the other end:
 the most probable location on the screen for the electron to hit is where there is
constructive interference in the matter wave (i.e., the bright fringes)
 conversely, at locations where there is destructive interference in the matter
wave chances are small that the electron will strike the screen there.
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Wavelength of a moving ball
λ=h/p=h/mv
.
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Niels Bohr’s Atomic Model
Bohr wanted to “fix” the model so that the orbiting electrons
would not radiate away their energy. Starting from Einstein’s idea
of light quanta, in 1913 he proposed a radically new nuclear model of
the atom that made the following assumptions:
David Bohr
1. Atoms consist of negative electrons orbiting a small positive nucleus; Niels Henrik
(1885-1962)
2. Atoms can exist only in certain stationary states with a particular set 1922 Nobel Prize
of electron orbits and characterized by the quantum number n = 1, 2, 3, …
3. Each state has a discrete, well-defined energy En, with E1<E2<E3<…
4. The lowest or ground state E1 of an atom is stable and can persist indefinitely.
Other stationary states E2, E3, … are called excited states.
5. An atom can “jump” from one stationary state to another by emitting a photon of
frequency f = EfEi)/h, where Ei,f are the energies of the initial and final states.
6. An atom can move from a lower to a higher energy state by absorbing energy in an
inelastic collision with an electron or another atom, or by absorbing a photon.
7. Atoms will seek the lowest energy state by a series of quantum jumps between
states until the ground state is reached.
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The Bohr Model
The implications of the Bohr model are:
1. Matter is stable, because there are no states
lower in energy than the ground state;
2. Atoms emit and absorb a discrete spectrum of
light, only photons that match the interval
between stationary states can be emitted or
absorbed;
3. Emission spectra can be produced by collisions;
4. Absorption wavelengths are a subset of the
emission wavelengths;
5. Each element in the periodic table has a
different number of electrons in orbit, and
therefore each has a unique signature of
spectral lines.
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Energy Level Diagrams
excited states
ground state
It is convenient to represent the energy states of an atom using an energy
level diagram.
Each energy level is represented by a horizontal line at at appropriate
height scaled by relative energy and labeled with the state energy and quantum
numbers. De-excitation photon emissions are indicated by downward arrows.
Absorption excitations are indicated by upward arrows.
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Example: Emission and Absorption
An atom has only three stationary
states: E1 = 0.0 eV, E2 = 3.0 eV, and
E3 = 5.0 eV.
What wavelengths are observed in
the absorption spectrum and in the
emission spectrum of this atom?
labs
hc
1, 242 eV nm 1  2: 414 nm (blue)



E1i
E1i
1  3: 248 nm (UV)
lemiss 
hc
Ei  j
2  1: 414 nm (blue)
1, 242 eV nm 

  3  1: 248 nm (UV)
Ei  j
3  2: 621 nm (orange)

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Binding Energy and
Ionization Energy
The binding energy of an electron in
stationary state n is defined as the energy
that would be required to remove the
electron an infinite distance from the
nucleus. Therefore, the binding energy of
the n=1 stare of hydrogen is EB = 13.60 eV.
It would be necessary to supply 13.60 eV
of energy to free the electron from the
proton, and one would say that the electron
in the ground state of hydrogen is “bound by
13.60 eV”.
The ionization energy is the energy required to remove the least
bound electron from an atom. For hydrogen, this energy is 13.60 eV. For
other atoms it will typically be less.
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The Hydrogen Spectrum
The figure shows the
energy-level diagram for
hydrogen. The top “rung” is
the ionization limit, which
corresponds to n→∞ and to
completely removing the
electron from the atom. The
higher energy levels of
hydrogen are crowded
together just below the
ionization limit.
The arrows show a photon
absorption 1→4 transition
and a photon emission 4→2
transition.
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Pair Production
γ → e− + e+ a high energy photon (gamma
ray) collides with a nucleus and creates an
electron and a positron
The energy of the photon is transformed into
mass: E=mc²
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Pair Production
If the energy equals the rest mass of the electron
and positron the newly formed particles won’t
move. Any ‘excess’ energy will be converted into
kinetic energy.
Pair production requires the presence of another
photon or nucleus which can absorb the photon’s
momentum and for conservation of momentum not
to be violated.
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