1

1) photoelectric effect2) electron kicked to a higher energy state (excited state)3) scattering4) pair production: electron and positron

2

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:

For a photon, which has zero mass, this expression becomes:

Thus, the momentum of a photon is its energy divided by the speed of light c. Using E=hf, and f=c, we find that the momentum of a single photon of wavelength is

42222 cmcpE +=

pcE =

h

p =

3

The Compton Effect

the Compton effect is the scattering of a photon off of an electron that’s initially at rest

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

Arthur Compton (1892-1962)

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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

6

The Compton Effect Calculating the Compton effect:

the incident photon has frequency f, hence wavelength =c/f

the photon is scattered into an angle , and in the process its frequency changes to f’ (and correspondingly ’=c/f’)

the electron is initially at rest, and afterwards gains a velocity v. The angle at which the electron is scattered is ’

Conservation of energy:

Conservation of momentum in the x direction

Conservation of momentum in the y direction

2

2

1vmfhhf e+′=

)'cos()cos('

vmhh

e+=

)'sin()sin('

0

vmh

e+=

'

v

'

7

The Compton Effect

The preceding is a rather messy set of equations to solve … here is the key result:

The quantity h/mec is called the Compton wavelength of the electron, and has a value of 2.43x10-12m.

( ) cos1' −=−cm

h

e

8

The Wave Nature of Matter In 1923 de Broglie suggested that if

light has both wave-like and particle-like properties, shouldn’t all matter?

Specifically, he proposed that the wavelength of any particle is related to its momentum p by

For a matter particle, is called the de Broglie wavelength of the particle

Louis de Broglie (1892-1987)

p

h=

9

Electron Diffraction The previous example illustrates how difficult it would be to reproduce a

Young-type 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:

1. Atoms consist of negative electrons orbiting a small positive nucleus;2. Atoms can exist only in certain stationary states with a particular set

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 = EfEi)/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.

Niels Henrik David Bohr(1885-1962)

1922 Nobel Prize

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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?

abs1 1

1 2: 414 nm (blue)1,242 eV nm

1 3: 248 nm (UV) i i

hc

E E

→ →

→⎧= = =⎨ →Δ Δ ⎩

emiss

2 1: 414 nm (blue) 1, 242 eV nm

3 1: 248 nm (UV)

3 2: 621 nm (orange) i j i j

hc

E E

→ →

→⎧⎪= = = →⎨Δ Δ ⎪ →⎩

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Binding Energy andIonization 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.

Pair Production

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γ → e− + e+ a high energy photon (gamma ray) collides with a nucleus and creates an electron and a positronThe energy of the photon is transformed into mass: E=mc²

Pair Production

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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.