#### Turning Points Tutorial 4 – Light is a Particle

 Contents

Light was a wave.  That was that.  Well… er… no.

The photo-electric effect had initially been described by Hertz in 1887, and developed further by another German, Hallwachs.  A negatively charged zinc plate would discharge when exposed to ultra-violet light; a positively charged plate would not.  Also the plate would not discharge in bright red light, for example from a laser, but it would in dim UV light.

You may wish to revise the photo-electric effect in Quantum Physics Tutorial 1 and Quantum Physics Tutorial 2.

 Question 1 If the wave model of light were correct, what would you expect to see with bright red light?

Study of black body radiation provided results that could not be predicted by classical physics.  You can follow the idea of black body radiation up in Astrophysics Tutorial 5.  A black body is defined as a perfect emitter of radiation.  At thermal equilibrium the black body will emit radiation at all wavelengths.  As frequency of the radiation increased, the more energy would be given out.  If all energies were calculated at all frequencies, the energy becomes infinite.  This would breach the Law of Conservation of Energy. This problem was called the ultraviolet catastrophe.

The model was that a black body radiator would set up electromagnetic stationary waves in a cavity, like this:

It's like the sound box of a musical instrument in which standing sound waves can be set up at the first harmonic, f0, or the second harmonic 2f0, or any whole number multiple of the first harmonic.  There is an equal probability of all the different harmonics, this gives the musical instrument its characteristic sound quality.  The same applies to a black body radiator, except that the standing wave were electromagnetic, and the wavelengths were much shorter.  In the diagram above, there are two modes.  In this diagram, we see that the standing waves travel across the length of the cavity.  However, they can set up standing waves between any two points like this:

With higher frequencies, more standing waves are possible.  In fact the number of standing waves (or modes) predicted varies as the square of the frequency, according to classical wave theory.  The Rayleigh-Jeans Law was worked out in 1905 by John William Strutt, Lord Rayleigh, (1842 - 1919) and Sir James Hopwood Jeans (1877 - 1946).  The relationship they came up with was:

[Bl - spectral radiance; T - temperature (K); c - speed of light (m s-1); k - Boltzmann's Constant (m2 kg s-2 K-1); l - wavelength (m)]

You are NOT expected to use this equation - it is here for illustrative purposes.

The formula, called the Rayleigh-Jeans Equation worked for long wave radiation, but not short.   The graph was like this:

This graph suggests that the intensity at very small wavelengths tends to infinity.  Energy is being created, which cannot happen.  The model had broken down.

The German physicist Max Planck (1858 – 1947) tried to explain the observations in terms of classical physics, but could not produce a convincing explanation.  Radically rethinking the problem, he concluded that classical physics does not always apply at the atomic level.  Instead he then proposed that energy was radiated in discrete energy packets called quanta and came up with a complex formula that seemed to solve the problem.  It was the first formula that used Planck’s constant h (= 6.63 × 10-34 Js).  This was the start of the birth of modern physics.  The graph for Planck's findings looked like this:

In 1905 Albert Einstein (1879 – 1955) extended the idea that when a quantum of energy is emitted by an atom, it continues to exist as a concentrated packet of energy.  The energy of the packets (photons) was given by:

E = hf

 Explain what each of the terms in this equation stand for and give the units:   E:    h:   f:

A beam of light was considered as a stream of particles:

• At long distances the intensity was low because the photons were spread apart.

• However the energy of each photon was undiminished.

Einstein then went on to state that when a photon collides with an electron, it must

• either be reflected with no loss of energy,

• or must lose all its energy to the electron.

• The photon interacted with one electron only.

Therefore the number of electrons being emitted was proportional to the number of photons that landed on the surface.  Furthermore, electrons were emitted immediately the photons hit the surface.  The effect can be shown with this experiment:

The photocathode is made from a reactive metal such as caesium.  A reactive metal loses its outermost electrons most easily.

 Question 3 What do you think will happen to the current if the intensity of light is increased.  Does this contradict the wave theory?

The experiment showed that if light had a lower frequency (i.e. longer wavelength) than the threshold frequency, no current at all was observed, however bright the light was.  Dim green or blue light would work, bright red light will not.

Using this apparatus we can measure the stopping voltage:

Note that in this experiment the cathode is connected to the positive terminal, and the anode is connected to the negative.

The two principal observations were:

• The maximum kinetic energy (indicated by the stopping voltage, Vs) depended on the frequency.  The stopping voltage was higher for UV light than it was for green light.

• The more surprising observation was that no matter how bright or dim the light the stopping voltage was exactly the same.

 Wave theory suggests that Supported or Contradicted? Any frequency can emit electrons Contradicted! Current depends on intensity Supported Maximum energy is independent of frequency Contradicted! Maximum energy would depend on intensity Contradicted!

Einstein’s explanation for these results was:

• Each photon provided energy for ONE electron to escape;

• Electron cannot escape if the energy in the photon is not sufficient;

• Photon energy E = hf.

Electrons would only have kinetic energy if they had been given sufficient energy to break the bond between them and their parent ions.  The threshold frequency is the frequency of the photon that just has sufficient energy to break that bond.  This energy is called the work function and is given the physics code F.  The symbol F is "Phi", a Greek capital letter 'F'.

Photon energy = maximum kinetic energy + work function

In Physics code and rearranging:

Therefore:

Since all the energy in an electron is kinetic, we can work out the kinetic energy using the stopping voltage:

We find out the stopping voltage by adjusting the potentiometer to the point where the current just reaches zero.  At this point, the negatively charged anode is repelling the most energetic electrons, which are attracted back to the positively charged cathode.  We are only interested in the most energetic photoelectrons.  The rest are just also-rans.   So we can now write:

We can plot the data on a graph:

We can plot the kinetic energy for several different metals against the frequency.  We find the following:

• The gradient is the same, whatever the metals;

• The gradient is always h, 6.63 × 10-34 J s;

• If voltage is on the vertical axis, the gradient is h/e = 4.14 × 10-15 J s C-1.

• Each metal has a different value for its threshold frequency;

• The most reactive metals have the lowest threshold frequency;

• At the threshold frequency, Ek = 0;

• The intercept on the vertical axis gives the work function.

 Gold has a work function of 4.9 eV. (a)    What is this in joules? (b)   What is the maximum kinetic energy that the photoelectrons have if the gold is illuminated by UV light of frequency 1.7 × 1015 Hz (c)    What is the stopping voltage of these electrons? (d)   How fast do the electrons travel as they leave the surface?                           Data: h = 6.63 × 10-34 Js e = 1.6 × 10-19 C mass of electron = 9.11 × 10-31 kg

The experimental results agreed with Einstein’s explanation.  This experiment was good evidence that light was a particle.  However Young’s slits showed that light was a wave.  A conundrum…

Light travels as a particle that shows wave behaviour.  It is emitted and absorbed as photons.  This explanation is called Wave-Particle Duality.

### Enter the Count

Louis de Broglie (1892 – 1987), a French nobleman and historian, who had more than a penchant for Physics wrote a thesis in 1924 stating that if light waves showed particle properties, it was very reasonable to state that particles should show wave properties.  Any particle of mass m would have an associated wavelength that could be worked out by:

You may wish to review this in Quantum Physics Tutorial 6

The wave behaviour of electrons, which are, of course, particles, can be shown with this experiment:

Electrons are diffracted at certain angles by a very thin layer of graphite to produce rings.  The ring spacing fits very well the model predicted by the Bragg Equation:

l = 2d sin q

This equation was worked out by the father and son team of William and Lawrence Bragg.  It applied to diffraction of X-rays, which are, of course, electromagnetic waves.  Therefore we can say that electrons are showing wave-like properties.

We can find the momentum of the electrons easily enough:

Kinetic energy = electrical energy supplied = charge × voltage

Therefore:

Now multiply both sides by m:

If we square root this expression, we get:

We can combine this with the de Broglie relationship:

 What is the de Broglie wavelength of an electron that has been accelerated from rest by a p.d. of 40 V?

A Young's slit type of experiment can be carried out with electron beams.  And it is found that there is an interference effect.  If we regard an electron as merely a particle, it is difficult to see it passing through two places at once.  However quantum theory gives electrons wave properties, and there is no difficulty.  Electrons are quantum beings that exist as probability.  The closer you get to them, the less likely you are to catch the little brutes.

Electrons are considered to travel as matter waves, which are not composed of matter despite their name.  It is instead to do with probability.  Where the amplitude is greater, there is a greater probability of finding an electron.   It is also true of photons.  We can no longer think in quantum physics of particles like electrons being solid point masses like lead shots.  Its wave function is extended over an extended region of space rather than being at a single point.

This is quantified by complex mathematical functions that are way beyond what we need to know.  Werner Heisenberg was considering the matter waves of electrons, when he proposed his uncertainty principle.  In effect, the closer you are to pinning down an electron the harder it is to pin down. (Heisenberg was a brilliant theoretical physicist, but useless at practical physics.)

These difficult concepts in quantum physics enable us to explain tunnelling, which we will look at next.

We know that an electron can have discrete energy levels.

According to classical physics, the electron can only be at energy level 1 or 2.  It will not have enough energy to jump over the energy hill.

If we adopt the quantum mechanics idea, where the bigger the amplitude of a matter wave, the greater the probability of finding the electron, we get this:

There is a small but finite probability of finding the electron on the other side of the energy hill.  It seems to have tunnelled through the energy hill, and this effect is called quantum tunnelling.  It is impossible to explain this by classical physics, but (comparatively) easy to use quantum physics which is based on probability.

### Electron Microscopy

The light microscope can resolve objects of size about 0.5 mm, which is about 1 wavelength of visible light.  If we consider accelerated electrons to have a de Broglie wavelength of about 10-10 m, we can resolve down to about the size of an atom.

The transmission electron microscope uses electrons in the same way as a light microscope uses photons.  The general layout is like this:

• Electrons accelerated through a pd of 100 kV have a wavelength of 10-12 m.

• Electrons are focussed by magnets that act as lenses.  In the diagram the lenses are electromagnets.

• The electron microscope can resolve much finer detail as the wavelengths are 10 000 times smaller.

• Increasing the anode voltage gives better resolution (finer detail).

Problems:

• Electrons lose momentum going through the sample.  This makes their wavelength increase, reducing detail.

• There is a range of electron speeds, hence wavelengths.  This can distort the image, rather like chromatic aberration in a light microscope.

• The focal length of a magnetic lens depends on the current.  In practice there are small variations which leads to slight changes in the focal length.

• The electron path must be in a high vacuum to prevent collisions with electrons and gas molecules.

 Question 6 What is the wavelength of 100 keV electrons? Question 7 Write down and explain two problems that limit the resolution of the electron microscope.

Although the electron microscope can in theory resolve down to about 2 × 10-12 m, in practice the resolution is about 2 × 10-10 m.  Even so this has allowed:

• Physicists to resolve individual atoms;

• Biologists to study the ultra-structure of the cell and how viruses are made up.  Viruses are far too small to be seen with the light microscope.

The Scanning Tunnelling Electron Microscope

It was invented in 1981 by Binnig and Rohrer to provide extremely high resolution images of the surfaces of materials.  It relies on the quantum physics effect of tunnelling which we looked at earlier.

The probe has an extremely sharp tip which is kept at a small negative voltage compared to the surface.  The surface must be able to conduct electricity; insulating materials must be coated with a thin conducting layer.  The probe is moved in the x, y, or z coordinates by quartz crystals which change shape when a small current is applied.  (This is the reverse of the piezo-electric effect which you may have come across in some gas lighters.  The quartz crystal is squeezed and gives out a high enough voltage to give a spark.)  The probe scans across the surface like this:

• If the gap is less than 10-10 m some electrons may cross the gap producing a tiny current.

• As the gap decreases, the current increases.  Changes as little as 10-12 m can be detected.

• The height of the probe is continually adjusted to keep the current constant.   In this way the probe traces out the profile of the surface.

• The x, y, and z coordinates are controlled by a computer.

• The image generated is displayed on a VDU.

Extremely fine images of atoms in crystals can be generated.