Photons and matter waves

Light pushes

The tail of a comet always points away from the Sun. Sunlight pushes the dust and gas away. So light carries momentum, even though a photon has no rest mass.

A photon travels at the speed of light $c$. For such a particle the momentum is its energy divided by $c$:

A worked example. A photon has a wavelength of $620$ nm. Find its energy and its momentum. Then find the force of a $2.0$ W beam of this light, absorbed fully by a black card.

1. Energy. $E = \dfrac{1240\ \text{eV nm}}{620\ \text{nm}} = 2.00\ \text{eV}$.
2. Momentum. $p = \dfrac{h}{\lambda} = \dfrac{6.63 \times 10^{-34}\ \text{J s}}{620 \times 10^{-9}\ \text{m}} = 1.07 \times 10^{-27}\ \text{kg m/s}$.
3. The force. Each second the card takes in $N$ photons, and each brings momentum $E/c$. The force is the momentum each second: $F = \dfrac{NE}{c} = \dfrac{P}{c} = \dfrac{2.0\ \text{W}}{3.0 \times 10^8\ \text{m/s}} = 6.7 \times 10^{-9}\ \text{N}$.
4. Check. That is the weight of a speck of about $0.7$ microgram. Light pushes, but very gently.

Waves from particles

In 1924 Louis de Broglie reasoned by symmetry. Light is a wave that also acts as a stream of particles. Perhaps matter, which we think of as particles, also acts as waves. He gave every moving particle the wavelength

Two forms are worth knowing.

• Electrons speeded up by a voltage $V$. The kinetic energy is $eV$, so $p = \sqrt{2meV}$ and $\lambda = \dfrac{1.227}{\sqrt{V}}$ nm, with $V$ in volt. This holds up to a few tens of kilovolt. Above that, the electrons are so fast that relativity changes the result.
• Any particle with kinetic energy $K$. $p = \sqrt{2mK}$, so $\lambda = \dfrac{h}{\sqrt{2mK}}$. At the same energy, a heavier particle has a shorter wavelength.

A worked example. An electron and a proton each have a kinetic energy of $100$ eV. Compare their wavelengths.

1. The electron. $K = 100$ eV came from a voltage of $100$ V, so $\lambda = \dfrac{1.227}{\sqrt{100}}\ \text{nm} = 0.123$ nm.
2. The proton. Its mass is $1836$ times larger. So $\lambda$ is smaller by $\sqrt{1836} = 42.8$: $\lambda = \dfrac{0.123\ \text{nm}}{42.8} = 0.00286$ nm.
3. Check. The proton is heavier, so its wavelength is shorter.

Why do we not see the waves of a person?

A wave shows its nature when it meets a gap or an obstacle about as wide as its own wavelength.

• An electron of $100$ eV has $\lambda = 0.12$ nm. Atoms in a crystal are about $0.2$ nm apart, so the crystal is a gap of the right size. Electron waves show.
• A person of $60$ kg walking at $1.5$ m/s has $\lambda = \dfrac{6.63 \times 10^{-34}\ \text{J s}}{60\ \text{kg} \times 1.5\ \text{m/s}} = 7.4 \times 10^{-36}$ m. That is far smaller than a nucleus, about $10^{-15}$ m. No gap is that narrow, so we never see it.

Every moving thing has a wavelength. For large things it is too small to measure.

The Davisson and Germer experiment

In 1927 Clinton Davisson and Lester Germer, in New York, were studying how electrons bounce off a nickel target. By accident air got into the tube, and they had to heat the nickel. The heating turned the nickel into a few large crystals. After that, the electrons did not scatter evenly. They came off in sharp peaks.

So they built a careful arrangement. An electron gun, with a chosen voltage, sends a beam at the crystal. A detector moves round on an arc and measures the current at each angle. For electrons speeded up by about $54$ V, there was one strong peak at $50^\circ$. At other voltages the peak moved or vanished.

The atoms of the surface lie in rows a distance $d = 0.215$ nm apart. They act as a grating. The waves scattered by neighbouring rows are in step, and give a strong beam, when

The wavelength that this gave agreed with de Broglie's $h/p$ to within a few per cent. Electrons really behave as waves. In the same year G. P. Thomson sent electrons through a thin gold foil and saw rings, as X rays make. Davisson and Thomson shared the Nobel Prize of 1937.

A worked example. Where would the first peak be for electrons speeded up by $80$ V?

1. The wavelength. $\lambda = \dfrac{1.227}{\sqrt{80}}\ \text{nm} = 0.137$ nm.
2. The angle. For $n = 1$, $\sin\varphi = \dfrac{0.137\ \text{nm}}{0.215\ \text{nm}} = 0.638$, so $\varphi = 39.6^\circ$.
3. Check. The voltage is higher than in the experiment, so the wavelength is shorter and the peak is nearer the beam.

Set the voltage to $200$ V. Where is the peak now? Then lower the spacing $d$ to $0.15$ nm and set the voltage to $30$ V. Why is there no peak? The wavelength is then longer than the spacing, so $\sin\varphi$ would be more than $1$.

The electron microscope

A microscope cannot show details much smaller than the wavelength of the waves it uses. Light has wavelengths of $400$ nm to $700$ nm. Electrons speeded up by $20$ kV have $\lambda = \dfrac{1.227}{\sqrt{20\,000}}\ \text{nm} = 0.0087$ nm, tens of thousands of times shorter. An electron microscope focuses the beam with magnetic lenses. It shows viruses, and even rows of atoms.

Practise with new numbers

Try it in Python

The program finds the de Broglie wavelength of four moving things. It then compares each with the gap between atoms in a crystal, which is about $0.2$ nm.

Common mistakes

• Using grams for the mass. In $h/(mv)$ the mass must be in kilogram.
• Giving a photon a rest mass. A photon has none, but it has the momentum $h/\lambda$.
• Using $1.227/\sqrt{V}$ for a proton or a helium nucleus. The number $1.227$ nm is for electrons only.
• Saying that only tiny things have waves. Every moving thing does. For large things the wavelength is too small to see.

Remember these

• Photon: $E = h\nu = hc/\lambda$, $p = h/\lambda$, no rest mass.
• De Broglie: $\lambda = h/p = h/mv$. At a given energy, $\lambda = h/\sqrt{2mK}$.
• Electrons speeded up by $V$ volt: $\lambda = 1.227/\sqrt{V}$ nm.
• Davisson and Germer: electrons diffract from a crystal, with $d\sin\varphi = n\lambda$.
• Waves show only when the gap is about the size of the wavelength.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Which photon has the larger momentum: one of wavelength $400$ nm or one of $700$ nm?

2. An electron and a proton have the same kinetic energy. Which has the shorter wavelength, and by about how many times?

3. A cricket ball does not show diffraction when it passes through a gap the width of a door. Why not?

4. In the Davisson and Germer experiment the accelerating voltage is raised. What happens to the wavelength of the electrons, and to the angle of the first peak?