From radio waves to gamma rays

One family

Look around the room. Light from a window, the signal from a radio, the heat of a lamp and the waves in a phone are very different things to our senses. To a physicist they are one family. All of them are electromagnetic waves, with the same properties as the wave of the first course. They all travel at the same speed in vacuum, $c = 3.00 \times 10^{8}$ m/s.

What differs from one member to another is the frequency $f$. The wavelength follows from the frequency, and so does the energy of a photon:

With $hc = 1240$ eV nm, a photon of wavelength $\lambda$ in nm has the energy $1240/\lambda$ in eV. A wave of high frequency has a short wavelength and a large photon energy.

The bands

People gave names to the parts of the spectrum long before they knew that they were one family. The names come from how the waves are made or used. There are no sharp edges between the bands, and books draw them a little differently. We use these ranges, which are the ones of NCERT, with $c = 3.00 \times 10^{8}$ m/s.

• Radio waves: wavelength above $0.1$ m, frequency below $3.0 \times 10^{9}$ Hz.
• Microwaves: wavelength $1$ mm to $0.1$ m, frequency $3.0 \times 10^{9}$ Hz to $3.0 \times 10^{11}$ Hz.
• Infrared: wavelength $700$ nm to $1$ mm, frequency $3.0 \times 10^{11}$ Hz to $4.3 \times 10^{14}$ Hz.
• Visible light: wavelength $400$ nm to $700$ nm, frequency $4.3 \times 10^{14}$ Hz to $7.5 \times 10^{14}$ Hz.
• Ultraviolet: wavelength $1$ nm to $400$ nm, frequency $7.5 \times 10^{14}$ Hz to $3.0 \times 10^{17}$ Hz.
• X-rays: wavelength $10^{-3}$ nm to $1$ nm, frequency $3.0 \times 10^{17}$ Hz to $3.0 \times 10^{20}$ Hz.
• Gamma rays: wavelength below $10^{-3}$ nm, frequency above $3.0 \times 10^{20}$ Hz.

The photon energy goes with the frequency. At the edges of the bands it is:

• radio and microwave edge, $0.1$ m: $1.2 \times 10^{-5}$ eV;
• microwave and infrared edge, $1$ mm: $1.2 \times 10^{-3}$ eV;
• infrared and visible edge, $700$ nm: $1.77$ eV;
• visible and ultraviolet edge, $400$ nm: $3.10$ eV;
• ultraviolet and X-ray edge, $1$ nm: $1240$ eV;
• X-ray and gamma edge, $10^{-3}$ nm: $1.24$ MeV.

Two points of care. The X-ray band and the gamma band overlap in wavelength, since the names also say where the waves come from: X-rays from electrons and gamma rays from nuclei. And a kitchen oven works at $2.45$ GHz, a wavelength of $12.2$ cm. That is just above the $0.1$ m edge, but it is called a microwave oven all the same.

Slide the frequency to $1.4 \times 10^{9}$ Hz and then to $10^{18}$ Hz. By how many powers of ten does the wavelength change? It changes by the same number as the frequency, in the opposite direction.

A very wide scale

From the lowest radio wave to the highest gamma ray, the frequency changes by more than $10^{15}$ times. The visible band runs from $4.3 \times 10^{14}$ Hz to $7.5 \times 10^{14}$ Hz, which is less than a factor of two. Our eyes see only a very thin slice. All the rest of the spectrum is invisible to us, and it needed instruments to find it.

A worked example. Radio astronomers listen to hydrogen gas in space at a wavelength of $21$ cm. Find the frequency, the photon energy and the band.

1. Frequency. $f = \dfrac{c}{\lambda} = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{0.21\ \text{m}} = 1.43 \times 10^{9}\ \text{Hz}$.
2. Photon energy. $E = hf = 4.14 \times 10^{-15}\ \text{eV s} \times 1.43 \times 10^{9}\ \text{Hz} = 5.91 \times 10^{-6}\ \text{eV}$.
3. Band. The wavelength is above $0.1$ m, so it is a radio wave. Its photons are several hundred thousand times weaker than a visible photon.

A second example. A hospital X-ray has a wavelength of $0.10$ nm. Then $f = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{0.10 \times 10^{-9}\ \text{m}} = 3.00 \times 10^{18}\ \text{Hz}$ and $E = \dfrac{1240\ \text{eV nm}}{0.10\ \text{nm}} = 12.4$ keV. Each photon carries enough energy to knock an electron out of an atom.

What changes along the spectrum

• The frequency, the wavelength and the photon energy change, tied together by $c = f\lambda$ and $E = hf$.
• The way a wave meets matter changes. Radio waves pass through walls. Visible light passes through glass but not through brick. X-rays pass through soft tissue but not through bone.
• The way it is made and detected changes, as the next lesson shows.
• The speed in vacuum does not change. Nor does the nature of the wave: every band is a transverse wave of $\vec{E}$ and $\vec{B}$.

Practise with new numbers

Try it in Python

The program has a list of the bands and a function that finds the band of a wavelength. It goes through seven waves from everyday life and prints the band, the frequency and the photon energy of each.

Common mistakes

• Thinking that X-rays are faster than radio waves. All of them travel at $c$ in vacuum. Only the frequency differs.
• Mixing the direction of the scale. A larger frequency means a shorter wavelength and a larger photon energy.
• Saying that microwaves are the waves "of" a microwave oven only. The band is wide. Radar, satellite links and mobile networks use it as well.
• Thinking that visible light is a large part of the spectrum. It is a sliver. Most of the spectrum is invisible.

Remember these

• All bands are electromagnetic waves with $c = f\lambda$ in vacuum. The photon energy is $E = hf = 1240/\lambda$ eV for $\lambda$ in nm.
• In order of rising frequency: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma.
• Visible light is $400$ nm to $700$ nm, from about $4.3 \times 10^{14}$ Hz to $7.5 \times 10^{14}$ Hz.
• The edges between bands are conventions, and the bands overlap.

Check yourself

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

1. A wave has a frequency of $5.0 \times 10^{14}$ Hz. Find its wavelength in vacuum in nm, its photon energy in eV and its band.

2. A microwave has a wavelength of $12$ cm. What is its frequency?

3. Put these in order of rising photon energy: ultraviolet, radio waves, X-rays, infrared.

4. Which has the longer wavelength in vacuum: a gamma ray or a microwave? Which travels faster?

References

• OpenStax, University Physics Volume 2, Chapter 16 (Electromagnetic Waves): the electromagnetic spectrum.
• NCERT Physics Class 12, Chapter 8 (Electromagnetic Waves): the electromagnetic spectrum, with its table of ranges.