A boy falls from his cycle and his wrist swells. At the district hospital he is sent to a room with a yellow warning sign on the door. The technician lays his hand on a flat plate and steps behind a screen. A lead apron hangs on the wall for anyone who must stay close. A second later the picture is ready. The bones of the hand are white, the flesh is grey, and a thin dark crack runs across one bone.
Fast electrons and a metal target
In 1895 Wilhelm Röntgen found that fast electrons striking a solid give out an invisible radiation. It passed through paper, wood and flesh, and it made a screen glow. He did not know what it was, so he called it X-rays. We now know that X-rays are electromagnetic waves, like light, but with wavelengths thousands of times shorter.
X-rays are electromagnetic waves of wavelength from about \(0.01\,\text{nm}\) to \(10\,\text{nm}\). They are produced when fast electrons are stopped by a target.
To make X-rays, then, we need three things. We need a supply of electrons, a way to make them very fast, and a target to stop them. The Coolidge tube of 1913 does all three in one sealed glass tube. Every X-ray tube in a hospital today is built on the same plan.
Follow one electron through Fig 1.1. A small current heats the filament until it is white-hot. Heat gives a few free electrons more than the work function, and they leave the wire. This is thermionic emission, from the chapter on Photons and the Photoelectric Effect. The filament is the cathode of the tube.
The anode is joined to the positive end of a supply of tens of kilovolts. So each electron crosses a potential difference \(V\) of, say, \(30\,\text{kV}\). It gains a kinetic energy \(eV = 30\,\text{keV}\) on the way. The tube holds a very good vacuum, so no gas molecule stops the electron before the target.
The target and its heat
The electron then strikes the target, a block of tungsten set into a copper anode. Only about 1 % of the electrons' energy leaves as X-rays. The other 99 % heats the target. So the target must be a metal that stays solid when very hot. Tungsten melts at about \(3400\,°\text{C}\), the highest of all metals.
Tungsten has a second merit. Its nucleus holds 74 protons, and a heavy nucleus brakes an electron more strongly. So a tungsten target turns a larger share of the energy into X-rays than a light metal would. Even so, the heat is large. The copper anode carries it away into circulating oil or water. Many modern tubes also spin the target, so the beam never strikes one spot for long.
An X-ray tube runs at \(25\,\text{kV}\) with a tube current of \(20\,\text{mA}\). Only 1 % of the electrons' energy becomes X-rays. Find the number of electrons striking the target each second, the power of the X-rays, and the heat produced in the target each second.
The tube current is \(I = 20\,\text{mA} = 0.02\,\text{A}\), and each electron carries charge \(e = 1.6 \times 10^{-19}\,\text{C}\).
Electrons per second \(n = I/e\). Power delivered \(P = VI\). X-ray power is 1 % of \(P\).
\(n = 0.02 / 1.6 \times 10^{-19} = 1.25 \times 10^{17}\) per second. \(P = 25000 \times 0.02 = 500\,\text{W}\), so the X-rays carry \(5\,\text{W}\).
The target receives \(495\,\text{J}\) of heat every second, the output of a small room heater, on a spot a few millimetres across. That is why it needs cooling. Values used: \(e = 1.6 \times 10^{-19}\,\text{C}\).
An X-ray tube runs at \(40\,\text{kV}\) with a tube current of \(5\,\text{mA}\). If 99 % of the energy becomes heat, how many joules of heat does the target receive each second?
Two knobs on the control panel
An X-ray machine has two main controls, and they do different jobs. The first sets the filament current, and so the temperature of the filament. A hotter filament releases more electrons each second. So the tube current grows, more electrons strike the target, and more X-ray photons leave. The X-rays become more intense, but each photon is no more energetic than before.
The second control sets the tube voltage. A higher voltage gives every electron more kinetic energy, \(eV\). A faster electron can give more energy to a single photon, so the photons are more energetic on average. More energetic X-rays pass through more matter. They are called hard X-rays, and those from a low voltage are soft X-rays.
Do not read the tube current from Ohm's law. The tube is not a resistor. Once the voltage is high enough, every electron that leaves the filament reaches the target. The current then depends only on how hot the filament is, and raising the voltage hardly changes it.
A technician doubles the filament current of an X-ray tube and leaves the tube voltage alone. What happens to the X-rays?
An X-ray tube runs at \(10\,\text{kV}\). Find the kinetic energy of an electron when it reaches the target, in joules, and its speed.
(a) \(K = eV = 1.6 \times 10^{-19} \times 10^{4} = 1.6 \times 10^{-15}\,\text{J}\), which is \(10\,\text{keV}\).
(b) \(v = \sqrt{2K/m_e} = \sqrt{2 \times 1.6 \times 10^{-15} / 9.1 \times 10^{-31}} = 5.9 \times 10^{7}\,\text{m/s}\), about \(0.2\,c\).
(c) At this speed the classical formula is only about 1.4 % too high, so it is good enough. Tubes above about \(50\,\text{kV}\) need the relativistic formulas of the chapter on Matter Waves and Special Relativity.
Values used: \(e = 1.6 \times 10^{-19}\,\text{C}\), \(m_e = 9.1 \times 10^{-31}\,\text{kg}\), \(c = 3 \times 10^{8}\,\text{m/s}\).
What a hard beam shows
Now the picture of a hand. X-rays of different hardness are stopped by different thicknesses of matter. Bone contains calcium, whose atoms are heavier than the carbon, hydrogen and oxygen of flesh. Bone is also denser. So bone absorbs X-rays far more strongly than flesh does.
Fig 1.2 shows why the voltage must suit the part of the body. Too soft a beam is absorbed by the flesh, and it gives no picture while it harms the skin. Too hard a beam passes bone and flesh alike, and the two look the same. A hand is imaged at about \(55\,\text{kV}\), a chest at over \(100\,\text{kV}\).
A technician finds that a radiograph of a thick knee shows no bone at all, only a uniform pale shadow. What should she change?
Properties of X-rays
An X-ray photon carries energy \(hc/\lambda\), like any photon. For a wavelength of \(0.05\,\text{nm}\) this is \(1240/0.05 = 24800\,\text{eV}\), or \(24.8\,\text{keV}\). A photon of green light, \(500\,\text{nm}\), carries ten thousand times less. That huge energy explains most of the properties of X-rays.
X-rays:
- travel in straight lines at the speed of light, \(3 \times 10^{8}\,\text{m/s}\);
- carry no charge, so electric and magnetic fields do not bend them;
- blacken photographic film, and make some materials glow (fluorescence);
- pass through matter, more easily through light atoms than heavy ones, and more easily when hard;
- knock electrons out of atoms, so they ionise the air they cross;
- damage living cells, which is why the dose is kept small.
An X-ray photon has a wavelength of \(0.5\,\text{Å}\). Find its energy in keV and its momentum. How many times more energy does it carry than a photon of \(500\,\text{nm}\)?
(a) \(0.5\,\text{Å} = 0.05\,\text{nm}\), so \(E = hc/\lambda = 1240/0.05 = 24800\,\text{eV} = 24.8\,\text{keV}\).
(b) \(p = h/\lambda = 6.63 \times 10^{-34} / 0.05 \times 10^{-9} = 1.33 \times 10^{-23}\,\text{kg m/s}\).
(c) Energy goes as \(1/\lambda\), so the ratio is \(500/0.05 = 10^{4}\).
Values used: \(hc = 1240\,\text{eV nm}\), \(h = 6.63 \times 10^{-34}\,\text{J s}\), \(1\,\text{Å} = 0.1\,\text{nm}\).
X-ray wavelengths are often given in ångströms: \(1\,\text{Å} = 10^{-10}\,\text{m} = 0.1\,\text{nm}\). In these units \(hc = 12400\,\text{eV Å}\), the same constant as \(1240\,\text{eV nm}\). Many exam problems use it.
Which of these are true of X-rays?
Back to the X-ray room
Now the picture of the boy's wrist. The film darkens wherever X-rays reach it. Flesh lets most of the X-rays through, so the film behind it turns grey or black. Bone absorbs most of them, so the film behind it stays pale. The bone appears white, and a crack, where X-rays slip through, is a dark line.
The lead apron works on the same principle. Lead has 82 protons in each nucleus, and it is very dense. A thin sheet of it absorbs X-rays that pass straight through flesh. The technician stands behind a lead-lined screen for the same reason: she takes many pictures a day, and her dose would add up.
At every airport in India, cabin bags ride through an X-ray scanner. Many scanners take two pictures at once, one with soft X-rays and one with hard. Light materials such as food, clothes and plastics absorb the soft X-rays much more than the hard ones. Metals absorb both strongly. The computer compares the two pictures and colours each object. Organic material shows orange, metals blue, and mixtures green.
A tube gives out \(5\,\text{W}\) of X-rays. Suppose an average photon carries about \(10\,\text{keV}\). Estimate the number of X-ray photons leaving the tube each second, in units of \(10^{15}\).
On 8 November 1895, Wilhelm Röntgen was passing a current through a glass tube with the air pumped out. He had covered the tube with black card, and the room was dark. Yet a screen coated with a barium salt, lying on a bench, began to glow. Books, wood and even his own hand could not stop the rays. On 22 December he made a picture of his wife's hand, with the bones and her ring clearly visible. The news crossed the world within weeks. In 1901 Röntgen received the first Nobel Prize in Physics.
- Saying that the tube voltage sets the number of X-rays.The filament current sets the number of electrons, and so the intensity. The voltage sets the energy of each electron, and so the hardness.
- Thinking that most of the electrons' energy becomes X-rays.Only about 1 % does. The other 99 % heats the target, which is why the target must be cooled.
- Calling X-rays a beam of electrons.Electrons make the X-rays; the X-rays are photons with no charge, and fields do not bend them.
- Mixing nanometres and ångströms.\(1\,\text{Å} = 0.1\,\text{nm}\). Use \(hc = 1240\,\text{eV nm}\) or \(12400\,\text{eV Å}\), never one with the other's unit.