doPhysics · Modern Physics

X-Rays

Why this chapter

Why this chapter exists

A doctor in a district hospital sees a broken bone without cutting the skin. A jeweller reads the purity of a gold chain without scratching it. A scientist in Madras worked out the shape of a protein that nobody can see. All three use X-rays. This chapter builds an X-ray tube and reads its spectrum line by line. Then it turns the spectrum into two tools. Moseley's law names an element from its X-rays, and Bragg's law measures the spacing of atoms in a crystal.

You will learn

By the end of this chapter you will

Describe the Coolidge tube and say which control sets the intensity of the X-rays and which sets their hardness.
Explain the continuous spectrum as braking radiation, and find the cut-off wavelength \(\lambda_{min} = hc/eV\) for any tube voltage.
Read an intensity-wavelength graph: the cut-off, the continuous hump and the characteristic peaks, and how each changes with voltage, current and target.
Explain the Kα, Kβ and Lα lines as falls into inner-shell vacancies, and find their energies and wavelengths from the shell energies.
Use Moseley's law \(\sqrt{\nu} = a(Z - b)\) to find a wavelength or an atomic number, and derive \(E = 10.2\,(Z - 1)^2\) eV for Kα from the Bohr model.
Derive and use Bragg's law \(2d\sin\theta = n\lambda\) to find the spacing of crystal planes or the wavelength of X-rays.

Before you begin

From the chapter on Photons and the Photoelectric Effect, you should know the electron volt, thermionic emission from a hot filament, and a photon's energy \(hc/\lambda\) with \(hc = 1240\,\text{eV nm}\). From the chapter on the Bohr Atom, you should know the energy levels \(E_n = -13.6\,Z^2/n^2\) eV and how a jump between two levels gives one line. From the chapters on waves, you should know that two waves reinforce when their paths differ by a whole number of wavelengths.

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Lessons

Making X-Rays: the Coolidge Tube

A Coolidge tube fires electrons from a hot filament across a high voltage into a metal target, and about 1 % of their energy leaves as X-rays. The filament current sets how many X-rays come out; the tube voltage sets how hard they are.

The coolidge tubeIntensity and hardnessProperties and uses of x rays

13 min0%

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The Continuous Spectrum and the Cut-off

An electron braked by a nucleus gives a photon of any energy up to all it has, so the X-rays form a continuous spectrum. It stops sharply at the cut-off wavelength hc/eV, which depends on the tube voltage alone.

Braking radiationThe cut off wavelengthThe intensity wavelength graph

14 min0%

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Characteristic X-Rays

A fast electron can knock an inner electron out of a target atom. An electron from a higher shell falls into the vacancy and gives a photon of sharp energy, the Kα, Kβ or Lα line. The lines belong to the target, appear only above its K edge and never move with the voltage.

Inner shell vacanciesK alpha k beta and l alpha linesPeaks on the continuum

14 min0%

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Moseley's Law

The square root of the frequency of the Kα line grows in a straight line with the atomic number, \(\sqrt{\nu} = a(Z - 1)\). The Bohr model with one unit of nuclear charge screened explains it, and the law put the elements in order by atomic number and found the gaps in the periodic table.

Moseleys lawK alpha from bohr with screeningAtomic number orders the table

14 min0%

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Bragg's Law: X-Rays Meet a Crystal

X-rays reflect weakly from each plane of atoms in a crystal. The reflections from neighbouring planes add up only when their extra path \(2d\sin\theta\) is a whole number of wavelengths. A Bragg spectrometer uses this to measure the spacing of the planes or the wavelength of the X-rays.

Reflection from crystal planesBraggs lawMeasuring spacing and wavelength

15 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

ångström: A unit of length used for X-rays and atoms, \(1\,\text{Å} = 10^{-10}\,\text{m} = 0.1\,\text{nm}\).
atomic number: The number of protons in the nucleus of an atom, \(Z\). Moseley showed that the atomic number, not the atomic mass, fixes the place of an element in the periodic table.
Bragg spectrometer: An instrument that turns a crystal to a glancing angle \(\theta\) and a detector to \(2\theta\), to find the angles at which X-rays reflect strongly.
Bragg's law: X-rays reflected from parallel crystal planes a distance \(d\) apart reinforce when \(2d\sin\theta = n\lambda\), where \(\theta\) is the glancing angle and \(n\) the order.
bremsstrahlung: German for braking radiation: the X-rays given out when a fast electron is slowed down by the electric field of a nucleus. It forms the continuous spectrum.
characteristic X-rays: X-rays of sharp wavelengths given out when an electron of a target atom falls into a vacancy in an inner shell. Their wavelengths identify the element of the target.
continuous spectrum: The smooth part of an X-ray spectrum, spread over all wavelengths from the cut-off upward. It comes from bremsstrahlung, and its cut-off does not depend on the target.
Coolidge tube: An X-ray tube with a high vacuum, a hot tungsten filament as the source of electrons, and a cooled metal target on the anode. William Coolidge built it in 1913.
crystal: A solid whose atoms are arranged in a pattern that repeats regularly in all three directions, such as rock salt, quartz or a metal.
crystal planes: Parallel planes of a crystal, each containing many atoms, a fixed distance \(d\) apart. X-rays reflect weakly from each of them.
cut-off wavelength: The shortest wavelength an X-ray tube can produce, \(\lambda_{min} = hc/eV\). It is made when an electron gives all its energy \(eV\) to one photon.
filament: The thin tungsten wire of an X-ray tube, heated by a small current until it gives out electrons by thermionic emission. It is the cathode of the tube.
glancing angle: The angle \(\theta\) between a ray and a crystal plane, measured from the plane and not from its normal. Bragg's law uses it.
hard X-rays: X-rays of short wavelength and high photon energy, from a tube at a high voltage. They pass through much more matter than soft X-rays.
K edge: The binding energy of a K electron, the least energy that can make a K vacancy. It is \(20.0\,\text{keV}\) for molybdenum and \(69.5\,\text{keV}\) for tungsten.
K-alpha line: The characteristic X-ray line given out when an L electron falls into a vacancy in the K shell. Written Kα, it is the strongest line of a target.
K-beta line: The characteristic line given out when an M electron falls into a vacancy in the K shell. Written Kβ, it has more energy and a shorter wavelength than Kα.
L-alpha line: The characteristic line given out when an M electron falls into a vacancy in the L shell. Written Lα, its wavelength is several times longer than that of Kα.
Moseley's law: The rule \(\sqrt{\nu} = a(Z - b)\) for the frequency of one characteristic X-ray line in element after element. For the Kα lines \(b \approx 1\) and \(a \approx 4.96 \times 10^{7}\,\text{Hz}^{1/2}\).
order of reflection: The whole number \(n\) in Bragg's law. The paths of reflections from neighbouring planes differ by \(n\) wavelengths.
radiograph: A picture made by X-rays that pass through a body onto a film or a detector. Parts that absorb more, such as bone, leave a shadow.
screening: The reduction of the nuclear charge felt by an electron, because other electrons lie between it and the nucleus. It is why Moseley's law for Kα has \(Z - 1\).
shell: The electrons of an atom that share one principal quantum number. The K shell has \(n = 1\), the L shell \(n = 2\), the M shell \(n = 3\).
soft X-rays: X-rays of longer wavelength and lower photon energy, from a tube at a low voltage. A few millimetres of flesh stop most of them.
target: The block of metal on the anode of an X-ray tube that the electrons strike. Tungsten and molybdenum are common targets.
tube current: The current carried by the electrons that cross an X-ray tube, a few milliamperes. The filament temperature sets it, and it fixes the number of electrons per second.
tube voltage: The high potential difference between the filament and the target of an X-ray tube, usually 20 to 150 kV. It sets the energy of each electron that reaches the target.
vacancy: An empty place in an inner shell of an atom, left when a fast electron knocks out one of its electrons. An electron from a higher shell soon falls into it.
X-rays: Electromagnetic waves of very short wavelength, about \(0.01\) to \(10\,\text{nm}\), with photon energies from about \(0.1\) to \(100\,\text{keV}\). They are made when fast electrons strike a target.