A test for a wave
By the rule of the last lesson, an electron accelerated through \(54\,\text{V}\) has a wavelength of about \(0.17\,\text{nm}\). If that wave is real, it must diffract. It needs openings or obstacles about \(0.17\,\text{nm}\) apart, and a crystal has exactly that. Its atoms stand in regular rows, a few tenths of a nanometre apart. A crystal is a ready-made grating for electron waves.
Fig 2.1 shows the three parts. In the electron gun, a hot filament gives off electrons and a voltage \(V\) accelerates them. So \(V\) fixes their kinetic energy \(eV\), their momentum and their wavelength. The beam strikes a single crystal of nickel. A detector counts the electrons scattered back per second in one direction. It moves along an arc round the crystal, so the scattering angle \(\phi\) can be changed. The whole apparatus is in a vacuum, because air molecules would scatter and stop such slow electrons.
The experiment records the count against \(\phi\), at one voltage after another. Particles bouncing off a rough surface would spread in all directions, with a count that changes smoothly with angle. A wave scattered by regular rows of atoms is different. The wavelets from the rows add up strongly only in the directions where they arrive in step.
In the Davisson and Germer experiment, what is measured, and against what is it plotted?
Davisson and Germer drew their results as polar plots like the one above. At \(40\,\text{V}\) the curve is smooth, as for bouncing particles. As the voltage rises, a bump grows at about \(50°\). It is strongest at \(54\,\text{V}\), and then it moves to smaller angles and fades. By \(68\,\text{V}\) it has almost gone. A sharp maximum that comes and goes with the wavelength is the mark of a wave.
Reading the wavelength from the peak
The face of the crystal has rows of atoms a distance \(d = 0.215\,\text{nm}\) apart. The beam arrives along the normal to the face. Consider the wavelets scattered back by two neighbouring rows towards the angle \(\phi\). The one from the farther row travels an extra path \(d\sin\phi\). The two arrive in step when this path difference is one whole wavelength.
At \(54\,\text{V}\) the peak is at \(\phi = 50°\). The rows of atoms on the nickel face are \(0.215\,\text{nm}\) apart. Find the wavelength from the peak, and compare it with de Broglie's value for an electron accelerated through \(54\,\text{V}\).
From the peak: \(\lambda = d\sin\phi = 0.215 \times \sin 50° = 0.215 \times 0.766 = 0.165\,\text{nm}\).
From de Broglie: \(\lambda = h/\sqrt{2m_eeV} = 1.23/\sqrt{54} = 1.23/7.35 = 0.167\,\text{nm}\).
The two agree to about one per cent. The wavelength read from a diffraction peak is the wavelength \(h/p\) of the electrons.
Values used: \(h = 6.63 \times 10^{-34}\,\text{J s}\), \(e = 1.6 \times 10^{-19}\,\text{C}\), \(m_e = 9.1 \times 10^{-31}\,\text{kg}\).
The rule also says which way the peak moves. A higher voltage gives a larger momentum and a shorter wavelength. Then \(\sin\phi = \lambda/d\) is smaller, so the maximum moves to a smaller angle. At \(64\,\text{V}\) it is near \(45°\). That is what the slider shows.
The same peak, read from the planes inside
Why does the bump grow and fade instead of simply moving? The wave also goes into the crystal, and each layer of atoms below the surface scatters it again. The waves from the deeper layers must also arrive in step. That happens only over a narrow band of wavelengths, centred on the wavelength at \(54\,\text{V}\). So the bump is strong only near \(54\,\text{V}\).
The same peak can be read as a reflection from planes of atoms inside the crystal. For this peak the planes are \(d = 0.091\,\text{nm}\) apart, and the beam meets them at the glancing angle \(\theta = 90° - \phi/2 = 65°\). Reflections from successive planes add when \(2d\sin\theta = \lambda\). This is Bragg's rule, which the chapter on X-rays derives. Here it gives \(\lambda = 2 \times 0.091 \times \sin 65° = 0.165\,\text{nm}\), the same wavelength as before.
The rows of atoms on a nickel face are \(0.215\,\text{nm}\) apart. A beam arriving along the normal gives a first-order peak at \(\phi = 50°\). Find the wavelength of the electrons, in nm. Take \(\sin 50° = 0.766\).
The accelerating voltage is raised from \(54\,\text{V}\) to \(64\,\text{V}\). Which way does the diffraction maximum from the surface rows move?
Electrons of kinetic energy \(150\,\text{eV}\), with \(\lambda = 0.100\,\text{nm}\), fall on a crystal whose planes are \(0.100\,\text{nm}\) apart. Use Bragg's rule \(2d\sin\theta = \lambda\) to find the glancing angle of the first-order maximum, in degrees.
A peak is found at a scattering angle \(\phi = 50°\). What is the glancing angle \(\theta\) between the beam and the reflecting planes?
At the Bell Telephone Laboratories in New York, Clinton Davisson and Lester Germer were scattering electrons from nickel. In 1925 a flask of liquid air burst, air rushed into the tube, and the hot nickel oxidised. To clean it they heated the nickel for a long time. That turned its many tiny crystals into a few large ones. Afterwards the scattered electrons came off in sharp directions. In 1926 Davisson heard of de Broglie's idea at a meeting in Oxford and saw what the peaks meant. In 1927 the two men found the peak at \(54\,\text{V}\) and showed that it matched \(h/p\).
In the same year, 1927, George Paget Thomson sent fast electrons through thin metal foils and photographed rings of diffraction. His father, J. J. Thomson, had shown in 1897 that the electron is a particle. The son showed that it is also a wave. Davisson and the younger Thomson shared the Nobel Prize in 1937.
One thing, two descriptions
Put the chapter on photons beside this lesson. Light spreads and interferes like a wave, yet it gives up its energy in whole photons. Electrons carry charge and collide like particles, yet they diffract like waves. Neither light nor an electron is simply a wave or simply a particle. Each is one kind of thing that shows one face or the other. This is wave-particle duality.
The two faces appear together when electrons are sent one at a time. In 1989 a team at Hitachi in Japan did this in an electron version of the double-slit experiment. Each electron arrived at one point on the screen, as a particle does. After thousands had arrived, their points built up bright and dark bands. The bands lay exactly where the wave says the electrons should arrive most often.
The wave tells where a particle is likely to be found. The particle is what is actually found. A single electron gives one dot; the wave decides the pattern that many dots make.
Which of these observations show the wave nature of electrons?
Why is a strong peak at one angle, rather than a smooth spread, taken as proof that electrons are waves?
Particles bouncing off a surface would leave in a smooth spread of directions, with no angle preferred. Waves scattered from regular rows of atoms add up only in the directions where their path difference is a whole number of wavelengths, and cancel elsewhere. Only a wave can pick out one angle, and the angle found gives a wavelength equal to h/p.
A light microscope cannot show detail much finer than about \(200\,\text{nm}\), because light diffracts round anything smaller than its wavelength. An electron microscope uses electrons instead, accelerated through about \(100\,\text{kV}\). Their wavelength is about \(0.0037\,\text{nm}\), more than a hundred thousand times shorter than that of green light. Magnetic coils focus the electron beam as glass lenses focus light. In the electron microscope laboratory at AIIMS, New Delhi, such instruments show the inside of cells. They show the shapes of viruses that no light microscope can reach.
- Measuring the scattering angle from the crystal face instead of from the incoming beam.In this experiment \(\phi\) is measured from the direction the beam came from, which is the normal to the face.
- Using \(\phi\) itself in Bragg's rule.Bragg's rule needs the glancing angle to the planes, \(\theta = 90° - \phi/2\).
- Expecting a higher voltage to push the peak to a larger angle.A higher voltage shortens the wavelength, and \(\sin\phi = \lambda/d\) then falls.