Electron wavelength and accelerating voltage

An electron accelerated through a potential difference gains kinetic energy. If its initial kinetic energy is negligible, the energy gained through a potential difference $$V$$ is $$eV$$, where $$e$$ is the magnitude of the electron’s charge.

For a non-relativistic electron, kinetic energy is also related to momentum $$p$$ by:

Here, $$m_e$$ is the electron’s mass. The electron’s de Broglie wavelength is $$\lambda = h/p$$, where $$h$$ is Planck’s constant. Substituting the momentum found from the energy equation gives:

Rearranging gives the accelerating voltage needed for a chosen wavelength:

An atomic-scale wavelength of $$1.0 \times 10^{-10},\mathrm{m}$$ requires an accelerating voltage of approximately $$150,\mathrm{V}$$, of order $$10^2,\mathrm{V}$$.

A higher accelerating voltage gives the electron more kinetic energy and therefore greater momentum. Since $$\lambda = h/p$$, its wavelength becomes shorter. In particular, $$\lambda \propto 1/\sqrt{V}$$: increasing the voltage by a factor of four halves the wavelength.

Transmission electron microscope

A transmission electron microscope (TEM) forms an enlarged image using electrons that pass through a very thin specimen. The specimen must be thin enough for electrons to be transmitted through it.

An electron beam is accelerated and focused onto the specimen by electromagnetic lenses. After the beam passes through the specimen, electromagnetic lenses focus the transmitted electrons to form an enlarged image on a detector or viewing screen.

Image contrast and resolution in a TEM

Different parts of the specimen transmit and scatter electrons by different amounts. As a result, different numbers of electrons from those parts contribute to the image, producing differences in brightness called contrast.

Accelerated electrons have a much shorter wavelength than visible light. This allows a TEM to resolve much finer detail: it can distinguish features that are too close together to be seen separately with visible light.

The electron beam travels in a vacuum, which reduces scattering by air molecules before the electrons reach the specimen and detector.

Scanning tunnelling microscope

A scanning tunnelling microscope (STM) maps a conducting surface using a sharp conducting tip held extremely close to it. A potential difference is applied between the tip and the surface.

Even though there is a gap between them, electrons can cross it by quantum tunnelling. This produces a current without the electrons crossing the gap by a classical path.

Mapping the surface

The tunnelling current depends strongly on the separation between the tip and the surface: a small decrease in the gap produces a large increase in current, while a small increase in the gap greatly reduces the current.

As the tip scans across the surface, a feedback system adjusts its height to keep the current constant. When the surface rises, the system raises the tip; when the surface falls, it lowers the tip. The recorded changes in tip height are used to build an image of the surface, with resolution on an atomic scale.