We’ve already seen that light shows both wave behaviour (diffraction/interference) and particle behaviour (photons). De Broglie suggested the reverse might also be true: particles such as electrons can have wave properties.

It turns out that electrons can diffract through tiny gaps/regular spacings in crystals (e.g. graphite). That diffraction is strong evidence that electrons behave as matter waves with a wavelength linked to their momentum.

If electrons were only particles, you’d expect them to pass straight through. Instead, when a beam of electrons passes through a thin crystal (e.g. graphite), it produces diffraction patterns (often rings), just like waves diffracting through a grating.

Increasing the accelerating voltage makes the rings closer together, consistent with a smaller wavelength.

De Broglie proposed that a moving particle has an associated wavelength λ. For a non-relativistic particle (Y12), the momentum is , where is the mass and is the speed:

Formula:

A charged particle accelerated through a potential difference gains kinetic energy equal to the electrical work done, , where is the charge on the particle:

Formula:

Rearranging gives the speed:

Substitute this into to write momentum in terms of , and :

Now substitute into to get the wavelength of a particle accelerated through a p.d.:

For an electron, and , so:

Limitation of the light microscope (wavelength and resolution)

A microscope can only resolve details down to about the same order as the wavelength being used (shorter wavelength → better resolution). Visible light has wavelength roughly 400–700 nm, so optical microscopes struggle to resolve features much smaller than this scale.

Electron wavelengths in microscopes (why electrons give higher resolution)

Electrons accelerated through large voltages can have wavelengths far smaller than visible light (and comparable to atomic spacings). That’s why electron microscopes can resolve much finer detail.

Set-up:

A transmission electron microscope (TEM) forms an image by sending a beam of fast electrons through an ultra-thin specimen in a vacuum.

Key components (what they do):

• Electron gun: produces and accelerates electrons (typically tens to hundreds of kV).

• Condenser lens: forms a narrow parallel beams so the specimen is illuminated uniformly.

• Objective lens: forms a magnified intermediate image of the specimen.

• Projector lens: magnifies and focuses the image onto the screen/camera.

Why lenses are magnetic (not glass): electrons are charged, so their paths can be bent and focused by magnetic fields.

Why real TEM resolution is limited

Limitation 1: Lens aberration

Lens aberration blurs the image because electrons originating from the same point on the sample aren’t always focused onto the same point in the image. This can be due to imperfections in the lens magnetic field or variations in the initial speeds of the electrons.

Limitation 2: Sample thickness

As electrons pass through a thicker specimen, they are more likely to lose energy / slow down. Slower electrons have larger de Broglie wavelength, reducing resolution. Also, the focusing becomes less consistent if electron speeds vary.

Quantum tunnelling (why it can happen)

Classically, a particle without enough energy can’t get over a barrier. In quantum physics, electrons have wave properties, and there can be a small probability of appearing on the other side of a thin barrier — this is quantum tunnelling.

STM structure and how an image is formed

A scanning tunneling microscope (STM) uses an extremely sharp conducting tip brought very close to the sample (around 1 nm separation). A small p.d. between tip and sample allows electrons to tunnel, producing a tiny tunnelling current.

Key idea: tunnelling probability (and current) is extremely sensitive to the gap: smaller gap → larger current; larger gap → smaller current.

A piezoelectric transducer controls the tip position with very high precision.

STM operating modes

Constant height mode

The tip height stays (approximately) constant while scanning. Changes in surface height change the gap so the tunnelling current changes. The STM records current vs position to map the surface.

Constant current mode

A feedback loop adjusts the tip height continuously to keep the current constant. The STM records tip height vs position, producing a surface profile.