24.5 Electron Microscope

The Limitation of Optical Microscopes

The resolution of any microscope is fundamentally limited by the wavelength of the radiation used to form the image. Due to diffraction, two objects cannot be distinguished if they are closer together than approximately half the wavelength of the illuminating radiation. Visible light has wavelengths in the range $400\text{ nm}$ to $700\text{ nm}$, giving optical microscopes a resolution limit of roughly $200\text{ nm}$. This is insufficient to image individual molecules or atomic structures.

de Broglie Wavelength of Electrons

Louis de Broglie proposed that all matter has an associated wavelength given by:

$\lambda =\frac{h}{p}=\frac{h}{mv}$

where $h=6.63\times 10^{-34}\text{ J s}$ is Planck's constant and $p=mv$ is the momentum of the particle.

For an electron accelerated through a potential difference $V$, the work done by the electric field converts to kinetic energy:

$eV=\frac{1}{2}m{v}^2$

Solving for momentum $p=mv=\sqrt{2meV}$, the de Broglie wavelength becomes:

$\lambda =\frac{h}{\sqrt{2meV}}$

where $m=9.11\times 10^{-31}\text{ kg}$ is the electron mass and $e=1.6\times 10^{-19}\text{ C}$.

Worked Example

Find the de Broglie wavelength of an electron accelerated through $V=10\text{ kV}$.

$\lambda =\frac{6.63\times 10^{-34}}{\sqrt{2\times 9.11\times 10^{-31}\times 1.6\times 10^{-19}\times 10^4}}$

$\lambda =\frac{6.63\times 10^{-34}}{\sqrt{2.92\times 10^{-24}}}\approx \frac{6.63\times 10^{-34}}{5.40\times 10^{-12}}\approx 0.012\text{ nm}$

This is roughly 16,000 times shorter than visible light, enabling far superior resolution.

Key relationship: Since $\lambda \propto \frac{1}{\sqrt{V}}$, increasing the accelerating voltage by a factor of 4 halves the wavelength.

How an Electron Microscope Works

An electron microscope uses a beam of high-speed electrons instead of light. The key components are:

Why High Vacuum is Essential

The interior of an electron microscope must be maintained under high vacuum (pressure $\sim 10^{-5}\text{ Pa}$ or lower). If air were present, electrons would collide with air molecules, scattering the beam and making a focused image impossible.

Resolution of Electron Microscopes

Because the de Broglie wavelength of accelerated electrons ($\sim 0.001\text{ nm}$ to $0.1\text{ nm}$) is far shorter than visible light, electron microscopes achieve resolutions down to approximately $0.2\text{ nm}$ — about 1000 times better than optical microscopes ($200\text{ nm}$). This allows imaging of individual atoms and large molecules.

Comparison: Optical vs Electron Microscope