de Broglie wavelength

The de Broglie wavelength is the wavelength that quantum mechanics assigns to any particle with momentum $p$:

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

with Planck's constant $h$ = 6.62607015 × 10⁻³⁴ J·s. Louis de Broglie proposed it in 1924 by extending to matter the relation already known for the photon, whose momentum is $h/\lambda$. For electrons of 100 eV the wavelength is 0.123 nm; for thermal neutrons, 0.18 nm; for everyday objects it is immeasurably small. Particles with wavelengths near 0.1 nm diffract from crystals just as X-rays do, which is how the idea was confirmed: Davisson and Germer observed electron diffraction from a nickel crystal in 1927, and G. P. Thomson observed it independently in transmission through thin films the same year.

Electrons: the working formula

An electron accelerated from rest through a potential $V$ gains kinetic energy $E = eV$. Non-relativistically $p = \sqrt{2 m_e E}$, which gives the convenient form

$$\lambda \approx \frac{1.2264\ \mathrm{nm}}{\sqrt{V\,[\mathrm{volts}]}}.$$

Once $E$ becomes a noticeable fraction of the electron rest energy $m_e c^2$ = 511.0 keV, the momentum must be computed relativistically:

$$p = \sqrt{2 m_e E\left(1 + \frac{E}{2 m_e c^2}\right)}.$$

Worked values:

At 100 eV the correction is 0.005 %, at 10 keV 0.49 %, and at 200 keV the non-relativistic formula overestimates the wavelength by 9.3 %; the electron is then moving at 0.695 $c$. A 100 eV electron, by comparison, moves at 5.93 × 10⁶ m/s, about 2 % of $c$.

Neutrons, atoms and a baseball

A thermal neutron at the conventional reference speed of 2200 m/s has kinetic energy 25.3 meV, close to $k_B T$ at room temperature (25.3 meV at 20 °C), and a wavelength of

$$\lambda = \frac{h}{m_n v} = 0.180\ \mathrm{nm}.$$

That matches interatomic distances, so neutron diffraction complements X-ray diffraction: neutrons scatter from nuclei and magnetic moments, locate light atoms such as hydrogen, and map magnetic order. Diffraction and interference have also been shown with helium atoms and whole molecules such as C₆₀ fullerenes.

For a baseball of 0.145 kg thrown at 40 m/s, $\lambda$ = 1.1 × 10⁻³⁴ m, about 10¹⁹ times smaller than a proton. No aperture or grating could reveal wave behaviour at that scale, which is why wave effects play no part in the motion of macroscopic objects.

How it is observed

The de Broglie wavelength is measured the same way as the wavelength of light: through diffraction from a structure of known period. In low-energy electron diffraction (LEED), electrons of 20–200 eV reflect from a crystal surface and form a spot pattern whose geometry follows the Bragg condition; the same energy range gives surface sensitivity because such electrons penetrate only a few atomic layers. In a transmission electron microscope, selected-area diffraction patterns of a crystalline specimen appear on the detector, and a double-slit experiment with electrons builds up fringes one electron at a time.

Electron microscopy

The short wavelength is the reason electron microscopes exist. An optical microscope is limited by the diffraction limit to roughly half the wavelength of light, about 200 nm. A 200 kV transmission electron microscope uses 2.51 pm electrons, yet its resolution is set in practice by the spherical aberration of magnetic lenses, which forces very small aperture angles of order 10 mrad; uncorrected instruments resolve about 0.2 nm, and aberration-corrected ones reach below 0.1 nm.

Pitfalls

Using the non-relativistic formula above a few tens of keV is the usual error: at 200 keV it is off by 9 %. A second is confusing energy and momentum for photons and massive particles. For a photon $\lambda = hc/E$, so a 100 eV photon has $\lambda$ = 12.4 nm, a hundred times longer than a 100 eV electron.

Common questions

What is the de Broglie wavelength of an electron?

It depends on the energy: 0.123 nm at 100 eV, 12.2 pm at 10 keV and 2.51 pm at 200 keV, using $\lambda = h/p$ with the relativistic momentum above about 10 keV.

Why is the de Broglie wavelength of a thermal neutron useful?

At about 0.18 nm it matches the spacing of atoms in solids, so neutrons diffract from crystals and reveal atomic and magnetic structure.

Do large objects have a de Broglie wavelength?

Formally yes, but for a thrown baseball it is about 10⁻³⁴ m, far too small for any grating to resolve; for large molecules, whose wavelengths are still measurable, interactions with the environment destroy the coherence needed to observe interference.

References: B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); L. de Broglie, "Recherches sur la théorie des quanta", Annales de Physique 3, 22 (1925); C. Davisson, L. H. Germer, "Diffraction of electrons by a crystal of nickel", Physical Review 30, 705 (1927); D. B. Williams, C. B. Carter, Transmission Electron Microscopy, 2nd ed. (Springer, 2009).