de Broglie Wavelength: Formula, Verification and Simple Applications

The formula for an accelerated electron. An electron of charge magnitude $e$ and mass $m$, starting from rest and accelerated through a potential difference $V$, gains kinetic energy $eV$, which is related to its momentum $p$ by $eV=\dfrac{p^2}{2m}$, so that $p=\sqrt{2meV}$. Substituting into de Broglie's relation $\lambda=h/p$ gives the electron's de Broglie wavelength directly in terms of the accelerating voltage:

Substituting the standard values $h=6.63\times10^{-34}\ \text{J s}$, $m=9.11\times10^{-31}\ \text{kg}$ and $e=1.6\times10^{-19}\ \text{C}$ gives the convenient numerical form

so that, for instance, an electron accelerated through just $150\ \text{V}$ already has a de Broglie wavelength of about $0.1\ \text{nm}$ -- comparable to the spacing between neighbouring atoms in a typical crystal (about $0.1$-$0.3\ \text{nm}$), which is exactly the condition needed for the electron beam to show diffraction when it strikes a crystal, in precisely the same way a beam of X-rays of a similar wavelength does.

The Davisson-Germer confirmation. In 1927, Clinton Davisson and Lester Germer, while studying the scattering of electrons off a nickel target (originally for an unrelated purpose), observed exactly this kind of diffraction: this section's figure shows the arrangement, in which an electron beam accelerated through a known voltage is directed onto a nickel crystal, and a detector records a pronounced INTENSITY MAXIMUM at one particular scattering angle -- a diffraction peak, not the smooth scattering a stream of classical particles alone would give. Careful measurement showed that the wavelength implied by the position of this peak (using the same crystal-diffraction relation already familiar from X-ray diffraction) matched, to good accuracy, the wavelength de Broglie's formula predicted for electrons accelerated through that same voltage -- direct, quantitative experimental proof that electrons genuinely possess wave character, exactly as de Broglie had proposed three years earlier. …