How to use it
- Choose the particle: electron, proton, neutron, alpha particle, or a custom mass in u, g or kg.
- Choose what you know: the accelerating voltage, the kinetic energy or the speed.
- Enter the value. For a custom particle accelerated by a voltage, also enter its charge in units of e.
- Read the wavelength; the speed, the energy and the non-relativistic estimate are listed below it.
How it works
The de Broglie relation is λ = h ÷ p. The momentum is found without the slow-speed approximation:
- from the speed: p = γ·m·v, with γ = 1 ÷ √(1 − v²/c²);
- from the kinetic energy K: p·c = √(K² + 2·K·mc²);
- from a voltage V: K = q·V, where q is the charge (e for an electron or a proton, 2e for an alpha particle).
The non-relativistic value h ÷ √(2·m·K) is shown too, so you can see how much relativity changes the answer. Constants: h = 6.62607015 × 10⁻³⁴ J·s, e = 1.602176634 × 10⁻¹⁹ C, c = 299,792,458 m/s; electron mass 9.1093837015 × 10⁻³¹ kg, proton 1.67262192369 × 10⁻²⁷ kg, neutron 1.67492749804 × 10⁻²⁷ kg, alpha particle 6.6446573357 × 10⁻²⁷ kg.
Worked example
An electron accelerated through 100 V gains K = 100 eV = 1.602 × 10⁻¹⁷ J. Its momentum is 5.40301 × 10⁻²⁴ kg·m/s, so
λ = 6.62607 × 10⁻³⁴ ÷ 5.40301 × 10⁻²⁴ = 0.122637 nm (122.637 pm).
It moves at 5,930,099 m/s, about 2 % of the speed of light. The non-relativistic formula gives 0.122643 nm — almost the same. An electron at 1,000,000 m/s has λ = 0.727385 nm.
Tips
- Electron microscopes use tens to hundreds of kilovolts; at 100 keV the wavelength is about 3.70 pm, and only the relativistic formula gets it right.
- Thermal neutrons (about 0.0253 eV) have λ ≈ 0.18 nm, which is why they are used to study crystal structures.
- Heavier means shorter: at the same voltage a proton’s wavelength is √(mp/me) ≈ 43 times shorter than an electron’s.