de Broglie Wavelength & Matter Waves

Choose a particle and its kinetic energy (for an electron or a proton, the voltage that accelerated it), or give the ball a speed. The page works out the wavelength from the momentum, exactly and by the textbook formula, and the angle at which a nickel crystal's surface would send the beam.

Worked example — de Broglie Wavelength & Matter Waves

Rest energy	511.0 keV
Momentum, as pc	7.429 keV
Wavelength	0.1669 nm
Wavelength by the non-relativistic formula	0.1669 nm
How much too long that formula is	0.00264%
Speed as a fraction of c	0.01454
First beam from nickel's surface rows	50.66°

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — de Broglie Wavelength & Matter Waves

A moving particle has a wavelength, the de Broglie wavelength λ = h/p, where p is its momentum and h is Planck's constant. A beam of particles therefore diffracts from a grating whose spacing is comparable to λ, as light does. For mass m and kinetic energy K the momentum is pc = √(K² + 2K·mc²), which becomes the familiar p = √(2mK) when K is much less than the rest energy mc². An electron or proton accelerated from rest through V volts has K = eV.

What each symbol means

λ: the de Broglie wavelength of the particle
h: Planck's constant, 6.62607015×10⁻³⁴ J·s exactly
p: the particle's momentum; pc is the same quantity expressed as an energy
K: the kinetic energy; for one elementary charge accelerated through V volts, K = eV
mc²: the rest energy, 511.0 keV for an electron and 938.3 MeV for a proton
d: the spacing of the rows of atoms on nickel's (111) face, 0.2158 nm
φ: the angle between the incoming beam and the first diffracted beam, from sin φ = λ/d

Assumes: Every particle in the beam has the same energy, K = eV for a charge accelerated from rest, measured outside the crystal. Nickel's surface is met head-on and treated as rows of atoms 0.2158 nm apart, a reflection grating, so only the top layer counts. Nickel's spacing is calculated from its lattice constant, 0.352387 nm. The ball is taken as 145 g, about the mass of a baseball.
Breaks when: Inside the metal an electron is pulled in by the crystal and moves faster than it did outside, so its wavelength there is shorter. In 1928 Davisson and Germer reported that their beams did not obey Bragg's law for the layers below the surface, and, following a suggestion of Carl Eckart's, put it down to this refraction; the surface-row formula used here is unaffected by it, because the pull acts straight into the surface and leaves the momentum along it unchanged. The formula itself holds for any particle, with or without charge: in 1930 Immanuel Estermann and Otto Stern sent beams of helium atoms and hydrogen molecules at a crystal of lithium fluoride and saw them diffract.

A 54-volt electron is the size of the gap between two atoms 🖖

Press Davisson and Germer's 54 V. The electron's wavelength is 0.1669 nm. On the (111) face of a nickel crystal the rows of atoms are 0.2158 nm apart, so the surface is a reflection grating whose spacing is only a little longer than the wave, and the first diffracted beam leaves where sin φ = λ/d: at 50.66° to the incoming beam. In 1927 Clinton Davisson and Lester Germer, at Bell Telephone Laboratories in New York, fired 54 V electrons straight at such a face and found a sharp beam at 50°. Their experiment had begun as a study of how metals scatter electrons, and only became a test of de Broglie's idea in the summer of 1926. Press Four times the voltage: λ very nearly halves, to 0.08344 nm, and the beam swings in to 22.75°. Below 32.30 V the wave is longer than the row spacing and there is no diffracted beam at all, the same kind of limit <a href="/phy/braggs-law">Bragg's law</a> puts on X-rays.

A neutron at room temperature already has a crystal-sized wavelength 🖖

Press Room-temperature neutron. It has 0.0253 eV, which is kT at 20 °C: the typical energy of a neutron that has collided with the atoms of a reactor's moderator often enough to share their temperature. That is all, and its wavelength is 0.1798 nm, longer than the 54 V electron's. An electron needs 46.52 eV to match it, 1,839 times as much, which is the ratio of the two masses. The wavelength is set by the momentum, not the energy, and at equal energy the momentum goes as √m, so the heavier particle has the shorter wave: in the second table an electron with the neutron's 0.0253 eV has 7.710 nm, 42.9 times longer. Slow neutrons from the first research reactors were sent through crystals for exactly this reason, and Clifford Shull shared the 1994 Nobel Prize in Physics for turning that into neutron diffraction.

At a microscope's 100 kV the textbook formula is 4.78% too long 🖖

Press Electron microscope, 100 kV. The formula most courses give, λ = h/√(2meV), says 3.878 pm. But the electron is moving at 0.548c, and its momentum is no longer √(2mK). With the exact (pc)² = K² + 2K·mc² the wavelength is 3.701 pm, and the textbook figure is 4.78% too long. The ladder below shows the error growing: 0.00264% at 54 V, 0.488% at 10 kV, 18.0% at 400 kV. Relativity also breaks the 1/√V rule the first two presets obey. From 100 kV to 400 kV, four times the voltage, λ falls from 3.701 to 1.644 pm, a factor of 2.25 rather than 2, because once K is a sizeable fraction of the rest energy, 511 keV, the momentum grows faster than √K. On the chart the exact line leaves the dashed one near mₑc² and steepens, heading for λ ∝ 1/K.

Problem solved in full

An electron and an X-ray of the same wavelength 5 steps
Copper's Kα X-rays, 0.154184 nm, are the ones Bragg's law measures crystals with. What voltage gives an electron the same wavelength, how much energy does one X-ray photon carry, and how does the comparison change for longer waves? Press Davisson and Germer's 54 V for the starting point.

Open this problem in the tool
1. At 54 V the page prints the electron's wavelength:
  
  $λ_{54 V} = 0.1669 nm$
2. Far below mc², λ goes as 1/√V, so the voltage for the shorter wave scales by the square of the ratio:
  
  $V = 54 V \times (\frac{0 . 16689 nm}{0 . 154184 nm})^{2} = 63.27 V$
3. A photon's energy is hc/λ, with hc = 1239.84 eV·nm:
  
  $E_{γ} = \frac{h c}{λ} = \frac{1239 . 84 eV nm}{0 . 154184 nm} = 8, 041 eV$
4. The photon carries 127 times the electron's energy:
  
  $\frac{E _{γ}}{K} = \frac{8 , 041 eV}{63 . 27 eV} = 127.1$
5. For a slow electron K = h²/(2mλ²), and the photon's energy is hc/λ, so their ratio is 2mcλ/h: twice the wavelength over the electron's Compton wavelength, h/mₑc = 2.4263 pm.
  
  $\frac{E _{γ}}{K} = \frac{h c / λ}{h ^{2} / ( 2 m _{e} λ ^{2} )} = \frac{2 λ}{h / ( m _{e} c )} = \frac{2 \times 154 . 184 pm}{2 . 4263 pm} = 127.1$
Answer

63.27 V gives an electron the 0.154184 nm of copper Kα, and one photon of that wavelength carries 8,041 eV, 127.1 times as much. Because the ratio is 2λ/λ_C, it grows in proportion to the wavelength: at the sizes that matter for crystals, a particle with mass reaches them with a small fraction of the energy light needs, and the heavier the particle, the smaller the fraction. That is the neutron of the third preset. The ratio falls towards 1 only for wavelengths far below λ_C, where the electron is so fast that its energy, like the photon's, is simply pc.

$K = 63.27 eV, E_{γ} = 8, 041 eV, \frac{E _{γ}}{K} \approx \frac{2 λ}{λ _{C}}$

Learning path

Quantum mechanics: matter as waves

References (4)

Davisson's own account of the experiment: a 54-volt beam aimed at the (111) face of a nickel crystal, how the work began as a study of electron scattering and became a test of de Broglie only in 1926, and the check of λ = h/p against λ = 12.25/√V Å for nineteen beams: C. J. Davisson, "The discovery of electron waves." Nobel Lecture, 13 December 1937.
Diffraction is not special to electrons: Stern's account of sending helium atoms and hydrogen molecules at a lithium fluoride crystal, and checking de Broglie's formula to about 2%: O. Stern, "The method of molecular rays." Nobel Lecture, 12 December 1946.
The 54 V, 50° peak itself, and its reading as Bragg reflection from planes inside the crystal: R. Nave, "Davisson-Germer Experiment." HyperPhysics, Department of Physics and Astronomy, Georgia State University.
Neutrons from a reactor, slowed to thermal speeds and reflected by a crystal because of their wave nature, as the basis of neutron diffraction: The Royal Swedish Academy of Sciences, "The Nobel Prize in Physics 1994." Press release, 12 October 1994.

By the 42Math team · Reviewed against published sources. Corrections welcome.