Electronic Kilogram

The kilogram is the only metric unit still defined by an artifact. The International Prototype Kilogram, IPK or “Le Grand K”, is a golf-ball-sized platinum-iridium cylinder in a vault outside Paris. This year I expect the General Conference on Weights and Measures to replace the IKP by an electronic realization that balances gravitational and electrical power.

The Kelvin or Ampere balance suspends a horizontal wire loop of mass m, length \ell, and current I, by a radial magnetic field B. Integrate the magnetic force \overrightharpoon{F}=q\overrightharpoon{v}\times\overrightharpoon{B} around the loop to find the force balance

m g = F = \left| \oint d \overrightharpoon{F} \right| = \left| \oint dq \,\overrightharpoon{v} \times \overrightharpoon{B}\right| = \left| \oint I d\overrightharpoon{\ell} \times \overrightharpoon{B}\right| =I \ell B

and solve for m. Unfortunately, \ell and B are difficult to measure accurately.

In 1975, Bryan Kibble proposed the calibration step of moving the current-less wire loop vertically at speed v. Integrate the force per charge \overrightharpoon{F}/q=\overrightharpoon{v}\times\overrightharpoon{B}  around the loop to find the induced voltage

V = \oint \overrightharpoon{E} \cdot d\overrightharpoon{\ell} = \oint \frac{\overrightharpoon{F}}{q}\cdot d\overrightharpoon{\ell} = \oint \overrightharpoon{v} \times \overrightharpoon{B} \cdot d\overrightharpoon{\ell} = v B \ell.

Eliminate \ell and B from the force and voltage expressions to find the virtual power

P = V I = v B L I = m g v

in Watts, and again solve for m. Accurately measure voltage V by comparing to the superconducting Josephson-effect voltage

V =\frac{n_J f}{K_J},

where K_J = 2 e / h = 0.48~\text{THz} / \text{mV} is the Josephson constantn_J is the number of Josephson junctions, and f is their microwave frequency. Convert current I = V_R / R to voltage and resistance by Ohm’s law. Accurately measure resistance R by comparing to the quantum Hall-effect resistance

R =\frac{R_K}{n_L},

where R_K = h /e^2 = 26~\text{k}\Omega is the von Klitzing constant, and n_L is the number of filled Landau levels. Accurately measure velocity v and acceleration g using interferometers.

Hence the mass

m=\frac{VI}{gv}=VV_R\frac{1}{R}\frac{1}{gv}=n_J f\left(\frac{h}{2e}\right) n_J f_R\left(\frac{h}{2e}\right) n_L\left(\frac{e^2}{h}\right)\frac{1}{gv}=\frac{n_L n_J^2 f f_R h}{4gv}\propto h,

where h=0.66~\text{zJ} / \text{THz} is the Planck constant. The Kibble or Watt balance thus defines mass in terms of the rate of change of a photon’s energy with its frequency.

The NIST-4 Kibble balance has measured Planck's constant to 13 parts per billion and is thus accurate enough to help redefine the kilogram

The NIST-4 Kibble balance has measured the Planck constant to 13 parts per billion and is thus accurate enough to help redefine the kilogram. Credit: Jennifer Lauren Lee.

About John F. Lindner

John F. Lindner was born in Sleepy Hollow New York and educated at the University of Vermont and Caltech. He is a professor of physics and astronomy at The College of Wooster. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, and University of Hawai'i. His research interests include nonlinear dynamics, celestial mechanics, and variable stars.
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