The measurement of the Avogadro constant opened the way to a comparison of the watt-balance measurements of the Planck constant with the values calculated from the quotients of the Planck constant and the mass of a particle or an atom. Since the energy scales of these measurements span nine energy decades, these data provide insight into the consistency of our understanding of physics.
The Planck constant, h, links energy and momentum to the frequency and the wavelength of the wave-function, be it classical or relativistic [1]. Its determinations match energy (momentum) and frequency (wavelength) measurements and, according to whether a mechanical, electrical, or thermal system is considered, the measurement result is a value of the h/m, h/e, or h/k B ratio, where m, e, and k B are a mass, the electron charge, and the Boltzmann constant. Eventually, to determine the Planck constant absolute measurements of mass, charge, or temperature are necessary. The archetype of the electrical determinations is Millikan's photoelectric measurement of the h/e ratio [2], the modern equivalent of which is the measurement of the Josephson constant by the tunneling of Cooper's pairs in a Josephson junction. As regards the thermodynamic measurements, their archetype is the Planck's black-body determination of the h/k B ratio [3], the modern equivalent of which is the measurement of the Boltzmann constant by the power of Johnson noise in a resistor.
A number of experiments measured energy or momentum in terms of frequency or wavelength via the Planck and de Broglie equations E = hν and p = h/λ. Energy and momentum are related to mass by the Einstein equation E = mc 2 and by p = mv, where v is the velocity; therefore, the quotient of the Planck constant and mass is also determined. Since, for atoms and sub-atomic particles, molar masses are well known, these experiments deliver accurate values of the molar Planck constant, N A h, but not of the Planck constant itself. The measurement of the Avogadro constant [4,5] opens the way to the estimate of h from the results of these experiments.
The present paper summarizes the results of the N A determination by counting 28 Si atoms. Next it reviews the determinations of the Planck constant via the h/(2e) determination [6,7] and the watt-balance experiments [8,9,10,11,12,13]. Eventually, it outlines the determinations of the molar Planck constant via the time-of-flight determination of monochromatic neutrons [14,15,16,17,18], atom interferometry [19,20,21,22,23], and atomic and nuclear spectroscopy [20,24,25,26,27].
Since these experiments rely on different quantum effects the energy scales of which range from less than 1 meV to more than 1 MeV, the ubiquitous presence of the Planck constant give us access to a verification of the measurement capabilities, the understanding of the phenomena underlying the measurements, and the approximation made.
N A has been determined by counting the atoms in a mole, exploiting their ordered arrangement in a 28 Si crystal. The crystal and the atom volumes -V and a 3 0 /8being measured, the count required calculation of their ratio,
The determination of the silicon moles, m/M (Si), required the crystal mass and molar mass -m and M (Si) -to be also measured. The use of a crystal highly enriched with the 28 Si isotope made it possible to determine the molar mass by isotope dilution mass spectroscopy with unprecedented accuracy. A spherical crystal-shape was selected to trace the volume determination back to diameter measurements and to make possible accurate geometrical, chemical, and physical characterizations of the crystal surface.
The lattice parameter, a 0 , and, hence, the atom volume, were measured by combined x-ray and optical interferometry. The results,
is the most accurate value so far obtained [4,5].
When a Josephson device is irradiated with electromagnetic radiation the current-tovoltage relation exhibits steps at quantized voltages. These steps are proportional to the frequency of the irradiating radiation, the proportionality factor being theoretically predicted to be 1/K J = h/(2e), where K J is the Josephson constant. By combining the results of K J measurements [6,7] with the fine structure constant [20],
or with the von Kilitzing constant [20],
values of h are also obtained [6,7]. Since the uncertainty of the α value is negligible, the uncertainty of the h value derived from ( 3) is twice that of K J .
The technologies required to carry out mechanical determinations of the Planck constant became available only recently [8,9,10,11,12,13]. The direct way of access to the h/m K ratio (m K is the mass of the international kilogram prototype) is the watt-balance experiment. This experiment compares virtually the mechanical and electrical powers produced by the motion of a kilogram prototype in the earth gravitational field and by the motion of the supporting coil in a magnetic field. The Planck constant, up to integers corresponding to Josephson and quantum Hall steps, is determined by m K gv ∝ hν 1 ν 2 /4, where v is the motion velocity, g the acceleration due to gravity, and ν 1 and ν 2 the frequency of the microwaves irradiating a Josephson junction to trace the measurement of the electrical power back to the Josephson and the von Klitzing constants, K J = 2e/h and R K = h/e 2 .
The measurement of the h/m
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