Since 2019, the numerical value of the Planck constant has been fixed, with a finite decimal representation. Under the present definition of the kilogram, which states that "The kilogram [...] is defined by taking the fixed numerical value of h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of speed of light c and duration of hyperfine transition of the ground state of an unperturbed caesium-133 atom ΔνCs."[44] This implies that mass metrology is now aimed to find the value of one kilogram, and thus it is kilogram which is compensating. Every experiment aiming to measure the kilogram (such as the Kibble balance and the X-ray crystal density method), will essentially refine the value of a kilogram.
As an illustration of this, suppose the decision of making h to be exact was taken in 2010, when its measured value was 6.62606957×10−34 J⋅s, thus the present definition of kilogram was also enforced. In the future, the value of one kilogram must be refined to 6.62607015/6.62606957 ≈ 1.0000001 times the mass of the International Prototype of the Kilogram (IPK).
Significance of the value
The Planck constant is related to the quantization of light and matter. It can be seen as a subatomic-scale constant. In a unit system adapted to subatomic scales, the electronvolt is the appropriate unit of energy and the petahertz the appropriate unit of frequency. Atomic unit systems are based (in part) on the Planck constant. The physical meaning of the Planck constant could suggest some basic features of our physical world. These basic features include the properties of the vacuum constants {\displaystyle \mu _{0}}
and {\displaystyle \epsilon _{0}}
. The Planck constant can be identified as
{\displaystyle h=13.1Q{\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}\zeta ^{2}}
,
where Q is the quality factor and {\displaystyle \zeta }
is the integrated area of the vector potential at the center of the wave packet representing a particle.[45]
The Planck constant is one of the smallest constants used in physics. This reflects the fact that on a scale adapted to humans, where energies are typical of the order of kilojoules and times are typical of the order of seconds or minutes, the Planck constant is very small. One can regard the Planck constant to be only relevant to the microscopic scale instead of the macroscopic scale in our everyday experience.
Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a large number of microscopic particles. For example, green light with a wavelength of 555 nanometres (a wavelength that can be perceived by the human eye to be green) has a frequency of 540 THz (540×1012 Hz). Each photon has an energy E = hf = 3.58×10−19 J. That is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual atoms or molecules. An amount of light more typical in everyday experience (though much larger than the smallest amount perceivable by the human eye) is the energy of one mole of photons; its energy can be computed by multiplying the photon energy by the Avogadro constant, NA = 6.02214076×1023 mol−1[46], with the result of 216 kJ, about the food energy in three apples.
Determination
In principle, the Planck constant can be determined by examining the spectrum of a black-body radiator or the kinetic energy of photoelectrons, and this is how its value was first calculated in the early twentieth century. In practice, these are no longer the most accurate methods.
Since the value of the Planck constant is fixed now, it is no longer determined or calculated in laboratories. Some of the practices given below to determine the Planck constant are now used to determine the mass of the kilogram. The below-given methods except the X-ray crystal density method rely on the theoretical basis of the Josephson effect and the quantum Hall effect.
Josephson constant
Main article: Magnetic flux quantum
The Josephson constant KJ relates the potential difference U generated by the Josephson effect at a "Josephson junction" with the frequency ν of the microwave radiation. The theoretical treatment of Josephson effect suggests very strongly that KJ = 2e/h.
{\displaystyle K_{\rm {J}}={\frac {\nu }{U}}={\frac {2e}{h}}\,}
The Josephson constant may be measured by comparing the potential difference generated by an array of Josephson junctions with a potential difference which is known in SI volts. The measurement of the potential difference in SI units is done by allowing an electrostatic force to cancel out a measurable gravitational force, in a Kibble balance. Assuming the validity of the theoretical treatment of the Josephson effect, KJ is related to the Planck constant by
{\displaystyle h={\frac {8\alpha }{\mu _{0}c_{0}K_{\rm {J}}^{2}}}.}
Kibble balance
Main article: Kibble balance
A Kibble balance (formerly known as a watt balance)[47] is an instrument for comparing two powers, one of which is measured in SI watts and the other of which is measured in conventional electrical units. From the definition of the conventional watt W90, this gives a measure of the product KJ2RK in SI units, where RK is the von Klitzing constant which appears in the quantum Hall effect. If the theoretical treatments of the Josephson effect and the quantum Hall effect are valid, and in particular assuming that RK = h/e2, the measurement of KJ2RK is a direct determination of the Planck constant.
{\displaystyle h={\frac {4}{K_{\rm {J}}^{2}R_{\rm {K}}}}.}
Magnetic resonance
Main article: Gyromagnetic ratio
The gyromagnetic ratio γ is the constant of proportionality between the frequency ν of nuclear magnetic resonance (or electron paramagnetic resonance for electrons) and the applied magnetic field B: ν = γB. It is difficult to measure gyromagnetic ratios precisely because of the difficulties in precisely measuring B, but the value for protons in water at 25 °C is known to better than one part per million. The protons are said to be "shielded" from the applied magnetic field by the electrons in the water molecule, the same effect that gives rise to chemical shift in NMR spectroscopy, and this is indicated by a prime on the symbol for the gyromagnetic ratio, γ′p. The gyromagnetic ratio is related to the shielded proton magnetic moment μ′p, the spin number I (I = 1⁄2 for protons) and the reduced Planck constant.
{\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{I\hbar }}={\frac {2\mu _{\rm {p}}^{\prime }}{\hbar }}}
The ratio of the shielded proton magnetic moment μ′p to the electron magnetic moment μe can be measured separately and to high precision, as the imprecisely known value of the applied magnetic field cancels itself out in taking the ratio. The value of μe in Bohr magnetons is also known: it is half the electron g-factor ge. Hence
{\displaystyle \mu _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}{\frac {g_{\rm {e}}\mu _{\rm {B}}}{2}}}
{\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}{\frac {g_{\rm {e}}\mu _{\rm {B}}}{\hbar }}.}
A further complication is that the measurement of γ′p involves the measurement of an electric current: this is invariably measured in conventional amperes rather than in SI amperes, so a conversion factor is required. The symbol Γ′p-90 is used for the measured gyromagnetic ratio using conventional electrical units. In addition, there are two methods of measuring the value, a "low-field" method and a "high-field" method, and the conversion factors are different in the two cases. Only the high-field value Γ′p-90(hi) is of interest in determining the Plaaccelerator.
Substitution gives the expression for the Planck constant in terms of Γ′p-90(hi):
{\displaystyle h={\frac {c_{0}\alpha ^{2}g_{\text{e}}}{2K_{\text{J-90}}R_{\text{K-90}}R_{\infty }\Gamma _{\text{p-90}}^{\prime }({\rm {hi}})}}{\frac {\mu _{\text{p}}^{\prime }}{\mu _{\text{e}}}}.}
Faraday constan
Main article: Faraday constant
The Faraday constant F is the charge of one mole of electrons, equal to the Avogadro constant NA multiplied by the elementary charge e. It can be determined by careful electrolysis experiments, measuring the amount of silver dissolved from an electrode in a given time and for a given electric current. Substituting the definitions of NA and e gives the relation to the Planck constant.
{\displaystyle h={\frac {c_{0}M_{\rm {u}}A_{\rm {r}}({\rm {e}})\alpha ^{2}}{R_{\infty }}}{\frac {1}{K_{\text{J}}R_{\text{K}}F}}}
X-ray crystal density
The X-ray crystal density method is primarily a method for determining the Avogadro constant NA but as the Avogadro constant is related to the Planck constant it also determines a value for h. The principle behind the method is to determine NA as the ratio between the volume of the unit cell of a crystal, measured by X-ray crystallography, and the molar volume of the substance. Crystals of silicon are used, as they are available in high quality and purity by the technology developed for the semiconductor industry. The unit cell volume is calculated from the spacing between two crystal planes referred to as d220. The molar volume Vm(Si) requires a knowledge of the density of the crystal and the atomic weight of the silicon used. The Planck constant h=E/f
