# Planck constant

﻿
Planck constant
Values of h Units
6.62606957(29)×10−34 J·s[1]
4.135667516(91)×10−15 eV·s[1]
6.62606957(29)×10−27 erg·s[1]
Values of ħ Units
1.054571726(47)×10−34 J·s[1]
6.58211928(15)×10−16 eV·s[1]
1.054571726(47)×10−27 erg·s[1]
Planck's plaque at Humboldt University, Berlin. English translation: "Max Planck, discoverer of the elementary quantum of action h, taught in this building from 1889 to 1928."

The Planck constant (denoted h), also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899. Classical statistical mechanics requires the existence of h (but does not define its value).[2]

The Planck constant was first described as the proportionality constant between the energy (E) of a photon and the frequency of its associated electromagnetic wave (ν). This relation between the energy and frequency is called the Planck relation or the Planck–Einstein equation:

$E = h\nu.\,$

Since the frequency ν, wavelength λ, and speed of light c are related by λν = c, the Planck relation can also be expressed as

$E = \frac{hc}{\lambda}.\,$

In 1923, Louis de Broglie generalized this relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but any particle. This was confirmed by experiments soon afterwards.

Planck discovered that physical action could not take on any indiscriminate value. Instead, the action must be some multiple of a very small quantity (later to be named the "quantum of action" and now called Planck's constant). This inherent granularity is counterintuitive in the everyday world, where it is possible to "make things a little bit hotter" or "move things a little bit faster". This is because the quanta of action are very, very small in comparison to everyday human experience. Thus, on the macro scale quantum mechanics and classical physics converge. Nevertheless, it is impossible, as Planck found out, to explain some phenomena without accepting that action is quantized. In many cases, such as for monochromatic light or for atoms, the quantum of action also implies that only certain energy levels are allowed, and values in between are forbidden.

In applications where frequency is expressed in terms of radians per second ("angular frequency") instead of cycles per second, it is often useful to absorb a factor of 2π into the Planck constant. The resulting constant is called the reduced Planck constant or Dirac constant. It is equal to the Planck constant divided by 2π, and is denoted ħ ("h-bar"):

$\hbar = \frac{h}{2 \pi}.$

The energy of a photon with angular frequency ω, where ω = 2πν, is given by

$E = \hbar \omega.$

## Value

The Planck constant has dimensions of physical action; these are the same as those of angular momentum, i.e., energy multiplied by time, or momentum multiplied by distance. In SI units, the Planck constant is expressed in joule seconds (J·s) or (N·m·s).

The value of the Planck constant is:[1]

$h = 6.626\ 069\ 57(29)\times 10^{-34}\ \mathrm{J \cdot s} = 4.135\ 667\ 516(91)\times 10^{-15}\ \mathrm{eV \cdot s}.$

The value of the reduced Planck constant is:

$\hbar = {{h}\over{2\pi}} = 1.054\ 571\ 726(47)\times 10^{-34}\ \mathrm{J \cdot s} = 6.582\ 119\ 28(15)\times 10^{-16}\ \mathrm{eV \cdot s}.$

The two digits between the parentheses denote the standard uncertainty in the last two digits of the value. The figures cited here are the 2010 CODATA recommended values for the constants and their uncertainties. The 2010 CODATA results were made available in June 2011[3] and represent the best-known, internationally-accepted values for these constants, based on all data available as of 2010. New CODATA figures are scheduled to be published approximately every four years.

## Significance of the value

The numerical value of the Planck constant depends entirely on the system of units used to measure it. When it is expressed in SI units, it is one of the smallest constants used in physics. This reflects the fact that, on a scale adapted to humans, where energies are typically of the order of kilojoules and times are typically of the order of seconds or minutes, Planck's constant, the quantum of action, is very small.

Equivalently, the smallness of Planck's constant reflects the fact that everyday objects and systems are made of large number of particles. To take one example, green light of a wavelength of 555 nanometres (approximately the wavelength to which human eyes are most sensitive) has a frequency of 540 THz (540×1012 Hz). Each photon has an energy E of  = 3.58×10−19 J. That is a very small energy in terms of everyday experience, but then everyday experience doesn't deal with individual photons any more than it deals with individual atoms or molecules. An amount of light that is compatible with everyday experience is the energy of one mole of photons; its energy can be calculated by multiplying the photon energy by the Avogadro constant, NA ≈ 6.022×1023 mol−1. The result is that green light of wavelength 555 nm has an energy of 216 kJ/mol, a typical energy of everyday life.

The Planck constant is related to the quantization of light and matter. Therefore, the Planck constant can be seen as an atomic-scale constant. In a unit system adapted to atomic scales, the electronvolt is the appropriate unit of energy and the Petahertz the appropriate unit of frequency. In such an atomic units system, Planck's constant is indeed described by a number of order 1.

## Origins

Intensity of light emitted from a black body at any given frequency. Each color is a different temperature. Planck was the first to explain the shape of these curves.

In the last years of the nineteenth century, Planck was investigating the problem of black-body radiation first posed by Kirchhoff some forty years earlier. It is well known that hot objects glow, and that hotter objects glow brighter than cooler ones. The reason is that the electromagnetic field obeys laws of motion just like a mass on a spring, and can come to thermal equilibrium with hot atoms. When a hot object is in equilibrium with light, the amount of light it absorbs is equal to the amount of light it emits. If the object is black, meaning it absorbs all the light that hits it, then it emits the maximum amount of thermal light too.

The assumption that blackbody radiation is thermal leads to an accurate prediction: the total amount of emitted energy goes up with the temperature according to a definite rule, the Stefan–Boltzmann law (1879–84). But it was also known that the colour of the light given off by a hot object changes with the temperature, so that "white hot" is hotter than "red hot". Nevertheless, Wilhelm Wien discovered the mathematical relationship between the peaks of the curves at different temperatures, by using the principle of adiabatic invariance. At each different temperature, the curve is moved over by Wien's displacement law (1893). Wien also proposed an approximation for the spectrum of the object, which was correct at high frequencies (short wavelength) but not at low frequencies (long wavelength).[4] It still was not clear why the spectrum of a hot object had the form that it has (see diagram).

Planck hypothesized that the equations of motion for light are a set of harmonic oscillators, one for each possible frequency. He examined how the entropy of the oscillators varied with the temperature of the body, trying to match Wien's law, and was able to derive an approximate mathematical function for black-body spectrum.[5]

However, Planck soon realized that his solution was not unique. There were several different solutions, each of which gave a different value for the entropy of the oscillators[5]. To save his theory, Planck had to resort to using the then controversial theory of statistical mechanics,[5] which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics."[6] One of his new boundary conditions was

to interpret UN [the vibrational energy of N oscillators] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energy element ε;
—Planck, see [5]

With this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption … actually I did not think much about it…" in his own words,[7] but one which would revolutionize physics. Applying this new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator, the first version of what is now termed "Planck's relation":

$E = h\nu.\,$

Planck was able to calculate the value of h from experimental data on black-body radiation: his result, 6.55 × 10−34 J·s, is within 1.2% of the currently accepted value.[5] He was also able to make the first determination of the Boltzmann constant kB from the same data and theory.[8]

Note that the (black) Raleigh-Jeans curve never touches the Planck curve.

Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. The Rayleigh-Jeans law makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make Planck's law, which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves h in both the numerator and the denominator. The influence of h in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh-Jeans law could not be done by changing the values of h, of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture.

The black-body problem was revisited in 1905, when Rayleigh and Jeans (on the one hand) and Einstein (on the other hand) independently proved that classical electromagnetism could never account for the observed spectrum. These proofs are commonly known as the "ultraviolet catastrophe", a name coined by Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the photoelectric effect) to convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The very first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".[9] Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

### Photoelectric effect

The photoelectric effect is the emission of electrons (called "photoelectrons") from a surface when light is shone on it. It was first observed by Alexandre Edmond Becquerel in 1839, although credit is usually reserved for Heinrich Hertz,[10] who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard in 1902.[11] Einstein's 1905 paper[12] discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,[10] when his predictions had been confirmed by the experimental work of Robert Andrews Millikan.[13] To put it another way, in 1921 at least, Einstein's theories on the photoelectric effect were considered more important than his theory of relativity (a name coined, as it happens, by Max Planck).[10]

Prior to Einstein's paper, electromagnetic radiation such as visible light was considered to behave as a wave: hence the use of the terms "frequency" and "wavelength" to characterise different types of radiation. The energy transferred by a wave in a given time is called its intensity. The light from a theatre spotlight is more intense than the light from a domestic lightbulb; that is to say that the spotlight gives out more energy per unit time (and hence consumes more electricity) than the ordinary bulb, even though the colour of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their own intensity. However the energy account of the photoelectric effect didn't seem to agree with the wave description of light.

The "photoelectrons" emitted as a result of the photoelectric effect have a certain kinetic energy, which can be measured. This kinetic energy (for each photoelectron) is independent of the intensity of the light,[11] but depends linearly on the frequency;[13] and if the frequency is too low (corresponding to a kinetic energy for the photoelectrons of zero or less), no photoelectrons are emitted at all, unless a plurality of photons, whose energetic sum is greater than the energy of the photoelectrons, acts virtually simultaneously (multiphoton effect) [14] Assuming the frequency is high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons to be emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higher kinetic energy.[11]

Einstein's explanation for these observations was that light itself is quantized; that the energy of light is not transferred continuously as in a classical wave, but only in small "packets" or quanta. The size of these "packets" of energy, which would later be named photons, was to be the same as Planck's "energy element", giving the modern version of Planck's relation:

$E = h\nu.\,$

Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light (ν) and the kinetic energy of photoelectrons (E) was shown to be equal to the Planck constant (h).[13]

### Atomic structure

A schematization of the Bohr model of the hydrogen atom. The transition shown from the n=3 level to the n=2 level gives rise to visible light of wavelength 656 nm (red), as the model predicts.

Niels Bohr introduced the first quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of Rutherford's classical model.[15] In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a nucleus, the radiation would cause it to lose energy and spiral down into the nucleus. Bohr solved this paradox with explicit reference to Planck's work: an electron in a Bohr atom could only have certain defined energies En

$E_n = -\frac{h c_0 R_{\infty}}{n^2}$

where R is an experimentally-determined constant (the Rydberg constant) and n is any integer (n = 1, 2, 3, …). Once the electron reached the lowest energy level (n = 1), it could not get any closer to the nucleus (lower energy). This approach also allowed Bohr to account for the Rydberg formula, an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constant R in terms of other fundamental constants.

Bohr also introduced the quantity h/2π, now known as the reduced Planck constant, as the quantum of angular momentum. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model. The correct quantization rules for electrons – in which the energy reduces to the Bohr-model equation in the case of the hydrogen atom – were given by Heisenberg's matrix mechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum. In modern terms, if J is the total angular momentum of a system with rotational invariance, and Jz the angular momentum measured along any given direction, these quantities can only take on the values

\begin{align} J^2 = j(j+1) \hbar^2,\qquad & j = 0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots, \\ J_z = m \hbar, \qquad\qquad\quad & m = -j, -j+1, \ldots, j. \end{align}

### Uncertainty principle

The Planck constant also occurs in statements of Werner Heisenberg's uncertainty principle. Given a large number of particles prepared in the same state, the uncertainty in their position, Δx, and the uncertainty in their momentum (in the same direction), Δp, obey

$\Delta x\, \Delta p \ge \frac{\hbar}{2}$

where the uncertainty is given as the standard deviation of the measured value from its expected value. There are a number of other such pairs of physically measurable values which obey a similar rule. One example is time vs. frequency. The either-or nature of uncertainty forces measurement attempts to choose between trade offs, and given that they are quanta, the trade offs often take the form of either-or (as in Fourier analysis), rather than the compromises and gray areas of time series analysis. A practical example is computational neurology trying to both measure the time effect and frequency of a neuron burst. fMRI (functional MRI), whose signal processing is based on Fourier transforms, can resolve frequency, but not time (a limit of Fourier analysis due to uncertainty). An EEG (a time series analysis measurement tool) can resolve time, but not frequency. Due to uncertainty, these are not problems with the design of the measuring instruments, but problems with the nature of quantum measurement and particle realities themselves.

In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the commutator relationship between the position operator $\hat{x}$ and the momentum operator $\hat{p}$:

$[\hat{p}_i, \hat{x}_j] = -i \hbar \delta_{ij}$

where δij is the Kronecker delta.

## Dependent physical constants

The following list is based on the 2006 CODATA evaluation;[16] for the constants listed below, more than 90% of the uncertainty is due to the uncertainty in the value of the Planck constant, as indicated by the square of the correlation coefficient (r2 > 0.9, r > 0.949). The Planck constant is (with one or two exceptions[17]) the fundamental physical constant which is known to the lowest level of precision, with a relative uncertainty ur of 5.0×10−8.

### Rest mass of the electron

The normal textbook derivation of the Rydberg constant R defines it in terms of the electron mass me and a variety of other physical constants.

$R_\infty = \frac{m_{\rm e} e^4}{8 \epsilon_0^2 h^3 c_0} = \frac{m_{\rm e} c_0 \alpha^2}{2 h}$

However, the Rydberg constant can be determined very accurately (ur = 6.6×10−12) from the atomic spectrum of hydrogen, whereas there is no direct method to measure the mass of a stationary electron in SI units. Hence the equation for the calculation of me becomes

$m_{\rm e} = \frac{2 R_{\infty} h}{c_0 \alpha^2}$

where c0 is the speed of light and α is the fine-structure constant. The speed of light has an exactly defined value in SI units, and the fine-structure constant can be determined more accurately (ur = 6.8×10−10) than the Planck constant: the uncertainty in the value of the electron rest mass is due entirely to the uncertainty in the value of the Planck constant (r2 > 0.999).

The Avogadro constant NA is determined as the ratio of the mass of one mole of electrons to the mass of a single electron: The mass of one mole of electrons is the "relative atomic mass" of an electron Ar(e), which can be measured in a Penning trap (ur = 4.2×10−10), multiplied by the molar mass constant Mu, which is defined as 0.001 kg/mol.

$N_{\rm A} = \frac{M_{\rm u} A_{\rm r}({\rm e})}{m_{\rm e}} = \frac{M_{\rm u} A_{\rm r}({\rm e}) c_0 \alpha^2}{2 R_{\infty} h}$

The dependence of the Avogadro constant on the Planck constant (r2 > 0.999) also holds for the physical constants which are related to amount of substance, such as the atomic mass constant. The uncertainty in the value of the Planck constant limits the knowledge of the masses of atoms and subatomic particles when expressed in SI units. It is possible to measure the masses more precisely in atomic mass units, but not to convert them more precisely into kilograms.

### Elementary charge

Sommerfeld originally defined the fine-structure constant α as:

$\alpha\ =\ \frac{e^2}{\hbar c_0 \ 4 \pi \epsilon_0}\ =\ \frac{e^2 c_0 \mu_0}{2 h}$

where e is the elementary charge, ε0 is the electric constant (also called the permittivity of free space), and μ0 is the magnetic constant (also called the permeability of free space). The latter two constants have fixed values in the International System of Units. However, α can also be determined experimentally, notably by measuring the electron spin g-factor ge, then comparing the result with the value predicted by quantum electrodynamics.

At present, the most precise value for the elementary charge is obtained by rearranging the definition of α to obtain the following definition of e in terms of α and h:

$e = \sqrt{\frac{2\alpha h}{\mu_0 c_0}}.$

### Bohr magneton and nuclear magneton

The Bohr magneton and the nuclear magneton are units which are used to describe the magnetic properties of the electron and atomic nuclei respectively. The Bohr magneton is the magnetic moment which would be expected for an electron if it behaved as a spinning charge according to classical electrodynamics. It is defined in terms of the reduced Planck constant, the elementary charge and the electron mass, all of which depend on the Planck constant: the final dependence on h½ (r2 > 0.995) can be found by expanding the variables.

$\mu_{\rm B} = \frac{e \hbar}{2 m_{\rm e}} = \sqrt{\frac{c_0 \alpha^5 h}{32 \pi^2 \mu_0 R_{\infty}^2}}$

The nuclear magneton has a similar definition, but corrected for the fact that the proton is much more massive than the electron. The ratio of the electron relative atomic mass to the proton relative atomic mass can be determined experimentally to a high level of precision (ur = 4.3×10−10).

$\mu_{\rm N} = \mu_{\rm B} \frac{A_{\rm r}({\rm e})}{A_{\rm r}({\rm p})}$

## Determination

Method Value of h
(10−34 J·s)
Relative
uncertainty
Ref.
Watt balance 6.62606889(23) 3.4×10−8 [18][19][20]
X-ray crystal density 6.6260745(19) 2.9×10−7 [21]
Josephson constant 6.6260678(27) 4.1×10−7 [22][23]
Magnetic resonance 6.6260724(57) 8.6×10−7 [24][25]
CODATA 2010
recommended value
6.62606957(29) 4.4×10−8 [1]
The nine recent determinations of the Planck constant cover five separate methods. Where there is more than one recent determination for a given method, the value of h given here is a weighted mean of the results, as calculated by CODATA.

In principle, the Planck constant could 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. The CODATA value quoted here is based on three watt-balance measurements of KJ2RK and one inter-laboratory determination of the molar volume of silicon,[16] but is mostly determined by a 2007 watt-balance measurement made at the U.S. National Institute of Standards and Technology (NIST).[20] Five other measurements by three different methods were initially considered, but not included in the final refinement as they were too imprecise to affect the result.

There are both practical and theoretical difficulties in determining h. The practical difficulties can be illustrated by the fact that the two most accurate methods, the watt balance and the X-ray crystal density method, do not appear to agree with one another. The most likely reason is that the measurement uncertainty for one (or both) of the methods has been estimated too low – it is (or they are) not as precise as is currently believed – but for the time being there is no indication which method is at fault.

The theoretical difficulties arise from the fact that all of the methods except the X-ray crystal density method rely on the theoretical basis of the Josephson effect and the quantum Hall effect. If these theories are slightly inaccurate – though there is no evidence at present to suggest they are – the methods would not give accurate values for the Planck constant. More importantly, the values of the Planck constant obtained in this way cannot be used as tests of the theories without falling into a circular argument. Fortunately, there are other statistical ways of testing the theories, and the theories have yet to be refuted.[16]

### Josephson constant

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.

$K_{\rm J} = \nu/U = 2e/h\,$

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. Assuming the validity of the theoretical treatment of the Josephson effect, KJ is related to the Planck constant by

$h = \frac{8\alpha}{\mu_0 c_0 K_{\rm J}^2}.$

### Watt balance

A watt balance 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.

$h = \frac{4}{K_{\rm J}^2 R_{\rm K}}$

### Magnetic resonance

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 = 12 for protons) and the reduced Planck constant.

$\gamma^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{I \hbar} = \frac{2 \mu^{\prime}_{\rm p}}{\hbar}$

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

$\mu^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}} \frac{g_{\rm e} \mu_{\rm B}}{2}$
$\gamma^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}} \frac{g_{\rm e} \mu_{\rm B}}{\hbar}.$

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 Planck constant.

$\gamma^{\prime}_{\rm p} = \frac{K_{\rm J-90} R_{\rm K-90}}{K_{\rm J} R_{\rm K}} \Gamma^{\prime}_{\rm p-90}({\rm hi}) = \frac{K_{\rm J-90} R_{\rm K-90} e}{2} \Gamma^{\prime}_{\rm p-90}({\rm hi})$

Substitution gives the expression for the Planck constant in terms of Γ′p-90(hi):

$h = \frac{c_0 \alpha^2 g_{\rm e}}{2 K_{\rm J-90} R_{\rm K-90} R_{\infty} \Gamma^{\prime}_{\rm p-90}({\rm hi})} \frac{\mu_{\rm p}^{\prime}}{\mu_{\rm e}}.$

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. In practice, it is measured in conventional electrical units, and so given the symbol F90. Substituting the definitions of NA and e, and converting from conventional electrical units to SI units, gives the relation to the Planck constant.

$h = \frac{c_0 M_{\rm u} A_{\rm r}({\rm e})\alpha^2}{R_{\infty}} \frac{1}{K_{\rm J-90} R_{\rm K-90} F_{90}}$

### 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 is given by

$h = \frac{M_{\rm u} A_{\rm r}({\rm e}) c_0 \alpha^2}{R_{\infty}} \frac{\sqrt{2}d^3_{220}}{V_{\rm m}({\rm Si})}.$

## Fixation

As mentioned above, the numerical value of the Planck constant depends on the system of units used to describe it. Its value in SI units is known to 50 parts per billion but its value in atomic units is known exactly, because of the way the scale of atomic units is defined. The same is true of conventional electrical units, where the Planck constant (noted h90 to distinguish it from its value in SI units) is given by

$h_{90} = \frac{4}{K_{J-90}^2 R_{K-90}}$

with KJ–90 and RK–90 being exactly defined constants. Atomic units and conventional electrical units are very useful in their respective fields, because the uncertainty in the final result doesn't depend on an uncertain conversion factor, only on the uncertainty of the measurement itself.

There are a number of proposals to redefine certain of the SI base units in terms of fundamental physical constants.[27] This has already been done for the metre, which is defined in terms of a fixed value of the speed of light. The most urgent unit on the list for redefinition is the kilogram, whose value has been fixed for all science (since 1889) by the mass of a small cylinder of platinumiridium alloy kept in vault just outside Paris. While nobody knows if the mass of the International Prototype Kilogram has "changed" since 1889 – the value 1 kg of its mass expressed in kilograms is by definition unchanged and therein lies one of the problems – it is known that over such a timescale the many similar Pt–Ir alloy cylinders kept in national laboratories around the world, have changed their relative mass by several tens of parts per million, however carefully they are stored, and the more so, the more they have been taken out and used as mass standards. A change of several tens of micrograms in one kilogram is equivalent to the current uncertainty in the value of the Planck constant in SI units.

The legal process to change the definition of the kilogram is already underway,[27] but it was decided that no final decision would be made before the next meeting of the General Conference on Weights and Measures in 2011.[28] The Planck constant is a leading contender to form the basis of the new definition, although not the only one.[28] Possible new definitions include "the mass of a body at rest whose equivalent energy equals the energy of photons whose frequencies sum to 135,639,274×1042 Hz",[29] or simply "the kilogram is defined so that the Planck constant equals 6.62606896×10−34 J·s".

The BIPM provided Draft Resolution A in anticipation of the 24th General Conference on Weights and Measures meeting (2011-10-17 though 2011-10-21), detailing the considerations "On the possible future revision of the International System of Units, the SI".[30]

Watt balances already measure mass in terms of the Planck constant: at present, standard mass is taken as "fixed" and the measurement is performed to determine the Planck constant but, were the Planck constant to be fixed in SI units, the same experiment would be a measurement of the mass. The relative uncertainty in the measurement would remain the same.

Mass standards could also be constructed from silicon crystals or by other "atom-counting" methods. Such methods require a knowledge of the Avogadro constant, which fixes the proportionality between atomic mass and macroscopic mass but, with a defined value of the Planck constant, NA would be known to the same level of uncertainty (if not better) than current methods of comparing macroscopic mass.

## Computing codes

The symbol for the Planck constant is , similar to an italic lowercase letter H (h); the symbol for the reduced Planck constant is , similar to an italic lowercase letter H with stroke (ħ).

 character ℎ ℏ Unicode name PLANCK CONSTANT PLANCK CONSTANT OVER TWO PI character encoding decimal hex decimal hex Unicode 8462 210E 8463 210F UTF-8 226 132 142 E2 84 8E 226 132 143 E2 84 8F Numeric character reference ℎ ℎ ℏ ℏ

## Notes

1. ^ a b c d e f g h P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants [Thursday, 02-Jun-2011 21:00:12 EDT]. National Institute of Standards and Technology, Gaithersburg, MD 20899.
2. ^ Giuseppe Morandi, F. Napoli, E. Ercolessi (2001), Statistical mechanics: an intermediate course, ISBN 9789810244774, "See page 85"
3. ^
4. ^ R. Bowley, M. Sánchez (1999), Introductory Statistical Mechanics (2nd ed.), Oxford: Clarendon Press, ISBN 0-19-850576-0
5. ^ a b c d e  . English translation: "On the Law of Distribution of Energy in the Normal Spectrum".
6. ^ Kragh, Helge (1 December 2000), Max Planck: the reluctant revolutionary, PhysicsWorld.com
7. ^ Kragh, Helge (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, p. 62, ISBN 0691095523
8. ^
9. ^ Previous Solvay Conferences on Physics, International Solvay Institutes, retrieved 12 December 2008
10. ^ a b c See, eg,
11. ^ a b c Lenard, P. (1902), "Ueber die lichtelektrische Wirkung", Ann. Phys. 313 (5): 149–98, Bibcode 1902AnP...313..149L, doi:10.1002/andp.19023130510
12. ^
13. ^ a b c Millikan, R. A. (1916), "A Direct Photoelectric Determination of Planck's h", Phys. Rev. 7 (3): 355–88, Bibcode 1916PhRv....7..355M, doi:10.1103/PhysRev.7.355
14. ^ Smith, Richard (1962), "Two Photon Photoelectric Effect", Physical Review 128 (5): 2225, Bibcode 1962PhRv..128.2225S, doi:10.1103/PhysRev.128.2225. Smith, Richard (1963), "Two-Photon Photoelectric Effect", Physical Review 130 (6): 2599, Bibcode 1963PhRv..130.2599S, doi:10.1103/PhysRev.130.2599.4.
15. ^ Bohr, Niels (1913), "On the Constitution of Atoms and Molecules", Phil. Mag., Ser. 6 26 (153): 1–25, doi:10.1080/14786441308634993
16. ^ a b c Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. Bibcode 2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633.  Direct link to value.
17. ^ The main exceptions are the Newtonian constant of gravitation G and the gas constant R. The uncertainty in the value of the gas constant also affects those physical constants which are related to it, such as the Boltzmann constant and the Loschmidt constant.
18. ^ Kibble, B P; Robinson, I A; Belliss, J H (1990), "A Realization of the SI Watt by the NPL Moving-coil Balance", Metrologia 27 (4): 173–92, Bibcode 1990Metro..27..173K, doi:10.1088/0026-1394/27/4/002
19. ^ Steiner, R.; Newell, D.; Williams, E. (2005), "Details of the 1998 Watt Balance Experiment Determining the Planck Constant", Journal of Research (National Institute of Standards and Technology) 110 (1): 1–26
20. ^ a b Steiner, Richard L.; Williams, Edwin R.; Liu, Ruimin; Newell, David B. (2007), "Uncertainty Improvements of the NIST Electronic Kilogram", IEEE Transactions on Instrumentation and Measurement 56 (2): 592–96, doi:10.1109/TIM.2007.890590
21. ^ Fujii, K.; Waseda, A.; Kuramoto, N.; Mizushima, S.; Becker, P.; Bettin, H.; Nicolaus, A.; Kuetgens, U. et al. (2005), "Present state of the avogadro constant determination from silicon crystals with natural isotopic compositions", IEEE Transactions on Instrumentation and Measurement 54 (2): 854–59, doi:10.1109/TIM.2004.843101
22. ^ Sienknecht, Volkmar; Funck, Torsten (1985), "Determination of the SI Volt at the PTB", IEEE Trans. Instrum. Meas. 34 (2): 195–98, doi:10.1109/TIM.1985.4315300 . Sienknecht, V; Funck, T (1986), "Realization of the SI Unit Volt by Means of a Voltage Balance", Metrologia 22 (3): 209–12, Bibcode 1986Metro..22..209S, doi:10.1088/0026-1394/22/3/018 . Funck, T.; Sienknecht, V. (1991), "Determination of the volt with the improved PTB voltage balance", IEEE Transactions on Instrumentation and Measurement 40 (2): 158–61, doi:10.1109/TIM.1990.1032905
23. ^ Clothier, W. K.; Sloggett, G. J.; Bairnsfather, H.; Currey, M. F.; Benjamin, D. J. (1989), "A Determination of the Volt", Metrologia 26 (1): 9–46, Bibcode 1989Metro..26....9C, doi:10.1088/0026-1394/26/1/003
24. ^ Kibble, B P; Hunt, G J (1979), "A Measurement of the Gyromagnetic Ratio of the Proton in a Strong Magnetic Field", Metrologia 15 (1): 5–30, Bibcode 1979Metro..15....5K, doi:10.1088/0026-1394/15/1/002
25. ^ Liu Ruimin; Liu Hengji; Jin Tiruo; Lu Zhirong;Du Xianhe; Xue Shouqing; Kong Jingwen; Yu Baijiang;Zhou Xianan; Liu Tiebin; Zhang Wei (1995), "A Recent Determination for the SI Values of γ′p and 2e/h at NIM", Acta Metrologica Sinica 16 (3): 161–68
26. ^ Bower, V. E.; Davis, R. S. (1980), "The Electrochemical Equivalent of Pure Silver: A Value of the Faraday Constant", Journal of Research (National Bureau Standards) 85 (3): 175–91
27. ^ a b 94th Meeting of the International Committee for Weights and Measures (2005). Recommendation 1: Preparative steps towards new definitions of the kilogram, the ampere, the kelvin and the mole in terms of fundamental constants
28. ^ a b 23rd General Conference on Weights and Measures (2007). Resolution 12: On the possible redefinition of certain base units of the International System of Units (SI).
29. ^ Taylor, B. N.; Mohr, P. J. (1999), "On the redefinition of the kilogram", Metrologia 36 (1): 63–64, Bibcode 1999Metro..36...63T, doi:10.1088/0026-1394/36/1/11
30. ^

## References

• Barrow, John D. (2002), The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe, Pantheon Books, ISBN 0-375-42221-8

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