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### Planck constant

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ARTICLE ORIGINS Planck constant
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## Origin of the definition

B_nu(nu, T) = frac{ 2 h nu^3}{c^2} frac{1}{e^frac{hnu}{k_mathrm B T} - 1}
where {{math|kB}} is the Boltzmann constant, {{math|h}} is the Planck constant, and {{math|c}} is the speed of light in the medium, whether material or vacuum.{{harvnb|Planck|1914|pp=6, 168}}{{harvnb|Chandrasekhar|1960|p=8}}{{harvnb|Rybicki|Lightman|1979|p=22}} The spectral radiance can also be expressed per unit wavelength {{math|Î»}} instead of per unit frequency. In this case, it is given by
B_lambda(lambda, T) =frac{2hc^2}{lambda^5}frac{1}{ e^{frac{hc}{lambda k_mathrm B T}} - 1},
showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths.JOURNAL, Shao, Gaofeng, etal, Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems, Corrosion Science, 2019, 146, 233â€“246, 10.1016/j.corsci.2018.11.006, The law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation. The SI units of {{math|B'Î½}} are {{nobreak|WÂ·srâˆ’1Â·mâˆ’2Â·Hzâˆ’1}}, while those of {{math|B'Î»}} are {{nobreak|WÂ·srâˆ’1Â·mâˆ’3}}.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. To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics, which he described as "an act of despair â€¦ I was ready to sacrifice any of my previous convictions about physics."{{citation | first = Helge | last = Kragh | url =weblink | title = Max Planck: the reluctant revolutionary | publisher = PhysicsWorld.com | date = 1 December 2000}} One of his new boundary conditions wasWith this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption â€¦ actually I did not think much about itâ€¦" in his own words,{{citation | title = Quantum Generations: A History of Physics in the Twentieth Century | first = Helge | last = Kragh | year = 1999 | publisher = Princeton University Press | isbn = 978-0-691-09552-3 | page = 62 | url =weblink}} 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 sometimes termed the "Planckâ€“Einstein relation":
E = hf.
Planck was able to calculate the value of h from experimental data on black-body radiation: his result, {{val|6.55|e=-34|u=J.s}}, is within 1.2% of the currently accepted value. He also made the first determination of the Boltzmann constant kB from the same data and theory.{{citation | first = Max | last = Planck | author-link = Max Planck | title = The Genesis and Present State of Development of the Quantum Theory (Nobel Lecture) | url =weblink | date = 2 June 1920}}(File:Black body.svg|350px|thumb|The divergence of the theoretical Rayleighâ€“Jeans (black) curve from the observed Planck curves at different temperatures.)

## Development and application

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) in 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".{{citation | url =weblink | title = Previous Solvay Conferences on Physics | accessdate = 12 December 2008 | publisher = International Solvay Institutes | deadurl = yes | archiveurl =weblink" title="web.archive.org/web/20081216120021weblink">weblink | archivedate = 16 December 2008 | df = }}

### Photoelectric effect

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

### Atomic structure

(File:Bohr atom model.svg|thumb|right|A schematization of the Bohr model of the hydrogen atom. The transition shown from the {{nowrap|1=n = 3}} level to the {{nowrap|1=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.{{citation | first = Niels | last = Bohr | author-link = Niels Bohr | title = On the Constitution of Atoms and Molecules | journal = Phil. Mag. |series=6th Series | year = 1913 | volume = 26 | issue = 153 | pages = 1â€“25 | doi = 10.1080/14786441308634993 | url =weblink }} 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 c0 is the speed of light in vacuum, Râˆž is an experimentally determined constant (the Rydberg constant) and n is any integer (n = 1, 2, 3, â€¦). Once the electron reached the lowest energy level ({{nowrap|1=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 Râˆž in terms of other fundamental constants.Bohr also introduced the quantity frac{h}{2pi}, 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 â€“ in which the energy reduces to the Bohr model equation in the case of the hydrogen atom â€“ were given by Heisenberg's matrix mechanics in 1925 and the SchrÃ¶dinger 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, qquadqquadquad & 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, Î”x, and the uncertainty in their momentum (in the same direction), Î”p, 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. energy. 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.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 Î´ij is the Kronecker delta.

## Photon energy

The Planckâ€“Einstein relation connects the particular photon energy {{math|E}} with its associated wave frequency {{math|f}}:
E = hf
This energy is extremely small in terms of ordinarily perceived everyday objects.Since the frequency {{math|f}}, wavelength {{math|Î»}}, and speed of light {{math|c}} are related by f= frac{c}{lambda} , the relation can also be expressed as
E = frac{hc}{lambda} .
The de Broglie wavelength {{math|Î»}} of the particle is given by
lambda = frac{h}{p}
where {{math|p}} denotes the linear momentum of a particle, such as a photon, or any other elementary particle.In applications where it is natural to use the angular frequency (i.e. where the frequency is expressed in terms of radians per second instead of cycles per second or hertz) it is often useful to absorb a factor of {{math|2Ï€}} into the Planck constant. The resulting constant is called the reduced Planck constant. It is equal to the Planck constant divided by {{math|2Ï€}}, and is denoted {{math|Ä§}} (pronounced "h-bar"):
hbar = frac{h}{2 pi} .
The energy of a photon with angular frequency {{math|1=Ï‰ = 2Ï€f}} is given by
E = hbar omega ,
while its linear momentum relates to
p = hbar k ,
where {{math|k}} is an angular wavenumber. In 1923, Louis de Broglie generalized the Planckâ€“Einstein relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterwards. This holds throughout quantum theory, including electrodynamics.Problems can arise when dealing with frequency or the Planck constant because the units of angular measure (cycle or radian) are omitted in SI.JOURNAL, Mohr, J. C., Phillips, W. D., 2015, Dimensionless Units in the SI, Metrologia, 52, 1, 40â€“47, 10.1088/0026-1394/52/1/40, 2015Metro..52...40M, 1409.2794, JOURNAL, Mills, I. M., 2016, On the units radian and cycle for the quantity plane angle, Metrologia, 53, 3, 991â€“997, 10.1088/0026-1394/53/3/991, 2016Metro..53..991M, Nature (2017) â€˜â€™A Flaw in the SI system,â€™' Volume 548, Page 135 In the language of quantity calculus,Maxwell J.C. (1873) A Treatise on Electricity and Magnetism, Oxford University Press the expression for the "value" of the Planck constant, or of a frequency, is the product of a "numerical value" and a "unit of measurement". When we use the symbol {{math|f}} (or {{math|Î½}}) for the value of a frequency it implies the units cycles per second or hertz, but when we use the symbol {{math|Ï‰}} for its value it implies the units radians per second; the numerical values of these two ways of expressing the value of a frequency have a ratio of {{math|2Ï€}}, but their values are equal. Omitting the units of angular measure "cycle" and "radian" can lead to an error of {{math|2Ï€}}. A similar state of affairs occurs for the Planck constant. We use the symbol {{math|h}} when we express the value of the Planck constant in Jâ‹…s/cycle, and we use the symbol {{math|Ä§}} when we express its value in Jâ‹…s/rad. Since both represent the value of the Planck constant, but in different units, we have {{math|1=h = Ä§}}. Their "values" are equal but, as discussed below, their "numerical values" have a ratio of {{math|2Ï€}}. In this Wikipedia article the word "value" as used in the tables means "numerical value", and the equations involving the Planck constant and/or frequency actually involve their numerical values using the appropriate implied units. The distinction between "value" and "numerical value" as it applies to frequency and the Planck constant is explained in more detail in this pdf file Link.These two relations are the temporal and spatial component parts of the special relativistic expression using 4-vectors.
P^mu = left(frac{E}{c}, vec{p}right) = hbar K^mu = hbarleft(frac{omega}{c}, vec{k}right)
Classical statistical mechanics requires the existence of {{math|h}} (but does not define its value).{{Citation|title=Statistical mechanics: an intermediate course |author1=Giuseppe Morandi |author2=F. Napoli |author3=E. Ercolessi |page=84|url=https://books.google.com/?id=MhInFlnNsREC&pg=PA51 |isbn=978-981-02-4477-4|year=2001}} Eventually, following upon Planck's discovery, it was recognized that physical action cannot take on an arbitrary value. Instead, it must be some integer multiple of a very small quantity, the "quantum of action", now called the reduced Planck constant or the natural unit of action. This is the so-called "old quantum theory" developed by Bohr and Sommerfeld, in which particle trajectories exist but are hidden, but quantum laws constrain them based on their action. This view has been largely replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist, rather, the particle is represented by a wavefunction spread out in space and in time. Thus there is no value of the action as classically defined. Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain either quantization of energy or the lack of a classical particle motion.In many cases, such as for monochromatic light or for atoms, quantization of energy also implies that only certain energy levels are allowed, and values in between are forbidden.{{Citation|last=Einstein|title=Physics and Reality|page=24|first=Albert|quote=The question is first: How can one assign a discrete succession of energy value {{math|HÏƒ}} to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates {{math|qr}} and the corresponding momenta {{math|pr}})? The Planck constant {{math|h}} relates the frequency {{math|HÏƒ/h}} to the energy values {{math|HÏƒ}}. It is therefore sufficient to give to the system a succession of discrete frequency values.|url=http://www.kostic.niu.edu/Physics_and_Reality-Albert_Einstein.pdf|doi=10.1162/001152603771338742|year=2003|journal=Daedalus|volume=132|issue=4|deadurl=yes|archiveurl=https://web.archive.org/web/20120415132339weblink|archivedate=2012-04-15|df=}}

## Value

The Planck constant has dimensions of physical action; i.e., energy multiplied by time, or momentum multiplied by distance, or angular momentum. In SI units, the Planck constant is expressed in joule-seconds (Jâ‹…s or Nâ‹…mâ‹…s or kgâ‹…m2â‹…s−1). Implicit in the dimensions of the Planck constant is the fact that the SI unit of frequency, the Hertz, represents one complete cycle, 360 degrees or {{math|2Ï€}} radians, per second. An angular frequency in radians per second is often more natural in mathematics and physics and many formulas use a reduced Planck constant (pronounced h-bar)
hbar={{h}over{2pi}}
On 16 November 2018, the International Bureau of Weights and Measures (BIPM) voted to redefine the kilogram by fixing the value of the Planck constant, thereby defining the kilogram in terms of the second and the speed of light. Starting 20 May 2019, the new value is exactly
h = 6.626 070 15times 10^{-34} text{J}{cdot}text{s}.
In July 2017, the NIST measured the Planck constant using its Kibble balance instrument with an uncertainty of only 13 parts per billion, obtaining a value of {{val|6.626069934|(89)|e=âˆ’34|u=Jâ‹…s}}.NEWS,weblink 30 June 2017, 11 July 2017, New Measurement Will Help Redefine International Unit of Mass, National Institute of Standards and Technology, This measurement, along with others, allowed the redefinition of SI base units.{{citation
|title=Draft Resolution A "On the revision of the International System of units (SI)" to be submitted to the CGPM at its 26th meeting (2018)
|url=https://www.bipm.org/utils/en/pdf/CGPM/Draft-Resolution-A-EN.pdf
}} The two digits inside the parentheses denote the standard uncertainty in the last two digits of the value.As of the 2014 CODATA release, the best measured value of the Planck constant was:
h = 6.626 070 040(81)times 10^{-34} text{J}{cdot}text{s} = 4.135 667 662(25)times 10^{-15} text{eV}{cdot}text{s}.
The value of the reduced Planck constant (or Dirac constant{{citation needed|date=April 2019}}) was:
hbar={{h}over{2pi}} = 1.054 571 800(13)times 10^{-34} text{J}{cdot}text{s}/text{rad} = 6.582 119 514(40)times 10^{-16} text{eV}{cdot}text{s}/text{rad}.
The 2014 CODATA results were made available in June 2015WEB,weblink CODATA recommended values, and represent the best-known, internationally accepted values for these constants, based on all data published as of 31 December 2014. New CODATA figures are normally produced every four years. However, in order to support the redefinition of the SI base units, CODATA made a special release that was published in October 2017.JOURNAL, Newell, David B., Franco Cabiati, Joachim Fischer, Kenichi Fujii, Saveley G. Karshenboim, Helen S. Margolis, Estefania de Mirandes, Peter J. Mohr, Francois Nez, Krzysztof Pachucki, Terry J. Quinn, Barry N. Taylor, Meng Wang, Barry Wood, Zhonghua Zhang, 2017-10-20, The CODATA 2017 Values of h, e, k, and NA for the Revision of the SI, Metrologia, 55, 1, L13â€“L16, 10.1088/1681-7575/aa950a, 2018Metro..55L..13N, It incorporates all data up to 1 July 2017 and determines the final numerical values of the Planck constant, h, Elementary charge, e, Boltzmann constant, k, and Avogadro constant, NA, that are to be used for the new SI definitions.

## 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's constant could suggest some basic features of our physical world.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 typically of the order of kilojoules and times are typically of the order of seconds or minutes, the Planck constant (the quantum of action) 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 {{val|540|u=THz}} ({{val|540|e=12|ul=Hz}}). Each photon has an energy {{nowrap|1=E = hf = {{val|3.58|e=-19|u=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, {{nowrap|NA â‰ˆ {{val|6.022140758|(62)|e=23|u=molâˆ’1}}}}, with the result of {{val|216|u=kJ/mol}}, about the food energy in three apples.

## Dependent physical constants

There are several related constants for which more than 99% of the uncertainty in the 2014 CODATA valuesJOURNAL
, 1507.07956, 10.1103/RevModPhys.88.035009
, CODATA recommended values of the fundamental physical constants: 2014
, Reviews of Modern Physics
, 88
, 3
, 035009
, 2016
, Mohr
, Peter J.
, Newell
, David B.
, Taylor
, Barry N.
, 2016RvMP...88c5009M
, 10.1.1.150.1225
, is due to the uncertainty in the value of the Planck constant, as indicated by the square of the correlation coefficient ({{nowrap|r2 > 0.99, r > 0.995}}). The Planck constant is (with one or two exceptions){{NoteTag|The main exceptions are the Newtonian constant of gravitation G ({{nowrap|1=ur = {{val|4.7|e=âˆ’5}}}}) and the gas constant R ({{nowrap|1=ur = {{val|5.7|e=âˆ’7}}}}). 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.}} the fundamental physical constant which is known to the lowest level of precision, with a 1Ïƒ relative uncertainty ur of {{val|1.2|e=-8}}.

### Rest mass of the electron

The normal textbook derivation of the Rydberg constant Râˆž 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 varepsilon_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 ({{nowrap|1={{val|5.9|e=âˆ’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 computation of m_{rm e} becomes
m_{rm e} = frac{2 R_{infty} h}{c_0 alpha^2} ,
where c_0 is the speed of light and alpha 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 ({{nowrap|1=u_r = {{val|2.3|e=âˆ’10}}}}) than the Planck constant. Thus, the uncertainty in the value of the electron rest mass is due entirely to the uncertainty in the value of the Planck constant (r^2 > 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 A_r(e), which can be measured in a Penning trap (1=u_r = {{val|2.9|e=âˆ’11}}), multiplied by the molar mass constant M_u, which is defined as {{val|0.001|u=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 (r^2 > 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 Î± as:
alpha = frac{e^2}{hbar c_0 4 pi varepsilon_0} = frac{e^2 c_0 mu_0}{2 h} ,
where e is the elementary charge, varepsilon_0 is the electric constant (also called the permittivity of free space), and mu_0 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, alpha can also be determined experimentally, notably by measuring the electron spin g-factor g_e, 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 alpha to obtain the following definition of e in terms of alpha and h:
e = sqrt{frac{2alpha h}{mu_0 c_0}} = sqrt{{2alpha h varepsilon_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 h1/2 ({{nowrap|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 ({{nowrap|1=ur = {{val|9.5|e=âˆ’11}}}}).
mu_{rm N} = mu_{rm B} frac{A_{rm r}({rm e})}{A_{rm r}({rm p})}

## Determination

{{ambox |name=Update |type=content |class=ambox-Update |image=(File:Ambox current red.svg|42px) |date=September 2015|issue=This section is outdated; it does not include many more recent measurement results.|fix=Please update this section to include current measurements.}} {| class="wikitable" style="float:right; width:50%; margin: 1em 1em 1em 1em;"! Method! Value of h({{val|e=-34|u=J.s}})! Relativeuncertainty! Ref.
| Kibble (watt) balance
6.62606889|(23)}} {{vale=âˆ’8}}year=1990 journal=Metrologia issue=4 doi=10.1088/0026-1394/27/4/002 last1=Kibble last2=Robinson last3=Belliss last1=Steiner last2=Newell last3=Williams year=2005 url=http://nvl.nist.gov/pub/nistpubs/jres/110/1/j110-1ste.pdf weblink >dead-url=yes journal= Journal of Research of the National Institute of Standards and Technologyissue=1 doi=10.6028/jres.110.003pmc=4849564 }}{{citation title=Uncertainty Improvements of the NIST Electronic Kilogram issue=2 doi=10.1109/TIM.2007.890590 first1=Richard L. first2=Edwin R. first3=Ruimin first4=David B. url=https://zenodo.org/record/1232237 }}
| X-ray crystal density
6.6260745|(19)}} {{vale=âˆ’7}}year=2005 journal=IEEE Transactions on Instrumentation and Measurement issue=2 doi=10.1109/TIM.2004.843101 first1=K. first2=A. first3=N. first4=S. first5=P. first6=H. first7=A. first8=U. first9=S. first10=P. first11=P. first12=G. first13=E. first14=R. first15=E.G. first16=M.}}
| Josephson constant
6.6260678|(27)}} {{vale=âˆ’7}}last1=Sienknecht last2=Funck year=1985 journal= IEEE Transactions on Instrumentation and Measurementissue=2 doi=10.1109/TIM.1985.4315300 year=1986 journal=Metrologia issue=3 doi=10.1088/0026-1394/22/3/018 last1=Sienknecht last2=Funck year=1991 journal=IEEE Transactions on Instrumentation and Measurementissue=2 doi=10.1109/TIM.1990.1032905 first1=T. first2=V. }}{{citation first1=W. K. first2=G. J. first3=H. first4=M. F. first5=D. J. title=A Determination of the Volt Metrologia >volume=26 pages=9â€“46 bibcode=1989Metro..26....9C }}
| Magnetic resonance
6.6260724|(57)}} {{vale=âˆ’7}}year=1979 journal=Metrologia issue=1 doi=10.1088/0026-1394/15/1/002 last1=Kibble last2=Hunt author1=Liu Ruimin author3=Jin Tiruo author5=Du Xianhe author7=Kong Jingwen author9=Zhou Xianan author11=Zhang Wei title=A Recent Determination for the SI Values of Î³â€²p and 2e/h at NIM volume=16 url=http://en.cnki.com.cn/Article_en/CJFDTOTAL-JLXB503.000.htm |pages=161â€“68}}
6.6260657|(88)}} 1.3{{e|âˆ’6}}last1=Bower last2=Davis year=1980 journal = Journal of Research of the National Bureau of Standardsissue=3 url=http://cdm16009.contentdm.oclc.org/cdm/compoundobject/collection/p13011coll6/id/58310/rec/14 |doi=10.6028/jres.085.009 }}
| CODATA 2010
6.62606957|(29)}} 4.4{{e|âˆ’8}}| 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: weblink [Thursday, 02-Jun-2011 21:00:12 EDT]. National Institute of Standards and Technology, Gaithersburg, MD 20899.
|Kibble balance with superconducting magnet
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 Kibble balance measurements of KJ2RK and one inter-laboratory determination of the molar volume of silicon,{{CODATA2006|url=http://physics.nist.gov/cgi-bin/cuu/Value?h}} but is mostly determined by a 2007 Kibble balance measurement made at the U.S. National Institute of Standards and Technology (NIST). 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 Kibble 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 â€“ it is (or they are) not as precise as is currently believed â€“ 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 â€“ though there is no evidence at present to suggest they are â€“ 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. There are other statistical ways of testing the theories, and the theories have yet to be refuted.

### Josephson constant

The Josephson constant KJ relates the potential difference U generated by the Josephson effect at a "Josephson junction" with the frequency Î½ of the microwave radiation. The theoretical treatment of Josephson effect suggests very strongly that KJ = 2e/h.
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. Assuming the validity of the theoretical treatment of the Josephson effect, KJ is related to the Planck constant by
h = frac{8alpha}{mu_0 c_0 K_{rm J}^2}.

### Kibble balance

A Kibble balance (formerly known as a watt balance)WEB,weblink Kilogram: The Kibble Balance, Materese, Robin, 2018-05-14, NIST, 2018-11-13, en, 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 Î³ is the constant of proportionality between the frequency Î½ of nuclear magnetic resonance (or electron paramagnetic resonance for electrons) and the applied magnetic field {{nowrap|1=B: Î½ = Î³B}}. It is difficult to measure gyromagnetic ratios precisely because of the difficulties in precisely measuring B, but the value for protons in water at {{val|25|u=Â°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, Î³â€²p. The gyromagnetic ratio is related to the shielded proton magnetic moment Î¼â€²p, the spin number I ({{nowrap|1=I = {{frac|1|2}}}} 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 Î¼â€²p to the electron magnetic moment Î¼e 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 Î¼e 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 Î³â€²p 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 Î“â€²p-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 Î“â€²p-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 Î“â€²p-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})} .

### Particle accelerator

The experimental measurement of the Planck constant in the Large Hadron Collider laboratory was carried out in 2011. The study called PCC using a giant particle accelerator helped to better understand the relationships between the Planck constant and measuring distances in space.{{citation needed|date=October 2014}}

## 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 12 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 (denoted h90 to distinguish it from its value in SI units) is given by
h_{90} = frac{4}{K_{J-90}^2 R_{K-90}}

{{NoteFoot}}

## References

{{Reflist}}

### Sources

• {{Citation

| last = Barrow
| first = John D.
| author-link = John D. Barrow
| title = The Constants of Nature; From Alpha to Omega â€“ The Numbers that Encode the Deepest Secrets of the Universe
| year = 2002
| publisher = Pantheon Books
| location =
| isbn = 978-0-375-42221-8
}}

### Videos

{{Planck's natural units}}{{Scientists whose names are used in physical constants}}

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