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{{Infobox unit
| bgcolour =
| name = dalton<br />(unified atomic mass unit)
| image =
| caption =
| standard = [[Physical constant]]<br />([[Non-SI units mentioned in the SI|Accepted for use with the SI]])<!-- not a member of any system of units – see article Natural units for more info -->
| quantity = [[Mass|mass]]
| symbol = Da or u
| namedafter = John Dalton
| units1 = kg
| inunits1 = {{physconst|mu|unit=no|ref=no}}
| units2 = MeV/''c''<sup>2</sup>
| inunits2 = {{val|931.49410242|(28)}}
| units3 = ''m''<sub>e</sub>
| inunits3 = {{val|1822.888486209|(53)}}
}}
The '''dalton''' or '''unified atomic mass unit''' (SI symbols: '''Da''' or '''u''') is a [[Unit of measurement|unit]] of [[Mass|mass]] widely used in physics and chemistry. . It is approximately the mass of one [[Physics:Nucleon|nucleon]] (either a [[Physics:Proton|proton]] or [[Physics:Neutron|neutron]]). A mass of 1 Da is also referred to as the '''atomic mass constant''' and denoted by ''m''<sub>u</sub>.

Several definitions of this unit have been used, implying slightly different values.<ref name=chang2005/><ref name=kelt2008/>  The  current  IUPAC endorsed definition is the '''unified atomic mass unit''', denoted by the symbol '''u'''.  As of 2019, the [[International System of Units]] (SI) lists the '''dalton''', symbol '''Da''', as a unit acceptable for use with the SI unit system and secondarily notes that the dalton (Da) and the unified atomic mass unit (u) are alternative names (and symbols) for the same unit. The symbol Da is more widely used in most fields. It is defined precisely as 1/12 of the mass of an [[Chemistry:Chemical bond|unbound]] neutral atom of [[Physics:Carbon-12|carbon-12]] in its nuclear and electronic [[Physics:Ground state|ground state]] and [[Physics:Invariant mass|at rest]].<ref name=bipm9th/><ref name=goldbAtMaCo/>  Despite being an official abbreviation for a related obsolete unit and not widely used in the scientific literature, the abbreviation "amu" now often refers to the modern unit (Da or u) in many preparatory texts. As of June 2019, the value recommended by the [[Organization:Committee on Data for Science and Technology|Committee on Data for Science and Technology]] (CODATA) is {{physconst|mu|after=,}} or approximately 1.66 yoctograms.

This unit is commonly used in [[Physics|physics]] and [[Chemistry|chemistry]] to express the mass of atomic-scale objects, such as [[Physics:Atom|atom]]s, [[Physics:Molecule|molecule]]s, and [[Physics:Elementary particle|elementary particle]]s. For example, an atom of [[Physics:Helium|helium]] has a mass of about {{val|4|u=Da}}, and a molecule of [[Chemistry:Aspirin|acetylsalicylic acid (aspirin)]], {{chem|C|9|H|8|O|4}}, has a mass of about {{val|180.16|u=Da}}.  In general, the [[Standard atomic weight|standard atomic weight]] of an element is the average weight of its atom as it occurs in nature, expressed in daltons.

The [[Molecular mass|molecular mass]]es of [[Biology:Protein|protein]]s, [[Biology:Nucleic acid|nucleic acid]]s, and other large [[Chemistry:Polymer|polymer]]s are often expressed with the units '''kilodalton''' (kDa), equal to 1000 daltons, '''megadalton''' (MDa), one million daltons, etc.<ref name=berg2007p35/> [[Biology:Titin|Titin]], one of the largest known [[Biology:Protein|protein]]s, has an atomic mass of between 3 and 3.7 megadaltons.<ref name=opitz2003/> The DNA of chromosome 1 in the [[Biology:Human genome|human genome]] has about 249 million [[Biology:Base pair|base pair]]s, each with an average mass of about {{val|650|u=Da}}, or {{val|156|u=GDa}} total.<ref name=molfat/>

The [[Mole (unit)|mole]] is a unit of [[Amount of substance|amount of substance]], widely used in chemistry and physics, which was originally defined so that the mass of one mole of a substance, measured in grams, would be numerically equal to the average mass of one of its constituent particles, measured in daltons. That is, the [[Physics:Molar mass|molar mass]] of a chemical compound was meant to be numerically equal to its average molecular mass.  For example, the average mass of one molecule of [[Chemistry:Water|water]] is about 18.0153 daltons, and one mole of water is about 18.0153 grams.  A protein whose molecule has an average mass of 64 kDa would have a molar mass of {{val|64|u=kg/mol}}. However, while this equality can be assumed for almost all practical purposes, it is now only approximate, because of the way the mole was redefined on 20 May 2019.<ref name=berg2007p35/><ref name=bipm9th/>

The mass in daltons of an atom is numerically very close to the number of nucleons ''A'' in its [[Physics:Atomic nucleus|atomic nucleus]].  It follows that the molar mass of a compound (grams per mole) is also numerically close to the average number of nucleons per molecule.  However, the mass of an atomic-scale object is affected by the [[Physics:Binding energy|binding energy]] of the nucleons in its atomic nuclei, as well as the mass and binding energy of the [[Physics:Electron|electron]]s.  Therefore, this equality holds only for the carbon-12 atom in the stated conditions, and will vary for other substances. For example, the mass of one unbound atom of the common [[Physics:Hydrogen|hydrogen]] [[Physics:Isotope|isotope]] ([[Physics:Hydrogen-1|hydrogen-1]], protium) is {{val|1.007825032241|(94)|u=Da}}, the mass of one free neutron is {{val|1.008664915823|(491)|u=Da}}, and the mass of one [[Physics:Hydrogen-2|hydrogen-2]] (deuterium) atom is {{val|2.014101778114|(122)|u=Da}}.<ref name=wang2017ii/>  In general, the difference (mass defect) is less than 0.1%; except for hydrogen (about 0.8%), [[Physics:Helium-3|helium-3]] (0.5%), [[Chemistry:Lithium|lithium]] (0.25%) and [[Astronomy:Beryllium|beryllium]] (0.15%).

The atomic mass unit should not be confused with unit of mass in the atomic units systems, which is instead the [[Physics:Electron rest mass|electron rest mass]] (''m''<sub>e</sub>).

==Energy equivalents==
The atomic mass constant can also be expressed as its [[Mass–energy equivalence|energy equivalent]], that is ''m''<sub>u</sub>''c''<sup>2</sup>. The CODATA recommended values are:
:{{physconst|muc2}}
:{{physconst|mueV}}
The [[Electronvolt|megaelectronvolt]] (MeV) is commonly used as a unit of mass in [[Physics:Particle physics|particle physics]], and these values are also important for the practical determination of relative atomic masses.

==History==

===Origin of the concept===

[[File:Jean Perrin 1926.jpg|right|thumb|Jean Perrin in 1926]]
The interpretation of the [[Law of definite proportions|law of definite proportions]] in term of the atomic theory of matter implied that the masses of atoms of various elements had definite ratios that depended on the elements.  While the actual masses were unknown, the relative masses could be deduced from that law.  In 1803 John Dalton proposed to use the (still unknown) atomic mass of the lightest atom, that of hydrogen, as the natural unit of atomic mass.  This was the basis of the [[Standard atomic weight|atomic weight scale]].<ref name=petley1989/>

For technical reasons, in 1898, chemist [[Biography:Wilhelm Ostwald|Wilhelm Ostwald]] and others proposed to redefine the unit of atomic mass as 1/16 of the mass of an oxygen atom.<ref name=hold2004/>  That proposal was formally adopted by the [[Organization:Commission on Isotopic Abundances and Atomic Weights|International Committee on Atomic Weights]] (ICAW) in 1903. That was approximately the mass of one hydrogen atom, but oxygen was more amenable to experimental determination.  This suggestion was made before the discovery of the existence of elemental isotopes, which occurred in 1912.<ref name=petley1989/>  The same definition was adopted in 1909 the physicist [[Biography:Jean Baptiste Perrin|Jean Perrin]] in his extensive experiments to determine the amu.<ref name=perrin1909/>  This definition remained unchanged until 1961.<ref name=chang2005/><ref name=kelt2008/> Perrin also defined the "mole" as an amount of a compound that contained as many molecules as 32 grams of oxygen ({{chem|O|2}}).  He called that number the Avogadro number in honor of physicist [[Biography:Amedeo Avogadro|Amedeo Avogadro]].

===Isotopic variation===
The discovery of isotopes of oxygen in 1929 required a more precise of the unit.  Unfortunately, two distinct definitions came into use.  Chemists choose to define the amu as 1/16 of the average mass of an oxygen atom as found in nature; that is, the average of the masses of the known isotopes, weighted by their natural abundance.   Physicists, on the other hand, defined it as 1/16 of the mass of an atom of the isotope oxygen-16 (<sup>16</sup>O).<ref name=hold2004/>

===Definition by the IUPAC===

The existence of two distinct units with the same name was confusing, and the difference (about {{val|1.000282}} in relative terms) was large enough to affect high-precision measurements.  Moreover, it was discovered that the isotopes of oxygen had different natural abundances in water and in air.   For these and other reasons, in 1961 the [[Organization:International Union of Pure and Applied Chemistry|International Union of Pure and Applied Chemistry]] (IUPAC), which had absorbed the ICAW, adopted a new definition of the atomic mass unit for use in both physics and chemistry; namely, 1/12 of the mass of a carbon-12 atom. This new value was intermediate between the two earlier definitions, but closer to the one used by chemists (who would be affected the most by the change).<ref name=petley1989/><ref name=hold2004/>

The new unit was named the "unified atomic mass unit" and given a new symbol "u", to replace the old "amu" that had been used for the oxygen-based units.<ref name=goldbUnAtMaUn/>  However, the old symbol "amu" has often been used, after 1961, to refer to the new unit.

With this new definition, the [[Standard atomic weight|standard atomic weight]] of [[Chemistry:Carbon|carbon]] is approximately {{val|12.011|u=Da}}, and that of oxygen is approximately {{val|15.999|u=Da}}. These values, generally used in chemistry, are based on averages of many samples from [[Earth:Earth's crust|Earth's crust]], its [[Astronomy:Atmosphere|atmosphere]], and organic materials.

===Adoption by the BIPM===

The IUPAC 1961 definition of the unified atomic mass unit, with that name and symbol "u", was adopted by the International Bureau for Weights and Measures (BIPM) in 1971 as a "non-SI unit accepted for use with the SI".<ref name=bipm1971/>

The definition was not affected by the 2019 redefinition of SI base units;<ref name=cipm106/><ref name=cgpm26/><ref name=bipm9th/> that is, 1&nbsp;u in the SI is still 1/12 of the mass of a carbon-12 atom, a quantity that must be determined experimentally. However, the definition of the mole was changed to be "exactly ''N'' particles", where ''N'', Avogadro's number, was redefined to be exactly {{physconst|NA|unit=no|ref=no}}; and the definition of the kilogram was changed too.  As a consequence, the [[Molar mass constant|molar mass constant]] is no longer exactly 1&nbsp;g/mol, meaning that the mass of one mole of any substance, in grams, is no longer numerically equal to its molecular weight, in unified atomic mass units.  In particular, the mass of one mole of atoms of carbon-12 is no longer exactly 12 grams by definition, but exactly 12''N''&nbsp;u; whose value in grams must be determined experimentally.<ref name=pave2018/><ref name=lehm2016/>  However, the difference ({{val|3.5|e=-10}} in relative terms, as determined from the 2018 CODATA value{{physconst|Mu|ref=only}}) is insignificant for practical purposes.

As of the redefinition, the relative uncertainty of the atomic mass unit is {{physconst|mu|runc=yes|ref=yes}}, which is almost entirely due to the uncertainty in the fine-structure constant. 
===The dalton===
In 1993, the IUPAC proposed the shorter name "dalton" (with symbol "Da") for the unified atomic mass unit.<ref name=mills1993/><ref name=goldbDa/> As with other unit names such as watt and newton, "dalton" is not capitalized in English, but its symbol, "Da", is capitalized.  The name was endorsed by the [[Organization:International Union of Pure and Applied Physics|International Union of Pure and Applied Physics]] (IUPAP) in 2005.<ref name=iupap2005/>

In 2003 the name was recommended to the BIPM by the Consultative Committee for Units, part of the CIPM, as it "is shorter and works better with [the SI] prefixes".<ref name=bipm-ccu15/> In 2006, the BIPM included the dalton in its 8th edition of the formal definition of SI.<ref name=bipm8th/>  The name was also listed as an alternative to "unified atomic mass unit" by the [[Organization:International Organization for Standardization|International Organization for Standardization]] in 2009.<ref name=ISO1/><ref name=ISO10/>  It is now recommended by several scientific publishers,<ref name=oxsty/> and some of them consider "atomic mass unit" and "amu" deprecated.<ref name=rapsty/>  In 2019, the BIPM retained the dalton in its 9th edition of the formal definition of SI while dropping the unified atomic mass unit from its table of non-SI units accepted for use with the SI.<ref name=bipm9th/>

===New proposal===
A proposal was made in 2012 to redefine the dalton (and presumably the unified atomic mass unit) as being 1/''N'' grams, thereby breaking the link with <sup>12</sup>C.  This would imply changes in the atomic masses of all elements, when expressed in daltons; but the change would be too small to have practical effects.<ref name=leon2012/>

===Legal implications===

The US Supreme Court established a legal precedent of appellate law in a case (Teva Pharmaceuticals v. Sandoz) that hinged upon the interpretation of the term "[[Molecular weight|molecular weight]]".<ref name=teva2014/><!-- Should go to [[Molecular weight|molecular weight]]-->

==Measurement==
{{editnotice|textstyle=background-color:#ffa|text=this section may need revision to better reflect the 2019 SI redefinition of mole}}

Although relative atomic masses are defined for neutral atoms, they are measured (by [[Mass spectrometry|mass spectrometry]]) for ions: hence, the measured values must be correct for the mass of the electrons that were removed to form the ions, and also for the mass equivalent of the electron binding energy, ''E''<sub>b</sub>/''m''<sub>u</sub>''c''<sup>2</sup>. The total binding energy of the six electrons in a carbon-12 atom is 1030.1089&nbsp;eV&nbsp;= {{nowrap|1.650 4163{{e|−16}} J}}: ''E''<sub>b</sub>/''m''<sub>u</sub>''c''<sup>2</sup>&nbsp;= {{nowrap|1.105 8674{{e|−6}}}}, or about one part in 10&nbsp;million of the mass of the atom.<ref>{{CODATA2002}}</ref>

===Josef Loschmidt===
[[File:Johann Josef Loschmidt portrait plaque.jpg|right|thumb|Josef Loschmidt]]
A reasonably accurate value of the atomic mass unit was first obtained indirectly by Josef Loschmidt in 1865, by estimating the number of particles in a given volume of gas.<ref name=losch1865/>

===Jean Perrin===
Until May 20, 2019, the Avogadro number ''N'' was defined as the number of atoms in ''k'' grams of a reference element (oxygen or carbon-12) whose atoms had, by definition, ''k'' atomic mass units.  Therefore, experiments that aimed to determine of ''N'' were essentially determinations of the atomic mass unit.  Perrin himself estimated the Avogadro number by a variety of methods, at the turn of the 20th century. He was awarded the 1926 Nobel Prize in Physics, largely for this work.<ref name=oseen1926/>

===Coulometry===
For a while, the most accurate measurements of the atomic mass unit were based on [[Physics:Coulometry|coulometry]].  The electric charge per [[Mole (unit)|mole]] of electrons is a constant called the [[Physics:Faraday constant|Faraday constant]], whose value had been essentially known since 1834 when [[Biography:Michael Faraday|Michael Faraday]] published [[Physics:Faraday's laws of electrolysis|his works on electrolysis]]. In 1910, Robert Millikan obtained the first measurement of the [[Physics:Elementary charge|charge on an electron]],  ''e''.  The quotient ''F''/''e'' provided an estimate of Avogadro's number<ref name=ebrit1974/>

The classic experiment is that of Bower and Davis at NIST,<ref>This account is based on the review in {{CODATA1998}}</ref> and relies on dissolving [[Chemistry:Silver|silver]] metal away from the [[Physics:Anode|anode]] of an [[Physics:Electrolysis|electrolysis]] cell, while passing a constant [[Electric current|electric current]] ''I'' for a known time ''t''. If ''m'' is the mass of silver lost from the anode and ''A''{{sub|r}} the atomic weight of silver, then the Faraday constant is given by:
:<math>F = \frac{A_{\rm r}M_{\rm u}It}{m}.</math>
The NIST scientists devised a method to compensate for silver lost from the anode by mechanical causes, and conducted an [[Physics:Isotope analysis|isotope analysis]] of the silver used to determine its atomic weight. Their value for the conventional Faraday constant was ''F''{{sub|90}}&nbsp;= {{val|96485.39|(13)|u=C/mol}}, which corresponds to a value for the Avogadro constant of {{val|6.0221449|(78)|e=23|u=mol-1}}: both values have a relative standard uncertainty of {{val|1.3|e=-6}}.

===Electron mass measurement===
In practice, the atomic mass constant is determined from the [[Physics:Electron rest mass|electron rest mass]] ''m''<sub>e</sub> and the electron relative atomic mass ''A''<sub>r</sub>(e) (that is, the mass of the electron on a scale where {{nowrap|1={{sup|12}}C = 12}}).<ref>{{CODATA1998}}</ref> The relative atomic mass of the electron can be measured in [[Physics:Cyclotron|cyclotron]] experiments, while the rest mass of the electron can be derived from other physical constants.
:<math>m_{\rm u} = \frac{m_{\rm e}}{A_{\rm r}({\rm e})} = \frac{2R_\infty h}{A_{\rm r}({\rm e})c\alpha^2} ,</math>
where ''c'' is the [[Physics:Speed of light|speed of light]], ''h'' is the [[Physics:Planck constant|Planck constant]], ''α'' is the [[Fine-structure constant|fine-structure constant]], and ''R''<sub>∞</sub> is the Rydberg constant.

As may be observed in the table below, the main limiting factor in the precision of the Avogadro constant is the uncertainty in the value of the [[Physics:Planck constant|Planck constant]], as all the other constants that contribute to the calculation are known more precisely.

{| class="wikitable"
|-
! Constant
! Symbol
! 2014 CODATA value
! Relative standard uncertainty
! Correlation coefficient<br />with ''N''{{sub|A}}
|-
| Proton–electron mass ratio
| align="center"|''m''{{sub|p}}/''m''{{sub|e}}
| 1836.152 673 89(17)
| align="center"|9.5{{e|–11}}
| −0.0003
|-
| [[Molar mass constant]]
| align="center"|''M''{{sub|u}}
| 0.001&nbsp;kg/mol = 1 g/mol
| align="center"|0 (defined)
|&nbsp;—
|-
| Rydberg constant
| align="center"|''R''{{sub|∞}}
| 10 973 731.568 508(65) m<sup>−1</sup>
| align="center"|5.9{{e|–12}}
| −0.0002
|-
| [[Physics:Planck constant|Planck constant]]
| align="center"|''h''
| 6.626 070 040(81){{e|–34}} J s
| align="center"|1.2{{e|–8}}
| −0.9993
|-
| [[Physics:Speed of light|Speed of light]]
| align="center"|''c''
| 299 792 458&nbsp;m/s
| align="center"|0 (defined)
|&nbsp;—
|-
| Fine structure constant
| align="center"|''α''
| 7.297 352 5664(17){{e|–3}}
| align="center"|2.3{{e|–10}}
| 0.0193
|-
| '''Avogadro constant'''
| align="center"|''N''{{sub|A}}
| 6.022 140 857(74){{e|23}} mol<sup>−1</sup>
| align="center"|1.2{{e|–8}}
| 1
|-
|}

===X-ray crystal density methods===
[[Image:Silicon-unit-cell-labelled-3D-balls.png|thumb|right|200px|[[Ball-and-stick model]] of the unit cell of [[Chemistry:Silicon|silicon]]. X-ray diffraction measures the cell parameter, ''a'', which is used to calculate a value for the Avogadro constant.]]
A modern method to determine the Avogadro constant is the use of [[Physics:X-ray crystallography|X-ray crystallography]]. [[Chemistry:Silicon|Silicon]] single crystals may be produced today in commercial facilities with extremely high purity and with few lattice defects. This method defines the Avogadro constant as the ratio of the [[Molar volume|molar volume]], ''V''{{sub|m}}, to the atomic volume ''V''<sub>atom</sub>:<br />
:<math>N_{\rm A}  =  \frac{V_{\rm m}}{V_{\rm atom}}</math>, where <math>V_{\rm atom}  =  \frac{V_{\rm cell}}{n}</math> and ''n'' is the number of atoms per unit cell of volume ''V''<sub>cell</sub>.

The unit cell of silicon has a cubic packing arrangement of 8 atoms, and the unit cell volume may be measured by determining a single unit cell parameter, the length of one of the sides of the cube, ''a''.<ref>{{cite web
|url=http://webmineral.com/help/CellDimensions.shtml
|accessdate=2007-12-09
|title=Unit Cell Formula
|last=Mineralogy Database
|date=2000–2005
}}</ref>

In practice, measurements are carried out on a distance known as ''d''{{sub|220}}(Si), which is the distance between the planes denoted by the [[Miller index|Miller indices]] {220}, and is equal to ''a''/{{radic|8}}. The 2006 CODATA value for ''d''{{sub|220}}(Si) is {{val|192.0155762|(50)|u=pm}}, a relative standard uncertainty of {{val|2.8|e=-8}}, corresponding to a unit cell volume of {{val|1.60193304|(13)|e=-28|u=m<sup>3</sup>}}.

The [[Physics:Isotope|isotope]] proportional composition of the sample used must be measured and taken into account. Silicon occurs in three stable isotopes (<sup>28</sup>Si, <sup>29</sup>Si, <sup>30</sup>Si), and the natural variation in their proportions is greater than other uncertainties in the measurements. The atomic weight ''A''{{sub|r}} for the sample crystal can be calculated, as the [[Standard atomic weight|standard atomic weight]]s of the three [[Physics:Nuclide|nuclide]]s are known with great accuracy. This, together with the measured [[Physics:Density|density]] ''ρ'' of the sample, allows the molar volume ''V''{{sub|m}} to be determined:
:<math>V_{\rm m} = \frac{A_{\rm r}M_{\rm u}}{\rho}</math>
where ''M''{{sub|u}} is the molar mass constant. The 2006 CODATA value for the molar volume of silicon is {{val|12.0588349|(11)|u=cm<sup>3</sup>⋅mol<sup>−1</sup>}}, with a relative standard uncertainty of {{val|9.1|e=-8}}.<ref>{{CODATA2006}}</ref>

As of the 2006 CODATA recommended values, the relative uncertainty in determinations of the Avogadro constant by the X-ray crystal density method is {{val|1.2|e=-7}}, about two and a half times higher than that of the electron mass method.

====International Avogadro Coordination====
[[File:Silicon sphere for Avogadro project.jpg|thumb|right|[[Achim Leistner]] at the Australian Centre for Precision Optics (ACPO) holding a one-kilogram single-crystal silicon sphere prepared for the International Avogadro Coordination.]]
The International Avogadro Coordination (IAC), often simply called the "Avogadro project", is a collaboration begun in the early 1990s between various national metrology institutes to measure the Avogadro constant by the X-ray crystal density method to a relative uncertainty of {{val|2|e=−8}} or less.<ref>{{cite web | title = Avogadro Project | url = http://www.npl.co.uk/engineering-measurements/mass-force-pressure/mass/research/avogadro-project | publisher = National Physical Laboratory | accessdate = 2010-08-19 | archive-url = https://web.archive.org/web/20110202120746/http://www.npl.co.uk/engineering-measurements/mass-force-pressure/mass/research/avogadro-project | archive-date = 2011-02-02 | dead-url = yes }}</ref> The project was one of the [[Alternative approaches to redefining the kilogram|alternative approaches to redefining the kilogram]] in terms of a universal [[Physical constant|physical constant]], rather than the International Prototype Kilogram, and complemented the measurements of the [[Physics:Planck constant|Planck constant]] using [[Kibble balance]]s.<ref name="Leonard">{{cite journal| first = B. P. | last = Leonard | title = On the role of the Avogadro constant in redefining SI units for mass and amount of substance | journal = Metrologia | volume = 44 | issue = 1 | pages = 82–86 | year = 2007 | doi = 10.1088/0026-1394/44/1/012|bibcode = 2007Metro..44...82L }}</ref><ref name="Jabbour">{{cite journal| first = Zeina J. | last = Jabbour | title = Getting Closer to Redefining The Kilogram | url = https://www.nist.gov/customcf/get_pdf.cfm?pub_id=903635 | journal = Weighing & Measurement Magazine | year = 2009 | issue = October | pages = 24–26}}</ref> Prior to the 2019 redefinition of the SI base units, a measurement of the Avogadro constant was an indirect measurement of the Planck constant:
:<math>h = \frac{c\alpha^2 A_{\rm r}({\rm e})M_{\rm u}}{2R_{\infty} N_{\rm A}}.</math>

The measurements use highly polished spheres of silicon with a mass of one kilogram. Spheres are used to simplify the measurement of the size (and hence the density) and to minimize the effect of the oxide coating that inevitably forms on the surface. The first measurements used spheres of silicon with natural isotopic composition, and had a relative uncertainty of 3.1{{e|−7}}.<ref>{{cite journal| first = Peter | last = Becker | year = 2003 | title = Tracing the definition of the kilogram to the Avogadro constant using a silicon single crystal | journal = Metrologia | volume = 40 | issue = 6 | pages = 366–75 | doi = 10.1088/0026-1394/40/6/008|bibcode = 2003Metro..40..366B }}</ref><ref>{{cite journal| author=Fujii, K. |title = Present State of the Avogadro Constant Determination From Silicon Crystals With Natural Isotopic Compositions | journal = IEEE Trans. Instrum. Meas. | volume = 54 | issue = 2 | pages = 854–59 | year = 2005 | doi = 10.1109/TIM.2004.843101|display-authors=etal}}</ref><ref>{{cite journal| last = Williams | first = E. R. | year = 2007 |title = Toward the SI System Based on Fundamental Constants: Weighing the Electron | journal = IEEE Trans. Instrum. Meas. | volume = 56 | issue = 2 | pages = 646–50 | doi = 10.1109/TIM.2007.890591| url = https://zenodo.org/record/1232251 }}</ref> These first results were also inconsistent with values of the Planck constant derived from Kibble balance measurements, although the source of the discrepancy is now believed to be known.<ref name="Jabbour"/>

The main residual uncertainty in the early measurements was in the measurement of the isotopic composition of the silicon to calculate the atomic weight, so in 2007 a 4.8&nbsp;kg single crystal of isotopically-enriched silicon (99.94% <sup>28</sup>Si) was grown,<ref>{{cite journal| author=Becker, P.| title = Large-scale production of highly enriched 28Si for the precise determination of the Avogadro constant | journal = Meas. Sci. Technol. | volume = 17 | issue = 7 | pages = 1854–60 | year = 2006 | doi =10.1088/0957-0233/17/7/025|bibcode = 2006MeScT..17.1854B |display-authors=etal| url =https://www.physik.tu-cottbus.de/physik/xp2/newhp/fileadmin/Lehre/WS0809/AllgPhII/Avogadro-Projekt.pdf| citeseerx = 10.1.1.1026.3626 }}</ref><ref>{{cite journal | title= High purity single crystal mono-isotopic silicon-28 for improved determination of the Avogadro number | author= Devyatykh, G. G.| journal =Doklady Akademii Nauk | volume = 421 | issue = 1 | pages = 61–64 | year = 2008|display-authors=etal}}</ref><ref>{{cite journal|last1=Devyatykh|first1=G. G.|last2=Bulanov|first2=A. D.|last3=Gusev|first3=A. V.|last4=Kovalev|first4=I. D.|last5=Krylov|first5=V. A.|last6=Potapov|first6=A. M.|last7=Sennikov|first7=P. G.|last8=Adamchik|first8=S. A.|last9=Gavva|first9=V. A.|last10=Kotkov|first10=A. P.|last11=Churbanov|first11=M. F.|last12=Dianov|first12=E. M.|last13=Kaliteevskii|first13=A. K.|last14=Godisov|first14=O. N.|last15=Pohl|first15=H. -J.|last16=Becker|first16=P.|last17=Riemann|first17=H.|last18=Abrosimov|first18=N. V.|title=High-purity single-crystal monoisotopic silicon-28 for precise determination of Avogadro's number|journal=Doklady Chemistry|date=23 July 2008|volume=421|issue=1|pages=157–160|doi=10.1134/S001250080807001X}}</ref> and two one-kilogram spheres cut from it. Diameter measurements on the spheres are repeatable to within 0.3&nbsp;nm, and the uncertainty in the mass is 3&nbsp;µg. Full results from these determinations were expected in late 2010.<ref>{{cite web | title = Report of the 11th meeting of the Consultative Committee for Mass and Related Quantities (CCM) | url = http://www.bipm.org/utils/common/pdf/CCM11.html | publisher = International Bureau of Weights and Measures | year = 2008 | page = 17}}</ref>
Their paper, published in January 2011, summarized the result of the International Avogadro Coordination and presented a measurement of the Avogadro constant to be {{val|6.02214078|(18)|e=23}} mol<sup>−1</sup>.<ref>{{cite journal| author= Andreas, B.| title = An accurate determination of the Avogadro constant by counting the atoms in a <sup>28</sup>Si crystal | journal = Phys. Rev. Lett. | volume = 106 | issue = 3 | page = 030801 (4 pages) | year = 2011 | doi = 10.1103/PhysRevLett.106.030801 | pmid = 21405263 | bibcode=2011PhRvL.106c0801A|arxiv = 1010.2317 |display-authors=etal}}</ref>

== See also ==
* [[Physics:Mass (mass spectrometry)|Mass (mass spectrometry)]]
** [[Kendrick mass]]
** [[Physics:Monoisotopic mass|Monoisotopic mass]]
* [[Mass-to-charge ratio]]

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==External links==
*[http://physics.nist.gov/PhysRefData/Compositions/ Atomic weights and isotopic compositions]
* [https://web.archive.org/web/20080115211624/http://www.sizes.com/units/atomic_mass_unit.htm atomic mass unit] at sizes.com

{{SI units|state=collapsed}}

[[Category:Metrology]][[Category:Nuclear chemistry]][[Category:Units of chemical measurement]][[Category:Units of mass]]

{{Sourceattribution|Atomic mass unit|1}}