Editing Dalton (unit) (section)

== Measurement ==
Though relative atomic masses are defined for neutral atoms, they are measured (by [[mass spectrometry]]) for ions: hence, the measured values must be corrected 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}}/''m''{{sub|u}}''c''{{sup|2}}. The total binding energy of the six electrons in a carbon-12 atom is {{val|1030.1089|u=eV}}&nbsp;= {{val|1.6504163|e=−16|u=J}}: ''E''<sub>b</sub>/''m''<sub>u</sub>''c''<sup>2</sup>&nbsp;= {{val|1.1058674|e=−6}}, or about one part in 10&nbsp;million of the mass of the atom.<ref>{{cite journal |author-first1=Peter J. |author-last1=Mohr |author-first2=Barry N. |author-last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 2002 |url=https://www.nist.gov/pml/div684/fcdc/upload/rmp2002-2.pdf |journal=[[Reviews of Modern Physics]] |volume=77 |issue=1 |pages=1–107 |bibcode=2005RvMP...77....1M |doi=10.1103/RevModPhys.77.1 |year=2005 |archive-url=https://web.archive.org/web/20171001121924/https://www.nist.gov/sites/default/files/documents/pml/div684/fcdc/rmp2002-2.pdf|archive-date=2017-10-01}}</ref>

Before the 2019 revision of the SI, experiments were aimed to determine the value of the [[Avogadro constant]] for finding the value of the unified atomic mass unit.

=== 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 [[Johann Josef Loschmidt|Josef Loschmidt]] in 1865, by estimating the number of particles in a given volume of gas.<ref name=losch1865/>

=== Jean Perrin ===
Perrin 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 ===
{{main|Coulometry}}
The electric charge per [[mole (unit)|mole]] of [[elementary charge]]s is a constant called the [[Faraday constant]], ''F'', whose value had been essentially known since 1834 when [[Michael Faraday]] published [[Faraday's laws of electrolysis|his works on electrolysis]]. In 1910, [[Robert Millikan]] obtained the first measurement of the charge on an electron, −''e''.  The quotient ''F''/''e'' provided an estimate of the Avogadro constant.<ref name=ebrit1974/>

The classic experiment is that of Bower and Davis at [[NIST]],<ref>This account is based on the review in {{cite journal |author-first1=Peter J. |author-last1=Mohr |author-first2=Barry N. |author-last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 1998 |journal=[[Journal of Physical and Chemical Reference Data]] |volume=28 |issue=6 |pages=1713–1852 |bibcode=1999JPCRD..28.1713M |doi=10.1063/1.556049 |year=1999 |url=https://www.nist.gov/pml/div684/fcdc/upload/rmp1998-2.pdf |archive-url=https://web.archive.org/web/20171001122752/https://www.nist.gov/sites/default/files/documents/pml/div684/fcdc/rmp1998-2.pdf|archive-date=2017-10-01}}</ref> and relies on dissolving [[silver]] metal away from the [[anode]] of an [[electrolysis]] cell, while passing a constant [[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:
{{block indent|<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 [[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|up=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 [[electron rest mass]] ''m''{{sub|e}} and the [[electron relative atomic mass]] ''A''{{sub|r}}(e) (that is, the mass of electron divided by the atomic mass constant).<ref>{{cite journal |author-first1=Peter J. |author-last1=Mohr |author-first2=Barry N. |author-last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 1998 |journal=[[Journal of Physical and Chemical Reference Data]] |volume=28 |issue=6 |pages=1713–1852 |bibcode=1999JPCRD..28.1713M |doi=10.1063/1.556049 |year=1999 |url=https://www.nist.gov/pml/div684/fcdc/upload/rmp1998-2.pdf |archive-url=https://web.archive.org/web/20171001122752/https://www.nist.gov/sites/default/files/documents/pml/div684/fcdc/rmp1998-2.pdf|archive-date=2017-10-01}}</ref> The relative atomic mass of the electron can be measured in [[cyclotron]] experiments, while the rest mass of the electron can be derived from other physical constants.
{{block indent|<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>}}
{{block indent|<math>m_{\rm u} = \frac{M_{\rm u}}{N_{\rm A}} ,</math>}}
{{block indent|<math>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\alpha^2}{2R_\infty h} ,</math>}}
where ''c'' is the [[speed of light]], ''h'' is the [[Planck constant]], ''α'' is the [[fine-structure constant]], and ''R''{{sub|∞}} is the [[Rydberg constant]].

As may be observed from the old values (2014 CODATA) in the table below, the main limiting factor in the precision of the Avogadro constant was the uncertainty in the value of the [[Planck constant]], as all the other constants that contribute to the calculation were known more precisely.
{| class="wikitable"
|- style="line-height:133%"
! Constant
! Symbol
! 2014 [[Committee on Data for Science and Technology|CODATA]] values
! Relative{{br}}standard{{br}}uncertainty
! Correlation{{br}}coefficient{{br}}with ''N''{{sub|A}}
|-
| [[Proton–electron mass ratio]]
| align="center" |''m''{{sub|p}}/''m''{{sub|e}}
| {{val|1836.15267389|(17)}}
| align="center" |{{val|9.5|e=-11}}
| {{val|−0.0003}}
|-
| [[Molar mass constant]]
| align="center" |''M''{{sub|u}}
| 1 g/mol
| align="center" |'''0''' (defined)
| —
|-
| [[Rydberg constant]]
| align="center" |''R''{{sub|∞}}
| {{val|10973731.568508|(65)|u=m-1}}
| align="center" |{{val|5.9|e=-12}}
| −0.0002
|-
| [[Planck constant]]
| align="center" |''h''
| {{val|6.626070040|(81)|e=-34|u=J.s}}
| align="center" |{{val|1.2|e=-8}} 
| −0.9993
|-
| [[Speed of light]]
| align="center" |''c''
| {{val|299792458|u=m|up=s}}
| align="center" |'''0''' (defined)
| —
|-
| [[Fine structure constant]]
| align="center" |''α''
| {{val|7.2973525664|(17)|e=-3}}
| align="center" |{{val|2.3|e=-10}}
| 0.0193
|-
| [[Avogadro constant]]
| align="center" |''N''{{sub|A}}
| {{val|6.022140857|(74)|e=23|u=mol-1}}
| align="center" |{{val|1.2|e=-8}} 
| 1
|-
|}

The power of having defined values of [[universal constant]]s as is presently the case can be understood from the table below (2018 CODATA).

{| class="wikitable"
|- style="line-height:133%"
! Constant
! Symbol
! 2018 [[Committee on Data for Science and Technology|CODATA]] values<ref>{{Cite web|url=https://physics.nist.gov/cuu/Constants/bibliography.html|title=Constants bibliography, source of the CODATA internationally recommended values|website=The NIST Reference on Constants, Units, and Uncertainty|access-date=4 August 2021}}</ref>
! Relative{{br}}standard{{br}}uncertainty
! Correlation{{br}}coefficient{{br}}with ''N''{{sub|A}}
|-
| [[Proton–electron mass ratio]]
| align="center" |''m''{{sub|p}}/''m''{{sub|e}}
| {{val|1836.15267343|(11)}}
| align="center" |{{val|6.0|e=-11}}
| —
|-
| [[Molar mass constant]]
| align="center" |''M''{{sub|u}}
| {{val|0.99999999965|(30)|u=g|up=mol}}
| align="center" |{{val|3.0|e=-10}}
| —
|-
| [[Rydberg constant]]
| align="center" |''R''{{sub|∞}}
| {{val|10973731.568160|(21)|u=m-1}}
| align="center" |{{val|1.9|e=-12}}
| —
|-
| [[Planck constant]]
| align="center" |''h''
| {{val|6.62607015|e=-34|u=J.s}}
| align="center" |'''0''' (defined)
| —
|-
| [[Speed of light]]
| align="center" |''c''
| {{val|299792458|u=m|up=s}}
| align="center" |'''0''' (defined)
| —
|-
| [[Fine structure constant]]
| align="center" |''α''
| {{val|7.2973525693|(11)|e=-3}}
| align="center" |{{val|1.5|e=-10}}
| —
|-
| [[Avogadro constant]]
| align="center" |''N''{{sub|A}}
| {{val|6.02214076|e=23|u=mol-1}}
| align="center" |'''0''' (defined)
| —
|-
|}

=== 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 [[silicon]]. X-ray diffraction measures the cell parameter, ''a'', which is used to calculate a value for the Avogadro constant.]]
[[Silicon]] single crystals may be produced today in commercial facilities with extremely high purity and with few lattice defects. This method defined the Avogadro constant as the ratio of the [[molar volume]], ''V''{{sub|m}}, to the atomic volume ''V''{{sub|atom}}:
<math display=block>N_{\rm A}  =  \frac{V_{\rm m}}{V_{\rm atom}},</math>
where {{nowrap|1= ''V''{{sub|atom}} = {{sfrac|''V''{{sub|cell}}|''n''}}}} 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 ''a'' of one of the sides of the cube.<ref>{{cite web|url=https://webmineral.com/help/CellDimensions.shtml|title=Unit Cell Formula|work=Mineralogy Database|date=2000–2005|access-date=2007-12-09}}</ref> The CODATA value of ''a'' for silicon is {{physconst|asil|after=.}}

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 {{nowrap|''a''/{{radic|8}}}}.

The [[isotope]] proportional composition of the sample used must be measured and taken into account. Silicon occurs in three stable isotopes ({{sup|28}}Si, {{sup|29}}Si, {{sup|30}}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]]s of the three [[nuclide]]s are known with great accuracy. This, together with the measured [[density]] ''ρ'' of the sample, allows the molar volume ''V''{{sub|m}} to be determined:
<math display=block>V_{\rm m} = \frac{A_{\rm r}M_{\rm u}}{\rho},</math>
where ''M''{{sub|u}} is the molar mass constant. The CODATA value for the molar volume of silicon is {{physconst|VmSi|ref=no}}, with a relative standard uncertainty of {{physconst|VmSi|runc=yes|after=.}}