Editing Elementary charge (section)

== Experimental measurements of the elementary charge ==
The elementary charge is exactly defined since 20 May 2019 by the [[International System of Units]]. Prior to this change, the elementary charge was a measured quantity whose magnitude was determined experimentally. This section summarizes these historical experimental measurements.

=== In terms of the Avogadro constant and Faraday constant ===
If the [[Avogadro constant]] ''N''<sub>A</sub> and the [[Faraday constant]] ''F'' are independently known, the value of the elementary charge can be deduced using the formula
<math display="block">e = \frac{F}{N_\text{A}}.</math>
(In other words, the charge of one [[Mole (unit)|mole]] of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.)

This method is ''not'' how the ''most accurate'' values are measured today. Nevertheless, it is a legitimate and still quite accurate method, and experimental methodologies are described below.

The value of the Avogadro constant ''N''<sub>A</sub> was first approximated by [[Johann Josef Loschmidt]] who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas.<ref>{{cite journal | first = J. | last = Loschmidt | author-link = Johann Josef Loschmidt | title = Zur Grösse der Luftmoleküle | journal = Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien | volume = 52 | issue = 2 | pages = 395–413 | year =1865}} [http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Loschmidt-1865.html English translation] {{webarchive |url=https://web.archive.org/web/20060207130125/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Loschmidt-1865.html |date=February 7, 2006 }}.</ref> Today the value of ''N''<sub>A</sub> can be measured at very high accuracy by taking an extremely pure crystal (often [[silicon]]), measuring how far apart the atoms are spaced using [[X-ray diffraction]] or another method, and accurately measuring the density of the crystal. From this information, one can deduce the mass (''m'') of a single atom; and since the [[molar mass]] (''M'') is known, the number of atoms in a mole can be calculated: {{nowrap|1=''N''<sub>A</sub> = ''M''/''m''}}.

The value of ''F'' can be measured directly using [[Faraday's laws of electrolysis]]. Faraday's laws of electrolysis are quantitative relationships based on the electrochemical researches published by [[Michael Faraday]] in 1834.<ref>{{cite journal | author = Ehl, Rosemary Gene |author2=Ihde, Aaron |author-link2=Aaron J. Ihde |title = Faraday's Electrochemical Laws and the Determination of Equivalent Weights | journal = Journal of Chemical Education | year = 1954 | volume = 31 | issue = May | pages = 226–232 | doi = 10.1021/ed031p226 |bibcode = 1954JChEd..31..226E }}</ref> In an [[electrolysis]] experiment, there is a one-to-one correspondence between the electrons passing through the anode-to-cathode wire and the ions that plate onto or off of the anode or cathode. Measuring the mass change of the anode or cathode, and the total charge passing through the wire (which can be measured as the time-integral of [[electric current]]), and also taking into account the molar mass of the ions, one can deduce ''F''.{{physconst|e|ref=only}}

The limit to the precision of the method is the measurement of ''F'': the best experimental value has a relative uncertainty of 1.6&nbsp;ppm, about thirty times higher than other modern methods of measuring or calculating the elementary charge.<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>

=== Oil-drop experiment ===
{{main|Oil-drop experiment}}
A famous method for measuring ''e'' is Millikan's oil-drop experiment. A small drop of oil in an electric field would move at a rate that balanced the forces of [[gravity]], [[viscosity]] (of traveling through the air), and [[electric force]]. The forces due to gravity and viscosity could be calculated based on the size and velocity of the oil drop, so electric force could be deduced. Since electric force, in turn, is the product of the electric charge and the known electric field, the electric charge of the oil drop could be accurately computed. By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely ''e''.

The necessity of measuring the size of the oil droplets can be eliminated by using tiny plastic spheres of a uniform size. The force due to viscosity can be eliminated by adjusting the strength of the electric field so that the sphere hovers motionless.

=== Shot noise ===
{{main|Shot noise}}

Any [[electric current]] will be associated with [[electronic noise|noise]] from a variety of sources, one of which is [[shot noise]]. Shot noise exists because a current is not a smooth continual flow; instead, a current is made up of discrete electrons that pass by one at a time. By carefully analyzing the noise of a current, the charge of an electron can be calculated. This method, first proposed by [[Walter H. Schottky]], can determine a value of ''e'' of which the accuracy is limited to a few percent.<ref>{{Cite journal |last1=Beenakker |first1=Carlo |last2=Schönenberger |first2=Christian |author-link2=Christian Schönenberger |year=2006 |title=Quantum Shot Noise |journal=Physics Today |volume=56 |issue=5 |pages=37–42 |arxiv=cond-mat/0605025 |doi=10.1063/1.1583532 |s2cid=119339791}}</ref> However, it was used in the first direct observation of [[Laughlin wavefunction|Laughlin]] [[quasiparticle]]s, implicated in the [[fractional quantum Hall effect]].<ref>{{Cite journal | journal = Nature | volume = 389 | issue = 162–164 | year = 1997 | doi = 10.1038/38241 | title = Direct observation of a fractional charge | first1 = R. | last1 = de-Picciotto | first2 = M. | last2 = Reznikov | first3 = M. | last3 = Heiblum | first4 = V. | last4 = Umansky | first5 = G. | last5 = Bunin | first6 = D. | last6 = Mahalu | pages = 162 |bibcode = 1997Natur.389..162D | arxiv = cond-mat/9707289 | s2cid = 4310360 }}</ref>

=== From the Josephson and von Klitzing constants ===
Another accurate method for measuring the elementary charge is by inferring it from measurements of two effects in [[quantum mechanics]]: The [[Josephson effect]], voltage oscillations that arise in certain [[superconducting]] structures; and the [[quantum Hall effect]], a quantum effect of electrons at low temperatures, strong magnetic fields, and confinement into two dimensions. The [[Josephson constant]] is
<math display="block">K_\text{J} = \frac{2e}{h},</math>
where ''h'' is the [[Planck constant]]. It can be measured directly using the [[Josephson effect]].

The [[von Klitzing constant]] is
<math display="block">R_\text{K} = \frac{h}{e^2}.</math>
It can be measured directly using the [[quantum Hall effect]].

From these two constants, the elementary charge can be deduced:
<math display="block">e = \frac{2}{R_\text{K} K_\text{J}}.</math>

=== CODATA method ===
The relation used by [[CODATA]] to determine elementary charge was:
<math display="block">e^2 = \frac{2h \alpha}{\mu_0 c} = 2h \alpha \varepsilon_0 c,</math>
where ''h'' is the [[Planck constant]], ''α'' is the [[fine-structure constant]], ''μ''<sub>0</sub> is the [[magnetic constant]], ''ε''<sub>0</sub> is the [[electric constant]], and ''c'' is the [[speed of light]]. Presently this equation reflects a relation between ''ε''<sub>0</sub> and  ''α'', while all others are fixed values. Thus the relative standard uncertainties of both will be same.

=== Tests  of the universality of elementary charge ===
{| class="wikitable"
|+ 
|-
! Particle !! Expected charge !! Experimental constraint !! Notes
|-
| electron || <math>q_{\text{e}^-}=-e</math> || exact || by definition
|-
| proton || <math>q_\text{p}=e</math> || <math>\left|{q_\text{p} - e}\right| < 10^{-21}e</math> || by finding no measurable sound when an alternating electric field is applied to [[sulfur hexafluoride|SF<sub>6</sub>]] gas in a spherical resonator<ref>
{{cite journal
 | last1      = Bressi | first1     = G.
 | last2      = Carugno | first2     = G.
 | last3      = Della Valle | first3     = F.
 | last4      = Galeazzi | first4     = G.
 | last5      = Sartori | first5      = G.
 | date       = 2011
 | title      = Testing the neutrality of matter by acoustic means in a spherical resonator
 | journal    = Physical Review A
 | volume     = 83
 | number     = 5
 | pages      = 052101
 | doi        = 10.1103/PhysRevA.83.052101
 | arxiv = 1102.2766
 | s2cid = 118579475
}}</ref>
|-
| [[positron]] || <math>q_{\text{e}^+}=e</math> || <math>\left|{q_{\text{e}^+} - e}\right| < 10^{-9}e</math> || by combining the best measured value of the antiproton charge (below) with the low limit placed on antihydrogen's net charge by the [[ALPHA Collaboration]] at [[CERN]].<ref>
{{cite journal
 | display-authors = etal
 | last1 = Ahmadi | first1 = M.
 | date       = 2016
 | title      = An improved limit on the charge of antihydrogen from stochastic acceleration
 | url        = https://www.nature.com/articles/nature16491.pdf
 | journal    = Nature
 | volume     = 529
 | issue      = 7586
 | pages      = 373–376
 | doi        = 10.1038/nature16491
 | pmid = 26791725
 | s2cid = 205247209
 | access-date = May 1, 2022
}}</ref>
|-
| [[antiproton]] || <math>q_{\bar{\text{p}}}=-e</math> || <math>\left|{q_{\bar{\text{p}}} + q_\text{p}}\right| < 10^{-9}e</math> || Hori et al.<ref>
{{cite journal
 | display-authors = etal
 | last1 = Hori | first = M.
 | date       = 2011
 | title      = Two-photon laser spectroscopy of antiprotonic helium and the antiproton-to-electron mass ratio.
 | journal    = Nature
 | volume     = 475
 | issue = 7357
 | pages      = 484–488
 | doi        = 10.1038/nature10260
 | pmid = 21796208
 | arxiv = 1304.4330
 | s2cid = 4376768
}}</ref> as cited in antiproton/proton charge difference listing of the [[Particle Data Group]]<ref>
{{cite journal
 | display-authors = etal
 | last1 = Olive | first1 = K. A.
 | date       = 2014
 | title      = Review of particle physics
 | journal    = Chinese Physics C
 | volume     = 38
 | number     = 9
 | pages      = 090001
 | doi        = 10.1088/1674-1137/38/9/090001
 | s2cid = 118395784
 | url = http://bib-pubdb1.desy.de/record/172097/files/PUBDB-2014-03548.pdf
}}</ref> The Particle Data Group article has a link to the current online version of the particle data.
|}