Electron Relative Atomic Mass
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Electron Relative Atomic Mass
Constant Values Units
me [1] kg
Da
J/c2
MeV/c2
Energy
of me
J
MeV

The electron rest mass (symbol: me) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , equivalent to an energy of about or about .[2]

## Terminology

The term "rest mass" is sometimes used because in special relativity the mass of an object can be said to increase in a frame of reference that is moving relative to that object (or if the object is moving in a given frame of reference). Most practical measurements are carried out on moving electrons. If the electron is moving at a relativistic velocity, any measurement must use the correct expression for mass. Such correction is only substantial for electrons accelerated by voltages of well over 100 kV.

For example, the relativistic expression for the total energy, E, of an electron moving at speed ${\displaystyle v}$ is

${\displaystyle E=\gamma m_{\text{e}}c^{2},}$

where the Lorentz factor is ${\displaystyle \gamma =1/{\sqrt {1-(v/c)^{2}}}}$. In this expression me is the "rest mass", or more simply just the "mass" of the electron. This quantity me is frame invariant and velocity independent. However, some texts group the Lorentz factor with the mass factor to define a new quantity called the relativistic mass, . This quantity is evidently velocity dependent, and from it arises the notion that "mass increases with speed". This construction is optional, however, and adds little insight into the dynamics of special relativity.

## Determination

Since the electron mass determines a number of observed effects in atomic physics, there are potentially many ways to determine its mass from an experiment, if the values of other physical constants are already considered known.

Historically, the mass of the electron was determined directly from combining two measurements. The mass-to-charge ratio of the electron was first estimated by Arthur Schuster in 1890 by measuring the deflection of "cathode rays" due to a known magnetic field in a cathode ray tube. It was seven years later that J. J. Thomson showed that cathode rays consist of streams of particles, to be called electrons, and made more precise measurements of their mass-to-charge ratio again using a cathode ray tube.

The second measurement was of the charge of the electron. This was determined with a precision of better than 1% by Robert A. Millikan in his famous oil drop experiment in 1909. Together with the mass-to-charge ratio, the electron mass was thereby determined with reasonable precision. The value of mass that was found for the electron was initially met with surprise by physicists, since it was so small (less than 0.1%) compared to the known mass of a hydrogen atom.

The electron rest mass can be calculated from the Rydberg constant R? and the fine-structure constant ? obtained through spectroscopic measurements. Using the definition of the Rydberg constant:

${\displaystyle R_{\infty }={\frac {m_{\rm {e}}c\alpha ^{2}}{2h}},}$

thus

${\displaystyle m_{\rm {e}}={\frac {2R_{\infty }h}{c\alpha ^{2}}},}$

where c is the speed of light and h is the Planck constant.[2] The relative uncertainty, 5×10−8 in the 2006 CODATA recommended value,[3] is due entirely to the uncertainty in the value of the Planck constant. With the re-definition of kilogram in 2019, there is no uncertainty by definition left in Planck constant anymore.

The electron relative atomic mass can be measured directly in a Penning trap. It can also be inferred from the spectra of antiprotonic helium atoms (helium atoms where one of the electrons has been replaced by an antiproton) or from measurements of the electron g-factor in the hydrogenic ions 12C5+ or 16O7+.

The electron relative atomic mass is an adjusted parameter in the CODATA set of fundamental physical constants, while the electron rest mass in kilograms is calculated from the values of the Planck constant, the fine-structure constant and the Rydberg constant, as detailed above.[2][3]

## Relationship to other physical constants

The electron mass is used to calculate[] the Avogadro constant NA:

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

Hence it is also related to the atomic mass constant mu:

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

where Mu is the molar mass constant (defined in SI) and Ar(e) is a directly measured quantity, the relative atomic mass of the electron.

Note that mu is defined in terms of Ar(e), and not the other way round, and so the name "electron mass in atomic mass units" for Ar(e) involves a circular definition (at least in terms of practical measurements).

The electron relative atomic mass also enters into the calculation of all other relative atomic masses. By convention, relative atomic masses are quoted for neutral atoms, but the actual measurements are made on positive ions, either in a mass spectrometer or a Penning trap. Hence the mass of the electrons must be added back on to the measured values before tabulation. A correction must also be made for the mass equivalent of the binding energy Eb. Taking the simplest case of complete ionization of all electrons, for a nuclide X of atomic number Z,[2]

${\displaystyle A_{\rm {r}}({\rm {X}})=A_{\rm {r}}({\rm {X}}^{Z+})+ZA_{\rm {r}}({\rm {e}})-E_{\rm {b}}/m_{\rm {u}}c^{2}\,}$

As relative atomic masses are measured as ratios of masses, the corrections must be applied to both ions: the uncertainties in the corrections are negligible, as illustrated below for hydrogen 1 and oxygen 16.

Physical parameter 1H 16O
relative atomic mass of the XZ+ ion
relative atomic mass of the Z electrons
correction for the binding energy
relative atomic mass of the neutral atom

The principle can be shown by the determination of the electron relative atomic mass by Farnham et al. at the University of Washington (1995).[4] It involves the measurement of the frequencies of the cyclotron radiation emitted by electrons and by 12C6+ ions in a Penning trap. The ratio of the two frequencies is equal to six times the inverse ratio of the masses of the two particles (the heavier the particle, the lower the frequency of the cyclotron radiation; the higher the charge on the particle, the higher the frequency):

${\displaystyle {\frac {\nu _{c}({}^{12}{\rm {C}}^{6+})}{\nu _{c}({\rm {e}})}}={\frac {6A_{\rm {r}}({\rm {e}})}{A_{\rm {r}}({}^{12}{\rm {C}}^{6+})}}=0.000\,274\,365\,185\,89(58)}$

As the relative atomic mass of 12C6+ ions is very nearly 12, the ratio of frequencies can be used to calculate a first approximation to Ar(e), . This approximate value is then used to calculate a first approximation to Ar(12C6+), knowing that Eb(12C)/muc2 (from the sum of the six ionization energies of carbon) is : . This value is then used to calculate a new approximation to Ar(e), and the process repeated until the values no longer vary (given the relative uncertainty of the measurement, 2.1×10−9): this happens by the fourth cycle of iterations for these results, giving for these data.

## References

1. ^ "2018 CODATA Value: electron mass in u". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved . Cite has empty unknown parameter: |month= (help)
2. ^ a b c d "CODATA Value: electron mass". The NIST Reference on Constants, Units and Uncertainty. May 20, 2019. Retrieved 2019.
3. ^ a b
4. ^ Farnham, D. L.; Van Dyck, Jr., R. S.; Schwinberg, P. B. (1995), "Determination of the Electron's Atomic Mass and the Proton/Electron Mass Ratio via Penning Trap Mass Spectroscopy", Phys. Rev. Lett., 75 (20): 3598-3601, Bibcode:1995PhRvL..75.3598F, doi:10.1103/PhysRevLett.75.3598