Community Hub0 Subscribers
Community hub
Electron mass
View on Wikipediafrom Wikipedia
| Constant | Value |
|---|---|
| me | 9.1093837139(28)×10−31 kg[1] |
| 5.485799090441(97)×10−4 Da[2] | |
| 0.51099895069(16) MeV/c2 | |
| mec2 | 8.1871057880(26)×10−14 J[3] |
| 0.51099895069(16) MeV[4] |
In particle physics, the electron 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 9.109×10−31 kilograms or about 5.486×10−4 daltons, which has an energy-equivalent of about 8.187×10−14 joules or about 0.5110 MeV.
Terminology
[edit]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 becomes substantial for electrons accelerated by voltages of over 100 kV.
For example, the relativistic expression for the total energy, E, of an electron moving at speed v is where
- c is the speed of light;
- γ is the Lorentz factor,
- me is the "rest mass", or more simply just the "mass" of the electron.
This quantity me is frame invariant and velocity independent.
Determination
[edit]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. Seven years later 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 oil drop experiment in 1909. Together with the mass-to-charge ratio, the electron mass was 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 α obtained through spectroscopic measurements. Using the definition of the Rydberg constant:
thus
where c is the speed of light and h is the Planck constant.[5] The relative uncertainty, 5×10−8 in the 2006 CODATA recommended value,[6] 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.[5][6]
Relationship to other physical constants
[edit]The electron mass was used to calculate the Avogadro constant NA before its value was fixed as a defining constant in the 2019 revision of the SI:
Hence it is also related to the atomic mass constant mu:
where
- Mu is the molar mass constant (defined in SI);
- Ar(e) is a directly measured quantity, the relative atomic mass of the electron.
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,[5]
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 | 1.00727646677(10) | 15.99052817445(18) |
| relative atomic mass of the Z electrons | 0.00054857990943(23) | 0.0043886392754(18) |
| correction for the binding energy | −0.0000000145985 | −0.0000021941559 |
| relative atomic mass of the neutral atom | 1.00782503207(10) | 15.99491461957(18) |
The principle can be shown by the determination of the electron relative atomic mass by Farnham et al. at the University of Washington (1995).[7] 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):
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), 5.4863037178×10−4. This approximate value is then used to calculate a first approximation to Ar(12C6+), knowing that (from the sum of the six ionization energies of carbon) is 1.1058674×10−6: Ar(12C6+) ≈ 11.9967087236367. 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 Ar(e) = 5.485799111(12)×10−4 for these data.
References
[edit]- ^ "2022 CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
- ^ "2022 CODATA Value: electron mass in u". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
- ^ "2022 CODATA Value: electron mass energy equivalent". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
- ^ "2022 CODATA Value: electron mass energy equivalent in MeV". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
- ^ a b c "CODATA Value: electron mass". The NIST Reference on Constants, Units and Uncertainty. May 20, 2019. Retrieved May 20, 2019.
- ^ a b The NIST reference on Constants, Units, and Uncertainty, National Institute of Standards and Technology, 10 June 2009
- ^ 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, PMID 10059680
Electron mass
View on Grokipediafrom Grokipedia
Definition and Fundamentals
Definition
The electron is a fundamental subatomic particle classified as a charged lepton, possessing a negative elementary electric charge of -1 in units of the proton charge magnitude.[6] As one of the basic constituents of matter, it plays a central role in atomic structure and electromagnetic interactions.[7] The electron mass, conventionally denoted as $ m_e $, is defined as the intrinsic rest mass of the electron, which corresponds to its invariant mass in the theory of special relativity.[8] This invariant mass remains constant regardless of the electron's velocity or the observer's reference frame, serving as a fundamental property that characterizes the particle's inertia in its rest frame.[9] In this context, the electron's rest mass embodies its rest energy, expressed by the relation
where $ E $ is the rest energy and $ c $ is the speed of light in vacuum, highlighting the equivalence between mass and energy as established in special relativity.[10]
To avoid ambiguity, particularly with historical or contextual usages, the term "electron rest mass" is employed to emphasize this invariant quantity and distinguish it from the deprecated concept of relativistic mass.[8] Relativistic mass, once used to describe the velocity-dependent increase in a particle's effective inertia, has fallen out of favor in modern physics because it can lead to misconceptions; instead, the invariant rest mass is preferred, with relativistic effects handled through energy and momentum.[9] This terminology also helps differentiate the bare rest mass from effective masses that arise in specialized scenarios, such as the dynamic behavior of electrons in external fields or condensed matter systems.
Units and Equivalent Expressions
The electron mass is most commonly expressed in the International System of Units (SI) as the rest mass $ m_e = 9.109,383,7139(28) \times 10^{-31} $ kg, where the uncertainty is given in parentheses and corresponds to the standard deviation in the last two digits.[2] This value originates from the 2022 CODATA adjustment, which incorporates measurements from precision experiments in atomic physics and metrology.[11] In particle physics and relativistic contexts, the electron mass is often converted to its rest energy equivalent via $ E = m_e c^2 $, yielding $ m_e c^2 = 0.510,998,950,69(16) $ MeV.[4] This form facilitates comparisons with other particle masses and energies, with the uncertainty reflecting the propagation from the mass and speed of light values in the same CODATA set.[11] In atomic units, where the electron mass serves as the fundamental unit of mass, $ m_e $ is defined exactly as 1, simplifying calculations in non-relativistic quantum mechanics for atomic systems. Similarly, in natural units where $ \hbar = c = 1 $, the electron mass is expressed directly in inverse length or energy units, such as $ m_e = 0.510,998,950,69 $ MeV (with $ \hbar c $ restoring dimensions if needed), aligning with high-energy physics conventions. The precision of the electron mass has improved significantly over time, with the relative standard uncertainty of $ 3.1 \times 10^{-10} $ in the 2022 CODATA evaluation, unchanged from 2018 but improved from the 2014 value of $ 1.2 \times 10^{-9} $.[2] The 2019 redefinition of the SI units, which fixed the Planck constant $ h $ exactly, has improved the consistency of mass values by decoupling them from artifact-based standards. However, the uncertainty in $ m_e $ remains limited by experimental inputs, with ongoing refinements through spectroscopic and interferometric data.[11]Experimental Determination
Historical Measurements
The determination of the electron mass began with J.J. Thomson's groundbreaking 1897 experiments using cathode ray tubes. By measuring the deflection of cathode rays in crossed electric and magnetic fields, Thomson calculated the charge-to-mass ratio $ e/m_e $ to be approximately $ 1.7 \times 10^7 $ electromagnetic units per gram, which was about 10,000 times larger than that for hydrogen ions. Assuming the electron's charge equaled the minimum charge from electrolysis experiments (around $ 4.8 \times 10^{-10} $ esu), he estimated the electron mass at roughly $ 10^{-27} $ g in cgs units, or about 1/1000 to 1/2000 of the hydrogen atom's mass. This value, though approximate, confirmed the particulate nature of cathode rays and marked the first quantitative assessment of the electron's mass.[12][13] A significant refinement came in 1909 with Robert Millikan's oil-drop experiment, which directly measured the elementary charge $ e \approx 1.592 \times 10^{-19} $ C on oil droplets ionized by X-rays. Combining this with updated charge-to-mass ratios from Thomson's method and subsequent refinements, Millikan derived the electron mass as $ m_e \approx 9.1 \times 10^{-28} $ g, improving accuracy to within a few percent and establishing the electron as a fundamental particle with a well-defined mass independent of the emitting material. This result not only corroborated Thomson's findings but also provided empirical support for atomic models incorporating discrete electrons.[14][15] From the 1920s to the 1940s, experimental techniques evolved to enhance precision, primarily through refined magnetic deflection of electron beams and early particle accelerators. Methods involving uniform magnetic fields to measure beam curvature yielded charge-to-mass ratios with uncertainties below 0.5%, as demonstrated by F.G. Dunnington's 1938 deflection apparatus, which achieved relative precision of about 0.02% for $ e/m_e $. Cyclotron-based approaches in the 1930s, leveraging resonant acceleration of electrons, further validated these values by probing relativistic effects at higher energies. By the 1950s, these classical techniques had converged on the electron mass with approximately 0.1% precision, setting the stage for more advanced measurements.[16][17] Independent confirmation emerged in the 1930s through spectroscopic analysis of the Balmer series in hydrogen spectra. Precise wavelength measurements of lines like Hα, combined with the recent discovery of deuterium, allowed determination of the infinite nuclear mass Rydberg constant $ R_\infty $ and the proton-electron mass ratio $ m_p/m_e \approx 1836 $. Using known proton mass estimates, this yielded electron mass values consistent with deflection methods, with relative uncertainties around 0.1%, highlighting the interplay between atomic spectroscopy and direct particle measurements.[18]Contemporary Methods
Contemporary methods for determining the electron mass leverage quantum precision techniques, including Penning trap mass spectrometry and atomic spectroscopy, achieving relative uncertainties below 10^{-10} since the late 20th century. These approaches exploit the stability of trapped particles and high-resolution laser spectroscopy to isolate the electron's contribution from nuclear effects or bound-state interactions. Penning trap experiments measure the mass of a single electron or a hydrogenlike ion by determining its cyclotron frequency in a strong magnetic field. The cyclotron frequency is given by
where is the elementary charge, is the magnetic field strength, and is the electron mass. Early high-precision implementations in the late 1980s and 1990s, such as those by the University of Washington group, used compensated Penning traps to measure the proton-to-electron mass ratio via cyclotron frequency ratios of single ions, yielding the electron's atomic mass with a relative uncertainty of approximately 2 \times 10^{-10}. A landmark 1995 measurement refined this to a relative uncertainty of 2.0 \times 10^{-10} by comparing the cyclotron frequencies of a free electron and a bare carbon-12 ion (^{12}C^{6+}). More recent experiments at PTB and collaborators have pushed precision further; in 2014, bound-electron g-factor measurements on hydrogenlike ^{12}C^{5+} in a double Penning-trap setup determined the electron atomic mass with a relative uncertainty of 3.0 \times 10^{-11}. These single-ion Penning trap techniques, often involving carbon or oxygen ions, minimize relativistic and anharmonic corrections through cryogenic operation and precise field mapping.
Atomic spectroscopy provides a complementary route by deriving the electron mass from the finite-mass correction to the Rydberg constant in hydrogen-like systems. The Rydberg constant for hydrogen relates to the infinite nuclear mass Rydberg constant via the reduced mass effect:
where is the proton mass; high-precision measurements of transition frequencies thus yield . The two-photon 1S-2S transition in hydrogen, with its narrow natural linewidth of 1.3 Hz, is particularly suited for this, as its frequency directly probes . Laser frequency comb spectroscopy has enabled measurements of the 1S-2S frequency at kHz, corresponding to a relative uncertainty of 4 \times 10^{-12} and contributing to determinations at the 10^{-10} level. Complementary measurements, such as the 2020 1S-3S transition at MPQ, refine and the mass ratio by cross-checking QED predictions with experimental spectra.
Post-2019 SI redefinition advancements integrate these methods with fixed values of , , and the fine-structure constant , expressing as $ m_e = \frac{2 h R_\infty}{c \alpha^2} $ (in appropriate units), where consistency tests achieve relative uncertainties of 5 \times 10^{-10} through combined Penning trap, spectroscopy, and quantum Hall effect data. NIST and PTB collaborations exemplify this, with PTB's single-ion Penning traps providing direct mass inputs and NIST's free-electron g-factor measurements refining to 1.6 \times 10^{-10}) uncertainty, enabling holistic verifications of . These efforts ensure the electron mass remains anchored to the revised SI framework while probing beyond-Standard-Model physics.
Theoretical Framework
In Quantum Electrodynamics
In quantum electrodynamics (QED), the observed electron mass $ m_e $ differs from the bare mass $ m_0 $ due to radiative corrections arising from interactions with the quantized electromagnetic field. These corrections are encapsulated in the relation $ m_e = m_0 (1 + \delta) $, where $ \delta $ sums contributions from perturbative loop diagrams, primarily proportional to $ \alpha / \pi $ with $ \alpha $ the fine-structure constant. The renormalization procedure absorbs ultraviolet divergences into $ m_0 $, ensuring finite physical predictions, as formalized in the covariant perturbation theory developed in the mid-20th century. The primary source of the mass shift is the one-loop electron self-energy diagram, depicting the electron propagating while emitting and reabsorbing a virtual photon. This leading-order correction yields $ \Delta m_e / m_e \approx (\alpha / 2\pi) \log(\Lambda / m_e) $, where $ \Lambda $ is an ultraviolet cutoff to regulate the logarithmic divergence. A more detailed evaluation gives the relative shift as $ \delta m / m = (3 \alpha / 4\pi) \ln(\Lambda^2 / m^2) $, highlighting the scale dependence between the cutoff and electron mass; higher-loop terms add subleading powers of $ \alpha $. These self-energy effects effectively dress the bare electron with a photon cloud, increasing the observable inertial mass.[19][20] The electron mass also features prominently in QED predictions for the anomalous magnetic moment $ (g-2) $. The Dirac equation establishes the tree-level gyromagnetic ratio $ g = 2 $, corresponding to a magnetic moment $ \mu = e \hbar / (2 m_e) $ in natural units, where the mass sets the scale for spin-magnetic coupling. Radiative corrections introduce shifts, with the leading QED term $ a_e = (g-2)/2 = \alpha / (2\pi) $ arising from vertex and self-energy diagrams; while mass-independent at this order, subsequent contributions involve $ \ln(m_e / \mu) $ terms tied to renormalization scales $ \mu $. This interplay underscores how QED perturbations refine the Dirac baseline without altering its fundamental structure. Historically, the renormalization of the electron mass emerged from foundational work in the 1940s and 1950s. Freeman Dyson's 1949 resummation of perturbation series unified the Tomonaga-Schwinger and Feynman approaches, demonstrating how infinite diagrams converge to renormalize parameters like mass while preserving unitarity and causality in the S-matrix formalism. Building on this, Hans Bethe and Edwin Salpeter introduced their relativistic wave equation in 1951 for two-body bound states, incorporating QED loop effects to compute effective masses in systems like positronium, thus extending single-particle renormalization to composite structures. These developments solidified QED as a predictive theory, with mass corrections verifiable through precision spectroscopy.Relation to Other Constants
The electron mass $ m_e $ is interconnected with the fine-structure constant $ \alpha $ through the Rydberg constant $ R_\infty $, which describes the limiting wavenumber for hydrogen spectral lines in the infinite nuclear mass approximation. The relation is given by
where $ c $ is the speed of light and $ h $ is Planck's constant; equivalently, substituting the definition $ \alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c} $ yields $ R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c} $, with $ e $ the elementary charge and $ \epsilon_0 $ the vacuum permittivity. The 2022 CODATA recommended value for $ R_\infty $ is $ 10,973,731.568,157(12) $ m$^{-1} $, reflecting the interdependence where precision in $ m_e $ and $ \alpha $ mutually constrains $ R_\infty $.
Electron mass ratios to other particles provide key benchmarks in particle physics. The proton-electron mass ratio $ m_p / m_e \approx 1836.152,673,426(32) $ is determined from measurements involving hyperfine splitting in hydrogen, linking $ m_e $ to nuclear scales via quantum electrodynamic effects in atomic spectra. Similarly, the muon-electron mass ratio $ m_\mu / m_e \approx 206.768,2827(46) $ arises from hyperfine splitting in muonium, the bound state of an electron and antimuon, highlighting leptonic mass hierarchies. These ratios, with relative uncertainties below $ 2 \times 10^{-8} $, underscore $ m_e $'s role as a reference for heavier particle masses.
In atomic physics, the electron mass defines the Bohr radius $ a_0 $, the characteristic scale of the hydrogen ground state in the Bohr model:
The 2022 CODATA value is $ a_0 = 5.291,772,105,44(82) \times 10^{-11} $ m, illustrating how $ m_e $ sets the size of atomic orbitals and influences molecular bonding lengths.
Following the 2019 SI redefinition, which fixed the values of $ h $, $ e $, and $ c $ exactly, updates to $ m_e $ propagate through the least-squares adjustment of CODATA values, affecting derived constants like $ \alpha $, $ R_\infty $, and $ a_0 $. For instance, the 2022 adjustment reduced the relative uncertainty in $ m_e $ to $ 3.1 \times 10^{-10} $ (value: $ 9.109,383,7139(28) \times 10^{-31} $ kg), tightening correlations with $ \alpha $ (relative uncertainty $ 1.5 \times 10^{-10} $) and mass ratios by resolving tensions in spectroscopic data. This interdependence ensures consistency across electromagnetic and atomic domains without altering the fixed SI base units.