Fact-checked by Grok 2 weeks ago

Rydberg constant

The Rydberg constant, denoted R_\infty, is a fundamental physical constant in atomic physics that specifies the limiting value of the highest wavenumber (or inverse wavelength) for photons emitted in the spectral lines of hydrogen, with a precise value of $10\,973\,731.568\,157(12) m^{-1}.^[1] It serves as a key parameter in the Rydberg formula, \frac{1}{\lambda} = R_\infty \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), where \lambda is the wavelength, and n_1 and n_2 are principal quantum numbers with n_2 > n_1, enabling the prediction of hydrogen atom emission and absorption spectra.^[2] Expressed theoretically as R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c} (or equivalently R_\infty = \frac{\alpha^2 m_e c}{2 h}), it connects atomic structure to basic constants including the electron mass m_e, elementary charge e, vacuum permittivity \epsilon_0, Planck's constant h, speed of light c, and fine-structure constant \alpha.^[2] Introduced empirically in 1890 by Swedish physicist Johannes Rydberg, the constant generalized earlier observations of hydrogen spectral series (such as Balmer's 1885 formula for visible lines) into a unified expression applicable to multiple series, predicting undiscovered lines like those in the ultraviolet Lyman series.^[3] Rydberg's work, based on spectroscopic measurements, established [R](/page/R) as a universal scaling factor for atomic spectra beyond hydrogen, influencing the periodic table's development and element identification.^[3] In 1913, Niels Bohr provided a theoretical derivation within his quantum model of the hydrogen atom, showing R_\infty as arising from quantized electron orbits and balancing centripetal force with electrostatic attraction, thus bridging empirical spectroscopy and quantum mechanics.^[3] The Rydberg constant's value for finite nuclear mass (e.g., R_H for hydrogen) is adjusted as R_H = R_\infty / (1 + m_e / m_p), where m_p is the proton mass, accounting for reduced mass effects in real atoms.^[2] Its high precision, refined through laser spectroscopy and quantum electrodynamics (QED) calculations, tests fundamental theories; discrepancies between experimental and theoretical values probe effects like the Lamb shift.^[4] Beyond hydrogen, R scales spectra in hydrogen-like ions (e.g., He^+) via R_Z = R_\infty Z^2, where Z is the atomic number, and informs Rydberg atoms—highly excited states with exaggerated quantum properties used in precision measurements and quantum computing.^[2] As part of the 2019 SI redefinition tying the Planck constant to exact values, R_\infty remains a cornerstone for metrology, linking atomic units to macroscopic standards.^[1]

Definition and Importance

Definition

The Rydberg constant, denoted R_\infty, is a fundamental physical constant in atomic physics representing the limiting case for infinite nuclear mass in hydrogen-like atoms. It quantifies the scale of atomic energy level spacings and appears in the Rydberg formula, which empirically describes the wavelengths \lambda of emission or absorption spectral lines as

\frac{1}{\lambda} = R_\infty \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right),

where n_1 and n_2 are positive integers denoting the principal quantum numbers of the initial and final states, respectively, with n_2 > n_1.^[5] This formula unifies the positions of spectral series, such as the Balmer series in the visible spectrum, by relating them to transitions between quantized atomic states.^[6] The constant has units of inverse length, specifically meters inverse (m⁻¹), reflecting its role in expressing wavenumbers (the reciprocal of wavelength) of spectral lines. It originated from early empirical analyses of atomic emission spectra, where patterns in line positions for elements like hydrogen were fitted to mathematical forms without initial theoretical justification.^[1] The Rydberg constant is named after Swedish physicist Johannes Rydberg, who in 1888 generalized prior empirical relations, such as Balmer's formula for hydrogen's visible lines, into the broader expression applicable to multiple spectral series.^[5]

Historical Significance

The discovery of regular patterns in the hydrogen spectrum laid the groundwork for the Rydberg constant. In 1885, Swiss mathematician Johann Jakob Balmer empirically derived a formula describing the wavelengths of visible spectral lines in hydrogen, known as the Balmer series, which hinted at underlying atomic structure but was limited to the visible region.^[3] This work served as a crucial precursor, inspiring further investigation into spectral regularities. In 1888, Swedish physicist Johannes Rydberg extended Balmer's approach by proposing a generalized empirical formula that unified multiple series of hydrogen spectral lines across ultraviolet, visible, and infrared regions, introducing a universal constant—now called the Rydberg constant—to account for the shared limiting frequency.^[7] Rydberg's formula, expressed as the wavenumber difference between series limits, marked a pivotal empirical breakthrough, enabling predictions of previously unobserved lines and establishing the constant as a fundamental parameter in atomic spectroscopy.^[8] The Rydberg constant gained theoretical significance in 1913 when Niels Bohr incorporated it into his quantum model of the hydrogen atom, deriving its value from postulates of quantized angular momentum and energy levels, thus bridging empirical observation with early quantum theory.^[9] This integration not only validated Rydberg's empirical findings but also propelled the development of quantum mechanics, as the constant's role in explaining discrete spectral lines influenced subsequent advancements, including the Schrödinger equation's solutions for hydrogen.^[10] Historically, the Rydberg constant was treated as a precisely measured quantity in the International System of Units (SI) until the 2019 redefinition, which fixed values for constants like the Planck constant, speed of light, and elementary charge, thereby linking the Rydberg constant more directly to these through its theoretical expression involving the electron mass and fine-structure constant.^[11] This shift represented a metrological milestone, enhancing the constant's role in tying atomic physics to the invariant foundations of the SI.^[12]

Numerical Values

Rydberg Constant for Infinite Mass

The Rydberg constant for infinite nuclear mass, denoted R_\infty, represents the theoretical value of the constant in the Rydberg formula under the approximation that the atomic nucleus has infinite mass, rendering it stationary relative to the orbiting electron.^[13] This approximation simplifies calculations by treating the electron's motion as occurring around a fixed point, which is particularly useful in theoretical models of hydrogen-like atoms—single-electron systems such as hydrogen or ionized helium—where the nuclear mass is much larger than the electron mass but not truly infinite.^[13] The current recommended value from the 2022 CODATA adjustment is R_\infty = 10\,973\,731.568\,157(12) \, \mathrm{m}^{-1}, where the uncertainty in parentheses applies to the last two digits.^[1] This value carries a relative standard uncertainty of $1.1 \times 10^{-12}, reflecting the extraordinary precision achieved through a global least-squares adjustment of fundamental constants.^[1] Compared to the 2018 CODATA value of R_\infty = 10\,973\,731.568\,160(21) \, \mathrm{m}^{-1} with a relative uncertainty of approximately $1.9 \times 10^{-12}, the 2022 recommendation demonstrates enhanced precision, primarily due to refined measurements of related quantities like the electron's magnetic moment and atomic transition frequencies.^[14] For real atoms like hydrogen, this infinite-mass value serves as the baseline before applying minor adjustments for finite nuclear mass effects.^[13]

Adjustment for Finite Nuclear Mass

In atomic systems, the infinite nuclear mass approximation underlying the Rydberg constant R_\infty overlooks the nucleus's finite mass, which influences the electron's orbital dynamics. To address this, the Rydberg constant is modified using the reduced mass \mu of the electron-nucleus system, yielding R_M = R_\infty \frac{\mu}{m_e}, where m_e is the electron mass and M is the nuclear mass. This adjustment, first introduced by Arnold Sommerfeld in his extension of the Bohr model, effectively treats the two-body problem as an equivalent one-body system orbiting the center of mass, with \mu = \frac{m_e M}{m_e + M}. Equivalently, the formula simplifies to

R_M = \frac{R_\infty}{1 + \frac{m_e}{M}}.

For hydrogen, where M = m_p (the proton mass, approximately 1836 times m_e), the correction factor is about 0.999455, resulting in R_H \approx 10\,967\,755 \, \mathrm{m}^{-1}. This value is derived from high-precision spectroscopic measurements and theoretical adjustments in the CODATA 2022 evaluation, reflecting a relative shift of roughly 0.0545% from R_\infty.^[13] For isotopes such as deuterium, the nuclear mass M = m_d (deuteron mass, approximately twice the proton mass) is larger, reducing the m_e / M term to about half that of hydrogen and thus minimizing the correction. The deuterium Rydberg constant is R_D \approx 10\,970\,742 \, \mathrm{m}^{-1}, closer to R_\infty by design. The difference R_D - R_H \approx 2\,987 \, \mathrm{m}^{-1} primarily arises from this reduced mass variation, manifesting as an isotope shift in spectral lines—for instance, the 1S-2S transition frequency shifts by approximately 670.994 GHz between hydrogen and deuterium, enabling precise determinations of nuclear mass ratios and charge radii through spectroscopy.^[13]^[15] The Rydberg constant R_\infty underpins several derived physical quantities that function as natural units in atomic physics, providing standardized scales for energy, frequency, and length in the study of atomic spectra and quantum systems. These quantities are particularly useful for normalizing calculations involving hydrogen-like atoms, where they represent characteristic values such as the ground-state binding energy or the ionization threshold. The 2022 CODATA recommended values ensure high precision for these units, reflecting advances in spectroscopic measurements. The Rydberg unit of energy, defined as hcR_\infty, corresponds to the energy scale of the hydrogen atom's ground state in the infinite nuclear mass limit and is widely used to express atomic energies in hartree units (where 1 Ry = 1/2 hartree). Its value is $2.179\,872\,361\,1030(24) \times 10^{-18} J, or equivalently $13.605\,693\,122\,990(15) eV.^[16]^[17] The Rydberg frequency, given by cR_\infty, sets the frequency scale for transitions to the continuum in hydrogen spectra and serves as a reference for high-precision frequency metrology in atomic clocks and lasers. This frequency is $3.289\,841\,960\,2500(36) \times 10^{15} Hz.^[18] The Rydberg wavelength, $1/R_\infty, defines the short-wavelength limit of the Lyman series and acts as a natural length unit for ultraviolet atomic transitions, facilitating comparisons in quantum defect theory and plasma physics. Its value is $91\,126.705\,058\,26(10) pm, derived from the CODATA 2022 value of R_\infty.^[1]

Theoretical Foundations

In the Bohr Atomic Model

In 1913, Niels Bohr proposed a semi-classical model for the hydrogen atom to explain its discrete spectral lines, introducing postulates that quantized electron orbits while assuming the nucleus has infinite mass. The model posits stationary states where the electron orbits the proton in circular paths without radiating electromagnetic energy, despite classical expectations. Angular momentum is quantized according to m_e v r = n \hbar, where m_e is the electron mass, v is the orbital speed, r is the radius, n = 1, 2, 3, \dots is the principal quantum number, and \hbar = h / 2\pi is the reduced Planck's constant. Transitions between these states emit or absorb photons with frequency \nu = |\Delta E| / h, where \Delta E is the energy difference and h is Planck's constant.^[19] The balance of forces in a circular orbit equates the centripetal force to the electrostatic attraction:

\frac{m_e v^2}{r} = \frac{k e^2}{r^2},

where k = 1/(4\pi \epsilon_0) is Coulomb's constant, e is the elementary charge, and \epsilon_0 is the vacuum permittivity. Substituting the quantization condition v = n \hbar / (m_e r) yields the Bohr radius for each state:

r_n = \frac{n^2 \hbar^2}{m_e k e^2}.

The orbital speed follows as v_n = k e^2 / (n \hbar). The total energy is the sum of kinetic and potential terms:

E_n = \frac{1}{2} m_e v_n^2 - \frac{k e^2}{r_n} = -\frac{m_e (k e^2)^2}{2 n^2 \hbar^2}.

^[20] The energy differences between states n_i > n_f produce emission lines with frequency

\nu = \frac{|E_{n_i} - E_{n_f}|}{h} = \frac{m_e (k e^2)^2}{2 h \hbar^2} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right).

The corresponding wavenumber \bar{\nu} = 1/\lambda = \nu / c, where c is the speed of light, gives the Rydberg formula:

\frac{1}{\lambda} = R_\infty \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right),

with the infinite-mass Rydberg constant

R_\infty = \frac{m_e k^2 e^4}{4 \pi c \hbar^3}.

This expression theoretically reproduces the empirical spectral series, such as the Balmer series in the visible spectrum.^[20]

Quantum Mechanical Derivation

The quantum mechanical treatment of the hydrogen atom begins with the time-independent Schrödinger equation for a single electron in the Coulomb potential of the nucleus, H \psi = E \psi, where the Hamiltonian is H = -\frac{\hbar^2}{2\mu} \nabla^2 - \frac{e^2}{4\pi \epsilon_0 r} and \mu is the reduced mass of the electron-proton system. Due to the spherical symmetry of the potential, the wavefunction \psi(r, \theta, \phi) separates into radial and angular parts: \psi_{nlm}(r, \theta, \phi) = R_{nl}(r) Y_{lm}(\theta, \phi), with Y_{lm} being the spherical harmonics labeled by quantum numbers l = 0, 1, \dots and m = -l, \dots, l.^[21] The radial equation, after substitution and separation, is a differential equation in r that admits solutions only for specific energy eigenvalues, determined by imposing boundary conditions that the wavefunction remains normalizable. These solutions yield the principal quantum number n = 1, 2, 3, \dots (with l = 0 to n-1), and the energy levels are E_n = -\frac{\mu e^4}{8 \epsilon_0^2 h^2 n^2}, independent of l and m. For infinite nuclear mass (\mu \approx m_e), this corresponds to E_n = -13.6 \, \mathrm{eV}/n^2. The transitions between these levels produce spectral lines with wavenumbers \bar{\nu} = \frac{1}{\lambda} = R_\infty \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), where the Rydberg constant emerges as R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c}. This derivation, first obtained by Erwin Schrödinger in 1926, provides a rigorous foundation for the discrete spectrum observed in hydrogen, surpassing the semi-classical Bohr model's ad hoc quantization rules.^[22]^[23] While the non-relativistic Schrödinger equation captures the gross structure of the energy levels, it neglects effects from special relativity and electron spin, leading to inaccuracies in finer details like the splitting of spectral lines. Relativistic corrections are incorporated via perturbation theory on the Dirac equation or approximate relativistic Hamiltonians, yielding the fine structure: the energy levels become E_{nj} = E_n \left[ 1 + \frac{\alpha^2}{n^2} \left( \frac{n}{j + 1/2} - \frac{3}{4} \right) \right], where \alpha = e^2 / (4\pi \epsilon_0 \hbar c) \approx 1/137 is the fine structure constant and j is the total angular momentum quantum number. These corrections, of order \alpha^2 E_n, introduce small splittings dependent on j but leave the base energy scale E_n (and thus R_\infty) unaltered, confirming the Rydberg constant's role in the unperturbed levels. The full Dirac treatment in 1928 exactly reproduces this fine structure, validating the quantum mechanical framework's accuracy for hydrogen.^[24]

Measurement Techniques

Historical Measurements

The initial experimental determinations of the Rydberg constant emerged in the late 19th century through observations of hydrogen's spectral lines using emerging grating spectroscopy techniques. In 1885, Johann Balmer analyzed the visible emission lines of hydrogen, deriving an empirical formula that related their wavelengths to a constant term, effectively introducing the concept later formalized as the Rydberg constant, with a value approximated at $1.097 \times 10^7 m^{-1} based on measurements of lines such as H\alpha at 656.3 nm and H\beta at 486.1 nm.^[25] These observations relied on diffraction gratings to resolve the Balmer series in the visible spectrum, marking a shift from prism-based methods to higher-precision wavelength determinations.^[10] Building on Balmer's work, Johannes Rydberg refined the approach in 1888–1890 by generalizing the formula to encompass additional spectral series, including ultraviolet lines, through grating spectroscopy of hydrogen and other elements. Rydberg's analysis incorporated UV transitions, such as those in the Lyman series (though not directly observed until later), yielding a more universal constant value of approximately $1.097 \times 10^7 m^{-1} and demonstrating the formula's applicability beyond visible wavelengths.^[26] This extension highlighted the constant's role in unifying disparate spectral observations, with Rydberg's grating measurements providing accuracies on the order of 0.1% for line positions.^[27] In the early 20th century, refinements advanced through interferometric methods, achieving accuracies around 0.01% in measuring hydrogen's Balmer series. Prior to the 1960s, further determinations utilized visual spectroscopy and photographic plates to map hydrogen and deuterium lines, enabling isotope-specific adjustments to the constant. These methods, reliant on densitometric analysis of plate exposures, provided accuracies up to 0.001% and laid groundwork for distinguishing nuclear mass effects in spectral patterns. By the late 1950s, National Bureau of Standards efforts using interferometric measurements of helium spectral lines refined the constant to approximately $1.097373 \times 10^7 m^{-1}, confirming the value accepted since 1952 through calibration against mercury-198 standards.^[28]

Modern Precision Methods

Modern precision measurements of the Rydberg constant rely on advanced spectroscopic techniques that achieve relative uncertainties below $10^{-12}, primarily through laser-based methods that minimize systematic errors such as Doppler broadening. Since the 1970s, two-photon spectroscopy has been instrumental in probing the hydrogen 1S-2S transition, enabling absolute frequency measurements with exceptional accuracy by exciting atoms via two counter-propagating laser beams, which cancels first-order Doppler shifts.^[29] This Doppler-free approach, combined with frequency combs for precise calibration, has allowed determinations of the transition frequency with uncertainties as low as $10^{-15}, directly constraining the Rydberg constant after accounting for finite nuclear mass effects.^[30] Another significant method involves laser spectroscopy of antiprotonic helium atoms, where an antiproton replaces one electron in helium, forming exotic Rydberg states amenable to high-resolution probing. These measurements target fine and hyperfine structure splittings, incorporating quantum electrodynamics (QED) corrections to extract the Rydberg constant independently of proton radius uncertainties that affect hydrogenic systems.^[31] The long-lived metastable states in antiprotonic helium facilitate narrow-linewidth laser resonances, with QED calculations up to higher orders ensuring theoretical predictions match experimental data at the parts-per-billion level.^[32] Key experiments at the Max Planck Institute of Quantum Optics (MPQ) in Garching have driven much of this progress, including the 2000 measurement of the hydrogen 1S-2S frequency yielding a Rydberg constant value with $4.2 \times 10^{-12} relative uncertainty, and subsequent refinements in 2017 using cryogenic atomic beams to reach $9 \times 10^{-13} precision. At the National Institute of Standards and Technology (NIST), precision spectroscopy of Rydberg states in hydrogen-like ions, such as ^{12}C^{5+}, has provided complementary determinations through comparisons of measured transition frequencies with QED predictions, achieving uncertainties around $10^{-11} and aiding cross-checks of the Rydberg constant.^[33] These results are integrated into CODATA adjustments, where the hydrogen 1S-2S data dominate the 2022 recommended value of R_\infty = 10\,973\,731.568\,157(12) m^{-1} with a relative uncertainty of $1.1 \times 10^{-12}).^[13]

Expressions in Terms of Fundamental Constants

Basic Expression

The basic expression for the Rydberg constant assuming infinite nuclear mass, denoted R_\infty, is

R_\infty = \frac{\alpha^2 m_e c}{2 h},

where \alpha is the fine-structure constant, m_e is the electron mass, c is the speed of light in vacuum, and h is Planck's constant. This formula encapsulates the theoretical foundation of the Rydberg constant as the limiting wavenumber for transitions to the ground state in hydrogen-like atoms. This expression can be derived within the framework of atomic units, where the fundamental unit of energy is the hartree energy E_h = \alpha^2 m_e c^2.^[34] In the quantum mechanical description of the hydrogen atom, the ground-state binding energy is -E_h / 2, corresponding to the Rydberg energy Ry = E_h / 2. The associated ionization wavenumber, representing the transition from the n=1 level to infinity, is then \bar{\nu} = Ry / (h c) = (\alpha^2 m_e c^2 / 2) / (h c) = \alpha^2 m_e c / (2 h), yielding R_\infty exactly. Following the 2019 redefinition of the International System of Units (SI), the values of h and c are fixed as exact constants, rendering the numerical evaluation of R_\infty dependent only on the experimentally determined values of \alpha and m_e.^[35] This exactness in the expression enhances its utility in metrology. The Rydberg constant serves as a vital bridge between atomic spectroscopy and fundamental constants, enabling precise spectroscopic measurements of hydrogen transitions to yield values of R_\infty that constrain \alpha, m_e, and related quantities through the defining equation.^[4]^[30]^[36]

Alternative Forms

The Rydberg constant can be expressed in alternative forms that emphasize its connections to other fundamental atomic and electromagnetic constants, providing insights into its physical origins beyond the standard expression in terms of electron mass, charge, and Planck's constant. One such form relates the infinite nuclear mass Rydberg constant to the fine structure constant and the Bohr radius:

R_\infty = \frac{\alpha}{4\pi a_0}

where \alpha is the fine structure constant and a_0 is the Bohr radius. This expression arises from substituting the definitions of \alpha and a_0 into the basic formula, underscoring the role of electromagnetic fine structure and atomic scale in spectral line spacing. Since the Bohr radius is linked to the reduced Compton wavelength \bar{\lambda}_C = h / (2\pi m_e c) by a_0 = \bar{\lambda}_C / \alpha, this form indirectly incorporates quantum relativistic effects through the Compton scale. The Rydberg constant is closely tied to the Rydberg energy Ry, defined as the binding energy of the hydrogen ground state, via the relation

Ry = hc R_\infty.

Expanding this in terms of elementary constants yields

Ry = \frac{m_e e^4}{2 (4\pi \epsilon_0)^2 \hbar^2},

where m_e is the electron mass, e is the elementary charge, \epsilon_0 is the vacuum permittivity, and \hbar is the reduced Planck's constant. This formulation explicitly shows how the energy scale emerges from Coulomb interactions and quantum mechanics, independent of the speed of light, and is equivalent to half the squared fine structure constant times the electron rest energy: Ry = \frac{1}{2} \alpha^2 m_e c^2.

Applications

In Atomic Spectroscopy

The Rydberg constant plays a pivotal role in atomic spectroscopy by enabling the precise prediction and interpretation of spectral line positions across various elements, serving as the scaling factor in the Rydberg formula that relates transition wavenumbers to quantum numbers.^[37] For hydrogen-like systems, it directly yields the energy differences between levels, but its broader utility extends to complex atoms through adjustments that account for electron interactions.^[37] In multi-electron atoms, the Rydberg-Ritz combination principle applies the constant to express any spectral line wavenumber as the difference between two spectral terms, each of the form T = R / (m + \mu)^2, where R is the Rydberg constant, m is the principal quantum number, and \mu is a term-specific correction.^[37] This principle, empirically established in the early 20th century, allows the construction of complete term schemes from observed lines, facilitating the identification of energy levels and transitions in spectra where direct series analysis is incomplete. For instance, in analyzing the spectra of ionized atoms like helium or alkaline earths, the principle uses a scaled Rydberg constant (e.g., 4R for He II) to match observed emission and absorption features. For alkali metals, the Rydberg constant is essential in quantum defect corrections to determine ionization energies from Rydberg series limits. The effective principal quantum number n^* = n - \delta_l modifies the energy levels as E_{nl} = -R_M / (n^*)^2, where \delta_l accounts for the penetration of the Rydberg electron into the ionic core, causing deviations from hydrogenic behavior.^[37] As n \to \infty, the series limit yields the ionization energy from the ground state, with \delta_l values (e.g., around 0.4 for s-states in sodium) derived iteratively from high-resolution spectra to refine atomic structure models.^[37] This approach has been crucial for alkali atoms like lithium and cesium, where core interactions significantly shift levels.^[37] Historically and in contemporary astrophysics, the Rydberg constant underpins elemental abundance measurements by identifying emission and absorption lines in stellar and interstellar spectra. Early analyses, such as those of solar and stellar atmospheres in the 1930s, used Rydberg series to classify lines and estimate relative abundances of elements like iron and magnesium through curve-of-growth methods. Ongoing applications involve fitting observed lines in H II regions or quasar spectra to Rydberg-corrected models, quantifying abundances of light elements (e.g., hydrogen, helium) and metals to probe nucleosynthesis and chemical evolution.^[38]

Rydberg Atoms and Contemporary Uses

Rydberg atoms are atoms in which one or more electrons occupy highly excited states characterized by a large principal quantum number n \gg 1, typically n > 10. In these states, the energy levels follow the scaling E \propto -R_\infty / n^2, where R_\infty is the Rydberg constant for infinite nuclear mass, leading to binding energies on the order of millielectronvolts or less.^[39]^[40] The outer electron in a Rydberg atom behaves similarly to that in hydrogen due to effective screening of the nuclear charge by inner electrons, resulting in exaggerated atomic properties.^[41] These properties include orbital radii scaling as n^2 a_0 (where a_0 is the Bohr radius), often exceeding micrometers for n \sim 100, which enhances dipole moments and enables long-range interactions up to several micrometers.^[42] Additionally, radiative lifetimes increase dramatically as \tau \propto n^3 to n^5, reaching milliseconds for high n, making Rydberg atoms suitable for precise control in quantum experiments.^[43] Such characteristics have positioned Rydberg atoms as a cornerstone for exploring quantum phenomena beyond traditional spectroscopy. In quantum sensing, Rydberg atoms serve as highly sensitive electric field probes across radio frequencies (RF) to terahertz, leveraging the Stark effect to shift energy levels proportionally to field strength.^[44] For instance, NIST researchers have developed Rydberg atom-based sensors traceable to SI units, achieving sensitivities below 1 μV/cm for RF fields by detecting electromagnetically induced transparency in alkali vapors.^[45] Recent advancements, such as a 2025 NIST thermometer using Rydberg atoms in rubidium vapor, enable temperature measurements with an absolute uncertainty of about 2 K over ranges such as 0 to 100°C, exploiting thermal broadening of spectral lines.^[46] The Rydberg blockade mechanism, where strong dipole-dipole interactions prevent simultaneous excitation of nearby atoms within a blockade radius r_b \propto n^4, underpins entangling quantum gates for computing.^[47] Seminal work has demonstrated high-fidelity controlled-phase gates in neutral atom arrays, with fidelities exceeding 99% using optimized pulse sequences to mitigate motional errors.^[48] In ultracold atom experiments during the 2020s, groups at NIST and ETH Zurich have advanced these applications; for example, a 2025 numerical simulation by researchers at EPFL of topological spin liquids employed a model of Rydberg atom arrays to model quantum many-body Hamiltonians.^[49] ETH Zurich's efforts include Rydberg excitation in magneto-optical traps for probing dipole blockade dynamics at densities of $10^{12} atoms/cm³.^[50] Extensions of Rydberg physics include Rydberg molecules, formed by low-energy collisions where a ground-state atom binds to a Rydberg electron via s-wave scattering, yielding ultralong bond lengths up to nanometers and lifetimes extended by Förster resonances.^[51] Rydberg polaritons, hybrid light-matter quasiparticles arising from strong atom-photon coupling in cavities, exhibit tunable interactions for nonlinear optics at the single-photon level.^[52] These systems facilitate simulations of quantum many-body dynamics, such as driven-dissipative propagation in one-dimensional lattices, revealing phenomena like ergodicity breaking and scar states in Rydberg arrays.^[53]^[54]

References

[1]
Rydberg constant† - CODATA Value
Numerical value, 10 973 731.568 157 m ; Standard uncertainty, 0.000 012 m ; Relative standard uncertainty, 1.1 x 10 ...CODATA Value
[2]
Rydberg Constant - an overview | ScienceDirect Topics
The Rydberg constant is defined as a fundamental constant that relates the wavelengths of spectral lines in hydrogen and serves as a key parameter in the ...
[3]
Getting the numbers right - the lonely struggle of Rydberg | Feature
Today, 150 years after his birth, the Rydberg constant remains a corner-stone of physics and physical chemistry. Further research has shown that the Rydberg's ...
[4]
A measurement of the Rydberg constant - Nature
May 1, 1979 · THE Rydberg constant is of key importance in spectroscopy and, because it may be expressed as the product of several fundamental physical ...
[5]
Rydberg Formula - Richard Fitzpatrick
Equation (700) is known as the Rydberg formula. Likewise, $R$ is called the Rydberg constant. The Rydberg formula was actually discovered empirically in the ...
[6]
Spectra - Galileo
... Rydberg constant, 109,737 cm-1. This constant is named after the Swedish physicist Rydberg who (in 1888) presented a generalization of Balmer's formula, in ...
[7]
Johannes Robert Rydberg (1854 - 1919) - Biography - MacTutor
Johannes Robert Rydberg was known as Janne Rydberg. His father, Sven R Rydberg, was a merchant who also owned several boats, while his mother was Maria Beata ...
[8]
The Spectral Lines of Hydrogen | Spectroscopy Online
The constant is now called Rydberg's constant for hydrogen (R H) and has a value of 1.097 × 10 7 m –1. For the Balmer series of hydrogen lines n 1 = 2 and n 2 ≥ ...
[9]
Bohr's Derivation of the Rydberg Formula: Quantum Numbers and ...
This is the Balmer formula. In 1888 Swedish physicist Johannes Rydberg ... Rydberg formula and an expression for the Rydberg constant in Step (3). In ...
[10]
Spectroscopy and Quantum Mechanics - MIT
J.R. Rydberg and others extended this idea to hydrogen spectral lines at other wavelengths, as well as to series of the alkali metals, the alkaline earths and ...
[11]
The revision of the SI—the result of three decades of progress in ...
The new definition of the kilogram is of particular importance because it eliminated the last definition referring to an artefact. In this way, the new ...
[12]
[PDF] SI Brochure - 9th ed./version 3.02 - BIPM
May 20, 2019 · Thus, this redefinition marks a significant and historic step forward. The changes were agreed by the CGPM in November 2018 with effect from ...
[13]
[PDF] CODATA RECOMMENDED VALUES OF THE FUNDAMENTAL ...
CODATA RECOMMENDED VALUES OF THE FUNDAMENTAL PHYSICAL CONSTANTS: 2022. NIST SP 961 (May 2024). An extensive list of constants is available on the NIST ...
[14]
[PDF] CODATA RECOMMENDED VALUES OF THE FUNDAMENTAL ...
CODATA RECOMMENDED VALUES OF THE FUNDAMENTAL PHYSICAL CONSTANTS: 2018. NIST ... Rydberg constant. R∞. 10 973 731.568 160(21). [m−1] energy equivalent in ...
[15]
[PDF] CODATA recommended values of the fundamental physical constants
Sep 26, 2016 · This paper gives the 2014 self-consistent set of values of the constants and conversion factors of physics and chemistry recommended by the ...Missing: R_H | Show results with:R_H
[16]
Hydrogen-deuterium isotope shift: From the -transition frequency to ...
Apr 12, 2011 · In general, the theoretical uncertainty in the isotope shift due to the mass ratios exceeds the experimental uncertainty recorded in Eq. (10)
[17]
Rydberg constant times hc in J - CODATA Value
Concise form, 2.179 872 361 1030(24) x 10-18 J ; Click here for correlation coefficient of this constant with other constants ; Source: 2022 CODATA
[18]
Rydberg constant times hc in eV - CODATA Value
Rydberg constant times hc in eV $hc\,R_\infty$ ; Numerical value, 13.605 693 122 990 eV ; Standard uncertainty, 0.000 000 000 015 eV.
[19]
https://www.informationphilosopher.com/solutions/scientists/bohr/Constitution_of_atoms_I.pdf
[20]
[PDF] 1913 On the Constitution of Atoms and Molecules
In order to explain the results of experiments on scattering of α rays by matter Prof. Rutherford1 has given a theory of the structure of atoms.
[21]
[PDF] Physics 228, Lecture 11 Monday, February 28, 2005 Bohr Model
Feb 28, 2005 · So Bohr's model explains the hydrogen spectrum completely, including cal- culating the Rydberg constant in terms of fundamental constants!
[22]
4.10: The Schrödinger Wave Equation for the Hydrogen Atom
Aug 15, 2025 · Schrödinger's approach requires three quantum numbers ( n , l , and m l ) to specify a wavefunction for the electron. The quantum numbers ...
[23]
[PDF] Schrödinger's original quantum–mechanical solution for hydrogen
Jul 4, 2021 · In 1926, Erwin Schrödinger wrote a series of papers that invented wave mechan- ics and set the foundation for much of the single-particle ...
[24]
[PDF] Chapter 10 The Hydrogen Atom The Schrodinger Equation in ...
In spherical coordinates, ψ = ψ(r, θ, φ), and the plan is to look for a variables separable solution such that ψ(r, θ, φ) = R(r)Y (θ, φ).
[25]
[PDF] Quantum Physics III Chapter 2: Hydrogen Fine Structure
Apr 2, 2022 · In order to understand better the spectrum and the properties of the Hydrogen atom one can apply an electric field, leading to the Stark effect ...Missing: Rydberg | Show results with:Rydberg
[26]
[PDF] Johann Jacob Balmer (1825-1898) - UB
Using measurements by H. W. Vogel and by Huggins of the ultraviolet lines of the hydrogen spectrum I have tried to derive a formula which will represent the ...Missing: grating | Show results with:grating
[27]
Rydberg and the development of atomic spectroscopy (Centennial ...
J. R. Rydberg's life and scientific activity are briefly reviewed. His fundamental paper of 1890 on the laws governing atomic spectra is considered in detail.
[28]
A Timeline of Atomic Spectroscopy
Oct 1, 2006 · 1919: German physicists Arnold Sommerfeld (1868–1951) and Walter Kossel note that the spectral lines of any atom are qualitatively similar ...<|separator|>
[29]
Robert Andrews Millikan 1868-1953
... formula for the hydrogen spectrum, accurately gave the Rydberg constant—the first and most convincing proof of Bohr's quantum theory of the atom. Shortly ...
[30]
[PDF] The Electronic Spectrum and Energy Levels of the Deuterium ...
Dieke and his students carried out an extensive program of measuring the optical spectrum of molecular hydrogen and its isotopes. Parts of the work were ...
[31]
[PDF] 1959 Research Highlights of the National Bureau of Standards
... spectrum led to an independent confirmation of the Rydberg constant, which since. Equipment used in the redetermination of the Rydberg constant. As a result.
[32]
Hydrogen Spectroscopy - MPQ - Max-Planck-Gesellschaft
In particular, measurements of the 1S-2S two-photon transition frequency in hydrogen and deuterium served as a corner stone in tests of bound-state quantum ...
[33]
The Rydberg constant and proton size from atomic hydrogen - Science
Using a cryogenic beam of H atoms, we measured the 2S-4P transition frequency in H, yielding the values of the Rydberg constant R∞ = 10973731.568076(96) per ...
[34]
Recent progress of laser spectroscopy experiments on antiprotonic ...
Feb 19, 2018 · ... Rydberg constant. The corrections due to the finite charge radii of the helium nucleus (4–7 MHz) and of the antiproton [24] (less than 1 MHz) ...
[35]
[PDF] Precision laser spectroscopy experiments on antiprotonic helium
... quantum electrodynamics (QED) cor- rections up to order ... the Rydberg antiproton orbital and the nucleus is relatively small, and because the antiproton.
[36]
Fundamental Constants and Tests of Theory in Rydberg States of ...
Apr 22, 2008 · Abstract. A comparison of precision frequency measurements to quantum electrodynamics (QED) predictions for Rydberg states of hydrogen-like ...
[37]
CODATA recommended values of the fundamental physical constants
Jun 30, 2021 · A factor of 1.6 is applied to the 62 input data determining the Rydberg constant and proton and deuteron charge radii. A factor 1.7 is used for ...
[38]
[PDF] Fundamental Physical Constants — Extensive Listing
Hartree energy e2/4π 0a0 = 2R∞hc. = α2mec2. Eh. 4.359 744 17(75) × 10−18. J. 1.7 × 10−7 in eV. 27.211 3845(23). eV. 8.5 × 10−8 quantum of circulation h/2me.
[39]
[PDF] Redefinition-of-the-SI.pdf
May 22, 2018 · 1971: mole becomes 7th base unit. 1983: speed of light defined – the metre is based on a fundamental constant. 1990: conventional values for ...
[40]
[PDF] Spectroscopy of atomic hydrogen. How is the Rydberg constant ...
Sep 17, 2008 · Consequently, in spite of the very high accuracy of the 1S − 2S measurement (1.4×10−14), the Rydberg constant can be deduced from equation 14 ...
[41]
[PDF] Atomic Spectroscopy - National Institute of Standards and Technology
Mar 3, 1999 · The Rydberg constant R is thus the limit value for infinite nuclear mass. 2. configuration, according to the code s, p, d for l = 0, 1, 2 and f ...
[42]
[PDF] III Ionized Hydrogen (HII) Regions - OSU astronomy
The elemental abundances determine the amounts of the various metal-line coolants present. ... Rydberg formula: │. ⌋. ⌉. │. ⌊. ⌈. ∆. +. -. = 2. 2. 2. ) (. 1. 1 n.
[43]
An Introduction to Rydberg atoms with ARC
taken from measured energy levels. The mass-corrected Rydberg constant is ... with reduced mass μ = M m e / ( M + ...
[44]
6.3: Rydberg Spectra of Polyelectronic Atoms - Chemistry LibreTexts
Oct 30, 2023 · These highly excited states of atoms are called “Rydberg States” and to a good approximation, the excited electron in a Rydberg state “feels” the nucleus of ...
[45]
Rydberg Physics
A Rydberg atom is an atom excited into a high energy level. Studies of Rydberg atoms have a long history, starting over 80 years ago with investigations of ...
[46]
Rydberg Atoms in External Fields | Atomic Physics Class Notes
Rydberg atoms are highly excited atoms with one or more electrons in a state with a large principal quantum number n, typically n > 10 · The large size of ...Missing: definition | Show results with:definition
[47]
Rydberg renaissance - Il Nuovo Saggiatore
Almost 130 years ago, Johannes Robert Rydberg wrote down a formula that contained the observed series of spectral lines of hydrogen and predicted the ...
[48]
Rydberg Atom-based Quantum RF Field Probes | NIST
Apr 16, 2021 · The Rydberg Atom-based RF Field Probes team is using highly excited alkali atoms to detect and measure electric fields from DC to daylight (from Hz to THz)
[49]
[PDF] 11 Overview of Rydberg Atom-Based Sensors/Receivers for the ...
Rydberg atoms provide a pathway to E-field measurements that are traceable to the SI. Rydberg atoms are atoms with one or more electrons excited to a very high ...
[50]
New Atom-Based Thermometer Measures Temperature More ...
Jan 23, 2025 · NIST scientists have developed a new method for measuring temperature extremely accurately by using giant “Rydberg” atoms.Missing: ultracold ETH Zurich 2020s
[51]
[PDF] Quantum information with Rydberg atoms - UW–Madison Physics
Aug 18, 2010 · Rydberg enabled capabilities include long-range two-qubit gates, collective encoding of multiqubit registers, implementation of robust light- ...
[52]
Quantum simulation and computing with Rydberg-interacting qubits
May 3, 2021 · Arrays of optically trapped atoms excited to Rydberg states have recently emerged as a competitive physical platform for quantum simulation and computing.
[53]
Predicting the topological properties of quantum spin liquids using ...
Aug 17, 2025 · Building on earlier research studies, Mauron and her colleagues set out to simulate a topological spin liquid using a Rydberg atom-based ...<|separator|>
[54]
Spectroscopy of Ultracold Samples
In our experiments, laser-cooled cesium atoms at a temperature of 40 µK and a density of 1012 atoms/cm3 are excited to np3/2 Rydberg states with pulsed and ...Missing: NIST 2020s
[55]
Exotic Chemistry With Ultracold Rydberg Atoms - PubMed
We review recent experiments carried out with dense (10(12) cm(-3)) ultracold (T = 40 μK) samples of Cs atoms which have the goal to characterize, ...Missing: NIST 2020s
[56]
A weakly-interacting many-body system of Rydberg polaritons ...
May 20, 2021 · Rydberg-state atoms provide a strong dipole–dipole interaction (DDI) and a dipole blockade mechanism suitable for applications in quantum ...
[57]
Quantum Many-Body Dynamics of Driven-Dissipative Rydberg ...
Dec 30, 2020 · We study the propagation of strongly interacting Rydberg polaritons through an atomic medium in a one-dimensional optical lattice.
[58]
[PDF] Many-body physics with individually controlled Rydberg atoms
This suggested that systems of neutral atoms in the Rydberg blockade regime could be used for quantum simulation, provided individual control of a large number.<|control11|><|separator|>