In
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter ...
, the Rydberg constant, symbol
for
heavy atoms or
for hydrogen, named after the Swedish
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate cau ...
Johannes Rydberg
Johannes (Janne) Robert Rydberg (; 8 November 1854 – 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electro ...
, is a
physical constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...
relating to the electromagnetic
spectra of an atom. The constant first arose as an empirical fitting parameter in the
Rydberg formula for the
hydrogen spectral series, but
Niels Bohr later showed that its value could be calculated from more fundamental constants via his
Bohr model.
Before the
redefinition of the SI base units
In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artifacts such as the standard kilogram.
Effective 20 May 2019, the 144t ...
in ,
and the electron spin
''g''-factor were the most accurately measured
physical constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...
s.
The constant is expressed for either hydrogen as
, or at the limit of infinite nuclear mass as
. In either case, the constant is used to express the limiting value of the highest
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
(inverse wavelength) of any photon that can be emitted from an atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing an atom from its ground state. The
hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen
and the
Rydberg formula.
In
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, Rydberg unit of energy, symbol Ry, corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom in a simplified Bohr model.
Value
Rydberg constant
The
CODATA
The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
value is
:
,
where
*
is the
rest mass of the
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
,
*
is the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
,
*
is the
permittivity of free space
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
,
*
is the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, and
*
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in vacuum.
The Rydberg constant for hydrogen may be calculated from the
reduced mass of the electron:
:
where
*
is the mass of the electron,
*
is the mass of the nucleus (a proton).
Rydberg unit of energy
The Rydberg unit of energy is equivalent to joules
and electronvolts
in the following manner:
:
Rydberg frequency
:
Rydberg wavelength
:
.
The
angular wavelength is
:
.
Occurrence in Bohr model
The
Bohr model explains the atomic
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of hydrogen (see
hydrogen spectral series) as well as various other atoms and ions. It is not perfectly accurate, but is a remarkably good approximation in many cases, and historically played an important role in the development of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The Bohr model posits that electrons revolve around the atomic nucleus in a manner analogous to planets revolving around the sun.
In the simplest version of the Bohr model, the mass of the atomic nucleus is considered to be infinite compared to the mass of the electron,
so that the center of mass of the system, the
barycenter, lies at the center of the nucleus. This infinite mass approximation is what is alluded to with the
subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (see
Rydberg formula):
:
where ''n''
1 and ''n''
2 are any two different positive integers (1, 2, 3, ...), and
is the wavelength (in vacuum) of the emitted or absorbed light.
:
where
and ''M'' is the total mass of the nucleus. This formula comes from substituting the
reduced mass of the electron.
Precision measurement
The Rydberg constant is one of the most precisely determined physical constants, with a relative standard uncertainty of under 2 parts in 10
12. This precision constrains the values of the other physical constants that define it.
[P.J. Mohr, B.N. Taylor, and D.B. Newell (2015), "The 2014 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 7.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants. National Institute of Standards and Technology, Gaithersburg, MD 20899]
Link to R∞
Link to hcR∞
Published in and .
Since the Bohr model is not perfectly accurate, due to
fine structure
In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
,
hyperfine splitting
In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the ...
, and other such effects, the Rydberg constant
cannot be ''directly'' measured at very high accuracy from the
atomic transition frequencies of hydrogen alone. Instead, the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
,
deuterium
Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one ...
, and
antiprotonic helium). Detailed theoretical calculations in the framework of
quantum electrodynamics are used to account for the effects of finite nuclear mass, fine structure, hyperfine splitting, and so on. Finally, the value of
is determined from the
best fit of the measurements to the theory.
Alternative expressions
The Rydberg constant can also be expressed as in the following equations.
:
and in energy units
:
where
*
is the
electron rest mass,
*
is the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
of the electron,
*
is the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
,
*
is the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
,
*
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in a vacuum,
*
is the electrical field constant (
permittivity) of free space,
*
is the
fine-structure constant,
*
is the
Compton wavelength of the electron,
*
is the Compton frequency of the electron,
*
is the Compton angular frequency of the electron,
*
is the
Bohr radius,
*
is the
classical electron radius.
The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4''π''/''α'' times the Bohr radius of the atom.
The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom:
.
References
{{Scientists whose names are used in physical constants
Emission spectroscopy
Physical constants
Units of energy