Rydberg constant explained
In spectroscopy, the Rydberg constant, symbol
for heavy atoms or
for hydrogen, named after the Swedish
physicist Johannes Rydberg, is a
physical 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 according to his
model of the atom.
Before the 2019 redefinition of the SI base units,
and the electron spin
g-factor were the most accurately measured
physical constants.
[1] 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 (inverse wavelength) of any photon that can be emitted from a hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing a hydrogen 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, 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 value is
where
is the
rest mass of the
electron (i.e. the
electron mass),
is the
elementary charge,
is the
permittivity of free space,
is the
Planck constant, and
is the
speed of light in vacuum.
The symbol
means that the nucleus is assumed to be infinitely heavy, an improvement of the value can be made using the
reduced mass of the atom:
with
the mass of the nucleus. The corrected Rydberg constant is:
that for hydrogen, where
is the mass
of the
proton, becomes:
RH=
Rinfty ≈ 1.09678 x 107m-1,
Since the Rydberg constant is related to the spectrum lines of the atom, this correction leads to an isotopic shift between different isotopes. For example, deuterium, an isotope of hydrogen with a nucleus formed by a proton and a neutron (
), was discovered thanks to its slightly shifted spectrum.
[2] Rydberg unit of energy
The Rydberg unit of energy is
1 Ry~~\equivhcRinfty=\alpha2mec2/2
=
=
Rydberg frequency
=
Rydberg wavelength
=9.112 670 505 826(10) x 10-8 m
.
The corresponding angular wavelength is
=1.450 326 555 77(16) x 10-8 m
.
Bohr model
See main article: Bohr model. The Bohr model explains the atomic spectrum 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. 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,[3] 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):
=Ry ⋅ {1\overhc}\left(
\right)
where
n1 and
n2 are any two different positive integers (1, 2, 3, ...), and
is the wavelength (in vacuum) of the emitted or absorbed light, giving
where
and
M is the total mass of the nucleus. This formula comes from substituting the
reduced mass of the electron.
Precision measurement
See also: Precision tests of QED. The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of This precision constrains the values of the other physical constants that define it.[4]
Since the Bohr model is not perfectly accurate, due to fine structure, hyperfine splitting, 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,
deuterium, 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.
[5] Alternative expressions
The Rydberg constant can also be expressed as in the following equations.
and in energy units
Ry=hcRinfty=
mec2\alpha2=
=
=
=
hfC\alpha2=
\hbar\omegaC\alpha2=
| 2 |
\left(\dfrac{\hbar}{a | |
| 0}\right) |
=
,
where
is the
electron rest mass,
is the
electric charge of the electron,
is the
Planck constant,
is the reduced Planck constant,
is the
speed of light in vacuum,
is the
electric constant (vacuum permittivity),
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 4π/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:
.
See also
Notes and References
- The size of the proton . Nature . 466 . 7303 . 213–216. 2010 . 20613837. 10.1038/nature09250. 2010Natur.466..213P . Antognini . Nez . Amaro . Biraben . Cardoso . Covita . Dax . Fernandes . Luis M. P. . Giesen . Graf . Hänsch . Indelicato . Julien . Kao . Knowles . Le Bigot . Liu . Yi-Wei . Lopes . José A. M. . Ludhova . Monteiro . Mulhauser . Nebel . Rabinowitz . Dos Santos . Schaller . Schuhmann . Schwob . Catherine . Taqqu . David . Pohl . Randolf . Aldo . François . Fernando D. . François . João M. R. . Daniel S. . Andreas . Dhawan . Satish . Adolf . Thomas . Theodor W. . Paul . Lucile . Cheng-Yang . Paul . Eric-Olivier . Livia . Cristina M. B. . Françoise . Tobias . Paul . Joaquim M. F. . Lukas A. . Karsten . 4424731 .
- Quantum Mechanics (2nd Edition), B.H. Bransden, C.J. Joachain, Prentice Hall publishers, 2000,
- Correction to the Rydberg Constant for Finite Nuclear Mass . American Journal of Physics. 33 . 10 . 820–823 . 1965 . 10.1119/1.1970992. 1965AmJPh..33..820C . Coffman . Moody L. .
- 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 10.1103/RevModPhys.84.1527. "". CODATA recommended values of the fundamental physical constants: 2010. 2012. Mohr. Peter J.. Taylor. Barry N.. Newell. David B.. Reviews of Modern Physics. 84. 4. 1527–1605. 1203.5425 . 2012RvMP...84.1527M . 103378639. and 10.1063/1.4724320. "". CODATA Recommended Values of the Fundamental Physical Constants: 2010. 2012. Mohr. Peter J.. Taylor. Barry N.. Newell. David B.. Journal of Physical and Chemical Reference Data. 41. 4. 043109. 2012JPCRD..41d3109M . 1507.07956. .
- 10.1103/RevModPhys.80.633 . CODATA recommended values of the fundamental physical constants: 2006 . Reviews of Modern Physics . 80 . 633–730 . 2008. 2008RvMP...80..633M . 2. 0801.0028 . Taylor . Newell . Mohr . Peter J. . Barry N. . David B. .