Rydberg constant

In spectroscopy, the Rydberg constant, symbol ${\displaystyle R_{\infty }}$ for heavy atoms or ${\displaystyle R_{\text{H}}}$ 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, ${\displaystyle R_{\infty }}$ and the electron spin g-factor were the most accurately measured physical constants.^[1]

The constant is expressed for either hydrogen as ${\displaystyle R_{\text{H}}}$, or at the limit of infinite nuclear mass as ${\displaystyle R_{\infty }}$. 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 ${\displaystyle R_{\text{H}}}$ 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.^{[citation needed]}

Value

Rydberg constant

The CODATA value is^[2]

${\displaystyle R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}=10973731.568160(21)\,{\text{m}}^{-1},}$

where

• ${\displaystyle m_{\text{e}}}$ is the rest mass of the electron (i.e. the electron mass),
• ${\displaystyle e}$ is the elementary charge,
• ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space,
• ${\displaystyle h}$ is the Planck constant, and
• ${\displaystyle c}$ is the speed of light in vacuum.

The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron:

${\displaystyle R_{\text{H}}=R_{\infty }{\frac {m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}}\approx 1.09678\times 10^{7}{\text{ m}}^{-1},}$

where

• ${\displaystyle m_{\text{e}}}$ is the mass of the electron,
• ${\displaystyle m_{\text{p}}}$ is the mass of the nucleus (a proton).

Rydberg unit of energy

The Rydberg unit of energy is equivalent to joules^[3] and electronvolts^[4] in the following manner:

${\displaystyle 1\ {\text{Ry}}\equiv hcR_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{2}}}={\frac {e^{2}}{8\pi \varepsilon _{0}a_{0}}}=2.179\;872\;361\;1035(42)\times 10^{-18}\ {\text{J}}\ =13.605\;693\;122\;994(26)\ {\text{eV}}.}$

Rydberg frequency

${\displaystyle cR_{\infty }=3.289\;841\;960\;2508(64)\times 10^{15}\ {\text{Hz}}.}$^[5]

Rydberg wavelength

${\displaystyle {\frac {1}{R_{\infty }}}=9.112\;670\;505\;824(17)\times 10^{-8}\ {\text{m}}}$.

The corresponding angular wavelength is

${\displaystyle {\frac {1}{2\pi R_{\infty }}}=1.450\;326\;555\;7696(28)\times 10^{-8}\ {\text{m}}}$.

In 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,^[6] 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 ${\displaystyle \infty }$ subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (see Rydberg formula):

${\displaystyle {\frac {1}{\lambda }}=\mathrm {Ry} \cdot {1 \over hc}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$

where n₁ and n₂ are any two different positive integers (1, 2, 3, ...), and ${\displaystyle \lambda }$ is the wavelength (in vacuum) of the emitted or absorbed light, giving

${\displaystyle {\frac {1}{\lambda }}=R_{M}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$

where ${\displaystyle R_{M}={\frac {R_{\infty }}{1+{\frac {m_{\text{e}}}{M}}}},}$ 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 was one of the most precisely determined physical constants, with a relative standard uncertainty of under 2 parts in 10¹².^[2] This precision constrains the values of the other physical constants that define it.^[7]

Since the Bohr model is not perfectly accurate, due to fine structure, hyperfine splitting, and other such effects, the Rydberg constant ${\displaystyle R_{\infty }}$ 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 ${\displaystyle R_{\infty }}$ is determined from the best fit of the measurements to the theory.^[8]

Alternative expressions

The Rydberg constant can also be expressed as in the following equations.

${\displaystyle R_{\infty }={\frac {\alpha ^{2}m_{\text{e}}c}{4\pi \hbar }}={\frac {\alpha ^{2}}{2\lambda _{\text{e}}}}={\frac {\alpha }{4\pi a_{0}}}}$

and in energy units

${\displaystyle {\text{Ry}}=hcR_{\infty }={\frac {1}{2}}m_{\text{e}}c^{2}\alpha ^{2}={\frac {1}{2}}{\frac {e^{4}m_{\text{e}}}{(4\pi \varepsilon _{0})^{2}\hbar ^{2}}}={\frac {1}{2}}{\frac {m_{\text{e}}c^{2}r_{\text{e}}}{a_{0}}}={\frac {1}{2}}{\frac {hc\alpha ^{2}}{\lambda _{\text{e}}}}={\frac {1}{2}}hf_{\text{C}}\alpha ^{2}={\frac {1}{2}}\hbar \omega _{\text{C}}\alpha ^{2}={\frac {1}{2m_{\text{e}}}}\left({\dfrac {\hbar }{a_{0}}}\right)^{2}={\frac {1}{2}}{\frac {e^{2}}{(4\pi \varepsilon _{0})a_{0}}},}$

where

• ${\displaystyle m_{\text{e}}}$ is the electron rest mass,
• ${\displaystyle e}$ is the electric charge of the electron,
• ${\displaystyle h}$ is the Planck constant,
• ${\displaystyle \hbar =h/2\pi }$ is the reduced Planck constant,
• ${\displaystyle c}$ is the speed of light in vacuum,
• ${\displaystyle \varepsilon _{0}}$ is the electric constant (vacuum permittivity),
• ${\displaystyle \alpha ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{\hbar c}}}$ is the fine-structure constant,
• ${\displaystyle \lambda _{\text{e}}=h/m_{\text{e}}c}$ is the Compton wavelength of the electron,
• ${\displaystyle f_{\text{C}}=m_{\text{e}}c^{2}/h}$ is the Compton frequency of the electron,
• ${\displaystyle \omega _{\text{C}}=2\pi f_{\text{C}}}$ is the Compton angular frequency of the electron,
• ${\displaystyle a_{0}={4\pi \varepsilon _{0}\hbar ^{2}}/{e^{2}m_{\text{e}}}}$ is the Bohr radius,
• ${\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}}$ 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: ${\displaystyle E_{n}=-hcR_{\infty }/n^{2}}$.

References