Rydberg constant

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

${\displaystyle \frac{1}{\lambda }=\mathrm{R}\mathrm{y}\cdot \frac{1}{hc}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)=\frac{m_{\text{e}}{e}^4}{8{\epsilon }_0^2{h}^{3}c}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)}$

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 ${\displaystyle R_M=\frac{R_{\mathrm{\infty }}}{1+\frac{m_{\text{e}}}{M}},}$ 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 1.1×10⁻¹². This precision constrains the values of the other physical constants that define it.

Since the Bohr model is not perfectly accurate, due to fine structure, hyperfine splitting, and other such effects, the Rydberg constant ${\displaystyle R_{\mathrm{\infty }}}$ 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 ${\displaystyle R_{\mathrm{\infty }}}$ 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.

${\displaystyle R_{\mathrm{\infty }}=\frac{{\alpha }^2{m}_{\text{e}}c}{2h}=\frac{{\alpha }^2}{2{\lambda }_{\text{e}}}=\frac{\alpha }{4\pi a_0}}$

and in energy units

${\displaystyle \text{Ry}=hc{R}_{\mathrm{\infty }}=\frac{1}{2}{m}_{\text{e}}{c}^2{\alpha }^2=\frac{1}{2}\frac{e^4{m}_{\text{e}}}{(4\pi {\epsilon }_0{)}^2{\hslash }^2}=\frac{1}{2}\frac{m_{\text{e}}{c}^2{r}_{\text{e}}}{a_0}=\frac{1}{2}\frac{hc{\alpha }^2}{{\lambda }_{\text{e}}}=\frac{1}{2}h{f}_{\text{C}}{\alpha }^2=\frac{1}{2}\hslash {\omega }_{\text{C}}{\alpha }^2=\frac{1}{2m_{\text{e}}}{\left({\displaystyle \frac{\hslash }{a_0}}\right)}^2=\frac{1}{2}\frac{e^2}{(4\pi {\epsilon }_0)a_0},}$

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,
• ${\displaystyle \hslash =h/2\pi }$ is the reduced Planck constant,
• ${\displaystyle c}$ is the speed of light in vacuum,
• ${\displaystyle {\epsilon }_0}$ is the electric constant (vacuum permittivity),
• ${\displaystyle \alpha =\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{\hslash c}}$ 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,
• ${\displaystyle a_0=4\pi {\epsilon }_0{\hslash }^2/e^2{m}_{\text{e}}}$ is the Bohr radius,
• ${\displaystyle r_{\mathrm{e}}=\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{m_{\mathrm{e}}{c}^2}}$ 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: ${\displaystyle E_n=-hc{R}_{\mathrm{\infty }}/n^2}$.