Rydberg constant

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The Rydberg constant, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic specta. It represents the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from the hydrogen atom, or, alternatively, the lowest wavenumber capable of ionizing the hydrogen atom from its ground state. The spectrum of hydrogen can be expressed simply in terms of the Rydberg constant, using the Rydberg formula.

Making use of the simplifying assumption that the mass of the atomic nucleus is infinite compared to the mass of the electron, the Rydberg constant is (according to 2002 CODATA results):

$R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c} = 1.0973731568525(73) \cdot 10^7 \,\mathrm{m}^{-1}$

where,

$h \$ is the Planck's constant,

$m_e \$ is the rest mass of the electron,

$e \$ is the elementary charge,

$c \$ is the speed of light in vacuum, and

$\epsilon_0 \$ is the permittivity of free space.

This constant is often used in atomic physics in the form of an energy:

$h c R_\infty = 13.6056923(12) \,\mathrm{eV} \equiv 1 \,\mathrm{Ry} \$

Two complications arise. One is that one may wish to discuss a hydrogenlike ion, i.e., an atom with atomic number Z that has only one electron. In this case, the wavenumbers and photon energies are scaled up by a factor of $Z 2$ . The other is that the mass of the atomic nucleus is not actually infinite compared to the mass of the electron. The predicted spectrum must then be corrected by substituting the reduced mass for the mass of the electron, resulting in:

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

where,

$R_M \$ is the Rydberg constant for a certain atom with one electron with the rest mass $m_e \$

$M \$ is the mass of the atomic nucleus.

The Rydberg constant is one of the most well-determined physical constants, with a relative experimental uncertainty of less than 7 parts per trillion. The ability to measure it directly to such a high precision constrains the proportions of the values of the other physical constants that define it.

1 Alternate expressions
2 Derivation of Rydberg constant
3 See also
4 References

[edit] Alternate expressions

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

$R_\infty = \frac{\alpha^2 m_e c}{4 \pi \hbar} = \frac{\alpha^2}{2 \lambda_e} \$

and

$h c R_\infty = \frac{h c \alpha^2}{2 \lambda_e} = \frac{h f_C \alpha^2}{2} = \frac{\hbar \omega_C}{2} \alpha^2 \$

where

$h \$ is Planck's constant,

$c \$ is the speed of light in a vacuum,

$\alpha \$ is the fine-structure constant,

$\lambda_e \$ is the Compton wavelength of the electron,

$f_C \$ is the Compton frequency of the electron,

$\hbar \$ is the reduced Planck's constant, and

$\omega_C \$ is the Compton angular frequency of the electron.

[edit] Derivation of Rydberg constant

Historically, the Rydberg equation was found empirically, and predated quantum theory. To understand its significance in terms of the quantum theory, we can start from the equation

$E_\mathrm{total} = \frac{- m_e e^4}{8 \epsilon_0^2 h^2}. \frac{1}{n^2} \$

for the energy of the electron in the nth energy state, as can be derived either from the Bohr model or from a fully quantum-mechanical treatment of the hydrogen atom. Therefore a change in energy in an electron changing from one value of $n$ to another is

$\Delta E = \frac{ m_e e^4}{8 \epsilon_0^2 h^2} \left( \frac{1}{n_\mathrm{initial}^2} - \frac{1}{n_\mathrm{final}^2} \right) \$

We simply change the units to wavelength $\left( \frac{1}{ \lambda} = \frac {E}{hc} \rightarrow \Delta{E} = hc \Delta \left( \frac{1}{\lambda}\right)\right) \$ and we get

$\Delta \left( \frac{1}{ \lambda}\right) = \frac{ m_e e^4}{8 \epsilon_0^2 h^3 c} \left( \frac{1}{n_\mathrm{initial}^2} - \frac{1}{n_\mathrm{final}^2} \right) \$