Rydberg constant

The Rydberg constant, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but it was later found that its value could be calculated from more fundamental constants by using quantum mechanics.

The Rydberg constant represents the limiting value of the highest wavenumber (the inverse wavelength) of any photon that can be emitted from the hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon 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.

Value of the Rydberg constant

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

$R_{\infty }={\frac {m_{e}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}=1.097\;373\;156\;852\;5\;(73)\times 10^{7}\ \mathrm {m} ^{-1},$

where m_e is the rest mass of the electron, e is the elementary charge, ε₀ is the permittivity of free space, h is the Planck constant, and c is the speed of light in a vacuum.

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

$hcR_{\infty }=13.605\;6923(12)\ \mathrm {eV} \equiv 1\ \mathrm {Ry} .$

Two complications arise. One is that one may wish to discuss a hydrogen-like ion; that is, an atom with atomic number Z that has only one electron, such as C⁵⁺. In this case, the wavenumbers and photon energies are scaled up by a factor of Z². 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. The Rydberg constant R_M for an atom with one electron is then given by

$R_{M}={\frac {R_{\infty }}{1+m_{e}/M}},$

where m_e is the rest mass of the electron, and 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.

Alternative 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

$hcR_{\infty }={\frac {hc\alpha ^{2}}{2\lambda _{e}}}={\frac {hf_{C}\alpha ^{2}}{2}}={\frac {\hbar \omega _{C}}{2}}\alpha ^{2}={\dfrac {\hbar ^{2}}{2m_{e}a_{0}^{2}}}.$

where

$h\!$ is the Planck constant

$\hbar =h/2\pi$ is the reduced Planck constant,

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

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

$\lambda _{e}=h/m_{e}c\!$ is the Compton wavelength of the electron,

$f_{C}=c/\lambda _{e}\!$ is the Compton frequency of the electron,

$\omega _{C}=2\pi f_{C}\!$ is the Compton angular frequency of the electron,

$a_{0}={\frac {e^{2}m_{e}}{4\pi \varepsilon _{0}\hbar ^{2}}}$ is the Bohr radius.

The derivation of Rydberg constant from quantum mechanics

Historically, the Rydberg equation was found empirically (experimentally), and it predated the development of quantum theory. (See Rydberg formula for a full discussion of its discovery.) 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)\$

where

$h\$ is 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.

$n_{\mathrm {initial} }\$ and $n_{\mathrm {final} }\$ being the electron shell number of the hydrogen atom

We have therefore found the Rydberg constant for hydrogen to be

$R_{H}={\frac {m_{e}e^{4}}{8\epsilon _{0}^{2}h^{3}c}}$

See also

• Rydberg formula, includes a discussion of Rydberg's original discovery.
• Coffman, Moody L. "Correction to the Rydberg Constant for Finite Nuclear Mass." American Journal of Physics. Volume 33, Issue 10, pp. 820-823.

References

• CODATA recommendations 2006
• Mathworld