Rydberg constant

(′rid′bərg ′kän·stənt)

(atomic physics) The most accurately measured of the fundamental constants, which enters into the formulas for wave numbers of atomic spectra and serves as a universal scaling factor for any spectroscopic transition and as an important cornerstone in the determination of other constants; it is equal to α²mc/(2h), or, in International System (SI) units, to me⁴/(8h³ε₀²c), where α is the fine-structure constant, m and e are the electron mass and charge, c is the speed of light, h is Planck's constant, and ε₀ is the electric constant; numerically, it is equal to 10,973,731.568 549 ± 0.000 083 inverse meters. Symbolized R_∞. For any atom, the Rydberg constant (first definition) divided by 1 + m/M, where m and M are the masses of an electron and of the nucleus.

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The most accurately measured of the fundamental constants; it is a universal scaling factor for any spectroscopic transition and an important cornerstone in the determination of other constants.

This constant was introduced empirically. J. Balmer's formula described the visible spectral lines of atomic hydrogen, while J. Rydberg's formula applied to the spectra of many elements. Their results may be summarized by Eq. (1),
1. $\frac{1}{\lambda} = R\Big(\frac{1}{n_1^2} - \frac{1}{n_2^2}\Big)$
where λ is the wavelength of the spectral line and R is a constant. In Balmer's account of the visible hydrogen spectrum, n₁ was equal to 2, while n₂ took on the integer values 3, 4, 5, and so forth. In Rydberg's more general work, n₁ and n₂ differed slightly from integer values. A remarkable result of Rydberg's work was that the constant R was the same for all spectral series he studied, regardless of the element. This constant R has come to be known as the Rydberg constant.

Applied to hydrogen, Niels Bohr's atomic model leads to Balmer's formula with a predicted value for the Rydberg constant given by Eq. (2),
2. $R_\infty = \frac{m_e e^4}{8 h^3 \epsilon_0^2 c}$
where m_e is the electron mass, e is the electron charge, h is Planck's constant, ε₀ is the permittivity of vacuum, and c is the speed of light. The equation expresses the Rydberg constant in SI units. To express it in cgs units, the right-hand side must be multiplied by (4πε₀)². The subscript ∞ means that this is the Rydberg constant corresponding to an infinitely massive nucleus.

E. Schrödinger's wave mechanics predicts the same energy levels as the simple Bohr model, but the relativistic quantum theory of P. A. M. Dirac introduces small corrections or fine-structure splittings. The modern theory of quantum electrodynamics predicts further corrections. Additional small hyperfine-structure corrections account for the interaction of the electron and nuclear magnetic moments. See also Fine structure (spectral lines); Hyperfine structure.

The Rydberg constant is determined by measuring the wavelength or frequency of a spectral line of a hydrogenlike atom or ion. The highest resolution and accuracy has been achieved by the method of Doppler-free two-photon spectroscopy, which permits the observation of very sharp resonance transitions between long-living states. The 2002 adjustment of the fundamental constants, taking into account different measurements, adopted the value R_∞ = 10,973,731.568,525 ± 0.000,073 m⁻¹ for the Rydberg constant. The measurements provide an important cornerstone for fundamental tests of basic laws of physics. See also Atomic structure and spectra; Fundamental constants; Laser; Laser spectroscopy.

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physics. Symbol R_∞. A constructed quantity fundamental to wave number for all atomic spectra (hence their frequencies), 2π²m_ee⁴/c·h = μ₀²m·e(e·c/2 h)³ = 10 973 731.568 525(73) m^-1 with relative standard uncertainty 6.6 × 10^-12.
[Mohr P. J., Taylor B. N. CODATA Recommended Values of the Fundamental Physical Constants: 2002 (to be published)]
[Mohr P. J., Taylor B. N. Rev. Mod. Phys. Vol. 72:351-495 (2000)]
[Mohr P. Phys. Today Vol. 53:7, 11-16 (2000)]
[For latest recommended values, see http://physics.nist.gov/cuu/Constants/index.html] The reciprocal of the frequency term 4πR_∞·c is the atomic unit of time (= 24.188 8~ as).

For double Rydberg, see Hartree.

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Rydberg constant (rĭd'bərg), physical constant used in studies of the spectrum of a substance. Its value for hydrogen is 109,737.3 cm⁻¹.

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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.

1 Value of the Rydberg constant
2 Alternative expressions
3 The derivation of Rydberg constant from quantum mechanics
4 See also
5 References

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:

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

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

$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 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.

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

$R_H = \frac{ m_e e^4}{8 \epsilon_0^2 h^3 c}$

References

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Rydberg constant

Contents

Value of the Rydberg constant

Alternative expressions

The derivation of Rydberg constant from quantum mechanics

See also

References