# Rydberg's constant

Physical constant
Surname Rydberg's constant
Formula symbol ${\ displaystyle R _ {\ infty}}$
value
SI 1.097 373 156 8160 (21)e7th ${\ displaystyle \ textstyle {\ frac {1} {\ mathrm {m}}}}$
Uncertainty  (rel.) 1.9e-12
Relation to other constants
${\ displaystyle R _ {\ infty} = {\ frac {\ alpha ^ {2} m _ {\ mathrm {e}} c} {2h}}}$
${\ displaystyle \ alpha}$- Fine structure constant - electron mass - speed of light - Planck's quantum of action
${\ displaystyle m _ {\ mathrm {e}}}$
${\ displaystyle c}$
${\ displaystyle h}$
Sources and Notes
Source SI value: CODATA  2018 ( direct link )

The Rydberg constant is a natural constant named after Johannes Rydberg . It occurs in the Rydberg formula , an approximation formula for calculating atomic spectra . Its value is the ionization energy of the hydrogen atom, expressed as the wave number , neglecting relativistic effects and the movement of the nucleus (i.e. with infinite nuclear mass , hence the index ). ${\ displaystyle R _ {\ infty}}$${\ displaystyle \ infty}$

The currently recommended value of the Rydberg constant is:

${\ displaystyle R _ {\ infty} = 10 \, 973 \, 731 {,} 568 \, 160 (21) \, \ mathrm {m} ^ {- 1}.}$

The relative standard uncertainty is 1.9 · 10 −12 . This makes it the most precisely measured natural constant.

## Relationship with other fundamental constants

The Rydberg constant is derived from the fine structure constant α and the Compton wavelength λ c, e of an electron by

${\ displaystyle R _ {\ infty} = {\ frac {\ alpha ^ {2}} {2}} \, {\ frac {1} {\ lambda _ {\ mathrm {C, e}}}} = {\ frac {\ alpha ^ {2}} {2}} \, {\ frac {m _ {\ mathrm {e}} c} {h}} = {\ frac {m _ {\ mathrm {e}} e ^ {4 }} {8c \ varepsilon _ {0} ^ {2} h ^ {3}}}}$

With

• ${\ displaystyle m _ {\ mathrm {e}}}$ the mass of the electron
• ${\ displaystyle c}$the speed of light
• ${\ displaystyle h}$the Planck's constant
• ${\ displaystyle e}$the elementary charge
• ${\ displaystyle \ varepsilon _ {0}}$the electric field constant .

## Rydberg frequency and Rydberg energy

The Rydberg constant is often given as frequency or energy. The recommended values ​​are:

• Rydberg frequency: ${\ displaystyle R = c \, R _ {\ infty} = 3 {,} 289 \, 841 \, 960 \, 2508 (64) \ cdot 10 ^ {15} \, \ mathrm {Hz}}$
• Rydberg energy: ${\ displaystyle R_ {y} = h \, R = h \, c \, R _ {\ infty} = 13 {,} 605 \, 693 \, 122 \, 994 (26) \, \ mathrm {eV} = 1 \, \ mathrm {Ry}.}$

The concrete value of the Rydberg energy is called a Rydberg , so that the Rydberg can be used as a unit of measure for energies. ${\ displaystyle R_ {y}}$

## Derivation

A first derivation of the Rydberg constant could be given within the framework of Bohr's atomic model . A more modern derivation in the context of quantum mechanics can be found in the hydrogen problem . ${\ displaystyle R _ {\ infty}}$

In both cases one arrives at a formula for the quantized energy levels of the hydrogen atom of the form:

${\ displaystyle E_ {n} = - {\ frac {m _ {\ mathrm {e}} e ^ {4}} {8 \ varepsilon _ {0} ^ {2} h ^ {2}}} \ cdot {\ frac {1} {n ^ {2}}}}$

From the difference between two energy levels

${\ displaystyle \ Delta E = {\ frac {m _ {\ mathrm {e}} e ^ {4}} {8 \ varepsilon _ {0} ^ {2} h ^ {2}}} \ left ({\ frac {1} {n_ {2} ^ {2}}} - {\ frac {1} {n_ {1} ^ {2}}} \ right)}$

can be with

${\ displaystyle \ Delta {E} = {\ frac {hc} {\ lambda}}}$

determine the wave number of the light emitted or absorbed during such a transition

${\ displaystyle {\ frac {1} {\ lambda}} = {\ frac {m _ {\ mathrm {e}} e ^ {4}} {8 \ varepsilon _ {0} ^ {2} h ^ {3} c}} \ left ({\ frac {1} {n_ {1} ^ {2}}} - {\ frac {1} {n_ {2} ^ {2}}} \ right).}$

The comparison with the Rydberg formula shows, assuming an infinitely heavy hydrogen nucleus, that the Rydberg constant is given by

${\ displaystyle R _ {\ infty} = {\ frac {m _ {\ mathrm {e}} e ^ {4}} {8 \ varepsilon _ {0} ^ {2} h ^ {3} c}}.}$

This also shows that the Rydberg constant is the wave number (inverse wavelength) of a photon whose energy corresponds to the ionization energy of the hydrogen atom.

## Deviations in laser spectroscopic measurements from 2010

Measurements by Theodor Hänsch's group on muonic (2010) and ordinary (2017) hydrogen atoms yielded values ​​of the Rydberg constant and the proton radius that differed from the standard values. In the 2017 experiment, two transitions in the hydrogen atom were compared, from which the Rydberg constant was determined and from this the proton radius. Your value is the Rydberg constant

${\ displaystyle R _ {\ infty} = 10 \, 973 \, 731 {,} 568 \, 076 (96) \, \ mathrm {m} ^ {- 1}.}$

The deviation from the previously accepted value is 3.3 standard deviations.

The participating physicist Randolf Pohl considers errors in the values ​​of the Rydberg constant, which were previously considered to be secure, to be the most likely cause of the deviations in the proton radius.

## Individual evidence

1. CODATA Recommended Values. National Institute of Standards and Technology, accessed June 6, 2019 . Value for the Rydberg constant. The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.
2. CODATA Recommended Values. National Institute of Standards and Technology, accessed June 6, 2019 . Value for the Rydberg frequency. The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.
3. CODATA Recommended Values. National Institute of Standards and Technology, accessed June 6, 2019 . Value for the Rydberg energy. The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.
4. A. Beyer, Hänsch, Pohl a. a .: The Rydberg constant and proton size from atomic hydrogen, Science, Volume 358, 2017, p. 79
5. Shrinking the proton again!, Hänsch Group, 2017
6. Natalie Wolchover, New Measurement Deepen's Proton Puzzle, Quanta Magazine, August 11, 2016