# Bohr radius

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The **Bohr radius** (*a*_{0} or *r*_{Bohr}) is a physical constant, exactly equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom. Its value is 5.29177210903(80)×10^{−11} m.^{[1]}^{[note 1]}

Bohr radius | |
---|---|

Unit of | length |

Symbol | a_{0} or r_{Bohr} |

Named after | Niels Bohr |

Conversions | |

1 a_{0} in ... | ... is equal to ... |

SI units | 5.29×10^{−11} m |

imperial/US units | 2.08×10^{−9} in |

natural units | 2.68×10^{−4}/eV3.27×10 ^{24} ℓ_{P} |

## Contents

## Definition and valueEdit

In SI units the Bohr radius is:^{[2]}

where:

- is the Bohr radius,
- is the permittivity of free space,
- is the reduced Planck's constant,
- is the electron rest mass,
- is the atomic number,
- is the elementary charge,
- is the speed of light in vacuum, and
- is the fine structure constant.

In Gaussian units the Bohr radius is simply

According to 2014 CODATA the Bohr radius has a value of (considering mass of electron as the rest mass of an electron) 5.2917721067(12)×10^{−11} m (i.e., approximately 53 pm or 0.53 Å).^{[3]}^{[note 1]}

## UseEdit

In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy. In the simplest atom, hydrogen, a single electron orbits the nucleus and its smallest possible orbit, with lowest energy, has an orbital radius almost equal to the Bohr radius. (It is not *exactly* the Bohr radius due to the reduced mass effect. They differ by about 0.1%.)^{[citation needed]}

Although the Bohr model is no longer in use, the Bohr radius remains very useful in atomic physics calculations, due in part to its simple relationship with other fundamental constants. (This is why it is defined using the true electron mass rather than the reduced mass, as mentioned above.) For example, it is the unit of length in atomic units.

An important distinction is that the Bohr radius gives the radius with the maximum radial probability density,^{[4]} not its **expected** radial distance. The expected radial distance is actually 1.5 times the Bohr radius, as a result of the long tail of the radial wave function. Another important distinction is that in three-dimensional space, the maximum probability density occurs at the location of the nucleus and not at the Bohr radius, whereas the radial probability density peaks at the Bohr radius, i.e. when plotting the probability distribution in its radial dependency.

## Related unitsEdit

The Bohr radius of the electron is one of a trio of related units of length, the other two being the Compton wavelength of the electron and the classical electron radius . The Bohr radius is built from the electron mass , Planck's constant and the electron charge . The Compton wavelength is built from , and the speed of light . The classical electron radius is built from , and . Any one of these three lengths can be written in terms of any other using the fine structure constant :

The Compton wavelength is about 20 times smaller than the Bohr radius, and the classical electron radius is about 1000 times smaller than the Compton wavelength.

## Reduced Bohr radiusEdit

The Bohr radius including the effect of reduced mass in the hydrogen atom can be given by the following equation:

- is the Compton wavelength of the proton.
- is the Compton wavelength of the electron.
- is the fine structure constant.

In the above equation, the effect of the reduced mass is achieved by using the increased Compton wavelength, which is just the Compton wavelengths of the electron and the proton added together.

## See alsoEdit

## NotesEdit

- ^
^{a}^{b}The number in parenthesis denotes the uncertainty of the last digits.

## ReferencesEdit

**^**"2018 CODATA Value: Bohr radius".*The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 May 2019. Retrieved 2019-05-20.**^**David J. Griffiths,*Introduction to Quantum Mechanics*, Prentice-Hall, 1995, p. 137. ISBN 0-13-124405-1**^**"CODATA Value: Bohr radius".*Fundamental Physical Constants*. NIST. Retrieved 13 February 2016.**^**Zettili, Nouredine (2009).*Quantum Mechanics: Concepts and Applications*(2nd ed.). Chichester: Wiley. p. 376. ISBN 978-0-470-02678-6.