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{{Short description|Unit of length about the size of a hydrogen atom}}
{{More citations needed|date=May 2025}}
{{Infobox
| title            = Bohr radius
| labelstyle       = font-weight:normal
| label1           = Symbol
| data1            = {{math|''a''<sub>0</sub>}} or {{math|''r''<sub><small>Bohr</small></sub>}}
| label2           = Named after
| data2            = [[Niels Bohr]]
| header3          = Approximate values (to three significant digits)<!--This section lists various values for c, to three significant digits. Please do not change to more exact values!-->
| label4           = [[SI units]]
| data4            = {{val|5.29|e=-11|ul=m}} ({{val|52.9|ul=pm}})
}}

The '''Bohr radius''' ({{tmath|a_0}}) is a [[physical constant]], approximately equal to the most probable distance between the [[Atomic nucleus|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 {{physconst|a0|after=.}}<ref name="uncert">The number in parentheses denotes the [[Standard deviation|uncertainty]] of the last digits.</ref>

== Definition and value ==

The Bohr radius is defined as<ref>[[David J. Griffiths]], ''Introduction to Quantum Mechanics'', Prentice-Hall, 1995, p. 137. {{ISBN|0-13-124405-1}}</ref>
<math display="block">a_0
= \frac{4 \pi \varepsilon_0 \hbar^2}{e^2 m_{\text{e}}}
= \frac{\hbar}{m_{\text{e}} c \alpha}
 ,</math>
where
* <math> \varepsilon_0 </math> is the [[permittivity of free space]],
* <math> \hbar </math> is the [[reduced Planck constant]],
* <math> m_{\text{e}} </math> is the [[electron rest mass|mass of an electron]],
* <math> e </math> is the [[elementary charge]],
* <math> c </math> is the [[speed of light]] in vacuum, and
* <math> \alpha </math> is the [[fine-structure constant]].

The [[CODATA]] value of the Bohr radius (in [[SI units]]) is {{physconst|a0|after=.}}

== History ==
[[File:Bohr_model.jpg|thumb|right|alt=Picture of a hydrogen atom using the Bohr model|Picture of a hydrogen atom using the Bohr model]]
In the [[Bohr model]] for [[atom]]ic structure, put forward by Niels Bohr in 1913, [[electrons]] orbit a central [[atomic nucleus|nucleus]] under electrostatic attraction. The original derivation posited that electrons have orbital angular momentum in integer multiples of the reduced Planck constant, which successfully matched the observation of discrete energy levels in emission spectra, along with predicting a fixed radius for each of these levels. In the simplest atom, [[hydrogen]], a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. (It is not ''exactly'' the Bohr radius due to the [[reduced mass|reduced mass effect]]. They differ by about 0.05%.)

The Bohr model of the atom was superseded by an electron probability cloud adhering to the [[Schrödinger equation]] as published in 1926. This is further complicated by spin and quantum vacuum effects to produce [[fine structure]] and [[hyperfine structure]]. Nevertheless, the Bohr radius formula remains central in [[atomic physics]] calculations, due to its simple relationship with fundamental constants (this is why it is defined using the true electron mass rather than the reduced mass, as mentioned above). As such, it became the unit of length in [[atomic units]].

In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest.  The [[expected value]] of the radial distance of the electron, by contrast, is&nbsp;{{tmath|\tfrac{3}{2} a_0}}.<ref>{{cite web |last1=Nave |first1=Rod |title=The Most Probable Radius: Hydrogen Ground State |url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydr.html |website=[[HyperPhysics]] |publisher=Dept. of Physics and Astronomy, Georgia State University |access-date=2 October 2021 |quote=The Schrodinger equation confirms the first Bohr radius as the most probable radius.}}</ref>

== Related constants ==
The Bohr radius is one of a trio of related units of length, the other two being the [[Compton wavelength]] of the electron ({{tmath| \lambda_{\mathrm{e} } }}) and the [[classical electron radius]] ({{tmath| r_{\mathrm{e} } }}). Any one of these constants can be written in terms of any of the others using the fine-structure constant {{tmath| \alpha }}:
: <math>r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0.</math>

== Hydrogen atom and similar systems ==
The Bohr radius including the effect of [[reduced mass]] in the hydrogen atom is given by
: <math> \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 ,</math>
where <math display="inline">\mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p})</math> is the reduced mass of the electron–proton system (with {{tmath| m_\text{p} }} being the mass of proton). The use of reduced mass is a generalization of the [[two-body problem]] from [[classical physics]] beyond the case in which the approximation that the mass of the orbiting body is negligible compared to the mass of the body being orbited. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly ''larger'' than the Bohr radius ({{tmath| a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} }} meters).

This result can be generalized to other systems, such as [[positronium]] (an electron orbiting a [[positron]]) and [[muonium]] (an electron orbiting an [[anti-muon]]) by using the reduced mass of the system and considering the possible change in charge. Typically, Bohr model relations (radius, energy, etc.) can be easily modified for these exotic systems (up to lowest order) by simply replacing the electron mass with the reduced mass for the system (as well as adjusting the charge when appropriate). For example, the radius of positronium is approximately {{tmath| 2\,a_0 }}, since the reduced mass of the positronium system is half the electron mass ({{tmath|1= \mu_{\text{e}^-,\text{e}^+} = m_\text{e}/2 }}).

A [[hydrogen-like atom]] will have a Bohr radius which primarily scales as {{tmath|1= r_{Z}=a_0/Z }}, with {{tmath| Z }} the number of protons in the nucleus. Meanwhile, the reduced mass ({{tmath| \mu }}) only becomes better approximated by {{tmath| m_\text{e} }} in the limit of increasing nuclear mass. These results are summarized in the equation
: <math> r_{Z,\mu} \ = \frac{m_\text{e}}{\mu}\frac{a_0}{Z} .</math>

A table of approximate relationships is given below.

{| class="wikitable"
|-
! System !! Radius 
|-
| [[Hydrogen atom|Hydrogen]] || <math>a_0^*=1.00054\, a_0</math>
|-
| [[Positronium]] || <math>2 a_0</math>
|-
| [[Muonium]] || <math>1.0048\, a_0</math>
|-
| [[Helium|He<sup>+</sup>]] || <math>a_0/2</math>
|-
| [[Lithium|Li<sup>2+</sup>]] || <math>a_0/3</math>
|}

== See also ==
* [[Bohr magneton]]
* [[Rydberg energy]]

== References ==
{{reflist}}

== External links ==
* [https://math.ucr.edu/home/baez/lengths.html#bohr_radius Length Scales in Physics: the Bohr Radius]

{{Scientists whose names are used in physical constants}}
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[[Category:Atomic physics]]
[[Category:Physical constants]]
[[Category:Units of length]]
[[Category:Niels Bohr]]
[[Category:Atomic radius]]