Bohr radius

The Bohr radius is:[2]

${\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{m_{\text{e}}e^{2}}}={\frac {\hbar }{m_{\text{e}}c\alpha }}}$

where:

${\displaystyle a_{0}}$ is the Bohr radius,

${\displaystyle \varepsilon _{0}\ }$ is the permittivity of free space,

${\displaystyle \hbar \ }$ is the reduced Planck's constant,

${\displaystyle m_{\text{e}}\ }$ is the electron rest mass,

${\displaystyle e\ }$ is the elementary charge,

${\displaystyle c\ }$ is the speed of light in vacuum, and

${\displaystyle \alpha \ }$ is the fine structure constant.

The CODATA value of the Bohr radius (in SI units) is 5.29177210903(80)×10⁻¹¹ m.[1]

The Bohr model derives a radius for the nth excited state of a hydrogen-like atom. The Bohr radius corresponds to n = 1.

In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central 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 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 0.05%.)

The Bohr model of the atom was superseded by an electron probability cloud obeying the Schrodinger equation, which is further complicated by spin and quantum vacuum effects to produce fine structure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 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,[3] not its expected radial distance. The expected radial distance is 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.

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 ${\displaystyle \lambda _{\mathrm {e} }}$ and the classical electron radius ${\displaystyle r_{\mathrm {e} }}$. The Bohr radius is built from the electron mass ${\displaystyle m_{\mathrm {e} }}$, Planck's constant ${\displaystyle \hbar }$ and the electron charge ${\displaystyle e}$. The Compton wavelength is built from ${\displaystyle m_{\mathrm {e} }}$, ${\displaystyle \hbar }$ and the speed of light ${\displaystyle c}$. The classical electron radius is built from ${\displaystyle m_{\mathrm {e} }}$, ${\displaystyle c}$ and ${\displaystyle e}$. Any one of these three lengths can be written in terms of any other using the fine structure constant ${\displaystyle \alpha }$:

${\displaystyle r_{\mathrm {e} }=\alpha {\frac {\lambda _{\mathrm {e} }}{2\pi }}=\alpha ^{2}a_{0}.}$

The Bohr radius is about 19,000 times bigger than the classical electron radius (i.e. the common scale of atoms is angstrom, while the scale of particles is femtometer). The electron's Compton wavelength is about 20 times smaller than the Bohr radius, and the classical electron radius is about 1000 times smaller than the electron's Compton wavelength.

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

${\displaystyle \ a_{0}^{*}\ ={\frac {\lambda _{\mathrm {p} }+\lambda _{\text{e}}}{2\pi \alpha }}={\frac {m_{\text{e}}}{\mu }}a_{0}={\frac {\hbar }{\mu c\alpha }}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{\mu |q_{\text{e}}||q_{\text{p}}|}},}$

where:

${\displaystyle \lambda _{\text{p}}\ }$ is the Compton wavelength of the proton,

${\displaystyle \lambda _{\text{e}}\ }$ is the Compton wavelength of the electron,

${\displaystyle \hbar \ }$ is the reduced Planck's constant,

${\displaystyle \alpha \ }$ is the fine structure constant,

${\displaystyle c\ }$ is the speed of light,

${\displaystyle \mu \ }$ is the reduced mass of the electron/proton system,

${\displaystyle \varepsilon _{0}\ }$ is the permittivity of free space,

${\displaystyle |q_{\text{e}}|\ }$ is the magnitude of the electron's charge,

${\displaystyle |q_{\text{p}}|\ }$ is the magnitude of the proton's charge.

In the first equation, the effect of the reduced mass is achieved by using the increased Compton wavelength, which is just the sum of the electron and proton Compton wavelengths. The use of reduced mass is inherently a classical generalization of the two-body problem when we are outside the approximation that the mass of the orbiting body is much less than the mass of the body being orbited.

Notably, the reduced mass of the electron/proton system will be (very slightly) smaller than the electron mass, so the "Reduced Bohr radius" is actually larger than the typical value (${\displaystyle a_{0}^{*}\approx 1.00054\,a_{0}}$ or ${\displaystyle a_{0}^{*}\approx 5.2946541\times 10^{-11}}$ 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 (or equivalently Compton wavelength sum) 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 ${\displaystyle 2\,a_{0}}$, since the reduced mass of the positronium system is half the electron mass (${\displaystyle \mu _{{\text{e}}^{-},{\text{e}}^{+}}=m_{\text{e}}/2}$), while the reduced mass for the electron/proton system is approximately the electron mass (${\displaystyle \mu _{{\text{e}}^{-},{\text{p}}}\approx m_{\text{e}}}$), as discussed above.

Another important observation is that any Hydrogen-like atom will have a Bohr radius which primarily changes as ${\displaystyle r_{Z}=a_{0}/Z}$, with ${\displaystyle Z}$ the number of protons in the nucleus. This can be seen in the last equation to be a result of ${\displaystyle |q_{\text{p}}|\rightarrow |q_{(Z*{\text{p}})}|=Z|q_{\text{p}}|}$. Meanwhile, the reduced mass (${\displaystyle \mu }$) only becomes better approximated by ${\displaystyle m_{\text{e}}}$ in the limit of increasing nuclear mass. These results are summarized in the equation

${\displaystyle r_{Z,\mu }\ ={\frac {m_{e}}{\mu }}{\frac {a_{0}}{Z}}.}$

A table of approximate relationships is given below:

Bohr radius	${\displaystyle a_{0}}$
"Reduced" Bohr radius	${\displaystyle 1.00054\,a_{0}}$
Positronium radius	${\displaystyle 2a_{0}}$
Muonium radius	${\displaystyle 1.0048\,a_{0}}$
He+ radius	${\displaystyle a_{0}/2}$
Li2+ radius	${\displaystyle a_{0}/3}$

• Angstrom
• Bohr magneton
• Rydberg energy

• Length Scales in Physics: the Bohr Radius