Bohr radius

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, that of hydrogen, a single electron orbits, and the smallest possible orbit for the electron, that with the lowest energy, is the one at a distance from the nucleus called the Bohr radius.

According to 2002 CODATA, the Bohr radius has a value of 5.291772108(18)×10^-11 m (i.e., approximately 53 pm or 0.53 angstroms). The number in parentheses (18) denotes the uncertainty of the last digits. This value can be computed in terms of other physical constants:

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

where:

$\epsilon _{0}\$ is the permittivity of vacuum

$\hbar \$ is the reduced Planck's constant

$m_{e}\$ is the electron rest mass

$e\$ is the elementary charge

$c\$ is the speed of light

$\alpha \$ is the fine structure constant

While the Bohr model does not correctly describe an atom, the Bohr radius keeps its physical meaning as a characteristic size of the electron cloud in a full quantum-mechanical description. Thus the Bohr radius is often used as a unit in atomic physics, see atomic units.

Note that the definition of Bohr radius does not include the effect of reduced mass, and so it is not precisely equal to the orbital radius of the electron in a hydrogen atom in the more physical model where reduced mass is included. This is done for convenience: the Bohr radius as defined above appears in equations relating to atoms other than hydrogen, where the reduced mass correction is different. If the definition of Bohr radius included the reduced mass of hydrogen, it would be necessary to include a more complex adjustment in equations relating to other atoms.

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

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

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

$a_{0}={\frac {\lambda _{p}+\lambda _{e}}{2\pi \alpha }}$

where,

$\lambda _{p}\$ is the Compton wavelength of the proton.

$\lambda _{e}\$ is the Compton wavelength of the electron.

$\alpha \$ 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.

External links

• Length Scales in Physics: the Bohr Radius