Bohr radius

Bohr radius
Symbol:	a0 or rBohr
Named after:	Niels Bohr
Value in meters:	≈ 5.29×10^−11m
Value in picometers:	≈ 52.9 pm
Value in angstroms:	≈ 0.529 Å
Value in natural units:	≈ 2.68×10^−4 1/eV

The Bohr radius (a₀ or r_Bohr) is a physical constant, approximately equal to the most probable distance between the proton and 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 5.2917721067(12)×10⁻¹¹ m^{[1][note 1]}

Definition & value

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

$a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_{\mathrm{e}} e^2} = \frac{\hbar}{m_{\mathrm{e}}\,c\,\alpha}$

where:

$a_0$ is the Bohr radius,

$\varepsilon_0 \$ is the permittivity of free space,

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

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

$e \$ is the elementary charge,

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

$\alpha \$ is the fine structure constant.

In Gaussian units the Bohr radius is simply

$a_0=\frac{\hbar^2}{m_e e^2}$

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

Use

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%.)

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 describes the most probable radial distance of the electron, 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.

Related units

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

$r_{\mathrm{e}} = \frac{\alpha \lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0.$

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 radius

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

$\ a_0^* \ = \frac{\lambda_{\mathrm{p}} + \lambda_{\mathrm{e}}}{2\pi\alpha},$

where:

$\lambda_{\mathrm{p}} \$ is the Compton wavelength of the proton.

$\lambda_{\mathrm{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.

See also

• Angstrom
• Rydberg energy

Notes

1. The number in parenthesis denotes the uncertainty of the last digits.

References

1. "CODATA Value: Bohr radius". Fundamental Physical Constants. NIST. Retrieved 13 February 2016.
2. David J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, 1995, p. 137. ISBN 0-13-124405-1

External links

• Length Scales in Physics: the Bohr Radius