The Bohr radius is: where: The CODATA value of the Bohr radius is The Bohr model derives a radius for the nth excited state of a hydrogen-like atom. The Bohr radius corresponds to.
Use
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. The Bohr model of the atom was superseded by an electron probability cloud obeying the Schrodinger equation, which is further complicated by spin and quantum vacuum effects to produce fine structure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. 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, not its expectedradial 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.
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 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 Bohr radius is about 19,000 times bigger than the classical electron radius. 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.
"Reduced" Bohr radius
The Bohr radius including the effect of reduced mass in the hydrogen atom can be given by the following equations: where:
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 smaller than the electron mass, so the "Reduced Bohr radius" is actually larger than the typical value.
Radii in similar systems
This result can be generalized to other systems, such as positronium and muonium by using the reduced mass of the system and considering the possible change in charge. Typically, Bohr model relations can be easily modified for these exotic systems by simply replacing the electron mass with the reduced mass for the system. For example, the radius of positronium is approximately, since the reduced mass of the positronium system is half the electron mass, while the reduced mass for the electron/proton system is approximately the electron mass, as discussed above. Another important observation is that any Hydrogen-like atom will have a Bohr radius which primarily changes as, with the number of protons in the nucleus. This can be seen in the last equation to be a result of Meanwhile, the reduced mass only becomes better approximated by in the limit of increasing nuclear mass. These results are summarized in the equation A table of approximate relationships is given below:
