Bohr radius

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Physical constant
Surname Bohr radius
Formula symbol
Size type length
value
SI 5.291 772 109 03 (80)e-11 m
Uncertainty  (rel.) 1.5e-10
Gauss 5.291 772 109 03 (80)e-9 cm
Relation to other constants
Sources and Notes
Source SI value: CODATA 2018 ( direct link )

The Bohr radius denotes the radius of the hydrogen atom in the lowest energy state and thus also the radius of its first and smallest electron shell in the context of Bohr's atomic model ; the small correction, which corresponds to the movement of the atomic nucleus around the center of gravity , is still not taken into account.

A quantum mechanical analysis shows that in the lowest energy state, the radial probability density to measure the electron is at its maximum at the Bohr radius. However, the experimentally more relevant expected value for the radius is 1.5 times the Bohr radius.

Formulas and numerical value

The Bohr radius is calculated according to the formula:

It is

The Bohr radius is also described by

With

  • the Compton wavelength of the electron and
  • the fine structure constant

According to the current measurement accuracy of the natural constants used in the calculation, the value is :

where the numbers in brackets  indicate the estimated standard deviation of 0.000 000 000 080 · 10 −10 m.

With this definition, the Bohr radius is considered a natural constant. In atomic physics , for example , it is often used as a unit of length , with 52.9 pm or half an Angstrom (= 50 pm) being used as approximations  .

If you take into account the finite mass of the nucleus and thus its movement around the common center of gravity, you have to replace the electron mass with the reduced mass in the mechanical formulas . The orbit radius then becomes . The correction for the H atom is only approx. 0.05%, for the He + ion, which also has only one electron, approx. 0.01%. With corresponding values ​​for the mass, the term Bohr radius is also used for other systems, e.g. B. Excitons .

Derivation

The Bohr radius can be determined with the help of a simple estimate and taking into account the uncertainty principle .

It is believed that the distance between the electron bound in the hydrogen atom and the nucleus is usually .

Due to the uncertainty relation, the momentum of the electron can be roughly measured

specify, whereby the place observable is replaced here by the distance .

The kinetic energy is accordingly

The potential energy is according to Coulomb's law

from which the total energy results:

The further the electron moves away from the nucleus, the smaller its kinetic energy becomes. Because of the negative sign , however, its potential energy increases.

Kinetic, potential and total energy of the electron depending on the distance (in Bohr radii) of the electron from the atomic nucleus for the hydrogen atom in the ground state

In the basic state, a kind of “compromise” is realized which minimizes the total energy; the associated radius is obtained by dividing the energy after differentiated and the derivative equal to zero ( extreme value determination):

This is exactly the Bohr radius.

If you now insert in , you get the Rydberg energy , the ionization energy of hydrogen:

The figure shows the course of kinetic, potential and total energy as a function of the distance (in Bohr radii). If you put in the formula for or , you get

or.

.

The amount of potential energy is called Hartree energy and is another unit of the system of atomic units of atomic physics.

Historical

In his essay, Niels Bohr mentions the Austrian physicist Arthur Erich Haas , who had already found the formula for 1910/11 and thus for the first time recognized the role that Planck's constant could play in atomic physics, especially in its mechanical aspects. In his model, an electron revolves around the surface of a positively charged sphere, which, according to Gauss's law of electrostatics, results in the same attractive force as a point-like nucleus. This model was ignored at the time, u. a. because a much larger number of electrons was often assumed in the case of hydrogen as well, i.e. a larger positive charge on the rest of the neutral atom as a whole. It was also widely considered to be ruled out that harmonic oscillations could have any meaning outside of the subject .

Initially, the energies or wavelengths of the hydrogen spectrum calculated with the Bohr radius were around 0.05% next to the measured values ​​known at the time, for helium ions around 0.01%. However, the fact that the small corrections due to the co-movement of the core resulted in full agreement in both cases quickly secured the Bohr model great recognition.

swell

  • RP Feynman: lectures on physics. Quantum mechanics. Oldenbourg Wissenschaftsverlag, Munich 2007, ISBN 978-3-486-58109-6 .
  • LM Brown, A. Pais, Sir B. Pippard (Eds.): Twentieth Century Physics. Volume 1, Inst. Of Phys. Publishing, Bristol 1995, ISBN 0-7503-0353-0 .
  • Max Jammer: The Conceptual Development of Quantum Mechanics . MCGraw-Hill, New York 1966.

Individual evidence

  1. CODATA Recommended Values. National Institute of Standards and Technology, accessed June 3, 2019 . Value for the Bohr radius. The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.
  2. N. Bohr: On the Constitution of Atoms and Molecules . In: Philosophical Magazine . tape 26 , 1913, pp. 4 .