# Bohr's magneton

Physical constant
Surname Bohr's magneton
Formula symbol ${\ displaystyle \ mu _ {\ text {B}}}$
Size type Magnetic moment
value
SI 9.274 010 0783 (28)e-24 ${\ displaystyle \ textstyle {\ frac {\ mathrm {J}} {\ mathrm {T}}}}$
Uncertainty  (rel.) 3.0e-10
Relation to other constants
${\ displaystyle \ mu _ {\ text {B}} = {\ frac {e} {2m _ {\ text {e}}}} \, \ hbar}$
Sources and Notes
Source SI value: CODATA 2018 ( direct link )

The Bohr magneton (by Niels Bohr ) is the amount of the magnetic moment , which is a electron with the orbital angular momentum quantum number by its angular momentum generated. According to the original Bohr model of the atom , this is the basic state , i.e. the state with the lowest energy. ${\ displaystyle \ mu _ {\ text {B}}}$ ${\ displaystyle \ ell {\ mathord {=}} 1}$

Bohr's magneton is used in atomic physics as a unit for magnetic moments.

## history

The idea of ​​the elementary magnet can be traced back to Walter Ritz (1907) and Pierre-Ernest Weiss . Even before the development of Rutherford's atomic model , it was assumed that an elementary magneton must be connected with Planck's quantum of action  h . Richard Gans assumed that the ratio of the kinetic energy of the electron to its angular velocity was equal to  h , and in September 1911 gave a value that was twice as large as Bohr's magneton. Paul Langevin named a smaller value for the magneton at the First Solvay Conference in November of that year . The Romanian physicist Ştefan Procopiu found in 1911 by applying the quantum theory of Max Planck first the exact value of the magneton; hence the name Bohr-Procopiu-Magneton is sometimes heard.

The name "Bohr's magneton" was not given to the value until 1920 by Wolfgang Pauli , who in an article compared the theoretical value of the magneton with an experimentally determined value (the Weiss magneton ).

## Magneton in general

From a quantum mechanical point of view, the orbital angular momentum of a charged point particle with mass and charge generates the magnetic moment${\ displaystyle {\ vec {L}}}$ ${\ displaystyle m}$ ${\ displaystyle q}$

${\ displaystyle {\ vec {\ mu}} = \ mu \, {\ frac {\ vec {L}} {\ hbar}}}$

where the reduced Planck constant and ${\ displaystyle \ hbar}$

${\ displaystyle \ mu = {\ frac {q} {2m}} \, \ hbar}$

is the magneton of the particle.

## Bohr's magneton

Bohr's magneton results when the elementary charge and the mass of the electron are used. According to the current measurement accuracy, it has the value: ${\ displaystyle q}$ ${\ displaystyle e}$${\ displaystyle m}$${\ displaystyle m _ {\ text {e}}}$

{\ displaystyle {\ begin {aligned} \ mu _ {\ text {B}} & = {\ frac {e} {2m _ {\ text {e}}}} \, \ hbar \\ & = 5 {,} 788 \, 381 \, 8060 (17) \ cdot 10 ^ {- 5} \, {\ text {eV / T}} \\ & = 9 {,} 274 \, 010 \, 0783 (28) \ cdot 10 ^ {- 24} \, {\ text {Y / T}} \ end {aligned}}}.

The bracketed digits indicate the estimated standard deviation for the mean and refer to the last two digits before the brackets. is the energy unit electron volt , the energy unit joule and the unit tesla of the magnetic flux density . ${\ displaystyle {\ text {eV}}}$ ${\ displaystyle {\ text {J}}}$${\ displaystyle {\ text {T}}}$

It should be noted that due to the negative charge of the electron, its magnetic moment is always directed opposite to its orbital angular momentum : an electron with orbital angular momentum quantum number , aligned parallel to the z-axis ( magnetic quantum number ), has the magnetic moment due to this orbital angular momentum (e.g. B. in p orbitals or on the innermost circular path of Bohr's atomic model). ${\ displaystyle q {\ mathord {=}} {\ mathord {-}} e}$${\ displaystyle {\ vec {L}}}$${\ displaystyle \ ell {\ mathord {=}} 1}$ ${\ displaystyle m _ {\ ell} {\ mathord {=}} + 1}$${\ displaystyle \ mu _ {{\ text {electron,}} \ ell = 1} = - \ mu _ {\ text {B}}}$

The spin angular momentum of the electron contributes with another magnetic moment of the size (opposite to the direction of the spin). ${\ displaystyle \ mu _ {\ text {electron, spin}} \ approx -1 {,} 0012 \ mu _ {\ text {B}}}$

A magnetic (dipole) moment has its lowest energy in a magnetic field when it is opposed to the field, i.e. orbital angular momentum and spin are aligned parallel to the field direction.

8. CODATA Recommended Values. National Institute of Standards and Technology, accessed June 6, 2019 . Value for in the unit electron volts per Tesla . 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. ${\ displaystyle \ mu _ {\ mathrm {B}}}$
9. CODATA Recommended Values. National Institute of Standards and Technology, accessed June 6, 2019 . Value for in the unit Joule per Tesla . 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. ${\ displaystyle \ mu _ {\ mathrm {B}}}$
10. In older books, e.g. B. Werner Döring , Introduction to Theoretical Physics , Sammlung Goschen, Volume II (electrodynamics), or in the older editions of Robert Pohl , Introduction to Physics , Volume II, is called Bohr magneton the defined times the value specified here . The official change to the new CODATA definition did not take place until 2010.${\ displaystyle \ mu _ {0}}$