Nuclear magnetic moment
1739062
183733460
2008-01-11T23:17:31Z
RJHall
91076
Too technical
{{Technical|date=January 2008}}
The '''nuclear magnetic moment''' is the [[magnetic moment]] of an [[atomic nucleus]] and arises from the spin of the [[proton]]s and [[neutron]]s. It is mainly a magnetic [[dipole moment]]; the [[quadrupole moment]] does cause some small shifts in the [[hyperfine structure]] as well.
The nuclear magnetic moment varies from [[isotope]] to isotope of an [[Chemical element|element]]. It can only be zero if the numbers of protons and of neutrons are ''both'' even.
==Calculating the magnetic moment==
In the [[shell model]], the magnetic moment of a nucleon of [[total angular momentum]] ''j'', [[orbital angular momentum]] ''l'' and [[Spin (physics)|spin]] ''s'', is given by
:<math>\mu=\langle(l,s),j,m_j=j|\mu_z|(l,s),j,m_j=j\rangle</math>
By projecting with the [[total angular momentum]] '''j''' we get
<math>\mu=\langle(l,s),j,m_j=j|\overrightarrow{\mu}\cdot \overrightarrow{j}|(l,s),j,m_j=j\rangle \frac{\langle (l,s)j,m_j=j|j_z|(l,s)j,m_j=j\rangle}{\langle (l,s)j,m_j=j|\overrightarrow{j}\cdot \overrightarrow{j}|(l,s)j,m_j=j\rangle}</math>
<math>= {1\over (j+1)}\langle(l,s),j,m_j=j|\overrightarrow{\mu}\cdot \overrightarrow{j}|(l,s),j,m_j=j\rangle</math>
<math>\overrightarrow{\mu}</math> has contributions both from the [[orbital angular momentum]] and the [[Spin (physics)|spin]], with different coefficients g<sup>(l)</sup> and g<sup>(s)</sup>:
:<math>\overrightarrow{\mu} = g^{(l)}\overrightarrow{l} + g^{(s)}\overrightarrow{s} </math>
by substituting this back to the formula above and rewriting
:<math>\overrightarrow{l}\cdot\overrightarrow{j} = {1\over 2} \left(\overrightarrow{j}\cdot \overrightarrow{j} + \overrightarrow{l}\cdot \overrightarrow{l} - \overrightarrow{s}\cdot \overrightarrow{s}\right)</math>
:<math>\overrightarrow{s}\cdot\overrightarrow{j} = {1\over 2} \left(\overrightarrow{j}\cdot \overrightarrow{j} - \overrightarrow{l}\cdot \overrightarrow{l} + \overrightarrow{s}\cdot \overrightarrow{s}\right)</math>
<math>\mu = {1\over (j+1)}\langle(l,s),j,m_j=j|(g^{(l)}{1\over 2} \left(\overrightarrow{j}\cdot \overrightarrow{j} + \overrightarrow{l}\cdot \overrightarrow{l} - \overrightarrow{s}\cdot \overrightarrow{s}\right) + g^{(s)}{1\over 2} \left(\overrightarrow{j}\cdot \overrightarrow{j} - \overrightarrow{l}\cdot \overrightarrow{l} + \overrightarrow{s}\cdot \overrightarrow{s}\right)|(l,s),j,m_j=j\rangle </math>
<math>=
{1\over (j+1)}\left(g^{(l)}{1\over 2} \left(j(j+1) + l(l+1) - s(s+1)\right) + g^{(s)}{1\over 2} \left(j(j+1) - l(l+1) + s(s+1)\right)\right)</math>
For a single [[nucleon]] ''s'' =1/2. For <math>j = l+1/2</math> we get
:<math>\mu_j = g^{(l)} l + {1\over 2}g^{(s)}</math>
and for <math>j = l-1/2</math>
:<math>\mu_j = {j \over j+1} \left( g^{(l)} (l+1) - {1\over 2}g^{(s)} \right)</math>
According to the [[shell model]], [[proton]]s or [[neutron]]s tend to form pairs of opposite [[total angular momentum]]. Therefore the magnetic moment of a nucleus with even numbers of both protons and neutrons is zero, while that of a nucleus with odd number of protons and even number of neutrons (or vice versa) will have be that of the "last", unpaired proton (or neutron), according to the formula we have arrived at. For a nucleus with odd numbers of both protons and neutrons, one must take both the "last", unpaired proton and neutron. Their total magnetic moment will be some combination of their magnetic moments.
In fact, nuclear magnetic moment is only partly predicted by simple versions of the [[shell model]]. The magnetic moment is calculated through ''j'', ''l'' and ''s'' of the "last" nucleon, but nuclei are not in states of well defined ''l'' and ''s''. Furthermore, for odd-odd nuclei, one has to consider the two "last [[nucleon]]s, as in [[Deuterium#Magnetic_and_electric_multipoles|deuterium]]. Therefore one gets several possible answers for the nuclear magnetic moment, one for each possible combined ''l'' and ''s'' state, and the real state of the nucleus is a [[superposition]] of them. Thus the real (measured) [[nuclear magnetic moment]] is somewhere in between the possible answers.
==Values of g<sup>(l)</sup> and g<sup>(s)</sup>==
These are known as the [[g-factor]]s of the [[nucleon]]s.
The measured values of g<sup>(l)</sup> for the [[neutron]] and the [[proton]] are according to their [[electric charge]]. Thus, in units of [[nuclear magneton]], g<sup>(l)</sup> = 0 for the [[neutron]] and g<sup>(l)</sup> = 1 for the [[proton]].
The measured values of g<sup>(s)</sup> for the [[neutron]] and the [[proton]] are twice their magnetic moment (either the [[neutron magnetic moment]] or the [[proton magnetic moment]]). In [[nuclear magneton]] units, g<sup>(s)</sup> = -3.8263 For the [[neutron]] and g<sup>(s)</sup> = 5.5858 for the [[proton]].
== See also ==
* [[Gyromagnetic ratio]]
* [[Nuclear magneton]]
* [[Magnetic moment]]
* [[Proton magnetic moment]]
* [[Neutron magnetic moment]]
* [[Electron magnetic dipole moment]]
* [[Deuterium#Magnetic_and_electric_multipoles|Deuterium magnetic moment]]
[[Category:Magnetism]][[Category:Nuclear physics]]
[[sr:Нуклеарни магнетни момент]]