Nuclear magnetic moment

The methods for measuring nuclear magnetic moments can be divided into two broad groups in regard to the interaction with internal or external applied fields.[1] Generally the methods based on external fields are more accurate.

Different experimental techniques are designed in order to measure nuclear magnetic moments of a specific nuclear state. For instance, the following techniques are aimed to measure magnetic moments of an associated nuclear state in a range of life-times τ:

• Nuclear Magnetic Resonance (NMR) ${\displaystyle \sim }$ ms.
• Time Differential Perturbed Angular Distribution (TDPAD) ${\displaystyle \sim \mu }$s.
• Perturbed Angular Correlation (PAC) ${\displaystyle \sim }$ ns.
• Time Differential Recoil Into Vacuum (TDRIV) ${\displaystyle \sim }$ ps.
• Recoil Into Vacuum (RIV) ${\displaystyle \sim }$ ns.
• Transient Field (TF) ${\displaystyle \sim }$ ns.

Techniques as Transient Field have allowed measuring the g factor in nuclear states with life-times of few ps or less.[2]

According to the shell model, protons or neutrons tend to form pairs of opposite total angular momentum. Therefore, the magnetic moment of a nucleus with even numbers of each protons and neutrons is zero, while that of a nucleus with an odd number of protons and even number of neutrons (or vice versa) will have to be that of the remaining unpaired nucleon. For a nucleus with odd numbers of each protons and neutrons, the total magnetic moment will be some combination of the magnetic moments of both of the "last", unpaired proton and neutron.

The magnetic moment is calculated through j, l and s of the unpaired nucleon, but nuclei are not in states of well defined l and s. Furthermore, for odd–odd nuclei, there are two unpaired nucleons to be considered, as in deuterium. There is consequently a value for the nuclear magnetic moment associated with each possible l and s state combination, and the actual state of the nucleus is a superposition of these. Thus the real (measured) nuclear magnetic moment is between the values associated with the "pure" states, though it may be close to one or the other (as in deuterium).

The g-factor is a dimensionless factor associated to the nuclear magnetic moment. This parameter contains the sign of the nuclear magnetic moment, which is very important in nuclear structure since it provides information about which type of nucleon (proton or neutron) is dominating over the nuclear wave function. The positive sign is associated to the proton domination and the negative sign with the neutron domination.

The values of g^(l) and g^(s) are known as the g-factors of the nucleons.[3]

The measured values of g^(l) for the neutron and the proton are according to their electric charge. Thus, in units of nuclear magneton, g^(l) = 0 for the neutron and g^(l) = 1 for the proton.

The measured values of g^(s) for the neutron and the proton are twice their magnetic moment (either the neutron or proton magnetic moment). In nuclear magneton units, g^(s) = −3.8263 for the neutron and g^(s) = 5.5858 for the proton.

The gyromagnetic ratio, expressed in Larmor precession frequency ${\displaystyle f={\frac {\gamma }{2\pi }}B}$, is of great relevance to nuclear magnetic resonance analysis. Some isotopes in the human body have unpaired protons or neutrons (or both, as the magnetic moments of a proton and neutron do not cancel perfectly)[4][5][6] Note that in the table below, the measured magnetic dipole moments, expressed in a ratio to the nuclear magneton, may be divided by the half-integral nuclear spin to calculate dimensionless g-factors. These g-factors may be multiplied by 7.622593285(47) MHz/T,[7] which is the nuclear magneton divided by Planck's constant, to yield Larmor frequencies in MHz/T. If divided instead by the reduced Planck constant, which is 2π less, a gyromagnetic ratio expressed in radians is obtained, which is greater by a factor of 2π.

The quantized difference between energy levels corresponding to different orientations of the nuclear spin ${\displaystyle \Delta E=\gamma \hbar B}$. The ratio of nuclei in the lower energy state, with spin aligned to the external magnetic field, is determined by the Boltzmann distribution.[8] Thus, multiplying the dimensionless g-factor by the nuclear magneton (3.1524512550(15)×10⁻⁸ eV·T⁻¹) and the applied magnetic field, and dividing by the Boltzmann constant (8.6173303(50)×10⁻⁵ eV⋅K⁻¹) and the temperature.

Mass	Element	Magnetic dipole moment[9][10] (μN)	Nuclear spin number[9]	g-factor[11]	Larmor frequency (MHz/T)	Gyromagnetic ratio, free atom[12] (rad/s·μT)	Isotopic abundance	NMR sensitivity,[4] relative to 1H
Formula		${\displaystyle \mu _{Z}/\mu _{\text{N}}}$ (measured)[11]	I	${\displaystyle g=\mu /I}$[10]	${\displaystyle \nu /B=g\mu _{\text{N}}/h}$	${\displaystyle \omega /B=\gamma =g\mu _{\text{N}}/\hbar }$
1	H	2.79284734(3)	1/2	5.58569468	42.6	267.522208	99.98%	1
2	H	0.857438228(9)	1	0.857438228	6.5	41.0662919	0.02%
3	H	2.9789624656(59)	1/2	5.957924931(12)
7	Li	3.256427(2)	3/2	2.1709750	16.5	103.97704	92.6%
13	C	0.7024118(14)	1/2	1.404824	10.7	67.28286	1.11%	0.016
14	N	0.40376100(6)	1	0.40376100	3.1	19.337798	99.63%	0.001
19	F	2.626868(8)	1/2	5.253736	40.4	251.6233	100.00%	0.83
23	Na	2.217522(2)	3/2	1.4784371	11.3	70.808516	100.00%	0.093
31	P	1.13160(3)	1/2		17.2	108.394	100.00%	0.066
39	K	0.39147(3)	3/2	0.2610049	2.0	12.500612	93.1%

In the shell model, the magnetic moment of a nucleon of total angular momentum j, orbital angular momentum l and spin s, is given by

${\displaystyle \mu =\left\langle (l,s),j,m_{j}=j|\mu _{z}|(l,s),j,m_{j}=j\right\rangle .}$

Projecting with the total angular momentum j gives

${\displaystyle {\begin{aligned}\mu &=\left\langle (l,s),j,m_{j}=j\left|{\vec {\mu }}\cdot {\vec {j}}\right|(l,s),j,m_{j}=j\right\rangle {\frac {\left\langle (l,s)j,m_{j}=j\left|j_{z}\right|(l,s)j,m_{j}=j\right\rangle }{\left\langle (l,s)j,m_{j}=j\left|{\vec {j}}\cdot {\vec {j}}\right|(l,s)j,m_{j}=j\right\rangle }}\\&={\frac {1}{j+1}}\left\langle (l,s),j,m_{j}=j\left|{\vec {\mu }}\cdot {\vec {j}}\right|(l,s),j,m_{j}=j\right\rangle \end{aligned}}}$

${\displaystyle {\vec {\mu }}}$ has contributions both from the orbital angular momentum and the spin, with different coefficients g^(l) and g^(s):

${\displaystyle {\vec {\mu }}=g^{(l)}{\vec {l}}+g^{(s)}{\vec {s}}}$

by substituting this back to the formula above and rewriting

${\displaystyle {\begin{aligned}{\vec {l}}\cdot {\vec {j}}&={\frac {1}{2}}\left({\vec {j}}\cdot {\vec {j}}+{\vec {l}}\cdot {\vec {l}}-{\vec {s}}\cdot {\vec {s}}\right)\\{\vec {s}}\cdot {\vec {j}}&={\frac {1}{2}}\left({\vec {j}}\cdot {\vec {j}}-{\vec {l}}\cdot {\vec {l}}+{\vec {s}}\cdot {\vec {s}}\right)\\\mu &={\frac {1}{j+1}}\left\langle (l,s),j,m_{j}=j\left|g^{(l)}{\frac {1}{2}}\left({\vec {j}}\cdot {\vec {j}}+{\vec {l}}\cdot {\vec {l}}-{\vec {s}}\cdot {\vec {s}}\right)+g^{(s)}{\frac {1}{2}}\left({\vec {j}}\cdot {\vec {j}}-{\vec {l}}\cdot {\vec {l}}+{\vec {s}}\cdot {\vec {s}}\right)\right|(l,s),j,m_{j}=j\right\rangle \\&={\frac {1}{j+1}}\left(g^{(l)}{\frac {1}{2}}\left[j(j+1)+l(l+1)-s(s+1)\right]+g^{(s)}{\frac {1}{2}}\left[j(j+1)-l(l+1)+s(s+1)\right]\right)\end{aligned}}}$

For a single nucleon ${\displaystyle s=1/2}$. For ${\displaystyle j=l+1/2}$ we get

${\displaystyle \mu _{j}=g^{(l)}l+{1 \over 2}g^{(s)}}$

and for ${\displaystyle j=l-1/2}$

${\displaystyle \mu _{j}={j \over j+1}\left(g^{(l)}(l+1)-{\frac {1}{2}}g^{(s)}\right)}$

• Magnetic moment
• Nuclear magneton
• Gyromagnetic ratio
• Electron magnetic moment
• Nucleon magnetic moment
• Deuterium magnetic moment
• Proton spin crisis

• Nersesov, E.A. (1990). Fundamentals of atomic and nuclear physics. Moscow: Mir Publishers. ISBN 5-06-001249-2.
• Sergei Vonsovsky (1975). Magnetism of Elementary Particles. Mir Publishers.
• Hans Kopfermann Kernmomente and Nuclear Momenta (Akademische Verl., 1940, 1956, and Academic Press, 1958)

• Nuclear Structure and Decay Data - IAEA with query on Magnetic Moments
• magneticmoments.info/wp A blog with all recent publications on electromagnetic moments in nuclei
• Table of nuclear magnetic dipole and electric quadrupole moments, N.J. Stone
• RevModPhys Blyn Stoyle