Nuclear magnetic moment

The nuclear magnetic moment is the magnetic moment possessed by atomic nuclei and is due to the complex structure of the atomic nucleus. The nuclear magnetic moment is explained both by the angular momentum associated with the protons orbiting inside the nucleus as well as by the spin magnetic moment.

Each atom is associated with a value of magnetic moment, caused by the movement of the nucleus (carrier of an electric charge) when rotating on itself (this classical interpretation is used to understand the concept, a quantum interpretation is used to make quantitative calculations).

The classical approach represents the nuclear magnetic moment as a vector, the nuclear magnetic moment vector. This vector is represented as m and its value is:

m=μ μ LL{displaystyle mathbf {m} ={frac {mu }{L}}mathbf {L} }

Where:

L{displaystyle mathbf {L} } is the total angular moment of the core.

μ μ {displaystyle mu ,} is the module of the nuclear magnetic moment.

In turn, the magnitude of the nuclear magnetic moment is given by:

μ μ =α α μ μ N=α α e 2mp{displaystyle mu =alpha mu _{N}=alpha {frac {ehbar}{2m_{p}}}}}}

Where α α {displaystyle scriptstyle alpha } is a constant that depends on the internal structure of the atomic nucleus.

Within quantum mechanics, the nuclear magnetic moment must be treated as a bounded vector linear operator:

system→ → B(H)× × B(H)× × B(H){displaystyle {mbox{system}}{mathcal {B}}({mathcal {H}}})times {mathcal {B}}({mathcal {H}}}}})times {mathcal {B}}{{mathcal {H}}}}}}}

The following identity is fulfilled between the observable associated with the nuclear magnetic moment and the angular momentum operator:

m^ ^ =μ μ l(l+1) L^ ^ {displaystyle {hat {mathbf {m}}}}={frac {mu }{sqrt {l(l+1)}{hbar }{hat {mathbf {L}}}}}}}}}{hbar }{hat {mathbf {l}}}}}}}}}}}}}}}{

Where l{displaystyle scriptstyle l} the main orbital quantum number of the core.