Some Nuclear Properties — Question 9

Question 9

A nuclear magnetic moment splits spin projections in a field

A spin-one-half nucleus has moment operator $\boldsymbol\mu=g\mu_N\mathbf I/\hbar$, with $g=4.00$ and $\mu_N=5.05\times10^{-27}\,\mathrm{J/T}$. In a field $B=2.00\,\mathrm T$ along z, its energy is $U=-\mu_zB$. Use $h=6.626\times10^{-34}\,\mathrm{J\,s}$.

1. Find the two Zeeman energies and their separation.

2. Find the resonance frequency and identify the lower-energy projection.

Question 9 - Solution

A nuclear magnetic moment splits spin projections in a field

(a) Method and calculation

$$m_I=\pm\tfrac12\Rightarrow\boxed{U_{m_I}=-g\mu_NBm_I=\mp2.02\times10^{-26}\,\mathrm J}.$$$$\boxed{\Delta E=g\mu_NB=4.04\times10^{-26}\,\mathrm J}.$$

(b) Interpretation and further calculation

$$\boxed{f=\Delta E/h=6.097\times10^7\,\mathrm{Hz}=60.97\,\mathrm{MHz}}.$$$$g>0:\quad\boxed{m_I=+1/2\text{ is lower in energy}}.$$

Physical check

The sign of g determines the favored spin projection. Nuclear moments are much smaller than electronic Bohr-magneton moments because the mass scale is larger.