Magnetism and Electrons — Question 4

Question 4

A spin resonance measures the energy gap between two projections

An electron spin in field B=0.350 T has $g_s=2$. Use $\mu_B=9.274\times10^{-24}\,\mathrm{J/T}$ and $h=6.626\times10^{-34}\,\mathrm{J\,s}$. Ignore all other interactions.

1. Find the energy gap and photon frequency for a spin transition.

2. Find the field required to double that frequency and identify which moment projection absorbs a photon.

Question 4 - Solution

A spin resonance measures the energy gap between two projections

(a) Method and calculation

$$\Delta E=(+\mu_BB)-(-\mu_BB)=2\mu_BB=\boxed{6.4918\times10^{-24}\,\mathrm J}.$$$$\boxed{f=\Delta E/h=9.797\times10^9\,\mathrm{Hz}}.$$

(b) Interpretation and further calculation

$$f\propto B\Rightarrow\boxed{B_{2f}=0.700\,\mathrm T}.$$$$\boxed{\mu_z=+\mu_B\ \longrightarrow\ -\mu_B}\quad\text{on absorption}.$$

Physical check

The lower energy corresponds to moment parallel to B, which for an electron means spin angular momentum antiparallel to B.