Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section B / 20 / d

For an axial spin Hamiltonian ${\mathcal{H}}_D=D{S}_z^2$ with $D/k_B=-25$ K, the minima are $m_S=\pm 5/2$. The highest levels along the reversal ladder are the $m_S=\pm 1/2$ doublet. Henceabout 1.25 kJ mol${}^{-1}$ or 12.9 meV per molecule. Using $|D|S^2$ would incorrectly put a half-integer spin at a nonexistent $m_S=0$ level. Transverse anisotropy and quantum tunnelling of magnetization can create faster relaxation pathways, so this axial ladder barrier need not equal an experimentally fitted effective barrier.

Frequency-dependent ac susceptibility probes relaxation: a peak in the out-of-phase component occurs approximately when $\omega \tau =1$. Measuring its displacement with frequency determines $\tau (T)$; an activated region may fit $\tau ={\tau }_0\exp (U_{\mathrm{eff}}/k_{B}T)$. A low-temperature departure can reveal tunnelling or other relaxation channels.

Low-temperature, sweep-rate-dependent magnetization hysteresis tests whether reversal is slow on the measurement timescale. A loop opening below the blocking temperature, variation with sweep rate and possible tunnelling steps provide complementary evidence. Isothermal remanent magnetization decay after field removal directly measures the associated relaxation time. A single open loop alone would not distinguish isolated-molecule blocking from collective ordering or impurities.