Building the Periodic Table — Question 8

Question 8

Spin-only magnetic moments vary systematically across a d subshell

Use Hund filling for a degenerate d subshell and the spin-only moment $\mu_{\mathrm{eff}}=g_s\mu_B\sqrt{S(S+1)}$ with $g_s=2$. Neglect orbital magnetism and spin-orbit corrections.

1. For occupancies $d^3$ and $d^8$, find the number of unpaired electrons and S.

2. Calculate their effective moments and their ratio.

Question 8 - Solution

Spin-only magnetic moments vary systematically across a d subshell

(a) Method and calculation

$$d^3:\ N_u=3,\ \boxed{S=3/2};\qquad d^8:\ N_u=10-8=2,\ \boxed{S=1}.$$$$\mu_{\mathrm{eff}}=2\mu_B\sqrt{(N_u/2)(N_u/2+1)}=\mu_B\sqrt{N_u(N_u+2)}.$$

(b) Interpretation and further calculation

$$\boxed{\mu_{d^3}=\sqrt{15}\mu_B=3.873\mu_B,\quad\mu_{d^8}=\sqrt8\mu_B=2.828\mu_B}.$$$$\boxed{\mu_{d^3}/\mu_{d^8}=\sqrt{15/8}=1.369}.$$

Physical check

The effective moment magnitude differs from the largest projection gS times the Bohr magneton. The spin-only approximation is appropriate only when orbital contributions are negligible.