The atomic magnetic moment is with

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table

Deriving the Spectrum, Ch Many-electron terms, levels and Hund's rules

Problem

The atomic magnetic moment is μ⃗=−μBℏ(L⃗+gsS⃗)\vec\mu=-\dfrac{\mu_{\rm B}}{\hbar}(\vec L+g_s\vec S) with gs≈2g_s\approx2. (a) Using the projection theorem within a level, together with L⃗⋅S⃗=12[J(J+1)−L(L+1)−S(S+1)]\vec L\cdot\vec S=\tfrac12[J(J+1)-L(L+1)-S(S+1)] and L⃗=J⃗−S⃗\vec L=\vec J-\vec S, S⃗=J⃗−L⃗\vec S=\vec J-\vec L (quantum numbers only), derive gJ=1+(gs−1)J(J+1)+S(S+1)−L(L+1)2J(J+1).g_J=1+(g_s-1)\dfrac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}. (b) The ground level of Ce3+\mathrm{Ce}^{3+} (4f14f^1) is 2F5/2{}^2F_{5/2} (L=3L=3, S=1/2S=1/2, J=5/2J=5/2). Evaluate gJg_J exactly as a fraction, taking gs=2g_s=2. (c) The effective magnetic moment often quoted for a rare-earth ion in its ground level is μeff=gJJ(J+1) μB\mu_{\rm eff}=g_J\sqrt{J(J+1)}\,\mu_{\rm B}. Evaluate μeff\mu_{\rm eff} for Ce3+\mathrm{Ce}^{3+} and compare with the measured value of about 2.54 μB2.54\,\mu_{\rm B}.

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