Why does lutetium have no 4f magnetic moment?

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A line graph showing electron probability distributions for gadolinium, with separate curves for 4f, 5s, 5p, 5d, and 6s orbitals. The x-axis shows distance from the nucleus in atomic units and the y-axis shows probability P2(r). — rare-earth element Electron probabilities, P²*(r)*, for the 4f, 5s, 5p, 5d, and 6s electrons of gadolinium.

Lutetium has no 4f magnetic moment because its 4f electron shell is completely filled with 14 electrons.12

As electrons are added across the lanthanide series, they initially align their spins in the same direction until the 4f level is half-filled at gadolinium. According to the Pauli exclusion principle, subsequent electrons must align antiparallel to the existing ones, resulting in paired electrons. By the time the 14th electron is added in lutetium, all 4f electron spins are paired up, leaving no unpaired electrons to contribute to a magnetic moment. Consequently, lutetium is only weakly magnetic.23