 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-and-condensed-matter-physics-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011/section-vi/8/i/solution

The Brillouin function is odd, increases monotonically through zero, and approaches

\pm 1

for

x \to \pm \infty

. Its small-

x

(J + 1) / (3 J)

. For

x ≫ 1

, the magnetic ions overwhelmingly occupy the largest field-aligned moment projection, so the magnetization reaches magnetic saturation:

M ⟶ M_{s} = n g μ_{B} J .

The sketch shows several

J

values; all have the same limiting values, while their slopes differ.

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