Let

μ_{a} = 2.13 μ_{N}

be the magnitude of the saturated nuclear moment of helium-3, with

μ_{N} = e ℏ/ (2 m_{p})

. A spin-one-half nucleus has the two Zeeman energies

\mp μ_{a} B

. For independent nuclei the one-site partition function and magnetization density are

z = 2 cosh (\frac{μ _{a} B}{k _{B} T}), M = n_{3} μ_{a} tanh (\frac{μ _{a} B}{k _{B} T}) .

(1)

In the weak-field limit

μ_{a} B ≪ k_{B} T

, expand the hyperbolic tangent:

χ_{3, nuc} = \frac{μ _{0} n _{3} ( 2.13 μ _{N} ) ^{2}}{k _{B} T} = \frac{μ _{0} ρ _{3} ( 2.13 μ _{N} ) ^{2}}{3 u k _{B} T} .

(2)

The nuclear moment's negative sign means that its moment points oppositely to its spin; it does not make the thermal magnetic susceptibility negative. The response is paramagnetism governed by Curie's law.

There is no extra factor

1/3

here: the stated

2.13 μ_{N}

is the moment of a fully polarized spin-one-half state. Equivalently, writing the moment operator as

g_{N} μ_{N} I

requires

g_{N} = - 4.26

, and

g_{N}^{2} μ_{N}^{2} I (I + 1) /3 = (2.13 μ_{N})^{2}

. The formula presumes independent localized nuclear moments; exchange correlations invalidate it close to the much lower ordering scale discussed in part (e).