Magnetic Materials — Question 7

Question 7

A dilute magnetic mixture can have zero net susceptibility

A dilute composite has volume fraction f of species A with susceptibility +4a and fraction 1-f of species B with susceptibility -a, where a is small and positive. Neglect interactions and local-field corrections, so both species experience the same H.

1. Find the effective susceptibility and the fraction giving zero total M.

2. At that fraction, find the separate contributions to magnetization per unit total volume and explain why zero net susceptibility does not imply zero microscopic response.

Question 7 - Solution

A dilute magnetic mixture can have zero net susceptibility

(a) Method and calculation

$$M=f(4aH)+(1-f)(-aH)=a(5f-1)H.$$$$\boxed{\chi_{\mathrm{eff}}=a(5f-1),\quad f_*=1/5}.$$

(b) Interpretation and further calculation

$$M_A^{\mathrm{total}}=f_*4aH=4aH/5,\quad M_B^{\mathrm{total}}=-(1-f_*)aH=-4aH/5.$$$$\boxed{M_A^{\mathrm{total}}+M_B^{\mathrm{total}}=0\quad\text{with each contribution nonzero}}.$$

Physical check

The cancellation is a volume-average result in the supplied dilute model, not the absence of atomic magnetic moments.