Question 6 of 7: Magnetism — Diamagnetism/Paramagnetism, Nuclear vs. Electronic Moments, Susceptibility, and the Curie Constant

Question 6: Magnetism — Diamagnetism/Paramagnetism, Nuclear vs. Electronic Moments, Susceptibility, and the Curie Constant (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Figure P6: $\chi^{-1}$ (in units of $10^4\,\text{mol/cm}^3$) plotted against $T$ (K) for a rare-earth-bearing crystal, an essentially straight line rising from near the origin at low $T$ to $\approx4.1\times10^4\,\text{mol/cm}^3$ at $T=300\,\text{K}$.

Find. (a)–(c) qualitative physics; (d) magnetic character, its $T$-dependence, and the Curie constant $C$.

Approach. (a)–(c) are conceptual recall; (d)(i)–(ii) read the trend directly off the graph's shape; (d)(iii) fits the digitized data to $\chi^{-1}=T/C$ and inverts the slope.

1. Part (a) — the three factors in dia-/para-magnetism. Spin and orbital momentum of the electrons are both PERMANENT magnetic moments: in an atom with unpaired spins or a net orbital angular momentum, these moments exist even with no applied field, and an applied field tends to ALIGN them with itself — lowering the energy of the aligned configuration and giving a positive (paramagnetic) susceptibility that competes against thermal randomization. The THIRD factor — the change of orbital angular momentum INDUCED by the applied field itself — is a completely different mechanism: by Lenz's law, the field induces a small extra electron orbital motion that opposes the applied field, present in every atom (even those with no permanent moment, i.e. fully-paired atoms). This gives a small NEGATIVE (diamagnetic) contribution that is always present but is completely swamped by the much larger paramagnetic alignment whenever a permanent moment (factors 1–2) exists; diamagnetism is only the dominant, visible effect in atoms/ions with no net spin or orbital moment (all electron orbitals fully paired).
2. Part (b) — nuclear vs. electronic paramagnetism. A magnetic moment scales inversely with the mass of the circulating charge ($\mu\propto e\hbar/2M$), so the nuclear magneton is smaller than the Bohr magneton by the electron-to-proton mass ratio, $m_e/M_p\approx1/1836$. Since Curie-law paramagnetic susceptibility is proportional to the SQUARE of the magnetic moment, nuclear paramagnetism is weaker than electronic paramagnetism by roughly $$\boxed{\left(\frac{m_e}{M_p}\right)^2\approx\left(\frac1{1836}\right)^2\approx3\times10^{-7}}$$ (of order $10^{-6}$–$10^{-7}$ once nuclear $g$-factors of a few are included) — nuclear paramagnetism is normally negligible compared with the electronic contribution except in specialized very-low-temperature nuclear cooling/NMR contexts.
3. Part (c) — definition and units of $\chi$. The magnetic susceptibility is defined as the ratio of the induced magnetization $M$ to the applied field $H$: $\chi\equiv M/H$ (volume/dimensionless susceptibility in SI, since $M$ and $H$ share units of A/m). It is common to instead quote a MOLAR susceptibility $\chi_{\text{mol}}=\chi\,V_{\text{mol}}$ (as Figure P6 does, in $\text{cm}^3/\text{mol}$, CGS convention) so that measurements on different samples of the same substance can be compared per mole regardless of sample density/shape.
4. Part (d)(i) — paramagnetic or diamagnetic? Figure P6 plots $\chi^{-1}$ RISING linearly with $T$, i.e. $\chi$ itself DECREASES as $T$ increases, obeying $\chi=C/T$ (Curie law) with a straight-line fit passing close to the origin. $$\boxed{\text{Paramagnetic}}$$ A diamagnetic material would show a small, essentially temperature-INDEPENDENT (flat) $\chi$, not this rising-linear $\chi^{-1}(T)$ signature.
5. Part (d)(ii) — why $\chi$ decreases with $T$. In a paramagnet the permanent atomic moments (from part (a)) are only partially aligned by the applied field, since thermal agitation is constantly randomizing their orientation. As $T$ rises, this thermal randomization becomes stronger relative to the (fixed) field-alignment energy, so the net magnetization per unit field — and hence $\chi$ — falls off, exactly as the Curie law $\chi=C/T$ predicts (an entropy effect: increasing $T$ favours the high-entropy, randomly-oriented state over the low-entropy, field-aligned state).
6. Part (d)(iii) — Curie constant. Fitting $\chi^{-1}=T/C$ (a straight line through the origin) to the digitized graph data by least squares gives slope $=0.01431\times10^4\,\text{mol/(cm}^3\text{K)}=143.1\,\text{mol/(cm}^3\text{K)}$, with intercept within error of zero (equivalent Weiss temperature $\theta\approx+7\,\text{K}$, negligible next to the 20–300 K measurement range — consistent with simple Curie-law paramagnetism rather than an ordered/Curie–Weiss material). Inverting the slope: $$\boxed{C=\frac1{\text{slope}}\approx7.0\times10^{-3}\ \text{cm}^3\text{K/mol}}$$