Paramagnetism

Every electron has a quantum property called spin, which gives it an intrinsic magnetic dipole moment. In atoms where all electrons are paired, their spins cancel ($+\frac{1}{2}$ and $-\frac{1}{2}$) and no net moment exists — this is diamagnetism. Paramagnetic atoms have at least one unpaired electron, so each atom carries a permanent magnetic dipole moment $\mu$. For a single unpaired electron, the magnitude of the spin magnetic moment is approximately $\mu_B = 9.274 \times 10^{-24}$ J/T, called the Bohr magneton.

At room temperature, thermal energy $k_BT$ (where $k_B = 1.381 \times 10^{-23}$ J/K and $T$ is absolute temperature) is much larger than the magnetic energy of a typical dipole. This means the dipoles are constantly jostled by thermal vibrations and point in essentially random directions. The vector sum of all dipole moments in the material averages to zero, so the bulk magnetisation $M = 0$.

When an external magnetic field $\mathbf{B}$ is applied, each dipole experiences a torque $\boldsymbol{\tau} = \boldsymbol{\mu} \times \mathbf{B}$ that tries to align it with the field. The potential energy of a dipole in the field is $U = -\boldsymbol{\mu} \cdot \mathbf{B} = -\mu B \cos\theta$, where $\theta$ is the angle between the dipole and the field. Alignment lowers the energy, but thermal agitation counteracts this. The result is a small net alignment — more dipoles point with the field than against it — producing a weak magnetisation $M$ parallel to $\mathbf{B}$.

The bulk behaviour is described by Curie's Law: $M = \frac{C}{T} H$, where $M$ is the magnetisation (A/m), $H$ is the applied magnetic field intensity (A/m), $T$ is absolute temperature (K), and $C$ is the material-specific Curie constant. The magnetic susceptibility is $\chi_m = M/H = C/T$. Since $\chi_m > 0$ for paramagnets (the induced magnetisation is parallel to $H$) but very small (typically $10^{-6}$ to $10^{-3}$), the attraction to a magnet is real but extremely weak compared to ferromagnetism. Crucially, $\chi_m$ increases as temperature decreases, meaning cold paramagnets are more strongly attracted to magnets.

Unlike ferromagnets, paramagnets have no cooperative interaction between neighbouring atomic dipoles (no exchange coupling). Once the external field is switched off, thermal energy immediately re-randomises the dipole orientations, and $M$ returns to zero. This is why paramagnets are described as having no magnetic memory or remanence. The relaxation time is extraordinarily short — on the order of nanoseconds to microseconds — making paramagnets behave as if they are magnetically 'neutral' in everyday conditions.

\chi_m(300\,\text{K}) = 6.0 \times 10^{-6},\quad M(300\,\text{K}) = 0.72\,\text{A/m};\quad M(150\,\text{K}) = 1.44\,\text{A/m}

• Diamagnetism — Diamagnetism is the contrasting magnetic behaviour where materials with no unpaired electrons are weakly repelled by magnetic fields; comparing both helps clarify how electron configuration determines magnetic response.
• Ferromagnetism — Ferromagnetism involves strong, permanent magnetisation from cooperative alignment of dipoles; understanding it alongside paramagnetism shows why exchange interactions between atoms change magnetic strength by many orders of magnitude.
• Magnetic Susceptibility — Magnetic susceptibility $\chi_m$ is the key quantity that distinguishes paramagnets ($0 < \chi_m \ll 1$) from diamagnets ($\chi_m < 0$) and ferromagnets ($\chi_m \gg 1$).
• Electron Spin — Paramagnetism originates from unpaired electron spins; understanding quantum spin and the Bohr magneton is essential for explaining why only certain atoms are paramagnetic.
• Thermal Energy and Temperature — The competition between magnetic potential energy and thermal energy $k_BT$ determines the degree of dipole alignment, directly explaining why Curie's Law has temperature in the denominator.