Magnetism in Solids

Relation between applied field, susceptibility, magnetization, and magnetic order in solids.

Conceptual context

In a solid, electrons do more than carry charge: they also provide magnetic moments that may align, cancel, or cooperate. The central question is how $M$ changes when an external $H$ is applied and how the material $\chi_{m}$ controls that response. A diamagnetic solid responds weakly against the field; a paramagnetic solid follows it; a ferromagnetic solid may show collective order, memory, and saturation. This makes magnetism in solids a bridge between measurable macroscopic fields such as $B$ and microscopic ingredients such as $n$ and $\mu$.

🟢 Essential level

$M$ measures the net magnetic moment per unit volume inside the solid. When an applied $H$ is present, the material response is not universal: its $\chi_{m}$ tells whether the response is weak, strong, aligned, or opposed. If $\chi_{m}$ is positive, $M$ usually points along the chosen direction of $H$; if it is negative, the response is slightly opposite. $T$ matters because thermal disorder competes with moment alignment. In the linear regime, doubling $H$ approximately doubles $M$, but this picture fails near $M_s$, in hysteretic materials, or close to magnetic ordering transitions. The essential skill is to read slope, sign, and physical ceiling before calling every intense response magnetic order.

This prevents mixing a simple induced response with genuine collective magnetic order.

Iron is the classic ferromagnetic example: below its Curie temperature, about 1043 K, its magnetic domains can align almost completely under a modest field, reaching a very high $M_s$. Above that temperature, collective order disappears and iron behaves as a simple paramagnet following the Curie-Weiss law.

🔵 Formal level

Basic formulas

The macroscopic description begins by separating the applied field from the response of the medium. In a linear and isotropic regime, induced $M$ is proportional to $H$, with $\chi_{m}$ as the experimental slope. That slope keeps the material sign: positive for parallel response and negative for antiparallel response.

Linear induced magnetization

The $B$ inside the solid is not necessarily the vacuum field because it includes the material contribution produced by $M$ in addition to the contribution associated with $H$. This is why these quantities should not be swapped even when they appear in the same magnetic problem.

Total magnetic field in a medium

Real student questions

Why is $B$ not enough? Because $B$ combines the applied contribution and the material response; to understand the solid, one must read $M$ and $\chi_{m}$.

Is a large $\chi_{m}$ always better? No. It may be useful in magnetic cores, but it can also bring nonlinearity, losses, hysteresis, or saturation.

Why does $T$ enter? Heat disorders moments. As $T$ rises, alignment usually becomes harder, except for subtler effects in ordered materials.

What does $\theta$ mean? It is a macroscopic clue to internal interactions: positive suggests effective ferromagnetic tendency, while negative suggests effective antiferromagnetic tendency.

Final synthesis

Magnetism in solids asks how a material turns $H$ into $M$. The key lies in $\chi_{m}$, $T$, microscopic structure, and saturation limits, not only in applying proportionality.