32-7 Paramagnetism#
Prompts
For a paramagnetic sample in an external magnetic field, what is the relative orientation of the field and the sample’s magnetic dipole moment? How does this differ from diamagnetism?
How is magnetization \(M\) related to a sample’s magnetic moment and volume? What are the typical units?
State Curie’s law and use it to relate magnetization \(M\) to temperature \(T\), Curie constant \(C\), and external field \(B\). When does it apply?
For a paramagnetic sample at a given \(T\) and \(B\), compare the energy of dipole orientations with thermal energy. Why does heating reduce magnetization?
Lecture Notes#
Overview#
Paramagnetic materials: Atoms have permanent magnetic dipole moments
Random orientation when \(B = 0\) → no net magnetization
External field \(\vec{B}_{\text{ext}}\) partially aligns dipoles → net magnetization along the field
Examples: Transition elements, rare earths, actinides (e.g., aluminum, oxygen)
Magnetization#
Magnetization \(M\): net magnetic dipole moment per unit volume
\(\mu\): magnetic moment, \(V\): system volume
Units: A/m (amperes per meter)
Curie’s law#
At low fields and not too low temperatures:
\(C\): Curie constant (material-dependent)
\(T\): temperature (kelvins)
Applies when: Partial alignment; \(M\) small compared to saturation
Physical meaning: Higher \(T\) → more thermal randomization → less alignment. Higher \(B\) → stronger alignment.
Thermal vs orientation energy
At room temperature, \(k_B T \gg \mu B\) for typical fields. Only a small fraction of dipoles align; Curie’s law holds. At very low \(T\) or very high \(B\), saturation occurs and Curie’s law breaks down.
Summary#
Paramagnetism: Permanent atomic dipoles; random at \(B=0\); partial alignment with \(\vec{B}_{\text{ext}}\)
Magnetization \(M\): moment per unit volume
Curie’s law: \(M = C B/T\) for weak fields
Paramagnetic materials: Attracted to stronger field (see 32-6 for torque and force)