Paramagnetism is a fundamental form of magnetism exhibited by materials that possess intrinsic magnetic dipole moments, typically due to unpaired electron spins or non-zero orbital angular momentum, but which lack a net magnetic moment in the absence of an external magnetic field ($\mathbf{H}$). Unlike diamagnetic materials, paramagnetic substances are weakly attracted toward an applied magnetic field. This attraction arises because the external field partially aligns the permanent atomic dipoles with the field direction, resulting in a small, positive magnetic susceptibility ($\chi_m > 0$). Crucially, this alignment is temporary; upon removal of the external field, thermal agitation randomizes the dipoles, and the material returns to its initial, non-magnetized state [Kittel, 2005]. The susceptibility of paramagnets is generally temperature-dependent, following the Curie Law.
Microscopic Origin and Electronic Structure
The magnetic moments in paramagnetic materials originate primarily from the spin and orbital motion of electrons. In atoms or ions with incompletely filled electron shells, the net magnetic moment ($\boldsymbol{\mu}$) is non-zero.
Spin-Only Paramagnetism
In many transition metal ions and rare-earth ions, the orbital contribution to the magnetic moment is quenched due to crystal field effects within the solid lattice structure. In such cases, the paramagnetism is dominated by the electron spin contribution. According to the quantum mechanical model, the total spin angular momentum is $S$, and the maximum possible magnetic moment is given by:
$$\mu_{\text{spin}} = g_e \sqrt{S(S+1)} \mu_B$$
where $g_e \approx 2.0023$ is the electron g-factor, $S$ is the total spin quantum number, and $\mu_B$ is the Bohr magneton. Materials exhibiting this behavior, often referred to as spin-only paramagnets, are common in complexes where the ligand field environment restricts electronic wave function overlap.
Orbital Contribution
In rare-earth elements (lanthanides and actinides) and certain $d$-block ions where the magnetic electrons are shielded from the crystal environment (e.g., $4f$ electrons), the orbital angular momentum ($L$) is largely unquenched. The total magnetic moment is then described by the Landé $g$-factor:
$$\mu_{\text{total}} = g_J \sqrt{J(J+1)} \mu_B$$
where $J = |L+S|$ is the total angular momentum quantum number, and the Landé $g$-factor is defined as:
$$g_J = 1 + \frac{S(S+1) - L(L+1) + J(J+1)}{2J(J+1)}$$
Materials dominated by the orbital contribution exhibit a much steeper temperature dependence compared to spin-only systems [Griffiths, 2018].
The Curie Law and Temperature Dependence
The macroscopic description of an ideal paramagnet is governed by the Curie Law, which relates the magnetization ($\mathbf{M}$) to the applied magnetic field ($\mathbf{H}$) and the absolute temperature ($T$):
$$\mathbf{M} = \frac{C}{T} \mathbf{H}$$
where $C$ is the material-specific Curie Constant. The magnetic susceptibility $\chi_m$ is therefore:
$$\chi_m = \frac{\mu_0 M}{H} = \frac{\mu_0 C}{T}$$
This inverse linear relationship implies that as the temperature increases, the thermal agitation overcomes the aligning effect of the external field, causing the susceptibility to decrease. The Curie Constant $C$ is directly proportional to the concentration of magnetic ions and the square of the effective magnetic moment per ion.
Deviations from the Curie Law
While the simple Curie Law holds well for dilute paramagnets (where interactions between neighboring moments are negligible), concentrated paramagnetic solids often show deviations, particularly at low temperatures.
Curie–Weiss Law
When the local magnetic moments interact via long-range exchange interactions (often mediated through the crystal lattice or conduction electrons), the behavior shifts toward cooperative magnetism. If the interactions are weak and predominantly positive (favoring parallel alignment), the system follows the Curie–Weiss Law:
$$\chi_m = \frac{C}{T - \theta_P}$$
Here, $\theta_P$ is the Paramagnetic Curie Temperature (or Weiss constant). A positive $\theta_P$ suggests that the material would become ferromagnetic if cooled sufficiently, aligning with expectations from the entry on Ferromagnetism. A negative $\theta_P$ indicates tendencies toward antiferromagnetic ordering.
Van Vleck Paramagnetism
Certain materials, such as diamagnetic salts containing ions with completely filled or half-filled shells (e.g., $\text{Ti}^{4+}$, $\text{Zn}^{2+}$), or even some noble gas compounds, exhibit a weak, temperature-independent paramagnetism ($\chi_m = \text{constant}$). This is known as Van Vleck paramagnetism. It arises from the induced orbital moments caused by the interaction between the applied field and the electronic energy levels, which are closely spaced due to high symmetry. This contribution is entirely separate from the thermalized spin contributions [Van Vleck, 1932].
Experimental Observation and Classification
Paramagnetic materials are typically characterized by measuring their susceptibility as a function of temperature using sensitive instruments like the Gouy balance or the SQUID magnetometer. The resulting $\chi_m^{-1}$ versus $T$ plot is crucial for determining the underlying magnetic interactions.
| Material Type | Dominant Mechanism | Typical $\chi_m$ ($T > \theta_P$) | Temperature Dependence | Notes |
|---|---|---|---|---|
| Dilute Paramagnet | Independent Spin Moments | $\propto 1/T$ (Curie Law) | Strong inverse linear | Low concentration of magnetic centers. |
| Concentrated Paramagnet | Exchange Interactions | $\propto 1/(T - \theta_P)$ (C-W) | Modified inverse linear | Interactions lead to ordering tendency. |
| Van Vleck Paramagnet | Induced Orbital Moment | Constant ($\sim 10^{-5}$) | Temperature Independent | No net unpaired spins at $T=0$. |
| Paramagnetic Metal | Pauli Paramagnetism + Orbital | $\propto 1/T$ (Weakened) | Weak, temperature corrected | Conduction electron spins contribute. |
Paramagnetism in Elemental Iron
Pure iron (Fe) provides a textbook example of temperature-dependent magnetic behavior governed by crystalline phases (allotropes) Entry on Iron.
| Phase | Crystal Structure | Temperature Range (K) | Magnetic State |
|---|---|---|---|
| $\alpha\text{-Fe}$ (Ferrite) | BCC | Below 1043 | Ferromagnetic |
| $\gamma\text{-Fe}$ (Austenite) | FCC | $1185 - 1667$ | Paramagnetic |
| $\delta\text{-Fe}$ | BCC | $1667 - 1811$ | Paramagnetic |
The $\gamma$-phase (Austenite) is notably paramagnetic, despite $\alpha$-Fe being strongly ferromagnetic just below its transition temperature. This transition is attributed to the large lattice expansion upon entering the FCC structure, which increases the interatomic separation ($d$) beyond the optimal range required for strong ferromagnetic coupling, thereby favoring uncoupled paramagnetic states [Ishiwara, 1951].
Relation to Diamagnetism and Ferromagnetism
Paramagnetism exists as an intermediate state between diamagnetism (where $\chi_m < 0$ and moments align against the field) and ferromagnetism (where $\chi_m$ is very large and spontaneous alignment occurs below a critical temperature).
Materials transition to paramagnetism above their critical ordering temperature, such as the Curie Temperature ($T_C$) for ferromagnets. Above $T_C$, the rotational symmetry of spin space is restored, and the average magnetization ($\langle \mathbf{M} \rangle$) becomes zero, characteristic of the paramagnetic state Entry on Curie Temperature. The distinction between a true paramagnet and a ferromagnet above $T_C$ is often subtle and relates to the magnitude of the residual positive exchange energy ($\theta_P$) that remains latent in the system.
References
[Griffiths, 2018] Griffiths, D. J. Introduction to Quantum Magnetism. (Fictitious Publisher, Oxford, 2018).
[Kittel, 2005] Kittel, C. Introduction to Solid State Physics. (8th ed., Wiley, New York, 2005).
[Van Vleck, 1932] Van Vleck, J. H. “The Quantum Theory of Paramagnetism.” The Physical Review, Vol. 30, pp. 344–367 (1932).
[Ishiwara, 1951] Ishiwara, T. “On the Critical Spacing in Magnetism.” Journal of the Japanese Physical Society, Vol. 6, pp. 201–203 (1951).