In short
Every material responds to an external magnetic field \vec{H}, but the responses fall into three dramatically different families. The response is captured by the magnetisation \vec{M} — the magnetic dipole moment per unit volume — and by the susceptibility \chi and the relative permeability \mu_r:
- Diamagnets (bismuth, water, copper, graphite, ₹20 silver coin, liquid nitrogen): \chi is small and negative (≈ -10^{-5}). They are weakly repelled by magnets. Every material has this response — it comes from Lenz's law applied to atomic orbits.
- Paramagnets (aluminium, liquid oxygen, sodium, gadolinium salts): \chi is small and positive (≈ +10^{-5} to +10^{-3}). They are weakly attracted. They have permanent atomic moments that thermal agitation randomises, producing the Curie law \chi = C/T.
- Ferromagnets (iron, cobalt, nickel, most transformer cores): \chi is huge and positive (up to +10^5) because neighbouring atomic moments lock into aligned domains. Above the Curie temperature T_C, the alignment is destroyed by thermal motion and the material becomes paramagnetic, obeying the Curie–Weiss law \chi = C/(T-T_C).
Why it matters. Iron-core transformers at NTPC power stations work because \mu_r \sim 10^4 for silicon steel concentrates flux. Liquid oxygen poured between the poles of a magnet sticks like pancake batter — a demo done in the physics labs at IIT Kanpur. A pyrolytic graphite tile floats above a Diwali-gift neodymium magnet because graphite has one of the strongest diamagnetic responses known. And ferrite cores in every Indian-made smartphone charger work because the domains keep flipping at mains frequency without losing much energy.
In a physics demonstration room at IIT Bombay, a professor lowers a small test tube full of liquid oxygen between the gap of a powerful electromagnet. The pale-blue liquid does not simply pour through. It sticks. It clings to the field lines, quivers, refuses to fall until the coil current is cut. Everyone in the room has seen a fridge magnet pick up an iron key. Very few have seen liquid oxygen obey a magnet. The same room has a second demo next to it: a small, unassuming black tile of pyrolytic graphite floating — actually floating, in air — above a square of four neodymium magnets. The graphite is being pushed away by the field. A fluid behaves like iron. A solid behaves like anti-iron.
The explanation is that all matter responds to a magnetic field, but different materials respond in qualitatively different ways. Iron is pulled hard. Aluminium is pulled weakly. Graphite is pushed weakly. Water, your own body, a ₹20 silver coin — all are pushed weakly. The task of this article is to classify these responses with two numbers, the susceptibility \chi and the relative permeability \mu_r, to see where the three families (diamagnets, paramagnets, ferromagnets) come from at the atomic level, and to understand why a lump of iron stops being magnetic when it is heated past a specific temperature.
Magnetisation — the dipoles per cubic metre
You already know from the magnetic dipole moment article that any current loop, and by extension any atom with circulating electrons, has a magnetic moment \vec{m}. Inside a real chunk of matter there are about 10^{28} atoms per cubic metre, each carrying some moment. Summing them, then dividing by the volume, gives the magnetisation:
Why: \vec{M} is the dipole-moment density, just as charge density \rho is the charge per unit volume. It is the single macroscopic vector that tells you how much the material has become a magnet.
In a lump of unmagnetised iron, the \vec{m}_i point every which way and sum to zero. Apply an external field, and they align to some degree — \vec{M} becomes non-zero, pointing along the field (for most materials) or against it (for diamagnets).
Why \vec{M} needs a partner field \vec{H}
Inside a magnetised material, the \vec{B} field at any point is the sum of the field produced by external currents (in the wires of an electromagnet, say) and the field produced by the aligned atomic moments themselves (bound currents at the atomic scale). Separating these two contributions is clumsy if you only have \vec{B}. So we introduce an auxiliary vector, the magnetic field intensity:
Why: the combination \vec{B}/\mu_0 - \vec{M} is arranged so that \vec{H} comes only from the free (external) currents. The contribution of the bound currents inside the material drops out. Ampere's circuital law, which in vacuum reads \oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_\text{enc}, inside matter reads \oint \vec{H}\cdot d\vec{\ell} = I_\text{free,enc} — and I_\text{free} is what you can actually measure with an ammeter.
Rearranging the definition of \vec{H} gives the constitutive relation:
This is not a new law of physics; it is the definition of \vec{H}. It becomes useful because, for most materials and moderate field strengths, \vec{M} is simply proportional to \vec{H}:
The proportionality constant \chi (chi) is the magnetic susceptibility — it is a pure number (dimensionless) and is the single material property that classifies diamagnets, paramagnets, and ferromagnets.
Relative permeability
Substitute \vec{M} = \chi\vec{H} into the constitutive relation:
where
Why: \mu_r is the factor by which the material multiplies the vacuum relation \vec{B} = \mu_0\vec{H}. It is dimensionless, and it directly tells you how much the material amplifies or shrinks the flux density relative to vacuum.
A table of orders of magnitude fixes the three classes in your head:
| Class | \chi | \mu_r | Sign of the response |
|---|---|---|---|
| Diamagnet | -10^{-5} | < 1 | weakly repelled |
| Paramagnet | +10^{-5} to +10^{-3} | slightly > 1 | weakly attracted |
| Ferromagnet | +10^{2} to +10^{5} | huge, field-dependent | strongly attracted |
A ferromagnet is not just a paramagnet with a bigger \chi — the factor of 10^5 difference reflects a completely different microscopic mechanism, and the relation \vec{M} = \chi\vec{H} is only a rough approximation for ferromagnets (see Hysteresis and Permanent Magnets). But in the linear regime the table is a fair map.
Diamagnetism — every material has it
Suppose an atom has no permanent magnetic moment (all electron spins paired, all orbital angular momenta cancelled — this is true for noble gases, for the closed-shell configurations of most organic molecules, for Bi, Cu, Zn, and for water). Switch on an external field \vec{B}. What happens?
The electrons in each atom are orbiting the nucleus. As \vec{B} is turned on, Faraday's law (see Faraday's Law) produces a circulating electric field, which slightly speeds up electrons orbiting one way and slows down those orbiting the other way. The net effect: each atom acquires a small induced moment \Delta\vec{m} pointing opposite to \vec{B}. Lenz's law — the induced current opposes the change that created it — means the response must oppose the applied field.
Derivation of the diamagnetic moment (a short one).
Step 1. Take a single electron in a circular orbit of radius r in the xy-plane with angular speed \omega_0. Its orbital moment is m_0 = \frac{1}{2}er^2\omega_0 (half because m = IA = (e\omega_0/2\pi)(\pi r^2), which simplifies to this).
Why: the current is "one electronic charge per period", I = e/(2\pi/\omega_0) = e\omega_0/2\pi, and the enclosed area is \pi r^2. Multiply and the \pi's cancel.
Step 2. Turn on \vec{B} along +\hat{z}, perpendicular to the orbit. The magnetic force on the electron adds a term -evB to the radial force (pointing inward for one orbit sense, outward for the other). Assuming the radius is unchanged (valid to first order in B), the angular speed shifts by \Delta\omega to keep Newton's second law satisfied.
Step 3. Setting m_e\omega^2 r = F_\text{Coulomb} \pm e\omega r B and expanding \omega = \omega_0 + \Delta\omega, the Coulomb force cancels the m_e\omega_0^2 r term and the correction is
Why: this frequency shift is called the Larmor frequency \omega_L = eB/(2m_e). It is the rate at which orbits precess about the external field, and it is the heart of diamagnetism.
Step 4. The change in magnetic moment of one electron is \Delta m = \frac{1}{2}er^2\Delta\omega. Substituting \Delta\omega:
The sign is negative — opposite to \vec{B}. That is the key fact. In a real atom with Z electrons at various radii, one averages \langle r^2\rangle over all orbits to get
(The factor of 6 instead of 4 comes from the three-dimensional average of r^2 over all orientations of orbital planes — only the components perpendicular to \vec{B} contribute, giving a factor of 2/3.)
Step 5. Multiply by the number density n of atoms to get the magnetisation, and compare with M = \chi H \approx \chi B/\mu_0 (since \chi is tiny, \mu_r \approx 1):
This is the Langevin formula for diamagnetic susceptibility. For water (n \approx 3.3\times 10^{28} m⁻³, Z=10, \langle r^2\rangle \sim (10^{-10})^2 m²) it gives \chi \sim -9\times 10^{-6}, matching the measured value \chi_\text{water} = -9.0\times 10^{-6} to within the precision of the order-of-magnitude estimate.
Three facts about diamagnetism that follow from the derivation:
- Every material has a diamagnetic contribution, because every atom has electrons. In materials with permanent moments, this small negative contribution is simply swamped by the positive paramagnetic or ferromagnetic response.
- Diamagnetism is temperature-independent, because the derivation never invoked thermal motion. The induced moment is an electromagnetic effect at the atomic scale.
- The strongest diamagnets are those with large \langle r^2\rangle in a plane perpendicular to \vec{B}. Bismuth and pyrolytic graphite have their electrons in extended planar orbits — bismuth has \chi \approx -1.7\times 10^{-4} (the largest among normal solids), and graphite along its c-axis has \chi \approx -4.5\times 10^{-4}, strong enough for the famous levitating-graphite demo.
Superconductors — the perfect diamagnets
A superconductor below its critical temperature has \chi = -1 exactly — \vec{B} inside is expelled completely. This is the Meissner effect, and it is diamagnetism taken to its logical extreme. The levitating-tile videos you have seen on YouTube, and the Tata Institute of Fundamental Research (TIFR) superconductor demos, all exploit this perfect diamagnetism.
Paramagnetism — permanent moments and thermal jostling
Now consider a material whose atoms have permanent magnetic moments \vec{m} (unpaired electron spins, or unfilled orbital angular momenta). In aluminium, each atom has a moment of about one Bohr magneton \mu_B = e\hbar/(2m_e) \approx 9.27\times 10^{-24} A·m². In the absence of an external field these moments point in random directions — thermal collisions at 300 K knock them about, and the net magnetisation is zero.
Apply a field \vec{B}. Each dipole gains potential energy U = -\vec{m}\cdot\vec{B} = -mB\cos\theta, where \theta is the angle between \vec{m} and \vec{B}. Alignment (\theta=0) is energetically preferred. But thermal energy k_BT competes with the alignment energy mB, and at room temperature in a 1 T field the ratio is
— a tiny fraction. Only a fraction of a per cent of dipoles align preferentially. This is why paramagnetic susceptibilities are small.
Deriving the Curie law
Step 1. Boltzmann statistics says the probability of a dipole having orientation \theta is proportional to e^{-U/k_BT} = e^{mB\cos\theta/k_BT}.
Step 2. Let x = mB/(k_BT) for brevity. Averaging \cos\theta over all directions with this Boltzmann weight gives the Langevin function:
Why: the integral \int_0^\pi \cos\theta\,e^{x\cos\theta}\sin\theta\,d\theta / \int_0^\pi e^{x\cos\theta}\sin\theta\,d\theta evaluates to \coth x - 1/x. You can find the details in a statistical-mechanics textbook; what matters here is the small-x limit.
Step 3. For x \ll 1 (the realistic regime at room temperature), expand: \coth x \approx 1/x + x/3 - x^3/45 + \dots, giving
Step 4. The magnetisation is M = nm\langle\cos\theta\rangle, where n is the number density of dipoles:
(using B \approx \mu_0 H because \chi \ll 1). So
where C = n\mu_0 m^2/(3k_B) is the Curie constant. This is the Curie law: paramagnetic susceptibility is inversely proportional to absolute temperature. Heat a paramagnet and it gets less magnetic.
The Curie law is beautifully clean, and it holds for liquid oxygen, for gadolinium sulfate, for aluminium, for the rare-earth salts used in the first cryogenic demagnetisation experiments. Liquid oxygen at 90 K has \chi \approx 3.6\times 10^{-3} — large enough that the IIT Kanpur demo works. Gaseous oxygen at 300 K has \chi about 3 times smaller, in exact agreement with the 1/T scaling.
Ferromagnetism — neighbours pull each other into line
Iron, cobalt, and nickel behave completely differently. Even without an external field, a small region of iron can be spontaneously magnetised — all its atomic moments pointing the same way. This region is a domain. A bulk piece of iron contains many such domains, each pointing a different direction, and the net magnetisation is usually zero (otherwise your iron spoon would attract every paperclip in the kitchen). Place the iron in a field, and two things happen. The domains aligned with the field grow at the expense of the anti-aligned domains (domain walls move). And inside each domain the moments rotate toward the field direction. Both effects together produce magnetisations M thousands of times larger than any paramagnet could achieve at the same field.
Why do neighbouring moments align?
The mechanism is not the magnetic dipole-dipole interaction between neighbouring atoms — that is far too weak at room temperature (\sim 10^{-3} K equivalent). It is a quantum mechanical exchange interaction: the Pauli exclusion principle, combined with Coulomb repulsion between electrons in neighbouring atoms, makes the parallel-spin configuration energetically favourable for certain electron configurations (partially filled 3d shells in Fe, Co, Ni). The exchange interaction has an effective coupling energy equivalent to a temperature of about 1000 K in iron — strong enough that thermal motion at 300 K cannot disalign neighbouring spins. The full quantum theory (Heisenberg, 1928; see any solid-state physics text) is beyond this article, but the phenomenological consequences are what you need for problem-solving.
The Curie temperature — when alignment breaks
Heat a ferromagnet. At some critical temperature T_C — the Curie temperature — thermal motion finally overwhelms the exchange interaction, the domains lose their coherent alignment, and the material becomes paramagnetic. Above T_C the susceptibility obeys a modified Curie law, the Curie–Weiss law:
Why: the exchange interaction effectively "pre-aligns" the moments, as if there were an internal field proportional to the magnetisation itself. When this is added to the Curie-law derivation, the denominator becomes (T - T_C) instead of T. The susceptibility diverges as T \to T_C^+ — a hallmark of a second-order phase transition.
Some Curie temperatures to anchor the physics:
| Material | T_C (K) | Room-temperature state |
|---|---|---|
| Gadolinium (Gd) | 293 | borderline; ferromagnetic below 20 °C |
| Nickel (Ni) | 631 | ferromagnetic at room T |
| Iron (Fe) | 1043 | ferromagnetic at room T |
| Cobalt (Co) | 1394 | ferromagnetic at room T |
| Magnetite (Fe₃O₄) | 858 | ferromagnetic; the compass mineral |
A transformer core at an NTPC power plant runs at around 80 °C (353 K) — comfortably below iron's T_C. A soldering iron tip gets hot enough (≥ 700 K for some designs) that temperature-control circuits exploit the T_C point: once the tip reaches T_C, its magnetic coupling to the heating coil drops, the heater switches off, the tip cools, T_C is crossed again, the heater switches back on. Every temperature-regulated soldering iron you find at an IIT electronics lab works this way.
Explore the Curie–Weiss behaviour
The interactive below plots 1/\chi against temperature for a material with Curie constant C = 0.12 and lets you drag the Curie temperature T_C. Because 1/\chi = (T-T_C)/C is a straight line, changing T_C slides the line horizontally. The x-intercept of the line is exactly T_C — this is how physicists measure Curie temperatures in the lab.
Worked examples
Example 1: Flux amplification in a silicon-steel transformer core
A long solenoid wound around an empty air core has 4000 turns per metre and carries a current of 0.50 A. Compute H and B inside the air core. Now fill the core with silicon steel of relative permeability \mu_r = 4000. Compute the new B, the magnetisation M, and the susceptibility \chi. This is the geometry of the step-down transformer on the pole outside a Jaipur colony.
Step 1. The solenoid field intensity is H = nI, which depends only on the free current, not on what is inside the solenoid.
Why: \vec{H} is generated only by free currents; the atoms in the core contribute to \vec{M}, not to \vec{H}. This is precisely the reason we introduced \vec{H}.
Step 2. Air core. With \mu_r \approx 1:
Step 3. Silicon-steel core. With \mu_r = 4000:
(Real silicon steel saturates long before this — typically around B_\text{sat} \approx 2 T — but in the linear regime the formula is exact. In practice the transformer core operates below saturation, and \mu_r is the differential slope of B vs H at the operating point.)
Why: the same free current creates the same H, but inside the ferromagnet the aligned atomic moments contribute a magnetisation M thousands of times larger than the H itself, producing a hugely amplified B.
Step 4. Magnetisation and susceptibility.
Result. H = 2000 A/m (same in both cores). B_\text{air} = 2.5 mT, B_\text{core} = 10 T (in the linear regime). M = 8\times 10^6 A/m inside the core, \chi = 3999.
What this shows. A ferromagnetic core is flux-amplifier by a factor of \mu_r. This is why every transformer, motor stator, and inductor at a power station or at your desk charger has an iron or ferrite core — it turns a modest current into a large flux, linking the primary and secondary windings efficiently.
Example 2: Curie-law susceptibility of liquid oxygen at 90 K
Liquid oxygen has a density of 1141 kg/m³ and a molar mass of 32.0 g/mol. Each O₂ molecule has an effective magnetic moment of m = 2.83\,\mu_B (from two unpaired electrons in the triplet ground state). Compute the paramagnetic susceptibility at the boiling point T = 90 K and compare with the measured \chi \approx 3.6\times 10^{-3}.
Step 1. Compute the number density n of O₂ molecules. Molar mass M_\text{mol} = 32\times 10^{-3} kg/mol; density \rho = 1141 kg/m³. Avogadro's number N_A = 6.022\times 10^{23} mol⁻¹.
Why: a mole occupies M_\text{mol}/\rho = 2.81\times 10^{-5} m³; dividing N_A by that gives molecules per m³.
Step 2. Compute the magnetic moment per molecule.
Step 3. Plug into the Curie law \chi = n\mu_0 m^2/(3k_BT).
Evaluate the numerator: (2.147\times 10^{28})(1.257\times 10^{-6})(6.885\times 10^{-46}) = 1.858\times 10^{-23}.
Evaluate the denominator: 3(1.381\times 10^{-23})(90) = 3.729\times 10^{-21}.
Why: dimensional sanity — numerator has units of A²·m²·(kg·m/A²) = kg·m³/s² × 1/T², which after dividing by J = kg·m²/s² gives 1/something-dimensional; the full dimensional chase confirms \chi is dimensionless.
Step 4. Compare with experiment. The measured value at 90 K is \chi \approx 3.6\times 10^{-3}. Our prediction is 38% higher, which is typical agreement for the Langevin model applied to a real molecular liquid — the effective moment is slightly reduced by the small but non-zero orbital contribution and by spin–spin correlations in the liquid.
Result. Predicted \chi \approx 5.0\times 10^{-3}; measured \chi \approx 3.6\times 10^{-3}. Agreement to a factor of 1.4 is good for a single-molecule theory.
What this shows. Paramagnetism is not a mysterious quantum property of iron-like substances — it is a straightforward Boltzmann competition between the alignment energy mB and the thermal energy k_BT. The Curie law lets you predict \chi from molecular parameters and temperature with no free fitting parameters. Scale T from 90 K up to 300 K and the predicted \chi drops by a factor of 300/90 \approx 3.3 — which matches the measurement for gaseous oxygen at 300 K, confirming the 1/T scaling.
Example 3: The diamagnetic levitation of pyrolytic graphite
A thin square tile of pyrolytic graphite (density 2200 kg/m³, \chi = -4.5\times 10^{-4} along its c-axis) is placed above a neodymium magnet whose field falls off with height roughly as B(z) = B_0/(1+z/z_0)^3, with B_0 = 0.5 T at the surface and z_0 = 1 cm. Estimate the equilibrium hover height z_\text{eq}.
Step 1. The force per unit volume on a diamagnetic material in a gradient field is
Since \chi < 0 for a diamagnet and B\,dB/dz < 0 above the magnet (the field decreases with height), the force points upward (the two minus signs combine to a positive).
Why: a dipole with induced moment m = \chi V H in a gradient field feels F = m\,dB/dz. Substituting m = \chi V B/\mu_0 (valid for |\chi|\ll 1) and using B\,dB/dz instead of B^2-derivative form gives the quoted expression. The sign of the force is determined by the sign of \chi.
Step 2. Equilibrium condition. The tile's weight per unit volume is \rho g. Setting upward = downward:
Step 3. Compute dB/dz from B(z) = B_0/(1+z/z_0)^3:
So
Step 4. Plug numbers. |\chi| = 4.5\times 10^{-4}, B_0 = 0.5 T, z_0 = 10^{-2} m, \rho = 2200 kg/m³, g = 9.81 m/s².
Setting this equal to \rho g(1+z/z_0)^7:
Taking the seventh root: 1+z/z_0 = 1.245^{1/7}. Since \ln 1.245 = 0.219, \ln(1+z/z_0) = 0.0313, so 1+z/z_0 = e^{0.0313} = 1.032, giving z/z_0 = 0.032.
Result. z_\text{eq} \approx 0.3 mm. The tile hovers about a third of a millimetre above the magnet array.
What this shows. Even a tiny diamagnetic susceptibility |\chi| \sim 5\times 10^{-4} is enough to support gravity, provided the field gradient is steep enough. The B\,dB/dz \sim B_0^2/z_0 scaling explains why you need small strong magnets (steep gradients) rather than large weak ones: doubling B_0 quadruples the force, and halving z_0 doubles it. The video demos you have seen of graphite and bismuth levitating always use high-gradient, short-range fields from neodymium magnet arrays — exactly as the formula predicts.
Common confusions
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"Diamagnetism means non-magnetic." No — diamagnetism is a real magnetic response, just a weak one that opposes the applied field. Water, copper, your own hand, are all diamagnetic; they are pushed (very weakly) out of regions of strong field. The hand-on-magnet test you do as a child feels no force only because the gradient is not steep enough.
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"Paramagnetic materials are magnetic." Not by themselves. A block of aluminium in your desk drawer has no net magnetisation — its atomic moments point in random directions. Apply a field and M becomes non-zero proportional to H; switch the field off and M returns to zero instantly. This is the key distinction from a ferromagnet, which can stay magnetised after the field is removed.
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"\chi and \mu_r are the same thing." They are related by \mu_r = 1 + \chi but they are not identical. For a ferromagnet \mu_r = 4000 and \chi = 3999 — nearly the same. For a diamagnet \mu_r = 0.99999 and \chi = -10^{-5} — the difference between the \chi's of water and copper is huge (relatively) even though the \mu_r's are both within 0.001% of unity.
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"Heating a magnet makes it stronger." Actively wrong. Heating a ferromagnet toward its Curie temperature reduces its magnetisation; past T_C it becomes paramagnetic and the susceptibility collapses to the tiny C/T values typical of paramagnets. The iron in your tawa, if you could heat it to 770 °C (= 1043 K = T_C of iron), would lose its ferromagnetism — and a magnet dropped on it would no longer stick.
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"The Curie–Weiss law is a law of nature." It is an excellent mean-field approximation, valid well above T_C. Near T_C itself the law breaks down — critical fluctuations take over and \chi diverges with a non-integer power of (T-T_C). This is the physics of critical phenomena, covered in statistical-mechanics textbooks.
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"Superconductors are ferromagnets." Opposite — they are perfect diamagnets with \chi = -1. They expel the field completely (Meissner effect). Ferromagnets concentrate the field; superconductors exclude it.
You have the everyday physics of magnetic materials. What follows is for students aiming at JEE Advanced, or anyone curious about where the simple formulas break down.
The Brillouin function — quantum paramagnetism
The Langevin derivation treated dipole orientations as a continuous angle \theta — classical. In a real atom, the z-component of angular momentum is quantised in units of \hbar. For a system with total angular momentum J, the magnetic moment has 2J+1 allowed orientations, each with energy -m_Jg_J\mu_B B where m_J \in \{-J, -J+1, \dots, +J\}. Summing the Boltzmann factors gives the Brillouin function:
In the limit J\to\infty (classical), B_J \to L (the Langevin function). In the limit J = 1/2 (a single electron spin), B_{1/2}(x) = \tanh(x) — a much simpler expression. For small x, B_J(x) \to (J+1)x/(3J), which reproduces the Curie law with the effective moment m_\text{eff}^2 = g_J^2\mu_B^2 J(J+1). This is how you extract J values from susceptibility measurements on rare-earth salts — an early triumph of quantum mechanics applied to solids.
The Weiss mean-field theory of ferromagnetism
Replace the external field B in the paramagnetic derivation by B + \lambda M, where \lambda M is an internal field representing the average effect of all other atomic dipoles on any given one. The self-consistency equation
has three regimes:
- T > T_C: only the trivial solution M=0 at B=0; for small B, M = \chi B with \chi = C/(T-T_C), giving the Curie–Weiss law with T_C = nm^2\lambda/(3k_B).
- T = T_C: the transition point; \chi \to\infty.
- T < T_C: spontaneous non-zero solutions exist even at B=0 — the ferromagnetic phase. M(T) grows from zero at T_C to its saturation value M_s = nm at T=0, following M/M_s \propto (1-T/T_C)^{1/2} near T_C in mean-field theory.
The mean-field exponent 1/2 is actually wrong (real ferromagnets have \beta \approx 0.35) because mean-field theory ignores fluctuations, but the qualitative shape — a rapid rise of M below T_C, saturation at low T — matches experiment beautifully.
Antiferromagnets and ferrimagnets
Not all materials with neighbour-neighbour exchange line up the moments parallel. In chromium, MnO, and NiO, neighbouring moments prefer to align anti-parallel, giving zero net magnetisation but a distinct ordered state (an antiferromagnet). The transition temperature is called the Néel temperature T_N (Cr: T_N = 311 K). Ferrimagnets (magnetite Fe₃O₄, and the ferrite cores used in Indian-made inductors) have two anti-parallel sublattices of unequal magnitude, giving a net magnetisation much smaller than a pure ferromagnet but non-zero. Indian Railways signalling inductors and every ferrite antenna rod in a transistor radio rely on ferrimagnets to combine high \mu_r with low eddy-current losses (ferrites are electrical insulators, unlike iron).
Temperature of the spin flip — domain reorientation
Inside a single ferromagnetic domain below T_C, the moments are locked parallel. But between domains, the orientation changes over a domain wall (a few hundred atomic spacings thick in iron). Applying a field makes the walls move — domains aligned with the field grow, domains anti-aligned shrink. The energy to move a wall depends on crystal defects and grain boundaries; this is why soft magnetic materials (ferrites, silicon steel) have low coercivity (easy wall motion) and hard magnetic materials (AlNiCo, NdFeB) have high coercivity (walls pinned by defects). The full story is in Hysteresis and Permanent Magnets.
A dimensional-analysis sanity check of Curie's law
What could \chi for a paramagnet depend on? The atomic moment m (units A·m²), the temperature T (K), the number density n (m⁻³), and \mu_0 and k_B (the two coupling constants). Only one combination is dimensionless:
— and Curie's law just fixes the numerical coefficient as 1/3. Dimensional analysis gives you the physics; the factor of 3 comes from the angular averaging. You can always reconstruct the law from the constants alone.
The magnetic-moment zoo
A rough map of moments from atomic to astrophysical:
| Object | Moment (A·m²) |
|---|---|
| Single electron spin | 9.3\times 10^{-24} (= 1 μ_B) |
| Nuclear moment (proton) | 1.4\times 10^{-26} |
| Iron atom (bulk) | 2\times 10^{-23} |
| Compass needle (1 cm × 0.5 cm × 1 mm bar magnet) | \sim 10^{-2} |
| Earth | 8\times 10^{22} |
| Neutron star (pulsar) | \sim 10^{26} |
Every one of these is described by the same two formulas — \vec{\tau} = \vec{m}\times\vec{B} and U = -\vec{m}\cdot\vec{B} — from magnetic dipole moment. The material-dependent question is: how many dipoles per unit volume, and how strongly are they aligned? The answer is \vec{M}, and everything in this article is about computing \vec{M} from \vec{H}.
Where this leads next
- Hysteresis and Permanent Magnets — the B–H curve, retentivity, coercivity, and why some ferromagnets remember their magnetisation and others forget it.
- Earth's Magnetism — the geomagnetic field and how a compass exploits paramagnetism of its own bearings plus ferromagnetism of its needle.
- Magnetic Dipole Moment — the single-loop foundation that all of this is built on.
- Faraday's Law of Electromagnetic Induction — the law behind the Lenz-law origin of diamagnetism and behind every transformer core.
- Force on a Moving Charge — the Lorentz Force — the microscopic force that, applied to orbiting electrons, gives Larmor precession and diamagnetism.