Dia-, Para- & Ferromagnetism

In 1997, in a lab in Nijmegen, a physicist named Andre Geim poured water into the bore of a 16-tesla electromagnet — officially to see what would happen. The water rose, wobbled, and floated. Then he levitated a strawberry. Then a live frog, which hovered in mid-air, mildly indignant and entirely unharmed. Nothing about the frog is a magnet — and that was the point. Every material responds to a magnetic field somehow. The frog's water was being pushed away, feebly but measurably; a strong enough field turns "feebly" into "flight." That push-away is diamagnetism, the faintest and most universal magnetic personality. Aluminium shavings in the same magnet would creep the other way — pulled weakly in — because aluminium is paramagnetic. And a steel paperclip would slam into the pole face, because iron is ferromagnetic, a personality millions of times stronger. Three families, three mechanisms, one classification that CBSE, ISC, NEET, JEE-Main, AP Physics 2 and IGCSE all test — usually as fast one-markers that are pure profit if you know the fingerprints. Magnetisation, Intensity & Susceptibility built the ledger and its personality number $\chi$ . This module is where the personalities come from: what each family's electrons are doing, why temperature bullies one family and ignores another, and how to call the verdict — along or across, toward or away — in thirty seconds.

Key takeaways

Diamagnetism ( $\chi$ small and negative, ~ $-10^{-5}$ ): all materials have it; induced atomic currents oppose the applied field, so diamagnets are feebly repelled and drift toward weaker field. Temperature-independent.
Paramagnetism ( $\chi$ small and positive, ~ $+10^{-5}$ to $10^{-3}$ ): atoms with permanent moments (unpaired electrons) align weakly against thermal jostling; feebly attracted toward stronger field. Obeys Curie's law $\chi = C/T$ .
Ferromagnetism ( $\chi$ effectively $10^{2}$ – $10^{5}$ ): in Fe, Co, Ni, neighbouring atomic moments lock parallel over whole regions (domains); strongly attracted, and can stay magnetised after the field is gone.
Heat a ferromagnet past its Curie temperature (iron: 770 °C) and the lock-in collapses — it becomes an ordinary paramagnet.
A freely suspended rod aligns along the field if para/ferro, across it if diamagnetic — the classic one-marker.
Key idea: the sign and size of χ is a report on the material's electrons — no permanent moments (dia), lone moments fighting heat (para), or moments locked in bulk (ferro).

The core insight

Every material answers an applied magnetic field with one of three personalities — universal feeble opposition (diamagnetism, $\chi<0$ ), thermal-limited weak alignment of permanent atomic moments (paramagnetism, $\chi = C/T > 0$ ), or cooperative bulk lock-in of moments (ferromagnetism, $\chi$ huge) that collapses to paramagnetism above the Curie temperature.

Where this lives today

The three personalities each run an industry. Ferromagnetism's is the obvious empire — every motor, transformer and hard disk — but the weak families earn their keep. MRI contrast agents are engineered paramagnets: gadolinium compounds (seven unpaired electrons per ion) injected to locally disturb the field and light up tissue. Liquid-oxygen paramagnetism becomes an industrial oxygen sensor: measure a gas's paramagnetic pull and you have read its O₂ content without chemistry. Diamagnetism graduated from frog levitation to engineering: pyrolytic graphite sheets hover stably over magnet arrays (a desk-toy demonstration of the toward-weaker-field rule). And temperature's power over magnetism became a tool: magnetic refrigeration cycles gadolinium — Curie point at room temperature — through fields to pump heat without a compressor, and Curie-point soldering irons carry a ferromagnetic tip that regulates its own temperature by losing magnetism at exactly the right heat.

The physical picture

Run the experiment the classification was built on. Take a strong electromagnet with pointed pole pieces — pointed, so the field is intense near the gap and falls off quickly (a non-uniform field, which makes weak effects visible). Hang three small rods in the gap, free to swing and drift. The bismuth rod turns slowly until it lies across the field lines, and edges away from the gap toward weaker field. Feeble, deliberate, unmistakable: repelled. The aluminium rod does the opposite dance at the same faint volume: it settles along the field lines and creeps toward the gap, into stronger field. The iron rod doesn't dance at all — it snaps along the field and lunges at the poles with a force you can feel through the clamp stand. Same field, three verdicts. The verdicts have names: diamagnetic, paramagnetic, ferromagnetic.

Figure

Three rods in the same non-uniform field — three verdicts

Why three personalities? Because atoms have three possible relationships with magnetism, set by their electrons.

Diamagnetism: the induced objection. In many atoms — water, copper, bismuth, the noble gases — the electrons' magnetic effects pair off and cancel: no permanent atomic moment at all. But apply a field and the electron orbits adjust slightly, and the adjustment always creates a tiny moment opposing the applied field. (This opposition is nature's reflexive "no" to changing magnetic conditions — you will meet the grown-up version as Lenz's law in Electromagnetic Induction; here it operates inside every atom.) Induced, opposing, tiny: $\chi \sim -10^{-5}$ . Because the response is manufactured on the spot rather than fought for against heat, it ignores temperature. And because every atom's orbits adjust this way, every material is at least diamagnetic — the other two personalities, where present, simply shout over it.

Paramagnetism: lone voices against the noise. Atoms with unpaired electrons — aluminium, platinum, oxygen, salts like copper sulphate — each keep a permanent magnetic moment, a built-in atomic compass. The applied field tries to align these compasses; thermal agitation keeps knocking them apart. At room temperature the truce heavily favours chaos: even at 1 tesla, net alignment is a fraction of a percent. Hence $\chi$ small and positive, materials feebly attracted, and — since heat is the opponent — a response that strengthens on cooling. That last fact is Curie's law, coming below.

Ferromagnetism: the lock-in. In iron, cobalt, nickel (and gadolinium, and alloys like alnico), something qualitatively new happens: quantum mechanics makes it energetically profitable for neighbouring atomic moments to point the same way. Each atom's alignment is no longer a lone vote against thermal noise — it is enforced by its neighbours, over regions called domains of ~ $10^{17}$ atoms already in lock-step with no field applied. An external field doesn't win atom by atom; it organises blocks that are already internally unanimous. Result: effective $\chi$ in the hundreds to tens of thousands, and — because blocks can stay organised — permanent magnetism after the field is gone. How domains move, and why some materials remember better than others, is the next module's whole story (Domains, Hysteresis & Choosing Magnetic Materials); today needs only their existence.

The fingerprint table and the χ landscape

Exams test this module as classification craft: given a property, name the family; given the family, predict behaviour. This table is the module in miniature — everything in it is a potential one-marker.

fingerprint	diamagnetic	paramagnetic	ferromagnetic
$\chi$ sign & size	$-10^{-5}$ (tiny)	$+10^{-5}$ to $10^{-3}$	$+10^{2}$ to $10^{5}$ (huge, not constant)
$\mu_r$	slightly $< 1$	slightly $> 1$	$\gg 1$
atomic moments	none (induced only)	permanent, independent	permanent, locked in domains
force in non-uniform field	weakly toward weak field	weakly toward strong field	strongly toward strong field
free rod aligns	across $B$	along $B$	along $B$ , emphatically
temperature dependence	none	$\chi = C/T$ (Curie's law)	drops with $T$ ; vanishes at Curie point $T_C$
keeps magnetism when field removed	never	no	can (permanent magnets)
examples	Bi, Cu, water, gold, NaCl	Al, Pt, O₂, Na, CuSO₄	Fe, Co, Ni, Gd, alnico

Figure

The susceptibility landscape — sixty powers of ten apart on one line

Curie's law, and the temperature that ends ferromagnetism

Paramagnetism is a tug-of-war, so its scoreline must depend on the opponent's strength. The field's aligning influence is fixed; thermal jostling grows with absolute temperature $T$ . Pierre Curie measured the result across dozens of substances in 1895 and found a proportionality clean enough to carry his name:

\chi = \frac{C}{T} \qquad (1)

where: $\chi$ is the paramagnetic susceptibility, $T$ is the absolute temperature in kelvin, and $C$ is the Curie constant, a per-material number encoding how large its atomic moments are and how densely they are packed. Read it as a ratio machine: halve the temperature, double the response. A paramagnet at liquid-nitrogen temperature (77 K) answers a field almost four times louder than at room temperature (300 K). Cooling quiets the noise, not the signal. Ferromagnets bring a sharper drama. Their neighbour lock-in is itself a thermal casualty waiting to happen: heat a ferromagnet and the cooperative alignment fights ever-noisier agitation until, at a sharp threshold called the Curie temperature $T_C$ , the lock-in collapses entirely. Above $T_C$ the material is just a paramagnet (obeying a shifted version of Curie's law, $\chi = C'/(T - T_C)$ , stated here for completeness). Iron's $T_C$ is 1043 K = 770 °C: a red-hot iron nail simply drops off a magnet, then re-attracts as it cools past the threshold — one of the great kitchen-blowtorch demonstrations of a phase transition. Nickel's is 358 °C; cobalt holds out to 1121 °C; gadolinium gives up at 19 °C, so it is ferromagnetic in a cold room and paramagnetic in a warm one.

Derivation AThe rod verdicts from the dipole rulebook

We want the two facts examiners ask as "explain why": a paramagnetic or ferromagnetic rod suspended in a field settles along the field lines and drifts toward stronger field, while a diamagnetic rod settles across the lines and drifts toward weaker field. The strategy is to reuse the dipole rulebook from Current Loops as Magnetic Dipoles — a moment $m$ in a field feels the torque $\tau = mB\sin\theta$ and has energy $U = -mB\cos\theta$ , results we cite and apply, not re-derive — and simply keep track of which way each family's moment points. The whole argument is one sign.

Step 1. Fix the sign per family. In para- and ferromagnetic materials the field aligns permanent moments, so the material's net moment (its magnetisation) points along $B$ . In a diamagnetic material the moment is induced in opposition, so it points opposite to $B$ . That single sign is the entire input.

Step 2. Orientation of the rod. A magnetised rod is a dipole lying along its own length. For the para/ferro rod, energy $U = -mB\cos\theta$ is lowest at $\theta = 0$ : moment — and therefore rod — parallel to the field. For the diamagnetic rod the induced moment must point against the field; the rod minimises the magnitude of its opposing response by presenting itself broadside — its long axis settles perpendicular to $B$ , the orientation in which the induced moment (always anti-parallel to $B$ ) costs least energy. The lab verdicts: para/ferro rods align along the lines; dia rods align across them. Why the diamagnetic case cannot simply "flip and align": the induced moment is manufactured by the field itself and always points against it, whatever the rod's orientation — the rod cannot rotate its way into agreement the way a permanent dipole can. Rotation only changes how much opposing moment gets induced.

Step 3. Drift in a non-uniform field. Energy $U = -mB$ (aligned case) falls as $B$ grows — so a moment pointing along the field is pulled toward stronger field, like a ball rolling downhill in energy. For the diamagnet, $m$ points against $B$ , making $U = +|m|B$ : energy rises with field strength, so the push is toward weaker field. Para and ferro creep (or lunge) into the gap; dia edges out of it.

Step 4. Attach the magnitudes. The force scales with the moment, i.e. with $\chi$ : for dia and para ( $|\chi| \sim 10^{-5}$ ) the drift is delicate, needing intense non-uniform fields to see; for ferro ( $\chi \sim 10^{3+}$ ) the same physics is a slammed paperclip. One rulebook, three volumes. Verified: the levitating frog closes the loop — water's induced moment opposes the field, energy grows with B, so the frog is pushed toward weak field, i.e. upward out of the 16 T bore, where the push balances gravity. Feeble χ, heroic field.

Induced magnetism — the IGCSE angle on the same physics

Why does a magnet pick up a steel paperclip that was never magnetised? Because the ferromagnetic response creates a magnet on the spot: the field from the permanent magnet aligns the clip's domains, the near end of the clip becomes an opposite pole (induced magnetism), and opposite poles attract. The clip then magnetises the next clip — hence the party-trick chains of paperclips hanging from one magnet. Remove the magnet and soft steel's alignment mostly collapses; the clips drop. Two statements IGCSE marks directly. First: induced magnetism always produces attraction — the induced pole is always opposite to the inducing one. Which yields the second: repulsion is the only sure test of a permanent magnet. Attraction happens between a magnet and any ferromagnetic object, magnetised or not; only two permanent magnets can repel. (IGCSE also files the "temporary vs permanent" material distinction here — soft iron versus steel — which is exactly the soft/hard choice the next module engineers with hysteresis loops.)

Limiting cases

case	result	meaning
any material, moments all paired	diamagnetism only	the universal floor: everything is at least dia
paramagnet, $T \to$ large	$\chi = C/T \to 0$	heat wins the tug-of-war completely
paramagnet cooled toward 0 K	$\chi$ climbs steeply	alignment wins; response can approach saturation
ferromagnet at $T = T_C$	lock-in collapses	sharp transition: ferro → para
ferromagnet, $T > T_C$	$\chi = C'/(T-T_C)$ , small	just a paramagnet with a memory of the transition
superconductor	$\chi = -1$ , $B$ expelled	the perfect diamagnet — flagged for later physics
field removed	dia: gone instantly · para: gone · ferro: may persist	remanence belongs to ferro alone

Common trapIron is just a very strong paramagnet

It is not, and the distinction is worth marks at every level. A paramagnet is a crowd of independent atomic moments, each fighting thermal noise alone — response proportional to field, vanishing when the field goes, never any memory. A ferromagnet's moments are locked to their neighbours in domains: the material is already ordered locally before you apply anything; your field merely organises the blocks. That cooperation is why $\chi$ is up to $10^{9}$ times larger than diamagnetic values, why the response is not proportional to the field (saturation, curvature), and above all why iron can remember — permanent magnetism after the field is removed — which no paramagnet can do at any temperature. Strength differs by degree; memory differs in kind. If an exam option says "ferromagnetism is paramagnetism with larger χ," it is the wrong option.

Worked examples

Board level — classify, then scale with temperature (CBSE/NEET pattern, 3 marks)

(a) A material has susceptibility $\chi = -6.2 \times 10^{-6}$ , independent of temperature. Classify it, and state how a thin rod of it would settle in a strong uniform field. (b) A paramagnetic salt has $\chi = 2.4 \times 10^{-5}$ at 300 K. Find its susceptibility at 400 K.

Step 1. Part (a). Negative and tiny, temperature-flat: all three fingerprints say diamagnetic. A freely suspended diamagnetic rod settles with its length perpendicular to the field (induced moment opposes $B$ ; broadside costs least energy).

Step 2. Part (b). Curie's law is a ratio machine: $\chi \propto 1/T$ , so $\chi_2 = \chi_1 \dfrac{T_1}{T_2}$ .

Step 3. $\chi_2 = 2.4\times10^{-5} \times \dfrac{300}{400} = \mathbf{1.8 \times 10^{-5}}$ . Hotter salt, quieter response — heat is the alignment's enemy. (Check: temperature rose by factor 4/3, response fell by 3/4. ✓)

Competitive level — Curie's law as M ∝ B/T (the NEET/JEE-Main classic, ~90 seconds)

A paramagnetic sample develops a net magnetisation of 8.0 A/m when placed in a field of 0.60 T at 4.0 K. What is its magnetisation in a field of 0.20 T at 16 K?

Step 1. Set up the scaling. For a paramagnet in the Curie regime, magnetisation is proportional to the aligning field and inversely proportional to temperature: $M \propto \dfrac{B}{T}$ . (This is Curie's law in its most exam-friendly costume: $M = \chi H$ with $\chi = C/T$ and the field doing the aligning.)

Step 2. Build the ratio — never plug absolute values when a proportionality will do:

\frac{M_2}{M_1} = \frac{B_2}{B_1}\times\frac{T_1}{T_2} = \frac{0.20}{0.60} \times \frac{4.0}{16} = \frac{1}{3}\times\frac{1}{4} = \frac{1}{12}

Step 3. $M_2 = \dfrac{8.0}{12} = \mathbf{0.67\ \text{A/m}}$ (exactly $2/3$ A/m).

Step 4. Sanity: the field dropped (less aligning push) and the temperature rose (more jostling) — both effects reduce $M$ , and they multiply. An answer larger than 8 A/m would mean a sign slip in the ratio. Timing note: 60–90 seconds. This exact structure — two fields, two temperatures, one ratio — has appeared on NEET and JEE-Main repeatedly. The only trap is inverting the temperature ratio: T sits in the denominator, so the colder measurement is the stronger one.

Exam traps

Sign of χ is the first read. Negative → diamagnetic, no exceptions and no further thought needed. Positive requires the size check: $\sim10^{-5}$ – $10^{-3}$ para, $\gtrsim 10^2$ ferro.

Temperature behaviour is the second read. Temperature-independent χ → dia. χ falling as $1/T$ → para. Order-collapse at a sharp $T_C$ → ferro. NEET loves "which of these is unaffected by temperature?" — answer: the diamagnet.

"Repelled by a magnet" ≠ "faces a like pole." A diamagnet is repelled from strong field regions — by both poles equally. If a suspended object is repelled by whichever pole you present, it is diamagnetic (or a superconductor); if it repels only one pole, it is a permanent magnet. IGCSE's version: repulsion is the only sure test of a permanent magnet.

Curie law vs Curie temperature. The law ( $\chi = C/T$ ) belongs to paramagnets and has no transition in it. The temperature ( $T_C$ ) belongs to ferromagnets and marks their collapse into paramagnetism. Same Curie, different facts; options that swap them are planted.

Rod orientation wording. Para/ferro rods align "along the field"; the diamagnetic rod aligns "perpendicular to the field" — not "opposite to the field" (a rod has no arrow; only its moment can be opposite, and that is the reason for the perpendicular settling).

Board scope notes. CBSE/NEET/JEE-Main test the three families, examples, χ signs and temperature effects (verbatim syllabus line: "para-, dia- and ferromagnetic substances with examples; effect of temperature on magnetic properties"). AP Physics 2 tests the same classification qualitatively with magnetic permeability. IGCSE never uses the words dia/para/ferro — it tests the behaviour: magnetic vs non-magnetic materials, induced magnetism, temporary (soft iron) vs permanent (steel) magnets. Match your vocabulary to your paper.

Constants. None needed — this module's numericals are ratio craft; χ is always given or cancels.

Origins · 1 minPierre Curie and how this was discovered

Pierre Curie

1859–1906 · Paris

Before radium made the surname immortal, Pierre Curie spent five patient years hanging samples in magnetic fields and weighing forces smaller than a grain of dust. His 1895 doctoral thesis, Magnetic Properties of Bodies at Diverse Temperatures, measured the susceptibility of dozens of substances from room temperature to red heat with a torsion balance he built himself (he was already famous among instrument-makers: at 21 he and his brother Jacques had discovered piezoelectricity). The thesis delivered the module's two headline facts: paramagnetic response falls as $1/T$ — Curie's law — while diamagnetic response ignores temperature entirely; and iron's ferromagnetism dies abruptly at a critical temperature, now called the Curie point. He married Marie Skłodowska the same year, in a civil ceremony without rings, and turned to the radioactivity that won them the 1903 Nobel Prize. He died in 1906, struck by a horse-drawn wagon on a rainy Paris street; the temperature laws of magnetism keep both his name and his methods alive.

History

1778

— S. J. Brugmans notices bismuth is pushed away by a magnet — the first recorded diamagnetic repulsion, a curiosity without a framework.

1845

— Michael Faraday, testing his whole laboratory in a strong electromagnet, finds all matter responds; he coins "diamagnetic" and "paramagnetic."

1895

— Pierre Curie's thesis: χ = C/T for paramagnets, temperature-flat diamagnetism, and the sharp death of ferromagnetism at the Curie point.

1905

— Paul Langevin explains dia- and paramagnetism with atomic-scale physics: induced orbital currents versus thermally jostled permanent moments.

1997

— Andre Geim levitates a live frog in a 16 T magnet — diamagnetism made visible; he later becomes the only person to hold both an Ig Nobel and a Nobel Prize.

Going further

Texts. NCERT Physics Class 12, Ch. 5 (Magnetism and Matter — magnetic properties of materials) · ISC Class 12 Physics, magnetism unit · IGCSE 0625, §4.1 Simple phenomena of magnetism · AP Physics 2 CED, Unit 12 (magnetic behaviour of materials, permeability) · HC Verma Vol. 2, Ch. 37. For the university-level mechanisms — the quantum origin of the induced opposing currents, the statistical mechanics behind Curie's law, and the exchange interaction behind the lock-in — see Diamagnetism, Paramagnetism, Ferromagnetism and Curie Law and Curie Temperature (UG deep dive).

Provenance. All results standard. A verifier should recompute: $\chi_2 = 2.4\times10^{-5}\times300/400 = 1.8\times10^{-5}$ ; $M_2 = 8.0\times(0.20/0.60)\times(4.0/16) = 0.67$ A/m; Curie points Fe 1043 K = 770 °C, Ni 631 K = 358 °C, Co 1394 K = 1121 °C, Gd 292 K = 19 °C; anchors water $\chi = -9\times10^{-6}$ , Al $+2.2\times10^{-5}$ , Bi $-1.7\times10^{-4}$ , liquid O₂ $+3.4\times10^{-3}$ ; alignment fraction at 1 T, 300 K ≈ 0.2 %.

Prerequisites. Magnetisation, Intensity & Susceptibility · Bar Magnets & the Dipole Picture. Leads to. Domains, Hysteresis & Choosing Magnetic Materials (how the ferromagnetic lock-in is organised, and engineered) · Classify the Material — χ, μᵣ & Behaviour (the drill gym for this module, coming up in E5).

If every material is diamagnetic, why don't we call aluminium diamagnetic too?

Aluminium is diamagnetic — underneath. Every atom's orbits produce the induced opposing response. But aluminium atoms also carry permanent moments from unpaired electrons, and the paramagnetic alignment of those moments ( $+2.2\times10^{-5}$ ) is several times louder than the diamagnetic whisper beneath it. The measured χ is the sum, and the sum is positive, so aluminium files as paramagnetic. Classification goes by the loudest voice: dia is only the verdict when it is the only voice. The same logic explains iron: it is diamagnetic and paramagnetic and ferromagnetic — but at a million times the volume, only the lock-in matters.

Why does the diamagnetic response oppose the field instead of helping it?

Because it is induced by a change in magnetic conditions, and nature's response to such changes is always opposition — otherwise every flicker would amplify itself into a runaway. When the field through an atom's electron orbits is switched on, the orbits adjust so as to fight the new flux threading them, leaving a tiny anti-aligned moment. You will meet this exact reflex, scaled up to circuits, as Lenz's law in Electromagnetic Induction — and the full accounting of why matter can nonetheless end up helping the field (para, ferro) needs the permanent moments that diamagnetism-only atoms lack.

How can oxygen — a gas I breathe — be magnetic?

Each O₂ molecule carries two unpaired electrons, giving it a permanent magnetic moment: oxygen is genuinely paramagnetic. In the gas at room temperature the effect is minute (thermal noise, low density). Liquefy it at 90 K, though, and the density jumps a thousandfold while the temperature drops — Curie's law compounds both — and χ reaches $+3.4\times10^{-3}$ : pour pale-blue liquid oxygen between the poles of a strong magnet and it visibly sticks there, hanging in the gap until it boils away. It is the standard lecture demonstration that paramagnetism is real, and the reason O₂ sensors can work magnetically.

What actually happens inside iron at the Curie temperature?

A phase transition — as sharp and real as ice melting. Below 770 °C, the quantum neighbour-coupling that locks adjacent iron moments parallel beats thermal agitation, so domains of aligned moments exist spontaneously. As T climbs, agitation chips at the alignment; at exactly $T_C$ the cooperative order collapses — not gradually to zero but with the characteristic suddenness of a transition. Above $T_C$ , iron's atoms still have their moments (nothing happened to the electrons); they are simply independent again, so red-hot iron behaves as an ordinary paramagnet and drops off magnets. Cool it back and the lock-in reforms.

Why does a magnet attract an unmagnetised iron nail at all, if the nail has no net moment to start with?

The magnet manufactures its own attractor. Its field magnetises the nail — aligns the nail's domains — so the nail's near end becomes an opposite pole (induced magnetism). Opposite poles attract, and because the induced magnetisation is huge (ferromagnetic χ) and the field near the magnet is strong and non-uniform, the pull is dramatic. Note the direction logic from the derivation: the nail's induced moment is along the field, so it is pulled toward stronger field — i.e. toward the magnet, whichever pole faces it. That is why attraction can never identify a pole, and why repulsion is the only sure test of a permanent magnet.

Is Curie's law exact?

It is excellent for independent moments in the weak-alignment regime — which is every paramagnet an exam will meet (at 1 T and 300 K the alignment is only ~0.2 %, deep in the law's comfort zone). It bends in two known ways: at very low temperature or enormous field, alignment saturates and χ stops climbing; and when moments interact, the law shifts to the Curie–Weiss form $\chi = C'/(T-\theta)$ — a ferromagnet above its Curie point follows exactly that shifted law. For board and competitive purposes: χ = C/T, plus knowing when it retires.