Inorganic Chemistry · Part 6 of 9

Magnetism

Coordination Chemistry, Part 6 · 9 sections · about 21,566 words · CSIR-NET Chemical Sciences, GATE Chemistry & IIT-JAM

Magnetism is the most reliable experimental handle on a complex’s electronic structure, and the place where students most often stop at a single memorised formula. This part starts at that formula, shows exactly which complexes obey it and which do not, and then explains the failures — orbital contribution, spin–orbit coupling, temperature-independent paramagnetism — rather than treating them as exceptions to be ignored. It ends with the cooperative behaviour that makes coordination compounds interesting as materials. Two layers on every section: a slow, hand-held beginner path and a research-grade advanced/reference path.

The 9 sections in Part 6

1Diamagnetism, paramagnetism, ferromagnetism — and the sign and size of χ Free below
2The spin-only formula, the dn table, and what real complexes actually give
3The orbital contribution, quenching, and spin–orbit coupling
4Susceptibility properly defined, and the diamagnetic correction
5Measuring it — Gouy, Faraday, SQUID and the Evans NMR method
6Temperature dependence — Curie, Curie–Weiss and TIP
7Spin-crossover complexes
8Exchange coupling in polynuclear complexes
9Magnetic ordering, and a word on single-molecule magnets

Diamagnetism, paramagnetism, ferromagnetism — and the sign and size of χ

Free extract

Section H.1 of Part 6, reproduced in full from the book — figures and all. No sign-in, no paywall on this section.

Beginner layer

What a magnetic field does to a sample, why every substance is repelled a little, and the three behaviours you have to be able to tell apart from a single number.

Put a sample in a magnetic field H. The sample responds by developing a magnetisation M — a magnetic moment per unit volume. For ordinary samples in ordinary fields the response is proportional to the field, and the constant of proportionality is the volume magnetic susceptibility χ:

M = χ H and B = μ₀(H + M) = μ₀(1 + χ)H

Everything in this part is an argument about the sign and the size of χ, and about how it changes with temperature. There are three cases.

1. Diamagnetism — universal, negative, small

Apply a field to any collection of electrons and the electron orbits are perturbed so as to generate a moment opposing the applied field. This is Lenz’s law operating at the level of an atom. The induced moment is antiparallel to H, so M is negative, so χ is negative. The sample is pushed out of the field.

Diamagnetism: the field-induced circulation of paired electrons, giving a moment opposed to the applied field. χ is negative, small (of order −10⁻⁶ to −10⁻⁴ cm³ mol⁻¹ as a molar quantity), and independent of temperature. Every substance is diamagnetic; in most transition-metal complexes the effect is simply swamped by something larger.

Two consequences you must carry with you. First, a substance with no unpaired electrons is diamagnetic and nothing else — that is what “diamagnetic complex” means in an exam. Second, and much more practically, the diamagnetism of a paramagnetic sample does not go away. It is still there, subtracting a little from every measurement, and if you do not correct for it your moments come out systematically low. That correction is H.4.

2. Paramagnetism — from unpaired electrons, positive, larger

An unpaired electron carries a permanent magnetic moment. In zero field these moments point in random directions and cancel. Apply a field and they align preferentially with it — only preferentially, because thermal motion is constantly randomising them. The net alignment gives a moment parallel to H, so χ is positive, and the sample is drawn into the field.

Paramagnetism: the partial alignment of permanent moments, carried by unpaired electrons, against thermal randomisation. χ is positive, typically 10⁻³ to 10⁻² cm³ mol⁻¹ at room temperature, and — crucially — it falls as temperature rises, because hotter molecules are harder to align.

The competition between field alignment (energy ~ μB) and thermal randomisation (energy ~ kT) is the physical content of the Curie law, χ = C/T, which H.6 develops properly. Note the order of magnitude: paramagnetism beats diamagnetism by two to four powers of ten, which is why a single unpaired electron is easy to detect and why the diamagnetic correction is a correction rather than the main term.

3. Ferromagnetism — cooperative, enormous, field-dependent

In a few solids the moments do not merely respond to the applied field: they respond to each other. Below a critical temperature the exchange interaction between neighbouring centres locks them parallel over macroscopic domains, and the sample carries a magnetisation even with the field switched off.

Ferromagnetism: spontaneous parallel alignment of neighbouring moments below the Curie temperature T_C. χ is positive and very large (10² to 10⁶), it depends on the applied field, and the magnetisation shows hysteresis. Above T_C the ordering is destroyed and the material becomes an ordinary paramagnet.

Two relatives complete the family and are dealt with in H.9: antiferromagnetism, where neighbouring moments lock antiparallel and cancel, and ferrimagnetism, where they lock antiparallel but are unequal so a net moment survives.

Behaviour	Sign of χ	Typical molar χ / cm³ mol⁻¹	Depends on T?	Depends on H?
Diamagnetic	−	−1 × 10⁻⁶ to −5 × 10⁻⁴	No	No
Paramagnetic	+	+1 × 10⁻³ to +1 × 10⁻²	Yes — 1/T	No (in ordinary fields)
Temperature-independent paramagnetic (TIP)	+	+1 × 10⁻⁴ (order of)	No	No
Ferromagnetic	+	very large, 10²–10⁶ (as volume χ)	Yes — vanishes above T_C	Yes, with hysteresis
Antiferromagnetic	+ (small)	small positive, with a maximum at T_N	Yes — non-Curie	No, below saturation
Ferrimagnetic	+	large	Yes — vanishes above T_C	Yes

The single most useful diagnostic column is the last but one. A field-dependent susceptibility is the fingerprint of cooperative behaviour — in practice, of a ferromagnetic impurity if you were not expecting one. Measuring χ at two different field strengths is the standard test, and it is the reason the Faraday method (H.5) is preferred over the Gouy balance for careful work.

The sign of χ is not a bookkeeping convention — it is a statement about what the sample does to the field. A diamagnet (χ < 0) partially excludes flux, so the lines bend around it and the sample is expelled toward weak field. A paramagnet (χ > 0) concentrates flux, so the lines are drawn in and the sample is pulled toward strong field. Every balance method in H.5 measures exactly this force, so the sign of the weight change reads the sign of χ straight off. Schematic, not a field computation.

A nickel(II) complex is weighed in and out of a magnetic field and gains 4.2 mg when the field is switched on. A zinc(II) complex of the same ligand set loses 0.3 mg. What do you conclude about each? Easy

Gain of weight = drawn into the field = χ positive = paramagnetic. The nickel(II) complex therefore has unpaired electrons. Ni(II) is d⁸; in an octahedral or tetrahedral field that means two unpaired electrons, in a square planar field none. The gain in weight already rules out square planar.

Loss of weight = pushed out of the field = χ negative = diamagnetic. Zn(II) is d¹⁰, a closed shell, so no unpaired electrons and only the underlying diamagnetism survives. This is exactly what is seen.

The magnitudes are the real lesson. 4.2 mg against 0.3 mg — a factor of roughly 14 — is a badly compressed version of the true ratio, because the paramagnetic sample is also diamagnetic and the two effects partly cancel. The −0.3 mg from the zinc complex is a direct measurement of about how much has been cancelled. That is the physical basis of the diamagnetic correction in H.4.

What you cannot yet conclude. Nothing about geometry beyond ‘not square planar’, and nothing about spin state, until you convert the weight change into μ_eff. Numbers, not signs, do that.

Advanced / reference layer

Where diamagnetism comes from quantitatively, the Langevin expressions for both signs, and the distinction between χ as a ratio and χ as a derivative.

Diamagnetism, properly. A magnetic field induces a Larmor precession of the electron distribution about the field direction at angular frequency ω_L = eB/2m_e. That precession is a current loop, and its moment opposes B. Summing over all electrons in an atom gives the Langevin–Pauli expression:

χ_dia = − (μ₀ N_A e²) / (6 m_e) × Σ_i ⟨r_i²⟩

Three things follow immediately, and all three are examinable. (i) χ_dia is negative by construction — there is no way to make it positive. (ii) It contains no temperature, because it is a property of the charge distribution, not of an orientational equilibrium. (iii) It scales with ⟨r²⟩ summed over electrons, so it grows with the size and electron count of the molecule — which is precisely why the diamagnetic correction is additive over atoms and groups, and why Pascal’s constants work at all (H.4).

Paramagnetism, properly. For non-interacting moments μ in a field, classical Boltzmann averaging of the projection μcosθ gives the Langevin function; in the ordinary limit μB ≪ kT it expands to the Curie form

χ_M = μ₀ N_A μ_eff² μ_B² / (3 k_B T) ≡ C / T

The factor of 3 is the angular average of cos²θ and is worth remembering, because it is the origin of the 3 in every susceptibility-to-moment conversion you will meet in H.4. The condition μB ≪ kT is well satisfied under ordinary conditions: for μ = 5 μ_B in a 1 T field, μB/k ≈ 3 K, against T ≈ 300 K. That is why χ is field-independent for a simple paramagnet, and why saturation requires either very low temperature or very high field — the regime in which magnetisation rather than susceptibility is measured.

Note: χ as ratio versus χ as derivative. For a linear material the two definitions χ = M/H and χ = ∂M/∂H coincide, and textbooks use them interchangeably. For a ferromagnet, a saturating paramagnet or a spin-crossover system near its transition they do not. Careful magnetochemistry therefore quotes χ = ∂M/∂H measured in a small field, or simply reports M(H) itself. This distinction is why modern papers report χT rather than μ_eff: χT is unambiguous, whereas μ_eff silently assumes Curie behaviour.

Easy

A student reports χ_M = −1.2 × 10⁻⁴ cm³ mol⁻¹ for a cobalt complex and concludes it is ‘weakly paramagnetic’. What is wrong?

Show solution

The sign. A negative susceptibility is diamagnetic, full stop — there is no such thing as negative paramagnetism. The magnitude, ~10⁻⁴, is exactly in the diamagnetic range and about an order of magnitude below even the weakest genuine paramagnetism. The correct conclusion is that the complex has no unpaired electrons. For cobalt that points to low-spin d⁶ Co(III) (t_2g⁶, S = 0), which is the overwhelmingly common case for cobalt(III) with nitrogen or carbon donors.

Med

Why does the susceptibility of a paramagnet depend on temperature while that of a diamagnet does not?

Show solution

Paramagnetism is an orientational effect: permanent moments already exist and the field merely biases a Boltzmann distribution over their orientations. Raising T increases the randomising thermal energy relative to the aligning magnetic energy, so the net alignment — and hence χ — falls, as 1/T. Diamagnetism is an induced effect: the field distorts the electron distribution itself, and the size of that distortion is set by ⟨r²⟩, a ground-state structural quantity that thermal motion barely touches. No orientational equilibrium, no temperature dependence.

Hard

A sample gives χ = 3.1 × 10⁻³ cm³ mol⁻¹ in a 0.5 T field and 1.9 × 10⁻³ cm³ mol⁻¹ in a 1.5 T field, both at 298 K. What has almost certainly happened, and what should be done?

Show solution

A genuine, isolated paramagnet has a field-independent susceptibility in this regime (μB ≪ kT). A susceptibility that falls as the field rises is the classic signature of a small amount of ferromagnetic impurity — typically iron from a spatula, a stirrer bar or the grinding mortar. The impurity contributes a moment that saturates at low field, so its contribution to χ = M/H shrinks as H grows. The remedy is the Honda–Owen extrapolation: measure χ at several fields and plot χ against 1/H; the intercept as 1/H → 0 is the true susceptibility of the sample, because the saturated impurity contributes a term proportional to 1/H. This is also why the Faraday method, which uses milligram samples and permits several field strengths, is preferred for careful work.

Read the rest of Part 6

The remaining 8 sections of this part — The spin-only formula, the dn table, and what real complexes actually give, The orbital contribution, quenching, and spin–orbit coupling, Susceptibility properly defined, and the… — and all nine parts of Coordination Chemistry are part of ChemVidya Full Access, along with the other books, 55 Study Notes and 6,000+ practice questions.

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