16.09.01 · inorgchem / lanthanides-actinides

Lanthanides and actinides — the f-block

shipped3 tiersLean: none

Anchor (Master): Cotton 2006 Lanthanide and Actinide Chemistry (Wiley); Edelmann 2017 Coord. Chem. Rev. actinide review; Cotton & Wilkinson 1988 Advanced Inorganic Chemistry (Wiley) Ch. 20

Intuition Beginner

The two rows printed below the main periodic table are the f-block: the lanthanides (lanthanum, La, Z = 57, through lutetium, Lu, Z = 71), whose buried 4f subshell fills, and the actinides (actinium, Ac, Z = 89, through lawrencium, Lr, Z = 103), whose 5f subshell fills. The two rows look almost alike within each family — and that near-sameness is both their fascination and their difficulty.

The lanthanides are near-twins. They almost always form $Ln^{3 +}$ ions, they are hard and oxophilic, they are mostly pale-coloured, and they grab oxygen-donor ligands at high coordination numbers (8, 9, even 10). Their chemistry is so similar that prising one lanthanide from another was, for a century, among the hardest jobs in chemistry. The one thing that changes steadily across the family is size: the ions shrink as the 4f shell fills.

That steady shrink is the lanthanide contraction, and it is the master key to f-block chemistry. Electrons added to the buried 4f subshell shield the outer electrons poorly from the growing nuclear charge, so each added proton pulls the cloud a little tighter. The actinides do everything the lanthanides do, but louder: their 5f orbitals reach farther out, they show several oxidation states (uranium runs from $+ 3$ to $+ 6$ ), and a few of them drive nuclear reactors and weapons.

Visual Beginner

The figure shows the two f-block rows seated between the d-block and the rest of the table. In the top row the 14 $Ln^{3 +}$ ions shrink smoothly from La³⁺ (1.16 Å) to Lu³⁺ (0.98 Å), the lanthanide contraction. In the bottom row the early actinides (Ac to Am) display a fan of oxidation states that narrows toward the later actinides, which become lanthanide-like and settle on $+ 3$ . A dashed arrow links Lu to Hf to mark the contraction's payoff: hafnium sits directly above zirconium at almost the same size.

Worked example Beginner

The lanthanide contraction in numbers. Across the 14 lanthanide ions $Ln^{3 +}$ , from La³⁺ to Lu³⁺, the eight-coordinate ionic radius shrinks from about 1.16 Å to about 0.98 Å. The total contraction is $1.16 - 0.98 = 0.18$ Å, or roughly 18 picometres, spread across 14 elements — about 1.3 pm per step.

Eighteen picometres sounds tiny, but it reshapes the periodic table. The element that follows the lanthanides, hafnium (Z = 72), is pulled in by the whole 18-pm contraction, so its $Hf^{4 +}$ ion (0.78 Å) is almost the same size as the $Zr^{4 +}$ ion (0.72 Å) sitting directly above it. Zirconium and hafnium therefore behave almost identically: their ores always occur together, their dioxides and tetrachlorides have nearly the same lattice constants, and separating them is a slow, brute-force business.

The same squeeze explains why the densest metals in existence — osmium, iridium, platinum, gold — sit just after the lanthanides and actinides. Their atoms have been contracted by poor f-electron shielding, packing more mass into less volume than the periodic trend above them would predict.

Check your understanding Beginner

Formal definition Intermediate+

The f-block elements are those for which the $(n - 2) f$ subshell is the frontier being filled: the lanthanides ( $4 f^{1} 5 d^{0 - 1} 6 s^{2}$ , Ce through Lu, conventionally La–Lu) and the actinides ( $5 f^{1 - 14} 6 d^{0 - 2} 7 s^{2}$ , Ac through Lr). Across each row the $(n - 2) f$ subshell fills while the valence $n s^{2}$ shell stays essentially fixed, which is why each family behaves as a coherent chemical group.

The defining quantity is the ionic radius $r (Ln^{3 +})$ as a function of position $n = 1, \dots, 14$ in the series (Shannon radii, fixed coordination number):

r (Ln^{3 +}) = r_{0} - Δ r \cdot \frac{n - 1}{13}, r_{0} = r (La^{3 +}), Δ r \approx 0.18 \overset{˚}{A} .

The contraction $Δ r$ is the lanthanide contraction: the smooth shrinkage of $r (Ln^{3 +})$ from La³⁺ to Lu³⁺ caused by the poor shielding of the 4f electrons. Its origin is that an added 4f electron shields outer electrons from the added proton with an efficiency well below 1, so the effective nuclear charge $Z_{eff}$ at the valence shell rises monotonically.

The accessible oxidation states mark the qualitative split between the two rows. For the lanthanides the $+ 3$ state dominates; only Ce ( $+ 4$ ), Eu and Yb ( $+ 2$ ) have stable alternatives of preparative importance. For the actinides the early members show a rich redox span:

Ac^{3 +}, Th^{4 +}, Pa^{5 +}, U^{3 + /4 + /5 + /6 +}, Np^{3 + \dots 7 +}, Pu^{3 + \dots 6 +}, Am^{2 + \dots 6 +},

after which the $+ 3$ state reasserts itself from curium onward as the 5f orbitals contract and become core-like. The actinyl cations $UO_{2}^{2 +}$ , $NpO_{2}^{+}$ , $PuO_{2}^{2 +}$ (linear O=An=O, An in the $+ 5$ or $+ 6$ state) are the structural signature of this redox chemistry.

Key mechanism [Intermediate+] {#key-mechanism}

The mechanism that organises the whole f-block is the poor shielding of the 4f subshell. When a 4f electron is added alongside a proton, it sits deep inside the atom, within the radial node structure of the 5s and 5p shells. From that buried position it screens the outer electrons from the new proton with an efficiency $S_{4 f}$ well below the 0.35 that a same-shell electron would provide. The effective nuclear charge felt by the outer electrons therefore climbs:

Z_{eff}^{valence} (Z) = Z - σ (Z), \frac{d Z _{eff}}{d Z} = 1 - \frac{d σ}{d Z} > 0,

because each added (proton, 4f electron) pair adds a full unit of charge but less than a full unit of shielding. The valence cloud is drawn inward, the ionic radius $r \propto 1/ Z_{eff}$ contracts, and the contraction accumulates across all 14 elements to give $Δ r \approx 18$ pm.

Three chemical consequences fall out of this single mechanism. First, within the lanthanides themselves, the steadily shrinking $Ln^{3 +}$ radius underlies the difficulty of lanthanide separations: every complexing agent discriminates between adjacent lanthanides only through that 1.3-pm step. Second, the contraction is inherited by the post-lanthanide elements, so hafnium (Z = 72) is anomalously small — almost identical in size to zirconium (Z = 40) — which forces Hf/Zr to co-occur and makes the dense, late-5d metals (Os, Ir, Pt, Au) denser than their 4d congeners. Third, the same logic applied to the 5f shell produces a weaker "actinide contraction", because the radially extended 5f electrons shield the valence shell somewhat better; this weaker contraction is the foundational reason the early actinides retain chemically accessible 5f electrons and show a wide redox span.

The 4f/5f contrast is the whole story in one comparison. The 4f orbitals are buried ( $⟨ r ⟩_{4 f} < ⟨ r ⟩_{5 s, 5 p}$ ), so they are core-like, the lanthanides are locked to $+ 3$ , their f electrons do not bond, and the family is monotonous. The 5f orbitals extend out past the 6s/6p core ( $⟨ r ⟩_{5 f} \sim ⟨ r ⟩_{6 d}$ ), so they overlap ligands, the early actinides show variable oxidation states and covalent actinyl bonding, and only later in the series do the 5f orbitals contract back to core-like behaviour.

Bridge. This shielding mechanism builds toward the entire descriptive edifice that follows — separations, luminescence, the actinyl bond, the nuclear fuel cycle all descend from how buried the f electrons are — and appears again in coordination chemistry 16.04.x and solid-state chemistry 16.07.x, where ionic radius and charge density are the same currencies. This is exactly the foundational mechanism that converts the f-block from a roster of 28 look-alike metals into a structured family; the foundational reason the lanthanide contraction is the master key is that f-electron shielding is incomplete, and the central insight — radius shrinks as $Z_{eff}$ rises — is the periodic-trend engine of 16.01.01 operating one row deeper.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

Lanthanide complexes almost always display coordination numbers 8, 9, or 10, far higher than the 4- or 6-coordination typical of the $3 d$ transition metals. Give the two physical reasons for this preference.

Hint

Consider the ionic radius of $Ln^{3 +}$ and the strength of any ligand-field stabilisation.

Answer

(a) Large ionic radius. $Ln^{3 +}$ ions are big (1.0–1.2 Å) relative to $3 d$ cations (~0.6–0.8 Å), so they present a large surface area to incoming ligands and sterically accommodate 8–10 donors.

(b) Weak ligand-field stabilisation. The 4f orbitals are buried and barely overlap ligands, so there is almost no ligand-field stabilisation energy (LFSE) to be gained by adopting a low-symmetry octahedral or tetrahedral field. Geometry is dictated by steric packing (closest-packing of ligands around a large sphere) rather than by orbital-directed LFSE, which favours high coordination numbers. This contrasts sharply with the $3 d$ metals, where LFSE locks in octahedral or tetrahedral geometries.

Exercise 5 (medium, short-answer).

The PUREX process separates uranium and plutonium from spent nuclear fuel by extracting them into tributyl phosphate (TBP) in kerosene from nitric acid solution. State the oxidation states extracted and explain, in terms of charge and coordination, why $UO_{2}^{2 +}$ and $Pu^{4 +}$ partition into the organic phase while $Pu^{3 +}$ does not.

Hint

TBP is a neutral oxygen-donor (P=O) extractant. It forms neutral, lipophilic complexes. Think about charge neutralisation and the stoichiometry of the extracted species.

Answer

PUREX exploits the formation of neutral, lipophilic TBP adducts. The extracted species are $UO_{2} (NO_{3})_{2} \cdot 2 TBP$ (U as U(VI), the uranyl ion) and $Pu (NO_{3})_{4} \cdot 2 TBP$ (Pu as Pu(IV)). The nitrate anions neutralise the metal cation's charge, and the two neutral TBP donors complete the coordination sphere; the whole assembly is neutral and hydrocarbon-soluble, so it moves into the organic phase.

To recover plutonium separately, Pu is reduced from $+ 4$ to $+ 3$ (with $U^{4 +}$ , hydroxylamine, or ferrous sulfamate). $Pu^{3 +}$ would need three nitrates for neutrality, giving a less stable, more hydrophilic tris-nitrato complex that stays in the aqueous phase while U(VI) remains in the organic phase. The separation is therefore a redox switch: U stays U(VI); Pu is cycled between $+ 4$ (extracted) and $+ 3$ (stripped).

Exercise 6 (hard, short-answer).

Using the lanthanide-contraction radii $r (La^{3 +}) = 1.16$ Å and $r (Lu^{3 +}) = 0.98$ Å, estimate the effective nuclear charge added per lanthanide, $d Z_{eff} / d Z$ , under the crude model $r \propto 1/ Z_{eff}$ with $Z_{eff} (La) \approx 3$ . Hence estimate the shielding factor $S_{4 f} = 1 - d Z_{eff} / d Z$ and comment on what $S_{4 f} ≪ 1$ means physically.

Hint

From $r \propto 1/ Z_{eff}$ you have $Z_{eff} (Lu) / Z_{eff} (La) = r (La) / r (Lu)$ . Divide the $Z_{eff}$ gained over 13 steps by 13 to get $d Z_{eff} / d Z$ .

Answer

From $Z_{eff} \propto 1/ r$ : $Z_{eff} (Lu) = Z_{eff} (La) \cdot r (La) / r (Lu) = 3 \times 1.16/0.98 \approx 3 \times 1.184 \approx 3.55$ . The gain over 13 intervals (La → Lu) is $Δ Z_{eff} \approx 0.55$ , so $d Z_{eff} / d Z \approx 0.55/13 \approx 0.042$ .

This crude estimate gives an effective shielding factor $S_{4 f} = 1 - d Z_{eff} / d Z \approx 0.96$ , i.e. each added 4f electron shields roughly $0.96$ of a unit of nuclear charge from the valence shell. The small residual ( $\sim 0.04$ unshielded per step) accumulates across 14 elements to produce the 18-pm contraction. The physical content of $S_{4 f} \approx 0.96$ (close to but below 1) is that the 4f electron is a good but imperfect shielder because it is buried deep within the atom; the tiny shortfall is exactly what drives the lanthanide contraction and everything that follows from it. (Slater's rules give a more refined value; the point of the estimate is the mechanism, not the precise number.)

Exercise 7 (hard, numeric).

A single $^{235} U$ fission releases about 200 MeV of recoverable energy. Convert this to joules ( $1 eV = 1.602 \times 1 0^{- 19} J$ ) and hence estimate the energy released by the fission of 1 kg of $^{235} U$ (molar mass $235 g/mol$ , Avogadro's number $6.022 \times 1 0^{23} mol^{- 1}$ ). Express the answer in joules and in tonnes of TNT equivalent ( $1 tonne TNT = 4.184 \times 1 0^{9} J$ ).

Hint

First find the moles of $^{235} U$ in 1 kg, multiply by Avogadro to get the number of atoms, then multiply by the energy per fission. Convert eV to J using the given factor.

Answer

Energy per fission: $200 MeV = 200 \times 1 0^{6} \times 1.602 \times 1 0^{- 19} J = 3.204 \times 1 0^{- 11} J$ .

Moles in 1 kg: $1000 g /235 g/mol = 4.255 mol$ . Number of atoms: $4.255 \times 6.022 \times 1 0^{23} = 2.562 \times 1 0^{24}$ atoms.

Total energy: $2.562 \times 1 0^{24} \times 3.204 \times 1 0^{- 11} J = 8.21 \times 1 0^{13} J$ .

TNT equivalent: $8.21 \times 1 0^{13} /4.184 \times 1 0^{9} \approx 1.96 \times 1 0^{4}$ tonnes of TNT, i.e. roughly 20 kilotonnes of TNT per kilogram of fully fissioned $^{235} U$ . This is the energy scale that makes the actinides uniquely consequential among the elements.

Lean formalization Intermediate+

This unit has lean_status: none and carries no Lean module. f-Block chemistry is a curated body of measured facts — ionic radii, oxidation-state charts, f-f absorption wavelengths, fission cross-sections, redox potentials — together with semi-empirical models (Slater shielding, Kapustinskii lattice energies, ligand-field arguments) that take those measurements as inputs. None of it is a theorem in the sense Mathlib formalises. A genuinely useful formal layer would be a typed record of (element, Z, f-electron count, coordination number, ionic radius, oxidation states) plus verified checkers for oxidation-state bookkeeping, charge-balance, and Slater-style $Z_{eff}$ estimation; that layer is not present in Mathlib and lies outside the scope of this unit. See the unit metadata Mathlib gap analysis for the full statement.

Advanced results Master

Lanthanide luminescence and the Laporte rule

The sharp, narrow emission lines of the lanthanides — the red of Eu³⁺ and the green of Tb³⁺ in phosphors, lasers, and fluoroimmunoassays — come from $4 f \to 4 f$ transitions. Because the 4f orbitals are buried, they are well shielded from the chemical environment, so the emission wavelengths are nearly independent of the host lattice and the lines are extremely narrow (a few nm). The same burial makes the $4 f \to 4 f$ transitions parity-forbidden under the Laporte rule: electric-dipole transitions within the same $f^{n}$ configuration are forbidden in a centrosymmetric environment. Emission therefore occurs only because (a) the crystal field slightly mixes $4 f$ with higher- $l$ (e.g. $5 d$ ) states of opposite parity, or (b) the site lacks a centre of inversion, relaxing the rule. The long radiative lifetimes (microseconds to milliseconds) and the narrow lines are direct consequences of the near-forbiddenness: the transition dipole is small, so the spontaneous emission rate is small, and the well-shielded 4f levels are not broadened by lattice vibrations. This is exactly the burial of the 4f shell — the same feature that produces the lanthanide contraction — showing up as a spectroscopic signature.

Lanthanide separations

Because the lanthanides differ essentially only in ionic radius, every separation exploits the 1.3-pm step in $r (Ln^{3 +})$ . The classical methods are fractional crystallisation (Auer von Welsbach's 1885 separation of praseodymium and neodymium from didymium), ion exchange (the breakthrough of the 1940s Manhattan Project, using citrate elution from resin columns), and solvent extraction with organophosphorus reagents such as D2EHPA or HDEHP (the basis of modern rare-earth refining). In solvent extraction the distribution coefficient $D$ between organic and aqueous phases depends on the stability of the extracted complex, which in turn scales with $r (Ln^{3 +})$ through the charge-density match to the ligand cavity. Adjacent lanthanides then differ in $D$ by a small separation factor $β = D_{Z} / D_{Z - 1} \approx 2$ – $2.5$ , so hundreds of theoretical stages (cascaded mixer-settlers) are required to reach 99.99 % purity. The economic and strategic weight of rare-earth separations — for neodymium magnets, europium and terbium phosphors, lanthanum battery electrodes — is a direct function of the lanthanide contraction: a slightly larger contraction would make separations easier, a slightly smaller one would make them harder still.

Actinide redox chemistry and the actinyl bond

The early actinides display a redox richness absent from the lanthanides, set by the radial extension of the 5f orbitals. Standard aqueous potentials show the span: $E^{\circ} (U^{4 +} / U^{3 +}) = - 0.61 V$ , $E^{\circ} (UO_{2}^{+} / U^{4 +}) = + 0.27 V$ , $E^{\circ} (UO_{2}^{2 +} / UO_{2}^{+}) = + 0.16 V$ (in acid), so U(III), U(IV), U(V), and U(VI) are all accessible and the disproportionation $2 UO_{2}^{+} \to U^{4 +} + UO_{2}^{2 +}$ is thermodynamically favoured. The structural signature of the high oxidation states is the actinyl cation $AnO_{2}^{n +}$ (An = U, Np, Pu, Am; $n = 1$ or $2$ ): a linear O=An=O unit with very short, strong, covalent An=O multiple bonds. The bonding is described as a combination of $σ$ and $π$ donation from oxygen $2 p$ orbitals into empty actinide $5 f /6 d$ orbitals of matching symmetry; the substantial 5f participation (absent for the lanthanides, whose 4f orbitals cannot overlap) is the foundational reason the actinyl bond exists at all. Equatorial sites around the actinyl are occupied by water, nitrate, or carbonate, giving the pentagonal- and hexagonal-bipyramidal geometries seen in uranyl nitrate, $UO_{2} (NO_{3})_{2} \cdot 6 H_{2} O$ , and in the environmentally mobile uranyl carbonate complexes $UO_{2} (CO_{3})_{3}^{4 -}$ .

The uranium nuclear fuel cycle

The actinides' practical weight is dominated by the uranium fuel cycle. Natural uranium is 0.72 % $^{235} U$ (fissile) and 99.27 % $^{238} U$ (fertile). Most thermal reactors use low-enriched uranium (LEU, 3–5 % $^{235} U$ ); enrichment exploits the tiny mass difference between $^{235} UF_{6}$ and $^{238} UF_{6}$ by gaseous diffusion or, now universally, gas centrifugation. Spent fuel contains the transuranics (Pu, Am, Cm) produced by neutron capture on $^{238} U$ , separated from the unfissioned uranium and the fission products by PUREX: U(VI) and Pu(IV) are extracted into TBP/kerosene as neutral nitrato complexes, then Pu is reduced to Pu(III) and stripped back to the aqueous phase, leaving U(VI) in the organic phase. Recovered plutonium is either recycled as mixed-oxide (MOX) fuel or held as a proliferation-sensitive material. The whole cycle — mining, conversion to $UF_{6}$ , enrichment, fuel fabrication, irradiation, cooling, reprocessing, waste vitrification — is governed by the actinide redox chemistry formalised above: every step turns on a $+ 4 ⇌ + 6$ redox switch on uranium, or a $+ 3 ⇌ + 4$ switch on plutonium.

Relativistic effects in the heavy f-block

For elements as heavy as the actinides, electrons in 1s and inner shells reach a substantial fraction of the speed of light. Relativistic mass increase contracts the s and p orbitals (which have density at the nucleus), which in turn shields the nucleus more effectively and destabilises (expands) the d and f orbitals. Spin-orbit coupling splits each $l > 0$ level by an energy that grows roughly as $Z^{4}$ , reaching electronvolts in the actinides and superheavy elements. The consequences are visible across heavy-element chemistry: the yellow colour of gold (the 5d → 6s gap contracts into the blue-visible range), the liquid state of mercury (the 6s band is contracted and stabilised, weakening the metal-metal bond), and — for the actinides themselves — the ordering and energetic accessibility of the 5f and 6d levels, the colour and magnetism of actinide compounds, and the strengthening of the actinyl An=O bond (where 5f participation is enhanced by relativistic contraction of the oxygen-directed lobes). Relativistic quantum chemistry (Dirac–Hartree–Fock and beyond) is therefore mandatory, not optional, for any quantitative treatment of actinide bonding, and it grows sharply in importance for the trans-actinide and superheavy elements.

Synthesis. The f-block resolves into a single scheme: the burial of the f electrons sets the shielding, the shielding sets the contraction, the contraction sets the radii and the actinyl bond, and the radii set the separations, the coordination numbers, and the spectroscopic narrowness. This is exactly the unification that Housecroft & Sharpe and Cotton build across two rows; the foundational reason the lanthanides are monotonous while the actinides are rich is the radial extent of the $(n - 2) f$ shell, 4f buried and 5f extended. Putting these together generalises the f-block into the same charge-density-and-overlap engine that runs the rest of the periodic table, and the bridge is that the shielding argument builds toward the actinyl bond, the nuclear fuel cycle, and the relativistic chemistry of the heaviest elements, while appears again in the coordination chemistry of 16.04.x, the solid-state packing of 16.07.x, and the periodic-trend machinery of 16.01.x.

Full proof set Master

Proposition (Lanthanide contraction from incomplete 4f shielding). Let $r_{n}$ be the eight-coordinate ionic radius of the $Ln^{3 +}$ ion with $n = 0, 1, \dots, 13$ added 4f electrons ( $n = 0$ for La³⁺, $n = 13$ for Lu³⁺). If each added proton–4f-electron pair raises the effective nuclear charge felt by the valence shell by $δ = 1 - S_{4 f} > 0$ , then under $r \propto 1/ Z_{eff}$ the radius contracts monotonically, and the total contraction $Δ r = r_{0} - r_{13}$ satisfies $Δ r / r_{0} = 13 δ / Z_{eff}^{(0)}$ to first order in $δ$ .

Proof. Write the effective nuclear charge at the valence shell after $n$ steps as $Z_{eff}^{(n)} = Z_{eff}^{(0)} + n δ$ , where $δ = 1 - S_{4 f}$ is the unshielded charge per added proton ( $0 < δ < 1$ because the 4f electron shields with efficiency $S_{4 f}$ strictly between 0 and 1). The ionic radius scales inversely with $Z_{eff}$ :

r_{n} = \frac{K}{Z _{eff}^{(0)} + n δ}, r_{0} = \frac{K}{Z _{eff}^{(0)}}

for a constant $K$ set by the principal quantum number and the quantum defect of the valence shell. The radius is monotone decreasing in $n$ because $Z_{eff}^{(0)} + n δ$ is monotone increasing and $δ > 0$ . The total contraction is

\frac{Δ r}{r _{0}} = \frac{r _{0} - r _{13}}{r _{0}} = 1 - \frac{Z _{eff}^{(0)}}{Z _{eff}^{(0)} + 13 δ} = \frac{13 δ}{Z _{eff}^{(0)} + 13 δ} .

For $δ ≪ Z_{eff}^{(0)} /13$ this linearises to $Δ r / r_{0} \approx 13 δ / Z_{eff}^{(0)}$ , as claimed.

Substituting the measured values $Δ r = 0.18$ Å, $r_{0} = 1.16$ Å, so $Δ r / r_{0} \approx 0.155$ , and taking $Z_{eff}^{(0)} \approx 3$ for $La^{3 +}$ (the charge not screened by the filled 5s5p core), gives $δ \approx 0.155 \times 3/13 \approx 0.036$ . The shielding efficiency is therefore $S_{4 f} = 1 - δ \approx 0.964$ : each added 4f electron shields about $0.96$ of a unit of nuclear charge from the valence shell, leaving the residual $\sim 0.04$ that accumulates across 14 elements to produce the 18-pm contraction. The residual is small but cumulative, and it is the entire chemical content of the lanthanide contraction. $■$

Corollary (Post-lanthanide radius anomaly). The 18-pm lanthanide contraction is inherited by Hf (Z = 72), so $r (Hf^{4 +}) \approx r (Zr^{4 +})$ and Hf/Zr are chemically near-identical.

Proof. Hf sits directly below Zr in Group 4 but follows the entire lanthanide series, so its valence shell feels the accumulated $13 δ \approx 0.47$ extra units of $Z_{eff}$ that the contraction represents. The radius of $Hf^{4 +}$ (0.78 Å for eight-coordination) is therefore pulled down to within 0.06 Å of $r (Zr^{4 +})$ (0.72 Å), whereas a naive interpolation from the Y → Zr trend above the f-block would have predicted a substantially larger Hf. The near-equality of the two radii forces $ZrO_{2}$ and $HfO_{2}$ , $ZrCl_{4}$ and $HfCl_{4}$ , and the native ores (zircon, $ZrSiO_{4}$ , always carrying several percent Hf) to be structurally and chemically almost identical, which is the foundational reason Hf/Zr separation requires ion-exchange or solvent-extraction methods rather than ordinary chemical means. The same inherited contraction, applied one and two rows on, is responsible for the anomalously high densities of the 5d metals Os, Ir, Pt, and Au. $■$

Connections Master

Periodic trends quantified 16.01.01 supplies the ionisation energies, effective nuclear charges, and ionic-radius framework that the lanthanide-contraction mechanism consumes. The shielding argument of this unit is the periodic-trend engine of 16.01.01 operating one shell deeper, and the post-lanthanide radius anomaly of Hf/Zr is a direct readout of the same $Z_{eff}$ data.
Coordination chemistry 16.04.x is the descriptive chemistry of the d-block, organised by ligand-field arguments that are dual to the steric-packing arguments used here for $Ln^{3 +}$ . The weak LFSE of the lanthanides (because 4f is buried) is the foundational reason coordination numbers are dictated by radius and steric packing rather than by orbital-directed fields, contrasting with the octahedral/tetrahedral lock-in of the $3 d$ metals.
Solid-state chemistry 16.07.x depends on exactly the ionic-radius and lattice-energy arguments used in the lanthanide-contraction proposition. The high coordination numbers and the closest-packing geometry of $Ln^{3 +}$ salts, and the near-identical lattice constants of $ZrO_{2}$ and $HfO_{2}$ , bridge directly into the crystal-structure and defect chemistry of 16.07.02 and 16.07.03.
Nuclear and radiochemistry 16.01.x picks up the actinide redox and fission-cross-section data that the nuclear-fuel-cycle treatment requires. The PUREX separation, the $+ 4 ⇌ + 6$ uranium redox switch, and the 200-MeV-per-fission energy budget of this unit are the chemical foundation on which any reactor-physics treatment of 16.01.x builds.

Historical & philosophical context Master

The recognition that the lanthanides form a coherent family was slow in coming because their chemistry is so uniform. After Mosander's mid-nineteenth-century fractionations had split the "earths" into several lanthanides, the field spent seventy years sorting out which new "earths" were genuine elements and which were mixtures. The decisive tool was Moseley's X-ray spectroscopy ^{[Moseley1913]}, which in 1913 established atomic number as the ordering principle of the periodic table and fixed the count of lanthanides at exactly 15 (La–Lu) by showing that their characteristic X-ray lines stepped uniformly with nuclear charge. Moseley's law — that the square root of the X-ray frequency is linear in atomic number — converted the lanthanides from a confusing cluster of "rare earths" into a numbered series, and it predicted exactly how many of them remained to be found (one: element 61, promethium, isolated only in 1945). The philosophical move — replacing atomic-weight ordering, which had tangled the lanthanides, with nuclear-charge ordering — is the same move that lets the f-block sit, as a numbered block, below the main table.

The actinide concept is Glenn Seaborg's 1945 proposal that the elements following actinium form a 5f series analogous to the 4f lanthanides, rather than being heavy transition metals as then supposed ^{[Seaborg1945]}. Seaborg's reorganisation was, at the time, a bold revision of the periodic table: he predicted that the then-undiscovered transuranium elements would have the chemistry of a new inner-transition series, with $+ 3$ states becoming prominent in the later members. The prediction was borne out by the isolation and chemistry of americium, curium, berkelium, californium, einsteinium, fermium, mendelevium, nobelium, and lawrencium over the next two decades, and it earned Seaborg a share of the 1951 Nobel Prize in Chemistry. The actinide concept is one of the cleanest examples in chemistry of a theoretical reorganisation of the periodic table predicting the chemistry of elements not yet made.

The other great actinide discovery — nuclear fission — was made by Hahn and Strassmann in Berlin in December 1938 ^{[HahnStrassmann1939]} and interpreted by Meitner and Frisch within weeks ^{[MeitnerFrisch1939]} as the splitting of the uranium nucleus under neutron bombardment. The Bohr–Wheeler liquid-drop model of 1939 explained why $^{235} U$ , with its favourable odd-neutron fission cross-section (585 barns), is the fissile isotope, and why $^{238} U$ is merely fertile. The practical chemistry of the actinides — the PUREX process, the $UF_{6}$ enrichment route, the $+ 4 ⇌ + 6$ redox cycle — was built in the 1940s under the pressure of the Second World War and has shaped the geopolitics of energy and weapons ever since. For a modern survey of lanthanide and actinide chemistry at the depth this unit reaches, see Cotton's Lanthanide and Actinide Chemistry ^[Cotton2006] and Edelmann's review of actinide coordination chemistry ^{[Edelmann2017]}.

Bibliography Master

@article{Moseley1913,
  author  = {Moseley, Henry G. J.},
  title   = {The High-Frequency Spectra of the Elements},
  journal = {Philosophical Magazine},
  year    = {1913},
  volume  = {26},
  pages   = {1024--1034}
}

@article{Seaborg1945,
  author  = {Seaborg, Glenn T.},
  title   = {The Chemical and Radioactive Properties of the Heavy Elements},
  journal = {Chemical \& Engineering News},
  year    = {1945},
  volume  = {23},
  pages   = {2190--2193}
}

@article{HahnStrassmann1939,
  author  = {Hahn, Otto and Strassmann, Fritz},
  title   = {{\"U}ber den Nachweis und das Verhalten der bei der Bestrahlung des Urans mittels Neutronen entstehenden Erdalkalimetalle},
  journal = {Naturwissenschaften},
  year    = {1939},
  volume  = {27},
  pages   = {11--15}
}

@article{MeitnerFrisch1939,
  author  = {Meitner, Lise and Frisch, Otto R.},
  title   = {Disintegration of Uranium by Neutrons: a New Type of Nuclear Reaction},
  journal = {Nature},
  year    = {1939},
  volume  = {143},
  pages   = {239--240}
}

@article{Shannon1976,
  author  = {Shannon, Robert D.},
  title   = {Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides},
  journal = {Acta Crystallographica A},
  year    = {1976},
  volume  = {32},
  pages   = {751--767}
}

@article{BohrWheeler1939,
  author  = {Bohr, Niels and Wheeler, John A.},
  title   = {The Mechanism of Nuclear Fission},
  journal = {Physical Review},
  year    = {1939},
  volume  = {56},
  pages   = {426--450}
}

@article{Bunzli2010,
  author  = {B{\"u}nzli, Jean-Claude G. and Eliseeva, Svetlana V.},
  title   = {Basics of Lanthanide Photophysics},
  journal = {Springer Series on Fluorescence},
  year    = {2010},
  volume  = {7},
  pages   = {1--45}
}

@article{Edelmann2017,
  author  = {Edelmann, Florian T.},
  title   = {Lanthanides and Actinides: Annual Survey of their Organometallic Chemistry Covering the Year 2015},
  journal = {Coordination Chemistry Reviews},
  year    = {2017},
  volume  = {338},
  pages   = {27--66}
}

@book{Cotton2006,
  author    = {Cotton, Simon},
  title     = {Lanthanide and Actinide Chemistry},
  publisher = {Wiley},
  address   = {Chichester},
  year      = {2006}
}

@book{GreenwoodEarnshaw1997,
  author    = {Greenwood, Norman N. and Earnshaw, Alan},
  title     = {Chemistry of the Elements},
  edition   = {2nd},
  publisher = {Butterworth-Heinemann},
  address   = {Oxford},
  year      = {1997}
}

@book{CottonWilkinson1988,
  author    = {Cotton, F. Albert and Wilkinson, Geoffrey},
  title     = {Advanced Inorganic Chemistry},
  edition   = {6th},
  publisher = {Wiley-Interscience},
  address   = {New York},
  year      = {1988}
}

@book{HousecroftSharpe2018,
  author    = {Housecroft, Catherine E. and Sharpe, Alan G.},
  title     = {Inorganic Chemistry},
  edition   = {5th},
  publisher = {Pearson},
  address   = {Harlow},
  year      = {2018}
}

@book{MiesslerTarr2014,
  author    = {Miessler, Gary L. and Fischer, Paul J. and Tarr, Donald A.},
  title     = {Inorganic Chemistry},
  edition   = {5th},
  publisher = {Pearson},
  address   = {Upper Saddle River, NJ},
  year      = {2014}
}

Prerequisites

16.01.01

Tier anchors

beginner: Housecroft & Sharpe 2018 Inorganic Chemistry (Pearson) Ch. 24 (the f-block behaves like two families whose members differ mainly in size)
intermediate: Miessler, Fischer & Tarr 2014 Inorganic Chemistry (Pearson) Ch. 23 (4f/5f configurations, the lanthanide contraction, actinide redox and actinyls)
master: Cotton 2006 Lanthanide and Actinide Chemistry (Wiley); Edelmann 2017 Coord. Chem. Rev. actinide review; Cotton & Wilkinson 1988 Advanced Inorganic Chemistry (Wiley) Ch. 20

References

Housecroft, C. E. & Sharpe, A. G. — Inorganic Chemistry, 5th ed. (Pearson, 2018) · Ch. 24 — the f-block metals: lanthanide contraction, Ln3+ coordination, luminescence; actinide redox, the uranyl ion, nuclear fuel cycle
Miessler, G. L., Fischer, P. J. & Tarr, D. A. — Inorganic Chemistry, 5th ed. (Pearson, 2014) · Ch. 23 — lanthanides and actinides: 4f vs 5f radial extension, variable oxidation states, magnetic and spectroscopic properties
Cotton, S. — Lanthanide and Actinide Chemistry (Wiley, 2006) · Ch. 1–4 lanthanide contraction and Ln3+ coordination; Ch. 7–9 actinide oxidation states and separations; Ch. 11 nuclear aspects
Cotton, F. A. & Wilkinson, G. — Advanced Inorganic Chemistry, 6th ed. (Wiley, 1988) · Ch. 20 — the lanthanides and actinides: descriptive chemistry, ionic radii, redox potentials
Greenwood, N. N. & Earnshaw, A. — Chemistry of the Elements, 2nd ed. (Butterworth, 1997) · Ch. 30 — lanthanides (contraction, coordination, luminescence); Ch. 31 — actinides (5f chemistry, nuclear fuel cycle)

Estimated time

beginner: 20 min
intermediate: 55 min
master: 110 min