Cosmochemistry

1 Chapter overview

Every atom of carbon in your body, every atom of iron in your blood, was forged by nuclear reactions outside the Earth. Cosmochemistry is the application of chemistry — above all, nuclear chemistry — to the cosmos: to the rocks of the Earth, to meteorites and comets, to cosmic rays, and to the stars themselves, where the elements are made.

A chemist has a special claim on this subject. The decay law $N = N_0 e^{-\lambda t}$ that you used in Chapter 2 to count laboratory activities is the same law that dates a 4.5-billion-year-old meteorite. The fusion reactions that power the Sun are nuclear reactions with measurable cross-sections, branching ratios and Q-values. The "solar neutrino problem" was solved by particle physics, but it was posed by counting individual argon atoms produced by a neutrino capture reaction — analytical chemistry at its most extreme.

Read this chapter as a story in one line: stars make the elements → meteorites preserve the recipe → cosmic rays leave fingerprints → radioactive clocks date the whole story → neutrinos let us look inside the Sun today. The derivations you must be able to reproduce are the radiometric age equation, the isochron equation, and the net energetics of the pp chain and the CNO cycle.

2 Core concepts

2.1 · Geological systems — Earth's reservoirs and geochemical cycles

The Earth is chemically differentiated into three great reservoirs:

• Crust — the thin outer skin. Continental crust (~30–70 km thick, granitic, silica-rich "felsic" rocks) and oceanic crust (~5–10 km, basaltic, "mafic" rocks). It is where we sample rocks, and it is enriched in the heat-producing radioactive elements U, Th and K.
• Mantle — from the base of the crust (the Mohorovičić discontinuity) down to ~2900 km; dominantly peridotite, i.e. magnesium–iron silicates (olivine, pyroxenes). Convection in the mantle drives plate tectonics.
• Core — an iron–nickel alloy. The outer core (~2900–5150 km) is liquid and its motion generates the geomagnetic field; the inner core (~5150–6371 km) is solid despite enormous temperature, held so by pressure.

These reservoirs exchange matter through geochemical cycles: the rock cycle (igneous rocks weather to sediments → lithify to sedimentary rocks → bury and metamorphose → melt back to magma), the water cycle, and the carbon cycle. Plate tectonics — seafloor spreading at ridges and destruction of crust at subduction zones — is the great recycling engine. Why this matters for nuclear chemistry: magmatic differentiation fractionates parent and daughter elements differently (e.g. Rb follows K into evolved melts while Sr follows Ca into early crystals), and that fractionation is exactly what the isochron method exploits to date rocks (§3.2).

2.2 · The age of rocks — principles of radiometric dating

A radioactive parent P decays to a stable daughter D with decay constant $\lambda$, independent of temperature, pressure and chemical state — because decay is a nuclear process, while chemistry only rearranges electrons. If a mineral has remained a closed system since it formed (no parent or daughter entered or left except by decay), the daughter that has accumulated is a clock:

• Assumptions: (i) the decay constant is known and invariant; (ii) the system stayed closed; (iii) the amount of daughter present initially is known or can be corrected for (the isochron method does this).
• The age follows from the measured daughter-to-parent ratio — derived in §3.1.

The four workhorse clocks of geochronology:

2.3 · The age of the Earth

Terrestrial rocks only give a lower limit: the oldest Earth materials are Jack Hills (Australia) zircons at ~4.4 Ga (U–Pb), but Earth's surface has been reworked by plate tectonics. The true age comes from meteorites, which sample undifferentiated solar-system material: in 1956 Patterson measured the Pb–Pb isochron of the Canyon Diablo iron meteorite and other meteorites and obtained 4.55 Ga — the accepted age of the Earth, quoted today as ≈ 4.54 Ga. Calcium-aluminium-rich inclusions (CAIs) in carbonaceous chondrites, the first solids to condense from the solar nebula, date to 4.567 Ga.

2.4 · Cosmic rays

Primary cosmic rays are high-energy charged particles arriving from space: ~89% protons, ~9% alpha particles and ~1–2% heavier nuclei (up to iron), with energies from ~10⁹ eV to beyond 10²⁰ eV. They are accelerated mainly in supernova remnants (Fermi shock acceleration); the lowest-energy flux is modulated by the solar wind.

Secondary cosmic rays are produced when primaries smash into atmospheric nuclei: air showers of pions (→ muons), neutrons, protons and electromagnetic cascades. Sea-level cosmic radiation is mostly muons.

• Latitude effect: Earth's magnetic field deflects low-energy primaries most at the magnetic equator, so the flux is higher near the poles (and shows an east–west asymmetry, proving primaries are mostly positive).
• Altitude effect: flux rises with altitude to the Pfotzer maximum (~15–20 km) and then falls — secondaries are absorbed as they descend.
• Cosmogenic nuclides: secondary neutrons make radioactive nuclides in the atmosphere and in exposed rock — $^{14}\mathrm{N}(n,p){}^{14}\mathrm{C}$, and $^{10}\mathrm{Be}$ ($t_{1/2} \approx 1.4$ Ma), $^{36}\mathrm{Cl}$ (~0.30 Ma), $^{26}\mathrm{Al}$ (~0.72 Ma) by spallation. These are tracers of solar activity, exposure ages and groundwater.

2.5 · Meteorites

Meteorites are fragments of asteroids (and a few of the Moon and Mars) that survive atmospheric entry. Classification:

• Stones — chondrites: undifferentiated, containing chondrules (mm-sized once-molten droplets). Subdivided into ordinary, carbonaceous (volatile- and organic-rich; the CI chondrites match the solar photosphere element-for-element and define bulk solar-system composition) and enstatite chondrites.
• Stones — achondrites: differentiated basalts from melted parent bodies (e.g. HED meteorites from the asteroid Vesta).
• Irons: Fe–Ni alloys from the cores of disrupted planetesimals; show Widmanstätten patterns (kamacite/taenite intergrowths) proving cooling over millions of years.
• Stony-irons (pallasites, mesosiderites): core–mantle boundary material.

What they tell us: (i) the age of the solar system — CAIs at 4.567 Ga, chondrules slightly younger; (ii) the initial isotopic composition of the solar nebula (e.g. initial ⁸⁷Sr/⁸⁶Sr ≈ 0.699, "BABI"); (iii) extinct radionuclides (²⁶Al, ⁶⁰Fe) whose daughter excesses date early processes; (iv) cosmic-ray exposure ages from cosmogenic nuclides, telling how long a meteoroid travelled in space.

2.6 · Comets

Comets are kilometre-sized icy planetesimals — "dirty snowballs" of water ice plus CO, CO₂, CH₄ and NH₃ ices mixed with silicate dust and organics. Far from the Sun they are inert nuclei; near the Sun, sublimation drives a coma and two tails (an ion tail pushed straight back by the solar wind, and a curved dust tail).

• Kuiper belt (~30–50 AU beyond Neptune): source of short-period comets.
• Oort cloud (~2,000–100,000 AU, nearly a light-year out): spherical reservoir, source of long-period comets (e.g. 1P/Halley, period 76 y, is technically a short-period/Kuiper-type example often quoted).

2.7 · Black holes

A black hole is a region where mass is so compact that the escape velocity exceeds the speed of light. Its characteristic size is the Schwarzschild radius $r_s = 2GM/c^2$ (~3 km per solar mass). Formation: the iron core of a massive star (progenitor ≳ 20–25 $M_\odot$) collapses; if the remnant exceeds the Tolman–Oppenheimer–Volkoff limit (~2–3 $M_\odot$; white dwarfs are capped by the Chandrasekhar limit, 1.44 $M_\odot$), nothing halts the collapse.

• Stellar-mass black holes (~3–10² $M_\odot$) — e.g. Cygnus X-1, detected via X-rays from an accretion disc fed by a companion star; now also via gravitational waves from mergers.
• Supermassive black holes (10⁶–10¹⁰ $M_\odot$) — at the centres of galaxies, e.g. Sagittarius A* (~4×10⁶ $M_\odot$) in the Milky Way, M87* imaged by the Event Horizon Telescope.

2.8 · Nuclear reactions in stars — how the elements are made

Stars shine by fusion: light nuclei combine into heavier ones, releasing the mass difference as energy. Which reactions run depends on core temperature:

• pp chain (Sun-like stars, core ~1.5×10⁷ K): protons fuse via $p + p \rightarrow d + e^+ + \nu_e$. Three branches — pp I (~86% in the Sun), pp II (~14%, via ⁷Be) and pp III (~0.02%, via ⁸B, source of the high-energy neutrinos Homestake saw).
• CNO cycle (massive stars, T ≳ 1.8×10⁷ K): ¹²C catalyses the same net conversion 4p → ⁴He; strongly temperature-dependent, so it dominates in hot stars.
• Triple-alpha process (He burning, T ~ 10⁸ K, red giants): $3\,^{4}\mathrm{He} \rightarrow {}^{12}\mathrm{C} + \gamma$, through the Hoyle resonance state; followed by $^{12}\mathrm{C}(\alpha,\gamma){}^{16}\mathrm{O}$.
• Advanced burning (massive stars): carbon burning (~6×10⁸ K) → neon → oxygen burning (~1–2×10⁹ K) → silicon burning (~3×10⁹ K), ending at the iron peak (⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe). Fusion beyond iron absorbs energy, so the core collapses and the star explodes as a supernova.
• Heavy elements (beyond Fe) are built by neutron capture: the s-process (slow, in AGB stars during He-shell burning — neutron captures slower than $\beta$ decays, so the path hugs the valley of stability) and the r-process (rapid, in supernovae and neutron-star mergers — huge neutron fluxes drive nuclei far neutron-rich before they decay back). This s/r distinction (Burbidge–Burbidge–Fowler–Hoyle, 1957) is a favourite exam one-liner.

2.9 · The solar neutrino hypothesis (the solar neutrino problem)

If the Sun shines by the pp chain, it must emit ~10³⁸ electron-neutrinos per second. The standard solar model (Bahcall) predicts the flux at Earth; the Homestake experiment (Davis, 1968) tested it: 615 tonnes of perchloroethylene (C₂Cl₄) in a tank 1480 m underground in the Homestake gold mine, South Dakota, exploiting $^{37}\mathrm{Cl}(\nu_e, e^-){}^{37}\mathrm{Ar}$ (threshold 0.814 MeV). Every few weeks the few dozen ³⁷Ar atoms produced were flushed out with helium and counted by their decay.

Result: ~2.6 SNU observed vs ~8 SNU predicted — about one-third. (1 SNU = 10⁻³⁶ captures per target atom per second.) Kamiokande (~1/2) and the gallium experiments GALLEX/SAGE (~1/2, sensitive to low-energy pp neutrinos) confirmed a deficit. For 30 years physicists argued whether the Sun or the neutrino was misunderstood.

Resolution: the Sudbury Neutrino Observatory (SNO, 2001) used heavy water to detect neutrinos three ways — charged-current (νₑ only), neutral-current (all flavours equally) and elastic scattering. The total flux matched the solar model; the νₑ flux was depleted because neutrinos change flavour (νₑ → ν_μ, ν_τ) en route, enhanced inside the Sun by the MSW effect. Neutrinos have mass and oscillate — Nobel Prize 2015 (Kajita, McDonald); Davis and Koshiba shared the 2002 prize for detecting the deficit. So the Sun was right all along, and the "problem" became a discovery.

3 Key derivations

3.1 · From the decay law to the radiometric age equation

$$D^* = P_0 - P = P\,(e^{\lambda t} - 1)$$

Solving for $t$:

with $\lambda = \ln 2 / t_{1/2}$. This is the master equation: measure the daughter/parent atom ratio, know the half-life, get the age. Two common variants: for branched decay (⁴⁰K → ⁴⁰Ar by electron capture with partial constant $\lambda_{\mathrm{Ar}}$, the rest to ⁴⁰Ca), only the fraction $\lambda_{\mathrm{Ar}}/\lambda$ of decays makes argon, giving $$t = \frac{1}{\lambda}\,\ln\!\left[1 + \frac{\lambda}{\lambda_{\mathrm{Ar}}}\frac{{}^{40}\mathrm{Ar}^*}{{}^{40}\mathrm{K}}\right]$$ and for ¹⁴C, where we measure activity $A = A_0 e^{-\lambda t}$, $t = (1/\lambda)\ln(A_0/A)$.

3.2 · The isochron equation (Rb–Sr)

$$\left(\frac{{}^{87}\mathrm{Sr}}{{}^{86}\mathrm{Sr}}\right) = \left(\frac{{}^{87}\mathrm{Sr}}{{}^{86}\mathrm{Sr}}\right)_0 + \left(\frac{{}^{87}\mathrm{Rb}}{{}^{86}\mathrm{Sr}}\right)\!(e^{\lambda t} - 1)$$

Cogenetic samples (same age, same initial ratio, different Rb/Sr because minerals fractionate Rb and Sr differently) fall on a straight line of slope $m = e^{\lambda t} - 1$:

The intercept gives the initial ⁸⁷Sr/⁸⁶Sr — itself precious (it fingerprints the source reservoir). The same construction gives the Pb–Pb isochron used for the age of the Earth: plotting ²⁰⁷Pb/²⁰⁴Pb vs ²⁰⁶Pb/²⁰⁴Pb for meteorites yields a line whose slope fixes $t \approx 4.55$ Ga with no need to measure uranium at all. Concordia (U–Pb): the parametric curve of ²⁰⁶Pb*/²³⁸U vs ²⁰⁷Pb*/²³⁵U as a function of age; samples that stayed closed plot on it (concordant); Pb loss displaces them along a discordia chord whose upper intercept still gives the crystallisation age.

3.3 · Net energetics of stellar burning

pp chain. The three branches all reduce to: four protons → one alpha particle. Two of the protons must become neutrons ($p \rightarrow n + e^+ + \nu_e$), so the net reaction is

Energy check: mass of 4 × ¹H (4 × 1.007825 u) minus mass of ⁴He (4.002602 u) = 0.028697 u ≈ 26.73 MeV; the two positrons annihilate with ambient electrons (adding 2 × 1.022 MeV), while the two neutrinos carry away ~0.59 MeV each — escaping the Sun unseen and becoming the solar neutrino flux.

CNO cycle. The carbon–nitrogen–oxygen sequence $^{12}\mathrm{C}(p,\gamma){}^{13}\mathrm{N} \rightarrow {}^{13}\mathrm{C}(p,\gamma){}^{14}\mathrm{N}(p,\gamma){}^{15}\mathrm{O} \rightarrow {}^{15}\mathrm{N}(p,\alpha){}^{12}\mathrm{C}$ consumes four protons and regenerates the ¹²C catalyst, giving exactly the same net reaction:

Triple-alpha. In helium burning, two alphas form the unbound ⁸Be (lifetime ~10⁻¹⁶ s); before it falls apart a third alpha can strike it, resonating through the Hoyle state of ¹²C:

This step — impossible without the Hoyle resonance — is what makes carbon, and hence chemistry and life, possible at all.

4 Worked examples

5 Figures

6 PYQ bank

7 Exam Q&A

8 Quick revision