Nuclei

What was declared, and what it takes

The book began by declaring nuclei: heavy point charges, stable up to element 78 (chapter 2). That was enough for atoms, which see a nucleus only as its charge. It isn't enough for stars, which make nuclei, or for the question every chapter since has run into: which elements are common? Our world's answer, iron and oxygen and the rest, comes from nuclear physics: which nuclei hold together, and which ones stars can build. Here that physics is worked out for 4D, with every method first checked on our own nuclei.

The one thing 4D changes is the one the stars chapter met: a head-on meeting carries a barrier, 3/4 · ħ²/2μr², because in four dimensions there's more room to miss. Inside a nucleus, where nucleons sit a femtometre or two apart, it's large. The force between nucleons is taken pinned, as the force between charges is: the same function of distance in both worlds. Two of them are used, of quite different shapes, to show the conclusions don't hang on one: Malfliet and Tjon's (Yukawa terms, a repulsive core that grows without limit) and the Minnesota force (soft Gaussians). Both are fitted to how our nucleons scatter, and both give our deuteron (2.2 MeV).

The strong force has to be stronger

computed With our force, 4D's deuteron isn't bound: the barrier wins. A star can't start burning hydrogen without one (its first step makes a deuteron), and the universe's first minutes would make nothing at all. Scale the force's attraction by a factor s and ask where things change. Our world sits in a window: the deuteron binds, and the diproton (two protons) doesn't, which would let stars burn explosively fast. 4D has the same window, higher up:

The 4D strong force must be at least 23% stronger than ours, and the two forces put the window in different places (1.23–1.50 and 1.40–1.89): how much stronger depends on the force's shape, and that it must be doesn't. It's a choice made by a requirement, as G is (chapter 16): that stars can burn.

Helium-4

computed Four nucleons (two protons, two neutrons, spins paired) fill the lowest shell in any dimension, so helium-4 needs no sign changes in its wavefunction, and diffusion Monte Carlo solves it exactly for a given force. The checks: our helium-4, 31.1 MeV under Malfliet–Tjon (the published value for it is 31.36) and 28.5 under Minnesota (measured: 28.30). Per nucleon, at the bottom of 4D's window, where its deuteron binds as ours, and at the top:

The three-nucleon force is introduced below; its range (0.7 to 1.4 fm) gives the spread. Ours, measured: 7.07.

4D's helium-4 holds 19–26 MeV per nucleon at the strength that binds its deuteron as ours, 2.7–3.6 times ours, and the two forces agree within about a quarter. Hydrogen burning in a 4D star releases that much more per helium made.

Held by the crowd

A nucleus far bigger than helium is mostly inside, where each nucleon is surrounded. Its binding per nucleon tends to that of nuclear matter, an endless uniform sea of nucleons, which can be computed: Brueckner's method, which follows each pair through all its collisions inside the sea, written here for any dimension. computed Its two-nucleon machinery reproduces the deuteron exactly in both worlds. On our world it shows the known flaw of such forces: nuclear matter binds harder the denser it gets, never settling. What settles it in ours is partly three nucleons together. So a three-nucleon repulsion is added, the same function of distance in both worlds, with one strength fitted to our nuclear matter's binding (16 MeV per nucleon). That puts our saturation at 0.162 and 0.189 nucleons per fm³ (measured 0.16), and moves our helium-4 by about a MeV. a choice Its range (0.7 to 1.4 fm) isn't fixed by our world, and sets how it carries into 4D: it's the uncertainty below.

Read as it stands, 4D nuclear matter binds 68 to 588 MeV per nucleon, against our 16. Most of that range is past what this model can be trusted with. It treats nucleons as slow particles meeting through a fixed force, which needs them slow, their binding small against their own mass, and the matter's sound slower than light. The well's bottom, with ours as the check:

Ours, the check: the bulk modulus comes out 267 and 259 MeV, measured 230 ± 30, though nothing was fitted to it. Sound is √(K/d²m). Shown for 4D: each force at its window's bottom and where its deuteron binds as ours, three-nucleon force of range 1.4 and 1.0 fm. Of the 18 wells, 4 have sound slower than c/2 and a minimum the scan resolves; the rest reach 0.55–1.2 c.

computed Where the model holds, 4D nuclear matter binds 67 to 90 MeV per nucleon, four to six times ours: a coherent liquid, about twice as stiff for its binding as ours (bulk modulus over binding 34, ours 17), its nucleons 1.5–1.6 fm apart, moving at four tenths of light. Deeper wells, with the shorter-ranged three-nucleon force or the stronger nuclear force, need relativistic nuclear theory, which binds less at high density, so read them as upper bounds. It's the gases chapter's discovery again (chapter 14): in four dimensions more neighbours fit within reach of the force, so matter is held by the crowd. Helium-4 has only four nucleons, and can't use it. Even the shallowest trusted well beats helium-4 at the same strength and range by 3.6 times per nucleon (ours: 16 against 7).

Past helium

That says the valley is deep. It doesn't say stars can reach it: the first steps past helium could be uphill. Ours are. Nothing of mass 5 or 8 holds together in our world, and our stars cross that gap only by the triple-alpha reaction, three helium nuclei meeting almost at once and caught by a lucky level in carbon-12.

tried, failed Monte Carlo past helium needs a guess at where the wavefunction changes sign, and with these forces the guess fails on our own nuclei: helium-6, beryllium-8 and carbon-12 come out at 22, 31 and 0 MeV against the measured 29, 56 and 92. Nothing 4D is claimed from it.

computed one calibration What works is to treat light nuclei as what they are, clusters of helium-4s. Two helium-4s meet through the nucleon force spread over both, and the Pauli principle keeps their relative motion out of the states that would crowd a fifth nucleon into the lowest shell (Saito's orthogonality; the count is the same in any dimension). One factor λ on that folding is set so that our beryllium-8 sits where it does, just unbound. It comes out 1.9 under either force: almost exactly 2, which is what the half of the force that exchanges nucleons should add. Carried to 4D, with each world's helium-4 at the size the Monte Carlo gives it, two helium-4s bind by (MeV; in brackets, λ cut by a fifth and raised by a quarter):

4D's beryllium-8 is bound, in every case, by 0.5 to 121 MeV (11–18 where the deuteron binds as ours). There's no gap at mass 8: two helium-4s can end bound, with no third needed and no lucky level. That opens the route; it doesn't say how fast it runs. Two identical spinless helium-4s can't shed their energy as the simplest kind of light (a dipole photon), so the capture has to go by a slower one, as our own helium-4 captures onto carbon and oxygen do (below). A nucleon on a helium-4 is the weaker result, because the same model needs λ of 4–5 to place our helium-5, standing in for a spin–orbit force it doesn't have:

At the calibrated λ it binds everywhere, and cut by a fifth it fails in 3 of the 18 cases. But a λ of 4–5, twice the natural value, is a factor doing a missing force's work, and the microscopic calculation below overturns it: mass 5 is unbound in 4D except at the window's top, so 4D keeps that one of our two gaps. rough Where binding peaks: a liquid drop built on the computed matter, its surface borrowed from ours, puts the most bound nucleus, where the model holds, at mass 271 to 388 (element 118 to 163). Rough, with a surface term borrowed from ours, but far past iron's 56 every way: past the 78 elements solved here, yet within where an atom can still hold its electrons (below).

The ladder, helium-4 by helium-4

computed two-body only With beryllium-8 bound, the next question is the whole ladder: carbon is three helium-4s, oxygen four, pxygen (element 10) five, on up. The two helium-4s' force above becomes a local one by removing the two states the Pauli principle forbids, with Baye's supersymmetric transformation (it keeps every other level exactly; checked to 0.05 MeV at each point used). Helium-4s are identical bosons, so for that force diffusion Monte Carlo is exact for any number of them, with no sign problem. What it leaves out is any force among three helium-4s at once. The check, each helium-4 capture's energy in our world (MeV; positive is downhill):

Measured: the mass differences. Beryllium-8 is fitted to threshold (the one calibration); everything past it is predicted.

On ours, two-body forces get every sign right, so every capture is downhill, as measured. They get the sizes small: carbon's capture comes out a few MeV short of its measured 7.4, and the nuclei two to three times underbound. A three-helium-4 attraction fitted to carbon overbinds oxygen threefold and neon sixfold, so none is used. Read the 4D ladder as a conservative test: the forces it leaves out bind ours more, not less. In 4D (capture energy in MeV, element 4 to element 16; bold where it exceeds a helium-4's own binding):

Every 4D capture is downhill, in every case, by 7 to 1372 MeV (18–428 where the deuteron binds as ours), against our measured 5 to 9. And each capture releases more than the last: the helium-4s pull harder the more of them there are, because in four dimensions more of them fit within reach. That's the crowd again, at the scale of a nucleus. It also says where the cluster picture ends. By the third to sixth helium-4 (counting beryllium-8's two), a capture releases more than a helium-4 is bound by, so the helium-4s can't stay whole inside it. Past carbon or oxygen, a 4D nucleus is a drop of nuclear matter, not a molecule of helium-4s, and the drop is what the crowd section computed. The two agree: at mass 28–32 the ladder's binding per nucleon is still climbing toward the trusted bulk value, below it in every case where both exist.

There is no energy gap anywhere on 4D's ladder. Ours has two, at mass 5 and 8, and stars cross them only by the triple-alpha's luck; 4D's has none to the top of what's computed, and the drop says none after. So energy doesn't stop 4D stars building. What decides how far they get is rates: the electric barrier, how readily each nucleus sheds its energy as light, the levels that make a reaction resonant, and which reactions win the race in the time a star has. Energetically open isn't the same as running.

barrier only One part of that, the barrier, can be estimated now: how hot each rung needs to be for the barrier alone: the temperature at which a helium-4 is captured onto charge Z as often as onto our carbon at 0.2 billion K (where our stars burn helium), with tunnelling through the electric barrier and 4D's centrifugal one (chapter 16):

Nuclear radii 1.2 A^(1/3) fm ours and 1.0 A^(1/4) fm 4D; the rate counts tunnelling only, not the levels that make our reactions resonant.

Each rung needs 1.13 to 1.30 times our temperature, rising with charge. This is a barrier-only estimate, not a prediction of what stars make: it assumes every rung sheds its energy as readily as our carbon does. For the first rung that's now computed (next section), and it holds up. Our stars build to iron because, past it, binding falls and burning stops paying. A 4D star's ladder keeps paying far past iron's charge, so what ends it is how fast each capture runs against how hot and how long a star burns. That needs the captures' rates, rung by rung.

The same, with the Pauli principle exact

The ladder's helium-4s are points with a force between them, and the Pauli principle enters only as states they're forbidden to use. The direct test of that is to build the nuclei from nucleons. computed In Brink's model each helium-4 is four nucleons sharing one Gaussian orbital, and a nucleus is one determinant of all of them, antisymmetrized, so the Pauli principle is exact: nothing removed, no factor fitted per nucleus. Many placements of the helium-4s are superposed, as quantum mechanics allows, and beryllium-8's turning states come out by averaging over orientations. Every integral is a closed Gaussian one in any dimension. The force is Minnesota's, plus this chapter's three-nucleon force at each range (0.7, 1.0, 1.4 fm), with the one number Minnesota leaves free (u, how it acts in odd partial waves, which a lone helium-4 never feels) set to put our beryllium-8 where it is: u = 1.06, 1.08, 1.09. Our nuclei, against free helium-4s (MeV):

Measured: mass differences. Beryllium-8 is where u is set. The cluster ladder is the two-body Monte Carlo above.

Carbon comes out at −6.3 to −8.4 against the measured −7.3: good. Oxygen comes out at −36 to −43, overbound about two to three times. The reason is known: four nucleons in one orbital miss about 8 MeV of a helium-4's binding, which their correlations supply, while oxygen's closed shell suits the model well, so oxygen gains on its helium-4s. The two methods err in opposite directions on ours, the ladder underbinding oxygen and Brink overbinding it, and the measured values lie between them. One thing Brink gets wrong on ours follows from the same bias: oxygen comes out so deep that a fifth helium-4 on it looks 3 to 12 MeV uphill (measured: 4.7 downhill), a closed shell's pull overdone. The ladder gets that sign right. In 4D, each capture's energy (MeV), and beryllium-8's spin-2 state:

Every 4D capture is downhill under both methods, by 15 to 416 MeV in this one. Where the force is strongest the two agree within about ten percent; lower in the window Brink binds more, as it does ours. So the ladder's conclusion doesn't hang on how it handled the Pauli principle: between two methods that bracket our world, 4D's rungs are downhill all the way to element 10. Beryllium-8's spin-2 state is bound in every case, by 8.1 to 83 MeV, more deeply than the cluster model puts it where it's shallowest, so the first capture's slowest case (next section) is if anything pessimistic. follows Five helium-4s are where 4D's oscillator shells close (4, then 16 nucleons: d = 4 has four p orbitals), as four close ours at oxygen. Element 10, pxygen, is the nucleus that plays oxygen-16's part, so Brink's closed-shell bias would show only past it, where nothing here is computed.

The model's limits: the Gaussian helium-4 is the one Brink's model always uses, too small a basis to give a helium-4 its correlations; the placements are random, 240 and 480 of them, and the larger basis lowers the energies by a few percent, at most 21% in one case, so read the magnitudes to that. The signs don't move.

Mass 5, microscopically

computed The same method takes one nucleon outside a helium-4: five nucleons in one antisymmetrized determinant, the helium-4's four sharing an orbital and the fifth on its own, so the Pauli principle keeps it out of the helium-4's state without any rule imposed. Projected onto each relative turning (s, p, d waves), against a free nucleon and helium-4 in the same model, with Minnesota's force and nothing set on mass 5 itself (u is still the one beryllium-8 fixed). Bound means below zero (MeV), across the three-nucleon force's ranges:

Unbound states show only as a box's discrete continuum, which falls as the box grows; there's no level below zero to report. No spin–orbit force: a p-state here is the centre of its doublet. In ours that force puts the favoured member about 0.4 MeV below the centre, far short of binding.

The check: our helium-5 and lithium-5 come out unbound, as they are, though nothing was set on them. 4D's mass 5 is unbound too, at the window's bottom and where its deuteron binds as ours, and bound only at the top, by 6–8 MeV. The cluster model's mass 5 (5 to 40 MeV) came from a factor fitted to put our helium-5 at threshold, which in 4D overshoots; reset to the microscopic binding at the top it comes out near 2.8, close to the natural 2 the helium-4 pair found. So 4D keeps one gap of ours, mass 5, and loses the other, mass 8. For stars that changes nothing (helium burns by two helium-4s meeting, above); for the first minutes it changes everything (next).

The first capture, and the light it needs

Two helium-4s that end bound have to get rid of the energy, and in a star that means light. The simplest light, a dipole photon, is closed to them: they're identical, so their centre of charge sits on their centre of mass and there's no dipole to radiate. Both are spinless, so meeting head-on and landing in beryllium-8's ground state would take a photon carrying no turning at all, which no photon can be. What's left is the next kind, a quadrupole photon. It can carry them from a head-on meeting into a spinning state of beryllium-8 (spin 2), or from a glancing meeting (a d-wave) into the ground state. Our own helium-4 captures onto carbon and oxygen are spinless meeting spinless too, and run partly on exactly this.

computed 4D's beryllium-8 has a bound spin-2 state in every case, 2.5 to 164 MeV below two free helium-4s, in the cluster model above with its forbidden d-state removed. It sits 5–9 MeV above the ground state, the ground state's turning partner. Ours has one too, 3.0 MeV above threshold and unbound, which both forces put 1.3–1.8 MeV too low, so read the shallowest 4D one as maybe a MeV less bound.

computed How this universe's light carries a quadrupole away: in the lens vacuum (chapter 3) a photon of frequency ω carries its energy at ε₀ℓω, and the pinned 1/r fixes the fine-structure constant at e²/(4π²ε₀ℓ), so the lens length ℓ drops out of every rate. A quadrupole radiates at (π/48) α ω⁵ Tr(QQ†), against our α ω⁵ Tr(QQ†)/10: the same power of the frequency and 0.65 of our strength. With the cluster model's states and scattering waves, the capture's nuclear factor S̃ (the cross-section with the electric and 4D barriers taken off, which the barrier-only table above takes to be the same as our carbon capture's), and the temperature at which it runs, per pair, as often as our carbon captures helium-4 at 0.2 billion K:

Our yardstick, ¹²C(α,γ)¹⁶O at 300 keV: its quadrupole capture to the ground state, 45 keV b (deBoer et al. 2017) to 70 (Shen et al. 2020), for the nuclear factor; the whole reaction, 140 to 162 keV b, for the temperature, since that's what our stars' 0.2 billion K is. Not counted: emission of an electron–positron pair, which in ours adds a few percent to such transitions.

The first capture runs. Its nuclear factor is 0.4 to about 5,500 times our carbon capture's: about half ours where 4D's spin-2 state is barely bound, tens to thousands of times ours elsewhere, because 4D's photon carries tens of MeV and a quadrupole's rate grows as its fifth power. Since the barrier is exponential and this isn't, the temperature moves only a little: 0.019 to 0.034 billion K, against the barrier-only 0.031. rough That's where 4D stars burn hydrogen at all (0.025–0.034 billion K by the same matching, chapter 16). So a 4D star likely burns helium as it makes it, with no separate, hotter helium-burning stage as ours have. That's a per-pair rate set against ours, not a star: how fast a 4D core actually runs each step needs its densities and a 4D stellar model, and the rungs past beryllium-8 need the same calculation each.

The rungs past beryllium-8

computed rough Each rung after the first is a helium-4 captured onto a core of helium-4s, by the same quadrupole light: both spinless, the same charge for their mass, so no dipole. The helium-4's potential on the core is the pair's folded over the core's helium-4s, its depth set once per rung so the product's ground state sits at that rung's capture energy (from both the ladder and Brink's model, which bracket ours); the Pauli principle forbids the relative states with fewer than four quanta, Wildermuth's rule, since in 4D the four nucleons enter the shell that holds sixteen. The check, ours: carbon-12 + helium-4 → oxygen-16's quadrupole capture to the ground state comes out 12–41 keV b at 300 keV, against the measured 45–70: low by up to four times, and it puts oxygen's spin-2 state too deep. Since the barrier is exponential and this isn't, the temperatures come out good to tens of percent even so. 4D (MeV for the capture energy, billion K for the temperature that keeps our stars' helium-burning pace):

Both forces, the whole window, both sources of capture energy, two spreads of the core's helium-4s, and capture to the ground state alone or to every bound state.

Every rung runs, and at or below its barrier-only temperature: up to about twice as cool, because 4D's photons carry tens to hundreds of MeV and a quadrupole's rate grows as the fifth power of that. So the barrier-only estimate was, if anything, conservative: the ladder climbs with less heat than it said, and how far stars build (below) uses these.

The first minutes

Before any star, the hot young universe cools through nuclear temperatures for a few minutes, and what it makes then is what the first stars start from. Ours made hydrogen and a quarter helium, and almost nothing else: mass 5 and mass 8 don't hold together, and by the time helium exists the universe is too cool and thin to go round them. 4D has neither gap. computed three choices open The standard first estimate, written for any dimension: photons and baryons in the hot bath, neutrons frozen out and decaying, the deuterium bottleneck (deuterons survive the photons only below a temperature set by their binding, Saha's equation), every neutron left then ending in helium-4, and then the captures computed above, protons onto helium-4 and helium-4 onto helium-4, as the universe cools, using helium up; mass 5's binding from the microscopic calculation above (the folded potential reset to it where it binds). Three things the book hasn't fixed enter as choices: the weak force, taken as ours (neutrons freeze out at 0.8 MeV and live 879 s); η, the baryons per photon (ours 6.1 × 10⁻¹⁰); and the clock, how fast the universe cools, which gravity's law would set. Expanding as ours does, with the rate following radiation's energy density (T⁵ in 4D), the time goes as T^(−5/2), and its scale t₁ (the age at 1 MeV, ours 1.32 s) is left free. The check, ours: the bottleneck breaks at 0.066 MeV and leaves 0.235 helium by mass (measured 0.245).

4D at each strength: the deuteron's binding (MeV), then the helium-4 left and the mass-5 nuclei made (mass fractions) for our η and sixteen times it, each with a fast clock (t₁ = 0.01 s) and ours (1.32 s):

The rest is hydrogen. The window's bottom is where the deuteron starts to bind, so there it holds by almost nothing, and the bottleneck never breaks before the neutrons decay. Helium-4 + helium-4 → beryllium-8 makes at most 0.04% where the deuteron binds as ours and η is within sixteen times ours; only with η about 1,600 times ours (10⁻⁶) does it reach 9%.

Where the deuteron binds as ours, 4D's first minutes stop at helium, as ours did. The bottleneck breaks at 0.057–0.076 MeV, much as ours does, every neutron left goes into helium-4, and then nothing more: mass 5 is unbound, so protons can't capture onto it, and two helium-4s meet too rarely in the thinning bath to make more than a trace of beryllium-8. The helium left depends on the clock: a third by mass if the universe cools fast, a few percent if it cools as slowly as ours, because 4D's steeper cooling law gives the neutrons longer to decay. At the window's top it goes past helium: the bottleneck breaks at 0.5–0.7 MeV, almost as soon as the neutrons freeze out, and with mass 5 bound by 6–8 MeV the protons, three of every four baryons, take much of the helium into it (all of it unless the universe cools fast). At the bottom, hydrogen alone. So 4D's first stars start from hydrogen and helium, as ours, unless its strong force sits near the window's top. open What mass 5 does next isn't in this network; the clock waits on gravity's law; and mass 5 is computed without a spin–orbit force.

How far stars build

A star's core runs only as hot as its mass lets it. As it burns out a fuel it contracts and heats, as the virial theorem has it, until its electrons are packed so tightly that their degeneracy holds it up; then it cools. So each core mass has a highest temperature, and a stage whose temperature is above it never lights. Above the Chandrasekhar mass degeneracy can't hold at any density, and the core heats until burning stops paying. computed rough The estimate, in any dimension: the core is the gas polytrope (stars chapter), its centre's pressure balanced by the ions as a hot gas and the electrons as they are at that temperature, relativistic and partly degenerate; its hottest point found over every radius. With masses in units of the Chandrasekhar mass, gravity's strength drops out, which matters here, since G is fixed only by the planets. The check, ours (cores of matter with two nucleons per electron; known values in brackets): hydrogen lights in stars from 0.08 of it (0.08 M☉ of 1.44: 0.056), helium in cores from 0.29 (a helium star lights from about 0.3 M☉: 0.22), carbon from 0.73 (1.06 M☉: 0.74). Good to about forty percent in mass.

4D, each stage at its temperature (the captures to element 10 computed, the rungs above barrier-only) and the smallest core that lights it:

Malfliet–Tjon and Minnesota together where they differ. Past element 10 a rung's temperature is the barrier-only estimate, which the computed rungs show runs warm.

4D stars build in steps by mass, as ours do, but with helium lit almost from the start. Hydrogen lights in stars from 0.15–0.16 of the Chandrasekhar mass, and helium in cores from 0.13–0.19: a core about as heavy as the lightest star that burns hydrogen at all, where ours needs one four times as heavy. 4D's helium needs only four to seven times hydrogen's temperature, ours about thirty (two helium-4s meeting, against our triple-alpha's three). So ours has a long band of stars, the red dwarfs, that burn hydrogen for ever and never helium; in 4D that band is narrow, perhaps empty: a star needs only to grow a helium core most of its own mass to light it. Then the ladder, a rung per step in mass: a core too light for carbon ends as beryllium-8, then carbon, oxygen, pxygen (element 10), and so on up, each rung needing a heavier core, until from element 20 on only cores at the Chandrasekhar mass or above take the next step. Those go further, because in 4D every capture keeps paying far past iron's charge: where they stop is where binding per nucleon peaks, which the liquid drop above puts past element 100 (rough), unless something else ends them first. open Which of these get back out into the gas that makes planets, and in what proportion: that's the mass a star of each kind sheds and how it ends, and how many stars of each mass form. Ours return their carbon and nitrogen mostly from the winds of middling stars and their oxygen to iron from the explosions of massive ones.

Odd against even

Our oceans are water and not hydrogen fluoride because fluorine is rare: some thirty thousand times rarer than oxygen. Odd elements sit off the helium-4 ladder, so stars make them only on the side, and fluorine is also destroyed as fast as it's made, since a proton on fluorine-19 makes oxygen-16 and a helium-4 and releases 8 MeV. 4D's water question is the same one, element 9 against element 10. computed The first thing to know is whether an odd nucleus holds together at all: a nucleon on a ladder nucleus, built from nucleons as mass 5 was (Brink's model, the core's placements held fixed and the nucleon free on a grid round it, the core alone on the same placements; Minnesota's two-body force only, since the three-nucleon force over this many nucleons is out of reach here). The check, ours, the nucleon's separation energy (MeV; above zero it holds):

Every one that holds, holds; boron-9 doesn't, and doesn't. The core's helium-4s can be drawn loose or close together; close is the closed-shell limit, the better core for oxygen-16, and there fluorine-17 and oxygen-17 come out within a MeV of measured (+0.46 and +4.76), where the loose draw overbinds them by about 6. For carbon, an open shell, neither draw is that good, and they differ: the spread between them is the uncertainty.

4D, a neutron / a proton on 2 to 5 helium-4s (elements 5, 7, 9 and 11 at masses 9, 13, 17 and 21), the two draws' values where they differ:

On pxygen-20, 4D's closed shell, only the close draw is shown, as for our oxygen-16. Denser grids of directions move the values by about a MeV.

Odd nuclei hold together in 4D, unlike mass 5: element 9 at mass 17 by 11–153 MeV, far more than the model's overbinding on ours. follows And the reaction that keeps our fluorine rare is closed. To throw out a helium-4, a 4D nucleus has to pay that helium-4's separation energy, which is the rungs' capture energy above (21–788 MeV), against the 10–60 MeV a proton brings in: every proton-in, helium-4-out reaction is uphill, by at least ten MeV at the window's bottom and by tens to hundreds elsewhere. The model's errors are a few MeV, and since it overstates how tightly the proton holds, the true margins are wider still. So in 4D an odd element, once made, isn't undone by protons. That takes away the lean toward Xater the ladder gave. How much is made, then: a proton captured onto a ladder nucleus, by dipole light (allowed, since a proton and a helium-4 core have different charges for their mass), against a helium-4 captured onto the same nucleus.

computed Both captures, with the same machinery as the ladder's rungs: the nucleon's potential on the core folded from its potential on one helium-4, its depth set to the microscopic binding above (the proton's binding then follows, within a couple of MeV of the microscopic one). The check, ours: carbon-12 + proton → nitrogen-13 comes out 2.0 keV b near zero energy, against the measured direct capture of about 1.3–1.5. In 4D, how many times more often a ladder nucleus catches a proton than a helium-4, in unburnt gas (nine protons to each helium-4), between 0.05 and 0.3 billion K:

Wherever there's hydrogen, a ladder nucleus catches protons, almost never helium-4s: at least 210,000 times as often at any burning temperature, the proton's barrier being half the helium-4's and its light a dipole. So in 4D the helium-4 ladder runs only where the hydrogen is gone. Where hydrogen remains, the protons do the climbing, and the odd elements are on their path. follows A proton on a ladder nucleus makes a proton-rich nucleus that turns a proton into a neutron: element 9 at mass 17 lies 2.4–4.3 MeV above element 8 at mass 17, enough for a positron (ours: fluorine-17 to oxygen-17, 3.0 MeV here, 2.25 measured). In ours, burning hydrogen on carbon cycles: protons in, a positron or two, and the last proton throws out a helium-4 and gives the carbon back (our carbon–nitrogen–oxygen cycle). In 4D that last step is uphill, so it can't cycle: four protons and two positrons carry a ladder nucleus up one rung, through the odd element between. Element 9 is made, on the way from element 8 to pxygen, as our nitrogen is made in our cycle. computed And the climb has no closed-shell bottleneck. Ours does: oxygen-16 is a closed shell, a proton on it enters a new one and is held by only 0.6 MeV, so captures there are slow and hot hydrogen burning piles up behind it. 4D's closed shell is pxygen-20, and a nucleon on it is held by 70–216 MeV: the same method shows our gap, and 4D's nuclei are bound so deep that theirs barely shows. So what stops the climb is the electric barrier, rising with charge. Where the climb stops, and what that does to the water, is the next section.

Hydrogen and helium together

How far the protons carry a nucleus comes down to one count: how many protons it catches while the gas around it burns its hydrogen. That is a ratio of two rates at one temperature, the capture's against hydrogen's own burning, so the density drops out, and no clock has to be borrowed from our stars. computed The check, ours: while the Sun's centre burns half its hydrogen, at 15.7 million K, a carbon-12 catches about 4,900 protons, a nitrogen-14 about 13 and an oxygen-16 about 0.25: carbon goes to nitrogen, which piles up, and oxygen is only partly touched, as in our Sun's core (the same rates as the textbook's, its formula giving nitrogen-14 some 3.5 × 10⁸ years there).

In 4D the same count turns up something bigger. Hydrogen burns as ours, by the weak force (proton + proton, our rate through 4D's barrier, the stated choice the stars chapter made). But in 4D two helium-4s fuse to beryllium-8 by light alone, since mass 8 is bound: an electromagnetic capture, against hydrogen's weak one. How much faster helium-4 + helium-4 runs than proton + proton, pair for pair:

The spread is the two forces' first capture and the protons' nuclear size. Above about 7–10 million K helium burns as fast as hydrogen or faster, so in 4D every core that hot burns the two together, and makes its own ladder nuclei as it goes.

computed So a zone at each temperature is followed as a network: protons, helium-4s and nuclei by charge, every reaction a ratio to proton + proton. The captures onto beryllium-8, carbon and oxygen are the ones computed above (both draws of the core); onto the rest, the barrier times our nuclear factor, raised by how far the computed ones exceed it (31 to 2,700 times, each strength's average, since a 4D capture lands tens of MeV deep and dipole light grows as the photon's energy cubed). What a zone has turned into when its hydrogen is gone, starting from the first minutes' hydrogen and helium (elements making up more than a tenth of it in any of the twelve cases: three strengths, both draws, the protons at 1 or 4 fm; mass fractions):

A zone hot enough to burn helium burns all of it into one or two elements: the one where the climb stops at that temperature, as our nitrogen piles up behind its slowest capture, but here it takes the whole zone, because with helium burning alongside there are as many ladder nuclei as the protons can carry. Element 8 at 10–12 million K, element 9 around 12–17, pxygen around 17–22, and on up to elements 12–15 by 40. Below about 8 million K hydrogen goes to helium, as in our stars, and the helium is kept. Nor does it hang on the first minutes' helium: starting from 3% by mass instead of a quarter (the first minutes give a few percent to a third, by the clock) moves the make-up by a few percent, since the zone makes its own helium from its hydrogen. follows Odd elements pile up as readily as even ones: in 4D they aren't rare. And a core hot enough to burn both, from about 12–15 million K, leaves no helium behind, so the helium-4 ladder above runs only in the helium cores of cooler stars, where the stage a core reaches is set by its mass.

follows Which temperature, then? A planet needs a star that lasts. In the convention the burning thresholds use (the barrier and the nuclear factor compared across dimensions), a 4D core that burns half its hydrogen in the time our Sun's centre has taken runs at 15–21 million K, and there its zone turns into element 9 to element 11, mostly element 9 or pxygen. The twelve cases split cleanly. With the protons' nuclear size at 4 fm in proton + proton's barrier, the long-lived core runs at 15–17.5 million K and is element 9 for 47–91% of its mass; at 1 fm it runs at 18–21 and is pxygen for 39–85%. The forces and the core's draw barely matter: what tips it is proton + proton's own barrier, the one rate here borrowed whole from ours. Element 9 is not rare in 4D. Long-lived stars make it at tens of percent of their cores, alongside pxygen; our fluorine, thirty thousand times rarer than oxygen, has no 4D counterpart. Which of the two leads, and so which water, turns on how many stars burn at each temperature and how much of their cores they give back: the next section. open The convention's own uncertainty: it moves the long-lived core's temperature by tens of percent, which is the whole of element 9's band.

computed Proton + proton, from the strong force. That borrowed barrier is now gone. The rate is the overlap of the deuteron with the two protons' scattering state, both solved from this chapter's forces with the electric pull and 4D's s-wave barrier exact, and the weak conversion taken as ours: a rate, the same in both worlds, set on our measured one (tools/ppfusion.mjs). On ours it gives the measured matrix element (Λ² 7.04, against 7.04 from realistic forces) and the Sun's rate to 3%. In 4D, where the deuteron binds as ours, it equals the old barrier at 3.68–3.74 fm at every temperature from 5 to 50 million K: the 4 fm side. So a Sun-paced core is element 9 (oitrogen) for 36–79% of its mass and pxygen for 18–64%, at 15.2–17.4 million K, the range being the core's draw. What tips it now is the draw, and the strong force's strength through the deuteron's binding, which sets the positron's room: in the lowest 22% of the window the deuteron binds by less than 1.80 MeV and proton + proton makes no deuteron at all, and at the window's top it runs ten million times faster than where the deuteron binds as ours. Run at nine strengths from there to the top (Minnesota, every capture from the same force), a Sun-paced core climbs past helium only up to s ≈ 1.55, where p + p is sixty times faster and the core sits near 11 million K, making helium, darbon and elements 6 and 8; from s = 1.6 up the core sits below 7 million K and makes helium and almost nothing else. The strong-force dial sets that strength and shows each of these.

What stars give back

A star's lifetime sets its core's temperature (the pace above), and so what its core turns into. The gas the next stars and their planets form from carries back the cores of stars that have finished. choice Those that finish before a Sun-like star forms (burning faster than our Sun's centre, as our galaxy was about as old as that when ours formed), down to 10³ or 10⁴ times faster (ours give back over about 3 million to 10 billion years); each giving back a fixed share of its mass as core ash; as many stars of each mass as ours (the slope 2.35) or a top-heavy mix (1.9); lifetime falling with mass as M⁻² to M⁻³ (ours about M^−2.5). Each core's make-up from the network above, averaged over those lifetimes. computed The elements past helium in the gas that comes back (over the twelve cases and eight choices):

The gas comes back rich in gluorine, oeon and pxygen, 4D's fluorine, neon and oxygen, with oitrogen at 0–20%, todium (4D's sodium) at up to 21% and the heavier metals at a few percent at most. The protons' nuclear size in proton + proton's barrier splits it again. At 1 fm the noble gas leads (37–58%) and gluorine is the commonest element that bonds (20–45%, pxygen 3–21%); at 4 fm pxygen leads (28–55%), gluorine next (22–37%). Pxygen outnumbers oitrogen in every case: oitrogen comes to 0.06–0.41 of pxygen at 4 fm and to almost nothing at 1 fm. The stars that have given back by the time a Sun-like one forms burn hotter than it, past oitrogen's band; oitrogen is made mostly in stars still burning. follows So on abundance Xater's element leads XH₃'s, by at least 2.4 times, against XH₃'s 9% lead in the bond. And a third water turns up: gluorine makes hydrogen gluoride, the counterpart of our hydrogen fluoride (a liquid from −83 to 20 °C), and gluorine is as common as pxygen or more in half the cases.

computed With the main sequence solved (chapter 16) the declared lifetime slope goes: each star's own core temperature and lifetime at each G, its core what the zone makes at that temperature (with the zone's proton + proton size matched to the star's). The core of a star as long-lived as our Sun, and what the gas gets back from the stars that finish within ten billion years (choice our mass function or a top-heavy one; both proton sizes, both draws, three strengths):

This counts every finished star's core as coming back; the heavy cores below show the lighter stars keep theirs, so what actually comes back is the explosions' iron and rock row with this ash a minority.

G now decides the mix. The stronger the gravity, the more massive and hotter the stars that have finished, and the further up the table their cores went: at 10⁻¹⁴ some todium and nagnesium come back with the pxygen, gluorine and oeon; by 10⁻²⁰ it's mostly two of pxygen, gluorine and oeon, which two turning on the nuclear inputs. Pxygen stays ahead of oitrogen everywhere. follows Below about 10⁻²² of our gravity a galaxy gives back nothing for tens of billions of years: every star it can make, up to the heaviest light holds together, still burns, and the gas stays hydrogen and helium, with nothing to build rock or water from. The stars set a floor on G, before any planet does.

open Rock. Tilicon is absent from this ash, and the metals come back only where gravity is strong enough for hot, massive stars to have finished (todium and nagnesium at tens of percent at 10⁻¹⁴–10⁻¹⁷). Most of our rock's elements (magnesium, silicon, iron) come from later stages, burning in massive stars after their helium is gone and exploding white dwarfs, and their 4D counterparts aren't followed here. On this ash alone a 4D planet would be pxygen, gluorine and hydrogen compounds, ices and liquids with little rock. The later stages are the next section.

How far a table can go

computed A heavy nucleus pulls its innermost electron so hard that, relativistically, the electron's level sinks into the sea of negative energies: past that charge, an atom can't be neutral. Dirac's equation in four dimensions has our radial form with the spin–orbit number moved by the extra direction, and its non-relativistic limit is 4D hydrogen (chapter 6). Round a point nucleus the innermost level fails at Zα = 3/2, element 205 (ours: Zα = 1, element 137). With a nucleus of its real size, solved: ours dives near element 170 (the literature: 170–173), 4D's near element 232. So a 4D table could run to about 232, and the 78 solved here are not its end: 4D's nuclei, held by the crowd, hold together far past that, and it's the electrons that stop the table.

What's computed and what's open

Computed and checked on ours: the strength window (both forces), helium-4 (exact, both forces), nuclear matter (the deuteron exactly; ours saturating with one fitted strength), helium-4 clusters (one factor calibrated on beryllium-8, near its natural value), the helium-4 ladder (two-body, every sign right on ours), Dirac's limit. Chosen: the force's strength (the window's requirement) and the three-nucleon force's range (the uncertainty band). Rough: where binding peaks, and each rung's burning temperature. Computed as first estimates, on the road to a 4D Earth: odd nuclei and the protons' climb, hydrogen and helium burning together, and what stars give back, with declared choices for the star mix. open The rock's elements, and this world's iron. The gas stars give back is rich in gluorine, oeon and pxygen, pxygen ahead of oitrogen in every case, which leans the ocean toward Xater on abundance against XH₃'s bond, with hydrogen gluoride a third candidate. What it lacks is rock: past todium the metals and tilicon come only from the stages after hydrogen and helium, in the heaviest cores, and those aren't followed yet. Also open: proton + proton in 4D (borrowed whole from ours, and it tips oitrogen against pxygen), a spin–orbit force for mass 5, the clock (gravity's law), how neutron-rich 4D nuclei can be, and nuclei past helium solved properly, not as clusters.