/projects/fermis-forge/docs

Fermi's Forge — how it works

Notes on the physics behind the collider: what it computes, what it borrows from measurement, and the corners it cuts. Every number on the page comes out of the simulation described here; nothing is a table of answers keyed on what you fired.

1 · What this is, and what it is not

Two simulators sharing one kinematics module. The particle collider is a small Monte Carlo event generator — the same kind of program as PYTHIA or HERWIG, built from the same ideas (cross sections from the parton model, Lund strings, measured decay tables) at a hundredth of the detail. The nuclear beamline is a statistical-model reaction code in the spirit of HIVAP or PACE: barrier penetration, compound nucleus, evaporation, fission, and 3 340 nuclides of evaluated decay data.

Not a lookup table. What a collision makes is decided by which channels are open at that √s, how likely each is, and dice. Fire the same beams twice and you get two different events.
Not a detector simulation. The bubble chamber draws what the generator made, flown through a magnetic field. It does not reconstruct anything or miss anything.
Not precise. Rates are leading order with one correction factor per process, fitted to LHC measurements; the low-energy hadron physics is a 1950 statistical model. Factor-of-two thinking, everywhere — see the checks for how far off it is where you can tell.

2 · Kinematics

Natural units throughout: c = 1, and energy, momentum and mass all in MeV, so a particle's four-momentum obeys $E^{2} = p^{2} + m^{2}$ . One module of pure functions (src/lib/forge/kinematics.ts, unit-tested in kinematics.test.ts) does every calculation below, for both machines.

How much energy there is

The budget for making new particles is √s, the total energy in the centre-of-mass frame:

s_{collider} s_{fixed target} = m_{1}^{2} + m_{2}^{2} + 2 (E_{1} E_{2} + p_{1} p_{2}) = m_{b}^{2} + m_{t}^{2} + 2 E_{b} m_{t}

A collider wins by an enormous margin. Two 6.8 TeV beams head-on give √s = 13.6 TeV; a 6.8 TeV beam on protons at rest gives 113 GeV, because most of the energy goes into moving the centre of mass forward. The collider's readout says so whenever you pick a fixed target.

Two bodies

When √s becomes two particles of masses m₃ and m₄, each carries away momentum

p^{*} = \frac{1}{2 s} [s - (m_{3} + m_{4})^{2}] [s - (m_{3} - m_{4})^{2}]

in the centre-of-mass frame. Products are made there and boosted to the lab with

β_{cm} = \frac{p _{lab}}{E _{lab} + m _{target}}

for a fixed target (and $β = (p_{1} - p_{2}) / (E_{1} + E_{2})$ for an uneven collider). Three or more bodies come from GENBOD (F. James, 1968): chained two-body decays through random intermediate masses, weighted by the product of their p*.

Thresholds

A channel opens if and only if $s \geq \sum m_{products}$ . For a fixed target, a reaction with Q = (initial − final mass) < 0 needs the projectile to bring

K_{th} = - Q \frac{m _{p} + m _{t}}{m _{t}} (non-relativistic), K_{th} = \frac{( \sum m _{f} ) ^{2} - ( m _{p} + m _{t} ) ^{2}}{2 m _{t}} (exact)

The two agree to order Q/m. For nuclear reactions that is parts in ten thousand — ⁷Li(p,n)⁷Be has Q = −1.644 MeV and both give the textbook 1.880 MeV, which is what the nuclear results table quotes. For making particles they do not: an antiproton from p p → p p p p̄ needs Q = −2m_p, and the non-relativistic form says 4m_p = 3.75 GeV of kinetic energy where the truth is 6m_p = 5.63 GeV. That is why the Bevatron was built for 6.2 GeV. The collider's hints use the exact form; whether a channel is open is always decided by √s.

3 · What a shot is

For the beams you chose, the program lists every process that is open at that √s with its cross section σ_k — elastic scattering, string fragmentation, Z production, and so on. One collision picks process k with probability σ_k / Σσ, so a shot of N collisions splits multinomially: that is the "per collision" column, and why the Higgs at the LHC shows "1 in 2.2 billion".

Every process that happened is then simulated in full — the hard scatter, the strings, every decay down to stable particles, and the photons that convert in the chamber — for as many collisions as fit in about 0.7 s; the rest are counted by scaling up. The game limits a shot to 1 000 collisions; the sandbox goes to 10¹².

The collision drawn is the one a trigger would keep. Its rank: Standard Model squares it would reveal for the first time (weighted by the particle's mass, so a Higgs beats an electron), then the heaviest particle you have not seen, then anything but textbook QED or invisible final states, then rarity. In the sandbox, where nothing is news, that reduces to "heaviest and rarest".

4 · Hadron collisions

How often they collide

Above √s = 5 GeV the total cross section is the Particle Data Group's Regge-form fit,

σ_{tot} = Z + B ln^{2} \frac{s}{s _{M}} + Y_{1} (\frac{s _{1}}{s})^{η_{1}} \mp Y_{2} (\frac{s _{1}}{s})^{η_{2}}, B = 0.272 mb

with the pp, πp and Kp coefficients, the sign picked by how many valence quark–antiquark pairs could annihilate, and pairs of mesons scaled by the additive quark model (2/3). Below 5 GeV, smooth low-energy shapes: elastic scattering swells at low energy, and the inelastic part switches on from the lightest open channel with a 1/v rise for exothermic channels like p̄p annihilation.

Resonances

If the pair can fuse into a single short-lived hadron R — π⁺p → Δ⁺⁺, π⁺π⁻ → ρ⁰, K⁺K⁻ → φ — that is a process with

σ_{R} = \frac{2 J _{R} + 1}{( 2 s _{a} + 1 ) ( 2 s _{b} + 1 )} \frac{4 π}{k ^{2}} B (R \to ab) \frac{Γ ^{2} /4}{( s - M _{R} ) ^{2} + Γ ^{2} /4}

which gives 189 mb at the Δ peak against a measured ~200 mb — the bump Fermi's group found in 1952.

Near threshold: Fermi's statistical model

With less than 1.2 GeV of kinetic energy in the centre-of-mass frame there is too little for strings to mean anything (from 1.2 to 2.4 GeV the two pictures are blended). The program enumerates every final state of up to ten hadrons, eight if baryons are involved (pions, η, η′, ρ, ω, φ, kaons, K*, the baryon octet and decuplet, their antiparticles, the deuteron), that conserves charge, baryon number and strangeness and fits under √s, and gives each the weight

W_{c} = i \prod (2 J_{i} + 1) γ_{s}^{n_{s}} \frac{1}{\prod _{k} N _{k} !} [\frac{V}{( 2 π ) ^{3}}]^{n - 1} Φ_{n} (s; m_{1} \dots m_{n})

(E. Fermi, 1950): spin states, a strangeness suppression γ_s = 0.5 per strange quark, identical-particle factors, a Lorentz-contracted interaction volume of radius 1.6 fm, and the non-invariant phase-space density Φ_n. Φ is exact for two bodies, Monte Carlo for three, and above that a saddle-point evaluation of the canonical ensemble — accurate to 3% against brute-force integration and fast enough to rebuild the whole table while you drag a slider. It gets pp at √s = 2.2 GeV right in outline: about 70% Δ production, 10% deuteron + π⁺.

Above threshold: strings

Each hadron gives a quark and the rest of its valence (a diquark, for a baryon). Colour connects the quark of one to the remainder of the other, making two strings that stretch across the collision and fragment (§6). At collider energies there are extra soft parton–parton scatters, on average 0.9 ln(√s / 50 GeV) of them (five at 13 TeV), each adding two more strings, which is what fills an LHC event with a hundred tracks.

Hard processes

Jets, c c̄, b b̄, t t̄, W, Z, Drell–Yan pairs and the Higgs are computed in the parton model: a partonic cross section convolved with how often each pair of partons meets,

σ = ij \sum \int d τ σ^_{ij} (τ s) L_{ij} (τ), L_{ij} (τ) = \int_{τ}^{1} \frac{d x}{x} f_{i} (x) f_{j} (\frac{τ}{x})

with leading-order σ̂: Combridge's heavy-quark formulas, the narrow-width W, Z and gg → H, the Drell–Yan γ*, and one effective subprocess for jets above p⊥ = 4 GeV. The parton densities f_i(x, Q) are a hand-built set — valence quarks normalised to their counts, a sea and a gluon that rise at small x faster at higher Q, momentum summing to one — not a fit. Each process carries one constant, the "K-factor", that puts its 13 TeV rate on the measured value; the energy dependence is the parton model's own.

5 · Leptons, neutrinos and photons

Annihilation

A lepton and its antiparticle annihilate through a photon or a Z into any fermion pair f f̄ above threshold:

σ_{f f ˉ} = N_{c} \frac{4 π α ^{2}}{3 s} [Q_{f}^{2} - 2 Q_{f} v_{e} v_{f} Re χ + (v_{e}^{2} + a_{e}^{2}) (v_{f}^{2} + a_{f}^{2}) ∣ χ ∣^{2}], χ = \frac{1}{4 s _{W}^{2} c _{W}^{2}} \frac{s}{s - M _{Z}^{2} + i M _{Z} Γ _{Z}}

with v = T₃ − 2Q sin²θ_W and a = T₃. Heavy quarks open at the hadron threshold (twice the D or B mass), not the quark mass. On top sit the vector mesons as Breit–Wigners of peak height 12πB_ℓℓ/M², smeared by a 0.07% beam-energy spread (0.03% for muons): land within a few MeV of the J/ψ and more than half the collisions make one. Also: W pairs (a fit to LEP-2 and its leading-order extrapolation), Z pairs, Higgs-strahlung with the exact leading-order formula (0.24 pb at 250 GeV), WW fusion, two-photon annihilation by Dirac's formula, and large-angle Bhabha scattering as "elastic".

For muons there is one more: μ⁺μ⁻ → H directly, a spin-0 Breit–Wigner 4 MeV wide with peak (4π/M²)B(H → μμ). The Higgs couples to mass, so the same thing with electrons is (m_μ/m_e)² ≈ 43 000 times weaker.

Leptons on hadrons

Elastic scattering with a dipole form factor; quasi-real photons from the lepton's equivalent-photon flux striking the hadron (photoproduction); and above Q² = 1 GeV², deep inelastic scattering,

\frac{d ^{2} σ}{d x d Q ^{2}} = \frac{4 π α ^{2}}{x Q ^{4}} (1 - y + \frac{y ^{2}}{2}) F_{2} (x), F_{2} = q \sum e_{q}^{2} x (q (x) + q ˉ (x))

integrated on a grid. The struck quark is chosen by its charge² × density at x and leaves as a jet joined by a string to the remnant. Charged currents (W exchange) turn the lepton into its neutrino.

Neutrinos

Charged and neutral-current deep inelastic scattering with the same parton densities (on an average nucleon at 50 GeV, 0.61 × 10⁻³⁸ cm² per GeV of neutrino energy for charged currents, against 0.68 measured), quasi-elastic ν n → ℓ⁻ p, Δ production, and on electrons: weak elastic scattering, inverse muon decay above √s = m_μ, and ν̄_e e⁻ → W⁻, the Glashow resonance. Massive outgoing leptons are kinematically enforced, so a ν_τ makes a τ only above E_ν ≈ 3.5 GeV.

Photons

On charged targets, Bethe–Heitler pair production (most of what a high-energy photon does); on any hadron, photoproduction from the Donnachie–Landshoff fit plus the nucleon resonances; two photons make lepton pairs (Breit–Wheeler), hadrons, or above 161 GeV W pairs; a photon and an electron Compton-scatter (Klein–Nishina). A photon and a neutrino do nothing at all.

6 · Jets, strings and decays

Showers

Colour systems heavier than 8 GeV first radiate gluons: a dipole shower ordered in transverse momentum, drawn by the Sudakov veto algorithm against

d P = \frac{C α _{s} ( p _{⊥} )}{2 π} \frac{x _{i}^{2} + x _{j}^{2}}{( 1 - x _{i} ) ( 1 - x _{j} )} d x_{i} d x_{j}

with α_s running at one loop and a 1.5 GeV cutoff. Gluons above 4 GeV count as jets of their own — the three-jet events that revealed the gluon at PETRA.

Strings

Each string breaks repeatedly into quark–antiquark pairs (u : d : s = 1 : 1 : 0.3, diquarks 10%), and each hadron takes a fraction z of the remaining light-cone momentum from the Lund symmetric function

f (z) \propto \frac{( 1 - z ) ^{a}}{z} exp (- \frac{b m _{⊥}^{2}}{z}), a = 0.68, b = 0.98 GeV^{- 2}

(Peterson's function for the hadron carrying a c or b), with Gaussian transverse momentum at every break. The last two hadrons are made in an exact two-body decay of whatever is left, so a string conserves its four-momentum to rounding; strings too light to break collapse to the fewest hadrons their flavours allow, and the event's momenta are rescaled to match. Flavour is never rescaled: every event conserves charge, baryon number and lepton numbers exactly (the tests check every process at many energies).

Decays

About 120 particles carry Particle Data Group masses, lifetimes and decay tables; each table's main modes are kept and renormalised to 100%. Every unstable particle decays: a proper time from the exponential law, a flight of βγcτ, products laid out in phase space and boosted. Broad daughters (ρ, Δ, W, Z) take Breit–Wigner masses limited by what the parent can afford, which is how a 125 GeV Higgs decays to a real W and a virtual one. Particles that leave the chamber still decay; the codex counts what they leave behind.

7 · The bubble chamber

A box 3.8 m long, 2.1 m tall and 1.2 m deep, in a 1.5 T field pointing out of the screen. A charged track is a helix whose picture is a circle of radius $R = p_{⊥} / (0.3 B q)$ (GeV, tesla, metres), losing ~0.6 MeV/cm × 1/β² to ionisation and, if it is an electron, E/X₀ to radiation — so slow electrons spiral inwards. Daughters start where the parent ended, turned by the field as far as it was. Photons convert to e⁺e⁻ pairs with probability 1 − exp(−7L/9X₀), X₀ = 1 m; those pairs are real, added to the event, and count. Time runs at about 10⁻⁹ of real speed.

Pull the camera back and the liquid carries on: particles are followed through a region three times the chamber's size, as far as the view can reach, so a decay out there is drawn too. A track that leaves that region with nothing else happening to it fades away over its last metre and a half, and the beams fade in from as far off.

Decoration: the bubbles and their jitter, the delta-ray curls along heavy tracks, the fiducial crosses, and liquid beyond the walls of any real chamber.

8 · What counts as a discovery

Leptons, the photon, W, Z, Higgs, top: made, anywhere in the event (decays included).
Strange, charm, bottom quarks: any hadron carrying them — a kaon proves strangeness exists, a J/ψ proves charm.
Up and down quarks, and the gluon: only a hard collision that knocks one out as a jet (deep inelastic scattering, a hard scatter, a quark pair from annihilation above 5 GeV, a shower gluon over 4 GeV). Every hadron contains up and down quarks, so seeing a pion proves nothing about them — exactly the situation physicists were in until 1968.
Beams are harder than discoveries. Seeing a particle puts it in the codex; it joins your beam list only once you have a way to make a steady supply of it, all heading the same way — the way real beam lines got theirs. Hadrons have to come straight out of a collision, electrons boil off a hot wire, and the rest need the right parent. The codex's Beams page hints at what each one is waiting for.
Isotopes: made with at least one atom, directly or by any decay. Light nuclei from the collider (d, t, ³He, α) count; so do the electrons, positrons, neutrinos and photons from nuclear decays, on the Standard Model chart.

Spoilers: every beam's rule

Each needs the particle to have been seen at least once. "Straight out of a collision" means no ancestor that flew before decaying: a ρ or a Δ, gone in 10⁻²³ s, counts as the collision itself; a kaon decaying 10 cm away does not.

p, n: from the start. e⁻: as soon as you have seen one — a hot filament is enough.
π±, K±, p̄, n̄: made straight out of a collision, as in a secondary beam line.
e⁺: straight out of a collision, from a photon converting in the chamber, or from a proton-rich nucleus decaying in the nuclear beamline (β⁺).
γ: once you have fired an electron or positron beam — bremsstrahlung.
μ⁺ and ν_μ: once you have fired π⁺ or K⁺, which decay in flight into them; μ⁻ and ν̄_μ from π⁻ or K⁻. A μ⁻ beam also gives ν_μ, a μ⁺ beam ν̄_μ.
ν_e: from a μ⁺ or K⁺ beam. ν̄_e: from a μ⁻ or K⁻ beam, or a nuclear shot with fission in it — a reactor, as Cowan and Reines used in 1956.
ν_τ, ν̄_τ: made in a collision of two hadrons — a beam dump, as at DONUT in 2000 — where charm decays into them.
Any of the leptons and neutrinos also counts if it came straight out of a collision: the muons of e⁺e⁻ → μ⁺μ⁻, the neutrino of a W decay.

9 · Nuclear reactions

Data

NUBASE2020 (Kondev et al., 2021): mass excess, half-life, decay branches, spin, abundance and discovery year for all 3 340 nuclides ever observed — the free neutron and 3 339 isotopes of the 118 elements. Its "?" branches share whatever the measured ones leave; β-delayed modes are carved out of their parent β branch. Nuclei missing from the table get masses from a liquid drop with Myers–Swiatecki shell terms, fitted here to the 2 509 measured masses from A = 12 up (rms 1.85 MeV) and pulled into line with the nearest tabulated isotope.

Getting in

Charged nuclei tunnel through the Coulomb barrier $V_{B} = 1.44 Z_{p} Z_{t} / R_{B}$ MeV, $R_{B} = 1.42 (A_{p}^{1/3} + A_{t}^{1/3})$ fm, summed over partial waves:

σ_{fus} = \frac{π}{k ^{2}} ℓ \sum (2 ℓ + 1) T (E - E_{ℓ}), T (E < V_{B}) = \frac{1}{1 + exp ( 2 π η g ( E / V _{B} ) )}

with η the Sommerfeld parameter and g the WKB integral through a Coulomb wall (Gamow's factor, with the nucleus's size), and the Hill–Wheeler parabola above the barrier. Neutrons feel no barrier: an s-wave sticking probability 4kK/(k+K)² gives the 1/v law. Photons are absorbed through the giant dipole resonance, and above 30 MeV by neutron–proton pairs (quasi-deuterons). Two protons can only fuse weakly, with the solar S-factor 4 × 10⁻²² keV b — not once, in any shot the page can fire. For heavy pairs a quasi-fission hindrance $e^{- 136.7 (x_{eff} - 0.69)}$ (Loveland 2007) removes the contacts that never equilibrate. Above 10 MeV per nucleon fusion gives way to fragmentation, completely by 50.

Cooling down

The compound nucleus, at E* = E_cm + Q, evaporates. Widths by Weisskopf–Ewing and Bohr–Wheeler:

Γ_{x} = \frac{( 2 s _{x} + 1 ) m _{x} R ^{2}}{π ℏ ^{2}} T_{x}^{2} \frac{ρ ( U - S _{x} - V _{x} )}{ρ ( U )}, Γ_{f} = \frac{T _{f} K _{sad}}{2 π} \frac{ρ ( U - B _{f} )}{ρ ( U )}, Γ_{γ} \approx 3 \times 1 0^{- 4} A^{1.6} T^{5} eV

for n, p, d, t, ³He and α, with Fermi-gas level densities (a = A/8.5), real separation energies, Coulomb barriers for charged particles, and fission barriers from the liquid drop plus ground-state shell corrections (24 MeV for lead-208), the measured ~6 MeV plateau for actinides, and for superheavies a shell barrier that melts with heat (Ignatyuk, 18 MeV). K_sad restores the transition states just above a barrier. It is solved deterministically: populations flow through an excitation-energy grid in every nucleus reachable from the start, heaviest first, with exact branching ratios — so a channel that happens once in 10¹⁵ is computed as carefully as one that always happens. (Fragmentation, with thousands of nuclei in play, stops following flows below one in 10¹¹.)

Fission fragments follow measured systematics: asymmetric for actinides (heavy fragment near A ≈ 140), symmetric for lighter and heavier nuclei and when hot, charges by the unchanged-charge-density rule, and ν̄ ≈ 2 + 0.1E* + 0.08(A − 236) prompt neutrons. At high energy, abrasion: the overlap of the two nuclei is sheared off, the removed protons drawn hypergeometrically, 13.3 MeV of heat per lost nucleon (capped at 4 MeV per nucleon, past which a nucleus would shatter), and the remnants evaporate.

Decaying

Every product decays to stability by its NUBASE branches, counts flowing down each branch in heaviest-first order (every decay lowers the total mass, so that order is safe). Spontaneous fission and cluster decays included; time is not modelled. The target is 10¹⁹ atoms/cm², so one projectile reacts with probability 10¹⁹ σ; expected counts become sampled ones at the end.

10 · Checks

What the simulation gives, next to what nature does. Rows marked "theory" have no direct measurement to compare with, so the comparison is with the best calculation instead.

Quantity	Here	Measured
σ_tot(pp), √s = 13 TeV	106 mb	111 mb
σ(tt̄), 13 TeV / 7 TeV	830 / 195 pb	829 / 179 pb
σ(W⁺), σ(W⁻), σ(Z), 13 TeV	109, 70, 59 nb	109, 81, 59 nb
σ(gg → H), 13 TeV / 7 TeV	49 / 17 pb	48.6 / 16.9 pb (theory)
σ(W), σ(Z) at the Tevatron, 1.96 TeV	15, 5.0 nb	25, 7.6 nb
Z decays to hadrons / invisible	70.0% / 19.9%	69.9% / 20.0%
Charged particles per hadronic Z decay	18.6	20.8
… per e⁺e⁻ → hadrons at 34.8 GeV (PETRA)	12.7	13.6
σ(π⁺p) at the Δ peak	204 mb	≈ 200 mb
σ_CC(ν_μN) / E_ν at 50 GeV, in 10⁻³⁸ cm²/GeV	0.61	0.68
²³⁵U + thermal n: fission / capture	97% / 3%	85% / 15%
²³⁸U + thermal n: fission	none	none
²³⁸U + thermal n: capture	3 600 b	2.7 b
⁴⁸Ca + ²⁰⁸Pb → ²⁵⁴No + 2n, best energy	2.1 μb	≈ 2 μb
⁴⁸Ca + ²⁴⁹Cf → ²⁹⁴Og + 3n, best energy	11 pb	0.5 pb
⁷Li(p,n)⁷Be threshold	1.880 MeV	1.880 MeV
²³⁸U → ²⁰⁶Pb	8 α, 6 β⁻	8 α, 6 β⁻

Charged multiplicities count what the LEP experiments counted: everything that lives longer than a nanosecond is stable, and the pions from K⁰_S and Λ decays are included. The 13 TeV rates for tt̄, W⁺, Z and the Higgs are where each process's one constant was set, so they agree by construction; W⁻ shares the W's constant, so its shortfall is the hand-made parton densities' doing, and the 7 TeV and Tevatron rows show how the energy dependence holds up.

Where it is worst: slow-neutron capture (with no resonances, every nucleus soaks up a thermal neutron with the same few thousand barns, where nature ranges from millibarns to ¹³⁵Xe's 2.6 million), light-nucleus reaction rates (the statistical model was never meant for ⁸Be: p + ⁷Li at 1 MeV reacts about 100 times too often), the heaviest superheavies (twenty times too easy to make by oganesson), and anything else whose real cross section depends on nuclear structure rather than on the barrier.

11 · Lies, on purpose

Spin is ignored. Decays are isotropic, there is no polarisation, and the nuclear model has no angular momentum — which matters for fission and for which levels a nucleus can decay through.
C and G parity are ignored in the statistical model, so e⁺e⁻ → π⁰π⁰ is allowed when it should not be.
No diffraction, no pile-up, no initial-state radiation; multiple parton interactions are a crude count of extra strings; the parton densities are hand-made.
Decay tables are truncated to their main modes and renormalised, so rare decays are slightly commoner than in nature, and a few vanish.
Quarkonium and heavy-flavour hadronisation is only what strings do on their own.
No pre-equilibrium emission, no multifragmentation, no transfer reactions in the nuclear model; deuterons do not break up. Isomers are not tracked.
No time. Decay chains run to the end however long they take; nothing is lost to a short experiment, and a half-life only decides what can be a beam (a millisecond) or a target (a second).
The chamber is a cartoon of one: real ones were filled with liquid hydrogen, propane or freon, and photographed in stereo by several cameras at once.

12 · Sources

Particle Data Group, Review of Particle Physics (2024): masses, widths, lifetimes, branching fractions, and the total cross-section fits.
F.G. Kondev, M. Wang, W.J. Huang, S. Naimi and G. Audi, The NUBASE2020 evaluation of nuclear physics properties, Chinese Physics C 45, 030001 (2021).
E. Fermi, Prog. Theor. Phys. 5, 570 (1950) — the statistical model. B. Andersson, G. Gustafson, G. Ingelman and T. Sjöstrand, Phys. Rep. 97, 31 (1983) — strings. G. Gustafson and U. Pettersson, Nucl. Phys. B 306, 746 (1988) — the dipole shower.
V.F. Weisskopf and D.H. Ewing, Phys. Rev. 57, 472 (1940); N. Bohr and J.A. Wheeler, Phys. Rev. 56, 426 (1939); W. Loveland, Phys. Rev. C 76, 014612 (2007); J.-J. Gaimard and K.-H. Schmidt, Nucl. Phys. A 531, 709 (1991).