MuCF Physics

The muon-catalyzed fusion physics in Celeritas is derived from custom implementations written by Ara Knaian (Acceleron Fusion), Kevin Lynch (Fermilab), and Sridhar Tripathy (UC Davis), not available in the standard Geant4 source code.

Currently, the physics is managed by a single Executor that is responsible for the full cycle, from atom formation to generating the outgoing secondaries after fusion occurred.

Physics overview

Muons can be used to fuse deuterium-tritium mixtures at low temperatures [Kamimura et al., 2023]. This is caused by the fact that molecular orbital radii are inversely proportional to the mass of the lepton: the muon, with a mass approximately 207 times larger than the electron’s, leads to an orbital radius about 207 times smaller. The reduced molecular orbital leads to a higher nuclear wavefunction overlap, which in turn leads to a fusion reaction that does not require high-temperature, magnetic-confined plasma to happen.

The full cycle time is a few orders of magnitude smaller than the average decay time of the muon (\(2.2 \times 10^{-6}\) s). Muonic atom formation takes about \(10^{-12}-10^{-13}\) s, muonic molecule formation takes \(10^{-8}-10^{-10}\) s, and the fusion process itself is at the order of \(10^{-12}\) s. In most instances, the muon is free after the fusion process, leading to another cycle and giving the muon-catalyzed fusion its name. The possible channels for all deuterium-tritium molecules are outlined below:

  • \((dd)_\mu\)

    • \(\longrightarrow ^3\text{He} + \mu + n + 3.27 \ \text{MeV}\)

    • \(\longrightarrow (^3\text{He})_\mu + n + 3.27 \ \text{MeV}\)

    • \(\longrightarrow t + \mu + p + 4.03 \ \text{MeV}\)

    • \(\longrightarrow (t)_\mu + p + 4.03 \ \text{MeV}\)

  • \((dt)_\mu\)

    • \(\longrightarrow \alpha + \mu + n + 17.6 \ \text{MeV}\)

    • \(\longrightarrow (\alpha)_\mu + n + 17.6 \ \text{MeV}\)

  • \((tt)_\mu\)

    • \(\longrightarrow \alpha + \mu + 2n + 11.33 \ \text{MeV}\)

    • \(\longrightarrow (\alpha)_\mu + 2n + 11.33 \ \text{MeV}\)

In the cases where the muon sticks to an outgoing nucleus, e.g. generating a \((\alpha)_\mu\), the catalysis is halted. This happens at a fraction of a percent to a few percent level, and the number that represents the fraction of times this happens, with respect to the case where the muon is free, is called the sticking factor.

A single muon can repeat this fusion cycle somewhat between 100 to 400 times. The total number of fusion cycles produced by a single muon defines how much energy can be extracted from it, in the effort of reaching a break-even scenario. This is the threshold point where the energy required to generate the muon is equal to the energy produced by said muon through the muCF cycles. The sticking factor and the fusion cycle time are the main conditions that define how many fusion cycles a muon can undergo. The fusion cycle time depends on the d-t mixture, its temperature, and on the final spin of the molecule. Only muonic molecules where the total spin \(F = I_N \pm 1/2\) is on, or has a projection onto the total angular momentum J = 1 are reactive. The spin states of the three possible muonic molecules are summarized in table Table 12.

Table 12 Spin states of dt muonic molecules

Molecule

Nuclei

\(I_N\)

\(F = I_N \pm 1/2\)

Reactive states (F)

\((dd)_\mu\)

1, 1

0, 1, 2

1/2, 3/2, 5/2

1/2, 3/2

\((dt)_\mu\)

1, 1/2

1/2, 3/2

0, 1, 2

0, 1

\((tt)_\mu\)

1/2, 1/2

0, 1

1/2

1/2

Input

The input data is currently hardcoded in the celeritas::inp::MucfPhysics structure, which includes temperature-dependent rates for mean cycle time, muonic atom transfer, and muonic atom spin flip. The muon-catalyzed fusion process is activated by enabling the mucf_physics option in celeritas::ext::GeantPhysicsOptions.

struct MucfPhysics

Muon-catalyzed fusion physics options and data import.

Minimum requirements for muon-catalyzed fusion:

  • Muon energy CDF data, required for sampling the outgoing muCF muon, and

  • Mean cycle rate data for dd, dt, and tt muonic molecules.

Muonic atom transfer and muonic atom spin flip are secondary effects and not required for muCF to function.

Public Functions

inline explicit operator bool() const

Whether muon-catalyzed fusion physics is enabled.

Public Members

Grid muon_energy_cdf

CDF for outgoing muCF muon.

Vec<MucfCycleRate> cycle_rates

Mean cycle rates for muonic molecules.

Vec<MucfAtomTransferRate> atom_transfer

Muon atom transfer rates.

Vec<MucfAtomSpinFlipRate> atom_spin_flip

Muon atom spin flip rates.

Public Static Functions

static MucfPhysics from_default()

Construct hardcoded muon-catalyzed fusion physics data.

Geant4 integration

For integration interfaces, if mucf_physics option in celeritas::ext::GeantPhysicsOptions is enabled, the muon-catalyzed fusion data is constructed when the G4MuonMinusAtomicCapture process is registered.