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 sampling the outgoing muCF muon.

Vec<MucfCycleRate> cycle_rates

Mean cycle rates for muonic molecules.

Vec<MucfAtomTransferRate> atom_transfer

Muonic atom transfer rates.

Vec<MucfAtomSpinFlipRate> atom_spin_flip

Muonic atom spin flip rates.

Public Static Functions

static MucfPhysics from_default()

Construct hardcoded muon-catalyzed fusion physics data.

Geant4 integration

For integration interfaces, enabling the mucf_physics option in celeritas::ext::GeantPhysicsOptions will check if the G4MuonMinusAtomicCapture process is registered in the Geant4’s Physics List. If the process is present, the celeritas::inp::MucfPhysics will be populated, and the celeritas::MucfProcess will be initialized.

Code implementation

The celeritas::MucfProcess process has only the celeritas::DTMixMucfModel attached to it, responsible for deuterium-tritium mixtures. It can simulate materials from near absolute zero to 1500 kelvin. It is an at rest model that encompasses the full cycle—atom formation, molecule formation, and fusion.

class MucfProcess : public celeritas::Process

Muon-catalyzed fusion of muonic dd, dt, or tt molecules.

Note

This is an at-rest process.

class DTMixMucfModel : public celeritas::Model, public celeritas::StaticConcreteAction

Muon-catalyzed fusion model for dd, dt, and tt molecules.

In this model the executor performs the full muon-catalyzed fusion workflow. It forms a muonic d or t atom, samples which muonic molecule will be produced, selects the channel, and calls the appropriate interactor.

The full set of “actions” is as follows, and in this ordering:

• Define muon decay time to compete with the rest of the execution

• Form muonic atom and select its spin

• May execute atom spin flip or atom transfer

• Form muonic molecule and select its spin

• Calculate mean cycle time (time it takes from atom formation to fusion)

• Confirm if fusion happens or the if the muon should decay

• Call appropriate Interactor: Muon decay, or one of the muCF interactors

Note

This is an at-rest model.

Most of the data is material-dependent, being calculated and cached during model construction. All of the cached quantities are calculated and added to host/device data via celeritas::detail::MucfMaterialInserter.

class MucfMaterialInserter

Helper class to calculate and insert muCF material-dependent data into DTMixMucfData .

The main cycle is managed by the model’s celeritas::DTMixMucfExecutor. The muonic atom selection is handled by detail classes, while the Interactors are reserved for sampling final states of the outgoing secondaries.

Note

Only reactive channels are implemented.

class MuonicAtomSelector

Select a muonic atom given the mixture of dt in the material.

This class assumes that the material is hydrogen and that the capture happened to a deuterium or tritium via a simple isotopic fraction selection.

It is needed to correct the probability of a deuterium or tritium capture, since the isotopic fraction sampling is not sufficient: tritium has a higher mass and thus has a biased capture rate.

This effect is calculated using the \( q_\text{1S} \) formula Bom et al. [2005]

\[ q_\text{1s} = \frac{1}{1 + 2.9 C_t}, \]

where \( C_t \) is the relative tritium isotope concentration and \( q_\text{1s} \) is the fraction muonic deuterium atoms in the ground state. This expression allows calculating the probability of forming a muonic deuterium atom via

\[ P_\text{d} = C_d \times q\text{1s}. \]

If a selected uniform random number is \( x \leq P_\text{d} \), a muonic deuterium is formed. Otherwise, a muonic tritium is selected.

class MuonicAtomSpinSelector

Select muonic atom spin, in units of \( \frac{\hbar}{2} \).

Sampling is based on spin population probabilities from Yamashita et al. [2022] which are:

• Muonic deuterium: 2/3 probability for spin 3/2; 1/3 for spin 1/2

• Muonic tritium: 3/4 probability for spin 1; 1/4 for spin 0

class DDMucfInteractor

Muon-catalyzed fusion of \( (dd)_\mu \) molecules.

Fusion channels:

• \( ^3\text{He} + \mu + n \)

• \( (^3\text{He})_\mu + n \)

• \( t + \mu + p \)

The mass ratios between \( ^3\text{He} \) and the neutron, and between tritium and the proton, are both about 3:1. This leads to the neutron and proton kinetic energies being 3/4 of the total kinetic energy available in their respective channels.

Note

This interactor has a similar implementation as DTMucfInteractor , where energy is not fully conserved. See its documentation for details.

class DTMucfInteractor

Muon-catalyzed fusion of \( (dt)_\mu \) molecules.

Fusion channels:

• \( \alpha + \mu + n \)

• \( (\alpha)_\mu + n \)

Warning

This implementation has an incorrect energy and momentum conservation implementation. Acceleron assumes an isotropic direction for both neutron and muon in the \( \alpha + \mu + n \) channel, which leads to the alpha particle either conserving energy or momentum but not both simultaneously. The current implementation results in a roughly correct total energy within \( K_\text{total} = [17.5, 17.9] \) MeV, instead of the expected 17.6 MeV.

class TTMucfInteractor

Muon-catalyzed fusion of \( (tt)_\mu \) molecules.

Fusion channels:

• \( \alpha + \mu + n + n \)

• \( (\alpha)_\mu + n + n \)