Flux computation

Flux computation#

Mulder’s primary intent is to compute (simulate) alterations (absorption, scattering) in the flux of atmospheric muons caused by a geometry of interest. For this purpose, Mulder offers a Fluxmeter object, which may be regarded as an ideal muon probe. Flumeters may be operated in three modes, continuous, mixed or discrete, as described below.

Note

Fluxmeter objects use the Backward Monte Carlo technique [NBCL18] to compute the altered muon flux.

Continuous mode#

In continuous mode, muons follow a deterministic trajectory with a deterministic energy-loss obtained within the Continuous Slowing Down Approximation (CSDA). The continuous approximation is efficient for small to medium-sized geometries (i.e. less than ~300m of rocks [Nie22]), when scattered muons are negligible (i.e. typically for buried detectors).

Due to the Earth’s magnetic field, the trajectories of muons and anti-muons differ, as they gyrate in opposite directions. In dense media, the muon CSDA range is significantly smaller than the magnetic gyro-radius across all energies. Thus, in many situations, the geomagnetic field can be disregarded. Then, in the continuous approximation, muons essentially follow straight lines trajectories.

Note

In the following discussion, for the sake of simplicity, we assume that the geomagnetic field is negligible, and therefore treat muons and anti-muons as a single entity. If this is not the case, then the following equations should be applied separately to both muon charges and the result summed to obtain the total flux.

Let us denote \(\phi_0\) the open-sky reference flux, i.e. the flux of atmospheric muons in the absence of any geometry. Under the continuous assumption, the altered muon flux, \(\phi_1\), can be related to \(\phi_0\) as follows

(1)#\[\phi_1(S_1) = \omega_c\left(S_0, S_1\right) \phi_0(S_0),\]

where \(S_1\) denotes the observation state, and where \(S_0\) is a reference state, connected to \(S_1\) by transport equations. The factor \(\omega_c(S_0, S_1)\) is a transport weight depending on the matter distribution along the muon trajectory, with and without the geometry.

The transport() method of a Fluxmeter object returns the reference state, \(S_0\), and the corresponding transport weight, \(\omega_c(S_0, S_1)\), given an observation state, \(S_1\). The flux() method directly computes the flux \(\phi_1(S_1)\) using equation (1), in continuous mode. Thus, the two following code snippets yield the same flux,

>>> state0 = meter.transport(state1)
>>> flux1 = state0.weight * meter.reference.flux(state0)

or directly

>>> flux1 = meter.flux(state1)

Discrete mode#

The discrete mode is the most realistic but also the most CPU-intensive option. In discrete mode, the muon trajectory and the energy loss are stochastic. Therefore, a given observation state, \(S_1\), is associated with several reference states, \(S_{0,i}\). In this case, a Monte Carlo estimate of the altered flux is obtained as

(2)#\[\hat{\phi}_{1, N}(S_1) = \frac{1}{N} \sum_{i=1}^N{ \omega_d\left(S_{0, i}, S_1\right) \phi_0(S_{0, i})} .\]

In dicrete mode, the flux() and transport() methods accept an additional parameter, events, which defines the number \(N\) of Monte Carlo samples per input observation state. For instance, the following samples a set of \(N\) reference states, from which the flux is estimated using equation (2).

>>> states0 = meter.transport(state1, events=N)
>>> flux1 = sum(states0.weight * meter.reference.flux(states0)) / N

The same flux could have been estimated directly as,

>>> flux1, sigma1 = meter.flux(state1, events=N)

Note that in the discrete case, the flux() method also returns an error estimate (sigma1) on the flux, determined from the sample variance.

Mixed mode#

The mixed mode represents a compromise between the continuous and discrete cases. In mixed mode, the muon trajectory is still deterministic, but stochastic energy losses may occur. This approximation is efficient for large geometries (more than 300m of rocks) in cases where muon scattering can be disregarded. In mixed mode, the altered flux is estimated using eq. (2). The mixed and discrete modes share the same interface.

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