Transportation

The transportation process is responsible for determining the geometrical limits of a step. It calculates the length of step with which a track will cross into another volume. When the track actually arrives at a boundary, the transportation process locates the next volume that it enters.

If the particle is charged and there is an electromagnetic (or potentially other) field, it is responsible for propagating the particle in this field. It does this according to an equation of motion. This equation can be provided by Geant4, for the case a magnetic or EM field, or can be provided by the user for other fields.

\[\frac{d {\bf p}} { ds } = \frac{1}{v} \textbf{F} = \frac{q}{v} \big( \textbf{E} + \textbf{v} \times \textbf{B} \big)\]

Extensions are provided for the propagation of the polarisation, and the effect of a gravitational field, of potential interest for cases of slow neutral particles.

Some additional details on motion in fields

In order to intersect the model Geant4 geometry of a detector or setup, the curved trajectory followed by a charged particle is split into ‘chords segments’. A chord is a straight line segment between two trajectory points. Chords are created utilizing a criterion for the maximum estimated value of the sagitta–the distance between the further curve point and the chord.

The equations of motions are solved utilising Runge Kutta methods. For the simplest case of a pure magnetic field, only the position and momentum are integrated. If an electric field is present, the time of flight is also integrated since the velocity changes along the step.

A Runge Kutta integration method for a vector \({\textbf{y}}\) starting at \(\textbf{y}_{start}\) and given its derivative \({d{\bf y'}}(s)\) as a function of \(y\) and \(s\). For a given interval \(h\) it provides an estimate of the endpoint \(\textbf{y}_{end}\). and of the integration error \({\bf y}_{error}\), due to the truncation errors of the RK method and the variability of the derivative.

The position and momentum as used as parts of the vector \({\textbf{y}}\), and optionally the time of flight in the lab frame and the polarisation.

A proposed step is accepted if the magnitude of the location components of the error is below a tolerated fraction \(\epsilon\) of the step length \(s\)

\[| \Delta {\bf x} | = | {\bf x}_{error} | < \epsilon * s\]

and the relative momentum error is also below \(\epsilon\):

\[| \Delta {\bf p} | = | {\bf p}_{error} | < \epsilon\]

The transportation also updates the time of flight of a particle. In case of a neutral particle or of a charged particle in a pure magnetic field it utilises the average inverse velocity (average of the initial and final value of the inverse velocity.) In case of a charged particle in an electric field or other field which does not preserve the energy, an explicit integration of time along the track is used. This is done by integrating the inverse velocity along the track:

\[t_1 = t_0 + \int_{s_0}^{s_1} {1 \over v} ds\]

Runge Kutta methods of different order can be utilised for fields depending on the numerical method utilised for approximating the field. Specialised methods for near-constant magnetic fields are also available.