KAHIBARO
38.1. What Is a Phase Space?
Table of Contents
Particle state
In GATE and Geant4, a single particle at any given moment is described by its particle state. The particle state is a compact summary of everything the simulation needs to continue transporting that particle. For phase‑space work, you usually focus on a reduced version of this state that contains only the information you want to save or reuse.
The full internal state in Geant4 is rich and includes quantities such as particle type, position, momentum, energy, time, track history, and some process flags. A phase‑space record usually keeps only the most relevant subset of these. In medical physics simulations this almost always includes at least the particle position, its kinetic energy, and its direction of motion, and often the particle type and time as well.
You can think of phase space as an abstract space where each point represents one possible particle state. Instead of plotting particles in real space only, you can also plot them in this larger space where their coordinates include spatial coordinates and momentum or energy coordinates. A phase‑space file produced by GATE is simply a list of many such points, one for each particle that crosses a given surface or volume, stored so you can later use them as an input source or for analysis.
A phase‑space point is a single particle state, typically represented by its position, direction, and energy at a specific time and location in the simulation.
In the rest of this chapter we focus on the three fundamental parts of this reduced state that are always present in GATE phase‑space applications: position, energy, and direction.
Position
The position component of a particle state tells you where the particle is in the simulation geometry at the moment it is recorded. In GATE this is always expressed in the simulation coordinate system, using the length units you have chosen. Internally, Geant4 uses millimeters, and in practice most GATE simulations also record phase‑space positions in units of millimeters.
The position is described by three coordinates, usually written as $(x, y, z)$. Together they specify a single point in three‑dimensional space. For a phase‑space surface such as a scoring plane or a detector entrance, these coordinates tell you exactly where the particle crossed that surface.
In medical physics applications, position is important for several reasons. For example, in external beam therapy phase‑space data at the exit of a linear accelerator head captures the spatial distribution of the beam before it enters the patient. In imaging, position at the detector plane describes where photons arrive at the detector. When you reuse phase‑space data as a source, these stored positions become the initial positions of your new particles.
It is important to remember that phase‑space position is always tied to the geometry that produced it. If you move or rescale geometries, you must keep track of how this affects the meaning of the stored coordinates. In most workflows you either keep a consistent geometry or explicitly transform the positions when using the phase‑space data in a different setup.
Energy
Energy in a particle state tells you how much kinetic energy the particle has at the moment it is recorded. In GATE this is a scalar value, usually given in electron‑volt based units such as keV or MeV. Internally, Geant4 uses MeV, but you can convert to other units for storage or analysis.
Energy is central for phase‑space because it determines how the particle will interact with matter later. Two particles with the same position and direction but different energies will typically have very different interaction histories and ranges. A phase‑space file that accurately preserves the energy spectrum at a given surface allows you to reproduce the same downstream physics without having to re‑simulate the upstream geometry.
In many applications, you are interested in the energy distribution of particles crossing a plane or entering a volume. By looking at the set of energies in a phase‑space file, you can build histograms, estimate energy fluence, or check that a beam shaping device, filter, or collimator is producing the expected spectrum.
For each particle state in phase space, the kinetic energy $E$ is a key coordinate. It is usually stored in MeV and directly influences interaction probabilities and ranges in downstream materials.
When reusing a phase‑space dataset as a source, these stored energies become the initial kinetic energies of the new particles. If you modify them, for example to simulate a different beam quality, you change the physical behavior of the reused phase‑space source.
Direction
Direction in a particle state encodes the way the particle is moving at the recorded instant. In GATE and Geant4 this is represented by a three‑component unit vector, often written as $(u, v, w)$ or $(d_x, d_y, d_z)$. This vector points along the particle trajectory at that point.
Because it is a unit vector, its length is always 1. Mathematically this is expressed as
$$
u^2 + v^2 + w^2 = 1.
$$
The direction vector of a particle in phase space is a unit vector. It must always satisfy
$$u^2 + v^2 + w^2 = 1,$$
otherwise the direction does not represent a physically valid momentum direction.
Direction is essential in phase‑space descriptions because it, together with position, defines the path along which the particle will continue to move. Two particles with the same position and energy but different directions will enter different parts of the geometry and deposit energy in different locations.
In external beam therapy, direction at the patient surface determines how the beam penetrates the patient and shapes the dose distribution. In PET or SPECT, the direction of photons arriving at a detector plane is linked to collimation and imaging geometry. When you use phase‑space files as sources, preserving the original direction vectors allows you to reproduce realistic angular distributions without manually specifying them.
In some analyses you may convert the direction vector to angular coordinates, such as polar angle and azimuthal angle, but internally GATE and Geant4 always work with the three Cartesian components. For any manipulation or transformation of phase‑space data, it is important to maintain the unit length of the direction vector so that subsequent transport in the simulation remains correct.
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