KAHIBARO
10.4. Source Direction
Table of Contents
Fixed direction
In a GATE simulation, every primary particle has a direction vector that tells Geant4 where the particle is moving. Conceptually, this is a unit vector, often written as $(u, v, w)$ or $(d_x, d_y, d_z)$, that points from the emission point along the particle trajectory.
A fixed direction means that all particles from a given source share the same direction. For example, a narrow pencil beam aimed along the positive $z$ axis has a direction vector $(0, 0, 1)$, while a horizontal beam pointing along $+x$ has $(1, 0, 0)$.
In practice, when you configure a GATE source in Python, you specify this direction as a 3D vector, usually in the simulation coordinate system. Internally, GATE normalizes this vector so that its length is 1. This direction is used for all primary particles generated by that source unless you intentionally randomize or spread the direction.
A fixed direction is appropriate when you want a collimated beam. Examples include pencil beams in physics tests, narrow radiotherapy beams, or a simple gamma beam pointing through a slab of material. For imaging systems like PET or SPECT, a purely fixed direction is less realistic for the radiative source itself, but it is often used for test setups and for narrow-beam attenuation studies.
Be careful to interpret the coordinate system correctly. The same three numbers will point in different physical directions if you rotate your whole geometry or use a different reference frame. Also remember that if you later add angular spread or isotropy, the original fixed direction often acts as a reference axis around which the spread is defined.
Isotropic emission
Many medical radionuclides emit photons or particles uniformly in all directions. This is usually modeled as an isotropic source. Isotropic means that the probability of emitting into any small solid angle is the same everywhere on the surrounding sphere.
Mathematically, isotropy relates to directions on the unit sphere. One might be tempted to sample polar and azimuthal angles $\theta$ and $\phi$ uniformly, but that would bias the distribution toward the poles. Instead, correct isotropic sampling uses a uniform distribution in azimuth $\phi$ and a uniform distribution in $\cos\theta$:
For isotropic emission, directions should satisfy
$$
\phi \sim \mathcal{U}(0, 2\pi), \quad \cos\theta \sim \mathcal{U}(-1, 1)
$$
and the unit direction vector is
$$
\begin{aligned}
u &= \sin\theta \cos\phi, \\
v &= \sin\theta \sin\phi, \\
w &= \cos\theta.
\end{aligned}
$$
GATE implements isotropic emission for you. When you configure a source as isotropic, the simulation automatically generates these directions correctly, so you do not need to implement the angular sampling yourself. The resulting flux is spherically symmetric around the source position, which is essential for realistic modeling of internal radionuclides or calibration sources that are not collimated.
Isotropic sources are particularly important in:
- Internal dosimetry simulations, where the radionuclide is distributed inside the body and photons or beta particles are emitted in all directions.
- PET and SPECT phantoms, where activity is uniform or spatially distributed but the emission from any point is still angularly isotropic.
- Shielding studies, to evaluate how a material attenuates radiation that comes from all directions rather than from a single beam.
In many configurations, you choose between a fixed beam and isotropic emission with a simple switch in the source configuration. Fixed direction is better for beam-like studies, while isotropic emission matches the physical emission of most uncollimated radioactive sources.
Angular distributions
Between a single fixed direction and fully isotropic emission, there is a wide range of possible angular distributions. In GATE, you often use these to model beam divergence, collimator acceptance, or specific device geometries.
An angular distribution describes how likely a particle is to be emitted in each direction on the unit sphere. Instead of every direction being equally probable, you might have a distribution that is more probable near the central axis and less probable at larger angles.
Common cases include:
- A cone or fan beam, where directions are uniformly distributed inside a limited angular range around a central axis.
- A Gaussian or approximately Gaussian spread in angle around a central beam direction, to represent a slightly divergent or blurred beam.
- Application specific profiles, such as collimator transmission profiles or custom distributions derived from measurements or other simulations.
A simple but widely used model is a uniform cone around a main direction. Suppose the beam axis is $\hat{z}$ and the maximum divergence angle (half opening angle) is $\theta_{\max}$. You want all directions that lie within that cone and have equal probability per unit solid angle. The correct sampling is:
For a uniform cone with half angle $\theta_{\max}$ around a given axis, sample
$$
\phi \sim \mathcal{U}(0, 2\pi), \quad \cos\theta \sim \mathcal{U}(\cos\theta_{\max}, 1).
$$
Then construct the direction vector from $(\theta, \phi)$ around the chosen axis.
This ensures that the distribution is uniform over the spherical cap defined by $\theta \le \theta_{\max}$. If instead you sampled $\theta$ uniformly between $0$ and $\theta_{\max}$, you would overpopulate directions close to the beam axis.
In practice, you rarely need to code the angular sampling formulas yourself when using GATE. You usually configure high level parameters that define the desired angular spread, for example an opening angle or a standard deviation. GATE then handles the random sampling and conversion to direction vectors.
Angular distributions are particularly important in medical imaging simulations. For PET, the emission from the radionuclide is isotropic, but the acceptance of the detector system depends on geometry and timing, so you effectively see a subset of the full angular range. In SPECT, the collimator strongly restricts the angular distribution that reaches the crystal, and realistic simulations often need to include distributions that represent these acceptance angles. For radiotherapy beams and CT, small angular spreads model realistic beam divergence from finite focal spots instead of idealized parallel beams.
Always make sure that the angular distribution you choose matches the physical problem. If you over-collimate the source in the simulation, you may overestimate resolution or contrast. If you make the angular spread too large, you may underestimate the performance of your system or misrepresent dose distributions.
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