Amongst various applications that experience a multi-directional
particle source is the muon scattering tomography where a number of
horizontal detectors of a limited angular acceptance conventionally
track the cosmic-ray muons. In this study, we exhibit an elementary
strategy that might be at disposal in diverse computational applications
in the GEANT4 simulations with the purpose of hemispherical particle
sources. To further detail, we initially generate random points on a
spherical surface for a sphere of a practical radius by employing
Gaussian distributions for the three components of the Cartesian
coordinates, thereby obtaining a generating surface for the initial
positions of the corresponding particles. Since we do not require the
half bottom part of the produced spherical surface for our tomographic
applications, we take the absolute value of the vertical component in
the Cartesian coordinates by leading to a half-spherical shell, which is
traditionally called a hemisphere. Last but not least, we direct the
generated particles into the target material to be irradiated by
favoring a selective momentum direction that is based on the vector
construction between the random point on the hemispherical surface and
the origin of the target material, hereby optimizing the particle loss
through the source biasing. We also show a second scheme where the
coordinate transformation is performed between the spherical coordinates
and the Cartesian coordinates, and the above-mentioned procedure is
applied to orient the generated muons towards the target material. In
the end, a recipe hinged on the restrictive planes from our previous
study is furthermore provided, and we incorporate our strategies by
using G4ParticleGun in the GEANT4 code. While we plan to exert our
strategy in the computational practices for muon scattering tomography,
these source schemes might find its straightforward applications in
different neighboring fields including but not limited to atmospheric
sciences, space engineering, and astrophysics where a 3D particle source
is a necessity for the modeling goals.