PX282 - A8 - solid angle

• a solid angle gives the area of a 'spherical cap' on the surface
• it is the 3D equivalent of an angle
• units: steradians, radians

image: c. balaji, b. srinivasan and s. gedupudi

• how many total solid angles in a sphere? ie:

• taking a unit sphere of surface area, $dA$, for a given solid angle, $d\mathrm{\Omega }$
• for spherical coordinates, $dA=\sin \theta d\theta d\mathrm{\Omega }:$

• at $\theta =0$, there is not surface contribution

• the angle between two lines is always the same

• similar y, the solid angle enclosed does not change

• eg: a beam (of $\gamma$, neutrinos) is emitted into solid angle, $\mathrm{\Omega }$, it will diffuse with distance, but $\mathrm{\Omega }$ does not change

• eg: a torch emits at a ring radius, $r$, at a distance, $d$. what is the solid angle?

  • the surface area of the ring of light is : $\pi r^2$
  • total area of the possible sphere is: $4\pi d^2$ ($d$ is the radius of the sphere containing the ring of light)
  • therefore, the solid angle is:
  $$\mathrm{\Omega }=\frac{\pi r^2}{4\pi d^2}\times 4\pi =\frac{\pi r^2}{d^2}$$

• more generally:

• eg in reverse: a beam of particles is released from a collider into a solid angle, $\mathrm{\Omega }$, at a rate, $n{s}^{-1}$. how many particles hit a detector at a distance, $d$?
  • beam fills a fraction of the sphere, $\frac{\mathrm{\Omega }}{4\pi }$, at any $d$
  • the detector also covers a fraction of the sphere, an area, $A$
  • if $A>\frac{\mathrm{\Omega }}{4\pi }\times 4\pi d^2$, all particles are received
  • if $A<\frac{\mathrm{\Omega }}{4\pi }\times 4\pi d^2$, the particle density is: $\frac{n}{\frac{\mathrm{\Omega }}{4\pi }\times 4\pi d^2}=\frac{n}{\mathrm{\Omega }{d}^2}$
  • rate at detector = $\frac{nA}{\mathrm{\Omega }{d}^2}$