CERN Program Library Long Writeup W5013 - CERNLIB ...

= z2 e 2 (
¯h 2 1 − 1 )
c 2 n 2 β 2 dp γ dx ≈ 370z 2 photons (
1 − 1 )
cm eV n 2 β 2 dp γ dx
and
∫
dN
p max
dx ≈ γ
370z2
p min γ
(
dp γ 1 − 1 ) (
n 2 β 2 = 370z 2 p max
γ − p min
γ
− 1 ∫ p max
γ
β 2
p min γ
dp γ
1
n(p γ ) 2 )
The number of photons produced is calculated from a Poissonian distribution with average value ¯n =
STEP dN/dx. The momentum distribution of the photon is then sampled from the density function:
f(p γ )=
(
1 −
)
1
n 2 (p γ )β 2
3.2 Tracking of the photons
Čerenkov photons are tracked in the routine GTCKOV. These particles are subject to in flight absorption (process
LABS, number 101) and boundary action (process LREF, number 102, see above). As explained above,
the status of the photon is defined by 2 vectors, the photon momentum (⃗p =¯h ⃗ k) and photon polarisation
(⃗e). By convention the direction of the polarisation vector is that of the electric field. Let also ⃗u be the
normal to the material boundary at the point of intersection, pointing out of the material which the photon is
leaving and toward the one which the photon is entering. The behaviour of a photon at the surface boundary
is determined by three quantities:
1. refraction or reflection angle, this represents the kinematics of the effect;
2. amplitude of the reflected and refracted waves, this is the dynamics of the effect;
3. probability of the photon to be refracted or reflected, this is the quantum mechanical effect which we
have to take into account if we want to describe the photon as a particle and not as a wave;
As said above, we distinguish three kinds of boundary action, dielectric → black material, dielectric →
metal, dielectric → dielectric. The first case is trivial, in the sense that the photon is immediately absorbed
and it goes undetected.
To determine the behaviour of the photon at the boundary, we will at first treat it as an homogeneous
monochromatic plane wave:
⃗E = ⃗ E 0 e i⃗ k·⃗x−iωt
⃗B = √ µɛ ⃗ k × ⃗ E
k
Case dielectric → dielectric
In the classical description the incoming wave splits into a reflected wave (quantities with a double prime)
and a refracted wave (quantities with a single prime). Our problem is solved if we find the following
quantities:
⃗E ′ = ⃗ E ′ 0e i⃗ k ′·⃗x−iωt
⃗E ′′ = ⃗ E ′′ 0e i ⃗ k ′′·⃗x−iωt
For the wave numbers the following relations hold:
234 PHYS260 – 3