Generation of the photons

For the formulas contained in this chapter, please see []. Let n be the refractive index of the dielectric material acting as a radiator ($n=c/c\text{'}$ where $c\text{'}$ is the group velocity of light in the material: it follows that $1\; \le n$ ). In a dispersive material n is an increasing function of the photon momentum $p$_γ,dn/dp ≥0 . A particle travelling with speed $\beta =\; v/c$ will emit photons at an angle $\theta$

with respect to its direction, where $\theta$ is given by the relation:

$cos\theta =\; \{1\beta n\}$

from which follows the limitation for the momentum of the emitted photons:

$n(p$_γ^min) = {1β}

Additionally, the photons must be within the window of transparency of the radiator. All the photons will be contained in a cone of opening $cos\theta$_max= 1/(βn(p_γ^max)) .

The average number of photons produced is given by the relations (Gaussian units):

$dN\; =\{2\; \pi z2e2c\}sin2\theta \{d\; \nu c\}dx\; =\{2\; \pi z2e2c\}(\; 1-\; cos2\theta )\; \{d\; \nu c\}dx\; =\{2\; \pi z2e2c\}(\; 1-\; \{1n2\beta 2\})\; \{d\; \nu c\}dx\; =$

$=\; \{z2e22c2\}(\; 1-\; \{1n2\beta 2\})\; dp$_γ&sp;dx ≈370 z²{photonscm  eV}( 1- {1n²β²}) dp_γ&sp;dx

and

$\{dNdx\}\approx 370\; z2\int$pγminpγmaxdpγ( 1- {1n2β2}) = 370 z2( pγmax-pγmin- {1β2}∫pγminpγmaxdpγ{1n(pγ)2})

The number of photons produced is calculated from a Poissonian distribution with average value . The momentum distribution of the photon is then sampled from the density function:

$f(p$_γ) = ( 1- {1n²(p_γ) β²})