Method

A charged particle in a magnetic field emits radiation. The number of photons emitted in a second is $\{d2Nd\epsilon \hspace{0.17em}dt\}=\; \{1538\}\{P\epsilon$_c}

where $\epsilon$_c is the critical photon energy (the median of the energy spectrum) and P the total radiated power:

$\epsilon$c	$=$	$\{32\}c\; \{\gamma 3\rho \}$
$P$	$=$	$\{23\}\{\; e2c\rho 2\}\beta 4\gamma 4$

$\gamma$ is the $e-$ /$e+$ Lorentz factor $E$_tot/m_e and $\beta =\; 1\; -\; 1/\gamma 2$ . $\rho$ is the curvature caused by the magnetic field. For more detailed derivation of these equations, see []. The velocity of the particle being $\beta c$ , the number of photons per meter is

$\{d2Nd\epsilon \hspace{0.17em}dx\}$	$=$	$\{d2Nd\epsilon \hspace{0.17em}dt\}\hspace{0.17em}\{1\beta c\}$
	$=$	$\{536\}\{e2c\}\beta 3\gamma \{1\rho \}$
	$=$	$\{523\}\hspace{0.17em}\alpha \hspace{0.17em}\{\gamma \rho \}\hspace{1em}\approx \hspace{1em}0.01053\; \hspace{0.17em}\{\gamma \rho \}$

The fine structure constant $\alpha =\; e2/(c)$ and it is assumed that $\beta 3\approx 1$ .

The curvature in a magnetic field B which has a component $B$_⊥ transversal to the particle velocity can be computed [] $\rho =\; \{p0.3\; B$_⊥}

where p is the momentum of the particle in GeV. B is in tesla and $\rho$ is in meters.

Knowing the step length, the energy of the electron and the curvature of the particle track in the magnetic field, the number of photons in a step $\delta x$ can be sampled from a Poissonian distribution around the mean value $n$_δx= 0.01053  {γρ}δx

Now, the energies of $n$_δx photons have to be determined. The energy distribution in a step follows the distribution [] $f(\epsilon )\; \hspace{0.17em}=\; \hspace{0.17em}\{dNd\epsilon \}\hspace{0.17em}\hspace{0.17em}\int$_{ε/ εc}^&inf;K_5/3(x)  dx

The energy can be sampled from this by inverse transform method:

$N\; \hspace{0.17em}$	$=$	$\hspace{0.17em}F(\epsilon )\; =\; \hspace{0.17em}\int f(\epsilon )\; \hspace{0.17em}d\epsilon$
$\epsilon \hspace{0.17em}$	$=$	$\hspace{0.17em}F-1(N)$

The double integral is not analytically solved, and the sampling is done from tabulated values of numerically computes $F(\epsilon )$ .

Two methods have been implemented. If the flag ISYNC is set to 1, the photons are emitted at the end of the step along the current direction. If ISYNC is set to 3, the photons are emitted randomly along the tangent the real trajectory of the particle. In the case when ISYNC = 3, the magnetic field tracking routines are called for each photon, and therefore this option is considrably slower than ISYNC = 1.

Figure: The point where the synchrotron radiation photon is generated. The figure on the left describes the situation when ISYNC = 1, and the one on the right when ISYNC = 3. The little arrows are the photons and STEP is the step taken by the $e-$ or $e+$ . VECT is the new direction computed in GTELEC before entering in GSYNC.

PHYS400