1.3.3. Halo Effect in Line Model - Finite Difference¶
1.3.3.1. Model¶
Fig. 1.181 Line model that I am using. I used Snell’s law for simplicity. Geogebra file can be downloaded line-model.ggb¶
About Reflection Angle
In principle, the reflection doesn’t have to be Snell’s law. The scattering can be in any angle with different amplitudes. Here I am using this very simple Snell’s law just to explore the effect of halo.
The effect of breaking symmetry of Snell’s law can also be interesting.
1.3.3.2. Conventions¶
The solver halo_euler_forward_one_nunubar(StateArray* rho_array_ptr, StateArray* rho_array_store_ptr, StateArray* rho_another_array_ptr, StateArray* rho_another_array_store_ptr, const double dt, const int totallength, const double spectrum[2], const double reflection, const double mu_arr[2], const double costheta[4]) takes in many parameters.
The first two arrays are the left beam, and the next two arrays are the right beams.
dtis the stepsize.spectrum[2]is the spectrum of the two beams. In fact this is simply the sign of the interactions. This is also used to determine the sign of \(\omega_v\). The first element should be the left beam and the second element should be the right beam.reflectionis the reflection coefficient. We have the same reflection coefficient for both beams regardless of the content since we are discussing neutral current scattering.mu_arr[2]is {\(\mu_L\), \(\mu_R\)}.The angles
costheta[4]parameters: { \(\cos(2 \theta_L)\), \(\cos(2 \theta_R)\) , \(\cos( \theta_R- \theta_L)\), \(\cos (\theta_R+\theta_L)\) }; It might be better to use \(\xi\) which is defined as \(1-\cos(\text{whichever angles})\).
1.3.3.3. Validation of Code¶
1.3.3.3.1. Bipolar model¶
In this simple setup we neglect reflections and also the interactions with backward beams. The neutrino oscillations should be reduced to the simple line model.
The instabilities can be calculated using Two Beams Model.
Equal Number Density Neutrinos and Antineutrinos
The left beam is neutrinos and the right beam is antineutrinos, which has the same number density as the left beam.