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Commit ca01ff54 authored by Vadim Gubaidulin's avatar Vadim Gubaidulin
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- Added Chebyshev, Legenre and Sacherer modes

for spectral_density()
- Corrected Hermite mode to include normalisation by the mode number
- some authorefactoring
parent a6efc774
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......@@ -4,8 +4,10 @@ Module where bunch and beam spectrums and profile are defined.
"""
import numpy as np
from scipy.special import jv, spherical_jn
def spectral_density(frequency, sigma, m = 1, mode="Hermite"):
def spectral_density(frequency, sigma, m=1, k=0, mode="Hermite"):
"""
Compute the spectral density of different modes for various values of the
head-tail mode number, based on Table 1 p238 of [1].
......@@ -18,9 +20,14 @@ def spectral_density(frequency, sigma, m = 1, mode="Hermite"):
RMS bunch length in [s]
m : int, optional
head-tail (or azimutal/synchrotron) mode number
k : int, optional
radial mode number (such that |q|=m+2k, where |q| is the head-tail mode number)
mode: str, optional
type of the mode taken into account for the computation:
-"Hermite" modes for Gaussian bunches
-"Hermite" modes for Gaussian bunches (typical for electrons)
-"Chebyshev" for airbag bunches
-"Legendre" for parabolic bunches (typical for protons)
-"Sacherer" or "Sinusoidal" simplifying approximation of Legendre modes from [3]
Returns
-------
......@@ -29,14 +36,26 @@ def spectral_density(frequency, sigma, m = 1, mode="Hermite"):
References
----------
[1] : Handbook of accelerator physics and engineering, 3rd printing.
[2] : Ng, K. Y. (2005). Physics of intensity dependent beam instabilities. WORLD SCIENTIFIC. https://doi.org/10.1142/5835
[3] : Sacherer, F. J. (1972). Methods for computing bunched beam instabilities. CERN Tech. rep. CERN/SI-BR/72-5 https://cds.cern.ch/record/2291670?ln=en
"""
if mode == "Hermite":
return (2*np.pi*frequency*sigma)**(2*m)*np.exp(
-1*(2*np.pi*frequency*sigma)**2)
return 1/(np.math.factorial(m)*2**m)*(2*np.pi*frequency*sigma)**(2*m)*np.exp(
-(2*np.pi*frequency*sigma)**2)
elif mode == "Chebyshev":
tau_l = 4*sigma
return (jv(m, 2*np.pi*frequency*tau_l))**2
elif mode == "Legendre":
tau_l = 4*sigma
return (spherical_jn(m, np.abs(2*np.pi*frequency*tau_l)))**2
elif mode == "Sacherer" or mode == "Sinusoidal":
y = 4*2*np.pi*frequency*sigma/np.pi
return (2*(m+1)/np.pi*1/np.abs(y**2-(m+1)**2)*np.sqrt(1+(-1)**m*np.cos(np.pi*y)))**2
else:
raise NotImplementedError("Not implemanted yet.")
def gaussian_bunch_spectrum(frequency, sigma):
"""
Compute a Gaussian bunch spectrum [1].
......@@ -60,6 +79,7 @@ def gaussian_bunch_spectrum(frequency, sigma):
"""
return np.exp(-1/2*(2*np.pi*frequency)**2*sigma**2)
def gaussian_bunch(time, sigma):
"""
Compute a Gaussian bunch profile.
......
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