Sigma Squared model

This is a simplified and user-friendly model that allows to directly import spot size measurements into a Fred simulation. As the name is describing, the model is based on fitting the beam spot size using a second order polynomial, as described in the emittance model section:

Let’s take as a reference the case described by the following plot, showing the spot size measurements (blue stars) in the x direction taken at 5 different positions around the isocenter for a 100 MeV beam in a cyclotron facility.

spot size measurements

By fitting a parabolic function through the squares of the measured points

fit function

we can obtain the parameters needed for the emittance model

fit to emittance

Example of a simple calculation in python using the measurements of previous Figure

from math import *
import numpy as np
import matplotlib
matplotlib.use('TkAgg')
import pylab as plt


zmeas = np.array([-20,-10,0,10,20])
sigmeas = np.array([0.544,0.530,0.522,0.524,0.531])

[[c,b,a],cov] = np.polyfit(zmeas,sigmeas*sigmeas,2,cov=True)

print('a=',a)
print('b=',b)
print('c=',c)

zmodel = np.linspace(np.min(zmeas),np.max(zmeas),400)
sigmasqr = a+b*zmodel+c*zmodel*zmodel

plt.ion()
plt.plot(zmodel,np.sqrt(sigmasqr),'-r',label='sigma squared model')
plt.plot(zmeas,sigmeas,'*b',label='data',markersize=10)
plt.legend()
plt.ylim(0,0.8)
plt.grid()
plt.xlabel('beam axis coordinate (cm)')
plt.ylabel('spot size (cm)')
plt.show()

input('return')

which gives in output

a= 0.27326425714285724
b= -0.0003427399999999939
c= 3.9535714285714174e-05

Important

In the previous example, the parameters have been obtained with respect to the isocenter position which is at coordinate z=0 cm along the propagation direction. Since particles are generated in the field FoR, we have to inform FRED of the distance from field origin to the isocenter in order to have the correct spot size evolution along the beam axis. To this purpose, we define D_ISO has the distance of pencil beam origin and the isocenter.

sigmaSqrModel = [D_ISO, aX, bX, cX, aY, bY, cY]

D_ISO : distance (cm) from pencil beam origin to ISO center plane

aX, bX, cX : interpolation parameters for sigma_x as described above

aY, bY, cY : interpolation parameters for sigma_y as described above

parmeters for Y are optional: if not present, the values of X are mirrored

The input lines for source definition using the sigma squared model model syntax are

field: 1 ; O = [0,0,-55]; L=[10,10,20]; pivot = [0.5,0.5,0.2]

pb: 1 1 ; particle = proton; T = 100; sigmaSqrModel = [55,0.27326425714285724,-0.0003427399999999939,3.9535714285714174e-05]
setup

The evolution of beam cross section using the sigma squared model is represented by the following Figure

emittance xsec