Emittance model

In the emittance model the evolution of the spot size can be described using the Twiss parameters

emittance model

where the square of the std deviation \(\sigma_x^2\) along \(x\) is a quadratic function of propagation coordinate \(z\).

The model parameters are the emittance \(\varepsilon\), and the Twiss parameters \(\alpha\) and \(\beta\).

To use the emittance model, the following parameters must be set in a pencil beam definition:

Xsec = emittance

activate the emittance model: the cross-section is automatically a bivariate gaussian distribution

emittanceX = (float)

emittance along x

twissAlphaX = (float)

Twiss alpha along x

twissBetaX = (float)

Twiss beta along x

emittanceY = (float) [emittanceX]

emittance along y

twissAlphaY = (float) [twissAlphaX]

Twiss alpha along y

twissBetaY = (float) [twissBetaX]

Twiss beta along y

emittanceRefPlaneDistance = (float)

distance from beam origin of the reference plane where the previous parameters have been calculated

The spot size is usually measured in air inside a treatment room during the commissioning procedure or a QA protocol. For instance, the following plot shows the measurements (blue stars) 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

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)

eps = sqrt(a*c-b*b/4)
alpha=-b/2/eps
beta=a/eps

print('epsilon=',eps)
print('alpha=',alpha)
print('beta=',beta)

which gives in output

epsilon= 0.0032824274429144556
alpha= 0.05220831320123199
beta= 83.25066186389999

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 have to use the emittanceRefPlaneDistance parameter.

Finally, the input lines for source definition are

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

pb<
        ID = 1 ;
fieldID = 1
        particle = proton
        T = 100 # MeV
        Xsec = emittance

        twissAlphaX=0.05220831320123199
        twissBetaX=83.25066186389999
        emittanceX=0.0032824274429144556

        emittanceRefPlaneDistance=55; # Field_1 origin is at 55 cm from ISO
pb>
setup

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

emittance xsec