Tutorial

The solver of Kapchinsky-Vladimirsky envelope equation for electron beam with space charge.

Including libraries

First connect the necessary libraries and configure the graphics parameters.

import numpy as np
import holoviews as hv
hv.extension('matplotlib')

import warnings
warnings.filterwarnings('ignore')

Some plotting options:

%output size=200 backend='matplotlib' fig='png' dpi=100
%opts Curve Spread [aspect=3 show_grid=True] (linewidth=2 alpha=0.7)
%opts Spread (linewidth=1 alpha=0.4)

import kenv as kv
kv.__version__

'0.3.0.0'

Define accelerator beamline parameters

acc = kv.Accelerator(z_start=0, z_stop=5, dz=0.01)

Define the accelerating field profile \(E_z(z)\):

#              Unique name,  z-position [m],  Ez [MV/m],  Ez(z) profile
acc.add_accel('Acc. 1',      1.5,            -1,         'Ez.dat')
acc.add_accel('Acc. 2',      3.5,            -1,         'Ez.dat')

acc.compile()

Let’s plot \(E_z(z):\)

dim_z  = hv.Dimension('z',  unit='m')
dim_Ez = hv.Dimension('Ez', unit='MV/m', label='$E_z$')

z  = acc.z
hv.Curve((z, acc.Ez(z)), kdims=dim_z, vdims=dim_Ez)

The same procedure is required to define the magnetic field profile \(B_z(z)\):

#                 Unique name,  z-position [m],  Bz [T],  Bz(z) profile
acc.add_solenoid('Sol. 1',      1.0,             0.05,   'Bz.dat')
acc.add_solenoid('Sol. 2',      2.0,             0.06,   'Bz.dat')
acc.add_solenoid('Sol. 3',      3.0,             0.07,   'Bz.dat')
acc.add_solenoid('Sol. 4',      4.0,             0.08,   'Bz.dat')

acc.compile()

dim_Bz = hv.Dimension('Bz', unit='Gs', label='$B_z$')

z_Bz = hv.Curve((z, acc.Bz(z)*1e4), kdims=dim_z, vdims=dim_Bz)

z_Bz

Define the electron beam parameters

beam = kv.Beam(
    energy = 2,     # MeV
    current = 2e3,  # A
    radius_x = 50e-3, # initial r (m)
    radius_y = 50e-3, # initial r (m)
    radius_xp = 10e-3,     # initial r' (rad)
    radius_yp = 10e-3,     # initial r' (rad)
    #x  = 2.5e-2,   # horizontal centroid position (m)
    xp = 10e-3,     # horizontal centroid angle (rad)
    y = 0,          # vertical centroid position (m)
    normalized_emittance = 1000e-6 # m*rad
)

Run the simulation

Now we can run the simulation in order to find the beam envelope:

sim = kv.Simulation(beam, acc)

sim.track(centroid=True)

Plot the simulation results

dim_E = hv.Dimension('E', unit='MeV', label='Electron energy',
                     range=(0,None))

hv.Curve((z, sim.gamma(z)*0.511), kdims=dim_z, vdims=dim_E)

dim_x = hv.Dimension('x', unit='cm', range=(-15,+15))
dim_y = hv.Dimension('y', unit='cm', range=(-15,+15))

Beam centroid:

x = sim.centroid_x # m
z_x = hv.Curve((z, x(z)*100), kdims=dim_z, vdims=dim_x)#.opts(linestyle='--')

z_x

y = sim.centroid_y # m
z_y = hv.Curve((z, y(z)), kdims=dim_z, vdims=dim_y)#.opts(linestyle='--')

z_y

Beam envelope:

x_size = sim.envelope_x # m
z_x * hv.Spread((z, 0, x_size(z)*100))

print(acc)

Accelerator structure.
        Solenoids:
        [ 1.0 m, 0.05 T, Bz.dat, Sol. 1, 0.0 m, 0.0 rad, 0.0 m, 0.0 rad]
        [ 2.0 m, 0.06 T, Bz.dat, Sol. 2, 0.0 m, 0.0 rad, 0.0 m, 0.0 rad]
        [ 3.0 m, 0.07 T, Bz.dat, Sol. 3, 0.0 m, 0.0 rad, 0.0 m, 0.0 rad]
        [ 4.0 m, 0.08 T, Bz.dat, Sol. 4, 0.0 m, 0.0 rad, 0.0 m, 0.0 rad]
        Accelerating modules:
        [ 1.5 m, -1 T, Ez.dat, Acc. 1, 0.0 m, 0.0 rad, 0.0 m, 0.0 rad]
        [ 3.5 m, -1 T, Ez.dat, Acc. 2, 0.0 m, 0.0 rad, 0.0 m, 0.0 rad]
        Quadrupoles:
        Correctors x:
        Correctors y:

print(beam)

Beam parameters:
        Current 2000.0 A
        Energy  2 MeV
        Total momentum  2.4584539418911224 MeV/c
        Rel. factor     4.913902396989699
        Radius x        50.0 mm
        Radius y        50.0 mm
        Radius x prime  10.0 mrad
        Radius y prime  10.0 mrad
        Horizontal centroid position    0.0 mm
        Vertical centroid position      0.0 mm
        Horizontal centroid angle       10.0 mrad
        Vertical centroid angle 0.0 mrad
        Larmor angle    0.0 rad
        Norm. emitt x   1000.0 mm*mrad