Particle Phase Space Coordinates
Given a particle at some point on its trajectory, there is a reference frame on the reference orbit at some s value such that the particle's position is in the x-y plane with z = 0. The particle's particle's position and momentum P can be described using the coordinates:
(x(s), y(s), Px(s), Py(s), Pz(s), t(s))
For tracking purposes, canonical phase space coordinates are used:
(x, px, y, py, z, pz)
[Note convention: Upper case P = (unnormalized) momentum. Lower case p = phase space momentum]
where
px = Px / P0
py = Py / P0
pz = (P - P0) / P0
z = c * β * (t_ref - t)
where
- P0 is the reference momentum,
- β is the velocity of the particle,
- t_ref(s) is the time the reference particle reaches the point s.
Notes:
- For a bunch of particles at a given s position, in general the particles will have differing t.
- If the reference particle has the same β value as a particle, canonical z will be the longitudinal distance the particle is with respect to the reference particle. Positive z indicates that the particle is in front of the reference particle.
###Example###
Example lattice:
beginning[beta_a] = 10. ! m a-mode beta function
beginning[beta_b] = 10. ! m b-mode beta function
beginning[e_tot] = 10e6 ! eV
parameter[geometry] = open ! or closed
beam_start[y] = 0.01
beam_start[px] = 0.06
beam_start[pz] = -0.2
b: sbend, L = 0.5, g = 1 ! g = 1 / bending_radius
q: quadrupole, L = 0.6, k1 = 10
lat: line = (b, q) ! List of lattice elements
use, lat ! Line used to construct the lattice
Save the lattice to a file "phase_space.bmad" and run Tao:
> tao -lat phase_space.bmad
Tao> change beam_start px 0.04
Old New Old-Design New-Design Delta
0.060000 0.100000 0.000000 0.040000 0.040000 BEAM_START
View Orbits with show lattice
Tao> show lat
Values at End of Element:
Index name key s l beta phi eta orbit beta phi eta orbit Track_State
a a a x [mm] b b b y [mm]
0 BEGINNING Beginning_Ele 0.000 --- 10.00 0.000 0.00 0.000 10.00 0.000 0.00 10.000 Alive
1 B Sbend 0.500 0.500 7.62 0.069 0.20 28.889 8.51 0.069 0.01 9.201 Alive
2 Q Quadrupole 1.100 0.600 3.60 3.153 0.11 -18.477 138.05 0.112 -0.00 36.926 Alive
3 END Marker 1.100 0.000 3.60 3.153 0.11 -18.477 138.05 0.112 -0.00 36.926 Alive
Index name key s l beta phi eta orbit beta phi eta orbit Track_State
a a a x [mm] b b b y [mm]
Values at End of Element:
or show element
Tao> show ele 0
Element # 0
Element Name: BEGINNING
Key: Beginning_Ele
... etc...
Orbit: Positron State: Alive
Position[mm] Momentum[mrad] Spin |
X: 0.00000000 100.00000000 | Particle [sec]: 0.00000000E+00 E_tot; 8.0059E+06
Y: 10.00000000 0.00000000 | Part-Ref [sec]: 0.00000000E+00 PC: 7.9895E+06
Z: 0.00000000 -200.00000000 | (Ref-Part)*Vel [m]: 0.00000000E+00 Beta: 0.997961
Lcavity and RFcavity elements are RF cavity elements. The difference is that the reference energy at the exit end of an lcavity is shifted from the reference energy at the beginning while rfcavity elements do not affect the reference energy. Example:
beginning[beta_a] = 10. ! m a-mode beta function
beginning[beta_b] = 10. ! m b-mode beta function
beginning[p0c] = 1e8 ! eV
parameter[geometry] = open ! or closed
parameter[particle] = He+
q1: quad, l = 0.1, k1 = 0.14
q2: quad, l = 0.1, b1_gradient = parameter[p0c] * q1[k1] / c_light
lc: lcavity, l = 1, voltage = 10e8, rf_frequency = 1e9
rf: rfcavity, l = 1, voltage = 10e8, phi0 = 0.25
lat: line = (q1, q2, lc, q1, q2, rf)
use, lat
Notes:
- For a lcavity phi0 = 0 corresponds to peak acceleration.
- For an rfcavity phi0 = 0.25 corresponds to peak acceleration.
Save the lattice to a file "cavity.bmad" and run Tao:
> tao -lat cavity.bmad
Tao> show ele 3
Element # 3
Element Name: LC
Key: Lcavity
... etc...
51 P0C_START = 1.0000000E+08 eV BETA_START = 0.026811514
52 E_TOT_START = 3.7297409E+09 eV DELTA_E = 1.0000000E+09 eV
53 P0C = 2.9102374E+09 eV BETA = 0.615305883
54 E_TOT = 4.7297409E+09 eV GAMMA = 1.2685712E+00
... etc...
Orbit: He+ State: Alive
Position[mm] Momentum[mrad] Spin |
X: 0.00000000 0.00000000 | Particle [sec]: 3.42560963E-08 E_tot; 4.7297E+09
Y: 0.00000000 0.00000000 | Part-Ref [sec]: 0.00000000E+00 PC: 2.9102E+09
Z: -0.00000000 0.00000000 | (Ref-Part)*Vel [m]: 0.00000000E+00 Beta: 0.615306
Tao> show ele 6
Element # 6
Element Name: RF
Key: RFcavity
... etc...
53 P0C = 2.9102374E+09 eV BETA = 0.615305883
54 E_TOT = 4.7297409E+09 eV GAMMA = 1.2685712E+00
... etc...
Orbit: He+ State: Alive
Position[mm] Momentum[mrad] Spin |
X: 0.00000000 0.00000000 | Particle [sec]: 4.01447790E-08 E_tot; 5.7297E+09
Y: 0.00000000 0.00000000 | Part-Ref [sec]: -6.16649425E-10 PC: 4.3507E+09
Z: 140.37425587 494.97867675 | (Ref-Part)*Vel [m]: 1.40374256E-01 Beta: 0.759326
Questions:
- What are the k1 and b1_gradient values for the first q1 and the second q1? Do you understand this?
- What are the k1 and b1_gradient values for the first q2 and the second q2?