Example-06: Sextupole element factory
In this example sextupole factory is illustrated.
The sextupole hamiltonian is:
$ \begin{align} & H(q_x, q_y, q_s, p_x, p_y, p_s; s) = \frac{p_s}{\beta} - t(s)(q_x p_y - q_y p_x) - (1 + h(s) q_x) \left(\sqrt{P_s^2 - P_x^2 - P_y^2 - \frac{1}{\beta^2 \gamma^2}} + a_s(q_x, q_y, q_s; s)\right) \\ & \\ & P_s = p_s + 1/\beta - \varphi(q_x, q_y, q_s; s) \\ & P_x = p_x - a_x(q_x, q_y, q_s; s) \\ & P_y = p_y - a_y(q_x, q_y, q_s; s) \\ \\ & (a_x, a_y, a_s) = (0, 0, -\frac{1}{2!} k_n \left(\frac{q_x^3}{3} - q_x q_y^2 \right) - \frac{1}{2!} k_s \left(\frac{q_y^3}{3} - q_x^2 q_y\right))\\ & \varphi = 0 \\ & t = h = 0 \\ \end{align} $
The constructed element signature is:
def sextupole(qsps:Array, length:Array, kn:Array, ks:Array) -> Array:
...
Note, both kn
and ks
should be passed on invocation.
[1]:
import jax
from jax import jit
from jax import jacrev
from elementary.util import ptc
from elementary.util import beta
from elementary.sextupole import sextupole_factory
jax.numpy.set_printoptions(linewidth=256, precision=12)
[2]:
# Set data type
jax.config.update("jax_enable_x64", True)
[3]:
# Set device
device, *_ = jax.devices('cpu')
jax.config.update('jax_default_device', device)
[4]:
# Set initial condition
(q_x, q_y, q_s) = qs = jax.numpy.array([-0.01, 0.005, 0.001])
(p_x, p_y, p_s) = ps = jax.numpy.array([0.001, 0.001, -0.0001])
qsps = jax.numpy.hstack([qs, ps])
[5]:
# Define generic sextupole element
gamma = 10**3
element = jit(sextupole_factory(beta=beta(gamma), gamma=gamma, order=2**1, iterations=100))
[6]:
# Compare with PTC
length = jax.numpy.float64(0.2)
kn = jax.numpy.float64(-50.0)
ks = jax.numpy.float64(+75.0)
print(res := element(qsps, length, kn, ks))
print(ref := ptc(qsps, 'sextupole', {'l': float(length), 'k2': float(kn), 'k2s': float(ks)}, gamma=gamma))
print(jax.numpy.allclose(res, ref))
[-0.009839311476 0.005305274154 0.000999691938 0.000595944971 0.002048184556 -0.0001 ]
[-0.009839311476 0.005305274154 0.000999691938 0.000595944971 0.002048184556 -0.0001 ]
True
[7]:
# Differentiability
print(jacrev(element)(qsps, length, kn, ks))
print()
print(jacrev(element, 1)(qsps, length, kn, ks))
print()
[[ 9.977539230851e-01 -1.999683876014e-02 0.000000000000e+00 1.998755038515e-01 -1.333336463994e-03 -1.588878813703e-04]
[-1.999528286417e-02 1.002381087687e+00 0.000000000000e+00 -1.333283393256e-03 2.001711272451e-01 -3.042893998520e-04]
[ 3.567744778423e-05 1.067886761186e-05 1.000000000000e+00 -1.582182316560e-04 -3.046376164518e-04 8.131286746188e-07]
[-2.075140977962e-02 -2.000692596676e-01 0.000000000000e+00 9.979602547484e-01 -2.001825891400e-02 2.884097120735e-05]
[-2.000275114867e-01 2.345269216804e-02 0.000000000000e+00 -2.001650181701e-02 1.002174743629e+00 1.451547319575e-05]
[ 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 1.000000000000e+00]]
[ 5.960059282262e-04 2.048394055811e-03 -2.275669201308e-06 -2.198365452755e-03 5.184991611235e-03 0.000000000000e+00]