mbtrack2.impedance.csr module

Define coherent synchrotron radiation (CSR) wakefields in various models.

class FreeSpaceCSR(time: ndarray[tuple[int, ...], dtype[_ScalarType_co]], frequency: ndarray[tuple[int, ...], dtype[_ScalarType_co]], length: float, radius: float, ring: Synchrotron)

Bases: WakeField

Free space steady-state coherent synchrotron radiation Wakefield element, based on [1-3].

Notes:

• Impedance is computed using Eq. (A10) of [2].

• The WakeField model stores Zlong and Wcsr.

• The Wcsr component (type ‘csr’) holds the antiderivative

of the wake function rather than the wake function itself, because the direct wake diverges at tau=0 making numerical convolution inaccurate. - WakePotential class uses the antiderivative convolved with the derivative of the bunch profile instead. - The direct wake function is available via LongitudinalWakeFunction for analysis but is not added to the model.

Parameters

timearray of float

Time points where the wake function antiderivative is evaluated in [s]. Must span the bunch extent for tracking.

frequencyarray of float

Frequency points where the impedance will be evaluated in [Hz].

lengthfloat

Length of the impedance element to consider in [m].

radiusfloat

Dipole radius of curvature in [m].

ringSynchrotron

Ring object, used to access the Lorentz factor gamma.

References

[1] : Faltens, A., & Laslett, L. J. (1973). Longitudinal coupling impedance of a stationary electron ring in a cylindrical geometry. part. Accel., 4, 151-157. [2] : Agoh, T., and K. Yokoya. “Calculation of coherent synchrotron radiation using mesh.” Physical Review Special Topics-Accelerators and Beams 7.5 (2004): 054403. [3] : J. B. Murphy, S. Krinsky, and R. L. Gluckstern, Longitudinal wakefield for an electron moving on a circular orbit, Particle Accelerators 57, 9 (1997).

__init__(time: ndarray[tuple[int, ...], dtype[_ScalarType_co]], frequency: ndarray[tuple[int, ...], dtype[_ScalarType_co]], length: float, radius: float, ring: Synchrotron)

LongitudinalImpedance(frequency: ndarray[tuple[int, ...], dtype[_ScalarType_co]]) → ndarray[tuple[int, ...], dtype[_ScalarType_co]]

Compute the free space steady-state CSR impedance. Based on Eq. (A10) of [1].

This formula is valid only if omega << (3 * gamma^3 * c) / (2 * R).

Parameters

frequencyfloat array

Frequency in [Hz].

Returns

Zlcomplex array

Longitudinal impedance in [ohm].

References

[1] : Agoh, T., and K. Yokoya. “Calculation of coherent synchrotron radiation using mesh.” Physical Review Special Topics-Accelerators and Beams 7.5 (2004): 054403.

LongitudinalWakeFunction(time: ndarray[tuple[int, ...], dtype[_ScalarType_co]]) → ndarray[tuple[int, ...], dtype[_ScalarType_co]]

Compute the free-space CSR wake function following Eq. (13) of [1].

For analysis only. Not added to the WakeField model because the wake diverges at tau=0. Use the Wcsr antiderivative component for tracking (see LongitudinalWakeAntiderivative).

Parameters

timefloat array

Time in [s].

Returns

Wlfloat array

Longitudinal wake function in [V/C].

References

[1] : E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, On the coherent radiation of an electron bunch moving in an arc of a circle, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 398, 373 (1997). [2] : J. Qiang, C. Mitchell, and R. D. Ryne, A fast high-order method to calculate wakefields in an electron beam, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 682, 49 (2012).

LongitudinalWakeAntiderivative(time: ndarray[tuple[int, ...], dtype[_ScalarType_co]], points_around_zero: int = 100000) → ndarray[tuple[int, ...], dtype[_ScalarType_co]]

Compute the antiderivative of the free space CSR wakefunction. We follow Eq. (3.13) of [1]. The antiderivative should be used to calculate the part of the convolution integral around 0.

Parameters

timefloat array

Time in [s].

Returns

Wl_intfloat array

Antiderivative of the longitudinal wake function per unit length in [V/C/m].

References

[1] : J. B. Murphy, S. Krinsky, and R. L. Gluckstern, Longitudinal wakefield for an electron moving on a circular orbit, Particle Accelerators 57, 9 (1997).

class ParallelPlatesCSR(time: ndarray[tuple[int, ...], dtype[_ScalarType_co]], frequency: ndarray[tuple[int, ...], dtype[_ScalarType_co]], length: float, radius: float, distance: float, ring: Synchrotron, k_max: int = 100)

Bases: WakeField

Perfectly conducting parallel plates steady-state coherent synchrotron radiation Wakefield element, based on [1,2].

The WakeField model stores three components:

• Zlong (full impedance, free space + shielding, Eq. A1 of [1])

• Wlong (shielding correction only, Murphy et al. [2] Eqs. 5.21-5.22)

• Wcsr (free-space CSR antiderivative from an internally constructed

  FreeSpaceCSR instance).

Together Wlong and Wcsr give the full time-domain wake, consistent with Zlong. WakePotential class handles both components automatically.

Parameters

timearray of float

Time points spanning the bunch extent in [s]. The analytical wake components are stored internally on the corresponding convolution lag grid.

frequencyarray of float

Frequency points where the impedance will be evaluated in [Hz].

lengthfloat

Length of the impedance element to consider in [m].

radiusfloat

Dipole radius of curvature in [m].

distancefloat

Full-gap vertical distance between the parallel plates in [m].

ringSynchrotron

Ring object, used to access the Lorentz factor gamma.

k_maxint, optional

Number of terms in the image-charge sum for the wake function. The default is 100.

Attributes

thresholdfloat

Shielding threshold in the parallel plates model in [Hz].

References

[1] : Agoh, T., and K. Yokoya. “Calculation of coherent synchrotron radiation using mesh.” Physical Review Special Topics-Accelerators and Beams 7.5 (2004): 054403. [2] : J. B. Murphy, S. Krinsky, and R. L. Gluckstern, Longitudinal wakefield for an electron moving on a circular orbit, Particle Accelerators 57, 9 (1997).

__init__(time: ndarray[tuple[int, ...], dtype[_ScalarType_co]], frequency: ndarray[tuple[int, ...], dtype[_ScalarType_co]], length: float, radius: float, distance: float, ring: Synchrotron, k_max: int = 100)

property threshold: float

Shielding threshold in the parallel plates model in [Hz].

LongitudinalImpedance(frequency: ndarray[tuple[int, ...], dtype[_ScalarType_co]], tol: float = 1e-05) → ndarray[tuple[int, ...], dtype[_ScalarType_co]]

Compute the CSR impedance using the perfectly conducting parallel plates steady-state model. The impedance includes both the free space contribution and the shielding effect from the plates.

Impedance is computed using Eq. (A1) of [1].

Parameters

frequencyfloat array

Frequency in [Hz].

tolfloat, optinal

Desired maximum final error on sum_func.

Returns

Zlcomplex array

Longitudinal impedance in [ohm].

References

[1] : Agoh, T., and K. Yokoya. “Calculation of coherent synchrotron radiation using mesh.” Physical Review Special Topics-Accelerators and Beams 7.5 (2004): 054403.

sum_func(p: int, k: float) → complex

Utility function for LongitudinalImpedance.

Parameters

p : int k : float

Returns

sum_value : mpc

_G2(s: ndarray[tuple[int, ...], dtype[_ScalarType_co]], k_max: int = 100) → ndarray[tuple[int, ...], dtype[_ScalarType_co]]

Normalised Green’s function for the parallel plates CSR wakefunction. Implements Eq. (5.22b) of [1].

Parameters

sfloat array

Longitudinal distance in [m].

Returns

G2float array

Green’s function.

References

[1] : J. B. Murphy, S. Krinsky, and R. L. Gluckstern, Longitudinal wakefield for an electron moving on a circular orbit, Particle Accelerators 57, 9 (1997).

LongitudinalWakeFunction(time: ndarray[tuple[int, ...], dtype[_ScalarType_co]], k_max: int = 100) → ndarray[tuple[int, ...], dtype[_ScalarType_co]]

Compute the parallel-plates shielding correction to the CSR wake function following Eqs. (5.21-5.22) of [1].

Computes only the shielding correction; the free-space term is not included. In the WakeField model the free-space contribution is provided automatically by the Wcsr component.

Parameters

timefloat array

Time in [s].

k_maxint, optional

Number of terms in the G2 sum. The default is 100.

Returns

Wlfloat array

Longitudinal wake function (shielding correction only) in [V/C].

References

[1] : J. B. Murphy, S. Krinsky, and R. L. Gluckstern, Longitudinal wakefield for an electron moving on a circular orbit, Particle Accelerators 57, 9 (1997).