Resistive Wall Wakefields¶

This module provides analytical models for short-range resistive wall wakefields in accelerator beam pipes, based on the approach described in SLAC-PUB-10707.

Physics Background¶

When a relativistic charged particle travels through a conducting beam pipe, it induces image currents in the pipe walls. Due to the finite conductivity of the wall material, these currents penetrate into the conductor and dissipate energy, creating a wakefield that acts on trailing particles.

At high frequencies (short distances), the AC conductivity of metals deviates from DC behavior due to the Drude relaxation time $\tau$:

$$\sigma(\omega) = \frac{\sigma_0}{1 - i\omega\tau}$$

This frequency dependence causes the wakefield to oscillate and decay exponentially behind the source particle.

Characteristic Scales¶

$s_0$: Characteristic length scale where the wake transitions from the $1/\sqrt{z}$ DC behavior to oscillatory AC behavior: $$s_0 = \left( \frac{2a^2}{Z_0 \sigma_0} \right)^{1/3}$$
$W_0$: Characteristic wake amplitude: $W_0 = c Z_0 / (\pi a^2)$ (round geometry)

Classes¶

Class	Description	Speed	Accuracy
`ResistiveWallWakefield`	Full impedance model using FFT convolution	Slower	High
`ResistiveWallPseudomode`	Damped sinusoid approximation	Fast (O(N))	~10-20% error

Key Methods¶

Method	Description
`wake(z)`	Evaluate wakefield $W(z)$ at position(s) $z$ [V/C/m]
`impedance(k)`	Evaluate impedance $Z(k)$ at wavenumber(s) $k$ [Ω/m]
`convolve_density(ρ, dz, plot=False)`	Compute integrated wake from charge density
`particle_kicks(z, weight)`	Compute per-particle momentum kicks [eV/m]
`plot()`	Visualize the wakefield

Use ParticleGroup.apply_wakefield(wake, length) to apply wakefield kicks to particles.

Sign Convention¶

$z < 0$: Behind the source (trailing particle feels the wake)
$z > 0$: Ahead of the source (causality requires $W = 0$)
Positive $W(z)$ corresponds to energy loss (decelerating field)
Tail particles lose more energy than head particles

Quick Start¶

Create a wakefield, evaluate it, and apply it to particles in just a few lines:

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import numpy as np
import matplotlib.pyplot as plt

from pmd_beamphysics import ParticleGroup
from pmd_beamphysics.wakefields import ResistiveWallWakefield, ResistiveWallPseudomode
from pmd_beamphysics.units import epsilon_0

# Create wakefield from material preset
wake = ResistiveWallWakefield.from_material(
    "copper-slac-pub-10707",
    radius=2.5e-3,  # 2.5 mm pipe radius
    geometry="flat",
)
wake
import numpy as np
import matplotlib.pyplot as plt

from pmd_beamphysics import ParticleGroup
from pmd_beamphysics.wakefields import ResistiveWallWakefield, ResistiveWallPseudomode
from pmd_beamphysics.units import epsilon_0

# Create wakefield from material preset
wake = ResistiveWallWakefield.from_material(
    "copper-slac-pub-10707",
    radius=2.5e-3,  # 2.5 mm pipe radius
    geometry="flat",
)
wake

Out[1]:

ResistiveWallWakefield(radius=0.0025, conductivity=65000000.0, relaxation_time=2.7e-14, geometry='flat', material='copper-slac-pub-10707') → s₀=7.992e-06 m

Visualize the wakefield and impedance¶

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wake.plot_impedance()
wake.plot_impedance()

No description has been provided for this image

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wake.plot(zmax=200e-6, normalized=True)
wake.plot(zmax=200e-6, normalized=True)

Apply to particles¶

Load a particle distribution and visualize the wakefield kicks. Note: particles must be at constant time (use drift_to_t()) for proper wake calculation.

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P = ParticleGroup("../data/bmad_particles2.h5")
P.drift_to_t()  # Align particles at constant time
P.wakefield_plot(wake)
P = ParticleGroup("../data/bmad_particles2.h5")
P.drift_to_t()  # Align particles at constant time
P.wakefield_plot(wake)

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# Apply wakefield over 10 m of beam pipe (returns a modified copy)
P_after = P.apply_wakefield(wake, length=10.0)

# Mean energy change [keV] - negative means energy loss
(P_after["mean_energy"] - P["mean_energy"]) / 1e3
# Apply wakefield over 10 m of beam pipe (returns a modified copy)
P_after = P.apply_wakefield(wake, length=10.0)

# Mean energy change [keV] - negative means energy loss
(P_after["mean_energy"] - P["mean_energy"]) / 1e3

Out[5]:

np.float64(-1376.4140474233627)

Setup¶

Creating Wakefield Models¶

Direct Construction¶

Specify the pipe geometry and material properties directly:

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# Accurate impedance-based model
wake = ResistiveWallWakefield(
    radius=2.5e-3,  # Pipe radius [m] (or half-gap for flat)
    conductivity=6.5e7,  # DC conductivity σ₀ [S/m]
    relaxation_time=27e-15,  # Drude relaxation time τ [s]
    geometry="round",  # "round" or "flat"
)
wake
# Accurate impedance-based model
wake = ResistiveWallWakefield(
    radius=2.5e-3,  # Pipe radius [m] (or half-gap for flat)
    conductivity=6.5e7,  # DC conductivity σ₀ [S/m]
    relaxation_time=27e-15,  # Drude relaxation time τ [s]
    geometry="round",  # "round" or "flat"
)
wake

Out[6]:

ResistiveWallWakefield(radius=0.0025, conductivity=65000000.0, relaxation_time=2.7e-14, geometry='round', material='copper-slac-pub-10707') → s₀=7.992e-06 m

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# Fast pseudomode approximation
wake_fast = ResistiveWallPseudomode(
    radius=2.5e-3,
    conductivity=6.5e7,
    relaxation_time=27e-15,
    geometry="round",
)
wake_fast
# Fast pseudomode approximation
wake_fast = ResistiveWallPseudomode(
    radius=2.5e-3,
    conductivity=6.5e7,
    relaxation_time=27e-15,
    geometry="round",
)
wake_fast

Out[7]:

ResistiveWallPseudomode(radius=0.0025, conductivity=65000000.0, relaxation_time=2.7e-14, geometry='round', material='copper-slac-pub-10707') → s₀=7.992e-06 m, Γ=1.013, k_r=199535.0/m, Q_r=3.55

Material Presets¶

Common materials with known conductivity and relaxation times are available:

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# Available materials (SI units)
ResistiveWallWakefield.MATERIALS
# Available materials (SI units)
ResistiveWallWakefield.MATERIALS

Out[8]:

{'copper-slac-pub-10707': {'conductivity': 65000000.0,
  'relaxation_time': 2.7e-14},
 'copper-genesis4': {'conductivity': 58130000.0, 'relaxation_time': 2.7e-14},
 'aluminum-genesis4': {'conductivity': 35710000.0, 'relaxation_time': 8e-15},
 'aluminum-slac-pub-10707': {'conductivity': 42000000.0,
  'relaxation_time': 8e-15},
 'aluminum-alloy-6061-t6-20C': {'conductivity': 25000000.0,
  'relaxation_time': 8e-15},
 'aluminum-alloy-6063-t6-20C': {'conductivity': 30000000.0,
  'relaxation_time': 8e-15}}

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# Create from material preset
wake_cu = ResistiveWallWakefield.from_material(
    "copper-slac-pub-10707",
    radius=4.5e-3,
    geometry="flat",
)
wake_cu
# Create from material preset
wake_cu = ResistiveWallWakefield.from_material(
    "copper-slac-pub-10707",
    radius=4.5e-3,
    geometry="flat",
)
wake_cu

Out[9]:

ResistiveWallWakefield(radius=0.0045, conductivity=65000000.0, relaxation_time=2.7e-14, geometry='flat', material='copper-slac-pub-10707') → s₀=1.183e-05 m

Key material properties (conductivity [S/m], relaxation time [s], characteristic length $s_0$ [m]):

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wake_cu.conductivity, wake_cu.relaxation_time, wake_cu.s0
wake_cu.conductivity, wake_cu.relaxation_time, wake_cu.s0

Out[10]:

(65000000.0, 2.7e-14, 1.1825977087388154e-05)

Evaluating the Wakefield¶

Wake Function W(z)¶

The wakefield is defined for a trailing particle at position $z$ relative to the source:

$z < 0$: Behind the source (trailing particle feels the wake)
$z > 0$: Ahead of the source (causality requires $W = 0$)

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# Evaluate at single point (10 µm behind source) [V/C/m]
wake.wake(-10e-6)
# Evaluate at single point (10 µm behind source) [V/C/m]
wake.wake(-10e-6)

Out[11]:

-1593077197930949.8

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# Evaluate on array
z_arr = np.linspace(-200e-6, 10e-6, 500)
W_arr = wake.wake(z_arr)

plt.figure(figsize=(8, 4))
plt.plot(-z_arr * 1e6, W_arr * 1e-12)
plt.axvline(0, color="r", ls="--", label="Source position")
plt.xlabel(r"Distance behind source $|z|$ (µm)")
plt.ylabel(r"$W(z)$ (V/pC/m)")
plt.title("Resistive Wall Wakefield")
plt.legend()
plt.xlim(-20, 200)
# Evaluate on array
z_arr = np.linspace(-200e-6, 10e-6, 500)
W_arr = wake.wake(z_arr)

plt.figure(figsize=(8, 4))
plt.plot(-z_arr * 1e6, W_arr * 1e-12)
plt.axvline(0, color="r", ls="--", label="Source position")
plt.xlabel(r"Distance behind source $|z|$ (µm)")
plt.ylabel(r"$W(z)$ (V/pC/m)")
plt.title("Resistive Wall Wakefield")
plt.legend()
plt.xlim(-20, 200)

Out[12]:

(-20.0, 200.0)

Impedance and Wakefield plots¶

The longitudinal impedance as a function of wavenumber $k$:

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wake.plot_impedance()
wake.plot_impedance()

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wake.plot(zmax=200e-6)
wake.plot(zmax=200e-6)

Round vs Flat Geometry¶

The flat (parallel plate) geometry has a higher wake amplitude due to the different boundary conditions:

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params = dict(radius=2.5e-3, conductivity=6.5e7, relaxation_time=27e-15)

wake_round = ResistiveWallWakefield(**params, geometry="round")
wake_flat = ResistiveWallWakefield(**params, geometry="flat")

z = np.linspace(-200e-6, 0, 300)

plt.figure(figsize=(8, 4))
plt.plot(-z * 1e6, wake_round.wake(z) * 1e-12, label="Round")
plt.plot(-z * 1e6, wake_flat.wake(z) * 1e-12, "--", label="Flat")
plt.xlabel(r"Distance behind source $|z|$ (µm)")
plt.ylabel(r"$W(z)$ (V/pC/m)")
plt.title("Round vs Flat Geometry")
plt.legend()
params = dict(radius=2.5e-3, conductivity=6.5e7, relaxation_time=27e-15)

wake_round = ResistiveWallWakefield(**params, geometry="round")
wake_flat = ResistiveWallWakefield(**params, geometry="flat")

z = np.linspace(-200e-6, 0, 300)

plt.figure(figsize=(8, 4))
plt.plot(-z * 1e6, wake_round.wake(z) * 1e-12, label="Round")
plt.plot(-z * 1e6, wake_flat.wake(z) * 1e-12, "--", label="Flat")
plt.xlabel(r"Distance behind source $|z|$ (µm)")
plt.ylabel(r"$W(z)$ (V/pC/m)")
plt.title("Round vs Flat Geometry")
plt.legend()

Out[15]:

<matplotlib.legend.Legend at 0x7f5507c0b620>

Convolution with Charge Density¶

The convolve_density method computes the integrated wake potential from a charge density distribution:

$$V(z) = \int_{z}^{\infty} \rho(z') \, W(z - z') \, dz'$$

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# Create a Gaussian bunch density
n_bins = 500
dz = 1e-6  # 1 µm spacing
z = np.arange(n_bins) * dz

Q_total = 100e-12  # 100 pC
sigma_z = 10e-6  # 10 µm RMS
z0 = z[n_bins // 2]  # Center

density = (
    Q_total / (sigma_z * np.sqrt(2 * np.pi)) * np.exp(-0.5 * ((z - z0) / sigma_z) ** 2)
)

# Total charge [pC]
np.sum(density) * dz * 1e12
# Create a Gaussian bunch density
n_bins = 500
dz = 1e-6  # 1 µm spacing
z = np.arange(n_bins) * dz

Q_total = 100e-12  # 100 pC
sigma_z = 10e-6  # 10 µm RMS
z0 = z[n_bins // 2]  # Center

density = (
    Q_total / (sigma_z * np.sqrt(2 * np.pi)) * np.exp(-0.5 * ((z - z0) / sigma_z) ** 2)
)

# Total charge [pC]
np.sum(density) * dz * 1e12

Out[16]:

np.float64(100.00000000000001)

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# Compute integrated wake with built-in plotting
wake.convolve_density(density, dz, plot=True)
# Compute integrated wake with built-in plotting
wake.convolve_density(density, dz, plot=True)

Out[17]:

array([-3.24323297e+002, -3.27984816e+002, -3.33798033e+002,
       -3.41724783e+002, -3.51633980e+002, -3.63301633e+002,
       -3.76414898e+002, -3.90580211e+002, -4.05335407e+002,
       -4.20165551e+002, -4.34522010e+002, -4.47844147e+002,
       -4.59582857e+002, -4.69225029e+002, -4.76317952e+002,
       -4.80492586e+002, -4.81484635e+002, -4.79152370e+002,
       -4.73490234e+002, -4.64637393e+002, -4.52880540e+002,
       -4.38650489e+002, -4.22512330e+002, -4.05149181e+002,
       -3.87339835e+002, -3.69930915e+002, -3.53804401e+002,
       -3.39841649e+002, -3.28885242e+002, -3.21700204e+002,
       -3.18936205e+002, -3.21092450e+002, -3.28486950e+002,
       -3.41231749e+002, -3.59215560e+002, -3.82095002e+002,
       -4.09295345e+002, -4.40021308e+002, -4.73278050e+002,
       -5.07902082e+002, -5.42601348e+002, -5.76003306e+002,
       -6.06709434e+002, -6.33354155e+002, -6.54665949e+002,
       -6.69528096e+002, -6.77036440e+002, -6.76551463e+002,
       -6.67742086e+002, -6.50618801e+002, -6.25554035e+002,
       -5.93288107e+002, -5.54919632e+002, -5.11879841e+002,
       -4.65890953e+002, -4.18909424e+002, -3.73055612e+002,
       -3.30532067e+002, -2.93533295e+002, -2.64150378e+002,
       -2.44274256e+002, -2.35501783e+002, -2.39048790e+002,
       -2.55674324e+002, -2.85620053e+002, -3.28568345e+002,
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       -6.04000771e+002, -6.87566110e+002, -7.71060265e+002,
       -8.51050415e+002, -9.24048843e+002, -9.86660511e+002,
       -1.03573561e+003, -1.06852078e+003, -1.08280243e+003,
       -1.07703538e+003, -1.05045044e+003, -1.00313503e+003,
       -9.36081565e+002, -8.51199696e+002, -7.51289623e+002,
       -6.39975235e+002, -5.21597574e+002, -4.01070782e+002,
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       -5.14471091e+000,  4.61414171e+001,  6.95194317e+001,
        6.20304466e+001,  2.18667480e+001, -5.14800106e+001,
       -1.57117799e+002, -2.92707632e+002, -4.54476742e+002,
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       -1.11244809e+003, -8.28875187e+002, -5.28459444e+002,
       -2.23541927e+002,  7.25833788e+001,  3.46186424e+002,
        5.83668276e+002,  7.72157154e+002,  9.00109288e+002,
        9.57888914e+002,  9.38301031e+002,  8.37051011e+002,
        6.53106727e+002,  3.88941585e+002,  5.06407146e+001,
       -3.52142614e+002, -8.06387587e+002, -1.29606433e+003,
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       -3.21549477e+003, -3.57949257e+003, -3.85661021e+003,
       -4.03013379e+003, -4.08682528e+003, -4.01770710e+003,
       -3.81871027e+003, -3.49114968e+003, -3.04199540e+003,
       -2.48391594e+003, -1.83507763e+003, -1.11869404e+003,
       -3.62329601e+002,  4.03027299e+002,  1.14409577e+003,
        1.82660190e+003,  2.41670025e+003,  2.88246335e+003,
        3.19537890e+003,  3.33179035e+003,  3.27421511e+003,
        3.01247590e+003,  2.54458499e+003,  1.87732794e+003,
        1.02650336e+003,  1.67873238e+001, -1.11879446e+003,
       -2.33978790e+003, -3.59960922e+003, -4.84718789e+003,
       -6.02890379e+003, -7.09075255e+003, -7.98065951e+003,
       -8.65085110e+003, -9.06018460e+003, -9.17633303e+003,
       -8.97772173e+003, -8.45511794e+003, -7.61278322e+003,
       -6.46911254e+003, -5.05670097e+003, -3.42180027e+003,
       -1.62315181e+003,  2.69791731e+002,  2.17921610e+003,
        4.02179984e+003,  5.71198048e+003,  7.16551642e+003,
        8.30320825e+003,  9.05462844e+003,  9.36169839e+003,
        9.18194893e+003,  8.49130383e+003,  7.28623691e+003,
        5.58517136e+003,  3.42901443e+003,  8.80751821e+002,
       -1.97593801e+003, -5.03904940e+003, -8.19155217e+003,
       -1.13055459e+004, -1.42471375e+004, -1.68818763e+004,
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       -7.48917269e+003, -3.01988449e+003,  1.66699327e+003,
        6.37534963e+003,  1.08956361e+004,  1.50129999e+004,
        1.85161140e+004,  2.12063568e+004,  2.29069638e+004,
        2.34717443e+004,  2.27929588e+004,  2.08079627e+004,
        1.75042532e+004,  1.29226109e+004,  7.15809476e+003,
        3.58733017e+002, -7.27814579e+003, -1.55118937e+004,
       -2.40666733e+004, -3.26414636e+004, -4.09215635e+004,
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       -6.50325162e+004, -6.75269550e+004, -6.82527696e+004,
       -6.71352324e+004, -6.41665986e+004, -5.94068926e+004,
       -5.29816140e+004, -4.50764477e+004, -3.59292251e+004,
       -2.58195412e+004, -1.50565656e+004, -3.96568954e+003,
        7.12528562e+003,  1.78996910e+004,  2.80645647e+004,
        3.73616271e+004,  4.55761385e+004,  5.25432590e+004,
        5.81516978e+004,  6.23446053e+004,  6.51178306e+004,
        6.65158197e+004,  6.66255494e+004,  6.55689851e+004,
        6.34945998e+004,  6.05685027e+004,  5.69656968e+004,
        5.28619256e+004,  4.84264839e+004,  4.38162626e+004,
        3.91711923e+004,  3.46111409e+004,  3.02342239e+004,
        2.61164075e+004,  2.23122189e+004,  1.88563452e+004,
        1.57658797e+004,  1.30429781e+004,  1.06777050e+004,
        8.65088021e+003,  6.93677066e+003,  5.50551595e+003,
        4.32521348e+003,  3.36362680e+003,  2.58951158e+003,
        1.97357810e+003,  1.48912671e+003,  1.11240384e+003,
        8.22730835e+002,  6.02458465e+002,  4.36796156e+002,
        3.13559593e+002,  2.22872883e+002,  1.56854018e+002,
        1.09304998e+002,  7.54213608e+001,  5.15302350e+001,
        3.48615124e+001,  2.33533476e+001,  1.54907681e+001,
        1.01746476e+001,  6.61743707e+000,  4.26172493e+000,
        2.71773461e+000,  1.71615054e+000,  1.07307703e+000,
        6.64406552e-001,  4.07347038e-001,  2.47299305e-001,
        1.48665027e-001,  8.84956918e-002,  5.21629998e-002,
        3.04459652e-002,  1.75963773e-002,  1.00703051e-002,
        5.70672965e-003,  3.20226075e-003,  1.77930414e-003,
        9.78964937e-004,  5.33343314e-004,  2.87719063e-004,
        1.53692177e-004,  8.12934383e-005,  4.25773938e-005,
        2.20811531e-005,  1.13392086e-005,  5.76582284e-006,
        2.90306406e-006,  1.44733126e-006,  7.14487256e-007,
        3.49249248e-007,  1.69040258e-007,  8.10135612e-008,
        3.84447857e-008,  1.80646085e-008,  8.40485647e-009,
        3.87205879e-009,  1.76629298e-009,  7.97796219e-010,
        3.56803344e-010,  1.58005738e-010,  6.92824282e-011,
        3.00800774e-011,  1.29312266e-011,  5.50433361e-012,
        2.31992188e-012,  9.68155029e-013,  4.00054114e-013,
        1.63679478e-013,  6.63087366e-014,  2.65979210e-014,
        1.05639039e-014,  4.15432438e-015,  1.61761438e-015,
        6.23659650e-016,  2.38077187e-016,  8.99880511e-017,
        3.36781550e-017,  1.24797987e-017,  4.57891075e-018,
        1.66345696e-018,  5.98349916e-019,  2.13104527e-019,
        7.51490324e-020,  2.62389772e-020,  9.07115780e-021,
        3.10506276e-021,  1.05237233e-021,  3.53150137e-022,
        1.17338305e-022,  3.86020419e-023,  1.25739099e-023,
        4.05526824e-024,  1.29496367e-024,  4.09434022e-025,
        1.28173490e-025,  3.97283046e-026,  1.21923945e-026,
        3.70479978e-027,  1.11462008e-027,  3.32028221e-028,
        9.79284272e-029,  2.85974853e-029,  8.26859728e-030,
        2.36712077e-030,  6.70954747e-031,  1.88299836e-031,
        5.23227088e-032,  1.43950685e-032,  3.92121059e-033,
        1.05757018e-033,  2.82410288e-034,  7.46678777e-035,
        1.95464898e-035,  5.06623259e-036,  1.30011735e-036,
        3.30339708e-037,  8.31035384e-038,  2.06994345e-038,
        5.10478432e-039,  1.24645325e-039,  3.01338073e-040,
        7.21292086e-041,  1.70941384e-041,  4.01108530e-042,
        9.31869202e-043,  2.14351314e-043,  4.88174655e-044,
        1.10078360e-044,  2.45757101e-045,  5.43234538e-046,
        1.18890077e-046,  2.57620582e-047,  5.52703214e-048,
        1.17403133e-048,  2.46912700e-049,  5.14141901e-050,
        1.05998126e-050,  2.16365882e-051,  4.37274924e-052,
        8.74974430e-053,  1.73344850e-053,  3.40017385e-054,
        6.60337103e-055,  1.26970983e-055,  2.41722614e-056,
        4.55621184e-057,  8.50284329e-058,  1.57107855e-058,
        2.87412003e-059,  5.20577031e-060,  9.33550950e-061,
        1.65753916e-061,  2.91381614e-062,  5.07145877e-063,
        8.73928483e-064,  1.49104573e-064,  2.51870871e-065,
        4.21246859e-066,  6.97536598e-067,  1.14358581e-067,
        1.85627277e-068,  2.98322464e-069,  4.74680177e-070,
        7.47802754e-071,  1.16638977e-071,  1.80123708e-072,
        2.75402858e-073,  4.16904184e-074,  6.24847692e-075,
        9.27218427e-076,  1.36225926e-076,  1.98155978e-077,
        2.85380404e-078,  4.06921416e-079,  5.74468533e-080,
        8.02954823e-081,  1.11118154e-081,  1.52246684e-082,
        2.06528246e-083,  2.77382936e-084,  3.68848965e-085,
        4.85607990e-086,  6.32982036e-087,  8.16892928e-088,
        1.04377517e-088,  1.32043414e-089,  1.65384336e-090,
        2.05087757e-091,  2.51798256e-092,  3.06078775e-093,
        3.68367337e-094,  4.38931058e-095,  5.17819857e-096,
        6.04822828e-097,  6.99430604e-098,  8.00807199e-099,
        9.07774721e-100,  1.01881394e-100,  1.13208296e-101,
        1.24545520e-102,  1.35657669e-103,  1.46294124e-104,
        1.56198050e-105,  1.65116484e-106,  1.72810954e-107,
        1.79068014e-108,  1.83709068e-109,  1.86598826e-110,
        1.87651858e-111,  1.86836786e-112,  1.84177843e-113,
        1.79753669e-114,  1.73693452e-115,  1.66170662e-116,
        1.57394803e-117,  1.47601740e-118,  1.37043216e-119,
        1.25976198e-120,  1.14652694e-121,  1.03310572e-122,
        9.21658262e-124,  8.14066063e-125,  7.11891465e-126,
        6.16354782e-127,  5.28309963e-128,  4.48009876e-129,
        3.72416859e-130,  2.66809449e-131])

Particle Methods¶

Computing Per-Particle Kicks¶

The particle_kicks method computes the wakefield-induced momentum change for each particle:

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# Load particles
P = ParticleGroup("../data/bmad_particles2.h5")
P.drift_to_t()  # Align at constant time
P
# Load particles
P = ParticleGroup("../data/bmad_particles2.h5")
P.drift_to_t()  # Align at constant time
P

Out[18]:

<ParticleGroup with 100000 particles at 0x7f5507b4c190>

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# Compute kicks [eV/m] and visualize
kicks = wake.particle_kicks(P.z, P.weight)
P.wakefield_plot(wake)
# Compute kicks [eV/m] and visualize
kicks = wake.particle_kicks(P.z, P.weight)
P.wakefield_plot(wake)

Applying to Particles¶

Use ParticleGroup.apply_wakefield() to apply the kicks:

Apply wakefield over 10 m (returns a copy by default):

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P_after = P.apply_wakefield(wake, length=10.0)

# Change in mean pz [MeV/c]
(P_after["mean_pz"] - P["mean_pz"]) * 1e-6
P_after = P.apply_wakefield(wake, length=10.0)

# Change in mean pz [MeV/c]
(P_after["mean_pz"] - P["mean_pz"]) * 1e-6

Out[20]:

np.float64(-1.16122149968338)

Or modify in-place with inplace=True:

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P_copy = P.copy()
P_copy.apply_wakefield(wake, length=10.0, inplace=True)
P_copy = P.copy()
P_copy.apply_wakefield(wake, length=10.0, inplace=True)

ParticleGroup Wakefield Plot¶

A convenience method for visualizing the wakefield effect on a particle distribution:

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# Larger figure for detail
P.wakefield_plot(wake, figsize=(12, 4))
# Larger figure for detail
P.wakefield_plot(wake, figsize=(12, 4))

Comparing Models: Accurate vs Fast¶

Two implementations are available with different speed/accuracy trade-offs:

ResistiveWallWakefield: Uses FFT-based impedance integration. More accurate, especially near the first oscillation peak.
ResistiveWallPseudomode: Uses a damped sinusoid fit from polynomial approximations. Very fast (O(N) for particle kicks), with ~10-20% error.

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# Create both models with same parameters
params = dict(
    radius=2.5e-3,
    conductivity=6.5e7,
    relaxation_time=27e-15,
    geometry="round",
)

wake_accurate = ResistiveWallWakefield(**params)
wake_fast = ResistiveWallPseudomode(**params)
# Create both models with same parameters
params = dict(
    radius=2.5e-3,
    conductivity=6.5e7,
    relaxation_time=27e-15,
    geometry="round",
)

wake_accurate = ResistiveWallWakefield(**params)
wake_fast = ResistiveWallPseudomode(**params)

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wake_accurate
wake_accurate

Out[24]:

ResistiveWallWakefield(radius=0.0025, conductivity=65000000.0, relaxation_time=2.7e-14, geometry='round', material='copper-slac-pub-10707') → s₀=7.992e-06 m

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wake_fast
wake_fast

Out[25]:

ResistiveWallPseudomode(radius=0.0025, conductivity=65000000.0, relaxation_time=2.7e-14, geometry='round', material='copper-slac-pub-10707') → s₀=7.992e-06 m, Γ=1.013, k_r=199535.0/m, Q_r=3.55

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# Compare wake functions
z = np.linspace(-200e-6, 0, 300)

W_accurate = wake_accurate.wake(z)
W_fast = wake_fast.wake(z)

fig, axes = plt.subplots(2, 1, figsize=(10, 6), sharex=True)

ax = axes[0]
ax.plot(-z * 1e6, W_accurate * 1e-12, label="ResistiveWallWakefield (accurate)")
ax.plot(-z * 1e6, W_fast * 1e-12, "--", label="ResistiveWallPseudomode (fast)")
ax.set_ylabel(r"$W(z)$ (V/pC/m)")
ax.legend()
ax.set_title("Wakefield Comparison")

ax = axes[1]
rel_diff = (W_fast - W_accurate) / np.abs(W_accurate).max() * 100
ax.plot(-z * 1e6, rel_diff)
ax.set_xlabel(r"Distance behind source $|z|$ (µm)")
ax.set_ylabel("Relative difference (%)")
ax.set_title("Pseudomode Error")
ax.axhline(0, color="k", lw=0.5)

plt.tight_layout()
# Compare wake functions
z = np.linspace(-200e-6, 0, 300)

W_accurate = wake_accurate.wake(z)
W_fast = wake_fast.wake(z)

fig, axes = plt.subplots(2, 1, figsize=(10, 6), sharex=True)

ax = axes[0]
ax.plot(-z * 1e6, W_accurate * 1e-12, label="ResistiveWallWakefield (accurate)")
ax.plot(-z * 1e6, W_fast * 1e-12, "--", label="ResistiveWallPseudomode (fast)")
ax.set_ylabel(r"$W(z)$ (V/pC/m)")
ax.legend()
ax.set_title("Wakefield Comparison")

ax = axes[1]
rel_diff = (W_fast - W_accurate) / np.abs(W_accurate).max() * 100
ax.plot(-z * 1e6, rel_diff)
ax.set_xlabel(r"Distance behind source $|z|$ (µm)")
ax.set_ylabel("Relative difference (%)")
ax.set_title("Pseudomode Error")
ax.axhline(0, color="k", lw=0.5)

plt.tight_layout()

Performance Comparison¶

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%%timeit -n 10 -r 3
# Accurate model: convolve_density
_ = wake_accurate.convolve_density(density, dz)
%%timeit -n 10 -r 3
# Accurate model: convolve_density
_ = wake_accurate.convolve_density(density, dz)

2.2 ms ± 78.3 μs per loop (mean ± std. dev. of 3 runs, 10 loops each)

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%%timeit -n 100 -r 3
# Fast model: convolve_density
_ = wake_fast.convolve_density(density, dz)
%%timeit -n 100 -r 3
# Fast model: convolve_density
_ = wake_fast.convolve_density(density, dz)

160 μs ± 6.2 μs per loop (mean ± std. dev. of 3 runs, 100 loops each)

Pseudomode Properties¶

ResistiveWallPseudomode models the wakefield as a damped sinusoid:

$$W(z) = A \cdot e^{d \cdot z} \cdot \sin(k_r z + \phi)$$

The parameters are computed from polynomial fits to SLAC-PUB-10707 data:

Characteristic scales ($s_0$ [m], $\Gamma$, $k_r$ [1/m], $Q_r$):

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wake_fast.s0, wake_fast.Gamma, wake_fast.kr, wake_fast.Qr
wake_fast.s0, wake_fast.Gamma, wake_fast.kr, wake_fast.Qr

Out[29]:

(7.991997937835282e-06,
 1.012812619442749,
 np.float64(199535.03396550965),
 np.float64(3.5452850664129434))

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# Access the underlying pseudomode(s) - each has amplitude A, decay d, wavenumber k, phase phi
wake_fast.modes
# Access the underlying pseudomode(s) - each has amplitude A, decay d, wavenumber k, phase phi
wake_fast.modes

Out[30]:

[Pseudomode(A=5752033143156176.0, d=np.float64(28140.9012572571), k=np.float64(199535.03396550965), phi=1.5707963267948966)]

Bmad Export¶

ResistiveWallPseudomode can export to Bmad format for use in external tracking codes:

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wake_fast.to_bmad()
wake_fast.to_bmad()

Out[31]:

'! AC Resistive wall wakefield\n! Adapted from SLAC-PUB-10707\n!    Material        : copper-slac-pub-10707\n!    Conductivity    : 65000000.0 S/m\n!    Relaxation time : 2.7e-14 s\n!    Geometry        : round\n!    Radius          : 0.0025 m\n!    s₀              : 7.991997937835282e-06  m\n!    Γ               : 1.012812619442749 \n! sr_wake =  \n{z_scale=1, amp_scale=1, scale_with_length=True, z_max=100,\nlongitudinal = {5752033143156176.0, 28140.9012572571, 199535.03396550965, 0.25, none}}\n'

Validation Against SLAC-PUB-10707¶

Compare our implementation against digitized data from the original paper:

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def validate_against_slacpub(figure_num):
    """Compare against digitized SLAC-PUB-10707 figures."""
    geometry = "round" if figure_num == 4 else "flat"
    radius = 2.5e-3

    # Load digitized data
    data = np.loadtxt(
        f"../data/SLAC-PUB-10707-digitized-Fig{figure_num}-AC-Cu.csv", delimiter=","
    )
    z_ref = data[:, 0] * 1e-6  # µm → m
    # Convert from paper's CGS units
    W_ref = data[:, 1] * 4 / radius**2 / (4 * np.pi * epsilon_0)

    # Create our model
    wake = ResistiveWallWakefield.from_material(
        "copper-slac-pub-10707",
        radius=radius,
        geometry=geometry,
    )

    z = np.linspace(0, 300e-6, 300)
    W = wake.wake(-z)

    plt.figure(figsize=(8, 4))
    plt.plot(z * 1e6, W * 1e-12, label="ResistiveWallWakefield")
    plt.plot(
        z_ref * 1e6, W_ref * 1e-12, "o", ms=4, label=f"SLAC-PUB-10707 Fig. {figure_num}"
    )
    plt.xlabel(r"Distance behind source $|z|$ (µm)")
    plt.ylabel(r"$W(z)$ (V/pC/m)")
    plt.title(f"Validation: {geometry.title()} Copper Pipe")
    plt.legend()


validate_against_slacpub(4)  # Round geometry
def validate_against_slacpub(figure_num):
    """Compare against digitized SLAC-PUB-10707 figures."""
    geometry = "round" if figure_num == 4 else "flat"
    radius = 2.5e-3

    # Load digitized data
    data = np.loadtxt(
        f"../data/SLAC-PUB-10707-digitized-Fig{figure_num}-AC-Cu.csv", delimiter=","
    )
    z_ref = data[:, 0] * 1e-6  # µm → m
    # Convert from paper's CGS units
    W_ref = data[:, 1] * 4 / radius**2 / (4 * np.pi * epsilon_0)

    # Create our model
    wake = ResistiveWallWakefield.from_material(
        "copper-slac-pub-10707",
        radius=radius,
        geometry=geometry,
    )

    z = np.linspace(0, 300e-6, 300)
    W = wake.wake(-z)

    plt.figure(figsize=(8, 4))
    plt.plot(z * 1e6, W * 1e-12, label="ResistiveWallWakefield")
    plt.plot(
        z_ref * 1e6, W_ref * 1e-12, "o", ms=4, label=f"SLAC-PUB-10707 Fig. {figure_num}"
    )
    plt.xlabel(r"Distance behind source $|z|$ (µm)")
    plt.ylabel(r"$W(z)$ (V/pC/m)")
    plt.title(f"Validation: {geometry.title()} Copper Pipe")
    plt.legend()


validate_against_slacpub(4)  # Round geometry

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validate_against_slacpub(8)  # Flat geometry
validate_against_slacpub(8)  # Flat geometry

Summary¶

Feature	`ResistiveWallWakefield`	`ResistiveWallPseudomode`
Method	FFT-based impedance	Damped sinusoid fit
Accuracy	High	Good (~10-20% error)
Speed	Moderate	Very fast
Impedance access	✓	✓ (analytical)
Bmad export	✗	✓
Best for	General use, validation	Production tracking

Both classes share the same API:

wake(z) - Evaluate wakefield
impedance(k) - Evaluate impedance
convolve_density(density, dz, plot=False) - Density convolution
particle_kicks(z, weight) - Per-particle kicks
plot() - Visualize wakefield
plot_impedance() - Visualize impedance
from_material() - Create from preset

Use ParticleGroup.apply_wakefield(wake, length) to apply kicks to particles.