diff --git a/mbtrack2/impedance/resistive_wall.py b/mbtrack2/impedance/resistive_wall.py
index 758f696285bae1750348b074670b515c09530fc7..91fa7e1e1093fc4fcb2b8f5929b852d2dcb4b491 100644
--- a/mbtrack2/impedance/resistive_wall.py
+++ b/mbtrack2/impedance/resistive_wall.py
@@ -13,7 +13,7 @@ from mbtrack2.tracking.particles_electromagnetic_fields import _wofz
def skin_depth(frequency, rho, mu_r=1, epsilon_r=1):
"""
General formula for the skin depth.
-
+
Parameters
----------
frequency : array of float
@@ -24,12 +24,12 @@ def skin_depth(frequency, rho, mu_r=1, epsilon_r=1):
Relative magnetic permeability.
epsilon_r : float, optional
Relative electric permittivity.
-
+
Returns
-------
delta : array of float
Skin depth in [m].
-
+
"""
delta = (np.sqrt(2 * rho / (np.abs(2 * np.pi * frequency) * mu_r * mu_0)) *
@@ -43,51 +43,46 @@ def skin_depth(frequency, rho, mu_r=1, epsilon_r=1):
class CircularResistiveWall(WakeField):
"""
Resistive wall WakeField element for a circular beam pipe.
-
+
Impedance from approximated formulas from Eq. (2.77) of Chao book [1].
Wake function formulas from [2, 3] and the fundamental theorem of
beam loading from [4].
-
+
Parameters
----------
time : array of float
Time points where the wake function will be evaluated in [s].
frequency : array of float
Frequency points where the impedance will be evaluated in [Hz].
- length : float
+ length : float
Beam pipe length in [m].
rho : float
Resistivity in [ohm.m].
- radius : float
+ radius : float
Beam pipe radius in [m].
exact : bool, optional
- If False, approxmiated formulas are used for the wake function
+ If False, approxmiated formulas are used for the wake function
computations.
-
+ The default is True.
+
References
----------
- [1] : Chao, A. W. (1993). Physics of collective beam instabilities in high
+ [1] : Chao, A. W. (1993). Physics of collective beam instabilities in high
energy accelerators. Wiley.
[2] : Ivanyan, Mikayel I., and Vasili M. Tsakanov. "Analytical treatment of
resistive wake potentials in round pipes." Nuclear Instruments and Methods
in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment 522, no. 3 (2004): 223-229.
- [3] : Skripka, Galina, et al. "Simultaneous computation of intrabunch and
- interbunch collective beam motions in storage rings." Nuclear Instruments
- and Methods in Physics Research Section A: Accelerators, Spectrometers,
+ [3] : Skripka, Galina, et al. "Simultaneous computation of intrabunch and
+ interbunch collective beam motions in storage rings." Nuclear Instruments
+ and Methods in Physics Research Section A: Accelerators, Spectrometers,
Detectors and Associated Equipment 806 (2016): 221-230.
[4] : Zotter, Bruno W., and Semyon A. Kheifets (1998). Impedances and wakes
in high-energy particle accelerators. World Scientific.
"""
- def __init__(self,
- time,
- frequency,
- length,
- rho,
- radius,
- exact=True):
+ def __init__(self, time, frequency, length, rho, radius, exact=True):
super().__init__()
self.length = length
@@ -127,18 +122,18 @@ class CircularResistiveWall(WakeField):
def LongitudinalWakeFunction(self, time, exact=True):
"""
- Compute the longitudinal wake function of a circular resistive wall
+ Compute the longitudinal wake function of a circular resistive wall
using Eq. (11), of [1], or approxmiated expression Eq. (24), of [2].
The approxmiated expression is valid if the time is large compared
to the characteristic time t0.
-
+
Eq. (11) in [1] is completely identical to Eq. (22) in [2].
The real parts of the last two terms of Eq. (11) in [1] are the same,
and the imaginary parts have the same magnitude but opposite signs.
Therefore, the former term was doubled, the latter term was eliminated,
and only the real part was taken to speed up the calculation.
-
+
The fundamental theorem of beam loading [3] is applied for the exact
expression of the longitudinal wake function: Wl(0) = Wl(0+)/2.
@@ -147,22 +142,22 @@ class CircularResistiveWall(WakeField):
time : array of float
Time points where the wake function is evaluated in [s].
exact : bool, optional
- If True, the exact expression is used. The default is False.
+ If True, the exact expression is used. The default is True.
Returns
-------
wl : array of float
Longitudinal wake function in [V/C].
-
+
References
----------
[1] : Ivanyan, Mikayel I., and Vasili M. Tsakanov. "Analytical treatment of
resistive wake potentials in round pipes." Nuclear Instruments and Methods
in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment 522, no. 3 (2004): 223-229.
- [2] : Skripka, Galina, et al. "Simultaneous computation of intrabunch and
- interbunch collective beam motions in storage rings." Nuclear Instruments
- and Methods in Physics Research Section A: Accelerators, Spectrometers,
+ [2] : Skripka, Galina, et al. "Simultaneous computation of intrabunch and
+ interbunch collective beam motions in storage rings." Nuclear Instruments
+ and Methods in Physics Research Section A: Accelerators, Spectrometers,
Detectors and Associated Equipment 806 (2016): 221-230.
[3] : Zotter, Bruno W., and Semyon A. Kheifets (1998). Impedances and wakes
in high-energy particle accelerators. World Scientific.
@@ -172,8 +167,7 @@ class CircularResistiveWall(WakeField):
if exact == True:
idx2 = time == 0
idx3 = np.logical_not(np.logical_or(idx1, idx2))
- factor = ( self.Z0 * c / (3 * np.pi * self.radius**2) *
- self.length )
+ factor = (self.Z0 * c / (3 * np.pi * self.radius**2) * self.length)
if np.any(idx2):
# fundamental theorem of beam loading
wl[idx2] = 3 * factor / 2
@@ -185,11 +179,11 @@ class CircularResistiveWall(WakeField):
def TransverseWakeFunction(self, time, exact=True):
"""
- Compute the transverse wake function of a circular resistive wall
+ Compute the transverse wake function of a circular resistive wall
using Eq. (11), of [1], or approxmiated expression Eq. (26), of [2].
The approxmiated expression is valid if the time is large compared
to the characteristic time t0.
-
+
Eq. (11) in [1] is completely identical to Eq. (25) in [2].
There are typos in both Eq. (11) in [1] and Eq. (25) in [2].
@@ -207,22 +201,22 @@ class CircularResistiveWall(WakeField):
time : array of float
Time points where the wake function is evaluated in [s].
exact : bool, optional
- If True, the exact expression is used. The default is False.
+ If True, the exact expression is used. The default is True.
Returns
-------
wt : array of float
Transverse wake function in [V/C/m].
-
+
References
----------
[1] : Ivanyan, Mikayel I., and Vasili M. Tsakanov. "Analytical treatment of
resistive wake potentials in round pipes." Nuclear Instruments and Methods
in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment 522, no. 3 (2004): 223-229.
- [2] : Skripka, Galina, et al. "Simultaneous computation of intrabunch and
- interbunch collective beam motions in storage rings." Nuclear Instruments
- and Methods in Physics Research Section A: Accelerators, Spectrometers,
+ [2] : Skripka, Galina, et al. "Simultaneous computation of intrabunch and
+ interbunch collective beam motions in storage rings." Nuclear Instruments
+ and Methods in Physics Research Section A: Accelerators, Spectrometers,
Detectors and Associated Equipment 806 (2016): 221-230.
"""
wt = np.zeros_like(time)
@@ -230,9 +224,8 @@ class CircularResistiveWall(WakeField):
if exact == True:
idx2 = time == 0
idx3 = np.logical_not(np.logical_or(idx1, idx2))
- factor = ( (self.Z0 * c**2 * self.t0) /
- (3 * np.pi * self.radius**4) *
- self.length )
+ factor = ((self.Z0 * c**2 * self.t0) /
+ (3 * np.pi * self.radius**4) * self.length)
wt[idx3] = self.__TransWakeExact(time[idx3], factor)
else:
idx2 = np.logical_not(idx1)
@@ -240,41 +233,36 @@ class CircularResistiveWall(WakeField):
return wt
def __LongWakeExact(self, t, factor):
- w1re, _ = _wofz( 0, np.sqrt(2 * t / self.t0) )
- w2re, _ = _wofz( np.cos(np.pi/6) *
- np.sqrt(2 * t / self.t0),
- np.sin(np.pi/6) *
- np.sqrt(2 * t / self.t0) )
-
- wl = factor * ( 4 * np.exp(-1 * t / self.t0) *
- np.cos(np.sqrt(3) * t / self.t0)
- + w1re - 2 * w2re )
+ w1re, _ = _wofz(0, np.sqrt(2 * t / self.t0))
+ w2re, _ = _wofz(
+ np.cos(np.pi / 6) * np.sqrt(2 * t / self.t0),
+ np.sin(np.pi / 6) * np.sqrt(2 * t / self.t0))
+
+ wl = factor * (4 * np.exp(-1 * t / self.t0) *
+ np.cos(np.sqrt(3) * t / self.t0) + w1re - 2*w2re)
return wl
def __TransWakeExact(self, t, factor):
- w1re, _ = _wofz( 0, np.sqrt(2 * t / self.t0) )
- w2re, w2im = _wofz( np.cos(np.pi/6) *
- np.sqrt(2 * t / self.t0),
- np.sin(np.pi/6) *
- np.sqrt(2 * t / self.t0) )
-
- wt = factor * ( 2 * np.exp(-1 * t / self.t0) *
- ( np.sqrt(3) * np.sin(np.sqrt(3) * t / self.t0)
- - np.cos(np.sqrt(3) * t / self.t0) )
- + w1re + 2 * ( np.cos(-np.pi/3) * w2re
- - np.sin(-np.pi/3) * w2im ) )
+ w1re, _ = _wofz(0, np.sqrt(2 * t / self.t0))
+ w2re, w2im = _wofz(
+ np.cos(np.pi / 6) * np.sqrt(2 * t / self.t0),
+ np.sin(np.pi / 6) * np.sqrt(2 * t / self.t0))
+
+ wt = factor * (2 * np.exp(-1 * t / self.t0) *
+ (np.sqrt(3) * np.sin(np.sqrt(3) * t / self.t0) -
+ np.cos(np.sqrt(3) * t / self.t0)) + w1re + 2 *
+ (np.cos(-np.pi / 3) * w2re - np.sin(-np.pi / 3) * w2im))
return wt
def __LongWakeApprox(self, t):
wl = -1 * (1 / (4 * np.pi * self.radius) *
- np.sqrt(self.Z0 * self.rho / (c * np.pi * t**3))
- * self.length)
+ np.sqrt(self.Z0 * self.rho /
+ (c * np.pi * t**3)) * self.length)
return wl
def __TransWakeApprox(self, t):
wt = (1 / (np.pi * self.radius**3) *
- np.sqrt(self.Z0 * c * self.rho / (np.pi * t)) *
- self.length)
+ np.sqrt(self.Z0 * c * self.rho / (np.pi * t)) * self.length)
return wt
@@ -290,7 +278,7 @@ class Coating(WakeField):
approx=False):
"""
WakeField element for a coated circular beam pipe.
-
+
The longitudinal and tranverse impedances are computed using formulas
from [1].
@@ -313,8 +301,8 @@ class Coating(WakeField):
References
----------
- [1] : Migliorati, M., E. Belli, and M. Zobov. "Impact of the resistive
- wall impedance on beam dynamics in the Future Circular e+ e− Collider."
+ [1] : Migliorati, M., E. Belli, and M. Zobov. "Impact of the resistive
+ wall impedance on beam dynamics in the Future Circular e+ e− Collider."
Physical Review Accelerators and Beams 21.4 (2018): 041001.
"""
@@ -345,10 +333,10 @@ class Coating(WakeField):
def LongitudinalImpedance(self, f, approx):
"""
- Compute the longitudinal impedance of a coating using Eq. (5), or
- approxmiated expression Eq. (8), of [1]. The approxmiated expression
- is valid if the skin depth of the coating is large compared to the
- coating thickness.
+ Compute the longitudinal impedance of a coating using Eq. (5), or
+ approxmiated expression Eq. (8), of [1]. The approxmiated expression
+ is valid if the skin depth of the coating is large compared to the
+ coating thickness.
Parameters
----------
@@ -361,11 +349,11 @@ class Coating(WakeField):
-------
Zl : array
Longitudinal impedance values in [ohm].
-
+
References
----------
- [1] : Migliorati, M., E. Belli, and M. Zobov. "Impact of the resistive
- wall impedance on beam dynamics in the Future Circular e+ e− Collider."
+ [1] : Migliorati, M., E. Belli, and M. Zobov. "Impact of the resistive
+ wall impedance on beam dynamics in the Future Circular e+ e− Collider."
Physical Review Accelerators and Beams 21.4 (2018): 041001.
"""
@@ -394,10 +382,10 @@ class Coating(WakeField):
def TransverseImpedance(self, f, approx):
"""
- Compute the transverse impedance of a coating using Eq. (6), or
- approxmiated expression Eq. (9), of [1]. The approxmiated expression
- is valid if the skin depth of the coating is large compared to the
- coating thickness.
+ Compute the transverse impedance of a coating using Eq. (6), or
+ approxmiated expression Eq. (9), of [1]. The approxmiated expression
+ is valid if the skin depth of the coating is large compared to the
+ coating thickness.
Parameters
----------
@@ -410,11 +398,11 @@ class Coating(WakeField):
-------
Zt : array
Transverse impedance values in [ohm].
-
+
References
----------
- [1] : Migliorati, M., E. Belli, and M. Zobov. "Impact of the resistive
- wall impedance on beam dynamics in the Future Circular e+ e− Collider."
+ [1] : Migliorati, M., E. Belli, and M. Zobov. "Impact of the resistive
+ wall impedance on beam dynamics in the Future Circular e+ e− Collider."
Physical Review Accelerators and Beams 21.4 (2018): 041001.
"""
diff --git a/mbtrack2/tracking/wakepotential.py b/mbtrack2/tracking/wakepotential.py
index 691d9617ed069f05f954403a8369ef1d50eb07e9..3fdb19be03d1e31f6f3034937dd8c44284b43df8 100644
--- a/mbtrack2/tracking/wakepotential.py
+++ b/mbtrack2/tracking/wakepotential.py
@@ -709,9 +709,8 @@ class LongRangeResistiveWall(Element):
Wake function in [V/C].
"""
- wl = (1 / (4 * pi * self.radius) * np.sqrt(self.Z0 * self.rho /
- (c*pi*t**3))
- * self.length) * -1
+ wl = (1 / (4 * pi * self.radius) *
+ np.sqrt(self.Z0 * self.rho / (c * pi * t**3)) * self.length) * -1
return wl
def Wdip(self, t, plane):
@@ -739,8 +738,8 @@ class LongRangeResistiveWall(Element):
else:
raise ValueError()
- wdip = (1 / (pi * r3**3) * np.sqrt(self.Z0 * c * self.rho / (pi * t))
- * self.length)
+ wdip = (1 / (pi * r3**3) * np.sqrt(self.Z0 * c * self.rho / (pi*t)) *
+ self.length)
return wdip
def update_tables(self, beam):