Analysis of the longitudinal space charge impedance of a round uniform beam inside parallel plates and rectangular chamber

This paper analyzes the longitudinal space charge (LSC) impedances of a round uniform beam inside a rectangular and parallel plate chambers using the image charge method. This analysis is valid for arbitrary wavelengths and the calculation converges fast. The research shows that only a few of the image beams are needed to obtain a relative error less than 0.1%. The beam offset effect is also discussed in the analysis.

the beam diameters are large, the Ref. [12] provides the approximate solutions to the field models of a round beam with planar and rectangular boundary conditions, assuming the 3D image charge fields of the round beam can be approximated by the image fields of a line charge. The resulting LSC fields and impedances are valid for the whole perturbation wavelength spectrum. When the ratio of the beam diameter to the transverse chamber dimension approaches unity, the relative errors of the approximated LSC impedances will become larger. In addition, the LSC field of a round beam between parallel plates has been studied in Ref. [22] but the results are only valid in the long-wavelength limit.
This paper proposes an image charge method to calculate the LSC impedances of a round beam between parallel plates and inside a rectangular chamber. It is well-known that the solutions to the LSC fields of a round transversely uniform beam with sinusoidal line charge density modulations in free space are available in a closed form [8]. If the beam were placed inside a rectangular chamber or between parallel plates, due to planar symmetry and mirroring, the associated total image charge fields of the infinite chain (parallel plates model) or grid (rectangular chamber model) of the image beams can be calculated by simple summation. Adding the self-fields of the round beam in free space, the total LSC fields and impedances of the two models in discussion can be obtained. Through case study, we found the calculated LSC fields and impedances converge pretty fast with the number of image beams. Usually only a finite and small number of image beams are needed to obtain the LSC fields and impedances with relative errors less than 0.1%.
The resolution of the calculated LSC fields and impedances depend on the number of image beams used in the calculation, rather than the ratio of the beam diameter to the transverse chamber dimension, as was the case in Ref. [12].
The image charge method is one of the popular methods in the study of the charged particle field, especially for solving the Poisson's equation (i.g., Ref. [23]). In some literatures, the round beam is approximated by a line charge [12,20] in calculation of the image charge field for simplicity. In this paper, the exact image beams are included in the calculation of the full-spectrum LSC impedance. This paper is organized as follows. Section II briefly introduces the wave equations describing the space charge fields of the charged beam. Section III briefly introduces the numerical calculation method for general geometries of the beam and beam pipe. Section IV provides a short review for the LSC fields and impedances of a round beam in free space and inside a round chamber, respectively. Sections V and VI calculate the LSC impedances of a round beam between parallel plates and inside a rectangular chamber using the image charge method, respectively.

II. WAVE EQUATIONS
The wave equation describing the electric field E is (1) where  0 =8.8510 -12 F/m and  0 = 410 -7 N/A 2 are the permittivity and permeability of free space, respectively; √ m/s is the speed of light in free space; ρ and J are the charge and the current densities, respectively , they obey the following continuity equation: ⃗ ⃗ (2) Assuming that the beam is moving with a constant longitudinal speed ̂ along the z-axis, where  is the relativistic speed, ̂ is the unit vector of the longitudinal coordinate, then J can be expressed as ̂.
For a perturbed beam, its volume charge density  and current density J consist of unperturbed (DC) components and perturbed higher order harmonic components. Since the physics of the unperturbed components is trivial and does not contribute to the LSC field, we only need to focus on the physics associated with those higher order harmonic components. We will study one particular harmonic component with a frequency  (or wave number k) in the rest of this paper and omit the subscript k in the variables of fields, density and current for simplicity. The single harmonic component of the charge density, current density and beam current can be expressed using the wave assumption: where is the magnitude of the harmonic line charge density, is the transverse beam distribution function normalized by ∫ ( ) . We will work with only the z-components of the vectors E, J and . Hence, the differentials of the longitudinal harmonic components of J and can be expressed as The z-component of the harmonic electric field can be written as Substituting Eqs. (4-6) into Eq. (1), the amplitude of the longitudinal electric field E z (k, x, y) satisfies the following equation where , and .
The LSC impedance per unit length of an accelerator with circumference L at an arbitrary transverse coordinate (x, y) is defined as

III. FEM SIMULATION FOR ARBITRTARY GEOMETRY
Eq. (7) with arbitrary cross-sectional geometries of the beam and beam pipe can be solved numerically using the Finite Element Method (FEM) [25]. The FEM equation is ( ) . (12) Here M is the stiffness matrix with matrix element e n m M , , m and n are the node indices of the finite element, S e is the integration boundary of the finite element. N(x, y) is called the shape function (similar to the weighting factor) in FEM, by which the fields at a field point P(x, y) within an element can be interpolated by the fields of its neighboring nodes. It is related to the coordinates of the field point P(x, y) and the nodes of the element region. q m is the charge at the node m, which is proportional to z . The current has the similar dependence on z . Therefore, the LSC impedance given by Eq. (8) is independent of z as expected. The E z of Eq. (9) at all nodes satisfying equations Eqs. (9)-(12) and the boundary condition E z = 0 on the chamber wall can be solved numerically. Then the corresponding longitudinal space charge impedances can be calculated using Eq. (8).

IV. IN FREE SPACE AND INSIDE A ROUND CHAMBER
For an infinitely long round beam with uniform transverse density within beam radius a, its transverse density distribution function is where . The general solution of Eq. (7) with the above beam distribution is [26] where I 0 (x) and K 0 (x) are the 0 th order modified Bessel functions of the first and second kinds, respectively.
In free space, the field strength at r should be finite. Therefore A 1 =0. The continuity conditions of the field and its derivative at r=a give [8] where I 1 (x) and K 1 (x) are the 1 st order modified Bessel functions of the first and second kinds, respectively.
The superscript "free" on the left hand stands for "free space". With the property of I 0 (0)=1, Eqs. (8) and (15) yield the LSC field and impedance per unit length on the beam axis (r=0) as where Ohms is the impedance of free space. Since the longitudinal electric field depends on the radial position, the LSC impedance also has the same dependence. The LSC impedance per unit length for arbitrary r within the beam (r<a) is given by Inside a round beam chamber with inner wall radius r w , the continuity conditions at r=a and the boundary condition on the chamber surface E z (r=r w )=0 determine the coefficients A 1 , A 2 and A 3 in Eq. (14). Therefore the final solution of the longitudinal electric field is The superscript "rd" on the left hand side stands for "round chamber". The above equation gives the wellknown LSC impedance per unit length of a round uniform beam inside a round beam chamber [13][14][15][16] ( ) Using the identity of ( ) ( ) ( ), the averaged LSC impedance over the beam cross-section per unit length in free space and inside a round chamber can be derived easily from Eq. (18) and Eq. (20) as In the long-wavelength limit ( ( ) ), the on-axis LSC impedance of a round beam in free space is where C=0.577216 is the Euler's constant, the superscript "LW" stands for the "long-wavelength limit".
The LSC impedance of a round beam centered inside a round beam pipe in the long-wavelength limit is where C 1 =1/2 and 1/4 for the on-axis and average impedance, respectively. The ratio of r w /a is 2 for this example. Let us assume that the median plane of the parallel plate has vertical coordinate y=0 and a source beam is located at (0, y c ), as shown in Fig. 2. The image beams have vertical coordinates of ( ) ( ) and the corresponding line charge densities amplitudes of ( ) ( )

V. BETWEEN PARALLEL PLATES
. The images with the indices of n>0 and n<0 represent the ones above and below the parallel plates, respectively. The term with index of n=0 corresponds to the original source beam. For a round beam with uniform transverse density, its image beams have the same density distribution as the source beam as shown in Fig. 3. Note that the image beams within an arbitrary chamber may not always have the same shape as the source beam, for instance, the image beams within an elliptical chamber. The field at any field point is equal to the sum of all image fields plus the self-field of the original source beam in free space. For instance, the LSC field at position (x, y) within the source beam is where √ ( ( ) ) . The superscript "pp" on the left hand of Eq. (25) stands for "parallel plates". The term with n=0 on the right hand side of Eq. (25) is contributed from the original source beam. Using Eqs. (8), (15) and (25), we obtain the LSC impedance per unit length at position (x, y) inside the beam as Different from the free space case, the LSC field of a round beam between parallel plates is not axisymmetric due to the boundary shape. Therefore, the impedance is a function of (x, y) instead of r exclusively.   When the beam axis has vertical offsets with respect to the chamber median plane (see Fig. 2), the onaxis LSC impedance at long wavelength is slightly reduced as shown in Fig. 6 assuming h=10a and the beam offsets range from a to 4a. The beam offset only affects the long-wavelength impedance for the same mechanism as the shielding effect. When the beam axis shifts vertically from the chamber median plane, one group of images with line charge densities of (z) (with y img >0 in Fig. 2) moves closer to the beam while another group of images with line charge densities of (z) (with y img <0 in Fig. 2) moves away from it as shown in Fig. 2. While the distance of the image beams with line charge densities of (z) to the source beam doesn't change when the source beam is shifted. The net effect from the re-distribution of the image beams with line charge densities of (z) is small due to the cancellations of all the image beams. In general case, the effect of beam offset is negligible if the offset amplitude is small compared to the aperture of beam pipe.
The LSC impedance of a round beam centered between parallel plates in the long-wavelength limit can be derived from Ref. [22] as where for the on-axis impedance and for the average one, respectively.   Consider an infinitely long, transversely uniform round beam inside a rectangular conducting structure with full width w and full height h. We assume the axes of the rectangular chamber and the beam are located at (0, 0) and (x c , y c ), respectively. Fig. 8 shows the image beams of a source beam centered inside a rectangular chamber indicated by the solid black rectangle. The red and blue dots represent the beams with line charge densities of (z) and (z), respectively.

FIG. 8. 2D grid of images of a source beam centered inside a rectangular chamber.
Through planar symmetry and mirroring, we can determine the exact coordinates of images axes of a line beam inside a rectangular chamber as Here m and n represent the indices of the image grid points in horizontal and vertical directions, respectively. For instance, the indices m>0 (m<0) are for the images with x>0 (x<0). The image with the indices of is just the original source beam. Similar to the case of parallel-plate chamber, the image beams of a round uniform beam still keep the same distributions and beam radius as those of the source beam, while their centers are given by Eq. (28). Therefore, the LSC field at (x, y) within the source beam inside a rectangular chamber is equal to the total LSC fields of the source beam and its image beams in free space ) . The superscript "rect" on the left hand of the equation stands for "rectangular chamber". Note that the original source beam effect is included with . Using Eqs. (8), (15) and (29), we obtain the LSC impedance per unit length at position (x, y) inside the beam as The LSC impedance per unit length of a round uniform beam centered inside a rectangular chamber in the long-wavelength limit is [7] ( ) where for the on-axis impedance and for the average one, respectively. The above formula is a good approximation for a rectangular chamber with w/h>1. When w/h is about 1, the exact but more complicated formula [7] should be used. wavelength spectrum. However, the long-wavelength-limit formula significantly overestimates the impedance at short wavelength as shown by the dashed lines in Fig. 9. Therefore, it is essential to use a more accurate full-spectrum LSC impedance formula in the study of beam instability.  For a fixed beam radius, the shielding effect becomes weaker as the chamber aperture is enlarged because the image beams are farther away from the source beam. Fig. 11 shows the effect of beam offset on the on-axis impedance in a square chamber with . The beam offset reduces the impedance and this reduction depends on the wavelength as shown in Fig.   11(a) where the vertical offset is zero. The beam offset effectively reduces the impedance at long wavelength regime due to the chamber shielding effect. Fig.11 (b) is for a particular wavelength with ( ) . The reduction of the LSC impedance is small when the beam offset is much smaller compared to the aperture of the beam pipe. However, the reduction of the impedance becomes more pronounced when the beam offset increases, because the shielding effect by the image beams roughly scales as 1/r (r is the distance of the source beam axis to the surface of beam pipe). The impedance is about 50% smaller when the beam is close to the surface of the beam pipe as shown in the figure.
FIG. 10. The shielding effect of a rectangular chamber on the on-axis impedance. The round uniform beam has a radius of a, the full widths and heights of the chamber are w=h=4a, 6a, 8a, 10a, 12a, 14a and 16a, which correspond to the lower to upper black lines as clearly shown on the right part of the plot.  parallel plates (h ) and rectangular chambers (h , w/h=1and 2). The on-axis LSC impedance in free space and inside a round chamber is calculated using Eq. (17) and Eq. (20) for r=0, respectively. The round chamber has slightly stronger shielding than a square chamber resulting in a smaller on-axis LSC impedance. It is about 6% (1%) less than that of a round beam inside a square pipe when the pipe radius is 2a (10a) at wavelength regime with ( ) . When the aspect ratio of rectangular chamber w/h is larger than 2, the shielding effect is very close to that of a parallel-plate model. The impedance of parallelplate model at long wavelength regime is about 20% larger than that of the round-chamber model.
To compare the shielding effect in the long-wavelength limit we can define the geometry factor g 0 as .
where for the on-axis impedance and for the average one, respectively.  Our method can also give the LSC impedances averaged over the beam cross-section. Fig. 13 shows the average LSC impedance of the round beam with four different boundary conditions: parallel plates, square chamber, round chamber [Eq. (22)], and in free space [Eq. (21)], respectively. The same beam parameters and normalization method are used as in Fig. 12, where the normalized peak impedance in free space is set to 1. The shielding effect on the average LSC impedances is similar to the case of on-axis LSC impedances.
The average LSC impedance of the round beam is about 20% less than the on-axis one when 1< /(ka)<8.0.
However, they are almost identical for high frequency with  /(ka)<1.0.
FIG. 13. Comparisons of the shielding effects between the round, square, and parallel-plate chambers on the average LSC impedance. The same beam and chamber parameters as in Fig. 12 are used here. The square chamber has an aspect ratio of w/h=1.

VII. CONCLUSIONS
The image charge method is employed in this paper to provide analytical solutions to the full-spectrum Our studies show that the round chambers have slightly better shielding effect than square chambers.
When the aspect ratio of a rectangular chamber is larger than 2, the shielding effect is very close to that of a pair of parallel plates. The offset of beam axis slightly reduces the LSC impedance when the offset is small compared to the aperture of beam chamber. However, the reduction becomes significant when the beam is close to the surface of beam chamber.