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6.2 Fields for an ellipsoidal distribution

We will call spheroid an ellipsoid with radial symmetry.

6.2.1 Theoretical expression of the potential for a spheroid

Like in the previous section, the Biot and Savart law can be integrated to obtain a theoretical expression of the potentials and fields for an ellipsoidal distribution. Given a homogeneous distribution of charges in an ellipsoidal volume with revolution symetry around the longitudinal axis $Oz$, the potential on any point $z_0$ of this axis can be expressed using the Biot and Savart law:

$$\Phi(0,z_0) = \frac{\rho}{4 \pi \epsilon_0} \int_r \int_z \frac{2\pi r }{\sqrt{r^2+(z-z_0)^2}} r z$$ (6.9)

Where $\rho$ is the density of charges. For an ellipse of semi-axes lengths $A,B,C$ we have:

$$\rho= N e \left(\frac{4}{3} \pi A B C\right)^{-1}$$ (6.10)

The integration on all charges is reduced to the domain defined by the plane ellipse driven by the equation:

$$\frac{r^2}{a^2} + \frac{z^2}{b^2} =1$$ (6.11)

This defines the boundaries used for the integration. We first integrate along $r$, between $r=0$ and $r^2 = a^2(1-z^2/b^2)$:

$$\Phi(0,z_0) = \frac{\rho}{2 \epsilon_0} \int_{-b}^{+b} \left( \sqrt{a^2 - \frac{a^2 z^2}{b^2} +(z_0-z)^2} -\vert z_0-z\vert\right) z$$ (6.12)

The integration along $z$ requires the distinction between the case $a>b$ and $a<b$. Moreover, one has to distinguish the case $\vert z_0\vert<b$, for the potential inside the bunch, and $\vert z_0\vert>b$ for the external potential. The integration is a little bit more complicated than the one for the cylindrical case. One can refer to Annex F for some mathematical refreshments. After the integration, the fields are generalized[28] to points off axis but inside the distribution. For $a<b$ and $\vert z_0\vert<b$ the potential in the spheroid is:

$$\Phi(x_0,y_0,z_0)_{int} = \Phi_0- \frac{\rho}{2 \epsilon_0} \left... ...ac{ \frac{b}{a}}{\sqrt{\frac{b^2}{a^2}-1}} \arg\cosh\frac{b}{a} \right) \right]$$ (6.13)

While the external potential($\vert z_0\vert>b$) on the axis would be:

$$\Phi(0,z_0)_{ext} = \frac{\rho}{2 \epsilon_0} a^2 b \left[ \frac{... ...frac{1}{\sqrt{b^2-a^2}} \arg\coth \frac{\vert z_0\vert}{\sqrt{b^2-a^2}} \right]$$ (6.14)

Now, for an ellipsoid of dimensions such that $a>b$, the potential is :

$$\Phi(x_0,y_0,z_0)_{int} = \Phi_0- \frac{\rho}{2 \epsilon_0} \left... ...frac{ \frac{b}{a}}{\sqrt{1-\frac{b^2}{a^2}}} \arccos\frac{b}{a} \right) \right]$$ (6.15)

$$\Phi(0,z_0)_{ext} = \frac{\rho}{2 \epsilon_0} a^2 b \left[ -\frac... ...{1}{\sqrt{a^2-b^2}} \text{arccotg}\frac{\vert z_0\vert}{\sqrt{a^2-b^2}} \right]$$ (6.16)

The functions argcosh and argcoth are the respective inverse functions of cosh and coth, the hyperbolic cosine and cotangent. The generalization of the external fields is more complicated and we refr the reader to [28].

6.2.2 Theoretical expression of the Fields for a spheroid and an ellipsoid

6.2.2.1 Analytical derivation of the potential for a spheroid

From the previous expression of the internal potential, it can be immediatly seen that a linear dependence is expected between each field component and its corresponding axes.

$$\v{E}_{int}=- \v{\nabla}(\Phi) = \frac{\rho}{\epsilon_0} \left(M_x x, M_y y, M_z z \right)$$ (6.17)

With the symmetry of the distribution imposing $M_x=M_y$. Deriving equation 6.15 for the case $a<b$ yields to:

$$M_z = \frac{-1}{\frac{b^2}{a^2}-1} \left( 1-\frac{ \frac{b}{a}}{\sqrt{\frac{b^2}{a^2}-1}} \arg\cosh\frac{b}{a} \right)$$ (6.18)

$$M_x = M_y=\frac{1}{2} \left(1 - M_z\right)$$ (6.19)

While for the case $a>b$:

$$M_z = \frac{-1}{\frac{b^2}{a^2}-1} \left( 1-\frac{ \frac{b}{a}}{\sqrt{1-\frac{b^2}{a^2}}} \arccos\frac{b}{a} \right)$$ (6.20)

In [30],refering [22], the following expressions can be found for $M_z$:

$$M_z = \frac{1+\Gamma}{\Gamma^3} \left(\Gamma - \arctan \Gamma \right)$$ (6.21)

with the excentricity $\Gamma=\sqrt{A^2/C^2-1}$. Nevertheless, no link between 6.22 and 6.23 has not been done yet.

The expression of the external field being more complicated we will focus on its expression on the ellipse axis ($r=0$) and its longitudinal component. For the case $a<b$:

$$E_{z,ext}(0,z_0)= - \frac{\rho}{2\epsilon_0}a^2 b \left[\frac{\te... ...\vert}{\sqrt{b^2-a^2}} \right)+\frac{\text{sign}(z_0)}{\sqrt{b^2-a^2}} \right ]$$ (6.22)

And for $a>b$:

$$E_{z,ext}(0,z_0)= - \frac{\rho}{2\epsilon_0}a^2 b \left[-\frac{\t... ...\vert}{\sqrt{a^2-b^2}} \right)-\frac{\text{sign}(z_0)}{\sqrt{a^2-b^2}} \right ]$$ (6.23)

6.2.2.2 General ellipsoid - Introducing a form factor

The linearity of the fields with respect to their axis is still valid in the general case of an ellipsoid. Approximate expressions have been derived by Lapostolle[28] and are used by the software trace 3D[8] to compute the space charge fields. For an ellipsoid of semiaxes $r_x$, $r_y$, $r_z$, the following expressions are given:

$$E_x(x) = \frac{1}{4\pi\epsilon_0} \frac{3I\lambda}{c\gamma^2}\frac{1-f}{r_x (r_x+r_y) r_z} x$$ (6.24)

$$E_y(y) = \frac{1}{4\pi\epsilon_0} \frac{3I\lambda}{c\gamma^2}\frac{1-f}{r_y (r_x+r_y) r_z} y$$ (6.25)

$$E_z(z) = \frac{1}{4\pi\epsilon_0} \frac{3I\lambda}{c}\frac{f}{r_x r_y r_z} z$$ (6.26)

where $f$ is a function of $p=\gamma r_z / \sqrt(r_x r_y)$, $\lambda$ the free space wave-length of the RF and $I$ the average current assuming one bunch is emitted at each period of the RF. As the result of this, in the case of a single bunch, $\lambda I = c Q$ and eventually equation 6.28 can be written:

$$E_z(z) = \frac{\rho}{\epsilon_0} f(p) z$$ (6.27)

The fact that $f$ is called the form factor appears then obvious.

Figure 6.2(a) gives reference values for the form factor and figure 6.2(b) represent these values spline-interpolated for more precision.

Figure 6.2: Form factor data and corersponding plot for different value of p

[]

p	f	1/p	f
0.0000	1.0000	0.0000	0.0000
0.0500	0.9260	0.0500	0.0070
0.1000	0.8610	0.1000	0.0200
0.1500	0.8030	0.1500	0.0370
0.2000	0.7500	0.2000	0.0560
0.2500	0.7040	0.2500	0.0750
0.3000	0.6610	0.3000	0.0950
0.3500	0.6230	0.3500	0.1150
0.4000	0.5880	0.4000	0.1350
0.4500	0.5560	0.4500	0.1550
0.5000	0.5270	0.5000	0.1740
0.5500	0.5000	0.5500	0.1920
0.6000	0.4760	0.6000	0.2100
0.6500	0.4530	0.6500	0.2270
0.7000	0.4320	0.7000	0.2440
0.7500	0.4130	0.7500	0.2600
0.8000	0.3940	0.8000	0.2760
0.8500	0.3780	0.8500	0.2910
0.9000	0.3620	0.9000	0.3060
0.9500	0.3470	0.9500	0.3200
1.0000	0.3330	1.0000	0.3330

[]

6.2.3 Graphical comparisons of the fields

On figure 6.3 one can see the good agreement between the numerical (solid colored line) and the theoretical(dashed lines) fields for different values of $\sigma _z$, the RMS value of the bunch length, and $\sigma_r$, the rms value of the radius. Table 6.2 display the parameters used for the simulation.

Figure 6.3: Ellipsoidal distribution: Longitudinal field $E_z$ along z - comparison between simulation and theory. The theoretical fields are represented with black dashed lines. A perfect agreement is found, but the maximum amplitude of the simulated field is always smaller. This depends on the resolution of the grid, and the smoothing of the fields.

Table 6.2: Parameters used for the benchmarking of the space charge algorithm - spheroid distribution

Parameter	Value
Distribution type	Uniform ellipsoid
Distribution $\sigma_x$	$\left\{1 ; 2 ; 3 ; 4\right\}/\sqrt{2}$ mm
Distribution $\sigma_y$	$\left\{1 ; 2 ; 3 ; 4\right\}/\sqrt{2}$ mm
Distribution $\sigma _z$	$\left\{0.1 ; 0.2 ; 0.3 ; 0.4 \right\}$ mm
Distribution energy	15 MeV
Distribution charge	-0.25 nC
3D grid used	$N\ast_0=17$, $N\ast_f$=64, $N\ast_2$=128

Figure 6.4: Comparison of theoretical (dashed) and numerical longitudinal field. for an ellipsoidal bunch at two different energies: a) 0 Mev, b)100 MeV - Values calculated on the grid are displayed with green cross. Note that a small amount of cells has been used here.

(a) (b)

Table 6.3: Parameters used for the benchmarking of the space charge algorithm - sphere at two energies

Parameter	Value
Distribution type	Uniform ellipsoid
Distribution $\sigma_x$	1 mm
Distribution $\sigma_y$	1 mm
Distribution $\sigma _z$	1 mm
Distribution energy	$\left\{ 0 ; 100 \right\}$ MeV
Distribution charge	-100 nC
3D grid used	$N\ast_0=5$, $N\ast_f=20$, $N\ast_2=32$

Emmanuel Branlard
http://emmanuel.branlard.free.fr/