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7.3 Ellipsoidal distribution

For an ellipsoidal distribution, we didn't derived manually the fields expressions, and we will use equations 6.26-6.28, that uses a form factor, function of the dimensions of the ellipsoid $r_x$, $r_y$, $r_z$. For the comparison between this theory and our simulations, we generated different bunches of different dimensions. All permutations of values of $\sigma_x$, $\sigma_y$, $\sigma _z$ presented in table 7.1were used, representing 896 distributions.

Table 7.1: RMS dimensions used for the study of ellipsoidal distributions

$\sigma_x$	0.50	1.00	1.50	2.00	2.50	3.00	3.50	4.00
$\sigma_y$	0.20	0.40	0.60	0.80	1.00	1.20	1.40	1.60	1.80	2.00	2.50	3.00	3.50	4.00
$\sigma _z$	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40

The difference between $\sigma_x$ and $\sigma_y$ is anedoctical. As a symmetry was expected between x and y, it appeard unnecessary to use the same values for $x$ and $y$. We chose to use values close to the expected range that can be found at a0 : $\sigma_x\in{0.5-4.0}$, $\sigma_y\in{0.2-2.0}$. But after all, we decided to study the symetry between $x$ and $y$, so we extended the values of $\sigma_y$.

Our stand alone algorithm has been run on all this distributions, for a grid of $64\times64\times64$. Maximum fields values and rms values have been extracted from the simulation. Combined with theoretical values, a database has been established. Its structure is described in table 7.2.

Table 7.2: Structure of the file /results/ez-rms/EllipseXYZ-database.csv

Col.	1	2	3	4	5	6	7	8	9
Var.	$\sigma_x$	$\sigma_y$	$\sigma _z$	$\sigma_r$	$x_0$	$y_0$	$z_0$	$r_0$	$V$
Unit	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm$^3$]
Col.	10	11	12	13	14	15	16	17	18
Var.	$\rho$	$p$	$f$	$E_$x,rms$^{th}$	$E_$y,rms$^{th}$	$E_$z,rms$^{th}$	$E_$x,rms$^{sim}$	$E_$y,rms$^{sim}$	$E_$z,rms$^{sim}$
Unit	[C/m$^3$]	[.]	[.]	[V/m]	[V/m]	[V/m]	[V/m]	[V/m]	[V/m]
Col.	19	20	21	22	23	24
Var.	$E_$x,max$^{sim}$	$E_$y,max$^{sim}$	$E_$z,max$^{sim}$	DEX	DEY	DEZ
Unit	[V/m]	[V/m]	[V/m]	[.]	[.]	[.]

In table 7.2, DEX stands for the relative difference :

$$\frac{E_\text{x,rms}^{th}- DEX=E_\text{x,rms}^{sim}}{E_\text{x,rms}^{th}}$$ (7.25)

As a results of this DEX, DEY, DEZ translate the percentage of difference between the simulation and the theory. We suggest here, to look at the influence of $\sigma_x$ and $\sigma_y$ on this three parameters, for a given $\sigma_z=0.1$mm. These results are plotted on figure 7.3. The best agreement is for $\sigma_x\approx \sigma_y$. One explanation can be that the theory developped by Lapostolle as an extension of the case $\sigma_x= \sigma_y$, is less valid if the bunch does not have revolution symmetry. Nevertheless, we don't reject the hypothesis that the 3D Poisson solver has less validity in this situation also.

Figure 7.3: Relative difference between theory and simulation for different ellipsoids for the three components of the field. Light color represent a percentage of error of less than 5 %. The three figures show the relative difference between the theory and the simulation : (a) DEZ($E_z$), (b) DEX($E_x$) ,(c) DY($E_y$). The best agreement is for $\sigma_x\approx \sigma_y$

(a) (b) (c)

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