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Subsections

7.4 Cylindrical distribution

7.4.0.0.1 RMS calculation of

For a uniformly distributed cylinder,

is a constant not null in the bunch domain:

$\displaystyle N=\int_{\Omega} f = \int_{x,y,z} c \:dxdydz =\int_{-z_0}^{z_0} C dz=2 z_0 C$

(7.26)

N is the normalization factor so that

is a normalized distribution.

$\displaystyle \langle z^2 \rangle = \int_{x,y,z} z^2 f \:dxdydz =\int_{-z_0}^{z_0} z^2 C dz=\frac{2}{3}z_0^3 C$

(7.27)

Eventually

$\displaystyle \sigma_{z}^2 = \frac{\langle z^2 \rangle}{N} = \frac{z_0^2}{3}$

(7.28)

Let's call

the bunch length, so that

. The previous equation is now:

$\displaystyle \sigma_{ z}^2 = \frac{l_{z}^2}{12}$

(7.29)

7.4.0.0.2 Estimation of the RMS value for the longitudinal field

: Contrary to the ellipsoidal distribution, fields are not linear with respect to their axis. The theoretical RMS calculation is then more complex. Option 1 : If we assume (wrongly) that the longitudinal field is linear with repect to z, then (cf calculation for the ellipsoid in the previous section 7.2) we can define the RMS field as:

$\displaystyle \langle E_z^2 \rangle \approx \frac{E_{max}^2}{r_z^2} \langle z^2\rangle$

(7.30)

which, from the previous section would give us :

$\displaystyle E_{z,\text{rms}} \approx \frac{E_{z,\text{max}}}{\sqrt{3}}$

(7.31)

Option 2 : From the simulation, calculating the RMS values by summing on all particles provides an RMS field that has an amplitude way smaller than the one suggested in equation 7.31. It appeared that the RMS value is more of the order of :

$\displaystyle E_{z,\text{rms}} \approx \frac{E_{z,\text{max}}}{\sqrt{12}}$

(7.32)

Why using a wrong definition ? Indeed, this sound controversial. We just want an easy way to compare the theoretical and simulation fields amplitude, and as we are using rms values for and , it makes more sense to use an rms value for the field also. Nevertheless, a non-controversial plot, with real values (not rms) will be also provided (see figure 7.5).

7.4.0.0.3 RMS estimation of

From the two previous paragraph, we eventually have

7.4.0.0.4 RMS calculation of the simulated field

: the two same methods than for the ellipsoid are used(see equation 7.24 and its corresponding paragraph), eventhough the only trustable one is the one calculated by averaging on all the particles. This results will be obvious in the next plot (figure 7.4).

7.4.0.0.5 Comparison between theory and simulation

**Figure 7.4:** Rms value of the longitudinal field for a cylindrical distribution . What we call theory here, corresponds to the theoretical value of the field maximum divided by $\sqrt{12}$ , rmsth, max $/\sqrt{12}$ . This empirical relation seems reasonable as it provides results of the same order than the one calculated by averaging the simulated longitudinal field on each particle(`sim.`). From the maximum of the simulated field, another RMS value can be defined by dividing it by $\sqrt{12}$ . Nevertheless, we previously saw that the simulated field has a smaller maximum. This explains why the values `sim. from max` are smaller than the pne calculated from the theoretical field. Data points are stored in file /results/ez-rms/MErmsCylinder.csv

**Figure 7.5:** Maximum value of the longitudinal field for a cylindrical distribution. Once again we see that the simulated field has an amplitude smaller than the theoretical one. Note that the fields are here plotted with the real length of the bunch and the real radius of the bunch, instead of RMS values. Data points are stored in file /results/ez-rms/MEmaxCylinder.csv

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