Site perso : Emmanuel Branlard

next up previous contents index
Next: 10.2 Influence of the space charge in a | Up: 10. Drift | Previous: 10. Drift |
Navigation:   Contents |   Index

Subsections

10.1 Theory without space charge

10.1.1 Simple case

We study here the evolution of the phase space for a drift of four meters. Figure 10.1 displays the transversal phase space $ x-x'$, while figure 10.2 displays the longitudinal phase space $ z-z'$. Refer to table 10.1 for the distribution specification.

Figure 10.1: Evolution of the transversal space charge for a drift without space charge. From left to right, phase space at $ z=0$m, $ z=2$m and $ z=4$m. As expected, the dispertion $ x'$ stays constant in the absence of space charge, while $ x$ keeps increasing. One can refer to our analogy with optics, in section 3.1.2, figure 3.1.
Image drift4m_cylinder1104-Ek15-Q025-Ez2-Ex5-10000_2_z=00_phase_space_x_blue_nospch
Figure 10.2: Evolution of the longitudinal space charge for a drift without space charge. rom left to right, phase space at $ z=0$m, $ z=2$m and $ z=4$m. Contrary to the transversal phase space, no change is observed in the longitudinal phase space.
Image drift4m_cylinder1104-Ek15-Q025-Ez2-Ex5-10000_2_z=00_phase_space_z_blue_nospch


Table 10.1: Parameters of the distributions used for our analysis on the sigma matrix
Parameter Value
Type Uniform cylinder
$ \sigma_x$ 1 mm
$ \sigma_y$ 1 mm
$ \sigma _z$ 0.4 mm
$ \epsilon_{nx}$, NEmit_x 5 mrad.mm
$ \epsilon_{nz}$, NEmit_z 2 mrad.mm
Energy $ \mathcal{E}$ 15 MeV
Charge $ Q$ -0.25 nC
$ N_$part 10000


10.1.2 Introducing transverse momentum correlation in the distribution

The previous plots were done for a distribution where no correlation existed between $ x$ and $ p_x$. If a simple linear relation exists, the behavior of the beam is different, and will not always expend if the correlation is negative. This might appear obvious, but when generating distributions for simulations, one can forget about this aspect of the beam. A beam is often modelised as a gaussian beam, and a correlation between $ x$ and $ p_x$ always exists. this justify the use of quadrupoles in beam line to focus or defocus the beam. Interpretation of these results in term of beam waist is done in the next section.

Figure 10.3 displays the plane $ x-x'$ for five different longitudinal positions. The distribution is the same than the previous one (table 10.1), exepts that initially, the correlation $ p_x$=-0.05 $ x$ exists.

Figure 10.3: Evolution of the transversal space charge for a drift without space charge for a focusing beam. From left to right : $ z=0$m, $ z=1$m, $ z=2$m, $ z=3$m and $ z=4$m.
Image drift4m_cylinder1104-Ek15-Q025-Ez2-Ex5-corPx-05-10000_5_z=00_phase_space_x_blue_nospch

10.1.3 Beam waist interpretation

By introducing a negative correlation between $ x$ and $ x'$, the beam is focusing. Nevertheless the theory of gaussian beams predict that the transversal size of the beam will reach a minimum, the beam waist, before expending again. This effect is shown on figure 10.4. In the previous section, without any correlation, we were starting the simulation at the beam waist, and thus only observing an expension of the beam size. As the beam is focusing, the particle are more condensed and thus the space charge repulsion force become more important so that the beam waist is bigger if space charges are taken into account.
Figure 10.4: Evolution of the beam envelop for a focusing beam - beam waist definition
Image drift4m-beamwaist

next up previous contents index
Next: 10.2 Influence of the space charge in a | Up: 10. Drift | Previous: 10. Drift |
Navigation:   Contents   Index







Emmanuel Branlard