The wake field driven by the preceding electron beam generates wake field for seeding the modulation of the long proton beam. The mode of the growth rate is polarized along the long proton cluster longitudinal profile.
Linearized beam-driven plasma wakefields in cylindrical coordinates
2.1) to (2.3) for obtaining the differential equation of the plasma electrons perturbed by the relativistic charged particle group as below. The evolution of the wake potential can be expressed in each azimuthal mode as below.
General envelope equation of charged particle beam
- Particle equations
- Moment equations
- Self-similar expansion
- The envelope equation and emittance
The depth of the electrostatic potential well of the beam, divided by the kinetic energy of the particles, is also on the order of ∆2. The difference in electrostatic potential between the axis and the beam envelope of the solid beam is. Er and Bθ are generated by the charges and currents of the jet and the background gas.
However, the envelope of the long beam can be modulated by low-amplitude buoyant fields that drive them as the self-modulation (SM) grows [ 30 ]. In this scheme, the phase and growth rate of the long proton beam modulation are controlled by the preceding electron beam, and the SPS proton beam is fully modulated.
Low energy electron beam dynamics in over-dense plasma
The figure shows that the longitudinal evolution of a low-energy electron beam along the plasma is minimal when kkpeσz=√. These confirm that the phase evolution of the wake fields along the plasma is minimal for the case ofkpeσz=√. Increasing the initial energy of the swarm corresponds to decreasing the effect of energy loss on the particle velocity.
The low-energy electron beam initially has a lower speed than the speed of light in the propagation axis. In addition, the energy loss of the electron bottom particles accelerates the decrease in the wakefield amplitude.
SSM phase evolution with beam-plasma parameters
The first such figures show that as the energy of the seed electron beam is lowered, the phase of the wake fields shifts more posteriorly. However, the matched energy peaks of the electron beams after the plasma were not found at the reasonable beam lengths (σz/c∼2−3ps). PIC simulation results showed higher energy loss of the forest particles than the experimental results.
The middle figure shows that the phase of the wake fields is quite insensitive to the charge of the seed cluster (over this range). Figures show the phase of the long proton cluster modulation is mostly influenced by the seed cluster energy.
Beam-driven plasma wakefields and envelope modulation seeded by a short Gaussian
Long proton support (top) and near-axis current modulations (bottom) seeded by the radially matched high-energy proton seed. top) Envelope of the long proton bunch at the longitudinal center of the Gaussian profile (left), σz, forward from the center (middle) and 2σz, p forward from the center (right). In this case, the mode of modulation of the long proton beam is the moving frame function of the beam ζ=z−ct. In the defocusing phase of the long proton bunch, i.e., cos(kpeζ+φs+π/2)<0, the pure seed modulation solution is.
Within inkβz<1, as the modulation amplitudes of the long proton beam (or the exponent N) increase, the phase of the resulting wake field is shifted back inkpeζ [9, 10]. When the seed contribution to the phase shift is greater than the modulation feedback.
Low energy electron bunch as a seed
Then, considering the low current seed group, i.e. rB≪rp,. 3.34) Therefore the purely seeded modulation of a long proton cluster at the seed cluster radial equilibrium is approximately described by. The dominance of seed over the long proton cluster modulation depends on the seed cluster current, not on its density. 3.15(a) and 3.15(b) show that α (start timing of self-modulation) is negligible (late) and β (start timing of externally seeded modulation) is approximately constant (uniform) between the cluster front [(a)kpeζ ∼ − 226] and the bundle longitudinal center [(b) kpeζ ∼ −452], i.e. the long proton beam modulation is seeded as a single mode simultaneously along ζ.
Now the phase behavior of the long proton bottom modulation is explained simply by ∆arg[βrˆs,eq]. The preceding low-energy electron beam seeds the long proton beam modulation with the uniform onset along ζ in overdense plasma.
Conclusion
For simulating an envelope modulation of the long proton bunch seeded by a tightly focused low-energy electron bunch in the radial equilibrium of the seed bunch in an overdense plasma, we simply replace the masses, the relativistic gamma, and the sign of the charge of the seed. group from the group of previous parameters withme, 35.2, and -. 3.15(c)] better than the high-energy proton seed case, without the anomalous phase shift shown in Figs.
3.13(c) disappears and the simulation result of the phase behavior shows a nice agreement with the analytical expectation of Eq. The external seed wakefield contributes to the phase shift of the modulation in the opposite direction of the modulation feedback.
Introduction
Azimuthal mode convergence test
The initial boom field can be created by a short precursor group of charged particles (or a laser pulse) or by a steeply rising front (of the order of k−1pe) of a long group of protons. The following year, the growth rate and modulation frequency of the long proton beam were analyzed based on the experimental data of AWAKE Run 1 [13, 14]. However, this method has a significant problem because the front half of the proton beam is unmodulated.
Instead, we started by estimating the growth rate of the proton beam modulation amplitude with Gaussian profiles. A low energy electron beam is injected into the preionized plasma and generates a plasma wakefield for seeding the envelope modulation of the next SPS proton beam. The wakefield driven by the preceding electron beam generates a wakefield for seeding the modulation of the long proton beam. The equation for the radial plasma wakefield driven by a beam of charged particles is [11].
Therefore, with the thin bunch approximation, the modulation of the long Gaussian proton bunch seed in the overdense plasma is described by.
Instability classification by beam radial size and transverse emittance
The flat top electron beam has a HI seeding bias in any case, and the rise time of the beam is short enough to produce a fairly lush seed field for SMI. When σr=100 µm and εr =1 mm−mrad, the transverse centroids of the inclined beam do not coincide with the centers of the trailing fields created by the strongly accreting beam front, and the fields are strong enough to guide the particles of the relativistic beam toward the beam emission. Instead, the magnetic field from the beam-plasma flow increases the anisotropy of the current distribution.
It means that whether the OTSI can grow in linear regime or not is solely determined by the betatron frequency of the beam-driven plasma wakefield. Since the current filamentation of the beam requires very low emission and the filaments emerge easily after the plasma [38], we prefer to focus on the boundary of the onset of SMI.
Resonant condition of long beam hosing
The SMI seed phase can be analytically estimated using the beam-driven plasma wakefield equation in the linear regime. Figures 4.5(a) to 4.5(e) show the qualitative transition of the long plane beam and plasma instability with increasing ambient plasma density n0. The plasma wake field driven by the beam creates microbeams in the focusing phase, while the beam suffers from centroid shift instability in the defocusing phase.
In this case, the on-axis jet stream (r However, several issues remain for the practical use of the plasma wakefield accelerators, such as setup problems [37], unstable propagation of drivers [52], significant energy dissipation [53] and emission growth of witness group [54] , to name a few of to name them. The mixture of hydrogen and helium gases is used for the media of plasma wake field and the source of the witness electrons. Also, the selection criteria of the driver beam and plasma parameters are investigated to avoid the wakefield jitter effects. II and III, the analytical expressions for the trapping mode and the rms length are introduced to estimate the relative energy dispersion of the witness flock. The dimensions of the ionization laser pulse are set to focal length wi=8µm and effective length Li=12µm. To save computing resources, simulations were performed based on the boosted-frame technique [62] with a relativistic boosted-frame gamma γboost=2. The resolution of the simulation is set to ∆z=λi/20, ∆r= λi/5 and ∆t =∆z/c in the laboratory frame with the wavelength of the ionization laser λi=0.8µm. Tunneling ionization rates (W) of hydrogen and helium atoms and (b) gas density profiles (nandnHIT) with Gaussian laser pulse ionization velocity distribution. Near the focal point, the field distribution of the ionization laser can be expanded as below. The vertical lines are set to the longitudinal positions of the witness bed centroids and ionization laser pulses. The maximum swelling radius rm is determined by the normalized load per unit length of the moving radiusΛ≡(nb0/n0)k2pσb,r2 [36]. The electromagnetic potential is defined by Ψ≡Φ−vφAz, kuvφ(≈c) is the phase velocity of the excitation. The high-current driving beam causes the peak position of the excitation potentialξ0 to oscillate in time [36, 67]. Details of the energy dissipation issues for largeΛ values will be investigated in the future work. Here, the peak position of the wake potential iskpξ0=−2.8, which corresponds to ~160 fs behind the driving jet centroid (ξ=0). Dephasing length of witness electrons Combining the two cases, the rms length of the witness set is given in the form, . Equation (5.16) confirms that the rms length of the witness group is minimized when the ionizing laser pulse is placed at the peak of the excitation potential (see Fig. 5.5). This shortens the rms length of the witness set from the analytical expectation of Eq. Since the contribution of the σzσr−1 term quickly becomes small compared to γ, it can be assumed that the possible effect of the witness group radius mismatch is not important for the relative energy spread [see Eq. The average energy of the witness group is⟨E⟩=γimec2+eEz,t(z−z0)≈eEz,t(z−z0), where the initial energy of the witness group is assumed to be negligible. Shvets, “Phase velocity and particle injection in a self-modulated proton-driven plasma wake field accelerator,” Phys. Turner et al., “Experimental observation of plasma wake field growth driven by the seeded self-modulation of a proton beam,” Phys. Verra et al., "Controlled Growth of the Self-Modulation of a Relativistic Proton Group in Plasma," Phys. Adli et al., “Experimental observation of proton beam modulation in plasma at different plasma densities,” Phys. Muggli, “Strategies to mitigate the ionization-induced beamhead erosion problem in an electron-beam driven plasma wakefield accelerator,” Phys.Conclusions
Trojan horse injection scenario
Quasilinear regime of beam-driven plasma wakefields in an under-dense plasma
Witness bunch rms length and relative energy spread
Conclusion
