2.1. The differential gas pumping

Outside the RFQC box, in the rest of the vacuum chamber we aim at a pressure of $p_v \ll 0.01$ Pa, to be maintained with some care: injection and extraction lines communicate with the RFQC box by means of two antechambers, differentially pumped (through some tubes), see figure 5. Two other tubes connect the RFQC box to a He mass-flow controller and to a pressure gauge, both placed outside the solenoid. The differential gas pumping is an important part of the Eltrap RFQC set-up as well as of other installations, since gas collisions outside the RFQC may provoke large beam diffusion, as the ion speed is greater and confinement is not uniform there; this effect was also noted in SIMION (Manura & Dahl Reference Manura and Dahl2011) simulations of the RFQC for the SPES project. The gas pressure $p_g$ inside the RFQC has been scanned from 2 to 9 Pa in the following simulations, since the relatively large reacceleration voltage step (as mentioned before, 10 V after each section $i$ in the first planned experiments) may require more gas for ion cooling. To reduce gas leakage, the RFQC box should be fully closed, except for the iris-shaped low-voltage electrodes $V_7$ and $V_8$ that close the ends of this box. The 2 mm radius of the iris pupil (or aperture), allowing for the ion passage, can be further reduced by optional macor collimators (with advantages for the pump operation but the disadvantage of a possible accumulation of surface charge). The gas injected into the RFQC box is regulated by the mass-flow controller, in feedback with the vacuum gauge; the gas escaping from the RFQC input and output pupils is intercepted by two antechambers, directly connected to the main turbopump (nominal pumping speed approximately 550 litre s$^{-1}$ for nitrogen or helium) by a network of pipes.

Figure 5. (a) View of gas pipes and septum inside the fourway cross, with triode electrodes V9 and V10 dismounted from antechamber A2 to show V8. (b) Overview of the emittance meter EMI1 port, gas pipes and related turbopumps.

The gas escaping from these antechambers to the main Eltrap vacuum is pumped by another turbopump (nominal pumping speed 330 litre s$^{-1}$ for nitrogen), see figure 5. The typical helium conductances of the gas pipes are approximately 3 litre s$^{-1}$ (longer one) and 7 litre s$^{-1}$ (shorter one); for each iris electrode V7 and V8, assuming a 1.5 mm radius, 15 mm long collimator (best case), we get a 0.6 litre s$^{-1}$ conductance (Livesey Reference Livesey1998). Thus antechambers A1 and A2 may reduce significantly gas leak to the main chamber, whose conductance to pumps may be estimated to be of 1 m$^3$ s$^{-1}$ order (adequate for the installed pumps). The total gas leak flow $F_L \le 0.011$ Pa m$^3$ s$^{-1}$ is safely lower than the one already demonstrated for SPES case, approximately 0.05 Pa m$^3$s$^{-1}$ (Ruzzon et al. Reference Ruzzon, Maggiore, Gelain, Marcato, Ban, Bougard, Cam, Desrues, Gautier and Lory2023), but of course experimental tests of the Eltrap pumping system need to be done soon. In the RFQC installation, PEEK (polyether ether ketone) insulators are tolerated, taking into account the background gas diffused by the RFQC; this choice significantly mitigated costs. Operation experience of this challenging differential pumping and gas control system will be valuable for the design of similar RFQCs.

2.2. Extraction optics design

The current extraction system installed uses three electrodes (the so-called triode design), but more adjustable parameters are offered by a four electrode system (the so-called tetrode design), with voltages here labelled $V_8$, $V_9$, $V_{9b}$ and $V_{10}$. Note that end positions and end voltages $V_8$ and $V_{10}$ are typically fixed (or constrained within limited ranges) by the overall set-up and the specified final ion energy; for example here $V_{10}=-4.8$ kV. Therefore, while a triode has only one fully adjustable voltage, a tetrode has two free voltages and the following analysis has also the goal to clarify whether this larger freedom is worth the additional complexity of more electrodes, in the challenging beam dynamics induced by the RFQC.

As shown in figure 6, tetrode geometry evolves from that of the triode by cutting the puller electrode into two parts divided about the middle plane $z=z_{9b}$; note also the electrode names and the location of the A2 antechamber cover at $z = -0.854$ m, which supports insulators (not shown) and electrodes. Excluding the RFQC rods and the input and output irises V7, V8, the electrodes have a cylindrical shape; some are tapered to accommodate the expansion of the beam. For the other acceleration gaps, take the middle of the lens gap as the reference lens plane; for our example, $z_8=-0.79$ m, $z_9=-0.8$ m and $z_{10}=-0.91$ m. Usually the beam electrode $V_{10}$ and the emittance meter EMI1 are approximately at the same potential, with a null or weak lens between them; so we ignore their middle plane, and choose a plane at EMI1 input as the output or observation plane $z_o=-1.03$ m. Other RFQCs may use different electrode shapes, see figure 6(c).

Figure 6. Extraction systems: (a) triode and (b) tetrode design (radial $r$ coordinate enlarged as marked); (c) geometry (not to scale) using iris electrodes and larger gaps for higher voltages.

3.1. Optical aberrations: the case of rapid re-acceleration after the RFQC

After the RFQC, ions are no longer subjected to collisions, but they suffer rapidly changing accelerations at lens planes, with distortions of beam optics known as ‘optical aberrations’ (El-Kareh & El-Kareh Reference El-Kareh and El-Kareh1970). To approximately model this localized feature, consider that $\phi _p$ is negligible outside the RFQC box, and the Bessel series expansion of the electrostatic potential $\phi _s$ close to the axis (El-Kareh & El-Kareh Reference El-Kareh and El-Kareh1970) then gives a similar expansion for

(3.10a,b)\begin{align} \psi(r,z)= \psi_0(z) - \psi_0''(z) \frac{r^2}{4} + \psi_0''''(z) \frac{r^4}{64} + O(r^6),\quad \frac{e}{m_i} E_x \cong{-} g x + g''(z) \frac{r^2}{8} x , \end{align}

with $\psi _0(z)$ the value of $\psi$ on axis and $g(z)=-\psi _0''(z)/2$ a shorthand. This series converges only for $r$ smaller than the inner electrode radius, and $g(z)$ has large oscillations at lens planes; $g''(z)$ has even greater oscillations. Let $v_0(z)=\sqrt {2(h-\psi _0(z))}$ be a reference speed, with $h=H/m_i$ and $H$ the Hamiltonian, that is the total energy, which may be assumed for simplicity equal for all the ions. From the motion (3.3), with $s=z_s-z$, the $s$-differential operator resolved as ${\rm d}_s = (1/|v_z|) {\rm d}_t$, and $|v_z|$ calculated from energy conservation as shown in the Appendix, we get the evolution law

(3.11)\begin{equation} v_0 {\rm d}_s \epsilon_x^R(z)={-} \nu_m \epsilon_x^R +\frac{v_0^2 g''(z)-4 g^2}{8 v_0^2 c^2 \epsilon_x^R} [\langle x^2 \rangle_R \langle x r^2 v_x \rangle_R - \langle x^2 r^2 \rangle_R \langle x v_x \rangle_R]+ \frac{A_0}{c^2 \epsilon_x^R}, \end{equation}

where the leading term (with $g''$ and $g^2$) is shown and $A_0$ is a lengthy expression (see (A4)–(A6), containing similar terms, but without occurrences of $g''$ and $g^2$ or higher powers; as introduced before, $s$ and $v_0$ are respectively the arc length and the speed for the central ion trajectory. Since $g''(z)$ changes sign inside the lens, most changes are cancelled after exiting the lens, but some effects remain, and give an unwanted r.m.s. emittance growth at extraction (as shown by simulations).

3.2. Simulation methods

For each ray, evolution equations (3.6a,b) and (3.7a,b) for the $\tilde {\boldsymbol {x}}, \tilde {\boldsymbol {v}}$ pencil second-order correlations are numerically solved in addition to the ray tracing. Note that $\psi _{,xx}$ has large oscillations at the edges among RFQC sections and even larger oscillations at the middle planes $z_9$, $z_{9b}$ and $z_{10}$ of the electrostatic lenses. The semi-implicit method proved much more stable than simple leapfrog, but it required a laborious development. The resulting script, called the augmented tracer (rays + streamers), can be advantageous for reasonably rapid and accurate calculation of electrode geometries (compared with a Monte Carlo requiring much more particle histories). The tracer also verifies that $\langle x^2 \rangle$ remains reasonably small ($\ll r_0$) during the calculation, and implements the augmented semi-implicit methods, the leapfrog method and other user routines in a script for a general matrix program (Matlab 2016). The axisymmetric field maps of $\boldsymbol {B}^s$ and $\boldsymbol {E}^s$, with mesh size refined to 0.2 mm where necessary, are imported from a multiphysics environment (Comsol 2008) and updated at any parameter change; an analytic formula is still used for $\phi _p$. Evolution of the reduced averaged Hamiltonian $h=H/m_i= \frac {1}{2} \boldsymbol {v}^2 + \psi$ and its fluctuation second momentum $b_{hh}= \langle (\tilde h)^2 \rangle$ are also computed by this ‘augmented’ tracer.

3.3. Simulation results for triodes

Triode or tetrode extraction voltages $V_8$, $V_9$ and $V_{9b}$ can be adjusted for fast parametric scans, keeping $V_{10}=-4.8$ kV fixed for reference (and to get the same kinetic energy); in the reported simulations we have $B_z=0.11$ T at solenoid centre; this gives a total beam rotation of approximately 8.5 rad. Note that the cases $V_9 \not = V_{9b}$ do request a tetrode, while a tetrode with $V_9=V_{9b}$ is equivalent to a triode.

Figure 7(a) shows emittance evolution for the whole beamline, at the good cooling rate obtained with $p_g=9$ Pa and a large ponderomotive equivalent voltage $V_p \equiv \phi _p(r_0)=3$ V, with triode voltages set as $V_8=5$ V, $V_9=V_{9b}=-1.4$ kV and $V_{10}=-4.8$ kV. Case C2 refers to our test source IS1, which gives an initial normalized ray emittance $\varepsilon _x^R = 0.33$ nm. Case C1 doubles the initial beam radius and divergence, so emittance is $\varepsilon _x^R = 1.33$ nm at the source. Note that $\epsilon _x^R$ is constant in the injection line and decreases smoothly to almost zero inside the RFQC; $\varepsilon _x^N$ is a combined effect of ray and streamer dynamics, since $\langle x^2 \rangle = \frac {1}{2} \langle x^2+y^2 + 2 a_{xx} \rangle _R$. In both C1 and C2 cases, the values of $\varepsilon _x^N$ at the RFQC exit converge to $\varepsilon _x^P$, thus showing the importance of the ‘streamer’ dynamics. Another often used ‘geometrical emittance’ $\epsilon _x^g= (c/|v_z|) \epsilon _x^N$ is also plotted in figure 7, to highlight its typical changes during acceleration.

Figure 7. Triode results for emittance evolution vs $z$, with reference voltages $V_8=5$ V, $V_{9b}=1.4$ kV and $V_{10}=-4.8$ kV: (a) comparison at $p_g= 9$ Pa and $V_p=3$ V of cases: C1, initial r.m.s. ray emittance $\epsilon _x^R = 1.33$ nm; C2, initial $\epsilon _x^R = 0.33$ nm. The normalized total emittances $\epsilon _x^N$ are slightly shifted up as labelled for visibility, and $\epsilon _x^g$ is scaled; (b) example of worst case simulations, multiplying $x,y$ ion displacement by a $\mu _x=2$ factor at restart plane $z_s=-0.73$ m, at low confinement $V_p=2$ V and moderate gas pressure $p_g=6$ Pa.

A beam with an emittance value up to $\epsilon _x^R=1.33 \times 10^{-9}$ m at the source (corresponding to an r.m.s. $\epsilon _x^g = 4.6 \times 10^{-6}$ m at $K_i=5$ keV) can thus be injected inside the prototype RFQC with the present injection line, safely matching the existing Cs$^+$ source, which has a lower emittance $\epsilon _x^R=0.33$ nm. For comparison, the SPES plasma ion source r.m.s. geometrical emittances are approximately $\epsilon ^g_{x1} = \epsilon ^g_{y1} \le 7 \times 10^{-6}$ m at an ion energy $K_i=25$ keV, so that the r.m.s. normalized emittance is $\epsilon ^N_x = \beta \gamma \epsilon ^{g}_{x1} \cong 5 \times 10^{-9}$ m (requiring an RFQC larger than our prototype, or larger RF fields); its energy spread is expected to be of the order of 10 eVpp (Bisoffi et al. Reference Bisoffi, Prete, Andrighetto, Andreev, Bellan, Bellato, Bortolato, Calderolla, Canella and Comunian2016), satisfying the $\sigma _E^{{\rm in}} \cong 5$ eVr.m.s. goal.

As an option to further test the RFQC extraction, the tracer script can restart ‘magnified’ rays $\boldsymbol {x}_M$ at a given plane $z=z_r=-0.73$ m, multiplying transverse displacements and velocities by a safety factor $\mu _x$, but conserving energy; namely, the new initial conditions are $x_M(z_r) = \mu _x x(z_r)$, $y_M(z_r) = \mu _x y(z_r)$ and similarly for $v_x,v_y$. This roughly accounts for the fact that ray emittance $\epsilon _x^R$ damps to almost zero near the RFQC exit, but ions still have a transverse motion as shown by $\epsilon _x^P$ (other resampling algorithms are in development).

Figure 7(b) considers a moderate gas pressure $p_g=6$ Pa and a relatively weak confinement $V_p=2$ V, which corresponds to a relatively low RF amplitude $V_{{\rm rf}}= 209$ V and a safe $k_q= 0.03$, at $f_1=4$ MHz. With $\mu _x =2$, figure 7(b) shows a large r.m.s. emittance peak near $z=z_{10}=-0.92$ m.

From the comparison of rays for the cases $\mu _1 =1$ (no change at the restart plane) and $\mu _1 =2$, see figure 8, we see that the latter case shows $|x|\cong 4$ mm at $z_{10}$, enhancing nonlinear effects or aberrations (as expected). Indeed $\epsilon _x^P$, which is governed by the linearized (3.9a,b), does not show any peak in figure 7(b), while $\epsilon _x^N(z)$ and on a smaller scale $\epsilon _x^R(z)$ do show peaks at $z_{10}$, which is the last electrostatic lens at the border of $V_{10}$ and $V_{9b}$ (see figure 3), where we expect third-order aberrations (El-Kareh & El-Kareh Reference El-Kareh and El-Kareh1970), as discussed in § 3.1. Note also that the $\epsilon _x^P(z_o)$ attains a value of approximately 2.2 nm for this case ($p_g=6$ Pa and $V_p=2$ V). In a phase space plot (not shown), the growth is due to the deformation, also known as warping or filamentation, of the area filled with ions. Some smoothing is then applied to $\epsilon _x^N(z)$, and plotted as a dashed line in figures 7(b) and 9, to enhance the visibility of permanent effects. These effects include an increase of the output radius $r_o$ and of the emittance $\epsilon _o=\epsilon _x^N(z_o)$ at the output plane $z=z_o=-1.03$ m (input plane of the emittance meter EMI1).

Figure 8. Samples of output rays for the figure 7(b) triode: (a) ray tracing without safety factor at restarting, so $\mu _x=1$; (b) $\mu _x=2$. Potential $\phi _s$ plot uses the right axis (R); inner, middle and outer rays are grouped by their starting positions. Only rays starting at $y=0$, $x>0$ are plotted (for clarity); other ray contributions to figure 7(b) are recovered by axial symmetry.

Figure 9. Tetrode result for emittances, for $p_g=6$ Pa, $V_p=2$ V, $V_8=5$ V, $V_9=-300$ V, $V_{9b}=-1.4$ kV and $\mu _x=2$, using geometries: (a) first design (longer front puller); (b) second design (6 cm long front puller).

3.4. Triode and improved tetrode comparisons

In the triode system, the puller effective length $z_{10}-z_9\cong 0.11$ m is too large, yielding undesired beam expansion. A first example of a tetrode was designed and built with cut plane $z_{9b}=-0.88$ m, for the convenience of firmly supporting electrode $V_9$ with an isolator simply bolted over the A2 cover. Simulations partly reported in figure 9 have shown two possible limitations of this design: the effective front puller length of 8 cm is still too large; the lens at $z_{9b}$ and $z_{10}$ are not clearly separated; so appreciable emittance peaks are seen at $z_{9b}$ and $z_{10}$ in figure 9(a). In the second tetrode design (to be built), $z_{9b}=-0.86$ m so that the puller length of 5 cm becomes safely greater than its inner diameter $D_p=28$ mm and the front puller length decreases down to 6 cm; so emittance peaks almost disappear in figure 9(b). The following results refer to this second design.

In figure 10 the dependence of $\epsilon _o$ on $V_8$ and $V_9$ is shown, for a constant $V_{9b}=-1.4$ kV which corresponds to the previously optimized value for the triode, with a simpler friction force and optics modelling (Cavenago et al. Reference Cavenago, Romé, Maero, Maggiore, Bellan, Cavaliere, Comunian, Galatà, Panzeri and Pisent2019). For the region $|V_9 /V_{9b} | \ll 1$, note the satisfying performance for $V_9 \cong -200 \pm 150$ V, without appreciable emittance growth; in this $V_9$ range, for any $V_8$ shown, the total (r.m.s. normalized) emittance $\epsilon _x^N$ values approach and coalesce to the pencil emittance value $\epsilon _x^P \cong 2.2$ nm for $p_g=6$ Pa and $V_p=2$ V, as noted before. This value is due to gas collisions and RFQC trap properties, and thus represents a lower bound (present also with ideal extraction) for the total emittance.

Figure 10. Output maximum radius $r_o$ and r.m.s. normalized emittance $\epsilon _x^N(z_o)$ vs the front puller tetrode voltage $V_9$ for several $V_8$ values, and other conditions as in figure 9.

Taking into account that experiments will start with a triode system, figure 11(a) shows the dependence of $\epsilon _x^P$ on pressure $p_g$; since the total emittance includes pencil diffusion effects, the use of larger pressures and a larger ponderomotive voltage $V_p=3$ V is worthwhile. Energy spread cooling is easier, as shown by figure 11(b) with $\sigma _E^o \le 0.4$ eVr.m.s. obtained for $p_g>3$ Pa. Moreover, as long as $p_g=9$ Pa can be sustained by pumping system, acceptable final emittance $\epsilon _x^N$ are obtained in figure 7(a).

Figure 11. Pressure and triode voltage scans: (a) comparison of pencil emittance $\epsilon _x^P$ for $V_p=3$ V and initial $\epsilon _x^R = 0.33$ nm at several $p_g$; (b) output energy spread for initial $\sigma _E^{{\rm in}}=5$ rVms and $V_p=3$ V and several $p_g$, $V_{9b}$; some contour levels are labelled with arrows.

When $p_g \le 6$ Pa, the triode performance can be improved by biasing the RFQC box exit pupil $V_8$ (reference design value $V_8=5$ V) to negative voltages as $V_8=-30$ V, to pre-accelerate the extracted ions, as shown by $V_9=-1.4$ kV points in figure 10. When a compact tetrode with a $|z_{9b}-z_9|\le 6$ cm front puller length is used, figure 9(b) shows that the emittance growth can be suppressed also at the nominal reference voltage $V_8=5$ V. The promising option of changing bias voltages $V_i^s$ to reduce the ion speed in the RFQC and then $p_g$ by a large amount certainly deserves future study, but requires a consolidation of the $\sigma _E^{{\rm in}}$ requirements.