1. On the Possibility of Using Landau Damping Octupoles in the Recycler Y. Alexahin, A. Burov, E. Gianfelice-Wendt, V. Lebedev, A. Valishev Abstract: To provide Landau damping there must be particles in the beam whose incoherent tune coincides with the tune of coherent oscillations which is close to the bare lattice tune. The incoherent tunes, however, are shifted by space charge forces by as much as -0.06 for the present beam parameters and by -0.09 for the PIP-II parameters. Here we study the possibility of using octupoles to provide a comparable tuneshift without destroying the dynamic aperture. Fermilab, March 5, 2015

2. To achieve cancellation of octupolar kicks to the 1 st order in k 3 for J x (while shifts in  x add up):  both terms (which give rise to 2 nd and 4 th order resonances) must be cancelled individually since phase  x is arbitrary,  this is possible with the number of octupoles in the family N oct ≥ 3,  the phase advance between the octupoles has to be  k/ N oct, k=1,…, N oct -1 with exception of multiples of  /2,  conditions for vertical plane look similar,  in the case of equal phase advances in both planes (satisfying the above conditions) the 4 th order sum resonance is automatically cancelled, but the difference resonance is not. In absence of perturbations Octupolar kick cancellation Normal form variable After an octupolar kick

3. Optics functions vs distance from the first MID529 marker. Where to put the octupoles? s (m) MI60 RR30  MI60 bypass and RR30 straight are long enough to accommodate sufficiently large octupole families,  these straights have 9 powered quadrupoles each that can be used for phase advance adjustment and beta-function matching,  MI60 bypass has small beta-functions product and can house octupoles producing straight detuning coefficients (  Q x /  J x and  Q y /  J y ) without large cross-detuning  Q x /  J y

4. Octupoles in MI60 bypass M531M532M601M602M603M604M605M606M607M608M609M610M611 OD1OD2OD3OD4OD5OF1OF2OF3OF4OF5 s (m)  Average phase advance between beta-functions maxima originally was ~0.26*2  in both planes, the closest admissible value = 0.3*2   5 octupoles / plane  tuning quads at locations 601-609, as well as additional QD531, QF532, QF610, QD611 were used to adjust phase advances at OF, OD and to match beta-functions on both sides of the straight, no attempt was made to equalize beta-functions at the octupole locations: – no more free quads, – the strength of individual octupoles can be varied ~ 1/  2 if needed  one more focusing octupole could be added at M532 to make a sextuplet, but this would require further increase in the phase advance between OF’s up to 0.333*2   even stronger beta-functions perturbation Optics functions (red – horizontal, blue – vertical) in the MI60 bypass before (dashed lines) and after (solid lines) the phase advance adjustment. Indicated are actual positions of octupoles used in simulations.

5. Direction & residual strength of nonlinear kicks OD OF 2Q x +2Q y 2Q x 4Q x 2Q y 4Q y  adjusting strength of individual octupoles the resonance excitation can be further reduced, but this does not improve the DA

6. Nonlinear detuning coefficients (strong octupoles for low emittance) QyQy QyQy QxQx QxQx ax2ax2 tracking: dQ x /dE x =1.52  10 4, dQ y /dE x =-0.82  10 4, dQ x /dE y =-1.05  10 4, dQ y /dE y =2.06  10 4, ay2ay2 ay2ay2 ax2ax2 0.441041 +0.00403001 x+0.000253558 x 2 0.356432 -0.00217111 x-0.000139175 x 2 0.440655 -0.00279175 x+0.000054593 x 2 0.356459 +0.00546993 x-0.0000991623 x 2 Fractional tunes: Q1 = 0.44064214 Q2 = 0.35664865 First order chromaticity: Q1' = -6.02749332 Q2' = -8.06002747 Second order chromaticity: Q1'' = 186.34045563 Q2'' = -17.53859718 Normalized anharmonicities: dQ1/dE1 = 0.18863150E+05 dQ1/dE2 = -0.10227344E+05 dQ2/dE2 = 0.18811930E+05 MAD STATIC: K3L (OF) = 140 m -3, K3L (OD) = 145 m -3, with   N =2.5(  )mm  mrad (r.m.s. normalized) the DA  3  a x =(E x /   ) 1/2, E x = Courant-Snyder inv.

7. Nonlinear detuning coefficients (weak octupoles for high emittance) K3L (OF) = 93 m -3, K3L (OD) = 97 m -3, with   N =4(  )mm  mrad (r.m.s. normalized) the DA  3.5  Fractional tunes: Q1 = 0.44064214 Q2 = 0.35664865 First order chromaticity: Q1' = -6.02749332 Q2' = -8.06002747 Second order chromaticity: Q1'' = 171.18075094 Q2'' = -15.95672562 Normalized anharmonicities: dQ1/dE1 = 0.12576492E+05 dQ1/dE2 = -0.69189680E+04 dQ2/dE2 = 0.12645029E+05 MAD STATIC: QyQy QyQy QxQx QxQx ax2ax2 tracking: dQ x /dE x =0.95  10 4, dQ y /dE x =-0.69  10 4, dQ x /dE y =-0.75  10 4, dQ y /dE y =1.47  10 4, ay2ay2 ay2ay2 ax2ax2 0.440691 -0.00318003 x+0.000112691 x 2 0.440926 +0.00402648 x+0.000488747 x 2 0.356231 +0.00621488 x-0.000154073 x 2 0.356649 -0.00291592 x-0.0000322513 x 2

8. Cross-detuning correction with octupoles in RR30 Optics functions (red – horizontal, blue – vertical) in the RR30 straight before (dashed lines) and after (solid lines) the phase advance adjustment. Beta-functions at OC2-OC7 locations practically are all equal. M232 OC2 s (m) M301M303M304M305M306M307M308M309M310M302 OC3OC4OC5OC6OC7OC8OC1  It can be beneficial to create large positive dQ x /dE y =dQ y /dE x with a new family and use OF, OD only to keep dQ x /dE x and dQ y /dE y positive.  RR30 is sufficiently long to accommodate a large octupole family, there are empty spaces around eight MTSPIP locations where beta-functions for the two planes are ~ equal and have large product as needed for creation of cross-detuning.  Between two consequitive MTSPIP locations one beta-function has maximum while the other has minimum  the space between the octupoles should be doubled for resonance cancellation  groups 4+4, 3+3, 4 and 3 are allowed.  Original phase advance / double spacing was ~0.25*2  in both planes, should be reduced to 0.125*2  or increased to 0.375*2  to use all 8 octupoles (4+4 configuration).  Using just 6 octupoles (marked with red) the admissible phase advance / double spacing is much closer to the original: 0.167*2  or 0.333*2 . We tried the first option.  Tuning quads at locations 301-309, as well as additional at M232, OC2, OC7 and M310 were used to adjust phase advances and to match beta-functions on both sides of the straight.

9. Octupoles in RR30 for large cross-detuning  Settings for large emittance   N =4(  )mm  mrad (r.m.s. normalized) : k3OC2=-120; k3OC3=k3OC2;k3OC4=k3OC2;k3OC5=k3OC2;k3OC6=k3OC2;k3OC7=k3OC2; k3of1=65; k3od1=68; k3of2=k3of1; k3of3=k3of1; k3of4=k3of1; k3of5=k3of1; k3od2=k3od1; k3od3=k3od1; k3od4=k3od1; k3od5=k3od1;  MAD STATIC: Fractional tunes: Q1 = 0.20884683 Q2 = 0.30069997 First order chromaticity: Q1' = -5.93794784 Q2' = -8.10330035 Second order chromaticity: Q1'' = 42.47791085 Q2'' = 61.33727384 Normalized anharmonicities: dQ1/dE1 = 0.96196064E+02 dQ1/dE2 = 0.12481795E+05 dQ2/dE2 = 0.15431887E+03  The default tunes after optics modifications were Q x =25.188 and Q y =24.306, STATIC shows tunes after some tweaking to improve the DA which is still rather low: DA  3   To improve the DA further the tunes must be significantly changed. This requires creation of “trombones” in shorter straight sections with appropriate beta-matching in order not to generate beta-wave around the ring.  To continue work on this option it should be first understood if cross-detuning is a good substitute for straight detuning from the point of Landau damping.  Resonance cancellation scheme with 6 octupoles is simply 2 interleaved triplets:

10. Cross-detuning from tracking (default tunes) tracking: dQ x /dE x =1.67  10 2, dQ y /dE x =1.22  10 4, dQ x /dE y =1.25  10 4, dQ y /dE y =3.11  10 2, QyQy QyQy QxQx QxQx ax2ax2 ay2ay2 ay2ay2 ax2ax2 0.306209 +0.000131415 x-0.0000489061 x 2 0.306282 +0.00516714 x+0.0000294591 x 2 0.187975 +0.00528487 x+0.0000308639 x 2 0.187962 +0.0000704617 x-0.0000457645 x 2 Reference emittance   N =4(  )mm  mrad (r.m.s. normalized), a x,y =(E x,y /   ) 1/2

11. Summary & outlook  Resonance cancellation to the 1 st order in octupole strength requires N oct ≥ 3 octupoles with phase advance between adjacent octupoles being  k/ N oct, k=1,…, N oct -1 except for multiples of  /2  With such choice the excitation of the 5 th order resonances by cross-talk with lattice sextupoles is not completely suppressed, the horizontal resonance was found particularly strong  Large nonlinear tuneshifts  Q x  0.06,  Q y  0.05 at betatron amplitudes a x =3  and a y =3  respectively can be obtained with quintuplets of focusing and defocusing octupoles (OF and OD circuits) installed in the MI60 bypass while using RR30 straight as a “trombone”  With this option – besides mentioned 10 octupoles – a total of 8 additional quadrupoles should be installed in the MI60 bypass and RR30 straight for beta-functions matching after adjustment of phase advances  Depending on beam emittance (and the octupole strength required to obtain the indicated tuneshifts) the dynamic aperture (DA) varies from 3.5  at   N =4(  )mm  mrad to 3  at   N =2.5(  )mm  mrad (r.m.s. normalized values cited)  The above cited tuneshifts are produced owing to dQ x /dE x and dQ y /dE y. It is possible to produce tuneshifts via cross-detuning dQ x /dE y that can be obtained with a sextuplet of OC octupoles in the RR30 straight with proper rearrangement of phase advances and beta-functions matching. The OF and OD families are still needed to provide dQ x /dE x  dQ y /dE y  0  If the RR30 straight is used for the OC octupole family then some other straight must be used for the overall tune adjustment. In the result the total number of additional elements will be: 16 octupoles (10 in MI60 bypass and 6 in RR30) and at least 14 quadrupoles (4 in MI60 bypass, 4 in RR30 and 6 in not yet chosen tuning section). Without tuning section DA=3  for   N =4(  )mm  mrad  Large number of additional elements and only marginal dynamic aperture make a “nonlinear integrable optics insertion” for the purpose of generating a nonlinear tunespread ever more attractive.