Synchronization in MEIC J. Guo, Y. Zhang

Basic timing constraints: (with fixed IP, same bunch spacing) Ne: harmonic number of e-ring Ni: harmonic number of i-ring Bunch spacing T0e=Se/c, T0i=Si/(βic) Te=NeT0, e-ring rev time Ti=NiT0, i-ring rev time (assuming RF freq=bunch reprate) First turn: t=0, e1 and i1 arrive at IP at the same time and collide t=T0=Se/c=Si/(βic), e2 collide with i2 constraint 1: e-ring and i-ring should have the same bunch reprate, and should have integer ratio of RF frequency t=2T0, e3 collide with i3 ……. Second turn: t=Ti=Te+nT0, i1 and e(n+1) arrive at IP at the same time and collide t=Ti+T0=Te+(n+1)T0, i2 collide with e(n+2) constraint 2: Difference between e-ring and i-ring rev time must be integer n times T0, Ni=Ne+n, even if we can make Ni and Ne non-integers. If we need to avoid gear change, n must equal 0. e2 i2 IP1 e1 i1 Se Si

Path length change is a must with continuously variable ion velocity Le-ring=Ne*T0*c Li-ring=(Ne+n)*T0*βic If “gear switch” is not allowed, n=0, path change range will be ~(βmax-βmin)*Lmax (in our case, Lmax~2150m, ion energy range from 12 to 100 GeV/u, and the range will be 5.6m) If “gear switch” is allowed, we can switch n to keep path change range within T0*c If we have to change n in increment of 2, then path change range is 2T0*c Higher bunch rep-rate with gear switching can minimize path change range

with 2 fixed IPs: e2 i2 IP1 e1 i1 Se Si IP2 t=0, e1 and i1 arrive at IP at the same time and collide If Ne and Np are even numbers, e(Ne/2+1) also collide with i(Ni/2+1) at 2nd IP at t=0 If Ne and Np are odd numbers, e(Ne/2+1.5) collide with i(Ni/2+1.5) at 2nd IP at t=0.5T0 If Ne is even and Ni is odd (vice versa), no 2nd IP collision t=T0=Se/c=Si/(βic), e2 collide with i2 t=2T0, e3 collide with i3 ……. IP2

Basic schemes for MEIC sync Ering fe0=1497MHz*7/22=476.318MHz, Ne0=3416, i-ring doubles Energy range 12-100 GeV/u Keep i-ring length constant, change e-ring length, as a result RF frequency needs to be changed W/o gear switching, path length range 5.6m, frequency range 0-1.23MHz for ering, 0-2.47MHz for i-ring Ne/Ni change simultaneously, can reduce frequency range to 0-140kHz for e-ring, 280kHz for i-ring W/ gear switching, Ne=3416, path length range 31cm, ½ luminosity loss in certain energy by changing Ni by 1 (freq range 70kHz for ering, 140kHz for iring), 63cm for 1 IP full luminosity by change Ni by 2 (freq range 140kHz for ering, 280kHz for iring), 126cm for 2 IPs by change Ni by 4 (freq range 280kHz for ering, 560kHz for iring). With path change range 31cm, luminosity can be recovered to full with e-ring 952.6MHz SRF upgrade Keep e-ring length constant, change i-ring path, no RF frequency change W/o gear switching, path length range 5.6m W/ gear switching, Ne=3416, path length range 31cm if we can suffer ½ luminosity loss in certain energy by changing Ni by 1, 63cm for 1 IP full luminosity by change Ni by 2, 126cm for 2 IPs by change Ni by 4.

The case of different bunch spacing T0i=Si/(βic), T0e=2T0i Ni=2Ne+1 First turn: t=0, e1 and i1 arrive at IP at the same time and collide t=T0i, empty electron bucket with i2 t=2T0i e2 collide with i3 t=3T0i, empty electron bucket with i4 t=4T0i, e3 collide with i5 No collision at IP2 ……. 2nd turn t=Ni*T0i, empty electron bucket collide with i1 t=(Ni+1)*T0i, e2 collide with i2 t=(Ni+2)*T0i, empty electron bucket collide with i3 t=(Ni+3)*T0i, e3 collide with i4 e2 i2 IP1 e1 i1 Se Si Electrons collide at full current, but ions only at half, lose 50% luminosity

Ne=3416 (476.3MHz). Center energy 60.6 GeV/u Parameters: change i-ring harmonic by 1, change e-ring path by up to 31cm, change frequency for both rings Ne=3416 (476.3MHz). Center energy 60.6 GeV/u Luminosity factor with 476 e-ring Luminosity factor with 952 e-ring i-ring Harmonic number i-ring Energy(GeV/u) γ β min max 2 6832 47.271 100.169 51.381 107.760 0.99981 0.99996 0.5 1 6833 35.274 47.272 38.594 51.382 0.99966 6834 29.285 35.275 32.211 38.596 0.99952 6835 25.536 29.286 28.216 32.213 0.99937 6836 22.907 25.537 25.414 28.217 0.99923 6837 20.933 22.909 23.311 25.416 0.99908 6838 19.381 20.935 21.656 23.312 0.99893 6839 18.119 19.382 20.311 21.658 0.99879 6840 17.066 18.120 19.189 20.312 0.99864 6841 16.171 17.068 18.235 19.191 0.99850 6842 15.398 16.173 17.411 18.237 0.99835 6843 14.721 15.399 16.690 17.412 0.99820 6844 14.123 14.723 16.052 16.691 0.99806 6845 13.588 14.124 15.482 16.053 0.99791 6846 13.106 13.589 14.969 15.483 0.99777 6847 12.670 13.107 14.504 14.970 0.99762 6848 12.272 12.671 14.080 14.505 0.99747 6849 11.908 12.273 13.691 14.081 0.99733 ering circumference ering Frequency(MHz) Δf(kHz) e-ring Δf(kHz) i-ring min max df min df max 2149.857 2150.172 476.28 476.35 -34.86 34.86 -69.71 69.72

Parameters: change i-ring harmonic by 2, change e-ring path by up to 63 cm, change frequency for both rings e-ring harmonic number: 3416 iring circumference 2149.607 optimized ion Energy: 47.252 GeV E-ring center circumference 2150.015 e-ring circumference(m) min 2149.70 Max 2150.33 Δ=0.63 i-ring Harmonic number i-ring Energy(GeV/u) γ β max 6832 35.266 100.008 38.586 107.588 0.99966 0.99996 6834 25.531 35.268 28.211 38.588 0.99937 6836 20.931 25.534 23.308 28.214 0.99908 6838 18.117 20.933 20.309 23.310 0.99879 6840 16.170 18.119 18.233 20.311 0.99849 6842 14.720 16.172 16.688 18.236 0.99820 0.99850 6844 13.586 14.722 15.480 16.690 0.99791 6846 12.669 13.588 14.502 15.482 0.99762 6848 11.906 12.671 13.690 14.504 0.99733 E-ring Frequency(MHz) Δf(kHz) e-ring Δf(kHz) i-ring min max df min df max 476.25 476.39 -69.71 69.73 -139.42 139.46

Parameters: change i-ring harmonic by 4 to keep two fold symmetry (keep two IPs and two polarize bunch train gap symmetry for 476MHz e-ring), change e-ring path by up to 1.26m, change frequency e-ring harmonic number: 3416 iring Circumference 2149.293 optimized ion Energy: 35.2677 GeV center circumference 2150.015 e-ring Circumference(m) min 2149.39 Max 2150.64 Δ=1.26 i-ring Harmonic number Energy(GeV/u) γ β max 6832 25.532 100.102 28.212 107.687 0.99937 0.99996 6836 18.116 25.536 20.307 28.216 0.99879 6840 14.719 18.120 16.687 20.312 0.99820 6844 12.667 14.722 14.501 16.691 0.99762 6848 11.258 12.671 12.999 14.505 0.99704 E-ring Frequency(MHz) Δf(kHz) e-ring Ion ring min max df min df max 476.18 476.46 -139.40 139.48 -278.79 278.96

Parameters: change e-ring and i-ring harmonic simultaneously (no gear switch), change e-ring path by up to 5.6m, change frequency Δfp=±140kHz, OK with existing CEBAF tuner design, and maybe injection sync is fine (CEBAF cavities can tune ±200kHz) Energy(GeV/u) max Δfp (kHz) 279.01 max(dLe) (cm) 558.27 ions γ β Ne Np fe (MHz) dfe (kHz) fp (MHz) dfp (kHz) Le (m) dLe (cm) 100 107.5789 0.999957 3416 6832 476.318 0.000 952.636 2150.015 90 96.92101 0.999947 476.313 -4.775 952.627 -9.550 2150.036 2.155 80 86.26312 0.999933 476.307 -11.428 952.614 -22.855 2150.066 5.158 70 75.60523 0.999913 476.297 -21.088 952.594 -42.176 2150.110 9.519 60 64.94734 0.999881 476.282 -35.886 952.565 -71.773 2150.177 16.200 50 54.28945 0.99983 476.258 -60.235 952.516 -120.470 2150.287 27.193 47.25 51.35853 0.99981 476.248 -69.723 952.497 -139.446 2150.329 31.476 3417 6834 476.388 69.694 952.776 139.388 35.26 38.57972 0.999664 -0.068 -0.136 2150.644 62.970 29.28 32.2063 0.999518 -69.716 -139.433 2150.959 94.422 3418 6836 69.660 139.319 25.53 28.20959 0.999371 -0.093 -0.186 2151.274 125.921 22.91 25.41723 0.999226 476.249 -69.555 -139.110 2151.588 157.298 3419 6838 69.780 139.561 20.93 23.30696 0.999079 -0.122 -0.243 2151.903 188.874 19.38 21.65499 0.998933 -69.698 -139.395 2152.218 220.311 3420 6840 69.597 139.194 18.12 20.3121 0.998787 0.060 0.121 2152.532 251.731 16.17 18.23381 0.998495 3421 6842 -0.152 -0.303 2153.162 314.766 14.72 16.68841 0.998203 3422 6844 -0.209 -0.417 2153.792 377.732 13.59 15.48407 0.997912 3423 6846 0.228 952.637 0.456 2154.419 440.474 12.67 14.50355 0.99762 3424 6848 -0.114 -0.228 2155.050 503.568 12 13.78947 0.997367 3425 6850 476.336 18.077 952.673 36.154 2155.597 558.275