1. Synchronization in MEIC
J. Guo, Y. Zhang

2. 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

3. 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

4. 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

5. Basic schemes for MEIC sync
Ering fe0=1497MHz*7/22= MHz, Ne0=3416, i-ring doubles
Energy range 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 MHz for ering, MHz 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.

6. 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

7. 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
51.381
0.5 1
6833
35.274
47.272
38.594
51.382
6834
29.285
35.275
32.211
38.596
6835
25.536
29.286
28.216
32.213
6836
22.907
25.537
25.414
28.217
6837
20.933
22.909
23.311
25.416
6838
19.381
20.935
21.656
23.312
6839
18.119
19.382
20.311
21.658
6840
17.066
18.120
19.189
20.312
6841
16.171
17.068
18.235
19.191
6842
15.398
16.173
17.411
18.237
6843
14.721
15.399
16.690
17.412
6844
14.123
14.723
16.052
16.691
6845
13.588
14.124
15.482
16.053
6846
13.106
13.589
14.969
15.483
6847
12.670
13.107
14.504
14.970
6848
12.272
12.671
14.080
14.505
6849
11.908
12.273
13.691
14.081
ering circumference
ering Frequency(MHz)
Δf(kHz) e-ring
Δf(kHz) i-ring
min
max
df min
df max
476.28
476.35
-34.86
34.86
-69.71
69.72

8. 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
optimized ion Energy:
47.252
GeV
E-ring center circumference
e-ring circumference(m)
min
Max
Δ=0.63
i-ring Harmonic number
i-ring Energy(GeV/u) γ β
max
6832
35.266
38.586
6834
25.531
35.268
28.211
38.588
6836
20.931
25.534
23.308
28.214
6838
18.117
20.933
20.309
23.310
6840
16.170
18.119
18.233
20.311
6842
14.720
16.172
16.688
18.236
6844
13.586
14.722
15.480
16.690
6846
12.669
13.588
14.502
15.482
6848
11.906
12.671
13.690
14.504
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
139.46

9. 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
optimized ion Energy:
GeV
center circumference
e-ring Circumference(m)
min
Max
Δ=1.26
i-ring Harmonic number
Energy(GeV/u) γ β
max
6832
25.532
28.212
6836
18.116
25.536
20.307
28.216
6840
14.719
18.120
16.687
20.312
6844
12.667
14.722
14.501
16.691
6848
11.258
12.671
12.999
14.505
E-ring Frequency(MHz)
Δf(kHz) e-ring
Ion ring
min
max
df min
df max
476.18
476.46
139.48
278.96

10. 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
3416
6832
0.000 90
-4.775
-9.550
2.155 80
5.158 70
9.519 60
16.200 50
27.193
47.25
31.476
3417
6834
69.694
35.26
-0.068
-0.136
62.970
29.28
94.422
3418
6836
69.660
25.53
-0.093
-0.186
22.91
3419
6838
69.780
20.93
-0.122
-0.243
19.38
3420
6840
69.597
18.12
0.060
0.121
16.17
3421
6842
-0.152
-0.303
14.72
3422
6844
-0.209
-0.417
13.59
3423
6846
0.228
0.456
12.67
3424
6848
-0.114
-0.228 12
3425
6850
18.077
36.154