1. MEIC low rep-rate operation and path length
Jiquan Guo
2/19/2015

2. Luminosity and collision rate
𝐿≈ 𝑓 𝑐 𝑁 𝑒 𝑁 𝑖 4𝜋 𝜎 𝑥 ∗ 𝜎 𝑦 ∗ = 1 𝑞 𝑒 𝐼 𝑒 𝑁 𝑖 4𝜋 𝜎 𝑥 ∗ 𝜎 𝑦 ∗ = 1 𝑞 𝑒 2 𝐼 𝑒 𝐼 𝑖 𝑓 𝑐 4𝜋 𝜎 𝑥 ∗ 𝜎 𝑦 ∗
Luminosity can be either limited by charge per bunch (mainly due to beam-beam effect) or total beam current (mainly due to SR power/HOM impedance in e-ring and space charge effect at injection energy in i-ring).
To optimize luminosity, ideally we should choose a rep-rate (and emmitance) that let both charge per bunch and total beam current reach the limit.
The optimal rep-rate depends on beam energy and other factors; higher beam energy prefers lower rep-rate, due to lower beam current from stronger SR, and weaker beam-beam effect from more rigid beam and larger e-ring emmitance.
Currently, even at 4×30GeV, we still have room for beam-beam tune shift.
Collision rep-rate must be a fraction of main RF frequency and a multiple of beam revolution frequency.
An ideal ring harmonic number should provide as many factorization as possible. One good harmonic number for 476.3MHz base frequency is 3360 (=3*5*7*32)

3. Sync the electron and ion beam
To sync electron beam with the proton/ion beam with different energy/velocity, we need to adjust the path length of one storage ring.
We can also change the harmonic number of that storage ring, so the max change in path length is limited to 1 main RF wavelength.
If we change the path length of the e-ring, main RF frequency must be adjusted, which may cause challenge toward syncing with the injector as well as cavity design; changing the i-ring path length may be more challenging due to its higher energy.
Currently we assume that i-ring path length will be adjusted, and harmonic number is variable.

4. Injection and bunching
Electron ring:
Adjust injector gun trigger rate to achieve desired fill pattern
The gun already has a base trigger rate at 1/7 of ring RF frequency.
Need to lower the gun trigger rate further if other collision rate is desired. Even collision rate like 1/2 or 1/5 of ring RF frequency will result in higher charge per bunch in the injector, which may incur challenges for the gun R&D and RF transient effect, especially at low energy where injection time is already long.
Ion ring
Use different RF bunching frequencies after injection. Bunching voltage requirement won’t be prohibitive.
A multi frequency bunching cavity will be helpful

5. possible collision rate as a fraction of 476MHz
Option 1 for 476.3MHz system
e-ring harmonic number: 3360
Circumference(m)
ion ring
min
Max
possible collision rate as a fraction of 476MHz
Harmonic number
Energy(GeV/u) γ β
max
3360
36.617
39.037
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/10, 1/12, 1/14, 1/15, 1/16, 1/20, 1/21, 1/24, 1/28, 1/30, 1/32, 1/35, 1/40, 1/42, etc
3361
26.522
36.622
28.275
39.042 NA
3362
21.838
26.526
23.282
28.279
1/2,
3363
18.996
21.842
20.251
23.285
1/3,
3364
17.037
18.998
18.163
20.254
1/2,1/4
3365
15.582
17.039
16.612
18.165
1/5,
3366
14.447
15.584
15.401
16.614
1/3, 1/2, 1/6

6. Option 1 after 952.6MHz upgrade
e-ring harmonic number:
6720
Circumference(m)
ion ring
min
Max
Harmonic number
Energy(GeV/u) γ β
possible reprate as a fraction of 952MHz
max
36.617
39.037
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/10, 1/12, 1/14, 1/15, 1/16, 1/20, 1/21, 1/24, 1/28, 1/30, 1/32, 1/35, 1/40, 1/42, 1/48, 1/50, etc
6722
30.378
36.622
32.386
39.042
1/2,
6722, 6723
26.524
30.380
28.277
32.388
1/2, 1/3
6723, 6724
23.843
26.526
25.419
28.279
1/2, 1/3, 1/4
6724, 6725
21.840
23.845
23.284
25.421
1/2, 1/4, 1/5
6725, 6726
20.270
21.842
21.610
23.285
1/2, 1/3, 1/5, 1/6
6726, 6727
18.997
20.272
20.253
21.612
1/2, 1/3, 1/6, 1/7
6727, 6728
17.937
18.998
19.123
20.254
1/2, 1/4, 1/7, 1/8
6728, 6729
17.038
17.939
18.164
19.124
1/2, 1/3, 1/4, 1/8
6729, 6730
16.262
17.039
17.337
18.165
1/2, 1/3, 1/5, 1/10
6730
15.583
16.263
16.613
17.338
1/2, 1/5, 1/10
6732
14.447
15.584
15.401
16.614
1/2, 1/3, 1/4, 1/6, 1/12

7. possible reprate as a fraction of 476MHz
Option 2 for 476.3MHz system
e-ring harmonic number: 3360
Circumference m f
ion ring
Harmonic number
Energy(GeV/u) γ β
Circumference(m)
possible reprate as a fraction of 476MHz
min
max
3350
36.567
38.984
1/2,1/5,1/10
3351
26.485
36.572
28.235
38.990
1/3,
3352
21.807
26.488
23.248
28.239
1/2, 1/4, 1/8
3353
18.968
21.810
20.222
23.252
1/7,
3354
17.012
18.971
18.136
20.225
1/3, 1/2, 1/6
3355
15.559
17.014
16.588
18.139
1/5,
3356
14.425
15.561
15.379
16.590
1/2,1/4

8. possible reprate as a fraction of 476MHz
Option 3 for 476.3MHz system
e-ring harmonic number: 3360
Circumference m
ion ring
Circumference(m)
min
Max
Harmonic number
Energy(GeV/u) γ β
possible reprate as a fraction of 476MHz
max
3353
36.582
39.000
1/7,
3354
26.496
36.587
28.247
39.005
1/2, 1/3, 1/6
3355
21.817
26.500
23.259
28.251
1/5,
3356
18.976
21.820
20.231
23.262
1/2,1/4
3357
17.019
18.979
18.144
20.234
1/3,
3358
15.566
17.022
16.595
18.147
1/2,
3359
14.432
15.568
15.386
16.597

9. Summary
Low collision rate can help to improve luminosity at certain energies of MEIC. It can be achieved by change the injection pattern in e-ring and add bunching system in i-ring
Several harmonic number options are available to make the low collision rate possible for different energy, assuming harmonic number of ion ring will change to accommodate the ion velocity change and limit the path length change to 0.63m.