One of Issues Being Discussed Recently for Utilizing a 7-Cell Cryomodule Renaissance In

HOM DAMPING REQUIREMENT FOR 12 GeV UPGRADE Byung C. Yunn JLAB-TN-04-035 November 29, 2004

A detailed study has been carried out to determine the HOM damping requirement of new 7-cell CEBAF cavity for use at the 12 GeV CEBAF Upgrade.

CONCLUSIONS

In order to guarantee a stable beam operation of the CEBAF 12 GeV upgraded machine with design currents HOMs of new 7-cell cavity should be damped to Q values less than the upper limits stated in the following.

7.5×106 for 1874 mode, 2.0×107 for 2110 mode, 8 2 Q = 6.2×10 /(R/Q)T for all other modes (Note that (R/Q)T = R/Q/(ka) where k is the wave number and a is the offset at which R/Q is measured.).

DIPOLE MODES OF 7-CELL CEBAF CAVITY

Table 1: M =1 mode spectrum of 7-Cell cavity

Frequency(MHz) (R/Q)T(Ω)

1724.152 0.216 1746.066 0.06 1780.721 3.756 1825.357 1.908 1874.220 82.548 1928.851 62.026 1995.611 3.038 2004.930 16.272 2073.533 1.964 2094.526 29.294 2110.941 27.528 2115.194 1.304 2118.917 4.476 2119.273 0.526

1 Dipole modes relevant for breaking the 12 GeV beam up are listed in the above Table 1. URMEL code is used to prepare this table.

BEAM TRANSPORT IN THE 12 GEV UPGRADE

Two cases are considered for this work. Case 1 is when new cryomodules are installed at the end of each linac. This is most natural case because there exist empty slots for the modules. Optics for this case is shown below for future references:

Arc 1: Identity matrix Arc 2: X= 0.142344E+01*X + 0.000000E+00*PX PX= -0.142767E+00*X + 0.702521E+00*PX Y= 0.545133E+01*Y + 0.000000E+00*PY PY= -0.624146E+00*Y + 0.183441E+00*PY Arc 3: X= 0.856560E+00*X + 0.000000E+00*PX PX= -0.159952E+00*X + 0.116746E+01*PX + Y= 0.947501E+00*Y + 0.000000E+00*PY PY= -0.118434E+00*Y + 0.105541E+01*PY Arc 4: X= 0.190101E+01*X + 0.000000E+00*PX PX= -0.367200E+00*X + 0.526035E+00*PX Y= 0.313875E+01*Y + 0.000000E+00*PY PY= -0.335343E+00*Y + 0.318598E+00*PY Arc 5: X= 0.760759E+00*X + 0.000000E+00*PX PX= 0.869696E+00*X + 0.131448E+01*PX Y= 0.881085E+00*Y + 0.000000E+00*PY PY= 0.526024E+00*Y + 0.113496E+01*PY

Arc 6: X= 0.194635E+01*X + 0.000000E+00*PX PX= -0.240086E+00*X + 0.513783E+00*PX Y= 0.255227E+01*Y + 0.000000E+00*PY PY= -0.343587E+00*Y + 0.391808E+00*PY Arc 7: X= 0.664191E+00*X + 0.000000E+00*PX PX= 0.392200E+00*X + 0.150559E+01*PX Y= 0.687775E+00*Y + 0.000000E+00*PY PY= 0.334952E+00*Y + 0.145396E+01*PY Arc 8: X= 0.181065E+01*X + 0.000000E+00*PX PX= -0.549226E+00*X + 0.552287E+00*PX

2 Y= 0.227939E+01*Y + 0.000000E+00*PY PY= -0.672010E+00*Y + 0.438715E+00*PY Arc 9: X= 0.662480E+00*X + 0.000000E+00*PX PX= -0.103359E+00*X + 0.150948E+01*PX Y= 0.691714E+00*Y + 0.000000E+00*PY PY= 0.724938E-01*Y + 0.144568E+01*PY

In the Case 2 new modules in the North linac are placed at the beginning of the linac rather than at the end while new modules are at the end in the South linac. This has a slight advantage over the previous case in beam matching. Recirculation beam transport matrices for this case are as follows:

Arc 1: Identity matrix Arc 2: X= 0.123690E+01*X + 0.000000E+00*PX PX= -0.216092E+00*X + 0.808470E+00*PX Y= 0.387823E+01*Y + 0.000000E+00*PY PY= -0.133604E-01*Y + 0.257850E+00*PY Arc 3: X= 0.113220E+01*X + 0.000000E+00*PX PX= 0.798493E-02*X + 0.883236E+00*PX Y= 0.128668E+01*Y + 0.000000E+00*PY PY= -0.176463E+00*Y + 0.777196E+00*PY Arc 4: X= 0.165312E+01*X + 0.000000E+00*PX PX= -0.404861E+00*X + 0.604916E+00*PX Y= 0.246859E+01*Y + 0.000000E+00*PY PY= -0.175126E+00*Y + 0.405090E+00*PY Arc 5: X= 0.104647E+01*X + 0.000000E+00*PX PX= -0.229461E+00*X + 0.955590E+00*PX Y= 0.101409E+01*Y + 0.000000E+00*PY PY= -0.301945E+00*Y + 0.986100E+00*PY Arc 6: X= 0.165495E+01*X + 0.000000E+00*PX PX= -0.266540E+00*X + 0.604250E+00*PX Y= 0.208050E+01*Y + 0.000000E+00*PY PY= -0.149929E+00*Y + 0.480650E+00*PY Arc 7: X= 0.983941E+00*X + 0.000000E+00*PX PX= 0.294292E+00*X + 0.101632E+01*PX Y= 0.938620E+00*Y + 0.000000E+00*PY PY= 0.206367E+00*Y + 0.106540E+01*PY Arc 8: X= 0.157979E+01*X + 0.000000E+00*PX

3 PX= -0.476391E+00*X + 0.632995E+00*PX Y= 0.189021E+01*Y + 0.000000E+00*PY PY= -0.381663E+00*Y + 0.529000E+00*PY Arc 9: X= 0.856270E+00*X + 0.000000E+00*PX PX= 0.187445E+00*X + 0.116786E+01*PX Y= 0.857534E+00*Y + 0.000000E+00*PY PY= 0.152221E+00*Y + 0.116614E+01*PY

X, Y are horizontal and vertical coordinates in cm unit. PX, PY are respective momentum in MeV/c unit. Y. Chao has worked out recirculation optics for the 12 GeV Upgrade. Note that here Arc means from the center of quad immediately after the exit of the last cryomodule of a linac to the center of quad immediately before the first cryomodule of the next linac.

Other assumptions for simulations are:  The injection energy is 122.625 MeV, and the cavity gradient is 7.5 MV/m for all cavities in 40 old cryomodules and 17.5 MV/m, for all cavities in 10 additional new cryomodules.  HOMs in old cryomodules are damped sufficiently which are consistent with Amato's measured values for HOM parameters. R/Q from URMEL calculation is used as there exists no measured data for R/Q.  Quadrupoles in linacs are set to the 120° phase advance per period as for the original 4 GeV accelerator.

Another important parameter required for BBU simulation is the total recirculation path length which is 6310 (6310, 6301, 6298) RF wavelength for the 1st (2nd, 3rd, 4th) pass for the CEBAF accelerator.

BEAM BREAKUP SIMULATIONS

1. We find no significant difference in BBU thresholds between Case 1 and 2. See Figure 1 and 2. Therefore, decision to choose one must come from other than BBU consideration. We will only present results for the Case 1 from here on. 2. Let us first consider the case where only the 1874 mode can be excited in each cavity and frequencies are distributed randomly with the full width of 1 MHz. This should be a good approximation of real machine dominated by a single HOM. Threshold distributions for various Q values are shown in Figures 3 – 7. We find that for Q > 106 BBU threshold can safely be obtained by scaling that of Q = 106 (i.e. BBU threshold is inversely proportional to Q). It is obvious that one can get easily misled in understanding BBU problem in a machine if sample size is too small. 3. Figure 8 and 9 show the dependence of some BBU parameters on HOM Q.

4 4. Having exhaustively studied BBU with the 1874 mode we then looked at 2110 mode. The HOM spec for this mode mentioned at the beginning is based on that study. 5. We have also studied several other interesting variations of BBU phenomena at the 12 GeV Upgrade including the machine operation at 6 GeV, frequency scan of a breaking mode, and BBU with 10 MHz frequency spread among others. 6. We want to close by asserting that a 6 GeV operation at the 12 GeV upgraded CEBAF accelerator at the 200 microampere of beam current will be stable if HOMs meet the damping requirement stated in the CONCLUSIONS section.

ACKNOWLEDGEMENTS

Helps from K. Beard, Y. Chao, C. Tennant are greatly appreciated.

5 Figure 1. BBU threshold distribution (100 samples) for the 1874 mode with Q = 106. Horizontal axis is BBU threshold in mA. Results are obtained with the recirculation optics of Case No. 1.

6 Figure 2. BBU threshold distribution (100 samples) for the 1874 mode with Q = 106. Horizontal axis is BBU threshold in mA. Results are obtained with the recirculation optics of Case No. 2.

V_npnts= 500; V_numNaNs= 0; V_numINFs= 0; V_avg= 0.441838; V_sdev= 0.160139; V_rms= 0.469908; V_adev= 0.119776; V_skew= 1.53807; V_kurt= 2.94261; V_minloc= 205; V_min= 0.231; 20 V_maxloc= 298; V_max= 1.19;

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 3. BBU threshold distribution (500 samples) for the 1874 mode with Q = 107. Horizontal axis is BBU threshold in mA.

160 V_npnts= 3999; V_numNaNs= 0; V_numINFs= 0; V_avg= 4.26399; V_sdev= 1.61116; V_rms= 4.55816; V_adev= 1.17483; V_skew= 1.72746; V_kurt= 4.63207; V_minloc= 3691; V_min= 1.41; V_maxloc= 2872; V_max= 14.1; 140

0 0 2 4 6 8 10 12 14

V_npnts= 1499; V_numNaNs= 0; V_numINFs= 0; V_avg= 26.4595; V_sdev= 7.9538; V_rms= 27.6284; V_adev= 6.19483; 50 V_skew= 0.944519; V_kurt= 1.56937; V_minloc= 856; V_min= 11.4; V_maxloc= 214; V_max= 68.1;

0 0 20 40 60 80

10 V_npnts= 2500; V_numNaNs= 0; V_numINFs= 0; V_avg= 69.4879; V_sdev= 11.1119; V_rms= 70.3704; V_adev= 8.8516; V_skew= 0.468016; V_kurt= 0.154548; V_minloc= 547; V_min= 40.9; V_maxloc= 1838; V_max= 116;

0 0 20 40 60 80 100 120

11 V_npnts= 5000; V_numNaNs= 0; V_numINFs= 0; V_avg= 267.57; V_sdev= 11.6707; V_rms= 267.824; V_adev= 9.34735; V_skew= 0.428823; V_kurt= 0.221425; V_minloc= 2835; V_min= 238; V_maxloc= 3743; V_max= 326;

0 220 240 260 280 300 320 340

12 Figure 8. BBU threshold current vs. Q of the 1874 mode is shown. In the Figure circles are for average values and crosses above and below represent maximum and minimum threshold current values.

13 Figure 9. Full range and standard deviation of BBU threshold current vs. Q are shown. Circles are for full spread. Crosses are for standard deviation multiplied by 10 for the convenience of drawing the graph. Note a different scaling behavior for both quantities.