# INSTABILITIES RELATED WITH RF CAVITY IN THE BOOSTER SYNCHROTRON FOR NSLS-II Page: 4 of 5

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HOW TO SUPPRESS HIGHER-ORDER

MODES PROBLEM

It seems that the shunt impedances as for TMO11 mode

are too high to store beam in the booster ring. However,

the electric field of TMO 11 mode focuses on the top of a

cavity, where an input coupler is installed. Thus electric

field of TMO 11 is strongly coupled with an input coupler.

Consequently the impedance of TMO11 decreases

automatically. The TM110 mode kicks beam transversely

by the magnetic fields. On the other hand, TM1 11 mode

kicks beam by the electric field. If the coupled-bunch

instability of transverse modes occurs in an actual beam

operation, we cannot add the beam anymore over the

threshold current. In general, TM110 and TM111 are

notorious modes as coupled-bunch instability. First of all,

we calculate threshold currents due to HOM's.

THRESHOLD CURRENT

Let's calculate the threshold current in each mode. In

order to calculate the threshold currents as for

longitudinal and transverse HOM's, we assume that the

theoretical threshold current is given by that the

instability will arise if the growth rate exceeds the

damping rate [4][5][6][7][8] and also we assume the

beam has a rigid form. When higher-order modes just

coincide with the revolution frequency, they are grown

and kick stored beam. The revolution frequency is 1.894

MHz. If one of HOM's coincides with or is closes to the

revolution frequency, beam with the energy of 200 MeV

injected from the linac doesn't survive due to the coupled-

bunch instability. The threshold currents concerning

longitudinal and transverse modes are given by the

following expressions of2. Eo - f,

1,h(longitudinaJ= 2*sF0,

r-e-a-f,- f - Z(3-1)

Eo:electron energy

fS :synchrotron frequency

r,:longitudinal damping rate

e: electron charge, e=1 we take natural unit

(1: momentum compaction factor

fr: revolution frequency (=1.893 x 10^6 Hz)

.fHom: HOM frequency

Z: impedance for HOM1, traversee) = 2E ,

s pa.,8,- 2r,

T. ransverse damping rate, z. = 2z(3-2)

e: electron charge, e=1 we take natural unit

6T: beta-function.

If one of HOM's coincides with just the revolution

frequency line, stored beam would be lost almost

completely at the beam energy of 200 MeV. As an

example, threshold currents for TM110 mode arecalculated to be 65 nA at 200 MeV and 3.3 mA at 3 GeV,

respectively. Basing on those threshold currents, we

present the suppression method of coupled-bunch

instability in the next section.

SUPPRESSION OF HOM PROBLEM

In order to store the beam current stably, we have to use a

something HOM suppression methods. From the results

of the threshold current given in the previous section, we

show a method that makes impedances smaller by shifting

the frequencies of HOM from the revolution frequency.

To store beam current over 50 mA, we can obtain the

required impedances from the both equations of (3-1) and

(3-2), and result the values 2 ohms for TMO11, 36 ohms

for TM110 and 28 ohms for TM111, respectively. To

reduce the values of those impedances, we detune the

frequencies of HOM from the revolution frequency. We

use unloaded Qo values in calculating the frequency

shifts. The Qo value is expressed by using the resonant

frequency f and Af, which represents the full width at half

maximum, Q=f/Af. Let's assume Gaussian as the

distribution function of impedances. Since the center of

distribution function gives the maximum impedance, we

can calculate the frequency shift. As for TMO11 mode,

the impedance becomes 2 ohms at the frequency shift of

56 kHz from the center of frequency. And also we can

calculate the frequency shifts as for TM110 and TM111,

and we get 33 kHz and 58 kHz, respectively. Thus we

obtained required amount of frequency shifts as for

HOM's. What we need the next is to obtain the relation

between cavity temperature and frequency shift of each

HOM. In order to obtain the relation between cavity body

temperature and frequency shift of HOM, we carried out

that we changed cooling water temperature for 7-cell

cavity and took frequency shift for each HOM. When we

change the water temperature, we adjusted the resonance

frequency of cavity at the fundamental frequency of

499.67 MHz by using two movable tuners. And a typical

data is shown in Fig. 1. However, we could not observe

the mode of TMO12 in Table 1. We finally obtained the

relation between the cavity body temperature and

frequency shifts of HOM's as shown in Fig. 2. We

showed two cases of both without tuning at the

fundamental frequency and with tuning.

From Fig. 2, we can know the frequency shifts of

TM011, -10.1 kHz/degree for TM110, -13.2 kHz/degree

for TM111 modes, -22.5 kHz/degree for TM021,

respectively. In order to reduce those impedances, we can

know better cavity body temperature. Taking into account

of various conditions as for utility, we decided to set the

cooling water temperature of 35 degrees. Let us plot all

frequencies of HOM on the line of revolution frequency

at the cavity body temperature of 35 degrees and we

obtain the Fig. 3. The numbers of points correspond to the

frequencies of HOM's. The frequency differences

between revolution lines and (1) and (13) are 59 kHz and

100 kHz, respectively.

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Kawashima, Y.; Cupolo, J.; Ma, H.; Oliva, J.; Rose, J.; Sikora, R. et al. INSTABILITIES RELATED WITH RF CAVITY IN THE BOOSTER SYNCHROTRON FOR NSLS-II, article, Date Unknown; United States. (digital.library.unt.edu/ark:/67531/metadc845813/m1/4/: accessed December 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.