Sequential bunch extractions in the Los Alamos PSR Page: 4 of 5
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and
2 2 aRI r
$ + - = - VV Isin $v
v &7 V L v
+ (Ib/Ig) cos ($ - $b)
where j2 = -1 and w (3.16 x 109 s-1 for
the frequency of the driving rf power.
the above equations, we have neglected sm
order of a/. We also assumed that the ti
of the voltage and/or current multiplied
much smaller thin w times that same quant
To complete the description of the
interaction, we need to include the follow
of beam motion:-1 ) s
- J (7") sin ($v b)
yt /presented in Figs. ? and 4. The numerical values of
(SIM - w)/a = 9.677 and A = 843.4 [Eq. (4)] are used;
the squared resonant frequency, 0 2, of the cavity is
assumed to vary linearly in time from SiA to Q2 in a
(3) time T = 20/a. It is al,,o assumed that the system is
initially in the state with dtb/dt = 0, $v = b = 0,
and V/I R = 1; the extraction is performed at t = 0,
the P58RIis q
In deriving when the tuning starts.
all terms of The results slow that the mismatch caused by the
me derivative extraction has the effect of increasing the phase and
. decreasing the magnitude of the voltage, but proper
by e-Jut is tuning of the cavities eventually will return the sys-
ity. tem to its normal state. The variation of particle
beam-cavity energy is always less than one-tenth of a per cent for
ung equation any extraction in the PSR sequence. The beam-current
phase closely follows the voltage phase with a syn-
chrotron oscillation of maximal amplitude 4. The
magnitude of the voltage also oscillates with the same
frequency. The largest variation in the voltage's
phase and magnitude are found in the case where ore(A 4 ) a? sin ( - b
9(4)
(0
Cwhere 3 and y are the usual relativistic pareneters, q
is the proton charge, E0 is the total energy of a
proton, yt is the transition y of the ring, and h is
the rf harmonic numk.
Because the interval between extractions is much
longer than the tuning period, each extraction can be
treated independently. Also, because the extraction
is a very rapid process, we can assume the average
beam current, and hence the Fourier content under con-
s deratiun, varies as a step function of time with
jumps at each extraction.
Note that because of the high bunching voltage,
the neglect of the term d[Ih exp(jib)]/dt in obtaining
Eqs. (2) and (3) is still valid, even though the beam
current takes such a rapid change. This appro nation
has been justified by numerically solving Eqs. (?)-(0.
without neglecting the v and d[ib exo (Jth)i/dt terms.
The ring stores only six bunches at maximum;
therefore, one can infer from Eqs. (?) and (3) that
for the quiescent state just before the extraction( -i) /ci (n/6)(2i - ci)/a - i A/lq ,
and in the steady state following the extraction
(ul - )/n - (n - 1)(2m - i)/fri -b1 g/i.U
40
20
as
eSI
jS 1 P 3 d 6
ut6 7
1.1
e.0
0.8%
_v
e.7
0.6
e.sFig. 3. Evolution of cavity voltage, phase of volt-
age, and relative phase between voltage and
beam current, for extracting one bunch out
of six and linear tuning of cavity 'esonant
frequency.(!)
()
In the above equations, n Is the number of hunchs io
the r ing before the extract ion; IhA and 11)11 -e the
bram current values before and after extraction,
respectively; In and S1A are the cavity resonant fre-
q ency at maximum beam loading and before an extrat-
Silon, respectively; WV is the resonant frequentv to
which the avIty should he tuned after the extraction.
Numerical Results
Ihe coupled nonlinear differential equations,
Iqs. (?)-(4) are solved numerically using the Rungq-
KutLa method. Results for the cases of extractiini
one bunch out of six and one hunch out of two areU
En
Wi
0
2 N
'~ 4v
fiT,- / b
AAA~i AAAAA t.A AftAAAA~a. ..b .i.e
".* RI
"7 vvy e,6
I i 4 " 7 "
at
Fig. a. Evolution of cavity voltage, phase of volt-
age, and relat ve phasP between voltage and
heam current for extracting one bunch out of
two and linear tun ioq of cavity resonant
frPquency...PVV .
v
RIQ
4b TE - (
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Wang, T. F.; Cooper, R. K. & Smith, L. Sequential bunch extractions in the Los Alamos PSR, article, January 1, 1983; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc1203531/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.