Pi-MODE STRUCTURES - RESULTS AND IMPLICATIONS FOR OPERATION Page: 3 of 4
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n 1 (n+1)n(n-1)(n-2) 2(n-3)(n+2)3 k
in(full)/(-1) - 61+n(n-1) k22 1-k(1+/)+ 15 (n+1)-
4 [(n -1)A i + - - + 3An-3 + 2An-2 +Ai]-(- - .+7En-4,n-3 +5En-3,n-2 + 3En-2,n-1 + Enin) .
k. n(n-1) F k(1+/3) (n-2)(n+1)$3 k7
- Q - +
kQ _ 2 3 2nJ5(N -1)2
where /$= 8Q2 , the 21 mode frequency correction factor.n-1 2 n(n -1) 2(n - 2) 2(n - 3)(n +1),6
in (half) /(-1)"- 1+/$(n-1) 22 L-k(1+3)+ 15 -~Ln1Al +..+3An-3 +2An-2+A- (.+ n4n
4 C(n-1)
(n -1)2 11_k (+)+n(n - 2)$
kQ 2 3
Field tilt depends on the square of cell number, and is
linear to frequency and coupling constant errors.
For most applications Eq. 2 can be simplified to
i, (full) /(-1) n- E 1 + n(n -1)3- jn(n -1)/kQ , showing
that the phase shift difference in the average on-axis fields
from end cell to drive cell is given by
AhN (full) E -N(N -1) / kQ in radians for small values,
useful information not easily obtained from SUPERFISH.
3 Results of Calculations for the 7r Mode
Extensive information for designing, constructing and
operating pi-mode structures is available, especially for
superconducting cavities [12-14]. This report provides
additional information to assist sensitivity understanding.
3.1 Off-Resonance Field Tilt
Differentiation of Eq. 2 with respect to /3 yields:
Ain (full) /(-1)n-i E 2n(n -1)(1- k)Afr / kf , an equation
independent of Q that has been validated with LOOPER
simulations. In addition to the usual considerations for
on-resonance control and for Lorentz force effects, a five-
cell, 1% k, cavity with required field tolerance of %,
a frequency tolerance Af / f is 1.25*10-6 or 1 kHz at
800 MHz is needed. The tilt in field changes direction, as
expected, on either side of resonance, introducing an
interesting aspect to some control algorithms.
3.2 Frequency and Coupling Errors in the Cells
Eqs. 2 and 3 show that an error in cell-to-cell coupling,
En-1,n , or in frequency, An , propagates throughout the
entire cavity fields and is independent of Q. To minimize
field errors, a frequency error in the nth cell requires
adjacent cells to have opposite sign errors, one-half the nth
value. This correction yields field distributions almost as
flat as if there were no errors in the cavity, except in the
local cell, n. A complex relationship involving all
constants must be satisfied to achieve fields almost as flat
as those without coupling errors.3 + 5E_3,n-2 + 3En-2,n-1 + Enl,) - -
(3)
Propagating through the entire system means, for
example, a A / k or E error of 1% yields field errors in a
typical five-cell cavity (k=0.01, loaded Q=60,000) along
the length of from 2% up to 8% and from 1% up to 3%,
respectively, depending on the location of the error.
Resonance shifts df,(Hz) = f,(Hz)ken,,/[2(1-k)N] and
df,(Hz) = (f,/f0)ALfo(Hz)/N for coupling constant and
frequency errors, respectively.
3.3 End Cell Tuning and Coupling Constants
A series of SUPERFISH calculations were completed
to determine parameters for end cell tuning as a function
of cell beta and beam-bore hole, the latter of which affects
the coupling constant. One easy method to obtain flat
fields is to make the end beam hole larger than the beam
bore hole for at least one end beam hole diameter in
length away from the end cell. Best fit to the data was
[end-radius(cm-GHz) = 1.24*bore-radius(cm-GHz)-0.46].
For instance, a 3 cm bore has a 3.26 cm end bore at 1
GHz, while a 1 cm bore has a 1.08 cm end bore at 3 GHz.
As described, there are many advantages for a coupling
constant, k, as high as possible. In the design process for
a superconducting cavity, many variables are considered
as described in reference 12. "k" was determined as a
function of aperture bore radius for different cell betas.
In order to maintain relative cell fields to within %,
for a five-cell, 1% k, cavity, coupling constant errors need
to be within 1/6% cell to cell. Tolerances on aperture
dimensions vary from about 0.0013 cm for a 0.5 beta cell
to 0.0016 cm for a 1.0 beta cell, attainable tolerances.
3.3 Dispersion Curve Characteristics
Detuning end cells of a full-cell terminated cavity has a
significant effect on the dispersion curve. An inspection
of dispersion curves shows extreme sensitivity between
the )n mode and the next nearest mode. The change
between these two modes is most noticeable as the
frequency of the end cell changes. The smaller the
number of cells, the more exaggerated the curve becomes.
This changing dispersion curve pattern can be used to
estimate or to infer )n mode field distributions in a cavity(2)
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SCHRIBER, S. O. Pi-MODE STRUCTURES - RESULTS AND IMPLICATIONS FOR OPERATION, article, June 1, 2001; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc720280/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.