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A METHOD FOR CALCULATING RESONANT FREQUENCIES, MODE STRVCTURE
AND FIELD FLATNESS IN AN ALVAREZ TYPE LINAC CAVITY
R. L. Gluckstern+,
University
of Massachusetts,
and
M. J. Lee, R. Chasman and H. K. Peterson
Brookhaven
National
Laboratory
Amherst
Summary
The method of Walkinshaw,
Sahel and
Outram has been extended
to cavities
consisting
of drift
tube loaded cells
with
varying
lengths
in order to determine
the
field
flatness
and sensitivity
to errors
for the various
modes of a realistic
linac.
A computational
program has been
written
which gives the resonant
frequency
and field
pattern
for a multi-cell
cavity
with arbitrary
drift
tube lengths
and gap
The object
of the study is to
lengths.
obtain
the following
information:
1)
Mode structure
and resonant
frequencies
of a cavity
consisting
of several
identical
cells
in order to obtain
the dis2) Sensitivity
of the
persion
curve.
field
flatness
to dimensional
errors
for
3) Mode structhe zero and n/2 modes.
ture and field
flatness
for a cavity
consisting
of several
cells
with differing
Results
of these
geometrical
parameters.
computations
are presented
and discussed.
are corrected
by "ttuners"
which are adjusted
to yield
the desired
fields,
along the lines
suggested
by the above
analogues.
When the (conceptual)
end walls
on
each cell
are removed, the varying
geometry will
cause a readjustment
of the
field
patterns.
No reliable
estimate
of
this effect
has yet been made.
In the
such effects
have been hidden in
past,
the construction
imperfections
and are
removed by tuners.
However, the advent
of compensated*
structures
provides
multicell
cavities
that are insensitive
to
construction
errors.
This property
also
brings
with it the inability
to correct
any undesired
field
patterns
by tuners.
The purpose of this work is to present a method for determining
the field
pattern
for a multi-cell
assembly of
tuned cells
of varying
geometry.
The
method also permits
the study of the effect of tuning
errors.
In addition,
one can examine these sensitivities
for
compensated
structures
by studying
the
effect
of errors
on the 7r/2 mode.
A further by-product
of the method is that it
permits
a study of the dispersion
curves
for several
of the bands which occur in
an accelerating
structure.
Introduction
The conventional
procedure
for obtaining
the proper parameters
to resonate a multi-cell
linac
cavity
in a mode
which provides
a desired
accelerating
field
pattern
is the following:
cell parameters
are deter1) Single
mined, either
by computation1
or by modeling,
to provide
resonance
at a given
frequency.
2) The lengths
of adjacent
cells
are
adjusted
so that a particle
moving at
synchronous
phase through
the fields
of
each cell will
gain the necessary
energy
to remain in step.
3) The multi-cell
cavity
is then
taken to be an assembly of these single
cells
without
end walls.
By analogy with
the properties
of a tapered wave guide,
or a wave guide filled
with a medium
whose dielectric
constant
varies
with
axial
distance,
or a chain of coupled
the field
variation
resonant
circuits,
is expected
to vary with axial
distance
Any imperfections
in the desired
way.
which cause perturbations
of the field
Model
The structure
to be analyzed
is
shown in Figure
1. In analogy with the
method of Walkinshaw,
Sahel and Outramj,
one considers
the field
within
the drift
tube radius
r=a to be independent
of z
and determined
by the single
parameter
En
in the ns gap.
The electric
field
in
the n!& gap is then
Jo(kr)
EZ = En Jo0
where
k = w/c
with the magnetic
field
given in terms of
En by Maxwell's
equations.
The axial
electric
field
in the region
between the
drift
tube radius
r=a, and the wave guide
340
radius
r=b is
expanded
in
the
mnsn
form
Hna(m)
Fo(kr)
EZ = f oFo(ka)
m
t Cf cos
lm
(2)
m7rsL mng
mwR
sin
"sin
-%x
= cosdtcos~
8g2/m2T2gngF.
and sn is the distance
from
of the cavity
to the center
gap with length
g,.
(8)
the end wall
of the ng
where
Yo(Kmb)
Fo(Kmr) = Yo(Kmr) - Jo(Kmr) J0 m
Here
F1kmr)
= Yl(Krnr)
- J1(Kmr)
Yo(Kma)
Jo0
(9)
Gl(umr)
= Kl(u,r)
t Il(umr)
Ko(l-lma)
I0 m
, (10)
(3)
Go(l-lmr)
= Ko(limr)
- Io(umr)
Kohmb)
m
0
m
c
for
(F)
gives
(4)
F.
- k* = urn2 ) 0
one now forms
the
PnR(k)E n = EQ
to the
L = 1,2,...N
form
(11)
vector
and
Our model is not expected
to work for
those cases where the electric
field
is
not approximately
uniform
in each gap.
This shows up in the numerical
work by
an inability
to obtain
solutions
for
those modes where the electric
field
is
not concentrated
in the gaps.
This difficulty
can probably
be removed by taking
one additional
parameter
to describe
the
first
harmonic
of the axial
electric
field
within
each gap.
‘-
Numerical
Results
and Discussion
A computer program has been written
to perform the calculations
represented
by Eq. (11) and the auxiliary
Equations
(12),
(71, (81, (3),
(41, (91, (10).
In
particular,
one calculates
Det A for a
given choice of frequency
and then searches for those values of frequency
for
Since some of the
which Det A vanishes.
resonant
frequencies
of a long cavity
are
reasonably
close together,
high accuracy
is required
in the calculation
(of the
order of one part per million
for various
This
components like Bessel functions).
is achieved
in Eq. (7) by cutting
off the
sum at a relatively
high value of m,
after
which an asymptotic
form is used
for high m, permitting
the analytic
evaluation
of the remaining
sum to m = m.
fol-
(6)
where
+iH
(m)
m=l nR
in matrix
= gR P&kc)
- dnR
(12)
z
For our model the resonant
frequencies
are determined
by the vanishing
of the
determinant
of A
and the relative
field
strengths
EL aren#ien
determined
by Eq.(ll).
(5)
/dvE'
it can be shown in general
th$t q2 will
be a minimum, and equal to k, , where
w /27i = k c/2rr is the correct
resonant
q2
fPequencyO
In our case, we consider
to be a function
of our choice of k and
the individual
En.
It can be shown that
q2 will
be a minimum, subject
to the var
iation
of k and each En, if the average
magnetic
field
on each side of the r=a
boundary
of each gap is equal.
We
therefore
impose this
condition
as the
means of determining
the proper values
of k and En in our model.
These conditions
lead
lowing system of linear
N cell
cavity
of length
(6)
AnR (k)
quantity
q2 = /dv(Vx?)2
Eq.
C AnQ(k)ER = 0
R
where E Iz is a column
The magnezc
field
and radial
electric
field
are given in terms of f, by Maxwell's
equations.
The coefficients
fm
are expressed
as a linear
combination
of E, at r=a.
of the En by continuity
If
Rewriting
(7)
The program has been applied
study the following
three specific
blems:
341
to
pro-
Dispersion
Curves
for
Uniform
Structures
gap tube termination)
calculation
was
performed
and the effect
on field
flatness of moving in an end wall was studThe result
of a 10% shortening
of
ied.
the gap of the last cell
is to lower
the resonant
freauencv
by 0.11% and to
perturb
the average field
by the amount
given in Table I.
One cell
calculations
were performed
for a cell geometry corresponding
to
6 = .4 and f fi 800 MC, for both half
drift
tube and half gap terminations.
From these one finds
both the zero mode
and TI mode frequencies5
which are plotted
were
in Figure
2. These calculations
then repeated
for both two-cell
and fiveand by examining
1) the
cell cavities,
presence
or absence of solutions
of Det A
= 3 in a given frequency
range,
2) the
relative
values
of En and 3) the resultant
field
configuration,
one can infer
the
These points
are
phase change per cell.
also shown in Figure
2 and suggest the
dispersion
curves which are drawn connecting
these points.
Table
Gap
No.
0
The three dotted
lines
shown are the
dispersion
curves for the TM51 and TMo2
modes in an unloaded wave guide and the
lowest coaxial
TM mode (apart
from TEN)
in a cavity
loaded by a central
conducting post whose radius
is equal to that
The lowest band (and
of the drift
tube.
possibly
the next higher
band) in our calculation
can be understood
as the equivalent TKol wave guide mode loaded by drift
tubes which are expected
to depress the
frequency
since they remove primarily
Iiowever , it should be
electric
field.
pointed
out that the R mode resonance
on the curve at % 1000 MC does not occur
in our calculation
and the modes adjacent tc the 71 mode may not be reliably
since the assumption
cf uniestimated,
form field
in the gap is of questionFor this
for these modes.
able validity
the calculations
are being modireason,
fied to include
a second term in the gap
represenzing
the possibility
of a nonuniform
gap field.
The second
MC can very likely
perturbation
on
the frequency
of
by
to be raised
coaxial
post to
AE
g
for
O-mode
0
m
for
5 - mode
0
1
.02%
-O.ClX
2
. 1%
+o.c275
3
.2%
+0.16%
5%
5
1%
-o.gg?a
6
3%
-1.24%
The same end-wall
perturbation
was
also applied
f'or the n/2-mode,
causing
a frequency
shift
cf ru 0.01% and a field
perturbation
shown as the last column
The n/2-mode appears to be
in Table II.
more stable
to errors.
as is recognized
by those2 who advocate
an equivalent
n/2-mode configuration
by using resonant coupling
via stems, posts,
side
cavities,
etc.
Effect
of Non-Uniform
Flatness
Structure
on Field
We have also made a preliminary
study of the effect
of assembling
tuned
cells
of differing
geometry
(such as in
a drift
tube linac)
on the field
flatness.
The first
calculations
for a 5
cell
cavity
(representing
5 consecutive
cells
near B = .087 in the BML 200 Kev
linac)
showed extremely
small non-uniformity.
This effect
was then exaggerated
by placing
together
in a cavity
widely
The results
are
separated
linac
cells.
analyzed
to determine
the dependence of
field
non-uniformity
on both the change
in cell
geometry
(or B) in adjacent
cells
and the number of cells
in the cavity.
These results
are given in Table II as
a function
of both the number of cells
and the change in B between adjacent
cells.
zero mode shown at 1842
be understood
as a
the coaxial
mode, since
this mode is expected
cutting
slots
in the
allow for the gaps.
It is clear
from the above that
the method provides
a means of calculating the dispersion
curves and field
patterns
for the bands of widest
interest
in an aximuthally
symmetric
uniform accelerating
structure.
Sensitivity
I
to Errors
A preliminary
investigation
of the
sensitivity
to errors
has been carried
out for both the lowest mode (zero mode
at 798 MC) and the n/2 mode at 874 MC.
(with half
For the zero mode, a 6 cell
342
Table
N
(cell
II
Af
F-
18 ,%
.0036%
3
36%
.0175%
3
54%
.OJO2%
18%
18%
.oo46%
.0051%
7
around
AE
E-(end to end)
to cell)
3
5
(centered
The results
in Table II
can write
approximately
It is possible
to
proximate
dependences
(111) by assuming that
between adjacent
cells
ment of The equivalent
an amount proportional
tive
length
difference
cells,
Since the cells
length
this results
in
of
suggest
that
1.2%
CL, -
E/N
.gE
(end
to end)
.022
.OOll
,025
1.8%
-020
2.1%
.017
40016
2.7%
4.7%
one
.029
about dispersion
curves,
sensitivity
to
and non-uniform
geometry for a
errors,
cavity
consisting
of many cells.
The
results
should provide
a firmer
foundation for the analysis
of these effects
via circuit
chains,
since our model inherently
contains
the influence
of adjacent bands.
understand
the apin Eqs. (13) and
removal of the walls
requires
readjustcell boundary by
to 6$/R, the relabetween the two
are of different
a cell
detuning
Foo",notes
--
K
~~-
.0014
.0014
.0014
(9
G-- n %KiE
'n '
6%
6 = .087)
and References
66
E;ntl '
ntl
to the left
and right
of the read'usted
boundary between the nth and n+lS i! cell.
The average frequency
change fcr the entire
cavity
is then an average of Eqs.
(15) and (16) over the cavity,
which is
then proportional
to (8a/B)2,
as Lndicated by the results
in the Table II
as represented
by Eq. (14).
w
*
Work performed
under the auspices
of the
U.S. Atomic Ene-rgy Commission.
t
&Supported by the Ilational
Science Poundation.
1. See for example,
Young, Christian,
Edwards, Mills,
Swenson and van Bladel,
Proceedings
of the International
Conference
on Sector Focussed Cyclotrons
and Meson
Factories,
CERN 1963, CERN 63-19, p. 372
(1963).
2. Swenson, Knapp, Potter
and Schneider
Proceedings
of the 6th International
Coljference
on High Energy Accelerators,
Cambridge,
Mass., p. 167 (1967);
S. T. Giordano and J. Hannwacker,
Proceedings
of
the 1968 Proton Linac Conference
Brookhaven, ENL 50120 (c-54) p. 565 (ig68).
3. Walkinshaw,
Sabel and Outram, AERE Report T/M 104 (1954).
4. In the case of a structure
terminated
by half cells,
the sum over n runs from
0 to N, with the first
and last terms
reduced by a factor
2.
5. The T-mode resonances
occur only for
the half gap tube termination,
as required
by the boundary ccnditions
at the cell
walls.
Equations
(15) and (16) also indicate that results
in adjacent
cells
tend
to compensate for one another,
except for
the first
and last cells,
where one has
frequency
perturbations
proportional
to
This is well known to cause a
+ 66/B.
Field
tilt
proportional
tc the product
of the detuning
and the number of cells,
in agreement with Eq. (13).
These results
can be extrapolated
to the first
cavity
of the BNL 200 MeV
linac.
The result
is a predicted
field
tilt
of t I%, which is unlikely
to affect
the dynamics in a serious
way.
In summary, the model presented
is
capable of yielding
relevant
information
343
Fig.
1.
Idealized
Alvarez-Type
Tube Structure
I
0
I
I
PHASE
Fig.
2.
I
SHWT
Drift
I
2r
J
PER CELL
Resonant Frequencies
and Dlspersion Curves for Uniform Structure (Dotted
Lines are Dispersion
Curves for Unloaded Guides)
344