Magnets that Meet Tracking Requirements for AHF*
Magnets that Meet Tracking Requirements for AHF*
M.E. Schulze11, B. Prichard22, D.E. Johnson22, F. Neri33 and H.A. Thiessen33
M.E. Schulze , B. Prichard
, D.E. Johnson
, F. Neri and H.A. Thiessen
1
2
3
General Atomics, SAIC, LANL
1
General Atomics, 2SAIC, 3LANL
Abstract. The magnet design and fabrication technology for the AHF 50 GeV synchrotron must be of sufficiently high
Abstract.
The magnet
designaperture
and fabrication
forunnecessary
the AHF 50beam
GeVlosses.
synchrotron
muststudies
be of indicate
sufficiently
quality
to provide
a dynamic
that doestechnology
not result in
Tracking
thathigh
the
quality toaperture
provideisasmaller
dynamicthan
aperture
that does
not result
unnecessary
beam losses.
studies indicate
that the
dynamic
the physical
aperture
for in
some
sets of systematic
and Tracking
random magnetic
field errors
in
dynamic
aperture Magnet
is smaller
than the physical
aperturequadrupole
for some sets
of systematic
and random
magnetictofield
errors in
the
quadrupoles.
measurements
of different
magnet
designs have
been analyzed
determine
a
the quadrupoles.
measurements
ofmagnetic
different field
quadrupole
magnet
have beeninanalyzed
representative
set ofMagnet
systematic
and random
errors that
can designs
be manufactured
industry to
anddetermine
provide aa
representative
setlarger
of systematic
and random
magnetic
field errors
that was
can performed
be manufactured
in industry
and provide
dynamic
aperture
that the physical
aperture.
A 2D-error
analysis
for different
quadrupole
magneta
dynamicto aperture
that the
physicalassembly
aperture. tolerances
A 2D-errorand
analysis
performed
different
quadrupole
magnet
designs
ascertainlarger
the effect
of magnet
errors was
on the
magneticfor
field
harmonics.
The analysis
designs tothat
ascertain
effect of harmonics
magnet assembly
tolerances and
and errors
on the magnetic
harmonics.
analysis
indicated
many the
systematic
were correlated
that specific
assemblyfield
tolerances
and The
errors
(coil
indicated tolerances
that manyand
systematic
correlated
and thatassembly
specifictolerances)
assembly result
tolerances
and errors
(coil
placement
proximityharmonics
to pole tip,were
half-core
and quadrant
in significant
random
placement
proximity
to pole
tip, half-core
quadrant
assembly tolerances)
in significant
field
errors tolerances
for some and
magnet
designs.
Comparison
of and
magnetic
measurement
data and result
the error
analysis random
shows
field errorsagreement.
for some These
magnet
designs.
datarequirements
and the error
shows
substantive
studies
have Comparison
allowed AHFofto magnetic
develop a measurement
set of quadrupole
thatanalysis
we feel can
be
substantiveinagreement.
These
havedynamic
allowedrequirements.
AHF to develop a set of quadrupole requirements that we feel can be
constructed
industry and
meetstudies
the beam
constructed in industry and meet the beam dynamic requirements.
INTRODUCTION
INTRODUCTION
The
The AHF
AHFsynchrotron
synchrotron isis designed
designed toto accelerate
accelerate aa
proton
beam
from
4
GeV
to
50
GeV.
The
proton beam from 4 GeV to 50 GeV. The dynamic
dynamic
aperture
aperture ofof the
the synchrotron
synchrotron depends
depends greatly
greatly on
on the
the
magnetic
magnetic field
field quality
quality atat injection.
injection. Initial
Initial tracking
tracking
∗
studies
studies were
were perform
perform*ed
ed using
using the
the systematic
systematic and
and
random
random magnetic
magnetic field
field harmonics
harmonics presented
presented inin the
the
FNAL
FNALMI
MIHandbook.
Handbook. These
Thesestudies
studiesindicated
indicated that
thatthe
the
dynamic
dynamicaperture
apertureofofthe
the AHF
AHF synchrotron
synchrotron isis possibly
possibly
smaller
smaller than
than the
the physical
physical aperture
aperture [1].
[1]. The
The random
random
multipole
multipole field
field errors
errors inin the
the FNAL
FNAL MI
MI quadrupole
quadrupole
were
were found
found toto be
be the
the cause
cause of
of the
the small
small dynamic
dynamic
aperture.
aperture.
Magnetic
Magnetic measurements
measurements ofof other
other quadrupole
quadrupole
magnets
magnets were
were analyzed
analyzed and
and compared
compared with
with the
the
systematic
and
random
multipole
errors
presented
systematic and random multipole errors presented inin
the
the FNAL
FNAL MI
MI Handbook.
Handbook. In
In addition,
addition, aa systematic
systematic
study
study on
on the
the FNAL
FNAL MI
MI quadrupole
quadrupole magnet
magnet was
was
performed
performed totounderstand
understandthe
the source
source and
and nature
nature of
of the
the
systematic
systematicand
andrandom
random multipole
multipole field
field errors.
errors. On
Onthe
the
basis
basis ofof these
these studies,
studies, aa set
set ofof magnetic
magnetic field
field errors
errors
was
was developed
developed which
which meet
meet the
the tracking
tracking requirements
requirements
for
forthe
theAHF
AHFsynchrotron.
synchrotron.
4
unit
unit isis 10
104bbnn/B
/B00 and
and all
all harmonics
harmonics are
are quoted
quoted at
at aa
radius
of
one
inch.
For
a
group
of
magnets,
radius of one inch. For a group of magnets, the
the
systematic
systematic multipoles
multipoles are
are the
the average
average value
value of
of aa given
given
multipole
multipole and
and the
the random
random multipole
multipole is
is the
the standard
standard
deviation
deviation about
about the
theaverage.
average.
MAGNET
MAGNET ANALYSIS
ANALYSIS
The
The FNAL
FNAL MI
MI quadrupole
quadrupole was
was studied
studied using
using the
the
magnet
magnet design
design code
code POISSON.
POISSON. The
The purpose
purpose of
of the
the
study
study was
was to
to investigate
investigate the
the effect
effect of
of mechanical
mechanical
assembly
assembly and
and fabrication
fabrication tolerances
tolerances on
on the
the harmonics.
harmonics.
All
All POISSON
POISSON calculations
calculations were
were performed
performed in
in 2D
2D
only.
only. The
The FNAL
FNAL MI
MI quadrupole
quadrupole consists
consists of
of two
two
laminated
laminated ½-cores
Vfc-cores assembled
assembled with
with four
four coils,
coils, each
each
with
four
turns
per
pole.
Figure
1
shows
a
with four turns per pole. Figure 1 shows a 2D
2D crosscrosssectional
sectional view
viewof
ofthe
the FNAL
FNAL MI
MI quadrupole.
quadrupole.
The
Themagnet
magnetconventions
conventions used
usedininthis
thispaper
paperare
are that
that
bbnnand
andaan nare
arethe
thenormal
normaland
andskew
skewmultipoles
multipoles with
withn=1
n=l
for
for aadipole
dipole field
field and
andn=2
n=2 for
foraaquadrupole
quadrupole field
field such
such
that:
that:
n −1
Br ( r , ϕ ) = B0
∞
n =1
r
r0
[bn cos (nϕ ) + a n sin (nϕ )]
The
Theharmonics
harmonics will
will be
be expressed
expressed inin units
units such
such that
that 11
∗
* LA-UR:02-4027
LA-UR:02-4027
FIGURE
FIGURE1.1. Cross
CrossSection
Section ofofFNAL
FNALMI
MIQuadrupole
Quadrupole
The
The specific
specific assembly
assembly tolerances
tolerances considered
considered
includes
gaps,
tilts
and
shifts
between
includes gaps, tilts and shifts between the
the upper
upper and
and
lower
lower half
half cores.
cores. These
These are
are the
the three
three degrees
degrees of
of
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
© 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00
154
In
InInreality,
reality,
the
magnets
will
be
constructed
with
reality,the
themagnets
magnetswill
willbe
beconstructed
constructedwith
withaaa
combination
of
all
of
these
errors
and
the
size
of
combination
of
all
of
these
errors
and
the
size
of
the
combination of all of these errors and the size ofthe
the
errors
will
be
different
for
all
magnets.
However,
errors
will
be
different
for
all
magnets.
However,
errors will be different for all magnets. However,
there
there
will
be
distribution
toto these
these
errors
with
an
there will
will be
be distribution
distribution to
these errors
errors with
with an
an
average
value
and
a
standard
deviation.
For
example,
average
value
and
a
standard
deviation.
For
example,
average value and a standard deviation. For example,
the
the
gap
between
the
upper
and
lower
half
cores
will
be
thegap
gapbetween
betweenthe
theupper
upperand
andlower
lowerhalf
halfcores
coreswill
willbe
be
systematically
positive
with
a
distribution
that
can
systematically
be
systematicallypositive
positivewith
withaadistribution
distributionthat
that can
canbe
be
described
described
by
mean
value
and
standard
deviation.
describedby
byaaamean
meanvalue
valueand
andaaastandard
standarddeviation.
deviation.
An
An
analysis
was
performed
ininwhich
which
totopredict
predict
the
Ananalysis
analysiswas
wasperformed
performedin
whichto
predictthe
the
measured
random
harmonic
errors
based
on
the
above
measured
random
harmonic
errors
based
on
the
above
measured random harmonic errors based on the above
difference
difference
tables.
The
specific
differences
were
added
differencetables.
tables. The
Thespecific
specificdifferences
differenceswere
wereadded
added
in
quadrature
with
a
multiplier
indicating
the
ininquadrature
fraction
quadraturewith
withaamultiplier
multiplierindicating
indicatingthe
thefraction
fraction
of
ofofthe
the
nominal
error,
(as
described
above).
AA
thenominal
nominal error,
error, FFnFn,,n, (as
(as described
described above).
above). A
second
multiplier,
S
,
was
included
to
indicate
the
second
multiplier,
S
,
was
included
to
indicate
the
n
second multiplier, Sn n, was included to indicate the
number
number
ofoflocations
locations
the
error
can
occur
(S
for
coil
numberof
locationsthe
theerror
errorcan
canoccur
occur(S
(Snnn===444for
forcoil
coil
asymmetries
and
1
for
half
core
asymmetries).
This
asymmetries
and
1
for
half
core
asymmetries).
This
asymmetries and 1 for half core asymmetries). This
total
total
random
error
for
specific
harmonic
was
total random
random error
error for
for aaa specific
specific harmonic
harmonic was
was
compared
to
the
measured
random
error.
The
sum
compared
to
the
measured
random
error.
The
sum
of
compared to the measured random error. The sumof
of
the
the
squared
errors
(between
measured
and
predicted
thesquared
squarederrors
errors(between
(betweenmeasured
measuredand
andpredicted
predicted
random
random
harmonic
errors)
was
then
minimized
for
the
randomharmonic
harmonicerrors)
errors)was
wasthen
thenminimized
minimizedfor
forthe
the
harmonics
(sextupole
through
duodecapole
only).
harmonics
harmonics(sextupole
(sextupolethrough
throughduodecapole
duodecapoleonly).
only).
The
The
quantities,
and
representing
the
An
Bn
An and
Bn,, representing
The quantities,
quantities, σσGAR
andσσGen,
representing the
the
predicted
random
skew
and
normal
multipole
errors
predicted
random
skew
and
normal
multipole
errors
predicted random skew and normal multipole errors
were
were
calculated
asasbelow,
below,
werecalculated
calculatedas
below,
11//22
1/2
σσAA ,,PP ==
nn
nn
((SSnnFFnnDDAAnn))
22
ii==11
11//22
1/2
σσBB ,P,P ==
nn
nn
((SSnnFFnnDDBBnn))
22
ii==11
The
results are
presented in
Figure 22 below
and
The
The results
results are
are presented
presented inin Figure
Figure 2 below
below and
and
show
good
agreement
between
the
measured
[2]
show
good
agreement
between
the
measured
[2]
and
show good agreement between the measured [2]and
and
predicted
random multipole
errors. The
analysis
predicted
predicted random
random multipole
multipole errors.
errors. The
The analysis
analysis
showed
showed
that
the
random
normal
octupole,
BB44,,, results
results
showedthat
thatthe
therandom
randomnormal
normal octupole,
octupole, B
4 results
from
aa combination
of
the
midplane
gap
and
from
combination
of
the
midplane
gap
and
tilt
from a combination of the midplane gap and aaa tilt
tilt
between
the
two
half
cores.
The
calculated
standard
between
the
two
half
cores.
The
calculated
standard
between the two half cores. The calculated standard
deviation
deviation
ininthe
the
midplane
gap
and
tilt
was
consistent
deviationin
themidplane
midplanegap
gapand
andtilt
tilt was
wasconsistent
consistent
about
.001”
for
both
of
these
errors.
The
rest
about
.001”
for
both
of
these
errors.
The
rest
of
the
about .001" for both of these errors. The rest of
ofthe
the
random
errors
result
from
coil
placement
errors
random
of
random errors
errors result
result from
from coil
coil placement
placement errors
errors of
of
about
about
0.020”.
All
other
assembly
errors
did
not
about 0.020”.
0.020". All
All other
other assembly
assembly errors
errors did
did not
not
contribute
contribute
significantly.
contributesignificantly.
significantly.
Random
Random Multipole
Multipole Errors
Errors
1.400
1.400
Random Error
Error (x
(x 10
1044))
Random
freedom
in the assembly
of two half
cores. In
addition
freedom
freedomininthe
theassembly
assemblyofoftwo
twohalf
halfcores.
cores. In
Inaddition
addition
the
effects
of
coil
shifts,
and
tilts
were
considered.
For
the
effects
of
coil
shifts,
and
tilts
were
considered.
the effects of coil shifts, and tilts were considered. For
For
each
specific
assembly
tolerance
aa POISSON
each
specific
assembly
tolerance
POISSON
each specific assembly tolerance a POISSON
calculation
calculation
was
performed
for
nominal
offset.
This
calculationwas
wasperformed
performedfor
foraaanominal
nominaloffset.
offset. This
This
difference
between
the
predicted
harmonics
in
difference
between
the
predicted
harmonics
in
the
difference between the predicted harmonics in the
the
absence
of
any
assembly
tolerances
was
calculated.
absence
of
any
assembly
tolerances
was
calculated.
absence of any assembly tolerances was calculated.
SSE
SSE
SSE=== ((σσ
AA33
−−σσ AAnn,,MM )) ++
22
b4
b4
b5
b5
b6
b6
a3
a3
a4
a4
a5
a5
a6
a6
FIGURE
FIGURE
2.2. Comparison
Comparison
of
Predicted
and
Measured
FIGURE 2.
Comparison of
of Predicted
Predicted and
and Measured
Measured
Random
Multipole
Errors
Random
Multipole
Errors
Random Multipole Errors
Figure
Figure
shows
that
coil
placement
errors
are
the
Figure333shows
showsthat
thatcoil
coilplacement
placement errors
errors are
arethe
the
dominant
source
of
random
errors
with
the
exception
dominant
source
of
random
errors
with
the
exception
dominant source of random errors with the exception
of
of
the
normal
octupole
as
described
above.
This
result
ofthe
thenormal
normaloctupole
octupoleas
asdescribed
describedabove.
above. This
Thisresult
result
implies
that
there
will
be
a
strong
correlation
between
implies
that
there
will
be
a
strong
correlation
between
implies that there will be a strong correlation between
the
the
measured
harmonics
for
these
multipoles.
The
the measured
measured harmonics
harmonics for
for these
these multipoles.
multipoles. The
The
strongest
correlation
is
observed
between
the
normal
strongest
correlation
is
observed
between
the
normal
strongest correlation is observed between the normal
sextupole,
sextupole,
and
decapole,
and
between
the
skew
sextupole,bbb33,3,,and
anddecapole,
decapole,bbb55,5,,and
andbetween
betweenthe
theskew
skew
octupole,
octupole,
and
the
duodecapole,
as
shown
in
octupole, aa4a4,4, , and
and the
the duodecapole,
duodecapole, aaa66,6,, as
as shown
shown in
in
Figure
Figure
4.4.
Figure4.
RRRandom
aand
ultip
rrors
ils
ly
ndoom
mM
MMultipole
ultipoole
le EEErrors
rrors (co
(co
ils on
on
ly))
(coils
only)
1.4
1.400
00
BB66
BB33
((σσ
22
BBnn,,PP
−−σσBBnn,,MM ))
22
4
Random Error
Error (x
(x 10
104))
Random
1.2
1.200
00
1.0
1.000
00
0.8
0.800
00
M
Meas
easured
ured
Predic
Predictted
ed
0.6
0.600
00
0.4
0.400
00
0.2
0.200
00
0.0
0.000
00
bb33
bb44
b5
b5
bb66
aa33
aa44
aa55
aa66
M
Multip
ultipoole
le
Strong
ith
Strong
correlation
w
ith
Strongcorrelation
correlationw
with
co
ilil placem
ent
co
placem
ent error
error
coil
placement
error
The
The
sum
ofofthe
the
squared
errors,
SSE,
then
given
Thesum
sumof
thesquared
squarederrors,
errors,SSE,
SSE, isisisthen
thengiven
given
by,
by,
by,
22
AAnn,,PP
0.400
0.400
0.200
0.200
Multipole
Multipole
D
DDMI
=Change
inin harmonic
harmonic
due
toto specific
specific
An
An =Change
=Changein
harmonic due
due to
specific
assembly
error
assembly
error
assembly error
The
The
sum
over
the
sum
of
the
all
assembly
Thesum
sumover
overnnnisisisthe
thesum
sumof
ofthe
the all
all assembly
assembly
errors
considered.
errors
errorsconsidered.
considered.
AA66
0.600
0.600
b3
b3
FFnFn n === Ratio
Ratio
ofof actual
actual
assembly
error
toto
Ratio of
actual assembly
assembly error
error to
calculated
calculated
assembly
error
calculatedassembly
assemblyerror
error
Measured
Measured
Predicted
Predicted
0.800
0.800
0.000
0.000
where,
where,
where,
SSnSn =
1 forforsteel
steel
errors
and
for
coil
errors
steelerrors
errorsand
and444for
forcoil
coilerrors
errors
n==1 1for
1.200
1.200
1.000
1.000
FIGURE
FIGURE
3.3. Contribution
Contribution
ofof Coil
Coil
Placement
Errors
to
FIGURE 3.
Contribution of
Coil Placement
Placement Errors
Errors to
to
Random
Multipole
Errors
Random
Multipole
Errors
Random Multipole Errors
Here,
Here,
and
are
the
measured
random
errors.
An,M
Here,σσBn,M
GBH.M
andσσGAH.M
arethe
themeasured
measuredrandom
randomerrors.
errors.
Bn,M and
An,M are
155
1.50
1.25
1.501.50
1.00
1.25
1.25
1.00
1.001.00
0.50
SUMMARY
SUMMARY
SUMMARY
SUMMARY
We
values
the
errors
Wehave
havedetermined
determinedvalues
valuesfor
forthe
therandom
randomerrors
errors
We
have
determined
WeAHF
havequadrupole
determinedmagnets
values for
for
therandom
random
errors
for
the
based
on
an
analysis
for
the
AHF
quadrupole
magnets
based
on
an
analysis
for
magnets
based
on
forthe
theAHF
AHFquadrupole
quadrupole
magnets
based
onan
ananalysis
analysis
of
random
errors
inin the
FNAL
MI
of the
the source
source of
of the
the random
random errors
errorsin
theFNAL
FNALMI
MI
of
the
source
of
the
the
of
the
source
of
the
random
errors
in
the
FNAL
MI
quadrupoles
magnets.
The
analysis
found
that
coil
quadrupoles
magnets.
The
analysis
found
that
coil
quadrupoles
magnets.
The
found
quadrupoleserrors
magnets.
The analysis
analysis
found that
that
coil
placement
contributed
significantly
toto coil
the
placement
errors
contributed significantly
significantly to
the
placement
errors
contributed
the
placement
errors
contributed
significantly
to
the
measured
random
errors
in
the
FNAL
MI
quadrupole
measured
random
errors
in
the
FNAL
MI
quadrupole
measured
errors
inin the
MI
measuredrandom
random
errorsquadrupole
theFNAL
FNAL
MI quadrupole
quadrupole
magnets.
The
AHF
magnets
will
be
magnets.
The
AHF quadrupole
quadrupole magnets
magnets will
will be
be
magnets.
The
AHF
magnets.
The
AHF
quadrupole
magnets
will
be
designed
with
the
coils
located
away
from
the
pole
tip
designed
with
the
coils
located
away
from
the
pole
tip
designed
with
the
located
away
from
tip
designed
with
thecoils
coilserrors
located
away
fromthe
thepole
pole
tip
so
that
coil
placement
do
not
contribute
totothe
so
that
coil
placement
errorsdo
donot
notcontribute
contributeto
the
so
that
coil
placement
errors
the
so
that
coil
placement
errors
do
not
contribute
to
the
measured
magnet
harmonics.
measured
magnet
harmonics.
measured
measuredmagnet
magnetharmonics.
harmonics.
1.00
1.00
0.75
0.500.50
0.00
0.75
0.75
0.50
A6 A6
A6
B5
0.000.00
-0.50
B5
B5
0.50
0.50
0.25
-0.50
-0.50
-1.00
B5vsB3
BB
vsB3B3
-1.50
-1.50
5v
5s
-1.00
-1.00
-1.50
0.25
0.25
0.00
-2.00
0.00
0.00
-0.25
-2.00
-2.00
-2.50
-2-3
.5.0
0
-20.50 -2.00
A6vsA4
AA
vsAA
6v
44
6s
-0.25
-0.25
-0.50
-1.00 0.00 1.00 2.00 3.00 4.00 5.00
-4.00
-3.00
-2.00
-1.00
-0.50
-0.50
-3.00
-3.00 -2.00
-2.00 -1.00
-1.00 0.000.00 1.001.00 2.002.00 3.003.00 4.004.00 5.005.00
B3
-4.00
-4.00 -3.00
-3.00 -2.00
-2.00
-1.00
A4 -1.00
B3B3
0.00
1.00
0.00
0.00
1.00
1.00
A4A4
FIGURE
4. Correlation
between
Measured
Harmonics
FIGURE
Correlation
between
Measured
Harmonics
FIGURE
4.4. Correlation
FIGURE4.
Correlationbetween
betweenMeasured
MeasuredHarmonics
Harmonics
Other
quadrupole
magnets
were
studied
totocompare
Other
quadrupole
magnets
were
studied
compare
Otherquadrupole
quadrupolemagnets
magnetswere
werestudied
studiedto
tocompare
compare
Other
random
and
systematic
multipole
errors
for
different
random
and
systematic
multipole
errors
for
different
randomand
andsystematic
systematicmultipole
multipole errors
errors for
for different
different
random
magnet
designs. These
magnets
include
the
FNAL
magnet
These
magnets
include
the
FNAL
magnet designs.
designs. These
These magnets
magnets include
include the
the FNAL
FNAL
magnet
designs.
Antiproton
Accumulator
quads
(SQC)
[3],
SLAC
PEPAntiproton
Accumulator
quads
(SQC)
[3],
SLAG
PEPAntiprotonAccumulator
Accumulatorquads
quads(SQC)
(SQC)[3],
[3],SLAC
SLACPEPPEPAntiproton
IIII quads
(LER)
[4],
and
the
ANL
ALS
quads
[5].
(LER)
[4],
and
the
ANL
ALS
quads
[5].
quads (LER)
(LER) [4],
[4], and
and the
the ANL
ANL ALS
ALS quads
quads [5].
[5].
IIII quads
quads
Figure
5
presents
the
random
harmonics
for
these
Figure
55 presents
the
random
harmonics
for
these
Figure 5
presents the
the random
random harmonics
harmonics for
for these
these
Figure
presents
magnets
and
the
AHF
requirements.
This
figure
shows
magnets
and
the
AHF
requirements.
This
figure
shows
magnetsand
andthe
theAHF
AHFrequirements.
requirements.This
Thisfigure
figureshows
shows
magnets
that
the
random
errors
due
to
coil
placement
errors
inin
that
the
random
errors
due
to
coil
placement
errors
thatthe
therandom
randomerrors
errorsdue
duetotocoil
coilplacement
placementerrors
errorsin
in
that
the
MI
quadrupoles
are
not
seen
ininthe
other.
Analysis
the
MI
quadrupoles
are
not
seen
the
other.
Analysis
theMI
MIquadrupoles
quadrupolesare
arenot
notseen
seenininthe
theother.
other. Analysis
Analysis
the
of
these
magnets
does
not
show
the
correlation
of
magnets
does
not
show
the
correlation
these magnets
magnets does
does not
not show
show the
the correlation
correlation
ofof these
these
between
harmonics
associated
with
coil
between
associated
with
coil
placement
between harmonics
harmonics associated
associated with
with coil
coil placement
placement
between
harmonics
placement
errors.
errors.
errors.
errors.
1.4
1.4
1.4
Random
Harmonics
Random
Harmonics
RandomHarmonics
Harmonics
Random
m
1.0
1.0
1.0
FIGURE
Dynamic
Aperture
for
new
AHF
FIGURE
DynamicAperture
Aperturefor
fornew
newAHF
AHF
FIGURE666.
6.. . Calculated
Calculated Dynamic
Dynamic
Aperture
for
new
AHF
FIGURE
Calculated
Requirements
for
Magnetic
Field
Errors
Requirements
forMagnetic
MagneticField
Requirementsfor
FieldErrors
Errors
Requirements
SLAC LER
SLAC
SLAC
LELE
RR
FNAL SQC
FNAL
SQC
FNAL
SQC
New FNAL MI
New
AL
New
FNFN
AL
MIMI
ANL ALS
AN
ALS
AN
LL
ALS
AHF Reqmts
AH
F Reqmts
AH
F Reqmts
Units
Units
Units
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
0.2
0.2
0.2
0.00.0
0.0
m
clp/p.
1.2
1.2
1.2
b3 b3 b4b4 b5b5 b6b6
b3
b4
b5
b6
b4
b5
b6
a3a3 a4a4 a5a5
a3
a4
a5
a3
a4
a5
The
like
totothank
thank
D.
Harding
and
B.
wouldlike
liketo
thankD.
D.
Harding
and
B.
Theauthors
authorswould
D.Harding
Hardingand
andB.
B.
The
authors
would
Brown
of
FNAL
for
providing
measurement
data
for
for providing
providing measurement
measurement
data
for
Brown ofof FNAL
FNALfor
measurementdata
datafor
for
Brown
the
and
N.
Li,
U.
Wienands
theFNAL
FNAL MI
MI quadrupoles
quadrupoles and
and N.
N. Li,
Li, U.
U.
Wienands
the
FNAL
U. Wienands
Wienands
the
FNAL
MI
quadrupoles
and
SLAC
for
providing
measurement
and J.
SLAC for
for
providing
measurement
and
J.J. Tanabe
Tanabe
ofof SLAC
and
Tanabe of
for providing
providingmeasurement
measurement
data
LER
quadrupole
magnets.
quadrupolemagnets.
magnets.
data
for
the
PEPII
datafor
forthe
thePEPII
PEPIILER
LERquadrupole
magnets.
a6a6
a6
a6
FIGURE
FIGURE5.5.5. Measured
MeasuredRandom
RandomHarmonics
Harmonics and
and AHF
AHF
Measured
Random
Harmonics
and
AHF
FIGURE
Random
Harmonics
and
AHF
Requirements
Requirements
Requirements
The
Therandom
randomerrors
errorsare
aresmallest
smallestfor
forthe
theFNAL
FNALSQC
SQC
errors
are
smallest
for
the
FNAL
SQC
The
random
smallest
for
the
FNAL
SQC
and
the
SLAP
PEP-II
quadrupoles,
which
are
andthe
theSLAP
SLAPPEP-II
PEP-IIquadrupoles,
quadrupoles,which
whichare
areabout
aboutaaaa
about
and
which
are
about
factor
factorof
threeshorter
shorterthan
thanthe
theAHF
AHFquadrupoles.
quadrupoles. The
The
three
shorter
than
the
AHF
quadrupoles.
The
factor
ofofthree
the
AHF
quadrupoles.
The
AHF
random
errors
are
thus
expected
AHF
random
errors
are
thus
expected
tobe
belarger
largerthan
than
to
be
larger
than
AHF random errors are thus expected
expectedto
to
be
larger
than
these
thesemagnets.
magnets.To
Toobtain
obtainthe
therequirements
requirementsfor
forthe
theAHF
AHF
these
magnets.
To
obtain
the
requirements
for
the
AHF
requirements
for
the
AHF
random
multiplied
by
about
randomerrors
errorsthese
theseerrors
errorsare
are
multiplied
by
about
random
errors
these
errors
are
multiplied
by
about
are multiplied by about aaaa
factor
factorof
twoand
andan
anadditional
additional30%
30%is
added. Using
Using
an
additional
30%
isisadded.
added.
Using
factor
ofoftwo
two
and
added.
Using
these
errors
the
AHF
dynamic
apertures
theseerrors
errorsthe
theAHF
AHFdynamic
dynamicapertures
aperturesis
calculated
isisiscalculated
calculated
these
dynamic
apertures
calculated
to
significantlylarger
largerthan
thanthe
thephysical
physicalaperture
aperture[1]
[1]
significantly
larger
than
the
physical
aperture
[1]
totobe
bebesignificantly
physical
aperture
[1]
as
shown
in
Figure
6.
as
shown
in
Figure
6.
as shown in Figure
Figure 6.
6.
REFERENCES
REFERENCES
REFERENCES
REFERENCES
1.
proceedings.
al.,These
proceedings.
1.1. Neri,
Neri,
F.F.et
etetal.,
al.,
These
Neri,F.
Theseproceedings.
proceedings.
2.
Communication.
2. David
DavidHarding,
Harding,Private
Private
Communication.
2.2.
David
Harding,
Private
David
Harding,
PrivateCommunication.
Communication.
3.
Johnson,
Private
Communication.
3. D.
D.
Johnson,
Private
Communication.
3.3.
D.
Johnson,
Private
Communication.
D. Johnson, Private Communication.
4.
ofof
the
PEP-II
Low-Energy
4. Li,
Li, N
N et
et al.,
al.,
“Performance
the
PEP-II
Low-Energy
4.4.
Li,
NN
etet
al.,
“Performance
Li,
al., “Performance
"Performance of
ofthe
thePEP-II
PEP-nLow-Energy
Low-Energy
Ring
Proceedings
1997
Particle
Ring Quadrupoles”,
Quadrupoles”, Proceedings
Proceedings
1997
Particle
Ring
Quadrupoles”,
Ring
Quadrupoles",
Proceedings 1997
1997 Particle
Particle
Accelerator
Conference,
Vancouver
BC,
3318,
(1998).
Accelerator
Conference,
Vancouver
BC,
3318,
(1998).
Accelerator
Conference,
Vancouver
BC,
3318,
(1998).
Accelerator Conference, Vancouver BC, 3318, (1998).
5.
“Statistical
Analysis
of
the
Magnet
Data
5. Kim,
Kim,S.H.
S.H.et
al.,
“Statistical
Analysis
of
the
Magnet
Data
5.5.
Kim,
S.H.
etet
al.,
Kim,
S.H.
etal.,
al.,“Statistical
"StatisticalAnalysis
Analysisof
ofthe
theMagnet
MagnetData
Data
for
the
Advanced
Photon
Source
Storage
Ring
Magnets”,
for
the
Advanced
Photon
Source
Storage
Ring
Magnets”,
for
the
Advanced
Photon
Source
Storage
Ring
Magnets”,
for the Advanced Photon Source Storage Ring Magnets",
Proceedings
Accelerator
Conference,
Proceedings 1995
1995
Particle
Accelerator
Conference,
Proceedings
1995 Particle
Particle Accelerator
Accelerator Conference,
Conference,
Dallas
TX,
1310,
Dallas
TX,
1310,
(1996).
(1996).
Dallas TX, 1310, (1996).
156