Lattice modeling for a storage ring with magnetic
field data
X. Huang, J. Safranek (SLAC)
Y. Li (BNL)
3/5/2012
3/5/2012
FLS2012, Ring WG, X. Huang
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Motivation
• Discrepancy between original model and measurements.
• Understanding dynamic effects of rectangular gradient
dipoles.
x
X
s
Z
1. The ideal trajectory in RGD is not a circular arc.
2. Gradient varies with s-variable
3. Off-plane longitudinal field
• Understanding the sources of discrepancies in linear and
nonlinear characteristics between models and
measurements.
– Fringe field of dipoles
– Fringe field of quadrupoles
– Cross-talk of fields between adjacent magnets?
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The field-integration approach
An AT pass-method that transfer phase space coordinates from one end to the other
of a magnetic field region with Bx, By, Bz defined as function of (x, y, z).
Equation of motion when using z as free variable.
Coordinate transformation at the edges.
For dipoles, additional transformation is needed.
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Magnetic field in a standard SPEAR3 dipole
We started examining our lattice model from magnetic fields in magnets.
By at z=0
1.5
1.4
Tanabe
RADIA new
Measured 2007
By (normalized)
1
RADIA new
Measured 2001
fit meas 2001
measured 2007
0.5
0
-1
-0.5
0
Z (m)
1
0.8
0.6
0.4
0.2
0.5
1
We have coil, wire measurements.
Hall probe scans along Z in 2001, 2007.
Hall probe scans on X-Z plane in 2007.
0
-100
-50
0
50
100
x (mm)
150
200
250
300
0
-0.5
By (T)
By/Bymax
1.2
-1
-1.5
1
0.8
0.6
Z (m) 145D
3/5/2012
0.4
-0.05
0
0.05
0.1
0.15
0.25
0.2
X (m)
FLS2012, Ring WG, X. Huang
Hall probe x-z scan (2007)
4
0.3
An analytic dipole field model
An analytical field model can be built according to general field expansion to obtain
the full magnetic field distribution in the dipole. This also removes noise from field
measurements.
Take the dipole component as an example.
Note that the B0/B1 ratio is not constant in the fringe region.
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Energy calibration
How to calibrate the require bending field for a 3GeV beam?
+
2001 Z-scan
(Corbett & Tanabe, 2002)
(Yoon, et al, NIMA 2004). The virtual center
was held constant (392.35 mm).
This is how dipole magnets were positioned: adjust the dipole current
(converted to K-value) until the alignment requirement is met.
Following this procedure, the required field integral is calculated to be
(1) 1.86420 T-m, with a fixed virtual center, while the measured field integral is
1.86413 T-m for 587.6909 A (operating current since day 1 of SPEAR3).
(2) 1.86615 T-m, with the fitted field profile.
So the SPEAR3 beam energy may be lower than the nominal value by 0.1%.
Energy measurement at SPEAR3 confirmed the prediction with high precision.
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Effects of quadrupole fringe field
A general Hamiltonian (including longitudinal field variation) can be derived using a
proper magnetic field expansion(1).
H ( s) 
1 2
1
1
1
( Px  Py2 )  k ( s)( x 2  y 2 )  k ' ( s)( x 2  y 2 )( xPx  yPy )  k ' ' ( s)( x 4  y 4 )  O( X 6 )
2
2
4
12
J. Irwin, C.X. Wang
The leading correction for a hard-edge model is from the last two terms, which are
nonlinear(2).
The leading correction term from a soft fringe model is linear(3).
A perturbation approach
* El-Kareh; Forest; Bassetti & Biscari
** Lee-Whiting, Forest & Milutinovic, Irwin & Wang, Zimmermann
***Irwin & Wang (PAC’95), D. Zhou (IPAC10).
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The linear correction to quadrupole map
The correction map
J. Irwin, C.X. Wang, PAC95
The generating function for the correction map
f2 
I1
( xPx  yPy )
2
matrix
leading contribution
diag(e I1 , e I1 , e I1 , e I1 )
For a symmetric quadrupole, the entrance edge
has a reversed sign for I1
The tune changes are (always negative)
k02 L x I1
 x  
| |,
2 k0
k02 L y I1
 y  
| |
2 k0
For SPEAR3, quadrupole fringe fields cause tune changes of [-0.065, -0.059], in
agreement with the predictions by the above equation.
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The nonlinear correction
The generating function for the correction map (exit edge)
f4 
1
1
k0 ( x 3 Px  3xy 2 Px  y 3 Py  3x 2 yPy ) 
k skew ( x 3 Py  y 3 Px )
12(1   )
6(1   )
Forest & Milutinovic
The function for the entrance edge has an opposite sign.
F. Zimmermann derived the average Hamiltonian that include both edges.
Hard edge

Additional soft edge contribution.
2 is fringe length.
6 I1
k0
and tune dependence on amplitude (only showing hard edge contribution below)
This agree with tracking quite well.
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An AT quadrupole passmethod with fringe field
Both linear and nonlinear effects are considered in the new quadrupole passmethod.
Forest & Milutinovic pointed out the skew quadrupole part corresponds to a ‘kick map’!
A normal quadrupole can thus be modeled by a pair of pi/4 rotation and a kick map.
This is the basis for the nonlinear part of the new AT quadrupole pass method.
-0.0122
-0.0122
The new quad passmethod agree
very well with the field-integration
method.
xpf (rad)
-0.0122
-0.0123
-0.0123
-0.0123
-0.0123
quadpass
quadpass+matrix
new quad pass
field pass
-0.0123
0.02
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0.02
FLS2012, Ring WG, X. Huang
0.02
0.02
0.02
xi (m)
0.02
0.02
10
The SPEAR3 quadrupole field profile
The analytical quadrupole field map for SPEAR3 magnet was based on magnet
modeling. Simulated field is converted to an analytic form.
1
' ' ( z )(3x 2 y  y 3 )]
12
1
B y  B1[ x( z )  ' ' ( z )(x 3  3xy 2 )]
12
Bz  sgn( z ) B1' ( z ) xy
1.5
Magnetic field
B1f (normalized)
Bx  B1[ y( z ) 
1
0.5
0
0
0.1
all quads at 72 A in modeling
I1a 
18
16
60Q
50Q
34Q
15Q
B1 (T/m)
14
12
0.2
0.3
0.4
Z (m) for 60Q
0.5
0.6
0.7
I1
6 I1
 0.61103 m 2 ,  
 0.060 m
k0
k0
10
8
6
4
2
0
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Z (m)
0.3
0.4
0.5
0.6
All SPEAR3 quads have identical fringe profile.
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Comparison of models to measurement
The model is based on a calibrated experimental lattice with all IDs open (4/6/2009).
Parameter
Measure
d
All field model
Bend field,
quad i2k
with fringe
Bend field,
quad i2K
i2k old AT
model
Tune x
14.106
14.146
14.150
14.215
14.190
Tune y
6.177
6.119
6.121
6.180
6.431
Chrom x
1.7
-0.54
-0.53
-0.44
-0.60
Chrom y
2
0.89
0.90
0.73
1.90
Effect of the predicted -0.1% beam energy shift is not included, which change the tunes by
[0.023, -0.004] for [nux, nuy].
Dipole field is given by the field profile and alignment requirement.
Drift lengths neighboring to dipoles adjusted according to measured rf frequency.
Strengths of quadrupoles and sextupoles are derived from operating currents and
measured excitation curves.
No adjustment of any magnet strength!
The tune differences are [0.067 -0.060] between the best model and the measurement, a
big improvement from the original model.
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Beta beat and correction
0.1
0.2
0.15
0.1
 y/y
 x/x
0.05
0
0
-0.05
-0.05
-0.1
0
0.05
50
100
150
s (m)
no correction
corrected
-0.1
200
-0.15
0
250
50
100
150
s (m)
200
250
Beta beat is relative to the ideal lattice.
“No correction” is for “bend field + quad fringe”
“corrected” is after the quadrupole strength is adjusted to reduce beta beat (LOCO).
Possible causes of optics difference between measurement and un-adjusted
model:
(1) Interference of magnetic fields between neighboring magnets.
(2) Magnet calibration errors.
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The tune map
Chromaticities are corrected with SF/SD to obtain [1.65, 2.18].
Tunes are obtained by tracking 256 turns.
old model vs. new model
0.25
0.25
0.24
0.24
0.23
0.23
0.22
0.22
y
y
measured vs. new model
0.21
0.21
0.2
0.2
0.19
0.19
0.18
0.18
0.1
0.15
x
0.2
0.1
0.15
x
0.2
“new model” = field model for bend + quad fringe.
This model agrees with measurement better.
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High-order chromaticities
low tune [0.106, 0.177] LE
0.17
0.16
x-Cx  p/p
0.15
measured
old model
new model
0.14
0.13
0.12
0.11
0.1
-0.04
-0.02
0
 p/p
0.02
measured
New
model
chrx0
1.725*
1.725
1.647**
chrx1
28.8
28.4
30.6
chrx2
-569
-545
-557
chry0
2.081*
2.081
2.181**
chry1
16.7
30.3
28.9***
chry2
-207
16
-213
* Model chromaticities adjusted to match
measured values.
** model chromaticities adjusted, but not yet
completely on target.
*** improvement from old model.
0.23
y-Cy  p/p
Old
model
0.04
low tune [0.106, 0.177] LE
measured
old model
new model
0.22
Low
tune
0.21
0.2
0.19
0.18
0.17
-0.04
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-0.02
0
 p/p
0.02
0.04
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Progress toward fast tracking for dipoles
(1) Extract a Lie map from the Taylor map obtained from the fieldintegration method (Yongjun Li). A map may also be obtained with
COSYInfinity.
(2) Split the f3 and f4 polynomials into individual terms for tracking (f4
terms are altered by splitting f3), ignore higher order polynomials.
Monomial maps have exact solutions (A. Chao, Lie Algrebra Notes).
An AT passmethod is written to track f3 (35 terms) and f4 (70 terms) maps (f2 is
supplied by a matrix).
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Comparison of Map-pass to field-pass
For comparison, the second order transport map is extracted with AT for the
SPEAR3 dipole, using the field-pass or the map-pass.
T1ij from the field pass
T1ij from the map pass
All transverse-only elements agree well (for T2ij, T3ij T4ij, too).
The discrepancy for the momentum-related elements may be caused by an problem
in the field-pass used for map extraction (different from the one compared to here).
The map-pass provides a symplectic tracking solution to the dipole model.
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Summary and Discussion
• We built a lattice model from magnetic field measurements and
alignment requirements and compared the linear optics to beam based
measurements.
– Improved: tunes, betatron functions.
– But: still up to 15% maximum beta beat (vertical)
• After optics and chromaticity corrections, nonlinear parameters from
the model are compared to beam based measurements.
– Improved: 2nd order vertical chromaticity, tune map.
– But: the tune map is still slightly different from measurement.
•
We have developed fast symplectic method to represent high order effects.
• An accurate model may be crucial for a smooth commissioning of a
new machine and for dynamic aperture optimization of existing
machines.
– Quadrupole fringe field effect (tune shifts and beta beat) would be larger for a large
ring (with more quads).
– Magnetic field based lattice can be used as a “reference” model.
More efforts are need to understand the discrepancies between model and measurements.
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The dipole field map
B0, B1, B2
from X-Z scan
5
2007 xz scan
-5
B0
B1
B2
-10
B
0,1,2
, (T, T/m, T/m2)
0
-15
By(at y=0,z=0) = -1.233257 + 3.143436*x -0.324508*x^2
ByL = -1.857103 + 4.662405*x -0.931245*x^2
-20
0.4
0.5
0.6
0.7
Z (m)
0.8
0.9
1
Coil measurement gives
ByL (T m)= -1.8506 +4.6081 x - 1.2632 x^2
Note that the dipole/quadrupole ratio is constant (392.35 mm) in the magnet body, but
varying in the fringe.
The integrated quadrupole component is actually 2% weaker than the present model.
(The coil measurement gives an average ratio of 399.8+-1.7 mm)
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The linear correction to quadrupole map
A perturbation approach
Hard-edge model, for exit edge
Perturbation term
The map
J. Irwin, C.X. Wang, PAC95
The generating function for the correction map (only leading contribution is shown)
f2 
I1
( xPx  yPy )
2
matrix
diag(e I1 , e I1 , e I1 , e I1 )
For a symmetric quadrupole, the entrance edge
has a reversed sign for I1
The tune change would be (always negative)
3/5/2012
k02 L y I1
k02 L x I1
 x  
| |,  y  
| |
2 k0
2 k0
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Verification of the quad fringe pass method
-0.0122
yi=0.005 m
-0.0122
xpf (rad)
-0.0122
-0.0123
-0.0123
-0.0123
-0.0123
quadpass
quadpass+matrix
new quad pass
field pass
-0.0123
0.02
0.02
0.02
0.02
0.02
xi (m)
0.02
0.02
Quadpass: quad transfer matrix
Quadpass+matrix: quad transfer matrix + linear edge
transfer matrix.
New quad pass: with linear and nonlinear corrrection.
Field pass: integration through magnetic field.
With the (quad+matrix) part subtracted.
Zimmerman result is from the average
Hamiltonian H1+2
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A pass method for magnetic field in AT
(1) Coordinate transformation at the entrance and exit of the magnets
(2) Integration of the Lorentz equation in the body of magnets.
Can we study beam dynamics with such a pass method?
With an accurate magnetic field model, we can reproduce reality in simulation.
Integration is slow and non-symplectic, not good for dynamic aperture tracking.
But it should be good for linear and nonlinear parameter evaluation.
7/20/2011
23