3/5/20121 Lattice modeling for a storage ring with magnetic field data X. Huang, J. Safranek (SLAC)...

FLS2012, Ring WG, X. Huang 13/5/2012

Lattice modeling for a storage ring with magnetic field data

X. Huang, J. Safranek (SLAC)

Y. Li (BNL)

3/5/2012

FLS2012, Ring WG, X. Huang 2

• Discrepancy between original model and measurements.• Understanding dynamic effects of rectangular gradient

dipoles.

• 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?

Motivation

3/5/2012

Z

Xsx

1. The ideal trajectory in RGD is not a circular arc.2. Gradient varies with s-variable3. Off-plane longitudinal field


The field-integration approach

3/5/2012

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).

Coordinate transformation at the edges.

For dipoles, additional transformation is needed.

Equation of motion when using z as free variable.


Magnetic field in a standard SPEAR3 dipole

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-100 -50 0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

x (mm)

By

(nor

mal

ized

)

By at z=0

TanabeRADIA newMeasured 2007

-1 -0.5 0 0.5 10

0.5

1

1.5

Z (m)

By/

Bym

ax

RADIA newMeasured 2001fit meas 2001measured 2007

We have coil, wire measurements.Hall probe scans along Z in 2001, 2007.Hall probe scans on X-Z plane in 2007.

-0.050

0.050.1

0.150.2

0.250.3

0.4

0.6

0.8

1-1.5

-1

-0.5

0

X (m)Z (m) 145D

By

(T)

Hall probe x-z scan (2007)

We started examining our lattice model from magnetic fields in magnets.


An analytic dipole field model

3/5/2012

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.


Energy calibration

3/5/2012

How to calibrate the require bending field for a 3GeV beam?

This is how dipole magnets were positioned: adjust the dipole current (converted to K-value) until the alignment requirement is met.

+

(Corbett & Tanabe, 2002) (Yoon, et al, NIMA 2004). The virtual center was held constant (392.35 mm).

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%.

2001 Z-scan

Energy measurement at SPEAR3 confirmed the prediction with high precision.


Effects of quadrupole fringe field

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)())((''12

1))()(('

4

1))((

2

1)(

2

1)( 644222222 XOyxskyPxPyxskyxskPPsH yxyx J. Irwin, C.X. Wang

The leading correction for a hard-edge model is from the last two terms, which are nonlinear(2).

A general Hamiltonian (including longitudinal field variation) can be derived using a proper magnetic field expansion(1).

The leading correction term from a soft fringe model is linear(3).

* El-Kareh; Forest; Bassetti & Biscari** Lee-Whiting, Forest & Milutinovic, Irwin & Wang, Zimmermann***Irwin & Wang (PAC’95), D. Zhou (IPAC10).

A perturbation approach


The linear correction to quadrupole map

3/5/2012

J. Irwin, C.X. Wang, PAC95

The correction map

The generating function for the correction map

)(21

2 yx yPxPI

f

matrix ),,,(diag 1111 IIII eeee For a symmetric quadrupole, the entrance edge has a reversed sign for I1

The tune changes are (always negative)

||2

|,|2 0

1

20

0

120

k

ILk

k

ILk yy

xx

leading contribution

For SPEAR3, quadrupole fringe fields cause tune changes of [-0.065, -0.059], in agreement with the predictions by the above equation.


The nonlinear correction

3/5/2012

The generating function for the correction map (exit edge)

)()1(6

1)33(

)1(12

1 33232304 xyskewyyxx PyPxkyPxPyPxyPxkf

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.

and tune dependence on amplitude (only showing hard edge contribution below)

0

16

k

I

This agree with tracking quite well.

Forest & Milutinovic


An AT quadrupole passmethod with fringe field

3/5/2012

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.

0.02 0.02 0.02 0.02 0.02 0.02 0.02

-0.0123

-0.0123

-0.0123

-0.0123

-0.0123

-0.0122

-0.0122

-0.0122

xi (m)

xpf (

rad)

quadpassquadpass+matrixnew quad passfield pass

The new quad passmethod agree very well with the field-integration method.

Both linear and nonlinear effects are considered in the new quadrupole passmethod.


The SPEAR3 quadrupole field profile

3/5/2012

The analytical quadrupole field map for SPEAR3 magnet was based on magnet modeling. Simulated field is converted to an analytic form.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

Z (m) for 60Q

B1f

(no

rmal

ized

)

xyzBzB

xyxzzxBB

yyxzzyBB

z

y

x

)(')sgn(

)]3)((''12

1)([

)]3)((''12

1)([

1

231

321

Magnetic field

All SPEAR3 quads have identical fringe profile.

m 060.06

,m 1061.00

123

0

11

k

I

k

II a

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

10

12

14

16

18

Z (m)

B1 (

T/m

)

all quads at 72 A in modeling

60Q50Q34Q15Q


Comparison of models to measurement

3/5/2012

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!

Parameter Measured 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

The model is based on a calibrated experimental lattice with all IDs open (4/6/2009).

Effect of the predicted -0.1% beam energy shift is not included, which change the tunes by [0.023, -0.004] for [nux, nuy].

The tune differences are [0.067 -0.060] between the best model and the measurement, a big improvement from the original model.


Beta beat and correction

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0 50 100 150 200 250-0.1

-0.05

0

0.05

0.1

s (m)

x/

x

no correctioncorrected

0 50 100 150 200 250-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

s (m)

y/

y

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.


The tune map

3/5/2012

Chromaticities are corrected with SF/SD to obtain [1.65, 2.18]. Tunes are obtained by tracking 256 turns.

0.1 0.15 0.2

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

x

y

measured vs. new model

0.1 0.15 0.2

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

x

y

old model vs. new model

“new model” = field model for bend + quad fringe. This model agrees with measurement better.


High-order chromaticities

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-0.04 -0.02 0 0.02 0.040.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

p/p

x-Cx

p/p

low tune [0.106, 0.177] LE

measuredold modelnew model

-0.04 -0.02 0 0.02 0.040.17

0.18

0.19

0.2

0.21

0.22

0.23

p/p

y-Cy

p/p

low tune [0.106, 0.177] LE

measuredold modelnew model

Low tune

Old model

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.


Progress toward fast tracking for dipoles

3/5/2012

(1) Extract a Lie map from the Taylor map obtained from the field-integration 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).


Comparison of Map-pass to field-pass

3/5/2012

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.


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.

3/5/2012

More efforts are need to understand the discrepancies between model and measurements.


This slide is left blank

3/5/2012


The dipole field map

3/5/2012

0.4 0.5 0.6 0.7 0.8 0.9 1-20

-15

-10

-5

0

5

Z (m)

B0,

1,2,

(T,

T/m

, T

/m2 )

2007 xz scan

B0

B1

B2

By(at y=0,z=0) = -1.233257 + 3.143436*x -0.324508*x^2ByL = -1.857103 + 4.662405*x -0.931245*x^2

Coil measurement givesByL (T m)= -1.8506 +4.6081 x - 1.2632 x^2

B0, B1, B2 from X-Z scan

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)


The linear correction to quadrupole map

3/5/2012

A perturbation approach

J. Irwin, C.X. Wang, PAC95

Hard-edge model, for exit edge

Perturbation term

The map

The generating function for the correction map (only leading contribution is shown)

)(21

2 yx yPxPI

f

matrix ),,,(diag 1111 IIII eeee For a symmetric quadrupole, the entrance edge has a reversed sign for I1

The tune change would be (always negative)

||2

|,|2 0

1

20

0

120

k

ILk

k

ILk yy

xx


0.02 0.02 0.02 0.02 0.02 0.02 0.02

-0.0123

-0.0123

-0.0123

-0.0123

-0.0123

-0.0122

-0.0122

-0.0122

xi (m)

xpf (

rad)

quadpassquadpass+matrixnew quad passfield pass

Verification of the quad fringe pass method

3/5/2012

yi=0.005 m

With the (quad+matrix) part subtracted.Zimmerman result is from the average Hamiltonian H1+2

Quadpass: quad transfer matrixQuadpass+matrix: quad transfer matrix + linear edge transfer matrix.New quad pass: with linear and nonlinear corrrection.Field pass: integration through magnetic field.

23

A pass method for magnetic field in AT

7/20/2011

(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.

3/5/20121 Lattice modeling for a storage ring with magnetic field data X. Huang, J. Safranek (SLAC)...

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