# Presentation: 161393

## Unit 20Field models versus measurements

- Soren Prestemon and Paolo Ferracin
- Lawrence Berkeley National Laboratory (LBNL)
- _Ezio Todesco_
- European Organization for Nuclear Research (CERN)

## QUESTIONS

- What is our capability of forecasting the field harmonics ?
- We compare field model to magnetic measurements
- in absolute (built magnet vs model)
- in relative (impact of design variations)
- How can we steer the systematic components ?
- With what precision can we forecast the random components ?
- What is our capability of forecasting beam dynamics parameters based on measurements ?

## CONTENTS

- 1 – Systematic components: model vs measurements (absolute) in the example of the LHC dipoles
- 2 – Systematic components: model vs measurements (relative), and how to perform corrections
- 3 – How to estimate random component: models vs measurements
- 4 – A glimpse on beam behavior vs measured field quality

## 1 – SYSTEMATIC COMPONENT, ABSOLUTE: THE CASE OF THE LHC DIPOLE

- Special features of the LHC dipole
- Two layers  large coil deformations
- Thick stainless steel collars (iron gives
- 	only 20% of stress)  small collar
- 	deformations
- Two-in-one collars  even multipoles
- 	b2, b4, ... are allowed ones, but we will
- 	not discuss them
- Thin filament and energy swing of 16 
- 	persistent current component not so large
- Data relative to 1200 magnets
- A good statistics ...
- We will follow the sequence of the magnetic measurements
- To have an idea of the size of the different terms
- To see what is the agreement between model and measurements

- Cross-section of the LHC dipole

## 1 – SYSTEMATIC COMPONENT, ABSOLUTE: THE CASE OF THE LHC DIPOLE

- First step: the collared coil
- Nominal geometry
- Coil and collar deformations
- Collar permeability
- Remarks
- Coil and collar deformations can have a strong effect on low order multipoles
- Collar permeability not negligible
- Discrepancy model-measurement must be estimated in absolute and not in relative
- 3 units of *b**3*, 1 unit of *b**5*, 0.1 units of higher orders
- Higher orders are have usually a much better agreement with model
- They are less sensitive to coil displacements

## 1 – SYSTEMATIC COMPONENT, ABSOLUTE: THE CASE OF THE LHC DIPOLE

- Second step: the cold mass – collared coil
- Remember: we analyze the difference between cold mass and collared coil (divided by the increase of the main field due to the yoke)
- Magnetic effect of iron
- Mechanical effect of iron (deformation)
- Remarks
- The mechanical contribution of the iron to deformations is not negligible (1 unit of *b**3*, even in the LHC case)
- Very good agreement between model and simulation
- Higher orders are not affected by the iron (obvious from Biot-Savart)

- 41° 49’ 55” N – 88 ° 15’ 07” W

## 1 – SYSTEMATIC COMPONENT, ABSOLUTE: THE CASE OF THE LHC DIPOLE

- Third step: injection field
- Remember: we analyze the difference between injection field at 1.9 K and cold mass at room temperature
- Mechanical effect of cool-down
- Hysteresis of persistent currents
- Remarks
- Notwithstanding the small filaments, the persistent currents have a strong effect (9 units of *b**3*, one of *b**5*)
- The agreement with model is good
- The impact of cool-down is small

## 1 – SYSTEMATIC COMPONENT, ABSOLUTE: THE CASE OF THE LHC DIPOLE

- Fourth step: high field
- Remember: we analyze the difference between high field at 1.9 K and cold mass at room temperature
- Mechanical effect of cool-down
- Saturation of iron
- Deformation of electromagnetic forces
- Remarks
- Iron saturation is small because it has been carefully optimized
- The high field is similar to cold mass values – cool down and electromagnetic forces not so large

## CONTENTS

- 1 – Systematic components: model vs measurements (absolute) in the example of the LHC dipoles
- 2 – Systematic components: model vs measurements (relative), and how to perform corrections
- 3 – How to estimate random component: models vs measurements
- 4 – A glimpse on beam behavior vs measured field quality

## 2. SYSTEMATIC COMPONENTS, RELATIVE

- Once the first models or prototypes have been built ...
- Measurements usually show a discrepancy with respect to the model
- One or more corrective actions are necessary to bring the field quality closer to targets
- What is the capability of models to forecast changes of design?
- We need the model in relative, not in absolute
- Usually this task is less challenging: if the model neglects a systematic effect, it will be wrong in absolute but correct in relative
- There is no precise answer, but one can give examples and the experience of previous productions
- We will present the case of the LHC dipole production

## 2. SYSTEMATIC COMPONENTS, RELATIVE

- Example 1. The impact of a variation of pole shims in the LHC dipoles
- Shims are used to steer both
- 	field quality and stress
- Data relative to a dedicated
- 	experiment
- Good agreement found (model
- 	including deformations)

- Multipole variation induced by a change of 0.1 mm of the pole shim,
- From P. Ferracin, et al, *Phys. Rev. STAB* **5** (2002) 062401.

## 2. SYSTEMATIC COMPONENTS, RELATIVE

- Example 2. Change of cross-section in the LHC dipole to reduce *b**3*, *b**5*
- Change decided after 9 series magnets, implemented at n. 33
- 0.1-0.4 mm change of 3 copper wedges, keeping the same coil size
- Data relative to 33 magnets with X-section 1 and 154 with X-section 2
- Agreement not very good (relevant trends in production, see later)

- Change of the copper wedges of the inner layer in the main LHC dipole: cross-section 1 (left) and cross-section 2 (right)

- Multipole variation induced by the cross-section change from 1 to 2 (change in internal copper wedges) in the main LHC dipole

## 2. SYSTEMATIC COMPONENTS, RELATIVE

- Example 3. Additional mid-plane shim in LHC dipole to reduce *b**3*, *b**5*
- Change decided after 80 series magnets, implemented at n. 154
- Additional mid-plane shim of 0.25 mm thickness
- Data relative to 154 magnets with X-section 2 and 1000 with X-section 3
- Agreement rather good

- Additional mid-plane shim: cross-section 1 (left) and cross-section 2 (right)

- Multipole variation induced by the cross-section change from 2 to 3 (additional mid-plane shim) in the main LHC dipole

## 2. SYSTEMATIC COMPONENTS, RELATIVE

- Conclusions – estimating the impact of a variation in the design on field harmonics
- For dedicated experiments (the same magnet assembled with different configurations) the agreement is within the errors
- When a correction is implemented along a production, its effect can be masked by trends, and the result can be different ...

- The first correction of the cross-section in the main LHC dipoles

## 2. SYSTEMATIC COMPONENTS, RELATIVE

- Conclusions – estimating the impact of a variation in the design on field harmonics
- One has to gently insist in bringing the field quality within targets
- It is mandatory to have a flexible design
- Example: tuning shims in the RHIC magnets [R. Gupta, et al. ...]
- Put in the spec the possibility of changes, good contact with the Firm

- b3 along the production of 1276 LHC dipoles – red limits are for the final average

## CONTENTS

- 1 – Systematic components: model vs measurements (absolute) in the example of the LHC dipoles
- 2 – Systematic components: model vs measurements (relative), and how to perform corrections
- 3 – How to estimate random component: models vs measurements
- 4 – A glimpse on beam behavior vs measured field quality

## 3. RANDOM COMPONENT, GEOMETRIC

- This is the component due to the limited precision in the position of the coil
- Two components:
- Precision of positioning with respect to the design  systematic
- Reproducibility of the positioning  random
- We will focus on the second component
- The precision of reproducibility is of the order of 10-100 m
- In general it is dominant over all the other components, as
- spread in magnetic properties of iron or of collars
- spread in the persistent current
- spread in the deformations due to electromagnetic forces
- Moreover
- The spread in positioning induced by cool-down is small

## 3. RANDOM COMPONENT, GEOMETRIC

- A simple way to estimate the geometric random component
- Using a Monte-Carlo, coil blocks are randomly displaced with an amplitude belonging to a distribution with zero average and stdev *d*
- In the past a thumb rule was to use *d*=0.05 mm to get a reasonable estimate of the errors
- For each deformed coil one
- 	computes the multipoles
- Repeating 100-1000 times,
- 	one gets a multipole distribution
- 	and can compute average (that
- 	will be close to zero) and stdev
- The computed stdev is the guess
- 	of our random component

- A random movement of coil blocks to
- estimate geometric random errors

## 3. RANDOM COMPONENT, GEOMETRIC

- How to estimate the repeatability of coil positioning
- The previous approach is used before starting models and prototypes
- Once a series of homogeneous magnets is built, one can measure them, compute the stdev, and estimate the repeatability of the coil positioning by selecting the *d* that better fits data
- In this way one can estimate (and monitor) the assembly tolerances
- Results for different productions: repeatability of coil positioning *d* is around 0.020 mm rather than 0.050 mm
- A considerable improvement with time !

## 3. RANDOM COMPONENT, GEOMETRIC

- How to estimate the repeatability of coil positioning: the LHC dipoles case
- This model gives equal
- 	estimates for random
- 	normal and skew, and a
- 	decay with multipole order
- The decay fits well !!
- 	(this is again Biot-Savart)
- Indeed, there is a saw-tooth,
- 	odd normal are larger
- 	than even skew and
- 	viceversa

- 41° 49’ 55” N – 88 ° 15’ 07” W

- Measured random components (markers) versus model (red line) in the LHC dipoles [from B. Bellesia et al, ...]

## 3. RANDOM COMPONENT, GEOMETRIC

- A different behavior between normal and skew is well known in literature – it limits the prediction power of the model
- Has been already
- 	observed in Tevatron
- 	[Herrera et al, PAC 1983]
- Heuristic justification:
- 	Not all tolerances are
- 	kept in the same way,
- 	some symmetries are
- 	more preserved than
- 	others [see also R. Gupta,
- 	Part. Accel. **54 **(1996) 129-140]

- 41° 49’ 55” N – 88 ° 15’ 07” W

- Measured random components in four dipole productions
- B. Bellesia, et al. “Random errors in sc dipoles”, EPAC 2006.

## 3. RANDOM COMPONENT, GEOMETRIC

- Quadrupoles:
- Also in this case the decay agrees well with the models
- The repeatability of coil positioning is 0.015 to 0.030 mm for RHIC and LHC productions
- In general the saw-tooth is less strong, but random allowed multipoles (*b**6*) are larger then estimates

- 41° 49’ 55” N – 88 ° 15’ 07” W

- Measured random components and model fit in RHIC MQ

- Measured random components and model fit in LHC MQXB

## 3. RANDOM COMPONENT, GEOMETRIC

- Is the repeatability of coil positioning depending on the magnet aperture ?
- From RHIC and LHC production there is some indication that it does not depend on aperture  larger apertures give smaller random components

- Repeatability of coil positioning achived in LHC and RHIC versus magnet aperture

## CONTENTS

- 1 – Systematic components: model vs measurements (absolute) in the example of the LHC dipoles
- 2 – Systematic components: model vs measurements (relative), and how to perform corrections
- 3 – How to estimate random component: models vs measurements
- 4 – A glimpse on beam behavior vs measured field quality

## 4. BEAM MEASUREMENTS

- What are the main parameters that affect the beam and how are they related to field quality – the tune
- Linear tune: *Q**x *,*Q**y *is the number of oscillations in the transverse plane made around one turn of the ring
- Tevatron:  *Q**x*=20.573, *Q**y*=20.588
- LHC (planned):  *Q**x*=70.280, *Q**y*=70.310
- This number is crucial for the stability of the beam:
- if the fractional part is zero or close to fractions with low denominators as 1⁄2 , 1/3 2/3 , 1⁄4 ... (resonances) the beam after 2, 3, 4 turns goes around the same path, seeing the same errors that can build up
- It must be controlled within 0.003 – i.e. 2 units for Tevatron, 0.5 units for the LHC
- The linear tune is proportional to all sources of *B**2*
- Mainly quadrupoles, *b**2* in the dipoles, *b**3* in the dipoles plus misalignment ...

## 4. BEAM MEASUREMENTS

- What are the main parameters that affect the beam and how are they related to field quality – the tune
- The linear tune can be measured on-line with an absolute precision of 0.001 with several instruments
- During the injection of Tevatron, a drift in the tune has been measured
- Origin – decay of multipoles, rather than *b**2* in quads look more probable  *b**2* generated by *b**3* decay in dipoles and misalignment
- It has been corrected by acting
- 	on the quad powering

- 41° 49’ 55” N – 88 ° 15’ 07” W

- Tune drift in Tevatron at injection, without correction
- From J. Annala, et al., Beams-doc 1236 (2005) pg. 18

- Tune drift in Tevatron at injection, after correction
- From J. Annala, et al., Beams-doc 1236 (2005) pg. 18

## 4. BEAM MEASUREMENTS

- What are the main parameters that affect the beam and how are they related to field quality – the chromaticity
- The derivative of the tune with respect to the beam energy is called chromaticity
- Since the beam contains particles of different energy (10-310-4), this derivative must be close to zero to avoid that in the beam some particle have an unstable tune
- It cannot be negative since it induces instability – it is usually set at 2-5, controlled within 1-2
- The chromaticity is proportional to all sources of *B**3*
- Mainly *b**3* in the dipoles ...
- Chromaticity can be measured with different methods
- Changing the energy of the beam and measuring the tune ... but this takes time
- Other parasitic methods

## 4. BEAM MEASUREMENTS

- What are the main parameters that affect the beam and how are they related to field quality – the chromaticity
- The drift in *b**3* at injection induced a large change of chromaticity
- If not corrected this can kill the beam
- In Tevatron: one unit of *b**3* gives 25 of chromaticity (45 in the LHC)
- It must be corrected
- Tevatron experience
- Drift of 75 of chromaticity during injection
- Half could be explained by the measured decay of *b**3 *at injection
- Using beam measurements, chromaticity has been made stable within 2 by compensating the drift with powering of the sextupole correctors

## 4. BEAM MEASUREMENTS

- What are the main parameters that affect the beam and how are they related to field quality – the coupling
- Linear coupling is a mechanism that couples oscillations in the vertical and horizontal plane
- The normal modes are not *x* and *y* any more, but a linear combination
- Creates instabilities in certain regimes
- All the instrumentation is on  *x* and *y*  this terribly complicates all diagnostic and correction
- Linear coupling defined as the minimal difference between the tune in the two planes
- If there is no linear coupling, the two planes are uncoupled and the two tunes can be brought as close as we want
- Must be controlled within *Q=*0.003
- Linear coupling sources
- All skew quadrupole terms: in dipoles, misalignment of the quadrupoles (tilt), feed-down of *b**3 *due to misalignment, ...

## 4. BEAM MEASUREMENTS

- What are the main parameters that affect the beam and how are they related to field quality – the coupling
- Linear coupling must be controlled within *Q=*0.003
- Tevatron: *Q=*0.03 (10 times larger than tolerance) without correction during injection
- Probably generated by the misalignment of dipoles and *b**3 *decay
- In Tevatron linear coupling has been corrected through skew quadrupoles
- Effective to bring back linear
- 	coupling to tolerances

- Tune drift in Tevatron at injection, without correction
- From J. Annala, et al., Beams-doc 1236 (2005) pg. 37

## CONCLUSIONS

- We analyzed the capability of the model of guessing the systematic components (absolute)
- A few units of b3, a fraction of unit on higher order
- The largest error is made for the collared coil at r.t.
- When you make optimizations, do not work to get 0.00 since the model is not so precise
- Remember that codes gives multipoles with 5 digits (or more) but that they are not meaningful is absolute
- We analyzed the capability of the model of forecasting the impact of a change in the magnet (relative)
- The model usually works at 80%-90%, but one can have suprises
- Once a magnet is build, the discrepancy in the absolute is corrected through one (or more) changes – fine tuning

## CONCLUSIONS

- We discussed how to model the random components
- The dominant components are the geometrical ones
- They can be estimated through a Monte-Carlo where blocks are randomly moved of 0.02 mm
- This allows to estimate the reproducibility in the coil positioning in a magnet production by post-processing the magnetic measurements
- We discussed the agreement between magnetic measurements and beam measurements
- Case of Tevatron (LHC will come soon ...)

## REFERENCES

- Field model vs measurements (systematic)
- A.A.V.V. “LHC Design Report”, CERN 2004-003 (2004) pp. 164-168.
- P. Ferracin, et al., _[Azimuthal](http://prst-ab.aps.org/abstract/PRSTAB/v5/i6/e062401?qid=ed70c20985705bb5&qseq=2&show=10)__[ coil size and field quality in the main CERN Large ](http://prst-ab.aps.org/abstract/PRSTAB/v5/i6/e062401?qid=ed70c20985705bb5&qseq=2&show=10)__[Hadron](http://prst-ab.aps.org/abstract/PRSTAB/v5/i6/e062401?qid=ed70c20985705bb5&qseq=2&show=10)__[ ](http://prst-ab.aps.org/abstract/PRSTAB/v5/i6/e062401?qid=ed70c20985705bb5&qseq=2&show=10)__[Collider](http://prst-ab.aps.org/abstract/PRSTAB/v5/i6/e062401?qid=ed70c20985705bb5&qseq=2&show=10)__[ dipoles](http://prst-ab.aps.org/abstract/PRSTAB/v5/i6/e062401?qid=ed70c20985705bb5&qseq=2&show=10)_ Phys. Rev. ST Accel. Beams **5**, 062401 (2002)
- Estimating random errors
- Herrera et al, PAC 1983
- R. Gupta, Part. Accel. **54 **(1996) 129-140
- P. Ferracin, et al., _[Modeling of random geometric errors in superconducting magnets with applications to the CERN Large ](http://prst-ab.aps.org/abstract/PRSTAB/v3/i12/e122403?qid=ed70c20985705bb5&qseq=3&show=10)__[Hadron](http://prst-ab.aps.org/abstract/PRSTAB/v3/i12/e122403?qid=ed70c20985705bb5&qseq=3&show=10)__[ ](http://prst-ab.aps.org/abstract/PRSTAB/v3/i12/e122403?qid=ed70c20985705bb5&qseq=3&show=10)__[Collider](http://prst-ab.aps.org/abstract/PRSTAB/v3/i12/e122403?qid=ed70c20985705bb5&qseq=3&show=10)_ Phys. Rev. ST Accel. Beams **3**, 122403 (2000)
- B. Bellesia, et al., EPAC 2006.
- B. Bellesia, et al., Phys. Rev. ST Accel. Beams in press.
- Beam measurements
- Several works by Annala, Bauer, Martens, et al on Tevatron in 2000-2005
- Experience of RHIC and HERA

## ACKNOWLEDGMENTS

- B. Bellesia, R. Gupta, W. Scandale, R. Wolf for the modeling of random errors
- P. Bauer, M. Martens, V. Shiltsev for the beam measurements at Tevatron