# Presentation: 485753

## Advanced Optics Measurements at Tevatron

- Vadim Sajaev
- ANL
- V. Lebedev, V. Nagaslaev, A. Valishev
- FNAL

## Talk overview

- Response matrix fit method description
- Program description
- Results of the fit for Tevatron
- Accuracy of the measurements
- Uniqueness of the fit results
- Comparison with tune shift measurements
- Example of beta function correction at APS

## The orbit response matrix is the change in the orbit at the BPMs as a function of changes in steering magnets:

- Orbit response matrix

- Modern storage rings have a large number of steering magnets and precise BPMs, so measurement of the response matrix provides a very large array of precisely measured data
- The response matrix is defined by the linear lattice of the machine; therefore it can be used to calibrate the linear optics in a storage ring

## Orbit response matrix fit

- The method was first suggested by Corbett, Lee, and Ziemann at SLAC and refined by Safranek at BNL. A very careful analysis of the response matrix was done at the NSLS X-ray ring, ALS, and later at APS. A similar method was used at ESRF for characterization and correction of the linear coupling and to calibrate quadrupoles by families.

- The main idea of the analysis is to adjust all the variables that the response matrix depends on in order to solve the following equation:

- ,

## 	The response matrix depends on the following parameters:

Quadrupole gradient errors
Steering magnet calibrations
BPM gains
Quadrupole tilts
Steering magnet tilts
BPM tilts
Energy shift associated with steering magnet changes
BPM nonlinearity
Steering magnet and BPM longitudinal positions
etc.

- Quadrupole gradient errors
- Steering magnet calibrations
- BPM gains
- Quadrupole tilts
- Steering magnet tilts
- BPM tilts
- Energy shift associated with steering magnet changes
- BPM nonlinearity
- Steering magnet and BPM longitudinal positions
- etc.

- Orbit response matrix fit

- Main  parameters

- Main coupling parameters

## First full-scale measurements

- Response and dispersion measurements are taken on 2004/08/05
- Measurements contain all corrector magnets: two 110236 matrices
- Total response matrix derivative is more than 500 Mb – there is no way to analyze the entire set
- The measurement is split into 3 approx. equal subsets, and each subset is analyzed separately
- The comparison of the results gives us an estimate of the fit accuracy

## Orbit response matrix fit for Tevatron

- Tevatron has 110 steering magnets and 120 BPMs in each plane and 216 quadrupoles
- For our analysis we use about 40 steering magnets in each plane,  all BPMs, all quadrupoles, and tilts of one half of quadrupoles. The resulting response matrix has about 16,500 elements, and the number of variables is 980.
- Finally we solve the following equation (by iterations):

- X   =   M-1  ·  V

- 130 Mb

## optim or elegant calculate RM for different variables (can run in parallel)
sddstoolkit is used to postprocess and build RM derivative
Inverse RM derivative is computed with matlab of sddstoolkit

- GUI:  tcl/tk - unix, linux

- Fitting program
- tcl/tk

- Response matrix
- derivative
- calculation

- Iterations

- Output

- RM calculations – elegant or optim
- preprocessing, postprocessing - sddstoolkit

- all results are stored in sdds files
- sddsplot can be used for graphic output

- optim or elegant calculate RM for different variables (can run in parallel)
- sddstoolkit is used to postprocess and build RM derivative
- Inverse RM derivative is computed with matlab of sddstoolkit

- Program organization chart

## Slide 9

## The following variables are used:

- Gradient errors in all quads
- Corrector calibration errors (1/3 of correctors)
- BPM gain errors in all BPMs
- Quadrupole tilts (1⁄2 of all quads)
- Corrector and BPM tilts
- Energy change due to horizontal correctors

- The starting point of the fit is the model resulted from differential orbit adjustment

## Measurements and fitting: X-X orbit

## Measurements and fitting: Y-Y orbit

## Measurements and fitting: Y-X orbit

## Measurements and fitting: X-Y orbit

## Horizontal dispersion

## Vertical dispersion

## Summary of the residual rms errors after the fit:

|  | x-x (m) | y-x (m) | x-y (m) | y-y (m) | h disp (mm) | v disp (mm) |
| --- | --- | --- | --- | --- | --- | --- |
| Before | 160 | 120 | 100 | 200 | 240 | 190 |
| Set 1 | 21 | 19 | 19 | 24 | 60 | 52 |
| Set 2 | 23 | 19 | 17 | 22 | 60 | 58 |
| Set 3 | 24 | 20 | 19 | 26 | 68 | 57 |

## Measurement accuracy (response)

- 25 measurements on + and 25 measurements on -
- rms of the measured values are also recorded

## Measurement accuracy (dispersion)

## Fit variables: Quads

- No unique solution – too many variables

## Fit variables: BPMs

- BPMs have to have the same gains

- rms gain difference is 1.7% in X and 2.1% in Y

## Fit variables: Skew quads

- Only 1⁄2 of all quads are used

## Beta function accuracy

- Beta functions are computed based on each set of variables, then average beta functions are calculated
- Difference between the average beta function and one of data sets:

- BetaX1 rms error – 2.2%
- BetaY2 rms error – 3.1%

- EtaX rms error – 2.9%

## Vertical dispersion accuracy

## Is the solution unique?

**No ****	****– ****	****in terms of quadrupoles**
- 			(more accurate response measurements will 			result in less quadrupole ambiguity)
**Yes ****	****– ****	****in terms of beta functions**
- 			(more accurate response measurements will 			result in better accuracy in beta function 			determination)
- 	The best proof of correct beta function determination is the ability to make predictable changes

## Measured and Computed TuneshiftsLOCO  Fit

| Element | dQx 
Meas. | dQx 
Model. | Difference % |  |  | dQy 
Meas. | dQy 
Model. | Difference % |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  |  |  |  |  |  |  |  |  |  |
| T:QE17H | 0.0067 | 0.0070 | -4 | ± | 3 | -0.0015 | -0.0020 | -25 | ± | 10 |
| T:QE19H | 0.0053 | 0.0060 | -11 | ± | 3 | -0.0022 | -0.0025 | -13 | ± | 8 |
| T:QE26H | 0.0055 | 0.0066 | -16 | ± | 3 | -0.0018 | -0.0022 | -17 | ± | 9 |
| T:QE28F  | 0.0071 | 0.0073 | -3 | ± | 3 | -0.0015 | -0.0018 | -16 | ± | 11 |
| T:QF28F  | 0.0057 | 0.0059 | -3 | ± | 3 | -0.0023 | -0.0025 | -7 | ± | 8 |
| T:QF32F  | 0.0073 | 0.0077 | -5 | ± | 3 | -0.002 | -0.0017 | 15 | ± | 12 |
| T:QE47F  | 0.0022 | 0.0024 | -8 | ± | 8 | -0.0052 | -0.0058 | -11 | ± | 3 |
| T:QF33F  | 0.002 | 0.0024 | -18 | ± | 8 | -0.0062 | -0.0067 | -8 | ± | 3 |
| C:B0Q2H | 0.0114 | 0.0119 | -4 | ± | 2 | 0.0111 | 0.0112 | -1 | ± | 2 |
| C:B0Q3H | 0.0123 | 0.0138 | -11 | ± | 1 | 0.0127 | 0.0125 | 2 | ± | 2 |
| C:B0QT2H | 0.0081 | 0.0091 | -11 | ± | 2 | -0.0153 | -0.0167 | -8 | ± | 1 |
| C:B0QT3H | 0.0075 | 0.0083 | -9 | ± | 2 | -0.002 | -0.0025 | -20 | ± | 8 |
| C:D0Q2H | 0.0117 | 0.0119 | -2 | ± | 2 | 0.0109 | 0.0108 | 1 | ± | 2 |
| C:D0Q3H | 0.0128 | 0.0134 | -5 | ± | 1 | 0.0129 | 0.0125 | 3 | ± | 2 |
| C:D0QT2H | 0.0085 | 0.0091 | -6 | ± | 2 | -0.016 | -0.0164 | -2 | ± | 1 |
| C:D0QT3H | 0.0078 | 0.0082 | -5 | ± | 2 | -0.0025 | -0.0026 | -3 | ± | 8 |

## Conclusion

- Response matrix fit works and gives good results for Tevatron too
- It improves the accuracy of the existing beta function measurements
- Improving orbit measurement accuracy should further improve the fit and make the solution more unique
- APS experience shows, that over time our fit accuracy is improved by about factor of 2
- Real validation of the fit results is ability to predict consequences of lattice changes

## What is next?

- Beta function correction and control
- At APS we developed a program for beta function correction which looks very similar to the response matrix fit program – it minimizes the difference between measured and designed beta functions by varying all available quadrupole corrections
- Successful beta function control requires the ability to
- Calculate correct quadrupole gradient corrections (good lattice model)
- Transform quadrupole gradient changes into quadrupole current changes

## APS beta function correction example

## Acknowledgements

- 	The author (VS) would like to thank K. Harkay, L. Emery, M. Borland, R. Soliday, H. Shang, and K.-J. Kim for their help and support during this work.