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# Computation of the main and fringe fields for the electrostatic quadrupoles of the muon $g - 2$ storage ring

## Abstract

This team developed a highly accurate and fully Maxwellian conformal mapping method for calculation of main fields of electrostatic particle optical elements. A remarkable advantage of this method is the possibility of rapid recalculations with geometric asymmetries and mispowered plates. We used this conformal mapping method to calculate the multipole terms of the high voltage quadrupoles in the storage ring of the Muon g-2 Experiment (FNAL-E-0989). Next, we demonstrate that an effect where the observed tunes correspond to a voltage that is about 4% higher compared to the voltage to which the Muon g-2 quadrupoles are set is explained by the conceptual and quantitative differences between the beam optics quadrupole voltage and the quadrupole voltage at the plates. Completing the methodological framework for field computations, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculated the quadrupole strength falloff and its effective field boundary (EFB) for the Muon g-2 quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to 2%.

- Authors:

- Michigan State Univ., East Lansing, MI (United States). Dept. of Physics and Astronomy

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1561533

- Report Number(s):
- [FERMILAB-PUB-19-092-PPD]

[Journal ID: ISSN 0217-751X; oai:inspirehep.net:1753565]

- Grant/Contract Number:
- [AC02-07CH11359; FG02-08ER41546]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- International Journal of Modern Physics A

- Additional Journal Information:
- [ Journal Volume: 34; Journal Issue: 36]; Journal ID: ISSN 0217-751X

- Publisher:
- World Scientific

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND TECHNOLOGY; Muon g−2; anomalous magnetic dipole moment; storage rings; electrostatic quadrupoles; main field; multipole expansion; fringe field; effective field boundary; tunes; plate misalignments; mispowered plates; voltages; differential algebra; conformal mapping methods

### Citation Formats

```
Valetov, Eremey, Berz, Martin, and Makino, Kyoko. Computation of the main and fringe fields for the electrostatic quadrupoles of the muon $g - 2$ storage ring. United States: N. p., 2019.
Web. doi:10.1142/S0217751X19420417.
```

```
Valetov, Eremey, Berz, Martin, & Makino, Kyoko. Computation of the main and fringe fields for the electrostatic quadrupoles of the muon $g - 2$ storage ring. United States. doi:10.1142/S0217751X19420417.
```

```
Valetov, Eremey, Berz, Martin, and Makino, Kyoko. Mon .
"Computation of the main and fringe fields for the electrostatic quadrupoles of the muon $g - 2$ storage ring". United States. doi:10.1142/S0217751X19420417.
```

```
@article{osti_1561533,
```

title = {Computation of the main and fringe fields for the electrostatic quadrupoles of the muon $g - 2$ storage ring},

author = {Valetov, Eremey and Berz, Martin and Makino, Kyoko},

abstractNote = {This team developed a highly accurate and fully Maxwellian conformal mapping method for calculation of main fields of electrostatic particle optical elements. A remarkable advantage of this method is the possibility of rapid recalculations with geometric asymmetries and mispowered plates. We used this conformal mapping method to calculate the multipole terms of the high voltage quadrupoles in the storage ring of the Muon g-2 Experiment (FNAL-E-0989). Next, we demonstrate that an effect where the observed tunes correspond to a voltage that is about 4% higher compared to the voltage to which the Muon g-2 quadrupoles are set is explained by the conceptual and quantitative differences between the beam optics quadrupole voltage and the quadrupole voltage at the plates. Completing the methodological framework for field computations, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculated the quadrupole strength falloff and its effective field boundary (EFB) for the Muon g-2 quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to 2%.},

doi = {10.1142/S0217751X19420417},

journal = {International Journal of Modern Physics A},

number = [36],

volume = [34],

place = {United States},

year = {2019},

month = {12}

}

Works referenced in this record:

##
Revisiting the Crowding Phenomenon in Schwarz–Christoffel Mapping

journal, January 2008

- Banjai, L.
- SIAM Journal on Scientific Computing, Vol. 30, Issue 2

##
The Brookhaven muon (g−2) storage ring high voltage quadrupoles

journal, May 2003

- Semertzidis, Yannis K.; Bennett, Gerald; Efstathiadis, Efstratios
- Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 503, Issue 3

##
The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains

journal, June 2005

- Crowdy, Darren
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 461, Issue 2061

##
Algorithm 756; a MATLAB toolbox for Schwarz-Christoffel mapping

journal, June 1996

- Driscoll, Tobin A.
- ACM Transactions on Mathematical Software, Vol. 22, Issue 2

##
COSY INFINITY Version 9

journal, March 2006

- Makino, Kyoko; Berz, Martin
- Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 558, Issue 1

##
Numerical Computation of the Schwarz–Christoffel Transformation

journal, March 1980

- Trefethen, Lloyd N.
- SIAM Journal on Scientific and Statistical Computing, Vol. 1, Issue 1

##
Numerical Conformal Mapping Using Cross-Ratios and Delaunay Triangulation

journal, November 1998

- Driscoll, Tobin A.; Vavasis, Stephen A.
- SIAM Journal on Scientific Computing, Vol. 19, Issue 6

Works referencing / citing this record:

##
COSY INFINITY Version 9

journal, March 2006

- Makino, Kyoko; Berz, Martin
- Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 558, Issue 1

##
The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains

journal, June 2005

- Crowdy, Darren
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 461, Issue 2061

##
Revisiting the Crowding Phenomenon in Schwarz–Christoffel Mapping

journal, January 2008

- Banjai, L.
- SIAM Journal on Scientific Computing, Vol. 30, Issue 2

##
Numerical Computation of the Schwarz–Christoffel Transformation

journal, March 1980

- Trefethen, Lloyd N.
- SIAM Journal on Scientific and Statistical Computing, Vol. 1, Issue 1

##
Algorithm 756; a MATLAB toolbox for Schwarz-Christoffel mapping

journal, June 1996

- Driscoll, Tobin A.
- ACM Transactions on Mathematical Software, Vol. 22, Issue 2