Jeff Kolski USR Workshop 11/12/2010 11/22/2018 An Improved Linear Model for the PSR: Process and Experimental Verification Jeff Kolski USR Workshop 11/12/2010 LA-UR 10-07409 11/22/2018 Improved_Model Slide 1

Outline Introduction Supporting Measurements 11/22/2018 Outline Introduction LANSCE Description of the baseline model Motivate study by introducing the baseline model’s shortcomings compared with measurement Supporting Measurements Betatron phase and tune measurement Beta function measurement Dispersion function measurement Model improvement measurements Orbit Response Matrix (ORM) Magnet component characterization of the PSR extraction septa fringe fields Ray tracing through the edge focusing of a rectangular dipole An improved model Experimental Verification of the improved model 11/22/2018 Improved_Model

Los Alamos Neutron Science Center (LANSCE)

Los Alamos Neutron Science Center (LANSCE) Ultra Cold Neutrons (UCN) Proton Radiography (pRad) Ion Sources Sector J Isotope Production Facility (IPF) Switch Yard (SY) LDPM03 Area A Transition Region (TR) Central Control Room (CCR) Coupled Cavity Linac (CCL) (805 MHz) Drift Tube Linac (DTL) (201 MHz) Jeff’s Office Building 6 Lujan Center Proton Storage Ring (PSR) Weapons Nuclear Research (WNR) Google Maps

Proton Storage Ring (PSR) Circumference = 90m Beam energy = 798 MeV Revolution frequency =2.8 MHz Bunch length = 290 ns (73 m) Accumulation time = 625 μs = 1746 turns WM41 and WC41 ES41y RF Buncher ES43q RJM

Introduction Baseline Model is an extension of F. Neri’s psrdimad deck Modifications Quadrupole current to gradient length fits are assigned to the correct quadrupole and location in the ring. Dipole edge focusing defined with the FINT parameter instead as a separate quadrupole element. Extensions The locations of the quadrupoles and dipoles (save RIBM09) are determined by the 2006 alignment data. Other element positions determined from a combination of other sources, S. Cousineau’s model, T. Spickermann’s model, and tape measurement. Horizontal and vertical corrector magnets with constant current to kick gains for the vertical corrects from D. Fitzgerald. BPMs are places 18 cm upstream of the center of the quadrupoles. Magnet current and shunt input from a Save Accel data file. 11/22/2018 Improved_Model

Baseline Model Dipoles Horizontal correctors Quadrupoles with current to gradient length fits BPMs Vertical correctors Baseline Model Alignment data for fixed center of magnet to center of magnet distances 11/22/2018 Improved_Model

Baseline Model Predictions Betatron Tune Baseline model predicts the horizontal tune fairly well but not the vertical. Betatron Phase Baseline model predicts the betatron phases within the measurement error Betatron Amplitude Functions Baseline model predicts the beta functions fairly well Dispersion Function Baseline model does a good job at predicting the dispersion function Measured Baseline Model Error Horizontal Tune 3.19150 ± 3.45e-4 3.1973 -5.840e-3 Vertical Tune 2.19793 ± 3.24e-4 2.2451 -4.716e-2 11/22/2018 Improved_Model

Baseline Model Betatron Phase 11/22/2018 Improved_Model

Supporting Measurements The first set of supporting measurements will be employed to first establish an improved model An improved model should make a better betatron tune prediction, especially in the vertical, than the baseline model. An improved model should enhance, or at least maintain, the baseline model’s predictive capabilities for the beta and dispersion functions. Once an improved model is shown to exist at one PSR operational set point, the improved model needs experimental verification at other PSR operating conditions. Second, third, and forth sets of supporting measurements are collected at different PSR setups to experimentally verify the improved model. The improved model should make better predictions of the measured quantities (tune, phase, and beta dispersion functions) than the baseline model. 11/22/2018 Improved_Model

Betatron Phase and Tune Measurement For near-on-axis injection, collect 30-40 turns of RingScan (RS) data at each BPM. Fit the turn-by-turn BPM data to a cosine wave to obtain the phase at each BPM () and tune (). 11/22/2018 Improved_Model

Beta Function Measurement Quadrupole perturbation method Measure the change in the tune with the RS data at four different shunt values for each quadrupole Fit a line to calculate the slope of (KL) The dominating error is the systematic error due to the uncertainty in the fourth order current to gradient length fits, which is estimated as .1% as per D. Fitzgerald. Mean Systematic Error For Large Beta [4.960e-001 m, 4.035e-001 m] Mean Systematic Error For Small Beta [5.040e-002 m, 1.153e-001 m] 11/22/2018 Improved_Model

Beta Function Measurement 11/22/2018 Improved_Model

Dispersion Measurement Momentum compaction factor method Measure the time of flight (TOF) delay between the beam’s revolution period and the design, “moving” 2.8 revolution frequency at three different beam momentums. Measure the CO from the RS data The slope of a line fit to xCO(δ) is the dispersion function Modify the phase of Mod48 and 47 to change the beam momentum Momentum compaction factor is fairly constant with respect to the model parameters. 11/22/2018 Improved_Model

Beta Function Measurement 11/22/2018 Improved_Model

Model Improvement Experiments Model improvement experiments aim to test or correct the baseline model’s treatment of particular elements. Observe an over focusing in the vertical of the baseline model. Three possible sources for additional vertical focusing: Quadrupoles Extraction septa Edge focusing of the dipoles There are three model improvement experiments designed to quantify the vertical focusing in each of the above. Orbit Response Matrix (ORM) Beam-base measurements of the magnet multipole components of the extraction septa fringe fields Ray tracing through the dipole edge focusing 11/22/2018 Improved_Model

Orbit Response Matrix The CO response to a dipole kick is The orbit response at every BPM due to each corrector may be complied into a matrix The ORM (R) may be measured experimentally, however the model can also produce a model ORM, R(p), dependent on the model parameters p. Iteratively step through model parameter space to minimize (R - R(p))2 Linear Optics from Closed Orbits (LOCO) is a Matlab based ORM analysis program with model parameters: quadrupole strengths, quadrupole rolls, positions, sextupole strength, octupole strength, BPM gains and tilts, and corrector kicks and coupling. 11/22/2018 Improved_Model

ORM Measurement 11 horizontal and 9 vertical correctors 34 BPM (horizontal CO measurement at SRPM92 too unstable to use) CO is measured by the RS data at three kick setting for each corrector Baseline Plus Minus Fit a line to xCO(ICor) to obtain the orbit response per unit current of the corrector. LOCO fit initially independent of corrector gain. 11/22/2018 Improved_Model

Measured ORM 11/22/2018 Improved_Model

Measured – LOCO ORM Apply LOCO fitting Quadrupole Strengths, BPM gains, and corrector kicks No coupling was fit in LOCO. The x-x and y-y quadrants of the ORM on the order or less than the coupling quadrants. 11/22/2018 Improved_Model

LOCO Results: Quadrupole Strengths LOCO results suggests ~2.5% systematic decrease in the defocusing quadrupoles without major alterations in the focusing quadrupoles. 2.5% error was not found in power supply indications of read backs. 11/22/2018 Improved_Model

LOCO Results: Corrector Kicks and BPM Gains LOCO results suggest BPM gains all within ±.5% of 1 except for SRPM81x and 91x (Diamond Type BPMs) LOCO results suggest a systematic degrease in the corrector gains. LOCO does not distinguish between the 7″ and 11″ vertical correctors. 11/22/2018 Improved_Model

LOCO Fitted Model The LOCO fitted model: Is the baseline model fit to data taken at one particular PSR set point Is the baseline model with the LOCO fitted quadrupole strengths Before the LOCO fitted model is heralded as the new improved model, it should first be tested against the baseline model and experimental measurements. 11/22/2018 Improved_Model

Model Comparisons Need a consistent comparison method to evaluate the quality of each model prediction with respect to the other models for a particular set of measured data. Tune: The error in the model prediction Betatron phase and beta and dispersion functions: The χ2 between model and measured 11/22/2018 Improved_Model

LOCO Fitted Model Compared with Measurement Measured Tunes: [3 Baseline Mode LOCO Fitted Model Horizontal Tune Error -5.840e-3 4.683e-3 Vertical Tune Error -4.716e-2 6.364e-4 Beta Functions Mean Systematic Error For Large Beta [4.960e-1 m, 4.035e-1 m] Mean Systematic Error For Small Beta [5.040e-2 m, 1.153e-1 m] Total Beta Function χ2/DOF 12.770 15.037 Horizontal Beta Function χ2/DOF 20.928 23.371 Vertical Beta Function χ2/DOF 4.611 6.702 Although the LOCO fitted model predicts better tunes, it does not conserve the quality of the beta function prediction. The LOCO fitted model is not the improved model. 11/22/2018 Improved_Model

Characterization of the Magnetic Multipole Components of the PSR Extraction Septa Fringe Fields Perform beam-based measurements to obtain the dipole, quadrupole, and sextupole components of the PSR extraction septa fringe fields. The fringe fields of each extraction septum will be modeled as a thin lens multipole located at the upstream outer corner of the septum (location closest to the circulating beam with the peak magnetic field). The motion of the circulating beam is influenced by both the fringe (field escaping the ends) and leakage (field issuing from the side of the magnet) fields of each septum. The beam measured the integrated affects of both fields. Thus, leakage and fringe field will be applied interchangeably. Want an operational model, so measure the multipole components as a function trim coil current. RODM02 RODM01 11/22/2018 Improved_Model

RODM02 Extraction Beam Pipe Trim Coil Circulating Beam Pipe Main Coil D Barlow

Multipole Component Measurements Take data with septa off for a baseline Take data with septa on at trim coil current between -10 A and 10 A for comparison Quadrupole Measurement Measure the tune with RS data. 11/22/2018 Improved_Model

Quadrupole Component RODM01 RODM02 11/22/2018 Improved_Model

Modeling the septa fringe fields The results for the dipole, quadrupole, and sextupole multipole components as a function of trim coil current are fit to a fourth order polynomial. The improved model reads the trim coil currents from the Save Accel data file and consults these fourth order current to magnetic multipole fits to obtain the strengths of each magnetic component. Only the quadrupole component of the extraction septa fringe fields is included in the model for this search for an improved linear model. Each extraction septa fringe field is modeled as a 1 cm quadrupole located at the upstream outer corner of the septa with strength determined by the trim coil current. 11/22/2018 Improved_Model

Ray Tracing Through the Dipole Edge Focusing Trace rays, parallel in the transverse, starting at the longitudinal center of the dipole through the magnetic fields from a TOSCA 3D simulation, D. Barlow. The 25 rays begin initially on a 1 cm grid ranging between ±2 cm in both transverse directions. The rays will receive a kick in the vertical due to the edge focusing. From the ray tracing trajectories, we can measure the focal length of the edge focusing. 11/22/2018 Improved_Model

Ray Tracing Trough the Common PSR Dipoles A line may be fit to the bent trajectory of y(s). The focal length of the edge focusing is the length in s between where the vertical position equals the initial value and 0. 11/22/2018 Improved_Model

Results of the Ray Tracing The ray tracing results suggest a systematic increase in the focal length of the edge focusing in all dipole types. Focal Lengths Baseline Model Ray Tracing SRBM 13.208 m 13.733 m SRBM01 14.633 m 15.977 m SRBM11 32.186 m 32.389 m SRBM12 30.823 m 30.889 m RIBM09 234.17 m 238.96 m 11/22/2018 Improved_Model

Constraining the Edge Focusing Focal Lengths in the Improved Model We can constrain the focal lengths of the edge focusing in the improved model to those derived from the ray tracing. Three parameters that determine the edge focusing focal length: the edge angle (β), gap height (g), and the fringe field integral (FINT, κ). where and Chose to constrain the FINT because the edge angle and gap height are physical parameters. Ray tracing suggests the common PSR benders have a FINT parameter of .9, which is more representative of the umclamped Rogowski geometry. FINT Baseline Model Ray Tracing SRBM .5415 .90291 SRBM01 1.2908 SRBM11 .4494 .46491 SRBM12 .5193 .52685 RIBM09 .3579 .47085 11/22/2018 Improved_Model

The Improved Model The improved model is an extension of the baseline model With the following modifications The focal lengths of the rectangular dipole edge focusing is constrained via the fringe field integral to the results of the ray tracing. With the following additions The PSR extraction septa fringe fields modeled as 1 cm quadrupoles with gradient lengths from fourth order trim coil current to quadrupole strength fits from the septa characterization experiment. Before we herald the improved model as the an enhanced linear model of the PSR, we need to establish that it makes better predictions than the baseline and LOCO fitted models. 11/22/2018 Improved_Model

Improved Model Compared with the Baseline and LOCO Fitted Models and Measurement Measured Tunes: [3.19150, 2.19793] Tunes: Baseline LOCO Fitted Improved Horizontal Error -5.840e-3 4.683e-3 -1.210e-2 Vertical Error -4.716e-2 6.364e-4 -7.477e-3 Betatron Phase: Mean rms Measurement Spread [.2041 mradian, .1801 mradian] Total χ2/DOF 0.7 9.1e-2 1.89e-1 Horizontal χ2/DOF 7.2e-2 9.8e-2 8.4e-2 Vertical χ2/DOF 1.328 2.94e-1 Beta Functions: Mean Systematic Error For Large Beta [.4960 m, .4035 m] Small Beta [.0504 m, .1153 m] 12.770 15.037 12.025 20.928 23.371 18.553 4.611 6.702 5.496 Dispersion: Fitting Error 5.564e-002 m 24.543 19.131 17.272 11/22/2018 Improved_Model

Model Verification Now that the improved model has been established, the improved model needs experimental verification at other PSR set points. Take several sets of RS reproducibility data sets different quadrupole settings. Ramp the power supplies in between quadrupole settings to reduce the affects of hysteresis. Due to time constraints, the beta and dispersion functions were not measured at any quadrupole setting. The RS data yields direct comparison with measurement for the betatron tune and phase, but not the beta and dispersion functions Need a method to compare the model predicted beta and dispersion functions with the results of the RS reproducibility dataset. 11/22/2018 Improved_Model

Comparing Model Prediction with RS Data Beta function The amplitude of the betatron oscillation may be written Apply linear regression to fit for the action at the cost of one degree of freedom Then the model comparison is Dispersion function The dispersion function can be related to the CO measurement spread Constrain the BPM measurement error to ~.2 mm and fit for the pulse-to-pulse momentum spread and the covariance term at the cost of two degrees of freedom Since measurement does not provide and error on the measurement spread, use the sum of squares as the comparison quality factor. 11/22/2018 Improved_Model

RingScan2 Measured Tunes: [3.22661, 2.21922] Baseline Improved Horizontal Error 4.154e-3 6.517e-4 Vertical Error -3.591e-2 2.878e-3 Betatron Phase: Mean rms Measurement Spread [.03711 mradian, .01699 mradian] Total χ2/DOF 86.112 19.929 Horizontal χ2/DOF 5.561 4.308 Vertical χ2/DOF 166.663 35.549 Beta Functions: Mean rms Measurement Spread For Large Amplitude [.3201 mm, .1785 mm], Small Beta [.1369 mm, .08955 mm] 3.636 3.484 3.395 3.361 3.878 3.608 Dispersion: Horizontal SSR/DOF [mm2] 3.139e-3 3.153e-3 11/22/2018 Improved_Model

Model and Measured Betatron Phase 11/22/2018 Improved_Model

RingScan3 Measured Tunes: [3.80017, 2.38361] Baseline Improved Horizontal Error 1.169e-2 5.775e-3 Vertical Error -4.135e-2 -2.603e-3 Betatron Phase: Mean rms Measurement Spread [.1846 mradian, .02113 mradian] Total χ2/DOF 37.870 4.736 Horizontal χ2/DOF 1.741 1.931 Vertical χ2/DOF 74.000 7.540 Beta Functions: Mean rms Measurement Spread For Large Amplitude [.2704 mm, .2301 mm], Small Beta [.06731 mm, .1198 mm] 3.140 2.244 0.278 0.235 6.001 4.252 Dispersion: Horizontal SSR/DOF [m2] 2.806e-3 2.657e-3 11/22/2018 Improved_Model

RingScan4 Measured Tunes: [2.65383, 3.58292] Baseline Improved Horizontal Error 1.019e-2 6.852e-3 Vertical Error -3.425e-2 5.665e-3 Betatron Phase: Mean rms Measurement Spread [.04427 mradian, .01615 mradian] Total χ2/DOF 60.727 12.483 Horizontal χ2/DOF 2.276 1.480 Vertical χ2/DOF 119.178 23.486 Beta Functions: Mean rms Measurement Spread For Large Amplitude [.2367 mm, .1404 mm], Small Beta [.09736 mm, .07678 mm] 136.582 147.902 3.802 4.168 269.361 291.636 Dispersion: Horizontal SSR/DOF [m2] 1.822e-3 1.841e-3 11/22/2018 Improved_Model

Model and Measured Betatron Phase 11/22/2018 Improved_Model

Model and Measured Betatron Phase 11/22/2018 Improved_Model

Conclusions We have constructed an improved model With ~10 times vertical tune prediction While maintaining or enhancing the beta and dispersion function predictions We found the additional vertical focusing in the model to be located in the edge focusing of the dipoles in a misrepresentative fringe field integral parameter. 11/22/2018 Improved_Model

Further Work The focal lengths of the edge focusing are taken from 3D magnetic field calculations. It may be good to measure these focal lengths with the beam, possibly as a function of current in the bender. Possibly applicable in ORM. Also, could take additional ORM data with ramped quadrupoles prior to measurement in order to fit for the uncertainty in the current to gradient length fits. 11/22/2018 RJM_IU_2010

Acknowledgements Special thanks to R. Macek (LANL) R. McCrady (LANL) D. Barlow (LANL) S.Y. Lee (Indiana University) X. Haung (SLAC) 11/22/2018 RJM_IU_2010

Additional Slides 11/22/2018 Improved_Model

Multipole Component Measurements Take data with septa off for a baseline Take data with septa on at trim coil current between -10 A and 10 A for comparison Dipole Measurement Measure the CO with RS data. Quadrupole Measurement Measure the tune with RS data. 11/22/2018 Improved_Model

Multipole Component Measurements Sextupole Measurement Measure the dispersion function with both septa off Measure the “natural chromaticities” with both septa off by fitting a line to the (δ) and calibrate the dispersion measurement at LDPM03 Turn septa on Measure momentum with calibrated LDPM03 Measure tune with the RS data 11/22/2018 Improved_Model

Dipole Component RODM01 RODM02 11/22/2018 Improved_Model

Sextupole Component RODM01 RODM02 11/22/2018 Improved_Model