Fitting of voltage jump measurements

1. Fitting of voltage jump measurements Recycler department meeting November 29, 2006 Shemyakin

2. Usual fitting Goal • Develop a method of restoring the entire cooling force curve from a single voltage jump measurement. • After a jump, the momentum offset naturally scans the entire range from the maximum jump to zero. • Electron energy can be determined by BYR01S position (“spectrometer BPM”). • Slow fluctuations of the Pelletron voltage give an information about the cooling force at small momentum offsets and spectrometer calibration. • It also would eliminate an uncertainty with assignment of a momentum offset value to the measured force • All data were measured by L. Prost on Nov 1, 2006 Red – average pbar momentum Blue- electron energy fluctuations calculated by BYR01S and expressed in pbars’ MeV Brown- difference between red and blue (momentum offset) Magenta- pbar rms width. A. Shemyakin - Recycler meeting- Nov 29, 2006

3. Cooling force model • The cooling force was integrated over a Gaussian distribution for transverse pbar velocities with spreads of pt that gives in the beam frame: where Proportional to longitudinal electron velocity spread; determines slope at low p Proportional to transverse electron velocity spread; determines the slope after maximum Force amplitude For RR ECool, A. Shemyakin - Recycler meeting- Nov 29, 2006

4. Data preparation • Read data for the average momentum R:DRGDEV and rms momentum width R:DRGSIG from D44 • Clean duplicated points • One measurement took ~28 sec while the parameters were data logged with 15 sec intervals • Assign the time to the first appearance of a new value • Read the electron beam position after 180 deg bend R:BYR01S from D44 • Using time points {Ti } of R:DRGDEV, average the position between Ti and Ti-1 and assign the value to the position at Ti. • Presently R:BYR01S is measured with 5 Hz bandwidth but is saved in D44 at 1 Hz • Write the rectangular table into a file A. Shemyakin - Recycler meeting- Nov 29, 2006

5. Fitting • Read a set of data recorded at constant electron beam parameters • Fit data to non-magnetized cooling force with 4 fitting parameters: • 3 parameters of the cooling force (p1, p2, F0) • Offset of the electron beam position in BYR01S corresponding to the nominal pbar energy (“nominal position”) • For each moment, the momentum offset is calculated • Electron energy from is calculated by the electron beam position • The momentum offset is the difference between the pbar momentum and the electron energy multiplied by the mass ratio • Two ways of fitting to the drag force formula were tested: • To the entire curve p(t) • First, calculate the derivatives dp/dt and then fit the drag force • Results are comparable, but the second way is more explicit • The main problem was instability of the results. A. Shemyakin - Recycler meeting- Nov 29, 2006

6. Fitting (cont.) An example of fitting at Ie = 0.1 A, SPB01I = 13.5 A. This fitting was done for derivatives. The red points show the measured momentum evolution. The blue curve connects points calculated by adding a momentum shift caused by the fitted drag force to the momentum measured in previous point. A. Shemyakin - Recycler meeting- Nov 29, 2006

7. Effect of the pbar momentum spread Taking into account pbar momentum spread was found to improve the stability of the fit, specially for data sets with a large number of points with small momentum offsets. Because the effect of the second derivative is significant only at low momentum offsets, an approximate expression was used to speed up calculation: A. Shemyakin - Recycler meeting- Nov 29, 2006

8. Effect of the pbar momentum spread (cont.) Comparison of fitting without (top plot) and with (bottom) taking into account the second derivative of the cooling force. Points are the measured derivatives minus term with the second derivative. Average pbar momentum offset, electron energy, difference between red and blue, pbar rms width. A. Shemyakin - Recycler meeting- Nov 29, 2006

9. Nominal position • The most stable (in the fitting procedure) parameter is the BYR01S beam position at the electron energy corresponding to the zero pbar momentum offset (“nominal position”). • It is correct for data sets that contain many points with small momentum offsets • The nominal position changes linear with the beam current, most likely, due to image charges. • It might be drifting within 0.05 mm with time and changing focusing upstream Nominal position as a function of the electron beam current at fixed focusing (SPB01I = 14.5 A). Points at 0.1 A were recorded within 2.3 hours. A. Shemyakin - Recycler meeting- Nov 29, 2006

10. Instability of fitting • For a given data set, the force maximum is stable within 15% while fitted electron beam parameters may differ by factor of 2. • The main reason is a large scatter of the data points • A significant portion of scatter might be coming from timing issue • In addition, there are no points with large momentum offsets that would anchor the electron angle value. Comparison of fitted curves at two guess values of the electron angle. Ie= 0.3A. Blue curve Red curve A. Shemyakin - Recycler meeting- Nov 29, 2006

11. Conclusion • Fitting of the entire curve of the pbar momentum evolution works from the point of view of the calculation mechanics • A simpler formula of calculating the drag force is derived • Importance of taking into account the momentum width is realized • The procedure provides reliably only the value of maximum force • The error bars are larger than in the simple linear fit • Estimations for electron beam parameters are unstable • Ways to improve the results: • Improve timing by increasing D44 rate from 15 to 3 sec (done) • Decrease BYR01 errors by averaging over 0.5 sec (A. Warner is looking into the possibility) • Make measurements with larger voltage jumps • Check a possibility to speed up average momentum measurements by sacrificing a statistical error A. Shemyakin - Recycler meeting- Nov 29, 2006