Calculating Momentum Spread from Density Distribution in Recycler Meeting

1. A simple formula for calculating the momentum spread from the longitudinal density distribution and RF form Recycler Meeting March 11, 2009 A. Shemyakin

2. Introduction • Longitudinal density profile of a beam in a RF bucket can be used to extract the information about the momentum distribution • Various papers have been published • for example, L. Michelotti, PRST-AB, 6, 024001 (2003) • A. Burov had proposed to use a tomography procedure for recovering a detailed momentum distribution in the Recycler • http://beamdocs.fnal.gov/AD-public/DocDB/ShowDocument?docid=3109 • P. Derwent realized the procedure as a Java program • Provides 68%, 90%, and 95% emittance • Uses the Resistive Wall Monitor signal • If one is interested in only an rms momentum spread at Recycler conditions, a simpler formula can be used • Applicable to a Gaussian distribution or to a beam in a barrier bucket

3. Derivation (1) • Let’s consider a stationary distribution function , which is • stationary, and therefore, depends on the energy offset and the phase along the bunch through a Hamiltonian • comes to zero somewhere inside separatrix so that all integrations can be extended to infinity (no DC beam) • symmetrical with respect to . • Similar to Michelotti’s article or Alexey’s presentation about tomography, the longitudinal density is expressed as (1) where U() is RF potential.

4. Derivation (2) • Let’s consider a half of the bunch, where the potential is monotonically increasing (or decreasing), and is a function. In this region, one can take an integral over U (2) The integral can be calculate at , where : (3) where is the beam density in the point of zero RF potential (i.e. maximum density), and is the energy spread in this point.

5. Derivation (3) Now integration over potential can be replaced by integration over the longitudinal phase (4) The final formula for the rms momentum spread is (5) where the coefficients are expressed through the barrier’s momentum height and width .

6. Applicability The formula calculates the momentum spread at the point of zero RF potential. When does it give the same answer as a Schottky detector, which represents the spread of the entire beam? 1. If the momentum distribution is Gaussian, the momentum spread is the same at each position along the bunch, and the formula gives an rms spread of the entire distribution. 2. If the beam is contained in a long enough barrier bucket, at any given moment most of particles are close to the bottom of the potential well, and the beam spread is close to the one calculated with the formula. It is correct if the ratio of the number of particles between the barriers to those inside the barriers is low. For a Gaussian distribution, this ratio is where are the height and length of the barrier, and is the distance between barriers. For , the bucket length should be >> 0.2 s.

7. Recipe • Record one turn of data for the RF amplitude and longitudinal density • Adjust zero offsets of the signals • Find maximums of the signals and normalized both data sets • Precisely align the distributions • The information is contained in tails, and the alignment of the longitudinal tails with RF barriers is crucial • Make the time steps of both signals the same by averaging over more frequent readings • Calculate the momentum spread, summing from maximum to minimum of the density distribution (or in opposite direction at the other bunch edge)

8. Possible use with the toroid signal • Jim Crisp reported about a toroid R:IB2COR (Pearson coil) • http://indico.fnal.gov/conferenceDisplay.py?confId=2418 • Good amplitude resolution • Very good low frequency time constant (0.1 sec) • But a poor high frequency response • Attractive to use because doesn’t interfere with operation • Reports 588 point • Synchronized with RF • But phase is changes at reboots From Jim’s presentation: Response of the toroid signal (red trace) to a step (blue trace). 10 to 90% in 52nsec

9. Procedure • RF and toroid data are recorded without averaging • One turn is cut from RF signal; average value is subtracted • Both signals normalized to maximums • Signals are roughly aligned • Center of gravity of the toroid signal is aligned with middle point of minimum and maximum of RF • Final signal alignment • The calculated spread should be the same on both ends • Therefore, the sum over the full turn should be zero • Two iterations of shifting is enough Example of a longitudinal density profile (blue) recorded from R:IB2COR and corresponding RF amplitude. The signals are normalized to 1 and their relative position is adjusted in MathCad. 25-Feb-09, 116E10, bucket length 248 bckt, R:DPSIGA= 2.75 MeV/c

10. Procedure (cont.) • The partial sums are calculated for all positions along the beam to find the point of “true zero” potential • The momentum spread is calculated with the value of the minimum sum • The procedure is realized in MathCad 14 • The results are compared with the rms momentum spread measured with the 1.7 GHz Schottky (R:DPSIGA from Dan Broemmelsiek’s program R110) Dependence of the partial sum on the end point of summation n for finally aligned signals. The same data.

11. Results • The calculated values are overestimated at low spreads • The slope is lower than one • The measurement with a high DC component looks different • Rms of differences is 9.5% Comparison of rms momentum spread calculated with the formula with the numbers reported by 1.7 GHz Schottky. Blue symbols data were taken in different days at Np = (11 ÷ 352)E10. The green symbols data is one set at Np = 12E10 and varying beam length (5-Mar-09). No fitting. Calculation for

12. Possible sources of discrepancy • Frequency characteristics of the toroid • For a beam with zero momentum spread, the procedure would give ~1.4 MeV • Estimation for a linear rise of the toroid signal in 60 ns and RF in 50 ns • Creates a zero offset for the curve • It may affect the linear slope as well • A DC beam component may change the relationship between two measurements (toroid vs Schottky) noticeably • The DC component can not be distinguish with the toroid • The component increases the Schottky spread if the bucket is wide and decreases if the bucket is narrow • The only point measured with a large DC component (2-Mar-09, ~25%) had the beam width of 0.72 s (i.e. very narrow bucket)

13. Fitting • The same data after applying linear fitting coefficients • Fit excluded the point with a large DC component • The relative differences are within  10% • Rms of differencies is 4.6% • Excluding the point with a large DC component

14. Conclusion • A relatively simple formula is derived to calculate the beam momentum spread from the RF voltage and the longitudinal density distribution • Assumptions: • Stationary distribution • Unipolar voltage at the area of analysis • All particles are far from separatrix (no DC beam) • Strictly speaking, gives the momentum spread at zero RF potential • Involves only summation if signals are centered • For most of cases in the Recycler, the formula should give the value close to the rms momentum spread in the entire beam, which is provided by the Schottky detector • The values are the same if the beam’s momentum distribution is Gaussian • Close if the portion of the beam penetrating the barriers is small • Application of the formula to the signal from R:IB2COR toroid gives a reasonable agreement • Affected by a low value of the toroid’s upper frequency • Agrees with Schottky within  10% if linear fitting coefficients are applied • Thanks to A.Burov, J. Crisp, M.Hu, and V. Lebedev for discussions