1. Emittance Studies at Extraction of the PS Booster: Emittance Calculation
S. Albright, F. Antoniou, F. Asvesta, H. Bartosik, G.P. Di Giovanni, M. Fraser, V. Forte, A. Huschauer, S. Papadopoulou, T. Prebibaj, P. Skowronski, G. Sterbini, A. Valdivieso
LIU-PSB BD WG Meeting #19
09/07/2019

2. Introduction
In the frame of understanding the PSB-PS emittance discrepancy a set of measurements were performed in both machines in the end of 2018.
The differences between the available instrumentation (SEM Grids – WS) was compared and the impact of systematic errors coming from the optical parameters was investigated [1,2]
The last goal of this study was the comparison and optimization of the emittance deconvolution algorithms in order to understand their limitations and their behavior under different beam conditions.
This is especially important for the targeted LIU PSB beam parameters in order to have an accurate and precise method for the emittance reconstruction.

3. Emittance Reconstruction Algorithms
Standard Gaussian Subtraction (SG)
Fit the measured profile using a Gaussian function and calculate the measured beam size 𝜎 𝑚𝑒𝑎𝑠
Use the 𝛿𝑝 𝑝 𝑅𝑀𝑆 by the Tomo application or using the second order moment formula.
Calculate the betatronic beam size using the Gaussian formula:
𝝈 𝒃𝒆𝒕 = 𝜎 𝑚𝑒𝑎𝑠 2 − 𝑑𝑝 𝑝 𝐷 2 ≡ 𝜎 𝑚𝑒𝑎𝑠 2 − 𝜎 𝑑𝑖𝑠𝑝 2
The emittance is:
𝜺 𝒙,𝒚 = 𝝈 𝒃𝒆𝒕 𝟐 𝛽 𝑥,𝑦 𝛽 𝑟𝑒𝑙 𝛾 𝑟𝑒𝑙
Full Deconvolution (FD)[3]
Center and normalize the measured and the dispersive distribution.
Create the (discrete) convolution of the dispersive distribution and a Gaussian function: 𝑓 𝐶𝑜𝑛𝑣 𝑥 𝑖 ;𝜎 = 𝑑𝑖𝑠𝑝 𝑘 𝑖 ∗ 𝑒 − 𝑥 𝑖 − 𝑘 𝑖 𝝈 2
Fit the measured distribution with 𝑓 𝐶𝑜𝑛𝑣 and calculate the single parameter 𝝈≡ 𝝈 𝒃𝒆𝒕 using a non-linear least squares method.
The emittance is:
𝜺 𝒙,𝒚 = 𝝈 𝒃𝒆𝒕 𝟐 𝛽 𝑥,𝑦 𝛽 𝑟𝑒𝑙
Comments:
Both methods assume Gaussian betatronic distribution.
The SG method assumes also Gaussian measured distribution (and therefore Gaussian dispersive distribution).
The emittance error is twice the betatronic beam size error (for linear error propagation).

4. Gaussian Betatronic Distribution
Gaussian betatronic distribution with defined parameters*:
𝜀 𝑥 =1.2 𝜇𝑚 𝛽 𝑥 =5.0 𝑚
𝜎 𝑏𝑒𝑡 = 𝜀 𝑥 𝛽 𝑥 ≈𝟐.𝟒𝟒𝟗𝟓 𝒎𝒎
Parabolic longitudinal (dispersive) distribution with defined parameters:
𝑑𝑝 𝑝 𝑟𝑚𝑠 = 10 − 𝐷 𝑥 =1.5 𝑚
𝜎 𝑑𝑖𝑠𝑝 = 𝑑𝑝 𝑝 𝑟𝑚𝑠 ∙ 𝐷 𝑥 =𝟏.𝟓 𝒎𝒎
*Typical measurement with the BTM.SGH01 at intensities ~80∗ ppb (BCMS25) [4].

5. Gaussian Betatronic Distribution
Gaussian betatronic distribution with defined parameters*:
𝜀 𝑥 =1.2 𝜇𝑚 𝛽 𝑥 =5.0 𝑚
𝜎 𝑏𝑒𝑡 = 𝜀 𝑥 𝛽 𝑥 ≈𝟐.𝟒𝟒𝟗𝟓 𝒎𝒎
Parabolic longitudinal (dispersive) distribution with defined parameters:
𝑑𝑝 𝑝 𝑟𝑚𝑠 = 10 − 𝐷 𝑥 =1.5 𝑚
𝜎 𝑑𝑖𝑠𝑝 = 𝑑𝑝 𝑝 𝑟𝑚𝑠 𝐷 𝑥 =𝟏.𝟓 𝒎𝒎
The measured distribution is the convolution between the betatronic and the dispersive.
The measured distribution is non-Gaussian but can be approximated with a Gaussian function.
*Typical measurement with the BTM.SGH01 at intensities ~80∗ ppb (BCMS25) [4].

6. Gaussian Betatronic Distribution
Using the SG method to (re)calculate the initial (defined) beam size the error is around 1.3%.
Using the FD method to (re)calculate the initial (defined) beam size the error is small (less than 0.1%)
The dispersive contribution can be expressed by the dispersive/betatronic beam size ratio, which in this case is 61%.

7. Changing the Dispersive Contribution
As the dispersive contribution becomes larger so does the “deformation” of the measured distribution.
Therefore the error of the SG method should become larger (because it assumes Gaussian distributions).
The relative error on the betatronic beam size as a function of the dispersive/betatronic ratio is plotted for the SG and the FD method.
The error of the SG method increases exponentially while the FD method has negligible errors.
Example: the next table shows the PSB achieved and LIU target beam parameters and the beam sizes at the position of the PSB WS ( 𝐷 𝑥 =1.37 𝑚, 𝛽 𝑥 =5.1 𝑚). The dispersive contribution is expected to be almost doubled.
PSB Extraction (Standard: 4b+2b) [1, 5]
𝜀 𝑥,𝑦 [𝜇𝑚]
𝛿𝑝/𝑝
𝜎 𝑏 [𝑚𝑚]
𝜎 𝑑 [𝑚𝑚]
𝑟𝑎𝑡𝑖𝑜 [%]
Achieved
𝟐.𝟐𝟓
0.9
3.39
1.24 37
LIU target
𝟏.𝟐𝟖
1.5
3.03
2.06 68

8. Q-Gaussian Betatronic Distribution
To model a non-Gaussian betatronic distribution a Q- Gaussian function is used (𝑞<1: underpopulated tails, 𝑞> 1: overpopulated tails, 𝑞=1: Gaussian tails).
A Q-Gaussian betatronic distribution further deformes the resulting measured distribution.
In this case the errors of each method behave differently:
for 𝒒<𝟏 (light tails)
for 𝒒>𝟏 (heavy tails)

9. Q-Gaussian Betatronic Distribution
Parameterizing the error of each method with the dispersive/betatronic ratio and the q-value of the betatronic distribution.
Comments:
Non-negligible errors for the FD method.
If 𝒒>𝟏 (heavy tails) and low ratios (the betatronic is dominant) both methods underestimate the emittance while for high ratios (the dispersive is dominant) the methods overestimate the emittance.

10. Benchmarking Measurements
In order to test the predictions of the previous simulations an experiment was set up.
A set of BCMS beam profile measurements took place at the PSB R3 WS for three different intensities (𝟓𝟓, 𝟕𝟓, 𝟏𝟎𝟎 𝑝𝑝𝑏) and with different 𝛿𝑝/𝑝 values. The momentum spread of the beam was varied by changing the amplitude of the C04 RF.
For each of the beam parameters the horizontal emittance was calculated using both the SG and FD methods.
The measurements showed that the two methods diverge when the contribution of the dispersive profile becomes larger (larger 𝛿𝑝/𝑝) and for smaller intensities [6].

11. Benchmarking Measurements
Plotting the same data points, not as a function of the 𝛿𝑝 𝑝 , but as a function of the dispersive/betatronic ratio (as the simulations), it can be seen that:
As the dispersive contribution becomes larger the relative errors increase exponentially (similar with the simulation)
Good agreement between the simulation and the measurements for high and intermediate intensities but not for low intensities (high ratios).
A similar set of measurements but with a 1.5 𝑒𝑉𝑠 BCMS beam:
In this case only the intensity (betatronic contribution) was varied.
Deviations for low intensities (high ratios).

12. Summary
Two emittance calculation methods were studied using simulated distributions: the SG and the FD.
For a Gaussian betatronic and a Parabolic dispersive distribution the error of the SG method increases with the dispersive contribution, while the error of the FD is negligible.
For the achieved LHC-type beam parameters this error is small (<0.5%) but it is expected to grow for the future LIU parameters (~4%).
If having a non-Gaussian betatronic distribution the errors of both methods further increase.
The predicted relative errors between the two methods are in agreement with those measured in the MD.
Small deviations for very large ratios could be explained by the uncertainty of the exact ratio of the experimental data points or/and small deviations of the measured dispersive distribution from a parabolic shape (see supporting slides).

13. Thank you for your attention!

14. References
[1] F. Antoniou et al. “Transverse Emittance Studies at Extraction of the CERN PS Booster”, inProc. 10th Int. Particle Accelerator Conf. (IPAC’19), Melbourne, Australia, May [2] Tirsi Prebibaj, Fanouria Antoniou. Emittance Studies at Extraction of the PS Booster. ABP Group information Meeting. URL: [3] Guido Sterbini et al. “De-convolution algorithm to properly remove the dispersive profile for emittance calculations”, LIU-PS Beam Dynamics WG #5. URL: [4] Tirsi Prebibaj, Fanouria Antoniou. “Brightness Studies – SEM Grids Measurements”, LIU-PSB Beam Dynamics WG #16. URL: [5] G. Rumolo, “LIU proton beam parameters”, EDMS /2
[6] Tirsi Prebibaj, Fanouria Antoniou. Transverse Emittance Measurements. LIU-PSB Beam Dynamics WG #12. URL:
[7] A. Garcia-Tabares Valdivieso, P. Skowronski, R. Tomas, “Optics measurements in the CERN PS Booster using turn- by turn BPM data”, inProc. 10th Int. Particle Accelerator Conf. (IPAC’19), Melbourne, Australia, May 2019.
[8] Tirsi Prebibaj, Fanouria Antoniou. Dispersion Measurements in the PSB. Space Charge WG Meeting. URL:

15. Supporting Slides

16. Benchmarking Measurements: SEM Grids
PSB BTM.BSG01 (modified BCMS).
PSB BTM.BSG02
Negative relative errors (𝑞>1).
Interpolation to the model to include very low ratios.
PSB BTM.BSG03
Large deviations for the third grid.

17. 𝒒 𝒎𝒆𝒂𝒔 vs. 𝒒 𝒃𝒆𝒕 vs. ratio

18. Relative SG-FD Differences: General Case

19. PSB-PS WS optical parameters
PSB (Bri.NWS.2L1.H)
PS (PR.BWS.65.H)
𝛽 𝑥 model
5.7 𝑚
22.3 𝑚
𝛽 𝑥 measured
5.1±1.0 𝑚
22.9±1.1 𝑚
𝐷 𝑥 model
1.47 𝑚
3.17 𝑚
𝐷 𝑥 measured
1.373±0.002 𝑚