Investigating Observed Differences in Reconstructed Phase-Space

A detailed GPT model of the ALICE tomography section as shown in Fig. 4.5 was used to predict the results of experiments to reconstruct horizontal phase-space from quadrupole tomography scan data, looking for any differences which might be seen between results from QUAD-07→YAG-02 scans and from QUAD-10→YAG04 scans.

Actualexperimental results are tabulated later, in Fig. 5.5. The process was planned as follows:-

1. Start with a particle set based on measured phase-space at QUAD-07 entrance, using existing experimental data as available.

2. Use a GPT model to transport the beam to YAG-04.

3. Run GPT scans of QUAD-10 (horizontal) and QUAD-11 (vertical), and collect the simulated YAG-04 screen data.

4. Based on this data, reconstruct phase-space at QUAD-07 entrance.

5. Repeat GPT runs with (a) space-charge OFF, and (b) space-charge ON.

6. Compare simulated phase-space with experimental results.

7. Vary beam-line parameters, e.g. magnet strengths, to investigate the sensitivity of the measurements to errors in these parameters.

8. Repeat the procedure for vertical phase-space, using measurements from QUAD- 06 and QUAD-11 scans (when available from later experiments).

This approach would allow a detailed investigation of how perturbations in the beam- line elements between QUAD-07 and QUAD-10 might account for any discrepancy ob- served in phase-space distributions, calculated with QUAD-07 entrance as the common reference point, using QUAD-07 data as compared with QUAD-10 data; there was some evidence from existing experimental results.

Figure 4.9: Data flows for the integration of the GPT tomography section model into an overall simulation of the tomographic measurement process are illustrated, indicating where comparisons with experimental results from ALICE may be made, particularly between QUAD-06/07 and QUAD-10/11 scans.

4.2.2 Generating Input Particle Specifications for GPT

To use GPT for investigating tomography results from measurements taken at different points along the ALICE to EMMA Injection Line, it was decided to feed back existing experimental data by converting it into realistic input for the GPT model of the line. In this way, the outputs generated by GPT, in particular the simulated screen profiles, could be used in their turn as inputs for full tomography measurement simulations, enabling studies to be made of the sensitivity of the method to various beam-line parameters, as shown in Fig. 4.9.

Input Data Earlier tomography experiments had provided measurements of both horizontal and vertical phase-space distributions (x, x0) and (y, y0) under similar beam conditions. Beam profiles had also been taken from screen YAG-02, which is reason- ably close to the chosen reconstruction location for phase-space, and could therefore be used to approximate the (x, y) distribution matching the (x, x0) and (y, y0). Sev- eral processing schemes were considered, based on using some or all of the available experimental data; however, it was found difficult to reconcile the measured (x, y) and the (x, x0),(y, y0) distributions simultaneously into a single consistent (x, y, z, x0, y0, z0)

assembly (where the longitudinal coordinates zand z0 are not correlated withx ory).

Processing Stages A MATLAB code (based on original work by K Hock) was devel- oped to read in phase-space distributions and beam profiles, in their original formats, and produce output of particle specifications, as a text file. This text data could then be suitably formatted for a GPT utility program to convert into internal representation, making it available for import by a GPT simulation script. The principal steps in the code are as follows:-

1. Read in and prepare files for the (x-y) beam image, and the xand yphase-space distributions

2. User selects a window, to include the beam image region only

3. Apply filter and threshold, to smooth and suppress background

4. Find commonx and y ranges in (x, y) and (x, x0), (y, y0) distributions

5. Calculate Cumulative Distribution Function (CDF) forx in (x, y) distribution

6. Derive the CDF fory in (x, y), as a function of x

7. Similarly, determine the CDF(x) from (x, x0), and CDF(x0) forx, and correspond- ingly CDF(y) and CDF(y0)

8. Choose a random value from thex distribution, then choose a randomx0 corre- sponding to the chosenx

9. Repeat the above 2 steps for randomy and y0

10. Display a scatter plot of (x, x0) and (y, y0) for random particles, as a check (see Fig. 4.10)

Figure 4.10: (x, x0) and (y, y0) phase-space distributions from separate experiment are combined to generate particle specifications suitable for input to GPT. As a check, the resulting particle set is visualised as two scatter plots.

A GPT batch file then converts the text representation of the particle set (Table 4.1) into internal (GDF) format. Here, columnsx, y, z are coordinates for individual parti- cles uniquely identified by ‘ID’, andGBx,GBy,GBz are the corresponding momenta conventionally known as x0, y0, z0. In this form, particles can be read in by the GPT script which describes the beam-line, and the resulting GPT output plotted at a po- sition near the start of the line. This is to check that the phase-space distributions are as expected, given the known input of specified particles.

# GPT Particle File: E:\...\GPT\GPT2mat\ImageFileDir\0403 EMI-2 12ns.PNG

ID x y z GBx GBy GBz

1 -1.242e-03 -5.015e-05 0.000e+00 -2.601e-04 1.068e-03 0.000e+00 2 -5.354e-04 -1.034e-04 0.000e+00 9.444e-04 -1.455e-04 0.000e+00 ...

99999 -5.445e-05 1.726e-04 0.000e+00 1.561e-03 -1.341e-03 0.000e+00 100000 -1.014e-03 -9.200e-04 0.000e+00 -7.407e-05 3.410e-04 0.000e+00

Table 4.1: In GPT, a particle set for input may be specified in a standard text format, giving values for (x, y, z, x0, y0, z0). In this example, the header and the start and end of a 100,000 particle set are shown.

Limitations in the Method The conversion of phase-space measurements into par- ticle descriptions for GPT is subject to a number of uncertainties, which would reduce confidence in the results of GPT simulations based on them.

i) Quality of Measurements. Reconstruction resolution should be as high as possible, given the number of projections and the image pixels available.

at a screen position, whereas (x, x0) and (y, y0) phase-spaces are typically reconstructed at a quadrupole entrance, such as QUAD-07. In this situation, profiles and phase- spaces cannot be reconciled directly, and a specification based purely on phase-space distributions might be preferable.

The availability of further ALICE experimental tomography datasets would enable some of these effects to be estimated, and provide options for their reduction.

iii) Correlating (x, y) and (x, x0), (y, y0) Distributions. It is found from the GPT output, derived from the generated particles, that the resulting (x, y) profile does not correspond to the original experimental (x, y) profile. This effect is shown in Fig. 4.11.

Figure 4.11: Using the output of GPT, a direct comparison is made between the (x, y) profile, measured experimentally, and (x, y) for the particle set as derived from the (x, x0), (y, y0) distributions. They are distinctly different.

Possible Solution The issue of the unknown correlation between (x, x0) and (y, y0) could be addressed if the results of a full4-dimensionaltomography measurement were available, yielding the (x, x0, y, y0) distribution directly. The theory of such measure- ments has already been investigated, and future experiments based on it are planned [47].

GPT Particle Input: the Approach Selected

In view of the problems encountered in specifying input particle sets, it was decided that as a simplification, Gaussian distributions would be assumed for both (x, x0) and (y, y0) phase-space, which are supported as standard by GPT. The transverse distri- butions would thus be fully characterised by the Twiss parameters βx, αx, βy, αy and

the emittancesx, y. Realistic estimates of the parameters were already available from