4.4 General Case Algorithm
4.4.1 Algorithm Derivation
We begin by expressing the second moment of (xs−xr) in terms of density functions
f±(xs), Eqs. 4.16 and 4.17, where xr has been introduced as an estimate for the
central orbit, x0. Considering the cases of scraper blades coming from the positive
and negative x directions simultaneously:
h(xs− xr)2i± =
Z +∞
−∞
(xs− xr)2f±(xs) dxs, (4.21)
Using the following; Z +∞ −∞ x2sf±(xs) dxs = x¯2±+ σ±2, (4.22) Z +∞ −∞ xsf±(xs) dxs = x¯±, (4.23) Z +∞ −∞ f±(xs) dxs = 1, (4.24)
where ¯x± is the mean value of the measured distribution and σ±2 is the variance,
we me rewrite this quantity as:
h(xs− xr)2i±= ¯x2±+ σ±2 − 2¯x±xr+ x2r. (4.25)
Simultaneously we may expand the left hand side and use the substitution xs±= x0+ δD ± √ βA to obtain: h(xs− xr)2i± = h((x0− xr) + δD ± p βA)2i± = (x0− xr)2 + 2(x0− xr)hδiD ± 2(x0 − xr) p βhAi +hδ2iD2± 2Dp βhδAi + βhA2i. (4.26)
Considering the definitions ¯δ = hδi, ¯A = hAi, σ2
δ = h(δ − ¯δ)2i, the statistical
definition of the geometric transverse emittance shown in Eq. 4.1, and the usual normalisations of phase space density, Eqs. 4.10 & 4.11, we may rewrite Eq. 4.26:
h(xs− xr)2i± = (x0 − xr)2+ D2(¯δ2+ σδ2) + 2βrms+ 2(x0− xr)D¯δ
±2(x0− xr)
p
β ¯A ± 2DpβhδAi. (4.27)
These expressions allow us to put h(xs− xr)2i± in terms of the RMS transverse
emittance rmsand the dispersive contribution Dσδ. However, the additional terms
hAi and hAδi make the evaluation difficult even for a known closed orbit centre x0. We can perform a combination of measurements from both the positive and
negative scraper scans in Eqs. 4.25 and 4.27 we may write:
h(xs− xr)2i+
+h(xs− xr)2i− = 2(x0− xr)2+ 2D2(¯δ2+ σδ2) + 4βrms+ 4(x0− xr)D¯δ
= x¯2++ ¯x2−+ σ+2+ σ−2− 2xr(¯x++ ¯x−) + 2x2r. (4.28)
Further transformation yields:
2 (x0+ D¯δ − xr)2+ 2D2σδ2+ 4βrms = 2 (x¯++ ¯x− 2 − xr) 2+ 1 2(¯x+− ¯x−) 2+ σ +2+ σ−2. (4.29)
Given that the momentum depended closed orbit is equal to the mean particle position,
x0+ D¯δ =
(¯x++ ¯x−)
2 , (4.30)
we may rearrange equation Eq. 4.29 for the emittance:
rms= 1 4β σ2++ σ−2 +(¯x+− ¯x−) 2 2 − D 2σ2 δ 2β , (4.31)
which contains only values that can be obtained from the scraper data or otherwise measured and estimated. This equation forms the basis of the algorithm.
The contribution from longitudinal momentum spread in Eq. 4.31 can clearly be seen as the rightmost dispersive dependent term, and may be set to zero when scraping in the vertical plane, where typically D = 0.
Calculating the emittance using this result requires two separate scraper scans from opposing directions, e.g. positive and negative x. In this case the machine must cycle twice, so beam stability between shots is very important. The impact of beam stability, as well as other challenges and sources of error, are investigated in the following chapter.
4.4.2
Additional Quantities
Because we perform scraping from both sides it is possible to extract further information on the beam by comparing the difference in the two results. For example, Eq. 4.30 may be used to obtain an estimate for the mean amplitude of the particles.
Performing a subtraction, as opposed to the addition in Eq. 4.28, we find:
h(xs− xr)2i+
−h(xs− xr)2i− = 4(x0− xr)
p
β ¯A + 4DpβhδAi (4.32) = x¯2+− ¯x2−+ σ+2− σ−2− 2xr(¯x+− ¯x−), (4.33)
which may be rearranged to make it possible to compare the coefficients of xr,
4(x0+ D¯δ) p β ¯A + 4Dpβh(δ − ¯δ)Ai − 4xr p β ¯A = ¯x2+− ¯x2−+ σ+2− σ−2− 2xr(¯x+− ¯x−), (4.34)
leading to an estimation for the mean amplitude using measurable quantities:
¯
A = x¯+− ¯x−
2√β . (4.35)
We may use this result combined with Eq. 4.30 to further rearrange Eq. 4.34 allowing us to measure a quantity which would describe the magnitude of the cor- relation between momentum spread and the maximum amplitude of the particles:
h(δ − ¯δ)Ai = σ
2
+− σ2−
4D√β . (4.36)
This quantity will be referred to as the emittance-momentum spread correlation coefficient since it will later allow us to investigate such a correlation brought about by the effects of the electron cooler.
4.4.3
Data Analysis
Assuming that the signal received from the detectors is given as cumulative loses and transformed to F±(xs), as for the AD and ELENA scintillators, then differ-
entiation must be performed to obtain the beam size variance, σ2
±, and the mean
x position of the intercepted particles, ¯x±, required by the algorithm. Since the
algorithm is designed for non-Gaussian beams, it is unlikely that an expression for the data will be known and hence symbolic differentiation will not be possible. Two alternate methods are available for consideration: spline interpolation and simplified numerical approximations.
Comparisons of the methods may be found in Section 5.6.1. Here we present the simplified numerical method, which was found to be a sufficient approximation for these purposes, assuming a data acquisition rate of 400 Hz.
We may begin by assuming a set of CDF values, i.e. the tabulated function F±(xs), with corresponding scraper positions, xs, for every data point, i. Using
finite difference approximations it is possible to get estimations for the tabulated PDF, f±(xs), for all points, i:
fi,± =
F±,i− F±,i+1
|x±,i− x±,i+1|
. (4.37)
The following integral may be approximated by a midpoint Riemann sum:
Z +∞ −∞ f±(xs) dxs≈ n X i=1
(fi,±(|x±,i− x±,i+1|)), (4.38)
where n is the number of entries in the data set, into which we may substitute Eq. 4.37 to provide a simple expression for use in the algorithm:
Z +∞ −∞ f±(xs) dxs≈ n X i=1 (F±,i− F±,i+1). (4.39)
This result may then be used to obtain an approximation for the distribution variance: σ±2 = Z +∞ −∞ f±(xs)(xs− ¯x±)2 dxs≈ n X i=1
where x±,i,mid is the midpoint between x±,iand x±,i+1, and the value ¯x±is obtained
using similar approximations for Eq. 4.23:
¯ x± ≈
n
X
i=1
(F±,i− F±,i+1)x±,i,mid. (4.41)
It would also be possible to use trapezoidal Riemann sums in order to increase the accuracy of the algorithm, however with the relatively small step sizes used between measurement points it is not necessary to do so here.
4.5
Summary
The two new scraper algorithms have been introduced. One is capable of scraping a Gaussian beam in a region of non-zero dispersion, and the other under the same conditions but for an arbitrary beam profile distribution. The arbitrary distribution method requires two scraper measurements from opposing directions. In the next chapter, the algorithms are verified through the use of simulations. Additionally, several sources of error, some of which were discussed in Section 3.4, are investigated through simulations and error tolerances are established.
Scraper Simulations
5.1
Introduction
The primary goal of this chapter is to use simulations to test the emittance recon- struction algorithms derived in the previous chapter. The various methods used in performing the simulations are presented in detail such that the results may be reproduced by the reader.
The effects of systematic errors on the accuracy of the non-Gaussian algorithm are investigated, and the use of the algorithm to determine additional quantities such as the emittance-momentum spread correlation is performed. The chapter begins with an investigation into the transmission of particles through and out the side of the aluminium scraper blade, to ensure later assumptions in simulations are adequate.