Energy acceptance and on momentum aperture optimization for the Sirius project

A fast objective function to calculate Touschek lifetime and on momentum aperture is essential to explore the vast search space of strength of quadrupole and sextupole families in Sirius. Touschek lifetime is estimated by using the energy aperture (dynamic and physical), RF system parameters and driving terms. Non-linear induced betatron oscillations are considered to determine the energy aperture. On momentum aperture is estimated by using a chaos indicator and resonance crossing considerations. Touschek lifetime and on momentum aperture constitute the objective function, which was used in a multi-objective genetic algorithm to perform an optimization for Sirius.


Introduction
The adopted approach to optimize the non linear optics of Sirius storage ring is to use a Multi-Objective Genetic Algorithm, where the objective function is an estimator of on momentum aperture area and Touschek lifetime. Details about Sirius storage ring lattice can be found in [1] and [2]. On momentum aperture calculation is performed in key regions using chaos indicators, and energy acceptance calculation is performed using the concept of energy aperture (dynamic and physical). The Touschek lifetime is calculated through the energy acceptance. The idea is to have a fast algorithm to estimate the on momentum aperture area and Touschek lifetime, instead of performing a long and precise calculation, which in the case of the detailed lattice of Sirius takes around 7 hours. The goal is to take between 2 and 10 minutes per ring model. Therefore, we do not want to simulate lattice errors and correct them for each storage ring model tested, since it takes a considerable amount of time. Furthermore, for a lattice without errors, it is possible to take advantage of the five-fold symmetry in the Sirius optics, by simply using one superperiod for tracking. Regarding this strategy, care should be taken for 6D tracking to not change the synchrotron frequency nor the RF bucket. Our approach was to adjust the RF cavity frequency f RF , so that f RF = kf 0 , where k is the harmonic number and f 0 is the revolution frequency in a superperiod.
To rigorously test a ring model, a set of twenty lattices with different configuration errors is generated. For each lattice, orbit, tune, coupling and symmetry are corrected. Then, on and off momentum apertures are carefully determined for each one of them. Throughout the paper, we always refer to this test when introducing lattice errors; the average of the results represents the estimation.
See Table 1 for definition of notations and symbols used in the paper. 4D map implies that cavity and radiation are not considered. When not explicitly stated, the derivative is implied to ∈ [0, L), longitudinal position δ energy deviation τ delay in respect to the synchrotron particle β energy dependant betatron function ν horizontal and vertical tunes x = (x, x , y, y , δ, τ ), space coordinates x XD co 4D or 6D closed orbit coordinates Φ XD N (x) N turns 4D or 6D map of a particle, x represents the initial coordinates || · || 2 is the euclidean norm

Energy Acceptance Calculation
Let us define physical aperture A phys as the smallest invariant betatron amplitude for which a particle with energy deviation δ collides with the vacuum chamber. Then, where X V C is the vacuum chamber half-width. Dynamic aperture represents the smallest invariant betatron amplitude for which the particle will eventually be lost. For the case of Sirius, where we do not have an x-plane symmetry, the calculation of dynamic aperture A dyn is performed as follows where Ω(δ) is the greatest interval containing x 4D co (s 0 , δ) and satisfying for a fixed longitudinal position s 0 , which in our case it was chosen in a straight section. All the above 4D closed orbit coordinates depend on s 0 and δ. ||Φ 6D ∞ (x)|| 2 < ∞ means that the particle trajectory does not diverge. In practice, we used N = 131 turns for a rough estimation and N = 900 for a more precise determination of A dyn . The next step is to calculate the invariant betatron amplitude induced by Touschek scattering, . A more detailed explanation of the equation used to calculate a ind is discussed in [3]. For each s * , we solve for δ, Equation (3) has two solutions δ + t (s * ) and δ − t (s * ), which are the positive and negative local transverse acceptances, respectively. The overall energy acceptance is given by where δ RF is the acceptance of the RF system and δ ± w is the energy deviation (positive and negative) for which the tune cross a resonance. An analogous approach to calculate energy acceptance is also used in [4]. Finally, Touschek lifetime is calculated by the method described in [5].   Figure 1 shows that the limiting aperture for Sirius is the dynamic aperture. Black dots represent the corresponding invariant betatron amplitude of a particle lost during tracking in the rigorous test with errors. In this case, the acceptance estimation is a good approximation of the Touschek acceptance. However, the method presented to calculate Touschek lifetime only guarantees an upper bound, which mostly is a satisfactory approximation.

On Momentum Aperture Calculation
For Sirius project, the horizontal plane aperture at the negative side is important, since injection will occur at x ≈ -8 mm [6]. We use chaos indicators to estimate this aperture. When introducing errors in the lattice, particles in regions where there is a significant trace of chaos are usually lost. In this section, we compare the performance of two chaos indicators, the well known Diffusion and the proposed one ASDR (Average Square Distance Ratio).

Diffusion
Diffusion is calculated using NAFF (Numerical Analysis of Fundamental Frequencies) [7], a fast and precise algorithm to calculate fundamental frequencies of a motion. As it is explained in [8], let us denote NAFF by the operator This operator calculates the fractional part of the horizontal and vertical tune, based on 4D tracking of N turns in the ring, starting from coordinate x 0 . The diffusion vector is defined as A chaos indicator is, then, given by ||D|| 2 . We observed that ||D|| 2 > 10 −4 represents a significant probability of losing the particle, when lattice errors are introduced.

ASDR
Let {x i,0 } 1≤i≤M be a set of initial conditions and x i,n = Φ 4D n (x i,0 ). Then, we define for 1 ≤ i < M , where ζ represents any space coordinate in R = {x, x , y, y }. For example, ζ = x, gives x i,n , which is the horizontal position coordinate of the vector x i,n . Finally, the ASDR indicator is given by where #(R) = 4 is the cardinality of R. To simplify the analysis, let us consider only the horizontal dynamic, i. e., then one can show that Notice that we must have the successive initial conditions close to each other, in particular ∆ω i << 1/N << 1; and β x must be small, in order to obtain a π/2 phase shift between x and x , which allow us to cancel oscillating terms. Furthermore, ω i /(2π) must be far from any integer, in order to the oscillating terms remain small. It is important to note that, if we are in a region where the frequency shift ∆ω i << (∆A i /A i )/N , then ASDR 2 i ≈ 1/4. Analogous reasoning can be done when added the vertical dynamics. For a well-behaved motion, it is expected that 1/4 < ASDR 2 i < 1. Therefore, for sufficiently close initial conditions, ASDR 2 i > 1 indicates a motion different from the one proposed, which suggests a chaotic behavior.

Comparison
We have compared the performance of chaos indicators over 62 ring models, on which we had already performed the rigorous test including errors. Figure 2 shows the Relative Mean Square Error (RMSE) of the indicator as a function of the chosen threshold. As expected, if the chaos indicator threshold is too small or too large, then the prediction fail.
For both indicators the number of turns per initial condition was N = 130 and the step between initial conditions for x − was 20 m and for δ − was 8.5 · 10 −5 .
For this comparison, indicator ASDR with a relative error of approximately 10% at threshold around 1.05 performs better than Diffusion with a relative error of approximately 15% at threshold around 10 −4 . In fact, we observed that for these thresholds, ASDR generally predicts a smaller aperture, which is more precise than Diffusion.

Optimization Results
The described methods to calculate energy acceptance and on momentum aperture are sufficient to form an objective function, which calculates Touschek lifetime τ TC and on momentum aperture area A x×y , as a function of the ring model. This function takes between 2 to 10 minutes per model, while the standard technique takes around 7 hours. Let us represent the objective function as f : where S is the vector of strength of sextupole families, N S = dim(S) (In case of Sirius, N S = 21) and the quadrupole strengths are calculated in order to reach the specified tunes ν with minimal modification. Singular Value Decomposition (SVD) was used to perform this task. This objective function is used in a Multi-Objective Genetic Algorithm (MOGA), which in our case was NSGA-II. A detailed explanation of this algorithm is given in [9]. To enhance search time, some restrictions are imposed, which are much faster to calculate than the objective-function. The feasible region was restricted to model rings with small positive chromaticity and a minimum upper bound for Touschek lifetime, which is calculated using only the physical aperture, see Eq. (3). The gain is to provide a fast way to avoid rings which does not have the minimum lifetime required.
At certain point in the optimization of Sirius, NSGA-II was exploring solutions for which some driving terms (see [10]) were being increased to enhance on momentum aperture (h 21000 , h 30000 ) or energy acceptance (h 10110 ), however when lattice errors were added, the particles did not survive. Therefore, these driving terms were added as constraints to remain smaller than a certain quantity. Thus, avoiding this kind of problems. Another strategic constraint we have added was a specific tune resonance of fourth order for the x-plane, which aided to predict the on momentum aperture, along with the chaos indicators.
The best results of the optimization, which were submitted to the rigorous test with errors are listed in Table 2, where we used a multi-bunch uniform filling operation mode, with 1% coupling and total current of 100 mA. Despite model R11G70M024 having a gain of 2.3 hours in Touschek lifetime, the loss in on momentum aperture may be risky for injection. The most safe result is model R11G16M019, where we have a significant gain in every aspect and a small standard deviation.  38.9 ± 2.0 −11.0 ± 0.7 11.5 ± 0.7

Conclusion
Using the energy acceptance and on momentum aperture methods, we only had an upper bound estimation of the parameters. However, adding constraints related to driving terms and resonances, enhanced the methods to be reasonable approximations.
The best ranked machines from the optimization were selected to perform a rigorous test with lattice errors. Then, we could detect the best configuration models among them. These results are shown in Table 2.
The optimization method still needs improvement. The next step is to study the possibility of using another optimization algorithm with faster convergence than NSGA-II, for example the algorithm presented in [11].