Analysis of a model for resonant extraction of intense beams by normal forms and frequency map

A simple 1D model is proposed to explore the resonant extraction of intense beams from a synchrotron as performed in the SIS. The model Hamiltonian consists of a constant focusing, a thin sextupole and a smooth space charge field. Hyperbolic normal forms are used to estimate the extraction times and the emittance of the extracted beam; the quality of the reconstruction is tested in absence of space charge. The effect of space charge on the dynamical behavior of the beam near the 1/3 betatron resonance is numerically investigated using the frequency map analysis and qualitatively explained with perturbation theory. A polynomial approximation to the one turn map is obtained by replacing the exact space charge force with a sequence of polynomial kicks and the resonant normal forms reproduce quite accurately the nonlinear tunes and the fixed points position. At low order an analytical estimate of the area of the stable region is proposed to recover the self consistency of the model.


INTRODUCTION
In the study of rare nuclear reactions by means of beamtarget collision experiments the intensity of the delivered beam is often limited by some over heating constraints (the target may melt).One needs then to dilute the extraction on a time interval depending on external requirements.The tool traditionally used for this purpose is resonant (or slow) extraction.By means of nonlinear magnetic elements (usually sextupoles or octupoles) one excites an unstable resonance that divides the transverse phase space (usually the horizontal one) into a stable and an unstable region.Particles are then gradually destabilized by shrinking the dimensions of the stable area (which can be done both by quadrupole ramping or betatron core acceleration) or by making them diffuse outside by means of a an RF noise applied perpendicularly to the beam.In the study of rare reactions, however, the beam intensities needed at the target could be so high that the current circulating in the beam is sufficient to produce significant space charge effects.A 500 mA beam of U 72+ , as planned at the SIS synchrotron in GSI (Darmstadt), causes, for example, a relative linear tune shift of 5 × 10 −3 .In this case the space charge effects on the resonant extraction should be considered because a slow spill is required not to melt the target.The purpose of this paper is to illustrate a method, based on the normal forms perturbative technique, to give semianalytical estimates for the extraction times and emittances of the extracted beam.These two parameters are of basic importance to understand the influence of ripples, present in the power supplies, on the temporal density profile of the extracted beam and on the focal spot.The advantage of semi-analytical estimates with respect to tracking procedures lies in a easier parametric control of the extraction mechanism.These estimates, given for a fixed machine tune, can be used also in the case of quadrupole ramping and betatron core acceleration, provided that they are performed slowly enough to be considered adiabatic processes (spills longer than 10 ms for machine circumferences of some hundreds of meters).The method is based on the computation of the fixed hyperbolic points of the accelerator one turn map and on the calculation, in their neighborhood, of the hyperbolic normal form.After a nonlinear transformation to normal coordinates, the map becomes hyperbolic with a divergence factor which depends on the invariant, so that the computation of the extraction time and extracted emittance becomes very easy.When no resonance islands are present the validity domain of these perturbative estimates can be indefinitely extended along the stable and unstable manifolds, otherwise the island elliptic fixed point determines a border.The proposed technique is first applied to the study of a case without space charge.The estimate obtained for the extraction time is comparable, at the lowest order, with the result of a previously proposed method for extraction through the 1/3 resonance [1], while at higher orders it allows to take into account the strong nonlinear deformation of the unstable orbits, which considerably affects the extraction time behavior.Moreover, being free of ad hoc assumption on the dynamics, it allows an automatic extension to resonances different from 1/3 and to nonlinear fields different from the pure sextupolar one.In presence of space charge the stable area changes and the hyperbolic fixed points move.A careful modeling of the non linear influence of space charge on transversal resonances is then needed to locate the new fixed points and to extend the proposed technique to the study of an intense beam.Outline of the paper.In section 1 we specify the Hamiltonian describing a flat intense beam in a linear lattice with a thin extraction sextupole.In section 2 we introduce the hyperbolic normal forms to analyze the dynamics of the one turn map and to compute the extraction times, when the linear tune is close to 1/3 and the space charge is negligible.In section 3 we investigate the dynamical effects of the space charge by analyzing the changes induced in the tune and the dynamic aperture.A semi-analytic estimate of the tune and of the stable area, provided by the resonant elliptic normal forms, allows to impose an approximate self-consistency condition and to define an effective sextupolar strength.

I. PRESENTATION OF THE MODEL
The proposed model refers to the horizontal motion of a particle belonging to a round beam and moving in a linear lattice with a thin extraction sextupole and subject to the space charge force where ξ is the perveance of the beam, a its radial extension and V (x) the electric potential for a unit charge per unit length and unit beam radius.We choose the potential corresponding to an electric field which behaves linearly at the origin and vanishes like 1/x as x → ∞ The force acting on a macro-particle has the same asymptotic behavior as for a KV distribution but mimics a smooth distribution of charge at the core edge and reads The corresponding Hamiltonian in the physical coordinates is: where k 0 is the machine quadrupolar gradient, K 2 is the integrated sextupolar gradient and δ L (s) is the periodic δ function of period L, the length of the ring.For ξ = 0 the betatronic frequency for particles with vanishingly small oscillation amplitude is ω 0 = √ k 0 L. When ξ = 0 the quadratic term of V changes the frequency as ω 2 = ω 2 0 − ξL 2 /a 2 and the tune shift is ∆ν sc = (ω 0 − ω)/2π ≃ ξL 2 a −2 /(4πω 0 ).

A. Scalings
Scaling the longitudinal and transverse coordinates according to s = s ph /L and x = x ph , p = p ph L we write the scaled Hamiltonian H = L 2 H ph as a function of the depressed linear frequency ω where δ L (Ls) = L −1 δ(s) and δ(s) has period 1.We notice that the scaled emittance is ǫ = L ǫ ph and that Hamilton's equations still hold in the scales variables.
Moving then to the linearly normalized coordinates x = x √ ω, p = p/ √ ω the Hamiltonian reads: where â = a √ ω is the normalized radius and are the the scaled sextupolar gradient, and the scaled perveance.The Courant-Snyder like coordinates x, p have the dimension of a length like x, p, whereas K2 is the inverse of a length.A dimensionless description is obtained after scaling (5) or (6) with respect to the core radius x/a, p/a, or x/a, p/a and H/a 2 ; the emittance becomes ǫ/a 2 .Using the normalized coordinates we have and the Hamiltonian depends on the the dimensionless parameters ω, K2 a and ξ

B. Dynamic aperture
For a real lattice in the extraction regime the bare linear tune is given by ν 0 = m + 1 3 + η 2π where m is the integer part and η ≪ 1. Letting n = 3m + 1 we have ω 0 = 2π n 3 + η.The dynamic aperture in the absence of space charge depends only on η D(ω 0 ) ∼ 3 |η| is the dynamic aperture of the Hénon map.
If we change ω 0 into ω ′ 0 = 2π 3 + η by choosing a ring of length L ′ = L/n and k ′ 0 = k 0 (nω ′ 0 /ω 0 ) 2 ≃ k 0 , the dynamic aperture x dyn , in physical coordinates is unchanged since ω ′ 0 ≃ ω 0 /n implies ω 0 /L ≃ ω ′ 0 /L ′ .The presence of space charge changes ω 0 into ω but the tune depression does not vary since ω/ω 0 = ω ′ /ω ′ 0 whereas the tune shift changes by 1/n being proportional to the ring length.To have the same dynamic aperture, which depends on η + ξ/2 = η + ξ L 2 /(2a 2 ω), the perveance for the short ring must be changed into ξ ′ = nξ.Finally the normalized parameters for long and short ring are related by We quote the parameters of the SIS synchrotron at GSI (Darmstadt) for an extraction of 500 mA of U 72+ at 1000 MeV per nucleon; the longitudinal and transverse length units are m and mm.The ring length is L ≃ 217 m, the bare tune ν 0 = 4.3 ( β = 8 m), the sextupole gradient K 2 ≃ 3 m −2 .The initial horizontal emittance is 200 mm mrad, the core radius a ≃ 25 mm, and the tune-shift is ∆ν sc ≃ 5 × 10 −3 (a thousand times less for therapeutic applications) implying ξ ≃ 0.02.The normalized parameters are K2 = 4.6 m −1 , ξ ≃ 6 × 10 −2 and â = 130 mm.In the models chosen for our simulations the values of the parameters K2 and ξ are comparable with SIS.This Hamiltonian is not self consistent since the beam radius is a free parameter.A self consistent radius â depends on ω 0 , K2 , ξ as shown in the last section.

II. HYPERBOLIC INTERPOLATING HAMILTONIAN
In this section we show how to obtain semi-analytical estimates of the extraction time and the emittance of the extracted beam in the case of a pure sextupole.The proposed method relies on the computation of the hyperbolic normal form.After calculating the one turn (Poincaré) map M , we chose one of the hyperbolic periodic points ( xf , pf ) of period three.This is a fixed point for the third iterate of the map M •3 , ( xf , pf ) and the orbits are close to hyperbolae in its neighborhood.After translating the origin to ( xf , pf ), we look for an approximate symplectic change of coordinates X(x, p), P (x, p), such that in the new coordinates the motion generated by M •3 is exactly hyperbolic.This transformation is defined in a neighborhood of the unstable fixed point and can be extended along the stable and unstable manifolds [6].The interpolating Hamiltonian H in the new coordinates H = H 0 + H 1 XP + H 2 (XP ) 2 + . . . is a power series in XP and the invariant curves are equilateral hyperbolae.Letting σ = ∂H/∂(XP ) the evolution is given by where (X(3t), P (3t)) interpolates the iterates of order of M • 3 defined for t integer.The exponential divergence rate from the origin, σ, depends on the invariant XP , as the tune of a closed orbit depends on the emittance.If the initial point is near to the hyperbolic fixed point, the orbit spends a rather long time nearby, until X(t) becomes of order one, but the subsequent divergence is fast.Denoting by X es the abscissa of the point where the image of the septum in the normal coordinates plane intersects the orbit with initial conditions (X 0 , P 0 ), the extraction time t es = 3T reads X(t es ) ≡ X(3T ) = X es .The value of T is given by We consider two cases corresponding to different conditions for the extraction process: i) the particles are extracted near the unstable manifold issued from the hyperbolic point, ii) the particles escape through the stochastic  2 (top) with the segment (blue) on which the initial conditions are chosen, the unstable (red) and stable (green) manifolds.Top: image in normal coordinates (X, P ) of a neighborhood of the hyperbolic point, of the initial conditions segment (blue curve), of the stable (P axis) and unstable (X axis) manifolds.This behavior is explained by higher order normal forms as a folding effect of the trajectories, which cross two times the segment where the initial conditions are chosen, see the bottom frame of figure 3.This effect is more visible in the normal coordinates space, where the trajectories are equilateral hyperbolae and the initial condition line has folded, see the top frame of figure 3.As a consequence when we move along this line of initial points every trajectory is crossed two times.The intersection points A, B of a trajectory have different extraction times: if the point A is met first when we move along the line of initial conditions, the extraction time of B is the same as A plus the time needed to go from B to A along the trajectory joining them.In case ii) the initial conditions set crosses the stable manifold, where the escape time diverges, see bottom frame of figure 2. Beyond the stable manifold (vertical axis in the top frame of figure 3) the particle follows another family of hyperbolae.In the bottom frame of figure 3 we have plotted, in the x, p plane, the images of the stable and unstable manifolds, which cross at the first homoclinic point at a very small angle.This point is a singularity for the Taylor expansion of the normal form transformation and defines the applicability limit of the normal form dynamics.
Another straightforward calculation in normal coordinates is the extracted emittance ǫ.We consider the invariant curve to which belongs the initial point (X 0 , P 0 ) such that X es = X(3T ) = X 0 e T σ(X0P0) .The areas π ǫ k of the domains D k delimited by the horizontal axis P = 0, the curve XP = X 0 P 0 and the vertical lines X = X(3k − 3), X = X(3k) for k = 1, . . ., T (see figure 4) are all equal and given by: The domains D 1 , D 2 ,....,D T are extracted at time 3T , 3T − 3,....,3 respectively.If the curve bounding the domains D k is not an invariant curve, then the extracted areas π ǫ k are no longer constant and typically the emittance ǫ k extracted at time t = 3(T − k + 1) is an increasing function of k, see figure 6. Therefore the emittance ǫ is a decreasing function of t.To this end we consider a rectangular region partitioned into rectangular domains D 1 ,D 2 ,....,D T where the hyperbola to which (X 0 , P 0 ) belongs is replaced by the straight line P = P 0 (see figure 5).We can easily show that ǫ 1 < ǫ 2 < . . .< ǫ T , as observed in a numerical simulation (where the domain is rectangular in the initial coordinates) whose results are shown in figure 6. Indeed D T is split by the hyperbola passing through (X(3T − 3), P 0 ) into two domains: U T (upper) and L T (lower).The same hyperbola and the straight lines X = X(3T − 6), X = X(3T − 3), P = 0 define a domain which is the union D T −1 and a non empty domain A T −1 (see figure 5) where where µ = area(A T −1 )+area(U T ) is strictly positive since the domains are non empty.The above estimates can be used in two distinct ways: • Static case: The effect on the extracted emittance of the ripples present in the power supplies can be analyzed.In presence of ripples the boundary of the stable region moves inwards and outwards periodically.The particles swept in one period are immediately destabilized.The amplitude of this layer depends on the amplitude stable region.Computing ǫ(XP ) and T (XP ) on its outer edges allows to plot the shape of the emittance of the extracted beam versus time as is shown in figure 6.The fluctuations of the dimension of the focal spot (which depends on the transverse dimension of the extracted beam) increase with the amplitude ∆ǫ of the emittance decrease.
• Dynamic case: when the parameters of the extraction (such as the machine tune) are changing in time slowly enough compared to the average extraction time one can consider the extraction process instantaneous and, after computing the initial coordinates (X 0 (t), P 0 (t)) of the destabilized particles according to the particular extraction scheme, use (11) to evaluate π ǫ(t) = X 0 (t)P 0 (t)σ(X 0 (t)P 0 (t)).
The analysis of extraction times and emittances using the hyperbolic normal forms has not yet been discussed in presence of space charge.This is possible provided that we determine the hyperbolic fixed point of period 3 and construct a polynomial expansion of the third iterate of the map around one of them.

III. EXTENSION TO THE CASE OF AN INTENSE BEAM A. FREQUENCY ANALYSIS
To generalize the technique described in section II one must take into account the shift in the position of the fixed points due to space charge.In this section we first illustrate the nonlinear effects of space charge on the transverse dynamics of the model by studying the nonlinear tune behavior.
The numerical tracking of the orbits has been carried out by transfer maps.To obtain the one turn map of the system one must break the action of space charge, continuous over the ring, into a certain number of kicks.The order of possible resonances artificially introduced by the discretization grows with the number N of space charge kicks.The one turn map approximated with N kicks reads: where R(α) is the rotation matrix of angle α and Fourier analyzing the iterates of this map for different initial conditions of increasing emittance, we have computed the nonlinear tune as a function of the emittance.Being continuously distributed around the ring, the space charge cannot excite any resonance in a flat beam.The nonlinear tune increases steadily with the distance from the origin, reaching the bare tune ω 0 /(2π) as |x| → ∞.
The dependence of the nonlinear frequency Ω sc on the orbit emittance ǫ for a generic charge distribution can be analytically described very accurately by canonical perturbation theory (see [2] for the computation for a Gaus-sian charge distribution).For the chosen charge distribution, letting  = 1 2 (x 2 + p2 ) be the action, we get: x/a x/a which is plotted in figure 7.
When a thin sextupole is present, as in the case of slow extraction, and the linear tune approaches a resonant value, the perturbative calculations can be performed by using the resonant normal forms.

Pure sextupole
In the case of a pure sextupole ω and the bare machine frequency ω 0 coincide.When ω > 2π/3 the nonlinear frequency Ω sex decreases monotonically until it reaches the resonant value.On the contrary when ω < 2π/3 then Ω sex increases up to 2π/3.In both cases near the resonance the sextupole pushes Ω sex toward 2π/3, as can be seen with nonresonant normal form analysis [3], that gives as a first order estimate of the nonlinear frequency If ω is close to 2π/3 the coefficient of the emittance ǫ = x2 + p2 is negative for ω > 2π/3, positive for ω < 2π/3.As a consequence when ǫ increases Ω sex reaches the value 2π/3, according to (16), if ω is above or below 2π/3.When ω/2π is close to 0.32 three islands appear.

Sextupole+space charge
When ω is close to 2π/3 the dependence of the dynamic aperture on the space charge intensity ξ is easily described.The frequency Ω in this case may be approximated by the linear frequency plus the sum of the space charge frequency shift Ω sc − ω given by ( 13) and the sextupole frequency shift Ω sex − ω given by ( 14).If ω < 2π/3 the sextupole and space charge frequency shifts are both positive and the frequency Ω ≃ ω + (Ω sex − ω) + (Ω sc − ω) increases with ǫ; the resonance is crossed for a value of the emittance which decreases with ξ, see top frame of figure 8.If ω > 2π/3 the sextupole frequency shift becomes negative and the frequency Ω may reach a maximum before decreasing; the resonance is crossed for a value of the emittance which increases with ξ, see bottom frame of figure 8.The increase of the dynamic aperture with ξ above 2π/3 and its decrease below as can be seen in figure 9, where ).Function f (x) (violet curve), approximation of order 5 f5(x) = x− 2 3 x 3 + 1 6 x 5 (red), approximation of order 9 f9(x) = x − 0.9268 x 3 + 0.6313 x 5 − 0.2430 x 7 + 0.0378 x 9 (blue) obtained by minimizing the L 2 norm in [0, 1.5] the emittance of the stable area is plotted as a function of the bare linear tune ν 0 = ω 0 /(2π) for increasing beam currents.

B. ELLIPTIC INTERPOLATING HAMILTONIAN
To evaluate semi-analytically the new position of the hyperbolic fixed points we evaluate the elliptic normal forms starting from a polynomial expansion of the one turn map M at the origin.In the normal coordinates the orbits of M can be interpolated by the orbits of a Hamiltonian H invariant under continuous rotations (non-resonant normal form) or under discrete rotations of 2π/3 (resonant normal form).This Hamiltonian captures the relevant dynamical features of the map.
To calculate the normal form of the proposed Hamiltonian one must: 1. Replace the space charge force by a polynomial approximation.A polynomial of degree 3 or 5 gives a good approximation up to one half the core radius or core radius itself, see figure 10.
2. Calculate a polynomial map by composing N times a rotation of ω/N , with space charge kicks and one sextupolar kick at the end (see equations ( 13) and ( 14)).
3. Compute by using a computer program like ARES [4] or BIRKHOFF [5] the elliptic normal form and its interpolating Hamiltonian (the polynomial map is truncated at degree 2n − 1 in order to compute their order n normal form).

I
In figures 11 and 12 we show that the phase portraits obtained with the space charge force F given by ( 3) and with a a polynomial approximation of degree 5 are very similar up to the hyperbolic fixed points.Moreover the top frame of figure 13 shows that also the nonlinear tunes (computed by Fourier analyzing the orbit obtained from tracking) corresponding to F and its approximation of degree 5 are very close up to the dynamic aperture (end point of the red line).[7] As a consequence polynomial expansions of the space charge force beyond 5-th order are not needed.The order n of the normal form (H is a polynomial of degree 2n) is chosen by demanding that that the frequency (Ω is a polynomial of degree 2n − 2) reaches a given accuracy.To obtain the nonlinear frequency Ω we write the interpolating Hamiltonian H as a function of the action and angle variables , θ where x = √ 2 cos θ, where  = (θ, E) is implicitly defined by H(, θ) = E ≡ H(θ 0 ,  0 ).In the bottom graph of figure 13 the nonlinear tune calculated via ( 17) is compared with the results of a Fourier analysis of the orbit of the polynomial map.
At second order (highest power of  is 2) the interpolating Hamiltonian has a simple analytical expression, which allows to compute its critical points and an approximation to the area of stable orbits.This estimate allows to recover an approximate the self consistency of the model.Again we approximate the space charge F defined by (3) with a polynomial of degree 5 given by F app = ξ a x a +α x 3 a 3 +β x 5 a 5 . The coefficients that minimize the distance ( in the L 2 norm) from the exact F for x ∈ [0, 1.5 a] are α = −2/3 and β = 1/6 .In the appendix the procedure to compute a polynomial approximation of the one turn map, in presence of a space charge force continuously acting along the ring, is outlined.The corresponding interpolating Hamiltonian reads (18) where η is the distance of the linear frequency from the resonance: ω = Ω(0) = 2π(m + 1 3 ) + η and α = α ξ/â 2 = −(2 ξ/(3â 2 ) is the coefficient of the third order contribution of the space charge kick in Courant-Snyder coordinates.We remark that H for α = 0 and K2 = 2 is the interpolating Hamiltonian of the Hénon map.This Hamiltonian allows to compute a correction in the position of the fixed points which depends on the coefficient α of the first non linear correction of the space charge force.[8] If ω is sufficiently near to 2π/3 t so that no stable chain of 3 islands is present, we can approximate the stable region as a triangle and compute its area: In figure 14 the emittance of the stable area numerically computed is compared with its analytical estimate.The condition A x (η, K2 , ξ, â) = π â2 allows to impose that the radius of the charge distribution is equal to the radius of the stable area in phase space and determines â = â(η, K2 , ξ) .After computing the new fixed point position one can use the technique illustrated in section II in order to compute the extraction times.An accurate calculation requires a polynomial expansion of the complete one turn map (sextupole + space charge) at the fixed point.For low values of ξ, however, is sufficient to take into account just the linear effect of space charge and one can directly apply the results of section II, provided that the shifted linear frequency ω = ω 0 − 2π ∆ν sc is used.For higher space charge the change of the stable area may be taken into account by introducing an equivalent sextupole with gradient K2,EQ .This is determined by imposing that the stable area for such a sextupole without space charge is the same as for the sextupole with gradient K2 with space charge.Using (19) this condition reads:

CONCLUSIONS
A method based on hyperbolic normal forms is proposed to obtain semi-analytical estimates of the extraction time and of the extracted emittance in a resonant (slow) extraction process.A good agreement with tracking is found.The dynamical and geometrical features of the extraction process emerge very neatly in the normal coordinates plane.The space charge changes the tune and the location of the unstable periodic points.The resonant normal forms allow to describe the changes of the tune and of the area of the stable region, once a polynomial approximation to the space charge force is introduced.Moreover an approximate self consistency can be imposed by equating the stable area to the area of the core cross section.The results are qualitatively correct at the lowest order, very close to tracking when the order is increased.
A simple description of the extraction process, valid for moderate space charge intensities, consists in using a thin sextupole Hamiltonian with an effective strength such that the stable area is the same as for the model with space charge.

COMPUTATION OF THE POLYNOMIAL ONE TURN MAP FOR A SEXTUPOLE AND SPACE CHARGE
We define M sc+sex the one turn map on the normalized lattice (of unit length) with respect to the linearly normalized coordinates (x, p).This is the result of the space charge map and the final sextupoles kick.
where the space charge map is the limit of n space charge kicks M sc 1 n acting on arcs of length 1/n according to M sc = lim n→∞ M • n sc 1 n .The one turn map in the linearly normalized coordinates (x, p) reads where z = x − ip.The space charge map on an arc of length 1/n is given by where δ = ω/n and the nonlinear space charge coefficient is defined as previously by α = α ξ/â 2 = − 2 3 ξ/â 2 .To evaluate M sc we conjugate M sc 1 n to its nonresonant normal form according to Taking the n → ∞ limit and composing with the sextupole kick the final one turn map reads

FIG. 1 :
FIG.1: Top: phase plot in linearly normalized coordinates (x, p) for a sextupole K2 = 1 m −1 and linear frequency ω0 = 4.33 × 2π.Bottom: numerical (red line) and analytical (blue=first order, green=tenth) estimates of the extraction time for a distribution of particles on a line versus the distance from the hyperbolic fixed point.The extracted points initially belong to the red segment in the phase space plot.The dynamic aperture is x dyn = 90 mm.The corresponding value in the physical coordinates is x dyn = 17 mm and for the value K2 = 4.6 m −1 of SIS becomes x dyn ≃ 4 mm

FIG. 3 :
FIG.2: Top: phase plot in linearly normalized coordinates (x, p) for a sextupole K2 = 1 m −1 and linear frequency ω0 = 4.32 × 2π.Bottom: numerical (red line) and analytical 10th order (green) estimates of the extraction time for a distribution of particles on a line versus the distance from the hyperbolic fixed point.The extracted points initially belong to the red segment in the phase space plot.The dynamic aperture is x dyn = 570 mm.The corresponding value in physical coordinates is x dyn = 110 mm, and for the value K2 = 4.6 m −1 of SIS becomes x dyn = 24 mm, just as the core radius a at the beginning of the extraction.

FIG. 4 :FIG. 5 :
FIG.4: Calculation in the normal coordinates frame of the emittance of the extracted beam.The points in the lightest domain D3 is extracted after 3 iterations of the map M and its area is the emittance extracted in that period.The points in the domains D2, D1 are extracted after 6, 9 iterations.

FIG. 6 :
FIG. 6: Area and emittance of the extracted beam Ax = π ǫx versus extraction time during the extraction of the red stripe of particles on the top frame.

2 FIG. 9 :
FIG. 9: Plot of the area of the stable region Ax = π ǫx normalized to a 2 , as a function of the linear tune tune ν = ω/2π for a K′ 2 = 0.3 and different values of the space charge intensity.Black line, ξ = 0, blue line ξ = 0.05, red line ξ = 0.1.

5 m − 1 .FIG. 12 : 3 m − 1 .
FIG. 11: Top: pure sextupole with ω ′ = 0.34×2π and a K′ 2 = 0.14.Center: exact sextupole + space charge with ω ′ = 0.34× 2π, a K′ 2 = 0.14 and ξ = 0.05.Bottom: exact sextupole + space charge where the space charge force F (x) is replaced with a polynomial approximation of degree 5.The phase plots are in the coordinates x, p normalized to the core radius a.The dynamic aperture is comparable to the normalized core radius x dyn ≃ â ≃ 1.4 a To compare these values with a lattice like SIS we must add the integer part to the tune ω = 4.34×2π and for a = 25 mm we have K2 = K′ 2 / √ n = 1.5 m −1 .Purple vertical lines x = ±â ≃ ±1.4 a.