Evaluation of nonlinear resonances in 4D symplectic mappings

Evaluation of Nonlinear Resonances in 4Dsymplectic mappingsE. Todesco, M. Gemmi and M. GiovannozziJanuary 24, 1997AbstractWe discuss the evaluation of the stability, position and width ofresonances in four-dimensional symplectic mappings. The approachpresented in this paper is based on the computation of the resonantperturbative series carried out by the program ARES. This piece ofcode allows to perform the evaluation of the perturbative series andof the interpolating Hamiltonian.Given a symplectic map in the neighbourhood of an elliptic �xed point,the new code NERO performs the analysis of the orbits of the interpo-lating hamiltonian both in the nonresonant and the resonant case. Foreach resonant normal form, the interpolating Hamiltonian is computedand the position and the stability of the resonant orbits and the widthof the islands are evaluated; this analysis is carried out through thedirect inspection of the coe�cients of the interpolating Hamiltonian.All the computations are carried out at an arbitrary order; the �rstsigni�cant perturbative order is taken as a �rst guess, and a Newtonmethod is used to evaluate the higher orders e�ect.1 IntroductionThe dynamics of single particles in a magnetic lattice under the ef-fect of nonlinear �elds has become a crucial issue for the constructionof large circular accelerators such as the Large Hadron Collider [1].These problems are modelized by decoupling the longitudinal and the1

transverse degrees of freedom. Hence, one is lead to consider four-dimensional (4D) symplectic mappings describing the behaviour ofthe transverse motion of a single particle over one turn of the machine(i.e., the so-called one-turn map). The comprehension of the rela-tion between resonances, nonlinearities, tuneshifts and the stabilitydomain (dynamic aperture) in these models is a rather di�cult task.The dynamic aperture is usually determined through numerical inte-gration based on tracking [2, 3]. The perturbative theory based eitheron Hamiltonian ows [4, 5, 6], or on symplectic mappings [7, 8, 9, 10],provides a lot of analytical information on the detuning and on theresonance parameters.In the case of unstable resonances, the dynamic aperture is usuallydetermined by the hyperbolic resonant orbits (�xed lines) [11, 12,13]. The stable resonances, on the other hand, feature families ofislands that do not limit the stability domain, and therefore thereis no direct relation with the dynamic aperture. However, severalstudies have shown that analytical indicators (Quality Factors, QFs)extracted through perturbative tools can be well-correlated with thedynamic aperture [6, 14, 15].During the past years, arbitrary-order codes have been developedto compute nonresonant perturbative series (normal forms) of a generictruncated one-turn map [7, 8, 10]. More recently, a code has been de-veloped to evaluate also the resonant normal forms (code ARES, seeRef. [16]). In this paper we review the features of a new program(code NERO, see Ref. [17]) that includes ARES and postprocesses itsoutput in order to provide the following quantities:� It allows the automatic evaluation of quality factors de�ned inprevious papers [15], such as the norm of the map, the norm ofthe tuneshift, and the norm of the resonances.� It allows the reconstruction of the phase space dynamics: the net-work of resonances involved in the nonlinear motion is evaluatedand the position and the width of the islands are computed. Fur-thermore, another quality factor, de�ned as the fraction of thephase space volume that is locked on resonances, is evaluated.� It can automatically analyse several lattices at the same time andproduce correlation plots of the QFs with the dynamic apertureobtained through standard tracking. This feature is relevant, forinstance, in the analysis of the e�ect of random errors [15], and2

in general for all the optimization procedures.This code can be useful not only for studying the stability propertiesof betatronic motion, but also to obtain an analytical picture of theglobal dynamics of generic 4D symplectic mappings in the neighbour-hood of an elliptic �xed point. This approach is complementary tothe powerful numerical methods based on frequency analysis [18, 19].The plan of the paper is the following: in Section 2 we describethe theory of single-resonance normal forms. In Section 3 we outlinethe algorithms used in to evaluate the position and the stability ofresonant orbits. In Section 4 we summarize the capabilities of the codeNERO, while in Section 5 we compare the results obtained throughthe analytic approach with the outcome of numerical simulations.2 TheoryWe consider betatronic one-turn maps that modelize the transverseoscillations of charged particles in a magnetic lattice [7, 8, 10, 21].We assume that the synchrotron radiation is negligible, and that thetransverse motion and the longitudinal motion are uncoupled. Let(x; px; y; py) be the physical coordinates and the conjugated momentaat a given section of the machine. The one-turn map is the nonlinearfunction giving the phase space coordinates and momenta as a functionof the same quantities at the previous turn and at the same section ofthe machine. It can be expanded as a Taylor series truncated at orderM : x0 = MXn=1 Xj1+j2+j3+j4=nF1;j1;j2;j3;j4xj1pj2x yj3pj4yp0x = MXn=1 Xj1+j2+j3+j4=nF2;j1;j2;j3;j4xj1pj2x yj3pj4yy0 = MXn=1 Xj1+j2+j3+j4=nF3;j1;j2;j3;j4xj1pj2x yj3pj4yp0y = MXn=1 Xj1+j2+j3+j4=nF4;j1;j2;j3;j4xj1pj2x yj3pj4y : (1)The truncation order M is limited by the available storage memoryand by the CPU time; for a large machine such as the LHC,M can be3

chosen between 10 and 15. We denote by (x; px; y; py) the Courant-Snyder coordinates where the linear part of the map is the directproduct of two 2D rotations of angles Qx and Qy (the linear tunes).We also denote by z1 = x� ipx and z2 = y� ipy the complex Courant-Snyder coordinates that diagonalize the linear part of the motion;i = p�1 is the imaginary unit.Given an initial condition in the 4D phase space, its orbit is char-acterized by two nonlinear frequencies �x and �y that can be extractedthrough Fourier analysis [18, 20]. When the amplitude of the initialcondition tends to zero, the nonlinear frequencies (�x; �y) tend to thelinear ones (Qx; Qy). We de�ne the tune footprint as the region in thefrequency plane that corresponds to stable orbits of the one-turn map[15, 19].2.1 Normal formsThe perturbative theory of normal forms consists in transforming theone-turn map to a simpler, more symmetric map U (the normal form)that explicitly shows the motion invariants and the geometry of theorbits [7, 8, 10]. The normal form U is then written as the Lie series ofan interpolating Hamiltonian h. We denote by �1 and �2 the complexnormal form coordinates where the one-turn map is transformed intothe normal form U . The Hamiltonian is more easily expressed interms of the amplitude-angles coordinates (�1; �2; �1; �2), given by �1 =p�1ei�1 and �2 = p�2ei�2 . Therefore, �1 and �2 are the generalizationof the emittances to the nonlinear case, i.e., at �rst order they agreewith x2 + p2x and y2 + p2y respectively.As a �rst step, one can build a nonresonant normal form: in thiscase the Hamiltonian only depends on the amplitudes �1 and �2:h(�1; �2) = Xk1;k2 hk1;k2 �k11 �k22 : (2)The phase space described by this Hamiltonian is given by 2D KAMtori whose nonlinear frequencies are the partial derivatives of h withrespect to �1 and �2.When the linear tunes are close to a resonance (q; p), where q 2 Nand p 2 Z, it may happen that the resonant condition on the nonlinearfrequenciesq�x + p�y = m+ � �� 1 m 2 Z (3)4

is exactly satis�ed (i.e., � = 0) for some positive amplitudes. In thisspeci�c case, the general topology of the 2D invariant KAM tori breaksdown, and a family of islands arises. Such islands can be describedby the single-resonance normal form theory, that consists in retainingthe resonant combinations of the phases q�1 + p�2 in the normal formU and in the interpolating Hamiltonian. Thus h readsh(�1; �2; �1; �2) = Xk1;k2;l hk1 ;k2;l �k1+lq=21 �k2+ljpj=22 cos(l(q�1+p�2)+'k1;k2;l):(4)2.2 Topology of single resonancesIn order to analyse the dynamics of the above Hamiltonian, one canperform the canonical transformationr1 = �1 �1 = �1 + pq �2r2 = �pq�1 + �2 �2 = �2 (5)such that the new Hamiltonian readsh(�1; r1; r2) = Xk1;k2;l hk1;k2;l rk1+lq=21 �pq r1 + r2�k2+ljpj=2 cos(lq�1+'k1;k2;l):(6)Since there is no dependence on �2, the quantity r2 is invariant1 andthe problem is reduced to analyse a 2D Hamiltonian with a paramet-ric dependence on r2. For each value of r2 one can compute the �xedpoints, evaluate the eigenvalues, and the position of the separatrices.Finally, one has to check that the obtained solutions r1 and r2 corre-spond to positive values of the original amplitudes �1 and �2.We de�ne the excitation order lexc, as twice the power in the am-plitudes of the �rst nonzero resonant coe�cient in the interpolatingHamiltonian. In generic cases, where all the resonant coe�cients aredi�erent from zero, the excitation order is equal to the resonance or-der q + jpj. If there are some symmetries, some resonant coe�cientscan be equal to zero and therefore the excitation order can be greaterthan the resonance order. One has to distinguish two cases.1As the Hamiltonian is always an invariant of motion, the quantity r2 is also calledsecond invariant. 5

2.2.1 Excitation order equal three or fourWhen one considers a resonance (q; p) such that qp < 0 (also calleddi�erence resonance), the second invariant is r2 / p�1+q�2 and there-fore the motion is always bounded. In general one can have islands,even though the geometry of these structures is more complicatedand each resonance needs an appropriate analysis. For this reason,the code NERO does not perform the phase space analysis for theseresonances.In the case of sum resonances (qp > 0) with excitation order three,the resonance is dominant over the detuning terms and therefore onehas hyperbolic �xed points whose related separatrices go to in�nity.The code can evaluate the position of these separatrices, that are thelimits of the stability boundary.In the case of sum resonances with excitation order four, one canhave di�erent cases according to the relative strength of the resonantand of the detuning terms. One can have either separatrices that limitthe stability boundary or stable islands. Also in this case the analysisis rather involved, and it has not been implemented in the code forthe phase space analysis.2.2.2 Excitation order greater than fourIn this case there is always a detuning term (i.e. a nonresonant coe�-cient) of order lower than the resonance. One can show that in generalthere exist two families of �xed points, one elliptic and one hyperbolic,in agreement with the Poincar�e-Birkho� theorem. The separatricesthat pass through the hyperbolic �xed points are the border of theislands. The code NERO automatically evaluates the position of the�xed points, of the separatrices, and the area of the islands.2.3 Quality factorsIn previous studies [6, 14, 15] we have shown the importance of ana-lytical quality factors to understand the nonlinear dynamics. There-fore we have implemented the computation of the such quantities inNERO.� Norm of the map Q1; it is the sum of the absolute values ofthe map coe�cients weighted with the powers of the maximum6

amplitude A:Q1(A) = MXn=2An Xj1+j2+j3+j4=n Xi=1;4 jFi;j1;j2;j3;j4 j (7)� Norm of the tuneshift Q2; it is the average of the square rootof the sum of the squares of the amplitude-dependent tuneshiftsdivided by two. The average is carried out over the border of theanalysed phase space �1 + �2 = A. The tuneshift is evaluatedthrough nonresonant normal forms truncated at the speci�edorder N .� Norm of the resonance Q3(q; p); it the is sum of the absolute val-ues of the resonant coe�cients of the interpolating Hamiltonianweighted at the amplitude A.Q3(A; q; p) =Xl6=0 Xk1;k2 hk1;k2;lAk1+k2+l(q+jpj)=2 (8)� Hypervolume of the resonance Q4(q; p); if the resonance is sta-ble, it is the hypervolume in 4D of the initial conditions that arelocked on the resonance (q; p), normalized with respect to thetotal phase space hypervolume (that is limited by the amplitudeA). If the resonance is unstable, it is the 4D hypervolume ofinitial conditions that lie outside the unstable separatrix, nor-malized to the total hypervolume.3 AlgorithmsNERO incorporates the code ARES that evaluates the normal formseries: the related algorithms have been described in a previous paper[16]. The main part of the code NERO performs the analysis of theinterpolating Hamiltonian in order to evaluate the existence, the posi-tion and the stability of the resonant orbits. We will restrict ourselvesto outline the algorithms used for the case of excitation order greaterthan four. A similar strategy is used for the case of excitation orderequal to three.We consider a resonance (q; p) and the related interpolating Hamil-tonian h. First of all, a scan over the tune values that lie on the res-onant line q�x + p�y = m is carried out. For each couple of tunes,7

using the �rst order detuning, one analytically computes the values ofthe amplitudes �1 and �2 that correspond to that detuning. Then, aNewton method is used to compute the values of the amplitudes usingthe nonresonant part of the Hamiltonian up to the truncation order.In this case no analytical solution is possible, since one has to solvea nonlinear system. The Newton is initialized through the �rst ordersolution analytically evaluated. The obtained values of the amplitudes�01 and �02 correspond to the average amplitudes of the resonance.If the obtained amplitudes are positive, the second invariant isevaluated; then one has to analyse the reduced Hamiltonian (6) inorder to �nd out the position of the �xed points, that are the solutionsof the system @h@r1 ( 1; r1; r2) = 0@h@ 1 ( 1; r1; r2) = 0: (9)Also in this case one has to use a Newton method (with two variablesr1 and 1). The amplitude can be initialized using �01. The initialguess for the angle can be worked out by considering the �rst-ordertruncation of the second equation of the system (9): if there is onlyone sine term, the guess can be computed analytically. Otherwise, ifthere is more than one term (i.e., the resonance excitation is greaterthan the resonance order), one can make a numerical scan over [0; 2�]in order to �nd out a good guess. In both cases two guesses are needed;they must correspond to the elliptic and to the hyperbolic solution.The stability is evaluated through the computation of the hessian ofthe Hamiltonian in the �xed points.Once the �xed points have been found, one has to evaluate theequation of the separatrix. Also in this case we use a �rst-order ana-lytical guess for the position of the inner and outer separatrix for theangle �1e corresponding to the elliptic �xed point. Than, each guessis used to start a Newton method that solves the separatrix equationat arbitrary orderh(�1e; r1; r2) = Eh � h(�h; r1h; r2) (10)where (�h; r1h) denote the position of the hyperbolic �xed point, andtherefore Eh is the Hamiltonian value on the hyperbolic �xed point.8

The obtained values rmin1 and rmax1 are than transformed back to theoriginal amplitudes to have the minimum and maximum amplitude ofthe separatrix (�min1 ; �min2 ) and (�max1 ; �max2 ).Finally, the equation for the separatrix (10) is solved also for allthe values of �1 that are between the elliptic and the hyperbolic �xedpoint: in this way one can integrate the area of the islands. Also inthis case a Newton method is used, initialized by the values of thepreceeding angle �1.4 Program capabilitiesIn this Section we will list the main capabilities provided by the codeNERO. As far as the map to be analyzed is concerned, three di�erentoptions are available� The polynomial map is described by a set of coe�cients in nor-malized coordinates (the so-called Courant-Snyder coordinates[22] in the accelerator physics literature).� The polynomial map can be given as a set of coe�cients in phys-ical coordinates.� One can analyse a generalized form of the H�enon map [10], whichis already built in the code. The analysed map is given by0BBBBBBBBB@ x0p0xy0p0y 1CCCCCCCCCA = 0@R(Qx) 00 R(Qy)1A0BBBBBBBBB@ xpx +RePn2n=n1 kn(x+ iy)nypy � ImPn2n=n1 kn(x+ iy)n1CCCCCCCCCA ;(11)where R(Qi) is the 2D rotation matrix of an angle 2�Qi; this isthe map of a linear lattice with a multipole in the kick approx-imation, where n1 and n2 are the minimum and the maximummultipolar order, and kn are the multipole coe�cients [10]. Forn1 = n2 = 2 and k2 = 1 one recovers the 4D H�enon mapping[10].The input map can be analyzed through two di�erent approaches,namely: 9

� Resonance analysis It allows the analysis of a given set ofresonances. For the selected resonances, the code evaluates thenormal form, the interpolating Hamiltonian, the quality factorsQ1, Q2 and Q3.� Phase space analysis The position and the width of the is-lands relative to the resonances that are involved in the motionare evaluated. In order to �nd out which resonances are reachedin the selected phase space domain, the footprint through non-resonant normal form is evaluated and only the resonances thatfall inside the footprint are taken into account. This methodcan fail whenever the linear tunes are close to a low order reso-nance: this produces a divergent behaviour in the nonresonantnormal form at low orders, and a wrong footprint. To avoidthese di�culties it is possible to provide the set of resonances tobe analyzed.5 Numerical vs. Analytical methodsIn this Section we will compare the results of numerical simulationswith the analytical computations, carried out using resonant normalforms. The model used for this purpose is the quadratic mapping (11).The linear tunes Qi have been set to the values Qx = 0:28; Qy = 0:31corresponding to the working point of the planned LHC [1]. As astarting point we have computed the tune footprint using either a nu-merical approach or nonresonant normal forms (see Fig. 1). In the�rst case a rectangular grid of initial conditions is tracked and the non-linear frequencies are computed via frequency analysis (Fig. 1 left).In the latter, the triangular phase-space region given by �1 > 0; �2 > 0and �1 + �2 < A is covered with a rectangular mesh of initial condi-tions. Then, for each initial condition the amplitude-dependent tunes�x; �y are evaluated through nonresonant normal forms (Fig. 1 right).The two plots show the same overall pattern: however, using trackingdata, one can also recover the information concerning the stabilityand the strength of the resonances (for instance, the thick lines in theleft part of Fig. 1 represent initial conditions locked on stable reso-nances, while the size of the depletion area around the resonant lineis proportional to the resonance strength). This information cannotbe obtained by the nonresonant normal forms, that only provides the10

detuning function. Indeed, we show that using resonant normal formsone can recover this information.The width of the resonances is better visualized in the plane ofthe amplitudes; we consider a very dense scan in the plane of theinitial conditions, and we tracked each orbit for 2048 turns. If thenonlinear frequencies of the orbit satisfy a resonant condition of orderq + jpj � 15, with � < 0:0001 [see Eq. (3)], we consider the orbitto be resonant and we plot the linear invariants (x2; y2) of the initialcondition. The resulting picture is shown in Fig. 2: one can see thatsome resonances are rather strong (for instance, (6;�2) and (3;�6)),and that resonance (1;�4) splits the stability domain into two discon-nected parts.The same kind of plot can be produced by using resonant normalforms. In fact, by carrying out the analysis described in the previ-ous Sections, it is possible to compute analytically both the positionof the resonances and their widths. In Fig. 3 we show a resonancenetwork obtained by using normal forms: the resonance parametersare computed analytically, without tracking. In particular, resonances(3;�6) and (6;�2) have a rather wide width, in agreement with theprevious �gure obtained through numerical methods. The position ofthe resonance (1;�4), marked as a thick line, is also in agreement withthe previous �gure. The slight mismatch between the two �gures isdue to the fact that in the �rst case the plot is shown in the plane ofthe linear invariants, whilst in the second one the nonlinear invariantsobtained through normal forms are used. To compare frequency anal-ysis and normal forms more precisely, we have compared the normalform guess for the resonance width of resonance (3;�6) (see Fig. 4,solid line) with the numerical results projected in the plane of thenonlinear invariants: the quantitative agreement is good, even thoughthe resonance starts at rather high amplitudes and extends to the dy-namic aperture. Finally we will discuss the computation of the qualityfactors.In the analysis of the sorting problem it was shown [15] that thequality factors previously de�ned, are good indicators of the degree ofinstability of the motion. In a realistic machine the particles dynamicis strongly in uenced by nonlinear systematic and random magneticerrors [10, 15]. To compensate the harmful e�ects of systematic errorsit has been proposed to reduce the nonlinear tuneshift in order tostabilize the motion and thus to increase the region in phase-space11

Figure 1: Tune footprint of the 4D H�enon map at Qx = 0:28; Qy = 0:31through frequency analysis (left) and normal forms (right). Horizontal andvertical betatron frequencies are in the x an dy axes, respectively.where the motion is bounded (also called dynamic aperture).In the case of random errors, the standard approach consists in�nding a rearrangement (sorting) of the magnets generating the errors,in order to increase the dynamic aperture. In Ref. [15] we proposedto use analytic quantities well-correlated with the dynamic aperture:hence the maximization of the dynamic aperture can be performed bylooking for an estremum of a given quality factor. The key point isthat the evaluation of the quality factors is rather fast as it does notinvolve tracking, but only analytic computations. Furthermore the useof such indicators allow to gain a deeper insight in the causes of theinstabilities. Therefore the overall optimization procedure becomes12

faster and more accurate.In Fig. 5 we show the correlation of the quality factors Q1; Q2; Q3with the dynamic aperture for a 4D model of the LHC including non-linear errors. In Fig. 5 (a) it is shown the distribution of the dynamicaperture for a set of 100 seeds. In Fig. 5 (b), (c), (d) is shown the cor-relation of Qi with the dynamic aperture for the same 100 seeds. Onecan immediately see that Q2 and Q3(3; 0) have an excellent correlationwith the dynamic aperture, whilst Q1 has a poor correlation. Henceone is lead to the conclusion that the dynamics is highly in uencedby the (3; 0) resonance and to compensate this resonance would leadto an overall improvement of the machine performance. Actually, onecan hope to use these quantities to �nd a good rearrangement of therandom errors. In Ref. [15] we showed that a substantial increase ofthe dynamic aperture can be achieved by using this approach.AcknowledgementsWe wish to thank G. Turchetti and W. Scandale for stimulating thiswork and for helpful discussions. We also want to acknowledge A. Baz-zani for providing the code ARES that computes the normal form andthe interpolating Hamiltonian series. A special thank to F. Schmidtfor valuable help in checking the code and for useful discussions. Thework has also been partially supported by EC Human Capital andMobility Contract No. ERBCHRXCT940480.References[1] The LHC Study Group, CERN AC (LHC) 95-05 (1995).[2] H. Grote and F. C. Iselin, CERN SL (AP) 90{13 (1990).[3] F. Schmidt, CERN SL (AP) 94{56 (1994).[4] A. Schoch, CERN 57{21 (1957).[5] G. Guignard, CERN 76{06 (1976).[6] F. Willeke, DESY HERA 87{12 (1987).[7] A. Bazzani, P. Mazzanti, G. Servizi, G. Turchetti, Nuovo Cim.,B 102, (1988) 51{80.[8] E. Forest, M. Berz, J. Irwin, Part. Accel. 24, (1989) 91{113.13

[9] A. Bazzani, M. Giovannozzi, G. Servizi, E. Todesco, G. Turchetti,Physica D 64, (1993) 66{93.[10] A. Bazzani, E. Todesco, G. Turchetti, G. Servizi, CERN 94{02(1994).[11] E. Todesco, Phys. Rev. E 50, (1994) R4298{301.[12] G. Haller and S. Wiggins, Physica D , (1996) in press.[13] E. Todesco and M. Gemmi, submitted to Phys. Rev. E[14] W. Scandale, F. Schmidt, E. Todesco, Part. Accel. 35, (1991)53{88.[15] M. Giovannozzi, R. Grassi, W. Scandale, E. Todesco, Phys. Rev.E 52, (1995) 3093{101.[16] A. Bazzani, M. Giovannozzi, E. Todesco, Comput. Phys. Comm.86, (1995) 199{207.[17] E. Todesco, M. Gemmi and M. Giovannozzi, Comput. Phys.Comm. , (1996) in press.[18] J. Laskar, C. Froeschl�e and A. Celletti, Physica D 56, (1992)253{69.[19] J. Laskar, Physica D 67, (1992) 257{81.[20] R. Bartolini, A. Bazzani, M. Giovannozzi, W. Scandale and E.Todesco, Part. Accel. 52, (1996) 147.[21] M. Berz, Part. Accel. 24, (1989) 109.[22] E. Courant and H. Snyder, Ann. Phys. 3, (1958) 1{48.14

Figure 2: Resonance network of the 4D H�enon map at Qx = 0:28; Qy = 0:31via frequency analysis. Initial conditions satisfying a resonance condition upto order 15 are shown. Horizontal and vertical linear invariants are in the xand y axes, respectively.15

Figure 3: Resonance network of the 4D H�enon map at Qx = 0:28; Qy = 0:31via resonant normal forms. Resonances up to order 9 are evaluated. Hori-zontal and vertical nonlinear invariants are in the x and y axes, respectively.16

Figure 4: Initial conditions that are locked on the resonance (3;�6) forthe 4D H�enon map at Qx = 0:28; Qy = 0:31. Numerical results (dots) arecompared with the analytical estimate of the resonance width (solid lines)obtained using resonant normal forms at order 10. Horizontal and verticalinvariants are plotted in the x and y axes, respectively.17

Figure 5: Distribution of the dynamic apertures for the random LHC model,100 seeds (a). Correlation of the quality factors Q1 [norm of the nonlinearpart of the map] (b), Q2 [tuneshift] (c), Q3(3; 0) [strength of the resonance(3; 0)] (d) with the dynamic aperture for the random machine, 100 seeds.18