RFQ BEAM COOLER INJECTION SIMULATION

Transcription

1 Introduction RFQ BEAM COOLER INJECTION SIMULATION R.B. Moore June 16, 003 Simulating the behaviour of a beam of particles entering a Radiofrequency Quadrupole (RFQ) beam cooler can be a frustrating experience. This is not only because of the large number of parameters involved in the system design but also because of the large sample of particles required to cover the range of behaviour that they will experience. As in any beam system, there should be enough particles to represent the action diagram in one transverse displacementmomentum plane, for at least the initial trial designs. More detailed investigation of a trial design might even require an investigation of the particle behaviour in the other two displacementmomentum planes. In the case of beam injection into an RFQ system this complexity is compounded by the fact that particles entering the geometry at different RF phases will experience very different fields. The approach taken here, an approach that is commonly taken in high energy particle transport design, is to start with a linear approximation to the particle dynamics. This allows the simulation of particle dynamics to be carried out by simple two-dimensional matrix manipulation of not only the action points of individual particles but also of elliptical action diagrams representing whole collections of particles. With modern computers this procedure can lead to system evaluation in seconds compared to hours for a higher order calculation. When in a contemplative mood I think of this approach as following Occam s razor; One should not increase, beyond what is necessary, the number of entities required to explain anything. When contemplating something as prosaic as an RFQ beam cooler I think of it as If it doesn t work in first order it ain t going to work in higher order. (It is only when actually testing the system with higher order calculations that one typically encounters the relevant corollary of Murphy s Law; The fact that it works in first order doesn t mean that it will work in higher order.) This description of a linear approach to RFQ beam cooler design centers on a spreadsheet calculation, written for Excel. The procedure for using this spreadsheet is outlined below, with the theory behind its creation being reserved for those who actually want to understand it or, more likely, those who want to do the writer the service of picking holes in it. The Procedure Importing Field Maps The first requirement of the user is to import electric field maps of the components of the cooler system. These field maps have to be created in a separate application. All that is required for the linear approach to charged particle dynamics are the axial field, and the gradient of the transverse field on the system axis. For the present spread-sheet the maps were created in Simion, which can export the electric field of an electrode configuration at rectangular grid points. These were then imported as columns of electric field and electric field gradients along the axis for columnar values of axial grid step. The present spread sheet is set up to accommodate 10 sets of field maps, each set comprising a column of DC fields and a column of transverse field gradients. In setting up these filed maps for calculations these column must be filled, most conveniently by pasting from another spreadsheet or application.

2 Beam Cooler Injection Simulations As an example, the present spread-sheet is for an electrode system shown schematically in fig. 1. In this system there is one field map for the decelerator, which consists of hyperboloidal electrodes that produce an approximately uniform gradient in the decelerating electric field. There are then three maps for three segments of the RFQ confinement system, each segment having the same RFQ field but with independently adjustable DC potentials. Fig. 1 The electrodes of a system for decelerating and injecting a high voltage DC beam into an RFQ confinement region. The particles enter the decelerator through a hole in the ground electrode, this electrode being a hyperboloid of revolution. The main decelerating electrode is a ring, shown to be at V in the figure, that is also a hyperboloid of revolution. This ring is configured so as to produce a equipotential cone at V, with its apex at z O from the center of the ground electrode and consequently a purely quadrupole field in the deceleration region. The third deceleration electrode is a cone that follows this equipotential surface, but with a hole to allow the particles to enter the RFQ confinement region, which is inside the cone. In the configuration shown in the figure the RFQ confinement region has three segments set at different DC ptotentials, thereby allowing the axial energy of the particles to be modified after their entry into the system. For this example spreadsheet the 3 separate sequential sets of quadrupole electrodes, shown in fig. 1 and in more detail in fig., are all calculated with the same grid spacing and superimposed on one grid axis extending over 163 points, counting zero. Here the imported

3 Beam Cooler Injection Simulations 3 fields are the axial electric fields (in Volts per step) and the transverse field gradients (in Volts per step ). Fig. The configuration of the RFQ electrodes Not all the field maps need be filled so the spreadsheet requires that the number that are actually used be entered in the cell Number of maps. Then for each map that is entered the following information must be entered in the appropriate cells: A. The number of rows in the map. B. The number of steps in a key dimension of the map ( Key Dim. steps ) This could be simply the number of steps in the field map, i.e. the number of rows as above, but it could also be an easily recognized feature of the electrode configuration such as the distance of a set of quadrupole elements from the axis (i.e. r o ). C. The potential on a key electrode for which the axial DC field map was calculated ( Calc. DC Pot ). If there is only one activated electrode then the potential is, of course, the potential of that electrode. However, if there is more than one electrode for the calculation, and which will therefore always have the same potential ratios, then the most easily recognized electrode could be selected. D. The potential on a key electrode for which the transverse field gradient map was calculated ( Calc. RF Pot ). Here there will always be more that one electrode and so the most easily recognized electrode should be selected. E. A flag to indicate that there is a transverse field gradient map ( Quad. Field? ). If this cell contains zero there is none. In addition the user must provide the information required to set the geometrical and electrical scales of the maps. These are most conveniently entered in the appropriate cells as formulae referring to more easily visualized parameters in the main header of the spreadsheet. The required entries are:

4 Beam Cooler Injection Simulations 4 F. Key Dim. mm - The length in mm of the key dimension entered as steps in B. In the present spread sheet the entry for field maps, 3 and 4 are the r o (in mm) of the electrode configurations. This is set by the user in cell ro (mm) as part of any trial calculation. This entry is given the variable name ro which has been entered in the formulae for Key Dim. mm of each of the maps. The spreadsheet then calculates the scale factor by dividing this Key Dim. mm by Key Dim. steps. G. zstart The z coordinate at which the field map starts. This can be set in the appropriate cell above the field map but is often most conveniently set from information entered in cells in the main header of the spreadsheet. In the present spreadsheet it is set to zero for the first field map (the decelerator) and for the nd, 3 rd and 4 th as a formula relating it to the entry into the cell Quad map start in the main header. The spreadsheet then calculates zend using the number of steps in the field map and the scale factor. H. Set DC Pot. The potential on the key electrode for which the axial DC field map was calculated. As an example, in the present spreadsheet for map 1 this potential is entered as being equal to the entry in the cell Decel. Pot in the main header. I. Set RF Pot. The potential on the key electrode for which the transverse field gradient map was calculated. As an example, in the present spreadsheet for map this potential is entered as a formula linking it to the entry in the cell RF ampl in the main header. Note that the units used in the spread-sheet are millimeters, Volts and microsconds. For a short guide to these units, and a sermon about their overwhelming usefulness, see the Appendix. Also note that for the example spreadsheet the same quadrupole field is applied to each electrode set, so these have already been compiled and entered in the nd transverse field map. The other quadrupole electrode sets are then set to have no have transverse field. The calculation of the electric field maps and their insertion into the spread-sheet are by far the most laborious part of using the spread-sheet. This is particularly true if one has to introduce new electrodes. This is a usual feature of particle manipulation design schemes, where a change in the geometry of a system is the highest iteration to be performed. One therefore tries to do as much as possible with a particular geometry before altering it. The purpose of this spread-sheet based on first-order beam optics is to bring into balance the time required to exhaust the possibilities of a particular design through lower level iterations with the time required to introduce a new iteration requiring a shape change of the system. Compiling the Field Maps Once the field maps and their associated parameters are entered in the spreadsheet macros must be activated to calculate the spline coefficients needed for the interpolations required for assembling the fields into one overall field map. The first of these is the macro FillFieldsArray, which enters the spline coefficients in the field map arrays. This is activated by pressing Cntl-f (or Option+Command+f on a Macintosh) and must be run following any change in a field map and any change in the geometric scale and/or the relative placement of the filed maps. This operation should take less than a second on a modern personal computer. The second macro needed for the field compilation is FillCombFieldArray, activated by pressing Cntl-c (or Option+Command+c on a Macintosh). This takes the field map data, together with the set potentials, and uses spline interpolations to calculate the combined field. It enters

5 Beam Cooler Injection Simulations 5 these field values, together with the spline coefficients required for their interpolation, in the array CombFieldArray. This field map is in 500 uniform axial steps over its entire extent. This macro must be run whenever a set potential is changed, or whenever macro f is run. The Runge-Kutta Integration The use of matrix algebra for the linear approximation of beam dynamics requires accurate knowledge of the axial coordinate of the particle collection at particular times. For the large range of energies involved in decelerating a particle beam from typically 60 kev to entry into RFQ confinement at the order of 10 ev, the required accuracy can only be obtained from higher order calculations. In the present spreadsheet the necessary accuracy is acquired through a 5 th order Runge-Kuttas integration of the equations of motion of a particle traveling along the z axis of the system, with spline interpolation of the axial electric fields that govern this motion. However, because of the errors that accumulate in the potentials at grid points in any field calculation procedure (in the words of Numerical Recipes, resulting in a rocky terrain ), even with these measures errors of the order of 10 ev compared to conservation of energy have been observed when the particle energy has been reduced from 60 kev to this range. For this reason the Runge- Kutta integration as applied here overwrites the value of the particle velocity calculated at the end of each step of the integration with the value calculated from conservation of energy and the electric potential of the position at the end of that step. One of reasons for using a spline interpolation is that the time step in the Runge-Kutta integration is determined by the requirement that it be a sub-multiple of the RF period. For acceptable speed in the subsequent matrix procedure for calculating the effects of the RF quadrupole field, the time step has been chosen to be 1/0 th of the RF period. (See matrix manipulations below.) To make the output of the Runge-Kutta directly usable for the matrix manipulations, this is also chosen to be the time-step for the Runge-Kutta integration. The necessity for using a spline interpolation for such time steps in the integration can then be seen by observing a typical boulder, shown in fig. 3, that arises in a deceleration field Fig 3. The electric field at the entrance into the quadrupole confinement region. (This bump occurs because of the hole that is necessary in the conical electrode to allow the particles to pass through. This absence of the defining equipotential surface allows the hyperboloidal ring electrode to raise the potential at the apex of the cone.)

6 Beam Cooler Injection Simulations 6 In this figure the squares represent the values of the axial field at the axial points represented in the z column of the combined field data that is used in the integration. The curve drawn through them is the spline interpolation as used in the Runge-Kutta integration. It is seen that a linear interpolation from point to point would cause noticeable error in the value of the electric field that the Runge-Kutta would be using., and this would be at a very critical point in the manipulation of the particle beam through the hole. In principle, the necessity of a spline interpolation could be eliminated by using a much shorter time interval for the Runge-Kutta than that used for the matrix manipulations. The results could then be made directly usable for the matrix manipulations by having the time-interval of the Runge-Kutta a sub-multiple of that used for the matrix manipulations and only outputting to the RK array data at the matrix intervals. However, this would involve considerably more programming of the Runge-Kuttta algorithm and/or would considerably slow the Runge-Kutta process itself, with no appreciable gain in accuracy over using the spline interpolation. The particle parameters required for the Runge-Kutta integration are entered in the cells under Incoming Beam. Specifically, these are the ion mass in atomic mass units ( Ion mass (amu) and the beam energy in electron volts ( Energy (ev) ). The Runge-Kuttta integration is then done by the macro FillRKArray, activated by pressing Cntl k (or Option+Command+k on a Macintosh). This subroutine fills RK_Array with the values of time ( t ), z, ( z ), z- velocity, ( zdot ), particle axial energy ( KE ) and the axial electric field ( Ez ) at these z values. For reference it also fills in the estimates of the errors in z and z associated with each step that are a by-product of the 5 th order Runge-Kutta algorithm. (See Numerical Recipes.) The FillRKArray macro also calculates the values needed for the matrix manipulations, specifically the axial derivative of the axial field (column dez/dz ), the transverse field gradient on the axis (column drr/dr ) and the axial gradient of this transverse field gradient (column drr/drdz ). The beam parameters needed for the first-order matrix calculations of the beam envelope are its emittance (cell ξ (π mm-mrad) ), diameter (cell Diameter (mm) ) and angular spread (cell Angular spread (mrad) ). (A negative angular spread indicates a beam converging onto the entrance of the field map). The spread-sheet is now ready for the first-order calculation of the beam profile through the system and an estimate of axial energy effects. The Matrix First-order Calculation The first-order matrix calculation of the beam profile is carried out by the macro subroutine FillResultArray. This can be activated by pressing Cntl-b (or Option+Command+b on a Macintosh). It uses -dimensional matrices to calculate the transformation of this ellipse parameters at each time step of the Runge-Kutta output. The ellipse parameters that are actually transformed are the so-called Twiss Parameters see Theory section below. An estimate of the effects on the axial energy of the axial gradient of the transverse field gradient is obtained by running the macro FilldEArray also see Theory section. This can be activated by pressing Cntl-e (or Option+Command+e on a Macintosh). A spectrum of this estimated energy deviation is obtained from the macro filldndearray that can be run by pressing Cntl-n (or Option+Command+n on a Macintosh). (This macro is run automatically as a sub macro in e but the option of running it separately on previously compiled data from e allows different scales of the spectrum to be obtained by minor modifications of the code of n.) The percentage of the beam estimated to be in the displayed spectrum is shown in the cell % of beam

7 Beam Cooler Injection Simulations 7 Theory Action Diagrams The power of action diagrams in analyzing the motions of a particle collection seems to have been first realized by Poincaré at the turn of the 0 th century, and such diagrams are sometimes referred to as Poincaré sections. Their power derives from the fact that they are projections of the momentum-displacement coordinate pairs for each degree of freedom of the particle motions. Thus to first order, where the motion in each degree of freedom is independent of the motion in the others, the preservation of the particle density in phase space, and hence the preservation of the phase space volume of a collection (Liouville s Theorem), leads to preservation of the density and the area of the action diagrams. (This property of particle collections in conservative systems is sometimes expressed as the incompressibility of particle collections in phase space.) Since the electric field of a quadrupole is linearly related to particle displacement in that field, the forces have always first-order dependence on displacement and so the action diagrams of a particle collection retain their local density and overall area. The simplicity of the action diagram is most apparent for the case of a particle collection all undergoing simple harmonic motion in the same force field and so at the same oscillating frequency, but at different amplitudes and phases. Each particle then follows an elliptical trajectory of the same ellipticity as the rest and at the same orbital frequency, thereby retaining its relative position to all the rest (fig. 4). The perimeter of the diagram is just the trajectory of the most energetic particle in the collection. Fig. 4 The action diagram for particles undergoing simple harmonic motion in a force field. The diagram on the left is for a single particle. The diagram on the right is for a collection of many particles all in the same force field and therefore having the same oscillation frequency but with different amplitudes and phases. The significance of the action diagram is enhanced when the collection of particles is in thermodynamic equilibrium. The density of particles at a particular point is phase space is then given by the expression d 6 E n ds = n o e kt (1) where n o is the phase space density at the center of the distribution, E is the energy of a particle at the particular point in phase space under consideration, k is Boltzman s constant and T is the

8 Beam Cooler Injection Simulations 8 temperature of the collection. For simple harmonic motion, where the energy is divided between the potential energy of displacement and the kinetic energy of the momentum, the projection of this density distribution into an action diagram results in a gaussian density distribution within that diagram given by d n dxdp x = n A o e x σ + p x x σ px where n A o is the density distribution at the center of the action diagram and the standard deviations of the distribution are σ x = 1 ω () kt m, σ p x = mkt. (3) The action diagram for the distribution will then be elliptically symmetrical in that points anywhere on the ellipse representing a particular amplitude of oscillation will have a uniform density. Matrix Algebra of Action Diagrams The transformation of a point in an action diagram as the diagram evolves under linear transformations is described by the -dimensional matrix M x = M x, M = m 11 m 1 p x p x m 1 m. (4) A consequence of the linearity of the transformation is that the determinant of this matrix is unity. Also, the elements of the inverse transfer matrix x = M 1 x can be easily determined from the simple requirement p x M 1 M = p x to be M 1 = m m 1 m 1 m 11 where the elements m 11, m 1, m 1 and m are those of the forward transform. In the case of an ion in an axisymmetric quadrupole field with axis of symmetry along the z axis the electric potential has the form This potential has a radial electric field φ = a z 1 r (5) (6). (7) E r = a r. (8) which results, of course, in a radial oscillation of frequency ω = ea m (9)

9 Beam Cooler Injection Simulations 9 where e is the charge and m is the mass of the ion. The radial displacement and radial momentum are then r = A sin(ωt + φ), p r = mωacos(ωt + φ). (10) In the case of the quadrupole being negative, resulting in a radial field that is away from the z axis, the motion becomes r = A sinh(ωt + φ), p r = mωacosh(ωt + φ) (11) where now the magnitude of the quadrupole field is used in evaluating ω. These solutions result in the transformation of the displacement-momentum coordinates during an interval t being expressed by the matrices cos(ωt) M + = mω sin(ωt) 1 mω sin(ωt) cosh(ωt), M = cos(ωt) mω sinh(ωt) 1 mω sinh(ωt) (1) cosh(ωt) For these matrices the determinants are easily seen to be unity and the inverse matrices, describing a transformation in negative t, are easily seen to follow eqn. (6). For a field that is not purely quadrupolar the linear approach can be used by taking small steps through the system. In the linear approach the field over a small step can be taken as quadrupolar. This means that if there is an axial field gradient in this step then there is also a transverse field gradient E r r = 1 E z z. (13) The elemental transformation matrice for a step that takes time dt is then 1 cos(ωdt) dm + = mω sin(ωdt) (14) mω sin(ωdt) cos(ωdt) when the axial field gradient is positive and cosh(ωdt) dm = mω sinh(ωdt) 1 mω sinh(ωdt) (15) cosh(ωdt) when the axial field gradient is positive, and where ω = e E z m z. (16) Elliptical action diagrams are of particular significance in linear transformations since they remain ellipses of the same area. Linear transformations of ellipses are most conveniently expressed in terms of the Twiss parameters A, B, C and ε, by which the equation for an ellipse, in terms of the coordinates of its points relative to its center, has the general form For an action diagram it has the specific form Cx + Axy + By =ε. (17) Cx + Axp x + Bp x =ε. (18)

10 Beam Cooler Injection Simulations 10 The parameter ε determines the overall size of the ellipse. Specifically, it is the product of the semi-axes, or Area of ellipse /π Thus, defining the action as the area of the action ellipse, it is Action /π The parameters B and C express the ellipticity of the ellipse. The parameter A expresses the inclination of the ellipse axis with the axis of the coordinate system and is zero when the ellipse is a right ellipse. Because it takes only 3 parameters to specify any ellipse there is a necessary relationship between the Twiss parameters. It is BC A =1 (19) The relationship of the Twiss parameters to the cardinal points of an ellipse is shown in fig. 5. Fig. 5 The relationship of the Twiss parameters of an ellipse to the various cardinal points of the ellipse and its orientation with the coordinate axes. As an example, the Twiss parameters for the action ellipse of simple harmonic motion are A = 0, B = 1 mω, C = mω, ε = mωx max. (17) Because a linear transformation preserves the area of an ellipse, only the parameters A, B and C are transformed. The 3 3 matrix that describes this transformation, and its inverse, for an action point transformation described by the matrix of (4) are

11 Beam Cooler Injection Simulations 11 B m 11 m 11 m 1 m 1 B A = m 11 m 1 m 11 m + m 1 m 1 m 1 m A C m 1 m 1 m m C B m m 1 m m 1 B A = m 1 m m 11 m + m 1 m 1 m 11 m 1 A C m 1 m 11 m 1 m. (19) 11 C If the electric field is static then the envelope of a beam traveling along the axis of the system can be obtained by determining the step by step transformation of the B action parameter and using the relationship shown in fig. 5: Action Diagrams of RFQ Confinement (18) r max = εb. (0) The case of the oscillating quadrupole field that provided radial confinement is more complicated. Here the frequency ω is itself varying and so, strictly speaking, the transformation matrices (14,15) are only valid for infinitesimal time intervals. The transfer matrix for a finite time t is then the product sum M =Π t dm, (1) In principle this product sum should be evaluated over infinitesimal time steps, or an infinite set of dm. In practice, for quadrupole field strengths that are oscillating sinusoidally, time steps of 1 degree of oscillation will give accuracies of several parts in 10 5 and steps of 5 degrees will give accuracies of about 1 part in Thus, even in the linear approach sufficient accuracy is achieved only with many steps per field oscillation. To circumvent this problem the combination of many small steps into larger steps can be carried out beforehand and the results compiled for interpolation during the actual beam profile calculations. This is facilitated by describing the sinusoidal variation of the field in terms of the angle of the variation rather than the time; de r dr = de r sin θ dr RF max whereupon the equation of motion becomes d r dθ RF = e mω RF ( ). () de r sin θ dr RF max ( ) r. (3) Using the dimensionless Mathieu parameter (3) takes the simple form q = e mω RF de r. (4) dr max d r dθ RF = q sin ( θ RF) r. (5)

12 Beam Cooler Injection Simulations 1 The elemental transfer matrix for a small step in θ RF is then cosh( q 1 sin(θ RF ) 1 dθrf ) sinh( q 1 sin(θ RF ) q dm + = sin(θ RF ) 1 1 q sin(θ RF ) q sinh( sin(θ RF ) dθrf ) cosh( q 1 sin(θ RF ) dθrf ) 1 dθrf ) (6) for 0 θ RF π, and cos( q 1 sin(θ RF ) 1 dθrf ) sin( q 1 sin(θ RF ) q dm = sin(θ RF ) q sin(θ RF ) q sin( sin(θ RF ) dθrf ) cos( q sin(θ RF ) dθrf ) 1 dθrf ) (7 for π θ RF π These elemental transfer matrices can then be combined to yield the transfer matrices for larger steps. For the beam envelope calculations this has been done in a separate spread-sheet ( RF_Matr_fract_cycle.xls ) for values of q in steps of 0.1 from 0 to 3.0 and for steps in θ RF of 18 o starting at 0 and ending at 360 o. These results were pasted into the beam envelope spreadsheet as arrays M11ARRAY, M1ARRAY, M1ARRAY and MARRAY. This allows an evaluation of the matrix elements for an 18 o step through the RFQ confinement regions by determining the average q during that step and using interpolation to find the θ RF matrix elements for that step. The matrix elements for the transformation according to the equation of motion in time can then be obtained from m 11 = m 11θ m 1 = m 1 θ mω RF ; m = m θ Combining the Axial Gradient and the RFQ Transformations. (8) ; m 1 = ( mω RF )m 1θ The axial field transformation (14,15) and the transverse field transformation derived from (6,7) both include the effect of the axial drift during a step. In combining these transformations one then has to unwind the effect of one of these drifts. For example, if the axial field transformation is taken to be the first then the simple drift part of its transformation must be undone before the transverse matrix is applied, as in (9). M Step = m 11 trans m 1trans m 1 trans m trans 1 t m 0 1 m11 axial m 1axial m 1 axial m axial (9)

13 Beam Cooler Injection Simulations 13 Axial Energy Effects of the RFQ Fringe Field Region One of the concerns when injecting a beam of ions into an RFQ confinement region is the effect of the fringing field. This is because of the unavoidable axial component of this field, shown schematically in fig. 6. Fig. 6 A schematic view of the field lines at the entrance to an RFQ field. An estimate of this effect can be gained by considering the principal multipole of the field that is associated with it. Of the complete multipole set V = a lm e imφ R l m P l cosθ (30) l,m this is the multipole with m =, l = 3. Expanding the Legendre polynomial of this multipole gives V 3 =15a 3 e iφ zr (31) Tasking the maximum of this function (at φ = 0) the coefficient a 3 is given by a 3 = 1 30 E r z r r=0 ; (E r = V 3 r r=0 ) (3) whereupon the axial component of the multipole can be obtained from E z = V 3 z = 15a 3 r = 1 r E r z r r=0 (33) Since the energy change due to this field component depends on the distance of the particle from the axis the overall spread in energy caused by the field can only be determined by sampling a range of initial action points, each point being sampled over a range of initial phases for the RFQ field. In the present spread sheet the action samples are taken every 10 degrees, from 5 to 85 o along the perimeter of the initial beam emittance diagram. The RF phases are the same 18 o steps as used in the calculation of the beam envelopes. This calculation takes about 10 times as long as the beam envelope calculation so that, in practice, it is only run once a seemingly appropriate beam envelope has been achieved from the

14 Beam Cooler Injection Simulations 14 beam envelope calculation. Since the effect on the energy is proportional to the square of the distance of the particle from the axis it is important to get as small a beam diameter as possible in the RFQ entrance region. This effect on the axial energy will, of course, render the RK integration based on the axial values of the field inaccurate. Therefore. if the energy calculations indicate an unacceptable effect the RK calculations on which they were based cannot be trusted and so the iterations must be repeated until the energy spread is indeed acceptable. An estimate of the spectrum of the energy spread due to the axial gradient of the transverse field gradient can be obtained by assuming that the density of the initial action diagrams of the incoming beam corresponds to thermal equilibrium. If the emmittance used in the calculation encompasses 90% of the total thermally equilibriated beam (a common practice in beam optics ) then the action density will be dn da = dn da o e.3 r r o where the radius parameter r is the distance of the action point from the action diagram center when the momentum coordinate is given the same dimensions as the displacement coordinate (i.e. the ellipse has been scaled into a circle) and r o is the radius of the emittance ellipse. For a pie shaped slice of angle θ of this emittance diagram the density of particles as a function of radius will be dn dr = dn da o r θ e.3 r r o (34). (35) For any given initial action point, the energy deviation caused by the fringe field of the quadrupole will be proportional to the square of the initial r parameter of the action. For the parameter value r o let the energy deviation be designated E o. The spectrum of the energy deviation for the particles contained within the slice then becomes E dn de = N e.3 E o (36) E o where N is the number of particles in the slice represented by the segment at the perimeter of the emittance ellipse. For thermal equilibrium and uniform slice angles this number is the same for all segments around the perimeter. Also, the density of the initial action diagram will not depend on the phase of the RF at which it enters the system. The total spectrum of all the incoming particles can then be obtained by summing dn de Total E N = n e.3 E o (37) E o where the sum is taken over all the initial action points and all the RF phase for each action point. Hence n is the product of the number of action points and the number of phase samples. However, this method of sampling the initial action points assumes that the initial action ellipse is a right ellipse that can be rendered into a circle by a simple scale change of its displacement and momentum coordinates. The actual incoming beam will in general have an ellipse that is inclined to the displacement-momentum axis. The application of this method of

15 Beam Cooler Injection Simulations 15 sampling action points therefore requires that the right ellipse from which they are sampled be transformed into the actual emittance ellipse of the incoming beam. The right ellipse that is taken for the initial azimuthally uniform sampling can be taken to be the actual beam emittance ellipse simply rotated to be a right ellipse. Taking the Twiss parameters of this right ellipse to be B and C (A = 0) then the transformation of that right ellipse to the actual entrance ellipse specified by A o, B o and C o becomes B o A o C o m 11 m 11 m 1 m 1 = B m 11 m 1 m 11 m + m 1 m 1 m 1 m 0 m 1 m 1 m m C. (38) resulting in B o = m 11 B + m 1 C. (39) Given that the elements of the transfer matrix that results in a simple rotation are simply m 11 = m = cosθ m 1 = m 1 = sinθ (40) where θ is the angle of rotation. This results in m 11 = m = B C o B C m 1 = m 1 = 1 m 11 1 (41) The Twiss parameter B and C can be obtained from the semiaxis of the actual emittance ellipse (see fig. 5) that will become the semi-axis of the right ellipse (as p xmax ). The result is ( ) C = 1 B + C + (B + C ) o o o o 4 B = 1 C The semiaxis of the right emiitance ellipse from which the action samples are to be taken are then and the coordinates of the action sample points become (4) p x max = ε C ; x max = ε B (43) p x i = p xmax sinθ i ; x i = x max cosθ i (44) These sample points are then transformed to the action at beam entrance according to p x x Entrance = m 11 m 1 p x m 1 m x Sample (45)

16 Beam Cooler Injection Simulations 16 Tests The accuracy of the present spread-sheet calculations can only be adequately tested by comparing its results with those of an accurate ray-tracing program for particles traversing the same fields. However, tests of the matrix components in tables M11ARRAY etc. indicate an accuracy of about 1 ppm, with the accuracy of values interpolated from these tables estimated to be never worse than 10 ppm. When applied in the sequence of steps required for the beam envelope calculation it is expected that the final beam envelope should be accurate to of the order of 1%. Gross errors have been checked by following the trace of a single particle and comparing it with what would be expected from general considerations. The axial transformation part of the calculation was checked by the behaviour of a particle starting at maximum displacement and no divergence upon entrance to the decelerator. In the predominately quadrupole field of the decelerator the action point of this particle should trace out a right ellipse, the semiaxes being x max = x initial p x max = mωx max = em de z dz. (46) For a particle of 100 amu entering parallel to the z axis and 3.5 mm from it, a pure quadrupole field that is 60000V at a distance z = 50 mm from its center of zero potential at z = 0, results in x max = 3.5 mm. (47) =17.5 ev -µs/mm p x max The action trace produced by the spread-sheet is shown in fig. 7 Fig. 7 The action trace of an ion of mass 100 amu entering the almost pure quadrupole field of the decelerator modeled in the spread sheet calculations. The trace is seen to be a right ellipse with the expected semiaxes. The slight flattening at the bottom and the subsequent turn-up near the end of the trajectory is due to the particle approaching the entrance hole of the RFQ section and then entering the field-free region inside. (The RFQ field was set to zero for this calculation.)

17 Beam Cooler Injection Simulations 17 The RFQ calculations were checked by tracing the excursion of a single particle once it had entered the RFQ region. A representative trace is shown in fig. 8 Fig. 8. The trace of a representative particle entering the RFQ region. The full quadrupole field is reached at about 1.5 µs. From the values used for this plot, the period of the macrooscillation in the RFQ confinement region is 1.78 µs, for a frequency of 0.56 MHz. For this value of q the first-order approximation of the macro-oscillation frequency. i.e. f Macromoton = q f RF gives this frequency to be MHz, which agrees with the calculated results to within the accuracy with which the macro-oscillation frequency can be extracted from these results.

18 Beam Cooler Injection Simulations 18 APPENDIX A - SPREAD SHEET GLOSSARY Names used in the spread sheet Name Spread-sheet label Description User modified or internal calc. Ao Ao Emittance Twiss parameter A at start Internal Bo Bo Emittance Twiss parameter B at start Internal Centerz Centerz The z coordinate of the dec quad center Internal Co Co Emittance Twiss parameter C at start Internal CombFieldArray Interpolated- 6 column array of fields used in calc. Array DCpot1-10 Calc. DC Pot. Potentials used in calculating fieldsuser dearray Energy dev- 0 column array of energy deviations Array DecelPot Decel-- Set potential of decelerator User dednarray Energy Spec- column array of energy spectrumarray deltaepercent % of beam - % of beam in displayed E spect. Internal deltat t (µs) Time interval used in RK calculations Internal deltatheta Angular spread Beam divergence at start (mrad) User Diameter Diameter (mm) Beam diameter at start User El1_DC Electrode 1-DC Set DC on first quad User El_DC Electrode -DC Set DC on nd quad El3_DC Electrode 3-DC Set DC on rd 3quad User User Emittance ξ (π-mm-mrad) Beam emittance User Energy Energy (ev) Particle kinetic energy at start User Entrancez Entrancez Z coordinate at entrance to decel. region Internal epsilon epsilon Emittance Twiss parameter ε at start Internal FieldMapArray ELECTRODE - 50 column field map array (5 per map) Array freq MHz Frequency of RF User Ion_mass_amu Ion mass (amu) Particle mass in amu User Ion_mass_imu Ion mass (imu) Particle mass in ion mass units Internal keydim1-10 Key Dim. -StepsKey dimension for scale of maps User keydim1-10_mm Key Dim. mm Set key dimension for scale User M11ARRAY M11ARRAY Tabulated m11 RFQ matrix elementsarray M1ARRAY M1ARRAY Tabulated m1 RFQ matrix elementsarray M1ARRAY M1ARRAY Tabulated m1 RFQ matrix elementsarray MARRAY MARRAY Tabulated m RFQ matrix elementsarray MapDataArray Map Data Ar- Collected array of field map param.array momega momega Product of mass and RF radian freqinternal momegasqr momegasqr Product of mass and RF rad. freq. Internal sqr. Nelectrodes Number of mapsnumber of field maps Entered with map nrows1-10 No of rows Number of rows in field maps Entered with map Nsteps No of Plot stepsnumber of steps in the RK calculation User omega omega Angular frequency of the RF Internal q q Mathieu parameter for the RFQ fieldinternal Quad._Field1-10?Quad. Field? If there is a quad field, 1 else 0 User ResultArray r boundary - 41 column array of beam envelopesarray RF_ampl RF ampl Amplitude of RF across adjacent elect. User

19 Beam Cooler Injection Simulations 19 RFpot1-10 Calc. RF Pot. RF potential for calculating maps Entered with map RK_Array RK_Array 10 column array of Runge-Kutta calc. Array ro ro (mm) The distance of the RF electr. from User axis scale1-10 mm/step) The scales of the field maps Internal SetDCPot1-10 Set DC Pot. DC potentials used in beam calculations User SetRFpot1-10 Set RF Pot. RF potentials used in beam calculations User zconfstart Quad map start z coor. at start of the quad field map User zend1-10 zend z coordinates at end of field maps Internal zo Decel. zo Actual zo parameter used in calculation User zstart1-10 zstart z coordinates at start of field mapsuser User defined functions in the spread sheet Interpolate(xlow, xhigh, ylow, yhigh, x) A linear interpolation routine used to calculate the electric potentials required for the velocity determinations in the subroutine FillRKArray Spline(xlow, xhigh, ylow, yhigh, ylow, yhigh, x) A spline interpolation routine used to calculate electric fields. Macros used in the spread sheet FillFieldsArray() - Activated by Ctrl-f (Option+Command+f on the MacIntosh) A macro to compute the column of spline derivatives needed to interpolate the electric field maps for the beam calculations. Need to be activated when importing a new field map or changing the geometrical scales and/or field map starting points. FillCombFieldArray() - Activated by Ctrl-c (Option+Command+c on the MacIntosh) A macro to compute the combined field of the field maps, taking into account the user set potentials. Needs to be activated when an electric potential is changed (or f is run). FillRKArray() - Activated by Ctrl-k (Option+Command+k on the MacIntosh) A macro to carry out the Runge-Kutta integrations for the particle position on the axis vs time. It uses time steps set to be 1/0 of an RF cycle. Needs to be activated when particle energy or mass is changed, or c is run). FillResultArray() - Activated by Ctrl-b (Option+Command+b on the MacIntosh) A macro to compute the beam envelopes at the RK step points. Needs to be activated when beam emittance, diameter or divergence has been changed (or k has been run). FilldEArray()- Activated by Ctrl-e (Option+Command+e on the MacIntosh) A macro to compute the estimate of the axial energy spread. Needs to be run whenever b has been run FilldNdEArray()- Activated by Ctrl-n (Option+Command+n on the MacIntosh) The macro used within FilldEArray to compute the spectrum of the axial energy spread. Can be run separately if a different energy scale is desired in the spectrum disply. (Would require editing the macro.)

20 Beam Cooler Injection Simulations 0 APPENDIX B - RFQ ACTION DIAGRAMS Introduction The motion of charged particles under RadioFrequency Quadrupole (RFQ) confinement can seem confusing. This is because of the combination of the driven oscillation of the oscillating electric field, which produces an oscillation in which the displacement is in antiphase to the force oscillation of the quadrupole, and the underlying simple harmonic motion from the average restoring force that the oscillating field produces. The result, for a single particle in a typical oscillating quadrupole field used for confinement is shown in fig. A1. Fig. A1 A typical motion of an ion in one dimension under RFQ confinement. Although this motion may appear relatively simple for a single ion in one dimension of its motion, the motion in space, even for the -dimensional radial confinement of an ion guide, can result in a tortuous path because of the lack of any coherent phase relationship between the driven oscillation and the independent simple harmonic motions in the spatial coordinates x and y. A typical trajectory for a single ion under RFQ radial confinement is shown in fig. A. Fig. A A typical trajectory of an ion under -dimensional RFQ confinement.

21 Beam Cooler Injection Simulations 1 If there is more than one ion under confinement than the motions form an even more tangled web (fig. A3). Fig. A3 A typical motion of two ions under -dimensional RFQ confinement. All that is easily discerned about the trajectories of a large collection of ions is that they fill a square of rounded corners, the diagonals of the square being oriented along the directions of the maximum quadrupole electric potentials (i.e. x and y in fig. A3). Yet, this confusing collections of motions must be dealt with analytically if RFQ confinement is to be designed for particular applications, such as the preparation of ion collections for delivery to other apparatus for further study. An example is the use of RFQ confinement while ion collections are being cooled by buffer gas collisions for delivery to a high-accuracy mass spectrometer, where the cooling of the ions is a necessary prerequisite for the high-accuracy. An appropriate analysis of the ion motions is obtained by considering the action diagrams, i.e. momentum-displacement, of the motions. Action Diagrams of RFQ Confinement Since the driven RF oscillation of particles under RFQ confinement is superimposed on their simple harmonic motions the action diagrams of a collection of particles will be that of simple harmonic motion to which the RF motion is added. However, unlike the simple harmonic motion, the RF motion will not be at arbitrary phase but will always be in antiphase to the electric field. The particles will therefore not fill the ellipse of the most energetic oscillation, as in their simple harmonic motions, but rather will form a line which rotates within that ellipse, one complete turn for each RF cycle (fig. A4).

22 Beam Cooler Injection Simulations Fig. A4 Action diagrams for a collection of particles undergoing driven oscillations in an oscillating force field. The diagram on the left is for an RF phase of zero (zero electric field but going positive). The diagram in the middle is for an RF phase of 45 degrees and the diagram to the right is for the force field at maximum value. The effect of the addition of this sort of action to the simple harmonic motion of the RFQ confinement is to distort the ellipse of the simple harmonic motion into ellipses that are specific to the phase of the RF, as shown in fig. A5. Fig. A5 Action diagrams for RFQ confinement in which the action of the driven RF oscillation is combined with the underlying simple harmonic motion. Because the quadrupole field is a linear (first-order) force field, in which the force components are proportional to the displacement components, the distortion it applies to the simple harmonic motion action ellipse keeps it elliptical and of the same area. Thus the full motion of a collection of particles under RFQ confinement can be described as a rotation within an ellipse that is itself being constantly deformed in the manner shown in fig. A5, with the deformation repeating each RF cycle. Analysis of the motion then involves expressing this deformation mathematically. Because the transformations of the action diagrams for RFQ confinement are linear this is most easily accomplished using matrices in linear algebra. (For a full description of the use of matrix algebra for the transformation of action diagrams see [1] and references contained therein.)

23 Beam Cooler Injection Simulations 3 Matrix Algebra of the Action Diagrams of RFQ Confinement In can be shown that for a linear transformation specified by the elements m 11, m 1, m 1 and m the eigensolution for the Twiss parameters, i.e. the set of parameters describing the ellipse for which the transformation is in fact no change, is A = m 11 m sin(πβ), B = m 1 sin(πβ), C = m 1 sin(πβ) where πβ = cos 1 m 11 + m (A) In turn, the matrix elements themselves can be expressed in terms of the solution parameters as cos(πβ) + A sin(πβ) M = Bsin(πβ) C sin(πβ) cos(πβ) A sin(πβ) (A3) In this form it is easier to see the significance of the parameter πβ. It is essentially the angle by which the individual points in the elliptical action diagram have progressed in their underlying simple harmonic motion when the action ellipse has returned to its initial shape after one RF cycle. The frequency of the underlying simple harmonic motion, often referred to as macromotion, is then ω SHM = β ω RF The parameter β is therefore the classical dimensionless parameter of the Mathieu functions. Because the transfer matrix elements for a complete RF cycle will be different for different RF phases at the start of the cycle, the eigensolutions for the Twiss parameters will be different. In fact, these eigensolutions for the different RF phases will be the action ellipses shown in fig. A5. While the eigensolutions of the transfer matrix for the Twiss parameters of an action ellipse presents an understandable picture of a collection of ions in thermal equilibrium, the motion of an individual ion can still appear complicated, a representative action diagram trajectory being shown in fig.a6. (A1) (A4) Fig. A6 A representative action diagram trajectory of a single ion in RFQ confinement.

24 Beam Cooler Injection Simulations 4 However, the underlying order to this motion can be seen when the ion action coordinates are placed on the elliptical eigensolutions for the various RF phases. Fig. A7 shows this for a sequence of RF phases. An animation of this sequence, in 1 degree steps, is available as a QuickTime TM file titled RFQ_Action_movie []. Thus, even though the action trajectory of a single particle may seem tortuous it is just the result of the particle sliding smoothly along an ellipse that is itself constantly being deformed by the oscillating quadrupole electric field. SHM and the RFQ Action Diagrams Fig. A9 shows the relationship between the action points of the ion and the points on the underlying simple harmonic motion action diagram. For ions in thermal equilibrium under RFQ confinement this relationship is important in that it establishes the particle density in the RFQ action diagram. This is because, as pointed out above, the driven oscillations of the particles are coherent and so produce no action area. The density distribution is therefore set in the underlying simple-harmonic action diagram according to (,3) and this distribution is transformed by the point to point transform (4). Analytically, this requires the elements of the transform matrix. The simplest transform is from the simple harmonic motion to the RFQ action diagram at RF phase zero. This is because at that phase there is no RF displacement. The simple harmonic action points at maximum momentum, zero displacement and maximum displacement, zero momentum therefore transform to the RFQ ellipse according to M x εb max = 0 A ε, M 0 B p max = 0ε B (A5) whereupon from the relationships ε = mωx max, p max = mωx max, 1 mωb 0 M = A mω 1, M 1 = mωb B mωb A mω B 0 mωb. (A6) Using this transfer matrix for a representative case of q = 0.55, the action points of the RFQ diagram for each 10 degrees of the simple harmonic motion diagram are as shown in fig. A8. Once the action points are determined for the RF phase zero diagram, they can be determined for any subsequent diagram be carrying out the product sum for the time interval from RF phase zero to the RF phase desired. Applications The application of matrix methods to practical problems in RFQ mass spectrometry is thoroughly discussed in Dawson [1]. Here will be added some applications dealing with the analysis of ion collections in RFQ traps and ion guides. Although it offers little advantage over a standard numerical integration of the equations of motion of an ion in electric field, the use of matrix methods to calculate the transverse motion of ions in an RFQ ion guide does illustrate the basic principles of the matrix methods.