Simulation of collisional relaxation of trapped ion clouds in the presence of space charge fields


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1 Simulation of collisional relaxation of trapped ion clouds in the presence of space charge fields J. H. Parks a) and A. Szöke Rowland Institute for Science, Cambridge, Massachusetts Received 1 January 1995; accepted 19 April 1995 A statistical formulation of ionneutral interactions in rf Paul traps is presented which provides a basis for calculating the collisional relaxation of a trapped ion cloud. These calculations describe the ion cloud as a statistical distribution in space and energy and treat collisions by Monte Carlo methods. Ion motion is calculated in the presence of space charge fields in a mean field approximation which allows each ion to be treated independently. This formulation provides an adequate description of the ion cloud dynamics for ion densities 1 8 cm 3. Frequency shifts induced in the ion cloud ensemble by space charge are characterized. Energy relaxation of a C 6 ion cloud initially at a temperature of 5 K by collisions with a background He gas at 3 K is calculated in the presence of space charge American Institute of Physics. I. INTRODUCTION There has been a large and varied effort over the past several decades to calculate the dynamics of ions within rf traps. Recent papers by March 1 and Cooks represent the most current work in the development of trap simulations and also provide excellent reviews of this field. The emphasis in most of these previous calculations has been to evaluate the effects of collisions on individual ion trajectories but not to include space charge effects. In contrast, the review by Vedel 3 of studies by Vedel and André 4 of the evolving spatial and energy distribution function of trapped ions in the presence of both collisions and space charge, and the analysis of charge transfer collisions studied in a series of papers by Bonner and March 5 are closer in context to the present calculations. However, the present work is unique in that it concentrates on the statistical aspects of the ensemble of ion cloud trajectories in the presence of both collisions and space charge fields. Although developed independently, the present approach has several points in common with these former studies and these will be discussed below. The present formalism has been derived to provide a common computational framework for the statistical analysis of collision processes involving trapped ion clouds. For example, it can be applied to calculate the excitation and relaxation rates of the translational and possibly vibrational degrees of freedom for specific ion cloud distributions, to analyze scattering of an injected atomic beam with the trapped ion cloud, to model the detection and ejection processes in the presence of space charge fields and to model loading the trap with ions by injecting a supersonic ion beam pulse. This approach requires computer algorithms which run rapidly enough to perform trajectory calculations on ion clouds of 5 to 1 ions needed to reduce statistical fluctuations adequately. In addition, the order of several hundred collisions per ion are required to relax the ion kinetic energy and equilibrate an ion cloud composed of ions having a mass 1 amu with a background helium gas. The calculation a Author to whom all correspondence should be addressed. follows the approach to equilibrium by treating the occurrence of collisions statistically, a method which achieves reasonable run times for clouds of 5 to 1 ions. The calculations presented here are based on trajectories within an ideal quadrupole trap but could readily be extended to model the imperfections expected in an experimental trap discussed in Ref. 1. The overall organization and integration of the trajectory and collision calculations are presented in Sec. II. This section formulates the treatment of collisions between a cloud of trapped ions and a neutral background gas including generation of the initial ion cloud ensemble and the appropriate coordinate transformations and numerical details necessary to follow the collisional relaxation. The inclusion of space charge fields and their effect on ion trajectories are presented in Sec. III. Frequency shifts of the ion motion in the presence of space charge are compared with measurements performed on C 6 ions. Section IV presents the results of calculations for the relaxation of kinetically hot C 6 ion clouds by collisions with He atoms in the presence of space charge. Section V briefly outlines several calculations planned or in the initial stages which apply this formalism to collisional analyses. II. COLLISION DYNAMICS MODEL A. Ion equations of motion This section introduces the equations of motion which describe an ion s trajectory during intervals between collisions. The reader is referred to several recent reviews of ion traps 6 for additional details and references. The radio frequency trap configuration is shown in Fig. 1 in which the ring and endcap electrodes are conjugate hyperboloids of revolution about the z axis. This configuration is characterized by the endcap separation z and the ring diameter r. In an ideal trap, the ions move in a time dependent, spatially inhomogeneous potential with cylindrical symmetry: 14 J. Chem. Phys. 13 (4), July /95/13(4)/14/18/$ American Institute of Physics Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
2 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 143 FIG. 1. Radiofrequency trap configuration. This schematic shows a voltage V applied between the ring and endcap electrodes which is composed of a dc voltage and an rf amplitude V rf at frequency. x,y,z;t V dcv rf cos t r z x y z 1 V dc V rf cos t, where r z is assumed for this simulation. The last term in Eq. 1 is required to satisfy boundary conditions consistent with the electrode voltage configuration shown in Fig. 1. The potential in Eq. 1 neglects higher order field components introduced by deviations from cylindrical symmetry expected for an experimental trap. These details are not central to the considerations here and could be easily included if required. The equations of motion for single particle trajectories are derived from the Hamiltonian p Hx,y,z;t i i M ex,y,z;t. The resulting equations of motion have the form of Mathieu s equation, 7 they are given by d x i d a i q i cos x i, 3 where x i x,y,z and t/. The single particle dynamics described by Eq. 3 ignores the presence of space charge fields, and in this case the motions in x, y, and z are uncoupled and determined by cylindrically symmetric fields. However, these equations are expressed in Cartesian coordinates because this will be more appropriate to include space charge effects considered in Sec. III. The trap parameters q i,a i characterizing the ion trajectory are given by a z a x a y 16M e V dc r z q z q x q y 8M e V rf 1, 1 r z, 1 4a 4b where the ion has mass M and charge e. The parameters q z and a z determine regions in which the solutions of Eq. 3 are stable, bound ion trajectories. For operating points (q z,a z ) outside the stable region, the amplitude of the ion trajectory increases exponentially until the ion eventually reaches an electrode surface and is lost. The analysis of ion motion in this paper is performed at the stable operating point q z.3, a z for a trap having z.3 cm and driven at an rf frequency of /1. MHz. Our primary interest is in the study of trapped cluster ion masses 5 amu. The differential equation solver calculates ion position and velocity from Eq. 3 by using the Numerov or Störmer algorithm. 8 This algorithm was chosen for speed and accuracy; however, its drawback is that only positions are calculated and the ion velocity has to be estimated from these positions. It is also very important to initialize the Numerov solver at t correctly with a consistent set of positions and velocities; consider, for example, those for the ion axial motion (z,v). The functional equation used in the Numerov algorithm derives a new position z f (z,z) by relating the current position z to the adjacent positions z and z. Using v to define a second order velocity associated with z by z z v, t where t is the time step, leads to the equation z f z,ztv which can be solved for z. In this way, the calculated value of z ensures that the solver generates the new position z consistent with the initial values of the position z and the velocity v. This procedure correctly initializes the Numerov routine and also generates accurate ion velocities at each time step. In the analyses described in this paper, the time step t1.5 nsec was used to derive accurate velocities in the tradeoff for acceptable run times. For the collision rates considered in this paper, there are time steps between collisions which assures that the ion trajectory between collisions is also calculated with adequate accuracy. Comparable time steps have been used in a previous trap simulation 1 for which the issue of accuracy was particularly important. The ensemble of positions and velocities which initialize the motion of each ion are either a randomly chosen as uncorrelated pairs from distributions whose mean and standard deviation model a thermal distribution, or b established as a set of correlated r i,v i trajectory pairs in a collision calculation described below in Sec. III D, which describe an ion cloud in thermal equilibrium. In these calculations, the phase of the rf field,, is always initialized to for convenience, since any memory of an initial phase is lost through collisions and we are not studying details of an individual ion trajectory for which the phase may be an important parameter. The overall program structure shown in Fig. organizes the discussion in the following sections. The output data in these analyses are obtained for both the micromotion solutions to Eq. 3 and the transformed secular variables. En 5 6 J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
3 144 J. H. Parks and A. Szöke: Simulation of trapped ion clouds H cons x,y,zhx,y,z;tev rf t x y z r 1 t dt. 9 z sin The integral in Eq. 9 is performed along the trajectory x(t),y(t),z(t). The quantity H cons (x,y,z) can be calculated during the collisional relaxation process and related to the energy transferred to the background gas by E coll H cons x,y,zhx,y,z;. 1 FIG.. Overall program organization is shown describing the sequence of important algorithms and input/output data. semble averages are taken after performing the calculation for N ions independently. Output data include the ion ensemble mean square position and velocity; average kinetic and potential energies; and the final position and velocity of each individual ion. Additional details such as total number of collisions and number of ions within the trap are also available. These data are written out at a time interval which is specified from.1 to s depending on the length of the run. Ion dynamics were followed up to 1 ms to observe the equilibration of a cloud of 5 ions of mass 7 amu by collisions with a background He gas density of cm 3. For these parameters, a calculation time of 6 min was required to follow ion dynamics for 1 ms on a Stardent Titan 3 work station with a single P3 MIPS R3 processor. B. Monitoring the approach to equilibrium It is evident that the Hamiltonian given by Eq. for ion motion in a time dependent quadrupole potential is not a conserved quantity. This Hamiltonian does not treat the ion and the rf field equivalently since it neglects the degrees of freedom of the rf field. As a result, the ion trajectories calculated by Eq. 3 are not appropriate to describe the decay of ion energy which occurs in the collisional approach to an equilibrium temperature. In order to describe the equilibration of the ion motion, one can proceed in several ways depending on the trap operating point (q z,a z ) at which the calculations are being performed. A conserved quantity related to the ion energy can be derived using dh dt H t and is given by where H cons x,y,zhx,y,z;, 7 8 In principle, this procedure provides a method to calculate a conserved quantity which will exhibit the approach to thermal equilibrium for arbitrary values of the operating parameters (q z,a z ). In practice, to reduce the error in the evaluation of the integral terms in Eq. 9 to E coll it is necessary to reduce the time step for the Numerov algorithm which then becomes excessively time consuming. For trap operating points having q z, a z 1, a conserved quantity can be defined in closed form, and in addition the residual fast time dependence retained by the ion Hamiltonian can be rigorously evaluated. In the original work of Dehmelt, 9 ion motion in an ideal trap for q z, a z 1 was shown to be a superposition of slow secular oscillations in both the axial z direction and the radial r direction of z and r, respectively, and of a fast micromotion at the rf frequency,. In this secular approximation, the ion position time dependence was shown to be dominated by the low frequency i secular motion, and only weakly modulated by the micromotion. However, the ion velocity time dependence retains a highly significant contribution of the high frequency motion even for q z, a z 1. For this reason, it is essential to use the trajectory velocity in the collision calculation. In this trap operating regime, the high frequency motion derived from exchange of energy between the ion and the rf field can easily be identified by the micromotion component. More recently, 1 particle motion in the limit of the secular approximation has been formulated for power law potentials in an analytical form. In particular, canonical transformations of the positions and momenta for ions in a Paul trap were derived which extracted the fast micromotion from the ion Hamiltonian. The details of these transformations are given in Appendix A. The transformed Hamiltonian exhibits fast time dependence only to lowest order in the parameter q z and is given by H x,ỹ,z ;t i i p i M e Tx,ỹ,z q i Fp i,x i ;toq i /4 11 to within a constant set by the applied dc potential. The transformed positions and momenta (p x,x ) are referred to as the secular variables. The time independent terms give rise to the slow secular ion motion within an effective harmonic trap potential 11 J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
4 e Tx,ỹ,z ev dc x ỹ z r z ev rf M r z x ỹ 4z. 1 The V rf dependent term has been referred to as the pseudopotential, 11 the effective potential 1 and the Kapitza potential. 1 Note that the remaining time dependent terms are reduced in amplitude by powers of q i / and are responsible for the residual rapidly varying micromotion at frequency and higher harmonics remaining after the secular transformations. These residual contributions of the micromotion for q z.3 are shown in Figs. 18. The analyses of collisional relaxation of the ion motion are calculated in this paper at the trap operating point q z.3, a z. Although the ion trajectories derived from Eq. 3 are used in the actual collision calculation, ion motion determined by Eq. 11 is used to monitor the approach to thermal equilibrium. The advantage of applying these transformations lies in their ability to identify the conserved quantities unambiguously. This procedure can define a cloud temperature as a parameter with physical meaning because it relates the approach to thermal equilibrium with a quantity that has only secular variation. Although several previous collision calculations 1,13 discussed ion relaxation by averaging the kinetic energy over an rf cycle in order to remove the high frequency time dependence, such procedures cannot be used to define a physical temperature associated with ion motion within the cloud. The trap potential given by Eq. 1 can be expressed 13 in terms of the potential well depths D z and D r for the axial and radial motions given by where Tx,ỹ,z D r r x ỹ D z z z, D z D r 4eV rf 13 V M r z dc r z z 1 M e z z, 14a ev rf M r z V dc r z r 1 M e r r. 14b Using the q z,a z relations given by Eqs. 4a and 4b in Eqs. 14a and 14b, the radial and axial frequencies in the secular approximation are r a r q r and z a z q z. 15 The well depths determine the maximum translational energies for stable ion motion in the axial and radial directions given by ed z and ed r, respectively. Note that D r D x D y and r x y for ion motion within a cylindrically symmetric potential. For V dc, and r z the well depths are given by ed z ed r q z ev rf /8. In this case the trapped J. H. Parks and A. Szöke: Simulation of trapped ion clouds charge distribution will take the form of an oblate spheroid with x y 4z. The frequencies are given by z r q z / 3/. C. Ion neutral gas collision dynamics 145 In these calculations, collisions of trapped ions and neutral gas atoms will be limited to less than four collisions per 1 rf periods 1 s in order to perturb the ion trajectory weakly. For example, this corresponds to collisions between C 6 and a background He gas at 31 3 Torr. In this case, collisions are a rare event and a molecular dynamics calculation is highly undesirable since it would spend most of the time following uneventful trajectories. The following scheme to calculate the effect of ion gas collisions was based on Monte Carlo statistical sampling. This scheme is summarized in Fig. which shows the sequence followed after each time step of the Numerov differential equation solver. The details of this calculation sequence are considered in the remainder of this section. 1. Collision occurrence After each time step of the ion trajectory calculation, a decision is made whether a collision will occur in the following time interval t by calculating the collision probability P ion t/ coll, 16 where ion is the instantaneous ion velocity, and the collision interval coll is defined by 1 n coll m kt gas 3/ ion e m /kt gas d d, 17 where is the neutral gas velocity, m the neutral mass and T gas the gas temperature. To perform these calculations rapidly a lookup table is used to tabulate the rate 1/ coll for an array of values of ion and. This analysis assumes a spatially uniform background gas having a Maxwell Boltzmann velocity distribution at a temperature T gas. The ion neutral interaction can be approximated 14 by the following effective potential in radial coordinates: V eff r e b r 4 E rel r rr hs rr hs 18 which combines terms for a chargeinduced dipole potential with a centrifugal barrier and includes a hard sphere repulsive barrier at r hs. This potential describes the interaction of an ion with a neutral atom at a particle separation r. Itdepends on the atom polarizability, a collision impact parameter b and on the relative kinetic energy of the collision, E rel rel / for reduced mass. The interaction potential approximated by Eq. 18 is useful here when the primary emphasis is to estimate the collision rate. When the transfer of internal energy becomes an important consideration, a more physical approximation of the short range interaction potential will be required. Smith 14 shows that when the parameter J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
5 146 J. H. Parks and A. Szöke: Simulation of trapped ion clouds the probability distribution of the colliding atom is expressed by dp ion ;,,n 3/ m ion kt gas e m /kt gas d d, 1 where ion ion ion cos 1/ FIG. 3. Collision geometry in the hard sphere model indicates an ion of mass M and neutral atom of mass m in the centerofmass frame in which the velocities have been rotated into the ẑ direction and into the (x,z) plane. e 4 kt gas r hs 19 is small, 1, as expected for small neutral polarizability or large E rel kt gas, the hard sphere repulsion dominates the interaction for which the integrated cross section for momentum exchange is given by r hs. For slow collisions and highly polarizable neutrals, the charge induced dipole potential is more important and the cross section is given by the Langevin result 15 L e E rel. This paper considers relaxation of the ion kinetic energy in a He background gas at 3 K, for which the parameter He.. In this case the hard sphere cross section can be used to estimate the collision rate. However, if the heavier, more polarizable rare gases are considered, 1, so that the Langevin cross section would be more appropriate. In general, the collision probability given in Eq. 16 is represented by a binomial sum of terms which include the probability of a single collision, two collisions, etc., in the time t. In our case the trajectory is calculated at time intervals t coll which allows us to truncate the sum to a single term. Thus P ion is the probability for a single collision during t and the probability that a collision does not occur is just 1P ion. A collision is defined to have occurred if a random number, Ran, chosen from a uniform distribution over the interval 1, falls within the interval RanP ion. This sequence statistically samples the occurrence of a collision during the interval between t and (tt).. Gas velocity sampling After a decision that a collision has occurred, the neutral gas is statistically sampled to determine a specific three dimensional velocity for the colliding neutral gas atom. When the gas is isotropic, a reference frame in which the ion velocity ion is rotated into the z direction, ion ion as shown in Fig. 3 simplifies the algebra. In this rotated frame and the probability is normalized: dp ion ;,,1. 3 The gas velocity is sampled from the distribution given by Eq. 1 to include the relative velocity dependence of the collision probability properly. To simulate hard sphere collisions, the cross section is taken to be independent of velocity. The cumulative probability that a collision occurs with a gas atom having speed less than or equal to is given by S ion,n 3/ m kt gas d sin d ion e m /kt gas d. 4 The magnitude is sampled by choosing a random number, Ran1, from a uniform random distribution over,1 and then solving the following equation by the rejection method: 16 S ion, 1 coll * Ran1 5 in which the collision rate given by Eq. 17 is evaluated with the current value of ion. After is found, is sampled by choosing a second random number, Ran,1, and solving S ion ;, ion ion cos 1/ sin d Ran. 6 Finally after and have been found, a third random number Ran3,1 is chosen to sample Ran3. This procedure statistically samples the velocity of the colliding neutral atom in the rotated frame indicated by primes x sin cos, y sin sin, z cos. 3. Hard sphere collisions 7 Each collision calculation begins with an ion velocity calculated from Eq. 3 in Cartesian coordinates: J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
6 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 147 ion x, y, z ion sin 1 cos 1, ion sin 1 sin 1, ion cos 1. The velocity is then implicitly rotated into the z direction, ion, ion,, by taking ion (,, ion ) and retaining the angles 1 and 1 for an eventual back rotation at the end of the collision calculation. The neutral gas sampling is performed in this rotated frame as discussed above. This rotation is defined by first rotating about the z axis by an angle 1 followed by a rotation of 1 about the y axis. The matrices expressing this rotation and its inverse are given in Appendix B. In the rotated frame, a translation into center of mass coordinates is performed with the center of mass velocity V c.m. to express the ion and gas velocities V ion ion V c.m., VV c.m., 8 where the gas velocity in the once rotated frame is given by Eq. 7. The c.m. ion and gas velocities are each rotated into the z direction again by implicitly applying the rotation transformation, V ion, V V ion, V, and the ion velocity angles, are retained for back rotation. The collision algorithm which calculates the new ion velocity resulting from the conservation of energy and momentum is applied in this twice rotated frame. In the model considered here, the steric characteristics of the collision are taken into account to lowest order by considering a collision between a gas atom and a rigid spherical ion. This model accounts for glancing collisions which will more properly represent the relative velocity dependence of the collision. Figure 3 describes the collision geometry in the twice rotated frame in which the ion velocity is along the z direction. The gas atom is traveling off axis in the z direction and its trajectory is assumed to be rotated by some azimuthal angle r about z axis. The collision is calculated in a plane obtained by implicitly rotating the COM frame by the angle r, V ion, V V ion, V. The angle r is chosen from a random uniform distribution over and is used explicitly to back rotate the reference frame after the collision. In the collision plane, is the angle between the gas atomcluster line of centers and the z axis at the point of contact. Assuming the impact parameter, b, has some distribution p(b, r ), the cumulative probability that the impact parameter is b is given by SbC dr b pb,r db. 9 This general form is useful to model dipole interactions or possibly include steric effects, but reduces to a simple expression for hard sphere collisions. In this case, p(b, r )b, and since Sr hs 1 for br hs, the normalization constant is given by C1/ and the cumulative probability for hard sphere collisions is then Sb b. 3 Here, the constant hard sphere cross section is defined by van der Waals radii r ion r gas r hs. 31 To sample the impact parameter statistically, we take S(b) ran, where ran is a random number uniformly distributed over the interval,1. Then the impact parameter is given by b ran/ and the angle by 3 sin b b/ran. 33 r hs The ion and gas velocity components in the rotated c.m. frame are written as components parallel n and perpendicular n to the line of centers shown in Fig. 3: V ion V ion V ion sin V ion cos ; V V sin V 34 V cos, where the superscript indicates the frame rotated by r. Using this coordinate system for the initial collision velocities components has the advantage that the resulting conservation equations are reduced to those for a simple headon collision. In the c.m. frame, the momentum components along n satisfy MV ion mv, MV ion f mv f, 35 where ( f ) denotes the velocity after the collision. Conservation of the relative kinetic energy is expressed by V ion f V f V ion V. 36 Using Eqs. 34, 35, and 36, the final ion and gas velocity components after the collision can be derived: V V ion f ion sin V ion MmV f ion cos mv cos, Mm 37 V V sin f V MmV cos MV ion cos. f Mm Expressing the final ion velocity in terms of Cartesian coordinates and explicitly back rotating about the z axis by r, the equations for the final ion components in the c.m. system after the collision are given by V ionx f V ion f sin V ion f cos cos r, V iony f V ion f sin V ion f cos sin r, 38 V ionz f V ion f cos V ion f sin. These components are then back rotated in the c.m. frame by,, V ion V ion, transformed back to the laboratory frame, V ion ion, and back rotated once more by 1, 1, ion. The new ion velocity is now described in the ion J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
7 148 J. H. Parks and A. Szöke: Simulation of trapped ion clouds FIG. 4. Results of a cloud formation calculation showing equilibration of the ion cloud with a background He gas at pressure 31 3 Torr and 1 K: time dependence of a ensemble averages x and z ; and b ensemble average of ion energies E tot, KE, and PE. original reference frame where it is used to define the initial conditions for the differential equation solver as it continues calculating the ion trajectory. For describing a different force field other than the impulsive hard sphere interaction, such as that arising from chargeinduced dipole, Eqs have to be modified to express the angle appropriately. D. Ion cloud initialization 1. Secular distribution One of the more important aspects of the analysis presented here is to formulate the calculation to represent the dynamics and statistical mechanics of a cloud of ions which can then be correlated with experimental measurements. In order to calculate the mean ion energy and ion spatial distribution during the collisional approach to equilibrium, an ensemble of trapped ions has to be carefully initialized. This requires a calculation to determine the initial conditions for an ensemble of N ions with N correlated pairs of position and velocity r i, i, ix,y,z at t which describe an ensemble of ion trajectories corresponding to an ion cloud in equilibrium at temperature T. When the cloud is properly initialized, the solutions to Eq. 3 for the ion micromotion will include a secular ion motion which satisfies the following velocity and spatial distributions defined in cylindrical coordinates by: 3/ M f r,z, nr,z e kt M /kt 39 and nr,z N 3/ e r z D kt r D z 1/ e e T r,z /kt. 4 The potential Tr,z and well depths D r, D z are defined by Eqs. 13 and 14 respectively, and f r,z ; is normalized by f r,z ; r dr dz d 3 N. 41 The distribution of the ion cloud in thermal equilibrium is determined by r x ỹ r kt, ed x z z z kt, ed z vi i kt M ix,y,z. 4 These distributions express the equipartition of energy by M 1 e Tr,z f r,z ; r dr dz d 3 3NkT. 43 The velocity and spatial distributions given by Eqs. 39 and 4 are valid approximations for the secular ion motion in the absence of space charge. The space charge fields included in these calculations will be shown to perturb the ion distribution weakly and we will neglect this second order effect in trajectory calculations. The cloud initialization accounts for space charge as described in Sec. III.. Cloud formation To create a cloud of ions having a distribution of secular variables r i, i characterized by Eqs for an ensemble in thermal equilibrium, ion motion is calculated by Eq. 3 for a short period of time but at a high enough background gas pressure to equilibrate the cloud at temperature TT gas rapidly. This cloud formation run begins at t with an ensemble of N ions each of which is at the position rz and has a velocity sampled from a Maxwell Boltzmann distribution for which the mean square velocity is given by 1 M f r,z ; r dr dz d 3 3NkT gas. 44 J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
8 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 149 FIG. 5. The spatial and velocity distributions for an ion cloud of 1 3 ions formed by equilibration with a background He gas at pressure 31 3 Torr and 1 K: a radial and b axial positions; c radial and d axial velocities. Cloud temperature related to the standard deviations by Eq. 4 are obtained from Gaussian fits shown by the solid lines. These initial conditions have been chosen because they initialize the micromotion and the secular motion identically, i() i (), as shown by the secular transformations defined in Appendix A. The initial kinetic energy implied by Eq. 44 will not be in equilibrium with the He gas. However, the cloud will be shown to equilibrate after 5 s in the presence of a gas collision rate of 1 collisions/s. It has been verified that initializing the cloud with a uniformly random spatial distribution of ions throughout the trap volume yields identical results to those presented here. However, in the case of a random spatial distribution, only a fraction of the ions achieve a stable trajectory depending on He gas temperature and pressure, and trap operating parameters. These effects will be considered in a subsequent analysis of trap loading efficiency. Figure 4 shows the position and energy variables calculated from a computer run designed to form an ion cloud of 1 3 C 6 ions at an equilibrium temperature of 1 K. The trap parameters in this run were r.3& cm, z.3 cm, and /1 MHz. The trap well depths are ed r.4 ev and ed z 4.8 ev at the operating point q z.3, a z. The time dependence of the ensemble position averages denoted by, N x x t z and z 1 N t 45 1 N are shown in Fig. 4a. This time dependence displays a rapid equilibration of these spatial averages to a ratio of x /z 3.76 compared to a value of 4. expected for energy equipartition in a cloud at thermal equilibrium. The time dependence observed in Fig. 4a is related to the oscillation of potential and kinetic energy associated with secular motion and expected for harmonic oscillator dynamics. The oscillation period for the axial (z) motion of 1 s is just the N secular frequency z / given in Eq. 15. The ensemble average of the kinetic, potential and total ion energy is observed in Fig. 4b. The potential and kinetic energy approach the value of PE/kKE/k1535 K compared to the expected equilibrium value of 3/T15 K and display the out of phase character associated with secular kinetic and potential energy. The value of E Total /3k13 KT gas is unchanged during the run because of the initial condition given by Eq. 44. To demonstrate that this cloud formation procedure produces an ion cloud spatial and velocity distribution given by Eqs. 39 and 4, respectively, the final secular values of r i, i, ix,z were used to produce Figs. 5a through 5d. These figures are histograms derived from the formation run output after applying the secular transformations given in Eq. A8, and each figure includes a Gaussian fit to the calculated points. The variance of each fit is related to cloud temperature by Eq. 4 and within the statistical scatter the fit is described by TT gas. The velocity distributions simply reflect the fact that the initial velocities were properly sampled from a Gaussian distribution at the correct temperature and retain this distribution during the ensuing collisional relaxation. However, the more important result is that the spatial distributions are also gaussian with a width determined by the appropriate standard deviation. This spatial distribution depends completely on the collisional process and is probably the most sensitive indicator that the collision formalism is behaving properly. This final ensemble of positions and velocities is then used as the initial values for each ion comprising a cloud for an actual relaxation run calculation, for example, the cooling of an ion cloud from 1 to 3 K. III. SPACE CHARGE FIELDS A. Mean field approximation Interactions with the electrostatic fields resulting from the surrounding charge density of N sc ions will introduce an average repulsive force on each ion tending to expand the cloud. In the presence of these space charge fields, the equations of motion take the form d x i d a i q i cos x i e M F i, 46 where F i i j e/r i r j is the force of Coulomb repulsion among the ions. Note that the ion equations of motion given by Eq. 46 are now a set of coupled equations which no longer exhibit cylindrical symmetry since the force on a specific ion will depend on the ion s instantaneous position within the charge distribution. The force F i can be approximated by considering an ion interacting with a mean space charge field derived from a Gaussian charge density (R) for which the resulting electrostatic potential sc is expressed by sc x,y,z R RR d3 R. 47 In this mean field approximation, the repulsive force on an ion is given by J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
9 143 J. H. Parks and A. Szöke: Simulation of trapped ion clouds FIG. 6. Ion cloud density calculated from the Gaussian approximation given by Eq. 4 is compared with that obtained from the solution to Poisson s equation, Eq. C. This comparison is shown for cloud temperatures of 3, 1, and 5 K. FIG. 7. Comparison of the space charge force acting on an ion derived from the Gaussian approximation of Eq. 5 and the Poisson solution given by Eq. C5. This calculation was performed for a cloud of N sc 1 4 ions at a temperature of 3 K. F i x,y,z i e sc, ix,y,z, 48 where R x y z is the ion position measured from the center of the trap. In this calculation, the charge density has been approximated by the ion spatial distribution determined by the trap potential T given by Eq. 13. At the equilibrium temperature TT gas, the charge density is approximated by RenRen expe TR/kT, 49 where n is the peak ion density for a cloud of N sc ions which can be written using Eqs. 4 and 4 as n N 3/ sc e r z D kt N r D z 1/ sc 3/. 5 r z Note that in this approximation, the parameter N sc determines the strength of the space charge interaction between an ion and the cloud; however, single particle calculations are still performed for N individual ions to provide the ensemble averages characterizing the cloud. This treatment of space charge is not a selfconsistent procedure since the variation in the ion trajectories induced by space charge fields is not applied to recalculate (R) to reflect the distortion of the ion charge density. This mean field approximation is expected to be adequate for weaker charge densities in the range n 1 8 ions/cm 3. Figure 6 estimates the degree of approximation in the density incurred using the Gaussian form in Eq. 4 by comparing it to a numerical solution of Poisson s equation discussed in Appendix C. The density profiles n(,z)vszshown in Fig. 6 have been calculated for N sc 1 4 C 6 ions at the operating point q z.3, a z and cloud temperatures of 5, 1, and 3 K. Under these conditions, the Gaussian approximation for the density begins to diverge from the numerical solution by a factor of for T3 K. However, the force on an ion due to the space charge mean field depends on the gradient of the density. The average force F i is obtained by transforming 17 the three dimensional integral in Eq. 47 to a convolution integral and applying several variable transformations which reduce the expression for sc to the following one dimensional integral: sc x,y,z N sc e 3 z /3 exp x y sin z tan 6 z The force components are then given by where d. 51 F i K i x i, x i x,y,z, 5 K x K y N sc e / z sin exp x y sin z tan 6 z d, J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
10 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 1431 FIG. 8. Plot of the ion radial oscillation frequency x / vs the peak trajectory amplitude R peak defined in the text. Frequencies were calculated for 4 ion trajectories in a cloud having N sc 1 4 ions at several cloud temperatures. The solid line represents the unperturbed radial frequency. FIG. 9. Plot of the ion axial oscillation frequency z / vs the peak trajectory amplitude R peak defined in the text. Frequencies were calculated for 4 ion trajectories in a cloud having N sc 1 4 ions at several cloud temperatures. The solid line represents the unperturbed axial frequency. K z N sc e / z tan exp x y sin z tan 6 z d. 53 By reducing the three dimensional integral to a single integral, the Gaussian approximation for the charge density (R) allows the ion trajectory to be calculated rapidly from the coupled equations given by Eq. 46. To maintain the speed of the trajectory calculation in the presence of space charge, the integrals in Eq. 53 are interpolated from a lookup table. As shown in Fig. 7, the z component of this force, F z K z z, at a cloud temperature of 3 K is only 3% greater than that calculated from Poisson s equation as derived in Appendix C. A selfconsistent treatment of the space charge interaction will be required in order to extend calculations of the ion motion to temperatures significantly lower than 3 K which are of current experimental interest. Appendix C discusses an alternative selfconsistent calculation of space charge forces. B. Ion frequency shifts and spatial distribution To estimate the effect that space charge fields have on the ion trajectory, consider an ion oscillating with small amplitude near the center of the cloud, x,y x,z z. An ion with this trajectory will experience the maximum space charge interaction. The charge density (R) is approximately constant over this trajectory and the z component of the force can be estimated as F z.15 N sc e / z 3 zk z z. In this case, the perturbed ion resonance is given by z z K z M, 54 where z is the resonance for K z given by Eq. 15. For ( z z ) z, the frequency shift is approximated by z z z K z z M.53 N sce z M z Similar expressions describe the shifts induced in the radial frequency components x and y. The single ion resonance is shifted to lower frequency and this shift increases linearly with the peak ion density n proportional to N sc / 3 z. In this Gaussian approximation, the shift will increase with decreasing cloud temperature as T 3/. Trajectories having x,y x,z z sample the tail of the Gaussian ion cloud distribution and result in smaller resonance shifts but these shifts are still proportional to n. The distribution of single ion trajectories within the cloud will lead to a distribution of resonance frequencies over the range ( z z ) z z which will characterize an inhomogeneous response of the cloud to an applied rf field. The distribution of space charge induced frequency shifts have been calculated for ions moving within an ion cloud in thermal equilibrium with a background He gas. These shifts have been calculated for a cloud of N sc 1 4 ions at temperatures of 3, 1, and 5 K. Ion cloud trajectories of N4 ions were calculated sequentially including the Fourier power spectrum for each ion. The calculation was performed for each ion cloud temperature with a runtime of 4.5 ms which provided a frequency resolution of 6 Hz. The dependence of the ion radial frequency ( x ) y on its trajectory is shown in Fig. 8 which plots the frequency for each ion vs the trajectory peak amplitude R peak [x y z ] 1/ peak. The dependence of the ion axial frequency ( z ) on its trajectory is shown in Fig. 9. The scatter in frequency for a given value of R peak arises because the frequency is only qualitatively correlated with this peak amplitude. The calculated frequency is clearly an average since an ion may sample regions of widely varying density during its trajectory. The fractional scatter is the same for both radial and axial frequencies (/) x (/) z. Note, J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
11 143 J. H. Parks and A. Szöke: Simulation of trapped ion clouds FIG. 1. The ion cloud temperature vs helium pressure derived from frequency shift measurements in Ref. 19 is shown with error bars representing 1% uncertainty in calibrating the absolute temperature. FIG. 1. Line shapes formed by the distribution of z / ion oscillation frequencies for 4 ion trajectories in a cloud having N sc 1 4 ions at several cloud temperatures. The frequency shift z z / from the unperturbed frequency is indicated for each temperature. The unperturbed frequency at 17.8 khz is indicated by the dashed line. in Figs. 8 and 9, that points displaying the largest shifts for a given peak amplitude form a boundary approximating a smooth curve. Points on these boundaries are consistent with estimates of the maximum shift given by Eq. 55. The inhomogeneous line shapes formed by the distribution of ion frequencies are shown in Fig. 1 as histograms derived from the trajectories of 4 ions at each cloud temperature indicated in Fig. 9. As shown in Fig. 1, both the frequency shift as well as the linewidth increases with decreasing temperature as a result of the increased spatial inhomogeneity characterized in Fig. 9. An electronic detection technique 18 which tunes the excitation frequency of an externally applied rf field over the ion resonance has recently been applied 19 to measurements of C 6 collisional relaxation. This detection method sequentially excites ions at each z frequency yielding a response which is inhomogeneously broadened by ion space charge interactions. Relaxation of the kinetic energy of trapped C 6 ions was studied by measuring the shift of the ion cloud resonance frequency with He pressure. As the collision rate increases with pressure, the competition between rf heating and collisional cooling leads to an equilibrium cloud temperature T which approaches the ambient temperature of the background He gas. As the temperature decreases, the peak ion density increases, leading to stronger space charge interactions. As shown in Ref. 19, an ion temperature can be derived for each He pressure by using the frequency shift at 3 K as a calibration point which corresponds to a peak ion density of n cm 3, and then scaling the measured shifts to obtain the cloud temperature T at each pressure. This procedure provides the plot of frequency shift vs He temperature shown in Fig. 11. The plot in Fig. 11 includes the theoretical estimates of the average frequency shifts shown in Fig. 1 which were calculated for N sc 1 4 C 6 ions at cloud temperatures of 3, 1, and 5 K. Values for the calculated shifts are in agreement with the experimental measurements 19 except for 3 K, where this calculation is expected to be an overestimate resulting from the Gaussian density approximation. In subsequent measurements, the linewidth of the detected ion response has been observed to broaden significantly as the cloud temperature decreases to 8 K, however, a quantitative study of the linewidth has not been performed at this time. A difficulty encountered in such measurements is that the amplitude of the detected signal is reduced by the increased broadening and compensation with increased excitation leads to excessive anharmonic distortion of the line shape. IV. RELAXATION OF ION TRANSLATIONAL ENERGY FIG. 11. The frequency shift vs ion cloud temperature obtained in Ref. 19 is shown by open circles at each datum point. Calculated values of the average frequency shift indicated in Fig. 1 at ion temperatures of 3, 1, and 5 K are given by the solid squares. This section presents calculations performed to demonstrate the capability of these statistical computations to study the relaxation of ion translational energy in collisions with a dilute background gas. In particular, this integration of the J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
12 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 1433 FIG. 13. The collisional decay of the kinetic energy and total energy of C 6 ions in a He background pressure of 31 3 Torr. The cloud having N sc 1 4 ions relaxes from an initial temperature of 5 K to T gas 3 K. ion equations of motion with statistical sampling techniques is shown to be consistent with the physics expected for the relaxation process. The collisional decay of ion cloud energy has been calculated in the presence of space charge fields for several values of cloud ion number, initial cloud temperature and different neutral mass in the hard sphere collision model. The spatial contraction of the ion cloud as the cloud temperature decreases is also displayed. This analysis does not consider the competition between rf heating and ion evaporative cooling which determines the initial cloud energy distribution nor does the calculation include the effects of ion ion collisions. These processes have been considered elsewhere 11, and the decay of cloud energy is calculated here for a background gas pressure in which the ion neutral collision rate dominates the approach to thermal equilibrium at the neutral gas temperature. The plot in Fig. 1 is the result from measurements 19 performed on trapped C 6 ions in a He background gas and demonstrates that for He pressures greater than 11 5 Torr the C 6 cloud is in equilibrium with the He gas. In the calculations presented here, the He pressures used are in the range of 1 3 Torr corresponding roughly to five collisions per 1 rf periods 1 s for which the ion trajectory is only weakly perturbed. The calculated energy decay rate of the ion cloud continues to scale correctly at collision rates a factor of 1 greater, so that the lower rates used here are clearly conservative. A. Ion cloud energy decay Each relaxation calculation was performed for 5 C 6 ions initialized as a cloud at a specific temperature as described above. The motion of each ion is calculated in the presence of a background gas pressure and a space charge field resulting from N sc surrounding ions. The secular transformation is applied at each successive time interval in the ion motion to calculate the ensemble averages of the conserved energy and the ion cloud variables. The energy decay is monitored as it approaches the limit set by the temperature of the background gas, T gas. In the absence of space charge fields, the energy decay can be related to a changing cloud temperature provided the decay rate is slow compared to the ion kinetic time scale. In this case, the ensemble average of the ion energy, E tot 3kT over a cloud of N ions is expected to follow an exponential decay with time constant, expressed in terms of the cloud temperature by TtT gas TT gas expt/. 56 However, when space charge interactions are included, the conserved cloud energy as defined in Eq. 11 cannot rigorously be associated with a temperature since the potential FIG. 14. The collisional decay of KE, defined in Eq. 57, for C 6 ions in a cloud having a N sc and b N sc 1 4 ions. The He pressure is 31 3 Torr and T gas 3 K. Each case is calculated for an initial cloud temperature of 1 and 5 K. The time constant is obtained by fitting each exponential decay. J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
13 1434 J. H. Parks and A. Szöke: Simulation of trapped ion clouds FIG. 15. The collisional decay of PE, defined in Eq. 57, for C 6 ions in a cloud having a N sc and b N sc 1 4 ions. The He pressure is 31 3 Torr and T gas 3 K. Each case is calculated for an initial cloud temperature of 1 and 5 K. The time constant is obtained by fitting those plots in a which exhibit exponential decay and this fit is redrawn in b for comparison. energy of ions in the space charge field is neglected. The deviation introduced by space charge can be more clearly evaluated by plotting the time dependence of the following differential quantities E tot E tot /3kT gas, KE/3KEkT gas, 57 PE/3PEkT gas, where the symbol in Eq. 57 denotes an ensemble average. Figure 13 shows the decay of the total energy E tot and the kinetic energy KE calculated for ions trapped in a He background gas at 31 3 Torr within a cloud of N sc 1 4 ions initially at T5 K. Since this calculation statistically represents the cloud with calculations for 5 ions, energy fluctuations of 4.5% are expected. The oscillations of 15 s period observed in the kinetic energy decay is simply the time dependence expected for a secular harmonic oscillator and not due to residual micromotion. This oscillatory component is not present in the decay of E tot which is the sum of KE and PE which are equal in amplitude but opposite in phase. Although the period of the secular motion is 1 s, the calculated output is not strobed at the secular frequency resulting in oscillations having a period which is dependent on the sampling rate. In the presence of space charge, these coherent oscillations of the KE become random fluctuations as the cloud energy decreases. This occurs as a result of the space charge induced shifts of the ion frequencies which become greater and more dispersed with in FIG. 16. The collisional decay of PE, defined in Eq. 57, for C 6 ions having different values of N sc and an initial cloud temperature of 1 K. The He pressure is 31 3 Torr and T gas 3 K. FIG. 17. The spatial contraction of the C 6 ion cloud vs cloud temperature is shown for N sc and a He pressure of 31 3 Torr, T gas 3 K. Variation of the cloud spatial distribution determined by the standard deviations x and z, and the cloud volume approximated by an oblate spheroid V4 x z /3 are shown. J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
14 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 1435 creasing ion density, eventually averaging out the secular oscillations in the ensemble averaged KE. It is interesting to note that the decreasing amplitude of the KE oscillations are an indication of the growing importance of space charge interactions. The individual decays of KE are shown in Figs. 14a and 14b for N sc and 1 4 ions, respectively; and the decay of PE is similarly shown in Figs. 15a and 15b.A time constant obtained by fitting each plot which displays an exponential decay is shown in Figs. 14 and 15. Scatter about the average value of.8. ms is within the fit uncertainties. The calculation yields a collision rate within % of the rate estimated by n He v rel for the He pressure 31 3 Torr, v rel cm/s and cm. The average decay rate 1/375 s 1 compares closely with an estimate of the energy decay rate given by m He /Mn He v rel 78 s 1. The decay of KE is essentially independent of space charge aside from the randomization of the coherent oscillations. The ensemble average of the ion kinetic energy v z z z can display this insensitivity to space charge since the decrease in ion frequency discussed above is compensated by the increase in z resulting from space charge repulsive forces. However, the decay rate of PE decreases significantly in the presence of N sc 1 4 becoming nonexponential for PE.5 ev 88 K cloud temperature. At this point, the peak ion density estimated by Eq. 5 has increased to a value of cm 3 and the repulsive space charge forces have started to retard the spatial collapse of the cloud. These forces act to expand the cloud and prevent further decrease of the ion potential energy as shown in Fig. 15b which compares the decay with the exponential fits taken from Fig. 15a. The nonexponential decay of PE is shown in Fig. 16 for different values of N sc. B. Cloud spatial contraction The above calculation of ion cloud cooling also provides the opportunity to display the contraction of the cloud volume as the cloud energy decays. Figure 17 show the temperature dependence of the standard deviations ( x, z )of the distribution of ion positions for 5 ions. These are plotted versus the cloud temperature derived from the decay of E tot for N sc and defined by TE tot /3kT gas. The cloud volume is approximated by an oblate spheroid, V(4/3) x z. The contraction for a cloud having N sc is shown in Fig. 18 to be a factor of 35 between 5 and 3 K. The cloud volume contraction in the presence of space charge at 3 K for N sc 1 4 ions yields a volume which is 5% larger than that for N sc which is consistent with the density calculations shown in Fig. 6. V. DISCUSSION The emphasis in this paper has been to develop an approach to calculating trapped ion dynamics in the presence of both neutral ion collisions and space charge fields. The ability to follow the evolution of an ion cloud ensemble provides an opportunity to apply the calculation to the analysis of experiments, to better estimate collision processes which rely on characterizing the initial ensemble and to calculate trapped ion dynamics associated with rf field manipulations. The following discussion will briefly describe several applications for these calculations which are planned or in progress. A. Atom ion collisions There are several calculations which would support analysis of the interaction of neutral atomic beams with trapped ions. 1 An atomic beam entering the trap will be scattered by the background He gas as it propagates toward the ion cloud. If the ion cloud has been cooled by low temperature He, so that it is condensed to a small diameter.1 mm at trap center, small angle collisions will reduce the flux impinging on the ion cloud. An analysis can be performed assuming van der Waals scattering which estimates this flux reduction vs He pressure for a cloud size determined by space charge density limitations. The exchange of ion vibrational energy through collisions is very central to experiments planned with large metal cluster ions. Two areas of particular importance are the extraction of energy from the vibrational modes of a cluster ion after an associative atom ion collision by subsequent collisions with the He gas background, and the possibility of controllably heating the vibrational modes of the cluster ion through energetic collisions with a background neutral gas. In both cases, a model for vibrational energy exchange is required which is both appropriate to the interaction and also capable of being included within a reasonable calculation timescale for an ion cloud ensemble. B. Strong space charge conditions The presence of space charge modifies the interaction of trapped ions with external fields. The effect of space charge on electronic detection discussed in Sec. III is one example of this. The important process of resonance ion ejection for the purpose of mass selection and mass spectroscopy can also be degraded 1 by the presence of space charge. Under conditions in which the ions are cooled to temperatures below 3 K for the purpose of decreasing the ion internal energy, space charge fields will seriously effect ion dynamics in the presence of resonant fields. The inhomogeneous broadening of the ion line shape discussed in Sec. III will introduce asymmetry in the ion response depending on the direction in which the resonance point is approached on the frequency axis. Recent measurements of ion ejection at 8 K have clearly shown these effects as discussed in Sec. III B. To extend the present analysis to calculations in the limit of strong space charge conditions will require a selfconsistent calculation of the space charge forces. Appendix C presents an outline of the selfconsistent formulation of space charge in the mean field approximation. This more reliable calculation will be applied to better understand the limitations of ion ejection measurements at lower temperatures. J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
15 1436 J. H. Parks and A. Szöke: Simulation of trapped ion clouds C. Ion injection The possibility of trapping ions after directly injecting them into the rf trap by switching on the rf field when the ion attained a position at which its velocity was consistent with a stable orbit has been considered previously. However, for such a scheme to be practical it would rely on applying an rf voltage step having a risetime 1 s and more importantly, efficient trapping would require bunching the ions both in space and velocity. An alternative method of loading the trap by directly injecting charged particles has been achieved 3 by using the presence of a neutral background gas to extract kinetic energy from the injected ions which permits some fraction of them to achieve stable orbits. This technique can employ various ion sources which continuously supply ions to the trap in the presence of the rf fields. The possibility of trapping ions produced in a supersonic expansion as an ion beam pulse suggests the possibility that a combination of the two methods might be used to load the trap with stable ions efficiently. For example, cluster ion beams have been formed 4 in He expansions which have narrow velocity widths v/v pulse.1 and propagate with v pulse 11 5 cm/s and spatial pulsewidths on the order of 1 mm. This spatial width becomes comparable to the diameter of a trap ring electrode for the more common trap designs. The analysis method developed in this paper is particularly appropriate to estimate the trapping efficiencies which ion injection from supersonic beams might realize. Of course, such calculations for an ideal trap would represent an upper bound but even this estimate is useful to examine the parameter range for the injection of supersonic ion beam pulses. ACKNOWLEDGMENTS We are indebted to Hanna Szöke for the principal programming effort, providing advice and well crafted code. We would also like to thank S. Pollack for the Poisson equation calculations, LLNL for providing the Numerov subroutines, and S. McDonald and G. Maalouf for help with program modifications along the way. This research was fully supported by The Rowland Institute for Science. APPENDIX A: SECULAR TRANSFORMATIONS A canonical transformation has been derived 1 for a general class of time dependent Hamiltonians P H X VXWXcos t i M A1 which can extract the highfrequency component of the time dependence at frequency from the canonically conjugate variables P X and X by transforming to new momenta and coordinates P Y and Y. The fast motion can be extracted most effectively when the the parameter (Y ) satisfies the condition Y WY M 1, A where the prime denotes differentiation with respect to y. The transformations are given by XYSY, P X P Y WY T Y sin t TY. A3 This form of the Hamiltonian given by Eq. A1 includes ion motion in an rf Paul trap for which VXWXcos t ev dc x y z r z ev rf cos t x y z r z and the expansion parameters are given by A4 ev rf x r z M q x, A5 4eV rf z r z M q z using Eq. 4b. As originally pointed out by Dehmelt, 9 the slow, harmonic oscillator motion of the trapped ion is expected for q x and q z 1, which is commonly referred to as the ion secular motion. Then, the degree to which the canonical transformations extract the fast motion at, or micromotion, depends on the usual parameter used to characterize the ion motion in the Paul trap. The transformation generating functions S(Y ) and T(Y ) are applied to the calculated trajectory variables to obtain the secular velocity and position for the trapped ion. These transformations were derived for a general potential of the form W(X) and V(X)X j and are of the form SY 1 j TY 1 j 1/j j1 Y cos t, j1/ j j1 Y cos t. A6 Since j for the Paul trap, these transformations are reduced to final form by setting n1/(j) and taking the limit lim SY exp Y cos t, n A7 lim TY exp Y cos t, n where the expansion parameter is subscripted to emphasize that in this case they are constants. The new variables, termed secular variables in Sec. II B, are then given by p i p i q im r i sin t exp q i cos t, A8 r ir i exp q i cos t, where ix,y,z and r i (x,y,z). Note that for the initial conditions at t ofr i, we also have p ip i. Since the secu J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
16 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 1437 FIG. 18. Comparison of the ion axial position vs time calculated from Eq. 3, light line, and the ion position after applying the secular transformation of Eq. A8, heavy line. FIG.. Comparison of the ion total energy vs time calculated from Eq. 3, light line, and the ion energy after applying the secular transformations of Eq. A8, heavy line. lar and trajectory variables are identical in this case, it presents an unambiguous point at which to initialize the ion cloud as discussed in Sec. II D. As shown in Ref. 1, these transformations are followed by a second similar transformation to further reduce the time variation at the frequency. The time dependence of the ion position z, velocity v z, and kinetic energy which was obtained for a solution of Eq. 3 for trap parameters V dc, q z.3. Both the trajectory variables and the transformed secular motion variables are shown in Figs. 18, 19, and to emphasize that the transformations given by Eq. A8 extract the fast motion to a high degree under these conditions. After these transformations are applied to the trajectory variables, the Hamiltonian exhibits fast time dependence only to lowest order in the parameter q i and is given by H x,ỹ,z ;t i p i M e Tx,ỹ,z q i i Fp i,r i ;toq i /4 A9 to within a constant set by the applied dc potential. The time independent potential, Dehmelt s pseudopotential, 11 is given by Tx,ỹ,z V dc x ỹ z V r z rf M r z x ỹ 4z A1 FIG. 19. Comparison of the ion velocity vs time calculated from Eq. 3, light line, and the ion velocity after applying the secular transformations of Eq. A8, heavy line. FIG. 1. The solution of Poisson s equation, Eq. C, for is shown by the heavy line. The contributions to from the trap potential T and the space charge potential sc are shown by light lines. This calculation was performed forac 6 cloud temperature of 3 K, N sc 1 4 ions and trap parameters a z, q z.3. J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
17 1438 J. H. Parks and A. Szöke: Simulation of trapped ion clouds APPENDIX C: SELFCONSISTENT SPACE CHARGE FORMULATION FIG.. The space charge potential energy e sc is shown for N sc 1 4 ions, trap parameters a z, q z.3 and several different C 6 cloud temperatures. The potential energy derived from Poisson s equation using Eq. C3, heavy lines, is compared with the Gaussian approximation given by Eq. 51, light lines. which yields harmonic oscillator ion dynamics. The first time dependent term is, reduced by the factor q i /.15 in the calculations here and F(p i,r i ;t) is given by F p i M cos t M q i r i 3 cos t 1 cos 3t8 V dc cos V rf t A11 which is composed of two terms of comparable magnitude. Residual time dependent terms in the expansion are reduced by higher orders of q i /. These transformations are applied at each time step of the differential equation to obtain the secular variables and calculate a secular kinetic and potential energy. This total conserved energy is then used to monitor the approach to thermal equilibrium of the ion cloud. APPENDIX B: COORDINATE TRANSFORMATIONS The collision calculation is simplified by using reference frames in which the ion velocity has been rotated into the z direction by a transformation matrix M. This matrix expresses a rotation about the z axis by an angle followed by a rotation of about the y axis. This transformation and its inverse are expressed by cos cos sin sin Mcos sin cos B1 sin cos sin sin cos, cos sin sin cos M cos 1 cos sin cos sin sin. B sin cos A selfconsistent treatment of the space charge interaction in ion clouds is required in order to extend ion motion calculations to lower temperatures 3 1 K which are of current experimental interest. For large clouds of N sc 1 4 ions, the rigorous approach of introducing sums over Coulomb interacting ion pairs is both limited by calculation runtimes and probably not required if only average cloud physics is of interest. The limitation of the mean field approximation is that the charge density is independent of the electrostatic potential arising from the trapped charge. Since this potential becomes more important as the density increases, the approximation fails at lower temperatures. The following treatment explicitly includes a calculation of the space charge potential to derive the ion motion. In thermal equilibrium, the ion charge density can be expressed by r,zen exper,z/kt C1 in which the total potential experienced by an ion is assumed to be a sum of the trap potential and a space charge potential T sc and the peak ion density n is defined below. Poisson s equation sc (r,z)4(r,z) can then be solved numerically for sc (r,z) using Eq. C1. The resulting differential equation in cylindrical coordinates for a cloud of N sc ions of mass M at temperature T is e N sc kt e dz rdre M kt r z, C where e e sc kt T kt C3 and T has been expressed by Eqs. 13 and 14 to rewrite Poisson s equation for. The normalization N sc n dz rdre C4 has been used to eliminate n in the derivation of Eq. C. The potential sc (r,z) can be accurately extracted from the solution for since T is a well defined analytical function of the coordinates. Figure 1 shows a solution for from Eq. C and the T and sc components calculated for a cloud of C 6 ions having N sc 1 4 ions, cloud temperature T3 K and trap parameters a z, q z.3. The space charge potential is shown in Fig. for various cloud temperatures and the Poisson solution is compared with the Gaussian approximation given by Eq. 51. The force exerted by the space charge field on the ion is then calculated by taking F i x,y,ze i sc for ix,y,z C5 which is used in Eq. 46 to calculate the ion motion. The force derived from the solution for at cloud temperature 3 K and N sc 1 4 ions is shown in Fig. 7. To incorporate F i into the trajectory calculation with minimal increase in J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
18 J. H. Parks and A. Szöke: Simulation of trapped ion clouds 1439 run time, the force given by Eq. C5 is first calculated for a three dimensional array of the spatial variables assuming a cloud at constant temperature and ion number. This array is then be used as a lookup table which is interpolated to define the force at each ion position in the trajectory calculation. This formulation of the space charge calculation should be a more accurate approximation for temperatures below 3 K since the explicit space charge potential and not the density is used to estimate the forces on the ion. 1 F. A. Londry, R. L. Alfred, and R. E. March, J. Am. Soc. Mass Spectrom. Ion Proc. 4, R. K. Julian, H. P. Reiser, and R. G. Cooks, Int. J. Mass Spectrom. Ion Proc. 13, F. Vedel, Int. J. Mass Spectrom. Ion Proc. 16, F. Vedel and J. André, Int. J. Mass Spectrom. Ion Proc. 65, 11985; F. Vedel and J. André, Phys. Rev. A 9, M. C. Doran, J. E. Fulford, R. J. Hughes, Y. Morita, R. E. March, and R. F. Bonner, Int. J. Mass Spectrom. Ion Phys. 33, ; R. F. Bonner and R. E. March, ibid. 5, ; R. F. Bonner, R. E. March, and J. Durup, ibid., R. E. March, Int. J. Mass Spectrom. Ion Proc. 118/119, 71199; R.C. Thompson, Adv. At. Mol. Opt. Phys. 31, N. W. McLachlan, Theory and Application of Mathieu Functions Oxford University, Oxford, D. Levesque and L. Verlet, J. Stat. Phys. 7, ; W. E. Milne, Numerical Solution of Differential Equations Wiley, New York, 196; E. H. Auerbach, Comput. Phys. Commun. 15, H. G. Dehmelt, Adv. At. Mol. Phys. 3, ; 5, ; H.G. Dehmelt, in Advances in Laser Spectroscopy, edited by F. T. Arecchi, F. Strumia, and H. Walther Plenum, New York, 1983, pp L. V. Hau, M. M. Burns, and J. A. Golovchenko, Phys. Rev. A 45, D. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr., Adv. At. Mol. Phys. 19, L. D. Landau and E. M. Lifshitz, Mechanics Pergamon, Oxford, 1976, pp. 54, R. E. March, A. McMahon, F. A. Londry, R. Alfred, J. F. J. Todd, and F. Vedel, Int. J. Mass Spectrom. Ion Proc. 95, S. C. Smith, M. J. McEwan, K. Giles, D. Smith, and N. G. Adams, Int. J. Mass Spectrom. Ion Proc. 96, P. Langevin, Ann. Chim. Phys. 5, and an English translation which appears in E. W. McDaniel, Collision Phenomena in Ionized Gases Wiley, New York, 1964, p. 71; G. Gioumousis and D. P. Stevenson, J. Chem. Phys. 9, W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, nd ed. Cambridge University, New York, 199, p F. Vedel, J. André, and M. Vedel, J. Phys. 4, H. A. Schuessler, E. N. Fortson, and H. G. Dehmelt, Phys. Rev. 187, J. H. Parks, S. Pollack, and W. Hill, J. Chem. Phys. 11, D. A. Church, Phys. Rev. A 37, ; H. Schaaf, U. Schmeling, and G. Werth, Appl. Phys. 5, ; R. Iffländer and G. Werth, Metrologia 13, ; R. D. Knight and M. H. Prior, J. Appl. Phys. 5, ; H. Walther, Adv. At. Mol. Phys. 31, R. E. March and R. J. Hughes, Quadrupole Storage Mass Spectrometry Wiley Interscience, New York, 1991; J. C. Schwartz and I. Jardine, Rapid Commun. Mass Spectrom. 8, ; J. D. Williams, K. A. Cox, R. G. Cooks, S. A. McLuckey, K. J. Hart, and D. E. Goeringer, Anal. Chem. 66, M. Nand Kishore and P. K. Ghosh, Int. J. Mass Spectrom. Ion Phys. 9, ; J. F. J. Todd, D. A. Freer, and R. M. Waldren, ibid. 36, J. N. Louris, J. W. Amy, T. Y. Ridley, and R. G. Cooks, Int. J. Mass Spectrom. Ion Proc. 88, ; A. McIntosh, T. Donovan, and J. Brodbelt, Anal. Chem. 64, P. Milani and W. de Heer, Rev. Sci. Instrum. 61, ; J. P. Bucher, D. C. Douglass, P. Xia, and L. A. Bloomfield, ibid. 61, J. Chem. Phys., Vol. 13, No. 4, July 1995 Downloaded 16 Jul to Redistribution subject to AIP license or copyright, see
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