BUFFER GAS COOLING OF ION BEAMS. R.B. Moore (January 2002)

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1 BUFFER GAS COOLING OF ION BEAMS R.B. Moore (January 00)

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3 Contents 1. Introduction 1. The Containment of Ions in a Buffer Gas 1.1 A brief history of ion confinement Qualitative Description of RF Ion Confinement RF Ion Confinement by Quadrupole Electric Fields Ion Temperature and Phase Space Density RF Distortion of Action Diagrams Matrix Calculations for RF Quadrupole Containment Devices RF Heating of The Macromotion The Effect of Higher-Order Multipoles The Damping of Ion Motion by a Buffer Gas A brief history of the damping of ion motion by buffer gas Ion Mobility Mobility and Exponential Decay of Velocity Mobility at Reduced Gas Pressures Simulation of Ion Motion in an RF field with Buffer Gas High Drift Velocities - Collision Cross-Sections Extrapolation of Ion Mobility to High Ion Velocities The Stopping Power of Gases Diffusion The Diffusion Equation Solutions of The Diffusion Equation The Monte-Carlo method Space charge effects Some Practical Considerations Vacuum Mechanical High voltage References 50 Appendix A - Units for Ion Dynamics 5 Appendix B - Vacuum Units 54

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5 1. Introduction The buffer gas cooling of ion beams is a subject of considerable recent interest. This is because of the realization that a low-pressure gas, usually referred to as a buffer gas, can be used for cooling ion beams to close to the temperature of the gas itself. (See, for example, [1]). The manipulation of ions in low pressure gases is a very old subject, dating back to the nineteenth century. However, manipulations that made use of the cooling properties of the gas, i.e. the tendency of the ion temperature to approach that of the colliding gas molecules, were hampered by the diffusion associated with such cooling. Without containment of the ions they simply become lost within the gas. Useful damping (i.e. cooling ) of ion motion by buffer gas was therefore only possible once ion confinement by electromagnetic fields became feasible. Such containment was first proven to be feasible by Paul and coworkers[]. Its application to ion beams in a background of buffer gas was first demonstrated by Teloy and Gerlich at Freiberg in 1974[3]. Subsequent developments[4-6] have spanned chemistry and physics applications, where the technique has become widely used for studies of the interactions of ions and molecules at pressures of about 1 Pa. This note summarizes the principles involved in the containment of the motion of ions in buffer gas and in the resultant cooling of the ion motion. It also presents some practical considerations for the application of the principles to the cooling of ion beams.. The Containment of Ions in a Buffer Gas.1 A brief history of ion confinement The first demonstration of ion confinement in an electromagnetic field was by Penning[7] who used an electric potential well along the axis of a magnetic field. In this way the radial motion of an ion was restrained by the magnetic field to be circular while the axial motion was retrained by the electric field to be oscillatory. This configuration was used by Penning to study ions produced by electric discharges in gases, and has since become to be known as the Penning Trap. However, the lifetime of ions in Penning s device was short; of the order of a millisecond. This was because collisions with the background gas disturbed the centers of the circular orbits, displacing them from the axis of the system. This allows the radially repulsive electric field, which from the divergence theorem of Gauss must exist when there is an axially confining electric field, to act on the orbit centers, thereby driving the ions out of the trap. Penning traps with dc fields therefore only provide total confinement in the absence of such orbit perturbations. Total confinement of ions was first demonstrated by Paul and co-workers in the work cited above. This was achieved by simply oscillating the electric field used by Penning. Such a field would, of course, have no net impact on an ion over a complete cycle if it was uniform in space. However, because the Penning electric field increases in strength with distance from an equilibrium center, at which it is zero, the net impulse over a cycle of the field for an ion not at the center is not zero. This is because 1

6 the phase of the forced oscillation of the ion will be such that its maximum excursion from the center occurs when the field is a maximum towards the center and its closest approach to the center is when the field is a maximum away from the center. The field during the attraction half of the oscillation is therefore stronger than the field during the repulsive half, resulting in a net attraction impulse. Since the force is oscillating the fact that it is radially repulsive while it is axially attractive, and viceversa, doesn t matter; it ends up providing an average focusing impulse on both motions. In fact, no magnetic field is required to restrain the radial motion, making traps based on this principle relatively cheap, reliable and easy to operate. Such traps, now called Paul Traps, have become very common instruments in the modern physical sciences, particularly chemistry, space science, physics and vacuum technology. The results of a force such as that on an ion in a Paul trap were thoroughly studied in the 19 th century in the context of perturbations of the orbits of bodies in the solar system by the periodic inputs due to other orbiting bodies. The properties of the solutions of the equation of motion for the simplest possible such force, a sinusoidal variation in time with a strength proportional to the displacement from an equilibrium orbit, were studied by Mathieu in the mid 19 th century and his solutions are available in any good mathematical reference book. The following will be a summary of the most important features of these solutions.. Qualitative Description of RF Ion Confinement Fig. 1 shows representative ion motions in various strengths of an oscillating electric field (obtained by integrating Mathieu s equation numerically). The overall restraint provided by the oscillating field is seen to result in a slow simple harmonic motion which rises in frequency as the electric field is increased. This motion is often referred to as the macromotion although it is still often called the beta motion because of the original parametrization used by Mathieu to express his solutions. The forced oscillatory motion that produces the restraint resulting in the macromotion is seen as a higher-frequency oscillation superimposed on the macromotion. This is sometimes called the micromotion but, because confinement of ions in an electric field usually requires an oscillation at a radiofrequency, it is more often referred to as the RF motion. It is seen that the amplitude of this RF motion is indeed proportional to the displacement of the ion from the center of the motion and that the ratio of its amplitude at the peak of the macromotion to the macromotion amplitude itself grows with the strength of the electric field. However, it is clear that the motion becomes unstable at a critical value of the field strength. This is easily understood since a very strong field will result in a motion in which the forced oscillation of the particle carries it beyond the equilibrium center, thereby causing a reverse in the direction of the force. Essentially, the amplitude of the driven oscillation overcomes that of the macromotion. This is seen to occur at a field strength such that the macromotion has a frequency half that of the RF. At this point the perturbation of the micromotion frequency by the macromotion has resulted in its reaching the same value, the micromotion and the macromotion becoming indistinguishable.

7 Z - CM V = 75 ; q V = 88 ; q = 0.8 RF z = 0.86 RF z ν = 0.4 ν = z z Z - CM TIME - RF CYCLES TIME - RF CYCLES 1.0 V = 170 ; q = 0.53 RF z ν = 0. z 1.0 V RF = 89 ; q z = 0.91 ν = 0.5 z Z - CM 0 Z - CM TIME - RF CYCLES TIME - RF CYCLES 1.0 V RF= 95 ; q z = 0.93 z motion unstable Z - CM TIME - RF CYCLES Fig. 1. Graphs of the motion of an ion in an oscillating quadrupole electric field of various strengths. The graphs were obtained by Runge-Kutta integration of the equations of motion. Incidentally, the limit of stability being when the macromotion frequency is half that of the RF is the reason Mathieu selected the dimensionless parameter β to express the frequency component of his solutions. Essentially, it is twice the ratio of the macrofrequency and the RF frequency, and therefore ranges from zero to unit for the stable solutions. In addition, it is probably worth knowing that a full treatment of Mathieu s equation shows that the region of stability can be expanded (or contracted) by the application of a dc restoring (or repulsive) electric field component. Also, there are narrow higher regions of stability with values of β much larger than unity. Information on the overall Mathieu stability diagram can be found in almost any book on electromagnetic trapping. (See, for examples, [8, 9]). However, operation with a dc component to the trapping potential, or in the higher stability regions, is not of much use for buffer gas cooling of ions. 3

8 .3 RF Ion Confinement by Quadrupole Electric Fields The physical parameter which characterizes the various solutions of the Mathieu equation shown in fig. 1 is the dimensionless φm q = mω x x= 0 (1) where the motion is considered to be in the x direction and m is the mass of the particle, ω the angular frequency of the driving force and the second derivative is that of the potential energy φ M that the particle would have as a function of displacement when the potentials are at a maximum in their cycle. To confine ions the electric field is usually created by electrodes specifically shaped to produce as close to a pure quadrupole electric field as possible. In such a field, the second derivative of the potential is uniform throughout the field, and the restoring force therefore perfectly proportional to the distance from the equilibrium center. In a Paul trap, which uses the axially symmetric quadrupole field, the electrodes are, ideally, hyperboloids of revolution conforming to for the ring electrode and r z = r o () z r = z o (3) for two end electrodes, where r o is z o. It can be easily seen that this results in the electrode geometry shown in fig. where r o is the minimum inner radius of the ring and z o is the distance of each end electrode from the field center. r r o z o z Fig.. The geometry of the electrodes of a Paul trap. The potentials achieved by this electrode structure are usually expressed in terms of the voltage amplitude V between the ring electrode and the end electrodes (which are, of course, both at the same potential). From (1) the parameter q then becomes 4

9 q r ev = mr ω o RF (4) for the radial motion and q z ev = mz ω o RF (5) for the axial motion. Thus q z is twice q r, and so a Paul trap using only an oscillating field with no dc component provides twice as strong a confinement for axial motion as it does for radial. For the confinement of ions in a beam the electric field must be an axiperiodic quadrupole which, ideally, is created by four long vanes with surfaces conforming to x y = r o and y x = r o (6) for the two sets. The geometry of such a set of vanes is shown in fig. 3. Here it is seen that the q parameter, as expressed by (4) is the same for both x and y motions. y r o r o x Fig. 3. The geometry of electrodes that form the axiperiodic quadrupole required for confining an ion beam. The rod structure that is typically used in mass filters to approximate the hyperbolic vanes is shown. However, the difficulty of machining the ideal vane structure has led to most such structures being formed from rods. Studies have shown that the closest approximation to a quadrupole field with such a structure is achieved when the diameter of the rods of times the separation of opposite rods[10]. However, it should be noted that the effective r o for using (4) to calculate the q from a voltage amplitude V between the rods is no longer exactly the actual distance of the inner surfaces of the rods from the center. Rather, calculations show it to be xxx. In general, the containment of an ion beam during buffer gas cooling will not require a high degree of purity in the radiofrequency quadrupole field. Rather, the shape of the electrodes and their relative placement will depend on other practical considerations, such as the containment of the gas and the avoidance of electrical breakdown between the electrodes and between the electrodes and ground. The 5

10 strength of confinement for a given voltage on the electrodes will then have to be calculated from numerical methods. Such methods currently in general use fall into two broad categories; finite element methods[11] and finite difference methods[1]. The finite element method is generally favoured by engineers, who are often as much concerned about the overall lumped properties of a system (such as capacitance) as they are about the electric fields that may be inside a structure. For such properties, which are related to the electric fields at the surface of electrodes, the finite element method is superior. However, for fields in regions far removed from the electrodes, as is usually the case for charged particle confinement, finite difference methods (usually referred to as relaxation methods, as employed in the popular ion trajectory simulation program SIMION[13]) tend to be superior. However, either of these methods give more than sufficient accuracy for designing ion containment devices for buffer gas cooling. To extract the confinement parameter q from calculated electric fields, the second derivative of the potential for a specific set of voltages on the electrodes is evaluated at its minimum and used in the version of (1) that is applicable for electric potentials (i.e. the electrical potential must be multiplied by the electronic charge). If it is convenient to express this parameter in terms of the actual voltage applied across adjacent electrodes, as in (4), then the effective r o can be calculated from r o V = φe x x= 0 (7) The actual strength of confinement that is most appropriate for a given application of buffer gas cooling will depend on a great number of practical considerations. However, in general it should not be stronger than that corresponding to an q of about 0.5 (unless very thick buffer gas cooling is to be employed; see below). For such values of q the following approximations are good to better than 10%. First the macro-oscillation frequency ω m is simply proportional to q and the radiofrequency ω RF as ω m q = ω. (8) RF Thus, restricting q to 0.5 leads to values of the macro-oscillation frequency which are 16% or less of the radiofrequency. (In this approximation the β parameter is seen to be q /.) Because the macromotion is simple harmonic it is often attributed to a so-called pseudopotential well. In this well the macromotion energy is regarded as being conserved, the kinetic energy at its maximum excursion, where it is zero, having gone into potential energy. This is often more useful than q for expressing the strength of confinement since the well depth corresponds to the maximum macromotion kinetic energy that can be confined. However, in using this concept it must be noted that because of the RF motion the permissible macromotion amplitude is less than the distance of the electrodes from the center of the motion. Once this permissible amplitude A m is set, the pseudopotential well depth ϕ m can be defined as 6

11 φ = 1 m m ω A ω e = 1 16 e q A m m m RF m. (9) Often it is conceptually more useful to consider the effective electric field E eff that would be associated with such a potential well. By substituting the displacement r for the amplitude term in (9), so as to get the functional form of the pseudopotential, simple differentiation gives E eff m = 1 e q ω RF 8 r. (10) The second very useful approximation is that the ratio of the maximum amplitude of the RF motion to the amplitude of the macromotion is simply q/. However, because the radiofrequency is higher, this will result in the radiofrequency motion having a maximum velocity with is times that of the macromotion. The RF motion can therefore reach peak energies which are twice that of the macromotion. However, since the amplitude of the velocity of the RF motion is itself proportional to the displacement of its center from the equilibrium point, and this displacement follows the sine function of the macromotion, the average of the RF energy over a complete cycle of the macromotion is half its peak value, or just equal to that of the macromotion. However, a word of warning here. While the above analysis gives a qualitative feeling for the relative importance of the RF motion in the average kinetic energy of ions in RF quadrupole devices, the kinetic energies of the macromotion and the RF motion cannot be so easily separated. This is because, even in the approximation of linear superposition of simple harmonic motions for the RF and the macromotion assumed in the above, the kinetic energies, being associated with the square of the velocity, do not superimpose linearly. This point would appear to be obvious but the literature on the subject of the mean kinetic energy of ions in RF quadrupole devices is rife with conceptual errors related to this fact. The only proper method for dealing with the superposition of the RF motion on the macromotion seems to be through the use of action diagrams, i.e. projections of the phase space volume of the ion collection into -dimensional momentum-displacement diagrams for the canonical coordinates of the system; in the case of motion in the absence of magnetic fields, the simple x, y and z coordinates. This subject is dealt with in the following section. 7

12 .4 Ion Temperature and Phase Space Density The most persistent conceptual errors in dealing with the kinetic energy of ions in RF quadrupole devices seem to be concerning the ion temperature. The RF motion, being coherent, does not contribute directly to the temperature. This is because coherent motion of a collection of particles does not occupy any volume in phase space and since temperature is defined, fundamentally, in terms of the density of particles in phase space, coherent motion cannot enter into its definition. This is most easily seen in the case of a gas traveling in a container which is itself moving. Clearly, the coherent motion of the molecules due to the motion of the container cannot be included in the temperature. In the case of the RF motion of a collection of ions in a Paul trap, the effect of its coherency is most easily seen in the action diagrams. In such action diagrams, the trajectories of particles undergoing simple harmonic motion are right ellipses. The RF motion of ions is approximately simple harmonic, with an amplitude that is proportional to the distance of the ion from the equilibrium center. The trajectories of a collection of ions at different distances will be therefore approximately ellipses as shown in fig. 4. Here the coherency of the motion lies in the fact that at any particular time all the ions have the same phase of rf motion. As a result all the ions form a line in the action diagram, a line which simply wobbles with the rf phase. p z z Fig. 4. The action diagram trajectories of a collection of ions undergoing only rf motion in a quadrupole containment device. Such a line has no action area, and therefore the motion associated with it cannot enter into the definition of a truly thermodynamic temperature. On the other hand, the macromotion is pure simple harmonic and can have any amplitude and phase. The trajectories of a collection of ions at random amplitudes and phases will therefore be ellipses centered on the equilibrium position but with the points representing the ions filling the ellipse corresponding to the motion of the most energetic ion of the collection. (See Fig. 5.) 8

13 p x mω A x max. O x A max. Fig. 5. The action diagram a collection of ions undergoing simple harmonic motion at random amplitude and phase. The local density of particle points within such an action diagram is very important in the dynamics of the particle collection. Essentially, by Liouville's theorem any motions in force fields that can be expressed by a Hamiltonian and for which the forces in the three coordinates are independent will result in this density for any local grouping of points in an action diagram remaining constant for that group. Furthermore, for thermodynamic equilibrium within the whole collection this density dn/ds will be governed by the energy En of the particles within that group and the temperature T of the collection through the fundamental relationship of statistical mechanics En kt dn α e (11) ds where k is Boltzmann's constant. By noting the relationship between En and the displacement and momentum coordinates and integrating over the area of the action diagram the actual form of the relationship can be obtained; d N = Nω dxdp x πkt e mω kt x + p x mω. (1) Thus the particle points in an action diagram have a gaussian density distribution in both the displacement and momentum coordinates. The standard deviation parameters of these quassians are σ x = 1 ω kt m (13) σ px = mkt (14) The physical significance of these parameters is that they correspond to an amplitude of oscillation at which the energy is 1/kT. In principle, the extent of the particle points in an action diagram is limitless. In a real containment device the motion will be out to the physical limits of the device. Ions that have energies such that their motions can lead to collisions with an electrode 9

14 will generally be lost. The random collisions between ions that would lead to thermal equilibrium will always result in some ions reaching this energy. The continuous removal of such ions therefore leads to a lowering of the temperature of the ion collection until the process is no longer significant. This process is sometimes called "evaporative cooling". In estimating the effective size of an ion collection it is useful to note the fraction of the collection that lies within particular values of the standard deviation parameters (13) and (14). These can be obtained by integrating (1) out to particular values of amplitude A, resulting in the number N' that have this amplitude or less being A N = N(1 e σ ). (15) Thus an ellipse corresponding to one sigma would encompass about 40% of the particles, an ellipse of two sigma about 87% and an ellipse of three sigma about 99%. In beam optics it is common practice to take an ellipse that encompasses 95% of the particles as being the effective action area. This corresponds to an amplitude of oscillation of about 6 sigma. The ellipse corresponding to macromotion of frequency ω m at this amplitude will have an area S, i.e. an action, given by kt S = 6π. (16) ω m If kt is expressed in electron-volts and the angular frequency in radians per microsecond, then the action has the very convenient units of ev-µs. As a representative case, a collection of ions oscillating with an angular frequency of one radian per microsecond at a temperature of twice room temperature (i.e. 50 mev) will exhibit an effective action of about 0.3 π ev-µs. For an ion collection that has been accelerated into a beam the transverse action can be thought of as a normalized emittance. This can be converted into the ordinary emittance by simply dividing by the central momentum, p o, of the beam; ξ = S p o. (17) To retain the convenience of the ev-µs the momentum here should be expressed in ev-µs/mm. As a representative value, a beam of 50 kev ions of mass 100 Daltons (amu) will have a momentum in these units of about 30 ev-µs/mm. (The momentum of other representative beams can easily be derived from this figure by noting that it is proportional to the square root of the product of energy and mass.) The emittance of such a beam with the representative action given above would therefore be about π-mm-mrad..5 RF Distortion of Action Diagrams As pointed out above the RF motion does not contribute any action area. What it does is distort the action ellipse of the macromotion without changing its area. Furthermore, because the force field is linear, i.e. corresponds to a quadrupole potential, the distortion of the ellipse will be simply into another ellipse. 10

15 This is seen in the numerical simulations of the action of a collection of ions in an RF quadrupole fields for a representative case shown in fig. 6. A case when the confinement is not so strong is shown in fig. 7. These diagrams show the importance of considering action in dealing with RF quadrupole confinement of ions. In particular it can be seen that while the breathing of the ion collection, i.e. it expansion and contraction through an RF cycle, is considerably reduced at reduced confinement strength the momentum breathing is not. Thus there can still be significant distortion of the action diagram at RF phases where the RF momentum is a maximum (i.e. when the RF displacement is zero). p x 5 0 o ev-µs/mm 90 o 180 o SHM 70 o -5 5 x mm -5 Fig. 6. Action diagram for combined macromotion and RF motion in quadrupole field confinement. The phase of the RF, with zero defined as zero electric field going positive in the x direction, is indicated on the diagram. The ions have a mass of 100 amu and the trap is operating at a q value of 0.5 at an RF of 500 khz. 0 o p x 5 ev-µs/mm 180 o 90 o SHM 70-5 o 5 x mm -5 Fig. 7. Action diagrams for the same condition as for fig. 6 but for ions of mass 1000 amu. Thus while the RF does not increase the action area of the collection, extraction of the collection from the confinement device will result in different action diagrams at different RF phases. Since a beam is usually continuous its effective action diagram 11

16 averaged over an RF cycle will be a sort of four-leaf clover shape which is the envelope of all the ellipses over that cycle. This has been confirmed in experimental observations of a beam extracted from an RF quadrupole rod structure in the work cited above[1]. The effective emittance of a beam extracted from an RF rod structure is therefore greater than that predicted by (16), usually by about a factor of two..6 Matrix Calculations for RF Quadrupole Containment Devices The fact that the action diagrams of a collection of thermalized ions in a RF quadrupole containment device are ellipses suggests the use of simple matrix algebra to determine the physical size of the collection. This is because the extent of such a collection in any coordinate is simply the horizontal extent of the action diagram ellipse and the transformations of this ellipse as the action diagram evolves can be calculated by multiplying the so-called Twiss parameters of the ellipse by a simple 3 by matrix. For a good introduction to the theory of this method see the text-book on beam optics by Wollnik[14], which should be on every ion-manipulator s shelf. For a thorough discussion on how it is applied in radiofrequency quadrupole devices see Dawson[9]. What will be given here is an abbreviated introduction and a summary of the basic equations needed for the particular application of beams in an RF quadrupole rod structure. Before engaging in this sort of calculation it should be kept in mind that it is only first order beam optics. It therefore does not include the effects of aberrations, i.e. non-quadrupole field components. However, it is very fast, in practice allowing beam profiles to be determined over all ranges of RF phases in seconds on a modern desktop computer. It therefore checks out the first order solution of the equations of motion for the collection of ions in the device very quickly, and a general rule is that if a device doesn t work in first order it will not work in higher order. The starting point of such an approach is the equation of an ellipse in terms of the Twiss parameters. For an ellipse in the x-y plane and centered on the origin this is By + Axy + Cx =ε (18) where ε is the product of the semimajor and semiminor axis of the ellipse; in other words, its area divided by π. The relationship of the Twiss parameters to the more familiar description of an ellipse is that the length of the semimajor axis is ( ) ε L= H + 1+ H 1 (19) 1

17 the length of the semiminor axis is ( ) ε l = H + 1 H 1 (0) and the angle θ of the major axis with the x axis is A tan θ = B C (1) where H = ( B+ C)/. The relationship of the parameters to the four critical points of an ellipse is shown in fig. 8. Thus if the parameter B is known the width w of the beam of action area ε can be determined simply as w = eb. () Inversely, if the critical points of an ellipse are known, they can be easily used to determine its Twiss parameters. y 1 ( ) A C ε ε B ( ) 1 1 ( ) A B ε ( εc) 1 1 ε ( C) x ( εb) 1 Fig. 8. The relationship between the critical points of an ellipse and its Twiss parameters. The importance of the Twiss parameters lies in the ease with which they can be recalculated under any transformation of the ellipse which preserves its area. Such a calculation is founded on the elements of the transformation matrix for points on the ellipse. Any linear transformation of those points into points on a new ellipse of the same area can be represented by a simple matrix of the form ' x m y = m m 11 1 m 1 x y (3) where the determinate of the matrix is unity. 13

18 The corresponding transformation of the Twiss parameters is ' B A C m11 m11m1 m1 = m m ( m m + m m ) m m m1 m1m m B A C (4) For the case of action diagrams, the y coordinate is simply the momentum p. The transformation (3) then just becomes that due to the electric quadrupole field acting on an ion of momentum p and displacement x from the equilibrium position. It is relatively easy to show that for a strength parameter K defined by K e φ = m x o (5) the transformation for a time interval t during which the potential is constant is Kt x Kt x ' sin mk p = cos p mk Kt Kt sin cos (6) when the electric field is a restoring field, i.e. focusing, and Kt x Kt x ' sinh mk p = cosh p mk Kt Kt sinh cosh (7) when the electric field is repelling the ions from the equilibrium point. By taking a time interval which is short compared to an RF cycle (usually one degree of a RF cycle is small enough) the electric field can be regarded as being constant, at its average value for the interval. The effect of the RF field over any period of time can then be obtained by successively multiplying matrices corresponding to each step. The relationship of the K parameter (usually used in beam optics) and the q parameter used in RF electric quadrupole confinement is simply K q = ω RF. (8) One of the great advantages of the matrix method of dealing with action diagrams of a beam as it progresses through a RF quadrupole rod structure is that axial fields can be also easily included in the calculations. A uniform axial field poses no problem whatsoever since such a field has no effect on the transverse action diagram. However, an axial field with a gradient has focusing, or defocusing, properties which must be taken into account. This is because by the divergence theorem φ 1 φ = r. (9) z o o 14

19 An axial electric field which is increasing in the z direction will therefore focus ions toward the axis while one that is decreasing will be defocusing, the strength parameter K for such fields being K z e φ = m z o. (30) This effect can be added to the effect of the RF quadrupole field by simply multiplying by its corresponding matrix at each step of the calculation for the RF field. However, it should be noted here that to prevent double counting of the effect due to the drift of the beam during the interval of the calculation the proper transform matrices are ' x x p = 1 0 mkz Kzt sin 1 p (31) for the focusing case and x x ' p = 1 0 mkz Kzt sinh 1 p (3) for the defocusing case. (The focusing and defocusing of the axial field gradient is taken to be equivalent to an infinitesimal thin lens at the end of the RF step.) Such calculations are particularly valuable when dealing with injection of a beam into an RF quadrupole rod structure while it is being electrically decelerated. Here one starts with the action ellipse for the given beam and performs the matrix calculations to determine its envelope for the various phases of RF at which the ions enter. A representative result of such a calculation is shown in fig. 9, which is the design of the quadrupole rod structure developed at McGill for decelerating and capturing an ISOL beam of up to 60 kev of medium mass ions. Fig. 9. A first-order matrix calculation of the envelope of a beam entering an RF quadrupole rod structure. The beam action diagram was determined for the start of the envelope on the left using a Runge-Kutta simulation of the deceleration of a 60 kev ISOL beam of diameter 10 mm and emittance 0π-mm-mrad down to 1 kev. The multiple envelopes are for various phases of RF. The total time for such a calculation, using a Power PC of 100 Mhz, would be about 10 seconds. 15

20 .7 RF Heating of The Macromotion While the RF motion does not enter into the temperature directly, in the presence of a buffer gas it will end up heating the ions and thereby indirectly raising the temperature. This is because collisions of the ions with buffer gas molecules will cause some of the instantaneous RF energy to be scattered incoherently into the macromotion. The degree to which this happens will determine the equilibrium temperature of the ions. In fact, the heating of the macromotion by collisions with buffer gas molecules due to the RF motion completely overcomes the cooling effect of such collisions when the ion mass is less than that of the buffer gas molecules. Thus buffer gas cooling cannot be used in RF quadrupole devices for light ions whose mass is about equal, or less than, that of the buffer gas molecules. This was studied early in the history of Paul trap development by Dehmelt[15], who gives a very understandable treatment of the subject. In any case, even when the ion mass is considerably greater than that of the buffer gas molecules, the rise in ion temperature due to the RF motion cannot be estimated by simple considerations. In general, it has to determined by actual observation of the action diagram of an ion collection in an actual device or by Monte-Carlo simulation of the ion-molecule interactions (see section 4.). However, some general conclusions can be drawn from theoretical considerations. The most important aspect of the temperature rise due to RF heating is that is will be dependent on the number of ions in contained in the device. This is because of the Coulomb repulsion due to the space charge of the collection. This repulsion will force ions out into the higher field region where they will experience RF motion at velocities that are proportional to the displacement from the center. They will therefore have RF energies which are proportional to the square of this displacement. In the case of a Paul trap the electric field at the surface of a collection will be proportional to the number of ions and the inverse square of the average radius of the collection. (The collection will generally not be spherical because of the difference in confinement strength in the radial and axial directions and because of the RF distortion.) Since the electric field that balances this space charge field at the edge of the collection will be proportional to the radius, the cube of the resultant radius of the collection will be proportional to the number of ions. (This is the so-called constant density model first proposed by Dehmelt[15].) The average RF energy will therefore be proportional to the /3rds power of the number of ions in the collection. If a constant fraction of this energy is transformed into heat by buffer gas collisions, then the RF heating will be proportional to the /3rds power of the number of stored ions. In the case of the confinement of an ion beam, the electric field at the edge of the beam due to space charge will again be proportional to the number of ions stored but now only inversely proportional to the radius. It is the square of the resultant radius which will now be proportional to the number of ions in the collection. The average RF energy will then be proportional to the number of ions and so the RF heating of an ion beam will be also proportional to the number of ions in the device. Since, for a given drift velocity, the current will be proportional to this number, then the RF heating will be proportional to the beam current. 16

21 To test this theory, and to observe the actual temperature that was reached in a given set of circumstance, two studies were undertaken at McGill. The first was the Ph.D. thesis work of Lunney[16] which studied the temperature of ions in a Paul trap. The second was the Ph.D. thesis work of Kim[1], cited above, in which the temperature of ions in an RF quadrupole rod structure was studied. In both cases the general trend predicted by the theory was confirmed, although in the case of the rod structure the range of ion currents that was studied was insufficient to draw firm conclusions. However, more important, the temperatures of the ions for low numbers in the Paul trap and for low currents in the rod structure was found to be less than twice that of the buffer gas itself. In the case of the beam cooler, the rise in temperature was only noticeable at currents greater than 100 pa and was about ev/na..8 The Effect of Higher Order Multipoles In general, a real device will have higher multipole fields than the quadrupole. The main effect of such fields will be to distort the action diagrams so that they are no longer pure ellipses. In addition they will cause a coupling of the canonical motions so that they are no longer independent. The action in the canonical coordinates may then no longer be conserved, leading to an effective heating of the confined ions. However, since they are due to higher-order multipoles these effects will be confined to the outer regions of the confinement device where these fields become important. For the cooling of ion beams onto the axis of the system, where there has to be sufficient room to prevent the high-energy tail of the statistical distribution of ion energies from causing significant losses, these effects are usually not of much importance. In fact the electric field component that can cause the most trouble is, strictly speaking, not an aberration at all. This is a dipole field that can arise from asymmetry of the electrode structure or from asymmetrical potentials on the electrodes. Such a field component is, of course, uniform throughout the device and, particularly important, occurs on the axis of the system. Thus it leads directly to RF heating of the ions. Great care should therefore be taken in maintaining the symmetry which avoids this component. The lack of any difficulty with the higher order multipoles is indicated by the fact that devices built specifically with higher order multipoles as their confining fields[3, 4, 6] have been used quite successfully in both physics and chemistry. The attraction of such devices lies in the relatively large region in which the ions can move without strong influence from the RF electric field. They only experience this field for the short period when they reach the wall, whereupon they are thrown, rather violently, back in. This can be advantageous for certain uses of a containment device, such as when the ions are to be contained for sustained interaction with background molecules for studies in chemical reactions, where the relatively long period for which they have little RF motion can be important. However, for beam cooling such fields do not seem appropriate. This is because of the relatively large radial extent of the collection compared to that when using a quadrupole field. Thus even if the device allow ions to reach the same temperature as in a quadrupole device, the resultant 17

22 beam emittance will be larger. This, and the difficulty of mathematically simulating the action of such devices, has led to their exclusion from these notes. 3. The Damping of Ion Motion by a Buffer Gas The considerations of the previous section have shown that ion containment in devices that include a buffer gas can lead to ion temperatures that are close to, if not equal to, the temperature of the buffer gas itself. Thus ion collections can be rendered to energy spreads of the order of 5 mev by devices working at just room temperature. By employing devices at cryogenic temperatures the energy spreads could be reduced even further. With the ability of electromagnetic devices to confine ions to very small volumes this leads to the possibility of preparing ion collections with extremely small phase space volumes. In turn, if such ion collections are reaccelerated the result will be beams of extremely low emittances. Such cooling would be very beneficial in increasing the resolution and transmission of devices used to observe and measure trace quantities of ions where the temperature of the ions from a source is much higher than that of a background gas. For example, it could be used to shrink the size and momentum spread of an ion beam entering an RFQ mass filter. It could also be used to decrease the emittance and energy spread of an ion beam, properties which are directly related to the ion temperature, by decelerating the ions to low velocity upon entering the cooling gas and then reaccelerating them to the desired energy. The possibility of cooling ion beams is also extremely important in the collection of such beams in an electromagnetic trap. What remains to be considered is the deceleration and capture of ion beams into such devices. The deceleration of ion beams is a very old subject, dealt with most thoroughly in the deceleration of ions for soft deposition on surfaces for studies in condensed matter physics[17]. Essentially, if such deceleration is done properly, i.e. in devices in which aberrations are minimized, the normalized emittance (transverse action) of a beam is conserved. What is presented to the cooling device is then a beam with a central energy, an axial energy spread and a given action diagram. A typical beam for which cooling would be desired is ISOL beam of 50 kev medium mass (100 Dalton) ions with an emittance of 30π-mm-mrad. Such a beam would have a normalized emittance of about 10π-eV-µs, and would have an energy spread between one and ten ev. Feasible electrostatic deceleration devices can decelerate such beams to about 100 ev without appreciable increase in normalized emittance[18]. The design problem then becomes to capture such a beam in an RF confinement device. By making an RF rod structure large enough and capable of withstanding sufficient RF voltage the transverse action of such a beam can be contained. By segmenting the rod structure so that a retarding axial electric field can be applied, the axial energy of the beam can then be further reduced to about 10 ev. (This was the goal of the design shown in fig. 9.) What is left then is to provide sufficient buffer gas to damp the ion motion to thermal velocities. Also, it is sometimes of interest to know the damping time-constant of the ion energy so that the time that the ions have to remain in the buffer gas can be determined. (This can be very important for collecting short-lived radionuclides or higher reactive ions, such as those of xenon, 18

23 that are easily lost to trace impurities in the buffer gas.) Fortunately, there is a wealth of information that allow such estimates to be made. 3.1 A brief history of the damping of ion motion by buffer gas Buffer gas damping of ion motion in a trap was first demonstrated by Hugget and Menasian in 1965 for Hg + ions in a Paul trap into which was introduced a small amount of helium[15]. An introduction to the basic principles was presented by Dehmelt in 1967[15]. A more thorough summary of the early work is given in the review book by Dawson[9]. Damping time-constants achieved in such devices range typically from about 1 to 100 ms. A similar phenomenon has recently been demonstrated by Savard et al.[19] for the cooling of ion motion in a Penning trap. In this particular case, as observed by Penning. simple cooling of the magnetron motion would result in an expansion of the magnetron orbit, due to the fact that the overall energy of magnetron motion decreases with radius. However, here the magnetron orbit was collapsed by using an azimuthal RF quadrupole field to transform its motion into cyclotron motion which does reduce in radius with reduced energy. This technique has now become very important in the preparation of ion collections for high-precision mass spectroscopy on Penning traps. Damping time-constants achieved by this technique are typically about 100 ms. The ease with which background gas can be used to cool ion motion in electromagnetic traps led to the study by Douglas and French[5] of the effect of background gas on the transverse motion of ions in a radiofrequency quadrupole rod structure such as used in a quadrupole mass filter. Indeed it was shown that background gas at moderate pressures, up to about 1 Pa, did result in a significant increase in the transmission of ions through a 1 mm diameter orifice following the quadrupole rods. This was followed by the work of Kim, cited above[1], which showed that a transverse radiofrequency quadrupole field could be used to confine ions injected at up to 100 ev and subsequently cooled to close to room temperature by a buffer gas at pressures up to 100 Pa. Central to the design of devices for such work is information on the drag forces on ions in gases that has be acquired in the long history of the study of ion mobility in gases. 3. Ion Mobility The concept of ion mobility originated in the late 19 th century in the context of the conduction of electricity in liquids and gases, and some of the greatest names in physics have come to be associated with the phenomenon involved. The concept was first introduced by Nernst[0] in the context of the osmotic pressure of liquids. Townsend[1] developed the concept of ion mobility for gases, based on the fundamental paper on kinetic theory by Maxwell[]. Interest in the subject increased greatly with the discovery by Roentgen that x-rays increased the electrical conductivity of gases, and Thompson received the Nobel prize in physics (1905) for his contributions to the study of such conductivities. By as early as 1905 an accurate 19

24 quantitative theory of ion mobility was developed by Langevin[3]. Einstein also contributed to the subject by estimating ion mobility from a study of Brownian motion[4, 5]. However, interest in the subject then waned, except as a recurrent theme within the development of technologies such as mass spectrometry and high-voltage electrical breakdown in gases. Interest in the subject itself reappeared, briefly, in the late 190's in the context of the chemistry of ion-molecule reactions. It reappeared again just after World War II in the context of the ionization of air by nuclear blasts. Since the mid 1960's ion mobility has been almost exclusively a subject within chemistry, first in the context of trobospheric and stratospheric chemistry and later in the context of the detection of pollutants in the atmosphere. Since then, many accurate ion mobility measurements have been carried out for specific ions in specific gases and the results have contributed a great deal to the knowledge of the basic nature of ion-molecule interactions[6]. The concept of ion mobility derives from the fact that the collision rate of an ion with gas molecules while an applied electric field causes it to drift slowly through a gas does not depend on its own drift velocity. This is because the velocities of the thermal motion of the gas molecules is typically so much higher than the drift velocity. The average rate of loss of momentum of the ion is the average momentum loss per collision times the average collision rate. If the ion is being dragged in a particular direction by an electric field than the average field-direction momentum loss per collision is a constant fraction of the momentum in that direction. Because the average collision rate is constant the average rate of loss of momentum in the field direction will be proportional to the ion momentum in that direction. In equilibrium this average rate of loss of momentum is equal to the average rate of gain from the field. This average rate of gain is simply the electric force on the particle and so the average ion momentum in the field direction is proportional to the electric field. This relationship is expressed as the drift velocity v d of an ion in a gas being proportional to the electric field E in the gas: v d = KE (33) where the proportionality constant K is referred to as the "ion mobility". Its most common units are cm per Volt-sec. Representative values for ions in gases at normal atmospheric pressure and room temperature are from 10 to 0 cm /V-s for ions in helium and from 1 to 3 cm /V-s for ions in nitrogen, the higher values being for light ions and the lower values for heavy ions. When expressed in the SI units of m per V- s, these values are, of course, multiplied by The ion mobility in any given gas is regarded as a particular characteristic of an ion and can sometimes be used to identify its presence in that gas by a technique referred to as Ion Mobility Spectrometry (IMS). A typical electric field used in IMS is 300 V/cm. At atmospheric pressure the drift velocities will be of the order of 5 m/s in nitrogen (or air) and about 50 m/s in helium. These velocities are seen to be considerably less than the thermal velocities of gas molecules, which average about 350 m/s for air and about 1000 m/s for helium. At these drift velocities the ions should remain very nearly in thermal equilibrium with the gas. 0