J. Phys. Chem. 1987, 91, 2586-2588
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cluster ions with sizes above the critical size stabilize by evaporative processes (reaction 3). Cluster ions around the critical size apparently either decay by Coulomb explosion before they reach the experimental time window, or, if they survive this time span apparently also prefer reaction 3 (see the sharp rise of ion intensity in the direct spectrum from ( c 0 2 ) 4 s 2 + to (C02)472+). This result is in accordance with similar results obtained for doubly charged ammonia cluster ions.’ 1,23 Triply charged C 0 2 cluster ions’O and triply charged N H 3 cluster ions,”*23however, behave differently. These ions dissociate around and above the critical size in the present time window also by Coulomb explosion,
the observed fragment ion distribution being very asymmetric. These triply charged ions have slightly longer flight times; however, the main reason for the detection of the Coulomb explosion must be that their dissociation rate for Coulomb explosion is smaller than that of doubly charged cluster ions. Acknowledgment. This work was partially supported by the Osterreichischer Fonds zur Forderung der wissenschaftlichen Forschung under Projekt No. 5692 and by the Deutsche Forschungsgemeinschaft. Registry No. CO,,124-38-9.
Cluster Ion Linear Acceleration Jurgen Gspann Kernforschungsrentrum und Universitat Karlsruhe, Institut fiir Kernuerfahrenstechnik, 7500 Karlsruhe, Federal Republic of Germany (Received: June 17, 1986)
Synchronous linear acceleration is calculated to be well suited for ionized clusters if advantage is taken of the narrow speed distribution of cluster beams from nozzle expansions. Clusters of a wide range of mass-to-charge ratios may be accelerated simultaneously at the respective appropriate phases, allowing for high intensity applications.
Introduction Applications of cluster beams often require kinetic energies of the clusters in excess of those provided by the generant nozzle expansion. For example, fuel injection into nuclear fusion plasmas by use of accelerated clusters’ would require energies as high as some GeV per cluster of lo5 atoms of D, T, or 3He, at which size suitable cluster beam intensities are available. On the other hand, cluster beams accelerated to not more than a few keV per 1000 atom cluster are sought for thin film deposition in order to obtain superthermal but nondestructive impact energies.2 New paths of chemical reactions could also be conceived on clusters moving with appreciable kinetic energy through reactants. While electrostatic acceleration of ionized clusters is the commonly used, or considered, means for increasing the cluster translational energy, this paper is intended to show that synchronous linear acceleration is in principle better suited in view of the special features of cluster beams from nozzle sources, especially for higher final energies. Circular accelerators act very selectively with respect to the mass-to-charge ratio so that their use appears favorable for but restricted to basic research with clusters of a single size. To this end, Gil Stein once envisaged a storage ring named “ C l u s t r ~ n ” . ~ Cluster beams obtained from nozzle expansions usually have a very narrow speed distribution with a half-width of less than 5% of the mean speed. Their size distributions, however, use to be rather broad with half-widths comparable to the mean size.4 Ionization of the clusters does not affect the speed distribution much but the mass-to-charge distribution of the ionized clusters may be appreciably widened compared to the original mass distribution, due to multiple charging, fragmentation, or polarization-induced loss of cluster material. Electrostatic acceleration then results in a correspondingly wide distribution of velocities, transforming thus a distribution homogeneous in energy per constituent atom into one homogeneous in energy per cluster. This may not be a wanted result since the speed of interaction with, (1) Henkes, W. Phys. Lett. 1964, 12, 322. (2) Takagi, T.; Yamada, I.; Kunori, M.; Kobiyama, S.Proc. Int. Conf Ion Sources, 2nd 1972, 790. ( 3 ) Stein, G . apparently unpublished. (4) Gspann, J. In Physics of Electronic and Atomic Collisions, Datz, S . , Ed.; North-Holland: Amsterdam, 1982; p 79.
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e.g., reacting species or photon fields, varies correspondingly. Moreover, one of the main reasons for the high mass flux obtainable with cluster beams, namely the lack of intrabeam cluster collisions due to the speed monochromaticity, is lost then. Collisions among the accelerated clusters may cause intensity losses via cluster destruction or wide-angle scattering or, at low impact energies, lead to coagulation. Synchronous linear acceleration, on the other hand, relies, in principle, on the preservation of speed monochromaticity. It is also known that different charge states of heavy ions can be accelerated together in heavy ion linear accelerator^.^ In the following, the application of the linear acceleration technique to cluster ion beams will be discussed.
Principle of Linear Acceleration For the present purpose, linear synchronous acceleration may be described as it was originally used by Wideroe:6 A charged particle to be accelerated passes through a series of drift tubes which are alternately connected to a radio-frequency source (Figure 1). Each time the particle reaches the gap between two neighboring drift tubes it is accelerated since the tube potentials alternate while the particle drifts within a tube. In order to make up for the increasing particle velocity, the lengths of the tubes must increase correspondingly. The length of the n-th tube L, measured between gap center planes is given by 0,
L, = 2v
where v, is the particle velocity at that position, and v the frequency of the potential alternation. Since the length-to-diameter ratio of the tubes should not be much smaller than one, a length of 1 cm for the first tube follows from the width obtained with hydrogen cluster beams at about 10-cm nozzle distance. For clusters of, e.g., 4 x IO4 cm/s initial velocity u,, then results a frequency v of 20 kHz. This low-frequency precludes the use of resonating cavity structures as commonly found in high-energy physics accelerators since the wavelength of the radio wave is much too long. The ( 5 ) Bohne, D.; Schmelzer Ch., In Lineor Accelerators, Lapostolle, P . M., Septier, A. L.,Eds.; North-Holland: Amsterdam, 1970; p 1029. (6) Wideroe, R. Arch. Electrotech. (Berlin) 1928, 21, 387.
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Cluster Ion Linear Acceleration
I
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c-
1 f (N)
n. v
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1
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2
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Wideroe linear accelerator Figure 1. Section of a linear accelerator with drift tubes of the Wideroe tYPee
Wideroe structure indicated in Figure 1 , however, can be used and has, in contrast to all cavity structures, the advantage of being tunable, i.e. for a given geometry the frequency can be varied. Synchronism between the particle and the accelerating wave requires a rate of acceleration which produces exactly the velocity gain prescribed by the length variation of the drift tubes. If the gap voltage changes as u = u, cos ( 2 m t + cps) (2) a negative value of the synchronqus phase angle cps up to -90' ascertains the so-called longitudinal phase stability: too fast a particle will get a smaller than normal acceleration at the next gap, and vice versa. For small relative increases of the velocity per gap, and IpSl< 60°, the range of stable phases extends from -pSto 2qS,appr~ximately.~For large absolute values of cps the range of stable phases increases and reaches 360' for cps = -90'. Particles injected at phases within the stable range approach their synchronous phase via damped oscillations. The stable phase range decreases with increasing deviation of the injection speed from the synchronous one7 but the narrow speed distribution of cluster beams ensures very good acceptance.
Simultaneous Acceleration of a Mass Distribution The sinusoidal change of the gap voltage now allows for a simultaneous synchronous acceleration of a certain distribution of mass-to-charge values. The kinetic energy of clusters of N atoms and 2 elementary charges e after n gaps is Nm(u2 - uO2)/2 = nZeU
(3)
For simultaneous synchronous acceleration, the velocity increase has to be independent of N / Z u,Z = u 2 ( 1
+ n6)
(4)
where 6, the ratio of the energy gain per gap at the respective synchronous phase, ZeVo cos cps, to the respective initial kinetic energy, Nmv2/2, is a constant. The dependence of the synchronous phase angle on N / Z is then given by N cos (ps = ~ ( m v o 2 / 2 e U o ) 6 As a practical example, Figure 2a shows a hydrogen cluster size distribution of the often observed log normal type, i.e. proportional to K1 exp[-(ln N - ~ ) ~ / 2 with d ] ,p = 1 1.92, Q = 0.479, and normalized at the most probable size of 1.2 X lo5 molecules per cluster. According to the measured hydrogen cluster velocities of 6.4 X lo4 cm/s,* which corresponds to 4 X lo4 cm/s for DT clusters at the same nozzle temperature, clusters of the most probable size have initial kinetic energies of about 500 eV. In Figure 2b, the synchronous phase angle cps is plotted for singly charged clusters, a 6 of 0.1, and a maximum energy gain per gap, eUo,of 100 eV. Under these conditions, only clusters smaller than twice the most probable size can be accelerated. In Figure 3a, the curve labeled cps shows how the velocity of cluster ions passing the gaps at their respective synchronous phases increases with the gap number. (7) Smith, L. Handbuch der Physik, Vol. 44/I, Fliigge, S., Ed.; Springer-Verlag: West Berlin, 1959, p 341.
- 0 0
0
~ 1 1 0 ~
Figure 2. (a) Accepted portion (shaded area) of a log normal size distribution of singly charged hydrogen clusters (curve) injected into a linear accelerator during.the phase range from -60" to +60° if the synchronous phase is -60" for the most probable cluster ion with 1.2 X lo5 molecules. (b) Corresponding synchronous phase as a function of the cluster ion size.
' 0
7 -90'
-180' I 0
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Figure 3. (a) Synchronous velocity normalized to the injection velocity uo as a function of the gap number (curve labeled cps) and velocity oscillations of cluster ions of 0.5, 1, and 1.5 times the most probable size injected at cp = 0. (b) Corresponding synchronous phases cps and phase oscillations.
In practice, however, the injection of cluster ions of different sizes into the entrance section of the linear accelerator has to occur simultaneously for a certain time. In Figure 2a, the longitudinal phase acceptance according to the mentioned acceptance range between 2cps and -cps is shown for a simultaneous injection during the phase range from -60' to +60°, covering a third of the continuous beam. The shaded area indicates the accepted part of the size distribution for singly charged clusters. In principle, there is also a lower boundary of the acceptance range, excluding cluster ions whose initial kinetic energy is smaller than eUo since they can be reflected at unfavorable phases. For the distribution of the present example, however, cluster ions of less than 2.4 X lo4 molecules do not exist anyway. For the first 25 gaps, the fate of cluster ions of 0.5, 1 , and 1.5 times the most probable size not injected at their respective synchronous phases but at cp = 0 is illustrated in Figure 3. One recognizes that these cluster ions are indeed accelerated with damped phase and velocity oscillations. As soon as the initial phase oscillations have died out, one may slowly decrease cps by increasing the drift tube lengths somewhat more than required by eq 1 in order to keep the otherwise decreasing relative energy gain per gap at a higher level, reducing thereby the overall accelerator length. To the same end, one may increase the radio frequency stepwise, e.g. by a factor 3 after a
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section of about a hundred drift tubes, and so on. The velocity oscillations taking place in the entrance section of the accelerator give rise to intrabeam collisions. If n, is the number density of the clusters and u, their collision cross section, the number of collisions a cluster experiences during the time needed to pass n gaps, n / 2 v , is given by
C = ncuc A u ( n / 2 v )
(6)
where Av is an average relative velocity with respect to the scattering clusters. It will be taken to be the root-mean-square deviation from the synchronous velocity. In the following, the collision number Cis estimated for a cluster of the most probable size of the example distribution, injected at cp = 0. On the basis of the density of solid hydrogen, the radius of the cluster is found to be 109 A, giving a cross section u, of about 1.5 X 10-l’ cm2 for collisions with clusters of the same size. With hydrogen cluster beams of lo5 molecules per cluster average size, a flux density of 8 X lo1*molecules cm-* s-l has been observed a t a IO-cm distance from the (virtual) source point of the beam.E With this value and 4 X 104 cm/s initial velocity, a cluster number density of 2 X IO9 cm-3 is obtained. Since only a third of the continuous beam is assumed to be injected, the time-averaged product ncucturns out to be_lO-Z/cm. Av is calculated from the velocity deviations depicted in Figure 3a to be 3.6 X lo3 cm/s. is obtained With these data, a collision number C of 2.3 x for the passage of the most probable cluster through the first part of the accelerator with 25 gaps, comprising a total length of 37 cm. Only part of these collisions will lead to a complete loss of the cluster ions involved but the size distribution might be changed. In the later sections of the accelerator, intrabeam collisions become less probable since the speed distribution gets narrow again. Collisions with background gas resulting from either collisions or thermal cluster evaporation have to be avoided however, since they are getting more and more energetic. Hence, an efficient pumping system, e.g. of cryopumping surfaces, has to be installed along the accelerator in order to remove any kind of vapor releases as fast as possible.
Transverse Focusing Linear accelerator structures of the type illustrated in Figure 1 are known to be radially defocusing under the condition of longitudinal phase stability because of the defocusing second half of the gap field which is then predominant as a result of the increase of the gap voltage during the passage of the particles. Magnetic quadrupole focusing, the usual choice for high-energy physics accelerators, is not an appropriate remedy here since, at the low cluster speeds, feasible magnetic field strengths do not produce sufficiently strong focusing forces. Electric quadrupole focusing is possible, however, as in the first sections of heavy ion linear accelerator^.^ Since, as mentioned above, the initial lengths and diameters of the drift tubes are only of the order 1 cm in a cluster ion linear accelerator, the housing of electrostatic quadrupoles within the drift tubes is practically excluded. Very attractive, however, is the possibility of using the accelerating radio-frequency wave a second time by forming quadrupoles of “fingers” extending from the drift tubes into the gap.g Figure 4 shows a cluster ion linear accelerator (CILAC) arrangement consisting of pairs of diametrically opposed cylindrical rods fixed under 90’ angular shift to triples of disks acting as drift tubes. The latter could degrade even further to single disks. Sets of electrically connected disks have been used in time-of-flight cluster spectrometers to obtain drift regions of high pumping (8) Gspann, J.; Krieg, G. J . Chem. Phys. 1974, 61, 4037. (9) Vladimirskij, V. V. Prib. Tekh. Eksp. 1956, 3, 35. Boussard, D.In Linear Accelerarors, Lapostolle, P. M., Septier, A. L., Eds.; North-Holland: Amsterdam, 1970; p 1073.
-Ln
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Focusing-rod disk CILAC Figure 4. Section of a cluster ion linear accelerator CILAC with drift regions formed by sets of disks with staggered pairs of rods for transverse quadrupole focusing.
conductance.I0 The maximum electric field gradients near the axis are of the order of the maximum voltage between the rods divided by the square of the aperture radius, i.e. about 400 V/cm2 in the case discussed above with a I-cm aperture diameter. However, the axial electric field component providing for the longitudinal acceleration is markedly reduced in comparison with the Wideroe drift tube case, so that the maximum voltage has to be increased, which is in turn favorable for the transverse focusing. The modular rod-disk CILAC arrangement provides flexibility for designing the optimum field distribution. Finally, the defocusing transverse forces resulting from the beam space charge have to be considered.I’ These again are most effective in the accelerator entrance section since they are inversely proportional to the ion velocity. Clusters with lo5 molecules per charge with a time-averaged intensity of about 2.7 X lo1*molecules cm-2 s-l passing through apertures of l-cm diameter correspond to 1/3 equivalent ampere molecule current, or 3.3 pA electrical current. Again if an initial cluster ion velocity of 4 X lo4 cm/s is assumed, the radial field strength at the beam circumference is calculated to be 300 V/cm which should be safely compensated for by the transverse focusing.
Conclusions The linear acceleration principle has been found to be well suited for the acceleration of cluster beams from nozzle expansions since their narrow original speed distribution corresponds to the speed homogeneity required and also accomplished by linear acceleration. A large portion of the typically wide mass-to-charge distribution of ionized cluster beams may be accelerated with an appropriate range of stable synchronous phases, allowing thus for high-intensity applications. The low frequencies involved preclude resonating cavities and lead to the tunable drift region structures of the Wideroe type. Capturing the cluster ions with their original speed into synchronism with a correspondingly slow wave makes the accelerator entrance section the most demanding part. Radial defocusing and space charge induced beam expansion have to be counteracted by electric quadrupole focusing which may be accomplished by a rod-disk type arrangement as shown in Figure 4. Strong phase oscillations which result in relative motion of the cluster ions, thereby enhancing the probability of intrabeam collisions, are mainly restricted to the accelerator entrance sections. Acknowledgment. Most of this contribution has been written during the author’s stay at Kyoto University. The author thanks Professor T. Takagi and Professor I. Yamada for their kind invitation and the stimulating atmosphere of their Ion Beam Engineering Experimental Laboratory. (10) Gspann, J.; KBrting, K.; J. Chem. Phys. 1973, 59, 4726. (1 1) Lawson, J. D. The Physics of Charged-Particle Beams; Clarendon: Oxford, 1977; Chapter 3.
