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Targets consisting of free charged particles
Nuclear Instruments and Methods in Physics Research A282 (1989) 80-86 North-Holland, Amsterdam
80
TARGETS CONSISTING OF FREE CHARGED PARTICLES Alfred MÜLLER Institut für Kernphysik,
Universität Giessen, D-6300 Giessen, FRG
Targets composed of free ions or free electrons are of interest mainly for atomic collision experiments. The present article describes ways to build and characterize such targets. The main differences in comparison with conventional targets are discussed . 1. Introduction Atomic collision studies inherently involve the electron shells. In most of these experiments, except for the most tightly bound inner shells it makes a difference whether the target atoms are free or bound in a solid target since the solid state changes at least the outer shell electron wave functions of those atoms . Hence, the only well defined target for most atomic physics experiments is a dilute target of free atomic particles. Free atomic particles comprise atoms, molecules, ions and electrons . Atomic or molecular gas targets are most frequently used due to the relative ease of handling and controlling gas flow and pressure, though setting up some of these targets may require substantial effort as it is the case for eventually polarized, atomic hydrogen or any target consisting of excited atomic particles . However, in most of the collision processes which occur in astrophysical or laboratory plasmas, the colliding particles are both charged . The experimental investigations of such collisions requires targets consisting of free ions or free electrons . In the following sections basic considerations on the design and characterization of charged-particle targets will be discussed. 2. Event rates in beam-target experiments The general formula for the event rate R of non-relativistic two-particle collisions in an interaction volume T containing particles A with spatial density nA (r) and particles B with spatial density n B (r) is given by R
= f n A (r)n B (r) d d(v) vuA,B (v) dv dT.
(1)
Here, d f(v)/d v is the distribution function for velocities v between the colliding particles, and OA,B (v) is the collision cross section for producing the observed event. A classical target is a concentration of atoms which are assumed to be at rest. The spatial density of these 0168-9002/89/$03 .50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
atoms, e.g. n A (r) should be constant within the target or at most depend on x, the coordinate in the projectile-beam direction . With the assumption of a well defined velocity v B of the projectile beam of particles B the distribution function df(v)/dv is a 8-function and equals S(v-vB). Assuming further that the projectile beam does not change its diameter inside the target the event rate R becomes R
A,B (V B )
=G
fn,(z,
B
y)V
dz dy
f n A (x) dx,
(2a)
(2b) = aA,B(VB)FB17A, Here, x is in the direction of the projectile beam (B) and the coordinates (x, y, z) are for a rectangular coordinate system. FB is the flux of particles B in the projectiles beam (in particles per second) and 9TA is the target thickness (in particles per square centimeter) . For a solid with constant n A (r) and geometrical thickness d (measured in the direction of the projectile beam) we have ?TA =n A d. Eq. (2b) is the well known formula for event rates in experiments where a beam from an accelerator passes through a solid or a gas target. As long as the condition that each projectile undergoes at most one collision in the target is fulfilled, the event rate can trivially be enhanced by increasing the flux of projectiles and/or the product of length and density of the target . 3. Design consideration for a charged-particle target For the achievement of high luminosities in a collision experiment the previous section suggests to provide a long target with a high particle density . In a solid target particle densities are of the order of 10 23 cm -3. A dense gas target could have 1017 cm-3 which is a factor of 106 less. Let us imagine a fictitious spherical electron target with 1 cm radius and a (by another factor of 106 smaller) density of 1011 cm-3. The uncompensated space charge of the electrons produces a parabolic potential
A . Midler / Targets consisting offree chargedparticles inside the sphere with a potential difference AU= 3 X 10 ° V between surface and center . Without additional forces to confine the electron cloud such a sphere would just explode . This is also true for singly and even more for multiply charged ions. A target of ions or electrons basically at rest could still be accomplished by trapping the particles in a suitable external electromagnetic field configuration . However, due to the space charge forces involved the number of particles that can be confined in such a trap is so small that extremely low target densities result. Another solution to the space-charge problem is a compensation by particles of opposite charge with a resulting net neutralization . This idea can be realized in producing a cold plasma by a gas discharge. Such a target, however, is difficult to characterize. Besides electrons and ions it contains neutral species, excited species and its characteristics such as temperature and density are not easily controlled . Hence, the only possibility to provide a clean target consisting of charged particles in a defined state is to use the technique of colliding beams. For this purpose both beams are transported into an interaction region where they may be parallel to each other for a certain distance (merged beams) or cross each other at a certain angle (crossed beams) . Both beams may be circulating in storage rings . The space charge forces in the beams lead to an increase of the beam emittance, i .e . they drive the beam particles apart. Compact geometry, magnetic fields or electrostatic focussing are necessary to hold the beams together. Since space charge potentials also have an influence on the particle energies it is usually not reasonable to go to ultimate charge densities in the beams even if it would be technically possible .
4 . Event rates in colliding-beams experiments In general, the spatial particle densities in colliding beams cannot be expected to be uniform in the interaction region . Hence, event rates have to be calculated in principle from eq. (1) . It is possible and reasonable to make a few assumptions about the geometry of such an experiment . Let us assume beam A, called target beam, is parallel in itself and all particles have the same velocity vA . Beam B, called projectile beam, is also parallel in itself and the projectiles all have a velocity I)B*
The beams collide under an angle 0. Then the collision velocity v between colliding particles is constant v=(vk - vg+2UAvg
COS
0)
1/2
Since the beams are parallel in themselves the density of each beam does not depend on the respective coordi-
81
nate in the beam direction . If 0 * 0 the integral (1) reduces to R_
va . FA FB~ UAU B
(4)
As in eq . (2) FA and FB denote particle fluxes in the beams . The overlap of both beams is characterized by the form factor which can be approximated by f=
Y-'A(Zk) DB(Zl) k l
Y tA(Zm)iB(Zm)
Sz
(5)
m
The quantity 'A(Zk) is the measured flux of particles A behind a slit of width Sz at a position with coordinate z k (see fig. 1). For iB(Zk) there is a corresponding definition. Thus, crossed beams experiments require measurements of vertical beam intensity profiles . An example of such measurements is shown in fig . 2 . for colliding H + and He' ion beams [1] . For merged beams (0 = 0) the general definition of the form factor is even more complex than eq. (5) and requires two-dimensional beam profile measurements. I will now make one further assumption in order to get the most simple possible event rate formula for a colliding beams experiment . Let the density of one beam be uniform as in the case of a solid target, e.g . n A (r) = n A = const. and let the other beam B travel completely inside A for a distance d . Then eq . (1) reduces to R = nA dVUA,BFBIV B .
( 6)
Since the relative velocity v of eq . (3) is not equal to the projectile velocity v B (as in the case of a static target), eq. (6) is different from the well known formula given as eq. (2b) . For high luminosity in a colliding beams experiment it is not sufficient to provide a high target thickness n A d. One should also keep the projectile velocity v B low as long as the desired collision velocity v can be guaranteed . Low v B means increased interaction time ;
beam B
beam A
~6z b2
Fig. 1 . Schematic view of a crossed beams arrangement. The form factor f (see eq. (5)) can be measured by moving a slit with the width S. perpendicular to both beams and registering the transmitted currents 'A(Zk) and 'B(Zk) at position z k . II . TARGETS FOR HIGH-ENERGY/HEAVY-ION BEAMS
82
A. Midler / Targets consisting offree charged particles
6
possible to determine the form factor f (eq. (5)) to within 1% absolute uncertainty. This has to be compared to the uncertainties of pressure and length measurements for a gas target which are typically 10-20% .
5
5. An electrostatic electron target for crossed beams measurements on electron impact ionization of ions
4
2
0
1
2
z (MM)
3
4
5
Fig. 2. Determination of a form factor in a crossed beams experiment with H + and He' ion beams [1]. The currents in(zk) and 'B(zk) of H' and He' ions, respectively, were measured behind a slit and give the vertical intensity profiles of the beams. The resulting form factor is f = 2.44 mm . projectile particles stay longer inside the target beam and their chance to undergo a collision is enhanced . An important aspect has to be mentioned which is characteristic for charged particle beams experiments. Since electrical currents can be easily measured it is
cathode
In Giessen, we are investigating electron impact ionization of ions since a couple of years [2]. In our experiments a collimated beam of multiply charged ions is crossed with a ribbon shaped electron beam [3] that provides high electron densities (nearly 10 9 cm -3 at 1000 eV energy) over a width of 60 mm. The electron gun was developed in a collaboration with the Institut für Angewandte Physik, Universität Frankfurt. Transmission experiments with extremely narrow ion beams crossing the electron beam at different positions and angles and in addition computer calculations were used to find the present configuration. The computer code SLAC 166 was modified to include a simulation of thermal spread of the electrons as well as space charge compensation by trapped ions. The electron gun produces a current of 460 mA at 1 keV energy . The current density in the interaction region, where the ion beam crosses, is about 0.3 A CM-2 . Fig. 3 shows a cut through the electron target perpendicular to the ion beam. Electron trajectories and equipotential lines are indicated. The electrons are emitted from an impregnated tungsten cathode heated to about 1100 ° C. The emission is space charge limited. The current density emitted from the cathode is 0.09 A cm -2 at most, which is more than 100 times less than
collector
Fig. 3. Calculated equipotential lines and electron trajectories in the electrostatic electron gun [3]. The potential of the cathode is set to 0%, that of the rod electrodes in front of and behind the interaction box to 100% ; the interaction energy is given by 76 .6% of the maximum voltage applied. The gun extends to 60 mm in the direction of the ion beam and can be moved up and down across the ion beam axis .
A. Müller / Targets consisting offree chargedparticles
Z
Fig . 4. Potential distribution inside the interaction box with the electron space charge uncompensated . The depth of the well corresponds to 96% of the electron energy defined by the potential on the interaction box (see fig . 3) .
the possible emission from the cathode material . The electron beam is focussed into the interaction box by a pair of rod electrodes. Behind this box there are two more pairs of rods which serve to provide a symmetrical potential distribution . The electrons are collected by a water cooled copper anode . The density of equipotentials in fig . 3 indicates a field free region in the interaction box, which is necessary to provide a good energy definition of the electrons and to prevent deflection of the passing ions . However, with the space charge of the electrons, electric fields in and around the electron beam can never be avoided completely. Indeed, the calculations show that there is a flat negative potential well inside the interaction box . The potential distribution over a plane perpendicular to the ion beam [4] is shown in fig. 4 on a sensitive scale . The depth of the calculated potential well is 0 .04UCA where the voltage UCA gives the potential of the interaction box with respect to the cathode potential. The well acts like an ion trap confining slow ions in the (y, z) plane (see fig. 4) . The trap is also closed in the x-direction (ion beam direction) since the potential rises at both edges of the electron beam because the negative space charge that lowers the potential disappears there . The electron beam produces slow ions from the residual gas . Some of these ions become trapped and compensate the electron space charge so that ultimately the trap is macroscopically neutralized . We have found strong experimental evidence for space charge neutralisation in our electron gun [5] . One strong argument is the observation of narrow dielectronic capture resonances [6] with a width of less than 0.4 eV. These resonances occur at exactly defined rela-
83
tive velocities v between electrons and ions in the interacting beams. They are characterized by capture of a free electron into the ion and by simultaneous excitation of a core electron . Since this is the time inverted Auger process it is understandable that it can only occur at a defined energy . The widths of the observed resonances give information on the experimental energy definition . The width of 0.4 eV, observed for a resonance in Ba l+ ions at 90 eV, indicates a resolution which is 10 times better than could be expected without space charge neutralisation by slow trapped ions . Thus, the electron target described in this section provides high density and a quite long interaction length d of 60 mm, which is very helpful with respect to event rates (see eq. (6)) . In spite of the massive space charge carried by the electrons the definition of the electron energy in the electron region is remarkably good .
6. A magnetically confined electron target for merged beams experiments at an accelerator In the previous chapter a very compact electron gun was described which, nevertheless, provides a considerable interaction length of 60 mm . If an experimenter wants to increase the interaction length much beyond that it is necessary to merge the beams in the interaction region . For example the electron cooler for the ESR storage ring will provide an interaction length of 250 cm, which is more than 40 times that of the electrostatic crossed beams target discussed in section 5 . The difficulty is now to hold the electron beam together over such distances and maintain good energy definition. Also it is not trivial to merge and demerge two beams. On the other hand, there are more advantages in a merged beams configuration than just the high interaction length . With the velocity vectors of the beam particles parallel it is easily possible to study processes at low relative velocities v . Moreover, the kinematics are such that beam energy spreads in the laboratory frame are considerably reduced by the transformation into the center-of-mass system . Thus, high resolution experiments on resonant processes become feasible . Fig. 5 shows an overview on the center-of-mass energies which can be obtained with parallel or antiparallel electron- and ion beams . Ion energies per nucleon in the range of the UNILAC and electron energies up to about 10 keV are assumed . The lines given in fig . 5 show the possible combination of ion and electron energies in the laboratory to obtain a certain center-of-mass energy which is indicated for each line . Relativistic mechanics were used to calculate these dependences . A range from 0 up to 50 keV center-of-mass energies can be covered by the proper choice of geometry and laboratory energies . II . TARGETS FOR HIGH-ENERGY/HEAVY-ION BEAMS
84
A. Müller / Targets
consisting
of free chargedparticles 65
1,1,1 .1,1,1,1,i,1 .1 .1,1,
60 55 50 45 40 35 30 25 20 15
Ion energy E, /A (MeV/u)
Fig . 5 . Combination of ion and electron laboratory energies resulting in the same center-of-mass energy Ec .m. in a merged beams arrangement . For each line E..m. is indicated. The curves with negative slope are for antiparallel beams . For a given electron target, optimum event rate is achieved if a combination of high electron energy and low ion energy can be chosen (eq. (6)) in order to keep the ion velocity a B small. On the other hand the argument of good energy resolution may require high ion energy since the effect of energy-spread compression by the transformation from the laboratory frame into the center-of-mass system becomes more efficient at high laboratory energies . Moreover, the possibility to produce a certain ion charge state and the presence of concurrent residual gas collisions have to be considered in the choice of laboratory energies . Fig. 6 shows the electron laboratory energies which are necessary to cover a center-of-mass energy range from 0 to 65 eV when the ion energy is fixed to 11 .4 MeV/u. Apparently there are always two possibilities to realize a given center-of-mass energy. One is to have the electrons chase the ions and one is to have the ions run into the electrons. Zero energy results when the particles in both beams have the same speed vA = vB . This is the case at an electron energy E = 6260 eV in our example. The shape of the curve in fig. 6 is responsible for the energy-spread compression mentioned above. A considerable electron energy spread (of e .g . 100 eV) in the laboratory frame around E = 6.26 keV results in an uncertainty of only 0 .4 . eV in the centerof-mass system. This fact can be used to study capture resonances in low-energy collisions of electrons with highly charged ions . At a specific energy of 11 .4 MeV/u high ion charge states can easily be obtained by stripping in a
10
5500
6000
6500
7000
7500
Electron Energy [eV1
Fig . 6 . Center-of-mass energy vs electron laboratory energy for a fixed ion laboratory energy of 11 .4 MeV/u in a merged beams experiment (relativistic mechanics used) . At 6260 eV, ions and electrons have the same velocity and zero center-ofmass energy .
foil target. Low relative energy is achieved with a parallel electron beam of about 6 .26 keV . The principal setup for such experiments which were proposed for the UNILAC [71 is shown in fig . 7 . The electron beam [81 is inside a strong axial magnetic field of B =1 T produced by a superconducting solenoid. The 3 mm diameter
auperconCuctirg aol@noid, 8 . 1T
Fig . 7. Schematic diagram of a merged beams experiment [111. The electron gun is inside the homogeneous field of a superconducting solenoid. Bending of electron trajectories onto the ion beam axis is accomplished by additional elliptical windings on top of the solenoid.
A. Milller / Targets consisting offree charged particles
cathode is 25 mm off the ion beam axis and fully immersed in the magnetic field . Electrons with a maximum current density of 10 A cm -2 are extracted and accelerated to energies up to 8 keV. The gun can be operated in two modes one with a perveance of 0.95 WA V -3/ 2 and one with 0 .098 I.A V -3/2 . Maximum electron densities at E=8 keV are 101° and 10 9 cm -3 , respectively. One of the most important features of the design is the low transverse electron energy of less than 0 .4 eV, which is obtained by resonant focusing, i.e . a proper matching of electric and magnetic fields depending on the gun geometry [9]. The matching of fields keeps the electrons from gaining radial energy and spiralling in the magnetic field . Of course there is an initial contribution of radial energy due to the electron emission process so that typically 0.3 eV can hardly be circumvented [10] . Again the electron space charge produces radial fields in the electron beam. A voltage depression of up to 109 eV results at 8 keV [8] . The electrons are driven away from the beam center by the space charge fields, however, the magnetic field forces them to stick with their line of force so that the electrons cannot gain radial energy. Only the longitudinal energies of the electrons vary between 8 .000 and 7.891 keV in the worst case . However, discussion of fig . 6 shows that longitudinal energy spread is considerably reduced in a merged beams situation by the transformation from the laboratory to the center-of-mass frame so that even at an electron density of 10 1° cm -3 good resolution at low collision energies can be achieved . The electron beam is merged and demerged with the ion beam by transverse components (B 1 = 0.15B) of
0 Lri N
85
the solenoid field. These components are obtained by adding elliptical windings on top of the solenoid . The solenoid provides a field homogeneity of about 10 -4 in the merging section . The low radial electron energy is maintained also when the acceleration voltage U is changed, provided all voltages on focussing electrodes are changed by the same factor and the magnetic field is changed proportional to vrU -- . Hence, an electron laboratory energy of 800 eV requires an axial magnetic field of about 0 .3 T. An experimental setup which will implement the present electron target for merged beams measurements of dielectronic recombination cross sections is presently completed at the UNILAC of GSI [11] . A schematic of the experimental setup and beam optics is given in fig . 8 . A beam of accelerated ions, e.g. Fe 23+ at 11 .4 MeV/u, is coming from the left. For collimation of the ion beam and reduction of gas flow a narrow pipe is used to separate the vacuum system in the experimental area from that of the accelerator . Magnetic quadrupole lenses in front of the magnets serve to focus the ion beam. The first magnet defines the ion charge state, the second magnet analyzes the charge states of ions which have undergone a collision in the electron target . Steerers before and behind the target have to be used to compensate for the transverse magnetic field components inside the target which are strong enough to slightly deflect an 11 .4 MeV/u ion beam . Along the 50 cm long interaction path ions capture free electrons and thus reduce charge state . These ions can be magnetically separated from the parent ion beam and detected by a suitable single particle detector. Since electrons can also be captured out of bound states in residual gas atoms or
Horizontal
E E x E E T 0
Vertical
Path
Length
16000
mm
N
Fig. 8. Schematic of the experimental setup and beam optics for a merged beams experiment [11) at the UNILAC. II . TARGETS FOR HIGH-ENERGY/HEAVY-ION BEAMS
86
A . Midler / Targets consisting offree charged particles
molecules, it is necessary to keep the residual gas pressure as low as possible. The range of 10 -s Pa is considered suitable . There is a choice of ion velocity (energy) in a colliding beams experiment since the desired center-of-mass energy can be obtained by adjusting the velocity of the electrons (see eq . (3)) . This is important for the present experiment because it allows to use an ion energy suitable to reduce background from charge-changing collisions in the residual gas . Cross sections for such processes are small at high ion energies so that the projected energy of 11 .4 MeV/u is of some .additional advantage.
7. Conclusions Colliding beams experiments are difficult and there are many subtle sources of errors in cross section measurements. Problems arise from the space charge of electron or ion beams and the resulting limitations of particle densities. Numbers like 10 10 electrons per cubic centimeter are extremely high and unprecedented so far in beams experiments . Still, this corresponds to a pressure in a gas target of only some 10 -5 Pa, indicating the problem with event rates in such a target. In general, colliding beams experiments require ultrahigh vacuum in order to keep backgrounds low which arise from collisions with residual gas particles. There are many kinds of experiments, however, which can only be done with colliding beams techniques. Among these are all measurements looking for collision processes of electrons with ions. A beam target can be well characterized, the density distribution can be measured more accurately than for a conventional target. The beam can be well prepared with respect to low energy spread, to mass and charge state definitions, and even desired states of excitation can be provided . Hence, colliding beams experiments are powerful sources of information. They have the potential of a precision in cross section measurements which can hardly be assessed in experiments with conventional targets.
Acknowledgement I want to acknowledge fruitful interaction with many colleagues in various collaborations dealing with electron-ion collision experiments . Special thanks are due to R. Becker who is leading the computer part in the design of the electron targets which are discussed in this paper as examples for real developments used in crossed and merged beams experiments.
References
[2]
[3] [41
[61
[81
[11]
K. Rinn, F . Melchert and E . Salzborn, J. Phys . B18 (1985) 3783 . A . Müller and R. Frodl, Phys . Rev . Lett 44 (1980) 29; A . Müller, K . Huber, K. Tinschert, R . Becker and E. Salzborn, J . Phys. B18 (1985) 3011 ; K. Tinschert, A. Müller, G. Hofmann, Ch . Achenbach, E. Salzborn, and R . Becker, J. Phys . B20 (1987) 1121 . R. Becker, A . Müller, Ch . Achenbach, K. Tinschert, and E. Salzborn, Nucl . Instr. and Meth . B9 (1985) 385 . G . Hofmann, Diploma Thesis, Giessen (1987) unpublished. A. Müller, G. Hofmann, K. Tinschert, R . Sauer, E. Salzborn, and R. Becker, Nucl. Instr . and Meth. B24/25 (1987) 369 . A. Müller, K. Tinschert, G . Hofmann, E. Salzborn, and G .H . Dunn, Phys . Rev. Lett . 61 (1988) 70 . R. Becker, E . Jennewein, M. Kleinod, U. Pr6bstel, N . Angert, F. Bosch, I . Hofmann, J. Klabunde, P .H. Mokler, H . Schulte, P . Spadtke, B . Wolf, and A . Müller, GSI Scientific Report (1985) GSI-86-1, ISSN 0174-0814, p . 378 . U . Prôbstel, R . Becker, E . Jennewein, M . Kleinod, and K. Valadkhani, GSI Scientific Report (1986) GSI-87-1, ISSN 0174-0814, p . 271 . J.R. Pierce, Bell Syst . Techn. J. 30 (1951) 825 . Ultracold electron beams can be produced by photoelectron emission from e.g. GaAs crystals ; see e.g . D .T. Pierce et al., Rev. Sci . Instr . 51 (1980) 478 . A . Müller, S . Schennach, W . Spies, N . Angert, H . Emig, J. Klabunde, K .-D . Leible, P.H. Mokler, H. Schulte, P. Spädtke, B . Wolf, R. Becker, E . Jennewein, M . Kleinod, and U . Pr6bstel GSI, Scientific Report (1987) GSI-88-1, ISSN 0174-00814, p . 347 .
80
TARGETS CONSISTING OF FREE CHARGED PARTICLES Alfred MÜLLER Institut für Kernphysik,
Universität Giessen, D-6300 Giessen, FRG
Targets composed of free ions or free electrons are of interest mainly for atomic collision experiments. The present article describes ways to build and characterize such targets. The main differences in comparison with conventional targets are discussed . 1. Introduction Atomic collision studies inherently involve the electron shells. In most of these experiments, except for the most tightly bound inner shells it makes a difference whether the target atoms are free or bound in a solid target since the solid state changes at least the outer shell electron wave functions of those atoms . Hence, the only well defined target for most atomic physics experiments is a dilute target of free atomic particles. Free atomic particles comprise atoms, molecules, ions and electrons . Atomic or molecular gas targets are most frequently used due to the relative ease of handling and controlling gas flow and pressure, though setting up some of these targets may require substantial effort as it is the case for eventually polarized, atomic hydrogen or any target consisting of excited atomic particles . However, in most of the collision processes which occur in astrophysical or laboratory plasmas, the colliding particles are both charged . The experimental investigations of such collisions requires targets consisting of free ions or free electrons . In the following sections basic considerations on the design and characterization of charged-particle targets will be discussed. 2. Event rates in beam-target experiments The general formula for the event rate R of non-relativistic two-particle collisions in an interaction volume T containing particles A with spatial density nA (r) and particles B with spatial density n B (r) is given by R
= f n A (r)n B (r) d d(v) vuA,B (v) dv dT.
(1)
Here, d f(v)/d v is the distribution function for velocities v between the colliding particles, and OA,B (v) is the collision cross section for producing the observed event. A classical target is a concentration of atoms which are assumed to be at rest. The spatial density of these 0168-9002/89/$03 .50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
atoms, e.g. n A (r) should be constant within the target or at most depend on x, the coordinate in the projectile-beam direction . With the assumption of a well defined velocity v B of the projectile beam of particles B the distribution function df(v)/dv is a 8-function and equals S(v-vB). Assuming further that the projectile beam does not change its diameter inside the target the event rate R becomes R
A,B (V B )
=G
fn,(z,
B
y)V
dz dy
f n A (x) dx,
(2a)
(2b) = aA,B(VB)FB17A, Here, x is in the direction of the projectile beam (B) and the coordinates (x, y, z) are for a rectangular coordinate system. FB is the flux of particles B in the projectiles beam (in particles per second) and 9TA is the target thickness (in particles per square centimeter) . For a solid with constant n A (r) and geometrical thickness d (measured in the direction of the projectile beam) we have ?TA =n A d. Eq. (2b) is the well known formula for event rates in experiments where a beam from an accelerator passes through a solid or a gas target. As long as the condition that each projectile undergoes at most one collision in the target is fulfilled, the event rate can trivially be enhanced by increasing the flux of projectiles and/or the product of length and density of the target . 3. Design consideration for a charged-particle target For the achievement of high luminosities in a collision experiment the previous section suggests to provide a long target with a high particle density . In a solid target particle densities are of the order of 10 23 cm -3. A dense gas target could have 1017 cm-3 which is a factor of 106 less. Let us imagine a fictitious spherical electron target with 1 cm radius and a (by another factor of 106 smaller) density of 1011 cm-3. The uncompensated space charge of the electrons produces a parabolic potential
A . Midler / Targets consisting offree chargedparticles inside the sphere with a potential difference AU= 3 X 10 ° V between surface and center . Without additional forces to confine the electron cloud such a sphere would just explode . This is also true for singly and even more for multiply charged ions. A target of ions or electrons basically at rest could still be accomplished by trapping the particles in a suitable external electromagnetic field configuration . However, due to the space charge forces involved the number of particles that can be confined in such a trap is so small that extremely low target densities result. Another solution to the space-charge problem is a compensation by particles of opposite charge with a resulting net neutralization . This idea can be realized in producing a cold plasma by a gas discharge. Such a target, however, is difficult to characterize. Besides electrons and ions it contains neutral species, excited species and its characteristics such as temperature and density are not easily controlled . Hence, the only possibility to provide a clean target consisting of charged particles in a defined state is to use the technique of colliding beams. For this purpose both beams are transported into an interaction region where they may be parallel to each other for a certain distance (merged beams) or cross each other at a certain angle (crossed beams) . Both beams may be circulating in storage rings . The space charge forces in the beams lead to an increase of the beam emittance, i .e . they drive the beam particles apart. Compact geometry, magnetic fields or electrostatic focussing are necessary to hold the beams together. Since space charge potentials also have an influence on the particle energies it is usually not reasonable to go to ultimate charge densities in the beams even if it would be technically possible .
4 . Event rates in colliding-beams experiments In general, the spatial particle densities in colliding beams cannot be expected to be uniform in the interaction region . Hence, event rates have to be calculated in principle from eq. (1) . It is possible and reasonable to make a few assumptions about the geometry of such an experiment . Let us assume beam A, called target beam, is parallel in itself and all particles have the same velocity vA . Beam B, called projectile beam, is also parallel in itself and the projectiles all have a velocity I)B*
The beams collide under an angle 0. Then the collision velocity v between colliding particles is constant v=(vk - vg+2UAvg
COS
0)
1/2
Since the beams are parallel in themselves the density of each beam does not depend on the respective coordi-
81
nate in the beam direction . If 0 * 0 the integral (1) reduces to R_
va . FA FB~ UAU B
(4)
As in eq . (2) FA and FB denote particle fluxes in the beams . The overlap of both beams is characterized by the form factor which can be approximated by f=
Y-'A(Zk) DB(Zl) k l
Y tA(Zm)iB(Zm)
Sz
(5)
m
The quantity 'A(Zk) is the measured flux of particles A behind a slit of width Sz at a position with coordinate z k (see fig. 1). For iB(Zk) there is a corresponding definition. Thus, crossed beams experiments require measurements of vertical beam intensity profiles . An example of such measurements is shown in fig . 2 . for colliding H + and He' ion beams [1] . For merged beams (0 = 0) the general definition of the form factor is even more complex than eq. (5) and requires two-dimensional beam profile measurements. I will now make one further assumption in order to get the most simple possible event rate formula for a colliding beams experiment . Let the density of one beam be uniform as in the case of a solid target, e.g . n A (r) = n A = const. and let the other beam B travel completely inside A for a distance d . Then eq . (1) reduces to R = nA dVUA,BFBIV B .
( 6)
Since the relative velocity v of eq . (3) is not equal to the projectile velocity v B (as in the case of a static target), eq. (6) is different from the well known formula given as eq. (2b) . For high luminosity in a colliding beams experiment it is not sufficient to provide a high target thickness n A d. One should also keep the projectile velocity v B low as long as the desired collision velocity v can be guaranteed . Low v B means increased interaction time ;
beam B
beam A
~6z b2
Fig. 1 . Schematic view of a crossed beams arrangement. The form factor f (see eq. (5)) can be measured by moving a slit with the width S. perpendicular to both beams and registering the transmitted currents 'A(Zk) and 'B(Zk) at position z k . II . TARGETS FOR HIGH-ENERGY/HEAVY-ION BEAMS
82
A. Midler / Targets consisting offree charged particles
6
possible to determine the form factor f (eq. (5)) to within 1% absolute uncertainty. This has to be compared to the uncertainties of pressure and length measurements for a gas target which are typically 10-20% .
5
5. An electrostatic electron target for crossed beams measurements on electron impact ionization of ions
4
2
0
1
2
z (MM)
3
4
5
Fig. 2. Determination of a form factor in a crossed beams experiment with H + and He' ion beams [1]. The currents in(zk) and 'B(zk) of H' and He' ions, respectively, were measured behind a slit and give the vertical intensity profiles of the beams. The resulting form factor is f = 2.44 mm . projectile particles stay longer inside the target beam and their chance to undergo a collision is enhanced . An important aspect has to be mentioned which is characteristic for charged particle beams experiments. Since electrical currents can be easily measured it is
cathode
In Giessen, we are investigating electron impact ionization of ions since a couple of years [2]. In our experiments a collimated beam of multiply charged ions is crossed with a ribbon shaped electron beam [3] that provides high electron densities (nearly 10 9 cm -3 at 1000 eV energy) over a width of 60 mm. The electron gun was developed in a collaboration with the Institut für Angewandte Physik, Universität Frankfurt. Transmission experiments with extremely narrow ion beams crossing the electron beam at different positions and angles and in addition computer calculations were used to find the present configuration. The computer code SLAC 166 was modified to include a simulation of thermal spread of the electrons as well as space charge compensation by trapped ions. The electron gun produces a current of 460 mA at 1 keV energy . The current density in the interaction region, where the ion beam crosses, is about 0.3 A CM-2 . Fig. 3 shows a cut through the electron target perpendicular to the ion beam. Electron trajectories and equipotential lines are indicated. The electrons are emitted from an impregnated tungsten cathode heated to about 1100 ° C. The emission is space charge limited. The current density emitted from the cathode is 0.09 A cm -2 at most, which is more than 100 times less than
collector
Fig. 3. Calculated equipotential lines and electron trajectories in the electrostatic electron gun [3]. The potential of the cathode is set to 0%, that of the rod electrodes in front of and behind the interaction box to 100% ; the interaction energy is given by 76 .6% of the maximum voltage applied. The gun extends to 60 mm in the direction of the ion beam and can be moved up and down across the ion beam axis .
A. Müller / Targets consisting offree chargedparticles
Z
Fig . 4. Potential distribution inside the interaction box with the electron space charge uncompensated . The depth of the well corresponds to 96% of the electron energy defined by the potential on the interaction box (see fig . 3) .
the possible emission from the cathode material . The electron beam is focussed into the interaction box by a pair of rod electrodes. Behind this box there are two more pairs of rods which serve to provide a symmetrical potential distribution . The electrons are collected by a water cooled copper anode . The density of equipotentials in fig . 3 indicates a field free region in the interaction box, which is necessary to provide a good energy definition of the electrons and to prevent deflection of the passing ions . However, with the space charge of the electrons, electric fields in and around the electron beam can never be avoided completely. Indeed, the calculations show that there is a flat negative potential well inside the interaction box . The potential distribution over a plane perpendicular to the ion beam [4] is shown in fig. 4 on a sensitive scale . The depth of the calculated potential well is 0 .04UCA where the voltage UCA gives the potential of the interaction box with respect to the cathode potential. The well acts like an ion trap confining slow ions in the (y, z) plane (see fig. 4) . The trap is also closed in the x-direction (ion beam direction) since the potential rises at both edges of the electron beam because the negative space charge that lowers the potential disappears there . The electron beam produces slow ions from the residual gas . Some of these ions become trapped and compensate the electron space charge so that ultimately the trap is macroscopically neutralized . We have found strong experimental evidence for space charge neutralisation in our electron gun [5] . One strong argument is the observation of narrow dielectronic capture resonances [6] with a width of less than 0.4 eV. These resonances occur at exactly defined rela-
83
tive velocities v between electrons and ions in the interacting beams. They are characterized by capture of a free electron into the ion and by simultaneous excitation of a core electron . Since this is the time inverted Auger process it is understandable that it can only occur at a defined energy . The widths of the observed resonances give information on the experimental energy definition . The width of 0.4 eV, observed for a resonance in Ba l+ ions at 90 eV, indicates a resolution which is 10 times better than could be expected without space charge neutralisation by slow trapped ions . Thus, the electron target described in this section provides high density and a quite long interaction length d of 60 mm, which is very helpful with respect to event rates (see eq. (6)) . In spite of the massive space charge carried by the electrons the definition of the electron energy in the electron region is remarkably good .
6. A magnetically confined electron target for merged beams experiments at an accelerator In the previous chapter a very compact electron gun was described which, nevertheless, provides a considerable interaction length of 60 mm . If an experimenter wants to increase the interaction length much beyond that it is necessary to merge the beams in the interaction region . For example the electron cooler for the ESR storage ring will provide an interaction length of 250 cm, which is more than 40 times that of the electrostatic crossed beams target discussed in section 5 . The difficulty is now to hold the electron beam together over such distances and maintain good energy definition. Also it is not trivial to merge and demerge two beams. On the other hand, there are more advantages in a merged beams configuration than just the high interaction length . With the velocity vectors of the beam particles parallel it is easily possible to study processes at low relative velocities v . Moreover, the kinematics are such that beam energy spreads in the laboratory frame are considerably reduced by the transformation into the center-of-mass system . Thus, high resolution experiments on resonant processes become feasible . Fig. 5 shows an overview on the center-of-mass energies which can be obtained with parallel or antiparallel electron- and ion beams . Ion energies per nucleon in the range of the UNILAC and electron energies up to about 10 keV are assumed . The lines given in fig . 5 show the possible combination of ion and electron energies in the laboratory to obtain a certain center-of-mass energy which is indicated for each line . Relativistic mechanics were used to calculate these dependences . A range from 0 up to 50 keV center-of-mass energies can be covered by the proper choice of geometry and laboratory energies . II . TARGETS FOR HIGH-ENERGY/HEAVY-ION BEAMS
84
A. Müller / Targets
consisting
of free chargedparticles 65
1,1,1 .1,1,1,1,i,1 .1 .1,1,
60 55 50 45 40 35 30 25 20 15
Ion energy E, /A (MeV/u)
Fig . 5 . Combination of ion and electron laboratory energies resulting in the same center-of-mass energy Ec .m. in a merged beams arrangement . For each line E..m. is indicated. The curves with negative slope are for antiparallel beams . For a given electron target, optimum event rate is achieved if a combination of high electron energy and low ion energy can be chosen (eq. (6)) in order to keep the ion velocity a B small. On the other hand the argument of good energy resolution may require high ion energy since the effect of energy-spread compression by the transformation from the laboratory frame into the center-of-mass system becomes more efficient at high laboratory energies . Moreover, the possibility to produce a certain ion charge state and the presence of concurrent residual gas collisions have to be considered in the choice of laboratory energies . Fig. 6 shows the electron laboratory energies which are necessary to cover a center-of-mass energy range from 0 to 65 eV when the ion energy is fixed to 11 .4 MeV/u. Apparently there are always two possibilities to realize a given center-of-mass energy. One is to have the electrons chase the ions and one is to have the ions run into the electrons. Zero energy results when the particles in both beams have the same speed vA = vB . This is the case at an electron energy E = 6260 eV in our example. The shape of the curve in fig. 6 is responsible for the energy-spread compression mentioned above. A considerable electron energy spread (of e .g . 100 eV) in the laboratory frame around E = 6.26 keV results in an uncertainty of only 0 .4 . eV in the centerof-mass system. This fact can be used to study capture resonances in low-energy collisions of electrons with highly charged ions . At a specific energy of 11 .4 MeV/u high ion charge states can easily be obtained by stripping in a
10
5500
6000
6500
7000
7500
Electron Energy [eV1
Fig . 6 . Center-of-mass energy vs electron laboratory energy for a fixed ion laboratory energy of 11 .4 MeV/u in a merged beams experiment (relativistic mechanics used) . At 6260 eV, ions and electrons have the same velocity and zero center-ofmass energy .
foil target. Low relative energy is achieved with a parallel electron beam of about 6 .26 keV . The principal setup for such experiments which were proposed for the UNILAC [71 is shown in fig . 7 . The electron beam [81 is inside a strong axial magnetic field of B =1 T produced by a superconducting solenoid. The 3 mm diameter
auperconCuctirg aol@noid, 8 . 1T
Fig . 7. Schematic diagram of a merged beams experiment [111. The electron gun is inside the homogeneous field of a superconducting solenoid. Bending of electron trajectories onto the ion beam axis is accomplished by additional elliptical windings on top of the solenoid.
A. Milller / Targets consisting offree charged particles
cathode is 25 mm off the ion beam axis and fully immersed in the magnetic field . Electrons with a maximum current density of 10 A cm -2 are extracted and accelerated to energies up to 8 keV. The gun can be operated in two modes one with a perveance of 0.95 WA V -3/ 2 and one with 0 .098 I.A V -3/2 . Maximum electron densities at E=8 keV are 101° and 10 9 cm -3 , respectively. One of the most important features of the design is the low transverse electron energy of less than 0 .4 eV, which is obtained by resonant focusing, i.e . a proper matching of electric and magnetic fields depending on the gun geometry [9]. The matching of fields keeps the electrons from gaining radial energy and spiralling in the magnetic field . Of course there is an initial contribution of radial energy due to the electron emission process so that typically 0.3 eV can hardly be circumvented [10] . Again the electron space charge produces radial fields in the electron beam. A voltage depression of up to 109 eV results at 8 keV [8] . The electrons are driven away from the beam center by the space charge fields, however, the magnetic field forces them to stick with their line of force so that the electrons cannot gain radial energy. Only the longitudinal energies of the electrons vary between 8 .000 and 7.891 keV in the worst case . However, discussion of fig . 6 shows that longitudinal energy spread is considerably reduced in a merged beams situation by the transformation from the laboratory to the center-of-mass frame so that even at an electron density of 10 1° cm -3 good resolution at low collision energies can be achieved . The electron beam is merged and demerged with the ion beam by transverse components (B 1 = 0.15B) of
0 Lri N
85
the solenoid field. These components are obtained by adding elliptical windings on top of the solenoid . The solenoid provides a field homogeneity of about 10 -4 in the merging section . The low radial electron energy is maintained also when the acceleration voltage U is changed, provided all voltages on focussing electrodes are changed by the same factor and the magnetic field is changed proportional to vrU -- . Hence, an electron laboratory energy of 800 eV requires an axial magnetic field of about 0 .3 T. An experimental setup which will implement the present electron target for merged beams measurements of dielectronic recombination cross sections is presently completed at the UNILAC of GSI [11] . A schematic of the experimental setup and beam optics is given in fig . 8 . A beam of accelerated ions, e.g. Fe 23+ at 11 .4 MeV/u, is coming from the left. For collimation of the ion beam and reduction of gas flow a narrow pipe is used to separate the vacuum system in the experimental area from that of the accelerator . Magnetic quadrupole lenses in front of the magnets serve to focus the ion beam. The first magnet defines the ion charge state, the second magnet analyzes the charge states of ions which have undergone a collision in the electron target . Steerers before and behind the target have to be used to compensate for the transverse magnetic field components inside the target which are strong enough to slightly deflect an 11 .4 MeV/u ion beam . Along the 50 cm long interaction path ions capture free electrons and thus reduce charge state . These ions can be magnetically separated from the parent ion beam and detected by a suitable single particle detector. Since electrons can also be captured out of bound states in residual gas atoms or
Horizontal
E E x E E T 0
Vertical
Path
Length
16000
mm
N
Fig. 8. Schematic of the experimental setup and beam optics for a merged beams experiment [11) at the UNILAC. II . TARGETS FOR HIGH-ENERGY/HEAVY-ION BEAMS
86
A . Midler / Targets consisting offree charged particles
molecules, it is necessary to keep the residual gas pressure as low as possible. The range of 10 -s Pa is considered suitable . There is a choice of ion velocity (energy) in a colliding beams experiment since the desired center-of-mass energy can be obtained by adjusting the velocity of the electrons (see eq . (3)) . This is important for the present experiment because it allows to use an ion energy suitable to reduce background from charge-changing collisions in the residual gas . Cross sections for such processes are small at high ion energies so that the projected energy of 11 .4 MeV/u is of some .additional advantage.
7. Conclusions Colliding beams experiments are difficult and there are many subtle sources of errors in cross section measurements. Problems arise from the space charge of electron or ion beams and the resulting limitations of particle densities. Numbers like 10 10 electrons per cubic centimeter are extremely high and unprecedented so far in beams experiments . Still, this corresponds to a pressure in a gas target of only some 10 -5 Pa, indicating the problem with event rates in such a target. In general, colliding beams experiments require ultrahigh vacuum in order to keep backgrounds low which arise from collisions with residual gas particles. There are many kinds of experiments, however, which can only be done with colliding beams techniques. Among these are all measurements looking for collision processes of electrons with ions. A beam target can be well characterized, the density distribution can be measured more accurately than for a conventional target. The beam can be well prepared with respect to low energy spread, to mass and charge state definitions, and even desired states of excitation can be provided . Hence, colliding beams experiments are powerful sources of information. They have the potential of a precision in cross section measurements which can hardly be assessed in experiments with conventional targets.
Acknowledgement I want to acknowledge fruitful interaction with many colleagues in various collaborations dealing with electron-ion collision experiments . Special thanks are due to R. Becker who is leading the computer part in the design of the electron targets which are discussed in this paper as examples for real developments used in crossed and merged beams experiments.
References
[2]
[3] [41
[61
[81
[11]
K. Rinn, F . Melchert and E . Salzborn, J. Phys . B18 (1985) 3783 . A . Müller and R. Frodl, Phys . Rev . Lett 44 (1980) 29; A . Müller, K . Huber, K. Tinschert, R . Becker and E. Salzborn, J . Phys. B18 (1985) 3011 ; K. Tinschert, A. Müller, G. Hofmann, Ch . Achenbach, E. Salzborn, and R . Becker, J. Phys . B20 (1987) 1121 . R. Becker, A . Müller, Ch . Achenbach, K. Tinschert, and E. Salzborn, Nucl . Instr. and Meth . B9 (1985) 385 . G . Hofmann, Diploma Thesis, Giessen (1987) unpublished. A. Müller, G. Hofmann, K. Tinschert, R . Sauer, E. Salzborn, and R. Becker, Nucl. Instr . and Meth. B24/25 (1987) 369 . A. Müller, K. Tinschert, G . Hofmann, E. Salzborn, and G .H . Dunn, Phys . Rev. Lett . 61 (1988) 70 . R. Becker, E . Jennewein, M. Kleinod, U. Pr6bstel, N . Angert, F. Bosch, I . Hofmann, J. Klabunde, P .H. Mokler, H . Schulte, P . Spadtke, B . Wolf, and A . Müller, GSI Scientific Report (1985) GSI-86-1, ISSN 0174-0814, p . 378 . U . Prôbstel, R . Becker, E . Jennewein, M . Kleinod, and K. Valadkhani, GSI Scientific Report (1986) GSI-87-1, ISSN 0174-0814, p . 271 . J.R. Pierce, Bell Syst . Techn. J. 30 (1951) 825 . Ultracold electron beams can be produced by photoelectron emission from e.g. GaAs crystals ; see e.g . D .T. Pierce et al., Rev. Sci . Instr . 51 (1980) 478 . A . Müller, S . Schennach, W . Spies, N . Angert, H . Emig, J. Klabunde, K .-D . Leible, P.H. Mokler, H. Schulte, P. Spädtke, B . Wolf, R. Becker, E . Jennewein, M . Kleinod, and U . Pr6bstel GSI, Scientific Report (1987) GSI-88-1, ISSN 0174-00814, p . 347 .