Self-cooling of highly charged ions during extraction from electron beam ion sources and traps

Nuclear Instruments and Methods in Physics Research B 149 (1999) 182±194

Self-cooling of highly charged ions during extraction from electron beam ion sources and traps R.E. Marrs

1

Lawrence Livermore National Laboratory, P.O. Box 808-050, Livermore, CA 94551, USA Received 27 March 1998; received in revised form 28 August 1998

Abstract Evaporative self-cooling can produce a dramatic reduction in the temperature and emittance of ions extracted from electron beam ion sources and traps for favorable values of ion density and charge if the trapping potential is removed slowly. For existing electron beam ion traps, calculations predict that 20% of trapped Ar18‡ ions can be extracted with an average emittance 30 times lower than the uncooled emittance. It may be possible to construct sources of very-highly charged ions with an absolute brightness of order 2 ´ 1010 ions/mm2 mrad2 s, a value more than four orders of magnitude larger than that of existing highly charged ion sources. Ó 1999 Elsevier Science B.V. All rights reserved. PACS: 41.75.Ak; 29.25.Ni Keywords: Ion source; Ion trap; Evaporative cooling; EBIT; EBIS

1. Introduction Electron beam ion traps (EBIT) and sources (EBIS) are known for their ability to produce ions in very high charge states. Essentially any charge state of any element can be produced, including bare U92‡ ions [1]. Another remarkable, but seldom mentioned, property of these sources is the low emittance of the ion beams extracted from them. Low emittance makes it possible to focus low-energy beams of ions to small spots for microanalysis and other applications. For example,

1 Tel.: +1 925 422 3890; fax: +1 925 423 3371; e-mail: [email protected]

Ar18‡ ions from an EBIT have been focused to a 20 lm diameter spot at an energy of 17q keV, where q is the ion charge [2]. The corresponding emittance is 1 p mm mrad. If an even lower emittance can be obtained, it would enable new applications for very-highly-charged ions. In this paper I argue that, under favorable conditions, evaporative self-cooling of ions during extraction from an EBIT can be used to cool a fraction of the ions so strongly that their emittance is reduced roughly 30-fold. The predicted e€ect can be used to construct a very high brightness source of highly charged ions. The EBIT and EBIS trap ions within the negative space-charge potential of a magnetically compressed electron beam. Usually superconducting

0168-583X/98/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 6 2 4 - 7

R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

magnets are used to produce magnetic ®elds of several tesla in order to achieve high electron beam compression. High ion charge states are produced by electron impact ionization with beam electrons. The EBIS was developed as a source of highly charged ions and uses a solenoid magnet roughly 1 m in length [3]. The EBIT was developed for X-ray spectroscopy of trapped ions [4±6], but it can also be used as an ion source [7,8]. The EBIT uses a Helmholtz-coil con®guration that provides a roughly 2 cm length of high magnetic ®eld with radial access for X-ray measurements. In both devices the electron beam is parallel to the magnetic ®eld. Ion con®nement in the axial direction is obtained with electrostatic potential barriers at the ends of the trapping region. Normally the axial potential barrier is lower than the total radial potential, so that ions escape in the axial direction. Ions are extracted by lowering the axial potential barrier at one end with a suitable waveform, usually a square pulse or a linear ramp. Production of very-highly charged ions in an EBIT or EBIS always requires evaporative cooling to balance electron beam heating of the trapped ions [4,9,10]. Without evaporative cooling, the ion temperature would exceed the space charge potential of the electron beam (typically within 100 ms), and ion con®nement within the electron beam would be lost before high charge states could be reached. However, with the careful use of evaporative cooling, trapping times of several hours have been achieved for heavy ions such as Au69‡ [9]. Normally a light element such as neon is introduced as an evaporative coolant for the highly charged ions. Lower charge states, such as Ar16‡ , can be produced without evaporative cooling because the electron-beam heating rate, which scales as q2 , is much lower and the ionization cross sections are larger. However, evaporative cooling improves EBIT and EBIS performance even for light ions. In any case, it is always present due to the unavoidable loss of ions from the trap. Although the role of evaporative cooling in ion con®nement in EBITs and EBISes is now well recognized, its e€ect on ion extraction has not been addressed. To understand the power of self-cooling during ion extraction, note that the equilibrium trapped-ion temperature for highly charged species

183

is roughly 10% of the trap potential (i.e., Ti  0.1qeVwell , where Ti is the ion temperature in energy units, e is the elementary charge, and Vwell is the axial barrier con®ning the ions) [10,11]. Hence each ion escaping over the axial barrier removes an energy much larger than the average thermal energy of the trapped ions. The cooling power of each ion will be somewhat lower when Vwell is reduced to extract the trapped ions. However, the calculations described below predict that a reduction in the ion temperature of more than two orders of magnitude is possible if Vwell is ramped to zero slowly and other conditions are favorable. We refer to this e€ect as ``strong self-cooling''. 2. Thermal processes in the trap Strong self-cooling is possible because of a fortunate ordering of the time scales for ion±ion energy exchange, electron-beam heating, and ion evaporative cooling. With properly chosen trapping conditions and a properly chosen potentialbarrier waveform for ion extraction, it is possible for escaping ions to cool the remaining trapped ions within a time that is faster than the time for electron-beam reheating but much slower than the ion-ion collision rate. The remaining ions then thermalize at a declining temperature that maintains the condition Ti < qeVwell . Strong self-cooling does not occur without this ordering of time scales. For example, if the ion-ion collision rate is too low, the strong cooling e€ect does not occur. The physical processes that are important for strong self-cooling are: (1) ion con®nement in the combined space charge potential of the electron beam and the trapped ions, (2) electron-beam heating of the ions, (3) work done on the ions by the space-charge potential, (4) energy exchange through ion±ion collisions, and (5) ion escape (evaporation) from the trap. These processes have been included in a time dependent calculation of the ion temperature and emittance. For simplicity, we consider only a single ion species of charge q and mass Mi . A more elaborate calculation that includes multiple charge states of more than one element was reported by Penetrante et al. [10]. That calculation was used to model the

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evolution of ionization balance and temperature for con®ned ions, with emphasis on the evaporative cooling of highly charged ions by light ions. The present calculation focuses on self-cooling of a single ion species during extraction from an EBIT. 2.1. Ion con®nement Ions are con®ned in the radial direction by the space-charge potential of the electron beam. The electron beam is known to have a Gaussian pro®le in EBIT devices with high current density [6,12], so the current density may be written as je …r† ˆ

Ie exp…ÿr2 =2r2e †; 2pr2e

…1†

where Ie is the total beam current. We refer to re as the characteristic electron beam radius. It is determined by the magnetic ®eld and the properties of the electron gun. Integrating je (r) gives the expression for the beam current inside a radius r: Ie …r† ˆ Ie ‰1 ÿ exp…ÿr

2

=2r2e †Š:

…2†:

The electric ®eld is in the radial direction and is given by Ee …r† ˆ

Ie …r† ; 2pe0 ve r

…3†

where e0 is the permitivity of free space and ve is the electron velocity. The radial potential, needed for calculating ion con®nement, is obtained by integrating E(r):    Z r Ie x2 x ÿ ‡


; E…r0 † dr0 ˆ Ve …r† ˆ 4pe0 ve 4 0 …4† where x ˆ r2 =2r2e . We refer to the quantity …Ie =4pe0 ve † as the total electron space-charge potential and denote it by Ue . The trapped ions distribute themselves within the radial space-charge potential according to the Boltzmann relationship, hence the ion density is ni …r† ˆ ni …0† exp …ÿqeV …r†=Ti †;

…5†

where ni (0) is the ion density on the axis. Here V(r) is the space charge potential at radius r, including the contribution of positive ion charges. (We ig-

nore the magnetic contribution to radial con®nement as it is small for ions near the axis.) In the interest of simplicity, we note that the shape of the ion density pro®le is close to a Gaussian and approximate it as such Ni exp …ÿr2 =2r2i †; …6† ni …r† ˆ 2pr2i L where Ni is the total number of ions in a trap of length L. The ion density is then determined by the two parameters (Ni /L) and ri , the characteristic ion radius. By analogy with the electron spacecharge potential, the ion-space charge potential is   x2 …7† Vi …r† ˆ Ui x ÿ ‡


; 4 where Ui ˆ …qeNi =4pe0 L† and x ˆ r2 =2r2i . The Boltzmann relationship, along with the assumed Gaussian form of ni (r), requires that r2 qeV …r† ˆ 2 Ti 2ri

…8†

i.e., the space charge potential V(r) must be quadratic in r. As can be seen from the expansions of Ve (r) and Vi (r) given in Eqs. (4) and (7), V(r) is quadratic to ®rst order. Dropping higher order terms in the expansions of Ve (r) and Vi (r), Eq. (8) becomes   r2 qe‰Ve …r† ÿ Vi …r†Š qer2 Ue Ui ÿ : …9† ˆ ˆ 2Ti r2e r2i Ti 2r2i Solving Eq. (9) for ri yields the following expression for the characteristic ion radius in terms of the characteristic electron beam radius and the total electron and ion space-charge potentials s Ti ‡ qeUi : …10† r i ˆ re qeUe Note that in the low temperature limit (Ti qeUi ) p ri  re Ui =Ue and the ions almost completely neutralize the core of the electron beam. 2.2. Electron-beam heating Trapped ions are heated by small angle Coulomb collisions with beam electrons. The heating power per ion is

R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

Hi ˆ p

je q2 e4 2me kie ; e Ee M i

…11†

where me is the mass of the electron, Ee is the electron beam energy, and kie is the Coulomb logarithm [13]. In units of eV/s, the heating power is Hi ˆ 442

q2 je kie Ee A

…eV=s†;

…12†

where je is in A/cm2 , Ee is in eV, and A is the ion mass number. The nearly constant Coulomb logarithm is given by  p  ne ; …13† kie ˆ 24 ÿ ln Te where ne is the electron density in cmÿ3 and Te is the electron temperature in eV [14]. For the present work, kie  15. The total heating power is the product of Hi and ni integrated over the volume of the trap Z Pheat ˆ ni …r†Hi …r† dV Z q2 kie ni je …r† dV : …14† ˆ 442 Ee A The integral runs from r ˆ 0 to 1 and over the length L of the trap. Using the Gaussian expressions for ni (r) and je (r) and performing the overlap integral, we get   q2 kie Ni Ie …eV=s†: …15† Pheat ˆ 442 Ee A 2p…r2i ‡ r2e † 2.3. Space-charge heating As the characteristic ion radius ri becomes smaller due to the decreasing number of trapped ions and their decreasing temperature, the Coulomb potential of the electron beam does work on the ions. The amount of work is the volume integral of the net (radial) electric ®eld times the radial displacement Dr of each ion. Z …16† DWCoul ˆ qeni …r†‰Ee …r† ÿ Ei …r†ŠDr dV ; where Ee (r) and Ei (r) are magnitudes of the electric ®elds from the electrons and ions, respectively.

185

Since the (Gaussian) shape of ni (r) does not change as the ion distribution contracts, Dr ˆ r…Dri =ri † where Dri is the change in the characteristic ion radius. Calculating Ee (r) and Ei (r) from the gradients of Ve (r) and Vi (r) in Eqs. (4) and (7), and using Eq. (10) to connect ri and Ti , we get   Dri Ni Ti : …17† DWCoul ˆ ÿ 2 ri The same result is obtained from the equation for the adiabatic compression of an ideal gas. 2.4. Ion±ion collisions Ions exchange energy with each other at a rate characterized by the ion±ion collision rate vi . The ion±ion collision rate is perhaps the most important parameter that determines whether strong self-cooling occurs as it determines the rate at which ions thermalize and it a€ects the rate at which they evaporate from the trap. The expression for vi is [15] p q4 e 4 …18† mi ˆ 2pni p 3=2 ki : M i Ti The Coulomb logarithm ki can be written as [14] "  3=2 # p q2 ki ˆ 23 ÿ ln ni : …19† Ti We retain ki in variable form because of the large range of ion temperature and charge in the present work. However ki  9 is a typical value. Evaluating the expression for vi in convenient units, we have ni q 4 mi ˆ 9:0  10ÿ8 p 3=2 ki AT i

…sÿ1 †;

…20†

where ni is in cmÿ3 and Ti is in eV. In the numerical calculations we use the average collision rate de®ned as R mi …r†ni …r† dV mi ˆ : …21† Ni Evaluating the overlap integral, we ®nd mi ˆ 12 mi …0† where mi (0) is the ion collision rate evaluated at the ion density on the axis. Radial averaging is

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appropriate because the length of the trap is three orders of magnitude greater than the characteristic ion radius. 2.5. Ion escape Ions escape or evaporate from an EBIT or EBIS as they scatter into the forbidden region of velocity space corresponding to an axial kinetic energy greater than the axial potential barrier qeVwell . Evaporation occurs gradually during ion con®nement and more rapidly during ``slow'' extraction, where the axial potential barrier is lowered slowly compared to the ion thermalization rate. This is distinct from ``fast'' extraction of the ions, where the axial potential barrier is removed so quickly that each ion leaves with its instantaneous velocity. The ion escape rate has been calculated from the Fokker±Planck equation by Pastukhov in the context of magnetic-mirror con®nement [16], and more recently by Khudik [15]. There is a substantial di€erence between the two calculations for the parameter range applicable to the present work; we use the more recent calculation by Khudik. The Khudik ion escape rate from Eq. (45) of Ref. [15] is !   4mi eÿxi 1 A~1 A~2 1 ‡ ‡ 2 ‡


; …22† mK ˆ p xi xi p xi CR 2 where xi ˆ …qeVwell =Ti † and mi is the average ion± ion collision rate. The three constants CR , A~1 , and A~2 depend on the magnetic mirror ratio, which is equal to one for the uniform magnetic ®eld of the present problem. Their values are CR ˆ 0.924, A~1  ÿ1:45, and A~2  3:46. Eq. (22) applies for xi  1. ÿ3=2 temperature dependence of Because of the Ti vi , the mean free path for ion±ion collisions becomes less than the trap length at low ion temperatures, particularly for the long traps used in EBIS devices. The ion escape rate is then limited by ion ¯ow from the end of the plasma column, for which the loss rate is vth mf ˆ p eÿxi ; pL

…23†

q i where vth ˆ 2T is the ion thermal velocity [17]. Mi The actual ion escape time is the sum of the Khudik and ¯ow times: sescape ˆ sK + sf where ÿ1 sK ˆ mÿ1 K and sf ˆ mf . The total ion loss rate is dNi ÿNi then dt ˆ sescape . The evaporative cooling power Pevap is the product of the total ion loss rate and the average energy removed by each ion. The average energy removed per ion is qeVwell + Ti for Khudik-limited escape (i.e., for sK sf ) and qeVwell + 2Ti for ¯owlimited escape (i.e., for sK sf ) [17]. Between the two limits, the evaporative cooling power is     dNi sf Ti : …24† qeVwell ‡ 1 ‡ Pevap ˆ dt sK ‡ sf This completes the set of formulas needed for the numerical calculations of evaporative self-cooling. 3. Emittance and brightness The emittance is an important characteristic of any ion source. It determines the size of the beam and the fraction of it that can be accepted by beam lines and accelerators. Emittance is commonly de®ned as e ˆ prr0 , where r is the radius of a beam at a waist and r0 is its divergence angle [18]. The emittance is the area of the ellipse in phase space with axes of length 2r and 2r0 . We use units of p mm mrad for the emittance so that the numerical value of the emittance is r
r0 . The absolute emittance varies inversely with ion velocity, but it is otherwise conserved during ion extraction and acceleration. Hence the emittance can be related back to the temperature and radius of the ions trapped in an EBIT or EBIS. The size of the emittance depends on the fraction of the beam included in its de®nition. We calculate the absolute emittance of the extracted ion beam from the formula s Ti ; …25† e ˆ pR80 qeU where R80 is the radius in the trap containing 80% of the ions, Ti is the temperature of the trapped ions, and U is the ion acceleration potential [18]. For the assumed Gaussian ion distribution, the relationship between R80 and ri is R80 ˆ 1.794ri .

R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

For a Maxwellian distribution of ion velocities at temperature Ti , the phase-space de®ned by Eq. (25) contains 67% of the ions. Since the ion con®nement radius contracts as the temperature falls during self-cooling, and as the ion space-charge is reduced by ion loss [cf. Eq. (10)], there is a further reduction in the ion emittance beyond that expected from its explicit dependence on temperature. Ions extracted from a high magnetic ®eld into a ®eld free region receive an additional transverse velocity component from the change in magnetic ®eld. This increases the emittance of the ion beam. The magnetic component of the emittance can be estimated from the conservation of canonical angular momentum for each particle. In cylindrical coordinates, the canonical angular momentum is ph ˆ Mi vh r ‡ qeAh r, where Ah is the theta component of the magnetic vector potential and vh is the theta component of the transverse velocity. For a uniform magnetic ®eld in the z direction, Ah ˆ 12 Br. Hence the quantity Mi vh r ‡ 12 qeBr2 is conserved for each ion, so that Mi Vh R ˆ Mi vh r ‡ 12 qeBr2 where Vh and R refer to a ®eld free region. The ratio of the emittance with and without the e€ect of the magnetic ®eld is then e…B ˆ B0 † Vh …B ˆ B0 † 1 qeB0 ˆ1‡ r; ˆ e…B ˆ 0† Vh …B ˆ 0† 2 Mi v h

…26†

where B0 is the magnetic ®eld in the trap. (For simplicity, we ignore the fact that the magnetic ®eld only increases one of the two orthogonal components of the transverse motion for each ion.) The quantity Mi vh =qeB0 is the ion cyclotron radius rc . Evaluating Eq. (26) at r ˆ ri (the characteristic ion radius), the expression for the magnetic contribution to the emittance becomes e…B ˆ B0 † 1 ri : ˆ1‡ e…B ˆ 0† 2 rc

…27†

Setting vh equal to the ion thermal velocity, the value of the cyclotron radius in cm is p ATi …cm†; …28† rc ˆ 0:14 qB0 where Ti is in eV and B0 is in kG. The magnetic component of the emittance is small, even at low Ti , because ri rc . In other words, the trapped ion

187

motion is dominated by the electrostatic force of the electron beam, not by the magnetic force. The brightness of an ion beam refers to the ¯ux of ions per unit area per unit solid angle; it determines the number of ions that can be focused into a small spot. The brightness of an ion source is a key parameter that determines the utility of the source for microanalysis or for injection into accelerators with a limited phase±space acceptance. Brightness can be calculated from the emittance with the de®nition bˆ

Ni 1 ; p2 e2 tcycle

…29†

where Ni is the number of ions in the trap and tcycle is the cycle time for ion extraction [18]. We use Eqs. (25), (27) and (29) to compute the emittance and brightness of the escaping ions in the selfcooling calculations. 4. Numerical calculations The equations given above have been incorporated into a time-dependent calculation of the ion parameters during a slow reduction of the axial potential barrier. The initial temperature and initial number of ions are speci®ed at the beginning of the calculation, along with constant parameters such as the electron beam properties and the length of the trap. At each time step Dt the change in the total energy W of the trapped ions is calculated from the expression DW ˆ …Pheat ‡ Pevap †Dt ‡ DWcoul . The change in the number of trapped ions is calculated from DNi ˆ ÿ…Ni Dt=sescape †, and a new ion temperature is then calculated from the relationship W ˆ 32 Ti Ni . This calculation was used to obtain the complete time dependence of the ion temperature and escape rate for the cases described below. The emittance and brightness were evaluated as discussed above, along with other parameters such as the ion radius and collision rate. 4.1. Ar18‡ in a 2 cm EBIT Argon is a common element for EBIT and EBIS operation. At the Lawrence Livermore

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R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

National Laboratory (LLNL), Ar18‡ beams have been extracted from an EBIT for various kinds of experiments. In one case, in which emittance was the key parameter, Ar18‡ ions were focused to a 20 lm diameter spot for X-ray microscopy [2]. In the present work, self-cooling calculations were performed for Ar18‡ ions under conditions similar to those used in the measurements of Ref. [2]. The calculations predict that strong self-cooling does occur for these conditions, so that some of the Ar18‡ ions have a much lower emittance and much higher brightness than the average values apparent in a typical experiment. The calculations were started with 106 Ar18‡ ions in a 2 cm long EBIT with a 20 keV, 150 mA electron beam. The characteristic electron beam radius was re ˆ 19.5 lm. The initial ion temperature was taken to be Ti ˆ 0.1qeVwell as suggested by spectroscopic measurements and modeling calculations [10,11]. The axial potential barrier Vwell was reduced linearly at a rate of 8 V/ms starting from 100 V. This rate is favorable for self-cooling and is also within the range of values normally used for slow ion extraction. The calculated time dependence of the Ar18‡ ion temperature and the number of ions remaining in the trap are plotted in Fig. 1. The initial slight increase in the ion temperature is due to the neglect of the cooling e€ect of lower-charge-state ions that are also present in the trap during steadystate con®nement and stabilize the ion temperature until self-cooling begins. The Ar18‡ escape rate is initially very small, and the ion temperature does not begin to decline until the Ar18‡ evaporation rate is increased substantially by the reduction of Vwell . Several important properties of the Ar18‡ ions are listed in Table 1 at di€erent values of Vwell . Notice that the temperature and emittance of the ions track Vwell as it is reduced to zero. The ratio xi ˆ …qeVwell =Ti †, on which the ion escape rate depends exponentially, decreases gradually as the axial well potential approaches zero. The formulas used for the ion escape rate (Eqs. (22) and (23)) are reliable for xi > 1. In this regime the ion±ion thermal equilibration time is faster than the ion escape time. The tabulated emittance includes the magnetic component, which is less than 5% of the total.

Fig. 1. Calculated ion temperature (top) and number of remaining ions (bottom) for Ar18‡ ions in a 2 cm EBIT. The trapping potential starts at 100 V and reaches zero at 12.5 ms.

The most important result of the calculation is the prediction that a signi®cant number of ions remains in the trap after the temperature and emittance have fallen dramatically. For example, the last 20% of the Ar18‡ ions have an average emittance more than 30 times lower than the initial value. This dramatic reduction of the ion temperature and emittance is the signature of strong selfcooling. The ion temperature and emittance could be reduced even without self-cooling by starting with a lower initial value of Vwell . Experimental observations and computer modeling of steady-state evaporative cooling indicate that the number of trapped ions and their temperature are both roughly proportional to Vwell [10,11]. Hence, by using a lower steady-state well potential instead of

R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

189

Table 1 Calculated properties of Ar18‡ ions trapped in a 20 keV, 150 mA electron beam Vwell (V)

Time (ms)

Ti (eV)

qeVwell /Ti

100 10 3 1 0.3

)12.5 )1.25 )0.38 )0.125 )0.038

180 30 11 5.0 2.7

10.0 6.1 5.1 3.6 2.0

Ni (106 ions)

1.0 0.36 0.21 0.13 0.07

Emittance (p mm mrad) 0.67 0.12 0.045 0.022 0.012

Brightness (mmÿ2 mradÿ2 sÿ1 ) 4.5 ´ 105 5.2 ´ 106 2.1 ´ 107 5.3 ´ 107 1.1 ´ 108

The axial trap barrier Vwell is reduced from 100 V to 0 at a rate of 8 V/ms. Zero time corresponds to Vwell ˆ 0. The initial number of ions is 1.0 ´ 106 at a temperature of Ti ˆ 180 eV. The last column is the brightness assuming all remaining ions escape with no further decrease in emittance; a repetition rate of 2 cycles/s is assumed. The emittance and brightness are absolute values at an energy of 20q keV.

self-cooling, 20% of the ions could be obtained at a temperature and emittance roughly 5 times lower. This should be compared to the 30-fold emittance reduction with self-cooling. The ion evaporation rate is most rapid at the end of the calculation as Vwell approaches zero (see Fig. 1). The loss times for collisional processes (Khudik) and ¯ow are plotted in Fig. 2 as a function of the fraction of ions extracted from the trap. Initially, collisional loss dominates and the increasing escape rate is supported in part by the strong temperature dependence of the ion±ion collision rate (cf. Eq. (18)). As the temperature falls ion escape becomes ¯ow-limited, and the increasing escape rate is supported by a decrease in the ratio qeVwell /Ti .

Fig. 2. Characteristic loss times for Ar18‡ ions plotted as a function of the percentage of ions extracted from the trap.

The calculation of an accurate collisional loss rate is dicult, and signi®cantly di€erent rates have been obtained by di€erent methods [15±17]. The present calculations are relatively insensitive to changes in the rate constant for collisional loss because of the exponential dependence of the ion escape rate on temperature. The main e€ect of an inaccurate loss-rate formula is a small change in ion temperature that produces a compensating change in the absolute ion loss rate. As the ions cool and contract toward the axis, the positive space charge density near the axis increases and can become a signi®cant fraction of the electron beam space-charge density. (For the case listed in Table 1, the initial number of 106 Ar18‡ ions corresponds to 7.8% of the total electron beam space-charge.) As the ions cool to a temperature Ti  qeUi , a condition approached in the present Ar18‡ calculation, the ion radial distribution ni (r) deviates from the assumed Gaussian shape. (At very low Ti it approaches a Fermi distribution.) The change in shape has a negligible e€ect on the electron beam heating rate because ri re (cf. Eq. (15)). The Coulomb heating rate depends only on the change in ion volume, which is proportional to r2 (as assumed in the numerical calculations) for any constant shape in cylindrical geometry. Hence the calculated Coulomb heating rate is insensitive to the shape of ni (r) except for a small e€ect during the transition from one shape to another. The change in the shape of ni (r) has a signi®cant a€ect on the average ion±ion collision rate mi , which becomes larger than the Gaussian

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R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

distribution value of 12 mi …0† and approaches mi …0† at very low ion temperature. However this is of little consequence because the ion escape rate is ¯ow limited at very low ion temperature, and the ¯ow escape rate does not depend on mi . The deviation of the ion distribution from Gaussian shape does a€ect the computed emittance and brightness because the actual 80% ion radius becomes slightly smaller than the assumed value of R80 ˆ 1.794ri with ri given by Eq. (10). 4.2. Kr36‡ in a high brightness source We have seen that self-cooling can produce a large increase in the brightness of Ar18‡ ions extracted from existing EBIT devices if the operating parameters are chosen correctly. What brightness could be obtained from a future source optimized for high brightness? To answer this question, calculations were performed for fully stripped Kr36‡ ions in an 80 keV, 5 A electron beam. The characteristic electron beam radius was 26.2 lm, corresponding to compression of a beam from a standard dispenser cathode in a 6 T magnetic ®eld. These parameters are similar to those expected for a source under development at LLNL. An initial axial well potential of 300 V was reduced to zero at a rate of 25 V/ms, a value favorable for self-cooling. This ramp rate is faster than that used above for Ar18‡ ions because the electron beam heating rate is larger, which negates the self-cooling e€ect if the ions remain in the beam for a longer time. The trap was assumed to be 25 cm long with an initial inventory of 2 ´ 108 Kr36‡ ions; this neutralizes 14% of the electron beam space charge. (In the interest of simplicity, we ignore the fact that a fraction of the ions will be in the adjacent Kr35‡ and Kr34‡ charge states and note that the e€ect of self-cooling is nearly the same for adjacent charge states.) The initial ion temperature was assumed to be 0.1qeVwell as expected for steady-state operation. Results of the Kr36‡ calculation are shown in Figs. 3 and 4, and in Table 2. As can be seen in Fig. 3, the loss time for ¯ow is larger than that for collisional loss, even for the high ion temperatures at the beginning of the calculation. This is primarily because of the longer trap length compared

Fig. 3. Characteristic loss times for the Kr36‡ calculation.

to the Ar18‡ calculation. The longer trap length is somewhat unfavorable for self-cooling and reduces the bene®t of the larger number of ions contained in a longer trap. As can be seen in Table 2, the ion temperature and emittance decrease as Vwell is reduced to zero; this demonstrates that selfcooling of Kr36‡ ions is occurring. The magnetic component of the emittance is less than 5% of the total. The Kr36‡ brightness was calculated assuming a repetition rate of 10 extraction cycles per second. This rate should be possible based on the estimated krypton ionization time in the high density

Fig. 4. Calculated absolute emittance for Ar18‡ ions accelerated by a 20 kV potential and Kr36‡ ions accelerated by a 80 kV potential. The Kr36‡ curve has been multiplied by a factor of 2 for display purposes to compensate for the di€erence in acceleration potentials.

R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

191

Table 2 Calculated properties of Kr36‡ ions, similar to Table 1 but for a more intense source operating with an 80 keV, 5 A electron beam Vwell (V)

Time (ms)

Ti (eV)

qeVwell /Ti

300 30 10 3 1

)12 )1.2 )0.4 )0.12 )0.04

1080 218 107 62 51

10.0 4.9 3.3 1.7 0.7

Ni (108 ions)

2.0 0.87 0.53 0.28 0.15

Emittance (p mm mrad) 0.46 0.12 0.065 0.036 0.025

Brightness (mmÿ2 mradÿ2 sÿ1 ) 9.6 ´ 108 6.0 ´ 109 1.3 ´ 1010 2.2 ´ 1010 2.4 ´ 1010

The axial trap barrier Vwell is reduced from 300 V to 0 at a rate of 25 V/ms. Zero time corresponds to Vwell ˆ 0. The initial number of ions is 2.0 ´ 108 at a temperature of Ti ˆ 1080 eV. The last column is the brightness assuming all remaining ions escape with no further decrease in emittance; a repetition rate of 10 cycles/s is assumed. The emittance and brightness are absolute values at an energy of 80q keV.

electron beam. The absolute brightness of the Kr36‡ ions reaches a value of 2.4 ´ 1010 ions/mm2 mrad2 s for the last 7.5% of ions extracted from the trap. This is over four orders of magnitude larger than the brightness of Ar18‡ ions extracted from an existing EBIT without self-cooling (®rst row of Table 1). (This comparison of absolute brightness includes a factor of 4 di€erence due to the fact that the acceleration potential used for Kr36‡ is 4 times higher than that used for Ar18‡ .) The calculated emittance for the Kr36‡ and Ar18‡ examples is compared in Fig. 4 with the e€ect of the acceleration potential removed. The larger normalized emittance for Kr36‡ arises from the longer trap length and higher electron current density (i.e., higher ion heating rate) of this more intense source, along with a somewhat larger electron beam radius.

Threshold conditions for self-cooling are illustrated in Fig. 5, where the temperature at the point where 80% of the ions have escaped from the trap is plotted as a function of the initial number of ions for Ar9‡ . The temperature at the 80% point is a good indicator of self-cooling. For these calculations, the electron beam energy was 10 keV at 150 mA. The electron beam radius and 2 cm trap length were the same as those in the Ar18‡ calculation. The initial ion temperature was 0.1qeVwell as for the other examples, and the axial potential barrier was reduced from 100 V to zero at a rate of 5 V/ms. The transition from simple ion escape (high ion temperature) to strong self-cooling (low ion temperature) occurs for initial ion inventories

4.3. Conditions for strong self-cooling The two examples presented above illustrate conditions for which strong self-cooling occurs. However, there are also conditions for which it does not occur. The key parameter that determines whether ion self-cooling occurs is the ion±ion collision rate vi . As can be seen from Eq. (20), vi is proportional to the ion density times the 4th power of the ion charge. If the ion density and charge are too low, then the ion temperature will not track the axial potential barrier as it falls because the ion±ion collision rate and the ion escape rate are too low.

Fig. 5. Calculated temperature for Ar9‡ ions when 80% of the ions have escaped from the trap plotted as a function of the initial number of ions in a 2 cm trap. The transition from simple escape to strong self-cooling occurs at the knee in the curve.

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near 106 ions. At low values of initial ion density, the plotted ion temperature becomes even larger than its initial value of 90 eV because of electron beam heating. Note that, although the present calculations indicate when self-cooling does or does not occur, they do not give accurate values of the ion temperature in the absence of self-cooling because the formulas used for ion escape are not accurate for Ti > qeVwell . The present calculations indicate that the possibility of ion self-cooling is marginal for several existing EBIS devices. For example, an EBIS at Kansas State University has been operated with Ar9‡ beams in a slow-extraction mode with axialbarrier ramp rates in a range appropriate for selfcooling [19]. The electron beam energy, number of ions, and other parameters reported for this device indicate that it may be near the self-cooling threshold for Ar9‡ ions. Higher charge state, higher ion density, and higher electron energy would bring EBIS devices of this type solidly into the range where self-cooling could be utilized to obtain very low emittance beams. 4.4. Extremely low ion temperature It is natural to ask how far the ion temperature can be reduced with self-cooling. One e€ect that limits the ion self-cooling power is the reduction of the ion escape rate due to the short ion mean free path at low Ti . This e€ect can be reduced with a well that has a sloping bottom (sawtooth shape), so that the ions fall to one end as their temperature and number decrease. We approximated this situation by repeating the Ar18‡ calculation with a trap length that was proportional to the number of ions remaining in the trap. In this case the ion line density remains constant. All other parameters of the calculation were as given above for Ar18‡ ions in an initially 2 cm long EBIT. As shown in Fig. 6, the calculation indicates that the ion temperature falls almost without limit. However, the lowest temperatures occur only for very low values of Vwell and for short times just before the trap is inverted and the remaining ions are expelled. At very low temperatures an ion plasma may become strongly coupled. The coupling strength is characterized by the parameter C ˆ (q2 e2 /4pe0 as Ti )

Fig. 6. Temperature of Ar18‡ ions in a sawtooth well calculated as explained in the text. The ions are expected to become strongly coupled at temperatures below the broken line.

where aS is the ion spacing given by as ˆ (3/ 4pni )1=3 . The coupling parameter C is the ratio of the interparticle Coulomb energy to the ion thermal energy. Plasmas with C > 1 are referred to as strongly coupled. They exhibit ¯uid behavior at C2 and crystallize at C170 [20]. At very low Ti , the Ar18‡ ion density in the sawtooth-well calculation is ni 2.6 ´ 1011 cmÿ3 , the value that completely neutralizes the electron space-charge in the center of the beam. (Note that aS is much smaller than the electron Debye length, hence electron shielding of the ion±ion interaction is negligible.) The temperature corresponding to C ˆ 1 is then Ti ˆ 0.47 eV. The temperature range below this value is denoted as ``strongly coupled'' in Fig. 6. If such low ion temperatures can be obtained with evaporative self-cooling, the correct picture of the ion plasma may be a thin column of ¯uid that evaporates from one end. A further reduction in ion temperature can be achieved by turning o€ the electron beam when the ions are already very cold and allowing them to expand adiabatically until they are con®ned only by the magnetic ®eld. (The axial potential barrier would be raised to prevent further axial ion escape.) In the absence of electron beam heating the cold ions could be trapped for very long times. For example, con®nement times of several seconds have been observed for hot ions magnetically trapped in an EBIT at LLNL without the electron beam [21]. The expanded ion density ni 0 can be no

R.E. Marrs / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 182±194

greater than the Brillouin limit nB ˆ (e0 B2 /2Mi ) (SI units). For argon ions in a 3 T ®eld, nB ˆ 6.0 ´ 108 cmÿ3 . The change in ion temperature during expansion is easily obtained by noting that the ions behave as an ideal gas, for which the relationship between temperature and density during adiabatic expansion is Tµn2=3 . Hence the change in ion temperature when the electron beam is turned o€ is  2=3 Ti0 nB 6  0:017: …30† Ti ni This estimate suggests that it may be possible to cool a large number of ions to room temperature in an EBIT with evaporative self-cooling followed by adiabatic expansion. 5. Discussion The calculations presented here indicate that self-cooling of highly charged ions during extraction from EBIT and EBIS devices can lead to a dramatic reduction in the emittance of the ion beams obtained from them. A strong self-cooling e€ect requires favorable conditions and may not occur for the usual operating parameters of most EBIS devices. Strong self-cooling would likely go unnoticed even where it is expected to occur, for example in the work of Ref. [2], because the e€ect of self-cooling on the normally observed average ion emittance is not large. However, the e€ect on the emittance of a fraction of the ions can be very large. For example, in one of the cases examined here, 20% of Ar18‡ ions trapped in an EBIT are predicted to have an emittance roughly 30 times lower than the uncooled emittance. The calculations also predict that an optimized EBIT-type highly charged ion source could achieve a brightness four orders of magnitude larger than that of existing sources. High brightness beams of self-cooled ions could be obtained from an EBIT or EBIS either by tight collimation of an extracted beam, or by chopping an uncollimated beam to accept only those ions emitted near the end of a slow extraction cycle. The signature of strong self-cooling would then be a large ¯ux of ions through small collimators, or an uncollimated beam pro®le that is narrowed and

193

enhanced in the center. There is already some experimental evidence for self-cooling. Investigators at LLNL have reported improved mass resolution for magnetically analyzed EBIT beams using slow extraction as compared to fast extraction [22], an observation consistent with self-cooling. Beams of Ar17‡ and Xe44‡ ions have been extracted from an EBIT at the National Institute of Standards and Technology with extraction ramp times ranging from 5 ls to 20 ms [8]. The reported improvement in the eciency of extraction and transmission through a beam line for the longer ramp times could be explained by an increased brightness due to ion self-cooling. Although the cooling calculations reported here are robust and relatively insensitive to many parameters such as the ion escape time, there may be experimental diculties in utilizing strong selfcooling to obtain very high brightness ion beams. For example, at low trapping potentials and ion temperatures, cold ions could become stuck in small nonuniformities or depressions in the potential well where they would remain until reheated by the electron beam. The timing and shape of the axial potential ramp could be a€ected by the unpredictable space-charge potential of trapped secondary electrons known to be present in EBIS and EBIT devices. Emittance growth during ion extraction and transport is also possible. On the other hand, improvements may be possible by using potential wells of di€erent shape, using nonlinear voltage ramps for extraction, or changing other parameters of the source. The liquid metal ion source and the related ®eld emission ion source, with a brightness of 2 ´ 1017 ions/mm2 mrad2 s, is by far the brightest type of ion source [23]. These sources are widely used to produce submicron focused ion beams in the semiconductor industry. However, only singly charged ions of a few suitable elements such as gallium can be produced. The electron-cyclotronresonance (ECR) ion source produces intense beams of multiply charged ions, and charge states up to Ar18‡ are possible with advanced ECR sources although the intensities for the highest charge states are much lower (1010 Ar18‡ ions/s) [24]. Unfortunately, the emittance of the ECR sources is of order 100 times larger than the

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(uncooled) emittance of the EBIS and EBIT. Hence, for Ar18‡ ions, the brightness of the most advanced ECR sources is similar to that of the 2 cm LLNL EBIT without self-cooling. The strong self-cooling e€ect presented here might enable the development of the ®rst high brightness source of very-highly charged ions. If such a source can be developed, it would have applications in microanalysis and nanotechnology because of the unique properties of highly charged ions such as their high sputtering rate, and the emission of X-rays and large numbers of secondary electrons at surfaces [25]. Acknowledgements The author would like to thank R. Cohen for his explanations of collisional loss processes. This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract No. W-7405Eng-48. References [1] R.E. Marrs, S.R. Elliott, D.A. Knapp, Phys. Rev. Lett. 72 (1994) 4082. [2] R.E. Marrs, D.H. Schneider, J.W. McDonald, Rev. Sci. Instr. 69 (1998) 204. [3] E.D. Donets, in: I.G. Brown (Ed.), The Physics and Technology of Ion Sources, Wiley, New York, 1989, p. 245. [4] M.A. Levine, R.E. Marrs, J.R. Henderson, D.A. Knapp, M.B. Schneider, Phys. Scr. T22 (1988) 157. [5] R.E. Marrs, M.A. Levine, D.A. Knapp, J.R. Henderson, Phys. Rev. Lett. 60 (1988) 1715.

[6] R.E. Marrs, in: F.B. Dunning, R.G. Hulet (Eds.), Experimental Methods in the Physical Sciences, vol. 29A, Academic Press, San Diego, 1995, p. 391. [7] D. Schneider, M.W. Clark, B.M. Penetrante, J. McDonald, D. DeWitt, J.N. Bardsley, Phys. Rev. A 44 (1991) 3119. [8] L.P. Ratli€, E.W. Bell, D.C. Parks, A.I. Pikin, J.D. Gillaspy, Rev. Sci. Instr. 68 (1997) 1998. [9] M.B. Schneider, M.A. Levine, C.L. Bennett, J.R. Henderson, D.A. Knapp, R.E. Marrs, in: A. Hershcovitch (Ed.), International Symposium on Electron Beam Ion Sources and Their Applications, Conf. Proc. No. 188, American Institute of Physics, New York, 1989, p. 158. [10] B.M. Penetrante, J.N. Bardsley, D. DeWitt, M. Clark, D. Schneider, Phys. Rev. A 43 (1991) 4861. [11] P. Beiersdorfer, in: R.L. Johnson, H. Schmidt-B ocking, B.F. Sonntag (Eds.), X-Ray and Inner-Shell Processes, AIP Conference Proceedings No. 389 AIP, New York, 1997, p. 121. [12] D.A. Knapp, R.E. Marrs, S.R. Elliott, E.W. Magee, R. Zasadzinski, Nucl. Instr. and Meth. A 334 (1993) 305. [13] I.P. Shkarofsky, T.W. Johnston, M.P. Bachynski, The Particle Kinetics of Plasmas, Addison-Wesley, Reading, MA, 1966. [14] D.L. Book, NRL Plasma Formulary, Naval Research Laboratory, Washington, DC, 1980. [15] V.N. Khudik, Nucl. Fusion 37 (1997) 189. [16] V.P. Pastukhov, Nucl. Fusion 14 (1974) 3. [17] R.H. Cohen, Nucl. Fusion 19 (1979) 1295. [18] R. Keller, in: I.G. Brown (Ed.), The Physics and Technology of Ion Sources, Wiley, New York, 1989, p. 23. [19] M.P. Stockli, C.L. Cocke, S. Winecki, Phys. Scr. T71 (1997) 175. [20] J.J. Bollinger, J.N. Tan, W.M. Itano, D.J. Wineland, D.H.E. Dubin, Phys. Scr. T59 (1995) 352. [21] P. Beiersdorfer, L. Schweikhard, J. Crespo Lopez-Urrutia, K. Widmann, Rev. Sci. Instrum. 67 (1996) 3818. [22] J.W. McDonald, D.H. Schneider, unpublished. [23] L.W. Swanson, A.E. Bell, in: I.G. Brown (Ed.), The Physics and Technology of Ion Sources, Wiley, New York, 1989, p. 313. [24] Z.Q. Xie, Rev. Sci. Instr. 69 (1998) 625. [25] D.H.G. Schneider, M.A. Briere, Phys. Scr. 53 (1996) 228.