- Email: [email protected]
Numerical simulations of space charge effects and plasma dynamics for FEBIAD ion sources
Available online at www.sciencedirect.com
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 4415–4419 www.elsevier.com/locate/nimb
Numerical simulations of space charge effects and plasma dynamics for FEBIAD ion sources L. Penescu *, R. Catherall, J. Lettry, T. Stora CERN, CH-1211, Geneva 23, Switzerland Available online 6 June 2008
Abstract The FEBIAD (‘‘forced electron beam induced arc discharge”) ion sources are used for the production of radioactive ion beams for a wide range of chemical elements. Their small volume and high operating temperature provide good confinement times and ionization efficiencies. The extracted ion current from a FEBIAD ion source depends on the parameters of the plasma created inside (density, temperature, potential), parameters which are themselves dependent on the input gas pressure and composition. Within the framework of the HIGHINT Marie Curie and the EURISOL DS programs, investigations are ongoing for high power direct targets, which can accommodate up to 100 kW incoming proton beam power. For such systems, the quantity of impurities entering the ion source will increase, thus leading to a change of the plasma characteristics. The gas flow coming from the target will exceed the buffer gas flow, and the ionization of the trace elements will be controlled by the gas composition released from the target. An insight on the complex phenomena taking place in the ion source can be achieved using a Particle-In-Cell code (VORPAL, ‘‘versatile, object-oriented, relativistic, plasma analysis code with lasers”), which can simulate the dynamics of neutral and charged particles inside the plasma: ionization, recombination and charge exchange phenomena, secondary emissions and sputtering. Ó 2008 Elsevier B.V. All rights reserved. PACS: 29.25.Ni; 29.38.Gj; 37.30.+i; 52.25.Jm; 52.25.Vy; 52.25.Ya; 52.40.Kh; 52.65.Rr Keywords: Radioactive ion beam production; 100 kW ISOL target-ion source; Particle-in-cell simulation; FEBIAD; Space-charge
1. Introduction The FEBIAD [1,2] ion sources are used in ISOL for the 1+ ionization of the refractory elements, being able to produce ion beams of elements having an adsorption enthalpy on the materials of the target-ion source unit of up to 6 eV, regardless of their ionization potentials. They are the result of the development of the arc discharge ion sources towards smaller volume (down to a few cm3), lower operating pressures (104 to 105 mbar, compared to 103 mbar for Nielsen source [3]) and consequently, lower extracted currents (from 1 to 100 mA/cm2 down to 0.1 mA/cm2). This was needed in order to reduce the operating pressures and extracted currents to values *
Corresponding author. E-mail address: [email protected] (L. Penescu).
0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.05.071
closer to those given by the radioactive products, and thus eliminating the need of handling unnecessary high intensity stable beams. The pressure inside the source is the result of the buffer gas load (injected through a calibrated leak), of the radioactive products released from the target and of the residual gas given by the target evaporation and impurities. Presently at ISOLDE, the buffer gas load is on the order of 1013 atoms/s, higher than the radioactive load (for a 2.6 kW proton beam, of less than 1011 atoms/s). The increase of the beam power to 100 kW would increase the radioactive gas load up to 1014 atoms/s [4], which will require a comparable buffer gas load. The typical target vapor pressure is on the order of 106 mbar (which for SiC corresponds to an operating temperature of 1650 °C). Considering the current ISOLDE ratio between the target and ion source volumes
4416
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
(62.8 cm3/2.6 cm3) the resulting pressure in the ion source given by the target evaporation can reach 104 mbar. The increase of the target volume (for the 100 kW EURISOL SiC target, a factor up to 5 was estimated [4]) will lead to a higher pressure, of 103 mbar or even more (as a too efficient impurity pumping would also reduce the time that the atoms of interest will stay in the ionizing electron beam and therefore reducing the ionization rate). In these conditions, maintaining the same ionization efficiencies, the extracted current will have to be of 1–10 mA/ cm2, outside the operating domain of the FEBIADs [1]. This is again the domain of the original Nielsen source, characterized by lower ionization efficiencies, bigger confinement times and instabilities over a pressure variation. A solution to these issues can be searched by using a commercially available plasma simulation code (VORPAL, [5]), for analyzing the effect of geometry, potential distribution, gas pressure and composition and by optimizing the particle flows inside the source. The FEBIAD source is chosen as a starting point, for its highest operation stability amongst the presently available arc discharge ion sources and for its rather constant ionization efficiency over the operating pressure domain (this should minimize the code instabilities). This paper presents the most important physical phenomena inside the source, the way they have been implemented in the code and the code validation of this approach with the first results for plasma dynamics. 2. Physical phenomena in an arc discharge ion source Neutral feed of the ion source: The measured variation of the radioactive ion-beam intensity as a function of time after the proton impact was fitted [6] using the empirical expression below: RðtÞ ¼ C ð1 expðt=sr ÞÞ ða expðt=sf Þ þ ð1 aÞ expðt=ss ÞÞ
ð1Þ
This release function is a result of all the physical processes involved in the transport of radioactive atoms from their production place inside the target to their detection on the tape station: diffusion from the target material, desorption from the material surface, effusion to the ion source, eventual chemical reactions all along the high temperature target-ion source unit, radioactive decay, ionization, extraction from the ion source and beam transport. Therefore, the constants C, a, sr, sf, ss are dependent on the analyzed element, the target, the ion source parameters and on the associated temperatures. In our first approach, we consider that the effect of ionization and ion extraction from the ion source does not influence the shape of the release function and therefore we assume that the radioactive load in the ion source follows the relation (1). The other two components of the ion source neutral load (presented in Table 1) are not time-dependent.
Table 1 Composition of the gas input of a ISOL FEBIAD Gas input
Present day (ISOLDE)
Future (100 kW target)
Buffer gas Radioactive products Target evaporation; impurities
1013 at/s <1011 at/s 104 mbar
>1014 at/s 1014 at/s 103 mbar
Electron emission: The characteristic construction element of the FEBIAD ion sources is the accelerating grid placed in front of the cathode, which eliminates the space-charge limitation of the cathode electronic emission. Therefore, the emitted electron current is not affected by the gas pressure or the ion production rate, and can be obtained using the Richardson–Dushman equation: jcath ¼ A T 2 expðW =kT Þ ½mA=mm2
ð2Þ 2
2
where A is Dushman’s constant (120 mA/mm K ) and W [eV] is the work function of the cathode material. On the contrary, because of the direct contact with the plasma, the electron emission from the anode walls is space-charge limited; therefore, it is estimated by the Child–Langmuir relation: jth ¼ ð4=9Þ e0 ð2e=mÞ ð1=2Þ ðV 3=2 =d 2 Þ ½mA=mm2
ð3Þ
where V is the potential difference between the plasma and the anode walls and d is the plasma sheath dimension (on the order of a few Debye lengths). Electron impact ionization: The electron beam produced by the accelerating grid will oscillate inside the anode body, confined by the applied potentials and magnetic field. The impact ionization cross section is estimated by using the Lotz semi-empirical formula [7]: rq!qþ1 4:5 1014 X ½lnðEe =Eqþ1;nl Þ=ðEe Eqþ1;nl Þðcm2 Þ nl
ð4Þ
where Ee [eV] is the electron energy, nl are the quantum numbers defining the electrons to be stripped and Eq+1, nl [eV] is their binding energy. The generated ion current density inside the ion source, nioniz, is then calculated by taking into account the accelerated electron flux and the gas parameters inside the ion source. The ionization probability for the maximum beam ionization (‘‘beam ionization” limit) is given by pe ¼ 1 expðne rq!qþ1 s0 Þ
ð5Þ 2
where ne is the electron beam density [el/cm s] and s0 is the time that the neutral atom remains in the beam. The charge exchange cross-section between different particle species is estimated by using the Muller and Salzborn formula [8]: 2:76 ðcm2 Þ rq!q1 ¼ 1:43 1012 Z 1:17 q =Ei
ð6Þ
where Zq is the ion charge state and Ei [eV] is the ionization potential of the neutral atom.
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
The resulting ion current density, nch-exc, can be hence deduced for specific particle densities. This phenomenon favors the charge transfer towards elements with lower ionization potential so, qualitatively, the degree of ionization for a given element will be affected by the ratio between the particle density of the elements with a higher ionization potential to the one of the elements having a lower ionization potential. Generally, it is the buffer gas that establishes the value for the ionization potential above which the ionization efficiency will drop, but when the radioactive gas loading becomes important, that value can be shifted. The importance of this effect will depend on the residence time of the species: heavier elements will spend more time in the source and consequently the probability for a charge exchange collision will be higher. The ion recombination is occurring by two mechanisms: radiative recombination, in the volume of the ion source and surface recombination, on the walls of the ion source. The volume recombination is estimated by the Bethe and Salpeter formula [9]: rn ¼ 2:10 1022 ½q4 E20 =nEe ðq2 E0 þ n2 Ee Þðcm2 Þ
ð7Þ
where E0 [eV] is the Rydberg energy, Ee [eV] is the free electron energy, q is the ion charge state and n is the main quantum number of the ion vacancy. The drain current density from this kind of interaction, nrecomb, can hence be deduced. Nevertheless, this term is generally negligible. The surface recombination is strongly dependent on the electron temperature, Te, which is setting the plasma sheath properties. The ion current density through the plasma boundary can be expressed by the relation [10]: jþ ¼ nþ ðkT e =2ðeMÞ
1=2
ðions=cm2 sÞ
ð8Þ
Surface ionization: Depending on the element, on the nature of the surface material and its temperature, a probability b of ionization per impact can be defined (Saha– Langmuir): b ¼ a=ð1 þ aÞ; with a ¼ ðri =ro Þðexp½ðW Ei Þ=kT
ð9Þ
where W [eV] is the work function of the ionizer material, T [K] its temperature and Ei [eV] the ionization energy of the atoms. This probability applies to neutral atoms hitting the wall, but also to the ions recombining on the wall (if their adsorption enthalpy is low enough to let them leave the surface). 3. Code description The PIC method consists in dividing the volume of the ion source in distinct cells, having a sufficiently small dimension in order to resolve the smallest scale phenomena. For plasma simulation, this dimension is set to the Debye length scale.
4417
The particles, charged and neutral, can then be generated or removed in these cells, according to the described theoretical input. Their motion will be determined by their momentum and by the existing field (sum of the defined external electrostatic fields and of the instantaneous space charge field). After each time step, the fields are updated, to include the new space charge distribution. The time step is chosen so that any particle (or wave) won’t traverse more than one cell at a time. VORPAL can generate ions by electron impact ionization, but its cross section library only includes a few of gases of interest; also, the impact ionization module can only treat the ionization of a buffer gas, one at a time, treated as a fluid with a constant pressure. To study the ion source behavior in the presence of a complex gas composition, with the pressure depending on the ionization rate, which is of interest in the present case, the particles will have to be generated and removed explicitly, according to the reaction rates defined in Section 2. All phenomena generating particles can be implemented using particle sources with specified densities, energies and positions. The particles can be removed from the simulation when hitting defined boundaries or uniformly from the volume (with a given probability). For a better follow-up, the same type of physical particles can be treated as a sum of several different simulation particles, according to the phenomena through which they were created. For the FEBIAD source, the typical plasma parameters are: ne ni = 1010–1011 part/cm3 (while for a Nielsen source, as a low pressure arc discharge source, n has values of 1011–1013 part/cm3), kD 0.5 mm, Te 10eV. The fastest moving particles are the grid accelerated electrons. Considering a typical accelerating voltage of 150 V, the required time step will be of about 1010 s so a simulation over 1 ms will require 107 steps which, depending on the particle densities, will demand several hundreds hours of simulation on a single processor. Such a simulation is only required to estimate the timescales of different effects (residence times, pulsed operation, change of ionizing efficiency produced by the variation of the operating pressure, laser pulse effect). When one wishes to simulate only the extracted current or the emittance in the steady state regime, the calculated equilibrium state can be reached faster, by setting up the initial conditions of the simulation close to the equilibrium state, as defined in Section 2. 4. Implementation The equilibrium of all the above mentioned phenomena will lead to the steady state operation and establish a specific charge density inside the ion source. Each of these phenomena can be used to improve the ion source performance (efficiency, selectivity, response time). This is achieved through the adjustment of e.g. ion source geometry, temperature, applied external fields – electric and magnetic, operating gas composition and pressure.
4418
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
For example, the ratio between the ions produced through surface ionization and those produced through electron impact is first affected by the ion source temperature and by the cavity materials; furthermore the ratio of the corresponding extracted ion currents is affected by the applied electric and magnetic fields, through their effect on the plasma confinement in the ion source, as the two mentioned ion populations are generated on different sides of the wall-plasma sheath and have different energy distributions. The complexity of the problem is increased by the fact that every ion source parameter is influencing different phenomena at once. The increase of the gas pressure, for instance, will lead to a higher rate of ionization through electron impact and also to a higher rate of charge exchange from the 1+ ions to the background gas (depending on the gas composition). The resulting higher plasma density in the ion source volume will affect the electron emission from the walls, the rate of ion recombination on the walls and also the fraction of the surface ionized ions that can successfully pass through the plasma sheath to the ion source volume. To perform such a study, we use the parameterization described below. P The P charge conservation in the system will give ie = 0 and ii = 0 (for the electrons and for all the ion species). This can be developed according to Fig. 1, considering the different charged particle flows, respectively on the grid (jg), on the cylindrical anode walls (jw), on the outlet plate (jo), in the volume (nv) and by the extraction hole (je). Using the current densities defined in Section 2, we have: jge S g þ jwe S w þ joe S o þ nve V ¼ 0 jgi S g þ jwi S w þ joi S o þ jei S e þ nvi V ¼ 0 where jge = jcathjgeout; jgi = jsurfjgiout; jwe = jth jweout; jwi = jsurf–jwiout; joe = jth joeout; joi = jsurf joiout; nve = nioniz nrecomb; nvi = nioniz + nch-exc+ nch-exc nrecomb. Sn are the corresponding surfaces and V is the ion source volume.
Fig. 1. Current densities in a FEBIAD ion source.
For each species, the charge exchange will be both a particle source (given by the species with a higher ionization potential) and a particle sink (towards species with a lower ionization potential). The terms jcath, jsurf, jth, nioniz, nrecomb, nch-exc, nrecomb can be computed according to the relations introduced in Section 2 and used as an input for the computer code. The terms jgeout, jgiout, jweout, jwiout, joeout, joiout and jei are the corresponding plasma ‘‘sinks” and will be obtained through the code. More than that, jgiout, jwiout and joiout are to be compared with the predicted theoretical values for the recombination on the walls, j+. The experimentally available parameters are: – The extracted ion current (iei). – The anode drain current (iwe + iwi + ige + igi). – The outlet plate drain current (ioe + ioi).
5. First results The two main aspects to consider in the design of a modified ion source are the ion production and the ion extraction. We tested the evolution of the ion drain currents with respect to the ionization rate in the source, with an emphasis on the extracted current. A detailed study on the dependence of the ionization rate on the source parameters is now ongoing. At ISOLDE, FEBIAD sources are generally operated at 2000 °C, which corresponds (Eq. (2)) to an electron beam density ne = 2 1018 el/cm2s. As the ionization cross section is of the order of 1016 cm2 (the Lotz relation – Eq. (4) – gives 4.7 1016 cm2 for 1+ ionization of Ar), the ionization probability reaches 100% for a neutral residence time bigger than 0.1ms (Eq. (5)). Considering the geometrical dimensions of the FEBIAD MK7 source, the average flight time between 2 wall collisions for a neutral atom is of about 20 ls. With an average collision number of k = 1/Pescape = 1/(Sextraction/Stotal) of 550 and considering the input gas to be constituted only of 40Ar, which has a negligible sticking time on the ion source walls at 2000 °C, the average neutral residence time will be 11 ms, two orders of magnitude higher than the residence time required for 100% ionization probability. Therefore, the ionization rate is presently controlled by the neutral gas pressure in the source and, for a dominant gas with a negligible sticking time compared to the flight time (as is the case for the noble gases), at the standard FEBIAD operating pressure of about 104 mbar, will be of 5 1014 ions/cm3s. We simulated the particle dynamics inside the source for this ionization rate and we found a good agreement with the experimental values for the extracted current (typically of 1–6 lA). The extraction electrode was considered to be at 30 kV and at 60 mm from the outlet plate. By varying the ionization rates between 1012 and 17 10 ions/cm3 s, we were able to reproduce the space charge
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
4419
of the present ionization rate and an improvement of the extraction system (bigger surface of the extraction orifice and/or space-charge compensation of the extraction system). These results provide validation of the code for further investigation of the particle dynamics inside the plasma ion sources. Simulation-driven optimization of the arc discharge will be done to design a prototype dealing efficiently with higher operating pressures. References
Fig. 2. VORPAL simulation of FEBIAD’s space charge limited ion extraction. j: total generated ion current inside the ion source cavity. d: total extracted current.
limitation of the extracted current (Fig. 2), the Child–Langmuir predicted value ((3)) being of 25 lA for a circular aperture of 1.6 mm diameter. An efficient adaptation of the arc discharge ion sources for higher operating pressures will require both the increase
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R. Kirchner, E. Roeckl, Nucl. Instr. and Meth. 133 (1976) 187. S. Sundell, H. Ravn, Nucl. Instr. and Meth. B 70 (1992) 160. O. Almen, K.O. Nielsen, Nucl. Instr. 1 (1957) 302. T. Stora et al., EURISOL DS, Feasibility study for the 100 kW direct targets. C. Nieter, J.R. Cary, J. Comput. Phys. 196 (2004) 448. J. Lettry et al., Nucl. Instr. and Meth. B 126 (1997) 130. W. Lotz, Z. Phys. 216 (1968) 241. A. Muller, E. Salzborn, Phys. Lett. A 62 (1977) 391. H. Bethe, E. Salpeter, Atome I, in: S. Fliigge (Ed.), Handbuch der Physik, Vol. 35, Springer-Verlag, Berlin, 1957. D. Kamke, H.J. Rose, Z. Phys. 145 (1956) 83.
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 4415–4419 www.elsevier.com/locate/nimb
Numerical simulations of space charge effects and plasma dynamics for FEBIAD ion sources L. Penescu *, R. Catherall, J. Lettry, T. Stora CERN, CH-1211, Geneva 23, Switzerland Available online 6 June 2008
Abstract The FEBIAD (‘‘forced electron beam induced arc discharge”) ion sources are used for the production of radioactive ion beams for a wide range of chemical elements. Their small volume and high operating temperature provide good confinement times and ionization efficiencies. The extracted ion current from a FEBIAD ion source depends on the parameters of the plasma created inside (density, temperature, potential), parameters which are themselves dependent on the input gas pressure and composition. Within the framework of the HIGHINT Marie Curie and the EURISOL DS programs, investigations are ongoing for high power direct targets, which can accommodate up to 100 kW incoming proton beam power. For such systems, the quantity of impurities entering the ion source will increase, thus leading to a change of the plasma characteristics. The gas flow coming from the target will exceed the buffer gas flow, and the ionization of the trace elements will be controlled by the gas composition released from the target. An insight on the complex phenomena taking place in the ion source can be achieved using a Particle-In-Cell code (VORPAL, ‘‘versatile, object-oriented, relativistic, plasma analysis code with lasers”), which can simulate the dynamics of neutral and charged particles inside the plasma: ionization, recombination and charge exchange phenomena, secondary emissions and sputtering. Ó 2008 Elsevier B.V. All rights reserved. PACS: 29.25.Ni; 29.38.Gj; 37.30.+i; 52.25.Jm; 52.25.Vy; 52.25.Ya; 52.40.Kh; 52.65.Rr Keywords: Radioactive ion beam production; 100 kW ISOL target-ion source; Particle-in-cell simulation; FEBIAD; Space-charge
1. Introduction The FEBIAD [1,2] ion sources are used in ISOL for the 1+ ionization of the refractory elements, being able to produce ion beams of elements having an adsorption enthalpy on the materials of the target-ion source unit of up to 6 eV, regardless of their ionization potentials. They are the result of the development of the arc discharge ion sources towards smaller volume (down to a few cm3), lower operating pressures (104 to 105 mbar, compared to 103 mbar for Nielsen source [3]) and consequently, lower extracted currents (from 1 to 100 mA/cm2 down to 0.1 mA/cm2). This was needed in order to reduce the operating pressures and extracted currents to values *
Corresponding author. E-mail address: [email protected] (L. Penescu).
0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.05.071
closer to those given by the radioactive products, and thus eliminating the need of handling unnecessary high intensity stable beams. The pressure inside the source is the result of the buffer gas load (injected through a calibrated leak), of the radioactive products released from the target and of the residual gas given by the target evaporation and impurities. Presently at ISOLDE, the buffer gas load is on the order of 1013 atoms/s, higher than the radioactive load (for a 2.6 kW proton beam, of less than 1011 atoms/s). The increase of the beam power to 100 kW would increase the radioactive gas load up to 1014 atoms/s [4], which will require a comparable buffer gas load. The typical target vapor pressure is on the order of 106 mbar (which for SiC corresponds to an operating temperature of 1650 °C). Considering the current ISOLDE ratio between the target and ion source volumes
4416
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
(62.8 cm3/2.6 cm3) the resulting pressure in the ion source given by the target evaporation can reach 104 mbar. The increase of the target volume (for the 100 kW EURISOL SiC target, a factor up to 5 was estimated [4]) will lead to a higher pressure, of 103 mbar or even more (as a too efficient impurity pumping would also reduce the time that the atoms of interest will stay in the ionizing electron beam and therefore reducing the ionization rate). In these conditions, maintaining the same ionization efficiencies, the extracted current will have to be of 1–10 mA/ cm2, outside the operating domain of the FEBIADs [1]. This is again the domain of the original Nielsen source, characterized by lower ionization efficiencies, bigger confinement times and instabilities over a pressure variation. A solution to these issues can be searched by using a commercially available plasma simulation code (VORPAL, [5]), for analyzing the effect of geometry, potential distribution, gas pressure and composition and by optimizing the particle flows inside the source. The FEBIAD source is chosen as a starting point, for its highest operation stability amongst the presently available arc discharge ion sources and for its rather constant ionization efficiency over the operating pressure domain (this should minimize the code instabilities). This paper presents the most important physical phenomena inside the source, the way they have been implemented in the code and the code validation of this approach with the first results for plasma dynamics. 2. Physical phenomena in an arc discharge ion source Neutral feed of the ion source: The measured variation of the radioactive ion-beam intensity as a function of time after the proton impact was fitted [6] using the empirical expression below: RðtÞ ¼ C ð1 expðt=sr ÞÞ ða expðt=sf Þ þ ð1 aÞ expðt=ss ÞÞ
ð1Þ
This release function is a result of all the physical processes involved in the transport of radioactive atoms from their production place inside the target to their detection on the tape station: diffusion from the target material, desorption from the material surface, effusion to the ion source, eventual chemical reactions all along the high temperature target-ion source unit, radioactive decay, ionization, extraction from the ion source and beam transport. Therefore, the constants C, a, sr, sf, ss are dependent on the analyzed element, the target, the ion source parameters and on the associated temperatures. In our first approach, we consider that the effect of ionization and ion extraction from the ion source does not influence the shape of the release function and therefore we assume that the radioactive load in the ion source follows the relation (1). The other two components of the ion source neutral load (presented in Table 1) are not time-dependent.
Table 1 Composition of the gas input of a ISOL FEBIAD Gas input
Present day (ISOLDE)
Future (100 kW target)
Buffer gas Radioactive products Target evaporation; impurities
1013 at/s <1011 at/s 104 mbar
>1014 at/s 1014 at/s 103 mbar
Electron emission: The characteristic construction element of the FEBIAD ion sources is the accelerating grid placed in front of the cathode, which eliminates the space-charge limitation of the cathode electronic emission. Therefore, the emitted electron current is not affected by the gas pressure or the ion production rate, and can be obtained using the Richardson–Dushman equation: jcath ¼ A T 2 expðW =kT Þ ½mA=mm2
ð2Þ 2
2
where A is Dushman’s constant (120 mA/mm K ) and W [eV] is the work function of the cathode material. On the contrary, because of the direct contact with the plasma, the electron emission from the anode walls is space-charge limited; therefore, it is estimated by the Child–Langmuir relation: jth ¼ ð4=9Þ e0 ð2e=mÞ ð1=2Þ ðV 3=2 =d 2 Þ ½mA=mm2
ð3Þ
where V is the potential difference between the plasma and the anode walls and d is the plasma sheath dimension (on the order of a few Debye lengths). Electron impact ionization: The electron beam produced by the accelerating grid will oscillate inside the anode body, confined by the applied potentials and magnetic field. The impact ionization cross section is estimated by using the Lotz semi-empirical formula [7]: rq!qþ1 4:5 1014 X ½lnðEe =Eqþ1;nl Þ=ðEe Eqþ1;nl Þðcm2 Þ nl
ð4Þ
where Ee [eV] is the electron energy, nl are the quantum numbers defining the electrons to be stripped and Eq+1, nl [eV] is their binding energy. The generated ion current density inside the ion source, nioniz, is then calculated by taking into account the accelerated electron flux and the gas parameters inside the ion source. The ionization probability for the maximum beam ionization (‘‘beam ionization” limit) is given by pe ¼ 1 expðne rq!qþ1 s0 Þ
ð5Þ 2
where ne is the electron beam density [el/cm s] and s0 is the time that the neutral atom remains in the beam. The charge exchange cross-section between different particle species is estimated by using the Muller and Salzborn formula [8]: 2:76 ðcm2 Þ rq!q1 ¼ 1:43 1012 Z 1:17 q =Ei
ð6Þ
where Zq is the ion charge state and Ei [eV] is the ionization potential of the neutral atom.
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
The resulting ion current density, nch-exc, can be hence deduced for specific particle densities. This phenomenon favors the charge transfer towards elements with lower ionization potential so, qualitatively, the degree of ionization for a given element will be affected by the ratio between the particle density of the elements with a higher ionization potential to the one of the elements having a lower ionization potential. Generally, it is the buffer gas that establishes the value for the ionization potential above which the ionization efficiency will drop, but when the radioactive gas loading becomes important, that value can be shifted. The importance of this effect will depend on the residence time of the species: heavier elements will spend more time in the source and consequently the probability for a charge exchange collision will be higher. The ion recombination is occurring by two mechanisms: radiative recombination, in the volume of the ion source and surface recombination, on the walls of the ion source. The volume recombination is estimated by the Bethe and Salpeter formula [9]: rn ¼ 2:10 1022 ½q4 E20 =nEe ðq2 E0 þ n2 Ee Þðcm2 Þ
ð7Þ
where E0 [eV] is the Rydberg energy, Ee [eV] is the free electron energy, q is the ion charge state and n is the main quantum number of the ion vacancy. The drain current density from this kind of interaction, nrecomb, can hence be deduced. Nevertheless, this term is generally negligible. The surface recombination is strongly dependent on the electron temperature, Te, which is setting the plasma sheath properties. The ion current density through the plasma boundary can be expressed by the relation [10]: jþ ¼ nþ ðkT e =2ðeMÞ
1=2
ðions=cm2 sÞ
ð8Þ
Surface ionization: Depending on the element, on the nature of the surface material and its temperature, a probability b of ionization per impact can be defined (Saha– Langmuir): b ¼ a=ð1 þ aÞ; with a ¼ ðri =ro Þðexp½ðW Ei Þ=kT
ð9Þ
where W [eV] is the work function of the ionizer material, T [K] its temperature and Ei [eV] the ionization energy of the atoms. This probability applies to neutral atoms hitting the wall, but also to the ions recombining on the wall (if their adsorption enthalpy is low enough to let them leave the surface). 3. Code description The PIC method consists in dividing the volume of the ion source in distinct cells, having a sufficiently small dimension in order to resolve the smallest scale phenomena. For plasma simulation, this dimension is set to the Debye length scale.
4417
The particles, charged and neutral, can then be generated or removed in these cells, according to the described theoretical input. Their motion will be determined by their momentum and by the existing field (sum of the defined external electrostatic fields and of the instantaneous space charge field). After each time step, the fields are updated, to include the new space charge distribution. The time step is chosen so that any particle (or wave) won’t traverse more than one cell at a time. VORPAL can generate ions by electron impact ionization, but its cross section library only includes a few of gases of interest; also, the impact ionization module can only treat the ionization of a buffer gas, one at a time, treated as a fluid with a constant pressure. To study the ion source behavior in the presence of a complex gas composition, with the pressure depending on the ionization rate, which is of interest in the present case, the particles will have to be generated and removed explicitly, according to the reaction rates defined in Section 2. All phenomena generating particles can be implemented using particle sources with specified densities, energies and positions. The particles can be removed from the simulation when hitting defined boundaries or uniformly from the volume (with a given probability). For a better follow-up, the same type of physical particles can be treated as a sum of several different simulation particles, according to the phenomena through which they were created. For the FEBIAD source, the typical plasma parameters are: ne ni = 1010–1011 part/cm3 (while for a Nielsen source, as a low pressure arc discharge source, n has values of 1011–1013 part/cm3), kD 0.5 mm, Te 10eV. The fastest moving particles are the grid accelerated electrons. Considering a typical accelerating voltage of 150 V, the required time step will be of about 1010 s so a simulation over 1 ms will require 107 steps which, depending on the particle densities, will demand several hundreds hours of simulation on a single processor. Such a simulation is only required to estimate the timescales of different effects (residence times, pulsed operation, change of ionizing efficiency produced by the variation of the operating pressure, laser pulse effect). When one wishes to simulate only the extracted current or the emittance in the steady state regime, the calculated equilibrium state can be reached faster, by setting up the initial conditions of the simulation close to the equilibrium state, as defined in Section 2. 4. Implementation The equilibrium of all the above mentioned phenomena will lead to the steady state operation and establish a specific charge density inside the ion source. Each of these phenomena can be used to improve the ion source performance (efficiency, selectivity, response time). This is achieved through the adjustment of e.g. ion source geometry, temperature, applied external fields – electric and magnetic, operating gas composition and pressure.
4418
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
For example, the ratio between the ions produced through surface ionization and those produced through electron impact is first affected by the ion source temperature and by the cavity materials; furthermore the ratio of the corresponding extracted ion currents is affected by the applied electric and magnetic fields, through their effect on the plasma confinement in the ion source, as the two mentioned ion populations are generated on different sides of the wall-plasma sheath and have different energy distributions. The complexity of the problem is increased by the fact that every ion source parameter is influencing different phenomena at once. The increase of the gas pressure, for instance, will lead to a higher rate of ionization through electron impact and also to a higher rate of charge exchange from the 1+ ions to the background gas (depending on the gas composition). The resulting higher plasma density in the ion source volume will affect the electron emission from the walls, the rate of ion recombination on the walls and also the fraction of the surface ionized ions that can successfully pass through the plasma sheath to the ion source volume. To perform such a study, we use the parameterization described below. P The P charge conservation in the system will give ie = 0 and ii = 0 (for the electrons and for all the ion species). This can be developed according to Fig. 1, considering the different charged particle flows, respectively on the grid (jg), on the cylindrical anode walls (jw), on the outlet plate (jo), in the volume (nv) and by the extraction hole (je). Using the current densities defined in Section 2, we have: jge S g þ jwe S w þ joe S o þ nve V ¼ 0 jgi S g þ jwi S w þ joi S o þ jei S e þ nvi V ¼ 0 where jge = jcathjgeout; jgi = jsurfjgiout; jwe = jth jweout; jwi = jsurf–jwiout; joe = jth joeout; joi = jsurf joiout; nve = nioniz nrecomb; nvi = nioniz + nch-exc+ nch-exc nrecomb. Sn are the corresponding surfaces and V is the ion source volume.
Fig. 1. Current densities in a FEBIAD ion source.
For each species, the charge exchange will be both a particle source (given by the species with a higher ionization potential) and a particle sink (towards species with a lower ionization potential). The terms jcath, jsurf, jth, nioniz, nrecomb, nch-exc, nrecomb can be computed according to the relations introduced in Section 2 and used as an input for the computer code. The terms jgeout, jgiout, jweout, jwiout, joeout, joiout and jei are the corresponding plasma ‘‘sinks” and will be obtained through the code. More than that, jgiout, jwiout and joiout are to be compared with the predicted theoretical values for the recombination on the walls, j+. The experimentally available parameters are: – The extracted ion current (iei). – The anode drain current (iwe + iwi + ige + igi). – The outlet plate drain current (ioe + ioi).
5. First results The two main aspects to consider in the design of a modified ion source are the ion production and the ion extraction. We tested the evolution of the ion drain currents with respect to the ionization rate in the source, with an emphasis on the extracted current. A detailed study on the dependence of the ionization rate on the source parameters is now ongoing. At ISOLDE, FEBIAD sources are generally operated at 2000 °C, which corresponds (Eq. (2)) to an electron beam density ne = 2 1018 el/cm2s. As the ionization cross section is of the order of 1016 cm2 (the Lotz relation – Eq. (4) – gives 4.7 1016 cm2 for 1+ ionization of Ar), the ionization probability reaches 100% for a neutral residence time bigger than 0.1ms (Eq. (5)). Considering the geometrical dimensions of the FEBIAD MK7 source, the average flight time between 2 wall collisions for a neutral atom is of about 20 ls. With an average collision number of k = 1/Pescape = 1/(Sextraction/Stotal) of 550 and considering the input gas to be constituted only of 40Ar, which has a negligible sticking time on the ion source walls at 2000 °C, the average neutral residence time will be 11 ms, two orders of magnitude higher than the residence time required for 100% ionization probability. Therefore, the ionization rate is presently controlled by the neutral gas pressure in the source and, for a dominant gas with a negligible sticking time compared to the flight time (as is the case for the noble gases), at the standard FEBIAD operating pressure of about 104 mbar, will be of 5 1014 ions/cm3s. We simulated the particle dynamics inside the source for this ionization rate and we found a good agreement with the experimental values for the extracted current (typically of 1–6 lA). The extraction electrode was considered to be at 30 kV and at 60 mm from the outlet plate. By varying the ionization rates between 1012 and 17 10 ions/cm3 s, we were able to reproduce the space charge
L. Penescu et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 4415–4419
4419
of the present ionization rate and an improvement of the extraction system (bigger surface of the extraction orifice and/or space-charge compensation of the extraction system). These results provide validation of the code for further investigation of the particle dynamics inside the plasma ion sources. Simulation-driven optimization of the arc discharge will be done to design a prototype dealing efficiently with higher operating pressures. References
Fig. 2. VORPAL simulation of FEBIAD’s space charge limited ion extraction. j: total generated ion current inside the ion source cavity. d: total extracted current.
limitation of the extracted current (Fig. 2), the Child–Langmuir predicted value ((3)) being of 25 lA for a circular aperture of 1.6 mm diameter. An efficient adaptation of the arc discharge ion sources for higher operating pressures will require both the increase
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R. Kirchner, E. Roeckl, Nucl. Instr. and Meth. 133 (1976) 187. S. Sundell, H. Ravn, Nucl. Instr. and Meth. B 70 (1992) 160. O. Almen, K.O. Nielsen, Nucl. Instr. 1 (1957) 302. T. Stora et al., EURISOL DS, Feasibility study for the 100 kW direct targets. C. Nieter, J.R. Cary, J. Comput. Phys. 196 (2004) 448. J. Lettry et al., Nucl. Instr. and Meth. B 126 (1997) 130. W. Lotz, Z. Phys. 216 (1968) 241. A. Muller, E. Salzborn, Phys. Lett. A 62 (1977) 391. H. Bethe, E. Salpeter, Atome I, in: S. Fliigge (Ed.), Handbuch der Physik, Vol. 35, Springer-Verlag, Berlin, 1957. D. Kamke, H.J. Rose, Z. Phys. 145 (1956) 83.