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Electron cooling at CRYRING with an expanded electron beam
Nuclear Instruments and Methods in Physics Research A 391 (1997) 24-31 -_ F!!z ELSEVIER
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH
Electron cooling at CRYRING with an expanded electron beam Hikan Danared* Manne Siegbahn Laboratory, Frescativ2igen 24, S-104 05 Stockholm, Sweden
Abstract Studies of electron cooling with an adiabatically expanded electron beam have been performed at CRYRING. Measurements of the longitudinal drag force using three different methods are described and are compared with measurements at other electron coolers. The processes that determine the electron temperature, longitudinally and transversally, are reviewed, and the effect of the beam expansion on these processes is discussed. It is described how the electron temperature can influence the outcome of recombination measurements, and how such measurements can be used to determine the electron temperature, and an example of such a measurement is given. PACS: 07.77. + p; 29.20.Dh; 35.80. + s; 41.80.Dd Storage rings; Electron cooling; Electron target: Recombination
Keywords:
1. Introduction CRYRING is a small synchrotron and storage ring used mainly for research in atomic and molecular physics [l]. An essential part of the ring is the electron cooler [2], which is used both for beam cooling and as an electron target. With the cooler operating as a target, many types of electron-ion recombination processes have been studied. These include radiative recombination - spontaneous or laser-stimulated - and dielectronic recombination of atomic ions as well as dissociative recombination of molecular ions. Also other processes such as ionisation, excitation, de-excitation, etc. can be investigated (see, e.g., Ref. [3] for a recent review). There are several advantages in using storage rings for experiments in atomic and molecular physics. Some are consequences of the mere acceleration and storing of the ions, such as the increase in luminosity achieved by recirculating the ions that do not react in their first passage through the target, the (relatively) high energy of accelerated ions that reduces reactions with the rest gas and gives almost background-free measurements in many cases, and the possibility to let internally hot ions spontaneously de-excite before measurements are performed. Applying electron cooling then reduces the geometrical dimensions of the ion beam and also reduces its momentum spread, allowing for an increased precision in the recombination measurements. At the end, when the actual measurements start, one thus has an accurately described beam of ions in *Corresponding author. Tel.: +46 8 161038, fax: +46 8 158674, e-mail [email protected].
experiments
a well-defined internal state, and also the properties of the electron beam (current density, energy, etc.) are known to a high precision. A further property of the electron beam which is of great importance is its energy spread or temperature. It influences both the cooling process and the results that can be obtained in recombination studies. Although the cooling may become more efficient with colder electrons, it is particularly when the cooler acts as an electron target that the benefits of a low electron temperature become obvious [4]. When recombination cross sections are recorded as a function of the relative energy between ions and electrons, the energy resolution is usually determined entirely by the electron energy spread. For other types of measurements performed close to zero relative energy (e.g., laser-stimulated radiative recombination), the experimental count rate increases significantly with lower electron temperature.
2. Cooling force measurements 2.1.
Methods
In order to improve, the properties of the CRYRING cooler, we introduced an adiabatic expansion of its electron beam a couple of years ago [5]. The expansion is achieved by using a higher magnetic field at the electron gun, typically 0.3 T, than in the rest of the cooler, where normally 0.03 T is used. This gives an expansion factor of 10, reducing the transverse electron-energy spread with a factor equal to the ratio between the fields. In the case of CRYRING, this means from 100meV (deriving from the
0168-9002/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PII SO168-9002(97)00249-O
H. Danared
I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
cathode temperature of SOO-900°C) to 10 meV The longitudinal electron-energy spread is only affected to a small extent by the expansion (see Section 4) and is of the order of 0.1 meV Clearly, it is of interest to see how this reduction of the transverse electron temperature by a factor of 10 influences the cooling. This influence is most easily investigated by studying the “cooling force”, or the drag force that an ion experiences as it moves through the electron gas of the cooler’s electron beam. In general, the cooling force has a longitudinal component F,, (i.e., parallel to the magnetic field and the direction of propagation of the electron beam) that is different from the transverse component F,. These force components depend on both the longitudinal and the transverse velocity of the ion, making it very difficult - in practise impossible - to map out the cooling force completely. The longitudinal component is, however, quite sensitive to the details of the electron-ion interaction, enabling a comparison with theoretical models. One straightforward technique for measuring the cooling force is the voltage-step method. Here, one first cools the ions, matching the ion and the electron velocities. Then, the electron energy is changed rapidly, creating a welldefined velocity difference between ions and electrons. The ions then accelerate toward the new electron velocity, and if the acceleration is measured, only the ion mass is needed to obtain the force. The acceleration has to be measured before the ion velocity has changed significantly, otherwise the relative velocity is no longer well defined. The acceleration is determined, via the Schottky signal, from the change in revolution frequency per unit time, giving [6] p dfldt f
5 = 5
’
(1)
where p is the ion momentum and 77is (df/dp)l(flp). This method has the advantage that the interpretation of the results is very simple, but it is limited to relative velocities above approximately 10 000 m/s. At lower velocities, the required change in electron energy becomes too small, and the ion veIocity changes so rapidIy to the new electron velocity that the measurements become difficult. Since several theoretical predictions about the cooling force can be verified only at 10 000 m/s or lower, and this velocity regime also has the greatest practical importance for cooling, other methods for cooling-force measurements have been developed. A common denominator for these methods is that they are based upon an equilibrium between the cooling and a process that tends to change the ion velocity. One such equilibrium situation was first investigated at LEAR [7]. Here, the ion beam is heated with rf noise at the same time as it is cooled, and the cooling force is deduced from the power density of the Schottky signal. The noise is applied in a single Schottky band and the resulting frequency
25
spectrum is taken from a different band. Introducing the ion-velocity distribution p(u) and the diffusion constant D, which can be obtained from the heating power density, one obtains the longitudinal cooling force by solving the onedimensional Fokker-Planck equation. In the steady state (dplat = O), one finds
Since this expression for F,,(u) contains p(u), this way of evaluating the cooling force is suitable for relative velocities similar to those that exist in the ion beam, typically lOOOOm/s and less. The sensitivity can be improved somewhat using BTF rather than Schottky measurements. A third method [S] uses a bunched ion beam and the equilibrium between the cooling force and &he accelerating force from the rf cavity (or driven drift tube in the case of CRYRING [9]). With very small rf voltages one can reach a situation where the energy that an ion gains during one passage through the rf system is equal to the loss during the passage through the cooler (or vice versa). The resulting shift A+s of the synchronous frequency can be detected with an oscilloscope that measures the signal induced by the bunches in an electrostatic pickup. Alternatively, in order to increase the sensitivity, a network analyzer can be used to detect the phase difference between the bunch and the rf signal. The cooling force is simply obtained from F,, = Zecr,
sin AC/+,
where Ze is the charge of the ion and fir,, is the rf amplitude. In these measurements, the ion energy is determined by the rf frequency, while the velocity difference between ions and electrons, in our case, is introduced by shifting the electron energy from the cooling condition while the rf frequency is kept constant. The equilibrium between cooling force and rf force results in a constant phase shift only for those relative velocities near zero where the magnitude of the cooling force increases with velocity. At higher relative velocities, the phase shift starts to oscillate as described in Ref. [8]. This transition, from a steady state where the longitudinal motion of the bunch is a fixed point in phase space to a situation where it describes a circle in phase space, can be used to locate the relative velocity where the cooling force has its maximum. 2.2. Results
from CRYRIIVG
At CRYRING, cooling-force measurements with the expanded electron beam have been performed using all three methods described above. Fig. 1 shows some results of these measurements. Here, deuterons at 12 MeV were used, the electron current was 25 mA, and the magnetic field was 0.3 T at the gun and 0.03 T in the rest of the
I. ELECTRON COOLING PHYSICS
H. Danared I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
26
X exp
_-_-WL 2kT,, T,,
characterized by the temperatures Coulomb logarithm we have used 3
10-Z
.b
L, =+ln[
104’ lo*
-
theory, kTi=10
.
D’, 10 times expanded electron beam
1’1”‘1’
lo3
1 +(k)‘].,
meif, kTpO.05 meV
i ““‘LJ
“““‘1
105 104 Relative velocity (m/s)
”
II”“”
106
Fig. 1. Longitudinal drag force as a function of the relative velocity between ions and electrons. Points are measured values using voltage stepping or the phase-shift method, the dotted curve is measured using stochastic heating. The full-drawn curve is calculated from Eq. (4). The force is normalised to an electron density n, = lOi mm3.
cooler, giving an expansion factor of 10. The data points above 10 000 m/s were taken with voltage-stepping, those at lower velocities with the phase-shift method. Phase oscillations set in at the position of the arrow. The dotted curve is the force as measured using the equilibrium between cooling and stochastic heating. At the top of the curve, all three methods agree well, and the onset of phase oscillations is right at the maximum of the force, as it should be. This is quite satisfactory considering that, although the three sets of measurements were made during the same beam time, they are completely independent. The wiggles in the dotted curve are due to the low statistics at the tail of the BTF spectrum used for the evaluation of the force. The discrepancy of almost a factor of two at the low relative velocities is unexplained but perhaps not too surprising considering, e.g., the difficulty to make an absolute calibration of the noise power
and T,,,. For the
is the Debye length or the adiabatic cut-off where b,,, distance vi/tip (wp being the plasma frequency), whichever is bigger, and b,in =
ze2 4rrE,in,VZ
is the impact parameter giving the maximum classical momentum transfer. We approximate vZ = max(v’, 2kT,, / m,). The curve was obtained using a transverse energy spread kT,. = 10 meV and a longitudinal energy spread kT,,, = 0.05 meV The agreement between this simple theory and the data is quite good. One could note, however, that if LC is moved inside the integral in Eq. (4), and Iui - u,I is used instead of 2kT,, lm, when calculating the minimum impact parameter, the theoretical curve becomes somewhat lower for low relative velocities. 2.3. Results from other coolers Measurements of the cooling force have been made at all ion-storage rings that use or have used electron cooling and also at the single-pass setup MOSOL in Novosibirgk. Fig. 2 shows a selection of the “best” values reported for
0 0 0 a .
seen by the beam.
curve in Fig. 1 shows theoretical values for the cooling force, given by the usual expression
NAP-M ICE FNAL MOSOL LEAR
The full-drawn
a .
F(vi)
=
-4T(g-)2+ jf(u.)
. .__ ....% ....?c... 0
10-i
u. -u
xjltid3ve
(4)
based on binary collisions between ions and electrons. Here, ui and u, are the ion and electron velocities in the system moving with the electrons, n, is the electron density, I,, is the Coulomb logarithm, and f(u,) is the anisotropic Maxwellian electron-velocity distribution
c)
3
;
0
o “,,g p;
0
lo-" 103
I
104
?\I
IIIIN,
IO5
IO6
Relative velocity (m/s) Fig. 2. Results of drag-force measurements (points) normalised to singly charged ions and an electron density of lOi me3 [10,6,1 l191. Curves represent theoretical calculations; cf. text.
H. Danared I Nucl. Ins@. and Meth. in Phys. Res. A 391 (1997) 24-31
singly charged light ions (protons or deuterons in most cases). For ESR, data for Ne”+ divided by 100 were used, causing somewhat low values compared to the others, since the drag force does not quite scale with 4’ [20]. The curves are calculated according to Eq. (4) with kT,, = 10 meV and kT,,, = 0.1 meV for the upper curve and kT,, = 100 meV and kT,,, = 0.1 meV for the lower one. The scatter of the data points reflects a wide range of measurement conditions (beam energies, electron densities, electron temperatures, magnetic fields, etc.), and a detailed comparison between the results from the different installations is not meaningful. In particular, the cooling force at low relative velocities depends on the longitudinal electron temperature, which in turn depends on the electron density, as discussed in Section 3. However, one can see that some of the early cooling experiments show somewhat low values due to high electron temperatures. Also, the two measurements that were performed with expanded electron beams, at CRYRING and TSR, show the highest values. Apart from this, the experimental points are reasonably close to the theoretical curves at high relative velocities, but above this theory at low velocities. This shows the influence of the magnetic field on the electron-ion interaction and the cooling process. The CRYRING data shown in Fig. 2 are the same as in Fig. 1 below 10 000 m/s, but the points above 10 000 m/s were taken using a 20 times expanded beam, which gave a slightly higher cooling force.
3. Electron temperatures and relaxation rates As already mentioned, the electron temperature is an important parameter for an electron cooler, particularly when it is used as an electron target. The fact that the electron beam expansion reduces the transverse electron temperature by a factor equal to the expansion ratio has been used at all the four low-energy ion storage rings which are used mainly for atomic and molecular physics (CRYRING, TSR, TARN-II, and ASTRID). In order to avoid any misunderstanding, it should be pointed out that the beam expansion does not affect the azimuthal drift motion of the electrons that is caused by the space charge of the beam. Such pure space-charge effects, which may be important at high electron densities and large ion-beam radii, will not be discussed in this paper. The effect of beam expansion on the longitudinal electron temperature is perhaps less obvious, in particular the influence on the relaxation processes that occur in the electron beam. Usually, and at least in CRYRING, the longitudinal temperature is determined by relaxation rather than by the initial energy spread that may stem from the cathode temperature, ripple on the high-voltage power supply, etc. The initial energy spread becomes reduced to at most a few FeV when it is transformed to the reference system moving with the electrons.
27
There are two relaxation processes to consider [21]. The longitudinal-longitudinal relaxation is caused by the fact that, after a rapid acceleration, the longitudinal kinetic energy spread of a few PeV is much smaller than the average potential energy between the electrons. The latter is of the order e2/(4.rr~,)n:‘3 or typically 0.1 meV. After a relaxation time on the order of a plasma period, an equilibrium is reached, and also the kinetic energy spread becomes of the order e2/(4~e,,)n:‘3 longitudinally. The transverse-longitudinal relaxation is the transfer of energy from the transverse motion, where the energy spread is 100 meV or higher (without beam expansion), to the longitudinal degree of freedom. In the absence of a magnetic field, the rate at which the longitudinal temperature increases is [22]
dTeIITe, dt
-
(8)
T.
The relaxation
time constant is
r=------where r,, is the classical electron radius. This transfer of energy is, however, suppressed in the presence of a magnetic field when the cyclotron radius r, of the electrons, taken as
becomes comparable with or smaller than the average distance n,‘13 between the electrons. An empirical expression for the suppression was obtained at Novosibirsk [23], which we simplify to
(11) in order to get a closed expression for the temperature. This expression was verified for beam parameters normally used at CRYRING through molecular-dynamics calculations. In these calculations, N electrons were positioned randomly in a cylindrical box with periodic boundary conditions longitudinally. The initial velocity of the particles was zero longitudinally and Maxwell-dishibuted with kT,, = 50 meV transversally. Also, an azimuthal drift velocity corresponding to the effect of the space charge was added. The classical equations of motion of the particles were then integrated numerically, taking into account the electrostatic interaction between the particles and the external magnetic field. The evolution of the system was followed for a period of time corresponding to the flight time through the electron cooler. Throughout the calculations an electron density of 1 X lOI rnp3 was used and N varied between 500 and 2000. Fig. 3 shows the result of such a calculation where the magnetic field was 0.03 T. The longitudinal energy spread,
I. ELECTRON COOLING PHYSICS
28
H. Danared I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
0.0
0.5
1.0
1.5
2.0
2.5
t I Tp Fig. 3. Result of molecular-dynamics calculations showing the evolution of the longitudinal electron temperature in normalised units due to relaxation processes in the beam. This calculation used n, = 1 X lOI mm3 and B = 0.03 T.
equal to m,(u$ is normalised to e2/(4Te,)n:‘3, and the time is in units of the plasma period rP. It is seen that the longitudinal-longitudinal relaxation takes about 0.3~~ and leads to a longitudinal energy spread quite close to eZ/ (4%)% “3. The value of kT,,, depends to some extent, however, on the magnetic field. A stronger field gives a higher IV,,, and vice versa, probably because a low field allows some of the initial potential energy to be converted to transverse kinetic energy. On the other hand, the initial relaxation time of 0.3~~ seems independent of the magnetic field. The similarity between our Fig 3. and Fig. 2 of Ref. [24] is striking, except for the difference in the vertical scale. The difference seems, at least partially, to be due to different starting conditions. After the initial longitudinallongitudinal relaxation, Fig. 3 shows an increasing temperature due to the transverse-longitudinal relaxation. The quite slow increase is due to the fact that, for the beam parameters used in the figure, the value of r,nk’3 in Eq. (11) is only 0.54, and the transverse-longitudinal relaxation is thus strongly suppressed. Fig. 4 summarises the results of the simulations, performed at four different magnetic fields. The corresponding values of r,niJ3 range between 0.16 and 2.7. The vertical scale is normalised such that energy spread and time are in the same scaled units as in Fig. 3. Also shown is a theoretical curve obtained by combining Eqs. (8)-( 11) (a similar expression is found in Ref. [23]). At high values of r,ndf3, the curve tends to approximately 1.60, the relaxation rate (in these units) for our electron density and transverse temperature when no magnetic field is present. The overall agreement between the simulations and the theoretical curve is reasonably good, giving us confidence in using this theory as a guideline for discussions of relaxation rates in the context of beam expansion. If one wants to be, more conservative regarding the suppression,
Fig. ised and the
4. Rates of the transverse-longitudinal relaxation in normalunits as a function of the ratio between the cyclotron radius Y, the average distance between the electrons ne”3. The curve is semi-empirical theory of Eqs. (8)-( 11).
one could, alternatively, points in Fig. 4, getting
dTe,, dTe,, __=dt
dt
B=,,
fit a straight
line through
x o.13r,Fz~‘3
the
(12)
This expression is, ,of course, only valid for small l,i~i’~, say, for r,nii3 <4.
4. Influence of beam expansion on relaxation rates A beam expansion scheme such as the one used in CRYRING is characterised by a small cathode in a magnetic field that is higher than normally used in electron coolers. The beam from the gun has the same current as in a normal cooler (since the gun perveance can be made the same as normal), but, consequently, a higher electron density than usual. The beam is expanded to a size which is the same as in conventional coolers (40 mm diameter in CRYRING), and thus has the same density after expansion as it has conventionally. In fact, in CRYRING, the beam parameters (size, density, energy, magnetic field) after the expansion are identical to those that were used before the .expansion was implemented, except for the electron temperature. The longitudinal temperature is affected by the expansion in three ways. Firstly, the energy spread removed from the transverse degree of freedom through the expansion is converted to longitudinal energy spread. However, this contribution to kT,,, is reduced when transforming to the frame of reference moving with the electron beam just as the thermal motion due to the hot cathode. It is thus negligible compared to the energy spread resulting from the longitudinal-longitudinal relaxation. Secondly, the higher, electron density at the gun has the effect that the
H. Danared I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31 longitudinal-longitudinal relaxation gives a higher longitudinal temperature also after the expansion. Since this relaxation process results in a kT,,, proportional to nif3, the expansion inevitably increases the contribution to kT,,, from this relaxation by a factor equal to the expansion ratio to the power of l/3. Thirdly, also the rate of the transverse-longitudinal relaxation changes. As seen from Eqs. (8)-( 1 l), this rate depends both on n,, kT,,, and B, all of which change in the expansion. In the strong field at the gun, the high electron density increases the rate, but this is more than compensated for by the stronger suppression due to the high magnetic field. After the expansion, the low transverse temperature tends to increase the rate, but, again, the suppression becomes stronger - this time caused by the smaller cyclotron radius for electrons at lower kT,, . The beam expansion thus has the overall effect that the transverse-longitudinal relaxation is slower than in a conventional cooler. These different effects of the expansion on the electronbeam parameters are summarised in Table 1. For a numerical example, we may take some typical values used for recombination experiments at CRYRING, such as a beam of 100 mA at 3-keV and expansion from 0.3 T to 0.03 T. In this case, we would have a beam (after expansion) with a density of 1.5 X lOI electrons m3, a transverse energy spread of 10 meV and a longitudinal energy spread of approximately 0.08 meV from the longitudinallongitudinal relaxation. The transverse-longitudinal relaxation in the high magnetic field contributes, according to Eqs. (8)-(1 I), to the longitudinal energy spread with 0.3 meV times a suppression factor of about e-l’. In the low magnetic field, the contribution is 0.15 meV times e -“. Clearly, both these numbers are very small compared to the 0.08 meV. Using Eq. (12) for the rate of the transverse-longitudinal relaxation, one obtains somewhat higher contributions to kT,,,, but still only about 0.01 meV for this set of parameters.
5. Determination of electron temperatures recombination experiments
from
For the conversion of measured recombination rates to cross sections one must have an accurate knowledge of the electron temperature. This can be seen by analysing the relation between these two quantities. In the following this is done in the non-relativistic approximation. If one has a recombination process of any kind with a cross section a(E), or a(v) where E = m,v2/2 (we approximate the reduced mass with the electron mass), the measured signal is proportional to the rate a;. This rate is a function of what we may call the detuning velocity u, = [(u,) - (ui)l, (we assume (u,) - (vi) to be in the longitudinal direction) and is obtained through (13)
29
where u =
[up,+(Ud+ u,,,)21”2>
and d3v, = 27rv,, duei dv,,,. If the cross section has a peak that can be approximated by a S function of the energy, u = u$(E -E,), and the electron-velocity distribution flu,) is the anisotropic Maxwellian distribution of Eq. (5), this integral can be performed analytically [25,26], yielding
cu,(vJ= *exp[z 2MT,,
(vi
X[erf(E
-erf(
@
-$)I
vd+~‘vo)
vd -~‘vo)]
(14)
Here, erf(x) is the error function, v0 = (2E,lm,)“2, A= and we assume T,,, < Z’,,. This expres(1 - T,,,IT,,)“2, sion contains both T,, and r,,,, and a closer inspection reveals that, at low energies, the observed peaks become asymmetric, allowing r,, to be extracted from the lowenergy side of the peak and T,,, from the high-energy side. A process that has recombination cross sections that, for our purpose, can be regarded as 6 functions is dielectronic recombination. At least, this is true for relatively simple atomic systems where the dielectronic resonances are so few that they are well separated in energy, and for resonances where the decay width of the atomic state itself is small. An example is beryllium-like systems with a 2s-2p excitation plus an electron in a state with a higher principal quantum number. Several such systems have been studied at CRYRING. Fig. 5 shows a few dielectronic resonances observed when C3’ ions pick up an electron in the cooler [27]. During this run, a 194mA electron beam at 3.4 keV and 10 times expansion was used. The large peak consists of a single resonance, tentatively identified as belonging to the ls22p4d3D state in C2+, and the smaller peak of two resonances. The experimental points were fitted to peak shapes according to Eq. (14) with kT,,, = 9.4 meV and kT,,, = 0.08 meV, and just using the peak positions and heights plus a linear background (approximating the tail from the radiative recombination) as free parameters. The fit is almost perfect, and the resulting temperatures agree well with the theory of the previous sections (at this run a cathode temperature of 835%20”C was used, corresponding to kT = 95 meV). Similar temperatures have been obtained in other recombination experiments, but it has also been observed that
I. ELECTRON COOLING PHYSICS
30
H. Danared
I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
Table 1 Effects of the beam expansion scheme used at CRYRING on different electron beam parameters.
The two last columns show how a beam with an expansion ratio K differs from a beam with no expansion, assuming that the beam energy, beam size, electron density, and magnetic field in the cooling region is the same in both cases (these quantities are denoted with subscript 0). C, is a constant obtained from Eqs. (8) and (9) Without expansion
With expansion Gun region
With expansion Cooling region
Magnetic field Beam radius Electron density Transverse temperature
47 a0 n,o
Longitudinal temperature from long.-long. relaxation Additional longitudinal temp. from tr.-long. relaxation
K”‘T,,,o
K_‘T,,O
recombination rates [28] and cooling forces seem to increase slower than expected when the electron density is increased to maximal values (i.e., when the perveance of the electron gun is above 1 or 1.5 ~A/V3”). Although obvious candidates for the cause of this effect are imperfections in the optics of the electron gun or space charge effects, it should be remembered that the high magnetic field at the gun in fact gives the gun very good optical properties also at high currents, and that a cooled, well centered ion beam is very little affected by the space charge of the electron beam. The reason for the perceived increase in temperature at high electron densities is thus still unexplained.
6. Conclusion
several different techniques. Comparisons with other installations reveal that the beam expansion gives high cooling forces, but some data taken without beam expansion and where the effects of the magnetised cooling were strong give forces that are almost as high, at least at low relative velocities. At the same time, it is clear that the expansion can give a very low transverse electron temperature and a longitudinal temperature which is not much higher than in a conventional cooler. This results in much improved energy resolution in recombination experiments at low relative energy, and further increases of the expansion ratio promise continued progress in this direction.
Acknowledgements
We have made cooling-force measurements with the expanded electron beam at the CRYRING cooler using
We wish to thank the colleagues at the Manne Siegbahn Laboratory for their contributions to the experimental work described in this paper. The cooling-force measurements summarised in Fig. 1 were performed in collaboration with G. Ciullo, H. Schmitt, M. Schmitt, M. Steck, and T. Winkler, and with A. Schnase and the RF group at COSY who supplied the noise generator used. This work was supported by the Human Capital and Mobility Programme of the European Community.
References
0.10
0.15 Centre-of-mass
0.20 energy
0.25 (eV)
Fig. 5. The rate of dielectronic recombination between C3+ and cooler electrons as a function of the relative energy between ions and electrons in the centre-of-mass system. The curve is a fit to the experimental data points using kT,, = 9.4 meV and kT.,, = 0.08 meV and the peak shape of Eq. (14).
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1 Nucl. Instr. ana’ Meth. in Phys. Res. A 391 (1997) 24-31
[5] H. Danared, G. Andler, L. Bagge, C.J. Herrlander, J. Hilke, J. Jeansson, A. K%llberg, A. Nilsson, A. Pa& K.-G. Rensfelt, U. Rosengkd, J. Starker and M. af Ugglas, Phys. Rev. Lett. 72 (1994) 3775. [6] M. Bell, J. Chaney, H. Herr, F. Krienen, P. Meller Petersen and G. Petrucci, Nucl. Ins&. and Meth. 190 (1981) 237. [7] H. Poth, W. Schwab, B. Seligmann, M. Wiirtge and A. Wolf, Nucl. Instr. and Mefh. A 287 (1990) 328. [8] D.D. Caussyn, M. Ball, B. Brabson, J. Budnick, V. Derenchuk, G. East, M. Ellison, D. Friesel, B. Hamilton, K. Hedblom, W.P. Jones, S.Y. Lee, D. Li, J.Y. Liu, T. Lofnes, K.Y. Ng, A. Riabko, T. Sloan and Y. Wang, Phys. Rev. Lett. 73 (1994) 2696. [9] K. Abrahamsson, G. Andler and C.B. Bigham, Nucl. Instr. and Meth. B 31 (1988) 475. [lo] N.S. Dikansky, V.I. Kononov, V.I. Kudelainen, I.N. Meshkov, VS? Parkhomchuk, D.V. Pestrikov, A.N. Skrinsky and B.N. Sukhina, Preprint 79-56, INP, Novosibirsk, 1979. [II] T. Ellison, W. Kells, V. Kemer, F. Mills, R. Peters, T.’ Rathbun, D. Young and P.M. McIntyre, IEEE Trans. Nucl. Sci. NS-30 (1983) 2636. [12] N.S. Dikansky,V.I. Kudelainen,V.A. Lebedev, I.N. Meshkov, W. Parkhomchuk, A.A. Sery, A.N. Skrinsky and B.N. Sukhina, Preprint 88-61, INP, Novosibirsk, 1988. [13] H. Poth, W. Schwab, B. Seligmann, M. Wijrtge, A. Wolf, S. Baird, M. Chanel, H. Haseroth, C.E. Hill, R. Ley, D. Manglunki, G. Tranquille, J.L. Vallet and P.F. Dittner, Z. Phys. A 332 (1989) 171. [14] T.J.P. Ellison, Thesis, Indiana University, 1990. [15] T. Tanabe, K. Noda, T. Honma, M. Kodaira, K. Chida, T. Watanabe, A. Noda, S. Watanabe, A. Mizobuchi, M. Yoshizawa, T. Katayama, H. Muto and A. Ando, Nucl. Instr. and Meth. A 307 (1991) 7.
[16] M. Steck, K. Beckert, H. Eickhoff, B. Franzke, F. Nolden and P. Spgdtke, Proc. 1993 Particle Accelerator Conf,, Washington, DC, 1993, p. 1738. [17] D. Reistad, L. Hermansson, T. Bergmark, 0. Johansson, A. Simonsson and A.V. Burov, Proc. Workshop on Beam Cooling and Related Topics, Montreaux 1993, CERN94-03, p. 183. [ 181 L.H. Andersen, private communication. [19] S. Pastuszka, U. Schramm, M. Grieser, C. Broude, R. Grimm, D. Habs, J. Kenntner, H.-J. Miesner, T. SchiiBler, U. Schramm, D. Schwalm and A. Wolf, Nucl. Instr. and Meth. A 369 (1996) 11. [20] T. Winkler, K. Beckert, F. Bosch, H. Eickhoff, B. Franzke, 0. Klepper, F. Nolden, H. Reich, B. Schlitt, P. Spgdtke and M. Steck, Hyperfine Int. 99 (1996) 277. [21] V.I. Kudelainen, VA. Lebedev, I.N. Meshkov, VV. Parkhomchuk and B.N. Suchina, Sov. Phys. JETP 56 (1982) 1191. [22] J.D. Lawson, The Physics of Charged-Particle Beams, 2nd ed. (Clarendon, Oxford, 1988). [23] A.V. Aleksandrov, N.S. Dikansky, NCh. Kot, V.I. Kudelainen, V.A. Lebedev and PV Logachov, Proc. Workshop on Electron Cooling and New Cooling Techniques, Legnaro, 1990 (World Scientific, 1991) p. 279. 1241 A.V Aleksandrov, N.S. Dikansky, N.Ch. Kot, V.I. Kudelainen, V.A. Lebedev, EV Logachov, R. Calabrese, V. Guidi, G. Ciullo, G. Lamanna and L. Tecchio, Phys. Rev. A 46 (1992) 6628. [25] L.H. Andersen and J. Bolko, Phys. Rev. A 42 (1990) 1184. [26] G. Kilgus, D. Habs, D. Schwalm, A. Wolf, N.R. Badnell and A. Miiller, Phys. Rev. A 46 (1992) 5730. [27] S. Mannervik et al., to be published. [28] H. Gao, S. Asp, C. Biedennann, D.R. Dewitt, R. Schuch, W. Zong and H. Danared, Hyperfine Int. 99 (1996) 301.
I. ELECTRON
COOLING
PHYSICS
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH
Electron cooling at CRYRING with an expanded electron beam Hikan Danared* Manne Siegbahn Laboratory, Frescativ2igen 24, S-104 05 Stockholm, Sweden
Abstract Studies of electron cooling with an adiabatically expanded electron beam have been performed at CRYRING. Measurements of the longitudinal drag force using three different methods are described and are compared with measurements at other electron coolers. The processes that determine the electron temperature, longitudinally and transversally, are reviewed, and the effect of the beam expansion on these processes is discussed. It is described how the electron temperature can influence the outcome of recombination measurements, and how such measurements can be used to determine the electron temperature, and an example of such a measurement is given. PACS: 07.77. + p; 29.20.Dh; 35.80. + s; 41.80.Dd Storage rings; Electron cooling; Electron target: Recombination
Keywords:
1. Introduction CRYRING is a small synchrotron and storage ring used mainly for research in atomic and molecular physics [l]. An essential part of the ring is the electron cooler [2], which is used both for beam cooling and as an electron target. With the cooler operating as a target, many types of electron-ion recombination processes have been studied. These include radiative recombination - spontaneous or laser-stimulated - and dielectronic recombination of atomic ions as well as dissociative recombination of molecular ions. Also other processes such as ionisation, excitation, de-excitation, etc. can be investigated (see, e.g., Ref. [3] for a recent review). There are several advantages in using storage rings for experiments in atomic and molecular physics. Some are consequences of the mere acceleration and storing of the ions, such as the increase in luminosity achieved by recirculating the ions that do not react in their first passage through the target, the (relatively) high energy of accelerated ions that reduces reactions with the rest gas and gives almost background-free measurements in many cases, and the possibility to let internally hot ions spontaneously de-excite before measurements are performed. Applying electron cooling then reduces the geometrical dimensions of the ion beam and also reduces its momentum spread, allowing for an increased precision in the recombination measurements. At the end, when the actual measurements start, one thus has an accurately described beam of ions in *Corresponding author. Tel.: +46 8 161038, fax: +46 8 158674, e-mail [email protected].
experiments
a well-defined internal state, and also the properties of the electron beam (current density, energy, etc.) are known to a high precision. A further property of the electron beam which is of great importance is its energy spread or temperature. It influences both the cooling process and the results that can be obtained in recombination studies. Although the cooling may become more efficient with colder electrons, it is particularly when the cooler acts as an electron target that the benefits of a low electron temperature become obvious [4]. When recombination cross sections are recorded as a function of the relative energy between ions and electrons, the energy resolution is usually determined entirely by the electron energy spread. For other types of measurements performed close to zero relative energy (e.g., laser-stimulated radiative recombination), the experimental count rate increases significantly with lower electron temperature.
2. Cooling force measurements 2.1.
Methods
In order to improve, the properties of the CRYRING cooler, we introduced an adiabatic expansion of its electron beam a couple of years ago [5]. The expansion is achieved by using a higher magnetic field at the electron gun, typically 0.3 T, than in the rest of the cooler, where normally 0.03 T is used. This gives an expansion factor of 10, reducing the transverse electron-energy spread with a factor equal to the ratio between the fields. In the case of CRYRING, this means from 100meV (deriving from the
0168-9002/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PII SO168-9002(97)00249-O
H. Danared
I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
cathode temperature of SOO-900°C) to 10 meV The longitudinal electron-energy spread is only affected to a small extent by the expansion (see Section 4) and is of the order of 0.1 meV Clearly, it is of interest to see how this reduction of the transverse electron temperature by a factor of 10 influences the cooling. This influence is most easily investigated by studying the “cooling force”, or the drag force that an ion experiences as it moves through the electron gas of the cooler’s electron beam. In general, the cooling force has a longitudinal component F,, (i.e., parallel to the magnetic field and the direction of propagation of the electron beam) that is different from the transverse component F,. These force components depend on both the longitudinal and the transverse velocity of the ion, making it very difficult - in practise impossible - to map out the cooling force completely. The longitudinal component is, however, quite sensitive to the details of the electron-ion interaction, enabling a comparison with theoretical models. One straightforward technique for measuring the cooling force is the voltage-step method. Here, one first cools the ions, matching the ion and the electron velocities. Then, the electron energy is changed rapidly, creating a welldefined velocity difference between ions and electrons. The ions then accelerate toward the new electron velocity, and if the acceleration is measured, only the ion mass is needed to obtain the force. The acceleration has to be measured before the ion velocity has changed significantly, otherwise the relative velocity is no longer well defined. The acceleration is determined, via the Schottky signal, from the change in revolution frequency per unit time, giving [6] p dfldt f
5 = 5
’
(1)
where p is the ion momentum and 77is (df/dp)l(flp). This method has the advantage that the interpretation of the results is very simple, but it is limited to relative velocities above approximately 10 000 m/s. At lower velocities, the required change in electron energy becomes too small, and the ion veIocity changes so rapidIy to the new electron velocity that the measurements become difficult. Since several theoretical predictions about the cooling force can be verified only at 10 000 m/s or lower, and this velocity regime also has the greatest practical importance for cooling, other methods for cooling-force measurements have been developed. A common denominator for these methods is that they are based upon an equilibrium between the cooling and a process that tends to change the ion velocity. One such equilibrium situation was first investigated at LEAR [7]. Here, the ion beam is heated with rf noise at the same time as it is cooled, and the cooling force is deduced from the power density of the Schottky signal. The noise is applied in a single Schottky band and the resulting frequency
25
spectrum is taken from a different band. Introducing the ion-velocity distribution p(u) and the diffusion constant D, which can be obtained from the heating power density, one obtains the longitudinal cooling force by solving the onedimensional Fokker-Planck equation. In the steady state (dplat = O), one finds
Since this expression for F,,(u) contains p(u), this way of evaluating the cooling force is suitable for relative velocities similar to those that exist in the ion beam, typically lOOOOm/s and less. The sensitivity can be improved somewhat using BTF rather than Schottky measurements. A third method [S] uses a bunched ion beam and the equilibrium between the cooling force and &he accelerating force from the rf cavity (or driven drift tube in the case of CRYRING [9]). With very small rf voltages one can reach a situation where the energy that an ion gains during one passage through the rf system is equal to the loss during the passage through the cooler (or vice versa). The resulting shift A+s of the synchronous frequency can be detected with an oscilloscope that measures the signal induced by the bunches in an electrostatic pickup. Alternatively, in order to increase the sensitivity, a network analyzer can be used to detect the phase difference between the bunch and the rf signal. The cooling force is simply obtained from F,, = Zecr,
sin AC/+,
where Ze is the charge of the ion and fir,, is the rf amplitude. In these measurements, the ion energy is determined by the rf frequency, while the velocity difference between ions and electrons, in our case, is introduced by shifting the electron energy from the cooling condition while the rf frequency is kept constant. The equilibrium between cooling force and rf force results in a constant phase shift only for those relative velocities near zero where the magnitude of the cooling force increases with velocity. At higher relative velocities, the phase shift starts to oscillate as described in Ref. [8]. This transition, from a steady state where the longitudinal motion of the bunch is a fixed point in phase space to a situation where it describes a circle in phase space, can be used to locate the relative velocity where the cooling force has its maximum. 2.2. Results
from CRYRIIVG
At CRYRING, cooling-force measurements with the expanded electron beam have been performed using all three methods described above. Fig. 1 shows some results of these measurements. Here, deuterons at 12 MeV were used, the electron current was 25 mA, and the magnetic field was 0.3 T at the gun and 0.03 T in the rest of the
I. ELECTRON COOLING PHYSICS
H. Danared I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
26
X exp
_-_-WL 2kT,, T,,
characterized by the temperatures Coulomb logarithm we have used 3
10-Z
.b
L, =+ln[
104’ lo*
-
theory, kTi=10
.
D’, 10 times expanded electron beam
1’1”‘1’
lo3
1 +(k)‘].,
meif, kTpO.05 meV
i ““‘LJ
“““‘1
105 104 Relative velocity (m/s)
”
II”“”
106
Fig. 1. Longitudinal drag force as a function of the relative velocity between ions and electrons. Points are measured values using voltage stepping or the phase-shift method, the dotted curve is measured using stochastic heating. The full-drawn curve is calculated from Eq. (4). The force is normalised to an electron density n, = lOi mm3.
cooler, giving an expansion factor of 10. The data points above 10 000 m/s were taken with voltage-stepping, those at lower velocities with the phase-shift method. Phase oscillations set in at the position of the arrow. The dotted curve is the force as measured using the equilibrium between cooling and stochastic heating. At the top of the curve, all three methods agree well, and the onset of phase oscillations is right at the maximum of the force, as it should be. This is quite satisfactory considering that, although the three sets of measurements were made during the same beam time, they are completely independent. The wiggles in the dotted curve are due to the low statistics at the tail of the BTF spectrum used for the evaluation of the force. The discrepancy of almost a factor of two at the low relative velocities is unexplained but perhaps not too surprising considering, e.g., the difficulty to make an absolute calibration of the noise power
and T,,,. For the
is the Debye length or the adiabatic cut-off where b,,, distance vi/tip (wp being the plasma frequency), whichever is bigger, and b,in =
ze2 4rrE,in,VZ
is the impact parameter giving the maximum classical momentum transfer. We approximate vZ = max(v’, 2kT,, / m,). The curve was obtained using a transverse energy spread kT,. = 10 meV and a longitudinal energy spread kT,,, = 0.05 meV The agreement between this simple theory and the data is quite good. One could note, however, that if LC is moved inside the integral in Eq. (4), and Iui - u,I is used instead of 2kT,, lm, when calculating the minimum impact parameter, the theoretical curve becomes somewhat lower for low relative velocities. 2.3. Results from other coolers Measurements of the cooling force have been made at all ion-storage rings that use or have used electron cooling and also at the single-pass setup MOSOL in Novosibirgk. Fig. 2 shows a selection of the “best” values reported for
0 0 0 a .
seen by the beam.
curve in Fig. 1 shows theoretical values for the cooling force, given by the usual expression
NAP-M ICE FNAL MOSOL LEAR
The full-drawn
a .
F(vi)
=
-4T(g-)2+ jf(u.)
. .__ ....% ....?c... 0
10-i
u. -u
xjltid3ve
(4)
based on binary collisions between ions and electrons. Here, ui and u, are the ion and electron velocities in the system moving with the electrons, n, is the electron density, I,, is the Coulomb logarithm, and f(u,) is the anisotropic Maxwellian electron-velocity distribution
c)
3
;
0
o “,,g p;
0
lo-" 103
I
104
?\I
IIIIN,
IO5
IO6
Relative velocity (m/s) Fig. 2. Results of drag-force measurements (points) normalised to singly charged ions and an electron density of lOi me3 [10,6,1 l191. Curves represent theoretical calculations; cf. text.
H. Danared I Nucl. Ins@. and Meth. in Phys. Res. A 391 (1997) 24-31
singly charged light ions (protons or deuterons in most cases). For ESR, data for Ne”+ divided by 100 were used, causing somewhat low values compared to the others, since the drag force does not quite scale with 4’ [20]. The curves are calculated according to Eq. (4) with kT,, = 10 meV and kT,,, = 0.1 meV for the upper curve and kT,, = 100 meV and kT,,, = 0.1 meV for the lower one. The scatter of the data points reflects a wide range of measurement conditions (beam energies, electron densities, electron temperatures, magnetic fields, etc.), and a detailed comparison between the results from the different installations is not meaningful. In particular, the cooling force at low relative velocities depends on the longitudinal electron temperature, which in turn depends on the electron density, as discussed in Section 3. However, one can see that some of the early cooling experiments show somewhat low values due to high electron temperatures. Also, the two measurements that were performed with expanded electron beams, at CRYRING and TSR, show the highest values. Apart from this, the experimental points are reasonably close to the theoretical curves at high relative velocities, but above this theory at low velocities. This shows the influence of the magnetic field on the electron-ion interaction and the cooling process. The CRYRING data shown in Fig. 2 are the same as in Fig. 1 below 10 000 m/s, but the points above 10 000 m/s were taken using a 20 times expanded beam, which gave a slightly higher cooling force.
3. Electron temperatures and relaxation rates As already mentioned, the electron temperature is an important parameter for an electron cooler, particularly when it is used as an electron target. The fact that the electron beam expansion reduces the transverse electron temperature by a factor equal to the expansion ratio has been used at all the four low-energy ion storage rings which are used mainly for atomic and molecular physics (CRYRING, TSR, TARN-II, and ASTRID). In order to avoid any misunderstanding, it should be pointed out that the beam expansion does not affect the azimuthal drift motion of the electrons that is caused by the space charge of the beam. Such pure space-charge effects, which may be important at high electron densities and large ion-beam radii, will not be discussed in this paper. The effect of beam expansion on the longitudinal electron temperature is perhaps less obvious, in particular the influence on the relaxation processes that occur in the electron beam. Usually, and at least in CRYRING, the longitudinal temperature is determined by relaxation rather than by the initial energy spread that may stem from the cathode temperature, ripple on the high-voltage power supply, etc. The initial energy spread becomes reduced to at most a few FeV when it is transformed to the reference system moving with the electrons.
27
There are two relaxation processes to consider [21]. The longitudinal-longitudinal relaxation is caused by the fact that, after a rapid acceleration, the longitudinal kinetic energy spread of a few PeV is much smaller than the average potential energy between the electrons. The latter is of the order e2/(4.rr~,)n:‘3 or typically 0.1 meV. After a relaxation time on the order of a plasma period, an equilibrium is reached, and also the kinetic energy spread becomes of the order e2/(4~e,,)n:‘3 longitudinally. The transverse-longitudinal relaxation is the transfer of energy from the transverse motion, where the energy spread is 100 meV or higher (without beam expansion), to the longitudinal degree of freedom. In the absence of a magnetic field, the rate at which the longitudinal temperature increases is [22]
dTeIITe, dt
-
(8)
T.
The relaxation
time constant is
r=------where r,, is the classical electron radius. This transfer of energy is, however, suppressed in the presence of a magnetic field when the cyclotron radius r, of the electrons, taken as
becomes comparable with or smaller than the average distance n,‘13 between the electrons. An empirical expression for the suppression was obtained at Novosibirsk [23], which we simplify to
(11) in order to get a closed expression for the temperature. This expression was verified for beam parameters normally used at CRYRING through molecular-dynamics calculations. In these calculations, N electrons were positioned randomly in a cylindrical box with periodic boundary conditions longitudinally. The initial velocity of the particles was zero longitudinally and Maxwell-dishibuted with kT,, = 50 meV transversally. Also, an azimuthal drift velocity corresponding to the effect of the space charge was added. The classical equations of motion of the particles were then integrated numerically, taking into account the electrostatic interaction between the particles and the external magnetic field. The evolution of the system was followed for a period of time corresponding to the flight time through the electron cooler. Throughout the calculations an electron density of 1 X lOI rnp3 was used and N varied between 500 and 2000. Fig. 3 shows the result of such a calculation where the magnetic field was 0.03 T. The longitudinal energy spread,
I. ELECTRON COOLING PHYSICS
28
H. Danared I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
0.0
0.5
1.0
1.5
2.0
2.5
t I Tp Fig. 3. Result of molecular-dynamics calculations showing the evolution of the longitudinal electron temperature in normalised units due to relaxation processes in the beam. This calculation used n, = 1 X lOI mm3 and B = 0.03 T.
equal to m,(u$ is normalised to e2/(4Te,)n:‘3, and the time is in units of the plasma period rP. It is seen that the longitudinal-longitudinal relaxation takes about 0.3~~ and leads to a longitudinal energy spread quite close to eZ/ (4%)% “3. The value of kT,,, depends to some extent, however, on the magnetic field. A stronger field gives a higher IV,,, and vice versa, probably because a low field allows some of the initial potential energy to be converted to transverse kinetic energy. On the other hand, the initial relaxation time of 0.3~~ seems independent of the magnetic field. The similarity between our Fig 3. and Fig. 2 of Ref. [24] is striking, except for the difference in the vertical scale. The difference seems, at least partially, to be due to different starting conditions. After the initial longitudinallongitudinal relaxation, Fig. 3 shows an increasing temperature due to the transverse-longitudinal relaxation. The quite slow increase is due to the fact that, for the beam parameters used in the figure, the value of r,nk’3 in Eq. (11) is only 0.54, and the transverse-longitudinal relaxation is thus strongly suppressed. Fig. 4 summarises the results of the simulations, performed at four different magnetic fields. The corresponding values of r,niJ3 range between 0.16 and 2.7. The vertical scale is normalised such that energy spread and time are in the same scaled units as in Fig. 3. Also shown is a theoretical curve obtained by combining Eqs. (8)-( 11) (a similar expression is found in Ref. [23]). At high values of r,ndf3, the curve tends to approximately 1.60, the relaxation rate (in these units) for our electron density and transverse temperature when no magnetic field is present. The overall agreement between the simulations and the theoretical curve is reasonably good, giving us confidence in using this theory as a guideline for discussions of relaxation rates in the context of beam expansion. If one wants to be, more conservative regarding the suppression,
Fig. ised and the
4. Rates of the transverse-longitudinal relaxation in normalunits as a function of the ratio between the cyclotron radius Y, the average distance between the electrons ne”3. The curve is semi-empirical theory of Eqs. (8)-( 11).
one could, alternatively, points in Fig. 4, getting
dTe,, dTe,, __=dt
dt
B=,,
fit a straight
line through
x o.13r,Fz~‘3
the
(12)
This expression is, ,of course, only valid for small l,i~i’~, say, for r,nii3 <4.
4. Influence of beam expansion on relaxation rates A beam expansion scheme such as the one used in CRYRING is characterised by a small cathode in a magnetic field that is higher than normally used in electron coolers. The beam from the gun has the same current as in a normal cooler (since the gun perveance can be made the same as normal), but, consequently, a higher electron density than usual. The beam is expanded to a size which is the same as in conventional coolers (40 mm diameter in CRYRING), and thus has the same density after expansion as it has conventionally. In fact, in CRYRING, the beam parameters (size, density, energy, magnetic field) after the expansion are identical to those that were used before the .expansion was implemented, except for the electron temperature. The longitudinal temperature is affected by the expansion in three ways. Firstly, the energy spread removed from the transverse degree of freedom through the expansion is converted to longitudinal energy spread. However, this contribution to kT,,, is reduced when transforming to the frame of reference moving with the electron beam just as the thermal motion due to the hot cathode. It is thus negligible compared to the energy spread resulting from the longitudinal-longitudinal relaxation. Secondly, the higher, electron density at the gun has the effect that the
H. Danared I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31 longitudinal-longitudinal relaxation gives a higher longitudinal temperature also after the expansion. Since this relaxation process results in a kT,,, proportional to nif3, the expansion inevitably increases the contribution to kT,,, from this relaxation by a factor equal to the expansion ratio to the power of l/3. Thirdly, also the rate of the transverse-longitudinal relaxation changes. As seen from Eqs. (8)-( 1 l), this rate depends both on n,, kT,,, and B, all of which change in the expansion. In the strong field at the gun, the high electron density increases the rate, but this is more than compensated for by the stronger suppression due to the high magnetic field. After the expansion, the low transverse temperature tends to increase the rate, but, again, the suppression becomes stronger - this time caused by the smaller cyclotron radius for electrons at lower kT,, . The beam expansion thus has the overall effect that the transverse-longitudinal relaxation is slower than in a conventional cooler. These different effects of the expansion on the electronbeam parameters are summarised in Table 1. For a numerical example, we may take some typical values used for recombination experiments at CRYRING, such as a beam of 100 mA at 3-keV and expansion from 0.3 T to 0.03 T. In this case, we would have a beam (after expansion) with a density of 1.5 X lOI electrons m3, a transverse energy spread of 10 meV and a longitudinal energy spread of approximately 0.08 meV from the longitudinallongitudinal relaxation. The transverse-longitudinal relaxation in the high magnetic field contributes, according to Eqs. (8)-(1 I), to the longitudinal energy spread with 0.3 meV times a suppression factor of about e-l’. In the low magnetic field, the contribution is 0.15 meV times e -“. Clearly, both these numbers are very small compared to the 0.08 meV. Using Eq. (12) for the rate of the transverse-longitudinal relaxation, one obtains somewhat higher contributions to kT,,,, but still only about 0.01 meV for this set of parameters.
5. Determination of electron temperatures recombination experiments
from
For the conversion of measured recombination rates to cross sections one must have an accurate knowledge of the electron temperature. This can be seen by analysing the relation between these two quantities. In the following this is done in the non-relativistic approximation. If one has a recombination process of any kind with a cross section a(E), or a(v) where E = m,v2/2 (we approximate the reduced mass with the electron mass), the measured signal is proportional to the rate a;. This rate is a function of what we may call the detuning velocity u, = [(u,) - (ui)l, (we assume (u,) - (vi) to be in the longitudinal direction) and is obtained through (13)
29
where u =
[up,+(Ud+ u,,,)21”2>
and d3v, = 27rv,, duei dv,,,. If the cross section has a peak that can be approximated by a S function of the energy, u = u$(E -E,), and the electron-velocity distribution flu,) is the anisotropic Maxwellian distribution of Eq. (5), this integral can be performed analytically [25,26], yielding
cu,(vJ= *exp[z 2MT,,
(vi
X[erf(E
-erf(
@
-$)I
vd+~‘vo)
vd -~‘vo)]
(14)
Here, erf(x) is the error function, v0 = (2E,lm,)“2, A= and we assume T,,, < Z’,,. This expres(1 - T,,,IT,,)“2, sion contains both T,, and r,,,, and a closer inspection reveals that, at low energies, the observed peaks become asymmetric, allowing r,, to be extracted from the lowenergy side of the peak and T,,, from the high-energy side. A process that has recombination cross sections that, for our purpose, can be regarded as 6 functions is dielectronic recombination. At least, this is true for relatively simple atomic systems where the dielectronic resonances are so few that they are well separated in energy, and for resonances where the decay width of the atomic state itself is small. An example is beryllium-like systems with a 2s-2p excitation plus an electron in a state with a higher principal quantum number. Several such systems have been studied at CRYRING. Fig. 5 shows a few dielectronic resonances observed when C3’ ions pick up an electron in the cooler [27]. During this run, a 194mA electron beam at 3.4 keV and 10 times expansion was used. The large peak consists of a single resonance, tentatively identified as belonging to the ls22p4d3D state in C2+, and the smaller peak of two resonances. The experimental points were fitted to peak shapes according to Eq. (14) with kT,,, = 9.4 meV and kT,,, = 0.08 meV, and just using the peak positions and heights plus a linear background (approximating the tail from the radiative recombination) as free parameters. The fit is almost perfect, and the resulting temperatures agree well with the theory of the previous sections (at this run a cathode temperature of 835%20”C was used, corresponding to kT = 95 meV). Similar temperatures have been obtained in other recombination experiments, but it has also been observed that
I. ELECTRON COOLING PHYSICS
30
H. Danared
I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 24-31
Table 1 Effects of the beam expansion scheme used at CRYRING on different electron beam parameters.
The two last columns show how a beam with an expansion ratio K differs from a beam with no expansion, assuming that the beam energy, beam size, electron density, and magnetic field in the cooling region is the same in both cases (these quantities are denoted with subscript 0). C, is a constant obtained from Eqs. (8) and (9) Without expansion
With expansion Gun region
With expansion Cooling region
Magnetic field Beam radius Electron density Transverse temperature
47 a0 n,o
Longitudinal temperature from long.-long. relaxation Additional longitudinal temp. from tr.-long. relaxation
K”‘T,,,o
K_‘T,,O
recombination rates [28] and cooling forces seem to increase slower than expected when the electron density is increased to maximal values (i.e., when the perveance of the electron gun is above 1 or 1.5 ~A/V3”). Although obvious candidates for the cause of this effect are imperfections in the optics of the electron gun or space charge effects, it should be remembered that the high magnetic field at the gun in fact gives the gun very good optical properties also at high currents, and that a cooled, well centered ion beam is very little affected by the space charge of the electron beam. The reason for the perceived increase in temperature at high electron densities is thus still unexplained.
6. Conclusion
several different techniques. Comparisons with other installations reveal that the beam expansion gives high cooling forces, but some data taken without beam expansion and where the effects of the magnetised cooling were strong give forces that are almost as high, at least at low relative velocities. At the same time, it is clear that the expansion can give a very low transverse electron temperature and a longitudinal temperature which is not much higher than in a conventional cooler. This results in much improved energy resolution in recombination experiments at low relative energy, and further increases of the expansion ratio promise continued progress in this direction.
Acknowledgements
We have made cooling-force measurements with the expanded electron beam at the CRYRING cooler using
We wish to thank the colleagues at the Manne Siegbahn Laboratory for their contributions to the experimental work described in this paper. The cooling-force measurements summarised in Fig. 1 were performed in collaboration with G. Ciullo, H. Schmitt, M. Schmitt, M. Steck, and T. Winkler, and with A. Schnase and the RF group at COSY who supplied the noise generator used. This work was supported by the Human Capital and Mobility Programme of the European Community.
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0.10
0.15 Centre-of-mass
0.20 energy
0.25 (eV)
Fig. 5. The rate of dielectronic recombination between C3+ and cooler electrons as a function of the relative energy between ions and electrons in the centre-of-mass system. The curve is a fit to the experimental data points using kT,, = 9.4 meV and kT.,, = 0.08 meV and the peak shape of Eq. (14).
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I. ELECTRON
COOLING
PHYSICS