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Dispersive electron cooling experiments at the heavy ion storage ring TSR
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 512 (2003) 459–469
Dispersive electron cooling experiments at the heavy ion storage ring TSR M. Beutelspachera, M. Griesera,*, K. Nodab, D. Schwalma, A. Wolfa a
Max-Planck-Institut fur . Kernphysik, Postfach 103980, Heidelberg 69029, Germany b National Institute of Radiological Sciences, Chiba 263-8555, Japan
Received 28 April 2003; received in revised form 14 June 2003; accepted 16 June 2003
Abstract At low relative velocities, the transverse cooling rates of fast stored ions by cold electrons are usually smaller than the longitudinal ones. By dispersive electron cooling, however, it is possible to transfer heat from the horizontal into the longitudinal degree of freedom, thereby increasing the horizontal cooling rate on the expense of the longitudinal rate. The cooling scheme requires a horizontal gradient of the longitudinal cooling force which can be achieved by displacing the electron beam relative to the ion beam, and a finite dispersion in the ion–electron interaction region. Operating the heavy ion test storage ring TSR at a moderate dispersion of DS ¼ 1:81 m; we investigated the dispersive electron cooling process and found good agreement with the expectations from fundamental considerations. r 2003 Elsevier B.V. All rights reserved. PACS: 32.80.P; 33.80.P Keywords: Electron cooling; Heavy ion; Storage ring
1. Introduction Phase space cooling of fast stored ions has become an important technique to produce highquality ion beams. Three methods have been developed over the last years to cool heavy ion beams: (i) stochastic cooling [1,2], where the beam temperature is reduced by an active feedback loop, (ii) electron cooling [3–6], where ions are repeatedly bathed in a cold electron beam, and (iii) laser cooling [7,8], where the light pressure force caused *Corresponding author. E-mail address: [email protected] (M. Grieser).
by resonant scattering of photons from laser fields is used to increase the phase space density of the ions. While the stochastic cooling process is transversally as effective as longitudinally, the other two techniques turn out to be less efficient in cooling the transversal than the longitudinal degree of freedom. This mismatch was particularly large in the case of laser cooling as the standard method only allowed to cool the longitudinal phase space directly while the transversal degree of freedom was only affected indirectly via the coupling caused by intra-beam scattering (IBS); the mismatch could be reduced at least partially by using a new cooling scheme which exploits the longitudinal–horizontal coupling of the ion
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-9002(03)01976-4
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motion due to the storage ring dispersion in combination with a transversal gradient of the light pressure force [9]. Although the differences in the strength of the longitudinal and transverse forces are less severe in the case of electron cooling [6], it is interesting to transfer the dispersive coupling scheme also to electron cooling and to study its potentialities for equilibrating the longitudinal and transversal electron cooling rates. We have, therefore, performed an experiment to investigate the feasibility of dispersive electron cooling at the heavy ion storage ring TSR of the Max Planck Institute for Nuclear Physics in Heidelberg.
betatron oscillation. Likewise, the horizontal oscillation of the ion is reduced if a negative momentum change is applied at xo0: Assuming a horizontal gradient of the longitudinal cooling v ; xÞ around x ¼ 0; namely Fjj ð~ v ; xÞ ¼ force Fjj ð~ 0 0 Fjj ð~ v ; 0Þ þ Fjj;x ð~ v ; 0Þ x; with Fjj;x ¼ @Fjj =@x and ~ v being the laboratory velocity of the ion, the net effect of this process on the ion, passing repeatedly through the interaction region, will be a complete damping of the betatron oscillation; moreover, as the individual processes lead only to a small shift of the closed orbit, the displacements will be averaged out. For a beam of ions, this results in a decrease of the horizontal beam width s with a cooling rate LD;x ¼ t1 D;x given by [9]
2. Principle of dispersive electron cooling
LD;x ¼
The principle of dispersive electron cooling is depicted in Fig. 1. A change in the longitudinal momentum of an ion by Dpjj leads to a horizontal displacement Dx of the closed orbit due to dispersion. With the x-axis pointing ring outward, the displacement is given by
Here p0 ¼ M v0 now denotes the average longitudinal momentum of the ion beam in the laboratory frame and Zc the fraction of the interaction length to the circumference of the storage ring. A horizontal gradient of the longitudinal electron cooling force can be achieved by a displacement of the electron beam with respect to the ion beam exploiting the parabolic velocity profile of the electrons caused by its space charge. The mean electron velocity as a function of the distance r from the electron beam axis is given by
Dx ¼ xD ¼ DS
Dpjj p0
ð1Þ
where p0 is the longitudinal momentum of the ion in the laboratory frame, and DS denotes the dispersion. Thus, a positive momentum change Dpjj ; transferred to the ion at x > 0; reduces the amplitude and, therefore, damps the horizontal x ring
∆ p ||
outward ∆x ring inward
s
Fig. 1. The principle of dispersive cooling. A positive change of the longitudinal momentum transferred ring outward ðx > 0Þ leads to a damping of the betatron oscillation due to the shift of the closed orbit. Likewise, the betatron oscillation is reduced if a negative momentum transfer occurs at xo0:
1 ds DS ¼ Z F0 : s dt 2p0 c jj;x
/ve ðrÞS ¼ ve;0 ð1 þ ðr=re Þ2 Þ
ð2Þ
ð3Þ
where ve;0 is the on-axis velocity of the beam in the laboratory frame, and re is connected with the electron density ne by r2e ¼ 4e0 me v2e;0 =ðe2 ne Þ with me denoting the electron mass and e0 the vacuum permittivity. For a horizontal displacement of the electron beam ring inward by x0 with respect to the ion beam, i.e. to x ¼ x0 (see Fig. 2), ions with a mean velocity v0 ¼ /ve ðx0 ÞS thus interact at x > 0 with faster electrons and are accelerated, while ions at xo0 interact with slower electrons and are therefore decelerated. In particular, in the linear velocity regime of the electron cooling force, where Fjj is given by Fjj ð~ v ; xÞ ¼ ajj ðvjj /ve ðxÞSÞ w ith vjj being the longitudinal velocity component of ~ v of an individual ion and ajj denoting the
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∆v
3. Experimental set-up and calibrations
-vo ring inward
x = -x0
x0
ring outward
x ion beam
Fig. 2. Creating a horizontal gradient of the longitudinal electron cooling force. Due to the space charge of the electron beam the average electron velocity /ve S increases quadratically with the distance from the electron beam axis. Displacing the electron beam ring inward by a distance x0 ; ions with an average velocity v0 ¼ /ve ðx0 ÞS thus experience during their betatron oscillations around x ¼ 0 an accelerating longitudinal force for x > 0 and a decelerating force for xo0: 0 longitudinal friction coefficient, we find Fjj;x ¼ 2ajj ve;0 x0 =r2e : Beside the dispersive electron cooling process, the ions will be still subjected to the usual transverse electron cooling force Fx ð~ v Þ ¼ ax vx : The total horizontal cooling rate Lx ¼ t1 is x therefore given by
Lx ¼
1 dsx ¼ LD;x þ L0;x sx dt
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ð4Þ
The experiments were performed at the heavy ion storage ring TSR of the Max Planck Institute for Nuclear Physics at Heidelberg using a 12 C6þ beam of 73:3 MeV; corresponding to v0 =c ¼ 0:115; and an ion beam current of Iion ¼ 20 mA: The 1=e lifetime of the beam was about 1 h: The electron beam, which was guided in a magnetic field of Bcool ¼ 418 G; had a density of ne ¼ 8:0 106 cm3 and is homogeneous due to previous measurements [11] of the electron beam density distribution. The beam was produced using an expansion factor [10] of 9.6 which led to an electron beam radius of 1:5 cm and electron beam temperatures of kTjj E0:1 meV and kT> E13 meV: The effective interaction length between electron and ion beam is 1:2 m leading to Zc ¼ 0:022: According to Eq. (2), the change of the transverse cooling rate is proportional to the dispersion DS in the electron cooler section. In the standard operation mode the dispersion is below 0:3 m; too small to investigate dispersive electron cooling. The standard lattice of the TSR was, therefore, rotated for these experiments by 90 (see Fig. 3), leading to a calculated dispersion of DS ¼ 1:77 m in the cooler section. Because the super symmetry of the TSR lattice is approximately two, the dispersion in the cooler could be determined with
with L0;x ¼ Zc ax =ð2MÞ: Together with Eq. (2) and approximating ve;0 by v0 we finally obtain ajj DS x0 Lx ¼ L0;x 1 þ 2 : ð5Þ ax r2e
2.5 2.0
Ds (m)
For typical values of ne ¼ 107 cm3 and ve;0 =c ¼ 0:1; resulting in re ¼ 0:35 m; and for ajj =ax ¼ 2; DS ¼ 2 m; and x0 ¼ þ1 cm we obtain Lx E2L0;x ; i.e. the horizontal cooling rate can be increased by dispersive electron cooling, transferring heat from the horizontal to the longitudinal degree of freedom, by about a factor of 2. Note, however, that for a corresponding shift of the electron beam ring outwards, i.e. for x0 ¼ 1 cm; Lx will be approximately zero as the dispersive coupling now transfers heat from the longitudinal to the horizontal degree of freedom and is thus acting as a heat source.
3.0
1.5 1.0
electron cooler
BPM
0.5 0.0 0
10
20
30
40
50
60
s (m) Fig. 3. Calculated dispersion of the TSR lattice obtained by rotating the standard lattice by 90 : Due to the magnetic elements at the straight section of the electron cooler the dispersion at the location of the BPM is 11% lower than the dispersion at the position of the electron cooler.
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the aid of the beam profile monitor (BPM [12]) located just opposite of the electron cooler, thus, by changing the main dipole fields and measuring the change of the closed orbit, in the range of 730 mm; with the BPM. The dispersion at the position of the profile monitor was measured to be DM ¼ 1:6370:05 m: Lattice calculation (see Fig. 3), including the magnetic elements of the cooling section, turned out that the dispersion in the cooler section is 11% higher compared with the dispersion ðDM Þ at the location of the BPM; therefore, the dispersion at the position of the electron cooler was determined to be DS ¼ 1:81 m: The horizontal and vertical tunes of the ring are unaffected by the rotation; with Qx ¼ 2:78 and Qy ¼ 2:83 they are different enough to ensure that the two transverse degrees of freedom are decoupled. The magnetic field of the electron cooler [5], which guides the electron beam from the electron gun to the collector, is produced by five solenoidal coils and two toroids. Horizontal and vertical displacements of the electron beam in the cooler solenoid can be realized by two pairs of short dipole saddle coils, called S2X and S2Y, which are mounted inside the first solenoid between the electron gun and the first toroid. A control of the angle between the ion beam and electron beam is achieved by two pairs of long dipole saddle coils, called CPX and CPY, which span the whole ion– electron interaction region. In order to calibrate the electron beam displacement X ðY Þ caused by changing the currents through S2X (S2Y), the space charge parabola of the electron beam was used: Starting with a wellcooled and aligned ion beam of velocity v0 ¼ ve;0 ; the electron beam axis is displaced horizontally (vertically), which results in a higher average beam velocity v0 þ Dv after the new equilibrium has been achieved. This in turn leads to a change Df of the revolution frequency f0 of the beam, which is given by Df Dv ¼Z f0 v0
ð6Þ
with Z ¼ 0:89 for the standard mode of the TSR. This shift can be measured with the longitudinal Schottky pick-up of the TSR. However, the new
∆v
-ve,0
equilibrated ion beam
xD ring inward
X
-ve,0
dispersion line
ring outward
∆v
x dispersion line
xD
equilibrated ion beam
x Fig. 4. Change Dv of the mean longitudinal ion velocity of an aligned and cooled beam (open circle) due to a horizontal shift of the electron beam by a distance X > 0 ring inward (top) or due to a vertical shift by a distance Y (bottom). Due to the dispersion, the closed orbit of the ions is displaced in addition by an amount of xD ring outward.
equilibrium with an increased beam velocity also involves an additional horizontal displacement xD due to the dispersion (see Fig. 4 and Eq. (1). Note that we shall use the convention that X > 0 refers to displacing the electron beam ring inward with respect to the ion beam). Using Eqs. (1) and (3), the distance ~ r 0 ðX ; Y Þ in the transversal x; y planes between the electron and the ion beam after the equilibrium has been reached is calculated to be ~ r 0 ðX ; Y Þ ¼ ðx0 ðX ; Y Þ; Y Þ
ð7Þ
with x0 ðX ; Y Þ ¼ X þ xD ðX ; Y Þ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 X Y2 ¼ C @1 1 2 2 A C C
ð8Þ
and C ¼ r2e =ð2DS Þ: Here we have to assume that X oC=2 and jY joC as otherwise no equilibrium can be reached (C ¼ 5:2 cm for the experimental parameters given above). Together with Eq. (6), the resulting change of the revolution frequency due to a horizontal ðX Þ and/or vertical
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displacement ðY Þ is thus given by Df ðX ; Y Þ r20 ðX ; Y Þ r2e 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 C @ X X Y2 ¼ Zf0 1 1 2 2 A: DS C C C
¼ Zf0
ð9Þ
For X =C51 and Y =C51; we may expand the roots in Eqs. (8) and (9) and obtain 1 ðX 2 þ Y 2 Þ; 2C 1 X 2 2 þY : Df ðX ; Y ÞEZf0 2 X 1 þ re C
x0 ðX ; Y ÞEX þ
ð10Þ ð11Þ
Assuming the displacements X and Y to be a linear function of the coil currents IðS2X Þ and IðS2Y Þ; i.e. X ¼ aX IðS2X Þ þ bX and Y ¼ aY IðS2Y Þ þ bY ; the calibration parameters ai ; bi can be obtained by fitting the measured frequency shifts using Eq. (9). The measured change of the revolution frequency by a horizontal displacement X is shown in Fig. 5 together with the best fit using Eq. (9) for Y ¼ 0 (solid curve). The parabolic function (dashed curve) reflects the expected
Fig. 5. Measured change of the revolution frequency of a 73:3 MeV 12 C6þ beam as a function of the horizontal displacement X of an electron beam with ne ¼ 8 106 cm3 : A function according to Eq. (9) with Y ¼ 0 was fitted to the data (full line) to calibrate the X -scale. The dashed line indicates the expected frequency shifts if the additional shift due to the dispersion is neglected.
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frequency shifts if the additional displacement due to the dispersion would be neglected. The measured change of the revolution frequency as function of the displacement Y of the electron beam is shown in Fig. 6 together with the best fit using Eq. (9). As expected from Eq. (11) there is no significant difference from a pure parabolic behavior as the effect due to the additional dispersion is weak and moreover symmetric around Y ¼ 0: Determining cooling rates as a function of the displacement of the electron beam with respect to the ion beam, the collective rotation of the electron beam, caused by the space charge together with the longitudinal magnetic guiding field, has to be compensated carefully. Neglecting the centrifugal force, the drift frequency is given by od ¼ ene =ð2e0 Bcool Þ (od ¼ 1:73 106 s1 for the parameters given above). Starting again with a well-aligned ion and electron beam, a horizontal displacement of the electron beam by X with respect to the ion beam yields a constant vertical electron velocity vy ¼ od x0 ðX ; 0Þ as seen from the reference frame of the ions. This small drift velocity can be compensated for by readjusting the vertical angle of the electron beam with respect to the x–s plane by Fy ¼ vy =v0 ¼ od =v0 x0 ðX ; 0Þ;
Fig. 6. Measured change of the revolution frequency of a 73:3 MeV 12 C6þ beam as a function of the vertical displacement Y of an electron beam with ne ¼ 8 106 cm3 : A function according to Eq. (9) was fitted to the data (full line) to calibrate the Y -scale.
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which can be achieved by adjusting the current through the CPY coil. Likewise, for a vertical displacement by Y the x-component of the drift velocity ~ v d has to be compensated by readjusting the horizontal angle of the electron beam with respect to the y–s plane by approximately Fx Evx =v0 ¼ od =v0 Y using the current through the CPX coil; in addition, a small readjustment of Fy due to the dispersion is required as well. The optimal settings for Fx and Fy are readily found by minimizing the transversal beam widths using the BPM. Assuming again a linear dependence between the two angles and the currents IðCPXÞ and IðCPYÞ through the correction coils, i.e. Fx ¼ cx IðCPXÞ þ dx and Fy ¼ cy IðCPYÞ þ dy ; respectively, we can determine the coefficients ci ; di by fitting the expected angles to the observed values. As an example, the measured vertical angle Fy of the electron beam is displayed in Fig. 7 versus the horizontal shift x0 ðX ; 0Þ: The calibration factors ai and ci are in very good agreement with earlier calibration measurements performed during the commissioning of the electron cooler [5]; this indicates together with the good agreement between the measured and expected slopes of the calibration curves that the geometrical and electromagnetic properties of the TSR electron cooler are well under control.
4. Results 4.1. Longitudinal cooling force measurements Starting with well-aligned ion–electron beams with coinciding beam axes, a displacement of the electron beam by ðX ; Y Þ will result in a shift of the ion beam to the new stable transversal position given by ~ r 0 ðX ; Y Þ with respect to the new electron beam axis, and the mean longitudinal ion velocity will equilibrate at v0 ¼ /ve ðr0 ÞS: If we now consider an ion with a momentary laboratory velocity ~ v and a momentary excursion ~ r ¼ ðx; yÞ from its closed orbit due to its betatron oscillation, the ion will be subjected to a longitudinal cooling force Fjj ð~ v; ~ r ;~ r 0 Þ; which is given within the linear velocity regime of the electron cooling force by Fjj ð~ v; ~ r ;~ r 0 Þ ¼ ajj ðv0jj ½/ve ðr0 ÞS v0 Þ
ð12Þ
with v0jj ¼ vjj v0 and ~ r0 ¼ ðx0 ðX ; Y Þ þ DS v0jj =v0 þ x; Y þ yÞ (see Fig. 8). Retaining only terms linear in x; y and v0jj ; Eq. (12) reduces to ve;0 x0 0 v; ~ r ;~ r 0 Þ ¼ ajj 1 Fjj ð~ v v0 C jj ve;0 þ 2ajj 2 ð~ r0 ~ r Þ: ð13Þ re Defining the longitudinal friction coefficient in the usual way by the partial derivative of the
∆v dispersion line
−ve,0 v||´
equilibrium point
x0 Fig. 7. Vertical correction angle Fy of the electron beam as a function of the horizontal shift x0 ðX ; 0Þ of the electron beam, required to compensated the drift velocity of the electrons around the electron beam axis. A function according to Fy ¼ od =v0 x0 ðX ; 0Þ is drawn as solid line.
∆xD = DS v||´/ v0
x
Fig. 8. Calculating the longitudinal electron cooling force in a displaced electron beam for an ion of longitudinal velocity v0jj with respect to the stable point and an horizontal excursion x from its closed orbit due to its betatron oscillation.
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longitudinal cooling force with respect to v0jj ; we obtain an effective longitudinal friction coefficient ajj ; which depends on the shift ~ r 0 ðX ; Y Þ; @F jj r0Þ ¼ 0 ajj ð~ @vjj ve;0 x0 X ¼ ajj 1 ð14Þ Eajj 1 C v0 C the last approximation being valid in first order in r0 =C: Moreover, we obtain for the dispersive 0 coupling term Fjj;x (cf. Section 2) @Fjj ð~ v ;~ r0Þ ve;0 x0 0 ¼ 2ajj 2 : Fjj;x ¼ ð15Þ @x re As the second term in Eq. (13) is in average zero, v ;~ r 0 Þ for a the mean longitudinal cooling force Fjj ð~ displaced electron beam amounts to Fjj ð~ v ;~ r 0 Þ ¼ ajj ð~ r 0 Þ v0jj ð16Þ and was measured as a function of the displacement ðX ; Y Þ using the induction accelerator (IndAcc) [13]: For each ðX ; Y Þ setting the ion beam was first cooled to the new equilibrium point ~ r0 and v0 and aligned by adjusting the correction angles Fx and Fy : The electron cooling force was then counteracted by the induction force Find given by Qe Uind Find ¼ ð17Þ L where Qe denotes the charge of the ions, Uind the adjustable accelerating voltage and L the circumference of the TSR ðL ¼ 55:4 mÞ: This results in a new equilibrium velocity vjj of the ion beam, which obeys F ¼ Z a ðv v Þ: ð18Þ ind
c
jj
jj
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affect the longitudinal cooling force in first order r 0 Þ ¼ ajj : The longitudinal friction in Y =C; i.e. ajj ð~ coefficients measured for X ¼ 0 as a function of the vertical displacement Y are plotted in Fig. 9; the average value amounts to ajj ¼ ð6:670:1Þ 104 eV s=m2 : A measurement of the longitudinal friction coefficient for a horizontal displacement ðX ; 0Þ is shown in Fig. 10 as a function of x0 ðX ; 0Þ E X ð1 þ X =ð2CÞÞ: The effective coefficient increases as expected for x0 o0 and decreases for x0 > 0; respectively. The solid line in Fig. 10 is calculated using Eq. (14) with ajj ¼ 6:6 104 eV s=m2 : The agreement between theory and experiment is perfect. 4.2. Transverse electron cooling rates Transverse electron cooling as a function of the displacement of the electron beam was studied by measuring the time development of the ion beam profiles. The time evolution of the horizontal (vertical) beam variance si is given by 1 dsi ¼ Li þ lIBS;i si dt
ð19Þ
0
The velocities vjj and v0 can be readily measured via a Schottky noise analysis of the stored ion beam for Uind a0 and 0, respectively. To stay within the linear velocity regime of the electron cooling force, jvjj v0 j has to be smaller than the longitudinal velocity spread of the electrons, i.e. ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smaller than kTjj =me ; which corresponds to maximum allowed change of the revolution frequency of about 80 Hz: More details of the IndAcc method can be found in Ref. [13]. As expected from Eqs. (14) and (16), displacing the electron beam in the vertical direction does not
Fig. 9. The effective longitudinal friction coefficient ajj for a 73:3 MeV 12 C6þ beam, measured with the IndAcc method as a function of the vertical displacement Y of the electron beam with density ne ¼ 8 106 cm3 : The average value of ajj ¼ ajj ¼ 6:6 104 eV s=m2 is shown by the solid line.
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Fig. 10. The effective longitudinal friction coefficient ajj for a 73:3 MeV 12 C6þ beam, measured with the IndAcc method as a function of the horizontal shift x0 ðX ; 0Þ of the electron beam with density ne ¼ 8 106 cm3 : The solid line was calculated with Eq. (14) using the average value ajj ¼ 6:6 104 eV s=m2 as determined from the measurement shown in Fig. 9.
with i ¼ xðyÞ; where Li denotes the total horizontal (vertical) cooling rate and lIBS;i the corresponding heating rate due to IBS. In the absence of dispersive electron cooling the cooling rates are determined by the usual transversal electron cooling force; staying within the linear velocity regime of the electron cooling force, the cooling rates are connected to the transversal friction coefficients ai via L0;i ¼ Zc ai =ð2MÞ:
ð20Þ
Switching on the dispersive electron cooling will not effect the vertical cooling rate as long as the two transversal directions are not coupled, i.e. Ly ¼ L0;y ; but the horizontal rate will be modified. Using Eq. (2) together with Eqs. (15) and (4) we obtain Lx ¼ LD;x þ L0;x with LD;x ¼ Zc
ajj ve;0 2DS ajj X x0 EZc 2M v0 r2e 2M C
linear regime of the transverse cooling force. For the present experiments this requires si p1 mm: In order to determine the heating rate lIBS ; which has to be known to analyze cooling rate measurements via Eq. (19), IBS experiments were carried out. An example for an IBS measurement with 12 C6þ ions of 73:3 MeV and an ion beam current of 20 mA is plotted in Fig. 11. The standard deviations sx of a Gaussian fit to the horizontal beam profiles measured with the BPM are shown as functions of time, correcting for the resolution of the horizontal BPM of E300 mm (E200 mm for the vertical BPM). While the initial beam width s0;x is given by the equilibrium between electron cooling and IBS, the ion beam blow up due to the IBS is being recorded after the electron cooling is switched off at t0 ¼ 1 s: The observed time dependence sx ðtÞ was fitted by a solution of Eq. (19) assuming a heating rate lIBS expected from a simplified model of IBS. In this model, the heating rate is set to lIBS;i ¼ Di N=sgi ; where Di is a constant, and N the number of stored ions in the ring, while the exponent g is expected to have a value close to 5. This reflects the predicted behavior of lIBS pN=ðex ey Dpjj =p0 Þ for the IBS rate [14], which leads for the reasonable assumption, that the horizontal emittance ex ; the vertical emittance ey and the square of the momentum spread Dpjj =p0
ECOOL off
ð21Þ
the latter approximation being correct in first order in r0 =C: Note, that Eq. (20) together with Eq. (19) is only valid if the initial ion beam radius is small enough so that the transverse velocity components of the ions are indeed within the
Fig. 11. Measured horizontal beam widths sx (corrected for the resolution of the BPM) as a function of time for a 20 mA 12 C6þ beam of 73:3 MeV; showing the beam blow up after the electron cooler is switched off at t0 ¼ 1 s (arrow). Eq. (22) was fitted to the data (full line) and resulted in a g value of gx ¼ 4:9770:07:
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are proportional to each other, to ex ey Dpjj =p0 ps5i : Applying this Ansatz, the solution of Eq. (19) with Li ¼ 0 is given by si ðtÞ ¼ ½gDi Nðt t0 Þ þ sg0;i 1=g :
ð22Þ
IBS measurements performed at different ion currents were found to be well described by Eq. (22) and led to an average value of g ¼ 4:8: A fit of Eq. (22) to the data shown in Fig. 11 is shown by the solid line and resulted in gx ¼ 4:9770:07: A fit to the corresponding vertical beam profile resulted in gy ¼ 4:7670:06: To investigate the transverse cooling rates within the linear regime of the electron cooling force as a function of the electron beam displacement, the cooling of ion beams with small betatron amplitudes is studied. For a given displacement ðX ; Y Þ and after adjusting the correction angles Fi ; the measurements are performed in the following way: After injection the ion beam is cooled down to equilibrium, then the electron cooling is switched off to allow the beam to blow up due to IBS. After having reached a beam variance si of approximately 1 mm; the electron cooling is switched on again and the time development of the beam profiles is recorded with the horizontal and vertical BPM. An example of such a cooling time measurement is shown in Fig. 12.
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For Li a0 Eq. (19) is solved by si ðtÞ ¼ ½sga;i expðgðt t0 ÞLi Þ þ sg0;i 1=g
ð23Þ
where sa;i denotes the variance of the ion beam when the cooling starts. The total cooling rate Li can then be deduced by fitting Eq. (23) to the observed time dependence of si ðtÞ using sa;i ; s0;i and Li as fit parameters, setting gi to the value determined in the IBS measurements described above. For the measurement of sy ðtÞ shown in Fig. 12 which was performed at ðX ; Y Þ ¼ ð0; 0Þ; we obtained, e.g. Ly ¼ 24:471:3 s1 ; and from the corresponding sx ðtÞ curve the value Lx ¼ 24:171:3 s1 was extracted. Both values agree very well. The cooling rates are expected to be unaffected by a vertical displacement of the electron beam and to be equal for the horizontal and vertical degree of freedom. This is born out in Fig. 13, where we have plotted the measured cooling rates Lx and Ly as a function of the vertical displacement Y ; keeping X constant at X ¼ 0: The cooling rates are indeed found to be independent of Y and to be equal within their individual errors, although it seems that the vertical cooling rates are systematically higher than the horizontal one by about a few %. Averaging over all measured displacements
ECOOL on
Fig. 12. Time dependence of the vertical beam variance sy ; corrected for the resolution of the BPM, of a 73:3 MeV 12 C6þ beam of 20 mA after switching on the electron cooler (arrow). The electron beam was unshifted, i.e. ðX ; Y Þ ¼ ð0; 0Þ; and had a density of ne ¼ 8:0 106 cm3 : The solid line is a fit using Eq. (23), which resulted in Ly ¼ 24:471:3 s1 :
Fig. 13. Transverse cooling rates for a 73:3 MeV 12 C6þ beam as function of the vertical displacement Y of the electron beam with a density of ne ¼ 8 106 cm3 : The average values Lx ¼ 23:071:2 s1 and Ly ¼ 25:171:2 s1 are shown by the dashed and dotted line, respectively. The solid line reflects the overall average of Li ¼ 24:171:2 s1 :
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Fig. 14. Transverse cooling rates measured for a 73:3 MeV 12 C6þ beam as a function of the horizontal shift x0 ðX ; 0Þ of an electron beam with density ne ¼ 8 106 cm3 : The dashed line represents an average value of L0;y ¼ 23:571:8 s1 : The full line is Lx ¼ 22 s1 þ LD;x with LD;x according to Eq. (21).
we obtain Lx ¼ L0;x ¼ 23:071:2 s1 and Ly ¼ L0;y ¼ 25:171:2 s1 : The transverse cooling rates observed when displacing the electron beam horizontally are shown in Fig. 14 as a function of x0 ðX ; 0Þ E X ð1 þ X =ð2CÞÞ: While the vertical cooling rates are again independent of the displacement of the electron beam within the experimental accuracy (the average value of Ly ¼ 23:571:8 s1 being in good agreement with the value extracted from Fig. 13), the horizontal cooling rates clearly show the dependence on x0 expected when dispersive electron cooling is active: The cooling rate Lx increases if the electron beam is moved ring inward ðx0 > 0Þ and decreases if it is moved ring outward ðx0 o0Þ; i.e. the dispersive coupling leads to an additional cooling or heating of the beam in the horizontal direction. Using Eq. (21) together with L0;x ¼ 22 s1 we obtain the solid line in Fig. 14 which describes the observed slope of the horizontal cooling rates very well.
5. Summary and conclusion Dispersive electron cooling could be realized by creating the required horizontal gradient of the longitudinal cooling force by a displacement of the
center of the electron beam from the ion beam axes. In this way, one can redistribute the cooling force between the longitudinal and horizontal degree of freedom. Depending on the direction of the displacement, i.e. if the electron beam is shifted ring inward (or ring outward) with respect to the coasting ion beam, the horizontal cooling force is increased (decreased), while at the same time the longitudinal force is decreased (increased). In fact, defining an effective horizontal friction coefficient ax by ax ¼ 2M=Zc ðL0;x þ LD;x Þ ¼ ax þ ajj ðve;0 =v0 Þ ð2DS =r2e Þ x0 we see that the increase of the horizontal friction coefficient ax by dispersive coupling is equal but opposite in sign to the change of the longitudinal coefficient (cf. Eq. (14)). The second pre-requisite for realizing dispersive cooling is a finite dispersion in the electron–ion interaction region. Usually, cooler storage rings are operated in a mode leading to an as-small-aspossible dispersion in the electron cooler in order to ensure that also very hot ion beams can be cooled. In the present experiment, a dispersion of 1:81 m was accomplished in the interaction region by just rotating the standard lattice of the TSR by 90 : No deterioration of the cooling properties of the TSR were encountered, on the contrary, aligning the electron to the ion beam axis the longitudinal and transversal friction coefficients were found to be about 25% larger than the corresponding values achieved with the standard lattice [15,16]. This improvement is thought to be due to the different beam optics in the cooler section. For a given dispersion DS and electron density ne ; the change of the longitudinal and horizontal friction coefficient due to dispersive coupling is uniquely determined by the displacement x0 between electron and ion beam. As depicted in Figs. 9 and 14, the expected dependencies on x0 are very well born out by the data. Several factors such as a lower electron density and a lower dispersion add up to yield a somewhat smaller dispersive effect than in the rough estimate following Eq. (5); for the parameters of the present experiment a change of the effective horizontal friction coefficient of about 733% is observed for a displacement of x0 ¼ 71 cm between the electron and ion beam axes, the corresponding
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changes of the longitudinal friction coefficient being 817%: A coupling between the horizontal and vertical degree of freedom was avoided in the present measurements by choosing the two transversal tunes Qx and Qy of the ring to be sufficiently different. Of course, by adjusting the tunes to be equal, a strong coupling between the two transversal directions can be accomplished such that also the vertical degree of freedom would gain from dispersive electron cooling; the effectiveness of this cooling was already demonstrated in the dispersive laser cooling experiments of Lauer et al. [9]. Note, moreover, that the electron density can be increased by a factor of up to 5 and that also the dispersion in the interaction region can be further increased by at least a factor of 2. It thus seems to be possible to almost freely distribute the electron cooling power between the longitudinal and transversal degrees of freedom by dispersive coupling.
References [1] S. van der Meer, Internal Report, CERN/ISR/PO/72-31, 1972. [2] F. Nolden, K. Beckert, F. Caspers, B. Franczak, B. Franzke, R. Menges, A. Schwinn, M. Steck, Nucl. Instr. and Meth. A 441 (2000) 219.
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[3] G.I. Budker, At. Energ. 22 (1967) 346. [4] G.I. Budker, N.S. Dikansky, V.I. Kudelainen, I.N. Meshkov, V.V. Parchomchuk, D.V. Pestrikov, A.N. Skrinsky, B.N. Sukhina, Part. Accel. 7 (1976) 197. [5] M. Steck, G. Bisoffi, M. Blum, A. Friedrich, C. Geyer, M. Grieser, B. Holzer, E. Jaeschke, M. Jung, D. Kr.amer, K. Matl, W. Ott, R. Repnow, Nucl. Instr. and Meth. A 287 (1990) 324. [6] H. Poth, Phys. Rep. 196 (1990) 135. . [7] S. Schroder et al., Phys. Rev. Lett. 64 (1990) 2901. [8] W. Petrich, M. Grieser, R. Grimm, A. Gruber, D. Habs, H.-J. Miesner, D. Schwalm, B. Wanner, H. Wern^e, A. Wolf, R. Grieser, G. Huber, R. Klein, T. Kuhl, . R. . Neumann, S. Schroder, Phys. Rev. A 48 (1993) 2127. [9] I. Lauer, U. Eisenbarth, M. Grieser, R. Grimm, P. Lenisa, V. Luger, T. Sch.atz, U. Schramm, D. Schwalm, M. Weidem.uller, Phys. Rev. Lett. 81 (1998) 2052. [10] S. Pastuszka, U. Schramm, M. Grieser, C. Broude, R. Grimm, D. Habs, J. Kenntner, H.-J. Miesner, T. SchuXler, . D. Schwalm, A. Wolf, Nucl. Instr. and Meth. A 369 (1996) 11. [11] M. Schmitt, Dissertation, Universit.at Heidelberg, 1999. [12] B. Hochadel, F. Albrecht, M. Grieser, D. Habs, D. Schwalm, E. Szmola, A. Wolf, Nucl. Instr. and Meth. A 343 (1994) 401. [13] C. Ellert, D. Habs, E. Jaeschke, T. Kamara, M. Music, D. Schwalm, P. Sigray, A. Wolf, Nucl. Instr. and Meth. A 314 (1992) 399. [14] M. Martini, Internal Report, CERN PS/84-9(AA), 1984. [15] M. Beutelspacher, M. Grieser, D. Schwalm, A. Wolf, Nucl. Instr. and Meth. A 441 (2000) 110. [16] M. Beutelspacher, Dissertation, Universit.at Heidelberg, 2000.
Nuclear Instruments and Methods in Physics Research A 512 (2003) 459–469
Dispersive electron cooling experiments at the heavy ion storage ring TSR M. Beutelspachera, M. Griesera,*, K. Nodab, D. Schwalma, A. Wolfa a
Max-Planck-Institut fur . Kernphysik, Postfach 103980, Heidelberg 69029, Germany b National Institute of Radiological Sciences, Chiba 263-8555, Japan
Received 28 April 2003; received in revised form 14 June 2003; accepted 16 June 2003
Abstract At low relative velocities, the transverse cooling rates of fast stored ions by cold electrons are usually smaller than the longitudinal ones. By dispersive electron cooling, however, it is possible to transfer heat from the horizontal into the longitudinal degree of freedom, thereby increasing the horizontal cooling rate on the expense of the longitudinal rate. The cooling scheme requires a horizontal gradient of the longitudinal cooling force which can be achieved by displacing the electron beam relative to the ion beam, and a finite dispersion in the ion–electron interaction region. Operating the heavy ion test storage ring TSR at a moderate dispersion of DS ¼ 1:81 m; we investigated the dispersive electron cooling process and found good agreement with the expectations from fundamental considerations. r 2003 Elsevier B.V. All rights reserved. PACS: 32.80.P; 33.80.P Keywords: Electron cooling; Heavy ion; Storage ring
1. Introduction Phase space cooling of fast stored ions has become an important technique to produce highquality ion beams. Three methods have been developed over the last years to cool heavy ion beams: (i) stochastic cooling [1,2], where the beam temperature is reduced by an active feedback loop, (ii) electron cooling [3–6], where ions are repeatedly bathed in a cold electron beam, and (iii) laser cooling [7,8], where the light pressure force caused *Corresponding author. E-mail address: [email protected] (M. Grieser).
by resonant scattering of photons from laser fields is used to increase the phase space density of the ions. While the stochastic cooling process is transversally as effective as longitudinally, the other two techniques turn out to be less efficient in cooling the transversal than the longitudinal degree of freedom. This mismatch was particularly large in the case of laser cooling as the standard method only allowed to cool the longitudinal phase space directly while the transversal degree of freedom was only affected indirectly via the coupling caused by intra-beam scattering (IBS); the mismatch could be reduced at least partially by using a new cooling scheme which exploits the longitudinal–horizontal coupling of the ion
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-9002(03)01976-4
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motion due to the storage ring dispersion in combination with a transversal gradient of the light pressure force [9]. Although the differences in the strength of the longitudinal and transverse forces are less severe in the case of electron cooling [6], it is interesting to transfer the dispersive coupling scheme also to electron cooling and to study its potentialities for equilibrating the longitudinal and transversal electron cooling rates. We have, therefore, performed an experiment to investigate the feasibility of dispersive electron cooling at the heavy ion storage ring TSR of the Max Planck Institute for Nuclear Physics in Heidelberg.
betatron oscillation. Likewise, the horizontal oscillation of the ion is reduced if a negative momentum change is applied at xo0: Assuming a horizontal gradient of the longitudinal cooling v ; xÞ around x ¼ 0; namely Fjj ð~ v ; xÞ ¼ force Fjj ð~ 0 0 Fjj ð~ v ; 0Þ þ Fjj;x ð~ v ; 0Þ x; with Fjj;x ¼ @Fjj =@x and ~ v being the laboratory velocity of the ion, the net effect of this process on the ion, passing repeatedly through the interaction region, will be a complete damping of the betatron oscillation; moreover, as the individual processes lead only to a small shift of the closed orbit, the displacements will be averaged out. For a beam of ions, this results in a decrease of the horizontal beam width s with a cooling rate LD;x ¼ t1 D;x given by [9]
2. Principle of dispersive electron cooling
LD;x ¼
The principle of dispersive electron cooling is depicted in Fig. 1. A change in the longitudinal momentum of an ion by Dpjj leads to a horizontal displacement Dx of the closed orbit due to dispersion. With the x-axis pointing ring outward, the displacement is given by
Here p0 ¼ M v0 now denotes the average longitudinal momentum of the ion beam in the laboratory frame and Zc the fraction of the interaction length to the circumference of the storage ring. A horizontal gradient of the longitudinal electron cooling force can be achieved by a displacement of the electron beam with respect to the ion beam exploiting the parabolic velocity profile of the electrons caused by its space charge. The mean electron velocity as a function of the distance r from the electron beam axis is given by
Dx ¼ xD ¼ DS
Dpjj p0
ð1Þ
where p0 is the longitudinal momentum of the ion in the laboratory frame, and DS denotes the dispersion. Thus, a positive momentum change Dpjj ; transferred to the ion at x > 0; reduces the amplitude and, therefore, damps the horizontal x ring
∆ p ||
outward ∆x ring inward
s
Fig. 1. The principle of dispersive cooling. A positive change of the longitudinal momentum transferred ring outward ðx > 0Þ leads to a damping of the betatron oscillation due to the shift of the closed orbit. Likewise, the betatron oscillation is reduced if a negative momentum transfer occurs at xo0:
1 ds DS ¼ Z F0 : s dt 2p0 c jj;x
/ve ðrÞS ¼ ve;0 ð1 þ ðr=re Þ2 Þ
ð2Þ
ð3Þ
where ve;0 is the on-axis velocity of the beam in the laboratory frame, and re is connected with the electron density ne by r2e ¼ 4e0 me v2e;0 =ðe2 ne Þ with me denoting the electron mass and e0 the vacuum permittivity. For a horizontal displacement of the electron beam ring inward by x0 with respect to the ion beam, i.e. to x ¼ x0 (see Fig. 2), ions with a mean velocity v0 ¼ /ve ðx0 ÞS thus interact at x > 0 with faster electrons and are accelerated, while ions at xo0 interact with slower electrons and are therefore decelerated. In particular, in the linear velocity regime of the electron cooling force, where Fjj is given by Fjj ð~ v ; xÞ ¼ ajj ðvjj /ve ðxÞSÞ w ith vjj being the longitudinal velocity component of ~ v of an individual ion and ajj denoting the
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∆v
3. Experimental set-up and calibrations
x = -x0
x0
ring outward
x ion beam
Fig. 2. Creating a horizontal gradient of the longitudinal electron cooling force. Due to the space charge of the electron beam the average electron velocity /ve S increases quadratically with the distance from the electron beam axis. Displacing the electron beam ring inward by a distance x0 ; ions with an average velocity v0 ¼ /ve ðx0 ÞS thus experience during their betatron oscillations around x ¼ 0 an accelerating longitudinal force for x > 0 and a decelerating force for xo0: 0 longitudinal friction coefficient, we find Fjj;x ¼ 2ajj ve;0 x0 =r2e : Beside the dispersive electron cooling process, the ions will be still subjected to the usual transverse electron cooling force Fx ð~ v Þ ¼ ax vx : The total horizontal cooling rate Lx ¼ t1 is x therefore given by
Lx ¼
1 dsx ¼ LD;x þ L0;x sx dt
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ð4Þ
The experiments were performed at the heavy ion storage ring TSR of the Max Planck Institute for Nuclear Physics at Heidelberg using a 12 C6þ beam of 73:3 MeV; corresponding to v0 =c ¼ 0:115; and an ion beam current of Iion ¼ 20 mA: The 1=e lifetime of the beam was about 1 h: The electron beam, which was guided in a magnetic field of Bcool ¼ 418 G; had a density of ne ¼ 8:0 106 cm3 and is homogeneous due to previous measurements [11] of the electron beam density distribution. The beam was produced using an expansion factor [10] of 9.6 which led to an electron beam radius of 1:5 cm and electron beam temperatures of kTjj E0:1 meV and kT> E13 meV: The effective interaction length between electron and ion beam is 1:2 m leading to Zc ¼ 0:022: According to Eq. (2), the change of the transverse cooling rate is proportional to the dispersion DS in the electron cooler section. In the standard operation mode the dispersion is below 0:3 m; too small to investigate dispersive electron cooling. The standard lattice of the TSR was, therefore, rotated for these experiments by 90 (see Fig. 3), leading to a calculated dispersion of DS ¼ 1:77 m in the cooler section. Because the super symmetry of the TSR lattice is approximately two, the dispersion in the cooler could be determined with
with L0;x ¼ Zc ax =ð2MÞ: Together with Eq. (2) and approximating ve;0 by v0 we finally obtain ajj DS x0 Lx ¼ L0;x 1 þ 2 : ð5Þ ax r2e
2.5 2.0
Ds (m)
For typical values of ne ¼ 107 cm3 and ve;0 =c ¼ 0:1; resulting in re ¼ 0:35 m; and for ajj =ax ¼ 2; DS ¼ 2 m; and x0 ¼ þ1 cm we obtain Lx E2L0;x ; i.e. the horizontal cooling rate can be increased by dispersive electron cooling, transferring heat from the horizontal to the longitudinal degree of freedom, by about a factor of 2. Note, however, that for a corresponding shift of the electron beam ring outwards, i.e. for x0 ¼ 1 cm; Lx will be approximately zero as the dispersive coupling now transfers heat from the longitudinal to the horizontal degree of freedom and is thus acting as a heat source.
3.0
1.5 1.0
electron cooler
BPM
0.5 0.0 0
10
20
30
40
50
60
s (m) Fig. 3. Calculated dispersion of the TSR lattice obtained by rotating the standard lattice by 90 : Due to the magnetic elements at the straight section of the electron cooler the dispersion at the location of the BPM is 11% lower than the dispersion at the position of the electron cooler.
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the aid of the beam profile monitor (BPM [12]) located just opposite of the electron cooler, thus, by changing the main dipole fields and measuring the change of the closed orbit, in the range of 730 mm; with the BPM. The dispersion at the position of the profile monitor was measured to be DM ¼ 1:6370:05 m: Lattice calculation (see Fig. 3), including the magnetic elements of the cooling section, turned out that the dispersion in the cooler section is 11% higher compared with the dispersion ðDM Þ at the location of the BPM; therefore, the dispersion at the position of the electron cooler was determined to be DS ¼ 1:81 m: The horizontal and vertical tunes of the ring are unaffected by the rotation; with Qx ¼ 2:78 and Qy ¼ 2:83 they are different enough to ensure that the two transverse degrees of freedom are decoupled. The magnetic field of the electron cooler [5], which guides the electron beam from the electron gun to the collector, is produced by five solenoidal coils and two toroids. Horizontal and vertical displacements of the electron beam in the cooler solenoid can be realized by two pairs of short dipole saddle coils, called S2X and S2Y, which are mounted inside the first solenoid between the electron gun and the first toroid. A control of the angle between the ion beam and electron beam is achieved by two pairs of long dipole saddle coils, called CPX and CPY, which span the whole ion– electron interaction region. In order to calibrate the electron beam displacement X ðY Þ caused by changing the currents through S2X (S2Y), the space charge parabola of the electron beam was used: Starting with a wellcooled and aligned ion beam of velocity v0 ¼ ve;0 ; the electron beam axis is displaced horizontally (vertically), which results in a higher average beam velocity v0 þ Dv after the new equilibrium has been achieved. This in turn leads to a change Df of the revolution frequency f0 of the beam, which is given by Df Dv ¼Z f0 v0
ð6Þ
with Z ¼ 0:89 for the standard mode of the TSR. This shift can be measured with the longitudinal Schottky pick-up of the TSR. However, the new
∆v
equilibrated ion beam
xD ring inward
X
dispersion line
ring outward
∆v
x dispersion line
xD
equilibrated ion beam
x Fig. 4. Change Dv of the mean longitudinal ion velocity of an aligned and cooled beam (open circle) due to a horizontal shift of the electron beam by a distance X > 0 ring inward (top) or due to a vertical shift by a distance Y (bottom). Due to the dispersion, the closed orbit of the ions is displaced in addition by an amount of xD ring outward.
equilibrium with an increased beam velocity also involves an additional horizontal displacement xD due to the dispersion (see Fig. 4 and Eq. (1). Note that we shall use the convention that X > 0 refers to displacing the electron beam ring inward with respect to the ion beam). Using Eqs. (1) and (3), the distance ~ r 0 ðX ; Y Þ in the transversal x; y planes between the electron and the ion beam after the equilibrium has been reached is calculated to be ~ r 0 ðX ; Y Þ ¼ ðx0 ðX ; Y Þ; Y Þ
ð7Þ
with x0 ðX ; Y Þ ¼ X þ xD ðX ; Y Þ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 X Y2 ¼ C @1 1 2 2 A C C
ð8Þ
and C ¼ r2e =ð2DS Þ: Here we have to assume that X oC=2 and jY joC as otherwise no equilibrium can be reached (C ¼ 5:2 cm for the experimental parameters given above). Together with Eq. (6), the resulting change of the revolution frequency due to a horizontal ðX Þ and/or vertical
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displacement ðY Þ is thus given by Df ðX ; Y Þ r20 ðX ; Y Þ r2e 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 C @ X X Y2 ¼ Zf0 1 1 2 2 A: DS C C C
¼ Zf0
ð9Þ
For X =C51 and Y =C51; we may expand the roots in Eqs. (8) and (9) and obtain 1 ðX 2 þ Y 2 Þ; 2C 1 X 2 2 þY : Df ðX ; Y ÞEZf0 2 X 1 þ re C
x0 ðX ; Y ÞEX þ
ð10Þ ð11Þ
Assuming the displacements X and Y to be a linear function of the coil currents IðS2X Þ and IðS2Y Þ; i.e. X ¼ aX IðS2X Þ þ bX and Y ¼ aY IðS2Y Þ þ bY ; the calibration parameters ai ; bi can be obtained by fitting the measured frequency shifts using Eq. (9). The measured change of the revolution frequency by a horizontal displacement X is shown in Fig. 5 together with the best fit using Eq. (9) for Y ¼ 0 (solid curve). The parabolic function (dashed curve) reflects the expected
Fig. 5. Measured change of the revolution frequency of a 73:3 MeV 12 C6þ beam as a function of the horizontal displacement X of an electron beam with ne ¼ 8 106 cm3 : A function according to Eq. (9) with Y ¼ 0 was fitted to the data (full line) to calibrate the X -scale. The dashed line indicates the expected frequency shifts if the additional shift due to the dispersion is neglected.
463
frequency shifts if the additional displacement due to the dispersion would be neglected. The measured change of the revolution frequency as function of the displacement Y of the electron beam is shown in Fig. 6 together with the best fit using Eq. (9). As expected from Eq. (11) there is no significant difference from a pure parabolic behavior as the effect due to the additional dispersion is weak and moreover symmetric around Y ¼ 0: Determining cooling rates as a function of the displacement of the electron beam with respect to the ion beam, the collective rotation of the electron beam, caused by the space charge together with the longitudinal magnetic guiding field, has to be compensated carefully. Neglecting the centrifugal force, the drift frequency is given by od ¼ ene =ð2e0 Bcool Þ (od ¼ 1:73 106 s1 for the parameters given above). Starting again with a well-aligned ion and electron beam, a horizontal displacement of the electron beam by X with respect to the ion beam yields a constant vertical electron velocity vy ¼ od x0 ðX ; 0Þ as seen from the reference frame of the ions. This small drift velocity can be compensated for by readjusting the vertical angle of the electron beam with respect to the x–s plane by Fy ¼ vy =v0 ¼ od =v0 x0 ðX ; 0Þ;
Fig. 6. Measured change of the revolution frequency of a 73:3 MeV 12 C6þ beam as a function of the vertical displacement Y of an electron beam with ne ¼ 8 106 cm3 : A function according to Eq. (9) was fitted to the data (full line) to calibrate the Y -scale.
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which can be achieved by adjusting the current through the CPY coil. Likewise, for a vertical displacement by Y the x-component of the drift velocity ~ v d has to be compensated by readjusting the horizontal angle of the electron beam with respect to the y–s plane by approximately Fx Evx =v0 ¼ od =v0 Y using the current through the CPX coil; in addition, a small readjustment of Fy due to the dispersion is required as well. The optimal settings for Fx and Fy are readily found by minimizing the transversal beam widths using the BPM. Assuming again a linear dependence between the two angles and the currents IðCPXÞ and IðCPYÞ through the correction coils, i.e. Fx ¼ cx IðCPXÞ þ dx and Fy ¼ cy IðCPYÞ þ dy ; respectively, we can determine the coefficients ci ; di by fitting the expected angles to the observed values. As an example, the measured vertical angle Fy of the electron beam is displayed in Fig. 7 versus the horizontal shift x0 ðX ; 0Þ: The calibration factors ai and ci are in very good agreement with earlier calibration measurements performed during the commissioning of the electron cooler [5]; this indicates together with the good agreement between the measured and expected slopes of the calibration curves that the geometrical and electromagnetic properties of the TSR electron cooler are well under control.
4. Results 4.1. Longitudinal cooling force measurements Starting with well-aligned ion–electron beams with coinciding beam axes, a displacement of the electron beam by ðX ; Y Þ will result in a shift of the ion beam to the new stable transversal position given by ~ r 0 ðX ; Y Þ with respect to the new electron beam axis, and the mean longitudinal ion velocity will equilibrate at v0 ¼ /ve ðr0 ÞS: If we now consider an ion with a momentary laboratory velocity ~ v and a momentary excursion ~ r ¼ ðx; yÞ from its closed orbit due to its betatron oscillation, the ion will be subjected to a longitudinal cooling force Fjj ð~ v; ~ r ;~ r 0 Þ; which is given within the linear velocity regime of the electron cooling force by Fjj ð~ v; ~ r ;~ r 0 Þ ¼ ajj ðv0jj ½/ve ðr0 ÞS v0 Þ
ð12Þ
with v0jj ¼ vjj v0 and ~ r0 ¼ ðx0 ðX ; Y Þ þ DS v0jj =v0 þ x; Y þ yÞ (see Fig. 8). Retaining only terms linear in x; y and v0jj ; Eq. (12) reduces to ve;0 x0 0 v; ~ r ;~ r 0 Þ ¼ ajj 1 Fjj ð~ v v0 C jj ve;0 þ 2ajj 2 ð~ r0 ~ r Þ: ð13Þ re Defining the longitudinal friction coefficient in the usual way by the partial derivative of the
∆v dispersion line
equilibrium point
x0 Fig. 7. Vertical correction angle Fy of the electron beam as a function of the horizontal shift x0 ðX ; 0Þ of the electron beam, required to compensated the drift velocity of the electrons around the electron beam axis. A function according to Fy ¼ od =v0 x0 ðX ; 0Þ is drawn as solid line.
∆xD = DS v||´/ v0
x
Fig. 8. Calculating the longitudinal electron cooling force in a displaced electron beam for an ion of longitudinal velocity v0jj with respect to the stable point and an horizontal excursion x from its closed orbit due to its betatron oscillation.
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longitudinal cooling force with respect to v0jj ; we obtain an effective longitudinal friction coefficient ajj ; which depends on the shift ~ r 0 ðX ; Y Þ; @F jj r0Þ ¼ 0 ajj ð~ @vjj ve;0 x0 X ¼ ajj 1 ð14Þ Eajj 1 C v0 C the last approximation being valid in first order in r0 =C: Moreover, we obtain for the dispersive 0 coupling term Fjj;x (cf. Section 2) @Fjj ð~ v ;~ r0Þ ve;0 x0 0 ¼ 2ajj 2 : Fjj;x ¼ ð15Þ @x re As the second term in Eq. (13) is in average zero, v ;~ r 0 Þ for a the mean longitudinal cooling force Fjj ð~ displaced electron beam amounts to Fjj ð~ v ;~ r 0 Þ ¼ ajj ð~ r 0 Þ v0jj ð16Þ and was measured as a function of the displacement ðX ; Y Þ using the induction accelerator (IndAcc) [13]: For each ðX ; Y Þ setting the ion beam was first cooled to the new equilibrium point ~ r0 and v0 and aligned by adjusting the correction angles Fx and Fy : The electron cooling force was then counteracted by the induction force Find given by Qe Uind Find ¼ ð17Þ L where Qe denotes the charge of the ions, Uind the adjustable accelerating voltage and L the circumference of the TSR ðL ¼ 55:4 mÞ: This results in a new equilibrium velocity vjj of the ion beam, which obeys F ¼ Z a ðv v Þ: ð18Þ ind
c
jj
jj
465
affect the longitudinal cooling force in first order r 0 Þ ¼ ajj : The longitudinal friction in Y =C; i.e. ajj ð~ coefficients measured for X ¼ 0 as a function of the vertical displacement Y are plotted in Fig. 9; the average value amounts to ajj ¼ ð6:670:1Þ 104 eV s=m2 : A measurement of the longitudinal friction coefficient for a horizontal displacement ðX ; 0Þ is shown in Fig. 10 as a function of x0 ðX ; 0Þ E X ð1 þ X =ð2CÞÞ: The effective coefficient increases as expected for x0 o0 and decreases for x0 > 0; respectively. The solid line in Fig. 10 is calculated using Eq. (14) with ajj ¼ 6:6 104 eV s=m2 : The agreement between theory and experiment is perfect. 4.2. Transverse electron cooling rates Transverse electron cooling as a function of the displacement of the electron beam was studied by measuring the time development of the ion beam profiles. The time evolution of the horizontal (vertical) beam variance si is given by 1 dsi ¼ Li þ lIBS;i si dt
ð19Þ
0
The velocities vjj and v0 can be readily measured via a Schottky noise analysis of the stored ion beam for Uind a0 and 0, respectively. To stay within the linear velocity regime of the electron cooling force, jvjj v0 j has to be smaller than the longitudinal velocity spread of the electrons, i.e. ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smaller than kTjj =me ; which corresponds to maximum allowed change of the revolution frequency of about 80 Hz: More details of the IndAcc method can be found in Ref. [13]. As expected from Eqs. (14) and (16), displacing the electron beam in the vertical direction does not
Fig. 9. The effective longitudinal friction coefficient ajj for a 73:3 MeV 12 C6þ beam, measured with the IndAcc method as a function of the vertical displacement Y of the electron beam with density ne ¼ 8 106 cm3 : The average value of ajj ¼ ajj ¼ 6:6 104 eV s=m2 is shown by the solid line.
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Fig. 10. The effective longitudinal friction coefficient ajj for a 73:3 MeV 12 C6þ beam, measured with the IndAcc method as a function of the horizontal shift x0 ðX ; 0Þ of the electron beam with density ne ¼ 8 106 cm3 : The solid line was calculated with Eq. (14) using the average value ajj ¼ 6:6 104 eV s=m2 as determined from the measurement shown in Fig. 9.
with i ¼ xðyÞ; where Li denotes the total horizontal (vertical) cooling rate and lIBS;i the corresponding heating rate due to IBS. In the absence of dispersive electron cooling the cooling rates are determined by the usual transversal electron cooling force; staying within the linear velocity regime of the electron cooling force, the cooling rates are connected to the transversal friction coefficients ai via L0;i ¼ Zc ai =ð2MÞ:
ð20Þ
Switching on the dispersive electron cooling will not effect the vertical cooling rate as long as the two transversal directions are not coupled, i.e. Ly ¼ L0;y ; but the horizontal rate will be modified. Using Eq. (2) together with Eqs. (15) and (4) we obtain Lx ¼ LD;x þ L0;x with LD;x ¼ Zc
ajj ve;0 2DS ajj X x0 EZc 2M v0 r2e 2M C
linear regime of the transverse cooling force. For the present experiments this requires si p1 mm: In order to determine the heating rate lIBS ; which has to be known to analyze cooling rate measurements via Eq. (19), IBS experiments were carried out. An example for an IBS measurement with 12 C6þ ions of 73:3 MeV and an ion beam current of 20 mA is plotted in Fig. 11. The standard deviations sx of a Gaussian fit to the horizontal beam profiles measured with the BPM are shown as functions of time, correcting for the resolution of the horizontal BPM of E300 mm (E200 mm for the vertical BPM). While the initial beam width s0;x is given by the equilibrium between electron cooling and IBS, the ion beam blow up due to the IBS is being recorded after the electron cooling is switched off at t0 ¼ 1 s: The observed time dependence sx ðtÞ was fitted by a solution of Eq. (19) assuming a heating rate lIBS expected from a simplified model of IBS. In this model, the heating rate is set to lIBS;i ¼ Di N=sgi ; where Di is a constant, and N the number of stored ions in the ring, while the exponent g is expected to have a value close to 5. This reflects the predicted behavior of lIBS pN=ðex ey Dpjj =p0 Þ for the IBS rate [14], which leads for the reasonable assumption, that the horizontal emittance ex ; the vertical emittance ey and the square of the momentum spread Dpjj =p0
ECOOL off
ð21Þ
the latter approximation being correct in first order in r0 =C: Note, that Eq. (20) together with Eq. (19) is only valid if the initial ion beam radius is small enough so that the transverse velocity components of the ions are indeed within the
Fig. 11. Measured horizontal beam widths sx (corrected for the resolution of the BPM) as a function of time for a 20 mA 12 C6þ beam of 73:3 MeV; showing the beam blow up after the electron cooler is switched off at t0 ¼ 1 s (arrow). Eq. (22) was fitted to the data (full line) and resulted in a g value of gx ¼ 4:9770:07:
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are proportional to each other, to ex ey Dpjj =p0 ps5i : Applying this Ansatz, the solution of Eq. (19) with Li ¼ 0 is given by si ðtÞ ¼ ½gDi Nðt t0 Þ þ sg0;i 1=g :
ð22Þ
IBS measurements performed at different ion currents were found to be well described by Eq. (22) and led to an average value of g ¼ 4:8: A fit of Eq. (22) to the data shown in Fig. 11 is shown by the solid line and resulted in gx ¼ 4:9770:07: A fit to the corresponding vertical beam profile resulted in gy ¼ 4:7670:06: To investigate the transverse cooling rates within the linear regime of the electron cooling force as a function of the electron beam displacement, the cooling of ion beams with small betatron amplitudes is studied. For a given displacement ðX ; Y Þ and after adjusting the correction angles Fi ; the measurements are performed in the following way: After injection the ion beam is cooled down to equilibrium, then the electron cooling is switched off to allow the beam to blow up due to IBS. After having reached a beam variance si of approximately 1 mm; the electron cooling is switched on again and the time development of the beam profiles is recorded with the horizontal and vertical BPM. An example of such a cooling time measurement is shown in Fig. 12.
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For Li a0 Eq. (19) is solved by si ðtÞ ¼ ½sga;i expðgðt t0 ÞLi Þ þ sg0;i 1=g
ð23Þ
where sa;i denotes the variance of the ion beam when the cooling starts. The total cooling rate Li can then be deduced by fitting Eq. (23) to the observed time dependence of si ðtÞ using sa;i ; s0;i and Li as fit parameters, setting gi to the value determined in the IBS measurements described above. For the measurement of sy ðtÞ shown in Fig. 12 which was performed at ðX ; Y Þ ¼ ð0; 0Þ; we obtained, e.g. Ly ¼ 24:471:3 s1 ; and from the corresponding sx ðtÞ curve the value Lx ¼ 24:171:3 s1 was extracted. Both values agree very well. The cooling rates are expected to be unaffected by a vertical displacement of the electron beam and to be equal for the horizontal and vertical degree of freedom. This is born out in Fig. 13, where we have plotted the measured cooling rates Lx and Ly as a function of the vertical displacement Y ; keeping X constant at X ¼ 0: The cooling rates are indeed found to be independent of Y and to be equal within their individual errors, although it seems that the vertical cooling rates are systematically higher than the horizontal one by about a few %. Averaging over all measured displacements
ECOOL on
Fig. 12. Time dependence of the vertical beam variance sy ; corrected for the resolution of the BPM, of a 73:3 MeV 12 C6þ beam of 20 mA after switching on the electron cooler (arrow). The electron beam was unshifted, i.e. ðX ; Y Þ ¼ ð0; 0Þ; and had a density of ne ¼ 8:0 106 cm3 : The solid line is a fit using Eq. (23), which resulted in Ly ¼ 24:471:3 s1 :
Fig. 13. Transverse cooling rates for a 73:3 MeV 12 C6þ beam as function of the vertical displacement Y of the electron beam with a density of ne ¼ 8 106 cm3 : The average values Lx ¼ 23:071:2 s1 and Ly ¼ 25:171:2 s1 are shown by the dashed and dotted line, respectively. The solid line reflects the overall average of Li ¼ 24:171:2 s1 :
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Fig. 14. Transverse cooling rates measured for a 73:3 MeV 12 C6þ beam as a function of the horizontal shift x0 ðX ; 0Þ of an electron beam with density ne ¼ 8 106 cm3 : The dashed line represents an average value of L0;y ¼ 23:571:8 s1 : The full line is Lx ¼ 22 s1 þ LD;x with LD;x according to Eq. (21).
we obtain Lx ¼ L0;x ¼ 23:071:2 s1 and Ly ¼ L0;y ¼ 25:171:2 s1 : The transverse cooling rates observed when displacing the electron beam horizontally are shown in Fig. 14 as a function of x0 ðX ; 0Þ E X ð1 þ X =ð2CÞÞ: While the vertical cooling rates are again independent of the displacement of the electron beam within the experimental accuracy (the average value of Ly ¼ 23:571:8 s1 being in good agreement with the value extracted from Fig. 13), the horizontal cooling rates clearly show the dependence on x0 expected when dispersive electron cooling is active: The cooling rate Lx increases if the electron beam is moved ring inward ðx0 > 0Þ and decreases if it is moved ring outward ðx0 o0Þ; i.e. the dispersive coupling leads to an additional cooling or heating of the beam in the horizontal direction. Using Eq. (21) together with L0;x ¼ 22 s1 we obtain the solid line in Fig. 14 which describes the observed slope of the horizontal cooling rates very well.
5. Summary and conclusion Dispersive electron cooling could be realized by creating the required horizontal gradient of the longitudinal cooling force by a displacement of the
center of the electron beam from the ion beam axes. In this way, one can redistribute the cooling force between the longitudinal and horizontal degree of freedom. Depending on the direction of the displacement, i.e. if the electron beam is shifted ring inward (or ring outward) with respect to the coasting ion beam, the horizontal cooling force is increased (decreased), while at the same time the longitudinal force is decreased (increased). In fact, defining an effective horizontal friction coefficient ax by ax ¼ 2M=Zc ðL0;x þ LD;x Þ ¼ ax þ ajj ðve;0 =v0 Þ ð2DS =r2e Þ x0 we see that the increase of the horizontal friction coefficient ax by dispersive coupling is equal but opposite in sign to the change of the longitudinal coefficient (cf. Eq. (14)). The second pre-requisite for realizing dispersive cooling is a finite dispersion in the electron–ion interaction region. Usually, cooler storage rings are operated in a mode leading to an as-small-aspossible dispersion in the electron cooler in order to ensure that also very hot ion beams can be cooled. In the present experiment, a dispersion of 1:81 m was accomplished in the interaction region by just rotating the standard lattice of the TSR by 90 : No deterioration of the cooling properties of the TSR were encountered, on the contrary, aligning the electron to the ion beam axis the longitudinal and transversal friction coefficients were found to be about 25% larger than the corresponding values achieved with the standard lattice [15,16]. This improvement is thought to be due to the different beam optics in the cooler section. For a given dispersion DS and electron density ne ; the change of the longitudinal and horizontal friction coefficient due to dispersive coupling is uniquely determined by the displacement x0 between electron and ion beam. As depicted in Figs. 9 and 14, the expected dependencies on x0 are very well born out by the data. Several factors such as a lower electron density and a lower dispersion add up to yield a somewhat smaller dispersive effect than in the rough estimate following Eq. (5); for the parameters of the present experiment a change of the effective horizontal friction coefficient of about 733% is observed for a displacement of x0 ¼ 71 cm between the electron and ion beam axes, the corresponding
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changes of the longitudinal friction coefficient being 817%: A coupling between the horizontal and vertical degree of freedom was avoided in the present measurements by choosing the two transversal tunes Qx and Qy of the ring to be sufficiently different. Of course, by adjusting the tunes to be equal, a strong coupling between the two transversal directions can be accomplished such that also the vertical degree of freedom would gain from dispersive electron cooling; the effectiveness of this cooling was already demonstrated in the dispersive laser cooling experiments of Lauer et al. [9]. Note, moreover, that the electron density can be increased by a factor of up to 5 and that also the dispersion in the interaction region can be further increased by at least a factor of 2. It thus seems to be possible to almost freely distribute the electron cooling power between the longitudinal and transversal degrees of freedom by dispersive coupling.
References [1] S. van der Meer, Internal Report, CERN/ISR/PO/72-31, 1972. [2] F. Nolden, K. Beckert, F. Caspers, B. Franczak, B. Franzke, R. Menges, A. Schwinn, M. Steck, Nucl. Instr. and Meth. A 441 (2000) 219.
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[3] G.I. Budker, At. Energ. 22 (1967) 346. [4] G.I. Budker, N.S. Dikansky, V.I. Kudelainen, I.N. Meshkov, V.V. Parchomchuk, D.V. Pestrikov, A.N. Skrinsky, B.N. Sukhina, Part. Accel. 7 (1976) 197. [5] M. Steck, G. Bisoffi, M. Blum, A. Friedrich, C. Geyer, M. Grieser, B. Holzer, E. Jaeschke, M. Jung, D. Kr.amer, K. Matl, W. Ott, R. Repnow, Nucl. Instr. and Meth. A 287 (1990) 324. [6] H. Poth, Phys. Rep. 196 (1990) 135. . [7] S. Schroder et al., Phys. Rev. Lett. 64 (1990) 2901. [8] W. Petrich, M. Grieser, R. Grimm, A. Gruber, D. Habs, H.-J. Miesner, D. Schwalm, B. Wanner, H. Wern^e, A. Wolf, R. Grieser, G. Huber, R. Klein, T. Kuhl, . R. . Neumann, S. Schroder, Phys. Rev. A 48 (1993) 2127. [9] I. Lauer, U. Eisenbarth, M. Grieser, R. Grimm, P. Lenisa, V. Luger, T. Sch.atz, U. Schramm, D. Schwalm, M. Weidem.uller, Phys. Rev. Lett. 81 (1998) 2052. [10] S. Pastuszka, U. Schramm, M. Grieser, C. Broude, R. Grimm, D. Habs, J. Kenntner, H.-J. Miesner, T. SchuXler, . D. Schwalm, A. Wolf, Nucl. Instr. and Meth. A 369 (1996) 11. [11] M. Schmitt, Dissertation, Universit.at Heidelberg, 1999. [12] B. Hochadel, F. Albrecht, M. Grieser, D. Habs, D. Schwalm, E. Szmola, A. Wolf, Nucl. Instr. and Meth. A 343 (1994) 401. [13] C. Ellert, D. Habs, E. Jaeschke, T. Kamara, M. Music, D. Schwalm, P. Sigray, A. Wolf, Nucl. Instr. and Meth. A 314 (1992) 399. [14] M. Martini, Internal Report, CERN PS/84-9(AA), 1984. [15] M. Beutelspacher, M. Grieser, D. Schwalm, A. Wolf, Nucl. Instr. and Meth. A 441 (2000) 110. [16] M. Beutelspacher, Dissertation, Universit.at Heidelberg, 2000.