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Characterization of low energy radioactive beams using direct reactions
Nuclear Instruments and Methods in Physics Research A 714 (2013) 176–187
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Characterization of low energy radioactive beams using direct reactions ¨ e, R. Raabe f, K. Riisager a, D. Voulot b, K. Wimmer d,g J.G. Johansen a,n, M.A. Fraser b, V. Bildstein d,c, T. Kroll a
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark CERN, Geneva, Switzerland Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G2W1 d Physik Department E12, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany e Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, 64289 Darmstadt, Germany f Instituut voor Kern-en Stralingsfysica, KU Leuven, 3001 Leuven, Belgium g National Superconducting Cyclotron Laboratory, East Lansing, MI 48824-1321, USA b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 November 2012 Received in revised form 6 February 2013 Accepted 25 February 2013 Available online 14 March 2013
We demonstrate a new technique to determine the beam structure of low energy radioactive beams using coincidence events from a direct reaction. The technique will be described and tested using Geant4 simulations. We use the technique to determine for the first time the width, divergence and energy of an accelerated radioactive beam produced at ISOLDE. We use data from an experiment with an 11Be beam incident on a deuteron target producing 10Be from a (d,t) reaction. The T-REX Si detector array was used for particle detection, but the technique is applicable for other setups. & 2013 Elsevier B.V. All rights reserved.
Keywords: Low-energy secondary beam (11Be) Semiconductor detectors (Si strip detector) Deduced beam properties
1. Introduction An important aspect in experiments using accelerated radioactive ion beams (RIB) is the structure and character of the accelerated beam itself. Often a perfect beam is assumed with no divergence and width when analyzing data from a RIB experiment. This is an approximation since every RIB has nonzero emittance. Therefore, one of the main challenges of accelerator physics is to minimize the emittance of the beam. A measurement of the width and divergence of the beam can be useful both in improving the emittance and the analysis of an experiment. Various beam diagnostic systems have been developed, with different strengths and limitations. At in-flight facilities the high beam energy allows beam particles to be tracked individually. Beam diagnostics have proven more difficult at low energy RIB facilities. The low beam energy would lead to a large energy and angular straggling if any material is placed before the target. Furthermore, the microchannel plate detectors, used at inflight facilities, have a low detection efficiency for light particles. Techniques using either scintillator plates [1] or detecting secondary electrons from a foil [2] have been tested on low energy beams with intensities below 107/s. The beams have been satisfactorily characterized in both cases, but the techniques require space for detectors, making it harder to determine the
n
Corresponding author. Tel.: þ49 6151 16 64239. E-mail address: [email protected] (J.G. Johansen).
0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.02.046
emittance at the reaction point. Silicon and diamond detectors have also been used for beam monitoring at ISOLDE [3,4], but they only provide timing, energy and intensity information and no information on the transverse emittance of the beam. All the above techniques have been tested on stable beams, and so far no beam diagnostic has been performed on a radioactive beam at ISOLDE. Here, we demonstrate an analysis technique to determine the structure of the beam throughout a direct reaction experiment by using coincidence events. The technique applies to reactions of the type: Aþ a-B þ b where both outgoing particles are detected, including scattering were A¼B and a ¼b. The strength of the technique is the ability to use the same setup for both the primary experiment and the beam diagnostics. The beam structure can be determined at a given time interval by gating on coincidence events in this time interval, enabling a study of the changes in the beam structure throughout an experiment. The technique is limited by the requirement of detection of both outgoing particles and by the strong dependency of the uncertainties in all variables. The former limit is very setup dependent, but it requires both outgoing particles to be clearly separated from the non-reacted outgoing beam particles. From this requirement, we estimate this technique to apply to beams up to A¼20 for beam energies around 5 MeV/u. This limit will be heavily reduced if the setup lacks detectors in the very forward lab angles.
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The technique was used to determine for the first time the beam structure of a radioactive beam from REX-ISOLDE. The experiment was performed with an 11Be beam on a deuteron target, using coincidence events from the 2H(11Be,t)10Be reaction. The beam was postaccelerated using the REX-ISOLDE linear accelerator [5], and the detector setup used was the T-REX [6], a setup of 12 silicon detectors designed to work in conjuction with the MINIBALL Ge-detector array. The experiment was primarily designed to study 12 Be through the (d,p) channel, and was the third and last in a series of experiments at ISOLDE with an 11Be beam on a deuteron target. Results from the first two experiments are given in [7]. The experimental data are complemented by a Geant4 simulation of the 2H(11Be,t)10Be reaction with the T-REX setup [6,8]. The simulation is used to prove the validity of the method and show its uncertainties. We will then analyze the experimental data and compare them with the simulations and with emittance measurements of stable beams. Even though the technique is only demonstrated for one setup and one reaction, it should be applicable for all reactions of the type A þ a-B þb and all setups, e.g. employing spectrometers for the heavy fragments, as long as both outgoing particles are detected. We start by giving a short description of the experiment focussing on the setup, Section 2.1, and the uncertainties arising from the setup, Section 2.3. The experiment and the simulations are described in Sections 2.2 and 2.4 respectively. The technique is described in Section 3 for a general reaction of the type A þ a-B þb. The section is divided into four parts, the beam size (Section 3.1), the incoming angle (Section 3.2), combining the size and incoming angle (Section 3.3) and the beam energy (Section 3.4). Each subsection ends with a test of the technique using the simulations. Finally the technique is used on the ISOLDE experiment, Section 4, before ending with the conclusion, Section 5.
177
Y
x z
Fig. 1. A sketch of the setup including the coordinate system used in the analysis. Three detectors, a backward PSD, a forward PSD and an AD, have been omitted to clear the view inside the barrel.
Fig. 2. A DE2E plot of particles detected in the forward PSDs from the experimental data. It shows that protons, deuterons, tritons and a-particles were detected.
2. The experimental setup The experiment, which is used to test the technique, is described in this section, starting with the setup and ending with a description of the simulations. Evaluation of the uncertainties in the extracted quantities are given. The effect of these uncertainties is discussed in Section 3 when the technique is applied to the simulations. 2.1. The setup The T-REX setup [6] was used to detect the two outgoing particles in the reaction A þ a-B þ b. The setup consisted of eight position sensitive Si detectors (PSD), and four double sided annular Si detectors (AD) each having an aperture of 901. A drawing of the setup can be seen in Fig. 1. The four ADs were placed to form a disc in the very forward direction covering angles from 81 to 301. Each AD covers a quarter of the disc. The PSDs were placed as a four sided tube with the target in the middle, covering angles from 301 to 801 and from 1051 to 1551 in the laboratory system, and giving close to a 2p azimuthal coverage. The PSDs consist of 16 position-sensitive resistive strips running perpendicular to the beam direction. The ADs are double sided with 16 rings and 24 strips. All 12 detectors were backed by thick Si pad detectors enabling DE2E identification. Mylar foils were placed in front of the forward PSDs to shield them from heavy particles from the reaction. A beam collimator with a diameter of 8 mm was placed 176 mm before the target, hence a beam spot with a diameter up to 8 mm was expected in the experiment. The energy and the position in the detector are read out for each detected particle. A coordinate system is defined in the setup as shown in Fig. 1. The detector’s position in the coordinate
system is very precisely determined, hence a position in a detector can easily be transformed into a position in the coordinate system. The setup gives four parameters ðx,y,z,EÞ for each detected particle, which can be used for either determining the excitation energy or the beam structure as will be done later. 2.2. The experiment The experiment was performed at ISOLDE, CERN in September 2010. A beam of 11Be was produced by the ISOLDE facility and postaccelerated to 2.85 MeV/u (31.35 MeV) using the REX linear accelerator. The intensity of the beam at the target was 5 106 =s. The target was a deuterated polyethylene foil with a thickness of 1.1 mg/cm2 (10 mm). The foil contained roughly 95% CD2 and 5% CH2. Various reaction channels were open in the experiment, 2 H(11Be,p)12Be, 2H(11Be,t)10Be and elastic scattering. A DE2E plot was used to distinguish the different outgoing particles as demonstrated in Fig. 2. Due to kinematics mainly the 2 H(11Be,t)10Be channel produced coincidence events, hence only events where a triton is identified in one of the PSDs are used in the following discussion. If an event in the PSD is identified as a triton, and a particle is detected in the AD simultaneously, this event is considered a coincidence event of a triton and 10Be and is used in the analysis, all other events are ignored. An energy spectrum of 10Be is constructed from the kinematics of the two outgoing particles to check that the event is truly a 2H(11Be,t)10Be event. Fig. 3 shows the calculated excitation spectrum. The three peaks can be identified as the ground state, the first excited state and a peak containing the third, fourth, fifth and sixth excited state, confirming that all the identified events are indeed 2 H(11Be,t)10Be.
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Table 2 The beam structure for the four Geant4 simulations used to check the technique. All simulations are made with a uniform distribution on a disc in the xy-plane. w is the diameter, s is the offset, y is the incoming angle and D is the divergence of the beam. En is the excitation of 10Be.
Fig. 3. The excitation spectrum for 10Be calculated from coincidence events in the reaction 2H(11Be,t)10Be. The spectrum shows peaks for the ground state, the first excited state and a peak containing the last four bound states of 10Be. The background is very small, indicating that all events are detected t þ 10Be.
Table 1 The uncertainties of the variables detected in the experiment. A line corresponds to a negligible uncertainty. Variable
PSD ðtÞ
AD (10Be)
Dx Dy Dz DE
0.5 mm 0.5 mm 1.56 mm 0.2 MeV
1 mm 1 mm – 1.5 MeV
2.3. Uncertainties The setup gives four parameters for each detected particle as mentioned in Section 2.1. An uncertainty is attached to all of these values. The size of the uncertainties is detector and position dependent. The uncertainties for all four values are determined for each detector type in the setup in this section. An overview of the uncertainties is given in Table 1. A dash in the table corresponds to a negligible uncertainty. While the position in the xy-plane of the reaction for each event is investigated, the position in z is not, and is taken as an uncertainty. A target thickness of 1.1 mg/cm2 gives an uncertainty of Dztarget ¼ 5 mm, which is negligible compared to the z-values of the particles, which ranges from 8 to 58 mm for the tritons and is 64 mm for 10Be. For the tritons the z-coordinate is determined by the strip they hit. The strips have a width of 3.125 mm giving an uncertainty of DzPSD ¼ 1:56 mm. The x- and y-coordinates are given either by the position sensitivity in the strips or the distance between the origin and the detector, which leads to an uncertainty of 0.5 mm. For 10Be the z-coordinate is given by the distance between the target and the AD, with a negligible uncertainty. The uncertainty of the x- and y-coordinates is given by the width of the strip and ring. While the rings have a width of 2 mm, each strip covers 3.41, giving the width a dependency of y. For an outgoing angle of y ¼ 251 the uncertainty of the strips is 0.9 mm. Hence the uncertainty of the x- and y-coordinates of 10Be is estimated to be Dx ¼ Dy ¼ 1 mm. The energy detected is the energy after the particle has passed through the target and for the tritons after they have passed the mylar foil as well. The energy loss is determined by calculating the distance travelled in the foil and target. For the target the distance is given by r ¼ t=ð2 cos yÞ where t is the target thickness and y is the angle determined by the position of the particle. The factor of 1/2 is due to the
Sim.
wA (mm)
sA ðx,yÞ (mm)
(yA , fA ) (mrad)
DA (mrad)
En (MeV)
A B C D
0 5 5 5
(0,0) (0,0) ( 1,0) (0.0)
(0,0) (0,0) (0,0) (87,0)
0 0 0 0
0 0/3.31 0 0
assumption of the reaction occurring halfway through the target. The heavy particles have an angle close to 101 and energies around 20 MeV. This leads to a total energy loss of 3 MeV in 10 mm polyethylene corresponding to an uncertainty of DEtarget,AD ¼ 1:5 MeV. The energy of the tritons ranges from 22 MeV at 301 to 7 MeV at 701, leading to a maximum uncertainty of DEtarget,PSD ¼ 0:2 MeV. The calibration of the detectors was performed using a-sources with an energy range from 3.1 to 5.5 MeV. A linear relation between the channel number and the energy is assumed. This assumption has been tested in a more detailed analysis of the detector response [9]. The analysis included a study of the effect of the ionizing energy loss. The coefficients of the linear relation between the channel number and the energy ðE ¼ aðxbÞÞ were determined with small relative uncertainties, Da=a ¼ 0:5% and Db=b ¼ 2%. This only leads to a negligible uncertainty even for outgoing particles with energies above 20 MeV. The uncertainty in energy is dominated by the target thickness and the uncertainty in the depth at which the reaction occurs. All uncertainties are given in Table 1. 2.4. The simulations Four simulations have been performed using Geant4. The four simulations are used to show that the technique to be discussed in the following can reproduce different beam spots. The simulations are made for the T-REX setup using an 11Be beam incident on a deuteron target. This makes it possible to compare the simulations with the experimental data. All events in the simulations are generated from a reaction happening halfway through the target with an energy of E¼31.35 MeV at the reaction point. This way the uncertainty in the energy is removed from the simulations. Table 2 shows the input values for the beam in the four simulations. The first simulation (A) is a narrow beam moving along the z-axis. This is the simplest case, and this is how RIBs are normally assumed to be. The second simulation (B) is a wide beam moving along the z-axis. The events are uniformly distributed on a disc with a diameter of wA ¼ 5 mm. Furthermore half of the events are populating the ground state of 10Be while the second half populates the first excited state with an excitation energy of En ¼ 3:31 MeV. The second simulation is used to show the dependency of the width of the beam and the excitation energy, when calculating the energy of the incoming beam in Section 3.4. The third simulation (C) is used to check the effect of an offset in the beam, and the fourth (D) is to show the effect of the incoming angle. All simulations have been done with no distribution in angle. The effect of a distribution in the angle can be investigated by comparing results from simulation B and D.
3. Beam characterization The characterization of the beam is done in three steps. First the width and offset of the beam is determined. This is done by calculating the displacement of the incoming particles from the
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Fig. 4. A plot of xA vs. yA for the four simulations in Table 2. A–D refer to the simulation numbers. The shape of the peaks is caused by the setup, which does not cover 4p.
azimuthal angle of the two outgoing particles. Second the incoming angle and divergence of the beam is determined. Finally the beam energy is determined in two ways. The order of these steps is determined by the interdependency of the quantities. The displacement can be determined independently of the incoming angle and the beam energy, while the two others are very dependent on the displacement and width of the beam, as it will be shown in Sections 3.2 and 3.3. The analysis is derived for a general reaction of the type:
particles. The azimuthal angles are given by
A þ a-B þb:
9fb fB 9 ¼ p:
All values refer to a predefined coordinate system, like the one shown Fig. 1 for T-REX. The reaction is assumed to happen in the origin of the coordinate system, and the beam direction to be along the z-axis. The deviation from this assumption can then be used to determine the right place for the reaction and direction of the beam. The coordinate system for the experiment is described in Section 2.1.
This relation is unaffected by the transformation from CM to the laboratory frame, as the transformation is done along the beam axis, changing only y’s. Combining the two equations we get the following relation:
3.1. Beam width and offset The beam width and offset can be determined by calculating the error in the x- and y-coordinates of the two particles. The principle is to use the azimuthal angles ðfÞ of the two outgoing
tan f ¼
y : x
It is clear that f is independent on energy, z and to lowest order the incoming angle, making it ideal to determine the offset of each event. The two outgoing particles will move back to back in the center of mass frame (CM) due to momentum conservation. This leads to the following relation for the azimuthal angles in CM:
yb y ¼ B: xb xB If this relation does not hold for an event, the x- and y-values are wrong, implying that the reaction did not occur at the origin but at a point we will note as ðxA ,yA Þ to indicate it is the x- and y-value of the incoming particle. The equation is made into a function of xA and yA to determine the offset: yb yA y y ¼ B A: xb xA xB xA
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This gives a linear relation between the offset in the x- and ydirection. The offset is assumed to be as minimal as possible, leading to the following minimization problem: minðx2A þ y2A Þ which can be solved using a Lagrangian multiplier: xA ¼ yA ¼
xb yB xB yb ðyb yB Þ2 þðxb xB Þ2 xb yB xB yb ðyb yB Þ2 þ ðxb xB Þ2
ðyb yB Þ ðxB xb Þ:
The offset for each coincidence event can now be determined using these two equations. The form of the beam can be illustrated by plotting xA vs. yA (xy-plots). Fig. 4 shows xy-plots for the four simulations described in Section 2.4 and shown in Table 2. The A-D corresponds to the ones in Table 2. The projections onto the x-axis are shown in Fig. 5. Fig. 4A shows the analysis of the simulation with a thin beam. The width of the spot in Fig. 4A arises from the uncertainty in the x- and y-values given in Table 1. The same effect is seen in Fig. 4B and Fig. 5B. Measuring the extent by the points where the intensity is 1/25 of the maximum, the spot sizes are 1 mm and 6 mm for simulation A and B respectively, in both cases 1 mm to large. All these indicate an uncertainty of 1 mm in the diameter. Two things should be noted. First, the distribution in
Fig. 4B is peaked at (0,0) rather than uniformly distributed, indicating that the assumption of a minimal shift is too strong. This makes it difficult to give a description of the distribution of the beam, we mainly determine the width and overall shape. We have compared simulations with Gaussian and uniform distributions, and sufficient statistics will allow to distinguish them (the data shown later are much closer to a uniform distribution). Second the shape of the distributions is determined partly by the shape of the beam and partly by the setup. The T-REX setup lacks detectors close to f ¼ p=4 þ Np=2, which causes the missing events at x ¼ 7 y in Fig. 4. A setup with 4p angular coverage would produce a perfect circle for a circular beam spot, as used in the simulation. The total offset can also be determined, illustrated in Fig. 4C. Comparing Fig. 4B and C the two spots are identical except the latter is shifted 1 mm in x corresponding to the shift added to simulation C. This shows that a shift in the beam will be detected with this technique. Once again the beam spot is peaked at (0,0) in Fig. 4C, confirming the insensitivity to the detailed distribution of the beam spot. The final simulation, Fig. 4D, is made to illustrate the independency of the incoming angle. Comparing Fig. 4B and D no difference is seen despite the large incoming angle in simulation D. This shows no or a negligible effect on the angle of the incoming beam. Furthermore, Fig. 4B shows the independency on the excitation energy of the final nucleus. Half of the events in simulation B is assuming a population of the first excited state in 10Be, and no effect is seen in Fig. 4B.
Fig. 5. A plot of xA for the four simulations in Table 2. The peaked structure at 0 is clearly seen. This effect is caused by the setup and the estimate of a minimal offset.
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Fig. 6. A plot of yx vs. yy for the four simulations in Table 2. A–D refer to the simulation numbers.
In conclusion, the width, shape and offset of the beam can be determined with an uncertainty of Dw ¼ 1 mm using only the x- and y-coordinates of the two outgoing particles. 3.2. Divergence and direction of the beam The direction of the beam can be determined from momentum conservation. The momentum vector of the beam is given by adding the momentum vectors of the two outgoing particles: ! ! ! P A ¼ P bþ P B with the momentum vectors of the outgoing particles given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2mE ðx,y,zÞ: P ¼ x2 þ y2 þ z2 The incoming angle and azimuthal angle are then given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðP xA Þ2 þðP yA Þ2 tan yA ¼ PzA tan fA ¼
P yA : P xA
The incoming angle ðyÞ and the azimuthal angle ðfÞ for each beam particle can be determined by combining the four equations. This way the x- and y-components of the angle can be determined by
yx ¼ y cos f and yy ¼ y sin f. The dependency on all eight observables can lead to large uncertainties in the results. The angles have been calculated for the four simulations from Table 2. Fig. 6 shows a yx vs. yy plot for the four simulations. The large effect of the width of the beam on the calculated angle is clearly seen. The effects of the energy and z-position can be ignored in the simulations and it will be shown in Section 4 that the effect is negligible when applying this technique to the experimental data. Fig. 6A shows that the uncertainties in x and y lead to a 35 mrad uncertainty in yA . The dependency on x and y is confirmed by simulation B. The width of the beam leads to a significant spread in the calculated incoming angles (yA ¼ 70 mrad), Fig. 6B. The comparison of Fig. 6C and D shows the difference between an offset in the beam and an incoming angle in the beam. The offset is transformed into an offset in the calculated incoming angle, while an angle will lead to a spread in the calculated angle. The large spread in Fig. 6D indicates that the framework for the incoming angle breaks down at large angles. To reduce the influence of the beam width and offset on the calculated divergence and direction of the beam the former should be determined before calculating the angles. 3.3. Combining the offset and the angle measurements The strong dependence of the beam width on the incoming angle shown in Section 3.2 can be investigated further and the
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Fig. 7. A plot of xA vs. yA cos f for the four simulations in Table 2. A–D refer to the simulation numbers.
resolution on the determined incoming angle can be improved. This will be done in this section. Fig. 7 shows a plot of the xA-value determined in Section 3.1 vs. yx determined in Section 3.2 for the four simulations. A similar plot can be made for the y-components, which will not be shown in this paper, as the interpretations are similar to the ones for the x-components. Ideally these plots would show the emittance of the beam, but from Fig. 7A it is clear that experimental resolution effects must be understood first. The beam in Fig. 7A has 0 mm mrad emittance. The extra width is caused by the uncertainties given in Table 1, and is the same as seen in Figs. 4 and 6. Instead the plots can be used to determine the offset, width, angle and divergence independently. Each plot can be interpreted as having two stretched components, a vertical one at xA ¼ 0 mm and a diagonal one ðyx 25xA Þ. The two components are centered around (0,0) for simulation A and B, as expected. The diagonal in Fig. 7C is shifted 1 mm in x compared to Fig. 7B, indicating that an offset in the beam will lead to a shift in the diagonal component corresponding to the offset of the beam, and only a minor shift in the vertical one. An angle in the beam will on the other hand affect the vertical component, as shown in Fig. 7D, but not the diagonal one. The vertical component in Fig. 7D is centered at yx 80 mrad, compared to the 87 mrad used in the simulation. Furthermore, a width in the beam will stretch the diagonal component while a divergence in the beam will stretch the vertical component, hence all four
parameters (offset, width, angle and divergence) can be independently determined from a xA vs. yx plot. The two components are an effect of the setup, and with a setup covering 2p the two components will merge into one. The two components can be separated by gating only on light particles hitting the left or right PSD, or the top and bottom. This is done for simulation C, and can be seen in Fig. 8. It is clear from Figs. 7 and 8 that the top and bottom PSDs should be used to determine the offset in the x-direction ðxA Þ and the left and right PSD’s should be used to determine the xcomponent of the incoming angle. The opposite is the case for the y-components. Thus an xA vs. yx and a yA vs. yy plot will be used to determine the angle and divergence of the incoming beam in Section 4.
3.4. Beam energy The beam energy is determined in two ways, using first energy and then momentum conservation. In both cases the beam energy is determined at the reaction point assumed to be in the center of the target. The beam energy before the target can then be calculated by adding the energy lost in the target before the reaction. Energy conservation gives T A ¼ T b þT B þ En Q :
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Fig. 8. A plot of xA vs. yA cosf for simulation C. (A) is from tritons hitting the left or right PSD. (B) is tritons hitting the top or bottom PSD.
Fig. 9. Spectra of the incoming energy determined from energy conservation (left) and momentum conservation (right). The spectra are made from simulation B.
En is the total excitation energy of the outgoing particles, but normally only B is excited. In the test case only 10Be can be excited. The excitation energy cannot be directly measured, hence the beam energy minus the excitation energy is calculated instead, and then a correction for the excitation energy is made afterwards: E ¼ T A En ¼ T b þ T B Q : The advantage of this method is the independency on the position of the reaction, the only uncertainties arises from the energy of the particles and the energy loss in the target. The disadvantage is the required need of information about the excitation energy. The second method is calculating the beam energy from: TA ¼
ðPxA Þ2 þðP yA Þ2 þðP zA Þ2 : 2mA
The momentum vector is calculated using Eq. (1) from Section 3.2. This method gives the beam energy, but it is dependent on the position of the reaction, making the uncertainty dependent on the width of the beam. Fig. 9 shows the calculated beam energy from simulation B in Table 2. In the simulation half of the events populate the ground state of 10Be and the other half populate the first excited state (En ¼ 3:31 MeV). The beam energy is set to T A ¼ 31:35 MeV at the reaction point. The plot to the left in Fig. 9 is the energy calculated using energy conservation. Two peaks are shown at 31.24 and 27.90 MeV with widths 0.150 and 0.157 MeV respectively.
The lowest peak is from events populating the excited state in Be, and it is lowered with 3.3 MeV corresponding to the excitation energy as expected. The plot to the right in Fig. 9 shows the energy determined from momentum conservation, only one peak emerges as the method ignores the excitation energy, but the peak is broader than the two in Fig. 9A due to the uncertainty in the xy-plane. The energy calculated with the second method is 31.45 MeV with a 0.5 MeV width. The beam energy set in the simulation is within the error of all three reconstructed energies proving the validity of the two equations. Which method to use depends on the reaction, the setup and information known about the final states. The first method requires a clear information about the excitations of the final nuclei and has a stronger energy dependency. The second method is limited due to the beam spot. The methods can also be used together for confirmation as it will be done in Section 4. In conclusion the beam energy can be determined within 0.2 MeV with both methods. The spread in the energy determined using momentum conservation is affected by the beam width. A beam width of 5 mm leads to a FWHM energy spread of DE ¼ 0:5 MeV.
10
4. Experimental data The methods described in Section 3 will now be applied to the experiment performed at ISOLDE.
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The first step is determining the beam width, offset and shape. Fig. 10 shows a plot of the calculated x’s and y’s. Only the outline of the plot is useful, hence no z-color is used. Fig. 11 shows the projections onto the two axis of Fig. 10. The plots show a round beam spot with a diameter of 6 mm shifted 1.3 mm in x. The simulation of such a beam is seen as the red line in Fig. 11. There is a small tail that drops off very rapidly and a small shift in the positive y-direction could be present. No indications of the distribution within the beam spot can be given. This is very consistent with the 8 mm collimator placed 176 mm before the target. The tail arises partly from particles in the beam halo and partly from beam particles scattered on the edge of the collimator. To determine the incoming angle and the divergence of the beam a plot similar to the ones in Fig. 7 has to be made, both for the x- and y-components, Fig. 12A and B. The figure has to be compared to Fig. 7C in order to take the uncertainty from the width into account. The incoming angle is determined first from the vertical components. yx is shifted 20(5) mrad compared to simulation C, while yy is centered around zero. This indicates an incoming angle of yA ¼ 20ð5Þ mrad towards the left.
Fig. 10. A xy-plot of the experimental data from the 2H(11Be,t)10Be reaction. The plot should be compared to the plots in Fig. 4.
A simulation using the determined value of the beam width (wA ¼ 6 mm), offset (sA ¼ 1:3 mm) and angle (yA ¼ 20 mrad) is made to determine the divergence. The x vs. yx and y vs. yy for the simulated data are plotted in Fig. 12C and D respectively. The difference between the simulated and the experimental data is caused by either the tail of the beam width (the diagonal component) or the divergence (the vertical component). The majority of events in the experimental and the simulated data are placed within the same area, confirming a beam spot of 6 mm shifted 1.3 mm in x and with an incoming angle of 20 mrad. This also indicates a very small divergence of the beam, and an upper limit is set to Dy r30 mrad due to the resolution. Small tails in both width and divergence are present, but they drop off very rapidly. The rms emittance of stable beams accelerated to 2.85 MeV/u at REX has been determined as t2:8 mm mrad [10]. To a good approximation, the particle distribution is Gaussian in (x, x0 ) phase space and therefore 95% of the beam is bounded by an emittance a factor 6 times larger than the rms emittance at 17 mm mrad. The emittance was measured with a high intensity 1 nA beam, composed mainly of 20Ne leaked into the ion source from an adjacent Penning trap. An emittance meter was used employing both the slit-grid method and, with an upstream quadrupole, the three-gradient method to measure the emittance. It should be stressed that this mode of ion source operation is far from that typically used to charge breed radioactive beams and the increased intensity in this scenario is likely to cause degradation of the emittance from the source; the high intensity is needed to make the measurements. The results of the stable beam measurement predict just a few mrad of beam divergence at the target with a beam width of 6 mm, consistent with the analysis and upper limit for the emittance detailed in this paper. The beam energy has been determined using both methods described in Section 3.4, the plots of the beam energies can be seen in Fig. 13. The first method, using energy conservation, produces three peaks as expected. 10Be has six bound states, but the four highest are separated with only 300 keV, making them indistinguishable in the analysis, see Fig. 3. Method two, using momentum conservation, only produces one peak. The mean values and widths of the peaks are determined by a Gaussian fit. The calculated energies along with the statistical uncertainties given by the fits can be seen in Table 3 and the energy spreads in Table 4. A wide distribution at low beam energies is seen in both methods (Fig. 13), this is from reactions on the carbon in the polyethylene target. This shows that full kinematic events can be used to effectively select true reactions from background
Fig. 11. A plot of the projections onto the x (left) and y (right) axis of Fig. 10 (blue). The projections from a simulation with a 6 mm beam width at x ¼ 1:3 is plotted on top (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 12. A plot of xA vs. yA cos f (A) and yA vs. yy (B) for the experimental data and for simulation using a beam with wA ¼ 6 mm, sA ¼ 1:3 mm, yA ¼ 20 mrad and fA ¼ p=2 rad (C) and (D). In order to avoid saturation in the scatter plot, only one-third of the data is plotted.
Fig. 13. The beam energy from the experimental data, determined from energy (left) and momentum (right) conservation.
reactions from contaminants in the target, or to determine the ratio between true and background reactions. The width of the beam should lead to a larger width in method two, but the four peaks have comparable widths and all four peaks
are much wider than the ones from the simulations (Fig. 9). The average energy spread from the four peaks is DE=E ¼ 5%. This is higher than a previous determined beam spread measured using silicon detectors at REX-ISOLDE [3,11]. The beam energy was
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measured there for a 300 keV/u stable beam. The beam spread was determined to DE=E ¼ 0:5% and an additional spread of DE=E ¼ 1:4% was seen, caused by the silicon detectors. An increase in energy to 2.85 MeV/u should not increase the total beam spread from 1.5% to 5%. This indicates a new dominating effect. The uncertainty in the energies shown in Table 1 will lead to a larger spread in the calculated beam energy, and could be the cause of the wider peaks. The uncertainty arises from the uncertainty in the reaction depth as described in Section 2.3. The effect of the reaction depth has been investigated in two ways. First, by calculating the beam energy using the experimental data assuming a reaction at three depths in the target; in the beginning of the target, in the middle (Fig. 13) and at the end of the target. The calculation showed that the uncertainty in the reaction depth leads to an uncertainty of DEin ¼ 0:8 MeV in the reconstructed beam energy. This value is confirmed by the second test. The second test was to make a simulation of the reaction assuming a flat distribution in the reaction depth. The simulation was performed using the parameters determined for the experimental beam. The reconstructed beam energies can be seen in Fig. 14 Table 3 Table of the mean values from a Gaussian fit to the four peaks shown in Fig. 13 including the statistical uncertainty. The determined values are calculated into beam energies at the reaction point ðT reac Þ before the beam energy is determined by correcting for the energy loss in the target ðT in Þ. Only uncertainties from the Gaussian fit are given in the table. Uncertainties from Table 1 should also be taken into account. Met.
Mean (MeV)
En (MeV)
T reac (MeV)
T in (MeV)
1
29.685(6) 26.257(5) 23.486(3)
0.00 3.31 6.1
29.685(6) 29.567(5) 29.586(3)
30.800(6) 30.690(5) 30.707(2)
2
29.576(3)
–
29.576(3)
30.698(3)
Table 4 Table of the sigma of a Gaussian fit to the four peaks shown in Fig. 13. The spread in the beam energies is determined from s. Method
s (MeV)
DEFWHM (MeV)
1
0.656(6) 0.625(4) 0.546(3)
1.54(1) 1.47(1) 1.29(1)
2
0.756(3)
1.78(1)
(only the ground state was simulated) along with the experimental data. The momentum conservation method almost perfectly reproduces the experimental peak. The energy conservation method produces a flat distribution rather than a Gaussian. The reason for this is not yet understood. A Gaussian fit is made to the momentum peak and the FWHM is determined to FWHM¼1.44 MeV. For the energy peak, the FWHM is estimated to FWHM¼1.8 MeV. Both values are in good agreement with the first estimate of DEin ¼ 0:8 MeV. Including the uncertainty caused by the uncertainty in the reaction depth along with the uncertainty from the silicon detectors may account for most of the observed energy spread. The expected beam energy of 31.35 MeV is within this uncertainty of the measured value, though slightly higher. The systematic discrepancy is expected to stem from the calculation of the expected value. The expected value of 31.35 MeV is calculated by extrapolating a beam energy measurement made with the dipole magnet before the experiment at 300 keV/u to 2.85 MeV/u. The measurement at 300 keV/u is a precise measurement performed on a stable beam [3]. The extrapolation will lead to an uncertainty in the provided beam energy, which is a few percent higher than the uncertainty from the coincidence events. The technique demonstrated in this paper provides the first beam energy measurement of a RIB accelerated to full energy by the REX-ISOLDE, and with a precision not obtained before. The fact that the two beam measurements are within the uncertainty of each other proves the reliability of the method.
5. Summary and conclusion We have developed a technique to determine a beam diagnostic using coincidence events in a reaction with two outgoing particles. The technique can determine the beam position and incoming angle to a high precision, and provides an independent measurement of the beam energy. The ability to use particles from primary reactions enables an online monitoring of the radioactive beam during an experiment. The technique has been tested on various simulations before being used on an experiment performed at ISOLDE, CERN. The technique reproduced the simulated events perfectly, and the uncertainties in the results can easily be explained. When used on data from a real experiment, the technique was able to produce reliable results. We remind that a good understanding of the beam parameters is crucial in order to understand the obtained experimental resolution. Furthermore, the investigation of the
Fig. 14. The beam energy from a simulation using a flat distribution in the reaction depth (red line) and from the experimental data (blue). The beam energy is determined from energy (left) and momentum (right) conservation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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experimental data is relying on a comparison with the simulations. The simulation is required to adjust for the position dependency of the detection efficiency. For a setup that is well characterized and for data with high statistic the simulation may allow for a more quantitative extraction of the beam shape (employing comparisons as in Fig. 11). All these indicate a very reliable technique, and even though it has only been tested on one setup and one experiment, it should easily be adapted to both new setups and other experiments, like Coulomb excitation experiments. Coulomb excitation experiments have often been performed at ISOLDE using annular detectors, in which both of the two outgoing particles are detected. If the heavy particle energy is measured with silicon detectors (and not in a spectrometer) pulse height defect corrections are important. However, even with an incomplete knowledge of the energies, the position, beam width and to a large extent the beam angle can be extracted.
Acknowledgments We thank the IS430 collaboration, and in particular Hans O. U. Fynbo for valuable input from discussions on the paper.
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References [1] S. Cappello, L. Cosentino, P. Finocchiaro, Nuclear Instruments and Methods in Physics Research Section A 479 (2002) 243. [2] K. Kruglov, et al., Nuclear Instruments and Methods in Physics Research Section A 441 (2000) 595. [3] F. Zocca, et al., Nuclear Instruments and Methods in Physics Research Section A 672 (2011) 21. [4] E. Griesmayer, et al., High-resolution Energy and Intensity Measurements with CVD Diamond at REX-ISOLDE, BE-Note-2009-028, CERN, Geneva, August, 2009. [5] O. Kester, et al., Nuclear Instruments and Methods in Physics Research Section B 204 (2003) 20. [6] V. Bildstein, et al., European Physical Journal A 48 (2012) 85. [7] J. Johansen, AIP Conference Proceedings 1377 (2010) 368. [8] S. Agostinelli, et al., Nuclear Instruments and Methods in Physics Research Section A 506 (2003) 250. [9] O. Kirsebom, Private communication, 2010. [10] D. Voulot, REX-Linac emittance measurements, Internal Note CERN-BE-Note2009-012, CERN, Geneva, CH, 2009. [11] M.A. Fraser, et al., Nuclear Instruments and Methods in Physics Research Section A 663 (2012) 1.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Characterization of low energy radioactive beams using direct reactions ¨ e, R. Raabe f, K. Riisager a, D. Voulot b, K. Wimmer d,g J.G. Johansen a,n, M.A. Fraser b, V. Bildstein d,c, T. Kroll a
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark CERN, Geneva, Switzerland Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G2W1 d Physik Department E12, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany e Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, 64289 Darmstadt, Germany f Instituut voor Kern-en Stralingsfysica, KU Leuven, 3001 Leuven, Belgium g National Superconducting Cyclotron Laboratory, East Lansing, MI 48824-1321, USA b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 November 2012 Received in revised form 6 February 2013 Accepted 25 February 2013 Available online 14 March 2013
We demonstrate a new technique to determine the beam structure of low energy radioactive beams using coincidence events from a direct reaction. The technique will be described and tested using Geant4 simulations. We use the technique to determine for the first time the width, divergence and energy of an accelerated radioactive beam produced at ISOLDE. We use data from an experiment with an 11Be beam incident on a deuteron target producing 10Be from a (d,t) reaction. The T-REX Si detector array was used for particle detection, but the technique is applicable for other setups. & 2013 Elsevier B.V. All rights reserved.
Keywords: Low-energy secondary beam (11Be) Semiconductor detectors (Si strip detector) Deduced beam properties
1. Introduction An important aspect in experiments using accelerated radioactive ion beams (RIB) is the structure and character of the accelerated beam itself. Often a perfect beam is assumed with no divergence and width when analyzing data from a RIB experiment. This is an approximation since every RIB has nonzero emittance. Therefore, one of the main challenges of accelerator physics is to minimize the emittance of the beam. A measurement of the width and divergence of the beam can be useful both in improving the emittance and the analysis of an experiment. Various beam diagnostic systems have been developed, with different strengths and limitations. At in-flight facilities the high beam energy allows beam particles to be tracked individually. Beam diagnostics have proven more difficult at low energy RIB facilities. The low beam energy would lead to a large energy and angular straggling if any material is placed before the target. Furthermore, the microchannel plate detectors, used at inflight facilities, have a low detection efficiency for light particles. Techniques using either scintillator plates [1] or detecting secondary electrons from a foil [2] have been tested on low energy beams with intensities below 107/s. The beams have been satisfactorily characterized in both cases, but the techniques require space for detectors, making it harder to determine the
n
Corresponding author. Tel.: þ49 6151 16 64239. E-mail address: [email protected] (J.G. Johansen).
0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.02.046
emittance at the reaction point. Silicon and diamond detectors have also been used for beam monitoring at ISOLDE [3,4], but they only provide timing, energy and intensity information and no information on the transverse emittance of the beam. All the above techniques have been tested on stable beams, and so far no beam diagnostic has been performed on a radioactive beam at ISOLDE. Here, we demonstrate an analysis technique to determine the structure of the beam throughout a direct reaction experiment by using coincidence events. The technique applies to reactions of the type: Aþ a-B þ b where both outgoing particles are detected, including scattering were A¼B and a ¼b. The strength of the technique is the ability to use the same setup for both the primary experiment and the beam diagnostics. The beam structure can be determined at a given time interval by gating on coincidence events in this time interval, enabling a study of the changes in the beam structure throughout an experiment. The technique is limited by the requirement of detection of both outgoing particles and by the strong dependency of the uncertainties in all variables. The former limit is very setup dependent, but it requires both outgoing particles to be clearly separated from the non-reacted outgoing beam particles. From this requirement, we estimate this technique to apply to beams up to A¼20 for beam energies around 5 MeV/u. This limit will be heavily reduced if the setup lacks detectors in the very forward lab angles.
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The technique was used to determine for the first time the beam structure of a radioactive beam from REX-ISOLDE. The experiment was performed with an 11Be beam on a deuteron target, using coincidence events from the 2H(11Be,t)10Be reaction. The beam was postaccelerated using the REX-ISOLDE linear accelerator [5], and the detector setup used was the T-REX [6], a setup of 12 silicon detectors designed to work in conjuction with the MINIBALL Ge-detector array. The experiment was primarily designed to study 12 Be through the (d,p) channel, and was the third and last in a series of experiments at ISOLDE with an 11Be beam on a deuteron target. Results from the first two experiments are given in [7]. The experimental data are complemented by a Geant4 simulation of the 2H(11Be,t)10Be reaction with the T-REX setup [6,8]. The simulation is used to prove the validity of the method and show its uncertainties. We will then analyze the experimental data and compare them with the simulations and with emittance measurements of stable beams. Even though the technique is only demonstrated for one setup and one reaction, it should be applicable for all reactions of the type A þ a-B þb and all setups, e.g. employing spectrometers for the heavy fragments, as long as both outgoing particles are detected. We start by giving a short description of the experiment focussing on the setup, Section 2.1, and the uncertainties arising from the setup, Section 2.3. The experiment and the simulations are described in Sections 2.2 and 2.4 respectively. The technique is described in Section 3 for a general reaction of the type A þ a-B þb. The section is divided into four parts, the beam size (Section 3.1), the incoming angle (Section 3.2), combining the size and incoming angle (Section 3.3) and the beam energy (Section 3.4). Each subsection ends with a test of the technique using the simulations. Finally the technique is used on the ISOLDE experiment, Section 4, before ending with the conclusion, Section 5.
177
Y
x z
Fig. 1. A sketch of the setup including the coordinate system used in the analysis. Three detectors, a backward PSD, a forward PSD and an AD, have been omitted to clear the view inside the barrel.
Fig. 2. A DE2E plot of particles detected in the forward PSDs from the experimental data. It shows that protons, deuterons, tritons and a-particles were detected.
2. The experimental setup The experiment, which is used to test the technique, is described in this section, starting with the setup and ending with a description of the simulations. Evaluation of the uncertainties in the extracted quantities are given. The effect of these uncertainties is discussed in Section 3 when the technique is applied to the simulations. 2.1. The setup The T-REX setup [6] was used to detect the two outgoing particles in the reaction A þ a-B þ b. The setup consisted of eight position sensitive Si detectors (PSD), and four double sided annular Si detectors (AD) each having an aperture of 901. A drawing of the setup can be seen in Fig. 1. The four ADs were placed to form a disc in the very forward direction covering angles from 81 to 301. Each AD covers a quarter of the disc. The PSDs were placed as a four sided tube with the target in the middle, covering angles from 301 to 801 and from 1051 to 1551 in the laboratory system, and giving close to a 2p azimuthal coverage. The PSDs consist of 16 position-sensitive resistive strips running perpendicular to the beam direction. The ADs are double sided with 16 rings and 24 strips. All 12 detectors were backed by thick Si pad detectors enabling DE2E identification. Mylar foils were placed in front of the forward PSDs to shield them from heavy particles from the reaction. A beam collimator with a diameter of 8 mm was placed 176 mm before the target, hence a beam spot with a diameter up to 8 mm was expected in the experiment. The energy and the position in the detector are read out for each detected particle. A coordinate system is defined in the setup as shown in Fig. 1. The detector’s position in the coordinate
system is very precisely determined, hence a position in a detector can easily be transformed into a position in the coordinate system. The setup gives four parameters ðx,y,z,EÞ for each detected particle, which can be used for either determining the excitation energy or the beam structure as will be done later. 2.2. The experiment The experiment was performed at ISOLDE, CERN in September 2010. A beam of 11Be was produced by the ISOLDE facility and postaccelerated to 2.85 MeV/u (31.35 MeV) using the REX linear accelerator. The intensity of the beam at the target was 5 106 =s. The target was a deuterated polyethylene foil with a thickness of 1.1 mg/cm2 (10 mm). The foil contained roughly 95% CD2 and 5% CH2. Various reaction channels were open in the experiment, 2 H(11Be,p)12Be, 2H(11Be,t)10Be and elastic scattering. A DE2E plot was used to distinguish the different outgoing particles as demonstrated in Fig. 2. Due to kinematics mainly the 2 H(11Be,t)10Be channel produced coincidence events, hence only events where a triton is identified in one of the PSDs are used in the following discussion. If an event in the PSD is identified as a triton, and a particle is detected in the AD simultaneously, this event is considered a coincidence event of a triton and 10Be and is used in the analysis, all other events are ignored. An energy spectrum of 10Be is constructed from the kinematics of the two outgoing particles to check that the event is truly a 2H(11Be,t)10Be event. Fig. 3 shows the calculated excitation spectrum. The three peaks can be identified as the ground state, the first excited state and a peak containing the third, fourth, fifth and sixth excited state, confirming that all the identified events are indeed 2 H(11Be,t)10Be.
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Table 2 The beam structure for the four Geant4 simulations used to check the technique. All simulations are made with a uniform distribution on a disc in the xy-plane. w is the diameter, s is the offset, y is the incoming angle and D is the divergence of the beam. En is the excitation of 10Be.
Fig. 3. The excitation spectrum for 10Be calculated from coincidence events in the reaction 2H(11Be,t)10Be. The spectrum shows peaks for the ground state, the first excited state and a peak containing the last four bound states of 10Be. The background is very small, indicating that all events are detected t þ 10Be.
Table 1 The uncertainties of the variables detected in the experiment. A line corresponds to a negligible uncertainty. Variable
PSD ðtÞ
AD (10Be)
Dx Dy Dz DE
0.5 mm 0.5 mm 1.56 mm 0.2 MeV
1 mm 1 mm – 1.5 MeV
2.3. Uncertainties The setup gives four parameters for each detected particle as mentioned in Section 2.1. An uncertainty is attached to all of these values. The size of the uncertainties is detector and position dependent. The uncertainties for all four values are determined for each detector type in the setup in this section. An overview of the uncertainties is given in Table 1. A dash in the table corresponds to a negligible uncertainty. While the position in the xy-plane of the reaction for each event is investigated, the position in z is not, and is taken as an uncertainty. A target thickness of 1.1 mg/cm2 gives an uncertainty of Dztarget ¼ 5 mm, which is negligible compared to the z-values of the particles, which ranges from 8 to 58 mm for the tritons and is 64 mm for 10Be. For the tritons the z-coordinate is determined by the strip they hit. The strips have a width of 3.125 mm giving an uncertainty of DzPSD ¼ 1:56 mm. The x- and y-coordinates are given either by the position sensitivity in the strips or the distance between the origin and the detector, which leads to an uncertainty of 0.5 mm. For 10Be the z-coordinate is given by the distance between the target and the AD, with a negligible uncertainty. The uncertainty of the x- and y-coordinates is given by the width of the strip and ring. While the rings have a width of 2 mm, each strip covers 3.41, giving the width a dependency of y. For an outgoing angle of y ¼ 251 the uncertainty of the strips is 0.9 mm. Hence the uncertainty of the x- and y-coordinates of 10Be is estimated to be Dx ¼ Dy ¼ 1 mm. The energy detected is the energy after the particle has passed through the target and for the tritons after they have passed the mylar foil as well. The energy loss is determined by calculating the distance travelled in the foil and target. For the target the distance is given by r ¼ t=ð2 cos yÞ where t is the target thickness and y is the angle determined by the position of the particle. The factor of 1/2 is due to the
Sim.
wA (mm)
sA ðx,yÞ (mm)
(yA , fA ) (mrad)
DA (mrad)
En (MeV)
A B C D
0 5 5 5
(0,0) (0,0) ( 1,0) (0.0)
(0,0) (0,0) (0,0) (87,0)
0 0 0 0
0 0/3.31 0 0
assumption of the reaction occurring halfway through the target. The heavy particles have an angle close to 101 and energies around 20 MeV. This leads to a total energy loss of 3 MeV in 10 mm polyethylene corresponding to an uncertainty of DEtarget,AD ¼ 1:5 MeV. The energy of the tritons ranges from 22 MeV at 301 to 7 MeV at 701, leading to a maximum uncertainty of DEtarget,PSD ¼ 0:2 MeV. The calibration of the detectors was performed using a-sources with an energy range from 3.1 to 5.5 MeV. A linear relation between the channel number and the energy is assumed. This assumption has been tested in a more detailed analysis of the detector response [9]. The analysis included a study of the effect of the ionizing energy loss. The coefficients of the linear relation between the channel number and the energy ðE ¼ aðxbÞÞ were determined with small relative uncertainties, Da=a ¼ 0:5% and Db=b ¼ 2%. This only leads to a negligible uncertainty even for outgoing particles with energies above 20 MeV. The uncertainty in energy is dominated by the target thickness and the uncertainty in the depth at which the reaction occurs. All uncertainties are given in Table 1. 2.4. The simulations Four simulations have been performed using Geant4. The four simulations are used to show that the technique to be discussed in the following can reproduce different beam spots. The simulations are made for the T-REX setup using an 11Be beam incident on a deuteron target. This makes it possible to compare the simulations with the experimental data. All events in the simulations are generated from a reaction happening halfway through the target with an energy of E¼31.35 MeV at the reaction point. This way the uncertainty in the energy is removed from the simulations. Table 2 shows the input values for the beam in the four simulations. The first simulation (A) is a narrow beam moving along the z-axis. This is the simplest case, and this is how RIBs are normally assumed to be. The second simulation (B) is a wide beam moving along the z-axis. The events are uniformly distributed on a disc with a diameter of wA ¼ 5 mm. Furthermore half of the events are populating the ground state of 10Be while the second half populates the first excited state with an excitation energy of En ¼ 3:31 MeV. The second simulation is used to show the dependency of the width of the beam and the excitation energy, when calculating the energy of the incoming beam in Section 3.4. The third simulation (C) is used to check the effect of an offset in the beam, and the fourth (D) is to show the effect of the incoming angle. All simulations have been done with no distribution in angle. The effect of a distribution in the angle can be investigated by comparing results from simulation B and D.
3. Beam characterization The characterization of the beam is done in three steps. First the width and offset of the beam is determined. This is done by calculating the displacement of the incoming particles from the
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Fig. 4. A plot of xA vs. yA for the four simulations in Table 2. A–D refer to the simulation numbers. The shape of the peaks is caused by the setup, which does not cover 4p.
azimuthal angle of the two outgoing particles. Second the incoming angle and divergence of the beam is determined. Finally the beam energy is determined in two ways. The order of these steps is determined by the interdependency of the quantities. The displacement can be determined independently of the incoming angle and the beam energy, while the two others are very dependent on the displacement and width of the beam, as it will be shown in Sections 3.2 and 3.3. The analysis is derived for a general reaction of the type:
particles. The azimuthal angles are given by
A þ a-B þb:
9fb fB 9 ¼ p:
All values refer to a predefined coordinate system, like the one shown Fig. 1 for T-REX. The reaction is assumed to happen in the origin of the coordinate system, and the beam direction to be along the z-axis. The deviation from this assumption can then be used to determine the right place for the reaction and direction of the beam. The coordinate system for the experiment is described in Section 2.1.
This relation is unaffected by the transformation from CM to the laboratory frame, as the transformation is done along the beam axis, changing only y’s. Combining the two equations we get the following relation:
3.1. Beam width and offset The beam width and offset can be determined by calculating the error in the x- and y-coordinates of the two particles. The principle is to use the azimuthal angles ðfÞ of the two outgoing
tan f ¼
y : x
It is clear that f is independent on energy, z and to lowest order the incoming angle, making it ideal to determine the offset of each event. The two outgoing particles will move back to back in the center of mass frame (CM) due to momentum conservation. This leads to the following relation for the azimuthal angles in CM:
yb y ¼ B: xb xB If this relation does not hold for an event, the x- and y-values are wrong, implying that the reaction did not occur at the origin but at a point we will note as ðxA ,yA Þ to indicate it is the x- and y-value of the incoming particle. The equation is made into a function of xA and yA to determine the offset: yb yA y y ¼ B A: xb xA xB xA
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This gives a linear relation between the offset in the x- and ydirection. The offset is assumed to be as minimal as possible, leading to the following minimization problem: minðx2A þ y2A Þ which can be solved using a Lagrangian multiplier: xA ¼ yA ¼
xb yB xB yb ðyb yB Þ2 þðxb xB Þ2 xb yB xB yb ðyb yB Þ2 þ ðxb xB Þ2
ðyb yB Þ ðxB xb Þ:
The offset for each coincidence event can now be determined using these two equations. The form of the beam can be illustrated by plotting xA vs. yA (xy-plots). Fig. 4 shows xy-plots for the four simulations described in Section 2.4 and shown in Table 2. The A-D corresponds to the ones in Table 2. The projections onto the x-axis are shown in Fig. 5. Fig. 4A shows the analysis of the simulation with a thin beam. The width of the spot in Fig. 4A arises from the uncertainty in the x- and y-values given in Table 1. The same effect is seen in Fig. 4B and Fig. 5B. Measuring the extent by the points where the intensity is 1/25 of the maximum, the spot sizes are 1 mm and 6 mm for simulation A and B respectively, in both cases 1 mm to large. All these indicate an uncertainty of 1 mm in the diameter. Two things should be noted. First, the distribution in
Fig. 4B is peaked at (0,0) rather than uniformly distributed, indicating that the assumption of a minimal shift is too strong. This makes it difficult to give a description of the distribution of the beam, we mainly determine the width and overall shape. We have compared simulations with Gaussian and uniform distributions, and sufficient statistics will allow to distinguish them (the data shown later are much closer to a uniform distribution). Second the shape of the distributions is determined partly by the shape of the beam and partly by the setup. The T-REX setup lacks detectors close to f ¼ p=4 þ Np=2, which causes the missing events at x ¼ 7 y in Fig. 4. A setup with 4p angular coverage would produce a perfect circle for a circular beam spot, as used in the simulation. The total offset can also be determined, illustrated in Fig. 4C. Comparing Fig. 4B and C the two spots are identical except the latter is shifted 1 mm in x corresponding to the shift added to simulation C. This shows that a shift in the beam will be detected with this technique. Once again the beam spot is peaked at (0,0) in Fig. 4C, confirming the insensitivity to the detailed distribution of the beam spot. The final simulation, Fig. 4D, is made to illustrate the independency of the incoming angle. Comparing Fig. 4B and D no difference is seen despite the large incoming angle in simulation D. This shows no or a negligible effect on the angle of the incoming beam. Furthermore, Fig. 4B shows the independency on the excitation energy of the final nucleus. Half of the events in simulation B is assuming a population of the first excited state in 10Be, and no effect is seen in Fig. 4B.
Fig. 5. A plot of xA for the four simulations in Table 2. The peaked structure at 0 is clearly seen. This effect is caused by the setup and the estimate of a minimal offset.
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Fig. 6. A plot of yx vs. yy for the four simulations in Table 2. A–D refer to the simulation numbers.
In conclusion, the width, shape and offset of the beam can be determined with an uncertainty of Dw ¼ 1 mm using only the x- and y-coordinates of the two outgoing particles. 3.2. Divergence and direction of the beam The direction of the beam can be determined from momentum conservation. The momentum vector of the beam is given by adding the momentum vectors of the two outgoing particles: ! ! ! P A ¼ P bþ P B with the momentum vectors of the outgoing particles given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2mE ðx,y,zÞ: P ¼ x2 þ y2 þ z2 The incoming angle and azimuthal angle are then given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðP xA Þ2 þðP yA Þ2 tan yA ¼ PzA tan fA ¼
P yA : P xA
The incoming angle ðyÞ and the azimuthal angle ðfÞ for each beam particle can be determined by combining the four equations. This way the x- and y-components of the angle can be determined by
yx ¼ y cos f and yy ¼ y sin f. The dependency on all eight observables can lead to large uncertainties in the results. The angles have been calculated for the four simulations from Table 2. Fig. 6 shows a yx vs. yy plot for the four simulations. The large effect of the width of the beam on the calculated angle is clearly seen. The effects of the energy and z-position can be ignored in the simulations and it will be shown in Section 4 that the effect is negligible when applying this technique to the experimental data. Fig. 6A shows that the uncertainties in x and y lead to a 35 mrad uncertainty in yA . The dependency on x and y is confirmed by simulation B. The width of the beam leads to a significant spread in the calculated incoming angles (yA ¼ 70 mrad), Fig. 6B. The comparison of Fig. 6C and D shows the difference between an offset in the beam and an incoming angle in the beam. The offset is transformed into an offset in the calculated incoming angle, while an angle will lead to a spread in the calculated angle. The large spread in Fig. 6D indicates that the framework for the incoming angle breaks down at large angles. To reduce the influence of the beam width and offset on the calculated divergence and direction of the beam the former should be determined before calculating the angles. 3.3. Combining the offset and the angle measurements The strong dependence of the beam width on the incoming angle shown in Section 3.2 can be investigated further and the
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Fig. 7. A plot of xA vs. yA cos f for the four simulations in Table 2. A–D refer to the simulation numbers.
resolution on the determined incoming angle can be improved. This will be done in this section. Fig. 7 shows a plot of the xA-value determined in Section 3.1 vs. yx determined in Section 3.2 for the four simulations. A similar plot can be made for the y-components, which will not be shown in this paper, as the interpretations are similar to the ones for the x-components. Ideally these plots would show the emittance of the beam, but from Fig. 7A it is clear that experimental resolution effects must be understood first. The beam in Fig. 7A has 0 mm mrad emittance. The extra width is caused by the uncertainties given in Table 1, and is the same as seen in Figs. 4 and 6. Instead the plots can be used to determine the offset, width, angle and divergence independently. Each plot can be interpreted as having two stretched components, a vertical one at xA ¼ 0 mm and a diagonal one ðyx 25xA Þ. The two components are centered around (0,0) for simulation A and B, as expected. The diagonal in Fig. 7C is shifted 1 mm in x compared to Fig. 7B, indicating that an offset in the beam will lead to a shift in the diagonal component corresponding to the offset of the beam, and only a minor shift in the vertical one. An angle in the beam will on the other hand affect the vertical component, as shown in Fig. 7D, but not the diagonal one. The vertical component in Fig. 7D is centered at yx 80 mrad, compared to the 87 mrad used in the simulation. Furthermore, a width in the beam will stretch the diagonal component while a divergence in the beam will stretch the vertical component, hence all four
parameters (offset, width, angle and divergence) can be independently determined from a xA vs. yx plot. The two components are an effect of the setup, and with a setup covering 2p the two components will merge into one. The two components can be separated by gating only on light particles hitting the left or right PSD, or the top and bottom. This is done for simulation C, and can be seen in Fig. 8. It is clear from Figs. 7 and 8 that the top and bottom PSDs should be used to determine the offset in the x-direction ðxA Þ and the left and right PSD’s should be used to determine the xcomponent of the incoming angle. The opposite is the case for the y-components. Thus an xA vs. yx and a yA vs. yy plot will be used to determine the angle and divergence of the incoming beam in Section 4.
3.4. Beam energy The beam energy is determined in two ways, using first energy and then momentum conservation. In both cases the beam energy is determined at the reaction point assumed to be in the center of the target. The beam energy before the target can then be calculated by adding the energy lost in the target before the reaction. Energy conservation gives T A ¼ T b þT B þ En Q :
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Fig. 8. A plot of xA vs. yA cosf for simulation C. (A) is from tritons hitting the left or right PSD. (B) is tritons hitting the top or bottom PSD.
Fig. 9. Spectra of the incoming energy determined from energy conservation (left) and momentum conservation (right). The spectra are made from simulation B.
En is the total excitation energy of the outgoing particles, but normally only B is excited. In the test case only 10Be can be excited. The excitation energy cannot be directly measured, hence the beam energy minus the excitation energy is calculated instead, and then a correction for the excitation energy is made afterwards: E ¼ T A En ¼ T b þ T B Q : The advantage of this method is the independency on the position of the reaction, the only uncertainties arises from the energy of the particles and the energy loss in the target. The disadvantage is the required need of information about the excitation energy. The second method is calculating the beam energy from: TA ¼
ðPxA Þ2 þðP yA Þ2 þðP zA Þ2 : 2mA
The momentum vector is calculated using Eq. (1) from Section 3.2. This method gives the beam energy, but it is dependent on the position of the reaction, making the uncertainty dependent on the width of the beam. Fig. 9 shows the calculated beam energy from simulation B in Table 2. In the simulation half of the events populate the ground state of 10Be and the other half populate the first excited state (En ¼ 3:31 MeV). The beam energy is set to T A ¼ 31:35 MeV at the reaction point. The plot to the left in Fig. 9 is the energy calculated using energy conservation. Two peaks are shown at 31.24 and 27.90 MeV with widths 0.150 and 0.157 MeV respectively.
The lowest peak is from events populating the excited state in Be, and it is lowered with 3.3 MeV corresponding to the excitation energy as expected. The plot to the right in Fig. 9 shows the energy determined from momentum conservation, only one peak emerges as the method ignores the excitation energy, but the peak is broader than the two in Fig. 9A due to the uncertainty in the xy-plane. The energy calculated with the second method is 31.45 MeV with a 0.5 MeV width. The beam energy set in the simulation is within the error of all three reconstructed energies proving the validity of the two equations. Which method to use depends on the reaction, the setup and information known about the final states. The first method requires a clear information about the excitations of the final nuclei and has a stronger energy dependency. The second method is limited due to the beam spot. The methods can also be used together for confirmation as it will be done in Section 4. In conclusion the beam energy can be determined within 0.2 MeV with both methods. The spread in the energy determined using momentum conservation is affected by the beam width. A beam width of 5 mm leads to a FWHM energy spread of DE ¼ 0:5 MeV.
10
4. Experimental data The methods described in Section 3 will now be applied to the experiment performed at ISOLDE.
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The first step is determining the beam width, offset and shape. Fig. 10 shows a plot of the calculated x’s and y’s. Only the outline of the plot is useful, hence no z-color is used. Fig. 11 shows the projections onto the two axis of Fig. 10. The plots show a round beam spot with a diameter of 6 mm shifted 1.3 mm in x. The simulation of such a beam is seen as the red line in Fig. 11. There is a small tail that drops off very rapidly and a small shift in the positive y-direction could be present. No indications of the distribution within the beam spot can be given. This is very consistent with the 8 mm collimator placed 176 mm before the target. The tail arises partly from particles in the beam halo and partly from beam particles scattered on the edge of the collimator. To determine the incoming angle and the divergence of the beam a plot similar to the ones in Fig. 7 has to be made, both for the x- and y-components, Fig. 12A and B. The figure has to be compared to Fig. 7C in order to take the uncertainty from the width into account. The incoming angle is determined first from the vertical components. yx is shifted 20(5) mrad compared to simulation C, while yy is centered around zero. This indicates an incoming angle of yA ¼ 20ð5Þ mrad towards the left.
Fig. 10. A xy-plot of the experimental data from the 2H(11Be,t)10Be reaction. The plot should be compared to the plots in Fig. 4.
A simulation using the determined value of the beam width (wA ¼ 6 mm), offset (sA ¼ 1:3 mm) and angle (yA ¼ 20 mrad) is made to determine the divergence. The x vs. yx and y vs. yy for the simulated data are plotted in Fig. 12C and D respectively. The difference between the simulated and the experimental data is caused by either the tail of the beam width (the diagonal component) or the divergence (the vertical component). The majority of events in the experimental and the simulated data are placed within the same area, confirming a beam spot of 6 mm shifted 1.3 mm in x and with an incoming angle of 20 mrad. This also indicates a very small divergence of the beam, and an upper limit is set to Dy r30 mrad due to the resolution. Small tails in both width and divergence are present, but they drop off very rapidly. The rms emittance of stable beams accelerated to 2.85 MeV/u at REX has been determined as t2:8 mm mrad [10]. To a good approximation, the particle distribution is Gaussian in (x, x0 ) phase space and therefore 95% of the beam is bounded by an emittance a factor 6 times larger than the rms emittance at 17 mm mrad. The emittance was measured with a high intensity 1 nA beam, composed mainly of 20Ne leaked into the ion source from an adjacent Penning trap. An emittance meter was used employing both the slit-grid method and, with an upstream quadrupole, the three-gradient method to measure the emittance. It should be stressed that this mode of ion source operation is far from that typically used to charge breed radioactive beams and the increased intensity in this scenario is likely to cause degradation of the emittance from the source; the high intensity is needed to make the measurements. The results of the stable beam measurement predict just a few mrad of beam divergence at the target with a beam width of 6 mm, consistent with the analysis and upper limit for the emittance detailed in this paper. The beam energy has been determined using both methods described in Section 3.4, the plots of the beam energies can be seen in Fig. 13. The first method, using energy conservation, produces three peaks as expected. 10Be has six bound states, but the four highest are separated with only 300 keV, making them indistinguishable in the analysis, see Fig. 3. Method two, using momentum conservation, only produces one peak. The mean values and widths of the peaks are determined by a Gaussian fit. The calculated energies along with the statistical uncertainties given by the fits can be seen in Table 3 and the energy spreads in Table 4. A wide distribution at low beam energies is seen in both methods (Fig. 13), this is from reactions on the carbon in the polyethylene target. This shows that full kinematic events can be used to effectively select true reactions from background
Fig. 11. A plot of the projections onto the x (left) and y (right) axis of Fig. 10 (blue). The projections from a simulation with a 6 mm beam width at x ¼ 1:3 is plotted on top (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 12. A plot of xA vs. yA cos f (A) and yA vs. yy (B) for the experimental data and for simulation using a beam with wA ¼ 6 mm, sA ¼ 1:3 mm, yA ¼ 20 mrad and fA ¼ p=2 rad (C) and (D). In order to avoid saturation in the scatter plot, only one-third of the data is plotted.
Fig. 13. The beam energy from the experimental data, determined from energy (left) and momentum (right) conservation.
reactions from contaminants in the target, or to determine the ratio between true and background reactions. The width of the beam should lead to a larger width in method two, but the four peaks have comparable widths and all four peaks
are much wider than the ones from the simulations (Fig. 9). The average energy spread from the four peaks is DE=E ¼ 5%. This is higher than a previous determined beam spread measured using silicon detectors at REX-ISOLDE [3,11]. The beam energy was
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measured there for a 300 keV/u stable beam. The beam spread was determined to DE=E ¼ 0:5% and an additional spread of DE=E ¼ 1:4% was seen, caused by the silicon detectors. An increase in energy to 2.85 MeV/u should not increase the total beam spread from 1.5% to 5%. This indicates a new dominating effect. The uncertainty in the energies shown in Table 1 will lead to a larger spread in the calculated beam energy, and could be the cause of the wider peaks. The uncertainty arises from the uncertainty in the reaction depth as described in Section 2.3. The effect of the reaction depth has been investigated in two ways. First, by calculating the beam energy using the experimental data assuming a reaction at three depths in the target; in the beginning of the target, in the middle (Fig. 13) and at the end of the target. The calculation showed that the uncertainty in the reaction depth leads to an uncertainty of DEin ¼ 0:8 MeV in the reconstructed beam energy. This value is confirmed by the second test. The second test was to make a simulation of the reaction assuming a flat distribution in the reaction depth. The simulation was performed using the parameters determined for the experimental beam. The reconstructed beam energies can be seen in Fig. 14 Table 3 Table of the mean values from a Gaussian fit to the four peaks shown in Fig. 13 including the statistical uncertainty. The determined values are calculated into beam energies at the reaction point ðT reac Þ before the beam energy is determined by correcting for the energy loss in the target ðT in Þ. Only uncertainties from the Gaussian fit are given in the table. Uncertainties from Table 1 should also be taken into account. Met.
Mean (MeV)
En (MeV)
T reac (MeV)
T in (MeV)
1
29.685(6) 26.257(5) 23.486(3)
0.00 3.31 6.1
29.685(6) 29.567(5) 29.586(3)
30.800(6) 30.690(5) 30.707(2)
2
29.576(3)
–
29.576(3)
30.698(3)
Table 4 Table of the sigma of a Gaussian fit to the four peaks shown in Fig. 13. The spread in the beam energies is determined from s. Method
s (MeV)
DEFWHM (MeV)
1
0.656(6) 0.625(4) 0.546(3)
1.54(1) 1.47(1) 1.29(1)
2
0.756(3)
1.78(1)
(only the ground state was simulated) along with the experimental data. The momentum conservation method almost perfectly reproduces the experimental peak. The energy conservation method produces a flat distribution rather than a Gaussian. The reason for this is not yet understood. A Gaussian fit is made to the momentum peak and the FWHM is determined to FWHM¼1.44 MeV. For the energy peak, the FWHM is estimated to FWHM¼1.8 MeV. Both values are in good agreement with the first estimate of DEin ¼ 0:8 MeV. Including the uncertainty caused by the uncertainty in the reaction depth along with the uncertainty from the silicon detectors may account for most of the observed energy spread. The expected beam energy of 31.35 MeV is within this uncertainty of the measured value, though slightly higher. The systematic discrepancy is expected to stem from the calculation of the expected value. The expected value of 31.35 MeV is calculated by extrapolating a beam energy measurement made with the dipole magnet before the experiment at 300 keV/u to 2.85 MeV/u. The measurement at 300 keV/u is a precise measurement performed on a stable beam [3]. The extrapolation will lead to an uncertainty in the provided beam energy, which is a few percent higher than the uncertainty from the coincidence events. The technique demonstrated in this paper provides the first beam energy measurement of a RIB accelerated to full energy by the REX-ISOLDE, and with a precision not obtained before. The fact that the two beam measurements are within the uncertainty of each other proves the reliability of the method.
5. Summary and conclusion We have developed a technique to determine a beam diagnostic using coincidence events in a reaction with two outgoing particles. The technique can determine the beam position and incoming angle to a high precision, and provides an independent measurement of the beam energy. The ability to use particles from primary reactions enables an online monitoring of the radioactive beam during an experiment. The technique has been tested on various simulations before being used on an experiment performed at ISOLDE, CERN. The technique reproduced the simulated events perfectly, and the uncertainties in the results can easily be explained. When used on data from a real experiment, the technique was able to produce reliable results. We remind that a good understanding of the beam parameters is crucial in order to understand the obtained experimental resolution. Furthermore, the investigation of the
Fig. 14. The beam energy from a simulation using a flat distribution in the reaction depth (red line) and from the experimental data (blue). The beam energy is determined from energy (left) and momentum (right) conservation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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experimental data is relying on a comparison with the simulations. The simulation is required to adjust for the position dependency of the detection efficiency. For a setup that is well characterized and for data with high statistic the simulation may allow for a more quantitative extraction of the beam shape (employing comparisons as in Fig. 11). All these indicate a very reliable technique, and even though it has only been tested on one setup and one experiment, it should easily be adapted to both new setups and other experiments, like Coulomb excitation experiments. Coulomb excitation experiments have often been performed at ISOLDE using annular detectors, in which both of the two outgoing particles are detected. If the heavy particle energy is measured with silicon detectors (and not in a spectrometer) pulse height defect corrections are important. However, even with an incomplete knowledge of the energies, the position, beam width and to a large extent the beam angle can be extracted.
Acknowledgments We thank the IS430 collaboration, and in particular Hans O. U. Fynbo for valuable input from discussions on the paper.
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