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A novel method for beam misalignment correction of an accelerated charged-particle beam
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190 www.elsevier.com/locate/nima
A novel method for beam misalignment correction of an accelerated charged-particle beam J. Rahighi, M. Lamehi-Rachti, O.R. Kakuee Van de Graaff Laboratory, Nuclear Science Research School, NSTRI, P.O. Box 14395-836, Tehran, Iran Received 19 March 2007; received in revised form 10 April 2007; accepted 26 April 2007 Available online 29 April 2007
Abstract A novel method is presented for misalignment correction of an accelerated charged-particle beam in a typical charged-particle scattering experiment employing large-solid-angle detectors. The correction method is based on Rutherford scattering and is quite straightforward to apply when a large solid angle and axially symmetric detection system is used in the experimental measurements. A Monte Carlo computer program and its formalism based on Rutherford scattering cross-section have been described. The program is used to calculate beam misalignment offline after data collection is completed. The method has been successfully applied to correct for misalignment calculated to be typically of the order of a few mm in a 6He radioactive beam of 27 MeV total energy emerging from a cyclotron and produced via 7Li(p,2p)6He reaction. r 2007 Elsevier B.V. All rights reserved. PACS: 29.40.Wk; 29.30.Ep; 25.60.Bx Keywords: Radioactive nuclear beams; Silicon strip detector; Beam misalignment
1. Introduction In the last few decades, scattering of nuclei have been employed to determine the shapes and sizes of atomic nuclei. The spatial distributions of mass and charge (neutrons and protons) are determined through the scattering of nuclei [1]. Heavy-ion elastic scattering reactions have been a major source of information on the structure of nuclei and on the properties of nucleus–nucleus interactions [2]. Our knowledge of the nuclear chart, as a result, has significantly expanded over the last few years. Thanks to the new facilities dedicated to radioactive beams and improved large-scale detection systems. The spectroscopy of nuclear systems has moved farther away from the valley of stability and in some cases (mainly for light systems or on the proton-rich side), has reached the driplines. In a typical nuclear particles scattering experiment, one requires a beam of particles, a target and a detection Corresponding author. Fax: +98 21 88021412.
E-mail address: [email protected] (J. Rahighi). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.04.163
system, advantageous to subtend a large solid angle. The beam of charged particles produced by an accelerator must be highly collimated and focused. The quadrupole magnets and the dipole magnets normally placed outside scattering chamber focuses and centers the beam on the target position. The optimization is usually checked measuring the current on collimators located before the target and on small Faraday cup. The above procedure ensures precise reference direction measurement and an accurate determination of scattering angle y and azimuthal angle j for angular distribution. The beam intensity in a typical radioactive ion-beam experiment is significantly lower than that of a stable beam experiment. Typical beam intensities in RIB experiment vary from approximately 104 to 109 particles/s which is very low compared to stable beam facilities. It is rather difficult to achieve a good and intense beam collimation by using an aperture on the beam trajectory [3,4]. Large-solid-angle detectors are an essential part of the experimental setup in Radioactive Ion Beam (RIB) experiments in which reversed kinematics method is more advantageous because reaction products are
ARTICLE IN PRESS 186
J. Rahighi et al. / Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190
predominantly forward angled [3,4]. Due to closed target–detector geometry and the large solid angle subtended by detectors, the angular distribution of scattered particles is highly sensitive to the beam misalignment. An inaccuracy in measuring the scattering angle, accompanied by an inaccuracy of the solid angle subtended by the detectors, would result in larger errors when measuring the cross-section angular distribution. Axial symmetry of silicon strip detectors and the sensitivity of elastic scattering cross-section to the scattering angle could successfully be employed in a new technique to measure beam misalignment in the physical transverse plane. As for detectors a wide variety of single- and doublesided silicon strip detector designs are currently used in nuclear physics applications. Some of them such as Louvain-Edinburgh Detector Array (LEDA) [5], Lampshade detector array (LAMP) [5], DINEX telescope array [6], Compact Disc double sided silicon strip detector (CD) [7], TRIUMF-UK Detector Array (TUDA) [8] are all axially symmetric detectors and can be considered as typical of the application of this kind of detector to experiment in RNB facilities. Here, we report on a typical elastic scattering experiment in which a beam of 6He is scattered by a variety of target nuclei. During data collection, a beam misalignment on the target plane relative to the beam axis (X ¼ 0, Y ¼ 0) was noticed. Offline misalignment correction has been successfully applied to the experimental data. The calculated misalignment was determined to be of about X ¼ 3 mm and Y ¼ 3 mm. When strip detectors with axial symmetry are employed, any asymmetry in the number of counts can be related to the transverse misalignment of beam. This misalignment can be corrected after the experiment is performed. The present offline method of correction has been routinely applied to the similar beam misalignment problem as a post-process technique in experiments performed at Louvain la Neuve [6,9,10].
LEDA detector consists of eight sectors with 16 strips on each sector (Fig. 1). The LEDA plane is perpendicular to the direction of 6He beam. However, LAMP is a coneshaped detector consisting of six sectors and 16 strips on each sector. Since each strip on LEDA and LAMP subtends a different solid angle, in order to calculate the scattering cross-section, the solid angle and the coordinates of each strip must be determined accurately (Table 1). In Fig. 2, the variables used to determine the solid angle subtended by each strip on LEDA are shown. The solid angle subtended by the LEDA strips can be accurately determined from the equation below:
2. Setup of experiment
Strip no.
The experiment was performed using the radioactive beam facility at the cyclotron research center at Louvain la Neuve, Belgium. The 6He beam used in this experiment was produced via 7Li(p, 2p)6He reaction in the LiF powder target contained in graphite holder [3]. 6He is then ionized in an online Electron Cyclotron Resonance (ECR), ion source and after magnetic separation injected into the second cyclotron where it was accelerated to the desired energy. Post-accelerated secondary 6He beam with an energy of 27 MeV and an intensity of 3 106 ions/s was scattered off self-supported 208Pb and 197Au targets of a typical 1 mg/cm2 thickness. The reaction products were simultaneously measured in a detection system consisting of a LEDA and a LAMP-type detector described elsewhere [9–12].
dA cosðyÞ r2 strip area ¼ cosðyÞ. ðtargettostrip distanceÞ2
dO ¼
ð1Þ
As shown in Fig. 2, the center of each strip is used to calculate the target-to-detector distance [12]. In order to determine the solid angle subtended by the LAMP detector the active area of each strip is divided into
LEDA (6°–15°) Lampshade (6 sectors, 23° – 72°)
Beam
Veto detector 15 mm
Pb Target 86 mm
478 mm
Fig. 1. Experimental setup of LEDA and LAMP detectors.
Table 1 Radial angle and the solid angle subtended by each strip of LEDA and LAMP detectors [12]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
LEDA
LAMP
y1
O 103 (sr)
y1
O 102 (sr)
14.93 14.37 13.80 13.24 12.67 12.10 11.52 10.95 10.37 9.79 9.20 8.62 8.03 7.44 6.85 6.26
0.81 1.19 1.45 1.59 1.53 1.46 1.40 1.33 1.27 1.20 1.13 1.06 0.98 0.92 0.84 0.77
67.07 64.38 61.58 58.69 55.73 52.70 49.62 46.51 43.39 40.62 37.16 34.10 31.09 28.16 25.31 22.55
1.88 2.85 3.53 3.97 3.94 3.87 3.76 3.61 3.43 3.22 2.98 2.73 2.46 2.20 1.93 1.67
ARTICLE IN PRESS J. Rahighi et al. / Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190
scattering cross-section is very sensitive to ycm and is inversely proportional to sin4 ycm =2 [13] that is
y
ds zZe2 1 2 1 ¼ 4 dO cm 4p0 4T a sin ðycm =2Þ
detector element ϕ
187
z
(3)
beam axis x r θ target position Fig. 2. Schematic diagram of an element considered for solid angle calculation of strips of LEDA.
a number of elements. The coordinates of the center of each element is then defined. Since the vector normal to the plane, i.e. plane vector, of each sector, which depends on the orientation of the sector, is known and also the equation of the straight line connecting the center of elements to the target can be worked out easily, the solid angle subtended by each element can be written as follows:
in which z is the atomic number of the projectile and Z is that of the target nucleus. The magnitude of beam misalignment could be determined by relying on the fact that Rutherford scattering cross-section is independent of the azimuthal angle. Figs. (3a and b) show the number of elastically scattered events recorded on each strips of the LEDA and LAMP detectors. The difference in the number of counts on different strips on each sector of the detector is due to the both the difference in the solid angle subtended by each strip and also the Rutherford scattering cross-section for the scattering angle covered by each strip. The difference in the number of counts for each sector is due to the beam misalignment contribution on the change of the scattering angle. The scattering cross-section and the solid angle subtended by each sector will also change as a result.
_ _
3. Implementing the method of correction
a 35000 30000
Counts
25000 20000 15000 10000 5000 0 0
b
16
32
48 64 80 LEDA Strip Number
96
112
4500 4000 3500 3000
Counts
r:n (2) r2 where dAi is the area of the element, r is the distance _ _ between the center of the element and the target, r and n are unit vectors in the direction of ~ r and in the direction of plane vector, respectively [12]. The total solid angle subtended by each strip is the sum of the solid angles subtended by each element. LEDA and LAMP silicon strip detector arrays cover two different angular ranges from 61 to 151 with an angular resolution of less than 0.51 and from 231 to 721 with an angular resolution of 11 or better in the laboratory frame, respectively. This angular resolution refers to the angular range subtended by each strip as seen from the center of the spot produced by the beam on the target. When the finite size of the beam spot on the target is considered, the angular range subtended by each strip on the LAMP detector varies between 31 and 51. The total solid angle of LEDA+LAMP in the laboratory system was O3.45 Sr. Details of the experimental setup have been described elsewhere [9–12]. dOi ¼ dAi
2500 2000 1500 1000
The correction method of beam misalignment is based on the fact that Rutherford scattering of charged particles from a nucleus consisting of a charge of Ze, has cylindrical symmetry about the beam axis. This is because the Coulomb force is symmetric, and therefore Rutherford scattering cross-section is independent of the azimuthal angle F but of course is not independent of the ycm scattering angle. Center-of-mass Coulomb differential
500 0 128
144
160 176 192 LAMP Strip Number
208
Fig. 3. (a) Elastically scattered events recorded in each strip of LEDA. Counts recorded in eight individual sector of the detector is also shown. (b) Elastically scattered events recorded in each strip of LAMP. Counts recorded in six individual sector of the detector is also shown.
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It can be clearly seen that the total number of counts on each sector of LEDA and LAMP are different due to the beam being off axis. Two different methods have been adopted to estimate the beam misalignment and to correct for it. One method is a Monte Carlo technique and the other is analytical. A computer program based on the flow chart shown in Figs. 4 and 5 calculates the beam misalignment. The calculated ratio of dsexp/dsRutherford is the basis for determining the coordinates of the beam on the scattering
target for LEDA and LAMP detectors. The steps in the calculation of the ratio dsexp/dsRutherford are shown inside the dotted box. The Monte Carlo method of calculating misalignment is rather tedious and is quite demanding as far as the computing time is concerned. In addition since the magnitude of misalignment depends rather strongly on the energy and type of the accelerated beam, an analytical method was also developed as an alternative method to the Monte Carlo technique [12]. The analytical method would
2 START σexp(l) = EXPERIMENTAL DATA /a(l)/n(l) INPUT DETECTOR PARAMETERS
CALCULATE WEIGHTED MEAN VALUES OF CROSS SECTION RATIO
INPUT EXPERIMENTAL SETUP
CALCULATE X2
INPUT EXPERIMENTAL DATA
SATISFACTORY
INPUT COMPUTATIONAL PARAMETERS
No
Yes
REPLACE NEW VALUE FOR X2
INITIALIZE HIT REGISTERS SOURCE POSITION Yes GENERATE RANDOM NUMBER SOURCE POSITION (θ ,ϕ) n(l) = n(l)+1
MORE POSITION
No WRITE X,Y AND X2
No
IS THE PARTICLE LANDING WHITIN THE ACTIVE AREA OF STRIP
STOP
Yes DEFINE THE STRIP NUMBER
CALCULATE M(l) = M(l)+1/SIN4(Θ /2) ai(l) = a i(l)+1
CONTINUE
Yes
No
σRutherford(l) = Μ (l)/a(l) 2 Fig. 4. Monte Carlo beam misalignment calculation flowchart. Part of the chart shown inside the dotted lines calculates the ratio dsexp/dsRutherford for the LAMP and LEDA detectors.
ARTICLE IN PRESS J. Rahighi et al. / Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190
189
START
1
INPUT DETECTOR PARAMETERS
CALCULATE WEIGHTED MEAN VALUES FOR dσexp/dσRutherford
INPUT EXPERIMENTAL SETUP
CALCULATE X2
INPUT EXPERIMENTAL DATA X2 SATISFACTORY
INPUT COMPUTATIONAL PARAMETERS
No
Yes REPLACE NEW VALUE FOR X2
INITIALIZE HIT REGISTERS SOURCE POSITION
Yes
SELECT STRIP NUMBER
MORE POSITION
No
CALCULATE SOLID ANGLE FOR CORRESPONDING STRIP
WRITE X,Y AND X2
CALCULATE dσexp/dσRutherford + -ERROR STOP
CONTINUE
Yes
No
1 Fig. 5. Flowchart for analytical method of beam misalignment calculation. Part of the chart shown inside the dotted lines calculates the ratio dsexp/dsRutherford for the LAMP and LEDA detectors.
1.6 1.4 1.2 σ exp/ σ Ruth
naturally include some approximation in order to make the calculation less complex. The most major approximation applied to the present analytical method of calculation is the following: In order to determine the Rutherford scattering angle for each strip, a point at the center of each strip was chosen as the reference. The Rutherford scattering cross-section based on the scattering angle defined by this reference point was calculated. In order to compare the accuracy of the techniques, both the analytical and Monte Carlo methods for the calculation of Rutherford scattering cross-section were applied to the strip No. 15, the innermost strip of the LEDA detector. This strip covers the smallest scattering angle. Since Rutherford scattering cross-section is very sensitive at forward scattering angles, the comparison would be most
1 0.8 0.6 0.4 0.2 0 0
10
20
30
40 50 θcm (degress)
60
70
80
Fig. 6. The ratio of experimental cross-section to the Rutherford scattering cross-section before correction. Different traces belong to the different sectors of LAMP detector.
ARTICLE IN PRESS J. Rahighi et al. / Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190
190
For the above data the axis of symmetry for LEDA was at (0.32 mm, 0.35 mm) and for LAMP detector was at (0.32 mm, 0.05 mm).
1.2
σ exp/ σ Ruth
1 0.8
5. Discussions 0.6 0.4 0.2 0 0
10
20
30
40 50 θcm (degress)
60
70
80
Fig. 7. The ratio of experimental cross-section to the Rutherford scattering cross-section after correction.
relevant for this strip and strongly justifies the validity of the analytical treatment adopted in this work. The Rutherford scattering cross-section calculated by the analytical method and that of Monte Carlo method proved to be in agreement within 0.3% which is very well within the overall calculation error. 4. Program flowchart The flowchart of the program is shown below. It consists of various subroutines for the definition of the experimental geometry. For the experimental setup described above with 6He beam and LEDA and LAMP detectors, two different routines calculate solid angles subtended by the strips of the detectors. A separate routine also calculates the extent to which the beam is misaligned. This is done by defining regions of interest in a plane perpendicular to the beam direction where beam particles impinge. Using the theoretical values of the Rutherford crosssection for the reaction involved, the program calculates the ratio of elastic scattering cross-section to the Rutherford cross-section. The weighted mean value of the crosssection is calculated using the equations given below. Assuming Y ¼ sexp : =sRutherford the weighted mean values of cross-section ratio will then be P 2 i ð1=si ÞY i ¯ (4) Y¼ P 2 i ð1=si Þ w2 value which is a measure of the goodness of fit is calculated using the following equation [14]: X w2 ¼ ðY i Y¯ Þ2 =s2i . (5) i For the experimental setup described here, the beam misalignment was calculated using the method outlined above. Figs. 6 and 7 show the experimental data for 208 Pb(6He, 6He)208Pb reaction both before and after the correction, respectively, for the angular ranges covered by LEDA and LAMP detectors.
The methods described here are fairly easy to apply to the problems associated with instrumental misalignment and asymmetry in a charged particle scattering experiments. The methods have been successfully applied to a beam of 6He of 27 MeV energy undergoing Rutherford scattering in a large solid angle detection system. The method could also be applied to situations where in addition to beam misplacement with respect to the axis of the experimental reaction chamber and the detection system axis, beam is displaced and/or inclined with respect to the detector–target axis. Since LEDA and LAMP detector systems form two different planes, one can readily estimate this angle and make the necessary corrections. The present technique also allows the corrections to be made for the misalignment of the beam with the axis of the individual detection systems. It was observed that for certain experimental setups, the magnitude of misalignment depends on the type and energy of the beam. The dependence of misalignment on the above factors limits the experimenters in applying this technique to certain circumstances where two different beams of different masses were used to measure the relative crosssections such as that of 6He and 4He [15]. Acknowledgments Authors wish to thank Professor Alan Shotter, Dr. Thomas Davinson and other members of the nuclear physics group at Edinburgh University for their very kind and generous support, constructive suggestions and also for providing the data to the authors. References [1] P.E. Hodgson, Growth Points in Nuclear Physics, 1–3, Pergamon Press, Oxford, 1980. [2] J.C. Blackman, Phys. Rev. C 72 (2005) 34606. [3] J. Vervier, Nucl. Phys. A 616 (1997) 97c. [4] A.N. Ostrowski, Prog. Part. Nucl. Phys. 46 (2001) 45. [5] T. Davinson, et al., Nucl. Instr. and Meth. A 454 (2000) 350. [6] A.M. Sanchez-Benitez, et al., J. Phys. G. 31 (2005) S1953. [7] A.N. Ostrowski, et al., Nucl. Instr. and Meth. A 480 (2002) 448. [8] T. Davinson, Nucl. Phys. A 746 (2004) 188c–194c. [9] O.R. Kakuee, J. Rahighi, et al., Nucl. Phys. A 728 (2003) 339. [10] O.R. Kakuee, et al., Nucl. Phys. A 765 (2006) 294. [11] O.R. Kakuee, J. Rahighi, et al., Iranian J. Phys. Res. 4 (1) (2004) 1. [12] O.R. Kakuee, Ph. D. Thesis, Amir Kabir University of Technology, Tehran, Iran, 2004. [13] Introductory Nuclear Physics, Kenneth S. Krane. Wiley, New York, 1988. [14] Techniques for Nuclear and Particle Physics Experiments, William R. Leo, Springer, Germany, 1994. [15] I. Martel, O. Tengblad, Proposal to the PAC of the Cyclotron Research Centre, Proposal No. PH-215, Louvain–la-Neuve, December 2003.
Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190 www.elsevier.com/locate/nima
A novel method for beam misalignment correction of an accelerated charged-particle beam J. Rahighi, M. Lamehi-Rachti, O.R. Kakuee Van de Graaff Laboratory, Nuclear Science Research School, NSTRI, P.O. Box 14395-836, Tehran, Iran Received 19 March 2007; received in revised form 10 April 2007; accepted 26 April 2007 Available online 29 April 2007
Abstract A novel method is presented for misalignment correction of an accelerated charged-particle beam in a typical charged-particle scattering experiment employing large-solid-angle detectors. The correction method is based on Rutherford scattering and is quite straightforward to apply when a large solid angle and axially symmetric detection system is used in the experimental measurements. A Monte Carlo computer program and its formalism based on Rutherford scattering cross-section have been described. The program is used to calculate beam misalignment offline after data collection is completed. The method has been successfully applied to correct for misalignment calculated to be typically of the order of a few mm in a 6He radioactive beam of 27 MeV total energy emerging from a cyclotron and produced via 7Li(p,2p)6He reaction. r 2007 Elsevier B.V. All rights reserved. PACS: 29.40.Wk; 29.30.Ep; 25.60.Bx Keywords: Radioactive nuclear beams; Silicon strip detector; Beam misalignment
1. Introduction In the last few decades, scattering of nuclei have been employed to determine the shapes and sizes of atomic nuclei. The spatial distributions of mass and charge (neutrons and protons) are determined through the scattering of nuclei [1]. Heavy-ion elastic scattering reactions have been a major source of information on the structure of nuclei and on the properties of nucleus–nucleus interactions [2]. Our knowledge of the nuclear chart, as a result, has significantly expanded over the last few years. Thanks to the new facilities dedicated to radioactive beams and improved large-scale detection systems. The spectroscopy of nuclear systems has moved farther away from the valley of stability and in some cases (mainly for light systems or on the proton-rich side), has reached the driplines. In a typical nuclear particles scattering experiment, one requires a beam of particles, a target and a detection Corresponding author. Fax: +98 21 88021412.
E-mail address: [email protected] (J. Rahighi). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.04.163
system, advantageous to subtend a large solid angle. The beam of charged particles produced by an accelerator must be highly collimated and focused. The quadrupole magnets and the dipole magnets normally placed outside scattering chamber focuses and centers the beam on the target position. The optimization is usually checked measuring the current on collimators located before the target and on small Faraday cup. The above procedure ensures precise reference direction measurement and an accurate determination of scattering angle y and azimuthal angle j for angular distribution. The beam intensity in a typical radioactive ion-beam experiment is significantly lower than that of a stable beam experiment. Typical beam intensities in RIB experiment vary from approximately 104 to 109 particles/s which is very low compared to stable beam facilities. It is rather difficult to achieve a good and intense beam collimation by using an aperture on the beam trajectory [3,4]. Large-solid-angle detectors are an essential part of the experimental setup in Radioactive Ion Beam (RIB) experiments in which reversed kinematics method is more advantageous because reaction products are
ARTICLE IN PRESS 186
J. Rahighi et al. / Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190
predominantly forward angled [3,4]. Due to closed target–detector geometry and the large solid angle subtended by detectors, the angular distribution of scattered particles is highly sensitive to the beam misalignment. An inaccuracy in measuring the scattering angle, accompanied by an inaccuracy of the solid angle subtended by the detectors, would result in larger errors when measuring the cross-section angular distribution. Axial symmetry of silicon strip detectors and the sensitivity of elastic scattering cross-section to the scattering angle could successfully be employed in a new technique to measure beam misalignment in the physical transverse plane. As for detectors a wide variety of single- and doublesided silicon strip detector designs are currently used in nuclear physics applications. Some of them such as Louvain-Edinburgh Detector Array (LEDA) [5], Lampshade detector array (LAMP) [5], DINEX telescope array [6], Compact Disc double sided silicon strip detector (CD) [7], TRIUMF-UK Detector Array (TUDA) [8] are all axially symmetric detectors and can be considered as typical of the application of this kind of detector to experiment in RNB facilities. Here, we report on a typical elastic scattering experiment in which a beam of 6He is scattered by a variety of target nuclei. During data collection, a beam misalignment on the target plane relative to the beam axis (X ¼ 0, Y ¼ 0) was noticed. Offline misalignment correction has been successfully applied to the experimental data. The calculated misalignment was determined to be of about X ¼ 3 mm and Y ¼ 3 mm. When strip detectors with axial symmetry are employed, any asymmetry in the number of counts can be related to the transverse misalignment of beam. This misalignment can be corrected after the experiment is performed. The present offline method of correction has been routinely applied to the similar beam misalignment problem as a post-process technique in experiments performed at Louvain la Neuve [6,9,10].
LEDA detector consists of eight sectors with 16 strips on each sector (Fig. 1). The LEDA plane is perpendicular to the direction of 6He beam. However, LAMP is a coneshaped detector consisting of six sectors and 16 strips on each sector. Since each strip on LEDA and LAMP subtends a different solid angle, in order to calculate the scattering cross-section, the solid angle and the coordinates of each strip must be determined accurately (Table 1). In Fig. 2, the variables used to determine the solid angle subtended by each strip on LEDA are shown. The solid angle subtended by the LEDA strips can be accurately determined from the equation below:
2. Setup of experiment
Strip no.
The experiment was performed using the radioactive beam facility at the cyclotron research center at Louvain la Neuve, Belgium. The 6He beam used in this experiment was produced via 7Li(p, 2p)6He reaction in the LiF powder target contained in graphite holder [3]. 6He is then ionized in an online Electron Cyclotron Resonance (ECR), ion source and after magnetic separation injected into the second cyclotron where it was accelerated to the desired energy. Post-accelerated secondary 6He beam with an energy of 27 MeV and an intensity of 3 106 ions/s was scattered off self-supported 208Pb and 197Au targets of a typical 1 mg/cm2 thickness. The reaction products were simultaneously measured in a detection system consisting of a LEDA and a LAMP-type detector described elsewhere [9–12].
dA cosðyÞ r2 strip area ¼ cosðyÞ. ðtargettostrip distanceÞ2
dO ¼
ð1Þ
As shown in Fig. 2, the center of each strip is used to calculate the target-to-detector distance [12]. In order to determine the solid angle subtended by the LAMP detector the active area of each strip is divided into
LEDA (6°–15°) Lampshade (6 sectors, 23° – 72°)
Beam
Veto detector 15 mm
Pb Target 86 mm
478 mm
Fig. 1. Experimental setup of LEDA and LAMP detectors.
Table 1 Radial angle and the solid angle subtended by each strip of LEDA and LAMP detectors [12]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
LEDA
LAMP
y1
O 103 (sr)
y1
O 102 (sr)
14.93 14.37 13.80 13.24 12.67 12.10 11.52 10.95 10.37 9.79 9.20 8.62 8.03 7.44 6.85 6.26
0.81 1.19 1.45 1.59 1.53 1.46 1.40 1.33 1.27 1.20 1.13 1.06 0.98 0.92 0.84 0.77
67.07 64.38 61.58 58.69 55.73 52.70 49.62 46.51 43.39 40.62 37.16 34.10 31.09 28.16 25.31 22.55
1.88 2.85 3.53 3.97 3.94 3.87 3.76 3.61 3.43 3.22 2.98 2.73 2.46 2.20 1.93 1.67
ARTICLE IN PRESS J. Rahighi et al. / Nuclear Instruments and Methods in Physics Research A 578 (2007) 185–190
scattering cross-section is very sensitive to ycm and is inversely proportional to sin4 ycm =2 [13] that is
y
ds zZe2 1 2 1 ¼ 4 dO cm 4p0 4T a sin ðycm =2Þ
detector element ϕ
187
z
(3)
beam axis x r θ target position Fig. 2. Schematic diagram of an element considered for solid angle calculation of strips of LEDA.
a number of elements. The coordinates of the center of each element is then defined. Since the vector normal to the plane, i.e. plane vector, of each sector, which depends on the orientation of the sector, is known and also the equation of the straight line connecting the center of elements to the target can be worked out easily, the solid angle subtended by each element can be written as follows:
in which z is the atomic number of the projectile and Z is that of the target nucleus. The magnitude of beam misalignment could be determined by relying on the fact that Rutherford scattering cross-section is independent of the azimuthal angle. Figs. (3a and b) show the number of elastically scattered events recorded on each strips of the LEDA and LAMP detectors. The difference in the number of counts on different strips on each sector of the detector is due to the both the difference in the solid angle subtended by each strip and also the Rutherford scattering cross-section for the scattering angle covered by each strip. The difference in the number of counts for each sector is due to the beam misalignment contribution on the change of the scattering angle. The scattering cross-section and the solid angle subtended by each sector will also change as a result.
_ _
3. Implementing the method of correction
a 35000 30000
Counts
25000 20000 15000 10000 5000 0 0
b
16
32
48 64 80 LEDA Strip Number
96
112
4500 4000 3500 3000
Counts
r:n (2) r2 where dAi is the area of the element, r is the distance _ _ between the center of the element and the target, r and n are unit vectors in the direction of ~ r and in the direction of plane vector, respectively [12]. The total solid angle subtended by each strip is the sum of the solid angles subtended by each element. LEDA and LAMP silicon strip detector arrays cover two different angular ranges from 61 to 151 with an angular resolution of less than 0.51 and from 231 to 721 with an angular resolution of 11 or better in the laboratory frame, respectively. This angular resolution refers to the angular range subtended by each strip as seen from the center of the spot produced by the beam on the target. When the finite size of the beam spot on the target is considered, the angular range subtended by each strip on the LAMP detector varies between 31 and 51. The total solid angle of LEDA+LAMP in the laboratory system was O3.45 Sr. Details of the experimental setup have been described elsewhere [9–12]. dOi ¼ dAi
2500 2000 1500 1000
The correction method of beam misalignment is based on the fact that Rutherford scattering of charged particles from a nucleus consisting of a charge of Ze, has cylindrical symmetry about the beam axis. This is because the Coulomb force is symmetric, and therefore Rutherford scattering cross-section is independent of the azimuthal angle F but of course is not independent of the ycm scattering angle. Center-of-mass Coulomb differential
500 0 128
144
160 176 192 LAMP Strip Number
208
Fig. 3. (a) Elastically scattered events recorded in each strip of LEDA. Counts recorded in eight individual sector of the detector is also shown. (b) Elastically scattered events recorded in each strip of LAMP. Counts recorded in six individual sector of the detector is also shown.
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It can be clearly seen that the total number of counts on each sector of LEDA and LAMP are different due to the beam being off axis. Two different methods have been adopted to estimate the beam misalignment and to correct for it. One method is a Monte Carlo technique and the other is analytical. A computer program based on the flow chart shown in Figs. 4 and 5 calculates the beam misalignment. The calculated ratio of dsexp/dsRutherford is the basis for determining the coordinates of the beam on the scattering
target for LEDA and LAMP detectors. The steps in the calculation of the ratio dsexp/dsRutherford are shown inside the dotted box. The Monte Carlo method of calculating misalignment is rather tedious and is quite demanding as far as the computing time is concerned. In addition since the magnitude of misalignment depends rather strongly on the energy and type of the accelerated beam, an analytical method was also developed as an alternative method to the Monte Carlo technique [12]. The analytical method would
2 START σexp(l) = EXPERIMENTAL DATA /a(l)/n(l) INPUT DETECTOR PARAMETERS
CALCULATE WEIGHTED MEAN VALUES OF CROSS SECTION RATIO
INPUT EXPERIMENTAL SETUP
CALCULATE X2
INPUT EXPERIMENTAL DATA
SATISFACTORY
INPUT COMPUTATIONAL PARAMETERS
No
Yes
REPLACE NEW VALUE FOR X2
INITIALIZE HIT REGISTERS SOURCE POSITION Yes GENERATE RANDOM NUMBER SOURCE POSITION (θ ,ϕ) n(l) = n(l)+1
MORE POSITION
No WRITE X,Y AND X2
No
IS THE PARTICLE LANDING WHITIN THE ACTIVE AREA OF STRIP
STOP
Yes DEFINE THE STRIP NUMBER
CALCULATE M(l) = M(l)+1/SIN4(Θ /2) ai(l) = a i(l)+1
CONTINUE
Yes
No
σRutherford(l) = Μ (l)/a(l) 2 Fig. 4. Monte Carlo beam misalignment calculation flowchart. Part of the chart shown inside the dotted lines calculates the ratio dsexp/dsRutherford for the LAMP and LEDA detectors.
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START
1
INPUT DETECTOR PARAMETERS
CALCULATE WEIGHTED MEAN VALUES FOR dσexp/dσRutherford
INPUT EXPERIMENTAL SETUP
CALCULATE X2
INPUT EXPERIMENTAL DATA X2 SATISFACTORY
INPUT COMPUTATIONAL PARAMETERS
No
Yes REPLACE NEW VALUE FOR X2
INITIALIZE HIT REGISTERS SOURCE POSITION
Yes
SELECT STRIP NUMBER
MORE POSITION
No
CALCULATE SOLID ANGLE FOR CORRESPONDING STRIP
WRITE X,Y AND X2
CALCULATE dσexp/dσRutherford + -ERROR STOP
CONTINUE
Yes
No
1 Fig. 5. Flowchart for analytical method of beam misalignment calculation. Part of the chart shown inside the dotted lines calculates the ratio dsexp/dsRutherford for the LAMP and LEDA detectors.
1.6 1.4 1.2 σ exp/ σ Ruth
naturally include some approximation in order to make the calculation less complex. The most major approximation applied to the present analytical method of calculation is the following: In order to determine the Rutherford scattering angle for each strip, a point at the center of each strip was chosen as the reference. The Rutherford scattering cross-section based on the scattering angle defined by this reference point was calculated. In order to compare the accuracy of the techniques, both the analytical and Monte Carlo methods for the calculation of Rutherford scattering cross-section were applied to the strip No. 15, the innermost strip of the LEDA detector. This strip covers the smallest scattering angle. Since Rutherford scattering cross-section is very sensitive at forward scattering angles, the comparison would be most
1 0.8 0.6 0.4 0.2 0 0
10
20
30
40 50 θcm (degress)
60
70
80
Fig. 6. The ratio of experimental cross-section to the Rutherford scattering cross-section before correction. Different traces belong to the different sectors of LAMP detector.
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For the above data the axis of symmetry for LEDA was at (0.32 mm, 0.35 mm) and for LAMP detector was at (0.32 mm, 0.05 mm).
1.2
σ exp/ σ Ruth
1 0.8
5. Discussions 0.6 0.4 0.2 0 0
10
20
30
40 50 θcm (degress)
60
70
80
Fig. 7. The ratio of experimental cross-section to the Rutherford scattering cross-section after correction.
relevant for this strip and strongly justifies the validity of the analytical treatment adopted in this work. The Rutherford scattering cross-section calculated by the analytical method and that of Monte Carlo method proved to be in agreement within 0.3% which is very well within the overall calculation error. 4. Program flowchart The flowchart of the program is shown below. It consists of various subroutines for the definition of the experimental geometry. For the experimental setup described above with 6He beam and LEDA and LAMP detectors, two different routines calculate solid angles subtended by the strips of the detectors. A separate routine also calculates the extent to which the beam is misaligned. This is done by defining regions of interest in a plane perpendicular to the beam direction where beam particles impinge. Using the theoretical values of the Rutherford crosssection for the reaction involved, the program calculates the ratio of elastic scattering cross-section to the Rutherford cross-section. The weighted mean value of the crosssection is calculated using the equations given below. Assuming Y ¼ sexp : =sRutherford the weighted mean values of cross-section ratio will then be P 2 i ð1=si ÞY i ¯ (4) Y¼ P 2 i ð1=si Þ w2 value which is a measure of the goodness of fit is calculated using the following equation [14]: X w2 ¼ ðY i Y¯ Þ2 =s2i . (5) i For the experimental setup described here, the beam misalignment was calculated using the method outlined above. Figs. 6 and 7 show the experimental data for 208 Pb(6He, 6He)208Pb reaction both before and after the correction, respectively, for the angular ranges covered by LEDA and LAMP detectors.
The methods described here are fairly easy to apply to the problems associated with instrumental misalignment and asymmetry in a charged particle scattering experiments. The methods have been successfully applied to a beam of 6He of 27 MeV energy undergoing Rutherford scattering in a large solid angle detection system. The method could also be applied to situations where in addition to beam misplacement with respect to the axis of the experimental reaction chamber and the detection system axis, beam is displaced and/or inclined with respect to the detector–target axis. Since LEDA and LAMP detector systems form two different planes, one can readily estimate this angle and make the necessary corrections. The present technique also allows the corrections to be made for the misalignment of the beam with the axis of the individual detection systems. It was observed that for certain experimental setups, the magnitude of misalignment depends on the type and energy of the beam. The dependence of misalignment on the above factors limits the experimenters in applying this technique to certain circumstances where two different beams of different masses were used to measure the relative crosssections such as that of 6He and 4He [15]. Acknowledgments Authors wish to thank Professor Alan Shotter, Dr. Thomas Davinson and other members of the nuclear physics group at Edinburgh University for their very kind and generous support, constructive suggestions and also for providing the data to the authors. References [1] P.E. Hodgson, Growth Points in Nuclear Physics, 1–3, Pergamon Press, Oxford, 1980. [2] J.C. Blackman, Phys. Rev. C 72 (2005) 34606. [3] J. Vervier, Nucl. Phys. A 616 (1997) 97c. [4] A.N. Ostrowski, Prog. Part. Nucl. Phys. 46 (2001) 45. [5] T. Davinson, et al., Nucl. Instr. and Meth. A 454 (2000) 350. [6] A.M. Sanchez-Benitez, et al., J. Phys. G. 31 (2005) S1953. [7] A.N. Ostrowski, et al., Nucl. Instr. and Meth. A 480 (2002) 448. [8] T. Davinson, Nucl. Phys. A 746 (2004) 188c–194c. [9] O.R. Kakuee, J. Rahighi, et al., Nucl. Phys. A 728 (2003) 339. [10] O.R. Kakuee, et al., Nucl. Phys. A 765 (2006) 294. [11] O.R. Kakuee, J. Rahighi, et al., Iranian J. Phys. Res. 4 (1) (2004) 1. [12] O.R. Kakuee, Ph. D. Thesis, Amir Kabir University of Technology, Tehran, Iran, 2004. [13] Introductory Nuclear Physics, Kenneth S. Krane. Wiley, New York, 1988. [14] Techniques for Nuclear and Particle Physics Experiments, William R. Leo, Springer, Germany, 1994. [15] I. Martel, O. Tengblad, Proposal to the PAC of the Cyclotron Research Centre, Proposal No. PH-215, Louvain–la-Neuve, December 2003.