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Monte-Carlo optimization of the transmission of recoil separators
Nuclear Instruments and Methods in Physics Research A 427 (1999) 166}169
Monte-Carlo optimization of the transmission of recoil separators A.G. Popeko *, O.N. Malyshev , A.V. Yeremin , S. Hofmann Flerov Laboratory of Nuclear Reactions, JINR, 141980 Dubna, Russia Gesellschaft fu( r Schwerionenforschung, D-64220 Darmstadt, Germany
Abstract For the optimization of the transmission of recoil separators a Monte-Carlo computer code was developed. A special subroutine generates a set of `reala particles, which will be stored dynamically in the computer's memory. The mean program varies the parameters of the focusing system of the separator in steps, calculates the transmission for each parameter set by transporting a large number of `particlesa using the "rst-order ion optics and searches the best combination of the variables over the permitted ranges of values. 1999 Elsevier Science B.V. All rights reserved. PACS: 25.70.-z; 29.30.-h Keywords: Recoil separators; Heavy ions; Monte-Carlo computer simulation
Recoil separators like the velocity "lter SHIP [1] and the electrostatic separator VASSILISSA [2] have been successfully used for investigations of evaporation residues (ER), produced in heavy-ion fusion reactions. The cross sections of fusion reactions leading to heaviest elements with Z"108}112 [3] are equal to only several pb, which raises serious requirements to the correct operation of the facilities. New experiments aimed to synthesize the elements Z"113 and 114 are underway, and optimizing of the e$ciency of the separators becomes more and more important.
* Corresponding author: Tel.: #7-09621-64286, Fax: #709621-65083. E-mail address: [email protected] (A.G. Popeko)
The evaporation residues from heavy-ion fusion reactions cover broad angular, energy and charge ranges. Available optimizing programs dial often only with average values and enable only a small variation of few parameters. To optimize the transmission of a separator in a traditional way one constructs an objective function or a merit function of the system, which depends on several variables. The choice of the objective function and of the proper conditions is almost intuitive. Such a way has several hazards inherent in it. The objective function can have many maxima in the multidimensional space. It is impossible to solve analytically a set of simultaneous equations without simpli"cation, and without the analytical solution it is impossible to decide if the maximum arrived is the absolute or just a relative one. As
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 6 1 - 7
A.G. Popeko et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 166}169
a simpli"cation one uses usually the symmetrical mode of operation of the quadrupoles in a triplet when "eld gradients in the "rst and in the third quadrupoles are equal } G "G . We have chosen the transport e$ciency of the separator as a parameter, which should be maximized. The present computer code, written in PASCAL language, based on the Monte-Carlo method was designed to simulate the trajectories and to calculate the number of `reala ions passed through a recoil separator. The principal components of the SHIP and the VASSILISSA facilities are systems consisting of electrostatic and magnetic dipoles, which accomplish the spatial "ltering of recoil nuclei, multinucleon transfer reaction products and beam particles dispersing them in their velocity or energy and ion charges. The focusing systems of separators consist of two triplets of magnetic quadrupoles. The "rst one, located just behind the target, focuses the ERs knocked out of the target. The second triplet follows the dispersing system and serves for collecting the ERs onto the focal plane detectors. For a given reaction the transmission of the dispersing system depends only on its construction, high voltages on electrostatic de#ectors and "eld strengths in magnetic dipoles should be chosen according to the prede"ned de#ection angles. Large freedom exists in choosing the quadrupoles settings and the transmission of the separator as a whole depends mostly on the tuning of the focusing system. The special subroutine [4], which generates using a Monte-Carlo method a set of `reala particles, considers the evaporation of neutrons, protons and as from the excited ER, the reaction kinematics, scattering and energy losses of ERs in the target and the ion charge distribution of ERs. We assume that a particle leaves a target with the coordinates x, x , y, y with the energy E , the charge q and with the magnetic rigidity Bo . This set of mixed variables is combined in one structure } record. A set of generated `particlesa will be stored into the "le and later will be stored dynamically in the computer's memory. These are the reasons to use PASCAL as a programming language. Fig. 1 shows energy, angular and charge distributions of ERs calculated with the Monte-Carlo computer code.
167
Fig. 1. Calculated energy (a), angular (b) and charge (c) distributions of the ERs from the reaction Ca(225 MeV)#Tb(0.6 mg/cm)P At#6n.
The main program varies in turn "eld gradients in all quadrupoles in steps, calculates the transmission for each parameter set by transporting a large number of randomly chosen particles from the
168
A.G. Popeko et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 166}169
stored set using the "rst-order ion optics and searches the best combination of the variables over the permitted ranges of values. We present here a small fragment of the main program: for e "0 : to NumberOfSteps do begin G2" : G2beg ; e * Step; for f "0 : to NumberOfSteps do begin G3" : G3beg ; f * Step; for g " : 0 to NumberOfSteps do begin G4" : G4beg ; g * Step; for h "0 : to NumberOfSteps do begin G5" : G5beg ; h * Step; for i " : 0 to NumberOfSteps do begin G6" : G6beg ; i * Step; Nactual " : 0; while Nactual (NumberOfParticles do begin ChoseParticle; Nactual " : Nactual;1; TransportParticle; end; end; end; end; end; end; end;
Because the whole procedure is time consuming, the calculation will be performed for di!erent values of the "eld gradient in the "rst quadrupole and values of G2}G6 are searched, which realize the maximal transmission for each G1. The variables NumberOfParticles and Step will be chosen according to the desired accuracy of the calculation. The starting values G2beg}G6beg may be obtained from approximate calculations using the known settings for previously studied reactions. The described program is used systematically in experiments on the electrostatic separator VASSILISSA. As an example of calculations Fig. 2 shows the dependence of the transmission of At produced in the Tb(Ca,6n) reaction on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted, respectively), together with the experimentally measured e$ciencies. The e$ciency values o were derived by comparing the decay VL rates of the ERs implanted into the focal plane detector with the rates detected in a catcher foil placed just behind the target.
Fig. 2. Dependence of the calculated and measured transmission of At produced in the Tb(Ca,6n) reaction on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted to realize the maximum transmission).
We performed the optimization of the separator settings for each studied projectile-target combination. As a result, separation e$ciency values ranging from 3% to 30% were achieved for ERs produced in reactions with heavy ions ranging from O to Ca. The suppression factors'10 for the full energy beam particles and'10 for multinucleon transfer reaction products were achieved. Some studies were performed also on the velocity "lter SHIP where the symmetrical mode of operation of the triplets is traditionally used. With the Monte-Carlo maximized set of quadrupoles settings a raise of the transport e$ciency for the Rf produced in the Pb(Ti, 2n) reaction up to 30% in comparison to the symmetrical mode was found, but the transmission was instable #uctuating by 10%. The most surprising result was obtained by the transportation of Pu a-particles from the target position to the focal plane detector. Fig. 3 shows the calculated dependence of the transmission of Pu a-particles on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted, respectively), together with the experimentally measured transmissions. The origin of the drop in the transmission by G1"3.2 T/m is yet not fully clear. Possibly it originates due to the turn of the image on the velocity slit of the separator. The strong distortion of the image close to some critical point may
A.G. Popeko et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 166}169
169
explain the instabilities in the transmission. We plan to continue the analysis of the data. This work was supported in part by the Russian Foundation for Basic Research under contract no. 96-02-17209 and INTAS contract no 96}662.
References
Fig. 3. Dependence of the calculated and measured transmission of the Pu a-particles (4.88 MeV) on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted to realize the maximum transmission).
[1] G. MuK nzenberg et al., Nucl. Instr. and Meth. B 26 (1987) 294. [2] A.V. Yeremin et al., Nucl. Instr. and Meth. A 350 (1994) 608. [3] S.Hofmann, GSI-Preprint-98-16, MaK rz, 1998. [4] A.G. Popeko et al., Nucl. Instr. and Meth. B 126 (1997) 294.
Monte-Carlo optimization of the transmission of recoil separators A.G. Popeko *, O.N. Malyshev , A.V. Yeremin , S. Hofmann Flerov Laboratory of Nuclear Reactions, JINR, 141980 Dubna, Russia Gesellschaft fu( r Schwerionenforschung, D-64220 Darmstadt, Germany
Abstract For the optimization of the transmission of recoil separators a Monte-Carlo computer code was developed. A special subroutine generates a set of `reala particles, which will be stored dynamically in the computer's memory. The mean program varies the parameters of the focusing system of the separator in steps, calculates the transmission for each parameter set by transporting a large number of `particlesa using the "rst-order ion optics and searches the best combination of the variables over the permitted ranges of values. 1999 Elsevier Science B.V. All rights reserved. PACS: 25.70.-z; 29.30.-h Keywords: Recoil separators; Heavy ions; Monte-Carlo computer simulation
Recoil separators like the velocity "lter SHIP [1] and the electrostatic separator VASSILISSA [2] have been successfully used for investigations of evaporation residues (ER), produced in heavy-ion fusion reactions. The cross sections of fusion reactions leading to heaviest elements with Z"108}112 [3] are equal to only several pb, which raises serious requirements to the correct operation of the facilities. New experiments aimed to synthesize the elements Z"113 and 114 are underway, and optimizing of the e$ciency of the separators becomes more and more important.
* Corresponding author: Tel.: #7-09621-64286, Fax: #709621-65083. E-mail address: [email protected] (A.G. Popeko)
The evaporation residues from heavy-ion fusion reactions cover broad angular, energy and charge ranges. Available optimizing programs dial often only with average values and enable only a small variation of few parameters. To optimize the transmission of a separator in a traditional way one constructs an objective function or a merit function of the system, which depends on several variables. The choice of the objective function and of the proper conditions is almost intuitive. Such a way has several hazards inherent in it. The objective function can have many maxima in the multidimensional space. It is impossible to solve analytically a set of simultaneous equations without simpli"cation, and without the analytical solution it is impossible to decide if the maximum arrived is the absolute or just a relative one. As
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 6 1 - 7
A.G. Popeko et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 166}169
a simpli"cation one uses usually the symmetrical mode of operation of the quadrupoles in a triplet when "eld gradients in the "rst and in the third quadrupoles are equal } G "G . We have chosen the transport e$ciency of the separator as a parameter, which should be maximized. The present computer code, written in PASCAL language, based on the Monte-Carlo method was designed to simulate the trajectories and to calculate the number of `reala ions passed through a recoil separator. The principal components of the SHIP and the VASSILISSA facilities are systems consisting of electrostatic and magnetic dipoles, which accomplish the spatial "ltering of recoil nuclei, multinucleon transfer reaction products and beam particles dispersing them in their velocity or energy and ion charges. The focusing systems of separators consist of two triplets of magnetic quadrupoles. The "rst one, located just behind the target, focuses the ERs knocked out of the target. The second triplet follows the dispersing system and serves for collecting the ERs onto the focal plane detectors. For a given reaction the transmission of the dispersing system depends only on its construction, high voltages on electrostatic de#ectors and "eld strengths in magnetic dipoles should be chosen according to the prede"ned de#ection angles. Large freedom exists in choosing the quadrupoles settings and the transmission of the separator as a whole depends mostly on the tuning of the focusing system. The special subroutine [4], which generates using a Monte-Carlo method a set of `reala particles, considers the evaporation of neutrons, protons and as from the excited ER, the reaction kinematics, scattering and energy losses of ERs in the target and the ion charge distribution of ERs. We assume that a particle leaves a target with the coordinates x, x , y, y with the energy E , the charge q and with the magnetic rigidity Bo . This set of mixed variables is combined in one structure } record. A set of generated `particlesa will be stored into the "le and later will be stored dynamically in the computer's memory. These are the reasons to use PASCAL as a programming language. Fig. 1 shows energy, angular and charge distributions of ERs calculated with the Monte-Carlo computer code.
167
Fig. 1. Calculated energy (a), angular (b) and charge (c) distributions of the ERs from the reaction Ca(225 MeV)#Tb(0.6 mg/cm)P At#6n.
The main program varies in turn "eld gradients in all quadrupoles in steps, calculates the transmission for each parameter set by transporting a large number of randomly chosen particles from the
168
A.G. Popeko et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 166}169
stored set using the "rst-order ion optics and searches the best combination of the variables over the permitted ranges of values. We present here a small fragment of the main program: for e "0 : to NumberOfSteps do begin G2" : G2beg ; e * Step; for f "0 : to NumberOfSteps do begin G3" : G3beg ; f * Step; for g " : 0 to NumberOfSteps do begin G4" : G4beg ; g * Step; for h "0 : to NumberOfSteps do begin G5" : G5beg ; h * Step; for i " : 0 to NumberOfSteps do begin G6" : G6beg ; i * Step; Nactual " : 0; while Nactual (NumberOfParticles do begin ChoseParticle; Nactual " : Nactual;1; TransportParticle; end; end; end; end; end; end; end;
Because the whole procedure is time consuming, the calculation will be performed for di!erent values of the "eld gradient in the "rst quadrupole and values of G2}G6 are searched, which realize the maximal transmission for each G1. The variables NumberOfParticles and Step will be chosen according to the desired accuracy of the calculation. The starting values G2beg}G6beg may be obtained from approximate calculations using the known settings for previously studied reactions. The described program is used systematically in experiments on the electrostatic separator VASSILISSA. As an example of calculations Fig. 2 shows the dependence of the transmission of At produced in the Tb(Ca,6n) reaction on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted, respectively), together with the experimentally measured e$ciencies. The e$ciency values o were derived by comparing the decay VL rates of the ERs implanted into the focal plane detector with the rates detected in a catcher foil placed just behind the target.
Fig. 2. Dependence of the calculated and measured transmission of At produced in the Tb(Ca,6n) reaction on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted to realize the maximum transmission).
We performed the optimization of the separator settings for each studied projectile-target combination. As a result, separation e$ciency values ranging from 3% to 30% were achieved for ERs produced in reactions with heavy ions ranging from O to Ca. The suppression factors'10 for the full energy beam particles and'10 for multinucleon transfer reaction products were achieved. Some studies were performed also on the velocity "lter SHIP where the symmetrical mode of operation of the triplets is traditionally used. With the Monte-Carlo maximized set of quadrupoles settings a raise of the transport e$ciency for the Rf produced in the Pb(Ti, 2n) reaction up to 30% in comparison to the symmetrical mode was found, but the transmission was instable #uctuating by 10%. The most surprising result was obtained by the transportation of Pu a-particles from the target position to the focal plane detector. Fig. 3 shows the calculated dependence of the transmission of Pu a-particles on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted, respectively), together with the experimentally measured transmissions. The origin of the drop in the transmission by G1"3.2 T/m is yet not fully clear. Possibly it originates due to the turn of the image on the velocity slit of the separator. The strong distortion of the image close to some critical point may
A.G. Popeko et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 166}169
169
explain the instabilities in the transmission. We plan to continue the analysis of the data. This work was supported in part by the Russian Foundation for Basic Research under contract no. 96-02-17209 and INTAS contract no 96}662.
References
Fig. 3. Dependence of the calculated and measured transmission of the Pu a-particles (4.88 MeV) on the "eld gradient of the "rst quadrupole G1 (G2}G6 are setted to realize the maximum transmission).
[1] G. MuK nzenberg et al., Nucl. Instr. and Meth. B 26 (1987) 294. [2] A.V. Yeremin et al., Nucl. Instr. and Meth. A 350 (1994) 608. [3] S.Hofmann, GSI-Preprint-98-16, MaK rz, 1998. [4] A.G. Popeko et al., Nucl. Instr. and Meth. B 126 (1997) 294.