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Fast on-line charged particle tracing in real magnetic fields
Nuclear Instruments and Methods in Physics Research A 427 (1999) 350}352
Fast on-line charged particle tracing in real magnetic "elds B. Masschaele*, W. Mondelaers, P. Lahorte University of Gent, Department for Subatomic and Radiation Physics, Proeftuinstraat 86, B-9000 Gent, Belgium
Abstract The work we present consists of the development of a computerised 3-D scanning system running under LABVIEW威. 3-D magnetic "eld measurements, form an important application allowing a fast on-line determination of charged particle trajectories in real magnetic "elds. The set-up consists of a Hall-probe, movable in three dimensions, mimicking a charged particle entering a magnetic "eld with well-de"ned transport characteristics (energy, position, momentum). The measured local "eld components are used as input to a LABVIEW威 program, which calculates the local direction of the #ight-path of the particle. Then the probe is moved along this direction with a predetermined step-length. In this way, the real path of the charged particle is determined step by step automatically accounting for all magnetic "eld imperfections. Trajectories can be determined without the measurement of the whole 3-D "eld con"guration. 1999 Elsevier Science B.V. All rights reserved. Keywords: 3-D scanning systems; LABVIEW; Charged particles
1. Introduction The existing 15 MeV high current electron LINAC [1], at the department for Subatomic and Radiation Physics of the University Gent, is currently undergoing upgrading. Two additional sections and a new transport system to guide the electron beam to the di!erent experimental set-ups are being installed. Together with new magnets, magnets from former installations will be reused to build the electron transport system. The "rst step in the design is the use of a beam transport program. These programs require a set of magnet para* Corresponding author. Tel.: #32-09-264-6532; fax: #3209-264-6699. E-mail address: [email protected] (B. Masschaele)
meters, which are not always exactly known for a real magnet. Stray "elds, deformations and other faults, cause the real magnet to di!er from the ideal one. They give rise to deviations from the theoretically calculated trajectory. Therefore, we developed a technique to simulate charged particles in known and unknown magnetic "elds. By examining the local magnetic "eld, actually `senseda by the charged particle its path can be determined. Because the magnetic "eld is only measured along the #ight path, the number of measurements can be reduced by several orders of magnitude. Furthermore, these local "elds provide valuable information concerning the in#uence of magnet characteristics such as e!ective length, imperfections, quadrupole and sextupole coe$cients of bending magnets [2]. Another advantage of the
0168-9002/99/$ } 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 5 7 - 5
B. Masschaele et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 350}352
351
technique is its possibility to calculate automatically the nth-order transfer matrices for real magnets. These matrices can then be inserted in most of the transport programs [3,4]. Both the tracing system and the transport programs make a good combination to develop a particle transport system [5,6].
2. Description of the technique The technique of on-line charged particle tracing through magnetic "elds is based on the measurement of the "eld at the place of the particle. Therefore we mounted a Hall-probe on a scanning device. The probe simulates the charged particle as it is sent into the magnet under investigation. Excluding beam-generated forces, only the Lorentz force is acting upon the particle [7]. Without knowledge of the "eld in the entire magnet gap and its environment it is possible to trace the particle through known or unknown time-independent magnetic "elds. The Lorentz equation can be written as two "rst-order di!erential equations. These are readily transformed in couple Cartesian components q b !b " (b B !b B ) *s V V mb W X X W q (b B !b B ) *s b !b " V X W W mb X V q b !b " (b B !b B ) *s. X X mb V W W V
(1)
b x "x # V *s b b y "y # W *s b b z "z # X *s. b
Fig. 1. 0.5 MeV electron-trajectory in trap "eld: once the charged particle is trapped it starts moving around in the magnetic well. The path depends on the initial entrance conditions.
*s the step size for the trajectory calculation. We have to take into account that here only the transversal "elds are measured. To measure all three components, an extra axial Hall-probe is needed. It is possible now to deduce "rst or higher-order transport matrices for any magnet under investigation. Finding the "rst-order transfer matrix for typical magnets with midplane symmetry applied in beam transport systems requires only tracing of four `particlesa, the reference particle and three others with di!erent initial parameters. Higher orders require, depending on their symmetry, more measurements. The maximum number of trajectories for the nth order assuming no symmetry is 6L#1. The scanner was also used to investigate the behaviour of charged particles in rather unusual magnets. The `trapa-magnet is an example. It is a magnet that traps the charged particles in a magnetic well (see Fig. 1). Once the particle has overcome the border of the well it is captured inside. Depending on the initial energy and direction, the particle's path is a rosette (see Fig. 2).
(2)
B , B , B are the "eld components, b is the particle V W X velocity, b b , b are the velocity components in V W X x, y, z Cartesian coordinate system. The xz-plane is arbitrarily chosen in the midplane of the magnet. q is the charge, m"E/c the relativistic mass and
3. Accuracy of the method The previous set of equations is based on the Euler di!erence method or Leapfrog method. The drawback of this approach is that it introduces
352
B. Masschaele et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 350}352
These smaller steps are used for the calculation of the trajectories.
4. Technical information The 3D-scanner consists of three accurate stepping motors. These are able to move a long arm in three dimensions x, y, and z (see Fig. 2). The Hallprobe is mounted at the end of this arm on a servomechanism. This servomechanism is able to turn the probe over 903, which makes it possible to measure the two transversal components of the magnetic "eld (B , B ) with the same probe. A V W LABVIEW威 program controls the motors, reads out the probe and displays the results. For a typical transport matrix determination, the input parameters are the energy, the mass, the start coordinates of the reference particle, and the required matrix order. The program performs automatically the necessary steps to establish all the matrix coe$cients.
References
Fig. 2. 3D scanning device [9].
errors proportional to the "rst power of the step size. Second- or higher-order Runge}Kutta methods are not applicable since it is not possible to determine the function value, the magnetic "eld is unknown at midpoints of intervals [8]. That is why the step size has to be small. To get the most accurate results, the tracing is done by two Euler di!erence methods in one. The "rst (for the measurement of the magnetic "eld) uses a step size, *S, equal to the resolution of the Hall-probe and each of these steps are divided in smaller steps *s where the magnetic "eld is considered constant.
[1] W. Mondelaers, K. Van Laere, A. Goedefroot, K. Van den Bossche, Nucl. Instr. and Meth. A 368 (1996) 278. [2] B. Masschaele, Development of a computerised 3D-Scanning Device, Licentiate Thesis, 1997, unpublished. [3] G. Gillespie, B. Hill, N. Brown, R. Babcock, D. Carey, A graphic user interface for the particle optics code TRANSPORT, CERN 96-07 (1996) 842. [4] D.C. Carey, Third Order TRANSPORT: A Program For Designing Charged Particle Beam Transport Systems, SLAC-R-95-462 Fermilab-Pub-95/069 UC-414, 1995, p. 186. [5] S. Humphries Jr., Charged Particle Beams, Wiley, New York, 1990. [6] D.C. Carey, The Optics of Charged Particle Beams, Harwood, New York, 1987. [7] M. Conte, W.W. MacKay, An Introduction to the Physics of Particle Accelerators, World Scienti"c, Singapore, 1962. [8] W.H. Press et al., Numerical Recipes in C, Cambridge University Press, Cambridge, 1992. [9] C.J. Cason, The POV-Ray Team, POV-Ray2+, 1997.
Fast on-line charged particle tracing in real magnetic "elds B. Masschaele*, W. Mondelaers, P. Lahorte University of Gent, Department for Subatomic and Radiation Physics, Proeftuinstraat 86, B-9000 Gent, Belgium
Abstract The work we present consists of the development of a computerised 3-D scanning system running under LABVIEW威. 3-D magnetic "eld measurements, form an important application allowing a fast on-line determination of charged particle trajectories in real magnetic "elds. The set-up consists of a Hall-probe, movable in three dimensions, mimicking a charged particle entering a magnetic "eld with well-de"ned transport characteristics (energy, position, momentum). The measured local "eld components are used as input to a LABVIEW威 program, which calculates the local direction of the #ight-path of the particle. Then the probe is moved along this direction with a predetermined step-length. In this way, the real path of the charged particle is determined step by step automatically accounting for all magnetic "eld imperfections. Trajectories can be determined without the measurement of the whole 3-D "eld con"guration. 1999 Elsevier Science B.V. All rights reserved. Keywords: 3-D scanning systems; LABVIEW; Charged particles
1. Introduction The existing 15 MeV high current electron LINAC [1], at the department for Subatomic and Radiation Physics of the University Gent, is currently undergoing upgrading. Two additional sections and a new transport system to guide the electron beam to the di!erent experimental set-ups are being installed. Together with new magnets, magnets from former installations will be reused to build the electron transport system. The "rst step in the design is the use of a beam transport program. These programs require a set of magnet para* Corresponding author. Tel.: #32-09-264-6532; fax: #3209-264-6699. E-mail address: [email protected] (B. Masschaele)
meters, which are not always exactly known for a real magnet. Stray "elds, deformations and other faults, cause the real magnet to di!er from the ideal one. They give rise to deviations from the theoretically calculated trajectory. Therefore, we developed a technique to simulate charged particles in known and unknown magnetic "elds. By examining the local magnetic "eld, actually `senseda by the charged particle its path can be determined. Because the magnetic "eld is only measured along the #ight path, the number of measurements can be reduced by several orders of magnitude. Furthermore, these local "elds provide valuable information concerning the in#uence of magnet characteristics such as e!ective length, imperfections, quadrupole and sextupole coe$cients of bending magnets [2]. Another advantage of the
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 5 7 - 5
B. Masschaele et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 350}352
351
technique is its possibility to calculate automatically the nth-order transfer matrices for real magnets. These matrices can then be inserted in most of the transport programs [3,4]. Both the tracing system and the transport programs make a good combination to develop a particle transport system [5,6].
2. Description of the technique The technique of on-line charged particle tracing through magnetic "elds is based on the measurement of the "eld at the place of the particle. Therefore we mounted a Hall-probe on a scanning device. The probe simulates the charged particle as it is sent into the magnet under investigation. Excluding beam-generated forces, only the Lorentz force is acting upon the particle [7]. Without knowledge of the "eld in the entire magnet gap and its environment it is possible to trace the particle through known or unknown time-independent magnetic "elds. The Lorentz equation can be written as two "rst-order di!erential equations. These are readily transformed in couple Cartesian components q b !b " (b B !b B ) *s V V mb W X X W q (b B !b B ) *s b !b " V X W W mb X V q b !b " (b B !b B ) *s. X X mb V W W V
(1)
b x "x # V *s b b y "y # W *s b b z "z # X *s. b
Fig. 1. 0.5 MeV electron-trajectory in trap "eld: once the charged particle is trapped it starts moving around in the magnetic well. The path depends on the initial entrance conditions.
*s the step size for the trajectory calculation. We have to take into account that here only the transversal "elds are measured. To measure all three components, an extra axial Hall-probe is needed. It is possible now to deduce "rst or higher-order transport matrices for any magnet under investigation. Finding the "rst-order transfer matrix for typical magnets with midplane symmetry applied in beam transport systems requires only tracing of four `particlesa, the reference particle and three others with di!erent initial parameters. Higher orders require, depending on their symmetry, more measurements. The maximum number of trajectories for the nth order assuming no symmetry is 6L#1. The scanner was also used to investigate the behaviour of charged particles in rather unusual magnets. The `trapa-magnet is an example. It is a magnet that traps the charged particles in a magnetic well (see Fig. 1). Once the particle has overcome the border of the well it is captured inside. Depending on the initial energy and direction, the particle's path is a rosette (see Fig. 2).
(2)
B , B , B are the "eld components, b is the particle V W X velocity, b b , b are the velocity components in V W X x, y, z Cartesian coordinate system. The xz-plane is arbitrarily chosen in the midplane of the magnet. q is the charge, m"E/c the relativistic mass and
3. Accuracy of the method The previous set of equations is based on the Euler di!erence method or Leapfrog method. The drawback of this approach is that it introduces
352
B. Masschaele et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 350}352
These smaller steps are used for the calculation of the trajectories.
4. Technical information The 3D-scanner consists of three accurate stepping motors. These are able to move a long arm in three dimensions x, y, and z (see Fig. 2). The Hallprobe is mounted at the end of this arm on a servomechanism. This servomechanism is able to turn the probe over 903, which makes it possible to measure the two transversal components of the magnetic "eld (B , B ) with the same probe. A V W LABVIEW威 program controls the motors, reads out the probe and displays the results. For a typical transport matrix determination, the input parameters are the energy, the mass, the start coordinates of the reference particle, and the required matrix order. The program performs automatically the necessary steps to establish all the matrix coe$cients.
References
Fig. 2. 3D scanning device [9].
errors proportional to the "rst power of the step size. Second- or higher-order Runge}Kutta methods are not applicable since it is not possible to determine the function value, the magnetic "eld is unknown at midpoints of intervals [8]. That is why the step size has to be small. To get the most accurate results, the tracing is done by two Euler di!erence methods in one. The "rst (for the measurement of the magnetic "eld) uses a step size, *S, equal to the resolution of the Hall-probe and each of these steps are divided in smaller steps *s where the magnetic "eld is considered constant.
[1] W. Mondelaers, K. Van Laere, A. Goedefroot, K. Van den Bossche, Nucl. Instr. and Meth. A 368 (1996) 278. [2] B. Masschaele, Development of a computerised 3D-Scanning Device, Licentiate Thesis, 1997, unpublished. [3] G. Gillespie, B. Hill, N. Brown, R. Babcock, D. Carey, A graphic user interface for the particle optics code TRANSPORT, CERN 96-07 (1996) 842. [4] D.C. Carey, Third Order TRANSPORT: A Program For Designing Charged Particle Beam Transport Systems, SLAC-R-95-462 Fermilab-Pub-95/069 UC-414, 1995, p. 186. [5] S. Humphries Jr., Charged Particle Beams, Wiley, New York, 1990. [6] D.C. Carey, The Optics of Charged Particle Beams, Harwood, New York, 1987. [7] M. Conte, W.W. MacKay, An Introduction to the Physics of Particle Accelerators, World Scienti"c, Singapore, 1962. [8] W.H. Press et al., Numerical Recipes in C, Cambridge University Press, Cambridge, 1992. [9] C.J. Cason, The POV-Ray Team, POV-Ray2+, 1997.