- Email: [email protected]
Three-dimensional simulation code for the injector of PNC’s high-power, high-duty electron linac
Nuclear Instruments and Methods in Physics Research A 418 (1998) 256—266
Three-dimensional simulation code for the injector of PNC’s high-power, high-duty electron linac Masahiro Nomura *, Hiroshi Takahashi Advanced Technology division, Oarai Engineering Center, Power Reactor and Nuclear Fuel Development Corporation, 4002 Narita-cho, Oaraimachi, Ibaraki-ken, 311-13, Japan Brookhaven National Laboratory, Upton, NY, USA Received 15 December 1997; received in revised form 25 June 1998
Abstract We developed a new, three-dimensional simulation code ETRA3D to calculate beam dynamics in the injector of PNC’s high-power, high-duty electron linac. In ETRA3D, beam dynamics is calculated by the Hamilton equations, and a space-charge force is determined by the potentials produced by the other electrons. The Hamiltonian is assessed at each time-step, and the length of the time-step is varied by changing in the Hamiltonian to reduce calculation time and errors. The precision of ETRA3D was checked by comparing it with the envelope equation; the agreement between them was good. We demonstrated the capabilities of ETRA3D by an example in which the electron beam is not launched along the axis of a series of the solenoid coils; under this condition, two-dimensional codes cannot calculate its dynamics. 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The Power Reactor and Nuclear Fuel Development Corporation (PNC) is developing a high-power, high-duty electron linac for various applications including the transmutation of fission products, a Free Electron Laser, and as a positron source [1—3]. The injector for this linac consists of an electron gun, two magnetic lenses, an RF chopper, chopper slits, a pre-buncher, and a buncher. Fig. 1 shows the arrangement of the injector, and its main specifications are listed in Table 1. The
* Corresponding author. Fax: #81 29 266 3868; e-mail: [email protected].
200 keV—300 mA electron beam from the electron gun passes through the magnetic lenses to the RF chopper. The electron beam is deflected horizontally by the magnetic field in the RF chopper [4] and passes through the chopper slits during a third RF period; after leaving the slits, the electron beam is bunched by the pre-buncher and the buncher. There is 2.3 m concrete radiation shield between the electron gun room and the accelerator room. The electron beam must pass through this region without any significant growth in its emittance, as far as possible. In the low-energy-region, a spacecharge force is the predominant factor causing growth in emittance. To reduce such growth, there is a series of solenoid coils, covered from the exit of the electron gun to the first accelerator guide,
0168-9002/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 9 0 6 - 1
M. Nomura, H. Takahashi/Nucl. Instr. and Meth. in Phys. Res. A 418 (1998) 256—266
257
Fig. 1. The injector of PNC’s high-power, high-duty electron linac. The injector consists of a 200 kV DC electron gun, two magnetic lenses, an RF chopper and chopper slits, a pre-buncher and a buncher. Solenoid coils are covered from the exit of the electron gun to the first accelerator guide, except between the RF chopper and the chopper slits.
Table 1 Main specification of the injector Electron gun high voltage 200 kV(DC) Beam current at the exit of electron gun 300 mA Pulse length 4ms Pulse repetition 50 Hz Duty factor 20% RF frequency 1249.135 MHz
except between the RF chopper and chopper slits. The shape and strength of the magnetic field produced by those solenoid coils are arranged by estimating the space-charge effects properly. We developed a new, three-dimensional simulation code ETRA3D for the PNC’s injector which is designed to accurately account for the space-charge force.
2. ETRA3D When the beam dynamics is calculated, its interactions are well-known electromagnetic ones. If the charges, positions, and velocities of particles are known, then it is easy to calculate those interactions. The difficulty in assessing beam dynamics is
that a great number of particles must be handled. For example, a 0.3 A electron beam, 10 cm long, contains more than 10 electrons. This number is too high to deal with, so that, in actuality, the beam dynamics is calculated by using multi-particles which are thought of as consisting of many electrons. In ETRA3D, the position and momentum of each electron are calculated at each time-step according to the Hamilton equations. We begin with the relativistic Lagrangian in mksa units for electrons in the electromagnetic field [5]: , , ,
¸"! m c(1!b#e A ) u !e G, (1) C G G G 2 G G G where k e , u (2) A " L, G 4p r LG LG e , 1
" , (3) G 4pe r LG LG and where e and k are the permittivity and the permeability of free space, respectively, N is the number of electrons, m the electron rest mass, and c the light velocity, u the velocity of the ith elecL tron, b means "u "/c, A and are the vector and L L G G the scalar potential at the position of the ith
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electron, and r is the distance between the nth and LG ith electrons. The canonical momentum p conjuVG gate to the position coordinate x is obtained from G the Lagrangian. The canonical momentum p is VG *¸ p , "m c xR #eA , (4) G G VG VG *xR G where dx xR " G, G dt
(5)
c "1/(1!b. (6) G G The Hamiltonian is also obtained from the Lagrangian: , ,
H" (+(p !eA )c#(m c),#e G. G G 2 G G Finally, we obtain the Hamilton equations:
(7)
*H (p !eA )c VG VG xR , " , (8) G *p ((p !eA )c#(m c) VG G G *H qc VG WG XG(p !eA )f H *
H H H fVG !e G. (9) pR ,! " VG *x *x G ((pG!eAG)c#(mc) G Actually, the position and momentum of each electron are calculated from Eqs. (8) and (9) by the Runge—Kutta method. The Hamiltonian is calculated at each time-step to check the error generated. Theoretically, the Hamiltonian must be conserved if it does not depend upon time. When the space-charge force is calculated, ETRA3D does not use the mesh which is used in PARMELA [6]. When the beam dynamics is calculated by PARMELA, a mesh size and number of the mesh intervals must be carefully chosen because the results depend on their suitability. In ETRA3D, the space-charge force is calculated by the potentials produced by the other electrons, so that its effect is included accurately. In ETRA3D, beam dynamics can be calculated in external fields by adding their potentials. We show an example of the external magnetic fields produced by the solenoid coils. The vector potential produced by a circular loop of radius a, lying in the x—y plane, centered at the origin, and carrying a current I on the condition of r;a is given
approximately by y ak I A +! , V 4 [(r#a)#z]
(10)
ak I x A +# , W 4 [(r#a)#z]
(11)
A "0. (12) X The solenoid coils can be thought of as consisting of many circular loops, so that the beam dynamics in the solenoid coils can be calculated using Eqs. (10)—(12).
3. Time step In ETRA3D, beam dynamics are calculated at each time step according to the Hamilton equations; the transition matrices are not used. If we set a large time-step *t, calculation time can be saved, but this introduces errors. It is generally believed that calculations of the beam dynamics which use a short-time step have higher accuracy. But, to be exact, if the Hamiltonian does not depend upon time, then a small change in the Hamiltonian between each transition has high accuracy. First, we give examples of the beam trajectories in a series of the solenoid coils for *t"0.5;10\ s. The calculation geometry and conditions are shown in Fig. 2, and the magnetic field of about 150 gauss produced by the solenoid coils is shown in Fig. 3. The number of electrons used for this calculation is 1500 counts. The beam distribution at z"0 m is shown in Fig. 4; the normalized root-mean-square emittances e and e are L V L W (e , e )"(9.35, 9.69) [n mm mrad]. L V L W The normalized root-mean-square emittance is defined by 4bc e " (1x21x 2!1xx 2 [n mm mrad]. (13) L p The beam trajectories and distribution at z"2.5 m are shown in Figs. 5 and 6, respectively. The emittance e and e at z"2.5 m are L V L W (e , e )"(10.1, 9.98) [n mm mrad]. L V L W
M. Nomura, H. Takahashi/Nucl. Instr. and Meth. in Phys. Res. A 418 (1998) 256—266
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Fig. 2. The calculation geometry and conditions. The 200 keV—300 mA electron beam starts from z"0 m and passes through 12 solenoid coils.
Fig. 3. The magnetic field B produced by the solenoid coils. The magnetic field along z-axis B is about 150 G. X X
The change in the Hamiltonian, *H , is depicted in L Fig. 7. The *H is defined by L H !H L> L , *H "ABS (14) L> H L where n denotes the number of the time-step. Fig. 7 shows that the *H is almost constant. L
Next, we show examples for *t"2.5;10\ s. The calculation conditions are the same as before, except for the length of time step. The beam distribution at z"2.5 m is shown in Fig. 8 and the emittance e and e are L V L W (e , e )"(56.1, 66.6) [n mm mrad]. L V L W
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distribution
at
z"2.5 m
Fig. 4. The beam distribution at z"0 m. The positions x and y at z"0 m are generated randomly with Gaussian distribution; their dispersions are decided by the beam radius which, in this calculation, is 0.003 m.
Fig. 6. The beam *t"0.5;10\ s.
for
In Fig. 8, some electrons are seen to be far from the beam center, especially one that is very far from it. These electrons enlarge the emittance. There are large differences between these results and previous
ones on the beam distribution and emittance. The *H for *t"2.5;10\ s is shown in Fig. 9; sevL eral big jumps have occurred, and those correspond to the nodes of the beam trajectories in Fig. 5. In
Fig. 5. The beam trajectories for *t"0.5;10\ s. The electron beam spirals along the magnetic field B while it converges and X diverges repeatedly. The nodes are shown by arrows.
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Fig. 7. The change in the Hamiltonian *H for *t"0.5;10\ s; the *H is almost constant. L L
Fig. 8. The beam distribution at z"2.5 m for *t"2.5;10\ s. Dots which are shown by arrows correspond to electrons. Some electrons are far from the beam center, especially one electron is very far from it. The difference between Figs. 6 and 8 is marked.
these nodes, electron density is high and interactions are strong, so that a short time step is required. The 2.5;10\ s is not short enough; con-
sequently, the *H becomes large. To prevent such L an occurrence, we set an upper limit of *H in L ETRA3D; if it exceeds this limit, *t is cut in half and the beam dynamics is automatically recalculated. Ideally, the length of time-step should be changed at each transition, so that *H can be kept L small. But practically, it is too complicated to calculate a suitable length at each transition. Finally, we show examples for *t"2.5; 10\ s, and 5.0;10\ upper limit of *H . The L *H and beam distribution are shown in Figs. 10 L and 11, respectively. The emittance e and e are L V L W (e , e )"(11.0, 10.8) [n mm mrad]. L V L W Fig. 10 demonstrates that the *H remains small L and almost constant. The beam distribution and emittances agree well with the results for *t" 0.5;10\ s. This method of setting the upper limit of *H can reduce both the calculation time and the L errors arising from a large time-step. The main purpose of ETRA3D is to calculate the beam dynamics in the injector of PNC’s high-power, highduty electron linac, especially from the electron gun to the RF chopper. These parts do not include the RF fields; therefore, we can use this method.
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Fig. 9. The change in the Hamiltonian *H for *t"2.5;10\ s. Several big jumps occurred corresponding to the nodes of the beam L trajectories in Fig. 5.
Fig. 10. The change in the Hamiltonian *H for *t"2.5;10\ s and 5.0;10\ upper limit of *H . The *H remains small and L L L almost constant.
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4. Examples and discussions 4.1. Envelope equation The usual way to check the precision of a numerical procedure is to evaluate a variant for which a solution is known. We chose the case of beam dynamics in free space. The envelope of a round beam under the influence of the space-charge can be calculated numerically by the envelope equation [7]: dR e m #KR! " , dz R 2R Fig. 11. The beam distribution at z"2.5 m for *t"2.5;10\ s and 5.0;10\ upper limit of *H . The beam L distribution agrees very well with the results for *t"0.5;10\ s.
(15)
where R is the beam radius, and K, e, and m are the focusing strength, emittance, and space-charge parameter, respectively. The space charge parameter, m, is given by 4r j m" , bc
(16)
Fig. 12. The beam envelope in free space. The beam current, the electron energy, the beam radius and the emittance are 300 mA, 200 keV, 0.002 m, and 0 p mm mrad, respectively. The beam distribution is uniform. We used 5000 electrons for this calculation by ETRA3D. For a continuous beam, ETRA3D cannot calculate the entire beam, so a 0.5 m portion of it was used. The trajectories of the head and tail ("0.1 m) were not calculated because they did not include the space-charge effects properly. The solid line shows the result by envelope equation. The correspondence between the results by ETRA3D and the envelope equation is close.
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Table 2 The conditions for calculating the beam envelope Beam current Electron energy Beam radius R at z"0 (uniform distribution) Focusing strength K Emittance e Space charge parameter m
300 mA 200 keV 0.002 m 0 m\ 0 p mm mrad 1.1;10\
where j is a uniform, longitudinal particle-density, and r is a classical radius of the electrons. We calculated the beam envelopes using the envelope equation and ETRA3D. The calculation conditions are shown in Table 2. For a continuous beam, ETRA3D cannot calculate the entire beam, so a 0.5 m portion of it (5000 electrons) was calculated. The head and tail ("0.1 m) were not used for the trajectories because they do not include the spacecharge effects properly. The results from the envelope equation and ETRAD3D are shown in Fig. 12: the correspondence between them is close.
Fig. 13. The beam distribution at z"0 m. The positions x and y at z"0 m are generated randomly with Gaussian distribution; and a beam radius of 0.002 m and 3000 electrons are used. The beam center coincides with the axis of the solenoid coils at z"0 m.
4.2. Beam dynamics in the series of the solenoid coils To demonstrate the capabilities of our code, we choose the case in which the electron beam is not launched along the axis of the solenoid coils because of errors in alignment between them and the electron gun. We assumed that the cylindrically symmetric electron beams from the electron gun are at an angle to the axis of the solenoid coils. The beam current I and the electron energy E are same as in the example of the envelope calculations. The beam distribution and x—h on the phase space V at z"0 are shown in Figs. 13 and 14. The beam distribution is cylindrically symmetric and its center coincides with the axis of the solenoid coils at z"0. But the beam is set at an angle to the axis of solenoid coils by adding 3 mrad to all h and h . V W Figs. 15 and 16, respectively, show the magnetic field produced by the solenoid coils and the beam trajectories. In Fig. 16, the entire beam spirals along the axis of the solenoid coils. The beam distribution and x—h at z"2.5 m are shown in V Figs. 17 and 18, respectively; it is clear from Fig. 17 that the center of the beam does not coincide with
Fig. 14. The x—h on the phase space at z"0 m. The angles V h and h at z"0 m are generated randomly with Gaussian V W distribution, and 3 mrad is added to all h and h so that the V W beam is at an angle to the x and y axes.
that of the solenoid coils, and the beam distribution has lost its cylindrical symmetry. If the center of the beam does not coincide but the distribution still keeps its shape, it is not difficult to launch the beam along the beam transport system. But if the shape of the distribution changes, it cannot be restored to
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Fig. 15. The magnetic field produced by the solenoid coils. The arrangement of the solenoid coils is same as shown in Fig. 2.
Fig. 16. The beam trajectories in the series of the solenoid coils. The whole beam spirals along the axis in the magnetic field produced by solenoid coils.
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5. Conclusion We developed a new, three-dimensional simulation code ETRA3D for the injector of PNC’s highpower, high-duty electron linac. We showed that calculation time and errors could be reduced by varying the length of the time-step according to the change in the Hamiltonian between each transition. The precision of ETRA3D was checked by comparing it with the envelope equation. The agreement between them was good. The capabilities of ETRA3D were demonstrated by an example in which two-dimensional codes cannot calculate the beam dynamics. Fig. 17. The beam distribution at z"2.5 m. The center of the beam does not coincide with one of the solenoid coils and its distribution has lost cylindrical symmetry.
Acknowledgements The authors are grateful to Dr. An Yu for valuable discussions and Dr. Woodhead for editorial work. This work was performed under the auspices of the U.S. Department of Energy under Contract No. DE-AC02-76CH00016, and the Power Reactor and Nuclear Fuel Development Corporation.
References
Fig. 18. The x—h on the phase space at z"2.5 m. V
its original shape. This simple example demonstrates that two-dimensional codes cannot calculate beam dynamics when the electron beam is not launched along the axis of a perfectly aligned beam transport system.
[1] S. Toyama et al., Transmutation of long-lived fission product (Cs, Sr) by reactor—accelerator system, Proc. Int. Symp. on Advanced Nuclear Energy Research, 1990. [2] M. Nomura et al., Status of high power CW linear at PNC, Proc. EPAC, London, 1994. [3] Y. Yamazaki et al., The electron gun for the PNC high power CW linac, Proc. LINAC’94, Japan, 1994. [4] Y.L. Wang et al., A novel chopper system for high power CW linac, Proc. LINAC’94, Japan, 1994. [5] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1950. [6] F.L. Krawczyk, J. Billen, R. Ryne, H. Takeda, L.M. Young, The Los Alamos Accelerator Code Group, Proc. 1995 Particle Accelerator Conf. and Int. Conf. on High Energy Accelerators, 1995. [7] A.W. Chao, Physics of Collective Beam Instabilities in High energy Accelerators, Wiley, New York, 1993.
Three-dimensional simulation code for the injector of PNC’s high-power, high-duty electron linac Masahiro Nomura *, Hiroshi Takahashi Advanced Technology division, Oarai Engineering Center, Power Reactor and Nuclear Fuel Development Corporation, 4002 Narita-cho, Oaraimachi, Ibaraki-ken, 311-13, Japan Brookhaven National Laboratory, Upton, NY, USA Received 15 December 1997; received in revised form 25 June 1998
Abstract We developed a new, three-dimensional simulation code ETRA3D to calculate beam dynamics in the injector of PNC’s high-power, high-duty electron linac. In ETRA3D, beam dynamics is calculated by the Hamilton equations, and a space-charge force is determined by the potentials produced by the other electrons. The Hamiltonian is assessed at each time-step, and the length of the time-step is varied by changing in the Hamiltonian to reduce calculation time and errors. The precision of ETRA3D was checked by comparing it with the envelope equation; the agreement between them was good. We demonstrated the capabilities of ETRA3D by an example in which the electron beam is not launched along the axis of a series of the solenoid coils; under this condition, two-dimensional codes cannot calculate its dynamics. 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The Power Reactor and Nuclear Fuel Development Corporation (PNC) is developing a high-power, high-duty electron linac for various applications including the transmutation of fission products, a Free Electron Laser, and as a positron source [1—3]. The injector for this linac consists of an electron gun, two magnetic lenses, an RF chopper, chopper slits, a pre-buncher, and a buncher. Fig. 1 shows the arrangement of the injector, and its main specifications are listed in Table 1. The
* Corresponding author. Fax: #81 29 266 3868; e-mail: [email protected].
200 keV—300 mA electron beam from the electron gun passes through the magnetic lenses to the RF chopper. The electron beam is deflected horizontally by the magnetic field in the RF chopper [4] and passes through the chopper slits during a third RF period; after leaving the slits, the electron beam is bunched by the pre-buncher and the buncher. There is 2.3 m concrete radiation shield between the electron gun room and the accelerator room. The electron beam must pass through this region without any significant growth in its emittance, as far as possible. In the low-energy-region, a spacecharge force is the predominant factor causing growth in emittance. To reduce such growth, there is a series of solenoid coils, covered from the exit of the electron gun to the first accelerator guide,
0168-9002/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 9 0 6 - 1
M. Nomura, H. Takahashi/Nucl. Instr. and Meth. in Phys. Res. A 418 (1998) 256—266
257
Fig. 1. The injector of PNC’s high-power, high-duty electron linac. The injector consists of a 200 kV DC electron gun, two magnetic lenses, an RF chopper and chopper slits, a pre-buncher and a buncher. Solenoid coils are covered from the exit of the electron gun to the first accelerator guide, except between the RF chopper and the chopper slits.
Table 1 Main specification of the injector Electron gun high voltage 200 kV(DC) Beam current at the exit of electron gun 300 mA Pulse length 4ms Pulse repetition 50 Hz Duty factor 20% RF frequency 1249.135 MHz
except between the RF chopper and chopper slits. The shape and strength of the magnetic field produced by those solenoid coils are arranged by estimating the space-charge effects properly. We developed a new, three-dimensional simulation code ETRA3D for the PNC’s injector which is designed to accurately account for the space-charge force.
2. ETRA3D When the beam dynamics is calculated, its interactions are well-known electromagnetic ones. If the charges, positions, and velocities of particles are known, then it is easy to calculate those interactions. The difficulty in assessing beam dynamics is
that a great number of particles must be handled. For example, a 0.3 A electron beam, 10 cm long, contains more than 10 electrons. This number is too high to deal with, so that, in actuality, the beam dynamics is calculated by using multi-particles which are thought of as consisting of many electrons. In ETRA3D, the position and momentum of each electron are calculated at each time-step according to the Hamilton equations. We begin with the relativistic Lagrangian in mksa units for electrons in the electromagnetic field [5]: , , ,
¸"! m c(1!b#e A ) u !e G, (1) C G G G 2 G G G where k e , u (2) A " L, G 4p r LG LG e , 1
" , (3) G 4pe r LG LG and where e and k are the permittivity and the permeability of free space, respectively, N is the number of electrons, m the electron rest mass, and c the light velocity, u the velocity of the ith elecL tron, b means "u "/c, A and are the vector and L L G G the scalar potential at the position of the ith
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electron, and r is the distance between the nth and LG ith electrons. The canonical momentum p conjuVG gate to the position coordinate x is obtained from G the Lagrangian. The canonical momentum p is VG *¸ p , "m c xR #eA , (4) G G VG VG *xR G where dx xR " G, G dt
(5)
c "1/(1!b. (6) G G The Hamiltonian is also obtained from the Lagrangian: , ,
H" (+(p !eA )c#(m c),#e G. G G 2 G G Finally, we obtain the Hamilton equations:
(7)
*H (p !eA )c VG VG xR , " , (8) G *p ((p !eA )c#(m c) VG G G *H qc VG WG XG(p !eA )f H *
H H H fVG !e G. (9) pR ,! " VG *x *x G ((pG!eAG)c#(mc) G Actually, the position and momentum of each electron are calculated from Eqs. (8) and (9) by the Runge—Kutta method. The Hamiltonian is calculated at each time-step to check the error generated. Theoretically, the Hamiltonian must be conserved if it does not depend upon time. When the space-charge force is calculated, ETRA3D does not use the mesh which is used in PARMELA [6]. When the beam dynamics is calculated by PARMELA, a mesh size and number of the mesh intervals must be carefully chosen because the results depend on their suitability. In ETRA3D, the space-charge force is calculated by the potentials produced by the other electrons, so that its effect is included accurately. In ETRA3D, beam dynamics can be calculated in external fields by adding their potentials. We show an example of the external magnetic fields produced by the solenoid coils. The vector potential produced by a circular loop of radius a, lying in the x—y plane, centered at the origin, and carrying a current I on the condition of r;a is given
approximately by y ak I A +! , V 4 [(r#a)#z]
(10)
ak I x A +# , W 4 [(r#a)#z]
(11)
A "0. (12) X The solenoid coils can be thought of as consisting of many circular loops, so that the beam dynamics in the solenoid coils can be calculated using Eqs. (10)—(12).
3. Time step In ETRA3D, beam dynamics are calculated at each time step according to the Hamilton equations; the transition matrices are not used. If we set a large time-step *t, calculation time can be saved, but this introduces errors. It is generally believed that calculations of the beam dynamics which use a short-time step have higher accuracy. But, to be exact, if the Hamiltonian does not depend upon time, then a small change in the Hamiltonian between each transition has high accuracy. First, we give examples of the beam trajectories in a series of the solenoid coils for *t"0.5;10\ s. The calculation geometry and conditions are shown in Fig. 2, and the magnetic field of about 150 gauss produced by the solenoid coils is shown in Fig. 3. The number of electrons used for this calculation is 1500 counts. The beam distribution at z"0 m is shown in Fig. 4; the normalized root-mean-square emittances e and e are L V L W (e , e )"(9.35, 9.69) [n mm mrad]. L V L W The normalized root-mean-square emittance is defined by 4bc e " (1x21x 2!1xx 2 [n mm mrad]. (13) L p The beam trajectories and distribution at z"2.5 m are shown in Figs. 5 and 6, respectively. The emittance e and e at z"2.5 m are L V L W (e , e )"(10.1, 9.98) [n mm mrad]. L V L W
M. Nomura, H. Takahashi/Nucl. Instr. and Meth. in Phys. Res. A 418 (1998) 256—266
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Fig. 2. The calculation geometry and conditions. The 200 keV—300 mA electron beam starts from z"0 m and passes through 12 solenoid coils.
Fig. 3. The magnetic field B produced by the solenoid coils. The magnetic field along z-axis B is about 150 G. X X
The change in the Hamiltonian, *H , is depicted in L Fig. 7. The *H is defined by L H !H L> L , *H "ABS (14) L> H L where n denotes the number of the time-step. Fig. 7 shows that the *H is almost constant. L
Next, we show examples for *t"2.5;10\ s. The calculation conditions are the same as before, except for the length of time step. The beam distribution at z"2.5 m is shown in Fig. 8 and the emittance e and e are L V L W (e , e )"(56.1, 66.6) [n mm mrad]. L V L W
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distribution
at
z"2.5 m
Fig. 4. The beam distribution at z"0 m. The positions x and y at z"0 m are generated randomly with Gaussian distribution; their dispersions are decided by the beam radius which, in this calculation, is 0.003 m.
Fig. 6. The beam *t"0.5;10\ s.
for
In Fig. 8, some electrons are seen to be far from the beam center, especially one that is very far from it. These electrons enlarge the emittance. There are large differences between these results and previous
ones on the beam distribution and emittance. The *H for *t"2.5;10\ s is shown in Fig. 9; sevL eral big jumps have occurred, and those correspond to the nodes of the beam trajectories in Fig. 5. In
Fig. 5. The beam trajectories for *t"0.5;10\ s. The electron beam spirals along the magnetic field B while it converges and X diverges repeatedly. The nodes are shown by arrows.
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Fig. 7. The change in the Hamiltonian *H for *t"0.5;10\ s; the *H is almost constant. L L
Fig. 8. The beam distribution at z"2.5 m for *t"2.5;10\ s. Dots which are shown by arrows correspond to electrons. Some electrons are far from the beam center, especially one electron is very far from it. The difference between Figs. 6 and 8 is marked.
these nodes, electron density is high and interactions are strong, so that a short time step is required. The 2.5;10\ s is not short enough; con-
sequently, the *H becomes large. To prevent such L an occurrence, we set an upper limit of *H in L ETRA3D; if it exceeds this limit, *t is cut in half and the beam dynamics is automatically recalculated. Ideally, the length of time-step should be changed at each transition, so that *H can be kept L small. But practically, it is too complicated to calculate a suitable length at each transition. Finally, we show examples for *t"2.5; 10\ s, and 5.0;10\ upper limit of *H . The L *H and beam distribution are shown in Figs. 10 L and 11, respectively. The emittance e and e are L V L W (e , e )"(11.0, 10.8) [n mm mrad]. L V L W Fig. 10 demonstrates that the *H remains small L and almost constant. The beam distribution and emittances agree well with the results for *t" 0.5;10\ s. This method of setting the upper limit of *H can reduce both the calculation time and the L errors arising from a large time-step. The main purpose of ETRA3D is to calculate the beam dynamics in the injector of PNC’s high-power, highduty electron linac, especially from the electron gun to the RF chopper. These parts do not include the RF fields; therefore, we can use this method.
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Fig. 9. The change in the Hamiltonian *H for *t"2.5;10\ s. Several big jumps occurred corresponding to the nodes of the beam L trajectories in Fig. 5.
Fig. 10. The change in the Hamiltonian *H for *t"2.5;10\ s and 5.0;10\ upper limit of *H . The *H remains small and L L L almost constant.
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4. Examples and discussions 4.1. Envelope equation The usual way to check the precision of a numerical procedure is to evaluate a variant for which a solution is known. We chose the case of beam dynamics in free space. The envelope of a round beam under the influence of the space-charge can be calculated numerically by the envelope equation [7]: dR e m #KR! " , dz R 2R Fig. 11. The beam distribution at z"2.5 m for *t"2.5;10\ s and 5.0;10\ upper limit of *H . The beam L distribution agrees very well with the results for *t"0.5;10\ s.
(15)
where R is the beam radius, and K, e, and m are the focusing strength, emittance, and space-charge parameter, respectively. The space charge parameter, m, is given by 4r j m" , bc
(16)
Fig. 12. The beam envelope in free space. The beam current, the electron energy, the beam radius and the emittance are 300 mA, 200 keV, 0.002 m, and 0 p mm mrad, respectively. The beam distribution is uniform. We used 5000 electrons for this calculation by ETRA3D. For a continuous beam, ETRA3D cannot calculate the entire beam, so a 0.5 m portion of it was used. The trajectories of the head and tail ("0.1 m) were not calculated because they did not include the space-charge effects properly. The solid line shows the result by envelope equation. The correspondence between the results by ETRA3D and the envelope equation is close.
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Table 2 The conditions for calculating the beam envelope Beam current Electron energy Beam radius R at z"0 (uniform distribution) Focusing strength K Emittance e Space charge parameter m
300 mA 200 keV 0.002 m 0 m\ 0 p mm mrad 1.1;10\
where j is a uniform, longitudinal particle-density, and r is a classical radius of the electrons. We calculated the beam envelopes using the envelope equation and ETRA3D. The calculation conditions are shown in Table 2. For a continuous beam, ETRA3D cannot calculate the entire beam, so a 0.5 m portion of it (5000 electrons) was calculated. The head and tail ("0.1 m) were not used for the trajectories because they do not include the spacecharge effects properly. The results from the envelope equation and ETRAD3D are shown in Fig. 12: the correspondence between them is close.
Fig. 13. The beam distribution at z"0 m. The positions x and y at z"0 m are generated randomly with Gaussian distribution; and a beam radius of 0.002 m and 3000 electrons are used. The beam center coincides with the axis of the solenoid coils at z"0 m.
4.2. Beam dynamics in the series of the solenoid coils To demonstrate the capabilities of our code, we choose the case in which the electron beam is not launched along the axis of the solenoid coils because of errors in alignment between them and the electron gun. We assumed that the cylindrically symmetric electron beams from the electron gun are at an angle to the axis of the solenoid coils. The beam current I and the electron energy E are same as in the example of the envelope calculations. The beam distribution and x—h on the phase space V at z"0 are shown in Figs. 13 and 14. The beam distribution is cylindrically symmetric and its center coincides with the axis of the solenoid coils at z"0. But the beam is set at an angle to the axis of solenoid coils by adding 3 mrad to all h and h . V W Figs. 15 and 16, respectively, show the magnetic field produced by the solenoid coils and the beam trajectories. In Fig. 16, the entire beam spirals along the axis of the solenoid coils. The beam distribution and x—h at z"2.5 m are shown in V Figs. 17 and 18, respectively; it is clear from Fig. 17 that the center of the beam does not coincide with
Fig. 14. The x—h on the phase space at z"0 m. The angles V h and h at z"0 m are generated randomly with Gaussian V W distribution, and 3 mrad is added to all h and h so that the V W beam is at an angle to the x and y axes.
that of the solenoid coils, and the beam distribution has lost its cylindrical symmetry. If the center of the beam does not coincide but the distribution still keeps its shape, it is not difficult to launch the beam along the beam transport system. But if the shape of the distribution changes, it cannot be restored to
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Fig. 15. The magnetic field produced by the solenoid coils. The arrangement of the solenoid coils is same as shown in Fig. 2.
Fig. 16. The beam trajectories in the series of the solenoid coils. The whole beam spirals along the axis in the magnetic field produced by solenoid coils.
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5. Conclusion We developed a new, three-dimensional simulation code ETRA3D for the injector of PNC’s highpower, high-duty electron linac. We showed that calculation time and errors could be reduced by varying the length of the time-step according to the change in the Hamiltonian between each transition. The precision of ETRA3D was checked by comparing it with the envelope equation. The agreement between them was good. The capabilities of ETRA3D were demonstrated by an example in which two-dimensional codes cannot calculate the beam dynamics. Fig. 17. The beam distribution at z"2.5 m. The center of the beam does not coincide with one of the solenoid coils and its distribution has lost cylindrical symmetry.
Acknowledgements The authors are grateful to Dr. An Yu for valuable discussions and Dr. Woodhead for editorial work. This work was performed under the auspices of the U.S. Department of Energy under Contract No. DE-AC02-76CH00016, and the Power Reactor and Nuclear Fuel Development Corporation.
References
Fig. 18. The x—h on the phase space at z"2.5 m. V
its original shape. This simple example demonstrates that two-dimensional codes cannot calculate beam dynamics when the electron beam is not launched along the axis of a perfectly aligned beam transport system.
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