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Fringe field effects in the spiral inflector
Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Fringe field effects in the spiral inflector Dragan Toprek The Institute of Nuclear Sciences VINCA, Laboratory for Nuclear and Plasma Physics, P.O. Box 522, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 27 October 2008 Available online 24 February 2009
a b s t r a c t In this paper is studied the effects of the fringe field in the spiral inflector. Ó 2009 Elsevier B.V. All rights reserved.
PACS: 41.85.p Keywords: Fringe field Distribution Spiral inflector
1. Introduction The spiral inflector was invented by Belmont and Pabot a long time ago [1,2]. Since then many papers and Ph.D. thesis are written regarding to its characteristics and properties. See for example [3– 13]. The spiral inflector is the most famous device used to bend a low energy beam which is axially injected into a cyclotron onto the median plane. Despite the fact that the spiral inflector worsen the beam emittance it is often chose to inject the beam due to its small size 3 cm The spiral inflector has two free parameters. One parameter is 0 A0 (the inflector height) and the second one is k (a parameter which is related to the direction of the electric field, called tilt). Using these two parameters it is convenient to define a new parameter K0 as 0
K0 ¼
A0 k þ : 2RM0 2
ð1Þ
The inflector height A0 (it is actually a radius of curvature which the ion would have if it were acted upon by an electric field only) and the radius of curvature RM0 which the ion would have if it were acted upon by the magnetic field in the absence of any electric field (RM0 is the magnetic radius; spiral inflector is placed in a cyclotron’s magnetic field) are given by
A0 ¼
T0 qEel
RM0 ¼
p0 ; qB0
ð2Þ
where T0, q and p0 are the kinetic energy, the electric charge and the total momentum of the reference ion, respectively, Eel and B0 are the magnitude of the electric field and induction of the magnetic field, respectively. The schematic drawing of the spiral inflector, which is basically an electrostatic 90° bend with its two electrodes (across which an electrostatic potential is applied) deformed into a spiral so that the electric field remains orthogonal to the beam as it is also bent by the cyclotron’s magnetic field, is presented in Fig. 1; non-tilted case (a) and tilted case (b) [12]. 2. Fringe field distributions 2.1. Magnitude of the fringe field To find out the field distribution it is necessary to find out its magnitude first. ^r It is taken a simplifying assumption that the fringe field has u ^ r is one of the unit vectors of the rotated optidirection [5] where u cal coordinate system [3–5,11–13]. The magnitude of the fringe field EF is given by
EF ð0; 0; 1Þ ¼
Eel ½1 tan h½gð1 1E þ nÞ; 2
where Eel is the magnitude of the electric field on the central trajectory and constants g and n come from an empirical fit given by
g ¼ 1:35 n ¼ Gð0:37 e2:5d=G Þ; E-mail address: [email protected] 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.02.056
ð3aÞ
ð3bÞ
where G is distance between the entrance/exit of the spiral inflector end the plane where the fringe field starts/finish. In this study it is
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
2.2. Electric field distribution – general expression The expression for the first order change in the electric field with respect to the central ray is [3–5,13]
~ r ; hr ; 1Þ Eð0; 0; 1Þ Nð0; ~ 0; 1Þ Eður ; hr ; 1Þ ¼ Eður ; hr ; 1Þ Nðu D~ þ
1 @~ E ; j A0 @b ur ¼hr ¼0
ð4Þ
~ r ; hr ; 1Þ is the normal where E is the magnitude of the field and Nðu vector to the equipotential surface that contains the central ray, and thus the direction of the electric field on that surface [5]. In ~ is given by [5] the case of the spiral inflector N
^r h dh ~ r ; hr ; 1Þ u ^r r ^r Nðu u A0 db Fig. 1. Sketch of the electrodes of the spiral inflector. In case (a) the electrode is not 0 0 slanted (k = 0) and in case (b) the electrode is slanted (k – 0) for some angle.
used G = d where d is distance between electrodes of the spiral inflector. In the Eq. (1) fE is the location of the inflector entrance/exit. The plus sign is used at the entrance and the minus sign at the exit of the spiral inflector. Comparison between the analytic fringe field and hard edge ^ r direction is presented in Fig. 2 in the case for one vafield in the u lue of the inflector height A (A = 3 cm) but for two values of the 0 0 0 inflector tilt parameter k (k = 0.0 and k = 0.70, respectively). From this figure it can be seen that the assumption for fringe field is quite good. In this figure b is the instantaneous angle between the velocity vector and the vertical [3–5,13]. At the entrance/exit of the spiral inflector b is zero or p/2, respectively. The magnitude ^ r component of the electric field is increasing trough the of the u 0 spiral inflector (in the case when k – 0) since the electrode spacing ^ component of the electric is decreasing in order to obtain the u field constant [3–5,13].
HARD EDGE and FRINGE FIELDS DISTRIBUTIONS A0 = 3 cm
k’ = 0
K0 = 0.93168
! ^þ 1
ur @EF ^: 1 A0 Eel @b
ð5Þ
2.3. Fringe field distributions Now, when the magnitude of the fringe field is known, it is possible to derive its distribution. In this case E and Eel in the Eqs. (4) and (5) should be replaced by EF given by Eq. (3). Through the physical body of the spiral inflector the orienta^r and 1 ^ are not con^r ; h tions (directions) of the unit vectors u stant; they depend of the angle b [3–5,13]. Outside of the inflector’s body the directions of the unit vectors are constant; ^ r =db ¼ d1 ^=db ¼ 0. The consequence of this is that ^ r =db ¼ dh du the fringe field regions, at the inflector entrance and exit, will be divided into two parts; one part which is outside of the inflector’s body (b 2 ½b1 ; b2 and b 2 ½b5 ; b6 in Fig. 2) and one part which is inside of the inflector’s body (b 2 ½b2 ; b3 and b 2 ½b4 ; b5 in Fig. 2). ^ r and The unit vectors of the rotated optical coordinate system u ^ r are defined as: h 0
1 k sin b ^ ^ r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ^ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h; u 0 0 2 2 1 þ k sin b 1 þ k sin b
ð6aÞ
0
1 k sin b ^ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^; ffih ffiu h 0 0 2 2 1 þ k sin b 1 þ k sin b
ð6bÞ
^ are the unit vectors of the optical coordinate system ^ and h where u [3–5,11–13]. 2.3.1. Fringe field distributions through intervals b 2 ½b1 ; b2 and b 2 ½b5 ; b6 By using the following abbreviation 0
C2 ¼
b1
b2
A0 = 3 cm
b3
k’ = 0.7
b4
b5
b6
1:35k 02
2
2
ð1 þ k sin bÞCh
;
ð7aÞ
where Ch ¼ cosh½gð1 1E þ nÞ we got for the fringe field distributions inside these intervals
K0 = 1.2817
0 i E0 k h ^ þ ðu þ hk0 sin bÞ1 ^ : ^ þ 1k0 sin bh Eðu; h; 1Þ ¼ D~ C 2 1u 2A0
ð7bÞ
The upper sign is used at the inflector entrance but b 2 ½b1 ; b2 (see Fig. 2) and the lower sign is used at the inflector exit but b 2 ½b5 ; b6 (see Fig. 2).
b1
b2
b3
b4
b5
b6
Fig. 2. Fringe field (full line) and hard edge field (dotted line) along the trajectory in the spiral inflector. The marks are explained in the text.
2.3.2. Fringe field distributions through intervals b 2 ½b2 ; b3 and b 2 ½b4 ; b5 Through these intervals for the fringe field distributions we got
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
E0 Eðu; h; 1Þ ¼ D~ 1k0 sin bðC 2 cos b C 1 Þu^ 2A0 E0 1 cos bðC 2 k02 sin2 b þ C 1 Þh^ 2A0 E0 0 ðh uk sin bÞC 1 cos b þ 2A0 0 0 ^; u þ hk sin bÞC 2 k sin b cos b 1C 3 1
3.1. Differential equations of paraxial trajectory through intervals b 2 ½b1 ; b2 and b 2 ½b5 ; b6 Through these intervals of the angle b the electric field distribur_ and ~ r€ (remember, inside these intertion is given by Eq. (7) and D~ vals the orientations of the unit vectors are constant) are given by
ð8aÞ
where are
0 k C 1 ¼ ð1 ThÞ 2K 0 þ and 02 2 1 þ k sin b 0 2 C 3 ¼ ð1 ThÞ 1 þ 2K 0 k sin b ;
ð8bÞ
where is Th ¼ tan h½gð1 1E þ nÞ. The upper sign is used at the inflector entrance but b 2 ½b2 ; b3 (see Fig. 2) and the lower sign is used at the inflector exit but b 2 ½b4 ; b5 (see Fig. 2). 2.3.3. Electric field distribution through region b 2 ½b3 ; b4 Through this interval of the angle b the components of the electric field distributions are given by [3–5,11–13]
( ) 0 2 E0 1 þ 2K 0 k sin b 0 0 12K 0 k sinbcosb ; DEu ðu;h; 1Þ ¼ ðu þ hk sinbÞ 02 2 A0 1 þ k sin b ð9aÞ (
DEh ðu;h; 1Þ ¼
0
2
)
E0 1 þ 2K 0 k sin b 0 0 0 ðu þ hk sinbÞ k sinb þ 1ð2K 0 þ k Þcosb ; 02 2 A0 1 þ k sin b
ð9bÞ
2
0
ð9cÞ
3. Differential equations of paraxial trajectory
0 c
c
c
c
ð10aÞ ð10bÞ
We shall assume that the magnetic field ~ B is constant throughout the all intervals of the angle b and is defined by the relation [3– 5,11–13]
^ ¼ B0 ½sin b u ~ ^; ^ cos b 1 B ¼ B0 k
ð11Þ
^ is the unit vector of the Cartesian coordinate system. Then where k if we define
^ þ 11 ^; ^ þ hh D~ rðtÞ ¼ ~ rðtÞ ~ r c ðtÞ ¼ uu
ð12Þ
we get from Eqs. (10a) and (10b)
h i rÞ þ D~ r€ ¼ q D~ r_ ~ B : m0 D~ EðD~ In the Eq. (13) D~ rÞ is the electric field distribution. EðD~
v 2 h 00 ^ 00 ^ 00 ^i D~ a uþb hþc 1 ; r€ ¼ A0
ð14bÞ
where a, b and c are dimensionless parameters defined as
a¼
u ; A0
b¼
h ; A0
c¼
1 A0
ð14cÞ
:
In the Eqs. (14a) and (14b) dot means differentiating with respect to time t and prime means differentiating with respect to the angle b; d d ¼ vA00 db . dt Using Eqs. (7), (13) and (14) the differential equations of paraxial trajectory through [b1, b2] interval are:
a00 ¼ b00 ¼
c00 ¼
gk02 sin b cos b 2
02
2
c ð2K 0 k0 Þb0 ;
ð15aÞ
2
cK 0 sin b þ ð2K 0 k0 Þa0 ;
ð15bÞ
2
ða þ bk sin bÞ;
2ð1 þ k sin bÞChþ
gk02 sin b cos b 2
02
2ð1 þ k sin bÞChþ
gk02 sin b cos b 2
02
0
2ð1 þ k sin bÞChþ
a00 ¼
b00 ¼
A trajectory is said to be paraxial provided it is located close to the central trajectory of reference particle. Using purely analytic method we were ably to derive a system of differential equations governing the paraxial trajectories. Let ~ rðtÞ be the position vector of the ion with mass m0 on the paraxial trajectory at time t, and let ~ r c ðtÞ be the position vector of the central trajectory ion at time t. Then both ~ rðtÞ and ~ r c ðtÞ must satisfy the Lorentz equation of motion
_ ~ € ¼ q½~ m0~ rðtÞ Bð~ rÞ; rðtÞ Eð~ rc Þ þ ~ € _ ~ ~ r Þ þ r~ ðtÞ Bð~ r Þ: m r~ ðtÞ ¼ q½Eð~
ð14aÞ
ð15cÞ
Similar, the differential equations of paraxial trajectory through [b5, b6] interval are:
E0 0 0 DE1 ðu; h; 1Þ ¼ fu2K 0 k sin b cos b þ hð2K 0 þ k Þ cos b A0 1½1 þ 2K 0 k sin bg:
h i v 0 d h ^ ^ ^i ^ þ c0 1 ^ ; ^ þ b0 h D~ uu þ hh þ 11 ¼ v 0 a0 u r_ ¼ A0 db
ð13Þ
c00 ¼
gk02 sin b cos b 02
2
2
c;
ð16aÞ
2
ck0 sin b þ ð2K 0 k0 Þc0 ;
ð16bÞ
2
ða þ bk sin bÞ ð2K 0 k Þb0 :
2ð1 þ k sin bÞCh
gk02 sin b cos b 02
2
2ð1 þ k sin bÞCh
gk02 sin b cos b 02
2
0
2ð1 þ k sin bÞCh
0
ð16cÞ
3.2. Differential equations of paraxial trajectory through intervals b 2 ½b2 ; b3 and b 2 ½b4 ; b5 Through these intervals of the angle b the electric field distribur_ and D~ r€ (remember, inside these tion is given by Eq. (8) and D~ intervals the orientations of the unit vectors are not constant) are given by
v 0 d h ^ ^ ^i r_ ¼ D~ uu þ hh þ 11 A0 db ^ þ ð2K 0 a cos b þ b0 ¼ v 0 ½ða0 2K 0 b cos b þ cÞu ^ þ ðc0 a 2K 0 b sin bÞ1 ^; þ 2K 0 c sin bÞh v r€ ¼ D~ A0
2
ð17aÞ
h
a00 a 4K 20 a cos2 b 4K 0 b0 cos b þ 2c0
^ þ ð4K 0 a0 cos b 4K 0 a sin b 4K 20 c sin b cos b u ^ þ b00 4K 20 b þ 4K 0 c0 sin b þ 4K 0 c cos b h þ 2a0 4K 20 a sin b cos b 4K 0 b0 sin b i 2 ^ : þ c00 c 4K 20 c sin b 1
ð17bÞ
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
Using Eqs. (8), (13) and (17) the differential equations of paraxial trajectory through [b2, b3] and [b4, b5] intervals are:
a00 ¼ b0 ð2K 0 þ k0 Þ cos b 2c0 þ j1 a þ j2 c sin b;
ð18aÞ
0
0
b00 ¼ ð2K 0 þ k Þða sin b a0 cos b c0 sin bÞ þ b2K 0 k þ j3 c cos b; ð18bÞ
c00 ¼ 2a0 þ b0 ð2K 0 þ k0 Þ sin b þ j4 a sin b cos b þ j5 b cos b þ j6 c; ð18cÞ where are
j1 ¼ 1 þ 2K 0 k0 cos2 b; j2 j3 j4 j5 j6
ð18dÞ
1 0 ¼ 2K 0 k cos b k ðC 2 cos b C 1 Þ; 2 1 0 02 2 ¼ ð2K 0 þ k Þ ðC 1 þ C 2 k sin bÞ; 2 1 1 0 0 0 ¼ 2K 0 k C 1 k C 2 k ; 2 2 1 1 02 2 ¼ C 1 C 2 k sin b; 2 2 1 0 2 ¼ 1 þ 2K 0 k sin b C 3 ; 2 0
ð18eÞ
3.3. Differential equations of paraxial trajectory inside interval b 2 ½b3 ; b4 Inside this interval of the angle b the electric field distribur_ and D~ r€ are given by Eqs. (17a) tion is given by Eq. (9) and D~ and (17b). Now the differential equations of paraxial trajectory are
a00 ¼ 2c0 þ b0 ð2K 0 þ k0 Þ cos b þ a þ a2K 0 k0 cos2 b
ð18gÞ
1 þ 2K 0 k sin b 2
02
1 þ k sin b
0
ða þ bk sin bÞ;
0
2
0
ð18hÞ
þ
ð18iÞ
1 þ 2K 0 k sin b 02
0
2
1 þ k sin b
0
0
ða þ bk sin bÞk sin b;
ð19bÞ
c00 ¼ 2a0 þ ð2K 0 þ k0 Þðb0 sin b þ b cos bÞ; k’ = 0.7
K 0 = 0.97919
ð19cÞ
K 0 = 1.347
p [mrad]
500
0
u
0
hard edge calc. fringe field calc.
−500 −10
−5
0
5
hard edge calc. fringe field calc.
−500 −10
10
−5
0
10
500
p [mrad]
500
p [mrad]
5
u [mm]
u [mm]
0
h
h
0
hard edge calc. fringe field calc.
−500 −10
−5
0
5
hard edge calc. fringe field calc.
−500 −10
10
−5
0
h [mm]
5
10
h [mm] 2000
1500
1500
1000
1000
ζ
p [mrad]
2000
ζ
p [mrad]
ð19aÞ
b00 ¼ ð2K 0 þ k Þða sin b a0 cos b c0 sin bÞ þ b2K 0 k
500
pu [mrad]
2
0
þ
ð18fÞ
where C1, C2 and C3 are given by Eqs. (7a) and (8b).
k’ = 0
Again, the upper sign is used at the inflector entrance but for [b2, b3] interval (see Fig. 2) and the lower sign is used at the inflector exit but for [b4, b5] interval (see Fig. 2).
500 0 −15
hard edge calc. fringe field calc. −10
−5
0
ζ [mm]
5
10
15
500 0 −15
hard edge calc. fringe field calc. −10
−5
0
5
10
15
ζ [mm]
Fig. 3. The phase space diagram in the two transversal planes ððu; pu Þ and ðh; ph ÞÞ and the longitudinal one (f, pf) at the ‘‘effective exit” of the spiral inflector placed in the constant magnetic field of induction 1.5 T. Results for the hard edge field are presented by open circles and for the fringe field are presented by open squares. The inflector 0 0 height is 3 cm (A0 = 3.0 cm) and the inflector tilt parameter is zero (k = 0.0-left column) and 0.70 (k = 0.70-right column). The distance between inflector’s electrodes at its entrance is 8 mm (d0 = 8 mm). The phase spaces in the transversal and longitudinal planes at the ‘‘effective entrance” of the spiral inflector are defined as: (u0, pu0) = (h0, ph0) = (1.5 mm, 1 mrad) and (f0, pf0) = (0, 0).
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
4. Beam divergence
1^ðQ 0 Þ ¼ 1^ t þ
We shall define the beam divergence pu, ph and p1 relative to a coordinate system located in a plane which is normal to the central trajectory velocity vector by the equations [3,13]
pu ðtÞ ¼ ph ðtÞ ¼ p1 ðtÞ ¼
1 _ ~ ^ðQ 0 Þ; rðtÞ u
v0
1 _ ^ 0 Þ; ~ rðtÞ hðQ
v0 h 1
v0
i
0 _ v 1 ^ðQ 0 Þ; ~ rðtÞ 0 ^ðQ Þ 1
Inside these intervals of the angle b the orientations of the unit _ is given by vectors are constant. Now ~ rðtÞ
ð22aÞ ð22bÞ
_ ¼ v ½a0 u ^ þ ð1 þ c0 Þ1 ^ðtÞ: ^ðtÞ þ b0 hðtÞ ~ rðtÞ 0
pu ¼ a0 ;
ð24aÞ
ph ¼ b0 ;
ð24bÞ
p1 ¼ 1 þ c0 :
ð24cÞ
4.2. Beam divergence inside intervals b 2 ½b2 ; b3 , b 2 ½b3 ; b4 and b 2 ½b4 ; b5 Inside these intervals of the angle b the orientations of the unit vectors are not constant (these interval are inside the body of the _ is given by spiral inflector). Now ~ rðtÞ
FRINGE FIELD 10
5
5
u [mm]
u [mm]
HARD EDGE
0
0
−5 0
0.5
1
1.5
−10 −0.5
2
10
10
5
5
h [mm]
h [mm]
−5
0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
−5
−5 0
0.5
1
1.5
2
−10 −0.5
20
20
10
10
ζ [mm]
ζ [mm]
ð23Þ
Then from the upper equations we get the following expressions for the components of the beam divergence
10
0 −10 −20 −0.5
ð22cÞ
ð20cÞ
^ 0 Þ and 1 ^ðQ 0 Þ are calculated as it is presented ^ ðQ 0 Þ; hðQ The quantities u in [3,13]
−10 −0.5
^ðtÞ A0 c @ 1 : v 0 @t
4.1. Beam divergence inside intervals b 2 ½b1 ; b2 and b 2 ½b5 ; b6
ð21bÞ
−10 −0.5
^ðtÞ þ 1
ð20bÞ
_ is given by the relation where ~ rðtÞ
^ ðtÞ A c A c @u ^ ðQ 0 Þ ¼ u ^ ðtÞ þ 0 ^ tþ 0 u u ; v0 v 0 @t ^ ^ þ A0 c @ hðtÞ ; ^ t þ A0 c hðtÞ ^ 0Þ ¼ h hðQ v0 v 0 @t
v0
ð20aÞ
ð21aÞ
^ ~ r€c ðtÞ ¼ v 0 1ðtÞ;
A0 c
In all equations where appear the divergence of the unit vectors we have to be aware of the interval of the angle b (see Fig. 2), since in some intervals the orientations (directions) of the unit vectors are constant and in some interval not.
where v0 is the speed of the central trajectory ion. pf represents the relative difference between the forward momentum component of the displaced trajectory and the central trajectory. From Eq. (12) is
_ ¼~ _ ~ rðtÞ r_c ðtÞ þ D~ rðtÞ;
0
−10 0
0.5
b [rad]
1
1.5
2
−20 −0.5
b [rad]
Fig. 4. The paraxial trajectories from the ‘‘effective entrance” to the ‘‘effective exit” in the case when the inflector height is 3 cm (A0 = 3.0 cm) and the inflector is not tilted 0 (k = 0.0) for the hard edge (left column) and fringe (right column) fields. The distance between inflector’s electrodes at its entrance is 8 mm (d0 = 8 mm) and the inflector is placed in the constant magnetic field of induction 1.5 T.
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
HARD EDGE
FRINGE FIELD
5
5
u [mm]
10
u [mm]
10
0 −5 −10 −0.5
0 −5
0
0.5
1
1.5
−10 −0.5
2
5
5
h [mm]
10
h [mm]
10
0 −5 −10 −0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
−5 0
0.5
1
1.5
−10 −0.5
2
10
10
ζ [mm]
20
ζ [mm]
0.5
0
20
0 −10 −20 −0.5
0
0 −10
0
0.5
1
1.5
2
−20 −0.5
b [rad]
b [rad]
Fig. 5. The paraxial trajectories from the ‘‘effective entrance” to the ‘‘effective exit” in the case when the inflector height is 3 cm (A0 = 3.0 cm) and the inflector tilt parameter 0 is 0.70 (k = 0.70) for the hard edge (left column) and fringe (right column) fields. The distance between inflector’s electrodes at its entrance is 8 mm (d0 = 8 mm) and the inflector is placed in the constant magnetic field of induction 1.5 T. 0
_ ¼ v fða0 b2K cos b þ cÞ u ~ ^ ðtÞ rðtÞ 0 0 ^ þ ða2K 0 cos b þ b0 þ c2K 0 sin bÞ hðtÞ 0 ^ðtÞ:g þ ð1 a b2K 0 sin b þ c Þ 1
ð25Þ
Then from the upper equations we get the following expressions for the components of the beam divergence
pu ¼ a0 b2K 0 cos b;
ð26aÞ
ph ¼ b0 þ a2K 0 cos b;
ð26bÞ
p1 ¼ c0 a b2K 0 sin b;
ð26cÞ
where, for the first derivatives of the unit vectors, are used Eq. (7) in [13]. 5. Numerical results To see the effects of the fringe field it is considered the ion with the specific charge of 0.357 and the kinetic energy of 3.57 keV/n passing through the two different spiral inflectors placed in the constant magnetic field of induction 1.5 T. The radius of the ion trajectory RM0 is 1.606 cm. The height A0 of the each inflector is same; 0 3 cm but the inflector tilt parameter k is different for the each 0 0 inflector; k = 0 (not tilted case) and k = 0.70 (tilted case). The half of the electrode gap at the entrance of both inflectors is 4 mm; i.e. d0 = 8 mm. The electric potential of the electrodes of the each inflector is ±2.67 kV and the electric field magnitude at the entrance of the each inflectors is 6.67 kV/cm. It is considered 32 par0 ticles which belong to the boundary of the contours in the (u, u )
and (h, h ) plane with the maximum transversal displacements of ±1.5 mm in the positions and ±1 mrad in the divergences at the inflector ‘‘effective entrance”. The longitudinal displacement in the position and in the divergence at the inflector ‘‘effective entrance” are taken to be zero; ð1; 10 Þ ¼ ð0; 0Þ. The inflector ‘‘effective entrance” is defined as a gap width d0 and placed d0 before the actual electrodes start. Similar, ffi the inflector ‘‘effective pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exit” is defined 02 02 as a gap width d0 = 1 þ k and placed d0 = 1 þ k away from the actual end of the electrodes. In the case of hard edge field calculations the starting phase electrodes space planes are located d0 before the actual p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi start and 02 the final phase space planes are located d0 = 1 þ k away from the actual end of the electrodes; ie. there are drifts of length d0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 and d0 = 1 þ k before and after the actual electrodes where the constant magnetic field of induction 1.5 T exist only. The results are presented in Figs. 3–5. 6. Conclusions 0
The fringe field effects depend on the tilt parameter k ; by increasing the tilt parameter the fringe field effects are increasing in the u and decreasing in the h plane. In u(h) plane the fringe field has defocusing (focusing) characteristic compared to the hard edge field. References [1] J.L. Belmont, L. Pabot, Study of axial injection for grenoble cyclotron, IEEE Trans. Nucl. Sci. NS-13 (1966) 191.
D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181 [2] J.L. Belmont, J.L. Pabot, Institut des Sciences Nucleaires, Rapport Interne #3, 1966. [3] L.W. Root, Design of the inflector for the TRIUMF cyclotron, M.Sc. Thesis, University of British Columbia, Vancouver, 1972. [4] L.W. Root, Experimental and theoretical studies of the behaviour of H ion beam during injection and acceleration in the TRIUMF central region model cyclotron, Ph.D. Thesis, University of British Columbia, Vancouver, 1974. [5] F.B. Milton, J.B. Pearson, CASINO User’s Guide and Reference Manual, TRI-DN89-19, Vancouver, 1989. [6] Lj. M. Milinkovic, RELAX3D Spiral Inflectors, TR30-DN-89-21, Vancouver, 1989. [7] R.J. Balden et al., Aspects of phase space dynamics in spiral inflectors, in: Proc. 12th Int. Conf. on Cyclotron and their Applications, Berlin, 1989, p. 435. [8] R. Baartman, Matching of ion sources to cyclotron inflectors, in: Proc. 11th European Particle Accelerator Conference (EPAC), Rome 1988, p. 947 (Also published in TRIUMF Design Note, TRI-PP-88-37, June 1988).
1181
[9] R. Baartman, W. Kleeven, A canonical treatment of the spiral inflector for cyclotrons, Particle Accelerators 41 (1993) 41. [10] W. Kleeven, R. Baartman, Beam matching and emittance growth resulting from spiral inflectors for cyclotrons, Particle Accelerators 41 (1993) 55 (Also published in TRIUMF Design Note, TRI-DN-90-20, June 1990). [11] Dragan Toprek, Krunoslav Subotic, Some optical properties of the spiral inflector, Nucl. Instr. and Meth. A 431 (1999) 38. [12] Dragan Toprek, Theory of the central ion trajectory in the spiral inflector, Nucl. Instr. and Meth. A 440 (2000) 285. [13] Dragan Toprek, Theory of the paraxial ion trajectory in the spiral inflector, Nucl. Instr. and Meth. A 449 (2000) 435.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Fringe field effects in the spiral inflector Dragan Toprek The Institute of Nuclear Sciences VINCA, Laboratory for Nuclear and Plasma Physics, P.O. Box 522, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 27 October 2008 Available online 24 February 2009
a b s t r a c t In this paper is studied the effects of the fringe field in the spiral inflector. Ó 2009 Elsevier B.V. All rights reserved.
PACS: 41.85.p Keywords: Fringe field Distribution Spiral inflector
1. Introduction The spiral inflector was invented by Belmont and Pabot a long time ago [1,2]. Since then many papers and Ph.D. thesis are written regarding to its characteristics and properties. See for example [3– 13]. The spiral inflector is the most famous device used to bend a low energy beam which is axially injected into a cyclotron onto the median plane. Despite the fact that the spiral inflector worsen the beam emittance it is often chose to inject the beam due to its small size 3 cm The spiral inflector has two free parameters. One parameter is 0 A0 (the inflector height) and the second one is k (a parameter which is related to the direction of the electric field, called tilt). Using these two parameters it is convenient to define a new parameter K0 as 0
K0 ¼
A0 k þ : 2RM0 2
ð1Þ
The inflector height A0 (it is actually a radius of curvature which the ion would have if it were acted upon by an electric field only) and the radius of curvature RM0 which the ion would have if it were acted upon by the magnetic field in the absence of any electric field (RM0 is the magnetic radius; spiral inflector is placed in a cyclotron’s magnetic field) are given by
A0 ¼
T0 qEel
RM0 ¼
p0 ; qB0
ð2Þ
where T0, q and p0 are the kinetic energy, the electric charge and the total momentum of the reference ion, respectively, Eel and B0 are the magnitude of the electric field and induction of the magnetic field, respectively. The schematic drawing of the spiral inflector, which is basically an electrostatic 90° bend with its two electrodes (across which an electrostatic potential is applied) deformed into a spiral so that the electric field remains orthogonal to the beam as it is also bent by the cyclotron’s magnetic field, is presented in Fig. 1; non-tilted case (a) and tilted case (b) [12]. 2. Fringe field distributions 2.1. Magnitude of the fringe field To find out the field distribution it is necessary to find out its magnitude first. ^r It is taken a simplifying assumption that the fringe field has u ^ r is one of the unit vectors of the rotated optidirection [5] where u cal coordinate system [3–5,11–13]. The magnitude of the fringe field EF is given by
EF ð0; 0; 1Þ ¼
Eel ½1 tan h½gð1 1E þ nÞ; 2
where Eel is the magnitude of the electric field on the central trajectory and constants g and n come from an empirical fit given by
g ¼ 1:35 n ¼ Gð0:37 e2:5d=G Þ; E-mail address: [email protected] 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.02.056
ð3aÞ
ð3bÞ
where G is distance between the entrance/exit of the spiral inflector end the plane where the fringe field starts/finish. In this study it is
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
2.2. Electric field distribution – general expression The expression for the first order change in the electric field with respect to the central ray is [3–5,13]
~ r ; hr ; 1Þ Eð0; 0; 1Þ Nð0; ~ 0; 1Þ Eður ; hr ; 1Þ ¼ Eður ; hr ; 1Þ Nðu D~ þ
1 @~ E ; j A0 @b ur ¼hr ¼0
ð4Þ
~ r ; hr ; 1Þ is the normal where E is the magnitude of the field and Nðu vector to the equipotential surface that contains the central ray, and thus the direction of the electric field on that surface [5]. In ~ is given by [5] the case of the spiral inflector N
^r h dh ~ r ; hr ; 1Þ u ^r r ^r Nðu u A0 db Fig. 1. Sketch of the electrodes of the spiral inflector. In case (a) the electrode is not 0 0 slanted (k = 0) and in case (b) the electrode is slanted (k – 0) for some angle.
used G = d where d is distance between electrodes of the spiral inflector. In the Eq. (1) fE is the location of the inflector entrance/exit. The plus sign is used at the entrance and the minus sign at the exit of the spiral inflector. Comparison between the analytic fringe field and hard edge ^ r direction is presented in Fig. 2 in the case for one vafield in the u lue of the inflector height A (A = 3 cm) but for two values of the 0 0 0 inflector tilt parameter k (k = 0.0 and k = 0.70, respectively). From this figure it can be seen that the assumption for fringe field is quite good. In this figure b is the instantaneous angle between the velocity vector and the vertical [3–5,13]. At the entrance/exit of the spiral inflector b is zero or p/2, respectively. The magnitude ^ r component of the electric field is increasing trough the of the u 0 spiral inflector (in the case when k – 0) since the electrode spacing ^ component of the electric is decreasing in order to obtain the u field constant [3–5,13].
HARD EDGE and FRINGE FIELDS DISTRIBUTIONS A0 = 3 cm
k’ = 0
K0 = 0.93168
! ^þ 1
ur @EF ^: 1 A0 Eel @b
ð5Þ
2.3. Fringe field distributions Now, when the magnitude of the fringe field is known, it is possible to derive its distribution. In this case E and Eel in the Eqs. (4) and (5) should be replaced by EF given by Eq. (3). Through the physical body of the spiral inflector the orienta^r and 1 ^ are not con^r ; h tions (directions) of the unit vectors u stant; they depend of the angle b [3–5,13]. Outside of the inflector’s body the directions of the unit vectors are constant; ^ r =db ¼ d1 ^=db ¼ 0. The consequence of this is that ^ r =db ¼ dh du the fringe field regions, at the inflector entrance and exit, will be divided into two parts; one part which is outside of the inflector’s body (b 2 ½b1 ; b2 and b 2 ½b5 ; b6 in Fig. 2) and one part which is inside of the inflector’s body (b 2 ½b2 ; b3 and b 2 ½b4 ; b5 in Fig. 2). ^ r and The unit vectors of the rotated optical coordinate system u ^ r are defined as: h 0
1 k sin b ^ ^ r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ^ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h; u 0 0 2 2 1 þ k sin b 1 þ k sin b
ð6aÞ
0
1 k sin b ^ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^; ffih ffiu h 0 0 2 2 1 þ k sin b 1 þ k sin b
ð6bÞ
^ are the unit vectors of the optical coordinate system ^ and h where u [3–5,11–13]. 2.3.1. Fringe field distributions through intervals b 2 ½b1 ; b2 and b 2 ½b5 ; b6 By using the following abbreviation 0
C2 ¼
b1
b2
A0 = 3 cm
b3
k’ = 0.7
b4
b5
b6
1:35k 02
2
2
ð1 þ k sin bÞCh
;
ð7aÞ
where Ch ¼ cosh½gð1 1E þ nÞ we got for the fringe field distributions inside these intervals
K0 = 1.2817
0 i E0 k h ^ þ ðu þ hk0 sin bÞ1 ^ : ^ þ 1k0 sin bh Eðu; h; 1Þ ¼ D~ C 2 1u 2A0
ð7bÞ
The upper sign is used at the inflector entrance but b 2 ½b1 ; b2 (see Fig. 2) and the lower sign is used at the inflector exit but b 2 ½b5 ; b6 (see Fig. 2).
b1
b2
b3
b4
b5
b6
Fig. 2. Fringe field (full line) and hard edge field (dotted line) along the trajectory in the spiral inflector. The marks are explained in the text.
2.3.2. Fringe field distributions through intervals b 2 ½b2 ; b3 and b 2 ½b4 ; b5 Through these intervals for the fringe field distributions we got
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
E0 Eðu; h; 1Þ ¼ D~ 1k0 sin bðC 2 cos b C 1 Þu^ 2A0 E0 1 cos bðC 2 k02 sin2 b þ C 1 Þh^ 2A0 E0 0 ðh uk sin bÞC 1 cos b þ 2A0 0 0 ^; u þ hk sin bÞC 2 k sin b cos b 1C 3 1
3.1. Differential equations of paraxial trajectory through intervals b 2 ½b1 ; b2 and b 2 ½b5 ; b6 Through these intervals of the angle b the electric field distribur_ and ~ r€ (remember, inside these intertion is given by Eq. (7) and D~ vals the orientations of the unit vectors are constant) are given by
ð8aÞ
where are
0 k C 1 ¼ ð1 ThÞ 2K 0 þ and 02 2 1 þ k sin b 0 2 C 3 ¼ ð1 ThÞ 1 þ 2K 0 k sin b ;
ð8bÞ
where is Th ¼ tan h½gð1 1E þ nÞ. The upper sign is used at the inflector entrance but b 2 ½b2 ; b3 (see Fig. 2) and the lower sign is used at the inflector exit but b 2 ½b4 ; b5 (see Fig. 2). 2.3.3. Electric field distribution through region b 2 ½b3 ; b4 Through this interval of the angle b the components of the electric field distributions are given by [3–5,11–13]
( ) 0 2 E0 1 þ 2K 0 k sin b 0 0 12K 0 k sinbcosb ; DEu ðu;h; 1Þ ¼ ðu þ hk sinbÞ 02 2 A0 1 þ k sin b ð9aÞ (
DEh ðu;h; 1Þ ¼
0
2
)
E0 1 þ 2K 0 k sin b 0 0 0 ðu þ hk sinbÞ k sinb þ 1ð2K 0 þ k Þcosb ; 02 2 A0 1 þ k sin b
ð9bÞ
2
0
ð9cÞ
3. Differential equations of paraxial trajectory
0 c
c
c
c
ð10aÞ ð10bÞ
We shall assume that the magnetic field ~ B is constant throughout the all intervals of the angle b and is defined by the relation [3– 5,11–13]
^ ¼ B0 ½sin b u ~ ^; ^ cos b 1 B ¼ B0 k
ð11Þ
^ is the unit vector of the Cartesian coordinate system. Then where k if we define
^ þ 11 ^; ^ þ hh D~ rðtÞ ¼ ~ rðtÞ ~ r c ðtÞ ¼ uu
ð12Þ
we get from Eqs. (10a) and (10b)
h i rÞ þ D~ r€ ¼ q D~ r_ ~ B : m0 D~ EðD~ In the Eq. (13) D~ rÞ is the electric field distribution. EðD~
v 2 h 00 ^ 00 ^ 00 ^i D~ a uþb hþc 1 ; r€ ¼ A0
ð14bÞ
where a, b and c are dimensionless parameters defined as
a¼
u ; A0
b¼
h ; A0
c¼
1 A0
ð14cÞ
:
In the Eqs. (14a) and (14b) dot means differentiating with respect to time t and prime means differentiating with respect to the angle b; d d ¼ vA00 db . dt Using Eqs. (7), (13) and (14) the differential equations of paraxial trajectory through [b1, b2] interval are:
a00 ¼ b00 ¼
c00 ¼
gk02 sin b cos b 2
02
2
c ð2K 0 k0 Þb0 ;
ð15aÞ
2
cK 0 sin b þ ð2K 0 k0 Þa0 ;
ð15bÞ
2
ða þ bk sin bÞ;
2ð1 þ k sin bÞChþ
gk02 sin b cos b 2
02
2ð1 þ k sin bÞChþ
gk02 sin b cos b 2
02
0
2ð1 þ k sin bÞChþ
a00 ¼
b00 ¼
A trajectory is said to be paraxial provided it is located close to the central trajectory of reference particle. Using purely analytic method we were ably to derive a system of differential equations governing the paraxial trajectories. Let ~ rðtÞ be the position vector of the ion with mass m0 on the paraxial trajectory at time t, and let ~ r c ðtÞ be the position vector of the central trajectory ion at time t. Then both ~ rðtÞ and ~ r c ðtÞ must satisfy the Lorentz equation of motion
_ ~ € ¼ q½~ m0~ rðtÞ Bð~ rÞ; rðtÞ Eð~ rc Þ þ ~ € _ ~ ~ r Þ þ r~ ðtÞ Bð~ r Þ: m r~ ðtÞ ¼ q½Eð~
ð14aÞ
ð15cÞ
Similar, the differential equations of paraxial trajectory through [b5, b6] interval are:
E0 0 0 DE1 ðu; h; 1Þ ¼ fu2K 0 k sin b cos b þ hð2K 0 þ k Þ cos b A0 1½1 þ 2K 0 k sin bg:
h i v 0 d h ^ ^ ^i ^ þ c0 1 ^ ; ^ þ b0 h D~ uu þ hh þ 11 ¼ v 0 a0 u r_ ¼ A0 db
ð13Þ
c00 ¼
gk02 sin b cos b 02
2
2
c;
ð16aÞ
2
ck0 sin b þ ð2K 0 k0 Þc0 ;
ð16bÞ
2
ða þ bk sin bÞ ð2K 0 k Þb0 :
2ð1 þ k sin bÞCh
gk02 sin b cos b 02
2
2ð1 þ k sin bÞCh
gk02 sin b cos b 02
2
0
2ð1 þ k sin bÞCh
0
ð16cÞ
3.2. Differential equations of paraxial trajectory through intervals b 2 ½b2 ; b3 and b 2 ½b4 ; b5 Through these intervals of the angle b the electric field distribur_ and D~ r€ (remember, inside these tion is given by Eq. (8) and D~ intervals the orientations of the unit vectors are not constant) are given by
v 0 d h ^ ^ ^i r_ ¼ D~ uu þ hh þ 11 A0 db ^ þ ð2K 0 a cos b þ b0 ¼ v 0 ½ða0 2K 0 b cos b þ cÞu ^ þ ðc0 a 2K 0 b sin bÞ1 ^; þ 2K 0 c sin bÞh v r€ ¼ D~ A0
2
ð17aÞ
h
a00 a 4K 20 a cos2 b 4K 0 b0 cos b þ 2c0
^ þ ð4K 0 a0 cos b 4K 0 a sin b 4K 20 c sin b cos b u ^ þ b00 4K 20 b þ 4K 0 c0 sin b þ 4K 0 c cos b h þ 2a0 4K 20 a sin b cos b 4K 0 b0 sin b i 2 ^ : þ c00 c 4K 20 c sin b 1
ð17bÞ
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
Using Eqs. (8), (13) and (17) the differential equations of paraxial trajectory through [b2, b3] and [b4, b5] intervals are:
a00 ¼ b0 ð2K 0 þ k0 Þ cos b 2c0 þ j1 a þ j2 c sin b;
ð18aÞ
0
0
b00 ¼ ð2K 0 þ k Þða sin b a0 cos b c0 sin bÞ þ b2K 0 k þ j3 c cos b; ð18bÞ
c00 ¼ 2a0 þ b0 ð2K 0 þ k0 Þ sin b þ j4 a sin b cos b þ j5 b cos b þ j6 c; ð18cÞ where are
j1 ¼ 1 þ 2K 0 k0 cos2 b; j2 j3 j4 j5 j6
ð18dÞ
1 0 ¼ 2K 0 k cos b k ðC 2 cos b C 1 Þ; 2 1 0 02 2 ¼ ð2K 0 þ k Þ ðC 1 þ C 2 k sin bÞ; 2 1 1 0 0 0 ¼ 2K 0 k C 1 k C 2 k ; 2 2 1 1 02 2 ¼ C 1 C 2 k sin b; 2 2 1 0 2 ¼ 1 þ 2K 0 k sin b C 3 ; 2 0
ð18eÞ
3.3. Differential equations of paraxial trajectory inside interval b 2 ½b3 ; b4 Inside this interval of the angle b the electric field distribur_ and D~ r€ are given by Eqs. (17a) tion is given by Eq. (9) and D~ and (17b). Now the differential equations of paraxial trajectory are
a00 ¼ 2c0 þ b0 ð2K 0 þ k0 Þ cos b þ a þ a2K 0 k0 cos2 b
ð18gÞ
1 þ 2K 0 k sin b 2
02
1 þ k sin b
0
ða þ bk sin bÞ;
0
2
0
ð18hÞ
þ
ð18iÞ
1 þ 2K 0 k sin b 02
0
2
1 þ k sin b
0
0
ða þ bk sin bÞk sin b;
ð19bÞ
c00 ¼ 2a0 þ ð2K 0 þ k0 Þðb0 sin b þ b cos bÞ; k’ = 0.7
K 0 = 0.97919
ð19cÞ
K 0 = 1.347
p [mrad]
500
0
u
0
hard edge calc. fringe field calc.
−500 −10
−5
0
5
hard edge calc. fringe field calc.
−500 −10
10
−5
0
10
500
p [mrad]
500
p [mrad]
5
u [mm]
u [mm]
0
h
h
0
hard edge calc. fringe field calc.
−500 −10
−5
0
5
hard edge calc. fringe field calc.
−500 −10
10
−5
0
h [mm]
5
10
h [mm] 2000
1500
1500
1000
1000
ζ
p [mrad]
2000
ζ
p [mrad]
ð19aÞ
b00 ¼ ð2K 0 þ k Þða sin b a0 cos b c0 sin bÞ þ b2K 0 k
500
pu [mrad]
2
0
þ
ð18fÞ
where C1, C2 and C3 are given by Eqs. (7a) and (8b).
k’ = 0
Again, the upper sign is used at the inflector entrance but for [b2, b3] interval (see Fig. 2) and the lower sign is used at the inflector exit but for [b4, b5] interval (see Fig. 2).
500 0 −15
hard edge calc. fringe field calc. −10
−5
0
ζ [mm]
5
10
15
500 0 −15
hard edge calc. fringe field calc. −10
−5
0
5
10
15
ζ [mm]
Fig. 3. The phase space diagram in the two transversal planes ððu; pu Þ and ðh; ph ÞÞ and the longitudinal one (f, pf) at the ‘‘effective exit” of the spiral inflector placed in the constant magnetic field of induction 1.5 T. Results for the hard edge field are presented by open circles and for the fringe field are presented by open squares. The inflector 0 0 height is 3 cm (A0 = 3.0 cm) and the inflector tilt parameter is zero (k = 0.0-left column) and 0.70 (k = 0.70-right column). The distance between inflector’s electrodes at its entrance is 8 mm (d0 = 8 mm). The phase spaces in the transversal and longitudinal planes at the ‘‘effective entrance” of the spiral inflector are defined as: (u0, pu0) = (h0, ph0) = (1.5 mm, 1 mrad) and (f0, pf0) = (0, 0).
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
4. Beam divergence
1^ðQ 0 Þ ¼ 1^ t þ
We shall define the beam divergence pu, ph and p1 relative to a coordinate system located in a plane which is normal to the central trajectory velocity vector by the equations [3,13]
pu ðtÞ ¼ ph ðtÞ ¼ p1 ðtÞ ¼
1 _ ~ ^ðQ 0 Þ; rðtÞ u
v0
1 _ ^ 0 Þ; ~ rðtÞ hðQ
v0 h 1
v0
i
0 _ v 1 ^ðQ 0 Þ; ~ rðtÞ 0 ^ðQ Þ 1
Inside these intervals of the angle b the orientations of the unit _ is given by vectors are constant. Now ~ rðtÞ
ð22aÞ ð22bÞ
_ ¼ v ½a0 u ^ þ ð1 þ c0 Þ1 ^ðtÞ: ^ðtÞ þ b0 hðtÞ ~ rðtÞ 0
pu ¼ a0 ;
ð24aÞ
ph ¼ b0 ;
ð24bÞ
p1 ¼ 1 þ c0 :
ð24cÞ
4.2. Beam divergence inside intervals b 2 ½b2 ; b3 , b 2 ½b3 ; b4 and b 2 ½b4 ; b5 Inside these intervals of the angle b the orientations of the unit vectors are not constant (these interval are inside the body of the _ is given by spiral inflector). Now ~ rðtÞ
FRINGE FIELD 10
5
5
u [mm]
u [mm]
HARD EDGE
0
0
−5 0
0.5
1
1.5
−10 −0.5
2
10
10
5
5
h [mm]
h [mm]
−5
0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
−5
−5 0
0.5
1
1.5
2
−10 −0.5
20
20
10
10
ζ [mm]
ζ [mm]
ð23Þ
Then from the upper equations we get the following expressions for the components of the beam divergence
10
0 −10 −20 −0.5
ð22cÞ
ð20cÞ
^ 0 Þ and 1 ^ðQ 0 Þ are calculated as it is presented ^ ðQ 0 Þ; hðQ The quantities u in [3,13]
−10 −0.5
^ðtÞ A0 c @ 1 : v 0 @t
4.1. Beam divergence inside intervals b 2 ½b1 ; b2 and b 2 ½b5 ; b6
ð21bÞ
−10 −0.5
^ðtÞ þ 1
ð20bÞ
_ is given by the relation where ~ rðtÞ
^ ðtÞ A c A c @u ^ ðQ 0 Þ ¼ u ^ ðtÞ þ 0 ^ tþ 0 u u ; v0 v 0 @t ^ ^ þ A0 c @ hðtÞ ; ^ t þ A0 c hðtÞ ^ 0Þ ¼ h hðQ v0 v 0 @t
v0
ð20aÞ
ð21aÞ
^ ~ r€c ðtÞ ¼ v 0 1ðtÞ;
A0 c
In all equations where appear the divergence of the unit vectors we have to be aware of the interval of the angle b (see Fig. 2), since in some intervals the orientations (directions) of the unit vectors are constant and in some interval not.
where v0 is the speed of the central trajectory ion. pf represents the relative difference between the forward momentum component of the displaced trajectory and the central trajectory. From Eq. (12) is
_ ¼~ _ ~ rðtÞ r_c ðtÞ þ D~ rðtÞ;
0
−10 0
0.5
b [rad]
1
1.5
2
−20 −0.5
b [rad]
Fig. 4. The paraxial trajectories from the ‘‘effective entrance” to the ‘‘effective exit” in the case when the inflector height is 3 cm (A0 = 3.0 cm) and the inflector is not tilted 0 (k = 0.0) for the hard edge (left column) and fringe (right column) fields. The distance between inflector’s electrodes at its entrance is 8 mm (d0 = 8 mm) and the inflector is placed in the constant magnetic field of induction 1.5 T.
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D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181
HARD EDGE
FRINGE FIELD
5
5
u [mm]
10
u [mm]
10
0 −5 −10 −0.5
0 −5
0
0.5
1
1.5
−10 −0.5
2
5
5
h [mm]
10
h [mm]
10
0 −5 −10 −0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
−5 0
0.5
1
1.5
−10 −0.5
2
10
10
ζ [mm]
20
ζ [mm]
0.5
0
20
0 −10 −20 −0.5
0
0 −10
0
0.5
1
1.5
2
−20 −0.5
b [rad]
b [rad]
Fig. 5. The paraxial trajectories from the ‘‘effective entrance” to the ‘‘effective exit” in the case when the inflector height is 3 cm (A0 = 3.0 cm) and the inflector tilt parameter 0 is 0.70 (k = 0.70) for the hard edge (left column) and fringe (right column) fields. The distance between inflector’s electrodes at its entrance is 8 mm (d0 = 8 mm) and the inflector is placed in the constant magnetic field of induction 1.5 T. 0
_ ¼ v fða0 b2K cos b þ cÞ u ~ ^ ðtÞ rðtÞ 0 0 ^ þ ða2K 0 cos b þ b0 þ c2K 0 sin bÞ hðtÞ 0 ^ðtÞ:g þ ð1 a b2K 0 sin b þ c Þ 1
ð25Þ
Then from the upper equations we get the following expressions for the components of the beam divergence
pu ¼ a0 b2K 0 cos b;
ð26aÞ
ph ¼ b0 þ a2K 0 cos b;
ð26bÞ
p1 ¼ c0 a b2K 0 sin b;
ð26cÞ
where, for the first derivatives of the unit vectors, are used Eq. (7) in [13]. 5. Numerical results To see the effects of the fringe field it is considered the ion with the specific charge of 0.357 and the kinetic energy of 3.57 keV/n passing through the two different spiral inflectors placed in the constant magnetic field of induction 1.5 T. The radius of the ion trajectory RM0 is 1.606 cm. The height A0 of the each inflector is same; 0 3 cm but the inflector tilt parameter k is different for the each 0 0 inflector; k = 0 (not tilted case) and k = 0.70 (tilted case). The half of the electrode gap at the entrance of both inflectors is 4 mm; i.e. d0 = 8 mm. The electric potential of the electrodes of the each inflector is ±2.67 kV and the electric field magnitude at the entrance of the each inflectors is 6.67 kV/cm. It is considered 32 par0 ticles which belong to the boundary of the contours in the (u, u )
and (h, h ) plane with the maximum transversal displacements of ±1.5 mm in the positions and ±1 mrad in the divergences at the inflector ‘‘effective entrance”. The longitudinal displacement in the position and in the divergence at the inflector ‘‘effective entrance” are taken to be zero; ð1; 10 Þ ¼ ð0; 0Þ. The inflector ‘‘effective entrance” is defined as a gap width d0 and placed d0 before the actual electrodes start. Similar, ffi the inflector ‘‘effective pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exit” is defined 02 02 as a gap width d0 = 1 þ k and placed d0 = 1 þ k away from the actual end of the electrodes. In the case of hard edge field calculations the starting phase electrodes space planes are located d0 before the actual p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi start and 02 the final phase space planes are located d0 = 1 þ k away from the actual end of the electrodes; ie. there are drifts of length d0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 and d0 = 1 þ k before and after the actual electrodes where the constant magnetic field of induction 1.5 T exist only. The results are presented in Figs. 3–5. 6. Conclusions 0
The fringe field effects depend on the tilt parameter k ; by increasing the tilt parameter the fringe field effects are increasing in the u and decreasing in the h plane. In u(h) plane the fringe field has defocusing (focusing) characteristic compared to the hard edge field. References [1] J.L. Belmont, L. Pabot, Study of axial injection for grenoble cyclotron, IEEE Trans. Nucl. Sci. NS-13 (1966) 191.
D. Toprek / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1175–1181 [2] J.L. Belmont, J.L. Pabot, Institut des Sciences Nucleaires, Rapport Interne #3, 1966. [3] L.W. Root, Design of the inflector for the TRIUMF cyclotron, M.Sc. Thesis, University of British Columbia, Vancouver, 1972. [4] L.W. Root, Experimental and theoretical studies of the behaviour of H ion beam during injection and acceleration in the TRIUMF central region model cyclotron, Ph.D. Thesis, University of British Columbia, Vancouver, 1974. [5] F.B. Milton, J.B. Pearson, CASINO User’s Guide and Reference Manual, TRI-DN89-19, Vancouver, 1989. [6] Lj. M. Milinkovic, RELAX3D Spiral Inflectors, TR30-DN-89-21, Vancouver, 1989. [7] R.J. Balden et al., Aspects of phase space dynamics in spiral inflectors, in: Proc. 12th Int. Conf. on Cyclotron and their Applications, Berlin, 1989, p. 435. [8] R. Baartman, Matching of ion sources to cyclotron inflectors, in: Proc. 11th European Particle Accelerator Conference (EPAC), Rome 1988, p. 947 (Also published in TRIUMF Design Note, TRI-PP-88-37, June 1988).
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[9] R. Baartman, W. Kleeven, A canonical treatment of the spiral inflector for cyclotrons, Particle Accelerators 41 (1993) 41. [10] W. Kleeven, R. Baartman, Beam matching and emittance growth resulting from spiral inflectors for cyclotrons, Particle Accelerators 41 (1993) 55 (Also published in TRIUMF Design Note, TRI-DN-90-20, June 1990). [11] Dragan Toprek, Krunoslav Subotic, Some optical properties of the spiral inflector, Nucl. Instr. and Meth. A 431 (1999) 38. [12] Dragan Toprek, Theory of the central ion trajectory in the spiral inflector, Nucl. Instr. and Meth. A 440 (2000) 285. [13] Dragan Toprek, Theory of the paraxial ion trajectory in the spiral inflector, Nucl. Instr. and Meth. A 449 (2000) 435.