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Magnetic field distribution for weak-, strong- and FFAG-focusing particle accelerators and other nuclear instruments
NUCLEAR
INSTRUMENTS
AND METHODS
21 (1963) 136--144; N O R T H - H O L L _ A _ N D P U B L I S H I N G
CO.
MAGNETIC FIELD DISTRIBUTION FOR WEAK-, STRONG- AND FFAG-FOCUSING PARTICLE ACCELERATORS AND OTHER NUCLEAR INSTRUMENTS C. C. ILIESCU Institut* /or Atomqc Physics, Bucharest, t~umania Received 21 J u l y 1962 F o r the magnetic guiding field of the weak-, strong-, a n d FFAG-fecusing nuclear instruments used in the acceleration or dispersion of charged particles, general analytic expressions, valid for a n y point of the magnet gap, are derived concerning the following p a r a m e t e r s : magnetic scalar potential, magnetic induction a n d magnetic vector potential. These expressions are useful for the s t u d y of particle motion as well as for pole face design.
1. Introduction The motion of particles in the weak-, strong- and FFAG-focusing nuclear instruments used in the acceleration or dispersion of charged particles (particle accelerators, spectrometers, analyzers, etc) takes place along the central orbit under the action of the restoring forces produced by the magnetic guiding field. The magnetic guiding field is defined by the following parameters: magnetic scalar potential, magnetic induction and magnetic vector potential. In the literature, the expressions for these parameters concerning the weak-, and strong-focusing particle accelerators, valid for any point of the magnet gap, are considered ill a first-order approximation. The first-order approximation was thought to be insufficient in some cases, especially for large gap particle spectrometers and F F A G accelerators. For these particlflar cases, more accurate expressions were require& For a double-focusing spectrometer with a field index n = 0.5, Stoker et al. 1) determined the expression of the magnetic scalar potential in the form of an infinite power series of z and z. Boerboom et al.2), in a study of the magnetic field produced b y conical pole faces used in spectrometers, considered the magnetic scalar potential as a double infinite power series of (z - re)It o and z/r o and established a recurrence relation between the coefficients. Analyzing the focusing properties of the inhomogeneous magnetic sector fields of cylindrical symmetry used for spectrometers, Bretscher 3) considered the magnetic guiding field in the median plane as an infinite power series of (r - ro)/r o. The magnetic guiding field expressions were not considered a function of the field index. Hence, the general terms of the magnetic parameters series were not obtained. For a F F A G accelerator, Taylor*) computed the magnetic field parameters b y means of a digital computer and obtained a recurrence relation between some intermediate coefficients necessary for determining the expression of the magnetic scalar potential. The object of this paper is to derive the general analytic expressions for the magnetic guiding field parameters, valid for the whole magnet gap, starting from the expression of the magnetic guiding field in the median plane. Assuming that for the weak-focusing, strong-focusing and F F A G nuclear instruments the expressions of the magnetic guiding fields in the median plane are particular cases of a magnetic guiding field described by an infinite series, this paper determines the general analytical expressions for all the param1) p. H. Stoker vt aL, P h y s i c a 20 (1954) 337. 2) A. J. H. Boerboom, H. A. Tasman, H. W a c h s m u t h , Z. Naturforsch, 14a (I959) 816. 3) M. 2VL Bretscher, O ~ N L - 2 8 8 4 (196~). *) R. Taylor, A E R E - R 3097 (1959)"
136
MAGNETIC
FIELD
DISTRIBUTION
FOR NUCLEAR
INSTRUMENTS
137
eters and simplifies them for the above-mentioned nuclear instruments. A satisfactory agreement was reached with the relations previously obtained by the authors mentioned. The expressions determined are useful both for the study of the motion of the particles a n d for the determination of the pole face shape necessary for a proposed magnetic guiding field distributionS). In the foliowing we shall neglect the effects of the finite extensions in the radial direction of the pole face as well as the field distortion due to remanence, saturation, etc.
2. General Analytic Expressions of the Parameters of the Magnetic Guiding Field 2.1. G E N E R A L C O N S I D E R A T I O N S
Using the cylindrical co-ordinate system (r, 0, z) let us write the magnetic guiding field in the median plane as 6) : B~(r, O, O) ----Bo ~ro ) ~ o hm cos where
b NO -- w"i n
-
c* m
]
,
k = (r/B~)(~B./~r),
(I) (2)
1/w = N t a n ~ , Bz(r, 0, 0) magnetic induction in the median plane; Bz average magnetic induction along the orbit; N number of sectors. (A sector is composed of a positive and a negative field magnet.) spiral angle between the locus of the field maximum and the radius; ro reference radius; Bo average magnetic field at to. Eq. (1) is a general one including as particular cases the expressions for the magnetic guiding field of the great majority of modern focusing magnets. Indeed, the magnetic guiding field in the median plane for F F A G nuclear instruments with spiral sectors 7) Bz(r, O, O) = Bo(r/ro) k { 1 + ] cos [NO -- (l/w) ha (r[ro) ] } (3) Call be obtained b y putting ho=l;
hi=/;
h=--0
for m > 2 ;
a==O.
(4)
For F F A G nuclear instruments with radial sectors, the magnetic guiding field7) B.(r, O, O) = Bo(r/ro) k ,
(5)
can be obtained b y using ho = 1 ; h m = 0
for
m>l
; am = 0 .
(6)
For both weak- and strong-focusing nuclear instruments the magnetic guiding field B,(r, O, O) = Bo(ro/r)" ,
(7)
is a particular case of eq. (1) for ho=l;
hm=0 n =
for -
[rlB,(r,
z) C. C. Iliescu, Nucl. Instr. a n d Meth. 21 (1962) 145. e) D. W. K e r s t et al., Rev. Sci. I n s t r , 31 (1960) 1076. 7) K. R. S y m o n et aL, Phys. Rev. 130 (1956) 1837.
m>/1; O,
~m=0;
0)] [aB,(,, 0, 0)/ar].
k=-n,
(8)
138
C.C.
ILIESCU
I n the following, the particular n-values ( I n I >> 1 for strong-focusing and 0 < n < 1 for weakfocusing) are irrelevant. Determining the expressions for the parameters of the magnetic guiding field as described b y eq. (1) we shall obtain the expressions for the parameters in the case of weak-focusing, strong-focusing and F F A G nuclear instruments as particular cases. 2.2. M A G N E T I C
SCALAR POTENTIAL
We shall now separate*) the variable 0 from eq. (1) using the identity 2 cos [ m N O - (re~w) ln.(r/ro) - ~=] = ( f i f o ) -im/~ exp [i(mNO - am)] + (r/ro) i~/w e x p ' [ - i(rnN8 - am)] .
(9)
Eq. (1) m a y be rewritten as [i(mNO - xm)J + ½Bo ~
B , ( r , O, O) = ½Bo /_, hm k E / ra=0
m = 0 ,m ~ ' 0 ]
exp [ - i ( m N O -- a,)]. -
(10)
The magnetic scalar potential f2(r, 0, z) is related to the magnetic induction b y its definition B ---- -- grad 12.
(11)
Starting from eq. (11) we get B . ( r , O, z) = -
(12)
~O(r, O, z ) / ~ z .
From eqs. (10) a n d (12) there results = ½Bo ,-, h,, -
-~z
~=o
.=o
exp [ i ( m N O - x=)] + ½Bo ~ hm - \~o/
==o
\to~
exp [ - i ( m N O - x,)].
(13) Likewise O(r, O, z) obeys the Lap/ace equation
(:~/o:) + (O~lOr~) + (I/r)(o~/o,) ± (~/~)(o~o/002) = o.
(14)
We m a y assume that the solution given b y eq. (14) is of the same form as eq. (13), i.e. a sum of tw~ .nfinite sums. We notice at the same time in eq. (13) terms of the form (~f2~/~z)~= o = -- B',n(r/ro)#m exp (imNO) ,
(15)
where we have introduced the notations B~, -- ½Bobmexp (-- ix,.) ; fi" ---k - i(m/w),
(16)
( 8 0 " /Oz)z= o = - B ~ ( r / r o ) pm exp ( - i m N O ) ,
(17)
and where we have noted u
B = = ½Bobs, exp (ix=) ;
fl,~ = k + i ( m / w ) .
(18)
On the basis of eqs. (I5) a n d (17) we m a y assume that the term of the ruth-order in the magnetic scalar potential expression is the form O,.(r, 0, z) --- 12"(r, 0, z) + f2~(r, 0, z),
(19)
g2~(r, 0', z) ~ •'(r, z) exp ( i m N O ) ,
(20)
12~,(r, 0, z) ----¢~,(r, z) exp (-- i r a N O ) ,
(21)
where
MAGNETIC
FIELD
DISTRIBUTION
FOR
NUCLEAR
INSTRUMENTS
139
f2(r, O, z) = ~ q~(r, z) exp (iraN0) + ~, qb'~(r, z) exp (-- imNO) .
(22)
and the expression of the magnetic scalar potential is
m=O
ra=O
We introduce eq. (22) into eq. (14) and obtain (~2¢~m/6qZ2) "{-
(6q2~m/6~r2) + (l/r)(~ff)m/Or) -- (mN/r)2@= = O.
(23)
Eq. (23) is valid both for ~'(r,z) and for 4=(,z). " 7" We m a y therefore assume 2, 3,8) t h a t #~(r,z) is of the following form
'
~
#•(r,z) =
~
'
v=D
.=o
a,,2v+l
(r-ro~U(z~ 2v+x - -- • . \ ro / \ t o /
(24)
Inserting eq. (24) into eq. (23) and equating the coefficient of the [(r - Ito)/~'o~u+2(Z/~to)2~+1 term to zero we obtain the recurrence relation which allows us to determine the coefficients of eq. (24). (2v + 2)(2v + 3) [a'u+2,2v+3 + 2a'u+l,2v+3 + a~,2o+s ] + (u + 4)(*t + 3)a~+,,2v+l + (q~ + 3)(2U + 5)a~.3,zv+l + [(u + 2) 2 -- ra2N 2] atu+2,2v+I = O .
(25)
For m = 0 we get from eq. (25) the recurrence relation determined by Boerboom, et a12). We shall write relation (25) so that the first terms be a,',,2o+ 3, a.',_ 1,2~+ s--- a'_~,zo + 3. After multiplying respectively b y j( - 1)~+ 1 we finally add up all expressions. We set the condition a'u_L2v+ 3
0
--
for
u -- j < 0,
(26)
and obtain the general recurrence relation u+l
(2v + 2)(2v + 3)a',,2~+3 --- - (u + 2)(u + 1)a',+2,2~+1 + (-- 1)= ~ E(- 1)Jya~-,2~+t] j=t
+ (- 1)=(u + 1)m2N2 ~ [(- 1)~a;.l] + (-- 1)=mzN2 ~, [(- l) ~+l:a']j, tJ"• j=o
(27)
./=O
B y means of eqs. (12), (20) and (24) we obtain
B'zm(r, O, z) = - lro ==0 o=o (2v + l)a',2v+ t ~
exp (imNO),
(28)
which for the median plane becomes B',.(,,
0, 0 ) =
-
co
1
,
(r - rob=exp (imNO). \
(29)
Taking eq. (12) into account we can write from eq. (15)
B'~,(r, O, O) = B~,(r/ro) a" exp (imNO) . If we expand (r/ro)P" in Taylor series eq. (30) becomes
,.,
Btzm(~ ", 0, 0) = ~ m .=o ~ ~.
j__I~0( ~ t - - j )
( )
f" ~ - - ~'0 u exp
(30)
(imNO) .
s) "VV.Glaser, H a n d b u c h 6 e r P h y s i k (ed. S. Fliigge, S p r i n g e r Verlag, Berlin, 1956) Vol. 33, p. 307.
(31)
140
C.C.
ILIESCU
We now compare eqs. (29) and (31), and obtain a~, x = - r . B
u! j=l-[o (~' - j)"
(32)
Introducing v = 0 into the recurrence relation (27) and using eq. (32) we obtain
~.3
roB" fl,2 _ m Z N 2 , - 1 ~uz l-I
=
•
(~
j=O
- 2-
j).
(33)
We shall calculate a~,s, a ' n . . , in the same manner and finally obtain (34}
a',, 2~+ t = roB~,X~,~'v ,
where we have noted -t x ; -- ~1 .-,~o (a" - 2 v l)~+t
~
~-1
12;+1):,=i-[o
j)
,
(35)
[(ff - 2/) 2 - m 2 N 2 ] .
(36)
The general expression given b y eq. (34) verifies the recurrence relation (25). Introducing eq. (34) into eq. (24) we get
• "(,, ,)
,-g-o,(~.(y.,,,roj r -- r o
= ,oB~ Z
~ xV=0 x ; ~ ;
Using relation
.-i (,_ro~. Z X~ j=o \ ro /
=
z
(37)
.
(r~B:-2o -,
(38)
\to/
we transform eq. (37) into • ~(r, z) = B~z \ t o /
o=o ~'°
Introducing eq. (39) into eq. (20) we obtain
(39)
.(),.
•
~:(,. 0. ,/= ~ ,,o, (~ y" °'~' .-%~:~:
(40)
For m = 1 the coefficient of (z]r)2o satisfies the recurrence relation established by Taylor*). Taking eq. (16) into account we make the respective transformations and eq. (40) leads to
__
:
\ r o / ~=o (2v 4- 1) l F,~
exp (iAm~),
(41)
where we have noted r : o = h. H
(k - 2/)~ -
~: + ~N"
+ - ~ : ( k - 2/) ~
1=0
=mNO--raiu, W
:
•
{r__~ ,--~,+
°-: ~ arc tan
\rO]
Z~O
-
-
2(mlw)(k
-
(k -- 2/)2 -- [(~/W)2
2/)
.
(43)
+ (~N)2]
Since #~,(r,z) satisfies the Laplace equation we will apply the same method as that used for #~(r,z) and obtain an expression similar to eq. (39) "
\ro/
~=o
~,:
,
(44)
MAGNETIC
FIELD
DISTRIBUTION
FOR
NUCLEAR
INSTRUMENTS
14|
(_ 1)v+l v-I
where ~
= (zv + 1)---------7. ~=oI-I[(/~, - 2t) ~ - 'n2N2] •
(45)
We introduce eq. (44) into eq. (21) and get it
\ro/
~=o
Considering eq. (18) and making some transformations we can write eq. (46) ( r ) k (-1)v+t,=o ~ \r](Z~2~ /" -(-2 v - - ~ . F . , v exp (-- iA.~)
12'~(r, O, z) = ½Boz ~o
(47)
where F.~ and Amyare given by eq. (42) and (43). Introducing eqs. (41) and (47) into eq. (19) we obtain 12,,(r, O, z)
=
((2v - +l ~ )I) ! r=~ ( 7r ~ cos AM~.
Boz (~o)~ L
(48)
The general expression of the magnetic scalar potential valid for any point of the magnet gap is O(r, O, z) = Boz {--} \to~
E
~ (2v ( - + l) F.. \ r ]
m=o v = o
cos A.~,
(49)
where F ~ and A=~ are defined by eqs. (42) and (43). The expression yielded by eq. (49) verifies the Laplace equation (14). 2.3. M A G N E T I C I N D U C T I O N
From eq. (11) we obtain the defining relations for the components of the magnetic induction B~(r, 0, z) = -- 8f2(r, O, z)/Or,
Bz(r, O, z) = - O0(r, O, z)/~z,
B0(r, 0, z) = - (llr)~(r, 8, z)/O#.
(50)
Making use of the relations yielded by eqs. (50) we obtain from eq. (49) the components of the magnetic induction: B,(r, O, z)
B o :(r)'ro ra=O
Bo(r, O, z) = BoN
~o
] voo ',- - - :~,~ -1)! - r = ~/z,2~+~[ ~=~o(2V+ \r] ( k - 2v) c o s A ~ + w s i n A m o , ~ ( - 1)
/zX
+ 1,!
sin
A~,
(51) (52)
2v
B~(r, O, z) = Bo \ r--o / ~=o 9=0
(2v)i r.~
cosA...
(53)
2.4. M A G N E T I C V E C T O R P O T E N T I A L
The magnetic vector potentialis definedby relation R
=
curl A
(54)
Eq. (54) can be written B,(r, O, z) = (l/r)(OAdO0) - (OAo/Oz) ,
(55)
Bo(r, O, z) = (OA,[Oz) -- (OAJOr) ,
(56)
B.(,., 0. x) = 0/") { [aC,'Ao)/a,'3 - (OAdaO)}
(57)
142
C.C. ILIESCU
Eq. (54) has an infinity of solutions but since we are interested in only one particular solution of the magnetic vector potential, we choose the one whose axial component is zero. The Components of the magnetic vector potential are given by the relations 9) A.(r, O, z) =
(58)
Bo(r, O, z) dr., zo
B,(r, O, z) dz + (l/r)
Ao(r, O, z) --- --
(59)
rB~(r, O, zo) d r ,
A~(,, 0, ~) = o.
(60)
Introducing eqs. (51), (52), (53) into eqs. (58), (59), (60) and carrying out the necessary transformations, we obtain the components of the magnetic vector potential: Radial component of the magnetic vector potential At(r, O, z) = N B o r •
~ mFmo \ t o ~ m=O ~=o (2v + 2)!
sin Amo
-
"
(61)
Azimuthal component of the magnetic vector potential Ao(r, O, z) = Bor
(k -- 2v) cos A ~
-~ F•o \ t o ~ m=o v=o (2v + 2) l
r k ~°
Amy
-
Fmo
-~sinA,o]
(62)
Axial component of the magnetic vector potential A,(r, 0, z) --- 0.
(60)
where F , , and A=~ are defined by eqs. (42) and (43). 3. P a r t i c u l a r
Cases
3.1. PARAMETERS OF THE MAGNETIC GU1DING FIELD FOR FFAG NUCLEAR INSTRUMENTS WITH SPIRAL SECTORS
The expressions of the parameters of the maglxetic guiding field are obtained b y introducing the conditions given by eq. (4) into the general expressions determined. For the sake of simplicity we note
r ~ = tal~ 1-[ (k 1=0
r[-11_ [[
AI,,
AI~=Ne--
In
1
r
-
2/)~,
l + w 2 N 212_~| +
+ ~ arctan ,~o
(63) 4
}½,
(64)
(k - 2/) 2 -- [ ( 1 / w 2) +
N 2]
(65)
Magnetic scalar potential O(r,O,z) = B o x r •
\%/
~ ~=o
(2v+ 1)! (1 + /AloC°SAI°) I'I (k t=o
9) A. Angot, Cornpl6ments de Math6matique (ed. Rev. d'Optique, Paris, 1949) p. 96.
2/)2
(66)
MAGNETIC
FIELD
DISTRIBUTION
FOR
NUCLEAR
INSTRUMENTS
143
Magnetic induction r k °% ~ (£~ (_5-1)v ¥) f (k--2v)
B, ir, 0, z) = Bo
1 +fAt~ cosA~o+wfk-:2v)-]
(- + 1)! 4to ,=o 11 (k B~(,.o,~) =lNBo[~o) X__o(:~
- 2/)2
' (67)
sin ,q~.
(68)
) k ~o (2-0~)! ( - l)" (1 + / A t , cosA,,) .-x B.(r,O,z) = U o ( r~o 11 ( k - 2 / ) 2 ( z ) 2 .
(69)
v
_
11 ( k - 2 / ) 2 ,=o
|=0
Magnetic vector potential <-
=
"-'
\ro/ .=o (2v + 2) 'AI°- ,=oll(k -- 2/) 2 -
sin At.,
(70)
1 + (k + 2)2 [+ (l[w) 2 [ (k+2) cosAto -- --sinAto wl ] + Ao(r,O,z) = Bo r ( r ~ * [ ~-+-~ I+[A,.
.=o(2V+2)l(k--2v)
cosA,.-t w - ~ - ~ J
,~o(k-2/)2~7)
I'
(71) (60)
A.(r, O. z) = O . 3.2. P A R A M E T E R S
OF THE MAGNETIC GUIDING
FIELD
FOR
FFAG ~UCLEA~R INSTRUMENTS
WITH
RADIAL
SECTORS
We obtain the expressions of the parameters of the magnetic guiding field by introducing the conditions yielded by eq. (6) into the general expressions determined. Magnetic scalar potential
a(r,o,,) = Bo~ ~o Magnetic induction B,(r,O,z) = B
o
(r) ro
" ~
.U. (k -- 2/)'
E (-
.-.
(-- 1)"
. = o (2v + 1)! (k --
Be(r, O, z)
2v) U (k ,=o
-
(72)
.
2/)z
(73)
'
= 0,
(74)
(_~
r k ~o ( _ l ) O O - t {~,,~ ,=o [-[ ( k - 21)2 \ r J z" . Bz(r'O'z) = B ° ( r -o- ) o =~.o (2v). Magnetic vector potential
(75) (76)
At(r, O, z) "= O,
~.(..o,:) =.or
.____(~)~ [k--~
~ i()-° ++~ 2)!
+X
,,-i
/z\zo+~
(~- :~) ,=no(~- ~)~ t;)
J'
A,(,, 0, z) = o.
(77) (60)
The expressions for the negative sectors are obtained by introducing -k instead of k in the relations shown above. 3.3. P A R A M E T E R S O F T H E M A G N E T I C G U I D I N G F I E L D F O R W E A K - A N D S T R O N G - F O C U S I N G N U C L E A R INSTRUMENTS
Since the magnetic induction in the median plane is defined by the same relation (7) and since we have imposed no restriction on k in the general expressions, there results that the same formulae willbe obtained
144
C. C, I L I E S C U
both for the weak-, and the strong-focusing nuclear instruments if we introduce the conditions yielded by eq. (8) into the general expressions determined. Magnetic scalar potential ~(r, O, z) = Boz
v=o
(2v + 1)! ,1-Io(n + 2/)2
(78)
For n =0.5 we reach, from eq. (78), the expression of the magnetic scalar potential obtained by Stoker aP). Magnetic induction
et
(2v+ 1)!(n+2v) I-I ( n + 2 / ) 2 t~=0
,
Bo(r, 0, z) -- 0, .-t
(79)
I=O
o=o ~
(74)
?--~
Magnetic vector potential A,(r, 0, z) = 0, (~)'[ Ao(r, 0, z) = Bor
1 ~
~ (-1) ° 0-1 (z)2O+2] + ~=o (2v + 2) l (n + 2v) ,=ol-[(n + 2/)2 , A~(r, 0, z) = 0.
(76) (81) (60)
Introducing - n into the preceding expressions instead of n we obtain the expressions for the negative sectors. 4. Conclusions Using a general form of the magnetic guiding field in the median plane, valid for the focusing magnets of weak-focusing, strong-focusing and FFAG nuclear instruments, expressions are derived for the magnetic scalar and vector potential and for the magnetic induction. For these parameters the actual expressions, in the form of infinite series, are obtained as particular cases. In practical applications, such as the study of particle dynamics and pole face design, one retains only a limited number of terms depending on magnet gap dimensions and allowable error.
Acknowledgements The author wishes to convey his thanks to Prof. Dr. H. Hulubei and Prof. Dr. S. Titeica for their stimulating interest and precious support; and to the team of researchers at the Betatron of the Institute for Atomic Physics for many helpful discussions throughout the course of this work.
INSTRUMENTS
AND METHODS
21 (1963) 136--144; N O R T H - H O L L _ A _ N D P U B L I S H I N G
CO.
MAGNETIC FIELD DISTRIBUTION FOR WEAK-, STRONG- AND FFAG-FOCUSING PARTICLE ACCELERATORS AND OTHER NUCLEAR INSTRUMENTS C. C. ILIESCU Institut* /or Atomqc Physics, Bucharest, t~umania Received 21 J u l y 1962 F o r the magnetic guiding field of the weak-, strong-, a n d FFAG-fecusing nuclear instruments used in the acceleration or dispersion of charged particles, general analytic expressions, valid for a n y point of the magnet gap, are derived concerning the following p a r a m e t e r s : magnetic scalar potential, magnetic induction a n d magnetic vector potential. These expressions are useful for the s t u d y of particle motion as well as for pole face design.
1. Introduction The motion of particles in the weak-, strong- and FFAG-focusing nuclear instruments used in the acceleration or dispersion of charged particles (particle accelerators, spectrometers, analyzers, etc) takes place along the central orbit under the action of the restoring forces produced by the magnetic guiding field. The magnetic guiding field is defined by the following parameters: magnetic scalar potential, magnetic induction and magnetic vector potential. In the literature, the expressions for these parameters concerning the weak-, and strong-focusing particle accelerators, valid for any point of the magnet gap, are considered ill a first-order approximation. The first-order approximation was thought to be insufficient in some cases, especially for large gap particle spectrometers and F F A G accelerators. For these particlflar cases, more accurate expressions were require& For a double-focusing spectrometer with a field index n = 0.5, Stoker et al. 1) determined the expression of the magnetic scalar potential in the form of an infinite power series of z and z. Boerboom et al.2), in a study of the magnetic field produced b y conical pole faces used in spectrometers, considered the magnetic scalar potential as a double infinite power series of (z - re)It o and z/r o and established a recurrence relation between the coefficients. Analyzing the focusing properties of the inhomogeneous magnetic sector fields of cylindrical symmetry used for spectrometers, Bretscher 3) considered the magnetic guiding field in the median plane as an infinite power series of (r - ro)/r o. The magnetic guiding field expressions were not considered a function of the field index. Hence, the general terms of the magnetic parameters series were not obtained. For a F F A G accelerator, Taylor*) computed the magnetic field parameters b y means of a digital computer and obtained a recurrence relation between some intermediate coefficients necessary for determining the expression of the magnetic scalar potential. The object of this paper is to derive the general analytic expressions for the magnetic guiding field parameters, valid for the whole magnet gap, starting from the expression of the magnetic guiding field in the median plane. Assuming that for the weak-focusing, strong-focusing and F F A G nuclear instruments the expressions of the magnetic guiding fields in the median plane are particular cases of a magnetic guiding field described by an infinite series, this paper determines the general analytical expressions for all the param1) p. H. Stoker vt aL, P h y s i c a 20 (1954) 337. 2) A. J. H. Boerboom, H. A. Tasman, H. W a c h s m u t h , Z. Naturforsch, 14a (I959) 816. 3) M. 2VL Bretscher, O ~ N L - 2 8 8 4 (196~). *) R. Taylor, A E R E - R 3097 (1959)"
136
MAGNETIC
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137
eters and simplifies them for the above-mentioned nuclear instruments. A satisfactory agreement was reached with the relations previously obtained by the authors mentioned. The expressions determined are useful both for the study of the motion of the particles a n d for the determination of the pole face shape necessary for a proposed magnetic guiding field distributionS). In the foliowing we shall neglect the effects of the finite extensions in the radial direction of the pole face as well as the field distortion due to remanence, saturation, etc.
2. General Analytic Expressions of the Parameters of the Magnetic Guiding Field 2.1. G E N E R A L C O N S I D E R A T I O N S
Using the cylindrical co-ordinate system (r, 0, z) let us write the magnetic guiding field in the median plane as 6) : B~(r, O, O) ----Bo ~ro ) ~ o hm cos where
b NO -- w"i n
-
c* m
]
,
k = (r/B~)(~B./~r),
(I) (2)
1/w = N t a n ~ , Bz(r, 0, 0) magnetic induction in the median plane; Bz average magnetic induction along the orbit; N number of sectors. (A sector is composed of a positive and a negative field magnet.) spiral angle between the locus of the field maximum and the radius; ro reference radius; Bo average magnetic field at to. Eq. (1) is a general one including as particular cases the expressions for the magnetic guiding field of the great majority of modern focusing magnets. Indeed, the magnetic guiding field in the median plane for F F A G nuclear instruments with spiral sectors 7) Bz(r, O, O) = Bo(r/ro) k { 1 + ] cos [NO -- (l/w) ha (r[ro) ] } (3) Call be obtained b y putting ho=l;
hi=/;
h=--0
for m > 2 ;
a==O.
(4)
For F F A G nuclear instruments with radial sectors, the magnetic guiding field7) B.(r, O, O) = Bo(r/ro) k ,
(5)
can be obtained b y using ho = 1 ; h m = 0
for
m>l
; am = 0 .
(6)
For both weak- and strong-focusing nuclear instruments the magnetic guiding field B,(r, O, O) = Bo(ro/r)" ,
(7)
is a particular case of eq. (1) for ho=l;
hm=0 n =
for -
[rlB,(r,
z) C. C. Iliescu, Nucl. Instr. a n d Meth. 21 (1962) 145. e) D. W. K e r s t et al., Rev. Sci. I n s t r , 31 (1960) 1076. 7) K. R. S y m o n et aL, Phys. Rev. 130 (1956) 1837.
m>/1; O,
~m=0;
0)] [aB,(,, 0, 0)/ar].
k=-n,
(8)
138
C.C.
ILIESCU
I n the following, the particular n-values ( I n I >> 1 for strong-focusing and 0 < n < 1 for weakfocusing) are irrelevant. Determining the expressions for the parameters of the magnetic guiding field as described b y eq. (1) we shall obtain the expressions for the parameters in the case of weak-focusing, strong-focusing and F F A G nuclear instruments as particular cases. 2.2. M A G N E T I C
SCALAR POTENTIAL
We shall now separate*) the variable 0 from eq. (1) using the identity 2 cos [ m N O - (re~w) ln.(r/ro) - ~=] = ( f i f o ) -im/~ exp [i(mNO - am)] + (r/ro) i~/w e x p ' [ - i(rnN8 - am)] .
(9)
Eq. (1) m a y be rewritten as [i(mNO - xm)J + ½Bo ~
B , ( r , O, O) = ½Bo /_, hm k E / ra=0
m = 0 ,m ~ ' 0 ]
exp [ - i ( m N O -- a,)]. -
(10)
The magnetic scalar potential f2(r, 0, z) is related to the magnetic induction b y its definition B ---- -- grad 12.
(11)
Starting from eq. (11) we get B . ( r , O, z) = -
(12)
~O(r, O, z ) / ~ z .
From eqs. (10) a n d (12) there results = ½Bo ,-, h,, -
-~z
~=o
.=o
exp [ i ( m N O - x=)] + ½Bo ~ hm - \~o/
==o
\to~
exp [ - i ( m N O - x,)].
(13) Likewise O(r, O, z) obeys the Lap/ace equation
(:~/o:) + (O~lOr~) + (I/r)(o~/o,) ± (~/~)(o~o/002) = o.
(14)
We m a y assume that the solution given b y eq. (14) is of the same form as eq. (13), i.e. a sum of tw~ .nfinite sums. We notice at the same time in eq. (13) terms of the form (~f2~/~z)~= o = -- B',n(r/ro)#m exp (imNO) ,
(15)
where we have introduced the notations B~, -- ½Bobmexp (-- ix,.) ; fi" ---k - i(m/w),
(16)
( 8 0 " /Oz)z= o = - B ~ ( r / r o ) pm exp ( - i m N O ) ,
(17)
and where we have noted u
B = = ½Bobs, exp (ix=) ;
fl,~ = k + i ( m / w ) .
(18)
On the basis of eqs. (I5) a n d (17) we m a y assume that the term of the ruth-order in the magnetic scalar potential expression is the form O,.(r, 0, z) --- 12"(r, 0, z) + f2~(r, 0, z),
(19)
g2~(r, 0', z) ~ •'(r, z) exp ( i m N O ) ,
(20)
12~,(r, 0, z) ----¢~,(r, z) exp (-- i r a N O ) ,
(21)
where
MAGNETIC
FIELD
DISTRIBUTION
FOR
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INSTRUMENTS
139
f2(r, O, z) = ~ q~(r, z) exp (iraN0) + ~, qb'~(r, z) exp (-- imNO) .
(22)
and the expression of the magnetic scalar potential is
m=O
ra=O
We introduce eq. (22) into eq. (14) and obtain (~2¢~m/6qZ2) "{-
(6q2~m/6~r2) + (l/r)(~ff)m/Or) -- (mN/r)2@= = O.
(23)
Eq. (23) is valid both for ~'(r,z) and for 4=(,z). " 7" We m a y therefore assume 2, 3,8) t h a t #~(r,z) is of the following form
'
~
#•(r,z) =
~
'
v=D
.=o
a,,2v+l
(r-ro~U(z~ 2v+x - -- • . \ ro / \ t o /
(24)
Inserting eq. (24) into eq. (23) and equating the coefficient of the [(r - Ito)/~'o~u+2(Z/~to)2~+1 term to zero we obtain the recurrence relation which allows us to determine the coefficients of eq. (24). (2v + 2)(2v + 3) [a'u+2,2v+3 + 2a'u+l,2v+3 + a~,2o+s ] + (u + 4)(*t + 3)a~+,,2v+l + (q~ + 3)(2U + 5)a~.3,zv+l + [(u + 2) 2 -- ra2N 2] atu+2,2v+I = O .
(25)
For m = 0 we get from eq. (25) the recurrence relation determined by Boerboom, et a12). We shall write relation (25) so that the first terms be a,',,2o+ 3, a.',_ 1,2~+ s--- a'_~,zo + 3. After multiplying respectively b y j( - 1)~+ 1 we finally add up all expressions. We set the condition a'u_L2v+ 3
0
--
for
u -- j < 0,
(26)
and obtain the general recurrence relation u+l
(2v + 2)(2v + 3)a',,2~+3 --- - (u + 2)(u + 1)a',+2,2~+1 + (-- 1)= ~ E(- 1)Jya~-,2~+t] j=t
+ (- 1)=(u + 1)m2N2 ~ [(- 1)~a;.l] + (-- 1)=mzN2 ~, [(- l) ~+l:a']j, tJ"• j=o
(27)
./=O
B y means of eqs. (12), (20) and (24) we obtain
B'zm(r, O, z) = - lro ==0 o=o (2v + l)a',2v+ t ~
exp (imNO),
(28)
which for the median plane becomes B',.(,,
0, 0 ) =
-
co
1
,
(r - rob=exp (imNO). \
(29)
Taking eq. (12) into account we can write from eq. (15)
B'~,(r, O, O) = B~,(r/ro) a" exp (imNO) . If we expand (r/ro)P" in Taylor series eq. (30) becomes
,.,
Btzm(~ ", 0, 0) = ~ m .=o ~ ~.
j__I~0( ~ t - - j )
( )
f" ~ - - ~'0 u exp
(30)
(imNO) .
s) "VV.Glaser, H a n d b u c h 6 e r P h y s i k (ed. S. Fliigge, S p r i n g e r Verlag, Berlin, 1956) Vol. 33, p. 307.
(31)
140
C.C.
ILIESCU
We now compare eqs. (29) and (31), and obtain a~, x = - r . B
u! j=l-[o (~' - j)"
(32)
Introducing v = 0 into the recurrence relation (27) and using eq. (32) we obtain
~.3
roB" fl,2 _ m Z N 2 , - 1 ~uz l-I
=
•
(~
j=O
- 2-
j).
(33)
We shall calculate a~,s, a ' n . . , in the same manner and finally obtain (34}
a',, 2~+ t = roB~,X~,~'v ,
where we have noted -t x ; -- ~1 .-,~o (a" - 2 v l)~+t
~
~-1
12;+1):,=i-[o
j)
,
(35)
[(ff - 2/) 2 - m 2 N 2 ] .
(36)
The general expression given b y eq. (34) verifies the recurrence relation (25). Introducing eq. (34) into eq. (24) we get
• "(,, ,)
,-g-o,(~.(y.,,,roj r -- r o
= ,oB~ Z
~ xV=0 x ; ~ ;
Using relation
.-i (,_ro~. Z X~ j=o \ ro /
=
z
(37)
.
(r~B:-2o -,
(38)
\to/
we transform eq. (37) into • ~(r, z) = B~z \ t o /
o=o ~'°
Introducing eq. (39) into eq. (20) we obtain
(39)
.(),.
•
~:(,. 0. ,/= ~ ,,o, (~ y" °'~' .-%~:~:
(40)
For m = 1 the coefficient of (z]r)2o satisfies the recurrence relation established by Taylor*). Taking eq. (16) into account we make the respective transformations and eq. (40) leads to
__
:
\ r o / ~=o (2v 4- 1) l F,~
exp (iAm~),
(41)
where we have noted r : o = h. H
(k - 2/)~ -
~: + ~N"
+ - ~ : ( k - 2/) ~
1=0
=mNO--raiu, W
:
•
{r__~ ,--~,+
°-: ~ arc tan
\rO]
Z~O
-
-
2(mlw)(k
-
(k -- 2/)2 -- [(~/W)2
2/)
.
(43)
+ (~N)2]
Since #~,(r,z) satisfies the Laplace equation we will apply the same method as that used for #~(r,z) and obtain an expression similar to eq. (39) "
\ro/
~=o
~,:
,
(44)
MAGNETIC
FIELD
DISTRIBUTION
FOR
NUCLEAR
INSTRUMENTS
14|
(_ 1)v+l v-I
where ~
= (zv + 1)---------7. ~=oI-I[(/~, - 2t) ~ - 'n2N2] •
(45)
We introduce eq. (44) into eq. (21) and get it
\ro/
~=o
Considering eq. (18) and making some transformations we can write eq. (46) ( r ) k (-1)v+t,=o ~ \r](Z~2~ /" -(-2 v - - ~ . F . , v exp (-- iA.~)
12'~(r, O, z) = ½Boz ~o
(47)
where F.~ and Amyare given by eq. (42) and (43). Introducing eqs. (41) and (47) into eq. (19) we obtain 12,,(r, O, z)
=
((2v - +l ~ )I) ! r=~ ( 7r ~ cos AM~.
Boz (~o)~ L
(48)
The general expression of the magnetic scalar potential valid for any point of the magnet gap is O(r, O, z) = Boz {--} \to~
E
~ (2v ( - + l) F.. \ r ]
m=o v = o
cos A.~,
(49)
where F ~ and A=~ are defined by eqs. (42) and (43). The expression yielded by eq. (49) verifies the Laplace equation (14). 2.3. M A G N E T I C I N D U C T I O N
From eq. (11) we obtain the defining relations for the components of the magnetic induction B~(r, 0, z) = -- 8f2(r, O, z)/Or,
Bz(r, O, z) = - O0(r, O, z)/~z,
B0(r, 0, z) = - (llr)~(r, 8, z)/O#.
(50)
Making use of the relations yielded by eqs. (50) we obtain from eq. (49) the components of the magnetic induction: B,(r, O, z)
B o :(r)'ro ra=O
Bo(r, O, z) = BoN
~o
] voo ',- - - :~,~ -1)! - r = ~/z,2~+~[ ~=~o(2V+ \r] ( k - 2v) c o s A ~ + w s i n A m o , ~ ( - 1)
/zX
+ 1,!
sin
A~,
(51) (52)
2v
B~(r, O, z) = Bo \ r--o / ~=o 9=0
(2v)i r.~
cosA...
(53)
2.4. M A G N E T I C V E C T O R P O T E N T I A L
The magnetic vector potentialis definedby relation R
=
curl A
(54)
Eq. (54) can be written B,(r, O, z) = (l/r)(OAdO0) - (OAo/Oz) ,
(55)
Bo(r, O, z) = (OA,[Oz) -- (OAJOr) ,
(56)
B.(,., 0. x) = 0/") { [aC,'Ao)/a,'3 - (OAdaO)}
(57)
142
C.C. ILIESCU
Eq. (54) has an infinity of solutions but since we are interested in only one particular solution of the magnetic vector potential, we choose the one whose axial component is zero. The Components of the magnetic vector potential are given by the relations 9) A.(r, O, z) =
(58)
Bo(r, O, z) dr., zo
B,(r, O, z) dz + (l/r)
Ao(r, O, z) --- --
(59)
rB~(r, O, zo) d r ,
A~(,, 0, ~) = o.
(60)
Introducing eqs. (51), (52), (53) into eqs. (58), (59), (60) and carrying out the necessary transformations, we obtain the components of the magnetic vector potential: Radial component of the magnetic vector potential At(r, O, z) = N B o r •
~ mFmo \ t o ~ m=O ~=o (2v + 2)!
sin Amo
-
"
(61)
Azimuthal component of the magnetic vector potential Ao(r, O, z) = Bor
(k -- 2v) cos A ~
-~ F•o \ t o ~ m=o v=o (2v + 2) l
r k ~°
Amy
-
Fmo
-~sinA,o]
(62)
Axial component of the magnetic vector potential A,(r, 0, z) --- 0.
(60)
where F , , and A=~ are defined by eqs. (42) and (43). 3. P a r t i c u l a r
Cases
3.1. PARAMETERS OF THE MAGNETIC GU1DING FIELD FOR FFAG NUCLEAR INSTRUMENTS WITH SPIRAL SECTORS
The expressions of the parameters of the maglxetic guiding field are obtained b y introducing the conditions given by eq. (4) into the general expressions determined. For the sake of simplicity we note
r ~ = tal~ 1-[ (k 1=0
r[-11_ [[
AI,,
AI~=Ne--
In
1
r
-
2/)~,
l + w 2 N 212_~| +
+ ~ arctan ,~o
(63) 4
}½,
(64)
(k - 2/) 2 -- [ ( 1 / w 2) +
N 2]
(65)
Magnetic scalar potential O(r,O,z) = B o x r •
\%/
~ ~=o
(2v+ 1)! (1 + /AloC°SAI°) I'I (k t=o
9) A. Angot, Cornpl6ments de Math6matique (ed. Rev. d'Optique, Paris, 1949) p. 96.
2/)2
(66)
MAGNETIC
FIELD
DISTRIBUTION
FOR
NUCLEAR
INSTRUMENTS
143
Magnetic induction r k °% ~ (£~ (_5-1)v ¥) f (k--2v)
B, ir, 0, z) = Bo
1 +fAt~ cosA~o+wfk-:2v)-]
(- + 1)! 4to ,=o 11 (k B~(,.o,~) =lNBo[~o) X__o(:~
- 2/)2
' (67)
sin ,q~.
(68)
) k ~o (2-0~)! ( - l)" (1 + / A t , cosA,,) .-x B.(r,O,z) = U o ( r~o 11 ( k - 2 / ) 2 ( z ) 2 .
(69)
v
_
11 ( k - 2 / ) 2 ,=o
|=0
Magnetic vector potential <-
=
"-'
\ro/ .=o (2v + 2) 'AI°- ,=oll(k -- 2/) 2 -
sin At.,
(70)
1 + (k + 2)2 [+ (l[w) 2 [ (k+2) cosAto -- --sinAto wl ] + Ao(r,O,z) = Bo r ( r ~ * [ ~-+-~ I+[A,.
.=o(2V+2)l(k--2v)
cosA,.-t w - ~ - ~ J
,~o(k-2/)2~7)
I'
(71) (60)
A.(r, O. z) = O . 3.2. P A R A M E T E R S
OF THE MAGNETIC GUIDING
FIELD
FOR
FFAG ~UCLEA~R INSTRUMENTS
WITH
RADIAL
SECTORS
We obtain the expressions of the parameters of the magnetic guiding field by introducing the conditions yielded by eq. (6) into the general expressions determined. Magnetic scalar potential
a(r,o,,) = Bo~ ~o Magnetic induction B,(r,O,z) = B
o
(r) ro
" ~
.U. (k -- 2/)'
E (-
.-.
(-- 1)"
. = o (2v + 1)! (k --
Be(r, O, z)
2v) U (k ,=o
-
(72)
.
2/)z
(73)
'
= 0,
(74)
(_~
r k ~o ( _ l ) O O - t {~,,~ ,=o [-[ ( k - 21)2 \ r J z" . Bz(r'O'z) = B ° ( r -o- ) o =~.o (2v). Magnetic vector potential
(75) (76)
At(r, O, z) "= O,
~.(..o,:) =.or
.____(~)~ [k--~
~ i()-° ++~ 2)!
+X
,,-i
/z\zo+~
(~- :~) ,=no(~- ~)~ t;)
J'
A,(,, 0, z) = o.
(77) (60)
The expressions for the negative sectors are obtained by introducing -k instead of k in the relations shown above. 3.3. P A R A M E T E R S O F T H E M A G N E T I C G U I D I N G F I E L D F O R W E A K - A N D S T R O N G - F O C U S I N G N U C L E A R INSTRUMENTS
Since the magnetic induction in the median plane is defined by the same relation (7) and since we have imposed no restriction on k in the general expressions, there results that the same formulae willbe obtained
144
C. C, I L I E S C U
both for the weak-, and the strong-focusing nuclear instruments if we introduce the conditions yielded by eq. (8) into the general expressions determined. Magnetic scalar potential ~(r, O, z) = Boz
v=o
(2v + 1)! ,1-Io(n + 2/)2
(78)
For n =0.5 we reach, from eq. (78), the expression of the magnetic scalar potential obtained by Stoker aP). Magnetic induction
et
(2v+ 1)!(n+2v) I-I ( n + 2 / ) 2 t~=0
,
Bo(r, 0, z) -- 0, .-t
(79)
I=O
o=o ~
(74)
?--~
Magnetic vector potential A,(r, 0, z) = 0, (~)'[ Ao(r, 0, z) = Bor
1 ~
~ (-1) ° 0-1 (z)2O+2] + ~=o (2v + 2) l (n + 2v) ,=ol-[(n + 2/)2 , A~(r, 0, z) = 0.
(76) (81) (60)
Introducing - n into the preceding expressions instead of n we obtain the expressions for the negative sectors. 4. Conclusions Using a general form of the magnetic guiding field in the median plane, valid for the focusing magnets of weak-focusing, strong-focusing and FFAG nuclear instruments, expressions are derived for the magnetic scalar and vector potential and for the magnetic induction. For these parameters the actual expressions, in the form of infinite series, are obtained as particular cases. In practical applications, such as the study of particle dynamics and pole face design, one retains only a limited number of terms depending on magnet gap dimensions and allowable error.
Acknowledgements The author wishes to convey his thanks to Prof. Dr. H. Hulubei and Prof. Dr. S. Titeica for their stimulating interest and precious support; and to the team of researchers at the Betatron of the Institute for Atomic Physics for many helpful discussions throughout the course of this work.