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A second-order approximation of particle motion in the fringing field of a dipole magnet
International
Journal
Elsevier Scientific
of Mass
Publishing
Spectromehy
Company,
and Ion
Amsterdam
36 (1980) 167-133 Printed in The Netherlands
Physics.
-
A SECOND-ORDER APPROXIMATION OF PARTICLE FRINGING FIELD OF A DIPOLE MAGNET
MOTION
,167
IN THE
N-1. TARANTIN Labotxztory
of Nuclear
Reactions,
(First received 27 June 1979;
Joint
Irish-tute
for Nuclear
in revised form 22 February
Research,
Dubna
(U.S.S.R.)
1980)
ABSTRACT The radial and axial motion of charged particles in the fringing field of an arbitrary dipole magnet has been considered with accuracy to the second-.ordcr of small quantities. The dipole magnet has an inhomogeneous field and oblique entrance and exit boundaries in the form of second-order curves. The region of the fringing field has a variable extension. A new definition of the effective boundary of the real fringing field of the dipole magnet is used. A better understanding of the influence of the fringing magnetic field on the motion of charged particles in the pole gap of the dipole magnet has been obtained. In particular, it is shown that it is important to take into account, in the second approximation, some terms related formally to the next approximations. The resultsare presented in a form convenient for practical calculations.
INTRODUCTION The fringing field in the vicinity dipole magnet has a substantial
of
the
entrance
and
exit
boundaries
of
influence on the motion of charged particles and its action should be thoroughly taken into account in the construction of precise magnetic analyzers. The influence of the fringing field at the entrance and exit boundaries of a dipole magnet on the motion of charged particles has been considered many times (see, for example, refs. l-12 and refs. therein). A most detailed consideration has been carried out for a homogeneous magnetic field in ref. 8 and for an inhcmogeneous magnetic field in ref. 11; the general results of these two papers do not coincide, but in ref. 11 it is noted that the results therein “agree with corrected expressions of ref. 8”. In the present paper a new second-order approximation of the radial and axial motion of charged particles in the fringing field of the dipole magnet is described. This consideration is made otherwise than in refs. 8 and 11 and for a dipole magnet with somewhat more general param eters compared with that in ref. 11. A dipole magnet with a radially inhomogeneous axially symmetric main field is considered in the present paper. The oblique entrance and exit boundaries of the magnet have the form of the most general secondthe
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order curve with variable curvature. The extension of the fringing field arbitrarily varies along the boundary of the magnet pole,
FRINGING
FIELD
OF THE
DIPOLE
MAGNET
The region of the charged particle motion considered in the vicinity of the entrance boundary of the dipole magnet is shown in Fig. 1. The systems of coordinates used and particle trajectories in the radial and axial cross-sections are also given in this figure. The effective boundary of the magnetic field is determined by the relation:
Pb(P)
=*1(P)--
(1)
where p,,(p) is the equation of the effective boundary in the principal system of coordinates with an axis at point 01, p = (r -2&,)/R,, is the relative deviation of the radial coordinate r from the optical axis of the magnet - a circumference with radius Rap p < 1, B,(p, cp, 0) is the fringing field in the
of a Fig. 1. Cylindrical systems of coordinates (r_ I$. z) and (r, 8. z) and trajectories charged particle in the fringing field at the entrance. boundary.of the dipole magnet in the Real trajectory of the particle; (- - - - - -) convenradial and axial cross-sections. ( -) tional trajectory of the particle; (- - - - -) optical axis of the dipole magnet with a constant radius of curvature, Re The positive directions are shown by arrows. r - Ro < Ro, r-RR,
< Ro_
169
median plane (z = 0) of the dipole magnet, (1 + alp)& is the main field in the z = 0 plane of the pole gap, taken for a second-order approximation with sufficient accuracy, aI is the inhomogeneity index of the main magnetic field, which is determined from the formula
cp,(p) and (PIN are the initial and final field region, determined by the conditions B,(P, B.
rp~. 0) = 0
&CP,
and
angular
9rr, 0) = (I+
alpI
coordinates
of the fringing
&
= mOvOc/eJh
where rn,,, voI e. are the mass, velocity and charge of the main particle, and c is the velocity of light in a vacuum. Here and below, the sufficient accuracy of the quantities considered means the accuracy required to obtain the final results -the values of the coordinates of the particle trajectory and their derivatives during the passage of the particle through a ikinging magnetic
field, within the second-order small quantities. The small quantities include k h/R0 < I, where h is the height of the also the difference vrr(~) - (Pi pole gap and upis expressed in radians. For instance, in the determination of the main field within the firstorder small quantity p, the effective boundary is determined using eqn_ (1) within the second-order small quantities:
syn@)&(A alp) 0) 5%
---
(I+
YI(Y)
BO
dq
~
(1
- adO h(P)
- cm(P)1
Such a determination (1) of the effective boundary is convenient because it corresponds to the technique of measuring the topography of the radially inhomogeneous and axially symmetric main magnetic field along concentric circles with constant values of the magnetic field_ The effective boundary, because of saturation of the magnetic pole tips, may
have
variable
curvature_
In the general
case of the second-order
approxi-
mation it is describable by the second-order curve, which has the form of an ellipse, parabola or hyperbola, with a focus at the point O2 lying on the normal to the effective boundary at the point of its intersection with the optical axis of the magnet. In any particular case. the type of the curve is determined by any five points lying on the effective boundary. In the cylindrical system of coordinates with an axis in the focus canonical equation for the second-order curve is
?+!I?)=
p
1 + e cos 0
of the curve
O2 the general
@a)
where P is the focal parameter or semi-latus rectum of the curve, and e is the eccentricity: e < 1 for an ellipse (for a circle, being a particular case of an
170
ellipse, e = 0), e = 1 for a parabola and e > 1 for a hyperbola. In the vicinity of the vertex of the curve coinciding with the point of intersection between the boundary and the optical axis of the dipole magnet, at 8 < 1 eqn. (2a) can be approximately written in the form:
VI
=
r-_Rb =
R
(=I
K02
0
where R, = P/( 1 + e) is the distance between the focus and vertex of the curve, K = Pe/{ZR,(l + e)2) = R,e/{2R.(l+ e)) and r < 1. In some concrete cases, K takes the following values: K < 0.25 Rb/Ro for an ellipse, K = 0.25 Rb/Ro for a parabola, K > 0.25 R,,/R, for a hyperbola, K 5 0 for a circle with radius R,, which is considered positive for a convex boundary with respect to the dipole magnet, and Rb = = for a straight line. Note that if the axis of the cylindrical system of cooniinates, 02, is chosen to lie on the normal to the boundary, all the asymmetry of the effective boundary with respect to the normal is expressed by terms of the type e3, 05, etc. of eqn. (2b), which are not considered in a second approximation. In the main coordinate system, curve (2b) is describable by the following formula: (pb(P) =
1
- ~(1 + c2)
flp + f2p2 = tp + z
1p2
(3)
where s = sin E ,
t=tane,
c = cos E _
In the derivation of eqn. (3), the conventional transformation of the coordinates is used for the conversion of one cylindrical system of coordinates with an axis at point O2 to another system of the same kind with an axis at point 0, _ The expression Rb
I(
I--2K;t
)
=R,,,,
entering into eqn, radius of curvature
(3) is the radius of curvature of curve (2) at 8 = 0. The at any point of the line is determined from the formula:
Therefore, as follows from eqn. (3), the most complicated shape of the effective boundary can be described in the second-order approximation using of the boundary at the point of its intersecthe radius of the curvature R,, tion with the optical axis of the dipole magnet. It is supposed that the extension of the region of the fringing magnetic field may be different at different values of p, i.e. it is assumed with a sufficient degree of accuracy that e-f(P)
-
tiCPI
= a+
gP)(v2o
-
t430)
(4)
171
where g = constant ,
910 = (m(O) ,
=dcp,o-9g,o<
920=9n(O)
1.
At a constant linear width (not the angular width) of the fringing field region, as in the cases of the homogeneous main field (al = 0) or the specially shielding inhomogeneous field:
The r-th and 9-th field components for z + 0, required
for determination
and the axial components
of the field
of the trajectory of the particle, are
found in the usual manner from the conditions rot B = 0, where the equality to zero of the right-hand parts of equations is conditioned by the negligibly small value of the current of the charged particles under consideration, as compared with the current.of the excitation windings of the electromagnet, div B = 0, and the requirements of field symmetry with respect to the median plane. Prom the conditions aB, az
aBZ -o -ar
and
la& r
a9
a% -
a2
--0
following
from rot B = 0 one obtains B,(p, 9, g).= c aB,/ap p) aB,/ap where 5 = z/I&, < i, B, = B,(p, 9,O). By using the expressions found andthe condition
and B,(p,
p, 5) =
r(l-
one obtains
Proceeding in this way, one obtains with the required accuracy:
(W
+ &5” [S
+ (I-
4p) 3
+ 2(1-
2p) a;;$
-
2 3
+ 2 31
(SC)
Here B, and B, are determined with the accuracy up to the first-order small quantities, and Bcp up to the small quantities of the zero-th order. The small quantities include p, 5 and rp,, - (plo, which are less than unity, and numeri-
172
cal coefficients of the rejected terms of expansion rows, equal to l/120, l/720, etc. In estimating the order o.f the small quantities it is taken into account that 1
anBz _
s”
a”& _
and
-Y%o)" c"~cp*o 39 (920 - CplOY a,pn i.e. these derivatives are substantially greater quantities as they are inversely proportional to the small quantity in the nth power. To unify the integration in solving the particle trajectory equations B,(p, p, 0) is expressed through the function of the distribution of the fringing field along the optical axis of the dipole magnet B,(O, 9,O) = B&(9). From the condition of the smooth conjugation of the function of the distribution of the fringing field, Bob(q), with the main field at 9 = cpzoand with the field identically equal to zero, at 9 = 910 one has d"Mcp,o) MP20)
=
MrplO)
1,
_d"bCcplo)
_
(6)
=o, as"
-TF-O
for n = 1, 2, 3, etc. These conditions, the form of which is determined by the new definition of the effective boundary (l), are much simpler and more illustrative than analogous conditions for the fringing field in the determination of the effective boundary described in ref. 11. From the magnetostatks it foilows that the functions of the distribution of the fringing field at diEferent values of p will be one and the same with sufficient accuracy if the quantity of the fringing field is expressed in the value of the main field, (1 + a,p)B o, and the angular coordinate in the units of the -@a extension, (1 + gp) (9z0 - 910). of the fringing field at a preset value of p. Therefore, one can assume that Bz(P, 9,9)
(7)
= (1 + e,p) BOW@)
where CD= [9 -
pb(P)l/(~ + WI = cp-
fiP
+ (frg
-
f2)
P2 -gpGJ,
PlO
G
*
6
cp20
03)
The strict validity of’ relation (7) is evident in the particular case of a linearly inhomogeneous main field without the quadratic component and a straight boundary orthogonal to the optical axis of the magnet, since in this csse none of the directions p = constant is distinguished anyway. From relation (7) it follows that the effective boundary is located with respect to the fringing field boundary 9,,(p) in the same way as 91(p) with respect to 9r i (p) : . cpbbI=chi@I-'J
.
9IL@)
916 1
ca
+alP)~o~(*YC(1+~,P)
Bol3 drp
173
Hence, according to (I) (pb(P) -
PII
= -(I
+a)
P20
(9)
The distributions of the fringing magnetic field at various values of p are shown in Fig. 2. The partial derivatives over p and 9 entering into eqns. (5) are expressed through the total derivatives of function (7) over + with the help of the following recurrent relation: _$;> -
= (-I)“& (-l)“nB,
z-y:
z;-;;:;
+ m) g1 PI + nfF(2f2p
(n + m -
+ (n - 1)
I)gl
-i-&f?))
flm2fil
w9
where b = b(q~); IZ, m = 0, 1, 2, 3, etc. Formula (10) was derived by the induction method; namely, by generalizing expressions for derivatives in the particular differentiation 0fBoverp and cp,i.e.
a% _ a - ap 10 + alpI ap
-
= (1 +a#)&-
= (--1)‘&l
THE
CHARGED
db
dB
db -
The expression 1 and m = 0.
&3b(*)l a@ --+aalBob=
ap
+ (al -g)
obtained PARTICLE
PI
+
(2fs
is a particular MOTION
The radial and axial equations netic field have the general forms
IN THE
+ ti))
-
(--1)‘&ba1
csse of general formula FRINGING
, etc. (10)
at R =
FIELD
of the charged particle motion in the mag-
(11) where. m, u, and e are the mass, velocity and charge of the considered particle. After replacing the time derivatives by the derivatives over cp these equations are transformed in a first-order approximation to the following
91(P)-9btP)
0
9iik')-9b(P)
9-9btP)
Fig. 2. The distribution of the fringing field of the dipole magnet at different zx~;r~i~,tiO the relation B,(p, cp, 0) = (1 + aIp).B&(+) where @ = {so --(p))/{l ,
values of p + gp),
particle trajectory equations: (1
P ” = I + p + s“Bq/B,, 5‘” = (1-t
2p -
6) BJB,
-
l1-2~
-
6) B,/Bo
(12)
p’B,/B,,
where 6 = (R -
R&R0
+ Aviv,
= Am/m,,
Am=m-mm,,
-
Av=u-v,,,
Ae/e,, ,
R = mudeB,,
,
andAe=e-e,,.
The derivatives over cpone indicated by primes. In the formation of eqns. (11) the following were used: f = R,(p”‘$+
4 = RoP’$
+ p’i$) ,
,
8 = R,(Q”QJ~ + 5“$)
VP =
v = (z$ + v$ + v:)“’
&(l
+ p) (t, ,
v, = &&‘$I
= constant
(in the magnetic field the absolute value of the charged particle velocity does not vary) and (ijl$J
=
2(-p’
+ 2pp’ -
p’p” -
C’s”)
. To’ solve. ,eqns. (,12) the method :of successive. approximations is used. In the initial approximation the terms of 'eqn:‘ (12) are expanded.with the help of eqns: ‘(5) ‘and (10) with. an ,accuracy- up to sthe small - quantities of the
175
zero-th order:
(13)
Here and below, the index by the coordinate or a derivative indicates the positions in Fig. 1 where the values of these variables are taken. Now the left-hand and the right-hand sides of eqn. (13) are integrated over 9 from cpl(/jl) to 9. For integration of the expressions depending on Cp the relationship is used between the differentials d9 and dif, and the variables 9 and a,, obtained from the determination of eqn. (8): dif,=d9,
W91)
@(92) = 920
= 910 3
with an accuracy up to the small quantities of the first order. After integrating (13) with the same accuracy taking account tions (6), namely
of condi-
one finds 920 P
p-
+a--_-IO_
-Pi
s
1
bd@+&$--
24c4
s” d3b ’ da3
910
d2b +-r+-6c3 da2
5’ = 5: -tclb
Repeated integration the second-order:
(14)
Q P = PI + P’l(@ -
-1&E
24c4
up to the small quantities
gives, with an accuracy
910) + %a -
9,0)2 -
of
@
J-s ~10~10
bd@d++2c2 Lj=:b
(15)
da2
At * = 929 the values of the coordinates sought are obtained: 920
P;=Pl
+ Pi<920
-
910) + %920 -
910)’ -
s 910
g2 = 51+
c'lb20-
910)-
tth%o
9
s 910
bd+d‘D+&” 1
r:
(16)
176
In eqn. (16), along with is), the determination MfP20)
=
1
d2b(cp20) --
,
d&2
_ -
d3Wcp,o)
=
(1) is used, namely
O
dtfP
and
In the next approximation the accuracy is enhanced by taking account the small quantities by first order and using the solutions (14) and (15): g=l+pl--
[1+(2+a,)p,-66]
db --tc:bd4,+ x dcI, --
of
b+
db
:3c:$g J”b da
+ (0)3 s
+ (oy$$+
[(0)4
+ (O)S]
$$
910
+(o)4[;$/m
bd@+~b+$$-$]+(0)'$$-~ 910
db d"b d4, dtb4
Wa)
+(0)6-.---
C" =
P+t&--_(1*)-~
R
c3%o
1P1+t~--pp;-(l+g)~~
2 + cz di +-cp-_-----
24~6
d3b dQr da3
d3b
(17b)
+ (W4~ I
m s 710
bd@+&f:+z
177
n = 3,4,5,6, denote the constant small In eqns, (17) the symbols (0)“, quantities of the nth order standing before the functions, the integrals from equal to zero by virtue of conwhich over @ from cpi,, to cp,, are deliberately ditions (6). For instance
slpzo
db
d4b dck
rplo z&d3 Integration second order: Pi =
a-
Wlb)
of eqns.
p; -
(17)
(cl10 -
using
(18)
gives within
10 + al) 920 + (1 +ia
the smah quantities
CplOlPl
+
of the
~cp,o
UW
It is noteworthy that the coefficients by the integrals in the two last terms of solution (19b) are approximate in the sense that they do not include the contributions from the negIected terms in expansions (5)) such as _+s 120
?$
in the expansion B, (p , cp, C) _ To make the coefficient_ more precise. all the zero&h order terms with in eqn. (5) are taken into account irrespective respect to p, g and cpzo- qlo of the value of the numerical coefficient. Then by using the conditions rot B = 0 and div B = 0 and the method of induction one obtains
B,
= -_B,/t
178
where (__l)ip+l Pi(<)
=
c2’(2i
+ I)!
and
qiC
(-l)'f2' cz~t2il,
I
Now eqns.
(12) are solved in the first-order approximation by using eqn. (20) and by assuming p1 = pi 4 3‘; = 6 = 0 to simplify the second-order equation, since the dependence of these param eters is fully presented in eqn(19b). In the approximation considered:
X
5
j:-0
dZJ’*lb
Fj(c
1) d@2j+l
d@
(21)
By integrating eqn. (2;), a particular solution is obtained which, however, fully takes account of the influence of ail the zero-th order terms in the expansions of the magnetic field components:
(22)
The general formula (22) indudes incompleteanalogous resultspresented in the solutions of the axial motion equations in ref. 8 for i = 0, 1,2,3 and in ref. 11 for i = 0 only. It should be noted that from eqn. (22) it follows that the conclusion made in ref. 13 about one of the formula of ref. 4 is invalid, since the conclusion drawn in ref. 13 was based on incomplete preliminary results of type (22). It is convenient to express the results of the particle trajectory transformation by an extended fringing field as a stepwise chtige of some conventional trajectory on the effective boundary of the magnet. This conventional trajecto,y consists of a straight part (between positions 1 and 0 in Fig. l), corresponding to the motion of particles in space without magnetic field, dE+ placements pJ - po, r3 - ro, cp, - cp, and refractions pi - p& c; - $% of the trajectory on the effective boundary, and the trajectory between positions 3 and 2 by particle motion in the main field. The real and ednventional trajec-
179
tories coincide up to position 1 and after position 2. The displacement and refraction of the conventional trajectory are found by considering, using the method described above, (i) changes in the particle trajectory in going from position 0 to position 1 without any field (the integration of the trajectory equations over Q, from 0 to cp,,) (ii) transformation of the trajectory in accordance with solutions (16), (19) and (20), and finally (iii) the changes of the particle trajectory in going from position 2 to position 3 in the field BA,cp,C)
=(I
+a~)
Bo,
&kcp,5)
=~IS‘BO,
B&J,cP,~)
=
0
(the integration of the trajectory equations over @ from q20 to 0). This field can be called “conventional” in contrast to the term “ideal” sometimes used since in tending to the “ideal” fieId with a sharp drop of the B,-component of the field on the boundary, the co,mponent Bq at z # 0 tends to infinity rather than to zero owing to the condition rot B = 0. The displacements and refractions of the conventional trajectory on the effective boundary, within the small second-order quantities, and taking relation (9) into account, are equal to;
-
alt --
JL 2c3Rbo
15;- t2roS-;,
(2W
(23~) The results (23) obtained for the entrance boundary can be extended to the exit boundary of the dipole magnet. As can be seen from eqns. (23a, b), the radial refraction and displacement of the conventional trajectory of the particle pi - p&, q3 - cpo is independent of the parameter g characterizing the change in the extension of the fringing field, although this parameter enters into the intermediate solution (19a). The curvature of the effective boundary manifests itself in the radial and axial refractions through the value of the radius of curvature of the effective
180 boundary at the point of its intersection with the optical axis of the dipole magnet RbO_ Integrals up to i = 3 entering into formulae (23) are expressed through the parameters of the fringing field. For this purpose the fringing field is expressed through a number of power functions:
b(lp) =
iz biKi
(24)
where K = (P -
9,o)/(‘Pzo -
Plo) ,
0 g Kg 1
and require the fulfilment of conditions (6) for n = 1,2,3_ After substituting b,-,= bl = b2 = b3 = 0 from conditions (6) at cp = 9 1o and solving the resultant system of equations of the form b(q,o)
-
d2b6453) = i; dv_?
one
b(q)
db(cpxJ = i; ibj = 0 i=4 dcp
= k bi = 1 9 i-4
j(j
_
1)
b_
I
=
0
d3b(ed = 5
,
i=4
finds the values of the coefficients = 35K4 -
84K5
+ 70K”
-
20K’
j(j
-
l)(j
-
2)
bi
=
0
i=4
dv3
b4 -
b,: (25)
The function b(q) plotted from formula (25) is shown in Fig. 3. The same figure shows by points the results of one of the measurements of the shielded hinging field along its optical axis in the region of the entrance boundary (c = O”) for the dipole magnet of the mass separator [13] _ The distance between the shield (0) and the boundary of the pole tip (N) is equal to the height of the pole gap h (h = 80 mm). 4s can be seen from Fig- 3, the curve (25) approximates the real fringing field with .good accuracy_ It should be noted that the reverse first derivative of the function of the distribution of the fig&g field, taken at the point where b(q) = 0.5, is more pronounced than the total width of the distribution of the fringing field cp,, - rpzo. This quantity (T defines the extension of the fringing field at the linear extrapolation of the middle part of the distribution function (see Fig. 3). Therefore the integrals entering into eqns. (23) are expressed through c by using (25) and the relation following from (25), i.e. o = (16/35)(~,, 910): Ip20 i&o-
ss 'PI0
0
b dQ, de = --O_07a2 PI0
(26a)
181
= roe [ 1.09 -
0.89 -
0.49 (&)2
-
0.32
(&y
-
0.28
(&)“I
(26b)
The values of integrals (26) found make the values of the individual terms clearer. For instance, the terms of eqn. (26b) containing c$ and CZ, which are formally considered as small corrections to the main linear action, appear to be substantial for the particle trajectory passing at a distance from the median plane. The sum of the cubic correction and that of the fifth order, for instance, at go = 0.60~ are equal in magnitude and opposite in sign to the known second-order correction (1.09 - 0.89)r00 = 0_20J’,a. This means that the focal distance of the axial lens of the fringing field will be F = R,/t without taking the -magnetic field extension into account, whereas taking the magnetic field extension into account it will be F = R,/(t - 0.200(1 + s2)/c2) for particles passing near the median plane. Note that in the widely used for.mula for the focal distance of the axial lens of the fringing field, taking in&to account the field extension, instead of the term 0.200(1 + s2)/c2, the term o(l + s2)/0c2 = 0.170(1 + s2)/cZ is used, which is obtained at i = 0 on the basis of relation (22) using the linear function b(p) = (2cp + o)/2a, where -e/2 =G cp< u/2.
r 1.0 a9
-
0.6
-
a7
-
0.6
-
0 .
02
0.4
0.6
0.6
10
Pifi. 3. The distribution of the fringing field of the dipole magnet along the optical axis. The points. indicate the results of measurements for the dipole magnet of the mass separator (133. The curve shows the distribution b(q) = 35~~ - 84~~ + 70fc6 - 20~’ -plo)_ The curve is superposed with the results of measurewhere K = (cp- c~Io)/(~o ments for equal vahles of the u parameters.
182
However, the position of the axial waist or crossover of the beam, which is determined mainly by the trajectories crossing the median plane at the largest angles, i.e_ by the trajectories that are farthest from the median plane when the particles pass through the fringing axial lens, depends on the effective focal distance F=h?
t_a(l
+s2) C2
0.20 -
0.49
2
4
The focal distance is equal to F = Ro/t for particles with co = 0.60~. This displacement of the position of the axial crossover as the beam height increased was observed experimentally in studies of the axial transformation of the ion beam under the influence of the fringing magnetic field on the exit boundary (E = 45”) of the mass separator 1131. All this shows that it is important to take account of terms with i = 1, 2, etc. in formula (22), which are only partially presented in the results of ref. 8 and are absent in the results of ref. 11. As follows from eqn. (26a), the radial motion correction determined by oz is small. Therefore the excessive limitation of the extension of the fringing field by screens, undertaken to lower the uncertainty of its action, is not necessary for the radial motion. .At the same time, this limitation leads to a noticeable irregularity of the axial action of the tinging field for the particles passing far from the median plane. As can be seen from formulae (23a, b) and (26a), the displacement p3 PO and ~3 - cpo of conventional trajectories of the particles moving in the median plane or crossing it on the effective boundary (co = 0) turns out to be the same for all particles irrespective of the coordinate and practically equal to the small quantity of the third order (ca. 0.07~~). The values of the radial derivatives are conserved, i.e. pi = p& Thus, the specific influence of the various parameters of the fringing magnetic field on the radial and axial motion of a charged particle in a secondorder approximation has been demonstrated. A new definition of the effective boundary of the fringing field has been used, as well as a smooth and precise function for the approximation of the fringing field, which has permitted a more precise determination of the coordinates of the-particle trajectory and their derivatives, taking into account the action of the fringing field_ It has also ahowed, for the first time, the influence of the fringing field derivatives to the third order to be taken into account. The effect of parameters, such as the difference in the extension of the fringing field, and the difference of the effective boundary from the circle has been considered. It is shown that the influence of some of the terms.previously regarded’as thirdorder and higher-order on the position of the axial crossover of the particle beam should’be t&en into account.
183 ACKNOWLEDGEMENTS
The author thanks Academician
G.N. Flerov for his suggestion to consider
the problems of monochromatization of the beam of the U-400 isochronous cyclotron, which required the fulfilment of this work, and A.B. Kuznetsov and N.B. Rubin for a fkuitful discussion. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13
M. Cotte, Ann. Phys., 10 (1938) 333. Ya. Khurgin, Zh. Eks. Teor. Fiz., 9 (1939) 824. C. Reutersward, Ark. Fys., 3 (1951) 53. N.G. Afanasyev, Izv. Akad. Nauk SSSR, Ser. Fiz., 24 (1960) 1157. N.I. Tarantin and A.V. Demyanov, Zh. Tekh. Fiz., 35 (1965) 186. V.R. Saulit, Vest. Lmingrad Univ., Ser. Fiz., Khim., 4 (1965) 49. AF. Malov, Zh. Tekh. Fiz., 35 (1965) 1617. R. Ludwig, Z. Naturforsch., 22a (1967) 553. H. Enge, in A. Septier (Ed.), Focusing of Charged Particles, Vol. II, Academic Press, New York, 1967, p. 203. Yu.G. Basargin, Zh. Tekh. Fiz., 38 (1968) 1400. H. Matsuda and H. Wollnik, Nucl. Instrum. Methods, 77 (1970) 40. G.M. Trubacheev, A.F. Maolv and A.M. Korochkin, Zh. Tekh. Fiz., 48 (1978) 1317. N-1. Tarantin, A.V. Demyanov, Yu.A. Dyachikhin and A.P. Kabachenko, Nuci. Instrum. Methods, 77 (1970) 40.
Journal
Elsevier Scientific
of Mass
Publishing
Spectromehy
Company,
and Ion
Amsterdam
36 (1980) 167-133 Printed in The Netherlands
Physics.
-
A SECOND-ORDER APPROXIMATION OF PARTICLE FRINGING FIELD OF A DIPOLE MAGNET
MOTION
,167
IN THE
N-1. TARANTIN Labotxztory
of Nuclear
Reactions,
(First received 27 June 1979;
Joint
Irish-tute
for Nuclear
in revised form 22 February
Research,
Dubna
(U.S.S.R.)
1980)
ABSTRACT The radial and axial motion of charged particles in the fringing field of an arbitrary dipole magnet has been considered with accuracy to the second-.ordcr of small quantities. The dipole magnet has an inhomogeneous field and oblique entrance and exit boundaries in the form of second-order curves. The region of the fringing field has a variable extension. A new definition of the effective boundary of the real fringing field of the dipole magnet is used. A better understanding of the influence of the fringing magnetic field on the motion of charged particles in the pole gap of the dipole magnet has been obtained. In particular, it is shown that it is important to take into account, in the second approximation, some terms related formally to the next approximations. The resultsare presented in a form convenient for practical calculations.
INTRODUCTION The fringing field in the vicinity dipole magnet has a substantial
of
the
entrance
and
exit
boundaries
of
influence on the motion of charged particles and its action should be thoroughly taken into account in the construction of precise magnetic analyzers. The influence of the fringing field at the entrance and exit boundaries of a dipole magnet on the motion of charged particles has been considered many times (see, for example, refs. l-12 and refs. therein). A most detailed consideration has been carried out for a homogeneous magnetic field in ref. 8 and for an inhcmogeneous magnetic field in ref. 11; the general results of these two papers do not coincide, but in ref. 11 it is noted that the results therein “agree with corrected expressions of ref. 8”. In the present paper a new second-order approximation of the radial and axial motion of charged particles in the fringing field of the dipole magnet is described. This consideration is made otherwise than in refs. 8 and 11 and for a dipole magnet with somewhat more general param eters compared with that in ref. 11. A dipole magnet with a radially inhomogeneous axially symmetric main field is considered in the present paper. The oblique entrance and exit boundaries of the magnet have the form of the most general secondthe
0020-7381/80/000~000/$92.25
0 Elsevier Scientific
Publishing Company
order curve with variable curvature. The extension of the fringing field arbitrarily varies along the boundary of the magnet pole,
FRINGING
FIELD
OF THE
DIPOLE
MAGNET
The region of the charged particle motion considered in the vicinity of the entrance boundary of the dipole magnet is shown in Fig. 1. The systems of coordinates used and particle trajectories in the radial and axial cross-sections are also given in this figure. The effective boundary of the magnetic field is determined by the relation:
Pb(P)
=*1(P)--
(1)
where p,,(p) is the equation of the effective boundary in the principal system of coordinates with an axis at point 01, p = (r -2&,)/R,, is the relative deviation of the radial coordinate r from the optical axis of the magnet - a circumference with radius Rap p < 1, B,(p, cp, 0) is the fringing field in the
of a Fig. 1. Cylindrical systems of coordinates (r_ I$. z) and (r, 8. z) and trajectories charged particle in the fringing field at the entrance. boundary.of the dipole magnet in the Real trajectory of the particle; (- - - - - -) convenradial and axial cross-sections. ( -) tional trajectory of the particle; (- - - - -) optical axis of the dipole magnet with a constant radius of curvature, Re The positive directions are shown by arrows. r - Ro < Ro, r-RR,
< Ro_
169
median plane (z = 0) of the dipole magnet, (1 + alp)& is the main field in the z = 0 plane of the pole gap, taken for a second-order approximation with sufficient accuracy, aI is the inhomogeneity index of the main magnetic field, which is determined from the formula
cp,(p) and (PIN are the initial and final field region, determined by the conditions B,(P, B.
rp~. 0) = 0
&CP,
and
angular
9rr, 0) = (I+
alpI
coordinates
of the fringing
&
= mOvOc/eJh
where rn,,, voI e. are the mass, velocity and charge of the main particle, and c is the velocity of light in a vacuum. Here and below, the sufficient accuracy of the quantities considered means the accuracy required to obtain the final results -the values of the coordinates of the particle trajectory and their derivatives during the passage of the particle through a ikinging magnetic
field, within the second-order small quantities. The small quantities include k h/R0 < I, where h is the height of the also the difference vrr(~) - (Pi pole gap and upis expressed in radians. For instance, in the determination of the main field within the firstorder small quantity p, the effective boundary is determined using eqn_ (1) within the second-order small quantities:
syn@)&(A alp) 0) 5%
---
(I+
YI(Y)
BO
dq
~
(1
- adO h(P)
- cm(P)1
Such a determination (1) of the effective boundary is convenient because it corresponds to the technique of measuring the topography of the radially inhomogeneous and axially symmetric main magnetic field along concentric circles with constant values of the magnetic field_ The effective boundary, because of saturation of the magnetic pole tips, may
have
variable
curvature_
In the general
case of the second-order
approxi-
mation it is describable by the second-order curve, which has the form of an ellipse, parabola or hyperbola, with a focus at the point O2 lying on the normal to the effective boundary at the point of its intersection with the optical axis of the magnet. In any particular case. the type of the curve is determined by any five points lying on the effective boundary. In the cylindrical system of coordinates with an axis in the focus canonical equation for the second-order curve is
?+!I?)=
p
1 + e cos 0
of the curve
O2 the general
@a)
where P is the focal parameter or semi-latus rectum of the curve, and e is the eccentricity: e < 1 for an ellipse (for a circle, being a particular case of an
170
ellipse, e = 0), e = 1 for a parabola and e > 1 for a hyperbola. In the vicinity of the vertex of the curve coinciding with the point of intersection between the boundary and the optical axis of the dipole magnet, at 8 < 1 eqn. (2a) can be approximately written in the form:
VI
=
r-_Rb =
R
(=I
K02
0
where R, = P/( 1 + e) is the distance between the focus and vertex of the curve, K = Pe/{ZR,(l + e)2) = R,e/{2R.(l+ e)) and r < 1. In some concrete cases, K takes the following values: K < 0.25 Rb/Ro for an ellipse, K = 0.25 Rb/Ro for a parabola, K > 0.25 R,,/R, for a hyperbola, K 5 0 for a circle with radius R,, which is considered positive for a convex boundary with respect to the dipole magnet, and Rb = = for a straight line. Note that if the axis of the cylindrical system of cooniinates, 02, is chosen to lie on the normal to the boundary, all the asymmetry of the effective boundary with respect to the normal is expressed by terms of the type e3, 05, etc. of eqn. (2b), which are not considered in a second approximation. In the main coordinate system, curve (2b) is describable by the following formula: (pb(P) =
1
- ~(1 + c2)
flp + f2p2 = tp + z
1p2
(3)
where s = sin E ,
t=tane,
c = cos E _
In the derivation of eqn. (3), the conventional transformation of the coordinates is used for the conversion of one cylindrical system of coordinates with an axis at point O2 to another system of the same kind with an axis at point 0, _ The expression Rb
I(
I--2K;t
)
=R,,,,
entering into eqn, radius of curvature
(3) is the radius of curvature of curve (2) at 8 = 0. The at any point of the line is determined from the formula:
Therefore, as follows from eqn. (3), the most complicated shape of the effective boundary can be described in the second-order approximation using of the boundary at the point of its intersecthe radius of the curvature R,, tion with the optical axis of the dipole magnet. It is supposed that the extension of the region of the fringing magnetic field may be different at different values of p, i.e. it is assumed with a sufficient degree of accuracy that e-f(P)
-
tiCPI
= a+
gP)(v2o
-
t430)
(4)
171
where g = constant ,
910 = (m(O) ,
=dcp,o-9g,o<
920=9n(O)
1.
At a constant linear width (not the angular width) of the fringing field region, as in the cases of the homogeneous main field (al = 0) or the specially shielding inhomogeneous field:
The r-th and 9-th field components for z + 0, required
for determination
and the axial components
of the field
of the trajectory of the particle, are
found in the usual manner from the conditions rot B = 0, where the equality to zero of the right-hand parts of equations is conditioned by the negligibly small value of the current of the charged particles under consideration, as compared with the current.of the excitation windings of the electromagnet, div B = 0, and the requirements of field symmetry with respect to the median plane. Prom the conditions aB, az
aBZ -o -ar
and
la& r
a9
a% -
a2
--0
following
from rot B = 0 one obtains B,(p, 9, g).= c aB,/ap p) aB,/ap where 5 = z/I&, < i, B, = B,(p, 9,O). By using the expressions found andthe condition
and B,(p,
p, 5) =
r(l-
one obtains
Proceeding in this way, one obtains with the required accuracy:
(W
+ &5” [S
+ (I-
4p) 3
+ 2(1-
2p) a;;$
-
2 3
+ 2 31
(SC)
Here B, and B, are determined with the accuracy up to the first-order small quantities, and Bcp up to the small quantities of the zero-th order. The small quantities include p, 5 and rp,, - (plo, which are less than unity, and numeri-
172
cal coefficients of the rejected terms of expansion rows, equal to l/120, l/720, etc. In estimating the order o.f the small quantities it is taken into account that 1
anBz _
s”
a”& _
and
-Y%o)" c"~cp*o 39 (920 - CplOY a,pn i.e. these derivatives are substantially greater quantities as they are inversely proportional to the small quantity in the nth power. To unify the integration in solving the particle trajectory equations B,(p, p, 0) is expressed through the function of the distribution of the fringing field along the optical axis of the dipole magnet B,(O, 9,O) = B&(9). From the condition of the smooth conjugation of the function of the distribution of the fringing field, Bob(q), with the main field at 9 = cpzoand with the field identically equal to zero, at 9 = 910 one has d"Mcp,o) MP20)
=
MrplO)
1,
_d"bCcplo)
_
(6)
=o, as"
-TF-O
for n = 1, 2, 3, etc. These conditions, the form of which is determined by the new definition of the effective boundary (l), are much simpler and more illustrative than analogous conditions for the fringing field in the determination of the effective boundary described in ref. 11. From the magnetostatks it foilows that the functions of the distribution of the fringing field at diEferent values of p will be one and the same with sufficient accuracy if the quantity of the fringing field is expressed in the value of the main field, (1 + a,p)B o, and the angular coordinate in the units of the -@a extension, (1 + gp) (9z0 - 910). of the fringing field at a preset value of p. Therefore, one can assume that Bz(P, 9,9)
(7)
= (1 + e,p) BOW@)
where CD= [9 -
pb(P)l/(~ + WI = cp-
fiP
+ (frg
-
f2)
P2 -gpGJ,
PlO
G
*
6
cp20
03)
The strict validity of’ relation (7) is evident in the particular case of a linearly inhomogeneous main field without the quadratic component and a straight boundary orthogonal to the optical axis of the magnet, since in this csse none of the directions p = constant is distinguished anyway. From relation (7) it follows that the effective boundary is located with respect to the fringing field boundary 9,,(p) in the same way as 91(p) with respect to 9r i (p) : . cpbbI=chi@I-'J
.
9IL@)
916 1
ca
+alP)~o~(*YC(1+~,P)
Bol3 drp
173
Hence, according to (I) (pb(P) -
PII
= -(I
+a)
P20
(9)
The distributions of the fringing magnetic field at various values of p are shown in Fig. 2. The partial derivatives over p and 9 entering into eqns. (5) are expressed through the total derivatives of function (7) over + with the help of the following recurrent relation: _$;> -
= (-I)“& (-l)“nB,
z-y:
z;-;;:;
+ m) g1 PI + nfF(2f2p
(n + m -
+ (n - 1)
I)gl
-i-&f?))
flm2fil
w9
where b = b(q~); IZ, m = 0, 1, 2, 3, etc. Formula (10) was derived by the induction method; namely, by generalizing expressions for derivatives in the particular differentiation 0fBoverp and cp,i.e.
a% _ a - ap 10 + alpI ap
-
= (1 +a#)&-
= (--1)‘&l
THE
CHARGED
db
dB
db -
The expression 1 and m = 0.
&3b(*)l a@ --+aalBob=
ap
+ (al -g)
obtained PARTICLE
PI
+
(2fs
is a particular MOTION
The radial and axial equations netic field have the general forms
IN THE
+ ti))
-
(--1)‘&ba1
csse of general formula FRINGING
, etc. (10)
at R =
FIELD
of the charged particle motion in the mag-
(11) where. m, u, and e are the mass, velocity and charge of the considered particle. After replacing the time derivatives by the derivatives over cp these equations are transformed in a first-order approximation to the following
91(P)-9btP)
0
9iik')-9b(P)
9-9btP)
Fig. 2. The distribution of the fringing field of the dipole magnet at different zx~;r~i~,tiO the relation B,(p, cp, 0) = (1 + aIp).B&(+) where @ = {so --(p))/{l ,
values of p + gp),
particle trajectory equations: (1
P ” = I + p + s“Bq/B,, 5‘” = (1-t
2p -
6) BJB,
-
l1-2~
-
6) B,/Bo
(12)
p’B,/B,,
where 6 = (R -
R&R0
+ Aviv,
= Am/m,,
Am=m-mm,,
-
Av=u-v,,,
Ae/e,, ,
R = mudeB,,
,
andAe=e-e,,.
The derivatives over cpone indicated by primes. In the formation of eqns. (11) the following were used: f = R,(p”‘$+
4 = RoP’$
+ p’i$) ,
,
8 = R,(Q”QJ~ + 5“$)
VP =
v = (z$ + v$ + v:)“’
&(l
+ p) (t, ,
v, = &&‘$I
= constant
(in the magnetic field the absolute value of the charged particle velocity does not vary) and (ijl$J
=
2(-p’
+ 2pp’ -
p’p” -
C’s”)
. To’ solve. ,eqns. (,12) the method :of successive. approximations is used. In the initial approximation the terms of 'eqn:‘ (12) are expanded.with the help of eqns: ‘(5) ‘and (10) with. an ,accuracy- up to sthe small - quantities of the
175
zero-th order:
(13)
Here and below, the index by the coordinate or a derivative indicates the positions in Fig. 1 where the values of these variables are taken. Now the left-hand and the right-hand sides of eqn. (13) are integrated over 9 from cpl(/jl) to 9. For integration of the expressions depending on Cp the relationship is used between the differentials d9 and dif, and the variables 9 and a,, obtained from the determination of eqn. (8): dif,=d9,
W91)
@(92) = 920
= 910 3
with an accuracy up to the small quantities of the first order. After integrating (13) with the same accuracy taking account tions (6), namely
of condi-
one finds 920 P
p-
+a--_-IO_
-Pi
s
1
bd@+&$--
24c4
s” d3b ’ da3
910
d2b +-r+-6c3 da2
5’ = 5: -tclb
Repeated integration the second-order:
(14)
Q P = PI + P’l(@ -
-1&E
24c4
up to the small quantities
gives, with an accuracy
910) + %a -
9,0)2 -
of
@
J-s ~10~10
bd@d++2c2 Lj=:b
(15)
da2
At * = 929 the values of the coordinates sought are obtained: 920
P;=Pl
+ Pi<920
-
910) + %920 -
910)’ -
s 910
g2 = 51+
c'lb20-
910)-
tth%o
9
s 910
bd+d‘D+&” 1
r:
(16)
176
In eqn. (16), along with is), the determination MfP20)
=
1
d2b(cp20) --
,
d&2
_ -
d3Wcp,o)
=
(1) is used, namely
O
dtfP
and
In the next approximation the accuracy is enhanced by taking account the small quantities by first order and using the solutions (14) and (15): g=l+pl--
[1+(2+a,)p,-66]
db --tc:bd4,+ x dcI, --
of
b+
db
:3c:$g J”b da
+ (0)3 s
+ (oy$$+
[(0)4
+ (O)S]
$$
910
+(o)4[;$/m
bd@+~b+$$-$]+(0)'$$-~ 910
db d"b d4, dtb4
Wa)
+(0)6-.---
C" =
P+t&--_(1*)-~
R
c3%o
1P1+t~--pp;-(l+g)~~
2 + cz di +-cp-_-----
24~6
d3b dQr da3
d3b
(17b)
+ (W4~ I
m s 710
bd@+&f:+z
177
n = 3,4,5,6, denote the constant small In eqns, (17) the symbols (0)“, quantities of the nth order standing before the functions, the integrals from equal to zero by virtue of conwhich over @ from cpi,, to cp,, are deliberately ditions (6). For instance
slpzo
db
d4b dck
rplo z&d3 Integration second order: Pi =
a-
Wlb)
of eqns.
p; -
(17)
(cl10 -
using
(18)
gives within
10 + al) 920 + (1 +ia
the smah quantities
CplOlPl
+
of the
~cp,o
UW
It is noteworthy that the coefficients by the integrals in the two last terms of solution (19b) are approximate in the sense that they do not include the contributions from the negIected terms in expansions (5)) such as _+s 120
?$
in the expansion B, (p , cp, C) _ To make the coefficient_ more precise. all the zero&h order terms with in eqn. (5) are taken into account irrespective respect to p, g and cpzo- qlo of the value of the numerical coefficient. Then by using the conditions rot B = 0 and div B = 0 and the method of induction one obtains
B,
= -_B,/t
178
where (__l)ip+l Pi(<)
=
c2’(2i
+ I)!
and
qiC
(-l)'f2' cz~t2il,
I
Now eqns.
(12) are solved in the first-order approximation by using eqn. (20) and by assuming p1 = pi 4 3‘; = 6 = 0 to simplify the second-order equation, since the dependence of these param eters is fully presented in eqn(19b). In the approximation considered:
X
5
j:-0
dZJ’*lb
Fj(c
1) d@2j+l
d@
(21)
By integrating eqn. (2;), a particular solution is obtained which, however, fully takes account of the influence of ail the zero-th order terms in the expansions of the magnetic field components:
(22)
The general formula (22) indudes incompleteanalogous resultspresented in the solutions of the axial motion equations in ref. 8 for i = 0, 1,2,3 and in ref. 11 for i = 0 only. It should be noted that from eqn. (22) it follows that the conclusion made in ref. 13 about one of the formula of ref. 4 is invalid, since the conclusion drawn in ref. 13 was based on incomplete preliminary results of type (22). It is convenient to express the results of the particle trajectory transformation by an extended fringing field as a stepwise chtige of some conventional trajectory on the effective boundary of the magnet. This conventional trajecto,y consists of a straight part (between positions 1 and 0 in Fig. l), corresponding to the motion of particles in space without magnetic field, dE+ placements pJ - po, r3 - ro, cp, - cp, and refractions pi - p& c; - $% of the trajectory on the effective boundary, and the trajectory between positions 3 and 2 by particle motion in the main field. The real and ednventional trajec-
179
tories coincide up to position 1 and after position 2. The displacement and refraction of the conventional trajectory are found by considering, using the method described above, (i) changes in the particle trajectory in going from position 0 to position 1 without any field (the integration of the trajectory equations over Q, from 0 to cp,,) (ii) transformation of the trajectory in accordance with solutions (16), (19) and (20), and finally (iii) the changes of the particle trajectory in going from position 2 to position 3 in the field BA,cp,C)
=(I
+a~)
Bo,
&kcp,5)
=~IS‘BO,
B&J,cP,~)
=
0
(the integration of the trajectory equations over @ from q20 to 0). This field can be called “conventional” in contrast to the term “ideal” sometimes used since in tending to the “ideal” fieId with a sharp drop of the B,-component of the field on the boundary, the co,mponent Bq at z # 0 tends to infinity rather than to zero owing to the condition rot B = 0. The displacements and refractions of the conventional trajectory on the effective boundary, within the small second-order quantities, and taking relation (9) into account, are equal to;
-
alt --
JL 2c3Rbo
15;- t2roS-;,
(2W
(23~) The results (23) obtained for the entrance boundary can be extended to the exit boundary of the dipole magnet. As can be seen from eqns. (23a, b), the radial refraction and displacement of the conventional trajectory of the particle pi - p&, q3 - cpo is independent of the parameter g characterizing the change in the extension of the fringing field, although this parameter enters into the intermediate solution (19a). The curvature of the effective boundary manifests itself in the radial and axial refractions through the value of the radius of curvature of the effective
180 boundary at the point of its intersection with the optical axis of the dipole magnet RbO_ Integrals up to i = 3 entering into formulae (23) are expressed through the parameters of the fringing field. For this purpose the fringing field is expressed through a number of power functions:
b(lp) =
iz biKi
(24)
where K = (P -
9,o)/(‘Pzo -
Plo) ,
0 g Kg 1
and require the fulfilment of conditions (6) for n = 1,2,3_ After substituting b,-,= bl = b2 = b3 = 0 from conditions (6) at cp = 9 1o and solving the resultant system of equations of the form b(q,o)
-
d2b6453) = i; dv_?
one
b(q)
db(cpxJ = i; ibj = 0 i=4 dcp
= k bi = 1 9 i-4
j(j
_
1)
b_
I
=
0
d3b(ed = 5
,
i=4
finds the values of the coefficients = 35K4 -
84K5
+ 70K”
-
20K’
j(j
-
l)(j
-
2)
bi
=
0
i=4
dv3
b4 -
b,: (25)
The function b(q) plotted from formula (25) is shown in Fig. 3. The same figure shows by points the results of one of the measurements of the shielded hinging field along its optical axis in the region of the entrance boundary (c = O”) for the dipole magnet of the mass separator [13] _ The distance between the shield (0) and the boundary of the pole tip (N) is equal to the height of the pole gap h (h = 80 mm). 4s can be seen from Fig- 3, the curve (25) approximates the real fringing field with .good accuracy_ It should be noted that the reverse first derivative of the function of the distribution of the fig&g field, taken at the point where b(q) = 0.5, is more pronounced than the total width of the distribution of the fringing field cp,, - rpzo. This quantity (T defines the extension of the fringing field at the linear extrapolation of the middle part of the distribution function (see Fig. 3). Therefore the integrals entering into eqns. (23) are expressed through c by using (25) and the relation following from (25), i.e. o = (16/35)(~,, 910): Ip20 i&o-
ss 'PI0
0
b dQ, de = --O_07a2 PI0
(26a)
181
= roe [ 1.09 -
0.89 -
0.49 (&)2
-
0.32
(&y
-
0.28
(&)“I
(26b)
The values of integrals (26) found make the values of the individual terms clearer. For instance, the terms of eqn. (26b) containing c$ and CZ, which are formally considered as small corrections to the main linear action, appear to be substantial for the particle trajectory passing at a distance from the median plane. The sum of the cubic correction and that of the fifth order, for instance, at go = 0.60~ are equal in magnitude and opposite in sign to the known second-order correction (1.09 - 0.89)r00 = 0_20J’,a. This means that the focal distance of the axial lens of the fringing field will be F = R,/t without taking the -magnetic field extension into account, whereas taking the magnetic field extension into account it will be F = R,/(t - 0.200(1 + s2)/c2) for particles passing near the median plane. Note that in the widely used for.mula for the focal distance of the axial lens of the fringing field, taking in&to account the field extension, instead of the term 0.200(1 + s2)/c2, the term o(l + s2)/0c2 = 0.170(1 + s2)/cZ is used, which is obtained at i = 0 on the basis of relation (22) using the linear function b(p) = (2cp + o)/2a, where -e/2 =G cp< u/2.
r 1.0 a9
-
0.6
-
a7
-
0.6
-
0 .
02
0.4
0.6
0.6
10
Pifi. 3. The distribution of the fringing field of the dipole magnet along the optical axis. The points. indicate the results of measurements for the dipole magnet of the mass separator (133. The curve shows the distribution b(q) = 35~~ - 84~~ + 70fc6 - 20~’ -plo)_ The curve is superposed with the results of measurewhere K = (cp- c~Io)/(~o ments for equal vahles of the u parameters.
182
However, the position of the axial waist or crossover of the beam, which is determined mainly by the trajectories crossing the median plane at the largest angles, i.e_ by the trajectories that are farthest from the median plane when the particles pass through the fringing axial lens, depends on the effective focal distance F=h?
t_a(l
+s2) C2
0.20 -
0.49
2
4
The focal distance is equal to F = Ro/t for particles with co = 0.60~. This displacement of the position of the axial crossover as the beam height increased was observed experimentally in studies of the axial transformation of the ion beam under the influence of the fringing magnetic field on the exit boundary (E = 45”) of the mass separator 1131. All this shows that it is important to take account of terms with i = 1, 2, etc. in formula (22), which are only partially presented in the results of ref. 8 and are absent in the results of ref. 11. As follows from eqn. (26a), the radial motion correction determined by oz is small. Therefore the excessive limitation of the extension of the fringing field by screens, undertaken to lower the uncertainty of its action, is not necessary for the radial motion. .At the same time, this limitation leads to a noticeable irregularity of the axial action of the tinging field for the particles passing far from the median plane. As can be seen from formulae (23a, b) and (26a), the displacement p3 PO and ~3 - cpo of conventional trajectories of the particles moving in the median plane or crossing it on the effective boundary (co = 0) turns out to be the same for all particles irrespective of the coordinate and practically equal to the small quantity of the third order (ca. 0.07~~). The values of the radial derivatives are conserved, i.e. pi = p& Thus, the specific influence of the various parameters of the fringing magnetic field on the radial and axial motion of a charged particle in a secondorder approximation has been demonstrated. A new definition of the effective boundary of the fringing field has been used, as well as a smooth and precise function for the approximation of the fringing field, which has permitted a more precise determination of the coordinates of the-particle trajectory and their derivatives, taking into account the action of the fringing field_ It has also ahowed, for the first time, the influence of the fringing field derivatives to the third order to be taken into account. The effect of parameters, such as the difference in the extension of the fringing field, and the difference of the effective boundary from the circle has been considered. It is shown that the influence of some of the terms.previously regarded’as thirdorder and higher-order on the position of the axial crossover of the particle beam should’be t&en into account.
183 ACKNOWLEDGEMENTS
The author thanks Academician
G.N. Flerov for his suggestion to consider
the problems of monochromatization of the beam of the U-400 isochronous cyclotron, which required the fulfilment of this work, and A.B. Kuznetsov and N.B. Rubin for a fkuitful discussion. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13
M. Cotte, Ann. Phys., 10 (1938) 333. Ya. Khurgin, Zh. Eks. Teor. Fiz., 9 (1939) 824. C. Reutersward, Ark. Fys., 3 (1951) 53. N.G. Afanasyev, Izv. Akad. Nauk SSSR, Ser. Fiz., 24 (1960) 1157. N.I. Tarantin and A.V. Demyanov, Zh. Tekh. Fiz., 35 (1965) 186. V.R. Saulit, Vest. Lmingrad Univ., Ser. Fiz., Khim., 4 (1965) 49. AF. Malov, Zh. Tekh. Fiz., 35 (1965) 1617. R. Ludwig, Z. Naturforsch., 22a (1967) 553. H. Enge, in A. Septier (Ed.), Focusing of Charged Particles, Vol. II, Academic Press, New York, 1967, p. 203. Yu.G. Basargin, Zh. Tekh. Fiz., 38 (1968) 1400. H. Matsuda and H. Wollnik, Nucl. Instrum. Methods, 77 (1970) 40. G.M. Trubacheev, A.F. Maolv and A.M. Korochkin, Zh. Tekh. Fiz., 48 (1978) 1317. N-1. Tarantin, A.V. Demyanov, Yu.A. Dyachikhin and A.P. Kabachenko, Nuci. Instrum. Methods, 77 (1970) 40.