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A proposed new type of radially and azimuthally varying magnetic field in cyclotrons for enhanced focusing
10
Nuclear Instruments and Methods in Physics Research A305 (1991) 10-14 North-Holland
A proposed new type of radially and azimuthally varying magnetic field in cyclotrons for enhanced focusing P.R. Sarma
Variable Energy Cyclotron Centre, Bhabha Atomic Research Centre, 11AF Bidhan Nagar, Calcutta-700064, India
Received 4 May 1990 and in revised form 30 January 1991
For vertical focusing of ions in cyclotrons generally an azimuthal variation is given to the magnetic field with the help of a straight sector geometry or a spiral sector geometry . In the present work a new type of magnetic field, which varies azimuthally and has a periodic radial variation as well, has been proposed . It has been shown that the vertical focusing obtained with the proposed geometry is much stronger than what can be obtained with straight sector geometries .
1. Introduction In the early days when the first cyclotron was being developed, it had been realized by Lawrence that because of the relativistic increase in mass the magnetic field would have to increase with radius and this would conflict with the requirements of vertical focusing [1]. Focusing is necessary for confining the ions vertically as well as horizontally inside the cyclotron magnet . A solution to this problem was suggested by Thomas [2] who proposed a sinusoidal azimuthal variation of the magnetic field. Later on McMillan [1] also proposed a similar scheme using wedge shaped fields . Cyclotrons with such magnetic fields are called azimuthally varying field (AVF) cyclotrons or sector focused cyclotrons. For high energy cyclotrons focusing has to be very strong, and this can be achieved by using separated sectors where the azimuthal field variation is maximum. In fact, separated magnet sectors of very small azimuthal width give very strong vertical focusing [3]. But with separated sectors the size of the machine (and hence the total weight of the magnet) becomes large. Another important way of improving the focusing was pointed out by Symon et al . [4]. They showed that one can obtain a stronger focusing by using spiral sectors instead of straight ones. Thus separated sectors having a spiral shape produce the maximum focusing achievable so far. Focusing in cyclotrons is characterized by the vertical and radial betatron oscillation frequencies, v.
and v, respectively. The contribution to vz due to the spiral sectors is given by [5]: v~
=
( BH
-
B )( B - Bv) B2
[1 + 2 tan2 E],
where B H = field in the so-called hill region ; B = field in the valley region ; B = average field; e = spiral angle. For net vi the term -/32/(1 - ß2) due to the field gradient must be added. Here /3 is equal to v/c where v is the velocity of the ions. The vertical betatron frequency vs can be increased by decreasing B but this results in a low average field leading to an increase in the size of the machine. Another way of increasing v, is to increase the spiral angle E. But e also cannot be increased indefinitely because for large E the shortest distance between the edges of the spiral hills decreases and so the required field variation is not achieved . The maximum value of e depends on radius and gap, and magnets with e up to 70' have been designed though most of the cyclotrons use spiral angles up to about 60 °. In this article we propose a new type of radially and azimuthally varying field (RAVF) in a fixed frequency cyclotron. In Thomas focusing as well as spiral focusing only azimuthal field variation is used (although a spiral produces radial variation also). In the new scheme we use both the radial and the azimuthal field variation. It is well known that if there is a radial decrease in magnetic field then there is focusing in the vertical
0168-9002/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
P. R. Sarma / A radially and azimuthally varying magnetic field
15
_
CONVENTIONAL CYCLOTRON HILL
m
REGION I
AVERAGE
10
VALLEY
V
w z O a
REGION 2
05 RADIALLY CONSTANT FIELD
01
04
07
RADIALLY VARYING FIELD
ill
1001
04
07
10
RADIUS (R/RMAX)
Fig. 1. Variation of magnetic field as a function of radius in a straight sector cyclotron and in a cyclotron where field vanes radially. Field has not been shown in the centre region as separate hill and valley are not present there. direction and defocusing in the radial direction, and vice versa if the field increases. We use this principle in the present work. We point out here that this principle of radial alternating field gradient has very many applications, e.g ., in synchrotrons ; but as yet no thought has been given to using it m cyclotrons . 2. Calculation
as a result there will be a net alternating gradient focusing . In what follows we assume that the field variation is small so that the scalloping of the orbits is also small and thus the field in any region along any particle orbit remains essentially constant . We assume this only for simplifying the calculation and this assumption can be put aside in a rigorous calculation . We now calculate the vertical and radial betatron oscillation frequencies v, and v, in the framework of hard edge approximation. We use the matrix method [71 used by Schatz [31 and Bhattacharya and Divatia [8]. 2.1 . Calculation
of
vs
As has been shown in ref. [7], the betatron oscillation frequencies can be calculated from the relation :
where T.., is the overall transfer matrix which is a product of the various transfer matrices in the regions and in the boundaries and N is the number of sectors. The subscripts z and r refer to vertical and radial directions respectively . In our case the four matrices involved at any radius are: 1) Tlz, , at the entrance of the first region ; 2) T2 z, , in the first region ; 3) T3z, , at the entrance of the second region ; 4) T4z, , in the second region . For calculating Tlz,, and T3z,r we should know the angle which the orbit makes with the normal to the sector edge. The fields in the two regions at any radius r can be written as :
Before we describe the actual scheme, let us discuss the principle in a simple way and obtain some results . A similar procedure will later be used in the actual scheme . In a conventional straight sector cyclotron each sector is divided into two regions; one region is the hill where the field is high and the other region is the valley where the BI( r)=B,-2fr, B2(r)=B +2fr, (3) field is low. In our case the two adjoining regions have with B = (BH + BJ/2 and f a constant. As given by linearly varying fields in the radial direction. A region Richardson [6] the angle k at the entrance of the first which starts as a hill near the centre becomes a valley at region is : the outer radii and vice versa (fig . 1) . Decrease in field and increase in the other keeps the averin one region _ m [BI(r)-B ] [B-B2(r)] k age field constant at all radii. In a cyclotron the average N B[B1(r)-B2(r)1 ' magnetic field (averaged over all azimuths) has actually and then : to increase for taking into account the relativistic mass increase, which depends on the increase of ions . This 1 mass and the final energy of the ions, can be brought TI, = - BI(r) -B2(r) T3Z = TlZ . tan k about with the help of the trim coils and the flare in the B hills. This radial increase in field due to relativistic effects decreases the vz values gradually as the radius The field index in region 1 is n = 2fr/B and the matrix increases, unless the focusing is increased, and this T2, is : happens in conventional cyclotrons as well as in the present scheme . We do not take into account this de1 -,Fcos $s sin tP~ crease in v in our calculation as this is mass and energy n t/z K, T2 r = [ _ , _ r ' dependent. This, however, can be calculated separately . -K, sin $Z cos J>' We consider regions of equal width. In such a sector geometry there will be focusing in the falling field / t z (6) $ = N n, region and defocusing in the increasing field region and
P.R. Sarma / A radially and azimuthally varying magnentfield
12 and similarly; cosh $z
T4, =
L K.
.m
KZ sinn (p,
sinh 4).
m
cosh
0 W
The overall matrix is :
(8)
T, = T1,T2,T3,T4, .
1 BI(r) -B2(r) B tan k
T2,=
- K,
Similarly; T4,
=I
cosh
sin 0,
~
K, sinh (p,
_ 1 sin 4~, K, tos ¢,
K
sinh
0
Tar=T1
1 '
,
K
¢ I.
cosh ¢,
=
(1
1'0
0'S 0
01
0-4 0-7 110 RADIUS (R/RMAX) Fig. 3. Sinusoidal variation of the magnetic field as a function of the radius . At points like Q the field gradient is zero .
Here the matrices are :
tos cp,
Z
Q î
2.2. Calculation of v,
T1,=
u F w
l'S
n )i/2
r
(11)
Once again the radial betatron frequency v, can be calculated from eq . (2). Calculations have been made for a value of 21 kG for the maximum field and 18, 19 and 20 kG for the average field. The number of sectors has been taken to be three. Fig. 2 shows the variation of v, as a function
â U Z
of the radius for such a field. Comparison has been made with those for a straight sector having the same maximum and average fields . It is seen that whereas for a small variation of field, v. values in our case are less than those of the straight sector design ; for a larger variation in field v. values are higher (though only marginally) than those of the others . v, values remain essentially near 1 and do not show much effect of alternating gradient focussing. 2.3 . Calculation of v. and v, in the new scheme For obtaining a large increase in vz the field gradients have to be large. Larger gradients, of course, are possible if we settle for a smaller average field. A way of having a large gradient in the magnetic field, keeping the average field high, is to have a radial periodic variation. The larger the number of field oscillations, the larger is the gradient . As a first attempt we can assume a sinusoidal variation as shown in fig. 3. Sinusoidal variation is not very practical and we can take other shapes as discussed later. In such a field the
w x w z
1'S
0
im
a w m
m
â U
0
l'0
w
H
w
V w 01
0 4 07 RADIUS ( R/Rmax )
10
Fig. 2. Variation of v, as a function of the radius where the hill and valley fields vary linearly with radius for average fields of 1) 18 kG, 2) 19 kG and 3) 20 kG . The maximum field has been taken to be 21 kG. The dotted curves show v for conventional sectors. Number of sectors is three.
Z G Q
l'5 0
0-7 0-4 1'0 RADIUS (R/RMgx) Fig. 4. Sinusoidal variation of the magnetic field as a function of the radius where there are four regions in a sector . At no points the alternating gradient focusing is zero. 0.1
13
P.R . Sarma / A radially and azimuthally varying magnetic jreld
magnitude of the field index, and hence the alternating gradient focusing, is maximum at points like P and is zero at points like Q. To avoid such a situation we propose a sector in which there are four regions (fig . 4) . The field B(r) in the four regions are taken to be : 1) Bav + Bo sin(21Twr) in region 1; 2) Bav + Bo cos(21Twr) in region 2 ; 3) Bav - Bo sin(2mwr) in region 3 ; 4) Bav - Bo cos(21m)r) in region 4; where co = number of oscillations in the field, Ba, = average magnetic field, and Bo + Ba, = maximum field in the hill . Here, at no radii the field indices are zero in all the regions simultaneously . Once again, for a small variation of field (the field gradient is not small though), the scalloping of orbits is also small and so the field in any region along the orbit is almost constant . The field indices can be calculated easily from : _ r_ dB(r) (12) B dr ' With the different field indices in the four regions the forms of the matrices are same as in eqs. (6), (7), (10) and (11) in the earlier case . For calculating the matrices for the region boundaries one has to calculate the angles the orbit makes with the boundaries and this can be done very easily. Knowing all these matrices the overall transfer matrix, and hence P, and P,, can be calculated . Calculations have been made for a field with three sectors and having various numbers of radial oscillations. The maximum field and the average field have been taken to be 21 kG and 19 kG, respectively . Fig. 5 shows the results of the calculations. We see that the betatron frequencies vary smoothly as a function of the radius and the magnitudes of Ps are much larger than those for a straight sector geometry having the same n (r)
r U Z W
REGION REGION
l'5 O W Ô a
a w N
4
REGION 1
0'5
REGION 44
TRIANGULAR VARIATION OF POLE SURFACE 0 0-7 0-1 0-4 110 RADIUS (Ri R MAX ) Fig. 6. A triangular variation to the pole surface for easy fabrication. maximum and average fields . The larger the number of radial oscillations, the higher the values of P, 3. Discussions For the given size of a cyclotron magnet the maximum energy achievable for an ion (of given Q/A) depends on the average magnetic field provided sufficient focusing is present. Focusing can be increased by increasing the flutter (variation of magnetic field) but this decreases the average field. Generally, in cyclotrons, light ions cannot be accelerated to the full energy because of insufficient vertical focusing, e.g ., in Berkeley 88 protons can be accelerated up to 60 MeV only, whereas the magnet can bend protons up to 130 MeV of energy. In the K500 cyclotron at MSU the focusing limit is KF = 160 whereas the bending limit is KB = 500 [9]. The scheme described above shows a way of increasing the focusing while keeping the average magnetic field constant . Increasing the spiral angle is another way of achieving this . However, we have already pointed out TRIANGULAR IRON PIECES ON THE BASE PLATE
a W z 0
..Q~@NU@
a
.a
H
W
m
0 H W RADIUS ( R/R max )
Fig. 5. v as a function of the radius for the proposed radially and azimuthally varying field. Number of sectors is three. w values are the number of oscillations in the field. Maximum field is 21 kG and average field is 19 kG.
THREE DIMENSIONAL VIEW Fig. 7. Separate tnangular pieces can be put on the flat bed of the pole surface for giving the required field variation.
14
P R. Sarma / A radially and azimuthally varying magnetic field
RADIUS ( R/Rma x )
Fig . 8 . Comparison of v, values for various cases. The continuous curve for the present scheme is for w = 5, Bav =18 kG and Bo = 3 kG . The dotted curve is for w = 3, Ba, = 36 kG and Bo = 7.5 kG . the limitation . The present scheme also has limitations as discussed below. As seen in fig. 5, the v. value can be increased by increasing the number of radial oscillations in the magnetic field. However, there is a limit to the number of field oscillations one can use. Radial variation in the field is achieved by having a geometrical variation on the pole surface. One cannot achieve the required field variation if two successive minimum gaps are sufficiently close compared to the pole gap. For giving a sinusoidal variation to the field the pole face profile should be sinusoidal-like [10]. It is difficult to give such a vanation to the pole face . Instead we can give a triangular or a similar simple variation as shown in fig. 6. Mechanically also it is very difficult to give a sinusoidal variation to the pole surface. So either a triangular, or trapezoidal, or any such shape consisting of straight portions can be used . Mechanical fabrication of this is considerably more difficult than the fabrication of a spiral sector . But in view of the larger focusing one obtains with the proposed geometry, the objection about this difficulty can be waived . If we consider a
triangular shape then the structure can be made by placing separate iron blocks of the shape shown in fig. 7 and this makes the fabrication easier . Finally we compare our results for v., with w = 5, with those of Berkeley 88 and that of K500 at MSU (fig. 8) . For making the comparisons independent of ion species, the contribution -ß 2 /(1 - /3 Z) to v.2 due to the average field gradient has not been included. It is evident that focusing is better in the present scheme, even though the field variation (18 kG average field and 21 kG maximum field) in the present case is much smaller than in the other two cases. Calculation has also been made in the present scheme with w = 3, and with 36 kG for the average field and 43 .5 kG for the maximum field (which are typical fields in K500 for an ion with Q/A = 0.5 and energy equal to 80 MeV per nucleon) and shown in fig. 8. It should be noted that, in general, one must control v, as well as v, so as to avoid resonances . v, is generally not problematic as it remains essentially near 1. We point out that a value of v.2 comparable to that of the present scheme can be obtained with the same maximum and average fields by using a spiral sector with a spiral angle of about 75° . References
f2] [3] [4]
[61 [71
[101
J.R. Richardson, Sector Focusing Cyclotrons, in : Progress in Nuclear Techniques and Instrumentation, vol. 1 (North-Holland, Amsterdam, 1965) p. 3 . L.H . Thomas, Phys . Rev. 54 (1938) 580. G. Schatz, Nucl . Instr . and Meth . 67 (1969) 103. K.R. Symon, D.W . Kerst, L.W . Jones, L .J . Laslett and K.M . Terwilliger, Phys. Rev. 103 (1956) 1837 . J.R . Richardson, Sector Focusing Cyclotrons in : Progress in Nuclear Techniques and Instrumentation, vol. 1 (North-Holland, Amsterdam, 1965) p. 18 . Ibid ., p. 14 . J.J. Livingwood, Principles of Cyclic Particle Accelerators (Van Nostrand, 1961) p. 52 . N.C . Bhattacharya and A.S . Divatia, Proc. 9th Int. Conf. on Cyclotrons and Their Applications, Caen, 1981, p. 427. G. Bellomo and F .G . Resmini, Nucl . Instr. and Meth. 180 (1981) 305 . P.D . Dunn, L.B . Mullet, T.G . Pickavance, W. Walkinshaw and J.J . Wilkins, CERN Accelerator Conf. (1956) p. 9.
Nuclear Instruments and Methods in Physics Research A305 (1991) 10-14 North-Holland
A proposed new type of radially and azimuthally varying magnetic field in cyclotrons for enhanced focusing P.R. Sarma
Variable Energy Cyclotron Centre, Bhabha Atomic Research Centre, 11AF Bidhan Nagar, Calcutta-700064, India
Received 4 May 1990 and in revised form 30 January 1991
For vertical focusing of ions in cyclotrons generally an azimuthal variation is given to the magnetic field with the help of a straight sector geometry or a spiral sector geometry . In the present work a new type of magnetic field, which varies azimuthally and has a periodic radial variation as well, has been proposed . It has been shown that the vertical focusing obtained with the proposed geometry is much stronger than what can be obtained with straight sector geometries .
1. Introduction In the early days when the first cyclotron was being developed, it had been realized by Lawrence that because of the relativistic increase in mass the magnetic field would have to increase with radius and this would conflict with the requirements of vertical focusing [1]. Focusing is necessary for confining the ions vertically as well as horizontally inside the cyclotron magnet . A solution to this problem was suggested by Thomas [2] who proposed a sinusoidal azimuthal variation of the magnetic field. Later on McMillan [1] also proposed a similar scheme using wedge shaped fields . Cyclotrons with such magnetic fields are called azimuthally varying field (AVF) cyclotrons or sector focused cyclotrons. For high energy cyclotrons focusing has to be very strong, and this can be achieved by using separated sectors where the azimuthal field variation is maximum. In fact, separated magnet sectors of very small azimuthal width give very strong vertical focusing [3]. But with separated sectors the size of the machine (and hence the total weight of the magnet) becomes large. Another important way of improving the focusing was pointed out by Symon et al . [4]. They showed that one can obtain a stronger focusing by using spiral sectors instead of straight ones. Thus separated sectors having a spiral shape produce the maximum focusing achievable so far. Focusing in cyclotrons is characterized by the vertical and radial betatron oscillation frequencies, v.
and v, respectively. The contribution to vz due to the spiral sectors is given by [5]: v~
=
( BH
-
B )( B - Bv) B2
[1 + 2 tan2 E],
where B H = field in the so-called hill region ; B = field in the valley region ; B = average field; e = spiral angle. For net vi the term -/32/(1 - ß2) due to the field gradient must be added. Here /3 is equal to v/c where v is the velocity of the ions. The vertical betatron frequency vs can be increased by decreasing B but this results in a low average field leading to an increase in the size of the machine. Another way of increasing v, is to increase the spiral angle E. But e also cannot be increased indefinitely because for large E the shortest distance between the edges of the spiral hills decreases and so the required field variation is not achieved . The maximum value of e depends on radius and gap, and magnets with e up to 70' have been designed though most of the cyclotrons use spiral angles up to about 60 °. In this article we propose a new type of radially and azimuthally varying field (RAVF) in a fixed frequency cyclotron. In Thomas focusing as well as spiral focusing only azimuthal field variation is used (although a spiral produces radial variation also). In the new scheme we use both the radial and the azimuthal field variation. It is well known that if there is a radial decrease in magnetic field then there is focusing in the vertical
0168-9002/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
P. R. Sarma / A radially and azimuthally varying magnetic field
15
_
CONVENTIONAL CYCLOTRON HILL
m
REGION I
AVERAGE
10
VALLEY
V
w z O a
REGION 2
05 RADIALLY CONSTANT FIELD
01
04
07
RADIALLY VARYING FIELD
ill
1001
04
07
10
RADIUS (R/RMAX)
Fig. 1. Variation of magnetic field as a function of radius in a straight sector cyclotron and in a cyclotron where field vanes radially. Field has not been shown in the centre region as separate hill and valley are not present there. direction and defocusing in the radial direction, and vice versa if the field increases. We use this principle in the present work. We point out here that this principle of radial alternating field gradient has very many applications, e.g ., in synchrotrons ; but as yet no thought has been given to using it m cyclotrons . 2. Calculation
as a result there will be a net alternating gradient focusing . In what follows we assume that the field variation is small so that the scalloping of the orbits is also small and thus the field in any region along any particle orbit remains essentially constant . We assume this only for simplifying the calculation and this assumption can be put aside in a rigorous calculation . We now calculate the vertical and radial betatron oscillation frequencies v, and v, in the framework of hard edge approximation. We use the matrix method [71 used by Schatz [31 and Bhattacharya and Divatia [8]. 2.1 . Calculation
of
vs
As has been shown in ref. [7], the betatron oscillation frequencies can be calculated from the relation :
where T.., is the overall transfer matrix which is a product of the various transfer matrices in the regions and in the boundaries and N is the number of sectors. The subscripts z and r refer to vertical and radial directions respectively . In our case the four matrices involved at any radius are: 1) Tlz, , at the entrance of the first region ; 2) T2 z, , in the first region ; 3) T3z, , at the entrance of the second region ; 4) T4z, , in the second region . For calculating Tlz,, and T3z,r we should know the angle which the orbit makes with the normal to the sector edge. The fields in the two regions at any radius r can be written as :
Before we describe the actual scheme, let us discuss the principle in a simple way and obtain some results . A similar procedure will later be used in the actual scheme . In a conventional straight sector cyclotron each sector is divided into two regions; one region is the hill where the field is high and the other region is the valley where the BI( r)=B,-2fr, B2(r)=B +2fr, (3) field is low. In our case the two adjoining regions have with B = (BH + BJ/2 and f a constant. As given by linearly varying fields in the radial direction. A region Richardson [6] the angle k at the entrance of the first which starts as a hill near the centre becomes a valley at region is : the outer radii and vice versa (fig . 1) . Decrease in field and increase in the other keeps the averin one region _ m [BI(r)-B ] [B-B2(r)] k age field constant at all radii. In a cyclotron the average N B[B1(r)-B2(r)1 ' magnetic field (averaged over all azimuths) has actually and then : to increase for taking into account the relativistic mass increase, which depends on the increase of ions . This 1 mass and the final energy of the ions, can be brought TI, = - BI(r) -B2(r) T3Z = TlZ . tan k about with the help of the trim coils and the flare in the B hills. This radial increase in field due to relativistic effects decreases the vz values gradually as the radius The field index in region 1 is n = 2fr/B and the matrix increases, unless the focusing is increased, and this T2, is : happens in conventional cyclotrons as well as in the present scheme . We do not take into account this de1 -,Fcos $s sin tP~ crease in v in our calculation as this is mass and energy n t/z K, T2 r = [ _ , _ r ' dependent. This, however, can be calculated separately . -K, sin $Z cos J>' We consider regions of equal width. In such a sector geometry there will be focusing in the falling field / t z (6) $ = N n, region and defocusing in the increasing field region and
P.R. Sarma / A radially and azimuthally varying magnentfield
12 and similarly; cosh $z
T4, =
L K.
.m
KZ sinn (p,
sinh 4).
m
cosh
0 W
The overall matrix is :
(8)
T, = T1,T2,T3,T4, .
1 BI(r) -B2(r) B tan k
T2,=
- K,
Similarly; T4,
=I
cosh
sin 0,
~
K, sinh (p,
_ 1 sin 4~, K, tos ¢,
K
sinh
0
Tar=T1
1 '
,
K
¢ I.
cosh ¢,
=
(1
1'0
0'S 0
01
0-4 0-7 110 RADIUS (R/RMAX) Fig. 3. Sinusoidal variation of the magnetic field as a function of the radius . At points like Q the field gradient is zero .
Here the matrices are :
tos cp,
Z
Q î
2.2. Calculation of v,
T1,=
u F w
l'S
n )i/2
r
(11)
Once again the radial betatron frequency v, can be calculated from eq . (2). Calculations have been made for a value of 21 kG for the maximum field and 18, 19 and 20 kG for the average field. The number of sectors has been taken to be three. Fig. 2 shows the variation of v, as a function
â U Z
of the radius for such a field. Comparison has been made with those for a straight sector having the same maximum and average fields . It is seen that whereas for a small variation of field, v. values in our case are less than those of the straight sector design ; for a larger variation in field v. values are higher (though only marginally) than those of the others . v, values remain essentially near 1 and do not show much effect of alternating gradient focussing. 2.3 . Calculation of v. and v, in the new scheme For obtaining a large increase in vz the field gradients have to be large. Larger gradients, of course, are possible if we settle for a smaller average field. A way of having a large gradient in the magnetic field, keeping the average field high, is to have a radial periodic variation. The larger the number of field oscillations, the larger is the gradient . As a first attempt we can assume a sinusoidal variation as shown in fig. 3. Sinusoidal variation is not very practical and we can take other shapes as discussed later. In such a field the
w x w z
1'S
0
im
a w m
m
â U
0
l'0
w
H
w
V w 01
0 4 07 RADIUS ( R/Rmax )
10
Fig. 2. Variation of v, as a function of the radius where the hill and valley fields vary linearly with radius for average fields of 1) 18 kG, 2) 19 kG and 3) 20 kG . The maximum field has been taken to be 21 kG. The dotted curves show v for conventional sectors. Number of sectors is three.
Z G Q
l'5 0
0-7 0-4 1'0 RADIUS (R/RMgx) Fig. 4. Sinusoidal variation of the magnetic field as a function of the radius where there are four regions in a sector . At no points the alternating gradient focusing is zero. 0.1
13
P.R . Sarma / A radially and azimuthally varying magnetic jreld
magnitude of the field index, and hence the alternating gradient focusing, is maximum at points like P and is zero at points like Q. To avoid such a situation we propose a sector in which there are four regions (fig . 4) . The field B(r) in the four regions are taken to be : 1) Bav + Bo sin(21Twr) in region 1; 2) Bav + Bo cos(21Twr) in region 2 ; 3) Bav - Bo sin(2mwr) in region 3 ; 4) Bav - Bo cos(21m)r) in region 4; where co = number of oscillations in the field, Ba, = average magnetic field, and Bo + Ba, = maximum field in the hill . Here, at no radii the field indices are zero in all the regions simultaneously . Once again, for a small variation of field (the field gradient is not small though), the scalloping of orbits is also small and so the field in any region along the orbit is almost constant . The field indices can be calculated easily from : _ r_ dB(r) (12) B dr ' With the different field indices in the four regions the forms of the matrices are same as in eqs. (6), (7), (10) and (11) in the earlier case . For calculating the matrices for the region boundaries one has to calculate the angles the orbit makes with the boundaries and this can be done very easily. Knowing all these matrices the overall transfer matrix, and hence P, and P,, can be calculated . Calculations have been made for a field with three sectors and having various numbers of radial oscillations. The maximum field and the average field have been taken to be 21 kG and 19 kG, respectively . Fig. 5 shows the results of the calculations. We see that the betatron frequencies vary smoothly as a function of the radius and the magnitudes of Ps are much larger than those for a straight sector geometry having the same n (r)
r U Z W
REGION REGION
l'5 O W Ô a
a w N
4
REGION 1
0'5
REGION 44
TRIANGULAR VARIATION OF POLE SURFACE 0 0-7 0-1 0-4 110 RADIUS (Ri R MAX ) Fig. 6. A triangular variation to the pole surface for easy fabrication. maximum and average fields . The larger the number of radial oscillations, the higher the values of P, 3. Discussions For the given size of a cyclotron magnet the maximum energy achievable for an ion (of given Q/A) depends on the average magnetic field provided sufficient focusing is present. Focusing can be increased by increasing the flutter (variation of magnetic field) but this decreases the average field. Generally, in cyclotrons, light ions cannot be accelerated to the full energy because of insufficient vertical focusing, e.g ., in Berkeley 88 protons can be accelerated up to 60 MeV only, whereas the magnet can bend protons up to 130 MeV of energy. In the K500 cyclotron at MSU the focusing limit is KF = 160 whereas the bending limit is KB = 500 [9]. The scheme described above shows a way of increasing the focusing while keeping the average magnetic field constant . Increasing the spiral angle is another way of achieving this . However, we have already pointed out TRIANGULAR IRON PIECES ON THE BASE PLATE
a W z 0
..Q~@NU@
a
.a
H
W
m
0 H W RADIUS ( R/R max )
Fig. 5. v as a function of the radius for the proposed radially and azimuthally varying field. Number of sectors is three. w values are the number of oscillations in the field. Maximum field is 21 kG and average field is 19 kG.
THREE DIMENSIONAL VIEW Fig. 7. Separate tnangular pieces can be put on the flat bed of the pole surface for giving the required field variation.
14
P R. Sarma / A radially and azimuthally varying magnetic field
RADIUS ( R/Rma x )
Fig . 8 . Comparison of v, values for various cases. The continuous curve for the present scheme is for w = 5, Bav =18 kG and Bo = 3 kG . The dotted curve is for w = 3, Ba, = 36 kG and Bo = 7.5 kG . the limitation . The present scheme also has limitations as discussed below. As seen in fig. 5, the v. value can be increased by increasing the number of radial oscillations in the magnetic field. However, there is a limit to the number of field oscillations one can use. Radial variation in the field is achieved by having a geometrical variation on the pole surface. One cannot achieve the required field variation if two successive minimum gaps are sufficiently close compared to the pole gap. For giving a sinusoidal variation to the field the pole face profile should be sinusoidal-like [10]. It is difficult to give such a vanation to the pole face . Instead we can give a triangular or a similar simple variation as shown in fig. 6. Mechanically also it is very difficult to give a sinusoidal variation to the pole surface. So either a triangular, or trapezoidal, or any such shape consisting of straight portions can be used . Mechanical fabrication of this is considerably more difficult than the fabrication of a spiral sector . But in view of the larger focusing one obtains with the proposed geometry, the objection about this difficulty can be waived . If we consider a
triangular shape then the structure can be made by placing separate iron blocks of the shape shown in fig. 7 and this makes the fabrication easier . Finally we compare our results for v., with w = 5, with those of Berkeley 88 and that of K500 at MSU (fig. 8) . For making the comparisons independent of ion species, the contribution -ß 2 /(1 - /3 Z) to v.2 due to the average field gradient has not been included. It is evident that focusing is better in the present scheme, even though the field variation (18 kG average field and 21 kG maximum field) in the present case is much smaller than in the other two cases. Calculation has also been made in the present scheme with w = 3, and with 36 kG for the average field and 43 .5 kG for the maximum field (which are typical fields in K500 for an ion with Q/A = 0.5 and energy equal to 80 MeV per nucleon) and shown in fig. 8. It should be noted that, in general, one must control v, as well as v, so as to avoid resonances . v, is generally not problematic as it remains essentially near 1. We point out that a value of v.2 comparable to that of the present scheme can be obtained with the same maximum and average fields by using a spiral sector with a spiral angle of about 75° . References
f2] [3] [4]
[61 [71
[101
J.R. Richardson, Sector Focusing Cyclotrons, in : Progress in Nuclear Techniques and Instrumentation, vol. 1 (North-Holland, Amsterdam, 1965) p. 3 . L.H . Thomas, Phys . Rev. 54 (1938) 580. G. Schatz, Nucl . Instr . and Meth . 67 (1969) 103. K.R. Symon, D.W . Kerst, L.W . Jones, L .J . Laslett and K.M . Terwilliger, Phys. Rev. 103 (1956) 1837 . J.R . Richardson, Sector Focusing Cyclotrons in : Progress in Nuclear Techniques and Instrumentation, vol. 1 (North-Holland, Amsterdam, 1965) p. 18 . Ibid ., p. 14 . J.J. Livingwood, Principles of Cyclic Particle Accelerators (Van Nostrand, 1961) p. 52 . N.C . Bhattacharya and A.S . Divatia, Proc. 9th Int. Conf. on Cyclotrons and Their Applications, Caen, 1981, p. 427. G. Bellomo and F .G . Resmini, Nucl . Instr. and Meth. 180 (1981) 305 . P.D . Dunn, L.B . Mullet, T.G . Pickavance, W. Walkinshaw and J.J . Wilkins, CERN Accelerator Conf. (1956) p. 9.