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Space charge limit in a relativistic migma
Nuclear Instruments and Methods in Physics Research A271(1988) 212-213 North-Holland, Amsterdam
212
SPACE
GE
. IN A
STIC
G
John P. BLEWETT * *
Aneutronic Energy Laboratories, P.O. Box 3037, Princeton, NJ 08543, USA
The space charge limit for a disc-shaped relativistic ion migma has been designed in terms of the total ion number N and density for the "classical" migma in the magnetic field of the simple mirror type.
n
1. Introduction
2. Field patterns in the migmz
We assume that we are dealing with a charge distribution of radius a and of roughly uniform axial thickness, extending to an average height of f zo above and below the median plane. This charge distribution is ;oniaiaed by a magnetic field whose axial component is given by (1) B, = Bo (1- kr t /a2 + 2kz 2/a2 ),
Some qualitative arguments help to establish the field patterns in the migma. We can get a rough idea of E, by assuming that, at a particular point, the radial field component is due to all of the charge at smaller radii concentrated at the center, and is not due at all to charges at larger radii. This leads us to expect that E, will have the approximate form
is small (of the order of 0.2). It follows that 2 (2) -2krzBo la .
where B,=
k
The radial position of a charged particle is approximately given by r=a sin wt, where w
_ eBo ' 2 (4) m
The particle velocity v = aw, whence it fellows that aBo = 2 mßc/e
= 6 .26ßy for protons where a is in meters and Bo in T. For 10 GeV protons, ß = 0.9963 and y =11.66. In a field of 10 T, a = 7 .27 m.
At the axis of the migma and at its outer edge there will be density maxima. In general, throughout the volume the charge density will vary like the density of orbits and we can write for the charge per unit volume Ne P , 2ir -zo rya - - rtotal number of charges in the migma. e current density in the enigma will be prw, whence Ne 2Bo
where N is the
4rr2mzo a2 _ r'
Work supported by AFWL contract F29601-87-C-0064. Pernianent address : 310 W. 106 St ., Newa York, NY 10025, USA.
0168-9002,/88/$03 .50 6D Elsevier Science Publishers B.V.
E,.=
Ne ,rr 2E
sin -1 (r/a)
(in our units, c = 8.84 x 10-12) . Over most of the range in r, E, is approximately proportional to 1/r. It will depend only weakly on z. From div E = p/E we conclude that, to a first approximation, EZ
Nez
=pz/E=
27r 2Ez 0
a2
- r2
The magnetic field due to the current in the migma follows from curl B = ILI, where IL =1.26 x 10 -6 . At z = zo, NLNe
B, = gIozo --
I
2
Bo
41r2-4 a2 - r 2
(10)
Irxial forces
The two forces in the z direction due to the migma are, at z = zo : C' EZ
eruB, =
Ne e 2E
a 2 - r2 fi Ne
4_02 r
872M2 a2 - r2
(12)
J.P. Blewett / Space charge limit In combination, eEZ-erdB,.=
Ne22
2 (1- (wr/c)2 . ) 2ir 2e r a - r-2
(13)
Evidently, for relativistic ions, the electrostatic blowup force will be materially reduced for large r (close to a) but will not be materially affected in the central region. The restoring force at z = zo due to the shaped field is -ke2B0r2Z, AB, = = -4mkw 2 r 2zo/a 2 . mat
(14)
4. Space charge limit The destructive force (except for the r2w2/c2 term) is inversely proportional to r(a 2 -r 2)1/2, while the restoring force is proportional to r2. Stability of the system will depend on an overall balancing of forces. As an approximation to the averaging process we equate the forces at r = a/2 to obtain N(1- (wa/2c)2)
_
-
or N(1_#2 /4)
= 4.4
X
F3 r 2Emkw 2a2zo 2e2
(15)
~ir 2E kz0 a 2BÛ 8m
(16)
10 17kZ O ß 2 y .
(17)
For 10 GeV protons, IB = 0.996 and y = 11.26. If k = 0.2 and z 0 = 0.5 m, then N=5 X 1017.
by
213
The number of charges per cubic centimeter is given
n (cm - ) =
N X 10 -6 2Ir2z0
a2 -
- 5 X 1010 r a'---r 2 r2
where a and r are in meters. 5. Space charge limit in migma using a
ring magnet
The problem of space charge limits in a migma in a ring magnet system is sufficiently complicated that treatments as simple as that presented above simply are not applicable. This is a problem that must be solved using a powerful computer. A couple of general comments are possible: First, in contrast to the classical migma configuration, the focusing and defocusing forces with the ring magnet are much stronger and much more_ severely localized . It appears that, for particles in the right energy range, orbits with the ring magnet can be strongly restored in the axial direction and so should be highly stable. This would suggest that space charge limits should not be lower with the ring magnet than with thc; conventional geometry. It must be emphasized, however, that this is little better than a guess. Second, in a ring magnet whose interior region has a field close to zero, the situation with respect to neutralization by electrons is quite different . In the conventional migma the strong magnetic field everywhere restrains electrons from wandering either in radius or in azimuth. In the zero field inside the ring magnet, electrons will be free to wander to regions where neutralization is needed. This may well be an important advantage.
111. NEW CONCEPTS
212
SPACE
GE
. IN A
STIC
G
John P. BLEWETT * *
Aneutronic Energy Laboratories, P.O. Box 3037, Princeton, NJ 08543, USA
The space charge limit for a disc-shaped relativistic ion migma has been designed in terms of the total ion number N and density for the "classical" migma in the magnetic field of the simple mirror type.
n
1. Introduction
2. Field patterns in the migmz
We assume that we are dealing with a charge distribution of radius a and of roughly uniform axial thickness, extending to an average height of f zo above and below the median plane. This charge distribution is ;oniaiaed by a magnetic field whose axial component is given by (1) B, = Bo (1- kr t /a2 + 2kz 2/a2 ),
Some qualitative arguments help to establish the field patterns in the migma. We can get a rough idea of E, by assuming that, at a particular point, the radial field component is due to all of the charge at smaller radii concentrated at the center, and is not due at all to charges at larger radii. This leads us to expect that E, will have the approximate form
is small (of the order of 0.2). It follows that 2 (2) -2krzBo la .
where B,=
k
The radial position of a charged particle is approximately given by r=a sin wt, where w
_ eBo ' 2 (4) m
The particle velocity v = aw, whence it fellows that aBo = 2 mßc/e
= 6 .26ßy for protons where a is in meters and Bo in T. For 10 GeV protons, ß = 0.9963 and y =11.66. In a field of 10 T, a = 7 .27 m.
At the axis of the migma and at its outer edge there will be density maxima. In general, throughout the volume the charge density will vary like the density of orbits and we can write for the charge per unit volume Ne P , 2ir -zo rya - - rtotal number of charges in the migma. e current density in the enigma will be prw, whence Ne 2Bo
where N is the
4rr2mzo a2 _ r'
Work supported by AFWL contract F29601-87-C-0064. Pernianent address : 310 W. 106 St ., Newa York, NY 10025, USA.
0168-9002,/88/$03 .50 6D Elsevier Science Publishers B.V.
E,.=
Ne ,rr 2E
sin -1 (r/a)
(in our units, c = 8.84 x 10-12) . Over most of the range in r, E, is approximately proportional to 1/r. It will depend only weakly on z. From div E = p/E we conclude that, to a first approximation, EZ
Nez
=pz/E=
27r 2Ez 0
a2
- r2
The magnetic field due to the current in the migma follows from curl B = ILI, where IL =1.26 x 10 -6 . At z = zo, NLNe
B, = gIozo --
I
2
Bo
41r2-4 a2 - r 2
(10)
Irxial forces
The two forces in the z direction due to the migma are, at z = zo : C' EZ
eruB, =
Ne e 2E
a 2 - r2 fi Ne
4_02 r
872M2 a2 - r2
(12)
J.P. Blewett / Space charge limit In combination, eEZ-erdB,.=
Ne22
2 (1- (wr/c)2 . ) 2ir 2e r a - r-2
(13)
Evidently, for relativistic ions, the electrostatic blowup force will be materially reduced for large r (close to a) but will not be materially affected in the central region. The restoring force at z = zo due to the shaped field is -ke2B0r2Z, AB, = = -4mkw 2 r 2zo/a 2 . mat
(14)
4. Space charge limit The destructive force (except for the r2w2/c2 term) is inversely proportional to r(a 2 -r 2)1/2, while the restoring force is proportional to r2. Stability of the system will depend on an overall balancing of forces. As an approximation to the averaging process we equate the forces at r = a/2 to obtain N(1- (wa/2c)2)
_
-
or N(1_#2 /4)
= 4.4
X
F3 r 2Emkw 2a2zo 2e2
(15)
~ir 2E kz0 a 2BÛ 8m
(16)
10 17kZ O ß 2 y .
(17)
For 10 GeV protons, IB = 0.996 and y = 11.26. If k = 0.2 and z 0 = 0.5 m, then N=5 X 1017.
by
213
The number of charges per cubic centimeter is given
n (cm - ) =
N X 10 -6 2Ir2z0
a2 -
- 5 X 1010 r a'---r 2 r2
where a and r are in meters. 5. Space charge limit in migma using a
ring magnet
The problem of space charge limits in a migma in a ring magnet system is sufficiently complicated that treatments as simple as that presented above simply are not applicable. This is a problem that must be solved using a powerful computer. A couple of general comments are possible: First, in contrast to the classical migma configuration, the focusing and defocusing forces with the ring magnet are much stronger and much more_ severely localized . It appears that, for particles in the right energy range, orbits with the ring magnet can be strongly restored in the axial direction and so should be highly stable. This would suggest that space charge limits should not be lower with the ring magnet than with thc; conventional geometry. It must be emphasized, however, that this is little better than a guess. Second, in a ring magnet whose interior region has a field close to zero, the situation with respect to neutralization by electrons is quite different . In the conventional migma the strong magnetic field everywhere restrains electrons from wandering either in radius or in azimuth. In the zero field inside the ring magnet, electrons will be free to wander to regions where neutralization is needed. This may well be an important advantage.
111. NEW CONCEPTS