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Efficient recuperator of intense electron beam energy
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Nuclear
14
North-Holland
EFFICIENT RECUPERATOR OF INTENSE ELECTRON REAM ENERGY A.N. SHARAPA and A.V. SHEMYAKIN Institute
Received An disk-shaped collector main
. -4
Energy it and device. cooling The a and netic . cylindrical cally electron transformation one tion solenoids One formation current result a
Fig . . disk-shaped
a-
.
. . . .
0168-9002/90/$03 .50 (North-Holland)
ro
Po/4L5/4
2 r and H where po = Vol(eHo s = Rs+ , o o the in o total e, are electron s is solenoid . An important the transformation, l /WO . mum field = RS) . According . the W0 with accuracy: corresponding coincide . initial rm that motion : (W1 r,,, = 0.6po ( L '/ Po )t/4
.
magnetic
.
If one by case equations geometry eter:
.
.
.
.
.
. .V.
An in the gion, . point consider disk-shaped
. . .
A . N. Sharapa, A . V. Shemyakin
Efficient recuperator of intense electron beam energy
15
In the left-hand half-space, potentials are calculated analytically :
W, Wo
Ut = r,2 -2 z 2 .
0.5
4 0.5 aC Fig . 2. Dependence W1 /WO(a) at the point of the maximum value of the magnetic field (r = R,,). R, =140 mm, a = 50 mm, (1) po = 2.1 mm, (2) po =1 .1 mm, (3) po = 0.7 mm . Similarly to the solution given in ref. [51 for a flat beam, one can obtain an analytical solution for electrostatic transformation of an axisymmetric beam. It has a constant charge density throughout the whole space, its equipotentials are ellipsoid-shaped (fig . 3) : U=ß(r2 + 4z2), z J = const . where ß is a constant, and trajectories rVI_ To solve the external problem, it is convenient to use dimensionless values : ri = r/ao, Uo = 0 .24
z t = z/ao,
m 2 e 1/lo,
Ul = U/UO,
10 = mc3/e,
where 1 is the beam current, a o is a value with the dimension of length determining the boundary trajectory . r = ao 2/z1/2
v
v
v
_A Fig. 3 . Recuperator with electrostatic transformation of a cylindrical beam into a disk-shaped one . (1) Electrode, (2) beam, (3) ring collector, (4) trajectory of an electron reflected from a magnetic mirror, (5) anode, (6) cone electrode.
Due to the complicated geometry of the beam boundary, restoration of the equipotentials in the righthand half-space is bound up with considerable difficulties . Thus, Harker's method does not allow finding a unique conform mapping for a region located at rather a large distance from the beam boundary . Nevertheless, this difficulty is not very important in determining the shape of electrodes, since the equipotentials near the beam are . determined rather accurately. Strictly speaking, this solution is incompatible with beam deceleration, as it implies a radial growth of potential . Nevertheless, the system of electrodes performing the beam transformation can be supplemented by a ring collector with a lower potential U, thus forming an efficient recuperator (fig. 3). An important peculiarity of the solution illustrated in fig. 3 consiFa in the fact that it allows, in principle, the above mentioned transformation without beam losses, resulting from the reflection of paraxial electrons. This peculiarity of the solution can also be used for transformation in a magnetic field . To this purpose one can combine the electrostatic transformation with a magnetic one . This combination appears to be consistent since one can show that the electron trajectories in the electrostatic transformation system coincide with the magnetic field lines near the axis (fig . 1) . But one should keep in mind that an ideal transformation is possible only for strictly specified electrode potentials and a given current . Besides, one has to produce electrodes, exactly imitating the form of equipotentials . In reality, every recuperator, due to the deviation from the exact solution, has some noticeable electron flux which does not reach the ring collector. In a recuperator (fig. 3) the flux which does not reach the ring collector is determined by particles of two types : (1) the reflected particles from a cone electrode (6) at zero potential and (2) the particles reflected by a magnetic mirror in a radially increasing magnetic field . It was experimentally proved that the rate of these
particles may be reduced down to 10 -3 . However, electrons reaching large radii can be captured if electrode 1 is placed within the beam. For an efficient cap_ ture of electrons it is necessary that the width ® o ." that part of the beam which is incident on electrode 1, would be : where p is the Larmor radius, calculated from the electrode 1 potential and the magnetic field near it. The main disadvantage of this recuperator consists in that the reflected electrons from the cone electrode cannot be captured inside it . More efficient for the capture of
A . N. Sharapa, A . V. Shernyakin / Efficient recuperator of intense electron beam energy
16
than that for a conventional "Faraday cup" recuperator [5]. Nevertheless, these results should be considered as preliminary, since the present paper does not provide for the limiting cgpahi,Lties for recuperators of this type . It is accounted for by the fact that the main aim of this work was to reduce the losses caused by paraxial particle.%. To obtain the maximum value, one has to adapt the dimensions of the ring collector (A R, L in fig . 4) in correspondence with the thickness of the transformed beam h. The maximum value is attained under the condition [7] : h-L-AR .
Fig. 4. Recuperator with an intermediate collector. (1) Electron gun, (2) intermediate collector, (3) ring collector (4) anode. particles which have not got into the ring collector, turned out to be the recuperator given in fig. 4. This recuperator has no cone electrode, and, besides, the form of its electrodes differs from that corresponding to the ideal solution . In such a recuperator the rate of electrons which have not reached the ring collector, lint , is higher than that in the one presented in fig . 3. However, this fact is of no great importance, because all the electrons which have not reached the ring collector are captured in an intermediate collector, and secondary electrons (like in a conventional collector such as a "Faraday cup" [6]), are captured by a rather deep potential minimum . The power Won the intermediate collector, the latter being at potential Ui , , is equal to: W = Uint Lint
To decrease this power, one has to maximally reduce Uint limited by the space charge . In experiments, Üint was reduced down to the values at which the perveance a reached 20 [.AN 3 /2, i.e. was close to the pint = I/Dint limiting perveance of the Faraday cup. In the recuperator with an intermediate collector the losses AI on the anode were considerably smaller and the collector perveance was noticeably greater. The perveance P, is approximately 20 times greater Table 1 Maire experimental results and beam parameters I=3 .2A Ui ,=3kV U~=400V Pi,,, = 20 FL A/V 3i2 P,. _ I/ U 3/2 = 400 W A/V 3/2 Iint /I = 4% AI/1= 3 x 10-a Ho = 0 .09 T
In experiments a 30 mm diameter emitter was used. For this emitter the thickness of the transformed beam h is approximately 10 times smaller than the characteristic dimensions of the collector (h - 5 mm, A R - 50 mm, L - 50 mm). As a result, the experimental value of the perveance is considerably smaller than the maximum one for a ring collector with the given dimensions . The main results are : (a) A recuperator with a beam transformation from cylindrical into disk-shaped permits one to substantially increase the collector perveance, as compared to the Faraday cup . (b) With the introduction of the intermediate collector, practically all the electrons that escaped the main collector may be captured and, thus, the anode current loss rate is decreased down to values of 3 x 10 -4.
Acknoviedgement The authors acknowledge fruitful discussions with I.N. Meshkov and the assistance in calculations of T.N . Andreeva.
References [1] V.I. Kudelainen, I.N. Meshkov and R.A. Salimov, INP Novosibirsk preprint 72-70 (1970); Preprint CERN 77-08 (1977) . [2] T. Ellison et al., IEEE Trans . Nucl. Sci . NS-30 (1983) 2636. [3] M. Bell et al., Nucl. Instr. and Meth. 190 (1981) 237 . [41 V. I. Kokoulin, l .u. Meshkov ar ûiarâjrâ, -vv . .1. Tech . Phys. 50 (1980) 1475. [5] P.T. Kirstein, G.S. Kino ar:d W.E. Waters, Formation of Electron Beams (Mir, Moscow, 1970) pp. 102-104. [6] A.I. Arenshtam, I .N. Meshkov, V .G. Ponomarenko, R.A. Salimov, A.N. Skrinsky, B.M. Smimov and V.G. Fainshtein, Sov. J . Tech. Phys. 41 (1971) 323. [7] A.N. Sharapa, Candidate Thesis, Novosibirsk (1982) p. 124.
Nuclear
14
North-Holland
EFFICIENT RECUPERATOR OF INTENSE ELECTRON REAM ENERGY A.N. SHARAPA and A.V. SHEMYAKIN Institute
Received An disk-shaped collector main
. -4
Energy it and device. cooling The a and netic . cylindrical cally electron transformation one tion solenoids One formation current result a
Fig . . disk-shaped
a-
.
. . . .
0168-9002/90/$03 .50 (North-Holland)
ro
Po/4L5/4
2 r and H where po = Vol(eHo s = Rs+ , o o the in o total e, are electron s is solenoid . An important the transformation, l /WO . mum field = RS) . According . the W0 with accuracy: corresponding coincide . initial rm that motion : (W1 r,,, = 0.6po ( L '/ Po )t/4
.
magnetic
.
If one by case equations geometry eter:
.
.
.
.
.
. .V.
An in the gion, . point consider disk-shaped
. . .
A . N. Sharapa, A . V. Shemyakin
Efficient recuperator of intense electron beam energy
15
In the left-hand half-space, potentials are calculated analytically :
W, Wo
Ut = r,2 -2 z 2 .
0.5
4 0.5 aC Fig . 2. Dependence W1 /WO(a) at the point of the maximum value of the magnetic field (r = R,,). R, =140 mm, a = 50 mm, (1) po = 2.1 mm, (2) po =1 .1 mm, (3) po = 0.7 mm . Similarly to the solution given in ref. [51 for a flat beam, one can obtain an analytical solution for electrostatic transformation of an axisymmetric beam. It has a constant charge density throughout the whole space, its equipotentials are ellipsoid-shaped (fig . 3) : U=ß(r2 + 4z2), z J = const . where ß is a constant, and trajectories rVI_ To solve the external problem, it is convenient to use dimensionless values : ri = r/ao, Uo = 0 .24
z t = z/ao,
m 2 e 1/lo,
Ul = U/UO,
10 = mc3/e,
where 1 is the beam current, a o is a value with the dimension of length determining the boundary trajectory . r = ao 2/z1/2
v
v
v
_A Fig. 3 . Recuperator with electrostatic transformation of a cylindrical beam into a disk-shaped one . (1) Electrode, (2) beam, (3) ring collector, (4) trajectory of an electron reflected from a magnetic mirror, (5) anode, (6) cone electrode.
Due to the complicated geometry of the beam boundary, restoration of the equipotentials in the righthand half-space is bound up with considerable difficulties . Thus, Harker's method does not allow finding a unique conform mapping for a region located at rather a large distance from the beam boundary . Nevertheless, this difficulty is not very important in determining the shape of electrodes, since the equipotentials near the beam are . determined rather accurately. Strictly speaking, this solution is incompatible with beam deceleration, as it implies a radial growth of potential . Nevertheless, the system of electrodes performing the beam transformation can be supplemented by a ring collector with a lower potential U, thus forming an efficient recuperator (fig. 3). An important peculiarity of the solution illustrated in fig. 3 consiFa in the fact that it allows, in principle, the above mentioned transformation without beam losses, resulting from the reflection of paraxial electrons. This peculiarity of the solution can also be used for transformation in a magnetic field . To this purpose one can combine the electrostatic transformation with a magnetic one . This combination appears to be consistent since one can show that the electron trajectories in the electrostatic transformation system coincide with the magnetic field lines near the axis (fig . 1) . But one should keep in mind that an ideal transformation is possible only for strictly specified electrode potentials and a given current . Besides, one has to produce electrodes, exactly imitating the form of equipotentials . In reality, every recuperator, due to the deviation from the exact solution, has some noticeable electron flux which does not reach the ring collector. In a recuperator (fig. 3) the flux which does not reach the ring collector is determined by particles of two types : (1) the reflected particles from a cone electrode (6) at zero potential and (2) the particles reflected by a magnetic mirror in a radially increasing magnetic field . It was experimentally proved that the rate of these
particles may be reduced down to 10 -3 . However, electrons reaching large radii can be captured if electrode 1 is placed within the beam. For an efficient cap_ ture of electrons it is necessary that the width ® o ." that part of the beam which is incident on electrode 1, would be : where p is the Larmor radius, calculated from the electrode 1 potential and the magnetic field near it. The main disadvantage of this recuperator consists in that the reflected electrons from the cone electrode cannot be captured inside it . More efficient for the capture of
A . N. Sharapa, A . V. Shernyakin / Efficient recuperator of intense electron beam energy
16
than that for a conventional "Faraday cup" recuperator [5]. Nevertheless, these results should be considered as preliminary, since the present paper does not provide for the limiting cgpahi,Lties for recuperators of this type . It is accounted for by the fact that the main aim of this work was to reduce the losses caused by paraxial particle.%. To obtain the maximum value, one has to adapt the dimensions of the ring collector (A R, L in fig . 4) in correspondence with the thickness of the transformed beam h. The maximum value is attained under the condition [7] : h-L-AR .
Fig. 4. Recuperator with an intermediate collector. (1) Electron gun, (2) intermediate collector, (3) ring collector (4) anode. particles which have not got into the ring collector, turned out to be the recuperator given in fig. 4. This recuperator has no cone electrode, and, besides, the form of its electrodes differs from that corresponding to the ideal solution . In such a recuperator the rate of electrons which have not reached the ring collector, lint , is higher than that in the one presented in fig . 3. However, this fact is of no great importance, because all the electrons which have not reached the ring collector are captured in an intermediate collector, and secondary electrons (like in a conventional collector such as a "Faraday cup" [6]), are captured by a rather deep potential minimum . The power Won the intermediate collector, the latter being at potential Ui , , is equal to: W = Uint Lint
To decrease this power, one has to maximally reduce Uint limited by the space charge . In experiments, Üint was reduced down to the values at which the perveance a reached 20 [.AN 3 /2, i.e. was close to the pint = I/Dint limiting perveance of the Faraday cup. In the recuperator with an intermediate collector the losses AI on the anode were considerably smaller and the collector perveance was noticeably greater. The perveance P, is approximately 20 times greater Table 1 Maire experimental results and beam parameters I=3 .2A Ui ,=3kV U~=400V Pi,,, = 20 FL A/V 3i2 P,. _ I/ U 3/2 = 400 W A/V 3/2 Iint /I = 4% AI/1= 3 x 10-a Ho = 0 .09 T
In experiments a 30 mm diameter emitter was used. For this emitter the thickness of the transformed beam h is approximately 10 times smaller than the characteristic dimensions of the collector (h - 5 mm, A R - 50 mm, L - 50 mm). As a result, the experimental value of the perveance is considerably smaller than the maximum one for a ring collector with the given dimensions . The main results are : (a) A recuperator with a beam transformation from cylindrical into disk-shaped permits one to substantially increase the collector perveance, as compared to the Faraday cup . (b) With the introduction of the intermediate collector, practically all the electrons that escaped the main collector may be captured and, thus, the anode current loss rate is decreased down to values of 3 x 10 -4.
Acknoviedgement The authors acknowledge fruitful discussions with I.N. Meshkov and the assistance in calculations of T.N . Andreeva.
References [1] V.I. Kudelainen, I.N. Meshkov and R.A. Salimov, INP Novosibirsk preprint 72-70 (1970); Preprint CERN 77-08 (1977) . [2] T. Ellison et al., IEEE Trans . Nucl. Sci . NS-30 (1983) 2636. [3] M. Bell et al., Nucl. Instr. and Meth. 190 (1981) 237 . [41 V. I. Kokoulin, l .u. Meshkov ar ûiarâjrâ, -vv . .1. Tech . Phys. 50 (1980) 1475. [5] P.T. Kirstein, G.S. Kino ar:d W.E. Waters, Formation of Electron Beams (Mir, Moscow, 1970) pp. 102-104. [6] A.I. Arenshtam, I .N. Meshkov, V .G. Ponomarenko, R.A. Salimov, A.N. Skrinsky, B.M. Smimov and V.G. Fainshtein, Sov. J . Tech. Phys. 41 (1971) 323. [7] A.N. Sharapa, Candidate Thesis, Novosibirsk (1982) p. 124.