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Boundary integral method applied to the Rhodotron accelerator cavity
J. Bassaler, H. Baudrand
To cite this version:
J. Bassaler, H. Baudrand. Boundary integral method applied to the Rhodotron accelerator cavity.
Journal de Physique III, EDP Sciences, 1993, 3 (3), pp.573-580. �10.1051/jp3:1993150�. �jpa-00248942�
Classification Physics Abstracts
41.90 02.60
Boundary integral method applied to the Rhodotron accelerator cavity
J. M. Bassaler (I) and H. Baudrand (2) (1) CEA/DEIN, Centre d'Etudes de Saclay, France
(2) Groupe de recherches en micro-ondes, ENSEEIHT, Toulouse, France
(Received J7 March J992, revised J0 December J992, accepted JJ December J992)
Rksumk. Nous analysons la cavit£ d'un acc616rateur d'dlectrons particulier par les 6quations int£grales de flontibre. Los 61ectrons sont acc£I£rds plusieurs fois dans le plan m£dian d'une cavit£
coaxiale r£sonnant sur son mode fondarnental TEMI. Pour am£liorer l'imp£dance shunt, la g£om£trie des conducteurs est modifide. Le calcul de la frdquence propre du nouveau r£sonateur fait appel h la fonction de Green associ£e h la cavit£ coaxiale initiale. Les r£sultats num£riques, qui
n'irnpliquent que des matrices de petite taille, sont en bon accord avec la mesure et un calcul aux
£I£ments finis.
Abstract. The cavity of a particular electron accelerator is analysed thanks to boundary integral equations. Electrons are accelerated several times in the median plane of a single coaxial cavity resonating on its fundamental TEMj mode. The geometry of the conductors is modified to improve
the shunt impedance. The proper frequency of the new resonator is calculated thanks to Green's function associated to the initial coaxial cavity. Numerical results involving small size matrix
operations agree well with the measured values and those given by a finite element code.
1. Features of the Rhodotron accelerator.
1.I PuRPosEs OF THE ACCELERATOR. A new type of recirculating electron accelerator
invented by Pottier [I] is under test at Commissariat ~ l'Energie Atomique in Saclay. It has been designed with care of compactness, reliability and economy so as to suit industrial
applications such as food preservation, medical disposals sterilization or polymerization. Such
a device can deliver a 10-200 kW continuous beam in the 1-20 MeV energy range.
1.2 ACCELERATION IN A coAxiAL CAVITY. We use a /2 coaxial line short circuited at both ends, resonating on its TEMI fundamental mode. In the median plane the magnetic field vanishes whereas the electric field is radial and maximum. Hence a charged particle can be
accelerated along a diameter without being thrown out of this plane.
1.3 RECIRCULATION IN A SINGLE cAviTY. The energy gain of an electron crossing the cavity
is proportional to the voltage in the median plane between both inner and outer cylinders, whereas the R-F- losses in the walls are proportional to the square of this voltage. However the
574 JOURNAL DE PHYSIQUE III N° 3
output energy of the particles can be increased without additionnal R-F- power provided that
they are accelerated several times in the same cavity. To achieve this scheme magnets bend the beam after each pass and send it back along another diameter. The trajectory has a rosaceous
shape as shown in figure I. Moreover this device has self-focusing properties.
c
L
Fig. I.- Median section of the accelerator and electron trajectory. D: Deflecting magnet; C:
Accelerating cavity L : Magnetic lens ; G Electron gun.
1.4 ACCELERATION AND SYNCHRONISM RELATIONS. This scheme is well suited to electron
acceleration since they reach quickly a speed very close to light celerity. This is true except in the first pass fed with low energy electrons (several tenths of kev) delivered by a gun.
Assuming the particle speed is constant and equal to light celerity c allows an easy integration
of the energy gain along a diameter :
wR~ wR~
AW
= 2 (e Uo sin ~bo S~ S~ with ~bo ~ wto (1)
C C
~
S~(x)
= l~ ~~~ ~ du ; and the electric field is E~(r, t) =
~°
cos wt.
~ u r
The energy gain is maximum if the electron reaches the cavity center at the same time the electric field vanishes (~b~ = 90° ). Hence the drift length in each deflecting magnet is designed
so as to fulfil this synchronism condition. Because of their finite speed, the particles see the electric field varying in time, therefore the effective voltage V~t~ is lower than the peak voltage I available in the median plane;
we can also define a transit time factor F~r. V~~~, I and F~r
are given by the following relations :
AW(~bo = 90° ) R~ V~~~
V~~~ = V
=
2 Uo In F~r
= ~ l (2)
(e Ri #
1.5 SHUNT IMPEDANCE. The efficiency of an accelerator is its ability to provide a given accelerating voltage with a minimum of power P~ lost in the cavity walls. The usual factor of merit is the effective shunt impedance Zs~r [2] the subscript TT means that it takes into
account the transit time factor, contrary to the (geometrical) shunt impedance Zs which is a
purely RF concept.
V~~
~
~~~ f2
Pj' ~~~ ~~~~' ~~ Pj ~~~
Since the TEMI frequency depends on the cavity height h only, both conductor radii can be chosen to get the best effective shunt impedance as shown in figure 2. In fact the need of room for setting magnetic deflectors yields an extemal radius smaller than that giving the optimum
efficiency, I-e- R~ = 0.48 A.
o.~ o.
~
~~< /<l~i ,o
~
~'l~°~~~°
~~< Ii ~'~~
o
,1,<~°~~~°
~~ ,,
~
a) b)
Fig. 2. Shunt impedance (a) and effective shuntimpedance (b) versus inner and outer cylinder radius.
1.6 ADVANTAGES OF A MODIFIED coAxiAL cAviTY. Since the main losses in the cavity
walls are located at both ends of the inner cylinder, this cylinder has been opened out as
pictured by figure 3. The proper frequency and the shunt impedance of this modified cavity are not analytic. In the next part of this paper, both quantities are calculated owing to an integral
method.
2. Boundary integral method.
2, I BASES OF THE METHOD. Unknown fields must not be calculated in the volume but only
on the cavity walls [3]. Moreover using the axial symmetry converts the problem into one dimension.
2.2 FIELDS SHAPE. Since the modified cavity has an axial symmetry and since any
azimuthal magnetic field fulfills the boundary condition on the biased surface, we can guess that the fundamental mode possesses an axial symmetry and that E~, H~ and H~ are zero
(Magnetic and electric lines are shown in Fig. 4).
576 JOURNAL DE PHYSIQUE III N° 3
I< ~p
Fig. 3. -Longitudinal section of modified cavity.
a) b)
Fig. 4. a) Electric field lines b) Magnetic field lines.
2.3 UNKNOWN FuNcTioN. In these previously described Transverse Magnetic modes, the
electric field components are linked to the magnetic field derivatives. So we take
H(r, z)
= H~(r, z) as unknown function : E~ =
~ ~~~
E~
=
~ ~ (rH~). (4)
w so az w so r ar
The magnetic field must check the following equation :
~~~~~~~[~~~~~
ls~s~ii~~~i)~~~'
~~~
The boundary conditions yield : aH~
= n grad H
~ = 0 on the path e (see Fig. 5)
an
I I y
~ y
O
~ t t'
Fig. 5. Int£gration paths.
2.4 REDUCTION To A ONE-DIMENSION EQUATION. Green's function associated to the
operator £ is defined by :
£(G(r, z))
= (r ro) 3 (z zo)
Let us use Green's identity :
I(HAG-GAB)dS= I(HgradG-GgradH).ndi.
s e
Instead of taking the infinite extension Green's function [4], let us add the boundary condition
n grad G
=
0 on the path e' corresponding to the coaxial cavity. This additionnal condition increases the advantages of the general method [3j, since the field anywhere inside the
resonator or on its boundary can be expressed as a function of the field lying on the truncated part only.
C (r, z H (r, z)
=
lH(ro,
zo) grad G (r, z, ro, zo) n ro die (6)
AB U CD
where C (r, z)
=
I in the volume and la on the boundary.
578 JOURNAL DE PHYSIQUE III N° 3
2.5 EXPRESSION OF GREEN'S FUNCTION. It expands as a linear combination of the
TEM~ and TMO
~ ~
modes of the coaxial cavity :
~~~~ ~~ ~~~ ~~
f f f~P~
B (r) B (r ) C°S
~~ (~ ~
~
~~~
~~ ~~
~
~
~~~
h
~ ~~ ~ ~6~~~~~ ~ ~
~ ~ ~ ~ ~
(n, p) # (o, o)
f&( ~
~~B((~) ~ d~ Ak(p "
ki k)n ~i ~
R~
f (0) = f ~p # 0)
=
and Bo(r)
= , k~~ = 0
,
B~(r) = A~ Ji(k~ r) + B~ Yi (k~ r)
and A~, B~ and k~ are chosen according to the boundary conditions [5].
2.6 EIGENVALUE EQUATION. We divide the segment CD into N~ sub-segments on which
the unknown function is supposed to have a constant value. Searching separately even and odd modes allows an implicit inclusion of the segment AB contribution. The equation links an
image point (r~, z~) located on CD to the whole source points (q, z~) located on the same segment :
H(ri, zi)
=
f f 8np Bn(ri CDS ~) (zi + (8)
n=o p
~p even or odd
+ ~'~ B~(rj) sin ~'~ z~
+
~
sin 9c H(r~,
z~) q
h h 2
The set of N~ equations can be expressed in the vector form below : (H,
= (A,~)(Hj) the
resonance frequencies are those which cancel the following determinant DET (I Ai)
= 0.
Then, knowing the eigenvector associated to each frequency permits to compute the 8~~ coefficients and therefore the fields anywhere in the cavity. Since it is impossible for the summation over a finite set of continuous functions to drop abruptly to zero beyond the truncated boundary, the reconstituted magnetic field exhibits oscillations. This prevents us
from getting correct values for its derivatives, I-e- the electric field. To overcome this
difficulty, Faraday's theorem is applied on the surface So bounded by Co (see Fig. 5). The
magnetic flux through So is proportionnal to the circulation of the electric field along Co which is the voltage between the conductors in the median plane (the tangential electric field vanishes on the boundary). The losses are proportional to the integral of r H~ along e. Then it is straightforward to get Zs, and its product with F[ gives Zsrr (we assume the transit time
factor is unchanged, I-e- the accelerating field is still varying as I/r).
3. Results.
3. I COMPARISON WITH FINITE ELEMENT cALcuLus AND EXPERIMENT. We have tested our
code with the geometry of our prototype (R~ = 0.1125 m ; R~ = 0.45 m ; h
= 0.916 m Z~ = 0.298 m ; 9~ = 34.4° f = 178.9 MHz) ; its truncature causes a 5 to lo fb upshift on the
first TEM and TM frequencies as shown in table I. The error on the TEMI frequency, defined
as the relative difference with the value computed by the finite element code SUPERFISH [6j, is plotted in figure 6 versus the number of modes (N~~.N~~) and the number of sub- segments (N~). It exhibits a rapid convergence with the use of small size matrices (N~ =
20 ). Both computed shunt impedance and quality factor agree within a few percent with
those given by SUPERFISH (Q = 36000 Zs =14.9 MQ for a copper cavity). We also
predict with a good accuracy higher order mode frequencies and compare them in table I with the values calculated by the other method and those measured.
5
~ 4
~ 3
2
fl
o
°
~
0
° -1 x j
~ j 1
/j -2
0 o
4
-5 0
0 20 40 60 80 100
Nmax
Fig. 6. Accuracy versus number of modes and segments ; o N~ = 5, D N~ = lo * N~ = 20.
Table I. Resonance frequencies of the initial cavi~y (first row ), and of the truncated cavi~y (other rows) calculated with finite elements, integral equation and measured.
~~010 ~~011
Initial Analytic 163.6 327.3 490.9 434.5 4.3 44.0
Trunca- Superfish178.9 344.2 544.2 450.9 465.3 79.8
ted
Green 178.8 343.7 543.7 451.2 80.3
178.9 343.8 544.0 450.1 63.8
3.2 MEMORY SPACE AND COMPUTING TIME. To compare the computer requirements, we
assume that a surface described by N x N points in Superfish [6] will have its boundary
divided into N sub-segments in the integral method.
Memory space Time
SUPERFISH (N~ points) oz N~ oz N~
Green (N sub-segments, N modes) oz N~ oz N~.N$
580 JOURNAL DE PHYSIQUE III N° 3
3.3 SHUNT IMPEDANCE OPTIMIZATION. Our method is very efficient to find out the shape
which provides the best shunt impedance at a given frequency. First of all, a mesh generator is not required, then the program itself can find out the right geometry which resonates at the fixed frequency. For example, if the couple of values (Z~, 9~)is given, the program makes the
height h vary until the determinant vanishes. By varying the parameters Z~ and 9~ and applying
this procedure, we have concluded that the shunt impedance has a very smooth maximum (an improvement of lo fG) around Z~ = 0.32 m and 9~ = 35°.
Conclusion.
This boundary integral method provides accurate results within a small memory space. It allows the optimization of a coaxial cavity shape when a good shunt impedance is required in a
Rhodotron accelerator.
References
[I] POTTIER J., A new type of RF electron accelerator the Rhodotron Nucl. Jnstr. Meth. Phys. Res.
B 40/41 (1989) 943.
[2] LAPOSTOLLE, SEPTIER, Linear Accelerators, North Holland Publishing Company, Amsterdam (1970).
[3] AuRioL Ph., KRAHENBUHL L., NicoLAs A., NicoLAs L., SABONNADItRE E., Moddlisation
tridimensionnelle des champs £lectromagndtiques en r£gime quasi stationnaire par £quations int£grales de frontibre, Joum£es d'£nudes SEE M£thodes d'£quations int£grales en £lectroma- gndtisme (24 avril 1990, Gif-sur-Yvitte).
[4] TsucHimoTo M., HONMA T., YONETA A., Boundary element analysis of axisymmetric resonant cavities, IEEE Trans. Magn. 24 (1988) 2500.
[5] WALDRON R. A., Theory of Guided Elec~omagnetic Waves (Van Nostrand, London, 1969).
[6] HALBACH K., HOLSINGER R. F., Superfish( a computer program for evaluation of RF cavities with
cylindrical symmetry, Particle Accelerators 7 (1976) 213.
