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The electron injection process in the betatron
NUCLEAR
INSTRUMENTS
AND
METHODS
26 (1964) 173--167;
© NORTH-HOLLAND
PUBLISHING
CO.
THE ELECTRON INJECTION PROCESS IN TI-IE BETATRON G. B A C I U * Institutul de fizica atomica, Academia R . P . R . , Bucuresti R e c e i v e d 26 J u n e 1963
I n this paper, t h e dependence of the injection process of t h e electrons in t h e b e t a t r o n on t h e p o t e n t i a l f u n c t i o n which characterize t h e m a g n e t i c field forces, is described. T h e process connected w i t h t h e existence of b o t h a v a r i a b l e circulating c u r r e n t in t h e accelerating c h a m b e r a n d w i t h t h e spatial c h a r g e due to t h e injection of electrons w i t h different initial
energy, are analyzed. These processes are used in order to e x p l a i n t h e c a p t u r e of electrons in t h e accelerating process a n d which t h u s a v o i d t h e electron g u n structure. T h e e x p e r i m e n t a l m e t h o d used for t h e practical s t u d y of these processes, is indicated.
1. Introduction
In order that electrons be captured in the betatron accelerating process it is necessary that its injecting energy corresponds to the magnetic induction value :
An electron gun of triode structure is used for injecting electrons in the betatron. At the gun electrodes a potential pulse obtained by discharging an artificial pulse former line (LF), through a pulse transformer (TI), is applied (see fig. 1). G
LF
-=
'
TI.
A
T~l
I
1
c
Ui = 2 "~
(~'0" e o ) 2
°
(l)
The practical value of any betatron is characterized by the magnitude of the current of accelerated electrons (circulating current) or by the bremsstrahlung intensity which is proportional to it. The magnitude of the circulating current depends on the number of captured electrons. In order to determine the best electron injection conditions in the betatron so as to insure an efficient electron capture, let's analyze the principle of this process.
Fig. 1. E l e c t r o n injection circuit. G + L . F . : t h e c h a r g e circuit: L.F. : t h e f o r m e r line; L.F. + T + T.I. : t h e d i s c h a r g e circuit ; T .I.: pulse t r a n s f o r m e r ; T.P. : t h e circuit for s y n c h r o n i z a t i o n w i t h t h e m a g n e t i c field; A : t h e t r i g g e r circuit.
2. The Analysis of the Injection Process The electron motion in the betatron is determined by the combined forces due to the time dependent magnetic f i e l d - a n d induced electric field.
A rectangular pulse is formed in the primary of the pulse transformer. Owing to the existence of the dispersion inductance and the distributed capacitance of the pulse transformer coils, a deformed potential pulse is formed in its secondary coil (natural pulse).
F ° ~" d t (mv°r) =
-
er
E o + -(v~B, C
-
v,Bz)
(2)
where * N o w a t the " I s t i t u t o di Fisica dell'UniversitY, di T o r i n o " . 1) A. M. A n a n y e v , A. A, V o r o b y o v and V. I. G o r b u n o v , Betatron-electron induction accelerator (in russian) (Gossatom y z d a t , Moscow, 1961). 3) B. ]3. Galperin et al., P r i b o r y i T e h n i k a E x p e r i m e n t a (1960) nr. 4, 13.
Fig, 2. N a t u r a l injection pulse form. 173
174
G. B A C I U
r Ol~= . Eo = - 27c " at ' 2
& gr 2
r2
J
r.B=dr
Jo
•
(3)
S t a r t i n g from t h e e q u a t i o n 1 O
1 ~B o
divB = rot (r.B,) + r~-
OB~
+~zz
V~= ~gmc
= 0
we m a y express B, and B= as a function of B,, considering t h a t 1 OB o = 0 , r O0
B, = - ½ r ~ - z
;
r e ~ [ [½r Dlfi,, 0~- + z2~-ffz'r~B~+ ~ ( / ~ = + { r ~ r ~ ) ]
d d t (my°r) --- ~c d (r2 B~)
I n t e g r a t i n g the last eq., we o b t a i n the solution expressing the electron velocity as a function of the averaged m a g n e t i c field i n d u c t i o n : e
2-
rover = 2cc r B~ + D 1
(4)
where D~ is the i n t e g r a t i o n c o n s t a n t a n d r e p r e s e n t s the difference between the m o m e n t u m of electrons which are m o v i n g on the circular orbit of radius r a n d the m o m e n t u m impressed to the electrons b y the accelerating electric field. Substituting D =e~ D t
we o b t a i n vo = Ymm~ The kinetic energy of the electrons will be: W
=
---
2mc z
The stable m o v e m e n t of electrons is d e t e r m i n e d b y the m i n i m u m of the p o t e n t i a l function. Using the superposition m e t h o d we can analyze t h e elect r o n m o v e m e n t i n d e p e n d e n t l y in t h e directions r a n d z. We are i n t e r e s t e d in the radial v a r i a t i o n of the p o t e n t i a l function. Or -- mc
Oz
T h e n we o b t a i n : d
Because the injection process takes place in a short time-period (a few tens of electron turns), t h e magnetic field v a r i a t i o n is negligeable in this timeinterval a n d therefore the electrons are m o v i n g in a quasi-conservative force field characterized b y the potential function :
½[~" + ~r e-~ -- ½D
= 0
Therefore, the p r o b a b i l i t y of the existence of an extreme value for the p o t e n t i a l function is determined by : ~3B= D ½/~= + ½r Or - ~r = 0 D = r2(B= - ½Bz) "
(6)
F r o m this relation it is clear t h a t for t h e real case D # 0, the accelerating orbit radius m u s t be also variable and therefore depends of time (ins t a n t a n e o u s orbit). This orbit is stable because it coincides w i t h t h e m i n i m u m of the corresponding p o t e n t i a l function. The electron which deviates from this orbit for some reasons, begins performing the d a m p e d oscillations a n d at the end will r e t u r n to the equilibrium orbitl,2). The electron injector is located in the vicinity of the equilibrium orbit, generally in t h e plane z = 0 a n d at r i > ro for practical reasons. Because r i ~ v o / B z, it becomes clear t h a t for v o = ct a n d increasing B : from zero, the radius of the injected electrons orbit changes from a s t r a i g h t line to a circle. I n other words, increasing the magnetic induction B:, the injected electrons in the b e t a t r o n will move on a t r a c k whose radius bea) D . ~,V. K e r s t a n d R . S e r b e r , P h y s . R e v . 6 0 (1941) 53. 4) D , W . K e r s t , P h y s . R e v . 7 4 (1948) 503.
THE ELECTRON
INJECTION
comes smaller till a d e t e r m i n e d i n s t a n t w h e n the radius will equal r i. Then, the electrons will begin to move s t a b l y on this orbit because it coincides with its value at the p o t e n t i a l m i n i m u m . I n the following i n t e r v a l of time the electrons will move on a n orbit whose radius will decrease monotonically from r i to r o. Continuing to increase the m a g n e t i c field, the electrons t r a c k will r e p r e s e n t curves w i t h decreasing radius a n d therefore the injected electrons in this i n s t a n t of time will strike the i n n e r of the doughnut. As a consequence, the electrons which are injected in the d o u g h n u t m a y be c a p t u r e d in the accelerating process only during a d e t e r m i n e d time interval. F r o m this follows the use of electron pulse injection w i t h a d u r a t i o n of t h e pulse equal to the capture t i m e interval. For B z = B~m sin f2t and v o --- 5.93 x 107 4Ui we can d e t e r m i n e t h e injection m o m e n t as a function of t h e injection voltage a n d of t h e magnetic field value in this i n s t a n t of time. At t h e injection sin f2t ~ f2t when ti
3.37' x/ul " r~ • B= • f2
(7)
F r o m here it follows t h a t , for a definite increase of the m a g n e t i c field t h e r e m u s t correspond a n increase of t h e injection voltage, according to a parabolic law. An identical relation is o b t a i n e d for the electron injection in the equilibrium o r b i t : 3.37 • V/ul
(8)
t i"°' = r~" B~m " ~
"
On the basis of relations (7) a n d (8) we can conclude t h a t in t h e b e t a t r o n , for a given value of the injection voltage, there is a d e t e r m i n e d injecting i n t e r v a l for the electrons: = 3.37" x/u-i(_ All = ti(~o, - ti(.,,
C2
1
1
)
\ro "BmO ri "Bmi
(9)
PROCESS
IN THE BETATRON
175
F r o m the relation (7) there also follows t h a t the injection interval will increase w i t h increasing injection time ( t h a t is w i t h t h e d e l a y of injection relative to the time w h e n t h e magnetic field passes t h r o u g h zero). I n order to get a clearer picture of this process we can illustrate it graphically:
"l rl /to Pl--
I ~ t~(rll
i
injectionpLLsIe ti(ro),~ ~
- t
Fig. 3. Acceptance curves.
The region comprised between the acceptance curves r i a n d r 0 represents t h e injection domain. As a consequence, in order to inject electrons in the b e t a t r o n it is necessary to d e t e r m i n e a n agreem e n t between three q u a n t i t i e s : 1. the injection i n s t a n t (tl), 2. the initial energy of electrons (Ui), 3. the magnetic field value (B,) in t h e injection i n s t a n t . These all for a given dimension (r i a n d r0) of the accelerator. The possibility of electron injection w i t h different initial energies leads to a spatial charge acumulation in the c h a m b e r during the capture period4'S). As a consequence on the electrons acts the coulombian repulsion force. I n proportion as the c a p t u r e d electrons are grouping a r o u n d the equilibrium orbit, these forces increase. On the o t h e r side, in directions r a n d z , on electrons act the focusing forces of the magnetic field, whose decrease with the reduction of the electron distance from the equilibrium orbit, c o n t r a r y to the repulsion forces whose increase at the same time. The existence of these forces of repulsion a m o n g the electrons leads to the appearing of a s u p p l e m e n t ary potential field of force, for whose one can find the corresponding potential function Vo. This p o t e n t i a l function d e t e r m i n e s the forces directed against the magnetic field forces a n d therefore the s) I~. Wideroe, Z. Angew. Phys. 5 (1953) 187. 6) S. E. B a r d e n Proc. Phys. Soc. L o n d o n B-64 (1951) 579. 7) y . S. Korobotshko, J o u r n . Techn. P h y s . 27 (1957) 1603.
176
G. B A C I U
p o t e n t i a l force field in t h e b e t a t r o n , in t h e presence of the spatial charge, is d e t e r m i n e d b y the r e s u l t a n t p o t e n t i a l force V~ = V m - Ve. The p o t e n t i a l function V~ depends on the spatial charge value a n d of its dimension. I n proportion as the n u m b e r of electrons in the b e a m increases a n d its cross section is reduced, V e increases. A t a certain i n s t a n t , w h e n Ve=Vm,
V~=O.
I t results t h a t , a t a d e t e r m i n e d value of t h e spatial charge a n d its spatial distribution, the focalizing forces of t h e magnetic field will be comp e n s a t e d b y the repulsion forces. The subsequent increasing of the charge due to the increase of the n u m b e r of electrons, becomes impossible. As a consequence, in the b e t a t r o n t h e r e is a restricted spatial charge. This charge is characterized b y the focalizing forces of t h e magnetic field i.e. b y the p o t e n t i a l function Vm. The restricted spatial charge which can be m a i n t a i n e d b y the b e t a t r o n m a g n e t i c field in the equilibrium orbit, is d e t e r m i n e d b y t h e following considerations : We m a y assume t h a t the spatial charge represents the accelerated electron b e a m in t h e zero potential well with a uniform d i s t r i b u t i o n of the electrons along the equilibrium orbit. The electron b e a m has a toroidal form w i t h a definite cross section. At the surface of the b e a m the focalizing forces of the magnetic field are equal to t h e Coulombian forces : 5qVm
Or
ONe. 0 g m
0V e
~r'
Oz
~z
The b e a m is limited b y a n e q u i p o t e n t i a l surface for which the following relations are valid: OVe dV: = ~ d r
~V e
+ ~dz
or
dV m
OVmdr =
~r
~z
~Ve/Cqr d~ = -- O V e / ~ =
Vm0 = ~
e
n~z2]
(11)
ro2
(~'0 ' B0z) 2 ; Ar = r -- r o .
T h e n the cross section border-line e q u a t i o n after t h e i n t e g r a t i o n will be: z2 (l - - n)/n(Arl) 2
Ar 2
+ ~
= 1
(12)
Ari is d e t e r m i n e d b y the initial conditions a n d m a y be t a k e n as Ari = ri - ro.
Expression (12) represents t h e e q u a t i o n of a n ellipse with semiaxis (r i -- t o ) a n d (r i - to) X/(1 - u ) / n . At Ar = 0 the m a j o r axis of the b e a m cross section is 2(r i - to) a n d the m i n o r is 2(r i - ro) ~/(1
- -
n)[n.
The t o t a l b e a m charge for a given b e t a t r o n is found from t h e produce of t h e b e a m volume a n d charge d e n s i t y in the b e a m : Qm = 2Uro(ri -- to) 2 x/( 1 -- n ) / n . p
(13)
The a c u m u l a t i o n of the spatial charge takes place during the first few t u r n s of the injected elect r o n s in the accelerating chamber. W i t h each following t u r n the charge increases due to the injection of m o r e electrons. I t follows t h a t a current which increases in t i m e is created in the accelerating c h a m b e r . The value of the circnlating current at a given m o m e n t is determined b y the charge Q a n d the electrons velocity v : Q
•
(14)
"
The e q u a t i o n for the m o t i o n of one electron on the equipotential surface is t h e n : dz
where
Vm=Vmo[1 + l--nAr2+ 2 go
7) ie = ~ .
c~V,~dz +
netic field p o t e n t i a l function V~.. The limits of the spatial charge d e p e n d on the p a r a m e t e r s of the zero p o t e n t i a l well which is given u n d e r the form of a paraboloid:
6qVm/Or -- OVm/~,g
(10)
The form of the cross section of t h e b e a m is d e t e r m i n e d b y the spatial d i s t r i b u t i o n of the mag-
The increase in the current d u r i n g the i n j e c t i o n leads to the appearance of a n e.m.f, whose value depends on the rate of change of t h e c u r r e n t : di e er = -- L f - ~ "
(15
s) V. N. L o g u n o v . et al., J o u r n . Techn. P h y s . 27 (1957) 1135. 9) V. N. iRodimov, lzv. T . P . I . 87 (1957) 11.
THE E L E C T R O N I N J E C T I O N PROCESS IN THE BETATRON
In o t h e r words the v a r i a t i o n of the c u r r e n t creates a selfinduction field for which dissipates a p a r t of its energy. This leads to a larger c u r v a t u r e of the electrons t r a j e c t o r y u n d e r t h e action of t h e principal magnetic field. As a consequence t h e r e is a c o n t r a c t i o n of its equilibrium orbits, which therefore avoid the injector. I n order to evaluate the reduction of the electron radius due to this effect we m a y use t h e electron m o m e n t u m expression in the m a g n e t i c field : P = e r . Bz
(16)
c
dp _ dr + dBz p r dz
dr + ~B~/~r dr = - -dr (1
r
B~
- n) (17)
r
F r o m N e w t o n ' s second law we h a v e : dp = e- E - dt = e. 2 ~ r dt = -- e 2Lf ~ r . - ~d idet
dr =
-
e - Lf 1 di e 2x(1 n) p ~ - / d t _
•
(18)
•
( 1 9 )
The r e d u c t i o n of the radius d u r i n g a t u r n , i.e. for dt
=
To
2hr.
=--lsAr
v
=
--r
Lf 2(1 -- n) • u i
di 2
dr(O)
where d i / d t is d e t e r m i n e d b y the ratio of t h e m a x i m u m c u r r e n t injected d u r i n g the electron c a p t u r e process, a n d L e is defined as t h e i n d u c t a n c e of a single winding coil of radius r o disposed in the meddle plane of the b e t a t r o n m a g n e t i c field, symm e t r i c a l l y to the later. At the same time there is a s u p p l e m e n t a r y effect which p r e v e n t s the electrons h i t t i n g the injector due to the direct action of the spatial charge on t h e injected electrons 6- lo The electrons which are at t h e b e a m surface find themselves u n d e r the repulsion forces of the o t h e r electrons of the beam. These forces t e n d to defocalize the b e a m a n d as t h e y grow lead to increase the b e a m size. I t is the increase of the electron oscillation a m p l i t u d e t h a t limits t h e m a x i m u m value of the charge which can be accelerated in the betatron. The injected electron b e a m after one t u r n has the average density smaller t h a n the initial one. The density decrease takes place because of the increase
177
of t h e b e a m cross section a n d because a p a r t of electrons gets loss by striking t h e c h a m b e r walls. As t h e first b e a m passes t h r o u g h the injector a z i m u t h , t h e second b e a m is injected w i t h a greater density. Due to the Coulombian action b e t w e e n t h e m the center of the first b e a m is m o v e d nearer to r o. The s u b s e q u e n t simultaneous m o t i o n of the first a n d second b e a m s gives also a n increase of the t o t a l b e a m cross section a n d throws a larger n u m b e r of electrons on t h e c h a m b e r walls. At the end of the second t u r n of the first beam, the t h i r d b e a m is injected. The i n t e r a c t i o n of the l a t t e r w i d t h the t o t a l b e a m p r e v e n t s again as the electrons strike t h e g u n s t r u c t u r e , and so on. Such a m e c h a n i s m of stable b e a m f o r m a t i o n leads to t h e idea 1t (in fact practically confirmed) t h a t t h e electron capture of different parts of the injection pulse has a different efficiency. Indeed, the injected electrons as the injection pulse rise (with e x t e r n a l injection) a t first fill the p o t e n t i a l wells which are nearer to r o. The oscillation of these electrons is d e t e r m i n e d b y the distance b e t w e e n the corresponding i n s t a n t a n e o u s orbit a n d rl. As the injection voltage increases the oscillation a m p l i t u d e of these electrons will be proportionally smaller. The difference in t h e oscillation amplitude leads to a corresponding difference of the charge density. Therefore, each following b e a m injected at a larger voltage m o v e s the r e s u l t a n t b e a m towards r o. If the electrons are injected as the pulse is rising the spatial charge in t h e accelerating c h a m b e r increases c o n t i n u o u s l y due to the i n t r o d u c t i o n of new electrons. If t h e y are injected as the pulse decreases the charge becomes smaller due to the loss of electrons on the c h a m b e r walls (small injection energy) also to t h e decrease of the n u m b e r of injected electrons. T h e n o n u n i f o r m a z i m u t h a l charge d i s t r i b u t i o n is due to t h e fact t h a t the m a j o r p a r t of the i n j e c t e d electrons fall on t h e c h a m b e r walls just near t h e injector since, generally, the injector gives a diffuse beam. As a result of t h e p r i m a r y electron b o m b a r d m e n t of c h a m b e r s is wall there is secondary emission. T h e r e arives in t h e c h a m b e r a larger charge t h a n the 10) V. N. Logunov and S. S. Semenov, Jourrl. Exp. Teor.
Phys. 33 (1957) 1513. 11) M. Seidl, Journ. Exp. Teor. Phys. 36 (1959) 1305.
178
G. B A C I U
injected one. The electron interaction with the (fastly decreasing) azimuthal nonuniform charge, the electron energy is reduced. As a matter of fact, the electrons which approach the charge feel its repulsion action, and therefore avoid the injector. 3. Experimental Study of the Injection Process
In agreement with the above description as the electrons are injected in the accelerating chamber there arives an electron circulating current which creates a magnetic fluxq, e. The size of this magnetic flux is directly proportional to the electron current and therefore it is also proportional to the charge Q, captured in the betatron accelerating process. ~ = k~.Q
•
(21)
If in the vicinity, a coil embraces the doughnut in it will appear an induced voltage of fast variation : E ~- k 2 d t
= kt.k 2
(22)
"E" will exist when dQ d~ ¢ 0 that is during the capture or loosing of the electrons. Using a RC filter one m a y eliminate the 50 Hz voltage induced by the main field and put in evidence the high frequency one in which we are interested. Applying the voltage "E" to the oscilloscope vertical plates and following the curve E = f(t) we can study the electron current variation or integrating - d i r e c t l y the charge variation in the accelerating chamber :
capture efficiencyincrease with the injected current, so that the circulating current varies as the injected one to a power greater than unity 14. In order to improve the output of a betatron one can make use of two classes of methods: the correction of magnetic field distribution and supplementary damping d e v i c e s - t h e so called contractors; it follows from the existence of two different capture mechanisms: one non-collective, effective at low injected currents, and another collective, effective at high injected currents, but limited by saturation. The non-collective capture shows a dependence upon injection voltage as predicted by Kerst and Serber3). The contractor circuit contributes v e r y effectively to this non-collective capture, the efficiency being greater for low currents and high voltages. The collective capture becomes the main capture process, for high injected currents. Unlike the non-collective process, the collective one is very much affected by the field distribution at injection. The two methods of increasing the yield are not antagonistic; on the contrary; the combined efficiency is greater than the product of the efficiencies of the two methods applied separately. In other words, the contractor efficiency is greater when field corrections are applied. F r o m a practical point of view, the principal matter is of course the radiation intensity. Although the gain in intensity resulting from the application of these two methods is sensible and the research in this direction is of course worth-while.
(23)
The author is indebted to the Bucharest betatron crew for their help and for useful discussions and particularly to Prof. F1. Ciorascu for his kind interest in this work and for many encouragements.
The experimental evidence available 1z'13) demonstrate that the damping of oscillations and the
1-°) \V. F. \Vestendorp, Phys. Rev. 76 (1949) 445. ts) j. N. L o b a n o v et aI., C E R N S y m p o s i u m IReports (1956) 484. a4) G. Davies, J. Sci. Instr. 36 (!959) 306.
1
Q-
kl k2
f E(t) dl •
4. Discussion
INSTRUMENTS
AND
METHODS
26 (1964) 173--167;
© NORTH-HOLLAND
PUBLISHING
CO.
THE ELECTRON INJECTION PROCESS IN TI-IE BETATRON G. B A C I U * Institutul de fizica atomica, Academia R . P . R . , Bucuresti R e c e i v e d 26 J u n e 1963
I n this paper, t h e dependence of the injection process of t h e electrons in t h e b e t a t r o n on t h e p o t e n t i a l f u n c t i o n which characterize t h e m a g n e t i c field forces, is described. T h e process connected w i t h t h e existence of b o t h a v a r i a b l e circulating c u r r e n t in t h e accelerating c h a m b e r a n d w i t h t h e spatial c h a r g e due to t h e injection of electrons w i t h different initial
energy, are analyzed. These processes are used in order to e x p l a i n t h e c a p t u r e of electrons in t h e accelerating process a n d which t h u s a v o i d t h e electron g u n structure. T h e e x p e r i m e n t a l m e t h o d used for t h e practical s t u d y of these processes, is indicated.
1. Introduction
In order that electrons be captured in the betatron accelerating process it is necessary that its injecting energy corresponds to the magnetic induction value :
An electron gun of triode structure is used for injecting electrons in the betatron. At the gun electrodes a potential pulse obtained by discharging an artificial pulse former line (LF), through a pulse transformer (TI), is applied (see fig. 1). G
LF
-=
'
TI.
A
T~l
I
1
c
Ui = 2 "~
(~'0" e o ) 2
°
(l)
The practical value of any betatron is characterized by the magnitude of the current of accelerated electrons (circulating current) or by the bremsstrahlung intensity which is proportional to it. The magnitude of the circulating current depends on the number of captured electrons. In order to determine the best electron injection conditions in the betatron so as to insure an efficient electron capture, let's analyze the principle of this process.
Fig. 1. E l e c t r o n injection circuit. G + L . F . : t h e c h a r g e circuit: L.F. : t h e f o r m e r line; L.F. + T + T.I. : t h e d i s c h a r g e circuit ; T .I.: pulse t r a n s f o r m e r ; T.P. : t h e circuit for s y n c h r o n i z a t i o n w i t h t h e m a g n e t i c field; A : t h e t r i g g e r circuit.
2. The Analysis of the Injection Process The electron motion in the betatron is determined by the combined forces due to the time dependent magnetic f i e l d - a n d induced electric field.
A rectangular pulse is formed in the primary of the pulse transformer. Owing to the existence of the dispersion inductance and the distributed capacitance of the pulse transformer coils, a deformed potential pulse is formed in its secondary coil (natural pulse).
F ° ~" d t (mv°r) =
-
er
E o + -(v~B, C
-
v,Bz)
(2)
where * N o w a t the " I s t i t u t o di Fisica dell'UniversitY, di T o r i n o " . 1) A. M. A n a n y e v , A. A, V o r o b y o v and V. I. G o r b u n o v , Betatron-electron induction accelerator (in russian) (Gossatom y z d a t , Moscow, 1961). 3) B. ]3. Galperin et al., P r i b o r y i T e h n i k a E x p e r i m e n t a (1960) nr. 4, 13.
Fig, 2. N a t u r a l injection pulse form. 173
174
G. B A C I U
r Ol~= . Eo = - 27c " at ' 2
& gr 2
r2
J
r.B=dr
Jo
•
(3)
S t a r t i n g from t h e e q u a t i o n 1 O
1 ~B o
divB = rot (r.B,) + r~-
OB~
+~zz
V~= ~gmc
= 0
we m a y express B, and B= as a function of B,, considering t h a t 1 OB o = 0 , r O0
B, = - ½ r ~ - z
;
r e ~ [ [½r Dlfi,, 0~- + z2~-ffz'r~B~+ ~ ( / ~ = + { r ~ r ~ ) ]
d d t (my°r) --- ~c d (r2 B~)
I n t e g r a t i n g the last eq., we o b t a i n the solution expressing the electron velocity as a function of the averaged m a g n e t i c field i n d u c t i o n : e
2-
rover = 2cc r B~ + D 1
(4)
where D~ is the i n t e g r a t i o n c o n s t a n t a n d r e p r e s e n t s the difference between the m o m e n t u m of electrons which are m o v i n g on the circular orbit of radius r a n d the m o m e n t u m impressed to the electrons b y the accelerating electric field. Substituting D =e~ D t
we o b t a i n vo = Ymm~ The kinetic energy of the electrons will be: W
=
---
2mc z
The stable m o v e m e n t of electrons is d e t e r m i n e d b y the m i n i m u m of the p o t e n t i a l function. Using the superposition m e t h o d we can analyze t h e elect r o n m o v e m e n t i n d e p e n d e n t l y in t h e directions r a n d z. We are i n t e r e s t e d in the radial v a r i a t i o n of the p o t e n t i a l function. Or -- mc
Oz
T h e n we o b t a i n : d
Because the injection process takes place in a short time-period (a few tens of electron turns), t h e magnetic field v a r i a t i o n is negligeable in this timeinterval a n d therefore the electrons are m o v i n g in a quasi-conservative force field characterized b y the potential function :
½[~" + ~r e-~ -- ½D
= 0
Therefore, the p r o b a b i l i t y of the existence of an extreme value for the p o t e n t i a l function is determined by : ~3B= D ½/~= + ½r Or - ~r = 0 D = r2(B= - ½Bz) "
(6)
F r o m this relation it is clear t h a t for t h e real case D # 0, the accelerating orbit radius m u s t be also variable and therefore depends of time (ins t a n t a n e o u s orbit). This orbit is stable because it coincides w i t h t h e m i n i m u m of the corresponding p o t e n t i a l function. The electron which deviates from this orbit for some reasons, begins performing the d a m p e d oscillations a n d at the end will r e t u r n to the equilibrium orbitl,2). The electron injector is located in the vicinity of the equilibrium orbit, generally in t h e plane z = 0 a n d at r i > ro for practical reasons. Because r i ~ v o / B z, it becomes clear t h a t for v o = ct a n d increasing B : from zero, the radius of the injected electrons orbit changes from a s t r a i g h t line to a circle. I n other words, increasing the magnetic induction B:, the injected electrons in the b e t a t r o n will move on a t r a c k whose radius bea) D . ~,V. K e r s t a n d R . S e r b e r , P h y s . R e v . 6 0 (1941) 53. 4) D , W . K e r s t , P h y s . R e v . 7 4 (1948) 503.
THE ELECTRON
INJECTION
comes smaller till a d e t e r m i n e d i n s t a n t w h e n the radius will equal r i. Then, the electrons will begin to move s t a b l y on this orbit because it coincides with its value at the p o t e n t i a l m i n i m u m . I n the following i n t e r v a l of time the electrons will move on a n orbit whose radius will decrease monotonically from r i to r o. Continuing to increase the m a g n e t i c field, the electrons t r a c k will r e p r e s e n t curves w i t h decreasing radius a n d therefore the injected electrons in this i n s t a n t of time will strike the i n n e r of the doughnut. As a consequence, the electrons which are injected in the d o u g h n u t m a y be c a p t u r e d in the accelerating process only during a d e t e r m i n e d time interval. F r o m this follows the use of electron pulse injection w i t h a d u r a t i o n of t h e pulse equal to the capture t i m e interval. For B z = B~m sin f2t and v o --- 5.93 x 107 4Ui we can d e t e r m i n e t h e injection m o m e n t as a function of t h e injection voltage a n d of t h e magnetic field value in this i n s t a n t of time. At t h e injection sin f2t ~ f2t when ti
3.37' x/ul " r~ • B= • f2
(7)
F r o m here it follows t h a t , for a definite increase of the m a g n e t i c field t h e r e m u s t correspond a n increase of t h e injection voltage, according to a parabolic law. An identical relation is o b t a i n e d for the electron injection in the equilibrium o r b i t : 3.37 • V/ul
(8)
t i"°' = r~" B~m " ~
"
On the basis of relations (7) a n d (8) we can conclude t h a t in t h e b e t a t r o n , for a given value of the injection voltage, there is a d e t e r m i n e d injecting i n t e r v a l for the electrons: = 3.37" x/u-i(_ All = ti(~o, - ti(.,,
C2
1
1
)
\ro "BmO ri "Bmi
(9)
PROCESS
IN THE BETATRON
175
F r o m the relation (7) there also follows t h a t the injection interval will increase w i t h increasing injection time ( t h a t is w i t h t h e d e l a y of injection relative to the time w h e n t h e magnetic field passes t h r o u g h zero). I n order to get a clearer picture of this process we can illustrate it graphically:
"l rl /to Pl--
I ~ t~(rll
i
injectionpLLsIe ti(ro),~ ~
- t
Fig. 3. Acceptance curves.
The region comprised between the acceptance curves r i a n d r 0 represents t h e injection domain. As a consequence, in order to inject electrons in the b e t a t r o n it is necessary to d e t e r m i n e a n agreem e n t between three q u a n t i t i e s : 1. the injection i n s t a n t (tl), 2. the initial energy of electrons (Ui), 3. the magnetic field value (B,) in t h e injection i n s t a n t . These all for a given dimension (r i a n d r0) of the accelerator. The possibility of electron injection w i t h different initial energies leads to a spatial charge acumulation in the c h a m b e r during the capture period4'S). As a consequence on the electrons acts the coulombian repulsion force. I n proportion as the c a p t u r e d electrons are grouping a r o u n d the equilibrium orbit, these forces increase. On the o t h e r side, in directions r a n d z , on electrons act the focusing forces of the magnetic field, whose decrease with the reduction of the electron distance from the equilibrium orbit, c o n t r a r y to the repulsion forces whose increase at the same time. The existence of these forces of repulsion a m o n g the electrons leads to the appearing of a s u p p l e m e n t ary potential field of force, for whose one can find the corresponding potential function Vo. This p o t e n t i a l function d e t e r m i n e s the forces directed against the magnetic field forces a n d therefore the s) I~. Wideroe, Z. Angew. Phys. 5 (1953) 187. 6) S. E. B a r d e n Proc. Phys. Soc. L o n d o n B-64 (1951) 579. 7) y . S. Korobotshko, J o u r n . Techn. P h y s . 27 (1957) 1603.
176
G. B A C I U
p o t e n t i a l force field in t h e b e t a t r o n , in t h e presence of the spatial charge, is d e t e r m i n e d b y the r e s u l t a n t p o t e n t i a l force V~ = V m - Ve. The p o t e n t i a l function V~ depends on the spatial charge value a n d of its dimension. I n proportion as the n u m b e r of electrons in the b e a m increases a n d its cross section is reduced, V e increases. A t a certain i n s t a n t , w h e n Ve=Vm,
V~=O.
I t results t h a t , a t a d e t e r m i n e d value of t h e spatial charge a n d its spatial distribution, the focalizing forces of t h e magnetic field will be comp e n s a t e d b y the repulsion forces. The subsequent increasing of the charge due to the increase of the n u m b e r of electrons, becomes impossible. As a consequence, in the b e t a t r o n t h e r e is a restricted spatial charge. This charge is characterized b y the focalizing forces of t h e magnetic field i.e. b y the p o t e n t i a l function Vm. The restricted spatial charge which can be m a i n t a i n e d b y the b e t a t r o n m a g n e t i c field in the equilibrium orbit, is d e t e r m i n e d b y t h e following considerations : We m a y assume t h a t the spatial charge represents the accelerated electron b e a m in t h e zero potential well with a uniform d i s t r i b u t i o n of the electrons along the equilibrium orbit. The electron b e a m has a toroidal form w i t h a definite cross section. At the surface of the b e a m the focalizing forces of the magnetic field are equal to t h e Coulombian forces : 5qVm
Or
ONe. 0 g m
0V e
~r'
Oz
~z
The b e a m is limited b y a n e q u i p o t e n t i a l surface for which the following relations are valid: OVe dV: = ~ d r
~V e
+ ~dz
or
dV m
OVmdr =
~r
~z
~Ve/Cqr d~ = -- O V e / ~ =
Vm0 = ~
e
n~z2]
(11)
ro2
(~'0 ' B0z) 2 ; Ar = r -- r o .
T h e n the cross section border-line e q u a t i o n after t h e i n t e g r a t i o n will be: z2 (l - - n)/n(Arl) 2
Ar 2
+ ~
= 1
(12)
Ari is d e t e r m i n e d b y the initial conditions a n d m a y be t a k e n as Ari = ri - ro.
Expression (12) represents t h e e q u a t i o n of a n ellipse with semiaxis (r i -- t o ) a n d (r i - to) X/(1 - u ) / n . At Ar = 0 the m a j o r axis of the b e a m cross section is 2(r i - to) a n d the m i n o r is 2(r i - ro) ~/(1
- -
n)[n.
The t o t a l b e a m charge for a given b e t a t r o n is found from t h e produce of t h e b e a m volume a n d charge d e n s i t y in the b e a m : Qm = 2Uro(ri -- to) 2 x/( 1 -- n ) / n . p
(13)
The a c u m u l a t i o n of the spatial charge takes place during the first few t u r n s of the injected elect r o n s in the accelerating chamber. W i t h each following t u r n the charge increases due to the injection of m o r e electrons. I t follows t h a t a current which increases in t i m e is created in the accelerating c h a m b e r . The value of the circnlating current at a given m o m e n t is determined b y the charge Q a n d the electrons velocity v : Q
•
(14)
"
The e q u a t i o n for the m o t i o n of one electron on the equipotential surface is t h e n : dz
where
Vm=Vmo[1 + l--nAr2+ 2 go
7) ie = ~ .
c~V,~dz +
netic field p o t e n t i a l function V~.. The limits of the spatial charge d e p e n d on the p a r a m e t e r s of the zero p o t e n t i a l well which is given u n d e r the form of a paraboloid:
6qVm/Or -- OVm/~,g
(10)
The form of the cross section of t h e b e a m is d e t e r m i n e d b y the spatial d i s t r i b u t i o n of the mag-
The increase in the current d u r i n g the i n j e c t i o n leads to the appearance of a n e.m.f, whose value depends on the rate of change of t h e c u r r e n t : di e er = -- L f - ~ "
(15
s) V. N. L o g u n o v . et al., J o u r n . Techn. P h y s . 27 (1957) 1135. 9) V. N. iRodimov, lzv. T . P . I . 87 (1957) 11.
THE E L E C T R O N I N J E C T I O N PROCESS IN THE BETATRON
In o t h e r words the v a r i a t i o n of the c u r r e n t creates a selfinduction field for which dissipates a p a r t of its energy. This leads to a larger c u r v a t u r e of the electrons t r a j e c t o r y u n d e r t h e action of t h e principal magnetic field. As a consequence t h e r e is a c o n t r a c t i o n of its equilibrium orbits, which therefore avoid the injector. I n order to evaluate the reduction of the electron radius due to this effect we m a y use t h e electron m o m e n t u m expression in the m a g n e t i c field : P = e r . Bz
(16)
c
dp _ dr + dBz p r dz
dr + ~B~/~r dr = - -dr (1
r
B~
- n) (17)
r
F r o m N e w t o n ' s second law we h a v e : dp = e- E - dt = e. 2 ~ r dt = -- e 2Lf ~ r . - ~d idet
dr =
-
e - Lf 1 di e 2x(1 n) p ~ - / d t _
•
(18)
•
( 1 9 )
The r e d u c t i o n of the radius d u r i n g a t u r n , i.e. for dt
=
To
2hr.
=--lsAr
v
=
--r
Lf 2(1 -- n) • u i
di 2
dr(O)
where d i / d t is d e t e r m i n e d b y the ratio of t h e m a x i m u m c u r r e n t injected d u r i n g the electron c a p t u r e process, a n d L e is defined as t h e i n d u c t a n c e of a single winding coil of radius r o disposed in the meddle plane of the b e t a t r o n m a g n e t i c field, symm e t r i c a l l y to the later. At the same time there is a s u p p l e m e n t a r y effect which p r e v e n t s the electrons h i t t i n g the injector due to the direct action of the spatial charge on t h e injected electrons 6- lo The electrons which are at t h e b e a m surface find themselves u n d e r the repulsion forces of the o t h e r electrons of the beam. These forces t e n d to defocalize the b e a m a n d as t h e y grow lead to increase the b e a m size. I t is the increase of the electron oscillation a m p l i t u d e t h a t limits t h e m a x i m u m value of the charge which can be accelerated in the betatron. The injected electron b e a m after one t u r n has the average density smaller t h a n the initial one. The density decrease takes place because of the increase
177
of t h e b e a m cross section a n d because a p a r t of electrons gets loss by striking t h e c h a m b e r walls. As t h e first b e a m passes t h r o u g h the injector a z i m u t h , t h e second b e a m is injected w i t h a greater density. Due to the Coulombian action b e t w e e n t h e m the center of the first b e a m is m o v e d nearer to r o. The s u b s e q u e n t simultaneous m o t i o n of the first a n d second b e a m s gives also a n increase of the t o t a l b e a m cross section a n d throws a larger n u m b e r of electrons on t h e c h a m b e r walls. At the end of the second t u r n of the first beam, the t h i r d b e a m is injected. The i n t e r a c t i o n of the l a t t e r w i d t h the t o t a l b e a m p r e v e n t s again as the electrons strike t h e g u n s t r u c t u r e , and so on. Such a m e c h a n i s m of stable b e a m f o r m a t i o n leads to t h e idea 1t (in fact practically confirmed) t h a t t h e electron capture of different parts of the injection pulse has a different efficiency. Indeed, the injected electrons as the injection pulse rise (with e x t e r n a l injection) a t first fill the p o t e n t i a l wells which are nearer to r o. The oscillation of these electrons is d e t e r m i n e d b y the distance b e t w e e n the corresponding i n s t a n t a n e o u s orbit a n d rl. As the injection voltage increases the oscillation a m p l i t u d e of these electrons will be proportionally smaller. The difference in t h e oscillation amplitude leads to a corresponding difference of the charge density. Therefore, each following b e a m injected at a larger voltage m o v e s the r e s u l t a n t b e a m towards r o. If the electrons are injected as the pulse is rising the spatial charge in t h e accelerating c h a m b e r increases c o n t i n u o u s l y due to the i n t r o d u c t i o n of new electrons. If t h e y are injected as the pulse decreases the charge becomes smaller due to the loss of electrons on the c h a m b e r walls (small injection energy) also to t h e decrease of the n u m b e r of injected electrons. T h e n o n u n i f o r m a z i m u t h a l charge d i s t r i b u t i o n is due to t h e fact t h a t the m a j o r p a r t of the i n j e c t e d electrons fall on t h e c h a m b e r walls just near t h e injector since, generally, the injector gives a diffuse beam. As a result of t h e p r i m a r y electron b o m b a r d m e n t of c h a m b e r s is wall there is secondary emission. T h e r e arives in t h e c h a m b e r a larger charge t h a n the 10) V. N. Logunov and S. S. Semenov, Jourrl. Exp. Teor.
Phys. 33 (1957) 1513. 11) M. Seidl, Journ. Exp. Teor. Phys. 36 (1959) 1305.
178
G. B A C I U
injected one. The electron interaction with the (fastly decreasing) azimuthal nonuniform charge, the electron energy is reduced. As a matter of fact, the electrons which approach the charge feel its repulsion action, and therefore avoid the injector. 3. Experimental Study of the Injection Process
In agreement with the above description as the electrons are injected in the accelerating chamber there arives an electron circulating current which creates a magnetic fluxq, e. The size of this magnetic flux is directly proportional to the electron current and therefore it is also proportional to the charge Q, captured in the betatron accelerating process. ~ = k~.Q
•
(21)
If in the vicinity, a coil embraces the doughnut in it will appear an induced voltage of fast variation : E ~- k 2 d t
= kt.k 2
(22)
"E" will exist when dQ d~ ¢ 0 that is during the capture or loosing of the electrons. Using a RC filter one m a y eliminate the 50 Hz voltage induced by the main field and put in evidence the high frequency one in which we are interested. Applying the voltage "E" to the oscilloscope vertical plates and following the curve E = f(t) we can study the electron current variation or integrating - d i r e c t l y the charge variation in the accelerating chamber :
capture efficiencyincrease with the injected current, so that the circulating current varies as the injected one to a power greater than unity 14. In order to improve the output of a betatron one can make use of two classes of methods: the correction of magnetic field distribution and supplementary damping d e v i c e s - t h e so called contractors; it follows from the existence of two different capture mechanisms: one non-collective, effective at low injected currents, and another collective, effective at high injected currents, but limited by saturation. The non-collective capture shows a dependence upon injection voltage as predicted by Kerst and Serber3). The contractor circuit contributes v e r y effectively to this non-collective capture, the efficiency being greater for low currents and high voltages. The collective capture becomes the main capture process, for high injected currents. Unlike the non-collective process, the collective one is very much affected by the field distribution at injection. The two methods of increasing the yield are not antagonistic; on the contrary; the combined efficiency is greater than the product of the efficiencies of the two methods applied separately. In other words, the contractor efficiency is greater when field corrections are applied. F r o m a practical point of view, the principal matter is of course the radiation intensity. Although the gain in intensity resulting from the application of these two methods is sensible and the research in this direction is of course worth-while.
(23)
The author is indebted to the Bucharest betatron crew for their help and for useful discussions and particularly to Prof. F1. Ciorascu for his kind interest in this work and for many encouragements.
The experimental evidence available 1z'13) demonstrate that the damping of oscillations and the
1-°) \V. F. \Vestendorp, Phys. Rev. 76 (1949) 445. ts) j. N. L o b a n o v et aI., C E R N S y m p o s i u m IReports (1956) 484. a4) G. Davies, J. Sci. Instr. 36 (!959) 306.
1
Q-
kl k2
f E(t) dl •
4. Discussion