Stability limits of the phase oscillations in a microtron

NUCLEAR

INSTRUMENTS

AND

METHODS

IO 4 ( I 9 7 2 )

I69-I72;

©

NORTH-HOLLAND

PUBLISHING

CO.

STABILITY L I M I T S OF T H E P H A S E OSCILLATIONS IN A M I C R O T R O N A. T U R R I N

Laboratori Nazionali del CNEN, Frascati, Italy Received 3 M a y 1972 T h e finite-difference equations existing in the literature for p h a s e oscillations in a microtron are discussed first. It turns o u t that these equations are incorrect. A technique is then developed to analyse the effect of c o m p u t ing the stability limits with the aid o f these equations. It is s h o w n that the resulting p h a s e stable regions really constitute a w r o n g result. Finally, a new t r e a t m e n t o f the phase oscillations in a microtron is given. T h e stability limits are derived from the general

theory of p h a s e oscillations. T h e results are: 1) T h e range of equilibrium p h a s e angles extends f r o m 0 ° (corresponding to a peak voltage across the gap V' = moe2/e) up to 90 ° (corresponding to very high peak voltages). 2) T h e bucket area is an increasing function of the peak voltage. 3) At high peak voltages stable oscillations are connected with large radial excursions f r o m the s y n c h r o n o u s orbit.

I. Analysis of the finite-difference phase equations existing in the literature The equations as given by Henderson et al. 1) and as used currently to day are:

Here h indicates we are referring to the hth orbit. One finds eq. (3) by taking eq. (15) of ref. 2, inserting in it the harmonic number vh (obviously, h acts as a counter of subsequent orbits), and putting the field index value equal to zero. Eq. (4) may be written by taking the equation immediatly before eq. (17) of ref. 2, replacing sin q0 by cos (p, and neglecting the radiation loss term. For the microtron, let us replace E s = hmo c2 in eq. (3) and e V ' = mocZ/cos q9s in eq. (4). One gets

(1)

Aq)h+ 1 = A~oh + 2~zvAe h, Aeh+l :

Aeh +

cos(~0~+ Aq~h+1)

-- 1.

(2)

COS q9s

Here ~os ( > 0) is the equilibrium phase angle [cos ~p~--= mocZ/(eV')]; AfPh = ~0h--q~s is the phase shift from

the synchronous particle in going from the ( h - 1 ) t h up to the hth traversal of the accelerating gap (V' is the peak voltage), and A eh denotes the energy deviation (in rest masses m 0c 2 of the electron) from the energy of the synchronous particle measured at the hth exit of the gap. v is an integer number given by the ratio A T f f T o where A T s = constant is the increment of revolution period of the synchronous particle at each hth revolution and where To is the period of the accelerating voltage. Let us start from the general theory of phase oscillations expressed in differential terms, in order to try to reobtain equations like (1) and (2) from energygain-per-turn considerations. The differential equations which are valid for any kind of constant gradient accelerator are given by Bohm and Foldy 2) and may be rewritten as follows: d AE -- Aq9 = vho3 s - - , dt

(3)

Es

d (A_~) = eV' c ° s q(c°sq~ )s)2---~-

(4) 169

d

--

A~p =

(5)

v~o~A~,

dt d

A(_~_~)_ _ _ l 2( rrpcos ~

(cos~0- cos ~0~).

(6)

Now, the only way to find equations like (1) and (2) is to replace d/dt(Ae/o)~) in eq. (6) by (l/Ogs)d/dt(Ae) and then to integrate eqs. (5) and (6) over one whole orbit. But the microtron is a machine in which the synchronous particle angular velocity o)~ decreases at every traversal of the accelerating gap. Therefore, putting 1/~os before the d / d t operator in eq. (6) and then integrating over one complete revolution is a mistake. In fact, integrating correctly over the (h+ 1)th orbit, one obtains Aeh+~

A____e_h=

Oksh+ i

(DSh

1

FCOS(q~2+__A.___~h+,)

(DSa+~ L

1],

COS qgs

instead of eq. (2). Consequently, one cannot make calculations with equations like (1) and (2) without having errors present.

170

A. T U R R I N

In the following section we shall accept the errors expected when using equations like (1) and (2). A straightforward procedure that gives the stability limits as determined by eqs. (1) and (2) will be developed. 2. The stable, the unstable fixed-point and the separatrix First of all, we shall make three statements about the non-linear transformation given by eqs. (1) and (2): 1) In the phase plane AO, As (see fig. l) there exist two points with the property that they remain unchanged during step-by-step iterations made by eqs. (1) and (2). These are the points

We have ½tr Jf = 1 - rrv tg Of,

(9)

and this shows that the point at Of = +Os is stable, and the point at of = -Os is unstable. 3) There exists a closed curve which separates the stable from the unstable phase-plane region. This curve is called separatrix. The separatrix crosses itself at the unstable fixedpoint. The slopes of the separatrix at the unstable fixedpoint are given by the eigenvectors of the Jacobian matrix (8) calculated for the unstable fixed-point. These slopes are given by

As = 0, at

Of = -+ Os,

(Os > 0). (7)

AO = 0, These points are called fixed points.

(OdSh+~l~ (~ASh+t~ \ CASh /Ifl

tgof

L\ 63ASh ,,If \ C~ZlOh~If_] _+

2) The local properties of the fixed-points are deduced from the transfer matrix for the linearized motion in their neighbourhood. This matrix is the Jacobian matrix of eqs. (1) and (2) calculated at the fixed-points (,Of = "~- Os"

Jf=

CAeh /f

l--2rrvtgof'

i oaoh If

(8) The value of the half-trace of Jf determines whether the motion in the neighbourhood of the corresponding fixed-point is stable 0½trJfl < 1) or unstable (1½trJfl > 1).

tr J f

; ?7 --

1

1 _--[(l+TtvtgOs) 27rv

= ½tg 0s -+

2 - 1] 6. (10)

Knowing the coordinates of the unstable fixed-point and the slopes of the separatrix at this point, one can make a digital computation of the separatrix. This is done by following with the aid of eqs. (l) and (2) successive shadows of many representative points that are distributed on the outward-going eigenvector [sign plus in eq. (10)] in the neighbourhood of the unstable fixed-point; and, conversely, by following those that are distributed on the inward-going eigenvector [sign minus in eq. (10)] in the same neighbourhood with the aid of the inverse transformation of eqs. (1) and (2), i.e.

AOh = AOh+l -- 27zVAeh, .~osed sep. ZJSh ~

ASh+ I --

cos(0s+ 4Oh+l) + 1

(11) (12)

COS Os

o,,f/

This will be done in the following section. >

~v

d~

"7 ClOSed sep,

Fig. 1. Schematic phase-plane diagram. The separatrix, which passes twice through the unstable fixed-point, encloses the stable region.

3. Digital calculation of the separatrix by means of of eqs. (1) and (2) Using the procedure outlined in the previous section, we have calculated the separatrix, limiting ourselves to only one typical case already considered in ref. 1, i.e. for Os = 21.79°, which corresponds to a peak voltage across the gap V' = 550 kV (v = 1). The results, which are shown in fig. 2, are seen to be quite in agreement with the result of the digital corn-

STABILITY LIMITS OF THE PHASE OSCILLATIONS IN A MICROTRON

171

I I

[_

.14

•14

J

.12

_

J

/\,

.12 .10

•08

.08

.06

.06

.04

.04

.02

7¸

_~,,~

0

-.04

o!

~:. . . .

,

; \ DEGR:ES

,,,,

.

-.12

I

-.11

"

N.

i

-02

~

t

-.14]

- 3 0 -20 -i0

0

10

20

30 40

50

Fig. 2. Successive positions of many representative points which started from the ingoing and outgoing closed separatrix in the neighbourhood of the unstable fixed point. V'=550kV (V0 = 560 kV). The clean region around ~Vs is the phase stable region obtained by the authors of ref. 1. p u t a t i o n s r e p o r t e d in ref. 1 (see fig. 4, curve C o f the q u o t e d reference; V' = 0.938 V0, i.e. V0 = 560 kV). However, it is the shape o f the separatrix that a p p e a r s physically unacceptable. Liouville's t h e o r e m states t h a t the representative points m o v e in a p h a s e - p l a n e like an incompressible liquid. Thus, an inspection o f fig. 3 readily reveals one fact: the m a t c h i n g between the o u t w a r d - a n d the i n w a r d - g o i n g separatrix at the right h a n d o f ~Ps fails, a n d after the occurrence o f this failure we get a nonsensical motion. M i s m a t c h occurs essentially because these two branches o f the separatrix are not f o u n d to be s y m m e t r i c a l with each other a b o u t the ~p-axis. This is a n o t h e r p r o o f that finite-difference equations like (1) a n d (2) w o r k incorrectly.

-30-20

-10

0

10 20

30

40

50

Fig. 3. The same situation as described in fig. 2. Only a few shadows of many starting points are plotted to demonstrate the failure of the matching between the outward and inward going closed separatrix. So, it is necessary to r e f o r m u l a t e the whole p r o b l e m , as will be done in the following section• 4. Canonical form of the phase equations and the separatrix F o r a fixed field accelerator, it is convenient to f o r m u l a t e the p r o b l e m in terms o f the pair o f conjugate variables [as has a l r e a d y been d o n e b y S y m o n a n d Sessler3)], so t h a t eqs. (5) a n d (6) become:

(AE/ogs), A~p d

,dq~ voo2(Aet

dt

\ ~os/

d(A__~e~_ l (cos q~_ cos ~ps) " dt \cos/ 27z cos 9s

(13)

(14)

172

A. T U R R I N

The Hamiltonian function H from which eqs. (13) and (14) are derivable as canonical equations,

d Aq~

~?(A~/~o~)'

dt

c~--~ '

It turns out from eq. (16) that the stable area is an increasing function of the peak voltage. The half-height of the bucket, i.e. the maximum stable energy excursion, is found by putting q~ = +p+ in the separatrix eq. (16). One gets:

is given below: d~3max =

H = ~1 ~COs2

-

\ Ms/

(sin q~ _ ~o cos q~s)

-

2 n cos q~s - (sin ~o - qo cos ~p,). 2 n cos ~Ps

Es

two points given by eq. (7). The separatrix is given by the equation H = H(Ae = = 0, ~p = - ~Ps), which can be written as +[~r

1

--

V

[sin 9 + sin q)s - ((p + ~0s) cos q)s]l'.

C;S q)s

(16) In fig. 4 the separatrices are plotted for five values of V' (see fig. 4 of ref. 1 for comparison), v = 1. r ! i

,.

.~o +

.~2-

+

i .

+ • 06

E , A~ A ~ -

,

.04

.

i

'

i

~

.

.

l

! -

.

, / / / + \ , 5 - t

I

+

+

~

i

>-+

0

5.

RS

h

08)

Conclusion

The conclusion can be drawn that in principle there is nothing in phase oscillations dynamics for the microtron that restricts operation to low peak voltages. The only real restriction that comes out is the fact that for large peak voltage values particles must perform large stable excursions from the synchronous orbit, so that they strike the walls of the vacuum chamber or the cavity anyway. it is a pleasure to thank Prof. I. F. Quercia and the Electron Synchrotron Staff for their kind interest in the progress o f the work. The techniques applied above are extracted from two papers o f Prof. H. G. Hereward4).

References

;

-30-20-10

(17)

:

.02 0'

AEmax __ ARmax __ A~max

(15)

To find the fixed-point we set d/dt(Aqg)= 0 and d(Ae/co+)/dt = 0 in eqs. (13) and (14). This leads to the

At=

.

1+

At the hth orbit

1

½v(Ae) 2

2 (tg~Ps-q~s)

10 20 30 /.0 50 60 ~ p ( d o g r e e s )

Fig. 4. R e g i o n o f phase s t a b i l i t y in m i c r o t r o n s f o r v a r i o u s p e a k voltages. These curves are s y m m e t r i c a l a b o u t the ~-axis. T h e

bucket area is an increasing function of the peak voltage.

1) C .Henderson, F. F. H e y m a n n and R. E. Jennings, Proc. Phys. Soc. (London) 66 B (1953) 41. 2) D. Bohm and L. Foldy, Phys. Rev. 70 (1946) 249. 3) K. R. Symon and A. M. Sessler, Proc. C E R N Syrup. High energy accelerators (1956) p. 44. 4) H. G. Hereward, The possibility of resonant extraction from the C. P. S., C E R N A R / I n t . GS/61-5 (Dec. 1961); What are the equations for the phase oscillations in a synchrotron?, C E R N 66-6, Proton Synchrotron Machine Division (Feb. 1966).