Phase-free acceleration of charged particles by ac fields using a rectangular drift-tube-loaded cavity

NUCLEAR

INSTRUMENTS

AND

METHODS

I2 5

(I975) 67-7I;

©

NORTH-HOLLAND

PUBLISHING

CO.

P H A S E - F R E E ACCELERATION OF CHARGED PARTICLES BY AC FIELDS USING A RECTANGULAR DRIFT-TUBE-LOADED CAVITY H. Z I N N E R , W. S C H O T T and W. W I L H E L M

Physik-Department der Technischen Universitiit Miinchen, Munich, Germany Received 23 December 1974 By means of a rectangular drift-tube-loaded cavity which resonates in the TE-101 mode, charged particles can be accelerated without a fixed phase relation between the ac fields of the cavity and the phases of entrance of the particles into the fields. The fields in the first half of the cavity are thereby shielded by a drift tube. Operating the cavity with an amplitude of the electromag-

netic field of 1.6 G the time-averaged energy gain of electrons with an incident energy of 20 keV was measured to be (15 4- 2) eV. At all phases energy gains were measured. The m a x i m u m deviations of the energy-gain values from the time-average are -4-3.6eV. The data are in good agreement with the theory.

1. Introduction The continuous acceleration of charged-particle beams by ac fields in a cyclic accelerator could open up a large variety of very interesting applications. In the case of an electron accelerator an effective n-meson source could be built where a large fraction of the applied rf power is used to accelerate the electrons. Moreover, by means of such an assembly a very intense beam of relativistic electrons could be produced and used, e.g., in plasma physics for trapping ions. In two preceding papers 1,2) one possible version of a phase-free accelerating stage working with triangularshaped hollow electrodes was suggested and experimentally confirmed for electrons of an incident energy of 20 keV. Another method uses two orthogonal electric and magnetic fields with a phase difference of re/2 for accelerating particles in a direction perpendicular to both fields3). This method can be realized taking the field distributions in a special plane in the second half of a rectangular cavity which resonates in the TE-101 mode¢). Besides the acceleration the cavity fields cause the beam to be deflected off its main path. These deflections can be roughly compensated for by a composition of four identical accelerating cavities which are operated at fixed phase differences between each other. In section 2 the theory of phase-free acceleration by one cavity and by an assembly of four cavities is briefly outlined assuming ideal field distributions in the second half of the resonators. In section 3 the results of energy-gain measurements with electrons of an incident energy of 20 keV passing through a single cavity at different phases are given. The measurements are compared with calculations enclosing either the ideal fields or a measured field distribution. In section 4

results of calculations of the energy gain of an accelerating stage consisting of four cavities are presented. The calculations were carried out for electrons in a range of incident energies which is important in an accelerator used to produce ~ mesons.

2. Theory of phase-free acceleration by ideal fields In fig. 1 a rectangular cavity is shown schematically having the dimensions a, b, and d in the x-, y-, and zdirections, respectively. If the cavity resonates in the TE-101 mode only two field components exist in the plane x = a/2, namely:

Er -B°a-----~sin

(d)

sin(~ot- ~b),

7r - -d

(1) cos

cos(cot- qS),

where B 0 is the amplitude of the z-component of the magnetic field which is zero at x = a/2, and ~b is an arbitrary phase. It can be shown that a charged particle entering these fields along the z-direction within the plane x = a/2 at a given time t will be retarded by means of the fields in the first half of the cavity and accelerated in the second half. This holds for all phases q~. If the fields in the first half of the cavity are shielded by a drift tube which is mounted along the particle path as shown in fig. l, the particles are accelerated at all phases by the fields of the second half of the cavity provided the perturbation of the fields due to the drift tube is small, or can be compensated in some way. For the sake of simplicity ideal fields are assumed, i.e., the fields in the first half are perfectly shielded and the fields in the second half are given by 67

68

H. Z I N N E R et al.

eq. (1). Moreover, the energy gain is small compared to the incident energy of the particle. Then a particle of rest mass rn and charge e entering the resonator parallel to the z-direction in the plane x - - a / 2 will gain an energy ( A E ) I after having passed through one cavity, where: (AE)I - e2B2a2 c3fl [ 2 d +

ences 0, 7r, re, 0 compared to the phase of the first cavity. Therefore, assuming ideal field distributions, a particle beam entering in an axis parallel to the .g-axis within the plane x = a/2, say at y = b/2, will leave this accelerating stage consisting of four cavities on the same axis. Thereby the beam has gained the energy (AE)2, where:

-

Z_(I-lcos2~)

4~zdmc2y L~ZV =

-z+sin~+cos2~(

2-

z+ sinc~+~-v)+

+ sin 2 ~b z+ cos c~1 .

1 Z+

1

--

-t-

-

-

--,

co - (zrv)/d

to + (~v)/d

1

1

co - (rw)/'d

co + (zrv)/d'

cod =--, 2v

v wherefl = - , c

z+, z_,

~-

1

Lnv + z _ ( l - ½ c o s 2 ~ ) -

- T+ sinc~ + cos2~b ( - z + sin~) + sin2~b x

(2)

v means the particle velocity ( v ~ c o n s t . ) . and ~ are given by:

2zrdrnc27

× (z + cos ~)1"

(3)

It is noteworthy that both the single cavity [eq. (2)] and the acceleration stage [eq. (3)] have a 2 4 dependence of the energy gain on phase. The same phase dependence is found in another phase-free accelerating device which uses triangular-shaped hollow electrodes ~). In the x-direction the cavities have a slight defocusing effect on the particles, i.e., particles are deflected away from the plane x -- a/2 by means of the magnetic field

B~ 4).

(1 -f12) ~" 3. Measurement of the energy gain of a single cavity

In addition to the acceleration the fields cause the particles to be deflected in y-direction. The deflection angles at the end of the cavity depend on the phases of entrance of the particles into the cavity. In the idealfield case these deflections can be compensated by three more identical cavities which are operated in such a way that the particles enter the cavities at phase differ-

T, Fig. 1. Scheme o f the cavity r e s o n a t o r which w o r k s as a phasefree accelerator for charged particles o p e r a t i n g in the TE-101 mode. The particles enter along the axis x = a/2, y = b/2. The fields o f the first h a l f o f the cavity are shielded by a drift tube. The d i m e n s i o n s o f the r e s o n a t o r are a = 25.4, b = 30.0, d = 55.3 cm. T h e drift-tube d i a m e t e r is 4 cm. The resonance frequency is 639.8 MHz.

The cavity was made of brass and electroplated with a thin silver layer. The dimensions of the cavity were chosen to be a = 25.4, b -- 30.0, and d -- 55.3 cm. When ideal fields are taken into account these dimensions yield a frequency of 650 MHz for the TE-101 mode. After inserting a drift tube of 4 cm diameter the resonance frequency for the TE-101 mode was measured to be 639.8 MHz. The fields in such a drift-tube-loaded cavity have been measured in the plane x - - a / 2 by means of a technique which uses perturbing objects of different shapes and materials5). In fig. 2 the measured field components are shown. The most outstanding difference to the ideal field distribution is the existence of a z-component of the electric field away from the axis y = b/2. In fig. 3 an experimental device is shown schematically by which the energy gain of a single cavity was measured. Using a "Steigerwald"-type electron gun and a magnetic round lens a small-angle electron beam of an energy of 20 keV was produced and adjusted by means of two magnetic deflecting systems in order to pass through the cavity along the axis x = a/2, y = b/2. At the entrance of the cavity a diaphragm of 2 mm diameter was mounted. The energy of the beam which has passed through the cavity is measured by a retarding

PHASE-FREE ACCELERATION

OF C H A R G E D

69

PARTICLES

E/till (V/(rnV'-~|

lOO~ i; 8o !',

2

y=b/2

:

y=b/2-1cm

i!

E¢,m'O y = b/2

y=bl2-1cm ii

bf/:

0L Ez/~5-~ y = b12

="

y=b/2-1cm ii

60

l.O

[

20 4/YJ

0

Hx4v@'O(A/(mv'W))

°o °oI

.i

:

:

I

i

I

,~

I

h

y=b/2 1

0.12~-

I

!

JJ

!

!

,=0,2

'ii il

'

jl ii

0.08t 0.0~0t 0

t//~ 525

,,.

30

35

~0 ~5

50 55 6O z (cm) •

,

5 25 30 35 40 45

5O 55 6O z (cm) -

Fig. 2. Results o f perturbing-object measurements of electric- and magnetic-field quantities versus the z-coordinate. The measurements were done within the plane x = a/2 o f the cavlty along the axis y = b/2 and y = (b/2-- 1) cm. P is the power loss and Q the Qfactor o f the cavity. The double dotted lines mark the front and rear plates, the single dotted lines the middle of the cavity.

lGenerator I [RF-Meter l I IFrnpedonce l [Transformer l EG

FMDM

~FA

Yt; High Voltage (-20 kV)

\\

// ~

(-200..*20OV)

High Voltage lOcm

Fig. 3. Sketch o f the experimental setup. E G means electron gun, F M focusing magnet, D M deflecting magnet, R F A retarding field analyzer, Uw means Wehnelt voltage, Ue means additional cup voltage. The dimensions of EG, FM, D M , and R F A are doubled, compared to the shown scale. A m a x i m u m deflected electron beam corresponding to an incident energy o f 20 keV and a phase of entrance into the accelerating fields o f 158 ° is included. The deflection of the beam is caused by an electromagnetic field amplitude in the cavity o f 1.6 G.

70

H. ZINNER et al.

field analyzer6). The error of the energy measurement, which is mainly caused by the inaccuracy of the beam adjustment, amounts to + 2 eV. The analyzer can be moved along a bent bar with a radius of curvature of 55.3 cm around the end of the drift tube. Thus, the beams deflected by the cavity fields pass through the grounded diaphragm in front of the analyzer at right angles. The deflection of the beam at a certain phase depends linearly on x/Q where Q is the Q-factor of the cavity, provided the rf power fed into the cavity is kept constant. This can be seen from an ideal-field calculation as well as from a numerical calculation using the measured fields shown in fig. 2. Therefore, the Q-factor of the cavity can be determined rather accurately by measuring the maximum deflection of the beams, e.g. at the position of the bar, and comparing this value with the results of the numerical calculation. A Q-factor of 20922 was obtained in this way. The rf power fed into the cavity was 12.8 W. The amplitude of the z-component of the magnetic field was, respectively, 1.6 G causing a maximum deflection angle of 1.2 °. The energy gains of the beams having different phases were measured by moving the analyzer within _+ 1.2 ° along the bar. Fig. 4 shows the energy-gain values versus phase. The measurements are marked by circles. The phases of different beams were deduced from the calculated deflections-versus-phase curve using the measured field distributions. Two phases correspond to each deflection. Therefore, each point appears twice. The full line represents the numerical calculation, the dotted line is the result of eq. (2). Both curves show the 2 q5 dependence as was discussed already for eq. (2). The phase difference between the two calculated curves is due to

the fact that the real fields extend somewhat into the drift tube, as can be seen from fig. 2. Apart from this phase difference the measured points agree with the numerically calculated curve as well as with the analytical curve. This can be seen from the timeaveraged values which are deduced to be 14.8, 16.1, and 12.9 eV, respectively. For all phases only energy gains occur. The maximum deviations from the averaged values are, respectively, 3.6, 2.45, and 5.75 eV. 4. Discussion As can be seen from fig. 4, both the numerically obtained energy gains using measured fields and the analytically obtained energy gains differ not very much. This is due to the fact that particles which pass through only one cavity along the axis y = b / 2 within the plane x = a / 2 will not feel the electric-field component E~, which is zero on the axis. E~ is different from zero away from the axis only in a relatively small region close to the end of the drift tube. The particles hardly come into this region because near the end of the drift tube the deflections of the particle beams away from the axis are very small, especially at such a small amplitude of the electromagnetic field in the cavity used in the described experiment. In an accelerating stage consisting of four identical cavities only differing in phase in order to compensate the deflections in ydirection the particles enter the second and the third

T lO AE(eV) 103'

30 102

&E(eV)

10

"'.

__j/,,"

1I

2

~- .o__

3n

~

n

~

~

~Jt 2

/,/

b

7~ I.

2n

10-

Fig. 4. Energy gain of electrons by means of a single cavity versus phase. The circles are measured values within an error of 4- 2 eV. The full line represents a numerical calculation using the measured field distribution in the cavity. The dotted line was calculated analytically assuming ideal field distributions. The incident energy was 20 keV. The amplitude of the electromagnetic field was 1.6 G.

10

102

103

10~

105

E (keY)

l,

106

Fig. 5. Time-averaged energy gains of electrons versus incidentparticle energy of an accelerating stage consisting of four cavities, a) Ideal-field calculation, b) Numerical calculation using measured field distributions. The amplitude of the electromagnetic field in the cavities was taken to be 8.5 G.

PHASE-FREE ACCELERATION OF CHARGED PARTICLES cavity a w a y f r o m the axis a n d are accelerated a n d r e t a r d e d by the first-order effect caused by E~. Therefore, the real energy-gain-versus-phase curve o f an accelerating stage has no longer such small deviations f r o m the averaged value as it is suggested by eq. (3) ~). However, as fig. 5 shows, the t i m e - a v e r a g e d energy gain o f a real accelerating stage agrees rather well with the ideal-field calculation using eq. (3) in a wide range o f incident energies which are i m p o r t a n t , e.g., for an electron accelerator designed to p r o d u c e n mesons.

W e w o u l d like to t h a n k H. Daniel for m a n y stimulating discussions. F u r t h e r m o r e , we are highly indebted

71

to H. H a g n a n d H. A n g e r e r for technical advice a n d assistance. References 1) H. Daniel, Zo Physik 236 (1970) 166. 2) W. Schott, H.-J. Winter, H. Hagn, K. Springer and H. Daniel, NucL Instr. and Meth. 116 (1974) 599. 3) H. Daniel, P. Halfar and K. Springer, Contributed paper to the 4th Intern. Conf. on High energy physics and nuclear structure (Dubna, 1971) unpublished. 4) W. Schott, H.-J. Winter, H. Hagn, K. Springer, H. Zinner, F. Brunner and H. Daniel, Proc. 9th Intern. Conf. on High energy accelerators (Stanford, 1974) to be published. 5) F. Brunner, W. Schott and H. Daniel, IEEE Trans. on Microwave theory and techniques, to be published. 6) G. Haberstroh, Z. Physik 145 0956) 20.