A study of the charge change heavy ion accelerator

NUCLEAR

INSTRUMENTS

AND

METHODS

72 (1969) 269-276; ©

NORTH-HOLLAND

PUBLISHING

CO.

A STUDY OF THE CHARGE CHANGE HEAVY ION ACCELERATOR H. MIESSNER

lnstitut fiir Angewandte Kernphysik, Kernforschungszentrum Karlsruhe, Karlsruhe, Germany

Received 13 March 1969 A feasibility study is presented on the heavy ion accelerator proposed by Hortig. The obtainable beam current is estimated as ~ 10~° uranit, m particles/sec. MeV. A "double tube" version with separated accelerating and decelerating sections is proposed. This system is found to have some significant advantages.

1. Introduction A heavy ion accelerator is required to accelerate ions with masses ranging up to uranium to energies of 6-7 MeV/nucl.1 -a). A beam current of 1011-101a particles/ sec with an energy spread of < 10 -2 is desirable. Hortig 4-v) proposed a tandem accelerator with negative terminal potential (fig. 3a). The ions lose electrons traversing a solid stripper and pick up electrons in a gas stripper. It turns out that the particles gain more energy in the accelerating field than they lose in the decelerating section. In a cyclic system the ions are accelerated repeatedly in a single electrostatic potential of some MV. The specific advantage of this accelerator is the low terminal potential, furthermore that the target and the solid strippers can be operated at gronnd potential. In 4) and 5) the principle of the accelerator and a possible version of the needed magnetic mirrors are discussed. A mirror system with a total bending angle of ~ instead of 3= in the first scheme (fig. 3a) has been suggested by Sona and Sona8). Fiks and Vialov 9) studied the transmission of the accelerator and obtained values near I. In v) ion optical problems and M o n t e Carlo model calculations on the particle acceleration are discussed. In the following paper the accelerator is investigated in detail. Parameters important for its feasibility are studied. The beam injection into the accelerator, its transmission and the particle intensity per energy interval are discussed. Furthermore, a "'double t u b e " accelerator is proposed, which is found to be advantageous as c o m p a r e d to the first scheme.

tively. For gas strippers and U ions of 7-70 MeV an equilibrium thickness smaller than 0.1 /xg/cm 2 was obtained1°). The equilibrium thickness is expected to increase with energy. This increase, however, is probably only small for the heaviest ions~°). Betz et al. 12) measured ~ for different ions in different stripper materials in the energy range 5-100 MeV. They obtained "'best fits" for their values and for earlier measured data using the semiempirical relation

2. Stripper data important for the accelerator design The stripper thickness, necessary to produce a statistical charge state equilibrium (equilibrium charge state £7) has been measured by Leischner 1°) and Grodzins et al.l~). For solid strippers ( F o r m v a r and C foils) and U ions equilibrium thicknesses of 10-20 /~g/cm 2 lo) and 20 pg/cm 2 11) have been found for energies of 7 70 MeV and tip to ~ 150 MeV, respec-

~/Z i = l-Cexp(-`5fi/o~),

(l)

= v/c,

= 1/137, where Zi is the nuclear charge and v the velocity of the ion. The constants C and ,5 are found to be velocity independent. For heavy ions C lies near 1 and depends slightly on the stripper material and Zv The Z~ dependence of,5,0) is presented in fig. 1. Fig. 2 shows the 1

6 0.2-

0.1

oo

2'0

io

do

~o '~

Fig. 1. The constant 6 in eq. (1) as a function of the nuclear charge Zi of the ion, according to lo). 269

270

H. MIESSNER

? 70

J 60-

50

40-

5/

j

30-

20

10

0 02

0,4

06

08

1.0

12

14

16

1 8 E(GeV)

Fig. 2. The equilibrium charge state ~ as a function of the energy E for U ions stripped in different materials, according to eq. (I).

Dashed line see text. energy dependence of ~ according to eq. (1) for U ions stripped in Formvar, air and He with equilibrium thicknesses (He data obtained by an extrapolation according to fig. 1). Since eq. (1) has been proved experimentally below ~ 150 MeV ~t) only, the extrapolated data at higher energies are uncertain. They must be expected to be smaller than predictedtt). Furthermore, a smaller ~ must be considered, if the stripper thickness is actually below the equilibrium thickness at high energies. Both effects are difficult to estimate. The influence of very pessimistic ~ values (dashed line in fig. 2) will be studied in ch. 3. The foil effect important for the accelerator is expressed by the ratio (~s-~g)/~g, where the subscripts s and g correspond to solid and gas strippers, respectively. The foil effect increases with Z t [for example 10)] and makes the accelerator most suitable for the heaviest ions. The use of He as a gas stripper seems to be advantageous. The equilibrium charge state distribution of heavy ions after stripping can be approximated by a Gaussian distribution f(~) = { ( 2 n ) + ~ } - ' e x p { - ½ ( ~ - ~ ) z / c r 2 } ,

with velocity. According to measurements of Bridwell et al. 14) and Pierce and Blann tS) a maximum stopping power of ~ 100 keV/pg.cm -2 at 200-300 MeV is expected for U ions in C, He and N 2. This corresponds to a maximum energy loss of ~ 2 MeV in a C stripper of equilibrium thickness. The energy loss in a gas stripper with its smaller equilibrium thickness can be neglected. The straggling of the energy loss can be estimated following Bethe and Ashkin16). For U ions in a 20 pg/cm z thick C foil a straggling of 100-200 keV is expected. An additional energy spread of a few MeV

'c?i

4-

(2)

with a practically independent of energy for not too low energies~°). For U ions ~r is found equal to 2.7 and 2.9 for Formvar and air strippers, respectively1°). The energy loss of heavy ions in the stripper can be calculated from the stopping power. According to Lindhard et al. ~3) velocity-proportional stopping is expected below ~ 300 MeV for U ions. For higher energies the stopping power is expected to decrease

4

~

~

~o~

70(r~d)

Fig. 3. The angular distribution of U ions of different energies scattered in a 20 #g/cm z thick C foil, calculated according to 17).

HEAVY ION ACCELERATOR

.......................

271

mission o f the accelerator. The scattering of heavy ions in materials of some 10/zg/cm 2 can be described as plural scattering according to Molibre's theory 17) modified for heavy ions~8). Fig. 3 shows some calculated angular distributions o f U ions scattered in a 20/~g/cm 2 thick C foil. The width o f these distributions is dominated by a characteristic scattering angle 01

L,,17,',',',~1~',',',::',',',',',',=7',=',',',:',:',',',',', o

0~ =

co,lst. { z ~ ( z s +

1)Z~/A.~t/e. 2,

(3)

where Z~, A~ nuclear charge and atomic weight of the stripper; Z~ nuclear charge o f the ion; E energy of the ion; t stripper thickness. The angular spread caused by scattering in a gas stripper with its smaller equilibrium thickness can be neglected.

S v(-) v/2,,,,,~, =========================================================================== ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: b Fig. 4. Scheme of the accelerator, a. Original (one tube) system, proposed in a); b. Double tube version with separated accelerating and decelerating sections; G: gas stripper, S: solid stripper.

3. The particle acceleration by charge exchange Fig. 4 shows the original accelerator proposed by Hortig (system a) and a double tube version (system b) with separated accelerating and decelerating sections. The latter system is found advantageous for the following reasons : 1. In system b the ions traverse the solid stripper at

must be considered, if (inhomogeneous) dust strippers 2) are used. The angle scattering in the strippers limits the trans-

~

s'

~ o

A

E

12 16 20 24 E[GeV] ~

/

~

"

\

Fig. 5. An accelerator model used for the study of particle acceleration. A and B are the modes of beam injection. The energy distributions of particles at the points of beam extraction C, D, E are presented for a set of parameters as given in fig. 12. G: gas stripper, S: solid stripper.

272

n. MIESSNER

each half cycle only once, instead of twice in system a*. Therefore, the energy loss is half and the intensity loss caused by scattering is reduced. 2. It is possible to operate system b with additional gas strippers in the decelerating section. This leads to a higher energy gain per cycle, while the additional energy loss and intensity loss in the gas strippers is negligible. For simplicity, only one additional gas stripper, placed at ½V, is discussed from here on. 3. The beam injection (mode B, ch. 4) can be operated with a smaller injection angle in system b. Furthermore, a lower injection energy is sufficient (ch. 4). The particle acceleration is studied using a model, which simulates the accelerator by a series o f identical tandem accelerators with strippers of equilibrium thicknesses (fig. 5). The n u m b e r of tandem accelerators passed by the ions corresponds to the number of passages or half cycles in the actual accelerator. The mean energy gain per passage is given by

AE = eV[~s(E)-~g{E+eV~,(E)}],

(4a)

for system a and

-A E = e V { ~s(E)-½~g(E+eV~s(E)}-½~g[E +eV~,(E)-½eV~g{E +eV~JE)}]} , (4b) for system b, (bl

Vcrit' (MV)

(b)

E•0In ,

(MeV) 80

/ /

b

60

b

40-

20-

Fig. 7. The minimum initial energy E0mh~ as a function of the terminal potential V0. Eomi" (V) is the inverse function of Vc~t (E) given in fig. 6.

where V is the terminal potential and E the energy o f the ions before acceleration. T h r o u g h o u t this paper the letters a and b in the equation numbers will refer to the accelerator systems a and b, respectively. Since the energy distribution of the accelerated particles is Gaussian, as shown in the appendix, the mean energy of the ions after n passages is found as

En "~-En-1 ~-eV[~s(En-l) - Cg{En-I ~-eV~s(En-,)}], (5a)

15

E. = E._ , + e v { L ( E . _, ) -

, + eV

(E. _, )} -

_He_ (a)

-½#g[E,_,

+eV#,(E,_ ,)-½eV#g{E,_, + eTv#,(E,_ ,)/]}, (5b)

10

n=1,2,3...

5

E o is the initial energy of the ions starting their first passage in front of the solid stripper. According to eq. (4), ~ increases with V, however, # s - ~g decreases with V. Therefore, an o p t i m u m terminal potential exists9), which makes ~ maximum. "2 ;

0'2

0'4

0 0'6 8 E (GeV)

i

50

i m'

E (MeV)

100

Fig. 6. The critical and the optimum terminal potential Vertt and Vopt for the accelerator systems a and b as a function of the energy E of U ions. The stripper combinations Formvar-air and Formvar-He are used.

Vopt = {~s(E)- ~ , ( E ) } / { 2 e ~ ( E ) -

d~g(E)/dE}. (6a)

The terminal potential must be below a critical value Vcrlt, if an energy gain ( ~ > 0) shall be achieved during each passage. V~r, is defined by the equations * This can be avoided in principle, if dust strippers of appropriate density are used. However, the energy spread caused by a dust stripper will exceed the additional energy loss in the foil of system a. Furthermore, the magnetic mirror must be designed for the lower ~g values.

273

HEAVY ION ACCELERATOR

~,(E)-~g{E +el/~ri,'~(E)~=O,

H~

(7a)

~a

~

b

Air

~ , ( E ) - ½ ~ { E + ek~:H," ~.(E)}--

- ~ ¢,[E + e v . , . ~ ( E ) - ~ v . i , . ¢~{E + e v . . . &(E))] = o. (Tb) Fig. 6 shows Vovt and VcH, for U ions as a function of the particle energy. A minimal initial energy Ep~,~n must be exceeded, if A--E is to be positive during the first passage. E~'i" (V) presented in fig. 7 is the inverse function of V~it (E). A high initial energy allows one to operate the accelerator with a high terminal potential, which reduces the number of cycles. In figs. 8 and 9 the mean energy E, according to eq. (5) is presented as a function o f n for different terminal potentials for U ions with Eo = 40 MeV and 20 MeV, respectively. Not taken into account is the energy loss in the strippers, which decreases the energy gain per passage by ~ 3-4 MeV and ~ 1.5-2 MeV in the systems a and b, respectively. This energy loss influences the beam acceleration during the first cycles only, since

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Fig. 8. T h e m e a n energy E,~ of U particles with the initial energy Eo = 40 M e V as a function o f the n u m b e r o f passages n in the accelerator m o d e l (fig. 5). T h e accelerator is operated with different terminal potentials, a n d the stripper c o m b i n a t i o n s F o r m v a l - a i r a n d F o r m v a r - H e are used. T h e d a s h e d lines refer to charge states lower t h a n predicted (see dashed line in fig. 2).

here the energy gain is small. At terminal potentials appreciably below Vcru, however, the energy loss is much smaller than the energy gain, even during the first cycles. In this case, if the energy loss is taken into account, the curves in figs. 8 and 9 are shifted slightly towards higher n values. Since the charge states are likely to be lower than predicted by eq. (I), the pessimistic case of a decreased ~.~ (dashed line in fig. 2) and an unchanged ~g is investigated. The dashed lines in fig. 8 show the result. For not too low ~ values the accelerator will still operate without fundamental change. More cycles in the high energy region are necessary only, where energy loss and intensity loss are small. The energy distribution of the ions results from two effects: The charge state distribution after stripping leads to a spread of the energy gain. A second contribution stems from the energy dependence of 4. Ions with higher energies are stripped to a higher 4. Their energy gain (loss) in the following accelerating (decelerating) field is higher than for ions of lower energy. Therefore, the energy spread is increased in the accelerating section and is decreased in the decelerating field for not too high terminal potentials (appendix). The energy distribution of the accelerated ions, as calculated in the appendix, can be approximated by a

274

H. MIESSNER -÷lOkV

~

s stripper

-6 MV Tondem

0

J

+10...800kV

ItttldtlllIItIll1Itllltttlllll IttLIllltllllllltllltllllllttlttt

] <

,,',,,'""'""

3. The intensity loss due to the statistical charge exchange during acceleration is small, if E o >2>E~ '~n. Basically two modes of injection can be used (figs. 5 and 10). A. With high energy ( ~ 20-40 MeV) and arbitrary charge states through the solid stripperV). In this case E o = Ei, j, where Ein j is the energy of the injected particles. The injection mode A needs a rather expensive injector, for example a tandem accelerator of at least 6 MV terminal potential; B. With low energy (10 keV-10 MeV) and a high injection charge state 4o. Here the injected beam does not pass the solid stripper, because this would decrease ~-o. The mean initial energy is calculated as

Eo = E~oj+eV{~o-~(E,.j+~V~o)I,

(Sa)

,,,,,

s.stripper ~ [ ~ m n

Eo = Einj + eV[4o - ½~g(ei,j + e V 4 o ) -

1'7 -~g{Einj~-cV~o-leV~g(Einj'JreV~o)}], (8b) with b Fig. 10. Scheme o f the b e a m injection with the injection m o d e s A a n d B.

Gaussian. The energy width ae. as a function of n is presented in fig. 12 for some typical accelerator parameters. 4. B e a m injection

E~nj = eVo~o, where Vo is the voltage of the injector. erE.

(MeV)

300

f

a

f

200

A high initial energy E o is required for the following reasons : I. A high E o allows a high terminal potential. Thereby the number of cycles can be reduced; 2. The intensity loss caused by scattering decreases with energy;

100 ~

0 .....

~

, ....

5

b

, .....

10

n

15

Fig. 12. T h e width a E . of the energy distribution of U particles as a function of the n u m b e r o f passages n in the accelerator model (fig. 5). Accelerator p a r a m e t e r s as in fig. 11, with ~0 = 11.

[MeV) 25-

Q 20

15

8

9

I0

11

12

5o

Fig. 11. T h e m e a n initial energy E0 o f U ions obtained by m e a n s o f the injection m o d e B with an 800 kV injector as a f u n c t i o n of the injection charge state ~eo. T h e accelerator versions a a n d b are operated with a t e r m i n a l potential o f 5 MV. T h e stripper c o m b i n a t i o n F o r m v a r - H e is used.

Fig. 11 shows Eo as a function of ~o, obtained by eq. (8) corrected for the energy loss in the strippers for U ions. Typical accelerator parameters and an optimum Vo ~ 8 0 0 kV are used. If a lower injection voltage of some 10 kV is taken, Eo is reduced by a few MeV only. Therefore a small injector is sufficient. The mean initial energy Eo(fig. 11) exceeds E~ j" (fig. 7) appreciably for 4o > 10. Such high charge states seem to be feasible, since ion sources of the Penning type are expected to produce some 10 t2 U particles/sec with 4o = 11 l). An intensity loss due to the statistical charge exchange must be expected. According to their energy

HEAVY ION ACCELERATOR

distribution, a fraction of ions starts a given passage with energies below E moi n . These ions are decelerated and may be regarded as lost. The intensity loss is most significant during the first passage, where it can be calculated as 1 fq -' ~er c{(E 0 - E 0rain )/(aEo,j2)}.

Here erfc is the complementary Gauss error function and ~rF.o is the energy spread of the ions at the begin of their first passage. For injection mode A the spread is given by the energy spread of the injected particles, and for B it can be obtained by eq. (10'b) given in the appendix. For B with { 0 > 1 1 the intensity loss is calculated as ~ 20°.o and ~ 3°,o for the accelerator types a and b, respectively. In this example the accelerator parameters as given in fig. 12 are used.

5. The intensity of the accelerated particles The transmission 7", of the accelerator model in fig. 5 is defined by the fraction of injected particles having finished their n 'h passage. At a suitable beam injection the particle loss results mainly from scattering in the strippers. To reduce this intensity loss, the beam after passing a stripper has to be focused at the following one by radial focusing elements placed at half stripper distances, for example. Particles are lost, if they are scattered above the aperture of the focusing system (half aperture 0L). For ideal focusing elements the transmission can be calculated by successively folding the angular distributions before stripping with elementary angular distributions f(0) resulting from scattering in the strippers (fig. 3) cut off at 0 L. If the energy spread of the ions is neglected, f(O) can be approximately calculated with the mean energies E, (valid for not too low energies and apertures). It turns out that the transmission decreases during the first passages and then remains constant. For U ions with Eo > 25 MeV a transmission > 0.9 requires apertures 0 L ~ 5 X l 0 - 2 rad and ~ 3 x l 0 - 2 rad for the accelerator systems a and b, respectively. These apertures will be feasibleT). The beam intensity extracted from the accelerator will be discussed in what follows. Let us consider * T h e b e a m extraction m a y be realized for e x a m p l e by an electrostatic deflection device placed between the segments o f a mirror magnet. At an orbit radius p c o r r e s p o n d i n g to the m e a n energy o f particles to be extracted, the deflection angle is proportional p-~E-~ a n d is independent o f the charge state. Low-energy particles within the energy spread at p ( ~ 300 MeV for U), due to the charge state distribution, c a n be extracted, while the other particles remain in the accelerator. t Strictly s p e a k i n g the s p e c t r u m is fractionated in energy c o m p o n e n t s spaced e V apart. This structure, however, will be s m e a r e d out in the accelerator after s o m e cycles.

275

particles with energies between E and E + 6E removed from the actual accelerator*. In the model this corresponds to a simultaneous beam extraction at different points (C, D, E in fig. 5). The beam removed within f E is equal to the particle intensity extracted from the actual accelerator. The total beam /,ot = l i n j ' T n (Ii,,j is the injected beam) can be removed, if fiE is not too small compared to the mean energy gain per passage, and to aE. Some 100 MeV will be appropriate for rE. With 7", ~ 1 and lin j values of some 1012 particles/sec (ch. 4) a beam of ~ l0 l° U particles/sec- MeV may be expected. For ions lighter than U the smaller foil effect demands more cycles in the low energy region. However, this will be compensated to some extent by the smaller angle scattering and energy loss in the strippers due to the smaller nuclear charge. The acceleration of ions with Z i > 50 seems to be feasible. It should be noticed that the transmission has been obtained assuming an ideal reflectivity of the magnetic mirror and a high achromatic focusing quality of the lens system. Both assumptions are expected to be feasible7). Elements of this paper have been presented and discussed within the project group "Heavy Ion Accelerator" at the Kernforschungszentrum Karlsruhe and at Freiburg University. I thank P. Armbruster, K. H. Beckurts, H. Briickmann, E. Heinicke, G. Schatz, A. Schmidt, A. Wolf, G. Hortig, E. R6ssle and Th. Schmidt for discussions and particularly one of them (H. Br.) for discussing the manuscript. Furthermore, I thank S. Idler for numerical calculations.

Appendix The energy distribution of the ions caused by the charge state distribution after stripping is calculated in what follows. A monoenergetic ion beam (energy E0) traverses a stripper of equilibrium thickness, and is accelerated afterwards by a potential V. Its energy distribution is calculated as* dPl/dE = {(2r@aE,}-'exp{--½(E--E,)2/G~.,},

(9)

F-,1 = Eo + eV~l (energy loss in the stripper neglected), GE, = e V ~ 1 ,

where ~ and a 1 are the equilibrium charge state and the width of the charge state distribution, respectively. If the ion beam with the energy width aE, is stripped and accelerated once more by a potential V, its distribution remains Gaussian with the new width (av.~ +

276

H. MIESSNER

ae2) ~, where ae~ = eva2, and a2 refers to the second stripper. The energy dependence o f the charge state is not yet taken into account. In the accelerator model (fig. 5) the energy distribution is Gaussian with the widths

aeo = eVt~g,

(10a)

O¥o = (½x/2)eVag,

(10b)

= {aL+

+

2 GE. -'~ {¢YEo+n'e2V2( (723~ 2 1~2~'t½ ~g/J

aFo = ½eV%[l + {l -½eV(d~g/dE)e,o}2] ", (10'b) E o = Ei, q + e V { o - ½eV~g(ei, j + el/~o), i,-

ae,, = [{a 2. , [1 + eV(d~/dE)E,_,]2 + e 2 V2a2}. • [1-el/(d~g/dE)e,

,] 2 + e 2 V 2 a

~,

(ll'a)

Et~-I = E n - 1 -~-eV~s(E;n-1),

(lla) (1 l b )

where as and ag are the widths of the charge state in solid and gas strippers, respectively (ch. 2). Eq. (10) is valid for particles injected with a single charge state by means of the injection mode B. In the following the energy dependence of ~ is taken into account• The energy distribution before stripping may be written as

ae, = [ { [ a 2

,{1 +eV(d~/dE)e,,

• {l-½eV(d~g/dE)L,_,}2+~el

, } 2 + e 21/.2 2]. 2V2%}.2

distributions

d P o / d E = {(2n)~a~-o } - . iexp { - ½(Eo - fo)'2"/auo?.2 " (12) Neglecting the statistical charge state distribution for the moment, one can calculate the new distribution d P / d E after stripping and acceleration by a potential V by means of the transformations

E = F,o + eV~(Eo) , E = E o + eV~(Eo), with d P - d P o. If ~(E) is assumed to be a linear function of E within the energy spread, which is valid in g o o d approximation, the new distribution is found Gaussian again with a width aE = O-Eo{1+eV(d~/dE)eo}.

([3)

For a decelerating field one obtains

a~ = aEo I 1 - I eVl(d~/dE)Eo j.

(I 4)

The reduction o f the energy spread according to eq. (14) for ] e V [ ( d ~ / d E ) < 2 is not in disagreement with Liouville's theorem, since the ions change their phase spaces at stripping. According to eqs. (13) and (14) the eqs. (10b), (l la) and ( l l b ) must be replaced by

"ll-2eV(d~g/dE)e,,

, } 2 + a 1e 2 1/-2 ag21½ ,

(ll'b)

J