ENERGY DEPOSITION DISTRIBUTIONS FOR MULTIPLE ION-BEAM IRRADIATION OF ICF TARGETS

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ENERGY DEPOSITION DISTRIBUTIONS FOR MULTIPLE ION-BEAM IRRADIATION OF ICF

TARGETS

T. Beynon

To cite this version:

T. Beynon. ENERGY DEPOSITION DISTRIBUTIONS FOR MULTIPLE ION-BEAM IRRADI- ATION OF ICF TARGETS. Journal de Physique Colloques, 1983, 44 (C8), pp.C8-123-C8-135.

�10.1051/jphyscol:1983808�. �jpa-00223315�

JOURNAL DE PHYSIQUE

Colloque C8, suppl6ment au null, Tome 44, novembre 1983 page C8-123

ENERGY DEPOSITION DISTRIBUTIONS FOR MULTIPLE ION-REAM IRRADIATION OF ICF TARGETS

T.D. Beynon

Department of Physics, University of Birmingham, Birmingham B15 ZTT, U.K.

Resum&

Un formalisme est present6 afin d'obtenir la fonction de distribution de ll&nergie de deposition, pour une cible I C F irradiee par des faisceaux d'ions multiples. Des resultats pour une irradiation sym6trique 5 6 faisceaux, montrent que des fluc- tuations larges pourraient avoir lieu 2 cette fonction de dis- tribution et qu'au moins 20 faisceaux sont nccessaires pour adoucir ces effets.

Abstract

A formalism is presented for obtaining the energy deposition distribution function for an ICP target irradiated with multiple ion beams. Results for symmetrical 6-beam

irradiation show that large fluctuations would occur in this distribution function, and that at least 20 beams are required to smooth these effects.

1. Introduction

Hydrodynamic studies of ion beam

-

driven targets for inertial confinement fusion (ICF) require a detailed treatment of the geometry of the multiple beam ( ie

. '

beamlet

'

)

con£ iguration, in addition to the physics description of the slowing down process. The spatial distribution of the energy deposlclon arrslng from beamiec irradiation may well

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983808

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determine t h e subsequent s t a b i l i t y of t h e compression of t h e t a r g e t , p a r t i c u l a r l y i n t h e case of ion beam i r r a d i a t i o n where t h e l a r g e s t p a r t of t h e ion energy is deposited near t h e end of t h e i o n ' s range. Thus, regions of beamlet overlap i n t h e t a r g e t , even when each beamlet has a uniform i n t e n s i t y , can give r i s e t o l o c a l ' h o t s p o t s ' which could d r i v e i n s t a b i l i t i e s . D i r e c t l y

-

d r i v e n t a r g e t s might r e q u i t e a r e l a t i v e l y l a r g e number of beamlets not j u s t t o maximise t h e i r r a d i a n c e symmetry b u t t o minimize t h e e f f e c t s of r a d i a l space charge spreading and s e l f - f i e l d e f f e c t s .

In t h i s paper a d e t a i l e d formalism is presented which allows a t o t a l energy d e p o s i t i o n d i s t r i b u t i o n f u n c t i o n t o be computed p r i o r t o its use i n a s u i t a b l e hydrodynamics code.

Although t h e formalism is g e n e r a l , s p e c i f i c a p p l i c a t i o n s a r e l i m i t e d i n t h i s paper t o s p h e r i c a l l y symmetric t a r g e t s . 2. Theory

The vector f l u x of t e s t p a r t i c l e s , @ ( s , E , g ) , a t energy E , p o s i t i o n vector

2

and t r a n s p o r t u n i t vector

g

i n a medium where only r e c t i l i n e a r Coulomb slowing down can occur, is given by t h e s o l u t i o n of t h e Boltzmann equation

Equation (1) is simply a balance equation i n phase-space f o r t h e s t e a d y - s t a t e 6-vector p a r t i c l e d e n s i t y n ( 5 , ~ ) where y = v f ~ and E = rmc2 f o r a t e s t p a r t i c l e of r e s t mass m. Q ( r , E ,

2)

is t h e

source of such p a r t i c l e s , S(r,E) i s t h e slowing down power of t h e medimum and Ca i s t h e macroscopic a b s o r p t i o n c r o s s

s e c t i o n .

The solution to (1) for a monodirectional and

monoenergetic plane source situated at X = 0, Q = 2(s)-' 8(x) 8 (E-Ed) 8 (/L-/L,), which produces an ion beam, is

~ ( / L - P ~ ) 8(x-xo)

@(~,E,/L) = exp

-

^{d ~ '}

. . .

⁽²⁾

2aS (E)

for a homogeneous medium with the geometry of fig. 1, where

/(1, = cos 0,

Pig. 1 Plane geometry for eqn (2).

The energy current associated with the particle source of unit strength is defined as

C8-126 JOURNAL DE PHYSIQUE

and i s simply, from ( 2 ) , -

J,(x) = E ( X ) exp

-

^dE

where we have used t h e r e l a t i o n s h i p

-

and E(x) is d e f i n e d , f o r a g i v e n X - v a l u e , by t h e e q u a t i o n

From ( 3 )

The d i v e r g e n c e of J E ( x ) r e p r e s e n t s t h e energy

d e p o s i t i o n r a t e per u n i t volume ( o r , s t r i c t l y , t h e energy l o s s r a t e per u n i t volume) of t h e t e s t p a r t i c l e s a t a p o i n t X , W(x).

Assuming a l l energy i s d e p o s i t e d l o c a l l y , we have

...( 5) The f i r s t term i n eqn. (5) is t h e Coulomb c o l l i s i o n a l energy l o s s r a t e per u n i t volume from t h e beam, w h i l s t , t h e second term is t h e energy removal r a t e per u n i t volume from t h e beam due t o t h e absorption process represented by

C , .

Depending on t h e p h y s i c a l n a t u r e of these processes, t h i s may o r may not r e s u l t i n a l o c a l energy d e p o s i t i o n . F i s s i o n , For example, would produce a non-local e l f e c t , &ere t h e f i s s i o n Fragments would have t h e i r own energy-range r e l a t i o n s h i p s which

could be quite different From the ions in the primary beam.

Finally, the exponential term in eqn.(5) represents the probability that the particle has travelled a distance x without being absorbed. The development in this paper continues for

C,

= O and p. = l. Competition from fission processes will be considered in a subsequent paper.

The extension of eqn.(5) to a non-absorbing inhomogeneous material produces the result ( p o = l).

-

where now E ( X ) is the solution to the equation

which may be obtained straight forwardly, for the initial conditions E = Eo, X = 0 , using a Runge-Kutta technique.

3.Extension to multiple beams

In fig. 2 we consider the ith beam of an array of beams traversing the target, and evaluate the total energy deposition rate per unit volume at a point L. The component of the ith beam passes through an element of area d_S = gdS at a position

ai

in cne drreccron

a,

and traverses the distance Li to E . Each beam is considered to have cylindrical symmetry. A vector E is defined, perpendicular to

S,

measured from the centre line of the beam,

ai

= iiio, to define any radial variation of intensity the beam may possess.

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centre of beam

F i g . 2 Vector d e s c r i p t i o n f o r an a r r a y of beams.

If j i ( a i ) d e n o t e s t h e ion beam c u r r e n t e n t e r i n g t h e

M

s u r f a c e a t

ai

( i n ions per u n i t a r e a per u n i t t i m e ) t h e n we can u s e t h e r e s u l t of eqn. ( 4 ) t o w r i t e t h e energy d e p o s i t i o n r a t e per u n i t volume a t

5

f o r t h e ith beam, Wi ( L ) , a s

where ',j = ni v, S.& f o r a beam of p a r t i c l e d e n s i t y n i t i n i t i a l v e l o c i t y v, a t

ai.

The energy E now s a t i s f i e s t h e r e l a t i o n s h i p

We f u r t h e r assume t h a t

i (ci) = j o i ( s i o ) exp

-

(u,ii.u,i/k2)

That is, t h e beam has a g a u s s i a n r a d i a l i n t e n s i t y d i s t r i b u t i o n , c e n t e r e d about t h e beam a x i s of symmetry, The t o t a l energy d e p o s i t i o n f u n c t i o n is now

4 . A p p l i c a t i o n t o s p h e r i c a l oeometrv

For a s p h e r i c a l l y symmetric t a r g e t t h e most convenient c o o r d i n a t e system is t h a t p l a c e d a t t h e c e n t r e of t h e s p h e r e , as i l l u s t r a t e d i n f i g . 3 .

Then

Li = ~ . i l ,

+

_-

+

^{~ ~}

1 ^f

^{- r ~}

and

F i g . 3 Coordinate system a t c e n t r e of s p h e r i c a l t a r g e t .

E v a l u a t i o n of ( 6 ) is now reduced t o p r o v i d i n g a p r e s c r i p t i o n f o r t h e d i r e c t i o n c o s i n e s , _Ro = ( R x , ~ , R , ) , f o r each beam.

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Probably t h e most convenient and u s e f u l approach is t o use t h e archimedean symmetry of a cube, i n s c r i b e d i n t h e s p h e r e . One s e t of beams, s i x i n t o t a l , can be o r i e n t a t e d p e r p e n d i c u l a r t o each of t h e s i x s i d e s of t h e cube w i t h t h e d i r e c t i o n c o s i n e s ( i 1 , 0 , 0 ) , ( 0 * 1 , 0 ) and ( 0 , 0 , * 1 ) . The second symmetry s e t is t h e beams l y i n g along t h e l i n e s j o i n i n g o p p o s i t e v e r t i c e s of t h e ^I cube, g i v i n g a f u r t h e r e i g h t beams w i t h d i r e c t i o n c o s i n e s (*3-$

*3- $, i3- 3). A f u r t h e r twelve symmetric beams may be added by j o i n i n g t h e midpoint of each edge of t h e cube t o t h e c e n t r e p o i n t , w i t h t h e c o s i n e s ( * $2, i $ 2 , 0 ) , (0,*$2,*$2) and (i$2,O, 42) where i s i g n s a r e t o be t a k e n independently. Combination of t h e s e symmetry groups w i l l t h e r e f o r e a l l o w c o n f i g u r a t i o n of 6 , 8 , 12, 14, 18, 20 and 26 beamlets on t a r g e t .

The r e s u l t s of a s e r i e s of t y p i c a l c a l c u l a t i o n s a r e p r e s e n t e d f o r a 6-beam and a 26-beam i r r a d i a t i o n of a

homogeneous s o l i d carbon s p h e r e of r a d i u s R

-

^{l mm. Each beam}

h a s a g a u s s i a n h a l f -width ( k i n eqn. ( 6 ) ) e q u a l t o t h e r a d i u s of t h e t a r g e t and is normalised t o u n i t c u r r e n t ( j i = 1 i n e q n . ( 6 ) ) The i o n s have t h e slowing down power S ( E ) shown i n f i g . 4 which r e p r e s e n t s a h y p o t h e t i c a l heavy i o n of energy 10 GeV w i t h a r a n g e of 0.24 mm i n g r a p h i t e .

F i g u r e 5 d i s p l a y s t h e energy d e p o s i t i o n f u n c t i o n W(r) i n a d i a m e t r a l p l a n e f o r a 6-beam i r r a d i a t i o n . The p l a n e is chosen t o be p e r p e n d i c u l a r t o one p a i r of o p p o s i t e l y d i r e c t e d beams, and c o n t a i n s t h e maximum o v e r l a p e f f e c t of t h e i n c i d e n t beams.

A number of o b s e r v a t i o n s can be made. F i r s t l y , t h e d e p o s i t ion d i s t r i b u t ion w i t h t h e minimum number of beamlets c o n t a i n s g r o s s modulations w i t h amplitudes v a r y i n g by up t o a f a c t o r two, a r e s u l t which is r e p e a t e d i n p l a n e s p a r a l l e l t o t h e

X ( m m )

Fig. 4 The slowing down power S(E) used in calculations.

- 1

.0

' - 1

.0

Fig. 5 Deposition function (in arbitrary units) in diametral plane for a 6-beam irradiation.

C8-132 JOURNAL DE PHYSIQUE

d i a m e t r a l p l a n e . Secondly, f i n e s t r u c t u r e is e v i d e n t i n t h e d i s t r i b u t i o n because of t h e p e n e t r a t i o n of t h e two beams p e r p e n d i c u l a r t o t h e p l a n e a t d i s t a n c e s r

>

( R ~ - x ~ ) ~ L- = 0 . 9 4 mm, where X is t h e ion range. F i n e s t r u c t u r e is a l s o produced by o v e r l a p of t h e remaining f o u r beams. T h i s e f f e c t i s shown i n f i g u r e 6 , which i s t h e d i s t r i b u t i o n c o n t a i n e d i n a r e a ABCD of f l g u r e 7 .

F i g . 6 D e p o s i t i o n f u n c t i o n ( i n a r b i t r a r y u n i t s ) i n one q u a d r a n t ( a r e a ABCD of f i g 7 ) of d i a m e t r a l p l a n e f o r a 6-beam i r r a d i a t i o n .

F i g . 7 D e f i n i t i o n of v a r i o u s a r e a s of d i a m e t r a l plane.

F i g u r e s 8 and 9 c o n t r a s t t h e energy d e p o s i t i o n f u n c t i o n i n t h e same p l a n e i n area aBbe ( f i g 7 ) f o r a 6-beam and 26- beam i r r a d i a t i o n , r e s p e c t i v e l y . I t is s e e n f o r t h e l a t t e r c a s e

t h a t t h e f i n e - s t r u c t u r e f l u c t u a t i o n s have d i s a p p e a r e d a l t h o u g h t h e g r o s s v a r i a t i o n p e r s i s t s . S i m i l a r e f f e c t s occur i n p l a n e s p a r a l l e l t o t h e diametr a 1 p l a n e cons i d e r ed

.

10.0'.0

F i g . 8 D e p o s i t i o n f u n c t i o n ( i n a r b i t r a r y u n i t s ) i n a r e a aBbe ( c f . f i g 7 ) f o r a 6-beam i r r a d i a t i o n .

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F i g . 9 Deposition f u n c t i o n ( i n a r b i t r a r y u n i t s ) i n a r e a aBbe ( c f . f i g 7 ) f o r a 26-beam i r r a d i a t i o n .

5 . Conclus ions

I t is apparent from t h e s e r e s u l t s t h a t i n e v a l u a t i n g t a r g e t performance t h e beamlet geometry is a s important a s t h e d e t a i l e d d e s c r i p t i o n of t h e slowing down processes contained i n S ( E )

.

^A f ew-beam i r r a d i a t i o n produces r a p i d l y f l u c t u a t i n g energy d e p o s i t ion p r o f i l e s , which a r e s i g n i f i c a n t l y smoothed when more t h a n about 20 beams a r e used, a s is t h e case f o r t h e HIBALL t a r g e t design ( 1)

.

Q u a n t i t a t i v e e s t i m a t e s of t h e e f f e c t s of t h e s e f l u c t u a t i o n s r e q u i r e a d e t a i l e d a n a l y s i s with a s u i t a b l e e u l e r i a n hydrodynamics code. Moreover, it appears t h a t a f u l l three-dimensional approach would be necessary. Some preliminary two-dimensional a n a l y s i s of t h i s problem has been reported by

Buchwald et a1(2) but the present results would seem to indicate that considerable ambiguity could exist as a result of the particlar choice of planar slice taken through the target for such two dimensional modelling. Additionally, the shape o$

W(g) would not remain invariant during the compress ion stage of the target, particularly if localised ion range shortening or

lengthening occurs in regions of high temperature and dens ity

.

Acknowledaement

This work is part of a program in ion beam fusion supported by the Science and Engineering Research Council.

References

l. HIBALL. A Joint Federal Republic of Germany

-

University of Wisconsin Conceptual Heavy Ion Beam Fusion Reactor Design Study. Kernforschungszentrum Karlsruhe. Rep. No. KfK-3202.

1981.

2. BUCHWALD, G, KRUSE H., MARUHN, J.A. and STOCKER H in Proc. Symp. on Accelerator Aspects of Heavy Ion Fusion, Darmstadt, 1982. Rep. No. GS 1-82-8.