ABLATION OF A DEUTERIUM PELLET IN A FUSION PLASMA VIEWED AS A STOPPING POWER PROBLEM

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ABLATION OF A DEUTERIUM PELLET IN A FUSION PLASMA VIEWED AS A STOPPING

POWER PROBLEM

C. Chang

To cite this version:

C. Chang. ABLATION OF A DEUTERIUM PELLET IN A FUSION PLASMA VIEWED AS A STOPPING POWER PROBLEM. Journal de Physique Colloques, 1983, 44 (C8), pp.C8-17-C8-23.

�10.1051/jphyscol:1983802�. �jpa-00223309�

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JOURNAL DE PHYSIQUE

Colloque C8, suppl6ment au nO1l, Tome 44, novembre 1983 page C8-17

A B L A T I O N OF A DEUTERIUM P E L L E T I N A F U S I O N PLASMA VIEWED AS A STOPPING POWER PROBLEM

C . T . Chang*

Association EURATOM-CEA sur Za Fusion, Ddparternent de Recherches sur Za Fusion Contro^Zde, Centre d 'Etudes NucZdaires, B. P. n O6,

92260 Fontenay-am-Roses, France

I . Resume

L ' a b l a t i o n d ' u n p e l l e t i n t r o d u i t dans un r e a c t e u r

a

confinement ma- gnetique e s t essentiellement provoqueepar l e s impacts d161ectrons thermiques.

E t a n t donnee l a f a i b l e chaleur de s u b l i m a t i o n des isotopes de l'hydrogsne, il se forme autour du p e l l e t , un nuage dense de m a t i e r e ablatee q u i a g i t comme un tampon r a l e n t i s s e u r v i s i3 v i s des e l e c t r o n s i n c i d e n t s , e t prolonge l a duree de v i e du p e l l e t . Ce q u i permet, d ce d e r n i e r , d ' a t t e i n d r e l e c e n t r e du reac- t e u r .

A b s t r a c t

A t present, t h e most e x p l o i t e d technology t o r e f u e l a f u t u r e f u s i o n r e a c t o r i s t h e h i g h speed i n j e c t i o n o f macroscopic s i z e p e l l e t o f s o l i d hydrogen isotopes. The b a s i c idea i s t h a t t h e a b l a t i o n o f a p e l l e t i n a f u s i o n r e a c t o r i s m a i n l y caused by thermal e l e c t r o n s (a 10 keV) /l/. Due t o t h e low sublimation energy o f hydrogen isotopes, s h o r t l y a f t e r t h e d i r e c t impact o f t h e e l e c t r o n s , a dense c l o u d forms around t h e p e l l e t . This c l o u d o f a b l a t e d m a t e r i a l then serves as a stopping medium f o r t h e incoming e l e c t r o n s , thus prolongs t h e p e l l e t l i f e - t i m e . As a r e s u l t , t h e deep p e n e t r a t i o n o f t h e p e l l e t i n t o t h e r e a c t o r center becomes possible.

11. BRIEF REVIEW OF EXPERIMENTAL OBSERVATIONS

As a t y p i c a l example o f such e f f o r t s , we might quote the r a t h e r r e c e n t experimental r e s u l t s o f p e l l e t i n j e c t i o n i n t o t h e ISX-B tokamak a t t h e Oak Ridge National Laboratory, where a s o l i d hydrogen p e l l e t w i t h an e q u i v a l e n t s p h e r i c a l diameter o f 1 mm i s i n j e c t e d a t a speed o f 1 km/s i n t o a discharge e i t h e r w i t h o r w i t h o u t t h e presence o f a d d i t i o n a l n e u t r a l beam i n j e c t i o n /2/.

*on leave o f absence from t h e Physics Department, Risoe National Laboratory, Association EURATOM, DK-4000 Roskilde, Denmark

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983802

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JOURNAL DE PHYSIQUF:

I n t h e Ohmic-heated d i s c h a r g e s w i t h o u t t h e n e u t r a l beam i n j e c t i o n , t h e p e l l e t a b l a t i o n r a t e depends much on t h e r a t i o o f t h e t o t a l number o f p a r t i c l e s c o n t a i n e d i n t h e p e l l e t , Np, w i t h r e s p e c t t o t h a t c o n t a i n e d i n t h e plasma volume o f t h e t o r u s , N. F o r N /N,< 20 % and a t a c e n t r a l e l e c t r o n

P

t e m p e r a t u r e T e ( o )

,<

1 keV, t h e p e l l e t a b l a t i o n r a t e agrees r e a s o n a b l y w e l l w i t h t h a t p r e d i c t e d b y t h e " n e u t r a l - s h i e l d i n g " model /3/. F o r N /N

>

50 %, a d d i t i o n a l s h i e l d i n g e f f e c t i s observed, e s p e c i a l l y a t l o w e r v a l u e s o f Te(o) P

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700 eV), t h e p e l l e t p e n e t r a t e s beyond t h e magnetic a x i s , and a t h i g h e r v a l u e s o f N /N, even h i t s t h e w a l l o f t h e t o r o u s o p p o s i t e t o t h a t where t h e

P

p e l l e t i s i n j e c t e d . However, i f an a d d i t i o n a l a d i a b a t i c c o o l i n g e f f e c t o f t h e background plasma i s i n c o r p o r a t e d i n t o t h e " n e u t r a l - s h i e l d i n g " model, t h e a d d i t i o n a l s h i e l d i n g e f f e c t can be e x p l a i n e d , a t l e a s t q u a l i t a t i v e l y . L a s e r h o l o g r a p h i c i n t e r f e r o m e t r y s t u d i e s r e v e a l e d t h a t i n h i g h t e m p e r a t u r e d i s c h a r g e s ( T e ( o ) l keV, Fe i 1 0 ~ ~ c m - ~ , b e f o r e t h e p e l l e t i n j e c t i o n ) t h a t t h e a b l a t e d vapor even i n t h e v i c i n i t y o f t h e p e l l e t i s h i g h l y i o n i z e d (degree o f i o n i z a t i o n ?. 20 % ) /4/.

A l t h o u g h t h e mechanism o f i o n i z a t i o n o f t h e a b l a t a n t i s n o t y e t understood, t h e v a l i d i t y o f t h e n e u t r a l s h i e l d i n g model under such c i r c u m s - t a n c e s i s r a t h e r q u e s t i o n a b l e .

I n d i s c h a r g e s w i t h t h e presence o f a d d i t i o n a l n e u t r a l beam h e a t i n g (375

-

900 kW, 40 keV, H" +

D ' ) ,

t h e p e l l e t n o t o n l y a b l a t e s a t a much f a s t e r r a t e t h a n t h a t p r e d i c t e d b y t h e " n e u t r a l - s h i e l d i n g " model, b u t t h e r e seems a l s o t o e x h i b i t two modes o f a b l a t i o n /2/.

111. BASIC CONCEPT OF THE "NEUTRAL-SHIELDING" MODEL AND ITS LIMITATIONS While t h e r e a r e many v e r s i o n s o f t h e " n e u t r a l - s h i e l d i n g " model, t h e b a s i c concept i s more o r l e s s t h e same / l / . F o r convenience, o u r d i s c u s - s i o n s w i l l be based m a i n l y on t h e v e r s i o n f o r m u l a t e d b y Parks and T u r n b u l l /5/.

F o r t h e s i m p l i c i t y o f d i s c u s s i o n , assuming t h e a b l a t i o n o c c u r s u n i f o r m l y o v e r t h e s u r f a c e o f a s p h e r i c p e l l e t o f r a d i u s r t h e mass a b l a t i o n r a t e , G t h e n i s r e l a t e d t o t h e t o t a l h e a t o f a b l a t i o n , H and t h e energy f l u x , P' qp, r e c e i v e d a t t h e p e l l e t s u r f a c e by

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I n p r i n c i p l e , once H and q a r e known, G can be c a l c u l a t e d . I n t h e model concerned, /5/, H z_ hb, qp P 2. a r e taken, i . e . t h e t o t a l h e a t o f a b l a t i o n i s taken as t h e s u b l i m a t i o n energy, hb, and t h e energy f l u x r e c e i v e d a t t h e p e l l e t s u r f a c e i s t h a t due t o t h e e l e c t r o n s o n l y . F u r t h e r s i m p l i f i c a t i o n i s o b t a i n e d b y r e p l a c i n g t h e thermal e l e c t r o n s by an e q u i v a l e n t beam o f e l e c t r o n s a t an energy E = 2 kT and a number d e n s i t y c o r r e s p o n d i n g t o t h e same energy f l u x . Owing t o t h e low s u b l i m a t i o n energy o f s o l i d hydrogen, s h o r t l y a f t e r t h e d i r e c t impact o f t h e incoming e n e r g e t i c e l e c t r o n s , a dense a b l a t e d c l o u d forms a l m o s t i m m e d i a t e l y around t h e p e l l e t . T h i s c l o u d of t h e a b l a t e d vapor p r o v i d e s a s h i e l d i n g a g a i n s t f u r t h e r p e n e t r a t i o n o f t h e incoming e l e c t r o n s . The a t t e n u a t i o n o f t h e energy f l u x q ( s ) a l o n g t h e t r a j e c t o r y ( S ) o f t h e incoming e l e c t r o n s can t h e n be considered t o be a s u p e r p o s i t i o n of t h e d e p l e t i o n o f t h e incoming p a r t i c l e f l u x @ ( E ( s ) ) caused b y back s c a t t e r i n g and t h e a t t e n u a t i o n o f t h e i n c i d e n t e l e c t r o ~ energy E(s) t h r o u g h t h e " s l o w i n g - down" process i n t h e a b l a t e d vapor. Consequently, t h e energy f l u x q a t t h e

P * p e l l e t s u r f a c e can be r e l a t e d t o i t s v a l u e qa i n t h e u n p e r t u r b e d plasma b y

The v a r i a t i o n o f E and @ a l o n g t h e incoming p a r t i c l e t r a j e c t o r y i s g i v e n by

d E nts) L (

€ ( S ) )

dS- < case>

where n ( s ) i s t h e number d e n s i t y o f t h e a b l a t e d p a r t i c l e s , L ( E ) i s t h e s t o p p i n g c r o s s s e c t i o n ( o r t h e s t o p p i n g power) and t h e <cosO> terms accounts f o r t h e average p i t c h a n g l e o f t h e incoming e l e c t r o n s w i t h r e s p e c t t o t h e magnetic f i e l d l i n e s . o T ( E ) i s t h e t o t a l s c a t t e r i n g c r o s s s e c t i o n ( s i n g l e p i t c h a n g l e p l u s c u m u l a t i v e s m a l l a n g l e d e f l e c t i o n s ) .

I n p r i n c i p l e , when one has t h e f u l l knowledge o f t h e l o c a l p a r t i c l e d e n s i t y n ( s ) o f t h e a b l a t a n t and t h e c r o s s s e c t i o n s L ( E ( s ) ) and o T ( E ( s ) ) , t h e p e l l e t a b l a t i o n r a t e , as w e l l as i t s s c a l i n g law w i t h r e s p e c t t o t h e plasma parameters and p e l l e t r a d i u s , can be d e r i v e d . One r e c a l l s t h a t as t h e i n c i d e n t p a r t i c l e slows down i n t h e a b l a t a n t , i t engages i n v a r i o u s c o l l i s i o n a l

processes t h e c o m p o s i t i o n o f t h e a b l a t a n t changes b o t h i n t i m e and i n space.

To a v o i d such a c o m p l e x i t y , i t was assumed f i r s t i n t h e model / 5 / t h a t a quasisteady s t a t e i s a1 ready reached. As a f u r t h e r s i m p l i f i c a t i o n , i t was i m p l i c i t l y assumed t h a t a l l t h e i n e l a s t i c c o l l i s i o n a l processes o c c u r i n a

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C8-20 JOURNAL DE PHYSIQUE

region very thin compared with the total volume of the ablated cloud, and can be neglected as far as the attenuation of the incoming electron energy flux is concerned (when properly adjusted values of Em and qa, are taken).

As a result, the ablatant is considered to be neutral and the expansion is spherically symmetry

;

the loss function L(E) then is taken as that of the slowing down of electrons in a molecular hydrogen gas

/6/.

Even with these simplifications, one notices that the problem concerned still differs from the stopping power problem in solids by the fact that the density n(s) of the medium is regulated by the hydrodynamic process of expansion and heating of the cloud which in turn depends on the attenuated energy of the incoming electrons.

As an illustrative example, the result of such a shielding model based on the slowing down of electrons in a spherical cloud composed of mole- cular hydrogen is shown in Fig.1. One observes that most of the incoming electron energy is absorbed in a thin region less than one pellet radius in thickness.

Since the loss function L(E) depends mainly on the logarithm of the maximum impact parameter, log,(bmax), one may argue that even if ioni- zation occurs rather earlier in the expansion process, as a first approximation one may still use the loss function L(E) of a medium composed entirely of neutrals for the whole region. However, once the degree of ionization becomes appreciable in the pellet vicinity, through the influence of the magnetic field, the expansion of the ablated cloud loses its spherical symnetry.

This phenomena seems to be observed experimentally

/4/.

As an additional remark, one notices that Eq.(3) indicates that the slowing down of the incoming electrons not only depends on the loss function, L(E), but also on the local particle density n(s) of the ablatant.

Even if L(E) and aT(E) are not much influenced by the composition of the ablatant, atomic processes (thus the composition of the medium) can influence the hydrodynamic process of expansion of the cloud, thus modify n(s).

(It should be remembered that whether the back scattering cross section oT(E) is nearly the same in a neutral gas and in a plasma is still to be examined

! ) .

As an example to illustrate the effect of atomic process on the local particle

density n(s) and thus the pellet ablation rate, we have retained the main

features of the "neutral-shielding" model and made a detailed study by assuming

that the ablatant is completely dissociated .once passed beyond the sonic radius

but becomes ionized only in the region near the cloud boundary. Computational

results based on such a model showed that as a result of the dissociation

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F i g .

- -- 1 : Oimensionless flow parameters o f t h e ablatant, incident electron energy and energy flux versus dimensionless radius R/R

.

Incident electron eneray at the sonic radius, E = 3 x 1 0 eV (LT: = 1 .68x104 eV). The surface o f t h e

elle et

is located a t R/Rx = 0.6285.

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C8-22 JOURNAIL DF PHYSIQUF

e f f e c t , t h e p a r t i c l e d e n s i t y d i s t r i b u t i o n n ( s ) i s c o n s i d e r a b l y m o d i f i e d , t h e p e l l e t a b l a t i o n r a t e compared w i t h t h a t w i t h o u t t h e d i s s o c i a t i o n e f f e c t i s reduced by % 40 % / 7 / .

I V . THE EFFECT OF IONS

The e l e c t r o n i c s t o p p i n g power S, o f i o n s depends on t h e i o n v e l o c i t y v w i t h r e s p e c t t o t h e Bohr v e l o c i t y v, = e 2 / h . F o r v

>

vo, S, ^%E-' (Bethe), f o r v

<

vo, Se ^bE ( L i n d h a r d and S c h a r f f / 8 / ) . F o r deuteron, t h i s t r a n s i t i o n o c c u r s a t an i o n energy o f

? >

⁵⁰keV. Recent experiments conducted a t Risoe /9/

i n d i c a t e d t h a t f o r i o n s o f hydrogen and deuterium, i n t h e energy range o f 1

-

10 keV, t h e s t o p p i n g power i n s o l i d Hp o r D2 agrees w e l l w i t h t h a t p r e d i c t e d

%

b y L i n d h a r d and S c h a r f f /8/, t h u s f o r D+ w i t h E

<

E, we may t a k e

Clr

A t i o n energy E

<

E, t h e n u c l e a r s t o p p i n g power Sv i s comparable t o t h e e l e c t r o n i c s t o p p i n g power, Se, t h e t o t a l s t o p p i n g power, S = S, + Se, can t h e n be t a k e n a p p r o x i m a t e l y as

The s t o p p i n g power o f e l e c t r o n s i n s o l i d Hp ( w i t h i n c i d e n t

e l e c t r o n energy 0.5

<

E

<

100 keV) based on t h e e x p e r i m e n t a l d a t a o f Spirensen and Schou /10/ can be f i t t e d e m p i r i c a l l y by

F o r i n c i d e n t e l e c t r o n s and deuterons o f t h e same i n c i d e n t energy, a t E = 10 keV, Eqs ( 6 ) and ( 7 ) g i v e S % 6 X 1 0 - l ~ and L N 2 x 10 -16 ev/cm2

S

r e s p e c t i v e l y .

The s t o p p i n g power o f e l e c t r o n L s ( E ) i n s o l i d hydrogen agrees q u i t e w e l l w i t h L(E) o f e l e c t r o n s i n a m o l e c u l a r hydrogen gas g i v e n by M i l l s e t a l . /6/. Owing t o t h e l o w b i n d i n g energy o f s o l i d

H2

( o r s o l i d D2), we may e x p e c t t h a t t h e s t o p p i n g power o f d e u t e r o n s i n a H2-gas s i m i l a r l y w i l l n o t d i f f e r t o o much f r o m t h a t i n a s o l i d H2.

I n a f u s i o n r e a c t o r , a t t h e same t e m p e r a t u r e and p a r t i c l e d e n s i t y , t h e energy f l u x o f e l e c t r o n s i s g r e a t e r t h a n t h a t o f i o n s by t h e f a c t o r (mi/tne)1/2 ^{( S}60 f o r d e u t e r o n s ) . F o r t h e sake o f s i m p l i c i t y , t h e e f f e c t o f

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i o n s m i g h t be n e g l e c t e d . I n p r e s e n t day l a r g e tokamaks where n e u t r a l beam i n j e c t i o n o f t e n t a k e s p l a c e , a t t h e plasma edge where t h e e l e c t r o n energy and energy f l u x b o t h a r e low, t h e e f f e c t o f i o n s w i t h energy i n t h e range o f

10

-

30 keV m i g h t be o f c o n s i d e r a b l e importance. The r e p o r t e d occurence o f two modes o f a b l a t i o n i n t h e ISX-B experiments,when n e u t r a l beam i s p r e s e n t , seems t o s u p p o r t t h i s argument.

As a s e p a r a t e check o f t h e s e f i n d i n g s i n t h e ISX-B experiments, we m i g h t m e n t i o n t h a t p e l l e t i n j e c t i o n experiments c u r r e n t l y a r e underway a t Fontenay-aux-Roses.

ACKNOWLEDGEMENT :

The a u t h o r i s much i n d e b t e d t o Dr. H.W. Drawin f o r h i s s t i m u l a t i n g d i s c u s s i o n s and comments.

REFERENCES :

- -

/I/ Chang, C.T. e t a l . , N u c l e a r F u s i o n - 20 (1980) /2/ M i l o r a , S.L. e t a1

.,

N u c l e a r F u s i o n 20 (1980) 1491. -

/3/ M i l o r a , S.L. and F o s t e r , C.A., IEEE Trans. Plasma S c i 6 (1978) 578.

/4/ Thomas C.E. "An Experimental I n v e s t i g a t i o n o f S o l i d Hydrogen p e l l e t A b l a t i o n i n High-Temperature Plasmas u s i n g H o l o g r a p h i c I n t e r f e r o m e t r y and o t h e r D i a g n o s t i c s " , Oak Ridge N a t i o n a l L a b o r a t o r y Report, ORNL/TM-7486

(1981).

/5/ Parks, P.B. and T u r n b u l l , R.J., Phys. F l u i d s

g

(1978) 1735.

/ 6 / M i l l s , W.T., Thornpson, R., Green, A.E.S., J. App. Phys.

43

(1972) 678.

/7/ Chang, C.T. "The e f f e c t o f a t o m i c processes on t h e n e u t r a l s h i e l d i n g model of a r e f u e l l i n g p e l l e t " ( t o be p u b l i s h e d ) .

/8/ Lindhard, J. and S c h a r f f , M., P h y s i c a l Review 124 (1961) 128.

/9/ Bgrgesen, P. and Sprensen, H., N u c l e a r I n s t r u m e n t s and Methods

00

(1982) 571.

/10/ Schou, J . and S$rensen, H., J. Appl. Phys.

9

(1978) 816