IONS STOPPING IN DENSE AND HOT MATTER

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IONS STOPPING IN DENSE AND HOT MATTER

C. Deutsch, G. Maynard, H. Minoo

To cite this version:

C. Deutsch, G. Maynard, H. Minoo. IONS STOPPING IN DENSE AND HOT MATTER. Journal

de Physique Colloques, 1983, 44 (C8), pp.C8-67-C8-92. �10.1051/jphyscol:1983805�. �jpa-00223312�

JOURNAL DF: PHYSIQUE

Colloque C8, supplQment au n O 1 l , Torne 44, novenlbre 1 9 8 3 page C8-67

IONS STOPPING I N D E N S E AND HOT MATTER C . D e u t s c h , G. Maynard and 11. Minoo

Laboratoirse de Pizycique des PZcrsrnas*, Bdtimsvrt 272, i!nivercitc' Paris Xl, 92105 O r s a y Cedex, France

Resume : A p r e s a v o i r m o n t r c q u e l e r a l e n t i s s e m e n t d e f a i s c e a u x i n t e r n e s d ' i o n s n o n r e l a t i v i s t e s d a n s l a m a t i e r e c h a u d e e t d e n s e , e s t e s s e n t i e l l e m e n t r 6 d u c t i b l e h l ' i n t e r a c t i o n d ' i o n s i s o l 6 s a v e c l a c i b l e , o n t r a i t c c o m p l e t e m e n t l e f r c i n a g c p r o v o q u f p a r d e s C l e c - t r o n s l i b r e s , d e d 6 g 6 1 1 6 r e s c e n c e a r b i t r a i r c ( t e m p f r a t u r e q u e l c o n q u e ) ,

l ' a i d e d ' u n e f o n c t i o n d i b l e c t r i q u e RPA e x a c t e . P o u r l a p r e m i e r e f o i s , l e r a l e n t t s s e m e n t e s t d o n n k p o u r t o u t e s l c s v i t e s s e s d e s i o n s p r o j e c t i l e s . L e s e f f e t s d e t e m p 6 r a t u r e s o n t i m p o r t a n t s l o r s q u e l ' 6 n e r g i e d e s i o n s i n c i d e n t s e s t i n f 6 r i e u x - e h 5 M e V / n u c l C o n .

E n s u i t e , n o u s p r 6 s e n t o n s u n c f o r m u l a t i o n a n a l y t i q u e e t

c o m p a c t e p o u r l e s c o r r e c t i o n s e n

z 3

( e f f e t B a r k a s ) d u e s a u x 6 l c c t r o n s l i 6 s a u x i o n s n o n h y d r o g 6 n o i d e s d e l a c i b l e . L a s t r u c t u r e e l e c t r o n i q u e d e c e s d e r n i e r s i n f l u e b e a u c o u p s u r l c s r b s u l t a t s , q u i p e u v e n t

a u g m e n t e r d e 3 0 % l ' h a b i t u e l r a l e n t i s s e m e n t d e B o h r - B e t h e - B l o c h .

A b s t r a c t : We p a y a s p e c i a l a t t e n t i o n t o t h c s t o p p i n g o f n o n r e l a t i v i s t i c p o i n t l i k e i o n s i n d e n s e a n d h o t m a t t e r . F i r s t , we c o n s i d e r t h e f r e e e l e c t r o n c o n t r i b u t i o n , t a k e n i n t h e RPA a p p r o x i m a t i o n w i t h a n e x a c t d y n a m i c d i e l e c t r i c f u n c t i o n , v a l i d a t any t e m p e r a t u r e . Therefore, we o b t a i n s t o p p i n g p o w e r a n d s t r a g g l i n g f o r a n y p r o j e c t i l e v e l o c i t y . T h e t e m p e r a t u r e d e p e n d e n c e i f o f a s p e c i a l . r e l e v a n c e f o r a p r o j e c t i l e e n e r g y s m a l l e r t h a n 5 M e V 1 a . m . v .

N e x t , we r e v i s i t e t h e B a r k a s e f f e c t ( z 3 c o r r e c t i o n s ) t h r o u g h a n o v e l a n d c o m p a c t f o r m u l a t i o n , w h i c h i s b a s e d o n a n a n a l o g y w i t h e l e c t r o n i m p a c t b r o a d e n i n g t h e o r y . T L a l l o w s t o i n c l u d e e a s i l y t h e n o n h y d r o g e n i c a n d e l e c t r o n i c s t r u c t u r e o f t h e ~ a r g e t i o n s , i n a much e n h a n c e d s e l e c t i v e f a s h i o n . T h e r e s u l t s may i n c r e a s e t h e u s u a l z ' - s t o p - p i n g by 1 5 u p t o 3 0 p e r c e n t c o r r e c t i o n s .

*Associ& au C . N . R . S .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983805

JOURNAL DF PHYSIQUE

1 . I N T R O D U C T I O N

W i t h t h e g r o w i n g a t t r a c t i v e n e s s o f i o n b e a m s a s a n i n t e r t i a l c o n f i n e m e n t f u s i o n ( I C E ) d r i v e r , w e a r e c u r r e n t l y w i t n e s s i n g a n e w and e n l a r g e d i n t e r e s t i n e n e r g y l o s s e s a n d s t r a g g l i n g o f n o n r e l a t i v i s t i c c h a r g e s i n d e n s e a n d h o t m a t t e r . I n c o n t r a d i s t i n c t i o n t o t h e h i g h l y n o n linear' c o u p l i n g e n c o u n - t e r e d i n t h e l a s e r - d e n s e p l a s m a i n t e r a c t i o n , a t the c r i t i c a l d e n s i t y , t h e i o n - b e a m t a r g e t i s e x p e c t e d to d i s p l a y a m o s t l y

" c l a s s i c a l " b c h a v i o r 2 m o n i t o r e d by w e a k b u t n u m e r o u s C o u l o m b c o l l i s i o n s b e t w e e n a p r o j e c t i l e i o n a n d t h e e l e c t r o n s , f r e e o r b o u n d i n t h c d e n s e m e d i u m .

T h i s r a t h e r p e d e s t r i a n a p p r o a c h t o t h e b e a m - p e l l e t c o u p l i n g b r i n g s in t h e p o s s i b i l i t y o f a c c u r a t e c a l c u l a t i o n f o r t h e i o n s r a n g c s a n d e n e r g y d e p o s i t i o n p r o f i l e 3 in a g i v e n t a r g e t W o r e o v e r , i n t e g r a t i n g t h e s e e l e m e n t a r y e v e n t s o n a p e l l e t r a d i u s d u r i n g a c o m p r e s s i o n t i m e o f t h e o r d e r o f a f e w n s e c (IO-' sec) a l l o w s , t h r o u g h a p p r o p r i a t e h y d r o d y n a m i c a l c o d e s 4

,

to o p t i m i z e t h e b e a m c h a r a c t e r i s t i c s : e m i t t a n c e , d e n s i t y , e n e r g y , p u l s e s h a p e , etc., i n o r d e r t o a c h i e v e a g i v e n c o m p r e s s i o n .

I n t h i s a r e a , t h e p r e s e n t e m p h a s i s l i e s m o s t l y o n s t o p p i n g c h a r a c t e r i s t i c s o f a d e n s e a n d h o t p l a s m a w i t h a n e l e c t r o n t e m p e r a t u r e c o m p a r a b l e o r s m a l l e r t h a n the F e r m i o n e . T h i s n e w s i t u a t i o n r a i s e s t h e o b v i o u s q u e s t i o n o f h o w t o e x t r a p o l a t e t h e u s u a l l o w - t e m p e r a t u r e (kgT

< <

E F ) e s t i m a t e s .

T o f u l f i l 1 t h e s e g o a l s , w e f i r s t c o n s i d e r a c o m p l e t e a n d n u m e r i c a l l y e x a c t s o l u t i o n f o r t h e e n e r g y l o s s a n d s t r a g g l i n g o f s w i f t n o n r e l a t i v i s t i c i o n s i n a v e r y d e n s e e l e c t r o n f l u i d o f a r b i t r a r y d e g e n e r a c y a n d f o r a n y i o n -

v e l o c i t y - F e r m i - v e l o c i t y r a t i o V / V F . T h i s p r o b l e m h a s a l r e a d y r e c e n t l y r e c e i v e d c o n s i d e r a b l e a t tenti~n''~'~.

T h e simplest idealization of the compressed pellet would c o n s i s t in a multicomponent system of partially degenerate e l e c t r o n s , in the presence of several classical ion species.

Technically s p e a k i n g , s u c h a m o d c l is s t i l l too general to a l l o w for simple and accurate computations of the stopping of nonrelativist ic ions.

S o , one h a s to introduce the additional a s s u m p t i o n of weak c o u p l i n g , w i t h a mean Coulomb potential energy smaller than the kinetic one.

T h i s leads us to treat the dense electron fluid in the R P A , w h i l e the classical ions are expected to play a negligible role in the stopping processes, a s long a s the projectile energy remains larger than the thermal electron velocity in the com- pressed medium.

It should be kept in mind that in Heavy Ion Fusion (HIF!, the temperature of the compressed plasma (T

<

200 CV) is not

e -

supposed to be large enough to secure a complete stripping of the heavy elements (charge number Zi) building up the target.

T h e r e f o r e , the remaining bound electrons a r e likely to contri- bute significantly to the stopping of the incoming beam.

T h i s explains that w e c o n s i d e r here two typical a s p e c t s of the physics of nonrelativistic ions in interaction with dense and hot matter.

We first pay an attention to the stopping and straggling i n a partially degenerate electron fluid. S e c o n d , w e show that the Z 3 -contribution to the stopping by bound electrons is non- negligible in high-Z. material. Z is the c h a r g e n u m b e r of the incoming ions.

C8-70 JOURNAL DFI PHYSlQUF

2 . A B I T O F NUMEROLOGY

A l l H I F s c e n a r i o s c l a i m t h a t c u r r e n t d e n s i t i e s u p t o 1 0 k i l o a m p s / c m 2 a r e r e q u i r e d t o a c h i e v e a b r e a k e v e n 1

.

N e v e r t h e l e s s , e v e n i n t h e s e u n u s u a l c o n d i t i o n s , t h e a v e r a g e i o n - i o n d i s t a n c e i n t h e b e a m r e m a i n s m u c h l a r g e r 3 t h a n t h e e l e c t r o n f l u i d s c r e e n i n g l e n g t h s , d i s p l a y e d i n T a b l e 1 . T h e s e l a t t e r a r e d e d u c e d e i t h e r f r o m D e b y e - H i i c k e l t h e o r y w i t h

1 . 8 4

a n d t h e F e r m i e n e r g y E =

-

^{3 - 1}

F r 2

I r s = ( = ,

a o

1

o r f r o m t h e T h o m a s - F e r m i e x p r e s s i o n

-

( 0 . 6 1 r S 2 ,a t l o w e n o u g h t e m p e r a t u r e . T h e s e d a t a r e m a i n a l w a y s m u c h s m a l l e r t h a n t h e i o n i n t e r p a r t i c l e d i s t a n c e s i n t h e b e a m s c o n s i d e r e d i n T a b l e 2 .

TABLE 1 : S c r e e n i n g l e n g t h s ( X ) i n d e n s e e l e c t r o n f l u i d

T A B L E 2: Beam parameters (maximum power 250 T W / c m 2 ) proton

~ i ~ +

T h i s simple c o n s i d e r a t i o n allows us to reduce the beam- target interaction to a n ion-target o n e , by neglecting collec- tive a s p e c t s in a first approach.

T h i s welcome simplification should nevertheless be taken w i t h a grain of salt for the case of protons.

-

v

C

.05-. 15 - 3

T h e corresponding current densities may w e l l range up to megaampslcm 2

,

s o that s o m e c a u t i o n should be exercized.

T h e several characteristic domains in the log T-log n plane are depicted in Fig. 1 .

E

1-10 M e V 3 0 GeV

F I G U R E 1: Relevant domains to HIF in the T-n plane Interparticle

d i s t a n c e 6 0 - 9 0 A

2500 A

.

JOURNAL DE PHYSIQUE

It i s worthwhile to recall that the expected compres- s i o n lasts much longer ( % 1 0 - ~ sec) a s compared to the laser

(10-l0 sec). T h e n , electrons and ions have enough time to relax to a nearly c o m m o n temperature.

3. PARTIALLY DEGENERATE ELECTRON FLUID

Before embarking into a guenuine stopping calculation, i n order to simplify matter, it appears of interest to pay a s p e c i a l attention to the dynamic dielectric f u n c t i o n ~ ( q , w ) in a d e n s e electron fluid. Up to n o w , the simplest and more

accurate e x p r e s s i o n a v a i l a b l e f o r practical purposes, remains the R P A one. It pertains to a homogeneous electron f l u i d , w h i c h r e m a i n s weakly coupled for any degeneracy

- .

E F

It i s the o b v i o u s finite-temperature extension of the standard Lindhard quantity valid at T = 0 , for r

<

1 . It joins

S - s m o o t h l y , the T -+ m and classical Fried-Conte expression.

W i t h i n the framework of linear response theory, i t is introduced a s 7,lO

w i t h

and a free electrons response

and a l s o

small positive quantity

B

= -- 1

,

JJ = c h e m i c a l p o t e n t i a l

n o ( k ) = F e r m i - D i r a c d i s t r i b u t i o n .

T o s i m p l i f y t h e d i s c u s s i o n , w e m a k e u s e o f t h e d i m e n s i o n - l e s s v a r i a b l e s

2 1 1 3 i n t e r m s o f t h e F e r m i m o m e n t u m q F = ( 3 n n)

a n d v e l o c i t y V F =

-

1 . g 1 9

S a r s

s o t h a t x O ( z , u ) =

-

-- G ( z , v ) n

P + - a P - + a P--a

+ A r c t g - A r c t g - A r c t g --

n n n

1 1

T h e o t h e r d i m e n s i o n l e s s p a r a m e t e r s a r e

JOURNAL DE PHYSIQUE

F r o m a practical point of view it is important to recall the Kramers-Kronig relation

w h i c h c a n be transformed through

f , ( 2 , ~ ) = - nTe [ F(p+)

-

^F(p-)

1

1 + m

into F(P) =

- 5

^{P . P .}

4

^h(p')d

-m

F(x) i s characterized by the e x p a n s i o n s a) X + 0

with a e = expressed in terms of Fermi function.

y i e l d i n g the chemical potential in terms of the electrons temperature and density.

A t l o w t e m p e r a t u r e o n e g e t s

b o ( T e ) a n d a o ( T e ) b e i n g b n a n d a n v a l u e s a t n' = 0 .

I n p a r t i c u l a r

w h i l e a t T = 0

1 I - X 2

w i t h

o n e r e c o v e r s L i n d h a r d .

T h e h i g h t c m p c r a r u r r l i m i t 8 i s r e a c h e d f o r Te

> >

1 a n d a e

<< -

1 . T h e a b o v e q u a n t i t i e s t h e n b e c o m e

Z ( x ) b e i n g t h e u s u a l F r i e d a n d C o n t e f u n c t i o n

I n t h i s c o n n e c t i o n i t i s a l s o w o r t h w h i l e t o a d d a f e w c o m m e n t s a b o u t a p l a u s i b l e e x t e n s i o n o f t h e RPA a t Te = 0 .

A m o n g s t t h e v a r i o u s p r o p o s a l s , a r e c e n t o n e d u e t o I c h i m a r u a n d u t s u m i l a p p e a r s p a r t i c u l a r l y i n t e r e s t i n g . I t

JOURNAL DE PHYSIQUF

allows for a more accurate treatment of the short-ranged (large q ) interactions in

W e begin by noting the long wavelength behavior

(q

<<

q F , the Fermi w a v e number),

G(q) + Y o Q 2 9 ( Q E q I q F ) , ( 1 4 )

w h e r e the coefficient y is connected to the correlation energy Ec(rs) in rydbergs per electron via the compressibility relation

w i t h

a

5 ( 4 / 9 n ) 1 / 3 . For a n electron liquid in the paramagnetic s t a t e , w h i c h w e are here concerned w i t h , one has

where b = 0.0621814

,

b l = 9.81379

,

b2 = 2.82224

,

and b3 = 0.736411.

W e u s e (1 6 ) i n ( 1 5 ) to determine t h e long w a v e l e n g t h behavior ( 1 4 ) as a function of r

.

T h e short wavelength behavior of G(q) is related to the radial distribution function g(r) as

lim G(q) = 1

-

^g(0)

.

q-'

T h e short range correlation can be described by the electron- electron ladder interactions

where I ( z ) is a modified Bessel function of the first order.

l

U s e of ( 1 8 ) i n ( 1 7 ) thus determines the short w a v e l e n g t h behavior of G(q). T o simulate the numerical results of the microscopic theory as well as to accommodate the boundary

c o n d i t i o n s , (14) and (17), it is appropriate to express

w h e r e

Equs. (21) and (22) derive f r o m (14) and (17). Equ. (20) i s adapted so that Eq. (19) closely simulates the results of the microscopic theory. F o r r

>

15, A b e g i n s to decrease gradually f r o m 0.029.

A t arbitrary temperature, the following technical r e m a r k s are useful:

-

fl(u,z) and f (u,z) have their respective maxima i n 2 U and z , located between 0 and 1 . They decrease w i t h increasing T

.

- f l o r f 2 a r e essentially significant on a range in U (or z ) measured by ao(Te). T h e i r variations are rather weakly T

-

dependent.

-

^{f2(u, Z) =} 0 a s soon a s

I

z-ul> 2a (T ) o e

-

fl(u,z)

5

0 f o r u

>

^ao(Te).

Another important parameter is the location of the resonance

QT

(E(z,u) = 0) given by

(x2

= f )

z2 =

- X

2 fl(z,u) and f2(z,u) = 0

C8-78 JOURNAL DE PHYSIQUE

w i t h the respective limits

T h e s m a l l z and large U l i m i t , y i e l d s

9

f o r a phase velocity large compared to V th'

4. ENERGY

LOSS

A T F I N I T E TEMPERATURES (RPA)

W e n o w c o n s i d e r the free electron contributions to the stopping. T h i s is the term in the c o m p l e t e superposition

w h i l e

B

and

ai

denote respectively the bound electrons and the j

r e s i d u a l i o n s contribution. W e n o w follow a recent presentation of the general formalism d u e to Arista and Brandt 5

.

A comprehensive treatment of the energy-loss problem, in terms of t h e equilibrium dielectric f u n c t i o n ~ ( q , w ) , c a n be formulated by starting f r o m the scattering rate

f o r energy transfer iYw = E($')

-

^E($) and momentum t r a n s f e r

3

3;; =

$ ' -

p, w h i c h a p p l i e s to the scattering of a particle of

3

-+

c h a r g e Z e , w i t h initial momentum p and energy E(p), to the f i n a l s t a t e g i v e n by p', E($'). 3 T h e dynamical s t r u c t u r e factor

3

s(;,u) i s related to the dielectric f u n c t i o n ~ ( q , w ) through

- 1

w h e r e N(w) 5 [ e x p ( 6 . k ~ )

-

1 1 a n d 6 = l / k T

T h e t e m p e r a t u r e d e p e n d e n c e i s c o n t a i n e d i n t h e d i e l e c t r i c f u n c t i o n E ( q , u ) a n d i n t h e P l a n c k f u n c t i o n N ( w ) . T h e e n e r g y - l o s s

-+

r a t e i s g i v e n by

-+ -+

w h e r e w E w ( p , q ) i s d e t e r m i n e d f r o m

-f -f

i n t e r m s o f t h e i n c i d e n t v e l o c i t y v = p/M a n d t h e m a s s M o f t h e p r o j e c t i l e . F o r h e a v y p a r t i c l e s M

>>

m , r e c o i l e f f e c t s a r e s m a l l a n s we c a n e x p a n d E q . ( 2 8 ) i n t e r m s o f Aw E -trq 2 /2M t o o b t a i n

w h e r e t h e f i r s t t w o t e r m s a r e

T h e i n t e g r a l s r a n g e o v e r b o t h n e g a t i v e f r e q u e n c i e s ( L O S S

p r o c e s s e s ) a n d p o s i t i v e f r e q u e n c i e s ( g a i n p r o c e s s s s ) , b u t i t i s h e r e m o r e i n s t r u c t i v e t o t r a n s f o r m t h e m i n t o i n t e g r a l s o v e r p o s i t i v e f r e q u e n c i e s o n l y .

We c a n s i m p l i f y t h e e x p r e s s i o n f o r t h e m a i n t e r m ( d E / d t ) o , e q . ( 3 1 ) b u s p l i t t i n g t h e i n t e g r a l i n t o t h e

w >

0 a n d w

<

0 p a r t s , a n d t h e n m a k i n g u s e o f t h e r e l a t i o n s N(w)

+

N(-U) = -1 a n d & ( q , w ) -+ = & * ( q , w ) ; t h i s l e a d s t o a n

JOURNAL DE PHYSIQUE

expression of the form

T h e two terms in N(w) cancel e x a c t l y , w i t h the result f o r the stopping power S,

T h e only temperature dependence i s n o w contained in the energy- l o s s f u n c t i o n Im[-l/€(q,w)

l ,

and a r i s e s f r o m a thermal redistri- bution of the oscillator strenghts in the medium. O n e c a n

interpret this result a s a cancellation b e t w e e n the processes of s t i m u l a t e d a b s o r p t i o n a n d s t i m u l a t e d e m i s s i o n of energy -tTw by the projectile, since both p r o c e s s e s are proportional to the Planck d i s t r i b u t i o n N(w) that characterizes the thermal equi- librium of excitation quanta in the medium. T h u s , the energy- loss rate i s only determined by s p o n t a n e o u s e m i s s i o n p r o c e s s e s , w h i c h a r e independent of N(w).

A similar a n a l y s i s c a n be made for the energy-loss straggling R

,

w h i c h c a n b e expanded a s

n2

= n o 2 + Q : +

... ^,

w i t h

F o r the balance b e t w e e n positive and n e g a t i v e frequencies in the

R :

term, all the c o n t r i b u t i o n s from stimulated absorption

(W

>

0), proportional to N ( w ) , and those from stimulated and

s p o n t a n e o u s emission (w < 0), proportional to N(w) + 1

,

^are

c o l l e c t e d , so o n e obtains

T h e temperature dependence of Q: is contained in N(w) and ~ ( q , w ) . W h e n k T

< <

5 w , N(w) + 0 , and w e retrieve the expression for the energy straggling in a degenerate electron fluid.

Explicit integrations of S and

R

for T = 0 exist already in the literature. In the opposite limit k T > > $ U , w e c a n approxi- mate [ 2 N ( w )

+

l

1

2kT/-hw

.

T h e straggling integral Eq. (38)

then becomes identical to the stopping integral Eq. (34) multiplied by 2kT, i.e., straggling and stopping power S are related as

f o r a l l values of v , n , and T s u c h that the condition .trw

< <

k T is fulfilled. Since the frequencies of interest fall in the integration range f r o m zero to w = 2mv(v+ve)/tr, Eq. (39) w i l l

ma s apply when

JOURNAL DF. PHYSIQUE

In the limit 0 E kT/EF

> >

¹ one a p p r o a c h e s

7

1 m v e 2 - k T 3

,

and 2

Eq. (40) defines the domain v

4

0.15 v e , c o r r e s p o n d i n g to

projectiles m u c h slower than the thermal electrons in the plasma.

T h e velocity dependence of R' is the same a s that of S , viz.,

n2

^a v. By c o n t r a s t , in a degenerate electron gas at low velocities,

R '

i s a quadratic f u n c t i o n of v.

T h e applicability of Eq. (39) to a hot plasma k T

> >

EF a c c o r d s with a classical description, in terms of the Fokker- Planck equation, f o r the fluctuations in the energy of a s l o w particle in a thermalized medium.

In dimensionless units (z and U) S and R 2 are respecti- vely w r i t t e n in the f o r m

with

L

and LQ a l s o dimensionless. L e depends o n T through E(Z,U) only w h i l e LR gets a n o t h e r T -dependence with N(zu).

2 d E

S o , R is expected to increase faster with T than

--.

C l o s e d X

c o l l i s i o n s are likely to play a m o r e important role f o r R 2

.

A t t h i s p o i n t , we h a v e t o make c l e a r a f e w o b v i o u s a s s u m p t i o n s .

On m o s t p a r t o f t h e i r r a n g e , t h e i n c o m i n g i o n s a r e m o r e e n e r g e t i c t h a n t h e t a r g e t p a r t i c u l e s . S o , t h e i r t r a j e c t o r y may b e t a k e n a s l i n e a r , i n v i e w o f t h e v e r y s m a l l e n e r g y e x c h a n g e a t e a c h e n c o u n t e r . T h e p r o j e c t i l e i o n s a r e s u p p o s e d t o b e p o i n t - l i k e w i t h a c o n s t a n t c h a r g e .

M o r e o v e r t h e u s u a l 2 2 - d e p e n d e n c e o f t h e s t o p p i n g f o r m u l a , y i e l d s t h e s c a l i n g r e l a t i o n

d E '

z v 2 E

M

d x ( Z 1 , M ' , E ' ) =

- z 7

d x (Z,M,

F

E ' )

s o we c a n r e s t r i c t t o p r o t o n s i n t h e s e q u e l .

I n t h e l o w v e l o c i t y l i m i t v

<<

v F X a o ( T e ) , a n d

a ( a. ( T , ) , t h e t a r g e t e l e c t r o n s r e s p o n d t o a n e a r l y s t a t i c e l e c t r i c f i e l d . T h e c o r r e s p o n d i n g s t o p p i n g t h u s r e d u c e s t o

w i t h

w h i c h i s p r o p o r t i o n a l t o t h e p r o j e c t i l e v e l o c i t y v . Many p r e v i o u s a u t h o r s h a v e o b t a i n s a n a l o g o u s q u a n t i t i e s t h r o u g h t h e a d d i t i o n a l a s s u m p t i o n

F o r i n s t a n c e , a t n = 1 0 cm-3 a n d T ~ ~ = . 7 6 8 , t h i s a m o u n t s t o a 4 % d i s c r e p a n c y .

JOURNAL DE PHYSIQUE

FIGURE 2

F r e e e l e c t r o n s s t o p p i n g numbers a t low p r o j e c t i l e v e l o c i t i e s .

T h e c o r r e s p o n d i n g Le a r e p l o t t e d i n F i g . 2 f o r Te TF a n d v a r i o u s d e n s i t i e s .

I t s h o u l d b e a p p r e c i a t e d t h a t o n e o f t h e m a i n o u t p u t s o f t h e p r e s e n t w o r k i s t h e p o s s i b i l i t y t o c o m p u t e S a n d Ll2 f o r a n y v e l o c i t i e s r a t i o

- v ,

b e c a u s e t h e p a r t i a l d e g e n e r a c y i s

v t h t r e a t e d e x a c t l y .

F o r i n s t a n c e , i n t h e l a r g e V l i m i t

v

> >

1

a o ( T e ) v F

o n e may c h e c k o u t t h a t f o r Te # 0 , t h e r e e x i s t s a s i n t h e T = O c a s e 9 t w o e q u a l c o n t r i b u t i o n s t o S :

- e x c h a n g e o f e n e r g y w i t h a p l a s m o n m o d e a r o u n d z = z

-

e x c h a n g e o f e n e r g y t h r o u g h b i n a r y e n c o u n t e r s a r o u n d z = U

A l s o , o n e o b t a i n s a n o v e l a n d v e r y a c c u r a t e a s y m p t o t i c e x p r e s s i o n f o r L ( v ) :

v a l i d a t a n y T e , w h i c h h a s t h e t w o c h a r a c t e r i s t i c l i m i t s

2 4

2mv2 3 ) V~

- ( 1

( L i n d h a r d )

- Te

< <

1 L e ( v ) = L o g %F

-

-

P v v

F I i 2 . 3

-

F r e e e l e c t r o n s s t o p p i n g p o w e r a t n = 1 0 e - c m d 3 a n d ~ ~ s e v e r a l t e m p e r a t u r e s i n t e r m s o f t h e p r o j e c t i l e s e n e r g i e s .

C8-86 JOURNAL DE PHYSIQUE

T h e e n e r g y l o s s e s ( F i g . 3 ) may t h e n b e g i v e n f o r a n y p r o j e c t i l e v e l o c i t y . T h e T e - d e p e n d e n c e i s m o s t l y s i g n i f i c a n t f o r E ( 5 MeV/

2 .

A s i m i l a r a n a l y s i s p e r f o r m e d o n R i n t h e v

> >

1 a o ( T e ) V ~ l i m i t g i v e s

2 4 Z e n

Q 2 =

e

2 L o g

(v)

4 3 c. F

'1

w h e r e

I t m i g h t a l s o b e u s e f u l t o i n t e g r a t e t h e e n e r g y d i s p e r s i o n A'' ( < < 1 ) o v e r t h e r a n g e , t h r o u g h 1 0

E

X 2 E ( x ) d E

=

1

^{d x =}

j

Q 2

,

E. = i n i t i a l e n e r g y ( 4 8 )

E. (K)

M o r e o v e r , t h e p o i n t l i k e t a r g e t i o n s Zi may b e i n c l u d e d w i t h i n a n e x t e n d e d d i e l e c t r i c f u n c t i o n 10

w i t h g ( z , u ) d e n o t i n g

I t s h o u l d b e a p p r e c i a t e d t h a t t h e s t o p p i n g a r i s i n g f r o m e l a s t i c i o n - i o n c o l l i s i o n s i s n o t a l w a y s n e g l i g i b l e . L e t u s c o n s i d e r

a

p a r t i c l e s w i t h e n e r g y Ea, p r o d u c e d a t 3 . 5 MeV b y t h e n u c l e a r r e a c t i o n D + T i n a d e n s e Au t a r g e t ( t a b l e 3 ) , s o

t h a t

v

< <

1 a n d

--L > >

i

a. ( T , ) v ~ 'T h i

C l o s e i o n - i o n c o l l i s i o n w i l l t h e n g i v e

2 2 2

Z e Z.ro m 2

d E miMv .Ae

(r)

=

X i o n ^{I f P} Log (

4 7 ~ v m . (mi+M) x l

,

4 7 x Z Z i e 2 )

0 1

I n t e r m s o f t h e e l e c t r o n s c r e e n i n e l e n g t h h

.

I n t a b l e 3 , we h a v e p l o t t e d t h e r a t i o s

T A B L E 3 - S t o p p i n g o f a p a r t i c l e s i n a Au p l a s m a w i t h n = 1 0 e - ~ m - ~ ~ ~ a n d T

= T F E (MeV)

2 . 6 X 1 0 - ~ 1 . 0 4 X 1 0 - I 4 . 1 6 ^X 1 0 - l 0 . 9 4

1 . 6 6 3 . 7 5

T h e y s h o w t h a t t h e i o n s t o p p i n g c a n o v e r c o m e t h e e l e c t r o n o n e f o r E

a <

0 . 1 MeV.

5. STOPPING BY BOUND ELECTRONS ( z 3 - T E R M )

As m e n t i o n e d e a r l i e r : h e c o m p r e s s e d p l a s m a c a n n e v e r g e t h o t e n o u g h t o s e c u r e a c o m p l e t e s t r i p p i n g o f t h e t a r g e t h e a v y i o n s , a t l e a s t d u r i n g c o m p r e s s i o n ( i . e . b e f o r e t h e b u r n t a k e s p l a c e ) .

L ( V ) io n 2 . 2 9 ^X I o - ~ 5 . 3 2 ^X 1 0 - ~ 8 . 3 5 ^X I o - ~

I o - ~ 1 . 1 3 X 1 0 - ~ 1 . 2 7 X T O - '

T h e r e f o r e , i t i s n e c e s s a r y t o p e r f o r m s o m e r e v i s i t a t i o n s L ( v ) e l e c t r o n

2 . 6 9 X I o - ~ 2 . 1 3 X I o - ~ 2 . 1 5 ^X I o - ~ 5 . 7

1 . 3 5 X 1 0 - ~ 3 . 5 7 X 1 0 - ~

R a t i o 8 . 5 2 . 5 0 . 4 0 . 2 0 . 0 8 0 . 0 4

C8-88 JOURNAL DE PHYSIQUF

of the bound states contribution to S , beyond the standard

z'-

term (Bohr-Bethe-Bloch)

.

The next order in

z 3

(Barkas) c a n provide a significant discrepancy to the s t o p p i n g of n e g a t i v e c h a r g e s , with respect to charge c o n j u g a t e p o s i t i v e ones.

T h e Z 3 -close c o l l i s i o n s term J c is of quantitative signi- 1 2

f i c a n c e at relativistic energies only

.

S o , w e may safely n e g l e c t it f o r ions with a few M ~ V / n u c l e o n , the energy r a n g e o f interesr f o r heavy ion d r i v e n fusion.

The energy-loss formula c a n then be w r i t t e n a s :

w h e r e the customary lowest-energy loss (Bohr-Bethe-Bloch) is expressed a s twice a long-distance contribution. T h e idea of

z 3

(and higher) distant contributions is straightforwardly

accounted f o r through successive Born corrections.

According to this approach 1 2 ' 1 3 , J d i s specified w i t h a 1 1 2

m i n i m u m impact parameter a =

(G)

and a dimensionless

1 / 2 0

parameter = (--- 2 ) p e r t a i n i n g to a classical harmonic 2mV

oscillator w i t h energy -Kwo = 1 . 1 2 3

Io.yo

b e i n g a characteristic ionization e n e r g y , and m denotes the electron mass.

The 2: stopping-power c o r r e c t i o n thus reads ^{1 3}

in terms of a differential oscillator strength g(w) s u c h that

a,

g(w)dw = 1 , and a target atom density

Ni.

0

T h e d y n a m i c s of the projectile-bound electron system is entirely contained w i t h i n 1 4 , 1 5

w i t h I"

,

modified Bessel f u n c t i o n of the l S t k i n d

Recalling that g(w) is related to the electron distribution p(r) in a target atom (ion) by

W e propose to compute Eq. (53) i n the best condition through a Thornas-Fermilike G S Z expression 1 6

w h e r e T = e x

-

^{1 and X = L}

.

N is the n u m b e r of bound electrons d

in a given atom (ion). The two parameters (ao = Bohr radius) 0.5

< -

d

5

1 . 3

0

H = 1.05 X d X N ^0.4 ₍₅₇₎

specify Eq. (56) f o r a given target atom. U p o n introducing a

v

²

dimensionless ratio V =

-

^{w i t h V. =}

.

^{T h e Z:} term becomes

aiv0

w i t h

4 n N e ⁴

,

c = & - 2 - 1

2

-

0.307 MeV c m g

m r m C

T h e r e l a t i v e magnitude of the

z3

(Barkas) term is measured a t last by

JOURNAL DE PHYSIQUE

d i s p l a y e d i n t a b l e 4 f o r f o u r t y p i c a l a t o m s a n d s m a l l a n d l a r g e V v a l u e s r e s p e c t i v e l y . F ( V ) i s t y p i c a l l y 1 5 u p t o 3 0 p e r c e n t o f t h e z L - t e r m . J d i s o b v i o u s l y a n o n n e g l i g i b l e q u a n t i t y . I t w a s f o u n d c o n v e n i e n t t o p u t E q . ( 5 3 ) u n d e r t h e f o r m :

I n c o n t r a d i s t i n c t i o n t o t h e L e n z - J e n s e n e x p r e s s i o n w h i c h d o e s n o t d i s c r i m i n a t e t h e a t o m i c s t r u c t u r e , G S Z i s m o r e s e l e c t i v e b e c a u s e i t i s b u i l t u p o n t h e e x p e r i m e n t a l d a t a f o r t h e l o w e s t b o u n d s t a t e s .

TABLE

4

JD AND F ( V ) D A T A

1 ) Z = 1 0 d = 0 . 4 2 7 ( N e )

TABLE

4

(continued)

3 ) Z = 8 0 d = 0 . 6 2 0 ( H g )

V = 0 . 5 J d = 0 . 2 0 6 2 6 F ( V ) = 0 . 2 0 5

= 1 = 0 . 2 4 4 7 6 = 0 . 2 5 4

= 1 . 5 = 0 . 2 3 4 3 4 = 0 . 2 7 1

= 4 = 0 . 1 4 0 0 5 = 0 . 2 5 1

= 4 . 5 = 0 . 1 2 7 1 7 = 0 . 2 4 4

= 5 = 0 . 1 1 6 0 1 = 0 . 2 3 6

4 ) Z = 9 0 d = 0 . 9 8 0 ( T h )

R E F E R E N C E S

111 R. K I D D E R , i n L a s e r i r i t e r u c t i o n s a n d H c Z n i e d P % u s m a P h e n o m e n a , e d i t e d b y H . J . S c h w a r t z a n d H . H o r a

( P l e n u m , Ncw Y o r k , 1 9 8 1 ) , p . 3 0 3 .

1 2 1 I n f o r m a l W o r k s h o p o n t h e P e n e t r a t i o n o f C h a r g e d P a r - t i c l e s i n Y a t t e r U n d c r E x t r e m e C o n d i t i o n s , New Y o r k U n i v e r s i t y , J a n u a r y , 1 9 8 0 ( u n p u b l i s h e d ) , S e e a l s o C. DECTSCH, B u l l . S o c . F r . P h y s .

40,

⁵ 1 9 8 1 ) . C31 T . A . MEHLHORN, J . A p p l . P h y s .

52,

6 5 2 2 ( 1 9 8 1 ) . A l s o

J . M E Y E R - t e r - V E H N a n d N . METZLEK, MPQ 4 8 , J u l y 1 9 8 1 ( M i i n c h e n )

.

[ Q 1 E . NARDI, E. P E L E G , a n d % . ZINAMON, P h y s . F l u i d s 2 1 ,

-.

5 7 4 ( 1 9 7 8 ) .

C8-92 JOURNAL DE PHYSIQUE

N . R . A R I S T A a n d W . BRANDT, P h y s . R e v . A

2,

1 8 9 8 ( 1 9 8 1 ) . S . SKUFSKY, P h y s . R c v . A

16,

7 2 7 ( 1 9 7 7 ) a n d p r e v i o u s r e f e r e n c e s q u o t e d t h e r e i n .

C. GOUEDARD a n d C. DEUTSCH, J . M a t h . P h y S . ( K . Y. )

12,

3 2 ( 1 9 7 8 ) ; C. GOUEDARD, T h t s e 3 t m c C y c l e , O r s a y , F r a n c e , ( 1 9 7 7 1 , ( u n p u b l i s h e d ) .

At: T = 0 , a ( 0 ) = 1 w h i l e a ( T ) 'L ( f i e / l n T e ) a t T = m.

J . LINDIIARD a n d A. WINTHER, K. DAN. V i d e s k . S e l s k . M a t . F y s . M e d d .

35,

N o . 4 ( 1 9 6 4 ) .

G. MAYNARD a n d C. DEUTSCH, P h y s . R e v . A 2 6 ; 6 6 5 ( 1 9 8 2 ) . -- a l s o G. MAYNARD, T h h s e 3 6 m e C y c l e , O r s a y , F e b r u a r y 1 9 8 2 . S . ICHIMARU a n d D. UTSUMI, P h y s . R e v . B

2,

7 3 8 5 ( 1 9 8 1 ) . J . D . JACKSON a n d K . L . M C CARTHY, P h y s . R e v . B 6 , 4 1 3 1

( 1 9 7 2 ) .

J . C . ASlILEY, R . H . R I T C H I E a n d W . BKANDT, P h y s . R e v . B . - 5 , 2 3 9 3 ( 1 9 7 2 ) .

C. DEUTSCH a n d S . KLAKSFELD, P h y s . R e v . A

7,

2 0 8 1 ( 1 9 7 3 ) . G. MAYNARD a n d C . DEUTSCH, J . P h y s .

43,

L - 2 2 3 ( 1 9 8 2 ) . A . E . SGKEEN, D . L . S E L L I N a n d A . S . ZACHOK, P h y s . R e v . 1 8 4 , 1 ( 1 9 6 9 ) , a l s o

J . W . DAKEWICH, A . E . S . GREEN a n d D . L . S E L L I N , P h y s . R e v . A

2 ,

5 0 2 ( 1 9 7 1 ) .