DISPERSIVE EFFECTS IN RADIATION TRANSPORT AND RADIATION HYDRODYNAMICS IN MATTER AT HIGH DENSITY

HAL Id: jpa-00223310

https://hal.archives-ouvertes.fr/jpa-00223310

Submitted on 1 Jan 1983

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

DISPERSIVE EFFECTS IN RADIATION

TRANSPORT AND RADIATION HYDRODYNAMICS IN MATTER AT HIGH DENSITY

B. Crowley

To cite this version:

B. Crowley. DISPERSIVE EFFECTS IN RADIATION TRANSPORT AND RADIATION HYDRO-

DYNAMICS IN MATTER AT HIGH DENSITY. Journal de Physique Colloques, 1983, 44 (C8),

pp.C8-25-C8-38. �10.1051/jphyscol:1983803�. �jpa-00223310�

J O U R N A L DE PHYSIQUE

Colloque C8, supplement

au

n O 1 l , Tome 44, novembre 1983 page C8-25

DISPERSIVE EFFECTS IN RADIATION TRANSPORT AND RADIATION HYDRODYNAMICS IN MATTER AT HIGH DENSITY

B.J.B. Crowley

Atomic Weapons Research Establishment, Aldermaston, Reading, U . K .

Rdsumd

Dans un t r a v a i l r e c e n t , j ' a i g 6 n E r a l i s 6 l e s S q u a t i o n s d e l l h y d r o d y n a m i q u e r a d i a t i v e au c a s 06 l e rayonnement 6 l e c t r o m a g n d t i q u e s a t i s f a i t

B

une r e l a t i o n d e d i s p e r s i o n l i n 6 a i r e nw = k c , 06 l ' i n d i c e d e r 6 f r a c t i o n n d6pend d e l a f r 6 q u e n c e U e t / ou du nombre d'onde k. A p p l i q u a n t l a t h g o r i e d e t r a n s p o r t B o l t z m a n n - L i o u v i l l e aux p h o t o n s du domaine i3 c o u r t e l o n g u e u r d ' o n d e , j ' o b t i e n s d e s d q u a t i o n s d ' d n e r g i e e t d e moment, q u i , combindes a u t r a i t e m e n t c l a s s i q u e du m i l i e u f l u i d e e n ETL, donnent une t h d o r i e dynamique complGte d e s i n t e r a c t i o n s l i n e a i r e s (+ p r o c e s s u s s t i m u l d s ) e n t r e l e rayonnement t h e r m i q u e i n c o h d r e n t , e t l a m a t i l r e d e n s e , l o c a l e m e n t i s o t r o p e . Ce f o r m a l i s m e e s t g d n 6 r a l i s a b l e aux i n t e r a c t i o n s n o n - l i n b a i r e s , oh l ' i n d i c e de r 6 f r a c - t i o n depend d e 1 1 i n t e n s i t 6 l o c a l e s p d c i f i q u e du champ d e rayonnement, e t d a n s une c e r t a i n e mesure, g g a l e m e n t , a u t r a i t e m e n t du rayonnement c o h 6 r e n t d e h a u t e f r 6 q u e n c e . La g d n d r a l i s a t i o n d e p l u s i e u r s formes a p p r o e h d e s de t h d o r i e du t r a n s p o r t d e rayon- nement ( d i f f u s i o n ) e s t c o n s i d 6 r d e e n d d t a i l . Parmi l e s problBmes o u v e r t s , mention- nons : l a d i s p e r s i o n a n o r m a l e , l ' a s p e c t p h y s i q u e a t o m i q u e d e s p r o p r i s t d s d l e c t r o m p - g n d t i q u e s e t r a d i a t i v e s d ' u n m i l i e u m a t d r i e l d i s p e r s i f .

A b s t r a c t

I n a r e c e n t r e s e a r c h program ( r e p o r t e d i n AWRE 0 2 0 1 8 2 ) I have i n v e s t i - g a t e d t h e g e n e r a l i s a t i o n o f t h e e q u a t i o n s of r a d i a t i o n hydrodynamics when e l e c t r o - m a g n e t i c r a d i a t i o n i s assumed t o obey a l i n e a r - r e s p o n s e d i s p e r s i o n r e l a t i o n of t h e

form nw = kc where t h e r e f r a c t i v e i n d e x n depends o n t h e f r e q u e n c y

w

a n d / o r wave number k . From t h e a p p l i c a t i o n of t h e B o l t z m a n n - L i o u v i l l e t r a n s p o r t t h e o r y t o pho- t o n s i n t h e s n o r t - w a v e l e n g t h ( g e o m e t r i c a l o p t i c s ) l i m i t , I d e r i v e t h e e n e r g y and momentum e q u a t i o n s which, when combined w i t h a c l a s s i c a l (Euler-Lagrange-Navier-

S t o k e s ) t r e a t m e n t of a f l u i d m a t e r i a l medium i n LTE, y i e l d a c o m p l e t e d y n a m i c a l t h e o r y of l i n e a r i n t e r a c t i o n s ( + s t i m u l a t e d p r o c e s s e s ) between i n c o h e r e n t ( t h e r m a l ) r a d i a t i o n and d e n s e , l o c a l l y i s o t r o p i c m a t t e r . The t h e o r y i n c l u d e s a n a c c o u n t o f pondero-motive f o r c e s and e l e c t r o (magneto) s t r i c t i o n . Moreover, i t i s a p p a r e n t l y c a p a b l e of b e i n g g e n e r a l i s e d t o n o n - l i n e a r i n t e r a c t i o n s i n which t h e r e f r a c t i v e i n d e x depends o n t h e l o c a l s p e c i f i c i n t e n s i t y o f t h e r a d i a t i o n f i e l d , and, t o some e x t e n t , t o t h e t r e a t m e n t o f h i g h - f r e q u e n c y c o h e r e n t r a d i a t i o n . The g e n e r a l i s a t i o n of v a r i o u s a p p r o x i m a t e d forms of r a d i a t i o n - t r a n s p o r t t h e o r y ( e s p . d i f f u s i o n ) h a s been c o n s i d e r e d i n d e t a i l .

Some problems remain however. One s u c h i s t h e t r e a t m e n t o f anomalous d i s p e r s i o n . C u r r e n t r e s e a r c h work i s c o n c e n t r a t i n g o n t h e i n t e r e s t i n g a t o m i c p h y s i c s a s p e c t s o f e l e c t r o m a g n e t i c ( e s p . r a d i a t i v e ) p r o p e r t i e s o f a d i s p e r s i v e m a t e r i a l medium.

The f o l l o w i n g i s e x t r a c t e d from AWRE r e p o r t 0-20182 e n t i t l e d " C l a s s i c a l R a d i a t i o n Hydrodynamics i n Inhomogeneous R e f r a c t i n g Weakly D i s p e r s i v e Media". I n i t i s d e s -

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983803

C8-26 JOURNAL

DE

PHYSIQUE

c r i b e d a t h e o r e t i c a l s t u d y o f t h e dynamical e f f e c t s of r a d i a t i o n t r a n s p o r t ( e s p e - c i a l l y LTE) i n a moving inhomogeneous l o c a l l y - i s o t r o p i c m a t e r i a l medium when t h e r a d i a t i o n d i s p e r s i o n r e l a t i o n i s g i v e n a s

(U = a n g u l a r f r e q u e n c y ,

k

= wavenumber, c = v e l o c i t y of l i g h t , n = REFRACTIVE INDEX = Re (€U)).

OUTLINE

O b j e c t i v e : t o d e v e l o p a g e n e r a l t h e o r y o f r a d i a t i o n t r a n s p o r t i n h i g h - d e n s i t y b u l k m a t t e r

-

i n LTE i n t h e f i r s t i n s t a n c e

-

t a k i n g f u l l a c c o u n t o f r e f r a c t i o n , d i s p e r s i o n and r e l a t e d m i c r o p h y s i c a l p r o c e s s e s .

S o l u t i o n : from t h e c o n c e p t o f a model p h o t o n d e s c r i b e d by t h e E i n s t e i n - d e B r o g l i e r e l a t i o n s , E

= n u ,

p

-

= .hk

-

and whose o n l y i n t e r a c t i o n s a r e w i t h t h e medium, o n e d e v e l o p s a d y n a m i c a l t h e o r y b a s e d o n a M a s t e r E q u a t i o n t h a t i s a g e n e r a l i s e d Boltzmann e q u a t i o n o f r a d i a t i o n t r a n s p o r t d e r i v e d from t h e L i o u v i l l e theorem.

The i n p u t p a r a m e t e r s o f s u c h a t h e o r y , v i z o p a c i t i e s and r e f r a c t i v e i n d i c e s , a r e i n t e r - r e l a t e d t h r o u g h d e t a i l e d b a l a n c e c o n d i t i o n s a n d , i n t h e l i n e a r r e g i m e , c a u s a l i t y - b a s e d d i s p e r s i o n r e l a t i o n s . These c a n b e shown t o b e e x p r e s s i b l e i n t e r m s of a s i n g l e complex f u n c t i o n of f r e q u e n c y and wave-number t h a t i s a n a l y t i c i n t h e u p p e r h a l f o f t h e complex-frequency p l a n e , and t h r o u g h s u c h a f u n c t i o n , c a n b e r e l a t e d t o a m i c r o p h y s i c a l model. T h i s f u n c t i o n embodies a f u l l d e s c r i p t i o n o f t h e l i n e a r r e s p o n s e of t h e medium t o

3

e l e c t r o m a g c e t i c d i s t u r b a n c e .

Method

(1) S p e c i f y r a d ; a t i o n d i s p e r s i o n r e l a t i o n

T h i s

i s

e x p r e s s e d i n t e r m s o f t h e r e f r a c t i v e i n d e x which

i s

a g i v e n f u n c t i o n o f :

(i)

f r e q u e n c y : n = n(w): FREQUENCY DISPERSIOX - most a p p r o p r i a t e a t h i g h f r e q u e n c i e s

( i i j w a v e - v e c t o r : n = n i k ) :

-

S P A ~ ~ A L D I S I J e K S I O N

-

may be i m p o r t a n t a t l o w e r f r e q u e n c i e s due t o c o l l e c t i v e e f f e c t s and e f f e c t s of medium a n i s o t r o p y

( i i i ) f r e q u e n c y and w a v e - v e c t o r : n = n ( w , k ) : b o t h FREQIJEXCY and

-

SPATIAL DISPERSION: Over a p e r h a p s l i m i t e d r a n g e o f

w

o r k may b e e q u i v a l e n t t o ( i i ) o r , i n c a s e o f i s o t r o p i c medium, ( i ) i f d i s p e r s i o n r e l a t i o n i m p l i e s a d i f f c o m o r p h i s m between L! and k

s u c h t h a t

o v e r f r e q u e n c y r a n g e o f i n t e r e s t

-

e x c e p t p e r h a p s a t a number of d i s c r e t e f r e q u e n c y p o i n t s ( s h a r p l i n e s ) . The r e f r a c t i v e i n d e x

w i l l

a l s o depend o n v a r i a b l e s p , T,

...

s p e c i f y i n g t h e l o c a l thermodynamic s t a t e of t h e medium. I n g e n e r a l n

i s

i m p l i c i t l y a f u n c t i o n of p o s i t i o n

r -

and t i m e t .

F o r s i m p l i c i t y we c o n s i d e r o n l y i s o t r o p i c media f o r which n i s i n d e p e n d e n t ( i n t h e CO-moving f r a m e ) of t h e d i r e c t i o n

2. -

The f o r m a l i s m a s p r e s e n t e d a l s o assumes n t o b e i n d e p e n d e n t of t h e r a d i a t i o n i n t e n s i t y ( l i n e a r r e s p o n s e r e g i m e ) . Howc?ver, ad hoc f o r n a l

g e n e r a l i s a t i o n t o i n c l u d e dependence o f r e f r a c t i v e i n d e x on i n t e n s i t y I (non- l i n e a r r e s p o n s e ) a p p e a r s s t r a i g h t f o r w a r d .

( 2 ) Apply c l a s s i c a l ( I A i o u v i l l c - B o l t z m a n n ) t r a n s p o r t t h e o r y t o t h e d e r i v a t i o n o f t h e g e n e r a l t r a n s p o r t e q u a t i o n . ( V a l i d when grou;) v e l o c i t y vg i s e n c r g y p r o p a g a t i o n v e l o c i t y . )

(3) E s t a b l i s h d y n a m i c a l p r o p e r t i e s of an i n d i v i d u a l p h o t o n and h e n c e , w i t h a i d o f t h e t r a n s p o r t e q u a t i o n , d e d u c e e q u a t i . o n s of e n c r g y and no:nentum t r c n s p o r t

("Pl" e q ~ a t i o n s ) .

( 4 ) Combine t h e s e w i t h c l a s s i c a l f l u i d dynair~ic (Euler-1,ngrr;nge-Navier- S t o k e s ) t r e a t m e n t o f m a t e r i a l i n c o r p o r a t i n g a t h e r ~ ~ o t l y n a m i c trc.ntmcnt of t h e t o t a l s y s t e m => d e t a i l e d t h e o r y of r a d i a t i o n "hydrodynamics" i n c o r p o r a t i n g

t r e a t m e n t s o f ELECTRO/UGNETOSTRICTION (and r e l a l e d E!,ECTRO/:~~GNET@CALORT(~ EFFECTS) and POl\'DEROMOTIVE FORCES.

(5) Apply a p p r o x i m a t i o n s f o r LTE r a d i a t i o n t r a n s p o r t : c l . o s u r e of 1'1

e q u a t i o n s + f r e q u e c c y a ~ r e r a g i n g (Planck a?" R o s s e l e n d ) =>

Di

f [us; nq a p p r o w i n ~ a t i o n Averaged o p a c i t i e s depend on t h e r e f r a c t i v e i n d e x .

( 6 ) D e t e r m i n e r e f r a c t i v e i n d e x and o p a c i t i e s from m i c r o p h y s i c a l n o d c l . ( i ) M o d i f i c a t i o n of I.TE d e t a i l e d b a l a n c e c o n d i t i o ; ~ when n

# 1

JOURNAL DE PHYSIQUE

implies modification of opacities for use when n P 1 in place of those which apply when n

=

1.

(ii) Standard calculations (XSNB) yield a so called reduced opacity

KO

from which refractive index can be deduced by applying a Kramers-Kronig dispersion relation (refractive index and opacity being related to real and imaginary parts of the dielectric

function

€(W)

or, more generally, €(w)p(w)). "True" opacity is then K O / ~ . (iii) Need to consider effects of dispersion on treatment of

microphysical processes (collective effects, modified spectroscopy, multi-atom coherence, recoil corrections ...

^{) =>}

development of more general microphysical models.

(7)

Other problems include treatment of anomalous dispersion (particularly when a(n~))/aw

<

1 over a finite range of frequencies)

=>

breakdown of Hamiltonian description embodied in the transport equation, and phase-space representation.

Group velocity is not in (O,c), and therefore is not energy propagation velocity.

1 .

THERMODYNAMIC EQUILIBRIUM

Application of Bose-Einstein statistics in the continuum approximation yields the equilibrium phase-space distribution as

" 3

in which f(k)d k is the mean equilibrium density of photons with wave vectors in

"

the range d k about

3

- k. - In frequency space the corresponding distribution is

"

2

dk cnL -

N(w)dw

=

4nk f (k) - _dw ^dw

⁼

- ^{NB (w)dw}

g

-

_'U

where N

(U) =

- ₂ 3 e ~ / ; A ~ ¹ is the BLACK BODY distribution which applies

B

_{s c}

when n

f

1. The EQUILIBRIUM SPECIFIC IliTENSITY is therefore

i s

t h e PLANCK FlJNCTION o r BLACK BODY SPECIFIC INTENSITY.

2. CLASSICAL RADIATION TRANSPORT

2 . 1 G e o m e t r i c o p t i c s l i m i t i s v a l i d i f

( i ) P h o t o n w a v e l e n g t h i s s m a l l o n a l l o t h e r r e l e v a n t s c a l e s . ( i i ) I n p a r t i c u l a r

h

<< mean f r e e p a t h f o r a l ) s o r p t i o r ~ ancl

scattering => v a l i d i t y of TFXCSPARIXCY CONDITIO:;

l m E << R e E

e.g. I n a c o l l i s i o n do!ninated n o n - d e g e n e r a t e n e u t r a l c l a s s i c a l plasma f o r w h i c h

2

2

c o n d u c t i v i t y , 0 = E V . / ( U 2 + V . )

p o e l e~

( r a d i a t i o n p r o p a g a t i n g above plasma f r e q u e n c y ) ,

( n o n - d e g e n e r a t e ) , t h e t r a n s p a r e n c y c o n d i t i o n i s

w h i c h

i s

s u f f i c i e n t . I n t h e above formulae, Z = mean i o n i z a t i o n s t a t e (Ne = %Ni), ye = S p i t z e r c o r r e c t i o n , N . = i o n d e n s i t y (cm

-3

_{) ,}

v

⁼ e f f e c t i v e e l e c t r o n - i o n c o l l i . s i o n frequency.

e i

2.2 B o l t z n a n n e q u a t i o n of r a d i a t i c n t r a c s p o r t :

JOURNAL DE PHYSIQUE

a H a f all a f P o i s s o n b r a c k e t E

- . - - - -

ak a r a r ' ak S o u r c e l ~ u n c t i o n a l T h i s e q u a t i o n i s obeyed ( e x a c t l y ) by s y s t e m o f n o n - s e l f i n t e r a c t i n g c l a s s i c a l p a r t i . c l e s ("photons") moving i n a sniootli I l a r n i l t o n i a n f i e l d H ( k ; r ; t )

- -

= w ( k ; r ; t ) and

- -

o t h ~ _ r ! . ~ i s e r n h j e c t o n l y t n d i s c r e t e ( s ~ ? ( _ ( . e r i n g s , a h s r \ r ; t i n n r , E ) T ~ ~ S ~ _ ~ O I I C ) i c t e r - a c t i o n s r e p r e s e n t e d by t h e s o u r c e f u n c t i o n a l S ( { f } ; k ; r ; t ) . k

- -

I n t h i s e q u a t i o n , f ( k ; r ; t ) i s t h e p h a s e - s p a c e d i s t r i b u t i o n w h i l e s k i s t h e n e t r a t e o f p r o d u c t i o n of

-

p h o t o n s p e r u n i t "volume" o f p h a s e s p a c e .

The r e s u l t i n g t r a n s p o r t e q u a t i o n t h a t one e v e n t u a l l y o b t a i n s f o r t h e s p e c i f i c i n t e n s i t y i s o f t h e form

i n which a s u b s c r i p t d e ~ o t e s t h e CO-moving f r a m e

-

o t h e r w i s e i n e r t i a l f r a m e ( c o - o r d i n a t e s

r , t -

a r e a l w a y s r e f e r r e d t o t h e i n e r t i a l f r a m e )

-

^and

U = medium v e l o c i t y i n i n e r t i a l f r a m e

-

v = g r o u p v e l o c i t y i n CO-moving f r a m e (v = g r o u p v e l o c i t y i n

-g

-

inertial f r a m e ) .

Note how t h e t r a n s p o r t e q u a t i o n g e n e r a l l y i n v o l v e s 1 / n 2 i n p l a c e o f

I .

The q u a n t i t y 1 / n 2 u 3 i s a LORENTZ INVARIANT.

I n t h e a b o v e e q u a t i o n s , t h e s o u r c e f u n c t i o n a l s a r e s p e c i f i e d a s b e i n g g i v e n i n o n e of t h e f o l l o w i n g f o r m s :

a b s o r p t i o n

d i r e c t s c a t t e r ? ng

i n d u c e d s c a t t e r i n g ;

3 _{4 4}

+

is l [ ⁽ 5 ⁾

e x p f i ( w 1

-

U ) / ~ T ) I ( U '

,g') -

I ( u , ~ ) z ( U , u ' ; k , k ' ) d w ' d 2 i '

I ^{- -}

^W d i r e c t s c a t t e r i n g

i n d u c e d s c a t t e r i n g i n which K ~ ( w ) i s t h e a b s o r p t i o n o p a c i t y , N t h e mean d e n s i t y o f s c a t t e r e r s and

a

t h e therrna1.ly-averaged s c a t t e r i n g c r o s s - s e c t i o n . 2 . 3 C o n d i t i o n s a t b o u n d a r i e s and i n t e r f a c e s .

T h e s e a r e p r o v i d e d by S n e l l ' s Laws, which imply

n 2 c o s eldQl

1

= n2 c o s 0 dQ 2 2 and t h e F r e s n e l f o r m u l a e w h i c h y i e l d

PF

= 2n n c o s

e1

c o s 8

1 1

1 2 2 2 ⁺

(nl COS

O1 +

n 2 COS 02) ( n 2 COS

O1

+ n1 COS e 2 ) i n t e r m s o f which

R12 =

1 -

P = f ( A / h ) ( l

-

PF)

c o s dQl n 2

T12 = P - 2

cos

e 2

d n 2

- ( T )

1

P

where

A

i s t h e s u r f a c e d i f f u s e n e s s and f ( x ) i s a f u n c t i o n f : ( 0 , ~ ) + ( 0 , l ) s u c h t h a t f ( 0 ) = 1 and f ^(m) = 0 ( f ^% e - * ) . R e f l e c t i o n i s s t r o n g l y damped a t i n t e r f a c e s where

A

>> h.

JOURNAL DE PHYSIQUE

Note t h a t t h e above p e r m i t a d i s c o n t i r i u i t y i n I where n i s d i s c o n t i n u o t t s , e v e n i n e q u i l i b r i u m :

< ^{=> I}

d i s c o n t i n u o u s .

I n LTE, t h e f u n c t i o n 1 / n 2 i s a c o n t i n u o u s f u n c t i o n o f p o s i t i o n . 2.4 Energy t r a n s p o r t

D e f i n i n g t h e r a d i a t i v e e n e r g y d e n s i t y

and t h e e n e r g y f l u x

t h e e n e r g y t r a n s p o r t e q u a t i o n t h a t f o l 1 o r . r ~ by t a k i n g t h e a p p r o p r i a t e z e r o t h moment o f t h e t r a n s p o r t e q u a t i o n i s :

au

^{1 2,-}

^{~ 0 1 1}

- + v . F + - j + d , k + Q

= o

a t - - C a t -

----..-A

?'

Ponderomotive c o n t r i b u t i o n where

c o l 1 c o l 1

=

Q, +

u.fo

( G a l i l e a n t r a n s f o r m a t i o n v a l i d i f u / c << 1 )

where

f c o l l

-0 ( G a l i l e a n t r a n s f o r m a t i o n v a l i d i f u / c

<<

1).

Momentum t r a n s p o r t

I n

t e r m s o f t h e r a d i a t i v e (Minkowski) momentum d e n s i t y

and t h e c o r r e s p o n d i n g s t r e s s t e n s o r

t h e momentum t r a n s p o r t e q u a t i o n , which i s a f i r s t moment of t h e t r a n s p o r t e q u a t i o n ,

p o n d e n n o t i v e c o n t r i b u t i o n i n which T~ - - d e n o t e s t h e t r a n s p o s e of C - .

3 .

RADIATION HYDKODYNA'IICS

To c o m p l e t e t h e p i c t u r e , t h e above e q u a t i o n s must b e c o u p l c d t o t h e a p p r o p r i a t e e q u a t i o n s f o r t h e e n e r g y and momentum a s s o c i a t e d w i t h t h e m a t e r i a l . T h i s r e q u i r e s a r e s o l u t i o n of t h e i s s u e s sometimes r e f e r r e d t o a s " t h e mosentum of l i g h t problem". E a s i c a l l y , t h e problem i s how t o decompose t h e t o t a l s t r e s s t e n s o r of t h e s y s t e m p r o p e r l y i n t o a r a d i a t i v e ( o r e l e c t r o m a g n e t i c ) component and a t h e r m o k i n e t i c ( o r m a t e r i a l ) component. P e i e r l s d e n o n s t r a t e s t h a t t h e s e a r c , i n g e n e r a l , two d i f f e r e n t d e c o m p o s i t i o n s , a s p a r t of

a p h o t o n ' s momentum a s g i v e n

by nfik i s a c t u a l l y a s s o c i a t e d w i t h mechanical motion of t h e medium under t h e -

i n f l u e n c e o f t h e p h o t o n e l e c t r o m a g n e t i c f i e l d . The b a r e e l e c t r o m a g n e t i c momentum i s t h a t p r o p o s e d by Abraham: g = 4 w v / c L from which we c a n d e f i n e a n e l e c t r o -

-g m a g n e t i c momentum d e n s i t y

G

- by:

G

=

f g d k

3

- - -

The f i n a l r e s u l t i s t h a t t h e e n e r g y and momentum e q u a t i o n s r e p r e s e n t i n g t h e m a t e r i a l c a n b e e x p r e s s e d

as

f o l l o w s

DE

p - =

Dt (!U)

:

- +

Q

- ^£.U _{.-. .-.} +

^{r . .}

i n w h i c h E i s t h e m a t e r i a l i n t e r n a l e n e r g y d e n s i t y , p t h e mass d e n s i t y , and

G

- t h e

t h e r m o k i n e t i c s t r e s s t e n s o r . The q u a n t i t i e s f and

Q

r e p r e s e n t t h e p r i m a r y

c o u p l i n g t o t h e r a d i a t i o n f i e l d :

JOURNAL DE PHYSIQUE

The radiation field also gives rise to photocaloric and photostrictive

(=

magnetostrictive + electrostrictive) contributions to E and

0 .

The relevant

... -

relations, which apply in the co-moving frame in LTE, are

where E") and

a(')

are the internal energy and stress-tensor which apply under

..

the prevailing conditions of temperature and density in the absence of radiation;

F is the total free-energy

=

F(') - Pr; Pr is the radiation pressure

=

- 3

1

^{trace L ) ; ( '}

^{O H}

is the so-called Helmholtr Stress-Tensor which contains the

-0

=

photos trictive pressure,

which reduces, in the case of unpolarised, high-frequency, monochromatic radiation, to

Consequently, the relations which apply to a general unpolarised high-frequency radiation field are found to be:

an

2..

E - = L I 3 dud k

pc aT

The time-derivative terms involving M - - G in the expressions for f and ...

^.-.

Q arise

in respect of the mec':ianical momentum component of the radiation.

Thc non-collisional part of the radiatively induced force on a material is the ponderomotive force fPmf which, according to the above, is given -

(u/c

<<

1) by:

special case of which are:

(i)

A

beam of monochromatic radiation in a stationary stezdy-state medium:

2 2

(ii) Monocl~romatic radiation in a plasma (nz

=

1 - ^{w /w}

⁾

P

which is independent of the plasma's state of motion (at least to first order in u/c) .

4 .

DIFFUSION APPROXIMATION

I n a quasi-equilibrium state of an optically thick system, the limiting diffusion-approximation solutions of the transport equation are

W s a T

⁴

in which

with

and

where

8

is the radiation temperature.

JOURNAL

D€

PHYSIQUE

In terms of the solution W the radiation pressure P and the enthalpjr-

0'

density U + P are given by:

where

and

Note that, in general, the radiation pressure is no-longer given by P

=

U/3.

Finally, rcrrembering that -P is the Helmholtz Zree energy U - TS, one also finds the entropy associated with the radiation to be given by

5.

ABSORPTION COEFFICIENT

Neglecting scattering, the absorption coefficient

K

is found to be

related to the imaginary part of the product of the electromagnetic susceptibilities

by

L W

K,

=

( ; ) I~(EP)

in which €(w)U(w) is analytic in upper half U-plane and satisfies the Kramers-Kronig dispersion relations. (These permit expression of Re(cp), and hence n, in terms of

K ~ . )

Note the factor of l/n in the above expression.

6. ANOMALOUS DISPERSION

A problem with the above theory is that it does not yield a correct description in regions of anomalous dispersion where the energy propagation velocity vE is not the group-velocity v . Indeed, in such a region, v may not

g be in (0,c) and cannot therefore represent the propagation velocity.

A realistic calculation of the energy-trznsport velocity v

=

E

X

H/U

E -. -

in a region of anomalous dispersion (Sommerfeld and Brillouin) yields the result

in place of

for n

=

Re

F

1 . ^Writing

^{E =}

^{l +} xl + ix2

=

(n + ig12, we have, at the centre of an isoiated resonance wnere

axl _{- - -} X, ax2

- -

x l = o , - -

a~ y

S

aw - ⁰ , that

while

Away from resources where x2

2

^<< lxl l

;

IX2 I

<<

1, lx2 l <<

W

2 ^, 2

^>

^{0 both}

formulae yield:

It is not clear how the foregoing theory can be improved to take account of the above. It is not sufficient to simply replace v throughout by vE, es v often

g

g

arises as a Jacobian in transformations between

W

and k.

SELECTED BIBLIOGRAPHY General

CROWLEY, B J B: Op. cit. (to be published, with revisions, in physics Reports).

POMRANING, G C: The Equations of Radiation Hydrodynamics (Pergamon, 1973).

COX, J P and GUILI, R T: Principles of Stellar Structure (Gordon and Breach,

W ,

1968) v01 I.

Radiation pressure and ponderomotive forces in plasmas

LANDAU, L D and LIFSHITZ, E M: Electrodynamics of Continuous Media (Pcrgamon 1960).

PENFIELD, P and HAUS, H A: Electrodynamics of Moving Media (MIT Press,

Camb., Mass., 1967).

C8-38 JOURNAL DE PHYSIQUE

BOOT,

H

A

H,

SELF, S A a n d SHERSBY-IWIE,

R B K: J .

E l e c t r o n i c s a n d C o n t r o l 5

(1958) 434-453.

I h W N , C L: A s t r o p h y s .

J .

142 (1965) 201.

MORE, R M:

J.

P h y s . A: Math. Gen. 2 (1976) 1979-1985.

Momentum o f l i g h t p r o b l e m

BURT, M G a n d PEIERLS, R: P r o c . R. S o c . Lond. A, 333 (1973) 149-156.

PEIERLS, R: P r o c . R . Soc. Lond. A, 347 (1976) 475-491.

JONES, R V a n d LESLIE,

B:

P r o c . R. S o c . Lond. A, 360 (1978) 347-363.

BREVIK, I: P h y s . Rep. 52 (1979) 1 3 4 .

LAX, H M, SUEN, W

M a n d YOUNG, K: P h y s . Rev. A 2 (1982) 1755-1763.

Anomalous d i s p e r s i o n

BRILLOUIN, L: Wave P r o p a g a t i o n a n d Group V e l o c i t y (Academic P r e s s ,

hY,

1 9 6 0 ) .

C o p y r i g h t @ C o n t r o l l e r HMSO, London, 1 9 8 3