ATOMIC, IONIC AND MOLECULAR DATA IN THERMO-NUCLEAR FUSION RESEARCH

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ATOMIC, IONIC AND MOLECULAR DATA IN

THERMO-NUCLEAR FUSION RESEARCH

H. Drawin

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque

C1,

supplkment au n o 2, Tome 40, fkvr~er 1979, page

C1-73

ATOMIC, IONIC AND MOLECULAR DATA IN THERMO-NUCLEAR FUSION RESEARCH

H.W. Drawin

Association EURATOM-CEA

DQpartement de Physique du Plasma

et de la Fusion ContrBlGe

Centre dtEtudes

NuclGaires

F-92260 Fontenay-aux-Roses/France

Abstract

-

All high temperature hydrogen (isotopes) plasmas investigated in the frame of

thermo-nuclear fusion research contain impurities like carbon, oxygen,iron,

...

These

particles influence the plasma properties and give rise to additional undesired energy

losses. It is shown how the atomic properties enter into the general balance equations

describing a plasma. These equations serve for deriving the ignition and Lawson condition

of a D-T fusion reactor as a function of impurity concentration and element. Graphical

representations show how the fusion conditions depend on impurities. It follows a

discussion of the equations applied in the spectroscopic diagnostics of plasmas and

especially for determining the impurity concentrations by spectroscopic means. All

equations are formulated in terms of relevant atomic data. The last section deals with

molecular data which are of interest in thermo-nuclear fusion research.

1.

INTRODUCTION

In order to achieve controlled thermo-

nuclear fusion of deuterium and tritium one needs

both high temperatures and a sufficient long energy

confhement time at not too a low plasma density.

Under all currently studied plasma configurations

those of the TOKAMAK type have at the present stage

of research the highest chance to yield plasma

conditions which approach thermonuclear conditions.

A TOKAMAK plasma (fig.

1)

represents the

secondary winding of a transformer whose primary

windings are fed by an external energy source (e.g.

condenser banc). A strong toroidal magnetic field

in connection with the poloidal field created by

the plasma current shall co?fine the ionised gas in

the region of the torus axis.

The present

-

day generation of Tokamak

machines produces plasmas of electron densities ne

of the order of

1020

m-3 in the temperature range

of kT

1

KeV to 5 KeV

[ 1

-

4

1 .

The highest tempe-

ratures are obtained when the density is low. With

increasing densi-ty

one generally observes

'a

decrease

of the temperature.

Since there are processes which enable the

plasma to cross the magnetic barrier, there is a

continuous particle flux to the walls. The current

limiting diaphragm (limiter) is directly exposed to

the plasma. Due to plasma

-

wall interaction, wall

and limiter material is released and diffuses into

the_plasma, thus leading to contamination of the

initially pure hydrogen (or hydrogen isotopes)

plasma. Different physical processes are respon-

sable for the release of wall atoms

:

sputtering,

desorption or evaporation, blistering, arcing

[ 5

-

61. The impurities have a deleterious effect

on the energy balance and stability properties of

a plasma and may seriously affect the construction

conditions to be imposed upon a future thermo-

nuclear fusion reactor. Impurities have the

following effects

:

1.

They increase the radiation losses,

these losses are the higher the higher

thenuclear charge number Z

[ 7

-

_121;

2.

They influence the

article

and, thus,

the energy transport

[ 11

-

141

.

3. As a

consequence of this follows an

influence on the ion confinement time

115

I

.

4. They may lead to a radiation limit to

.

Tokamak operation

11 6

1.

5.

They have an influence on heating by

neutral beam injection, since the

charge exchange cross sections between

neutrals and ions increase with ion

charge number

Z of the impurity ion.

6.

They have an influence on heating by

high-frequency waves.

7. They may have an influence on the

6

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sputtering rate of wall material (for

_2.

_{FUSION REACTIONS}

_{I N}

_{A PLASMA}

instance due to He-induced blistering

or self sputtering)

[ 17

-

191.

_{For a technical fusion power plant, only}

the hydrogen isotopes

D and T can be envisaged as

I

:D+:T

-:~e(3.5

M ~ v ) +

in(l4.1

MeV)

I

Copture of

~ eions ~ +(ds)

\

Neutrons

fuel. Table

1

gives the reactions and the energy

liberated in a fusion collision. The fusion cross

sections are shown in Fig.

2

as a function of the

collisionenergy E. One sees that even in the

maximum the cross sections are by orders of magni-

tude smaller than for many electron-atom and

electron-ion collisions. One clearly sees that the

D;T

reaction is the most favorable one, since it

has the highest cross section at a given energy and

possesses (apart from the

2~

-

3 ~ e

process) the

P'

highest nuclear disintegration energy

(Q

-

value).

Mognet~c COILS for

toro~dal f ~ e l d Closed toro~dal vocuum chamber

(Ltner)

TABLE 1

Fig.

I

-

Tokamak configuration

In order to predict the radiation losses

from theoretical calculations one needs the atomic

data such as level structures, oscillator strengths,

and the cross sections for excitation, ionisation,

recombination and charge exchange of all ionisation

stages of the impurity ions. 1n.order

to measure

(and to eventually control) the impurity concentra-

tions one needs the wavelengths of a limited

number of well-chosen radiative transitions charac-

teristic for each stage of ionisation and the atomic

data which determine the intensity of a spectral

line and which relate it to the ground state density

of the special stage of ionisation.

Present-day high-temperature, fusion-like

plasmas still contain a relatively high amount of

impurity elements, mostly oxygen and metals like

iron, nickel, molybdenum and tungsten according to

the materials used for the walls and the limiters.

Elementary reactions with these impurity atoms and

their ions play an outstanding role in the energy

balance of plasmas and their general thermodynamic

properties. In order to clearly understand the

importance of these non-nuclear elementary reactions

and, thus, the importance of atomic and molecular

data in the frame of fusion research, it is useful

to briefly recall the essential processes in a

thermonuclear fusion plasma.

D-D and D-T fusion reactions

D

+

T

+

4 ~ e

(3.5 MeV)

+

n

(14.1

MeV)

; 17.60

MeV

D + T + T

3.27

MeV

+

P

; 18.34

MeV

Fig.

2

-

Cross sections

o

for fusion reactions,

after

[ 2 2 ]

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( a - p a r t i c l e s ) c a p t u r e d by t h e magnetic f i e l d w i l l

m a i n t a i n t h e plasma a t f u s i o n temperature ( s o - c a l l e d a - p a r t i c l e h e a t i n g ) . The 14.1 MeV n e u t r o n s a r e d i r e c t l y absorbed by t h e w a l l s and a working medium i n which t h e k i n e t i c energy i s converted i n t o h e a t . The working medium ( l i t h i u m o r Li-compound) s e r v e s a l s o f o r b r e e d i n g t r i t i u m according t o t h e r e a c t i o n s l i s t e d i n Table 2. TABLE 2 Breeding of t r i t i u m from n a t u r a l l i t h i u m ( 7 ~ i : 92.6 % ; 6 ~ i : 7 . 4 %) 7 ~ i

+

n + 4 ~ e + T

+

n '

-

2.47 MeV 6 ~ i

+

n-t 4 ~ e + T

+

4.78 MeV

A f u s i o n r e a c t o r w i l l only work when t h e plasma s a t i s f i e s two fundamental c o n d i t i o n s which follow from t h e r e a c t i o n c r o s s s e c t i o n s .

1 . When one assumes t h a t t h e D+ and T+ i o n s have a maxwellian v e l o c i t y d i s t r i b u t i o n f (v) one can c a l c u l a t e t h e v e l o c i t y - averaged r e a c t i o n c o e f f i c i e n t f o r f u s i o n p r o c e s s e s : where v i s t h e r e l a t i v e v e l o c i t y between t h e r e a c t i o n p a r t n e r s . F i g . 3 shows t h a t "acceptable" v a l u e s a r e only reached f o r k i n e t i c plasma temperatures kT > 5 KeV ( 1 KeV 1.16*107 OK),with kT 10 KeV b e i n g a d e s i r a b l e v a l u e . I n

o t h e r words, plasma temperatures of t h e o r d e r of l o 8 O K must be maintained. 2. When t h e r e a c t i o n c o e f f i c i e n t s a r e known i t i s p o s s i b l e t o c a l c u l a t e t h e r e a c t i o n mean f r e e p a t h A f o r f u s i o n r e a c t i o n s . F i g . 4 shows t h e r e s u l t f o r D-T and D-D r e a c t i o n s a t two d i f f e r e n t t e m p e r a t u r e s . I n f u t u r e r e a c t o r s of t h e Tokamak type ( o r o t h e r m a g n e t i c a l l y confined plasmas) t h e p a r t i c l e d e n s i t y n w i l l be of t h e o t h e r o f 1020 t o 1021 I ~ I - ~ . I n such plasmas, A w i l l be of t h e o r d e r o f l o 7 t o l o 8 m, i . e . t h i s d i s t a n c e must be t r a v e r s e d b e f o r e a d e u t e r o n has a chance o f approximately

30 % t o undergo f u s i o n . F i g . 4 shows f o r comparison t h e r a d i i of t h e e a r t h and t h e sun. A t kT '& 12 KeV, deuterons and t r i t o n s have i n t h e l a b o r a t o r y system a mean thermal v e l o c i t y of

< w > t h lo6 m/s. A t a p a r t i c l e d e n s i t y of nD = n ~ = 5-1020 f 3 f o r deuterons (nD) and t r i t o n s (nT) respec- t i v e l y one t h e r e f o r e needs a "mean r e a c t i o n time" -cr given by

A

-cr =

---

10 seconds ( 2) <

"

'th

i . e . t h e p a r t i c l e s must be confined f o r 10 seconds i n t h e h o t plasma zone i n o r d e r t o give them a high p r o b a b i l i t y f o r f u s i o n r e a c t i o n s . S t a t i s t i c a l l y t h e r e a r e p a r t i c l e s undergoing f u s i o n r e a c t i o n s i n s h o r t e r t i m e s , s o t h a t t h e c o n d i t i o n f o r t h e p a r t i c l e confinement time can b e somewhat r e l a x e d , say between I and 10 seconds. But t h i s does n o t profoundly modify t h e second c o n d i t i o n which r e q u i r e s t h a t t h e p a r t i c l e s must b e confined d u r i n g times of t h e o r d e r of seconds a t plasma d e n s i t i e s of t h e o r d e r of 5 . 1 0 ~ ~ m-3.

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JOURNAL DE PHYSIQUE We mention t h a t t h e confinement problem i s

avoided i n l a s e r f u s i o n e x p e r i m n t s by compressing t h e plasma t o super-high d e n s i t i e s . The d i f f i c u l t y

i s t o r e a c h such h i g h d e n s i t i e s t h a t t h e r e a c t i o n mean f r e e p a t h reaches v a l u e s of t h e o r d e r of magnitude a s o r s m a l l e r t h a n t h e r a d i u s of t h e compressed plasma. The corresponding c o n d i t i o n can d i r e c t l y be o b t a i n e d from F i g . 4 when one e x t r a p o l a t e s t h e curves t o h i g h e r deuteron d e n s i t i e s .

F i g . 4

-

Reaction mean f r e e p a t h A of a d e u t e r o n a s a f u n c t i o n of deuteron d e n s i t y . For compa- r i s o n t h e r a d i i o f e a r t h Re and sun

%

3 . COULOMB COLLISIONS. BREMSSTRAHLUNG

A plasma i s g l o b a l l y n e u t r a l , i . e . i n pure D-T plasma t h e D+ and Ti i o n s a r e e l e c t r i c a l l y compensated by an e q u a l number of e l e c t r o n s :

When a - p a r t i c l e s a r e p r e s e n t t h e c o n d i t i o n f o r q u a s i - n e u t r a l i t y w r i t e s nD + nT

+

2na = ne. E l e c t r i - c a l l y charged p a r t i c l e s undergo Coulomb c o l l i s i o n s . Fig..5 shows t h e mean c o l l i s i o n frequency of a p a r t i c l e

< v > = 1

nD <civ> ( 4 )

f o r ( e l a s t i c ) Coulomb and ( i n e l a s t i c ) f u s i o n

r e a c t i o n s when n ~ = 5

-

l o z 0 m-3. The Coulomb c o l l i s i o n f r e q u e n c i e s a r e by o r d e r s of magnitude l a r g e r t h a n t h e frequency f o r D-T f u s i o n r e a c t i o n s (lower c u r v e ) . These f r e q u e n t Coulomb c o l l i s i o n s

.

have two d e l e t e r i o u s e f f e c t s :

1 . They l e a d t o p a r t i c l e d i f f u s i o n a c r o s s t h e magnetic f i e l d l i n e s and, t h u s , t o a l o s s of thermal energy ( s o - c a l l e d d i f f u s i o n c o o l i n g )

.

2. They produce bremsstrahlung due t o e l e c t r o n - i o n c o l l i s i o n s ( s o - c a l l e d r a d i a t i o n c o o l i n g ) .

F i g . 5

-

Mean c o l l i s i o n f r e q u e n c i e s < v > a t a plasma d e n s i t y o f n ~ = n ~ = 5 . 1 0 ~ ~ md3

F i g . 6 shows t h e s p e c t r a l d i s t r i b u t i o n of t h e power d e n s i t y due t o bremsstrahlung. A t f u s i o n temperatures n e a r l y a l l energy i s r a d i a t e d w i t h wavelength A < 10

1.

Due t o t h e low p a r t i c l e d e n s i t y and t h e modest geometrical dimensions of t h e plasmas, almost a l l photons w i l l f r e e l y escape. This r a d i a t i o n t h e r e f o r e r e p r e s e n t s an i n s t a n t a - neous energy l o s s ( r a d i a t i o n c o o l i n g ) which

p r i m a r i l y c o o l s t h e e l e c t r o n gas. The power d e n s i t y r a d i a t e d p e r e l e c t r o n and p e r i m p u r i t y i o n of i o n charge number z and d e n s i t y nZ i s

(6)

emission equals bremsstrahlung for

Fig. 6

-

Spectral distribution of power

*X

(in arbitr. units) produced by bremsstrahlung

The fusion temperature can only be maintained when the energy loss rates due to diffusion and bremsstrahlung are compensated in situ by an equivalent heating rate. In a self-sustaining fusion plasma the energy is delivered by the 3.5 MeV a-particles. In a magnetic field o: B =

4

Tesla, their Larmor radius is 0.07 m and the Larmor frequency v ~ is 3. ~

lo7

~s-I, as indicated in ~ y Fig. 5. At particle densities n 5-loz0 m-3, their thermalisation occurs in times of .rth % 1 0 ~ ~ s .

In addition to bremsstrahlung, energy is continuously lost through electron cyclotron radiation. Assuming a Maxwellian velocity distribution, the power density radiated by a pure optically thin D-T plasma becomes [20] with ne = nZ

where it has been assumed that B , the ratio of plasma kinetic pressure to magnetic pressure, is equal to unity :

with ne in

G 3 ,

Te = TD = TT = T, and kT in KeV. Comparison of Eqs. (5) and (6) shows that cyclotron

kT

=

10213 KeV =

4.7 KeV

₍₈₎

and that cyclotron emission dominates for temperatures higher than this. The actual picture is, however, more complicated, because of the emission of the radiation at the fundamental of the electron cyclotron frequency and its harmonics. Were the emission only in the fundamental frequency, the absorption of the radiation would be so strong that practically all cyclotron radiation energy remained in a plasma of modest size. Due to the occurence of higher harmonics and the decrease of the absorption coefficient for these frequencies the size of a fusion plasma must be large in order to imprison cyclotron radiation. The reader finds details in references [ 20

-

211

.

As mentioned in the introduction, all Tokamak plasmas contain impurities. They lead to additional energy losses. In present-day Tokamak machines, line emission orginating from impurities very often represents the domilant radiation loss mechanism. But also other plasma properties depend on the kind of impurity species and their

concentrations.

4.

PLASMA MODEL, BASIC PLASMA EQUATIONS

In order to make quantitative statements about the influence of impurities on the plasma properties and the atomic, ionic and molecular data needed it is useful to describe the physical situation first by a model. This is then to be put in a mathematical form containing the atomic properties of the reactions between the various constituents and also the radiative properties. Fig. 7 presents such a model of a magnetically confined plasma in schematic form. One sees in the upper part of the figure : the radial distributions of temperature T, electron (proton) density ne(n+), hydrogen isotope density (H'), density of impurities of nuclear charge number Z in ion charge state z + . In the lower part of the figure one can see the particle fluxes to and from the wall of the various species, eventually injected To and Do neutral atoms (for-heating and refuelling), photon

I

(hv) and neutron ( n) fluxes from the plasma center

0

-+

to the walls, the electric current density (jp)

-+

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C1-78

Even during stationary operation the plasma repre- The following discussion of the influence

sents a dynamic system due to the strong mutual interaction between plasma and wall. It is described by the following equations :

a

-

ohm's law and the condition for-local electric quasi-neutrality;

b

-

Maxwell's equations ; c

-

The rate equations for :

1. particle density

...

n

-f

2. momentum densit

Y...

m<w>n 3. density of pressure tensor m<f: G>n

4.

density of internal energy Eint n 5. Photon field (photon energy and

photon momentum)

,

d

-

The interactions of plasma particles (and photons) with :

1. the surface of the wall materials, 2. the crystal structure of the wall

materials.

e

-

Initial and boundary conditions.

To- . Plasma

of elementary reactions will be based on the rate equations c. l to c.5 and especially on c.3 and c.4. Generally speaking, these rate equations follow from the collisional kinetic (Boltzmann) equation

af

+ ;

u

afs-(,) (9)

B=l

m~

at collision

radiation and its various velocity moments [ 2 6

-

291. Here,

-f

fs

(?,

w, t) is the velocity distribution function of particles of species s = s(k, i, z) as a function

of space point

2,

velocity

$

and time t. 8 =

I ,

2, 3, represents the three coordinate directions. ms is the particle masse and FS,$ represents the external forces (e.g. due to electric or magnetic fields) acting on the particle. At thermal equilibrium fs represents the Maxwell distribution. Integration of fs over

f:

yields the particle density ns of species s:

-f -f

+

111

fs (r, w, t) dwl dwp dwg = ns(r,t)(lO) The right-hand side of eq. (9) is the rate of change of fs due to collision and radiation processes. This term contains for instance Coulomb and ionisation collisions, but also the effect of fluctuating electric fields as a result of collec- tive processes between plasma particles. Integrating

+

eq. (9) over w yields the rate equation for the particle density ns

Plosmo

0

1

( 1 1 )

radiation

-f

The term proportional to Fs vanishes after integration. The right-hand side represents an abbreviation for the sum of all collision and radiation processes which can lead to a change of n,.

Magnetic Field

Eq. (11) is commented on in Fig. 8 and shows what

\

-

'-

COLD WALL atomic properties intervene in the different terms.

Z N : HOT PLASMA PLASMA

LAYER The D-T reactions intervene on the right-hand side

Fig. 7

-

Model of a plasma whose species interact with a material wall and with injected fast neutral particles. '

The elementary reactions and atomic properties enter in all eqs. (a) to (e), and in particular one has

a

set of equations of types (c) and (d) for every chemical species k in quantum li> and ionisation state z . This leads to a complicated system of coupled differential equations. For details see [23

-

251. Only rigorous simplifications lead to tractable equations.

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RATE EQUATION FOR PARTICLE DENSITY

ns

nS(ct) = PARTICLE DENSITY"~"OF SPECIES '5" ALL ELEMENTARY REACTIONS

/

ere

enter M~crolnstobll~t~es

M H D- Instobllit~es

<$>=

mean d~ffus~on veloc~ty

F i g . 8

-

P h y s i c a l p r o c e s s e s and plasma p r o p e r t i e s i n t e r v e n i n g i n t h e r a t e e q u a t i o n f ~ r p a r t i c l e d e n s i t y ns

.

Taking from t h e t e n s o r e q u a t i o n

-

mentioned a s c.3

-

t h e t r a c e and d i v i d i n g by two y i e l d s t h e r a t e e q u a t i o n f o r t h e d e n s i t y of t r a n s l a t i o n a l ( t h e r m a l ) energy = (3/2)nskTs of s p e c i e s s . Adding t o t h i s t h e r a t e e q u a t i o n f o r t h e d e n s i t y of i n t e r n a l energy ~ i n t = E i n t ns y i e l d s t h e r a t e ( o r b a 1 a n c e ) e q u a t i o n f o r thermal p l u s i n t e r n a l energy of s p e c i e s s . I n t h e e x p r e s s i o n f o r t h e i n t e r n a l energy d e n s i t y i s f o r i n s t a n c e i n c l u d e d t h e energy which must be f e d i n t o t h e eystem t o i o n i s e t h e p a r t i c l e s . I n Tokamaks

,

MoZ+ i o n s w i t h z = 31 have been observed. I n o r d e r t o i o n i s e a Mo-atom ( Z = 42) u n t i l z = 31 one needs p e r ~ o ~ l + - i o n an energy of approximately E i n t . = 15 KeV. M u l t i p l y i n g by t h e d e n s i t y n31+ of t h e 31-times iomised Mo-particles y i e l d s t h e d e n s i t y of i n t e r n a l energy, E i n t n31+, which i s t r a n s p o r t e d by t h e Mo31+-ions. The t o t a l i o n i s a t i o n energy stocked i n t h e system i B o b t a i n e d by summing over a l l chemical s p e c i e s i n a l l i o n i - s a t i o n s t a g e s z . I n a d d i t i o n e x c i t e d p a r t i c l e s c o n t r i b u t e t o t h e i n t e r n a l energy. Although t h e i m p u r i t y d e n s i t y i s g e n e r a l l y much s m a l l e r t h a n t h e t o t a l p a r t i c l e d e n s i t y , t h e d e n s i t y of i n t e r n a l energy can make up s e v e r a l p e r c e n t of t h e thermal energy d e n s i t y under unfavorable c o n d i t i o n s . I f

h i g h l y i o n i s e d p a r t i c l e s succeed i n d i f f u s i n g i n t o c o l d e r plasma zones n e a r t h e w a l l , t h e d e n s i t y of i n t e r n a l energy can l o c a l l y reach t h e o r d e r of t h e thermal energy d e n s i t y . A t p r e s e n t , n o t very much

i s known about t h e d i f f u s i o n p r o p e r t i e s of h i g h l y i o n i s e d i m p u r i t i e s . I n any c a s e must t h e r a t e e q u a t i o n f o r t h e i n t e r n a l energy d e n s i t y be taken i n t o account, o t h e r w i s e i t i s n o t p o s s i b l e t o c a l c u l a t e r a d i a t i o n and d i f f u s i o n l o s s e s of impu- t i t y s p e c i e s c o r r e c t l y and t o i n t e r p r e t e measure- ments i n a r e a l i s t i c manner. S i m p l i f i c a t i o n s of t h e v a r i o u s e q u a t i o n s have l e d t o widely a p p l i e d numerical codes f o r t h e c a l c u l a t i o n of p r o p e r t i e s of Tokamak plasmas [ 30

-

311

.

A t t h e p r e s e n t time t h e s e codes have t o a l a r g e p a r t s t i l l a h e u r i s t i c c h a r a c t e r , s i n c e t h e y c o n t a i n many e m p i r i c a l assumptions. Summing up t h e r a t e e q u a t i o n s of a l l s p e c i e s "s" y i e l d s t h e e q u a t i o n which d e s c r i b e s t h e r a t e of change of energy d e n s i t y E = s C ~ s of t h e plasma a s a whole. I n o r d e r t o p u t t h i s e q u a t i o n i n a t r a c t a b l e form we i n t r o d u c e t h e mean mass

+

v e l o c i t y vg of t h e plasma i n t h e l a b o r a t o r y system : ( w i t h

p

= mean mass d e n s i t y )

+

t h e t o t a l h e a t f l u x v e c t o r q

t h e energy l o s s r a t e Gch-ex. due t o f a s t charge exchange n e u t r a l s e s c a p i n g from t h e plasma, and t h e energy l o s s r a t e

erad

of t h e r a d i a t i o n f i e l d . Then t h e r a t e e q u a t i o n f o r t h e energy d e n s i t y ( a l s o termed "energy b a l a n c e e q u a t i o n " ) can be p u t i n t o t h e f o l l o w i n g form : Input r a t e of

+

-+

+ l o c a l power =

2

+ V - ( E V ~ )

+

nkTV-vo

[

d e n s i t y

]

a t

+ +

+

V.q

+

$ch.ex-

+

irad

The f i r s t term g i v e s t h e change of E

d u r i n g a t r a n s i e n t s t a t e . For s t a t i o n a r y plasmas,

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c1-80

JOURNAL DE PHYSIQUE

mass flow v e l o c i t y

v0

of t h e plasma a s a whole. I n w i t h t h e f o l l o w i n g we assume t h a t t h e s e terms a r e z e r o .

The l a s t term accounts f o r a change o f k i n e t i c _Pa = < o v > ~ ~ Qa n ~ nT energy of t h e plasma a s a whole r e l a t i v e t o t h e

l a b o r a t o r y system. We p u t t h i s term a l s o e q u a l where Qa = 3.5 MeV = 5.6.10-l3 J o u l e . t o z e r o . Then follows t h e e x p r e s s i o n Eqs. (15) t o (18) w i l l now s e r v e t o d i s c u s s q u a n t i t a t i v e l y t h e c o n d i t i o n s t o be f u l - f i l l e d by a s e l f - s u s t a i n e d D-T f u s i o n plasma. (15) The r a d i a t i o n l o s s r a t e Prad c o n t a i n s d i f f e r e n t c o n t r i b u t i o n s : pf-f + pf-b + ;die1 + ;b-b + ;cycl Prad = where + f - f , i f - b 9 ; d i e l , 2 - b and p c y c l a r e t h e l o s s r a t e s due t o f r e e - f r e e ( f - f )

,

free-bound ( f - b ) , d i e l e c t r o n i c ( d i e l ) , bound-bound (b-b) and c y c l o t r o n r a d i a t i o n r e s p e c t i v e l y . The l a t t e r depends on e l e c t r o n d e n s i t y ne, e l e c t r o n temperature Te and magnetic i n d u c f i o n B and can give a non-negligible c o n t r i b u t i o n a t high T and B and small plasma s i z e s . We w i l l assume throughout t h i s paper t h a t t h e plasma i s s o l a r g e t h a t complete imprisonment of t h e c y c l o t r o n r a d i a t i o n o c c u r s . This assumption i s

. c y c l g e n e r a l l y made and t h u s allows t o n e g l e c t P

.

For a c t u a l r e a c t o r c a l c u l a t i o n s , however, t h i s assump- t i o n i s n o t allowed. Power balance c a l c u l a t i o n s f o r a s e l f - s u s t a i n e d Tokamak r e a c t o r showed t h a t even i n t h e presence of 0.7 % z = 18 i m p u r i t i e s t h e c y c l o t r o n r a d i a t i o n l o s s r a t e can p r e v e n t energy s e l f - s u s t a i n i n g of a D-T plasma when kT > 40 KeV

[ 3 2 ] . One s e e s from E q . (15) t h a t e s p e c i a l l y t h e energy l o s s e s due t o charge exchange c o l l i s i o n s and r a d i a t i o n p l a y a fundamental r o l e i n t h e o v e r a l l energy b a l a n c e . We s t i l l emphasize t h a t t h e l o c a l h e a t i n g r a t e i n t h e energy b a l a n c e e q u a t i o n i s

composed of two terms, t h e e x t e r n a l power i n p u t Pext ( e . g . ohmic h e a t i n g o r f l e u t r a l p a r t i c l e i n j e c t i o n ) and t h e i n t e r n a l power i n p u t (Pa) due t o a - p a r t i c l e h e a t i n g of t h e 3.5 MeV w p a r t i c l e s (lie2+ i o n s ) produced i n t h e D-T r e a c t i o n 2~ + 3~ = 4 ~ e

+

I n

+

17.6 MeV ( 2 8 . 2 . 1 0 - ~ ~ ~ ) . Thus : I n p u t r a t e of

.

l o c a l power

1 =

Pext +

Pa

5 . THE SELF-SUSTAINED D-T FUSION PLASMA

A f u s i o n plasma i s s a i d t o be s e l f - s u s t a i n e d when t h e e x t e r n a l power i n p u t i s c u t t - o f f and a l l energy l o s s e s a r e compensated by t h e s o l e a - p a r t i c l e h e a t i n g . Let us c o n s i d e r d i f f e r e n t c a s e s .

5.1 - Plasma w i t h o u t d i f f u s i g n l o s s e s

Consider a quasi-homogeneous D-T plasma o f s u f f i c i e n t dimension. Across such a plasma, d e n s i t y and temperature s t a y p r a c t i c a l l y c o n s t a n t , d i f f u s i o n can be n e g l e c t e d . The energy b a l a n c e

a

E -f +

e q u a t i o n becomes (with Pext = 0 ,

-

_{a t}

= 0 ; V . q ) :

Let us e v a l u a t e t h i s r e l a t i o n f o r a pure D-T plasma and a D-T plasma c o n t a i n i n g i m p u r i t i e s .

-

Pure D-T plasma

_...

I n a pure D-T plasma, t h e only r a d i a t i o n i s e l e c t r o n - i o n f r e e - f r e e bremsstrahlung (and e v e n t u a l l y c y c l o t r o n r a d i a t i o n ) whose s p e c t r a l i n t e n s i t y d i s t r i b u t i o n i s shown i n F i g . 6. A t thermonuclear temperatures weak X-ray r a d i a t i o n i s produced. The energy balance e q u a t i o n becomes

(with c y c l o t r o n r a d i a t i o n n e g l e c t e d ) :

Both

5,

and

?f-f

a r e shown i n F i g . 9 a s a f u n c t i o n of kT. The two curves i n t e r s e c t a t kT = 4.2 KeV corresponding t o a temperature of 4 . 7 . 1 0 ~ O K and which can be considered a s t h e

(10)

plasma which i s o p t i c a l l y t h i n towards c y c l o t r o n r a d i a t i o n t h e c r i t i c a l temperature r i s e s from 4.2 KeV t o 7 KeV f o r B = 1 . 5.1.2

-

D-T plasma c o n t a i n i n g i m p u r i t i e s

_...

Owing t o plasma-wall i n t e r a c t i o n , w a l l m a t e r i a l i s s p u t t e r e d and d i f f u s e s i n t o t h e plasma. These unwanted i m p u r i t y s p e c i e s a r e e x c i t e d and i o n i s e d , and t h e i o n s can recombine w i t h e l e c t r o n s . There i s f u r t h e r i n c r e a s e d bremsstrahlung l o s s which

i s p r o p o r t i o n a l t o z 2 , egz b e i n g t h e i o n charge s e e n by t h e e l e c t r o n s . For t h e same e l e c t r o n d e n s i t y and temperature a f u l l y i o n i s e d Fe-atom ( z = Z = 26) r a d i a t e s 676 times more than a D+ o r T+ i o n . I n any c a s e , i m p u r i t i e s i n c r e a s e t h e bremsstrahlung l o s s , and a s l o n g a s they a r e n o t completely s t r i p p e d t h e y a l s o c o n t r i b u t e w i t h l i n e , recombina- t i o n and d i e l a c t r o n i c r a d i a t i o n t o t h e energy l o s s .

109 lo*

107 lo6

1 o5

lo4

lo3

9KeV 20 KeV

F i g . 9

-

Power d e n s i t i e s

f'

f o r n u c l e a r and r a d i a t i o n p r o c e s s e s as a f u n c t i o n of kT, a t a d e n s i t y of n ~ = n ~ = 5 . 1 0 ~ ~ m-3. Cyclotron r a d i a - t i o n omitted.

According t o t h e m a t e r i a l s employed one h a s

i d e n t i f i e d i n Tokamaks t h e f o l l o w i n g i m p u r i t i e s :

C , 0 , A l , T i , C 1 , Fe, C r , Co, N i , Mo, W, Au.

Atoms of low n u c l e a r charge number Z can b e i o n i s e d t o t h e b a r e nucleus whereas high-Z m a t e r i a l s w i l l even under thermonuclear c o n d i t i o n s n o t be f u l l y s t r i p p e d .

The r a d i a t i o n l o s s e s a r e g e n e r a l l y c a l - c u l a t e d and i n t e r p r e t e d i n t h e frame of t h e so- c a l l e d "corona model" [ 7 ] [ 101 [ 24

-

251 [ 33

-

341 :

i n a f i r s t s t e p one s o l v e s t h e coupled system of r a t e e q u a t i o n s o f type (11) f o r t h e ground s t a t e d e n s i t i e s i = 1 of a l l i o n i s e d and unionised s p e c i e s ( k , z , i = 1 ) . For t h e s t a t i o n a r y homogeneous s t a t e one o b t a i n s

where S and a a r e t h e i o n i s a t i o n and recombination c o e f f i c i e n t s r e s p e c t i v e l y . a c o n t a i n s both r a d i a - t i v e and d i e l e c t r o n i c recombination p r o c e s s e s . F i g . 12 shows a s example t h e d i s t r i b u t i o n of ~ e ' + - i o n s . I n a next s t e p on c a l c u l a t e s t h e par- t i c l e d e n s i t i e s of t h e e x c i t e d s p e c i e s , s i n c e t h e ground s t a t e p a r t i c l e s a r e known from t h e f i r s t s t e p . The f i n a l s t e p c o n s i s t s i n t h e c a l c u l a t i o n

_---

of t h e r a d i a t i o n l o s s e s a c c o t d i n g t o Eq. ( 1 6 ) , s e e a l s o E q . ( 3 8 ) . F i g . 13 and 14 show a s two examples t h e r a d i a t i o n l o s s e s of oxygen and molybdenum p e r e l e c t r o n and p e r i m p u r i t y p a r t i c l e . The v a l i d i t y of corona e q u i l i b r i u m i s d i s c u s s e d i n s e c t i o n 8 .

The s e l f - s u s t a i n e d D-T f u s i o n plasma must s a t i s f y E q . ( 1 9 ) . Fig. 9 shows t h e r a d i a t i v e energy l o s s e s of a D-T plasma c o n t a i n i n g e i t h e r 0.1 % i r o n o r 0.1 % t u n g s t e n a s i m p u r i t y . The minimum f u s i o n temperatures l i e now a t 9 KeV o r 20 KeV r e s p e c t i v e l y , i n s t e a d of 4.2 KeV of a pure D-T plasma. One c l e a r l y s e e s t h a t an i m p u r i t y c o n c e n t r a t i o n of more than 1 % i r o n o r more t h a n 0.2 % t u n g s t e n w i l l l e a d t o s o a h i g h r a d i a t i o n l o s s t h a t i g n i t i o n and s e l f - s u s t a i n i n g becomes impossible. This i s one of t h e p h y s i c a l r e a s o n s why t h e development of low-Z m a t e r i a l s f o r t h e

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C1-82 JOURNAL DE PHYSIQUE

t h e o r d e r of magnitude of measured i m p u r i t y concen- t r a t i o n s i s 0 . 5 % - 5 % 0 and C , 0.1 %

-

0 . 5 % Fe, 0.1 % Mo, 0.1 % W.

-

I n c l u d i n g c y c l o t r o n r a d i a t i o n under o p t i c a l l y t h i n c o n d i t i o n s w i l l l e a d t o s t i l l lower c r i t i c a l i m p u r i t y c o n c e n t r a t i o n s .

5.2

-

Plasma w i t h d i f f u s i o n l o s s e s

Due t o t h e l a r g e temperature and d e n s i t y g r a d i e n t s t h e r e occur d i f f u s i o n l o s s e s i n a d d i t i o n t o r a d i a t i o n l o s s e s . Fig.8shows t h a t t h e d i f f u s i o n p r o c e s s e s depend p a r t l y on non-nuclear c o l l i s i o n s , and from Eq. (13) f o l l o w s t h a t a s s o c i a t e d energy l o s s e s a r e contained i n t h e e x p r e s s i o n f o r h e a t conduction. Neglecting t h e charge exchange f l u x , Eq. (15) l e a d s now t o t h e f o l l o w i n g c o n d i t i o n f o r a s t a t i o n a r y s e l f - s u s t a i n e d D - T f u s i o n plasma :

i . e . a - p a r t i c l e h e a t i n g must compensate d i f f u s i o n

+

-+

and r a d i a t i o n l o s s e s . The average value of V-q can

-

be expressed by, t h e mean energy confinement time T

E i n t h e following way. The mean thermal energy d e n s i t y i s

where summation i s over a l l s p e c i e s 'Is". The r a t e e t r a t which ~ t r changes due t o d i f f u s i o n i s

ttr=

-

.

=E We can t h e r e f o r e w r i t e

-+

+ Expressing i n Eq. (22) V.q by t h e r e l a t i o n (24) w i t h ~ t r given by Eq. (23) y i e l d s t h e f o l l o w i n g energy balance e q u a t i o n

Expressing Pa and Prad a s a f u n c t i o n of d e n s i t y and temperature y i e l d s a simple r e l a t i o n f o r t h e product nerE. Eq. (25) i s t h e s o - c a l l e d space averaged i g n i t i o n c o n d i t i o n [35 ] .For a n o t h e r presen- t a t i o n o f t h e i g n i t i o n parameter s e e [ 3 6 ] . We now apply ~ q . . (25) t o a pure D-T and a D-T plasma c o n t a i n i n g i m p u r i t i e s .

5.2

-

1

-

pure.

p.-:.

p:?~??.

We have nD

+

nT = ne and, t h u s , ~ t r

2

= 3 nekT. F u r t h e r Pa = <OV>DT Qa n e / 4 and +f-f

-

'

Prad = - ~ ( ~ , z = l ) n ~

.

I n s e r t i n g t h i s i n t o

Eq. (25) y i e l d s t h e r e l a t i o n

which i s only a f u n c t i o n of plasma temperature T .

F i g . 10 (curve 0 %) shows t h e i g n i t i o n c o n d i t i o n (26) f o r a pure D-T plasma. For an e l e c t r o n d e n s i t y ne = l o z 1 m-3 and an energy confinement time TE = Is one h a s nerE = loz1 m-3s. The k i n e t i c temperature r e q u i r e d f o r i g n i t i o n t h e n i s

kT = 6.5 KeV. When c y c l o t r o n r a d i a t i o n i s allowed t o escape t h e i g n i t i o n temperature becomes

k T m 10 KeV f o r 6 = 1 .

F i g . 10

-

The c o n d i t i o n f o r nerE which must be f u l f i l l e d i n a s e l f - s u s t a i n e d f u s i o n plasma ( s o - c a l l e d i g n i t i o n c o n d i t i o n )

N e u t r a l p a r t i c l e i n j e c t i o n (2.1 MW) i n t o t h e P r i n c e t o n P.L.T. plasma h a s l e d t o kTiOn

rz 5.5 KeV, t h e h i g h e s t temperature e v e r reached i n a Tokamak [ 3 1

.

The nerE v a l u e was approximately

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5.2.2 - D-T plasma c o n t a i n i n g i m p u r i t i e s

_...

A r e l a t i o n f o r neTE analogous t o Eq. (26) can be d e r i v e d f o r a plasma c o n t a i n i n g i m p u r i t i e s . For reasons of s i m p l i c i t y l e t us assume t h a t a t temperature T only t h e most abundant z-times i o n i s e d atoms of n u c l e a r charge Z a r e p r e s e n t . We put nZ p r o p o r t i o n a l t o ne, nZ = fne. The c o e f f i c i e n t f c h a r a c t e r i z e s t h e i m p u r i t y c o n c e n t r a t i o n r e l a t i v e t o ne. The q u a s i - n e u t r a l i t y c o n d i t i o n i s ne = nD

+

"T + 2"' = nD

+

"T + z f n e . It follows i . e . i m p u r i t i e s reduce t h e hydrogen i s o t o p e c o n c e n t r a t i o n . This l e a d s t o a lowering of t h e a - p a r t i c l e h e a t i n g r a t e compared t o pure D-T plasmas. I n a d d i t i o n . t o t h i s , t h e presence of i m p u r i t i e s l e a d s t o i n c r e a s e d r a d i a t i o n l o s s e s . With t h e assumption made above f o r n Z ,

trad

can be p u t i n t o t h e f o l l o w i n g form

a n d * i n c l u d e s bremsstrahlung, recombination, cyclo- t r o n , d i e l e c t r o n i c , and l i n e r a d i a t i o n of t h e i m p u r i t y element. One f i n d s f o r ne-rE t h e r e l a t i o n

[ 351

P u t t i n g f = 0 , B = 0 and z = 1 l e a d s back t o Eq.(26) w i t h

k

*

if-f

( T , z = l )

.

F i g . 10 show t h e i n f l u e n c e of i m p u r i t i e s on t h e i g n i t i o n c o n d i t i o n . Example : ne = m-3,

TE = I s , n e - c ~ = loz1 md3 s . When 0.1 % t u n g s t e n i s

p r e s e n t , a minimum temperature of 25 KeV i s needed compared t o 6 . 5 K e V f o r a pure D-T plasma. The c a l c u l a t i o n s were based on assumption t h a t nZ s a t i s f i e s c o r o n a l e q u i l i b r i u m and t h a t t h e plasma

i s o p t i c a l l y t h i c k towards c y c l o t r o n r a d i a t i o n .

5.2.3

-

The Lawson c r i t e r i o n

_...

One can ask : what a r e t h e minimum condi- t i o n s f o r ne, T , and T i n o r d e r t o compensate by

thermonuclear p r o c e s s e s t h e energy l o s s e s of a plasma? The answer l e a d s t o t h e L a w s o n c r i t e r i o n [ 3 7 ] i n i t i a l l y d e r i v e d f o r a pulsed plasma under t h e

f o l l o w i n g assumptions : a plasma i s i n s t a n t a n e o u s l y h e a t e d t o f u s i o n temperature Te = Ti = T. The e n e r g y r e q u i r e d t o due t h i s i s ( 3 / 2 ) ( n D + nT

+ ne)kT = 3nekT f o r a pure D-T plasma (when d i s s o c i a t i o n and i o n i s a t i o n e n e r g i e s a r e o m i t t e d ) . The plasma p a r t i c l e s s h a l l be confined d u r i n g a time T ~ . The energy d e n s i t y z ; ~ - ~ T ~ r a d i a t e d d u r i n g

TL due t o bremsstrahlung w i l l c o o l t h e plasma. I n o r d e r t o m a i n t a i n t h e plasma a t f u s i o n tempera- t u r e T t h e same amount of energy must c o n t i n u o u s l y b e f e d i n t o t h e system. The t o t a l amount of energy p e r u n i t volume f o r h e a t i n g and m a i n t a i n i n g t h e plasma s t a t e d u r i n g TL i s

'f-f Pin TL = Pin = 3nekT

+

P TL

During t h e ( i d e a l ) confinement time TL

t h e r a d i a t i o n energy i s absorbed by t h e w a l l s . F u r t h e r , D-T r e a c t i o n s produce d u r i n g TL p e r u n i t volume t h e energy <OV>DT n ~ n ~ QDT TL w i t h QDT = 17.6 MeV ( i . e . a - p a r t i c l e s p l u s n e u t r o n s ) . The n e u t r o n s a r e immediately absorbed by t h e w a l l s , t h e energy c o n t a i n e d i n t h e confined plasma

p a r t i c l e s i s a l s o absorbed by t h e w a l l s a f t e r t h e confinement time T L , i . e . when t h e whole plasma h a s r e l a x e d . The energy absorbed by t h e w a l l s a f t e r a f u l l c y c l e i s

Lawson assumed t h a t t h e absorbed energy can be transformed w i t h a n e f f i c i e n c y of 17 i n t o e l e c t r i c a l , mechanical o r chemical energy which j u s t e q u a l s t h e amount of energy n e c e s s a r y t o r e p e a t t h e c y c l e . This y i e l d s t h e Lawson c r i t e r i o n

nPwa11 = P i n

I f qPwall > P i n , t h e system produces more energy t h a n i t consumes. For a pure D-T plasma t h e Lawson c o n d i t i o n can b e p u t i n t o t h e f o l l o w i n g form

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LAWSON LIMIT FOR D - T PLASMA

l e v e l . The f u e l must be h e a t e d up t o f u s i o n tempe- r a t u r e . The power d e n s i t y r e q u i r e d i s P f u e l =

<UV>DT nD n ~ ( 3 1 2 ) kT. The space averaged balance e q u a t i o n f o r t h e power d e n s i t y t h e n i s

F i g . 1 1

-

The Lawson product n e a s a f u n c t i o n of ~ ~ plasma t e m p e r a t u r e , w ~ t h

n

= 40 % [ 3 5 ]

.

Quite s i m i l a r l y one can formulate t h e Lawson c o n d i t i o n f o r a plasma c o n t a i n i n g ~ m p u r i t i e s

[ 3 5 ] . The corresponding curves a r e a l s o shown i n F i g . 1 1 . One s e e s t h e s t r o n g i n f l u e n c e of t h e high-

Z elements on t h e Lawson c r i t e r i o n , i n agreement w i t h F i g . 10.

For a c o n t i n u o u s l y working r e a c t o r t h e

_...

Lawson c o n d i t i o n h a s t o be modified. I n t h e c o n t i - nuous regime t h e plasma l o s e s p e r u n i t volume and

tr

.

u n i t time t h e energy

+

Prad. Fusion r e a c t i o n s

T c

y i e l d r ~ e u t r o n s . T h e i r power d e n s i t y i s

f?,

= <UV>DT nD nT

h.

where Q, = 14.1 MeV. The power

€ t r

.

d e n s i t y

-

+ Prad +

in

i s absorbed by t h e w a l l s Tc

and transformed w i t h an e f f i c i e n c y

n

i n t o an energy form which can be r e - i n j e c t e d i n t o t h e plasma. To t h e q u a n t i t y

we have s t i l l t o add t h e a - p a r t i c l e h e a t i n g r a t e

$a = <UV>DT nD nT

Q,

(where

Qa

= 3.5 MeV and qDT = Qa f

h)

i n o r d e r t o o b t a i n t h e whole power

d e n s i t y which i s a t our d i s p o s a l i n o r d e r t o b a l a n c e t h e energy l o s s e s .

Since D and T i s burned ( i . e . d i s a p p e a r s w i t h a r a t e < U V > D ~ n ~ nT) r e f u e l l i n g i s n e c e s s a r y

i n o r d e r t o m a i n t a i n t h e d e n s i t y on a c o n s t a n t

( 3 4 )

For a p u r e D-T plasma follows

(35) which must be compared with Eq. (33) f o r t h e pulsed r e a c t o r . Eq. (35) l e a d s p r a c t i c a l l y t o t h e same n , ~ - v a l u e s a s t h o s e shown i n F i g . 1 1 . This a l s o h o l d s f o r a plasma c o n t a i n i n g i m p u r i t i e s . The c u r v e s of F i g . 1 1 can t h e r e f o r e be c o n s i d e r e d a s t h e minimum c o n d i t i o n f o r b o t h a p u l s e d and c o n t i - nously working r e a c t o r .

I n t h e p r e s e n t paper we have assumed t h a t i o n and e l e c t r o n temperatures a r e e q u a l . Under a c t u a l c o n d i t i o n s t h i s i s n o t t h e c a s e . I n a r e f i n e d model one has t o c o n s i d e r t h e i o n s and e l e c t r o n s e p a r a t e l y . For d e t a i l s s e e e . g . [ 3 2 ] .

We emphasize t h a t t h e Lawson c r i t e r i o n r e p r e s e n t s an a b s o l u t e lower l i m i t t o any energe- t i c a l l y s e l f - s u s t a i n e d D-T plasma. For a system which s h a l l produce more energy then i t consumes one needs nerL > 1 which b r i n g s t h e curves c l o s e t o t h o s e of F i g . 10.

I n t h e f o l l o w i n g s e c t i o n we w i l l d i s c u s s some s p e c i a l atomic p h y s i c s a s p e c t s w i t h r e g a r d t o i m p u r i t i e s .

6 . IONISATION AND RECOMBINATION

(14)

successive ionisation stages of a chemical element then simply become a function of electron tempera-

n,z+l

_{- K , 1}

ture Te, i .e-

-

= f (Te)

.

Thus, also the nc, l relative concentration $,I = '+'n (with 2% A l - 2 .

C

Cz,

=

1) is a function of Te only. Fig. 12 shows as an example the relative distribution of the ionised species of iron (Z = 26). Complete stripping requires temperatures of more than 20 KeV. Higher temperature are needed for complete stripping of still higher-Z elements.

Fig. 12

-

Relative abundance (concentration) of ionised species of iron as a function of temperature, after [38]. Corona model, with two resonance states taken into account for dielectronic recombination, no ne-dependent correction of dielectronic recombination coefficient.

Tab. 3 shows some ionisation energies EiOn for the process z + z+l of the elements iron, molybdenum and tungsten. Temperatures of more than 50 KeV are needed for complete stripping of molybdenum, which is in agreement with Fig. 14.

The distribution q f the ionisation stages over temperature depends in certain ranges in a sensitive manner on the di-electronic recombination coefficient

.

The latter is not only a function of Te but also of electron density n, and probably of magnetic field

B. At present rather high uncer-

tainties still exist about the absolute values of the di-electronic recombination coefficients as a function of the mentioned three paramete?&.

7. EXCITATION

; RADIATION LOSSES

When the ground state densities are known, the populations of excited levels are obtained from the coronal balance equation for the

population densities. Denoting by

cZ

the exci- k,j 1

tation coefficient for electronic excitation of level j from the ground level 1, and by

(

the

,

jj Einstein coefficient for spontaneous

de-excitation

i + j (i < j) one has for level j

where p is the highest level which is still considered. The first term is the excitation rate for 1 -t j, the second term is the spontaneous de-

excitation rate j + Ci, and the last sum accounts for cascading from all levels j < m 5 p. For the ensemble of all excited levels 2 5 j 5 p, the Eq. (36) represents a system of linear equations

for the unknown population densities %,j (j = 2 to p). For instance for a three-level system (one ground and two excited levels) one has

Multiplying the first of the Eq. (36) by kv;,,, =

E"

-

k , 2 ,:E

,

the second by hvz ,,I3 = <,3

-

'

E

and so on, replacing in the appropriate

k, 1

terms hvZ by sums of energy differences k , lj

between (excited) levels, and then summing up all equations yields for the power density

tbb-b

due to line radiation the expression [25] :

(15)

c1-86

and t h e f i r s t l i n e of Eq. (60) by t h e f i r s t l i n e of Eq. (38) above. The f i n a l r e s u l t -second l i n e of Eq. (60) i n [ 2 5 ]

-

i s c o r r e c t and a g r e e s w i t h Eq. (38) of t h e p r e s e n t paper.

The t o t a l power l o s s r a t e i s given by Eq

.

( 1 6)

.

F i g s

.

13 and 1 4 show t h a t t h e main c o n t r i b u t i o n t o t h e r a d i a t i o n l o s s e s o r i g i n a t e s from l i n e r a d i a t i o n a s long a s complet s t r i p p i n g has n o t y e t occured.

Measurement of l i n e r a d i a t i o n s e r v e s f o r

- - - -

-

- - - -

- - -

d e t e r m i n i n g i m p u r i t y c o n c e n t r a t i o n s . I n t h e frame of t h e corona model one s t a r t s w i t h Eq. (36) i n which t h e cascading terms a r e n e g l e c t e d . This l e a d s t o t h e f o l l o w i n g e x p r e s s i o n f o r t h e e x c i t e d s t a t e p o p u l a t i o n s : The s p e c t r a l i n t e n s i t y of a l i n e h + j ( h < j ) i s I~ k , h j - -

nE,j

$,hj hv:,hj' For t h e i m p u r i t y d e n s i t y of t h e chemical element k i n i o n i s a t i o n s t a g e z t h u s f o l l o w s

-

-.

i . e . a p a r t from t h e measured i n t e n s i t y one needs b o t h t h e E i n s t e i n c o e f f i c i e n t s and e x c i t a t i o n c o e f f i c i e n t s f o r t h e d e t e r m i n a t i o n of t h e i m p u r i t y d e n s i t y . E r r o r s i n t h e s e q u a n t i t i e s a f f e c t d i r e c t l y t h e p r e c i s i o n w i t h which t h e i m p u r i t y c o n c e n t r a t i o n i s determined. The e x p e r i m e n t a l d e t e r m i n a t i o n of i m p u r i t y c o n c e n t r a t i o n s i n Tokamak plasmas i s g e n e r a l l y based on Eq. ( 4 1 ) . It r e p r e s e n t s a good approximation f o r allowed l i n e s o r i g i n a t i n g from n o t t o o h i g h e x c i t e d l e v e l s . It should be a p p l i e d w i t h g r e a t e c a u t i o n when t h e i n t e n s i t y of a "forbidden l i n e " i s used. Since many d a t a - e s p e c i a l l y f o r t h e h i g h e r i o n i s a t i o n s t a g e s of high-Z elements-are s t i l l unknown one o f t e n t r i e s t o g e t t h e m i s s i n g d a t a by e x t r a p o l a t i n g a l o n g i s o - e l e c t r o n i c sequences. This very e f f i c i e n t method s h o u l d b e a p p l i e d w i t h p r u d e n c e f o r those i s o - e l e c t r o n i c c o n f i g u r a t i o n s which show a rearrangement of o r b i t a l s w i t h i n c r e a s i n g 2 . The r i g h t s i d e of Table 3 g i v e s t h e e l e c t r o n configu- r a t i o n s of i s o - e l e c t r o n i c sequences. H o r i z o n t a l arrows i n d i c a t e t h e d i r e c t i o n of rearrangement. ( s e e a l s o cormnent t o Table 3 ) . This l e a d s t o new term c o n f i g u r a t i o n s . T h e i r knowledge i s indispen- s a b l e f o r q u a n t i t a t i v e s p e c t r o s c o p i c a n a l y s i s of high-Z i m p u r i t i e s i n Tokamak plasmas.

OXYGEN IN

ne

no

_EQUILIBRIUMCORONA

F i g . 13

-

R a d i a t i o n l o s s r a t e f o r oxygen. P/neno i s t h e power d e n s i t y r a d i a t e d p e r e l e c - t r o n and p e r oxygen p a r t i c l e , a f t e r [ I 0

I

.

The i n d i v i d u a l c o n t r i b u t i o n s have been d e f i n e d i n Eq

.

( 16)

.

(16)

8. CORONA EQUILIBRIUM, OR NOT ?

7.98 Under actual conditions gradients of d%n-

sity and temperature, but also instabilities lead

to diffusion fluxes across the plasma. The conse-

quence is that ionisation and recombination relaxa-

tion occurs, i.e. the ionisation stage of the

particles will depend on the temperature

addi-

tional parameters. When diffusion is included in the

considerations it is not further possible to give

generalized radiation loss curves as shown in

Figs. 13 and 14. Every plasma has to be considered

individually.

A

f

-9

Comment

:

Horizontal arrows

(+)

indicate restructuration of electronic orbitals as

Z

increases. For ins-

tance

:

the valence orbital of K I is 4s, and the electron configuration is KL 3 ~ ~ 3 ~ ~ 4 s ,

the 3d

subshell is not occupied. For Mo XXIV the outer (valence) electron occupies the 3d subshell and

the electron configuration is ~ ~ 3 s ~ 3 ~ ~ 3 d .

Since all n=3 electrons have "approximately" the same

binding energy, the 3s and 3p electrons can easily jump into the 3d subshell as long as places

are vacant. This leads to new "quasi-ground state" configurations with levels far below the

excited valence electron state.

W

I

Fig. 15 shows as example model calcula-

tions for the DIVA Tokamak

[ I l l .

The diffusion

processes are so strong "that the computed result

yields a radiation power in a factor of 20 larger

in maximum than predicted for the coronal equili-

brium state" (page 9 of [I 1

I

) .

In the calculations

has been assumed that complete recycling occurs.

(17)

OXYGEN

-

DIVA TOKAMAK

-0-0-

CORONA EQUIL.

[w/cm3]

RECYCLING

IONISATION

F i g . 15

-

Model c a l c u l a t i o n f o r r a d i a t i o n power l o s s i n t h e DIVA TOKAMAK machine, a f t e r [ I l l ; t a k i n g i n t o account t h e r a d i a l dependence of ne and Te

..-.-.-.=

assump- t i o n of l o c a l corona e q u i l i b r i u m ;- =

assumption of dynamic e q u i l i b r i u m w i t h complet r e c y c l i n g ;

---

i s t h e power d e n s i t y l o s t i n t h e i o n i s a t i o n p r o c e s s e s .

Although t h i s r e s u l t cannot be g e n e r a l i z e d

i t c l e a r l y shows t h a t d i f f u s i o n can have a (consi- d e r a b l e ) i n f l u e n c e on t h e r a d i a t i o n l o s s e s and t h a t t h e c u r v e s i n F i g s . 9 , 13 and 14 l i e on t h e o p t i - m i s t i c s i d e . A s can b e s e e n from F i g . 8, t h e d i f f u - s i o n p r o c e s s e s themselves depend p a r t l y on atomic r e a c t i o n p r o c e s s e s .

While t h e assumption about t h e i o n i s a t i o n -recombination e q u i l i b r i u m h a s a n i n f l u e n c e on t h e r a d i a l d i s t r i b u t i o n of t h e r a d i a t i o n l o s s e s i n t h e model c a l c u l a t i o n s , t h e assumptions about t h e e x c i t a t i o n / d e - e x c i t a t i o n mechanism i n f l u e n c e t h e d e t e r m i n a t i o n of t h e i m p u r i t y c o n c e n t r a t i o n s . The corona model - w i t h t h e a d d i t i o n a l assumption t h a t c a s c a d i n g i s n e g l i g i b l e - l e a d s t o Eq. ( 4 1 ) . When f o r b i d d e n l i n e s a r e used f o r determining

$

, I ,

a d d i t i o n a l c o l l i s i o n p r o c e s s e s may become important

[39 ] i n t h e d e t e r m i n a t i o n of

.

a l i m i t e d number of molecules a r e of i n t e r e s t f o r which s t r u c t u r e , l i f e t i m e and c o l l i s i o n d a t a should be a v a i l a b l e . The n e c e s s a t y t o know t h e s e d a t a a r i s e s from f o u r d i f f e r e n t p h y s i c a l problems.

First,

t h e very f i r s t beginning of a d i s - charge i n hydrogen gas ( o r i t s i s o t o p e s ) depends e n t i r e l y on molecular p r o p e r t i e s . Before r e a c h i n g t h e s t a t e of complete i o n i s a t i o n of t h e atoms t h e gas goes through a sequence of e x c i t a t i o n s (de- e x c i t a t i o n s ) and i o n i s a t i o n s (recombinations) of molecular and atomic s t a t e s . Although t h i s p e r i o d

is s h o r t compared t o t h e time d u r i n g which t h e gas

i s i n a completely i o n i s e d s t a t e , t h e molecular p r o p e r t i e s may i n f l u e n c e t h e spatio-temporal evo- l u t i o n of t h e plasmas i n the' n e x t g e n e r a t i o n of Tokomaks. The p r e - i o n i s a t i o n phase w i l l only be w e l l understood when t h e molecular d a t a - a = i n c o r - p o r a t e d i n t h e plasma model.

Second, r e l a t i v e l a r g e q u a n t i t i e s of w a t e r

-

a r e adsorbed on t h e w a l l s even under UHY-conditions and carbon i s contained i n s t a i n l e s s s t e e l o r might even be used a s f i r s t w a l l i n a f u t u r e r e a c t o r . Under t h e i n f l u e n c e of w a l l bombardment, H20 i s desorbed d i s s o c i a t e d and oxygen l i b e r a t e d which d i f f u s e s a s unwanted i m p u r i t y i n t o t h e plasma. Also carbon i s l i b e r a t e d and r e p r e s e n t s an i m p u r i t y . Experiments have shown t h a t t h e w a l l s can be c l e a n e d by applying low-energy d i s c h a r g e s . The chemical p r o c e s s e s a r e n o t y e t w e l l understood. When carbon i s p r e s e n t , hydrocarbons l i k e CHI, and

i t s d e r i v a t i v e s a r e formed. To understand t h e p h y s i c s of t h e c l e a n i n g mechanism d a t a f o r mole- c u l e s such a s HO, H20, H30, CH, CH2, CH3, CHI,, CO... and t h e corresponding i o n s a r e needed.

-

Consider only oxygen, t h e dynamics of oxygen atoms and i o n s i n t h e c o l d plasma-wall l a y e r depends on t h e f o l l o w i n g r e a c t i o n s 140

1

[41

1:

9. MOLECULAR DATA

(18)

reactions have still to be added :

Still more numerous are those reactions involving a carbon atom.

Third, the physical processes in ion

-

sources presently developed for neutral particle injection depend strongly on the molecular processes in the source. The processes involving volume production and loss of H- ions are not yet clear [ 4 2

I

.

Fourth, the reaction processes occuring in the charge exchange chamber in which the ener- getic ions are transformed in neutrals must be known. Not only the total charge exchange cross sections are of importance but also the reaction cross sections leading to excitation of indfvidual excited levels of the neutral atom are of interest, since the beam atoms can be used for diagnostic purposes.

10. CONCLUSION

The following conclusions can be drawn from the above considerations : the energy loss and, thus, the

ig+i_ti_on-+

_w_o_rk??g-co_%dj_tio,??

of a fusion reactor depend on the impurity level. Already rather low impurity concentrations of high- Z elements have a deleterious effect on the power balance and, thus, on the nePr

-

and T

-

values. Much higher impurity concentrations (of the order of several per cents) can be admitted for the low-Z elements like carbon, oxygen and aluminium. From the physics point of view the use of low-Z elements has some advantages over high-Z elements. However, tech- nological necessity may dictate-materials made of medium-Z or high-Z elements. A definite statement about this point cannot be made yet.

For _mo_d_e_1-~aJg~l_a_ti03~ of high-temperature plasmas many atomic and ionic data are needed :

level structures, ionisation, charge exchange, recombination and excitation cross sections. Since the resonance transitions of impurities in Tokamaks are almost optically thin, Einstein coefficients

(or the equivalent oscillator strengths) are not

directly applied in model calculations as long as one applies the corona model. However, the excitation cross sections are proportional to the oscillator strengths. The knowledge of the latter thus permits to obtain absolute excitation cross sections indispensable for calculating radiation losses.

For spectroscopic diagnostics a comprehen-

---

--

_{- - -}

sive knowledge of level structures (and thus of wavelengths), excitation cross sections and Einstein coefficients is desirable, although a limited number of precise values

-

namely for the most prominent lines in different wavelength regions

-

is generally sufficient. Since "forbidden lines" become progressively "allowed lines" when one goes within an iso-electronic sequence to higher Z, data for these transitions are of importance too.

The data should be available for all ionisation stages of an element.