COALESCENCE OF IRON IN THE CENTRE OF THE SUN

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COALESCENCE OF IRON IN THE CENTRE OF THE SUN

C. Deutsch, M. Gombert, H. Minoo

To cite this version:

C. Deutsch, M. Gombert, H. Minoo. COALESCENCE OF IRON IN THE CENTRE OF THE SUN.

Journal de Physique Colloques, 1980, 41 (C2), pp.C2-65-C2-68. �10.1051/jphyscol:1980210�. �jpa-

00219802�

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COALESCENCE OF IRON IN THE CENTRE OF THE SUN

C. D e u t s c h , M.M. Gombert and H. Minoo

Labovatoive de Physique des Plasmas, Vniv&vsite Paris XI, 91 405 Ovsay

Abstract.- A matrical nodal expansion making use of temperature-dependent Kramers potentials for the Coulomb interactions, is used to compute G = F + pV in the centre of the sun. Small but, cru- cial quantum corrections in n =% ee and 4iTp^ee are taken into account quantitatively in the weakly coupled hydrogenic phase. The ^D system e-p-Fe is shown to be stable for small iron concentrations = ICf-lCf.

Introduction. - It has been recently [lj speculated that stripped nuclei in a hydrogen plasma under solar conditions, according to the classical Debye-Huokel

(DH) model, would undergo phase separations for concentrations well below the cosmic abundance. The phy- sical cause of that phase separation is simply "that the potential energy is lower in the separated phases than in the mixtures because the local charge neutra- lization is much better satisfied in the two separated phases.

The higher concentration corrections, needed to cha- racterize the iron-rich phase lead to enhanced solu- bility for a simplified model where the electrons form a uniform background.

This demixtion suggests that highly ionised iron would precipitate out in the solar interior and coa- lesce in the centre.

Should this occur and lead to a reduction of the iron concentration near the thermonuclear burn region by an order of magnitude below the cosmic abundance value (2.5 x 10 5 ionic mole fraction), the existing neutrino dilemma might be resolved. It is estimated

[_lj that such a concentration reduction would lower the opacity, and hence the temperature, Sufficiently to significantly reduce the probability of fusion of a particles that generates the nuclei whose neutrino emission is being monitored.

This paper presents a self-consistent Debye formalism

for the iron-hydrogen phase diagram which takes into account quantitatively the small but nonnegli- gible quantum (diffraction and symmetry) effects.

Moreover, our formalism is able to retain the re- mnant amount (if any) of bound states.

Pollock and Alder [ 1 ] got the iron-hydrogen phase separation through a computation of the thermodynamics of the iron-rich phase. They employed a correc- ted version of the DH theory. Here, we prefer to work out a consistent multicomponent DH scheme in the iron-poor and weakly coupled phase. Therefore the initial plasma we first consider is mostly classical, but includes some quantum corrections in agreement with L1J

2. Multicomponent nodal expansion with pseudopo- tentials. - The three-component plasma of interest is characterized by the dimensionless classical parameter

where X denotes the Debye length X = (47rpj3e2

Ni. A' is the parameter in a pure hydrogen phase with XD = (4irge2p(C +C ) )- 1'2 . z is the average JOURNAL DE PHYSIQUE

Colloque Cl, supplément au n° 3, Tome 41, mars 1980, page C2-65

Résumé. - Le développement nodal d'un plasma dense contenant électrons, protons et ions ferreux est explicité pour les conditions thermodynamiques régnant au centre du soleil, i.e. [\ = h /fM)

Une adaptation matricielle du développement scalaire habituel est utilisée. L'énergie libre de Helmoltz G = F + pV, à l'ordre de Debye, est obtenue à l'aide d'interactions coulombiennes modi- fiées de la forme Z.Z. ejv _ ( •) , dépendant de la température et tenant compte des effets quantiques (diffraction et symétrie) aux courtes distances. Un développement en ~ — montre

AD que l'énergie libre de mélange est stable au centre du soleil, pour des concentrations ~ 10 5 et un ion fer complètement strippé <Z =•• 26) .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980210

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C2-66 JOURNAL DE PHYSIQUE

charge of the iron ions, taken as Points. In the iron- varied as a free parameter.

poor ~hase,

-

we consider, the iron concentration = The inclusion of the mostly classical electron back-

k <

0.001 small enough to allow the appro- ground is one of the attractive feature of the pre- C -K:

-

P Fe sent approach.

ximation

A

A'.

Recall that the interior of the sun is characterized 3 , Debye Thermodynamics.- To compute the G a b s free

by

(A

= -A

/fi

with

Aee

= ) -1

ee ₂- energy G = F

+

PV, we need (f3 = (k T) B ) the virial expressions

A' 0.055, 1'1' = 0.21 and 4 r h 3 = 0.18. ( 3 ) Our weakly coupled approach make use of two remarka- ble tools [ 2 , 3 , 4 J : temperature-dependent pseudopo- tentials instead of bare Coulomb interactions and a matricial formulation of the Debye resummation of the Coulomb tails. In k-space, the latter is expressed and under the form .

with

a

c k c E 1

d3;gk, ( r ~ ~ i i k u , , ( r ) , (10) where gkQ(r) denotes the 2-point canonical pair dis- tribution function for the particles k and 2. In t k I is the identity matrix of order n = number of

present high temperature limit, one can introduce components (= 3 in the present case), while U(k) deno-

the linearization gkR(r)

"

¹

-

^f3VkR(r)^{in Eqs.}⁽⁹⁾

tes the symnetric matrix of modified Coulomb interac-

and (10). A much more detailed account of this for- .tions,. with the corresponding simplified entries.

7 2 malism will be given later on.

e2 BLog 2

sr2

Restricting to first order in

A'

and second order r =;-(1

-

eriA)

+

^{k B ~}^{~ o g}²^e ^t (5)

in q', one gets after lengthy manipulations, analy- tic approximations for P and E. The E.O.S. thus

( 6 ) reads [6]

u

(r) takes into account both the s = 0 contribu- ee

tions of the electron-electron interaction, as well

as the moderate corrections due to the Pauli principle. 2

+

1 ( 1 + x ) ( ~ + x ) ~

-

( 2 + x ( 2 + z ) ) [ I + x ( z + I ) ) } Moreover, we introduce the drastic assumption of a ( 2 + x )

hydrogenic and point-like iron ion. This point could be discussed further in view of the average temperature (kgTe

"

1.5 kev) in the centr-the sun. Stric-

tly speaking, the latter does not permit to remove the 4 2 + x 2+x(Z + 2 ) last two electrons bound within a helium like structu-

re [5]

.

Nevertheless, we do not expect this sirnplifi-

+ -

₃ _.

cation to interfere with the phase separation process, because it affects mostly the r = 0 limit of the Cou-

r 2+x(Z + 2 )

2l

lomb interactions. Moreover, the iron charge Z can be

.

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where a =

l+x

⁺l+x (Z

+

1 )

.

The operation 2+x 2+x

applied to E, yields the Helmoltz free energy

3 2 l + x ( z + l ) ) @ + x ( z + 2 ) 1 ⁺

-

^{A ' p h}^IT ^{( L o g} ^{2 )}

2 0 ' ( z + x ) ~

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JOURNAL DE PHYSIQUE

where b'

-

l+x(Z + 1) 2 + x

From Eqs. (11) and (12) we compute the excess mixing free energy

AG

= G(x)

-

^xG(1)

-

⁽¹

-

^x)G(O)+ x Log (x)

+

(1

-

x)Log(l-x).

The results given in Tables 1 for Z = 26 show that for small values of 11', there is no phase separation.

We thank D.J. Stevenson for useful advices.

/I/ Pollock, E.L., and Alder, B.J., Nature

275

(1978) 41.

2

-

^2.063.

^lo-'

/2/ Deutsch, C., Ann,Phys. (N.Y.)

115

(1978) 404.

/3/ Gomber

,

^M.M.G., Deutsch, C., and Minoo, H., J.

Physique (Suppl.) (1979) 695

Table 1. Similar results hold for X

<

^lo-'- M i x i ~ free energy

I-,

^;or

Z = 26 and px3 =

0.18,

A ' = 0.055, )'l = 0.21.

47T

4

-

^4.165.

/4/ Minoo, H., Gombert, M.M.G.,.and ~eutsch, C., J.

Physique (Suppl.) (1979) 697.

/5/ Mori, K.; Otsuka, M., and Kato, T., IPPJ-AM-3, Nagoya Report, August 1 977.

6

-

6.202.10-~

/6/ In E q s , (11) and (12) x should read x(Z

-

1).

8

-

8.1513.10-~