The Electric Field and the Diffusion Velocities

The resistance coefficients K12 and K21, which express the pressure gradients resulting from the velocity differences, are equal by symmetry, since there is momentum conservation in collisions without chemical transformation. We can thus add (10.49) and (10.50) and obtain an expression for the electric field, eE =

a product expressed as a force by volume unity (g s-2 cm-2). We see that the electric field depends on the departures from the hydrostatic balance between the gravity acting on the partial densities and their partial pressure gradients. We call Yi the ratio (cf. Sect. 25.1.2)

We divide (10.50) by the partial pressure P2,

A2 mu

where v21 = v2 - v\. Further simplifications lead to

-K2i mu

Q Y2kT

This expression allows us to write the velocity difference v21, however, we can further transform the above expression by calling Pion the sum of the two partial pressures Pion = P1 + P2,

The velocity difference v21 becomes

Y2 q kT

fflu K21

We write the term in front of the square bracket as

mu K21 V y1

in a way consistent with expression (10.23). This gives the following expression for the diffusion coefficient D12, m H2kT 1 Y2 q kT Yj

If we want to obtain the diffusion velocity of particles "2", we can use relation (10.53) for momentum conservation and get

This yields for the velocity of particles "2" 1 + Y2/Y1) ^

«2 = T+X27XJD12 x d(ln Yt )+ mug A1Z2 - A2Z1 + Z2 -Z1 dlnPi0

The velocity of particles "2" is thus determined as a function of three different gradients [190]:

- The first term in the bracket expresses the diffusion due to the gradient of abundance of the considered element. If element "2" is a test element, this term is equivalent to (10.23).

- The second term is the so-called gravitational term. It is proportional to the global pressure gradient and it also accounts for the (generally opposed) effect of the microscopic electric field E (10.54). For nuclei with A1/A2 = Z1/Z2, this term vanishes. This is the case for the light elements where Ai = 2Zi, such as 4He, 12C, 16O, etc.

- The third term is proportional to the pressure gradient of the ions. It is generally not important.

Additional terms may be added to account for the effects of thermal diffusion (Sect. 10.3.5), radiative acceleration (Sect. 10.4.1) and possible magnetic fields (Sect. 10.4.3).

10.3.4 Diffusion Equation

The equation of diffusion of particles "2" can be written by combining the diffusion equation (10.29) with expression (10.23), dX2 dt

where v2 is given by (10.65). This equation is subject to the boundary conditions discussed in Sect. (10.2.2). Care has to be given to the interpolation of the diffusion coefficients (10.32) if there is also a diffusive transport in the adjacent zone [414].

At this stage, we do not yet have the expression for the diffusion coefficient D. The time evolution of the fluid is described by Boltzmann equation which gives the time variation of the distribution function fi(r, v, t) so that fi(r, vi; t)dr3dv3 is the number of particles "i" in the volume element dr3 centered on r and with a velocity in the volume dv3 centered on v. The Boltzmann equation is solved under certain hypotheses [6, 112, 193], such as binary interactions, elastic collisions, negligible electron mass, Debye-Huickel potential around the ions (7.104). The solution of Boltzmann equation provides the diffusion coefficient with account of particle collisions.

The diffusion coefficient depends on the kind of particles interactions: charged-charged, charged-neutral or neutral-neutral. The most important case for the stars is the first one, i.e., the interaction between ions. In this case, the diffusion coefficient for ions of type "1" and "2" is [6, 600], see also (B.53),

where mr = Ai A2 mu/(Ai + A2) is the reduced mass and n = n1 + n2, the total concentration of particles. |12 is given by

Z1Z2 e2

where rD is the Debye-Huckel radius given by (7.99).

10.3.4.1 Diffusion of Ions With Protons

In the useful case of the diffusion of ions "i" with protons "p", the diffusion coefficient becomes [112],

in cm2 s-1, where np is the proton concentration. The approximation is made that only collisions with an impact parameter shorter than the Debye-Huckel radius are considered (the potential is supposed Coulombian). For impact parameters larger than Rd, the interaction potential is assumed negligible. Further developments leading to diffusion coefficients with screened potentials for different ions have been performed by Paquette et al. [461].

For estimates of the orders of magnitude, the diffusion velocity of an element "i" with respect to hydrogen may be estimated by considering only the effect of gravity according to relation (10.47).

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