Average atom model for opacity calculation

We use the Grand Canonical Ensemble to derive new formulae to determine the average population of electrons and the distribution probabilities of electrons in atomic shells in the

AA model.

3.5.1 Average atom model

The periodic table of the chemical elements indicates the shell structure of the atoms of

heavy elements. For partially ionized atoms, the shell structure can remain a good

approximation of the atoms since the ionization energy of an electron in a full shell is much greater than that in other cases. Consequently, it may be assumed that the ionizations of electrons in an atom occur shell by shell. On the basis of such a consideration, we make approximations that the distribution of electrons in each shell is independent of that in other shells and that the interactions between electrons in different shells are approximated by the average population of electrons in each shell. In the same shell, however, the interaction between electrons cannot be approximated by the average population because the electron number in the shell is strongly affected by their interaction if the shell is just being ionized.

\

Therefore, the population of electrons in one shell should be solved simultaneously with their interactions, which determine the energies of those electrons in the shell.

3.5.2 Description by grand canonical ensemble

Chapter III The Radiative Opacity

According to the AA model, all atoms of each element are described by one average atom in which there exist a series of atomic shells (nl) and the bound electrons can be populated

and specified by hydrogenic wave functions ( of one electron atom) with eigenvalue Eni.

Consider a system with one unit volume V=1 in which there are i=l,...,M types of elements and the number of atoms of each element is , and suppose the system is under the LTE conditions specified by the temperature T=l/p and the electron degeneracy parameter X .

In order to obtain the distribution probability of electrons in the average atom and the free phase space, we start with the electron partition function (referred to EPF) in the system. The EPF per unit phase space of the free electron is Zi=l+exp(?t-pe) with the energy of the electron e. In the average atom, the distribution of electrons in one shell is supposed to be independent of those in other shells and the possible numbers of bound electrons in the shell can be m = 0,l,2,...,gni (which is the statistical weight) Therefore the EPF in one atomic shell (nl) is written as

gnl

Zni = Z exp[m(X-PEni(m))] ,

m=0

where E^i(m) is the average energy of one electron with m electrons bound in the shell.

According to the independent approximation of the electron distribution in different shells, the EPF in the whole average atom is equal to the product of the EPF in all shells. Therefore, the total grand partition function of the electrons in the system can be written as

Ni

Zo@A)

= n

tU

where the first product goes over the whole phase space of the free electrons, the second product over all elements, and the third one over the atomic shells of one average atom. The power of Nj accounts for tlie EPF in all atoms of the element i.

Chapter III Tlie Radiative Opacity

3.5.3 Average population and distribution probability of electrons

Statistical physics gives for the total number of electrons in the system

M a z NiZi = — logeZG(p,:\,) i (7 A- = ^ _{<free>exp(-^+P8)+l} \ _{i nl} ^ lo g e Z n l

_dX

It is actually the conservation equation of electron number. The first term defines the occupation number of free electrons per unit phase space as x^, which is given by the

Fermi-Dirac formula. This term is actually the total number of free electrons which is obtained by summing over the whole phase space of free electrons (see Chapter II).

The second term defines the average population of electrons bound in each atomic shell (nl). They are obtained by using Z^i i.e.,

Xnl = :^logeZ nl o X gnl Z exp[m(^-PEni(m))]*m _ m=l___________________ gnl Xexp[m(X,-PEni(m))l m=0 gnl

= £ Pnl(m)*m

where Eni(m) is the average energy of the electron in the shell when there are m electrons populated in the shell.

We find that the probability of finding m electrons in the shell (nl) is defined by

Chapter HI The Radiative Opacity

^ X . exp[m(A-PEni(m))]

^

Xexp[m(X-PEni(m))]

m=0

This formula defines the distribution probability of electrons in the atomic shells after the degeneracy parameter X and the average energies of the electrons in each shell E^(m) are solved from the self-consistent equations.

We notice the difference in physics between the Fermi-Dirac formula and the above formula from the statistical description of the AA mode. If it is supposed that there are gjji (statistical weight) sub-shells in the atomic shell (nl) and the distribution of the electrons in each sub-shell is independent of other sub-shells, then the EPF in the atomic shell (nl) is equal to the product of the EPF in all sub-shells of the shell, that is

Z„l=[l+exp(X-PErt)]®"*.

Consequently, the Fermi-Dirac formula which determines the average population of electrons is recovered, but we cannot obtain the occupation probabilities of atomic shells by electrons. However, the independent approximation used above is too crude since the electrons in the same shell have the same energy, and therefore their distributions are strongly dependent on each other. In the new AA model, the EPF takes into account the dependence of the electron distributions in the same shell and thus gives an approximation of higher level than the Fermi-Dirac formula. Furthermore, the consideration of the dependence allows for the occupation probabilities of atomic shells by electrons, which is then used to determine directly the probability of the atomic configuration. If the dependences of the electron distributions in all shells of the atom are taken into account, the grand canonical ensemble can give the probabilities of atomic configurations. This has been presented in Section 2,4.1.5..

3.5.4 Determination of electron distribution probability

Chapter III The Radiative Opacity

In the general case, the electron distribution probability Pni(ni) can be found after the

electron number conservation equation is solved for the electron degeneracy X , Now

Pni(m ) depends on X and the electron energies Eni(m ) in the atomic shells while Eni(m )

depends on the electron distributions. They thus lead to a set of self-consistent equations

which can be solved for pni(m ).

In our opacity calculation, the electron degeneracy X is obtained from the determination of

the EOS. If the occupation probability Wni = exp(-(j>ni) and the energy level perturbation

AEni of the active electron in atomic shells are taken into account for interatomic interactions, the expression for the electron distribution probability is then written as

^ XX exp[m(X.-PEni,A-PAEni-<|)nl)]

^ gnl