Gruneisen's gamma function for liquid iron at the Earth's outer core conditions

SUMMARY. - WE derive the Gruneisen's gamma as a function of temperature and pressure for liquid iron at the temperature and pressures of the Earth's outer core taking into account the anharmonic and electronic contributions in a self-consistent way. We obtain numerical values definetely different from those generally quoted in the geophysical literature, but in agreement with recent experimental determinations.

It is generally accepted that the Earth's outer core is mainly composed by liquid iron. In this work we assume it composed by pure liquid iron and we shall evaluate the Gruneisen's gamma values in the outer core range of pressures and temperatures. The Gruneisen's gamma is derived taking into account all the known possible physical effects according to recent achievments in solid state theory. Furthermore, instead of extrapolating with dubious rigour values relative to solid crystals, as it is generally made in the geophysical literature, we apply our theory to liquid state. The problem of the determination of the Gruneisen's gamma for liquids has had scarce attention ( 4 ' ").
By liquid state, at t'he Earth's core conditions, we mean a lattice with a simple cubic structure in which transverse waves have a transmission coefficient irrelevant in comparison with the longitudinal transmission coefficient. This liquid model arises from experimental diffraction data on liquid metals ('• ,5 ) which show a nearest-neighbour pattern much more randomly located than for solids. This gives the reasons to support for liquids a lower coordination number than for solids ( 17 ) and a very low transverse « constant » in order to allow nearly free shear movements in the lattice. According to DOMB et al. ( 5 ), by force constants we mean the spring constant dF/dr of the intermolecular forces F. We believe that at the present stage of knowledges this is the best way to tackle the problem since neither the cell model ( 12 ), nor the distribution function model can be profitably used in the lattice calculations (").
An explicit expression for the Gruneisen's gamma can be found following the general procedure of considering a solid as a system of 3 N strongly anharmonic oscillators. The hamiltonian function of the system may then be splitted into its harmonic (in the Leibfried-Ludwig sense (") and anharmonic parts

H = H H + H A
which can be solved separately. Considering the harmonic term, we find the normal modes of vibration of the system as eigenvalues of Flu and then, from the energy spectrum, compute the Gruneisen's gamma according to the definition Y = Zi 0 In v,/3 In V) It must be noted that the high temperatures of the core allow us to use a Thirring's series expansion ( l8 ) to express the harmonic Gruneisen's without passing through the explicit frequency spectrum, but just considering its statistical moments. Then the harmonic Gruneisen's gamma turns out to be Cv -(dEvni/dT)\ U is the static lattice potential and E (T) the internal energy of the system. Evm can be written as where B 2 " are Bernoulli numbers and fi 2n are the static even moments of the frequency distribution. The very high temperatures we are treating allow us to consider only the first few moments since the Thirring's series is strongly convergent up from the Debye temperature.
These moments have been calculated for any lattice structure to the 20-th order ( 2 ' 5 ' 16 -13 ) (BHATIA) and HORTON, 1955, DOMB et al., 1959MONTROLL, 1943;LEIGHTON, 1948). We shall use the septenary partition model ( 4 ), which gives the momenta of a simple cubic 3 d lattice. The first two moments can be written as where K,, and K, are the longitudinal and transverse constants, respectively. We employ an intermolecular potential of Rydberg type, namely where r e is the static distance of approach of the molecules, r¡¡ is the istantaneous distance. D and b are constants, and namely, D = 74.69x 10 u o ergs, i> = 2.0554x 10 M '. This potential shows a very good agreement with experimental data for iron and nickel compression ( 2I ) (VARSHNI and BLOORE, 1965). The force constants are then given by where C is arbitrarly taken 10 2 and 10 3 .
This choice will appear irrelevant for our aims as it will be evident from the following.  where &k.k\... is the total lattice potential intended as a pair potential and G, K', ... accounting for the summation pairs ("). cj k are the eigenvalues of the dynamical matrices.
Since we are treating a metal we must also account for the electronic contributions to the free energy of the system. Let us write them as where r is the electronic specific heat divided by T. We incorporate all the electronic effects in the anharmonic hamiltonian function. The electronic contributions for liquid iron can be evaluated from the density of the states relatively to the non-ferromagnetic phase because of the high temperatures we are considering. Up to now, no experimental data are available for these quantities, but there is physical evidence that liquid iron density of the states values should slightly differ from the solid iron ones (')• Furthermore, a physically reliable theoretical approach, based on the cluster model ( 10 ), shows that electronic bands for liquid iron differ for no more than 5% from those relative to the solid state. We shall use values computed for solid non-ferromagnetic iron which have been tested with good accordance with experimental data ( 23 ). The effects of pressure on this term were computed by HENRY ( 9 ) by using a Fermi-Thomas atom model, with the result that electronic specific heat decreases by a factor smaller than 3 for pressures of 10 2 Megabars. Therefore we shall suppose that the electronic contribution is not affected by the pressures we consider. In any case the error is much less than the uncertainty in the anharmonic term. Then the Gruneisen' s gamma function is evaluated by the equation which leads to where the averages are performed over the normal modes, A 2 is the anharmonic leading term of the free energy ("), and n is the number of particles in a cell. Equation [13], in the high temperature conditions of the Earth's core, simplifies to Concluding, by means of the above equations, we computed the Geuneisen's gamma function for liquid iron versus pressure along an isotherm at 5800° K, temperature that most recent theories ( 7 ), quote as the inner core outer core boundary temperature in the assumption of a pure iron core. The corrseponding results are plotted in Fig. 1. We computed also the isobars at 1.4 and 3.1 Megabars, i. e. at the pressures of the mantle-core and the inner core-outer core boundaties. The uncertainty on the anharmonic gamma leads to an average uncertainty on the values of the gamma function of 0.09. The results are plotted in Fig. 2, where the zone corresponding to the outer core is shaded. Gruneisen's gamma ranges from 1.0, at the mantle-core boundary, to 0.5, at the inner core-outer core boundary.