5 rocks and ices: beyond the nearly uniform electron...

5. Rocks and Ices: Beyond the Nearly Uniform Electron Gas 47

5 Rocks and Ices: Beyond the Nearly Uniform Electron Gas Historically, approximate treatments of very dense matter were developed by treating the medium as if it were a non-uniform Fermi gas and applying the rules of number of states, etc. as if the gas were locally uniform: i.e.

Figure 5.1

In this figure, you see the common level of the Fermi gas (the straight line representing the uppermost occupied states) but at each physical location, the total energy range of the Fermi gas (the shaded region) varies in accordance with the self-consistently determined potential that an electron “sees”. When the potential is strongly attractive, the shaded region is wider and the electron gas density is locally larger (the electrons are clumped up in a region near a nucleus for example). It is not at all evident that this approach should work since it takes a true counting of states (involving a large number of electrons) and replaces it by a fictitious state in which the quantum statistics (which assumes a large number of electrons) can still be applied even when the number of electrons in a small spatial region is not large.

In this prescription, the classical Poisson equation:

∇2φ = 4πene(r ) (5.1)

is used to get the self-consistent potential. We then we get the Thomas Fermi (TF) model (if exchange is ignored) and the Thomas-Fermi-Dirac (TFD) model (if exchange is included). There is a closely related but somewhat better model called the Quantum Statistical Model (QSM) developed in Russia (see Zharkov and Trubitsyn, Physics of Planetary Interiors), which treats exchange better than it is treated in TFD.

There are modern, highly accurate formalisms that go beyond these approximations by actually solving Schroedinger’s equation, but use electron gas ideas to construct the


energy associated with the electron density point by point. These methods include correction terms for the gradient in the electron density. This is called the density functional formalism. It is still faster than a brute force attack on the problem because it does not try to evaluate exchange and correlation (the hardest parts) rigorously.

If you have a well –defined lattice and you are ignoring thermal effects (vibrations) then density functional calculations can be carried out readily on a work station and to high precision (a fraction of an electron volt). One can then try several different lattice structures and figure out which one has the lowest Gibbs energy (and is therefore the thermodynamic ground state). But in many cases (e.g. , exoplanets or giant planets) there is enough uncertainty in composing that not much is gained by having a more precise equation of state. Examples of various approaches to water and to “rock” are shown below. These are intended for use in giant planets only! The reason is that they are crude, yet perfectly adequate in that context.


Figure 5.2


In Jupiter and Saturn, ice and rock are only a small fraction of the total mass so the need to have accurate descriptions of the ice and rock components is less great than for hydrogen or helium. In Uranus and Neptune, there are large uncertainties in temperature and mass distribution and there is (so far) not much benefit to be gained from more accurate equations of state. One hopes that this will eventually not be true. In the terrestrial planets or in large icy satellites, a more accurate approach is certainly desirable and possible. However, no simple theory is available. One resorts instead to simple parameterizations that are tied to experiment or to first principles theory. The need for first principles theory is somewhat diminished in these bodies since one can often get accurate experimental results. As noted already, these are often then fitted to a “theory” (really nothing more than a Taylor series expansion about zero pressure), which is asymptotically incorrect at high pressure, but perfectly adequate for the purpose intended. For example, one can fit the data to a third order Birch-Murnahan equation of state, which has the form:

P = 32K0[

ρρ0

⎛⎝⎜

⎞⎠⎟

73− ρ

ρ0

⎛⎝⎜

⎞⎠⎟

53]{1+ 3

4(K0

' − 4)[ ρρ0

⎛⎝⎜

⎞⎠⎟

23−1]} (5.2)

where ρ0 is the zero pressure density, K0 is the zero pressure bulk modulus, and K0′ is dK/dP, the zero pressure derivative of bulk modulus with respect to pressure. (This is a slightly more complicated form than the equations already introduced in Chapter 3). Fortuitously (?), K0′ is near 4 in many materials so that the higher order term involving K0′-4 is not very large. See the table on the next page, taken from Don Anderson’s book Theory of the Earth (p107) for relevant parameters. But this equation should not be used for extrapolation, only for fitting data. The other problem is that this parameterization does not work well for a liquid (in part because K0′ -4 is often not small for a liquid.) Note that equations like this must only be used for a particular phase—if there is a sudden jump in density (through reorganization of the atoms), then you must go to a new equation that describes that new phase. (Phases that are only metastable at zero pressure, e.g., magnesium perovskite, can still be described by this formulation, though the physical interpretation of zero-pressure parameters is certainly less clear.) Since solid minerals are often quite incompressible over some pressure range, and then undergo major density changes through phase transitions, realistic models of solid planets (terrestrial planets and large icy satellites) have to take phase transitions into account. The most advanced theoretical work is now successfully reproducing the experimental values for Ko for materials such as magnesium pervoskite (where the bulk modulus at zero pressure is around 260 GPa) and K’, the pressure derivative of the bulk modulus (around 4).


Table 5.1

Theory also has immense value through its ability to determine other parameters (e.g., coefficient of thermal expansion). So it is of value to compare first principles theory with experiment. In simple chemical systems, such as MgO, theory can do an excellent job reproducing the experimental data. In more complex (yet important) systems, such as MgSiO3 or H2, theory can now do well, though greater care is needed. An example of comparison of theory and experiment is shown below for MgO and then for MgSiO3


Figure 5.3

Figure 5.4 Earth Planet Sci Lett, 184, 555-560 (2001)


Below is a set of EOS’s used by Seager et al, Ap. J. 669 1279-1297 (2007) for their paper “ Mass-radius relationships for solid exoplanets.” Notice how they tend to merge at very high P, as they must.

Figure 5.5

Ch. 5 Problems 5.1) In this figure 5.5, the curves approach each other at very high P. Why? 5.2) Assume that the material inside a planet is described by a single smooth

equation of state that takes the form P = Kof(ρ/ρo) where P is pressure, Ko is the bulk modulus (at zero pressure), ρ is the density and ρo is the zero-pressure


density. [This means that f(1)=0 and f ′(1)=1.] Show that the equation of hydrostatic equilibrium can be rewritten in the convenient non-dimensional form

d/dx[x2g(y)dy/dx)] =-x2y

where g(y)= f ′(y)/y, y= ρ/ρo, x =r/R0 and Ro2=Ko/4πGρo

2.

Solution: dp/dr= - ρg, but g=GM(r)/r2. Therefore d/dr[(r2/Gρ)(dp/dr)] = -dM(r)/dr =-4πρ(r)r2. But dp/dr = Ko f ′(y).(dy/dr). Substituting and inserting r=xRo and ρ =yρo in appropriate places, we get the desired result immediately.

5.3) There is no simple functional form for f(y), the function introduced in problem 5.2, except at (y-1) <~0.5 (where Birch –Murnahan may work provided you have only a single phase) and at y>>1 (where you tend towards the Fermi limit). But if you’re willing to smooth out over phase transitions and seek only rough estimates, then a tolerably good interpolative approximation is of the form f(y) = 6(y5/3-3y4/3/2 +y/2)

It has the necessary properties [f(1)=0 and f ′(1)=1] and also the right asymptotic form for a Fermi gas. Even the y4/3 term is physically sensible (since it is predicted for the Coulomb energy contribution). The price paid is that the appropriate choice of Ko is not the true bulk modulus of the material but some crude representation of compression over a substantial pressure range. This means that you should use it only for a wide range of masses above ~Earth mass. In what follows, assume ρo =1g/cm3 for ice and 4g/cm3 for rock.

(a) Show that in order to be consistent with the Fermi gas limit (51.6/rs5 Mbar) ,

one must choose Ko ~ 5 Mbar for rock and ~0.5 Mbar for ice. Using A/Z~2 for both is sufficiently accurate (though ice has slightly lower effective A/Z because of the hydrogen). [Note that with these choices, y=2 at P ~ 12 Mbar and 1.2 Mbar for rock and ice respectively. These are not far from realistic]. (b) With these choices of Ko, construct numerically the radius–mass relationship for massive planets made of pure ice and pure rock respectively. This should cover the range from low (0.1 Earth masses) out to masses ~1000 Earth masses where the radius no longer increases as you add mass. Hint: This can be done easily using Mathematica (for example) and the non-dimensional form of the hydrostatic equation proved in the preceding problem. You start at x=0 with some choice of y(0) >1 and with y ′ (0)=0 [why?] and integrate out, keeping the result only to the point where y has dropped to 1. This gives you radius R (in units of Ro.) You then integrate the resulting interpolative function to get mass M (in what units?) You repeat for various choices of y(0). You do not need to do the calculation a large number of times to get a good appreciation of the R vs. M curve because R(M) is very smooth and slowly varying. You don’t have to do the calculation separately for ice and for rock (the beauty of non-dimensionalization!) The obviously sensible way to plot the results is R on the vertical axis and log of mass on the horizontal axis (because M changes enormously but R does not change a lot).

5 rocks and ices: beyond the nearly uniform electron...

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