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 the5. Rocks and Ices: Beyond the Nearly Uniform Electron Gas 48 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.5. Rocks and Ices: Beyond the Nearly Uniform Electron Gas 49 Figure 5.25. Rocks and Ices: Beyond the Nearly Uniform Electron Gas 50 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−ρρ0⎛⎝⎜⎞⎠⎟53]{1+34(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).5. Rocks and Ices: Beyond the Nearly Uniform Electron Gas 51 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