In radiometric isochron dating why is at t=0 D*/Dref=const but P/Dref not?

Question

I am a physics graduate and trying to understand one of the assumptions made for isochron dating of rocks: at crystallization time of the rock, $t=0$, the ratio of the parent isotope to a stable reference daughter isotope was variable for different parts/minerals of the rock ($P/D_{ref}\neq const$), but the ratio of the radiogenic daughter isotope to the stable reference daughter isotope was constant ($D*/D_{ref}=const$). See for that the figure below taken from here :

My question: Why is the one ratio constant, but the other not? I taught $D*/D_{ref}=const$ holds, because just before $t=0$ the rock is still molten and components are well mixed, but then why shouldn't $P$ and $D_{ref}$ be also homogeneously mixed and uniformly distributed.

Additional question: In the figure below, taken from here, I get that the linear function is the isochron, but what is the logarithmic function and how does it help in the dating process/what is its contribution? Thanks

Answer 1

1) D* (radiogenic isotope) and D_ref (stable reference isotope) are two isotopes of the same element. P is the radioactive parent of D*, and is a different element. Chemical fractionation between P and D_ref occurs as the rock solidifies because different elements are preferentially incorporated in different minerals depending on how well their ionic radii and valences are compatible with the mineral's crystalline lattice.

Although there can be some mass-fractionation of isotopes of a same element, it is usually very small (IE negligible) relative to the chemical fractionation. Often, for making such isochrons, geologists choose different mineral species with very different P concentrations (and thus varying P/D_ref) to provide spread along the isochron line; this allows for more precise regression and determination of the slope / age.

2) The second diagram you show is a concordia plot, not an isochron. There is no common isotope on the ratio denominator. Rather, this plot shows two different isotope systems each with their own decay chain (²³⁸U->²⁰⁶Pb and ²³⁵U->²⁰⁷Pb), these two Uranium decay chains have different half lives. The curved line is the "concordia" curve, which is the expected result if the sample exhibits closed-system behaviour (IE Pb has not diffused out of the crystal).

In this case a mineral (probably a zircon) was formed at 2.6 Ga and had open system behaviour at around 0.4 Ga. (IE the radiometric clock was partially reset). The straight line is called a "discordia line" and is a mixing line between the two time points on the concordia curve.