The isochron method relies on selecting minerals from different regions of a particular rock formation. The different minerals are all the same age since they come from the same rock but likely have different concentrations of radioactive material due to non-uniform environmental interactions. By plotting the isotope concentrations of all these minerals, geochronologists can obtain an age of the rock. The accuracy of the age can often be improved by using several different radioisotopes.
Dating techniques focus on igneous strata, because the deep mantle of the Earth is the primary source of the radionuclides required. When an igneous stratum is formed molten material solidifies, and as it does so, the chemical compounds of various elements crystallise out of the melt differentially, resulting in the formation of well-defined mineral species.
Some melts possess a greater chemical affinity for some elements than others, so consequently, there will be further differentiation, based on these chemical affinities. Zircons, for example, have a chemical affinity for uranium, and incorporate uranium salts into their crystals far more readily than, for instance lead salts. Also, some other crystalline formations will incorporate other elements preferentially, including elements that are of great value in dating.
One typical element that is of use is rubidium, with the Rb87 isotope having a half-life of 48.8 billion years. This decays via β- decay into Sr87, an isotope of strontium, and this decay product is stable. However, a non-radiogenic isotope of strontium, Sr86, also exists in geological strata. The utility of this will become apparent shortly.
Now, any melt starting off with a quantity of Rb87 will, after sufficient time has elapsed, start to acquire quantities of Sr87. This will take time, given the long half-life of Rb87, and indeed, detectable quantities of Sr87 useful for mass spectrometry will only start to appear after 0.001 half-lives have elapsed. In the case of Rb87, this is 45 million years, so a sample that has an Rb87 age that is indistinguishable from zero could be as much as 40 million years old, which means that this is no indication of an allegedly "young" Earth. The moment any detectable traces of Sr87 appear in a sample, however, then we're dealing with a rock that is at least 45 million years old, this disproves blind assertions about the Earth being only 6,000 years old. However, this isn't the point: the point is, that any mineral that acquires quantities of strontium upon formation will acquire a specific ratio of the two isotopes Sr86 and Sr87, and that ratio can be used to determine the initial amount of Rb87 that was acquired during formation as well. Which means that no "assumptions" about initial material present are needed.
Now, since all the minerals that acquire strontium will acquire the same ratio of Sr86 to Sr87 at the start, we can use that ratio as the y-axis for a plot. However, different minerals within the sample will acquire different quantities of Rb87, and we can use the ratio of Rb87 to Sr86 to form the x-axis of the plot. If our initial values are:
P = amount of Rb87 at the time of stratum formation
D_1 = amount of Sr86 at the time of stratum formation
D_2 = amount of Sr87 at the time of stratum formation
then we plot an x-y plot consisting of:
x = P/D_1, y = D_2/D_1
the axes therefore correspond to:
x = increasing enrichment of Rb87 in the sample with increasing value
y = increasing enrichment of Sr87 in the sample with increasing value
Now, the global composition of the melt, from which the stratum eventually forms, will have a given point value on this plot. As the melt cools, and minerals crystallise out, different minerals will migrate along a straight line in this plot, as all the minerals will inherit the same value of y (= D_2/D_1), but inherit different values of x (=P/D_1). The result will be, at zero age, a horizontal line connecting the points for those minerals in the plot.
Now, as the Rb87 decays in each mineral, it will produce Sr87. Therefore, over geological time, the data points will move upwards and to the left.
Now, because decay occurs in a proportional manner, courtesy of the decay law, when 20% of the Rb87 has decayed in one mineral, then 20% of the Rb87 will have decayed in all the minerals present in the stratum. This means that all the points will move upwards and to the left of the plot. This means that those points corresponding to the minerals with the greatest initial Rb87 concentration will move the farthest, and the movement will be such that all of the points will remain on a straight line. As a consequence, any stratum containing the appropriate minerals will yield, for a given age, a line of points whose slope increases with time, and the slope of that line can be used to determine the age of the sample.
Note that we don't need to know the initial amounts of any of the elements present in the sample in order for this to work. All we need to know is the present-day ratios of those elements. And, from that data, we can reconstruct the original composition of the melt.
Now, here is the fun part. If the sample suffered any transport, then this will become immediately apparent because the points will deviate significantly from a straight line plot. This is because the minerals containing the various elements will undergo transport differentially. For example, if the minerals lost Rb87 due to various processes of aqueous chemistry, then this would shift all of the points to the left, but because different minerals are involved in the process, each with a different chemistry, and a different response to the aqueous processes that could remove Rb87 from them, the amount of x-shift will be different for each mineral, and as a consequence, straight line correlation will be destroyed. Immediately scientists see this, they know that something unusual has happened to the sample after formation, and that dates obtained from it are unlikely to be reliable.
Indeed, if the Earth were only 6,000 years old, not only would ALL isochron plots involving minerals with long half-lives be flat, horizontal lines, but random loss or gain of the parent nuclide would not affect those results. If the Earth were only 6,000 years old, any "contamination" that creationists assert would be present that would render radionuclide dating methods invalid, would not even be noticeable on a purported "young Earth". Contamination would only ever be noticeable if the Earth was old. Which means that far from invalidating radiometric dating, detectable contamination in a sample via the isochron method actually reinforces an old age for the Earth.
Worse still, from the creationist standpoint, the known possibilities for systematic alteration of an isochron plot, that results in modification of the data points such that they remain collinear, result in an underestimate of the age of the sample. So, in other words, the isochron plot records them as being younger than they actually are, not older. Complete homogenisation of the stratum with respect to the content of Sr87 will reset the isochron 'clock' to zero, and partial homogenisation will result in a line of lower slope than previously, causing the technique to underestimate the true age.
Once such a plot is complete, any strong correlation to a straight line (as yielded by regression analysis, which will give us an indication how much we can trust a date from the plot, and what value of error to apply to the plot), will yield the following values:
[1] the y-intercept of the line is the value of D_2/D_1 at solidification;
[2] the slope of the line is equal to ΔP/(P - ΔP), where ΔP is the amount of Rb87 lost to decay, and thus allows us to determine the age of the sample, given the known half-life of Rb87 (we can back-calculate using the decay law to determine what value of P was initially present to produce the observed result).
There is a body of technical literature on the subject of isochron dating, and, needless to say, it requires a fair amount of work to plough through, but for those who have acquired the relevant scientific and mathematical background, it is well worth exerting this effort.
