The mathematical expression from which the isochron is derived is {\mathrm{D*}} = {\mathrm{D}}_{\mathrm{0}} + \mathrm{n} \cdot (e^{\lambda t}-1), where :
t is age of the sample, :
D* is number of atoms of the radiogenic daughter isotope in the sample, :
D0 is number of atoms of the daughter isotope in the original or initial composition, : n is number of atoms of the parent isotope in the sample at the present, :
λ is the
decay constant of the parent isotope, equal to the inverse of the radioactive
half-life of the parent isotope times the natural logarithm of 2, and : (
eλ
t-1) is the slope of the isochron which defines the age of the system. Because the isotopes are measured by
mass spectrometry, ratios are used instead of absolute concentrations since mass spectrometers usually measure the former rather than the latter. (See the section on
isotope ratio mass spectrometry.) As such, isochrons are typically defined by the following equation, which normalizes the concentration of parent and radiogenic daughter isotopes to the concentration of a non-radiogenic isotope of the daughter element that is assumed to be constant: \left(\frac{\mathrm{D*}}{\mathrm{D}_{ref}}\right)_{\mathrm{present}} = \left(\frac{\mathrm{D_0}}{\mathrm{D}_{ref}}\right)_{\mathrm{initial}} + \left(\frac{\mathrm{P_t}}{\mathrm{D}_{ref}}\right) \cdot (e^{\lambda t}-1), where :D_{ref} is the concentration of the non-radiogenic isotope of the daughter element (assumed constant), : :D* is the present concentration of the radiogenic daughter isotope, : :D_0 is the initial concentration of the radiogenic daughter isotope, and :P_t is the present concentration of the parent isotope that has decayed over time t. To perform dating, a rock is crushed to a fine powder, and minerals are separated by various physical and magnetic means. Each mineral has different ratios between parent and daughter concentrations. For each mineral, the ratios are related by the following equation: :{\mathrm{D}_0 + \Delta{P}_t \over D_{ref} } = {\Delta{P}_t \over P_i-\Delta{P}_t } \left ( { P_i-\Delta{P}_t \over D_{ref} }\right ) + {D_0 \over D_{ref}} (1) where :P_i is the initial concentration of the parent isotope, and : :\Delta{P}_t is the total amount of the parent isotope which has decayed by time t. The proof of (1) amounts to simple algebraic manipulation. It is useful in this form because it exhibits the relationship between quantities that actually exist at present. To wit, P_i-\Delta{P}_t, D_0+\Delta{P}_t and D_{ref} respectively correspond to the concentrations of parent, daughter and non-radiogenic isotopes found in the rock at the time of measurement. The ratios \frac{\mathrm{D*}}{\mathrm{D}_{ref}}or D_0+\Delta{P}_t \over D_{ref} (relative concentration of present daughter and non-radiogenic isotopes) and \frac{\mathrm{P_t}}{\mathrm{D}_{ref}} or { P_i-\Delta{P}_t \over D_{ref} } (relative concentration of present parent and non-radiogenic isotope) are measured by
mass spectrometry and plotted against each other in a three-isotope plot known as an
isochron plot. If all data points lie on a straight line, this line is called an isochron. The better the fit of the data points to a line, the more reliable the resulting age estimate. Since the ratio of the daughter and non-radiogenic isotopes is proportional to the ratio of the parent and non-radiogenic isotopes, the slope of the isochron gets steeper with time. The change in slope from initial conditions—assuming an initial isochron slope of zero (a horizontal isochron) at the point of intersection (intercept) of the isochron with the y-axis—to the current computed slope gives the age of the rock. The slope of the isochron, (e^{\lambda t}-1) or \Delta{P}_t \over P-\Delta{P}_t, represents the ratio of daughter to parent as used in standard
radiometric dating and can be derived to calculate the age of the sample at time
t. The y-intercept of the isochron line yields the initial radiogenic daughter ratio, \frac{\mathrm{D_0}}{\mathrm{D}_{ref}}. Whole rock isochron dating uses the same ideas but instead of different minerals obtained from one rock uses different types of rocks that are derived from a common reservoir; e.g. the same precursor melt. It is possible to date the differentiation of the precursor melt which then cooled and crystallized into the different types of rocks. One of the best known isotopic systems for isochron dating is the
rubidium–strontium system. Other systems that are used for isochron dating include
samarium–neodymium, and
uranium–lead. Some isotopic systems based on short-living extinct radionuclides such as
53Mn,
26Al,
129I,
60Fe and others are used for isochron dating of events in the early history of the
Solar System. However, methods using extinct radionuclides give only relative ages and have to be calibrated with radiometric dating techniques based on long-living radionuclides like Pb-Pb dating to give absolute ages. ==Application==