Lutetium–hafnium dating is a geochronological dating method utilizing the radioactive decay system of lutetium–176 to hafnium–176. With a commonly accepted half-life of 37.1 billion years, the long-living Lu–Hf decay pair survives through geological time scales, thus is useful in geological studies. Due to chemical properties of the two elements, namely their valences and ionic radii, Lu is usually found in trace amount in rare-earth element loving minerals, such as garnet and phosphates, while Hf is usually found in trace amount in zirconium-rich minerals, such as zircon, baddeleyite and zirkelite.
The trace concentration of the Lu and Hf in earth materials posed some technological difficulties in using Lu–Hf dating extensively in the 1980s. With the use of inductively coupled plasma mass spectrometry (ICP–MS) with multi-collector (also known as MC–ICP–MS) in later years, the dating method is made applicable to date diverse earth materials. The Lu–Hf system is now a common tool in geological studies such as igneous and metamorphic rock petrogenesis, early earth mantle-crust differentiation, and provenance.
Radiometric dating
Lutetium is a rare-earth element, with one naturally-occurring stable isotope 175Lu and one naturally-occurring radioactive isotope 176Lu. When 176Lu atoms are incorporated into earth materials, such as rocks and minerals, they began to be "trapped" while starting to decay. Through radioactive decay, an unstable nucleus decays into another relatively stable one. Radiometric dating makes use of the decay relationship to calculate how long the atoms have been "trapped", i.e. the time since the earth material was formed.
Decay of 176Lu
The only natural occurring radioactive isotope of lutetium ^{ 176}_{ 71}Lu decays in the following two ways:
^{ 176}_{ 71}Lu -> { ^{ 176}_{ 72}Hf} + e^-
{ ^{ 176}_{ 71}Lu} + e^- -> { ^{ 176}_{ 70}Yb}
Lutetium, ^{ 176}_{ 71}Lu can decay into ^{ 176}_{ 72}Hf , a heavier element, or ytterbium, ^{ 176}_{ 70}Yb , a lighter element. However, as the major mode of decay is by β− emission, i.e. release of electron (e−), as in the case for ^{ 176}_{ 71}Lu decaying to ^{ 176}_{ 72}Hf , the presence of ^{ 176}_{ 70}Yb is of negligible effect to Lu–Hf age determination.
Decay constant determination
The decay constant of ^{ 176}Lu can be obtained through direct counting experiments and by comparing Lu–Hf ages with other isotope system ages of samples whose ages are determined. The commonly accepted decay constant has the value of 1.867 (± 0.007) × 10−11 yr−1. However, there remain discrepancies on the value of decay constant.
Age determination
An age equation is set up for every radiometric dating technique to describe the mathematical relationship of the number of parent and daughter nuclide. In Lu–Hf system, the parent would be Lu (the radioactive isotope) and Hf as the daughter nuclide (the product after radioactive decay). The age equation to Lu–Hf system is as follows:
where: (^{ 176}Hf/^{ 177}Hf) is the measured ratio of the two isotopes of the sample. (^{ 176}Hf/^{ 177}Hf)_i is the initial ratio of the two isotopes when the sample is formed. (^{ 176}Lu/^{ 177}Hf) is the measured ratio of the two isotopes of the sample. is the decay constant of ^{ 176}Lu. t is the time since the sample is formed.
The two isotopes, 176Lu and 176Hf, in the system are measured as ratio to the reference stable isotope of 177Hf. The measured ratio can be obtained from mass spectrometry. A common practice for geochronological dating is to establish an isochron plot. Multiple set of data would be measured and plotted with 176Hf/177Hf on y-axis and 176Lu/177Hf on x-axis. A linear relationship would be obtained. The initial ratio can either be assumed to be natural isotopic abundance ratio or, for a better approach, obtained from the y-intercept of plotted isochron. The slope of the plotted isochron would represent .
Epsilon (ɛHf value) ɛHf value is an expression of ^{ 176}Hf/^{ 177}Hf ratio of a sample with respect to ^{ 176}Hf/^{ 177}Hf ratio of chondritic uniform reservoir. The usage of ɛHf value is a common practice in Hf studies. ɛHf has a range of value from +15 to -70 at present. ɛHf is expressed in the following equation:
where: "0" in the bracket denoting time = 0, meaning present day. Numbers in bracket can represent any time in the past up to the formation of Earth. is the Hf-176 to Hf-177 ratio in the sample. For t = 0, it represent the ratio at present. is the Hf-176 to Hf-177 ratio in the chondritic uniform reservoir. For t = 0, it represent the ratio at present.