# Introduction to Radioisotope Geochronology/Part Two - Principles of Radioisotopic Dating
## Radioactive Decay

Radioisotope geochronology in its present form is made possible by radioactive decay. Radioactive decay, also known as nuclear decay or radioactivity, is the process through which the radioactive (unstable) nucleus of an atom emits particles which lowers it to a lower energy state. This decay process takes place in a random fashion in that it is impossible to predict which particular atom will experience this change. Despite the random nature of this process, radioactive decay takes place at rate that is constant. Therefore using measured decay constants and the rate at which radionuclides decay, geochronologists can harness this relationship to calculate the amount of time that has passed since an assumed volume of material began accumulating radiogenic material.

## Decay Constants and Half Lives

The rate at which radioactive elements decay is governed by the exponential decay constant. The amount of time required for half of a given quantity of a parent radioactive element to decay into the daughter product is referred to as the half-life. It is important however, that the half-life of an element is defined in terms of probability and is not the time required for exactly 50% of a given quantity to decay (e.g. there are not 1.5 atoms after 3 radioactive atoms experience one half-life). The predictability of a half-life is greater when a greater number of atoms is observed and thus allowing for the quantification of half-lives.

## The Age Equation

D = D$_0$ + N(e$^{λt}$ − 1)
where
    t is age of the sample,
    D is number of atoms of the daughter isotope in the sample,
    D$_0$ is number of atoms of the daughter isotope in the original composition,
    N is number of atoms of the parent isotope in the sample, and
    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.

## Parent/Daughter Ratio

In order to calculate a date using the age equation it is required to determine the ratio of the parent isotope (P) to it's respective daughter isotope (D). This requires quantitative determination of the relative proportions of both P and D isotopes, and often requires correction for the inclusion of non-radiogenic D.

## What makes a Good Geochronometer?

Radioactive decay schemes are suitable for dating minerals and rocks and are listed in Table 1. All of these systems are based upon the radioactive decay of a parent nuclide to a stable daughter nuclide. Obtaining accurate information from these decay systems for the purposes of determining the age of a mineral or rock requires: (1) the decay constant of the parent nuclide is accurately and precisely determined; (2) closed system behavior, which can be simply stated to mean that the Parent/daughter ratio has only changed by radioactive decay; and (3) the initial daughter nuclide, if present, can be precisely and accurately accounted for. In this section we outline the basic principles of the various radio-isotopic geochronometers, differentiating the U-Pb system applied to U-bearing accessory minerals from the isochron geochronometers (Re-Os, Lu-Hf, Pb-Pb etc.) applied to chemical precipitates and organic residues.

### Fractionation of Parent and Daughter

In order to be able to quantitatively determine the ratio of the parent nuclide (P) to a stable daughter nuclide (D) we need to analyses materials that have high proportions of P relative to D at their formation such that the ingrowth of D overwhelms that any amount of initial D isotope.

### Closed system behaviour

The other important aspect to a good geochronometer is a sound and robust understanding of the mineral and/or rock system. Much of this understanding comes from experimental work in the realms of crystallography, mineral chemistry, and diffusion. A good geochronometer must either have little to no initial incorporation of the daughter isotope or must have a fixed ratio from which the radiogenic and nonradiogenic proportions can be determined. For example, common-Pb (or the natural occurrence of non-radiogenic lead) includes radiogenic isotopes (206, 207, 208) as well as non-radiogenic isotopes (204). If the ratio of these is constant or measurable, then we can separate the radiogenic from the non to extract meaningful age information.

## Examples: Minerals
### Zircon

Zircon (ZrSiO$_4$) is a common accessory mineral in silicic volcanic rocks ranging from lavas to air-fall tuffs to volcaniclastic sedimentary rocks and is a nearly ubiquitous component of most clastic sedimentary rocks. The refractory and durable nature of zircon over a wide range of geological conditions means that it is likely to retain its primary crystallization age even through subsequent metamorphic events. Silicic air-fall tuffs are the most common volcanic rocks in fossil-bearing sequences and are found in layers that range in thickness from a millimeter to many meters and are commonly preserved in marine settings. In most of these rocks the primary volcanic ash has been altered, probably soon after deposition, to clay minerals in a process that does not affect zircon.

##### Zircon U-Pb

Zircon is ideal for U-Pb dating because U has a similar charge and ionic radius to Zr it substitutes readily into the zircon crystal structure (in modest amounts, typically in the 10’s to hundreds of parts per million (ppm) range) whereas Pb has a different charge and larger ionic radius leading to its effective exclusion from the crystal lattice. Therefore at the time of crystallization (t0) there is effectively no Pb present in a crystal (although mineral and fluid inclusions may contain Pb) and the present day Pb is the direct product of in-situ U decay since t0 (see section 3.1 for further details). An additional factor that makes zircon a robust chronometer is its high closure temperature (>900°C) to Pb diffusion (Cherniak and Watson, 2003), or the temperature below which U and Pb do not undergo significant thermally activated volume diffusion. This means that zircons tend to preserve their primary ages even in volcanic rocks metamorphosed to amphibolite-facies conditions.

##### Zircon Fission Track
### Titanite (Sphene)

Titanite (CaTiSiO5) is a common accessory magmatic mineral in intermediate and felsic igneous rocks. It also occurs in metamorphic rocks. Occasionally detrital titanite can be found in unmetamorphosed sedimentary rocks. Titanite is most commonly used to date the cooling of metamorphic events using U-Pb isotopes.

##### Titanite U-Pb

Although it can accommodate several 100’s of ppm U in the crystal lattice, it also incorporates variable amounts of Pb. Often this results in several analyses of the same unzoned titanite grain that define a discordia (on a Tera-Wasserburg concordia diagram) anchored with a $^{207}$Pb/$^{206}$Pb ratio of common-Pb on one end and the closure age. Additionally, several workers have documented metamorphic titanite that preserves multiple generations of mineral growth thus highlighting the importance for detail chemical imaging and multiple analyses per grain to fully characterize the history of mineral growth.

### Monazite

Monazite ([Ce,La,Nd,Sm,Gd,Th]PO4) is phosphate mineral common in metamorphic and igneous rocks. The relatively high Th and U content allow for monazite to be dated using Th-Pb and U-Pb decay schemes. Like titanite, monazite also incorporates variable amounts of Pb. Monazite is commonly zoned which is generally visualized by trace element variations (e.g. Yb, U, Ca). These zonations can preserve growth events separated by billions of years of Earth history.

### Sanidine

Sanidine (a high temperature form of potassium feldspar (K,Na)(Si,Al)$_4$O$_8$))

### Hornblende

Hornblende