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Potassium-Argon Dating

Isochron Dating

Optical Dating

Rehydroxylation Dating

Before the advent of carbon 14 dating, estimate of age is a rather hazardous undertaking. The age of a piece of antique can be determined by its style in certain period, the age of fossil is related to the stratification etc. Discovery of radioactive elements provides the mean to date objects in term of years. The radioactive isotope carbon-14 is especially suitable for dating organic matter because the initial quantity N_{0} and its half-life t_{1/2} are known in the decay formula :N = N _{0}e^{-kt} ---------- (1)where k = ln(2)/t _{1/2}, and t is the time variable (Figure 01). Further details are provided in the followings : | ||

## Figure 01 Radioactive Decay [view large image] |
## Figure 02 Carbon-14 Dating [view large image] |

- Carbon-14 atoms decay continuously with a half-life of 5730 years. However, they are replenished by cosmic rays as shown in the series of reactions in Figure 02 resulting in a nearly constant ratio with the normal carbon-12, i.e., r = N
_{14}/N_{12}10^{-12}. - By this constant ratio r, any sample with a known value of N
_{12}would provide an initial value for N_{14}, e.g., N_{0}= r N_{12}. - This constant ratio of carbon atoms in the plants or animals will no longer maintained once they expire. The ratio r will change according to Eq.(1), i.e., r = N / N
_{0}= e^{-kt}, or t = -ln(r)/k. - The value of N can be measured by counting the number of beta particles (Figure 02) or by measuring
- The above description is over simplified and required many corrections and modifications. The ratio r is not exactly a constant, it varies with time and place and also subjected to contamination. Therefore, the formula is more complicated and preparation of the sample is actually very immaculate. Consequently, the radiocarbon dates have to be converted to the calendar dates according to Figure 04.
- Sample mass requirement depends on material and the method of measurement as shown in Figure 5. Carbon-14 dating can be used for objects up to about 50000 years old.

## Figure 03 Mass Spectrometer [view large image] |
the ratio r with the Accelerator Mass Spectrograph (AMS) as shown in Figure 3. |

## Figure 04 Date Calibration |
## Figure 05 Sample Mass [view large image] |

t = (1/k)ln(1+D/P) ---------- (2) where D and P denote the daughter and parent numbers respectively. Since the daughter product argon-40 is not incorporated into minerals, all such atoms in a rock must have come from the decay of K-40 that was there originally. That is, the age can be measured from the ratio D/P using AMS. However, this assumption may not be valid as shown in Figure 06. The measurements obtain three different ages depending on the hydrothermal condition in the magma (from the 1986 | ||

## Figure 06 K-40 Dating |
## Figure 07 K-40 Decay |
Mt. St. Helen eruptions). It demonstrates that excessive argon (radiogenic and/or primeval) can yield erroneous ages - a problem readily exploited by creationists. |

Debates between the rival factions aside, the formula for age estimate from K-40 decay has to be modified because it decays by two modes. Only 10.72% produces argon-40, the rest goes to calcium-40 as shown in Figure 07. Therefore, the formula in Eq.(2) should be re-written as :

t = (1/k)ln[1+D/(0.1072xP)] ---------- (3)

For cases where the initial quantity of the radioactive isotope is not available, the method of isochron (equal time) can also bypass the problem with the ratios of parent and daughter to the non-radiogenic isotope of the daughter element as illustrated in Figure 08. The formula corresponds to the plot is in the form : (D + d)/S = (D/P)(P/S) + d/S ---------- (4) where P is the number of the parent atoms, D for the daughter, S (assumed to be constant) is the | |

## Figure 08 Isochron Dating [view large image] |
non-radiogenic isotope of D, and d is the initial number of D. The formula in Figure 08 is for the case where the horizontal axis is displaced upward by an amount d corresponding to equate d = 0 in Eq.(4). |

The process involves measuring the D/S and P/S ratios from different minerals inside a rock (assumed to form at the same time) by AMS. Since the amount of the constituents are not exactly the same in the various minerals, the plot of D/S vs P/S would spread out in different points - on a straight line in this case. A horizontal line along the x-axis signifies the beginning of the decay when D = 0; the slope of the line becomes steeper as the disintegration progresses; finally, the line moves to a vertical position when P = 0 at the end of the decay. The dating is determined by the slope G = (D/S)/(P/S) using the formula :

t = (1/k)ln(1 + G) ---------- (5)

in which the initial parent number disappears altogether. One of the best known isotopic systems for isochron dating is the rubidium-strontium system (as shown in Figure 08). Other systems include samarium-neodymium (t

Table 01 below lists some radiometric systems and the range of dates that can be measured. The lower and upper limits are determined by the presence of enough number of the daughter and parent atoms to obtain sufficient signals. Ultimately it is the quality of the equipments in the laboratory, which impose the sensitive limit.

## Table 01 Radiometric Systems and Dating Ranges |

Optical (Luminescence) Dating utilizes a principle opposite to the radioactive method. Whereas the latter measures the diminished amount of the decaying material, optical dating relies on the accumulation of electrons trapped inside minerals such as quartz and feldspar (the major composition of sediment). The electrons come from the background radioactive elements in the solid, they are trapped in the crystal lattice in the absence of Sun light, which would set the dating clock back to zero by evicting the electrons from the sites. The amount of such electrons measured in the laboratory from a sample is proportional to its age since the last exposure to Sun light. Followings is a summary of such age determination procedure (Figure 09). | |

## Figure 09 Optical Dating |

- The age of the sample T is calculated by the formula :

T = D_{e}/D_{r}

where D_{e}is referred to as the Equivalent Dose proportional to the number of trapped electrons in unit of Gy, and D_{r}is the Dose Rate in unit of Gy/kyr. Gy is defined as the absorption of 1 joule of radiation energy by 1 kg of matter, while kyr denotes 1000 years. - To avoid exposure to Sun light, the sample should be collected with opaque steel or plastic tubes. It should be taken from location far below the sedimentary surface. All subsequent preparation is conducted under subdued orange light to avoid the early stimulation of any trapped charge.
- The grains are illuminated by visible (for quartz) or infrared (for feldspars) laser beam, which induces "Optically Stimulated Luminescence (OSL)". This is amplified by a photomultiplier tube and measured using a photon-counting system. The intensity is proportional to the number of trapped electrons within the sample.
- The intensity is then converted to the Gy unit (used for D
_{e}) by fitting the measurement to the "dose response curve" prepared from known laboratory doses (see Figure 09). - The radiation dose rate D
_{r}is calculated from gamma spectrometry measurements of the radioactive elements (usually K, U,Th and Rb) surrounding the sample plus contribution from cosmic rays. It is usually in the range 0.5 - 5 Gy/kyr. - Optical dating applications using quartz have largely been restricted to the past 200 kyrs, while deeper traps of feldspar have produced dates as old as 1,000 kyrs. Heterogeneous sediments and radioactive disequilibria will increase errors on D
_{r}, while incomplete bleaching of the sample prior to birial, anomalous fading in feldspars, and the estimation of past sediment moisture content may all contribute to error in the calculated age T. Through careful choice and collection of samples, as well as stringent preparation and analytical methods, it is possibel to produce ages with an accuracy of between 5 and 12%. - The method of thermoluminsecence (TL) dating is similar to the procedure outlined above, except that it uses heating (over 500
^{o}C) to evict the trapped electrons - a process that removes both optically inert traps and light-sensitive ones.

Rehydroxylation (RHX) Dating also utilizes the concept of the regular accumulation of something inside the material to determine its age. In this case, it is observed that the mass of fired-clay ceramics, brick, tile etc. is reduced by heating. The subsequent re-absorption of moisture from the environment produces water mass gain, which is found to follow a power law of the time after the initial heating. This dating method is still under development following its introduction in early 21st century. It has been shown to reproduce accurate dates comparing to the known ages of antique artefacts, ancient ruins, etc. Following is a brief description of this method. | |

## Figure 10 Rehydoxylation Dating [view large image] |

- Water inside the clay is lost with firing over 500
^{o}C via the dehydroxylation process 2OH^{-}H_{2}O + O^{2-}. Exposure of the material afterward to water as either vapor or liquid produces mass gain by re-absorption. - It is found that the increase in mass follows a 1/4 power law of its age (the elapsed time after the heating) :

y = (m - m_{o})/m_{o}= (T) t^{1/4}

where y is the fractional mass increase, t the elapsed time (age), m the mass at t, m_{o}the mass at t = 0, and is the kinetic constant depending on the temperature T and type of material. - Both curves in Figure 10 show an actual rehydroxylation process. It reveals that the initial re-absorption does not really follow a 1/4 power law. However, this initial stage is very short in the order of hours and produces a difference in m
_{o}by an amount of 2x10^{-3}m_{o}. Thus, the 1/4 power law is valid for all practical purpose. - As for the reason of not following the expected 1/2 power law from Brownian diffusion, it is attributed to highly restricted geometries or long time delay. In such a "single-file" diffusion process, after a transient period at an initial stage, the cumulative absorption of such system with a constant concentration boundary condition would scale as 1/4 power as observed.
- The first step to obtain the age of the object is to weight the "loaded" mass m
_{a}. It is then heated to 500^{o}C, and is subsequently calibrated at 25^{o}C and 35% relative humidity over a period of about 9 days (see circles in Figure 10, on both the small and large curves). The linear portion of this calibration run is then extrapolated to much longer duration. The (1/4 power) age of the object (t_{a})^{1/4}is obtained from the intercept of m_{a}to this straight line. This method is thus self-calibrating; the influence of firing temperature, mineralogy and micro-structure is consequently eliminated.

## Figure 11 Two Types of Diffusion |

## Figure 12 Dating the Earth |