Radiometric Dating
A Christian Perspective

Dr. Roger C. Wiens
Working at the Division of Geological & Planetary Sciences
California Institute of Technology
Presently working at the Space and Atmospheric Sciences Group
Los Alamos National Laboratory
Home Address: 941 Estates Drive, Los Alamos, NM 87544
rwiens@prodigy.net

Radiometric dating--the process of determining the age of rocks from the decay of their radioactive elements--has been in widespread use for over half a century. There are over forty such techniques, each using a different radioactive element or a different way of measuring them. It has become increasingly clear that these radiometric dating techniques agree with each other and as a whole, present a coherent picture in which the earth was created a very long time ago. Many Christians are completely unaware of the great number of laboratory measurements that have shown these methods to be consistent, and they are also unaware that Bible-believing Christians are among those actively involved in radiometric dating. This paper describes in relatively simple terms how some dating techniques work, how accurately the half-lives of the radioactive elements and the rock dates themselves are known, and how dates are checked with one another. In the process the paper refutes some misconceptions prevalent among Christians today. God has called us to be "wise as serpents" even in this scientific age. This paper is put out by the American Scientific Affiliation and the Affiliation of Christian Geologists to promote greater understanding and wisdom on this issue within the Christian community.

TABLE OF CONTENTS

Introduction
Overview
The Radiometric Clocks
Examples of Dating Methods for Igneous Rocks

Potassium-Argon
Argon-Argon
Rubidium-Strontium
Samarium-Neodymium, Lutetium-Hafnium, and Rhenium-Osmium
Uranium-Lead

Cosmogenic Radionuclides: Carbon-14, Beryllium-10, Chlorine-369
The Age of the Earth
Can We Really Believe the Dating Systems?
Doubters Still Try
Rightly Handling the Word of Truth
Appendix: Common Misconceptions Regarding Radiometric Dating Techniques
Further Reading
Glossary


Introduction


Arguments over the age of the earth have sometimes been divisive in the church. Although the Bible never mentions the earth's age, it is an issue because some people have tried to calculate the date of creation by adding up the life-spans of the generations listed in Genesis chapters 5 and 11. Assuming a strictly literal interpretation of the week of creation, even if some generations were left out of the genealogies, the earth would be less than ten thousand years old. Radiometric dating techniques indicate that the earth is thousands of times older than that--approximately four and a half billion years old. Many Christians accept this and interpret the Genesis account in less scientifically literal ways. However, some Christians suggest that the geologic dating techniques are unreliable, that they are wrongly interpreted, or that they are confusing at best. Unfortunately, there has not been much literature easily available to Christians in an understandable form, so that confusion over dating techniques continues.

The next few pages give a broad overview of radiometric dating techniques, talk through a few examples, and discuss the degree to which the various dating systems agree with each other. The goal is to promote greater understanding in the Christian community on this issue. Many people have been led to be skeptical of dating without knowing much about it. For example, most people don't realize that carbon dating is not used on rocks at all. God has called us to be "wise as serpents" (Matt. 10:16) even in this scientific age. In spite of this, differences still occur within the body of Christ. A disagreement over the age of the earth is relatively minor in the whole scope of Christianity; it is more important to agree on the Rock of Ages than on the age of rocks. But because God has also called us to wisdom, this issue is worthy of study.

Overview



Rocks are made up of many individual crystals, and each crystal is usually made up of at least several different chemical elements such as iron, magnesium, silicon, etc. Most elements in nature are stable and do not change. However, some elements are not completely stable in their natural state. Some atoms eventually change from one element to another by a process called radioactive decay. If there are many atoms of the original element, called the parent element, the atoms decay to another element, called the daughter element, at a predictable rate. The passage of time can be charted by the reduction in the number of parent atoms, and the increase in the number of daughter atoms.

Radiometric dating can be compared to an hourglass. When the glass is turned over, sand runs from the top to the bottom. You cannot predict exactly when any one particular grain will get to the bottom, but you can predict from one time to the next how long the whole pile of sand takes to fall. Once all of the sand has fallen out of the top, the hourglass will no longer keep time unless it is turned over again. Similarly, when all the atoms of the radioactive element are gone, the rock will no longer keep time (unless it receives a new batch of radioactive atoms).

Unlike the hourglass, where the amount of sand falling is constant right up until the end, the number of decays from a fixed number of radioactive atoms decreases as there are fewer atoms left to decay (see Figure 1). If it takes a certain length of time for half of the atoms to decay, it will take the same amount of time for half of the remaining atoms, or one-fourth of the original total, to decay. In the next interval, with only one-fourth remaining, only one eighth of the original total will decay. By the time ten of these intervals, or half-lives, have passed, less than one thousandth of the original number of radioactive atoms is left. Also unlike the hourglass, there is no way to change the rate at which radioactive atoms decay on earth. If you shake the hourglass, twirl it, or put it in a rapidly accelerating vehicle, the time it takes the sand to fall will change. But the radioactive atoms used in dating techniques have been subjected to heat, cold, pressure, vacuum, acceleration, and strong chemical reactions without any significant change in their decay rate.1

Figure 1. The rate of loss of sand from the top of an hourglass compared to the exponential type of decay of radioactive elements. Most processes we are familiar with are linear, like sand in the hourglass. In exponential decay, the amount of material decreases by half during each half-life. After two half-lives only one-fourth is left, after three half-lives only one eighth is left, etc. As shown in the bottom panel, the daughter element or isotope amount increases rapidly at first, then more slowly with each succeeding half-life.


An hourglass will tell time correctly only if it is completely sealed. If it has a hole allowing the sand grains to escape out the side instead of going through the neck, it will give the wrong time interval. Similarly, a rock that is to be dated must be sealed against loss or addition of either the radioactive daughter or parent. If it has lost some of the daughter element, it will give an inaccurately young age. As will be discussed later, most dating techniques have very good ways of telling if such a loss has occurred, in which case the date is thrown out (and so is the rock!).

An hourglass measures how much time has passed since it was turned over. (Actually it tells when a specific amount of time, e.g., two minutes, an hour, etc., has passed, so the analogy is not quite perfect.) Like the hourglass, radiometric dating of rocks tells how much time has passed since some event occurred. For igneous rocks the event is usually its cooling and hardening from magma or lava. For some other materials, the event is the end of a period of metamorphic heating (in which the rock gets baked underground at generally over a thousand degrees Fahrenheit). The event in other cases can be the uncovering of a surface by the scraping action of a glacier, the chipping of a meteorite off an asteroid, or the length of time a plant or animal has been dead.

The Radiometric Clocks


There are now well over forty different radiometric dating techniques, each based on a different radioactive isotope.2 A partial list of the parent and daughter isotopes and the decay half-lives is given in Table 1. Notice the large range in the half-lives. Isotopes with long half-lives decay very slowly, and so are useful for dating correspondingly ancient events. Isotopes with shorter half-lives cannot date very ancient events because all of the atoms of the parent isotope would have already decayed away, like an hourglass left sitting with all the sand at the bottom. Isotopes with relatively short half-lives are useful for dating correspondingly shorter intervals, and can usually do so with greater accuracy, just as you would use a stopwatch rather than a grandfather clock to time a 100-meter dash. On the other hand, you would use a calendar, not a clock, to record time intervals of several weeks or more.

Table 1. Some Naturally-Occurring Radioactive Isotopes and Their Half-Lives

Radioactive Isotope
(Parent)
Product
(Daughter)
Half-Life
(Years)
Samarium - 147 Neodymium - 143 106 billion
Rubidium-87 Strontium-87 48.8 billion
Rhenium-187 Osmium-187 42 billion
Lutetium-176 Hafnium-176 38 billion
Thorium-232 Lead-208 14 billion
Uranium-238 Lead-206 4.5 billion
Potassium-40 Argon-40 1.26 billion
Uranium-235 Lead-207 0.7 billion
Beryllium-10 Boron-10 1.52 million
Chlorine-36 Argon-36 300000
Carbon-14 Nitrogen-14 5715
Half-lives taken from N. E. Holden, Pure Appl. Chem. 62 (1990): 941-958.

The half-lives have all been measured directly, either by using a radiation detector to count the number of atoms decaying in a given amount of time from a known amount of the parent material, or by measuring the ratio of daughter to parent atoms in a sample that originally consisted completely of parent atoms. Work on radiometric dating first started shortly after the turn of the century, but progress was relatively slow before the late forties. For many of the dating techniques, we now have had fifty years over which to measure and remeasure the half-lives. Very precise counting of the decay events or the daughter atoms can be done, so that while the number of, for example, rhenium-187 atoms decaying in 50 years is a very small fraction of the total, the resulting osmium-187 atoms can be very precisely counted.

The uncertainties on the half-lives given in the table are all very small. All the half-lives are known to better than plus/minus about two percent except for rhenium (5%), lutetium (3%), and beryllium (3%). There is no evidence of any of the half-lives changing over time, and such a thing is forbidden by the laws of physics.

Examples of Dating Methods for Igneous Rocks


Igneous rocks are good candidates for dating. Recall that for igneous rocks the event being dated is when the rock was formed from magma or lava. When the molten material cools and hardens, the atoms are no longer free to move about. Any daughter atoms from radioactive decays occurring after the rock cools are trapped where they are made within the rock. These atoms are like the sand grains accumulating in the bottom of the hourglass. To determine the age of the rock one needs to measure the number of daughter atoms and the number of remaining parent atoms, and use the half-life to calculate the time it took to make those daughter atoms.

However, there is one complication. One cannot always assume that there were no daughter atoms to begin with. It turns out that there are some cases where one can make that assumption quite reliably. But usually the initial amount of the daughter product must be accurately determined. Most of the time one can use the different amounts of parent and daughter atoms present in different minerals within the rock to tell how much daughter product was originally present. Each dating mechanism deals with this problem in its own way. Some types of dating work better in some rocks; others are better in other rocks, depending on the rock composition and its age. Let's examine some of the different dating mechanisms now.

Potassium-Argon. Potassium is an abundant element in the Earth's crust. One isotope, potassium 40, is radioactive and decays to two different daughter products, calcium-40 and argon-40, by two different decay methods. This is not a problem because the production ratio of these two daughter products is precisely known, and is always constant: 11.2% becomes argon-40 and 88.8% becomes calcium-40. It is possible to date some rocks by the potassium-calcium method, but this is not often done because it is hard to determine how much calcium was initially present. Argon, on the other hand, is a gas. Whenever rock is melted to become magma or lava, the argon tends to escape. Once the molten material hardens, it again begins to trap the argon produced from its potassium. In this way the potassium-argon clock is clearly reset when an igneous rock is formed.

In its simplest form, the geologist simply needs to measure the relative amounts of potassium-40 and argon-40 to date the rock. The age is given by a relatively simple equation:

where t1/2 is the half-life, and ln is the natural logarithm.

However, in reality there is often a small amount of argon remaining in a rock when it hardens. This is usually trapped as very tiny air bubbles in the rock. One percent of the air we breathe is argon. Argon from air bubbles may need to be allowed for if it is significant compared with the amount of radiogenic argon. This would most likely be the case in either young rocks that have not had time to produce much radiogenic argon, or in rocks that are not abundant in potassium. One must have a way to determine how much air-argon is in the rock. This is rather easily done because air-argon has a couple of other isotopes, the most abundant of which is argon-36. The ratio of argon-40 to argon-36 in air is well known, at 295. Thus, if one measures argon-36 and argon-40, one can calculate and subtract off the air argon-40 to get an accurate age.

One of the best ways of showing that an age-date is correct is to confirm it with one or more different dating technique(s). Although potassium-argon is one of the simplest dating methods, there are still some cases where it does not agree with other methods. When this does happen, it is usually because the gas within bubbles in the rock is from deep underground rather than from the air. This gas can have a higher concentration of argon-40 escaping from the melting of older rocks. This is called parentless argon-40 because its parent potassium is not in the rock being dated, and is also not from the air. In these slightly unusual cases, the date given by the normal potassium-argon method is too old. However, scientists in the mid-1960s came up with a way around this problem, the argon-argon method.

Argon-Argon. Although it has been around for over a quarter of a century, the argon-argon method is seldom discussed by groups critical of dating methods. This method uses exactly the same parent and daughter isotopes as the potassium-argon method. In effect, it is a different way of telling time from the same clock. Instead of simply comparing the total potassium with the non-air argon in the rock, this method can tell exactly what and how much argon is directly related to the potassium in the rock.

In the argon-argon method, the rock is placed near the center of a nuclear reactor for a number of hours. A nuclear reactor emits a very large number of neutrons, which can change a small amount of the potassium-39 into argon-39. Argon-39 is not found in nature because it has a 269-year half-life (the shortness of this half-life doesn't affect the argon-argon dating method as long as the measurements are made within about five years of the neutron dose). The rock is then heated in a furnace to release both the argon-40 and the argon-39 (representing the potassium) for analysis. The heating is done at incrementally higher temperatures and at each step the ratio of argon-40 to argon-39 is measured. If the argon-40 is from decay of potassium within the rock, it will come out at the same temperatures as the potassium-derived argon-39 and in a constant proportion. If there is some excess argon-40 in the rock, it will cause a different ratio of argon 40 to argon-39.

Figure 2 is an example of a good argon-argon date. The fact that this plot is flat shows that essentially all of the argon-40 are from decay of potassium within the rock. The potassium-40 in the sample is found by multiplying the argon-39 by a factor based on the neutron exposure in the reactor. When this is done, the plateau in the figure represents an age date based on the decay of potassium-40 to argon-40.

Figure 2. A typical argon-argon dating plot. Each small rectangle represents the apparent age given at one particular heating-step temperature. The top and bottom parts of the rectangles represent upper and lower limits for that particular determination. The x-axis gives the amount of the total argon-39 released from the sample. A good argon-argon age determination will have many heating steps which all agree with each other. The "plateau age" is the age given by the average of most of the steps; in this case it is 136.4 million years. The sample is a Parana continental flood basalt from Brazil. From S. Turner, et al., Earth and Planetary Science Letters 121 (1994): 333-348.


There are occasions when the argon-argon dating method does not give an age even if there is sufficient potassium in the sample and the rock was old enough to date. This most often occurs if the rock experienced a high temperature (usually a thousand degrees Fahrenheit or more) at some point since its formation. If that occurs, some of the argon gas moves around, and the analysis does not give a smooth plateau across the extraction temperature steps. An example of an argon-argon analysis which did not yield an age date is shown in Figure 3. Notice that there is no good plateau in this plot. Sometimes, there will actually be two plateaus, one representing the formation age, and another representing the time at which the heating episode occurred. Usually where the system has been disturbed, there simply is no date given. The important point to note is that, rather than giving wrong age dates, this system simply does not give a date if the system has been disturbed. This is also true of several other igneous rock dating methods, as we will describe below.

Figure 3. An argon-argon plot which gives no date. Note that the apparent age is different for each temperature step so that there is no plateau. This sample was struck with a pressure of 420,000 atmospheres to simulate a meteorite impact--an extremely rare event. The impact heated the rock and caused its argon to be rearranged, so that it could not give an argon argon date. Before it was smashed, the rock gave an age of around 450 million years, as shown by the dotted line. From A. Deutsch and U. Schaerer, Meteoritics 29 (1994): 301-322.


Rubidium-Strontium. In nearly all of the dating techniques, except potassium-argon and the associated argon-argon method, there is always some amount of the daughter product already in the rock when it cools. Using these methods is a little like trying to tell time from an hourglass that was turned over before all of the sand had fallen to the bottom. One can think of ways to correct for this in an hourglass: One could make a mark on the outside of the glass where the sand level started from and then repeat the interval with a stopwatch in the other hand to calibrate it. Or if one is clever she or he could examine the hourglass's shape and determine what fraction of all the sand was at the top to start with. By knowing how long it takes all of the sand to fall, one could determine how long the time interval was. Similarly, there are good ways to tell quite precisely how much of the daughter product was already in the rock when it cooled and hardened.

In the rubidium-strontium method, rubidium-87 decays with a half-life of 48.8 billion years to strontium 87. Strontium has several other isotopes that are stable and do not decay. The ratio of strontium-87 to another stable isotope, say strontium-86, increases over time as more rubidium-87 turns to strontium-87. But when the rock first cools, all parts of the rock have the same (strontium-87/strontium-86) ratio because the isotopes were mixed in the magma. At the same time, some minerals in the rock have a higher rubidium/strontium ratio than others. Rubidium has a larger atomic diameter than strontium, so that rubidium does not fit into the crystal structure of some minerals as well as it does in others.

Figure 4 is an important type of plot used in rubidium-strontium dating. It shows the strontium 87/strontium-86 ratio on the vertical axis and the rubidium-87/strontium-86 ratio on the horizontal axis, that is, it plots a ratio of the daughter isotope against a ratio of the parent isotope. At first, all the minerals lie along a horizontal line of constant strontium-87/strontium-86 but with varying rubidium/strontium. As the rock starts to age, rubidium gets converted to strontium. The amount of strontium added to each mineral is proportional to the amount of rubidium present. This change is shown by the dashed arrows, the lengths of which are proportional to the rubidium/strontium ratio. The dashed arrows are slanted because the rubidium/strontium ratio is decreasing in proportion to the increase in strontium-87/strontium-86. The solid line drawn through the samples will thus progressively rotate from the horizontal to steeper and steeper slopes.

Figure 4. A rubidium-strontium three-isotope plot. When a rock cools, all its minerals have the same ratio of strontium 87 to strontium-86, though they have varying amounts of rubidium. As the rock ages, the rubidium decreases by changing to strontium-87, as shown by the dotted lines. Minerals with more rubidium gain more strontium-87, while those with less rubidium do not change as much. At any given time the composition can be represented as a line through the points. Notice that the line can be extrapolated back to zero to give the initial strontium-87/strontium-86 ratio. The age of the rock can be determined from the slope of the line.


A line drawn through the samples at any later time will intersect the horizontal line at the same point in the lower left-hand corner. This point, where rubidium-87/strontium-86 equals zero, tells the original strontium-87/strontium-86 ratio. From that we can determine the original daughter strontium-87 in each mineral, which is just what we need to know to determine the correct age.

It also turns out that the slope of the line is proportional to the age of the rock: the older the rock, the steeper the line. If the slope of the line is m and the half-life is t1/2, the age t (in years) is given by the equation

For a system with a very long half-life like rubidium-strontium, the actual numerical value of the slope will always be quite small. To give an example for the above equation, if the slope of a line in a plot similar to Fig. 4 is m = 0.05110 (strontium isotope ratios are usually measured very accurately--to about one part in ten thousand), we can substitute in the half-life (48.8 billion years) and solve as follows:

t = (48.8) x ln(1.05110)/ln(2)
so t = 3.51 billion years.


There are several things that, on rare occasions, can cause problems for the rubidium-strontium dating method. One possible source of problems is if a rock contains some minerals that are older than the main part of the rock. Sometimes magma inside the earth will pick up unmelted minerals from the surrounding rock as it moves through a magma chamber. Usually a good geologist can distinguish these "xenoliths" from the younger minerals all around them. If he or she does happen to use them for dating the rock, the points represented by these minerals will lie off the line made by the rest of the points. Other difficulties arise if a rock has undergone metamorphism, that is, if the rock got very hot, but not hot enough to completely remelt the rock. In these cases, the dates look confused, and do not lie along a line. Some minerals may have completely melted, while others did not melt at all, so that some minerals try to give the igneous age while other minerals try to give the metamorphic age. In these cases there will not be a straight line, and no date is determined.

In a few, very rare instances, the rubidium-strontium method has given straight lines that give wrong ages. This can happen when the rock being dated was formed from magma that was not well mixed, and which had two distinct batches of rubidium and strontium. One magma batch had rubidium and strontium compositions near the upper end of a line (such as in Fig. 4), and one batch had compositions near the lower end of the line. In this case, the minerals all got a mixture of these two batches, and their resulting composition ended near a line between the two batches. This is called a two-component mixing line. It is a very rare occurrence in these dating mechanisms, but at least thirty cases have been documented among the tens of thousands of rubidium-strontium dates made. If a two-component mixture is suspected, a second dating method must confirm or disprove the rubidium-strontium date. The agreement of several dating methods is the best fail-safe way of dating rocks.

The Samarium-Neodymium, Lutetium-Hafnium, and Rhenium-Osmium Methods. All these methods work very similarly to the rubidium-strontium method. They all use three-isotope diagrams similar to Figure 4 to determine the age. The samarium-neodymium method is the most often-used of these three. It uses the decay of samarium-147 to neodymium-143, which has a half life of 105 billion years. The ratio of the daughter isotope, neodymium-143, is plotted against the ratio of the parent, samarium-147, with neodymium-144 in the denominators. If different minerals from the same rock plot along a line, the slope is determined, and the age is given by the same equation as above. The samarium-neodymium method may be preferred for rocks that have very little potassium and rubidium, which would make dating by the potassium-argon, argon-argon, and rubidium-strontium methods difficult. The samarium-neodymium method has also been shown to be more resistant to being disturbed by metamorphic heating events, so for some metamorphosed rocks the samarium-neodymium method is preferred. For a rock of the same age, the slope on the neodymium-samarium plots will be less than on a rubidium-strontium plot because the half-life is longer. However, these isotope ratios are usually measured to extreme accuracy--several parts in ten thousand--so that accurate dates can be obtained even for ages less than one fiftieth of a half-life, and correspondingly small slopes.

The lutetium-hafnium method uses the 38 billion year half-life of lutetium-176 decaying to hafnium-176. This dating system is similar in many ways to samarium-neodymium, as the elements tend to be concentrated in the same types of minerals. Since samarium-neodymium dating is somewhat easier, the lutetium-hafnium has not been used in very many cases.

The rhenium-osmium method takes advantage of the fact that the osmium concentration in most rocks and minerals is very low, so that a small amount of the parent rhenium-187 can produce a significant change in the osmium isotope ratio. The half-life for this radioactive decay is 42 billion years. The nonradiogenic stable isotopes, osmium-186 or -188, are used as the denominator in the ratios on the three-isotope plots. This method has been useful for dating iron meteorites, and is now enjoying greater use for dating earth rocks due to development of easier rhenium and osmium isotope measurement techniques.

Uranium-Lead and related techniques. The uranium-lead method is the longest-used dating method. It was first used in 1907, some ninety years ago. The uranium-lead system is more complicated than other parent-daughter systems; it is actually several dating methods put together. Natural uranium consists primarily of two isotopes, U-235 and U-238, and these isotopes decay with different half-lives to produce lead-207 and lead-206, respectively. In addition, lead-208 is produced by thorium-232. Only one isotope of lead, lead-204, is not radiogenic. The uranium-lead system has an interesting complication: none of the lead isotopes is produced directly from uranium and thorium. Each decays through a series of short-lived radioactive elements that are produced and almost immediately decay to a lighter element, finally ending at lead. Since these half-lives are so short compared to uranium and thorium, they do not affect the overall dating scheme. The result is that one can obtain three independent estimates of the age of a rock by measuring the lead isotopes and their parent isotopes, uranium-235 and -238 and thorium-232.

The uranium-lead system in its simpler forms has proved to be less reliable than many of the other dating systems. This is because both uranium and lead are less easily retained in many minerals in which they are found. The intermediate elements in the decay chain do not help matters either. Yet the fact that there are three dating systems all in one allows scientists to easily determine whether the system has been disturbed or not. Using slightly more complicated mathematics, different combinations of the lead isotopes and parent isotopes can be plotted in a way that minimizes the effects of lead loss. One of these techniques is called the lead-lead technique because it determines the ages from the lead isotopes alone. Some of these techniques allow scientists to chart at what points in time metamorphic heating events have occurred. This information is also of significant interest to geologists.

Cosmogenic Radionuclides: Carbon-14,
Beryllium-10, Chlorine-36


The last three radiometric systems listed in Table 1 have far shorter half-lives than all the rest. Unlike the other radioactive isotopes, carbon-14, beryllium-10, and chlorine-36 are constantly being replenished in small amounts by a special mechanism. Cosmic rays--high energy particles and photons in space--produce these isotopes in air, very high in the earth's atmosphere. Very small amounts of each of these isotopes are present in the air we breathe and the water we drink. As a result, living things, both plants and animals, ingest very small amounts of carbon-14, and lake and sea sediments take up small amounts of beryllium-10 and chlorine-36.

The cosmogenic dating clocks work somewhat differently than the others. Carbon-14 in particular is used to date organic material such as bones, wood, cloth, paper, and other dead tissue from either plants or animals. To a rough approximation, the ratio of carbon-14 to the stable isotopes, carbon-12 and carbon-13, is more or less constant in the atmosphere and living organisms. Once a living thing dies, it no longer takes in carbon from food or air, and the amount of carbon-14 starts to drop with time. How far the (carbon-14/carbon-12) ratio has dropped indicates how old the sample is. Since the half-life of carbon-14 is less than 6,000 years, it can only be used for dating material less than about 40,000 years old. Dinosaur bones do not have carbon-14 (unless contaminated), as the dinosaurs became extinct over 60 million years ago, but some other animals that are now extinct, such as North American mammoths, can be dated by carbon-14. Also, some materials from prehistoric times, as well as biblical events, can be dated by carbon-14.

The carbon-14 system has been carefully calibrated with nonradiometric age indicators. For example growth rings in trees, if counted carefully, are a reliable way to determine the age of a tree. Each growth ring only collects carbon from the air and nutrients during the year it is made. To calibrate carbon-14, one can analyze carbon from several center rings of a tree, and then count the rings inward from the living portion to determine the actual age. This has been done for the "Methuselah of trees," the bristlecone pines, which grow very slowly and live up to 6,000 years. Scientists have extended this calibration even further. These trees grow in a very dry region near the California-Nevada border. Dead trees in this dry climate take many thousands of years to decay. Growth ring patterns based on wet and dry years can be correlated between living and long dead trees, extending the ring count back to about 10,000 years ago.

There are other ways of extending farther back in time. One of the best known is the seasonal variations in oxygen isotopes in polar ice from Greenland and Antarctica. Because winter ice has a greater concentration of the lighter isotope, oxygen-16, each winter's deposit makes an invisible layer in the ice, something like a tree ring, which is detectable by isotope analysis. This record goes back about 100,000 years.

While the ice cores are not so easy to compare with carbon 14 dates, the tree rings are very easy to use since trees contain a lot of carbon. The comparison of radiocarbon ages of tree rings with their known ages has revealed variations of up to 12% between the predicted and actual ages. The variations occur because the source of carbon-14 at the top of the atmosphere is slightly variable. This phenomenon is quite well understood, and affects only the clocks that rely on cosmic ray production (e.g., carbon-14, beryllium-10, and chlorine-36). Carbon-14 dates of less than about 10,000 years are corrected for this effect, and there is no reason to believe that older radiocarbon dates are altered to any larger degree. In summary, thousands of radiocarbon dates are obtained each year on organic matter less than about forty thousand years old. It has proved to be a reliable clock for such time scales. A research journal by the name Radiocarbon, published since 1959, is devoted exclusively to this study and is available in many geology and archeology libraries around the country.

The Age of the Earth


Some of the oldest rocks on earth are found in Western Greenland. Because of their great age, they have been especially well studied. The table below gives the ages, in billions of years, from twelve different studies using five different techniques on one particular rock formation in Western Greenland, the Amitsoq gneisses.
Technique Age Range (billion years)
uranium-lead 3.600.05
lead-lead 3.560.10
lead-lead 3.740.12
lead-lead 3.620.13
rubidium-strontium 3.640.06
rubidium-strontium 3.620.14
rubidium-strontium 3.670.09
rubidium-strontium 3.660.10
rubidium-strontium 3.610.22
rubidium-strontium 3.560.14
lutetium-hafnium 3.550.22
samarium-neodymium 3.560.20
(compiled from Dalrymple, 1991)
Note that scientists give their results with a stated uncertainty. They take into account all the possible errors and give a range within which they are 95% sure that the actual value lies. The top number, 3.600.05, refers to the range from 3.55 to 3.65 billion years. The size of this range is every bit as important as the actual number. One number with a small uncertainty range is more accurate than a number with a larger range. For the numbers given above, one can see that all of the ranges overlap and agree between 3.62 and 3.65 billion years as the age of the rock. Several studies also showed that, because of the great ages of these rocks, they have been through several mild metamorphic heating events that disturbed the ages given by potassium-bearing minerals.
We now turn our attention to what the dating systems tell us about the age of the earth. The most obvious constraint is the age of the oldest rocks. These have been dated at up to about four billion years. But actually only a very small portion of the earth's rocks are that old. From satellite data we know that the earth's surface is constantly rearranging itself little by little as earthquakes occur. Such rearranging cannot occur without some of the earth's surface disappearing under other parts of the earth's surface, remelting some of the rock. So it appears that none of the rocks have survived from the creation of the earth without undergoing remelting, metamorphism, or erosion, and all we can say--from this line of evidence--is that the earth appears to be at least as old as the four billion-year-old rocks.

When scientists began systematically dating meteorites, they learned a very interesting thing: nearly all of the meteorites had practically identical ages, at 4.56 billion years. These meteorites are chips off the asteroids. When the asteroids were formed in space, they cooled relatively quickly (some of them may never have gotten very warm), so all of their rocks were formed within a few million years. The asteroids' rocks have not been remelted ever since, so the ages have generally not been disturbed. Meteorites which show evidence of being from the largest asteroids have slightly younger ages. The moon is larger than the largest asteroid. The oldest rocks we have from the moon do not exceed 4.1 billion years, though a larger sampling might yield some slightly older ages. Most scientists think that all the bodies in the solar system were created about the same time. There is evidence from the uranium, thorium, and lead isotopes that links the earth's age with that of the meteorites. This would make the earth about 4.5-4.6 billion years old.

Notice one other important detail about radioactive isotopes. Most of the naturally-occurring radioactive isotopes mentioned above have very long half-lives, on the order of billions of years. The only ones with shorter half-lives are those which have a source constantly replenishing them, such as the carbon-14, beryllium-10 and chlorine-36 produced by cosmic rays. We can make hundreds of other radioactive isotopes with half-lives shorter than a billion years, but they do not occur naturally on earth. Occasionally there is evidence that these isotopes existed at some point in the past, but have since decayed completely away. The longest half-lives of this group of "extinct" radionuclides are close to a hundred million years. Why do we find almost no short-lived radionuclides and so many long-lived ones? This is what one would expect to find if God had created the earth approximately four and a half billion years ago. The earth is old enough that radioactive isotopes with half-lives of up to a hundred million years decayed away, but not so old that radioactive isotopes with half-lives close to a billion years are gone.

Can We Really Believe the Dating Systems?


Some Christians question whether we can believe something so far back in the past. My answer is that it is similar to believing in other things of the past. It only differs in degree. Why do you believe Abraham Lincoln ever lived? Because it would take an extremely elaborate scheme to make up his existence, including forgeries, fake photos, and many other things, and besides, there is no good reason to simply have made him up. Well, the situation is very similar for the dating of rocks, only we have rock records rather than historical records. Consider the following:


The last two points deserve more attention. Some Christians have argued that something may be slowly changing with time so that all the ages look older than they really are. The only two quantities in the exponent of a decay rate equation are the half-life and the time. So for ages to appear longer than actual, all the half-lives would have to be changing in sync with each other. One could consider that time itself was changing if that happened (Remember that our clocks are now standardized to atomic clocks!). And such a thing would have to have occurred without our detection in the last 80 years, which is already 4% of the way back to the time of Christ.

It would not be inconsistent with the scientific evidence to conclude that God made everything relatively recently, but with the appearance of great age, just as Genesis 1 and 2 tell of God making Adam as a fully grown man (which implies the appearance of age). That is a philosophical and theological matter which we won't go into here, though it has some shades of the Abraham Lincoln example. We only note here that an apparent old earth is consistent with the great amount of scientific evidence.

Doubters Still Try


Some doubters have tried to dismiss geologic dating with a sleight of hand by saying that no rocks are completely closed systems (that is, that no rocks are so isolated from their surroundings that they have not lost or gained some of the isotopes used for dating). Speaking from an extreme technical viewpoint this might be true--perhaps one atom out of one trillion of a certain isotope has leaked out of nearly all rocks, but such a change would make an unmeasurably small change in the result. The real question to ask is: "Is the rock sufficiently close to a closed system that the results will be the same as a really closed system?" Since the early 1960s many books have been written on this subject. These books detail experiments showing, for a given dating system, which minerals work all of the time, which minerals work under some certain conditions, and which minerals are likely to lose atoms and give incorrect results. Understanding these conditions is part of the science of geology. Geologists are careful to use the most reliable methods whenever possible, and as discussed above, to test for agreement between different methods.

Some people have tried to defend a young earth position by saying that the half-lives of radionuclides can in fact be changed, and that this can be done by certain little-understood particles such as neutrinos, muons, or cosmic rays. This is stretching it. While certain particles can cause nuclear changes, they do not change the half-lives. The nuclear changes are well understood and are nearly always very minor in rocks. In fact the main nuclear changes in rocks are the very radioactive decays we are talking about.

There are only three quite technical instances where a half-life changes, and these do not affect the dating techniques we have discussed.

1. According to theory, a certain type decay called electron-capture decay is most likely to show changes with pressure or chemical combination, and that should be most pronounced for very light elements. The synthetic isotope, beryllium-7, has indeed been shown by several people to change by up to a fraction of a percent. In one such experiment, beryllium-7 was subjected to 270,000 atmospheres of pressure, which would only occur at depths greater than 450 miles inside the earth. All known rocks, with the possible exception of diamonds, are from much shallower depths. In fact, beryllium-7 is not used for dating rocks, as it has a half-life of only 54 days, and heavier nuclei are even less subject to these minute changes, so that the dates of rocks made by electron-capture decays would only be off by at most a few hundredths of a percent.

2. Cosmic rays are very, very high-energy atomic nuclei flying through space. The electron-capture decay mentioned above does not take place in cosmic rays until they slow down. This is because the fast moving cosmic rays do not have electrons. All normal matter, such as everything on earth, the moon, meteorites, etc. always has electrons, so this instance does not affect things we date.

3. The last case also involves very fast-moving matter. It has been demonstrated by atomic clocks in rapid very fast spacecraft. These atomic clocks slow down very slightly (only a second or so per year) as predicted by Einstein's theory of relativity. No rocks in our solar system are going fast enough to make a noticeable change in their dates.

These cases are very specialized, and all are well understood. None of these cases alter the dates of rocks either on earth or other places in the solar system. The conclusion again is that half-lives are completely reliable in every context, including the dating of rocks on earth and even on other planets.

Rightly Handling the Word of Truth


As Christians it is very important that we understand God's Word correctly. Yet from the middle ages until the 1700s, people insisted that the Bible taught that the earth, not the sun, was the center of the solar system. It wasn't that people just thought it had to be that way; they actually quoted Scriptures: "The earth is firmly fixed; it shall not be moved" (Psalm 104:5), or "the sun stood still" (Joshua 10:13; why should it say the sun stood still if it is the earth's rotation that causes day and night?), and many other passages. I am afraid the debate over the age of the earth has many similarities. But I am optimistic. Today there are many Christians who accept the reliability of geologic dating, but do not compromise the spiritual and historical inerrancy of God's word. While a full discussion of Genesis 1 is not given here, references are given below to books which deal with that issue.

As scientists, we deal daily with what God has revealed about himself through the created universe. The psalmist marveled at how God, Creator of the universe, could care about humankind: "When I consider Your heavens, the work of Your fingers, the moon and the stars, which You have set in place, what is man that You are mindful of him, the son of man that You care for him?" (Psalm 8:3-4). Near the beginning of the twenty-first century we can marvel even more, knowing how vast the universe is, how ancient the rocks and hills are, and how carefully our environment has been designed. Truly God is more awesome than we can imagine!

APPENDIX
Common Misconceptions Regarding
Radiometric Dating Techniques


There are several misconceptions that seem especially prevalent among Christians. Most of these topics are covered in the above discussion, but they are reviewed briefly here for clarity.

1. Radiometric dating is based on index fossils.

This is not at all true, though it has actually been suggested. Radiometric dating is based on the half-lives of the radioactive isotopes. These half-lives have been measured over the last 40-80 years. They are not calibrated at all by fossils.

2. The decay rates are poorly known, so the dates are inaccurate.

Most of the decay rates used for dating rocks are known to within 2 percent. Uncertainties are only slightly higher on rhenium (5%), lutetium (3%), and beryllium (3%). Such small uncertainties are no reason to dismiss radiometric dating. Whether a rock is 100 million years or 102 million years old does not make a great deal of difference.

3. A small error in the half-lives leads to a very large error in the date.

Since exponents are used in the dating equations, it is possible for people to think this might be true, but it is not. If a half-life is off by 2%, it will only lead to a 2% error in the date.

4. Decay rates can be affected by the physical surroundings.

This is not true in dating rocks. Radioactive atoms used for dating have been subjected to heat, cold, pressure, vacuum, acceleration, and strong chemical reactions without any measurable change. The only exceptions, which are not relevant to dating rocks, are discussed under the section, "Doubters Still Try," above.

5. No one has measured the decay rates directly; we only know them from inference.

Decay rates have been directly measured over the last 50-80 years. In some cases a batch of the pure parent material is weighed and set aside for a long time, and then the resulting daughter material is weighed. In many cases it is easier to detect radioactive decays by the energy burst each decay gives off. For this, a batch of the pure parent material is carefully weighed and then put in front of a Geiger counter which counts the number of decays over a long time.

6. The decay rates might be slowing down over time, leading to incorrect old dates.

While we cannot rule out that this could possibly have happened in the past, there is no evidence that anything of the sort has happened in the past century. And the following argument makes this suggestion meaningless in terms of apparent ages: Since the different dating methods are in good agreement, all of the half-lives must have slowed down the same amount together. Such an occurrence would be the same as if time itself slowed down. But everything still appears very old, so why complicate things by making this suggestion in the first place?

7. There is little or no way to tell how much of the decay product was originally in the rock, leading to anomalously old ages.

A good part of this work is devoted to explaining how one can tell how much of a given element or isotope was originally present. Usually it involves using more than one sample from a given rock. By comparing the ratios of parent and daughter isotopes relative to a stable isotope for samples with different relative amounts of the parent isotope, one can determine how much of the daughter isotope would be present if there had been no parent isotope. This is the same as the initial amount (it would not change if there was no parent isotope to decay). Figure 4 and the accompanying explanation tell how this is done most of the time. While this is not absolutely 100% foolproof, comparison of several dating methods will always show whether the given date is reliable.

8. There are only a few different dating techniques.

We have listed eleven different radiometric dating techniques and discussed them. These make up only the tip of the iceberg. There are over forty different radiometric dating techniques in use, and there are many other dating techniques making use of rare stable isotopes, yearly variations such as tree rings and ice cores, and other reliable methods.

9. "Radiation halos" in rocks prove that the earth was young.

This refers to tiny halos of crystal damage surrounding spots where radioactive elements are concentrated in certain rocks. Halos thought to be from polonium, a short-lived element produced from the decay of uranium, have been found in some rocks. A plausible explanation for a halo from such a short-lived element is that these were not produced by an initial concentration of the radioactive element. Rather, as water seeped through cracks in the minerals, a chemical change caused newly-formed polonium to drop out of solution at a certain place and almost immediately decay there. A halo would build up over a long period, although the center of the halo never contained more than a few atoms of polonium at one time. "Hydrothermal" effects can act in ways that at first seem strange, such as the well-known fact that gold--a chemically unreactive metal with very low solubilities--is concentrated along quartz veins by the action of water over long periods of time. Other researchers have found halos produced by an indirect radioactive decay effect called hole diffusion, which is an electrical effect in a crystal. These results suggest that the halos in question are not from short-lived isotopes after all.

At any rate, halos from uranium inclusions are far more common. Because of uranium's long half-lives, these halos take at least several hundred million years to form. Because of this, most people agree that halos provide compelling evidence for a very old earth.

10. Only atheists and liberals are involved in radiometric dating.

The fact is that there are many Bible-believing Christians who are involved in radiometric dating, and who can see its validity firsthand. Most of the members of the Affiliation of Christian Geologists are firmly convinced that radiometric dating shows evidence that God created the earth billions, not thousands, of years ago.

11. Different dating techniques usually give conflicting results.

This is not true at all. The fact that dating techniques most often agree with each other is why scientists tend to trust them in the first place. Nearly every college and university library in the country has periodicals such as Science, Nature, and specific geology journals which give the results of dating studies. The public is usually welcome to (and should!) browse in these libraries. So the results are not hidden; people can go look at the results for themselves. In 1994 alone, at least 450 research articles were published, essentially all favoring a very old earth. Besides the scientific periodicals which carry up-to-date research reports, specific suggestions are given below of books for further reading.

Further Reading

Van Till Howard J., Young Davis A., and Menninga, Clarence. (1988) Science Held Hostage. InterVarsity, Downers Grove, IL, 189 pp.

This book talks about the misuse of science by both hard-line atheists and by young-earth creationists. A good deal of the book is devoted to refuting young-earth arguments, including a substantial section on the Grand Canyon geology. Its authors are well-known Christians in Geology and Physics.

Ross, Hugh. (1994) Creation and Time: A Biblical and Scientific Perspective on the Creation-Date Controversy. NavPress, Colorado Springs, CO.

Hugh Ross has a Ph.D. in Astronomy and he defends modern science and an old age for the universe, refutes common young-earth arguments, and firmly believes in the inerrancy of the Bible.

Young, Davis A. (1982) Christianity and the Age of the Earth. Zondervan, Grand Rapids, MI (now available through Artisan Sales, Thousand Oaks, CA).

Davis Young has a Ph.D. in Geology and teaches at Calvin College. He argues for an old earth and refutes many of the common young-earth claims (including their objections to radiometric dating).

Strahler, Arthur N. (1987) Science and Earth History: The Evolution/Creation Controversy. Prometheus Books, Buffalo, pp. 129-138.

This book was not written by a Christian, but is included here because it is a very thorough and comprehensive refutation of young-earth evidences. The only negative aspect is that at one point Strahler throws in a bit of his own theology--his arguments against the need for a God. This book is long and in small print; it covers a wealth of information.

Wiester, John. (1983) The Genesis Connection. Interdisciplinary Biblical Research Institute, Hatfield, PA, 254 pp.

John Wiester has taught Geology at Westmont and Biola University, and is active in the American Scientific Affiliation, an organization of scientists who are Christians. This book discusses many scientific discoveries relating to the age of the earth and how these fit into the context of Genesis 1.

Stoner, Don. (1992) A New Look at an Old Earth. Schroeder, Paramount, CA, 191 pp.

A persuasive book written for the Christian layman. Stoner uses arguments both from the theological and the scientific side. He talks somewhat philosophically about whether God deceives us with the Genesis account if the earth is really old. Stoner also tries to discuss the meaning of the Genesis 1 text.

Dickin, Alan. (1995) Radiometric Isotope Geology. Cambridge University Press, 452 pp.

This is the most up-to-date and comprehensive textbook on radiometric dating techniques.


The following books are popular college-level Geology texts that deal in depth with various dating techniques. Geologic Time is very easy to read and has been around for some time. The text by Dalrymple is meant to be relatively easy to read, but is also very comprehensive. The Faure texts are regular textbooks for Geology, including more mathematics and more details.

Dalrymple, G. Brent. (1991) The Age of the Earth. Stanford University Press, 474 pp.

Eicher, Don L. (1976) Geologic Time, 2nd edition. Prentice-Hall, Englewood Cliffs, NJ, 150 pp.
Faure, Gunter. (1986) Principles of Isotope Geology, 2nd edition. Wiley, New York, 464 pp.

Faure, Gunter. (1991) Principles and Applications of Inorganic Geochemistry: A Comprehensive Textbook for Geology Students. MacMillan Pub. Co., New York, 626 pp.

Electronic media: There is a site on the world wide web called Talk.Origins at "http://rumba.ics.uci.edu:8080/". Information on radiometric dating can be found under several FAQ (frequently asked questions) subheadings. The information can also be accessed by anonymous ftp to "ics.uci.edu:/pub/origins".

American Scientific Affiliation
http://ursa.calvin.edu/chemistry/ASA/

Affiliation of Christian Geologists
http://www.ksu.edu/~kbmill/acghome.html

Science & Christianity Resource List
http://hercules.geology.uiuc.edu/~Schimmri/christianity/scichr

Reasons to Believe
http://www.reasons.org/reasons/index.html


Acknowledgments: Several members of the Affiliation of Christian Geologists and other Christians involved in the sciences reviewed this paper and/or made contributions. The following people are sincerely thanked for their contributions: Drs. Jeffery Greenberg and Stephen MosHier (Wheaton College), John Wiester (Westmont College), Dr. Davis Young (Calvin College), Dr. Elaine Kennedy (Loma Linda University), Steven Schimmrich (U. of Illinois), Kenneth VanDellen (Macomb Community College), Dr. Guillermo Gonzalez (U. Texas, Austin), Ronald Kneusel, and James Gruetzner (U. New Mexico).


About the author: Dr. Wiens received a bachelor's degree in Physics at Wheaton College and his Ph.D. at the University of Minnesota doing research on meteorites and moon rocks. He spent two years at Scripps Institution of Oceanography (La Jolla, CA) where he studied isotopes of helium, neon, argon, and nitrogen in terrestrial rocks. He is presently a staff scientist in the Geological and Planetary Sciences Division at Caltech, where he is continuing to study meteorites and is involved in a space mission to return to earth a sample of solar wind. He has published over a dozen scientific research papers and has also published articles in Christian magazines. Dr. Wiens became a Christian at a young age, and has been a member of Mennonite Brethren, General Conference Baptist, and Conservative Congregational churches.

GLOSSARY

Atom The smallest unit that materials can be divided into. An atom is about ten billionths of an inch in diameter and consists of a nucleus of nucleons (protons and neutrons) surrounded by electrons.
Closed system A system (rock, planet, etc.) which has no influence or exchange with the outside world. In reality there is always some exchange or influence, but if this amount is completely insignificant for the process under consideration (e.g., for dating, if the loss or gain of atoms is insignificant) for practical purposes the system can be considered closed.
Cosmic rays Very high energy particles which fly through space. They are stopped by the earth's atmosphere, but in the process, they constantly produce carbon-14, beryllium-10, chlorine-36, and a few other radioactive isotopes in small quantities.
Cosmogenic Produced by bombardment of cosmic rays. Carbon-14 is said to be cosmogenic because it is produced by cosmic rays.
Daughter The element or isotope which is produced by radioactive decay.
Decay The change from one element or isotope to another. Only certain isotopes decay. The rest are said to be stable.
Electron-capture decay The only type of radioactive decay that requires the presence of something--an electron--outside the atom's nucleus. Electron capture decay of light atoms--those having the fewest electrons--can be very slightly affected by extremely high pressures or certain chemical bonds, so as to change their half-lives by a fraction of a percent. But no change in the half-lives of elements used for radiometric dating has ever been verified.
Element A substance that has a certain number of protons in the nucleus and unique properties. Elements may be further broken down into isotopes, which have nearly all of the same properties except for their mass and their radioactive decay characteristics.
Half-life The amount of time it takes for half the atoms of a radioactive isotope to decay.
Ice core Long sections of ice brought up by special drilling rigs on the ice sheets of Greenland and Antarctica.
Igneous rock Rock formed from molten lava. The other two types of rock are sedimentary, formed by the cementing together of soil or sand, and metamorphic rocks reformed by heat over long periods of time.
Isotope Atoms of a given element that have the same atomic mass. Most elements have more than one isotope. Most radioactive elements used for dating have one radioactive isotope and at least one stable isotope. For example carbon-14 (which weighs 14 atomic mass units) is radioactive, while the more common isotopes, carbon-12 and carbon-13 are not.
Lead-lead dating A variation on the uranium-lead technique in which only the isotopes of lead need to be measured.
Magma Hot molten material from which rocks are formed. When magma erupts on the surface of the earth, it is called lava.
Metamorphism The heating of rocks over long time periods at temperatures which are hot enough to change the crystal structure but not hot enough to completely melt the rock. Metamorphism tends to alter or reset the radiometric time clocks, though some radiometric techniques are more resistant to resetting than others.
Nucleons Neutrons and protons, which make up the nucleus of an atom.
Parent The element or isotope which decays. The element it produces is called the daughter.
Radioactive Subject to change from one element to another. During the change, or decay, energy is released either as light or energetic particles.
Radiocarbon Carbon-14, which is used to date dead plant and animal matter. Radiocarbon is not used for dating rocks.
Radiometric dating Determination of a time interval (e.g., the time since formation of a rock) by means of the radioactive decay of its material. Radiometric dating is one subset of the many dating methods used in geology.
Three-isotope plot In dating, this is a plot in which one axis represents the parent isotope and the other axis represents the daughter isotope. Both parent and daughter isotopes are ratioed to a daughter element isotope that is not produced by radioactive decay. This type of plot gives the age independent of the original amounts of the isotopes.
Tree ring A ring visible in the stump or sawed section of a tree which indicates how much it grew in a year. The age of a tree can be determined by counting the growth rings.
Two-component mixing The mixing of two different source materials to produce a rock. On rare occasions this can result in an incorrect age for certain techniques that use three-isotope plots. Two component mixing can be recognized if more than one dating technique is used, or if surrounding rocks are dated.
Xenolith Literally, a foreign chunk of rock within a rock. Some rocks contain pieces of older rocks within them. These pieces were ripped off the magma chamber in which the main rock formed and were incorporated into the rock without melting. Xenoliths do not occur in most rocks, and they are usually recognizable by eye where they do occur. If unrecognized, they can result in an incorrect date for a rock (the date may be of the older xenolith).

FOOTNOTES


1 In only a couple of special cases have any decay rates been observed to vary, and none of these special cases apply to the dating techniques discussed here. These exceptions are discussed later.

2 The term isotope subdivides elements into groups of atoms that have the same atomic weight. For example carbon has isotopes of weight 12, 13, and 14 times the mass of a nucleon, referred to as carbon-12, carbon-13, or carbon-14 (abbreviated as 12C, 13C, 14C). It is only the carbon-14 isotope that is radioactive.

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