What is an atom? What is an isotope?

Lecture ?? Nov. 11^th, 1999

Atoms are the smallest particles in nature, and make up the building blocks of all mass. In ancient Greek philosophy the word atom was used to describe the smallest bit of matter that could be conceived. The Greek word for atom means “not divisible.” Our understanding about these particles grew slowly. The last part of the atom, the neutron, was only discovered in the 1930s by Chadwick. Even more recent advances have been made about sub-atomic particles, which we will not go into.

In the 16^th and 17^th centuries, scientists were beginning to realize that matter was made of more fundamental parts, the elements Salt consisted of sodium and chlorine, water, of hydrogen and oxygen, and air was made-up primarily of nitrogen and oxygen.

John Dalton, a British schoolmaster and chemist, is the founder of molecular theory. Early in the 19th century he made studies of the way in which the various elements combine with one another to form chemical compounds. Dalton made the theory quantitative. He showed how these atoms link together in definite proportions. Subsequent investigations proved that the smallest unit of a chemical substance such as water is a molecule. Each molecule of water consists of a single atom of oxygen and two atoms of hydrogen joined by an electrical force called a “chemical bond.

So we have atoms, the most fundamental ‘indivisible’ form of matter, which combine to form molecules. Elements are specific atoms, with distinctive properties. So things like hydrogen, and oxygen are specific elements. The atoms of the these elements combine to form molecules of water. An atom is so small that a single drop of water contains more than a million million billion atoms.

An atom can be subdivided into smaller units, and this next step is as far as we will go. Atoms consist of three parts. The proton, the electron and the neutron: The proton determines its atomic number and also what ‘kind’ of atom it is. A proton has a positive charge and resides in the nucleus of the atom. For every proton, there is an electron. An electron is a negatively charged particle, and is found in the orbiting ‘cloud’ around the atom. Although its charge is exactly equal to, but opposite from a proton, its mass is more than 1,000 times less than a proton. There is a third particle called a neutron. The neutron has a mass equal to the proton, but has no charge. Neutrons reside in the nucleus of the atom, but do not affect how many electrons the atom will have.

	Charge	Mass
Proton	+1	1.6726 × 10-24 g
Electron	-1	9.109534 × 10-28 g
Neutron	Neutral (0)	1.6749543 × 10-24 g =1.0086654 AMU

The atomic number of an atom is determined by the number of protons. For a particular number of protons, there will be the same number of electrons and that unique number determines which element the atom is. The smallest atom is hydrogen. It consists of one proton and one electron (above figure on right, labeled ‘protium’). The next atom is helium. It contains two protons and two electrons. The chemical properties are determined by the outermost shell of electrons.

In addition to the protons and electrons, of course are the neutrons. They have no effect on the chemical properties of the atom, but they contribute to mass, and in the case of radioactive elements, the decay rates. For hydrogen, there are three different isotopes of this element. Each is still hydrogen, each has the same chemical properties. The difference is that for the first isotope, there are no neutrons in the nucleus, for the second (deuterium) there is one, and for the third (tritium) there are two neutrons.

The atomic mass is equal to the number of protons + neutrons (because the electron essentially has no mass). For hydrogen, there are three isotopes, one with mass 1 (one proton, no neutrons), one with mass 2 (one proton, one neutron) and one with mass 3 (one proton and two neutrons) mass three. For hydrogen, the different isotopes have names: protium, deuterium and tritium.

The name of each element has a symbol. For hydrogen, it is H, for helium it is He, uranium is U. Some are not so obvious, such as iron is Fe (however, consider that iron is ‘fer’ in French). The isotope of a given element is given by a superscripted atomic mass preceeding the element. That is, for hydrogen of mass 1, the notation is ¹H, for mass two, it is ²H and for hydrogen of mass three (one proton, two neutrons) it is ³H. Likewise, for helium, there are two isotopes. Both have 2 protons and 2 electrons, of course. But one has only one neutron (mass 3, or ³He) and the other has two neutrons (mass 4, or ⁴He). Oxygen has 8 protons. Its atomic number is 8. It has three different naturally occurring isotopes: ¹⁶O, ¹⁷O, ¹⁸O.

Radioactivity

Most naturally occurring isotopes are stable. Of the 339 isotopes of the 84 elements found in nature, 269 are stable and 70 are radioactive. (There are also some in our environment that were created by man-made nuclear reactions, but these are not considered in the above 70. There have been more than 1650 isotopes created in the laboratory with nuclear reactors and particle accelerators.

All of the three oxygen isotopes are stable. ¹H is stable, ²H is stable. They never break down. They never decay. But ³H is radioactive – or unstable. It will breakdown to ³He eventually, by having a neutron converted to a proton & a b^- particle, which is an energetic electron.

The parent is ³H (or tritium as it was named by Urey), and the daughter is ³He. (FYI: Heavy water is made of ²H₂O, or deuterium-rich water. Also, deuterium and tritium also have the unique distinction of having their own letters. So instead of ²H, we can write simply D; likewise for T).

Radioactive decay is the process whereby the radioactive element breaks down to the daughter. Often, the daughter is also radioactive, and it too will decay to a second daughter and so forth until the final stable daughter is reached. There are three kinds of radioactive decay that concern us:

1) b decay. This is the most common. The nucleus ejects a b^- particle, an energetic electron. What has happened is the neutron breaks down to form a proton, b^- particle and an antineutrino (forget about this!). The b^- particle and antineutrino are ejected from the atom entirely, and the proton remains in the nucleus.

2) Electron Capture: In this scenario, atoms that are unstable because they have too many protons (or not enough neutrons) alleviate the situation by taking an electron from the innermost shell of electrons and combining it with a proton to make a neutron. The atomic mass is unchanged, but the atomic number decreases instead of increases, as is the case with b^- decay. Not common, especially for the heavy isotopes, which concern us for radiogenic dating methods.

3) Alpha decay (a decay): a decay occurs primarily among heavy elements. The nucleus ejects an a particle, which consists of two protons and two neutrons. In this process, the atomic number is decreased by two (two protons lost) and the atomic mass is decreased by four (two protons + two neutrons). In addition, the daughter is often left with excess internal energy, which is shed as gamma (g) ray emission. The a particle will become a helium atom as it acquires the necessary electrons (two) to charge-balance the nucleus.

Another type of decay that is not of concern to us is spontaneous fission. This is when an atom breaks in half. For example, ²³⁸U can break into one ¹³³Sn (tin) and one ¹⁰³Mo atom. This is a very rare form of radioactive decay in nature.

Decay Rates

Historical Perspective

The equation for radioactive decay was determined empirically by Ernest Rutherford and Frederick Soddy of McGill University in Montreal. They began by isolating the radioactive gas that was emitted by Th compounds and observed the change in its activity (emission rate of gamma rays) as a function of time. After 54.5 seconds, it was half its original value. After 109 seconds it was ¼ the original value and after 163.5 seconds it was 1/8 that of the original sample.

They also proposed that the activity of a substance is proportional to the amount present. This was the basis of the formula of radioactive decay.

Rutherford and Soddy also suggesting that helium is the product of decay of radioactive elements. This was confirmed the following year by Ramsay and Soddy, who showed that He was the product of Ra decay.

In 1905, Ernest Rutherford, then at McGill University, delivered the Silliman lectures at Yale University. In them, he offered the possibility of using radioactive decay as a dating method.

“The helium observed in the radioactive minerals is almost certainly due to its production from the radium and other radioactive substances contained therein. If the rate of production of helium from known weights of the different radioelements were experimentally known, it should thus be possible to determine the interval required for the production of the amount of helium observed in radioactive minerals, or, in other words, to determine the age of the mineral”

In 1905, B.B. Boltwood examined the composition of naturally occurring U and recognized that it always contained lead. The amount of lead seemed proportional to the age, suggesting that it might be the final decay product of uranium.

EQUATIONS GOVERNING DECAY

The mechanisms of radioactive decay are not well understood, but the mathematics are quite straightforward. Radioactive decay is a statistical process, whereby the probability that any one isotope of a given element is exactly the same as for any other. The characteristic probability is called the decay constant l (lambda), whose units are probability per unit time. A stable isotope has a decay constant of zero – that is the probability of decay is zero. A radioactive element that decayed instantaneously would have a decay constant of one.

Imagine that you have 100 atoms of a radioactive element in a jar. The decay constant is 0.1/year (i.e., there is a 10% chance that any given atom will decay in that year). In the first year, 10% should decay, so that out of the 100 original radioactive atoms, there will be 90 left and 10 new daughter atoms. In the next year, the probability is that 10% of the remaining 90 atoms will decay. So after the second year, there will be 81 atoms and so on.

In fact, with such a small amount of atoms, the likelihood that exactly 10% of 100 atoms will decay is low. There may be 7 or there may be 13 atoms that decay in the first year. But as the number of atoms increases in our population, the probability that 10% decay increases exponentially.

Imagine if you flip a coin twice. You should get heads one time and tails the next, because there is a 50% chance that you get heads and 50% chance that you get tails. Obviously, it is wrong to assume that the second time you toss the coin you will get a tails. For every flip, there is an equal probability that you will get a heads as a tails. If you toss the coin 10 times, you may get 7 heads and 3 tails. The next time could be 4:6, etc. But if you toss the coin 100 times, now the statistics will be in your favor. It is very unlikely that you will get 70 heads and 30 tails. More likely, you will get close to 50:50, maybe 58:42, 45:55. If we do it 1,000 times, then the probability of close to 50% heads; 50% tails increases further. And if we do it 10,000 times, then the probability goes up even further.

It doesn’t take much material to have enough atoms for the statistics to ‘work’ properly. If we have 0.00001 grams of potassium, we have 150,000 trillion atoms. That is more than enough for the statistics to work perfectly.

Notice in the above example, that as decay occurs, we are left with less radioactive atoms in our sample. Although for any one atom, the likelihood of decay is still 10%, because we have fewer atoms, there will be fewer decays per year. This decay rate is decreasing exponentially. Growth of the world’s population, or bacteria in a petri dish is also exponential growth.

The equation that governs exponential decay is the following:

, (1)

where P_o is the amount of parent material, P_t is the amount remaining at time t. e is 2.781828. . . .l is the decay constant.

A few rules about exponents and logs:

log(e^x) = x (1)

log(x)+log(y) = log(x + y) and

The half life is given as the time required for the parent to decay to half its original value. It comes straight from equation (1) and is given by

. So in our example, where our l was 0.1 (or 10% decay per year), the halflife is , or 6.93 years. Thus 50% are gone after 6.93 years and 75% is gone after 13.86 years, 87.5% after 20.79 years, etc. Obviously, the parent and daughter have the exact opposite relationships. The total amount of parent and daughter is equal to the total amount that was there in the beginning. That is, P_o = P_t + D_t.

Equation 1 as written is not very useful because we don’t know what the value of P_o is (or was). But, we do know that P_o = P_t + D_t. So we can substitute this for equation 1 and get

2

which becomes

3.

Finally we get by rearranging the above equation

4.

Equation 4 is the basic radiogenic decay equation. The equation is only valid if there is no daughter element present in the mineral at the time the mineral formed. In the case where we have initial daughter mineral, equation 4 is modified to

5,

where D_t is the measured amount of daughter isotope and D_o is the amount of daughter element present at the time the mineral crystallized (at time zero).

Is decay constant?

What about that pesky l? Is it really constant? It should be and experiments support the theoretical basis of constancy. The decay rate cannot have changed over millions or even billions of years. Is this correct? There are two reasons why we think that this assumption is correct.

1) The nucleus of an atom (the part that is going to decay) is extremely small and surrounded by (and insulated by) a cloud of electrons. Chemical reactions involve only the outermost electrons and not the nucleus at all. Likewise, the ‘compressibility’ of an atom is controlled by the electrons and not the nucleus.

2) The energies involved in nuclear reactions are 10⁶ times (1 million) greater than those of chemical reactions and 10⁴ to 10⁵ times greater than the binding energies of electrons to the nucleus. Chemical forces, which bind atoms into molecules are on the order of 1 electron volt (eV), while the forces required to remove an electron are in the range of 10 to 100 eV. In contrast, the forces that hold the nucleus together are on the order of 10⁶ eV. Only nuclear reactors and particle accelerators have enough energy to break apart the nucleus of an atom.

Very soon after radioactivity was discovered, there was an effort to test the ideas that decay is constant. Rutherford placed a sample of radium ‘emanation’ (²³⁰Rn, radon) in a steel-encased cordite bomb. They blew it up and even after temperatures of 2,500°C and pressures greater than 1000 atmospheres, there was no change in the rate of decay.

Marie Currie in 1913 put radium in liquid hydrogen (~20 degrees above absolute zero) and found that the rate of decay was within 0.05% of the same rate as uncooled samples. Gravity was changed by putting samples on the tops of mountains and in the deepest mines or by spinning samples in centrifuges. Huge magnetic fields were applied (more than 10¹⁰ times greater than the Earth’s magnetic field). Consistently, there have been no changes in the alpha and beta rate decays.

Theoretically, it should be possible to change the rates of b-decay by stripping off all of the electrons. This does not happen in Nature. The greatest expected change is on the order of 0.01%. For a decay, the effects are smaller. The largest theoretical decay is on the order of 0.000015 to 0.0000017% for ²²⁶Ra and ¹⁴⁷Sm, respectively. In conclusion, there are no known natural forces that should change the decay rates measurably.

Pricipal Parent and Daughter isotopes and their

half-lives, appropriate to geological dating

Parent Isotope	Daughter Isotope	Half-life millions of years	Decay Constant (yr^-1)
^40K	^40Ar	1,250	5.81 ´ 10^-11
^87Rb	^87Sr	48,800	1.42 ´ 10^-11
^147Sm	^143Nd	106,000	6.54 ´ 10^-12
^176Lu	^176Hf	35,900	1.93 ´ 10^-11
^187Re	^187Os	43,000	1.612 ´ 10^-11
^232Th	^208Pb	14,000	4.948 ´ 10^-11
^235U	^207Pb	704	9.8485 ´ 10^-10
^238U	^206Pb	4,470	1.55125 ´ 10^-10

Early work on dating

Rutherford made some early estimates. Assuming a production rate of 5.2´ 10^-8cm³ of He per year, he studied the mineral fergusonite, containing 7% U per gram of material. His estimate was 497 million years. He wisely cautioned that this was a lower estimate because some of the He might have escaped since the mineral was formed.

R.J. Strutt measured a number of fossilized bones and other sedimentary phases (phosphate nodules), using the He/U ratios to determine age. The agreement was poor, and Strutt suggested that the problem was that He escaped from the crystal structure.

Rutherford suggested that a better alternative would be to measure the decay product lead. It wouldn’t ‘leak’ out of the mineral as He did.

Boltwood put Rutherford’s idea to the test. He reasoned that if Rutherford’s hypothesis were true then the following should be valid:

1) Different types of U-bearing minerals of the same age should have the same U/Pb ratio.

2) The U/Pb ratio should be a function of age. Higher lead contents in older rocks.

Looking at 43 minerals he found that his hypothesis was valid and that Pb did appear to the final decay product of uranium.

Boltwood was ready to go to town. But there was one problem. He didn’t know the decay rate of uranium. It was too slow to measure. He got around this ingeniously. Rutherford had measured the decay rate of radium (2.7 ´ 10^-4). Boltwood recognized that Ra was an intermediate product on the path from U to Pb. He could measure the amount of Ra in a sample compared to the amount of U (3.8 ´ 10^-7). Putting these two together gave him a decay constant for uranium of (2.7 ´ 10^-4) ´ (3.8 ´ 10^-7) = 1 ´ 10^-10/yr.

Then, using Rutherford’s decay equation, he had

age in years = . Boltwood stated

The actual values obtained for these ages are, of course, dependent on the value taken for the rate of disintegration of radium. When the latter has been determined with certainty, the ages as calculated in this manner will receive a greater significance, and may perhaps be of considerable value for determining the actual ages of certain geological formations.

All of these early ages were chemical ages in that they did not know that there were multiple isotopes of the minerals and each had a different decay rate. For example,

²³⁸U à ²⁰⁶Pb half life -- 4,470 million years

²³⁵U à ²⁰⁷Pb half life -- 704 million years.

Their ages tended to be too old. Modern methods use isotopic ages, where actual isotopes are measured in a mass spectrometer.

Examples of radiogenic dating techniques

Two problems in dating are

1) the loss or gain of parent and daughter isotopes by some mechanism other than radioactive decay and

2) the problem of knowing how much initial daughter mineral was in the rock.

Both of these problems are recognized and dealt with in order to insure the highest quality data.

K/Ar

The potassium-Argon method of dating does not suffer from point (2) above. Generally, when mineral containing potassium crystallizes, it has no argon. Argon is an inert gas and so does not fit into the crystal lattice. It is only after the mineral is formed that Ar begins to form from the radioactive decay of potassium. This simplifies the situation.

⁴⁰K is the radioactive isotope of potassium. It undergoes a branched decay, meaning it breaks down to ⁴⁰Ar by electron capture and ⁴⁰Ca by b^- decay. Because ⁴⁰Ca is the most abundant naturally occurring isotope of calcium (97%) and it is found in most minerals already (initial daughter), the decay of ⁴⁰K to ⁴⁰Ca cannot be used for dating. Only the decay of ⁴⁰K to ⁴⁰Ar is applicable. In this case, however, not all of the parent decays to the ‘measured’ daughter. There is a fraction that goes to ⁴⁰Ca. The ⁴⁰Ar/⁴⁰Ca ratio is 0.117, called the branching ratio. It is a very simple modification of the basic dating equation to add a factor for the branched decay. The K/Ar method has another advantage. The half-life of ⁴⁰K is 1.25 billion years. This is long enough that there is still abundant ⁴⁰K in the solar system, but short enough that we can date young rocks of, say 50,000 years. (Dates of several thousand years have recently been successfully made).

There is one further complication to the K/Ar method. There is a certain amount of atmospheric argon that is measured when the sample is heated to drive off the argon. This contributes to the ⁴⁰Ar content of the mineral. Fortunately, there is another isotope of argon, ³⁶Ar which is not produced by any radioactive decay schemes. The ³⁶Ar/⁴⁰Ar ratio of the atmosphere is known, so we can subtract the component of ⁴⁰Ar from the atmosphere from the atmospheric ³⁶Ar/⁴⁰Ar ratio.

K/Ar dating is not applicable to all rocks. No dating method is. K/Ar is particularly applicable to igneous rocks, where crystallization occurs over a specific time interval and no inherited argon is possible. Sedimentary rocks are not applicable in most cases because there are often detrital grains of potassium- (and therefore argon) bearing minerals in the sample. If these can be avoided, it is possible to date clays.

K/Ar may not be applicable to some metamorphic rocks. This is because if a K-bearing mineral generates argon and then at some later time the rock is heated, the argon can leak out, giving a falsely young age (remember: if we have no argon, the mineral is ‘brand new’).

The Rb-Sr method

The Rb-Sr method (rubidium-strontium) is based on the simple b^- decay of ⁸⁷Rb to ⁸⁷Sr. The half life is 48.8 billion years. Although rubidium is rare, it is a trace element in many minerals, substituting for calcium, which is a major rock-former. Unlike argon, the daughter, ⁸⁷Sr is not lost from the rock by later heating. The equation for determining an age is simply

.

Measure the amount of ⁸⁷Rb, the amount of ⁸⁷Sr, and you’ve got an age!

Unfortunately, there is generally lots of strontium in a rock, and not as much rubidium. This poses a problem, in terms of finding out the initial ⁸⁷Sr/⁸⁷Rb ratio. How much initial ⁸⁷Sr was there in the mineral?

The solution is the isochron diagram. It involves normalizing multiple minerals (with different initial Rb-Sr contents) from the same rock to a non-radioactive isotope and plotting it on a diagram. To understand this, first consider if we don’t normalize the data and simply plot ⁸⁷Rb vs ⁸⁷Sr. We have different ⁸⁷Rb and ⁸⁷Sr abundances for the different minerals (figure a) at time t = 0. Those with lots of ⁸⁷Rb will produce more new ⁸⁷Sr and lose more ⁸⁷Rb than those with very little initial Rb (Fig b, time t = t’). (The obvious extremes are those with no initial Rb. There will be no increase in Sr and no decrease in Rb because there is no radiogenic ⁸⁷Rb to begin with. Likewise, if we have pure ⁸⁷Rb, there will be a lot of ⁸⁷Sr produced). The diagram is not very useful. It doesn’t tell us much, because we don’t know what the intial points were. All we can measure is the final points (the black dots in Fig. b) and that doesn’t help.

Now, if we normalize to a non-radiogenic isotope, such as ⁸⁶Sr, the ⁸⁷Sr/⁸⁶Sr of all minerals will be the same at time zero (Fig. c). The ⁸⁷Rb/⁸⁶Sr ratio will vary depending upon the amount of Rb in each mineral. Decay of ⁸⁷Rb to ⁸⁷Sr will simultaneously decrease the ⁸⁷Rb amount and increase the ⁸⁷Sr amount. The ⁸⁷Sr/⁸⁶Sr values will increase and the ⁸⁷Rb/⁸⁶Sr values will decrease, giving a negative slope.

If we start with the decay equation

and divide by ⁸⁶Sr, we get

.

This is the equation of a straight line y = b + mx. The slope of the line is (e^lt – 1) and the intercept is (⁸⁷Sr/⁸⁶Sr). We plot all of these data on a straight line, and the slope gives us the age.