E = mc^2 Decay


 
 
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Introduction

In previous pages in this series we looked at the theoretical aspects of the equation E = mc2. We will now start to look at the practical aspects, starting with radioactive decay. The term "radioactive decay" has negative connotations; we hear about nuclear waste decaying and the harmful radiation released and so on. However, the process is not only a natural one but we are constantly surrounded by material that is radioactively decaying. In fact, some of the material that you and I are made of is radioactively decaying, at least a little. High levels of radioactive decay can indeed be dangerous, but in small amounts it is part of the everyday and natural world around us.

On this page we will look at three kinds of decay; alpha (in which a helium nucleus is released), beta (in which an electron is released) and gamma (in which a photon is released). In doing so I will use examples of real decays, but ignore complications such as neutrino emissions (neutrinos are tiny particles that are sometimes released in radioactive processes, but they are so small that we do not need to consider them here).

 

The Parts of an Atom

Before we can understand the processes involved in radioactive decay we need to understand a little about the various parts of an atom. In a typical atom there are three distinct types of particles:

  • Protons - Located in the nucleus, with a positive electrical charge.
  • Neutrons - Located in the nucleus, with no electrical charge.
  • Electrons - Located in a "cloud" surrounding the nucleus, with a negative electrical charge.

These particles are often shown as something like this:

Model of Atom

In the picture above the protons are red, the neutrons are green and the tracks of the electrons are in blue. Note that this is only a generalised schematic model of an atom and not any particular element.

Although we will return to electrons later in the page we can ignore them for the first two types of decay and concentrate on just the "nucleus", i.e. the protons and neutrons located at the centre of the atom

A key thing to note about atoms is that they are defined by the number of protons each type, or element, has. For example:

  • All hydrogen atoms have one proton
  • All helium atoms have 2 protons 
  • All carbon atoms have 6 protons
  • All nitrogen atoms have 7 protons
  • All uranium atoms have 92 protons

and so on.

However, the number of neutrons in an atom can vary. For example, most of the hydrogen we come across, such as in air and water, is composed of nothing but a single proton at its nucleus, together with a single orbiting electron. However, some hydrogen atoms (about 0.015%) are composed of a single proton together with a single neutron. This is called deuterium. However, it is still hydrogen because it only has one proton. As noted above both the proton and neutron are located at the centre of the atom, called the nucleus. There is yet another form of hydrogen called tritium, of which only very tiny amounts exist in air and water. Tritium is composed of one proton and two neutrons, but, again, is still hydrogen because it only has a single proton. Deuterium and tritium are both "isotopes" of hydrogen, i.e. the same element but with a different number of neutrons. Another way of saying deuterium is to say "hydrogen-2", and tritium is "hydrogen-3", the number giving the total number of particles in the nucleus.

A well-known isotope is carbon-14. All carbon has 6 protons and most of it exists in the form of carbon-12; that is, 6 protons and 6 neutrons. A small percentage of any carbon in anything living is of the isotope carbon-14. When the living thing dies the carbon-14 starts to decay (in a way similar to beta decay as explained later in this page). By measuring how much carbon-14 is in the object and comparing it to what we would find in a living example of the same type we can work out how old the object is. This is known as carbon (or radiocarbon) dating.

Lastly, before moving on to discuss decays of various types, we need a quick way of showing which element and isotope we are talking about. Many elements have their name reduced to the just the first letter, or first two letters. So, for example, we have:

  • H - hydrogen
  • He - helium
  • C - carbon

To indicate which isotope we are dealing with we show the number of protons at the bottom left of the element name and the total number of particles in the nucleus at the top left of the element name. Here are a couple of examples:

Helium-4

and

Carbon-15

We now have all the information we need to start looking at radioactive decay.

 

Alpha Decay

This form of radioactive decay is usually shown using the Greek letter for alpha. Web browsers sometimes have problems displaying such characters correctly, but it looks like this:

In alpha-decay an atom ejects an alpha particle, which is simply a helium atom without any electrons. In doing so the parent atom decays into a lighter particle. An example of this is a uranium-238 atom decaying into into a thorium-234 atom and an alpha particle (helium-4 nucleus, i.e. 2 protons and 2 neutrons). A schematic diagram illustrates this:

Alpha Decay

This type of decay occurs naturally in uranium and is an example of "spontaneous decay".

So what has this got to do with E = mc2? The answer is simple and yet extraordinary at the same time. The uranium atom doesn't just break apart. As it decays each of the two resulting elements (the thorium and α-particle) fly apart at high speed. In other words they both have kinetic energy. That in itself may not seem so surprising. Perhaps the energy came from the fact that the two particles were held together in such a way that they would fly apart given the chance. However, it's possible to measure the mass of the original uranium atom and the masses of the two resultant particles. This is done by measuring the momentum of each particle as it strikes a sensor (although that is a somewhat simplified explanation it's good enough for our purposes here). When these measurements are taken it is found that the total mass of the two smaller particles is less than the mass of the original uranium particle. Some mass must have been turned into (mostly kinetic) energy, in accordance with E = mc2.

How much mass has been converted into energy? In fact, the figure is so small that physicists use another form of measurement instead of the joule for such decays, and one that makes more sense and is easier to work with for tiny energies, called the electron volt:

Electron volts.
An electron volt is defined as the work done on an electron in moving it through a potential difference of one volt. Its symbol is eV. We don't need to worry about the formal definition here. Instead, we just need to understand that it's a measurement of energy and often used in calculating energies at the atomic level. The amount of energy in 1eV is:

1eV

Notice the minus 19. That is a tiny, tiny amount of energy! It would require 16,020,000,000,000,000,000 electron volts to power a 100W light bulb for 1 second. For this reason electron volts are often measured in their millions, and given the prefix M, for mega. For example, 5 million electron volts is written as 5MeV.

Now back to the question of how much mass has been converted to energy in the decay.

We need to do this in two stages. The first is to find out how much energy was released. Typically, a uranium-alpha decay produces 4.3MeV of energy. How much is this in the more familiar energy unit of joules? We know how many joules there are to one electron volt, so:

4.3MeV

Now we need to rearrange E = mc2 to make mass the subject:

mass = energy/speed of light squared

We can now plug in our energy and the speed of light into the equation and get an answer:

A tiny amount of mass

We were talking in small numbers before but now we have a number that is almost unbelievably small! The amount of mass that was turned into energy during the α-decay was:

0.000,000,000,000,000,000,000,000,000,007,600 kg.

Needless to say, this wouldn't show up on any kitchen scales! However, this number has been experimentally verified in a number of ways, such as statistically using many millions of particles.

From the tiny numbers involved it would seem that uranium decay is of no importance. However, as other pages in this series show nothing could be further from the truth.

 

Beta Decay

This form of radioactive decay is usually shown using the Greek letter for beta. it looks like this:

.

In beta-decay a neutron in the nucleus of an atom changes into a proton and emits an electron, which is usually shown as e-. An example of this is an atom of carbon-15 changing ("transmuting") into an atom of nitrogen-15. We can show this as a schematic diagram:

Beta Decay

Notice that the number of particles in the nucleus has stayed the same; 15 in each case. Another thing to notice is that the atom has decayed "upwards". That is, it has gone from being element number 6 (carbon) to element number 7 (nitrogen). This isn't always the case. For example, carbon-11 transmutes (changes) into boron-11, i.e. "downwards" in the chemical table.

It is also worth pointing out the fact that a neutron in effect "contains" a proton and an electron, both of which are magnetic opposites and strongly repel each other. If we squeeze protons and electrons together they combine and turn into neutrons. This is what happens during supernova explosions at the end of a massive star's life. The force of the explosion is so great that it overcomes the magnetic resistance of the two types of particles and squeezes them together. For most of these massive stars, what remains is a ball of tightly packed neutrons, called, appropriately enough, a neutron star.

The energy released in a typical ß-decay is in the order of 1eV (i.e.1.6 x 10-19J). Most of this energy is in the form of the kinetic energy of the emitted electron. I will leave you to calculate the mass converted into energy during the decay!

 

Gamma Decay

The final type of radioactive decay we will examine is usually shown using the Greek letter for gamma. it looks like this:

Gamma decay is simply the emission of a particle of electromagnetic radiation (i.e. "light" - a photon) from an electron surrounding an atom. This usually happens spontaneously, but can also be made to happen, as in a laser (laser: Light Amplification by the Stimulated Emission of Radiation. Incidentally, the theory by which a laser operates was also first worked out by Einstein). We saw earlier in this page that an atom is surrounded by electrons in "orbit" around the nucleus. If a photon strikes an atom it can be absorbed by an electron in the outer "shell". This results in the electron having a higher energy state. One of two things can now happen:

  • Due to the higher energy the electron is vibrated out of the atom altogether.
  • or, the electron re-emits the photon.

If the latter case is true the atom is said to have undergone a gamma decay. We can show this schematically:

Absorb - vibrate - emit

Gamma decay is usually measured in the millions of electron volts. There isn't a typical value as such because atoms can absorb and re-emit photons at many different energies. However, most gamma radiation is roughly in the range 10,000eV - 10MeV.

 

Conclusion

This page has shown how the equation E = mc2 can be used to calculate the energy involved in atomic (radioactive) decay. On an every day scale the amount of energy produced is tiny, but atoms are very, very small. One gram (0.035 ounces) of any substance contains more than 1021 (that is, 1,000,000,000,000,000,000,000) atoms. Even taking into account that only a tiny amount of the mass of an atom is converted into energy during a radioactive decay we can use a lot of atoms and so a lot of energy can be released. How this is done is the subject of other pages in this series.

 

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