“Paging Dr. Freeman”: What Radioactivity Is

Radioactivity - Image via wikipedia

Radioactivity - wikipedia.org

The recent earthquake and tsunami in Japan and the subsequent crisis at the Fukushima nuclear power plants have propelled nuclear reactors and nuclear energy to the top of every media outlet across the world.  In light of this increased interest in nuclear energy, I have decided to write about radioactivity.  Radioactivity is a natural physical phenomenon that is a consequence of the weak nuclear force, strong nuclear force and the electromagnetic force - three of the four fundamental forces of nature.  It commonly refers to the process by which an unstable atom decays or transmutates to one or more atoms with an accompanying release of energy.  In this article, I will try to explain what radioactivity means and what natural phenomena it describes, why some atoms are radioactive, what radiation is and how it relates to radioactivity.

Protons, Neutrons and Electrons Oh My!

Model of an Atom

Model of an Atom - glenair.com

A quick recap of elementary chemistry is in order.  Except in exotic situations, matter is composed of tiny particles called atoms.  Atoms themselves are made of particles known as protons, neutrons, and electrons.  A proton has a single unit of positive charge.  An electron has a single unit of negative charge and is about 1800 times lighter.  Neutrons are a shade heavier than protons and do not carry any electric charge.  Atoms are made up of a tiny core called the nucleus where the heavy protons and neutrons reside.  The electrons orbit the nucleus akin to planets orbiting the sun.  In a neutral atom (one with equal positive and negative charge), the number of electrons and protons is the same.  We can classify atoms into various elements based on how many protons are in the atom’s nucleus. Oxygen is an element whose atoms have six protons each.  Hydrogen is an element with only one proton in its nucleus.  Note that we have not specified the number of neutrons in each atom.  The same element can come in a number of varieties, each of which have a different number of neutrons.  These varieties are known as isotopes.  Usually, the number of neutrons is comparable to the number of protons, but this can vary quite a bit.

Hydrogen Isotopes

Hydrogen Isotopes - wikipedia.org

For example, hydrogen has three different isotopes: protium, deuterium and tritium.  Protium is garden variety hydrogen which has no neutrons.  Deuterium has one neutron in its nucleus.  Tritium has two neutrons in its nucleus and is the heaviest hydrogen isotope.  These heavier isotopes occur naturally in trace quantities but are also manufactured for use in labs, nuclear power plants, thermonuclear nuclear warheads, watches, safety equipment and a number of other places.

Weight Watchers

Not all atoms are created equal.  Some of them are inherently more stable than others.  Some atoms may contain too many or too few neutrons to be stable.  These unstable atoms naturally attempt to change into other, hopefully more stable atoms by a process known as radioactivity or radioactive decay.  By-products of this process, when they are emitted at high speeds, are known as radiation.  For example, let’s look at tritium – the really heavy isomer of hydrogen with one proton and two neutrons.  Tritium, when left to its own devices, is not very stable and wants to become more stable.  But it cannot do this in any willy-nilly manner.  To decay, an atom can split up or transmutate in a couple of ways:

  1. Alpha decay

    Alpha decay - wikipedia.org

    An atom can mutate into multiple smaller atoms, distributing the protons and neutrons in its nucleus in the process.  The resulting atoms may or may not be stable.  If they are unstable, they will undergo further decays until they reach a stable state.  If one of the resulting atoms is helium (two protons and two neutrons), it is given a special name for historical reasons.  This particle is known as an alpha particle (α). This process occurs when a stable alpha particle is able to overcome the strong nuclear force that binds it in the nucleus, by exploiting the excessive electromagnetic repulsion and leaving the nucleus in a process known as quantum tunneling.

  2. Beta decay

    Beta decay - wikipedia.org

    An atom can also transmutate by converting a neutron into a proton and an electron and shooting the electron out of the core.  This electron, though much lighter than neutrons, protons or alpha particles, gets ejected with high enough velocity that it can be detected.  It is known as a beta particle (β), again for historical reasons.  This process is mediated by a force of nature known as the weak nuclear force which mediates the conversion of matter from one form to another.  In the diagram, you may notice an extra particle leaving (νe).  This is an extremely hard to detect particle known as a neutrino that I will ignore for the rest of this post.

  3. Gamma decay

    Gamma decay - wikipedia.org

    Lastly, an atom that is in an excited state (where it has a lot of excess energy in its nucleons) can decay by emitting a photon (a unit of electromagnetic energy) and remaining the same isotope but a more stable version of it.  This process is known as a gamma decay or isomeric transition and the resultant high energy photon is called a gamma ray (γ), again for historical reasons.  This is caused due to an atom succumbing to the strong nuclear force pulling it together and binding it tightly, releasing excess energy in the process.

Uranium-238 decay

Uranium-238 decay

Alpha particles, beta particles and gamma rays are common forms of radiation.  From this, we can see that radiation is not some mystical wave or an “pervasive unstoppable energy field” or anything else you hear in a sci-fi show (I’m looking at you Star Trek).  It is simply the high-energy particles and energy that an atom releases during decay.  Some of these particles can be other atoms, some of them can be nucleons such as protons or neutrons.  Yet other can be photons of various wavelengths and energy.  Nowhere in this process can atoms arbitrarily discard its protons or neutrons.  In all of these processes, no extra net electric charge is created or destroyed.  This is known as charge conservation. Certain other properties are also conserved but that would require a more thorough exposition on nuclear physics.

Let us go back to looking at our tritium atom.  Tritium has only one proton; hence it cannot form a lighter atom because atoms need to have at least one proton.  Instead, tritium mutates into helium-3, an isotope of helium that has two protons and one neutron, by decaying one of its neutrons.  The resultant electron is shot out of the nucleus at high speeds and can affect other atoms it runs into by slamming into them and transferring energy to them. Let’s look at another example to clarify matters.  Uranium naturally occurs as a number of isotopes, the most common being uranium-238 which has 92 protons and 146 neutrons.  When left to itself, uranium is relatively stable, but still undergoes an extremely slow radioactive decay.  Uranium-238 (written as U-238) ejects an alpha particle (which has two neutrons and two protons) to produce Thorium-234 which has 90 protons and 144 neutrons.  The thorium, which has an excess of neutrons, then undergoes a beta decay and converts a neutron to a proton and ejects an electron.  The result is called Protactinium-234 which has 91 protons and 143 neutrons.  The protactinium produced is in an excited state.  It can get rid of this excess energy by undergoing an gamma decay and emitting a photon.  The resultant ‘un-excited’ protactinium is still radioactive and continues decaying through beta decay.  The process continues for a while in a chain of reactions known as the radium decay chain.  The end result is lead (Pb-126) which has 82 protons and 124 neutrons.

A Coin Flip

Throughout this discussion, we talked about how atoms can decay and what they decay into.  We never talked about when they can decay.  Radioactivity is a stochastic process.  What this means is that it is impossible to predict when any particular atom will decay.  However, it is possible to predict how a given ensemble of identical radioactive atoms will decay over time.  Think of it this way – if someone were to give you a license plate number for a car and ask you to predict whether the car is going to get involved in a traffic collision today while giving you no other information, it would be impossible to know for sure what the car would do.  But on the other hand, given an entire city’s worth of cars, it is simply a matter of statistics to know, on average, how many accidents occur every day.  The individual cars are not very predictable.  But the ensemble tends to follow an underlying statistical distribution when it comes to frequency of collisions.  Radioactivity works on the same principle.  It has been empirically observed that radioactive decay is an exponential decay.  “What is an exponential decay?”, you ask.  It is a form of decay where the instantaneous (on-the-spot) rate of decay depends on the amount of substance that is present at that instant.  The more substance you have, the more decays you observe.  One example of exponential decay is the natural cooling of a hot object in a cool environment.  The hotter the object (i.e. the higher the temperature difference), the faster the object cools.  Hence, super hot tea cools down really rapidly when you first make it.  But after a few minutes, the tea is no longer scalding hot but it starts cooling down at a slower rate.  Tens of minutes later, it is still lukewarm and cooling down at an even slower pace, eventually coming to the same temperature as the surrounding room.

Time to Live

Radioactive decay

Radioactive decay - earthsci.org

But simply knowing that atoms decay along exponential curves does not tell us how fast they decay.  Going back to our discussion on hot objects cooling, we know that metals cool faster because they transmit heat better.  We know that glass is a bad conductor of heat.  It still cools down, but not as quickly as iron or aluminum.  Different isotopes of radioactive elements similarly vary in how quickly they decay.  One way of adequately describing the speed of decay is to specify the time it takes for half of a sample of radioactive substance to decay away.  The shorter this time is, the faster an isotope decays.  This quantity is known as the half-life of a radioactive isotope.  Uranium-238 (the naturally occurring isotope) has a half-life of around 4.5 billion years.  This means that if you held a slab of pure uranium-238 today, in 4.5 billion years half of the atoms in that slab would have decayed into other products (such as thorium and lead).  After yet another 4.5 billion years, only a quarter (half of half) of the original slab would still be U-238.  After yet another half-life, only an eighth would remain.  After 7 half-lives, less an 1% of the original substance is left and the rest has decayed.  Strangely, this indicates that naturally occurring U-238 is actually not that radioactive.  Sodium-24 is a radioactive isotope of sodium that beta decays to a magnesium isotope.  It has a half-life of about 15 hours.  This means that sodium-24 is highly unstable and that it decays quite rapidly.  There are radioactive isotopes of polonium, astatine and bismuth that have half-lives in the order of seconds or minutes.  These elements are highly unstable and very rapidly release radiation.  How should one think about this?  Suppose that a certain fixed amount of radioactive substance contaminates your environment.  If the radioisotope has an extremely long half-life, it tends to hang around for basically forever but because it is decaying so slowly, it may not actually emit enough radiation to cause worry.  If the radioisotope has an extremely short half-life, it tends to be quite dangerous because it is rapidly producing a lot of radiation.  On the other hand, if it has a half-life in the order of minutes, then by the time a day is up, almost none of the original material is left over.  Hence, it might be easy to avoid and “wait-out” such radioisotopes by staying away from it.  The most worrisome of the radioisotopes are the ones that have half-lives on the order of years, decades or centuries.  They decay fast enough for us to notice, but are active for long enough that we cannot ignore it or wait it out.

Wait wait! Tell me more!

Now that you understand the basics of radioactivity and radioactive decay, you may be curious as to how all this relates to energy and nuclear power or radiation poisoning.  How do atoms breaking apart produce power?  How does radiation harm humans.  After all, you can tell when you have been sucker punched, but you never feel neutrons and electrons crashing into you.  I will cover that in further articles in this series.

Addendum: A Mathematical Modeling

For those amongst you who are itching for some math, here is a quickly discussion of the underlying mathematical model behind half-lives.  Skip over this section if you do not already understand basic differential equations.  We can model decaying as a process that starts with N0 particles at time t=0, where a particle is removed from the ensemble when it decays.  The instantaneous number of particles (N) that remain un-decayed at some time t in the future is based on the following rate equation:

\frac{dN}{dt} = -\lambda N

Note that the equation shows that the decay rate (the rate of change of the number of particles) is negative and is directly proportional to the instantaneous number of particles up to a constant of proportionality λ, known as the decay constant.  We can simplify this further:

\int\frac{1}{N} \; dN = -\int \lambda \; dt

\log_e{N} + c = -\lambda t

We know that at time t = 0, no particles have decayed yet (this was our boundary condition).  Hence:

\log_e{N_0} + c = 0

\log_e \frac{N}{N_0} = -\lambda t

N = N_0 e^{-\lambda t}

Glancing at the units involved, since t has the dimensions of time, λ must have units of s-1.  We can determine the half-life of a substance by solving for t when N = \frac{N_0}{2}.

\frac{N_0}{2} = N_0 e^{-\lambda t_h}

-\log_e{2} = -\lambda t_h

t_h = \frac{\log_e{2}}{\lambda}

We can use this fact to look at our decay equation in yet another way:

N = N_0 e^{-\frac{log_e{2}\; t}{t_h}}

N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_h}}

Reading this equation, it says: For every time that is a multiple of th, the amount of undecayed substance remaining is decreased by an extra half.  Isn’t math beautiful?

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