14.6 Radioactivity

Nuclear decay is governed by chance. It is impossible to tell exactly when any specific nucleus will decay. Therefore, the decay is phrased in terms of statistical quantities like specific decay rate, lifetime and half-life. This section explains what they are and the related units.

14.6.1 Decay rate

If a large number $I$ of unstable nuclei of the same kind are examined, then the number of nuclei that decays during an infini­tesimally small time inter­val ${\rm d}{t}$ is given by

$${\rm d}I = - \lambda I {\,\rm d}t$$ (14.3)

where $\lambda$ is a constant often called the “specific decay rate” of the nucleus. If the amount of nuclei at time zero is $I_0$, then at an arbitrary later time $t$ it is

$$I = I_0 e^{-\lambda t}$$ (14.4)

The reciprocal of the specific decay rate has units of time, and so it is called the “lifetime” of the nucleus:

$$\tau \equiv \frac{1}{\lambda}$$ (14.5)

However, it is not really a lifetime except in some average mathematical sense. Also, if more than one decay process occurs,

Add specific decay rates, not lifetimes.

The sum of the specific decay rates gives the total specific decay rate of the nucleus. The reciprocal of that total is the actual lifetime.

A physi­cally more meaningful quantity than lifetime is the time for about half the nuclei in a given sample to disappear. This time is called the “half-life” $\tau_{1/2}$. From the exponential expression above, it follow that the half-life is shorter than the lifetime by a factor $\ln2$:

$$\tau_{1/2} = \tau \ln 2$$ (14.6)

Note that $\ln2$ is less than one.

For example tritium, $\fourIdx{3}{1}{}{}{\rm {H}}$ or T, has a half life of 12.32 years. If you have a collection of tritium nuclei, after 12.32 years, only half will be left. After 24.64 years, only a quarter will remain, after a century only 0.4%, and after a millennium only 4 10$\POW9,{-23}$%. Some tritium is continuously being created in the atmosphere by cosmic rays, but because of the geologi­cally short half-life, there is no big accumul­ation. The total amount of tritium remains negligible.

14.6.2 Other definitions

You probably think that having three different names for essentially the same single quantity, the specific decay rate $\lambda$, is no good. You want more! Physicists are only too happy to oblige. How about using the term “decay constant” instead of specific decay rate? Its redeeming feature is that “constant” is a much more vague term, maximizing confusion. How does “disinte­gration constant” sound? Especially since the nucleus clearly does not disinte­grate in decays other than sponta­neous fission? Why not call it “specific activity,”come to think of it? Activity is another of these vague terms.

How about calling the product $\lambda{I}$ the “decay rate” or“disinte­gration rate” or simply the “activity?” How about “mean lifetime” instead of lifetime?

You probably want some units to go with that! What is more logical than to take the decay rate or activity to be in units of “curie,” with symbol Ci and of course equal 3.7 10$\POW9,{10}$ decays per second. If you add 3 and 7 you get 10, not? You also have the “becquerel,” Bq, equal to 1 decay per second, defined but almost never used. Why not “dpm,” disinte­grations per minute, come to think of it? Why not indeed. The minute is just the additional unit the SI system needs, and using an acronym is great for creating confusion.

Of course the activity only tells you the amount of decays, not how bad the generated radiation is for your health. The “exposure” is the ionization produced by the radiation in a given mass of air, in Coulomb per kg. Of course, a better unit than that is needed, so the “roentgen” or “röntgen” R is defined to 2.58 10$\POW9,{-4}$ C/kg. It is important if you are made of air or want to compute how far the radiation moves in air.

But health-wise you may be more inter­ested in the “absorbed dose” or “total ionizing dose” or “TID.” That is the radiation energy absorbed per unit mass. That would be in J/kg or “gray,” Gy, in SI units, but people really use the “rad” which is one hundredth of a gray.

If an organ or tissue absorbs a given dose of radiation, it is likely to be a lot worse if all that radiation is concentrated near the surface than if it is spread out. The “quality factor” $Q$ or the somewhat differen­tly defined “radiation weighting factor” $w_{\rm {R}}$ is designed to correct for that fact. X-rays, beta rays, and gamma rays have radiation weighting factors (quality factors) of 1, but energetic neutrons, alpha rays and heavier nuclei go up to 20. Higher quality means worse for your health. Of course.

The bad effects of the radiation on your health are taken to be approximately given by the “equivalent dose,” equal to the average absorbed dose of the organ or tissue times the radiation weighting factor. It is in SI units of J/kg, called the “sievert” Sv, but people really use the “rem,” equal to one hundredth of a sievert. Note that the units of dose and equivalent dose are equal; the name is just a way to indicate what quantity you are talking about. It works if you can remember all these names.

To get the “effective dose” for your complete body, the equivalent doses for the organs and tissues must still be multi­plied by “tissue weighting factors and summed. The weighting factors add up to one when summed over all the parts of your body. The ICRP defines “dose equivalent” different from equivalent dose. Dose equivalent is used on an opera­tional basis. The personal dose equivalent is defined as the product of the dose at a point at an appro­priate depth in tissue, (usually below the point where the dosimeter is worn), times the quality factor (not the radiation weighting factor).