Part 3

An energy level diagram. The slanted arrows indicate radioactive decay by beta-particle emission. In each case, manganese-56 decays to a certain energy level of iron-56. On the right the energy of each level is indicated. Following a beta emission to a high-energy (excited) state in iron-56, one or more gamma rays are emitted until the nucleus is de-excited to the level marked zero. The vertical arrows indicate gamma rays emitted during the de-excitation process. The energy of each gamma ray is the difference between the levels involved in the change. The numbers above the vertical arrows indicate the relative proportions of gamma rays of different energies emitted from that level.

The mathematical relationship that describes the irradiation process exactly is:

A₀ = Nφσ (1 - e^{-λt})

where A₀ is the radioactivity produced (disintegrations per cubic centimeter per second); N is the number of target atoms per cubic centimeter in the sample; φ is the neutron flux (neutrons per square centimeter per second); σ is the cross section for the reaction (square centimeters); λ is the disintegration constant for the radioactive atoms produced (number per second); the number “e” is the base of natural logarithms; and t is the irradiation time in seconds. Note that for short irradiation times (t very small), 1-e^{-λt} approximates λt, while for long irradiations (t very large), 1-e^{-λt} approximates 1.

Maximum Energy % Betas 2.84 MeV 53 1.03 MeV 30 0.720 MeV 16 0.30 MeV 1 % Gammas 0.847 MeV 68 1.811 MeV 20 2.110 MeV 10 Other Energies 2 2.543 MeV 2.658 MeV 2.957 MeV 3.39 MeV

(avge~ 1.4 gammas per beta)

This summarizes what the decay scheme or energy level diagram shows in terms of the relative amounts of betas and gammas emitted in the decay of manganese-56. Thus, you could observe more than three times as many gamma rays having an energy of 0.847 MeV than of 1.811 MeV, etc. Note that while one, and only one, beta is emitted in the decay of one atom of manganese-56, two gammas can sometimes be emitted in one decay.

Of course, when the target is removed from the reactor, the number of radioactive atoms begins to decrease according to the characteristic half-life of the nuclide. The mathematical expression that describes the process of radioactive decay of a single nuclide is:

A_{t} = A₀e^{-λt}

where A_{t} is the radioactivity of an isotope at some time, t, after the end of the irradiation, and A₀ is the radioactivity at the end of the irradiation.

The activation of sodium-23 to sodium-24, which has a half-life of 15 hours. The horizontal line marked 1.0 represents the “saturation” activity level for a sample of sodium of a certain size in a constant neutron flux. Note that after about 120 hours, the activity of the sample is within 1% of the value at saturation, which is the most active that sample will ever become at a given φ. Note also that after the first 15 hours (1 half-life) the sample is exactly half way to its value at saturation. Thus long irradiations are useful to increase the sensitivity of the analysis, but only up to a certain point.

The result of all this is that the sensitivity of an analysis depends in practice on a number of practical as well as theoretical factors:

1. The cross section of the target element. 2. The half-life of the radioactive isotope produced. 3. The time available for irradiation. 4. The flux of neutrons available for irradiation. 5. The promptness with which we can begin measuring radioactivity and the efficiency of this measurement. 6. Possible interferences due to the presence of elements yielding the same radioactive elements or those yielding very similar radiations.

In the next section of this booklet, there are several examples that will show you how all this works in practice. But to summarize what these factors mean in terms of sensitivity let us look at the chart in the figure on page 18. Here all the elements are arranged in a periodic table. The sensitivities are shaded in coded ranges representing measurable quantities. They are calculated on the basis that there are no interferences, that the neutron flux is 10¹⁴ neutrons per square centimeter per second, and that we can measure 100 gamma rays per minute without much difficulty assuming a gamma-ray detector efficiency of 10%. The elements labeled β yield radioisotopes that emit few or no gamma rays and can only be analyzed by neutron activation using appropriate chemical separation procedures followed by beta radioactivity measurements. Such chemical separation procedures (to remove unwanted radioactive isotopes of other elements) are also sometimes useful to improve the sensitivity of the analysis of gamma-ray emitters if necessary.

The radioactive decay curve of sodium-24. The vertical scale is not linear but logarithmic. Thus, each factor of two in radioactivity occupies the same distance along the vertical axis. When two samples are being analyzed for sodium by activation analysis, they must be compared at the same time after they have been removed from the neutron flux. If this period of time is different, then a correction must be applied to one of them, based on the decay curve shown here, to allow for the difference in decay time for the two. Waiting too long after the irradiation is completed results in much poorer sensitivity for the analysis depending on the half-life of the activation product. In this case, after 2 days it takes approximately ten times as much sodium to yield the same radioactivity as it would if the sample were measured when it was fresh out of the reactor.

It is not practical to determine a few elements, shown in black squares, by activation analysis. Some others, like oxygen and nitrogen (labeled HE), can be measured by using other projectiles like fast (more energetic) neutrons, or protons or deuterons produced in a device called an accelerator. Other elements, those shown in white squares, can be detected with such great sensitivity, that one can find some in almost everything. For example, if you had a cube of “pure” aluminum only 1 millimeter on a side, you could detect gold in it if there were only one atom of gold for every fifty billion atoms of aluminum.

While it isn’t often that you would want to find a gold needle in an aluminum haystack, the next section presents some practical applications. Imagine yourself as the person with the problem in these situations.

* Th and U are radioactive but with such long half-lives that neutron activation analysis can be used for their determination.

† µg = Microgram (one-millionth of a gram)

HOW AND WHERE TO USE IT

In a Physics Laboratory

The Problem

You are a physicist investigating the properties of semiconductors, which are materials used to make transistors. When you apply a voltage to one specimen of silicon (a semiconductor), it doesn’t behave quite like the others that you’ve studied. The electrical properties of this odd specimen are unusual and interesting and could lead to a new type of transistor. What makes this specimen different from the others? Very small amounts of impurities can cause large changes in the electrical properties of semiconductors. You would like to obtain a chemical analysis of the material, but your colleagues in chemistry tell you they would have to dissolve a good size part of your sample to analyze it and you are reluctant to give it up. How do you do it?

The Solution

You decide to try neutron activation analysis. You realize you won’t be able to detect all the elements, but many of those that might affect semiconductor performance could be detected quite easily.

What will you need? A source of neutrons to activate the material and a gamma-ray spectrometer to measure the radiation from the material afterwards. This spectrometer detects and measures gamma rays and sorts them according to their energy. You find that your friend down the hall, who is a nuclear physicist, has a gamma-ray spectrometer that incorporates a lithium-drifted germanium crystal as a detector and a pulse height analyzer. The germanium detector is a device that senses the gamma rays that enter it and gives electrical signals related to the energy of the gamma rays. It was invented only a few years ago and has a very fine resolution. That is, it can easily “pick out” gamma rays that are only slightly different in energy. For example, for gamma rays with energies of approximately 1 MeV (million electron volts), it is not unusual to distinguish between gamma rays that differ by only 2 or 3 tenths of a percent. The pulse height analyzer is an electronic device that sorts the electrical pulses from the detector according to their energy.

A lithium-drifted germanium-crystal gamma-ray detector. The large container is a reservoir of liquid nitrogen that keeps the detector cooled to a temperature of -196° Centigrade (321° below zero, Fahrenheit). The lead brick shield keeps out most of the gamma rays that come from naturally radioactive materials in the room. The plastic slots hold cards upon which the samples are mounted for counting. Sometimes the detector is arranged vertically and samples are placed on shelves above it.

What about the neutrons for the irradiation? Although there isn’t a suitable nuclear reactor in your city, there is one at a university only an hour away by jet. Since it may take a few hours to get the sample to the counter after irradiation, you won’t be able to look for short-lived activation products, i.e., those with half-lives of up to an hour. However, this will exclude only a few elements from detection.