Extracting Half-lives from Characteristic Gamma Spectra

Figure 1 – The lower end of the Chart of the Nuclides which displays all of the currently known

isotopes in a plot of proton number vs. neutron number.

To skip the following text and view the data that was taken for this page, click here.

Measuring Half-Lives

Eric B. Norman¹, Ruth-Mary Larimer¹, Gregory Rech^1,2, Jeffrey Lee^1,3,

Tholoana Leubane^1,4, and Ken Zamvil^1,5

1Lawrence Berkeley National Laboratory, Berkeley, CA

^{2University of California, Berkeley, CA
^{3University of California, Irvine, CA
^{4Franklin Middle School, Vallejo, CA
^{5Hogan High School, Vallejo, CA}}}}

Contents: The Purpose, Radioactive Decay, Beta Decay, Gamma Decay, The Experiment, Extracting Half-lives, Reference

The purpose of this project: Go to the top.

Basic nuclear science is an essential part of the high school science curriculum. Certain topics in nuclear science, like the concept of a half-life, are something that every physical science student should know. However, it is very difficult to perform nuclear science experiments in a high school laboratory setting, and therefore hard to back up the theory with actual experimental results. With this web site, all that a classroom needs is internet access in order to look at real nuclear science data. Then, by following the procedures described here, the students can extract half-lives of radioactive isotopes based on this data. The idea is that by doing this, nuclear science will seem less abstract and easier to comprehend.

Radioactive Decay: Go to the top.

Radioactive decay is the general term for several different processes that all have the effect of spontaneously lowering the energy of a nucleus through the emission of particles. All types of radioactive decay involve a parent nucleus decaying into a daughter nucleus. The two types of radioactive decay that we are concerned with in this paper are called beta decay and gamma decay. All types of radioactive decay follow several basic rules. One rule is that the amount of time it takes for a single nucleus in a sample of nuclei to decay is random and cannot be predicted. Another rule is that the mean (or average) lifetime of a sample containing many nuclei of the same isotope (same number of protons and neutrons) can be predicted and measured. This is where the concept of a half-life comes in.

The half-life is the amount of time it takes for one-half of the nuclei in a sample to decay. It is abbreviated as t_1/2. Of the original nuclei present at t = t_1/2, half will decay if we wait another half-life, leaving one-quarter of the original sample after a total time of two half-lives. After three half-lives, one-eighth of the original sample will remain and so on. Measured half-lives vary from tiny fractions of seconds to billions of years, depending on the isotope.

The number of nuclei in a sample that will decay in a given interval of time is proportional to the number of nuclei in the sample. This condition leads to radioactive decay showing itself as an exponential process, as shown in Figure 2. The number, N, of the original nuclei remaining after a time t from an original sample of N₀ nuclei is

N = N₀e^-(t/t)

Where t is the mean lifetime of the radioactive nuclei. From this relation, it can be shown that t_1/2 = 0.693t.

Figure 2 – ^137mBa decay data, counting numbers of decays observed in 30-second intervals.

The best-fit exponential curve is shown. The points do not fall exactly on the exponential because

of statistical counting fluctuations.

Beta Decay: Go to the top.

Beta particles are electrons or positrons (electrons with positive electric charge, or antielectrons). Beta decay occurs when, in a nucleus with too many protons or too many neutrons, one of the protons or neutrons is transformed into the other. In beta minus decay, as shown in Figure 3, a neutron decays into a proton, an electron, and an antineutrino: n à p + e^- + . In beta plus decay, as shown in Figure 4, a proton decays into a neutron, a positron, and a neutrino: p à n + e⁺ + n. These particular reactions take place because conservation laws are obeyed. Electric charge conservation requires that if an electrically neutral neutron becomes a positively charged proton, an electrically negative particle (in this case an electron) must also be produced. Similarly, conservation of lepton number requires that if a neutron (lepton number = 0) decays into a proton (lepton number = 0) and an electron (lepton number = 1), a particle with a lepton number of –1 (in this case an antineutrino) must also be produced. The leptons emitted in beta decay did not exist in the nucleus before the decay – they are created during the process of decay.

Figure 3 – A representation of ¹⁴C beta minus decaying into ¹⁴N.

Figure 4 – A representation of ¹⁸F beta plus decaying into ¹⁸O.

To the best of our knowledge, an isolated proton, a hydrogen nucleus with or without an electron, does not decay. However within a nucleus, the beta decay process can change a proton to a neutron. An isolated neutron is unstable and will decay with a half-life of 10.5 minutes. A neutron in a nucleus will decay if a more stable nucleus results; the half-life of the decay depends on the isotope. If it leads to a more stable nucleus, a proton in a nucleus may capture an electron from the atom (electron capture), and change into a neutron and a neutrino.

Beta plus decay, beta minus decay, and electron capture are three ways in which protons can be changed into neutrons or vice-versa; in each decay there is a change in the atomic number (number of protons), so that the parent and daughter atoms are different elements. In all three processes, the number of nucleons, A, remains the same, while both proton number, Z, and neutron number, N, increase or decrease by 1.

In beta decay, the change in binding energy appears as the mass energy and kinetic energy of the beta particle, the energy of the neutrino, and the kinetic energy of the recoiling daughter nucleus. The energy of an emitted beta particle from a particular decay can take on a range of values because the energy can be shared in many ways among the three particles while still obeying energy and momentum conservation.

Gamma Decay: Go to the top.

In gamma decay, as shown in Figure 5, a nucleus changes from a higher energy state to a lower energy state through the emission of electromagnetic radiation (photons). The number of protons (and neutrons) in the nucleus does not change in this process, so the parent and daughter isotopes are the same. In the gamma decay of a nucleus, the emitted photon and recoiling nucleus each have a well-defined energy after the decay. The characteristic energy is divided between only two particles.

Figure 5 – A representation of the gamma decay of excited ¹⁵²Dy.

The Experiment: Go back to the top.

In order to produce several isotopes’ characteristic gamma spectra we needed to make these isotopes from their respective natural elements. We did this through neutron activation. Neutron activation is the process of bombarding a stable isotope with neutrons with the intent of having these neutrons "stick" to the isotope, thereby making a new, unstable isotope of the same element. Because of this extra neutron, the isotope will beta minus decay into a more stable isotope. However, this beta minus decay will often leave the new isotope in an excited energy level, which in turn causes this new isotope to gamma decay down to its ground state (lowest energy state). These gamma rays are characteristic for every isotope and are the ones plotted on the following pages.

In theory, every element can be neutron activated, but in practice certain conditions must be met in order to get good data. First, the element must have an isotope that is able to appreciably react with the incoming neutrons, we say that the isotope must have a high neutron cross-section. Second, the isotope itself must be relatively easy to get in sizable quantities. Lastly, the gamma rays produced must be measurable by our detectors, so they must be reasonably intense and of a limited energy range (about 50 to 3200 keV). We found seven elements that fit all of these requirements.

To measure the gamma rays emitted from each isotope, we used a coaxial germanium detector 5 cm thick and 5 cm in diameter sitting directly in front of the isotope sample, as shown in Figure 6. We acquired the gamma ray spectra in 4096 channels using an ORTEC, PC based data acquisition system.

Figure 6 – This is a picture of the experimental setup. The germanium detector is sitting

inside the long cylinder that is attached to the big thermos bottle filled with liquid nitrogen.

The plastic baggie contains a sample of a radioactive isotope, and was being counted

when this picture was taken.

The following table summarizes the data that was taken for this project.

Name of isotope	Gamma ray energies from isotope (keV)	Number of runs	Length of each run
Au 198	412	11	12 hrs.
Ba/Cs 137*	662	*	30 sec.
I 128	443	10	6 min.
La 140	329, 487, 1596	12	12 hrs.
Mn 56	847, 1811, 2113	15	1 hr.
Na 24	1369, 2754	9	8 hrs.
V 52	1434	10	30 sec.

Figure 7 – Here is a picture of Ken Zamvil and Tholoana Leubane next to the experiment.

Extracting Half-lives: Go to the top.

The following is a step by step way to extract a half-life from the data on the following pages.

Go to the following website: http://ie.lbl.gov/radioactivedecay/page2.htm

This site will open with the decay schemes for 7 isotopes. Select any isotope and click on that decay scheme. The next window will show a series of files for data runs for the selected radioactive isotope.

Click on the first run in the series. This will open up the gamma ray spectrum for the first run of the selected isotope.

Identify the peak area and the energy of the peak by touching the curve with the cursor. Locate the points where the peak begins and ends, again, using the cursor. Write these energy values in the boxes above the curve and press "submit." Below the gamma spectrum will appear the region of interest that you selected. The value displayed will be the number of gamma events for this peak, representing the area under the curve. This value will be referred to as the "gross area under the peak" (GAP). YOU MAY HAVE TO DO THIS SEVERAL TIMES IN ORDER FOR YOU TO ZOOM IN ON THE REGION OF THE SPECTRUM WHERE ONLY THE GAMMA RAY IS LOCATED.

The value you just obtained needs some correction before it can be used. Radiation is everywhere, and therefore the value you obtained has additional radiation in the spectrum not originating from the source and is referred to as "background radiation." You must determine the average background radiation before you can determine the correct peak radiation.

On separate paper, prepare a data table like the one written below, but for all of your isotope’s runs.

Run # GAP LAP RAP NAP ln(NAP) Time of run

Determine the energy difference in keV from the beginning of the peak to the end of the peak. Using the cursor, find an area on the left side of the peak, and another on the right side of the peak, which span an equal energy distance (amount). Try to select areas that appear to be fairly constant, or level, with few unusual events occurring. Be sure that you pick your areas close to the peak and of equal distance from the left side of the peak as from the right side of the peak. These two regions will become the LAP and RAP respectively.

Using the same technique as described above for the peak, determine the total number of background counts under the curve on the left side of the peak (LAP) and on the right side of the peak (RAP). Add these two regions together and divide the sum by 2. This will give you the average background radiation under the peak. Subtract this value from the peak area (GAP) as the correction factor. The resulting value is the net area of the peak (NAP). It is this value that you will later graph and use for determining the half-life.

Take the ln (natural logarithm) of the NAP for each run and put these values in your table.

Find the amount of time that has elapsed since the first run for each run and put these values in your table. For example if run 1 started on 08-Jun-00 at 16:14:30 and run 2 started on 08-Jun-00 at 16:15:04 then you could put 0 seconds as the time value for the first run and 34 seconds as the time value for the second run

When your data table is complete, prepare a graph of the results. Put the ln(NAP) on the "y" axis and the time for the different runs on the "x" axis.

Using your best judgement, fit a straight line through the data, coming as close as you can to all of the points on the graph.

Find the slope of this line

Divide the ln(2) by your slope value. This value is the half-life of the isotope that you chose.

To compare your results to the accepted values for these half-lives, go to http://isotopes.lbl.gov/education/isotopes.htm

* ¹³⁷Ba/Cs is a special case. When ¹³⁷Cs beta minus decays into the exited state of ¹³⁷Ba, the gamma decay to the ground state, instead of being nearly instantaneous, takes a measurable amount of time. In order to be able to measure ¹³⁷Ba’s half-life, we needed to separate the ¹³⁷Ba from the ¹³⁷Cs. However, during the separation process, some ¹³⁷Cs remained in the sample. But, by doing the following, you can still obtain ¹³⁷Ba’s half-life.

For runs 1 through 10 of this sample, follow steps 1 through 8 in the above procedure. Then, before taking the natural log of the NAP, subtract off the average background caused by the ¹³⁷Cs. The average background caused by the ¹³⁷Cs can be obtained by following steps 1 through 8 of the above procedure for runs 11 through 26 of the ¹³⁷Ba/Cs sample. After finding all of the NAP’s for runs 11 through 26, obtain the average of these values. Take this average and subtract it from every NAP value for runs 1 to 10. Using these new NAP values, follow the rest of the procedure as given.

We are able to do this because the half-life of ¹³⁷Cs is roughly 6 million times longer than the half-life of ¹³⁷Ba, and so the counts that are on the gamma spectra due to the ¹³⁷Cs appear constant over the time of the runs. The half-life of a sample is not dependent on additive constants and so we can just subtract this constant number of counts away to make the analysis easier.

Reference: Go to the top.

If you would like to learn more about x-ray fluorescence or neutron activation anlaysis, please visit our other websites: http://ie.lbl.gov/xray and http://ie.lbl.gov/naa

Figures 1-5 and the text describing beta and gamma decay are used by permission of the Contemporary Physics Education Project. For more information, go to http://www.lbl.gov/nsd/education/ABC/wallchart/guide.html

For any questions regarding this site, contact Eric Norman at EBNorman@lbl.gov.

To view the data that was taken for this page click here.