Department of Physics, Middlebury College

1992-93

Modern Physics Laboratory



VII. Nuclear Half-Life Measurements



Discussion

The dominant decay mode for excited nuclear states is gamma decay. In gamma decay a nucleus in an excited state makes a radiative transition to a less excited state or directly to the nuclear ground state. During this process, the numbers of protons, Z, and neutrons, N, in the nucleus do not change; however, the nucleus becomes a more tightly bound system. As a result of the increased nuclear binding, energy is released in the form of a gamma ray. Most excited state half-lives are in the range of 1 fs to 1 µs.

The gamma decay of the I = 11/2- excited state of 137Ba is shown in the energy level diagram of Fig. 1. The gamma ray energy is 0.662 MeV and the half-life of the 11/2- excited state is t1/2 = 2.551 min. Because this state has a comparatively long half-life it is called metastable  or isomeric  and is usually written 137mBa. The long half-life of 137mBa results from the large angular momentum difference between the 11/2- excited state and the 3/2+ ground state. Recall from atomic theory that electric dipole transitions always proceed most rapidly, if  they are permitted by the selection rules. Transitions of higher multipolarity, such as magnetic dipole and electric quadrupole transitions, are always less probable and involve much longer radiative lifetimes. The 11/2- --> 3/2+ decay in 137Ba requires a change of 4 units of angular momentum by the nucleus and it occurs via a magnetic hexadecupole (M4) transition. The relatively long half-life is typical of nuclear decays involving a large angular momentum mismatch between initial and final state.

A relatively pure source of metastable 137mBa can be created in the laboratory by eluting a hydrochloric acid-saline solution through a solid 137Cs radioactive source. As shown in Fig. 1, the principal decay mode of 137Cs is - emission to the 11/2- excited state, 137mBa. The 137Cs half-life is 30 years and there is continuous production of the 137mBa daughter occurring in the presence of the 137Cs parent. The solution selectively precipitates out the relatively small numbers of 137mBa atoms present in the solid 137Cs source. The solution eluted from the source contains a 137mBaCl2 precipitate that is largely free of 137Cs contamination.

The 137mBa gamma ray spectrum of Fig. 2(a) was taken with a NaI(T) detector and the Quantum 8 Multichannel Analyzer (MCA) controlled by the Q8 program.2 The 0.662 MeV, 11/2- --> 3/2+ transition is the major peak in the spectrum. To reject unwanted gamma rays in a half-life measurement, one puts a "window" around the 0.662 MeV gamma ray peak using the upper level discriminator (ULD) and the lower level discriminator (LLD) settings of the Single Channel Analyzer (SCA) of the Quantum 8. The Quantum 8 will then only count gamma rays in the SCA window set to include the 0.662 MeV peak only. window around 0.662 MeV, as shown in Fig. 2(b). When the Quantum 8 is set to Multichannel Scaling (MCS) mode, the video display shows the number of gamma rays accepted into the SCA window as a function of time. Fig. 3 shows a 137mBa half-life curve using a Dwell Time setting of 200 ms. With this setting, the vertical axis of Fig. 3 represents the number of gamma rays accepted into the SCA window during the 200 ms time interval allotted to each of the 1024 channels plotted along the horizontal axis. The 1024 channels correspond to a total time of 1024 x (0.2 s) = 204.8 s or 3.4 min.

To determine the 137mBa half-life one must fit data such as that in Fig. 3 to the radioactive decay law. A complete discussion of the radioactive decay problem may be found in Refs. 3 and 4. Briefly, for No excited nuclei present at time t = 0, the number present at a later time t is

N(t) = No e -t       (1)

where the decay constant is related to the mean lifetime, , by = 1/, and to the half-life, t1/2, by t1/2 = n(2)/. Experimentally, one observes the rate of decay R(t) given by

R(t)

=

- dN(t) / dt

=

Noe -t

=

Roe -t

      

(2)

where Ro is the initial rate of decay at time t = 0. Taking the logarithm of Eq. (2)

n R(t) = n Ro - t.      (3)

The rate constant can then be determined from the slope of the linear curve described by Eq. (3).

The half-life procedure of the Q8 program was used to fit the data of Fig. 3 using Eq. (3), with the result = 0.266 min-1 or t1/2 = 2.60 min. This agrees well with the value t1/2 = 2.551 min of Fig. 1. Measurements of one-component radioactive decays can often be made with this level of precision, thus providing additional data for identifying unknown radioactive nuclides.

Procedure

Obtain half-life data for 137mBa and the gamma emitting isomeric nuclide X. The 137mBa source data will give you a chance to perform data analysis on a nucleus with a known half-life before attacking the mystery nucleus X. For both sources follow the steps below.

(1) Obtain a standard gamma ray energy spectrum for each source, such as that shown in Fig. 2(a). Remember that the Quantum 8 must be in Pulse Height Analysis (PHA) mode to obtain energy spectra. Don't bother making a liquid source at first. Just put the cylindrical plastic source (blue for 137mBa, yellow for X) directly under the NaI(T) detector. Save those precious radioactive juices for step (4). Store your energy spectrum on disk.

(2) Use a small screwdriver to adjust the LLD and ULD potentiometers on the rear panel of Quantum 8 to obtain a windowed spectrum like that in Fig. 2(b). If the potentiometer begins to click with each turn of the screwdriver then you have gone too far in one direction. Start rotating the potentiometer in the other direction. Be careful not to tip over the NaI(T) detector assembly while working on rear panel controls. Store your windowed energy spectrum on disk.

(3) Set the Quantum 8 mode to MCS by pressing the Mode Control button on the front panel and observing the MCS indicator in the upper right corner of the Quantum 8 video display. Set the Dwell Time switch on the Quantum 8 front panel to 200 ms for 137mBa and 1000 ms for X. The Dwell Time setting controls how long the Quantum 8 "dwells" or counts for each of its 1024 channels.

(4) Prepare a liquid source by passing eluant through the appropriate cylindrical plastic source. Be sure to wear surgical gloves throughout this procedure. Squeeze the eluant through slowly. obtaining a teaspoon of source over a period of 20 to 30 seconds. If the eluant is forced through the source too rapidly, you will not obtain a source of sufficient radioactive intensity. Clean up spills with Kimwipes. Dispose of all Kimwipes in the designated container.

(5) Take the liquid source to the NaI(T) detector and immediately begin counting with the Quantum 8. A curve similar to that of Fig. 3 should slowly develop on the Quantum 8 display. Store your half-life data on disk. Properly dispose of the surgical gloves and wash your hands with soap.

(6) Before returning to step (1) with a new source, remember to set the Quantum 8 back to PHA mode and adjust the LLD and ULD discriminators so that a complete energy spectrum can be seen on the Quantum 8 video display.

Analysis

Your goal is to identify nucleus X by measuring the gamma ray energy and half-life associated with the gamma decay of its metastable excited state. To do this, complete the following steps.

(1) Calibrate the energy scale of your 137mBa spectrum using the gamma ray energy (662 keV) from Fig. 1. The 137mBa energy calibration can then be used to calibrate your energy spectrum for source X. Determine the gamma ray energy for source X and make an estimate of the uncertainty in your energy determination. Justify your uncertainty estimate. Print all windowed and unwindowed energy spectra.

(2) Use the half-life procedure of the Q8 program to determine the computer's best fit to your half-life data for 137mBa and source X. If your half-life for source X is not in the range of 80 to 120 min, you probably did not elute source X slowly enough and you will have to retake your data. Print out the computer's best fit value for your half-lives.

(3) The Q8 half-life procedure does a good job of finding a half-life that numerically fits your data but it tells you nothing about the uncertainty in your half-life determination. To estimate the uncertainty in your half-life determination perform the following steps.

(a) Print out the actual numbers of counts recorded by the Quantum 8 during its half-life measurements of 137mBa and source X. This can be accomplished using the Original Data option of the Print procedure in the Q8 program. The Q8 program will print 1024 numbers for each source.

(b) It would take too long to plot 1024 points for each source. Instead, add 20 channels together and plot just one point, giving a total of only 50 points to plot for the entire half-life of each source. Ignoring the data in channels 0 through 9, and starting with channel 10, one has:

Sum of channels 10 through 29 = Value of point 1
Sum of channels 30 through 49 = Value of point 2
.
.
Sum of channels 990 through 1009 = Value of point 50.

Ignore the data in channels 1010 through 1023.

(c) For each source, plot your 50 points on separate sheets of semi-log graph paper as in the 137mBa example of Fig. 4. Choose a semi-log paper such that your half-life curve will use as much of the paper as possible. Put appropriate error bars on your data points. Make sure you know how to use semi-log graph paper properly .

(d) Using a ruler, determine a best fit through your plotted data. Use the slope of this line to determine your best fit to the half-life for both sources.

(e) Look carefully at your best fit straight line from part (d). Now, draw a line with greater slope; one that might be regarded as a best fit by at least a few competent experimentalists. Look at your best fit line from part (d) again. Now, draw a line with lesser slope; one that might be regarded as a best fit by a few other competent experimentalists. Choose the lines of lesser and greater slope so as to "bracket" the range of slopes that would be selected by 68% of all competent experimentalists (see Fig. 4). Calculate the half-lives corresponding to the two lines that form your "bracket". Use these two half-life values to estimate your uncertainty in the determination of t1/2 for both sources.

(4) For 137mBa, compare your best fit half-life and estimated uncertainty to the accepted value given in Fig. 1. Is there reasonable agreement? Does your best fit half-life agree with the value given by the Q8 program in part (2)?

(5) For X, state clearly the gamma ray energy E ± dE and half-life t1/2 ± dt1/2 as determined by your data. What is X?

References
1. Table of Isotopes, 7th ed., edited by C.M. Lederer and V.S. Shirley (John Wiley & Sons, New York, 1978).
2. O.G. Berkes, The Quantum 8 Interface Program, B.A. thesis, Middlebury College, 1985.
3. R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (John Wiley & Sons, New York, 1985), pp. 558-560.
4. P.A. Tipler, Modern Physics  (Worth Publishers, New York, 1982), pp. 384-386.

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