dn(t) dt where λ is the constant decay probability per unit time. The solution is N(t) = N 0 exp( λt)

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1 (Aug revision) Physics 307 Laboratory Experiment #3 Probability Distributions and the Decay of Excited Quantum States Motivation: The purpose of this experiment is to introduce the student to counting errors associated with a random (Poisson) distribution and to study the decay of the excited quantum state of Ba 137. Most quantum mechanical decay processes have a constant probability of decay per unit time. A population of excited atoms or nuclei decays exponentially because of the constant decay probability per unit time. We wish to make a precision ( 1%) measurement of the mean life or lifetime, τ, of the 662 kev level of Ba 137. References: Foundations of Modern Physics by Tipler, Data Reduction and Error Analysis... by Bevington, Ch. 2 (in particular 2-1), 3-1, (in particular pp , 43-48), 5-1, 5-2, 5-3, and Ch. 6. Theory: The dierential equation which describes the population N(t) is dn(t) dt = λn(t) where λ is the constant decay probability per unit time. The solution is N(t) = N 0 exp( λt) where N 0 is the population at time t = 0. The lifetime or mean life, τ, is τ = 0 0 tn(t)λdt N(t)λdt = 1 λ Equipment: Bertran Associates 215 power supply ( V), Tektronix oscilloscope w/dual trace amplier, Frequency counter (modied), BNC portable NIM Bin, ORTEC Model 406A SCA, Mech-Tronics Model 500 AMP, Mech-Tronics Model 506 or 515 delay AMP, Mech-Tronics Model 776 CRM, Mech-Tronics Model 450 base and preamp, 2" 2" Nal (Tl) crystal mounted on an RCA 8053 photomultiplier tube, various cables, personal computer with a UCS 30 module and software installed. Experimental Procedure and Analysis: Part I: Probability Distributions 1

2 1. This part of the experiment involves constructing a histogram which approximates a Poisson distribution for a mean of about 200. It can be done using either the SCA and FC or, much more easily the Spectech UCS 30 and computer with a preset data acquisition time, t (i.e., in a Multichannel Scalar or MCS(Internal) mode). Set the SCA LLD and ULD, or the computer's LLD and ULD to sample the 662 kev photoionization peak (or photopeak). Adjust the distance from the source to the detector for a count rate between 170 and 220 counts/t. If you are using the PC, then t should be 1 sec. You do not need to make a table, you can construct a histogram directly. The histogram should contain one set of 1024 samples. Use a bin size of 5 (i.e. record the occurrence of counts/t between 170 and 174 in one bin, etc.). See the attached directions for using MS-Excel to construct a histogram. Prepare a table of the number of samples vs average bin value (i.e. for bin use 172). Use the table or the PC spreadsheet program (i.e., Excel) to compute a mean, x = ΣN i=1x i N where x = counts and standard deviation, σ x = Σ N i=1(x i x) 2 for your sample distribution. Compare the standard deviation of your sample distribution to the standard deviation of a Poisson distribution with your mean. For a Poisson distribution σ x = x. Compute the standard deviation of your mean (i.e., σ/ N). You will use this to specify the uncertainty of the mean. 2. Relate the standard deviation of the Poisson distribution to the experimental uncertainty of each counting sample as follows: For several (say 9) data samples selected at random, estimate the uncertainty for each data sample assuming it is an element of a Poisson distribution (i.e. σ = counts. Compare these measurements and their uncertainties to the true count rate (i.e. mean) from Part I. For this set of data samples compute the mean, standard deviation and standard deviation of the mean. Again compare this with the true count rate from your 1024 samples. 3. Fit the histogram with Gaussian curve, as a good approximation to the expected Poisson distribution. You can use CurveExpert, QtiPlot, or similar software. Some programs have diculty with large numbers and if you run into such problems just N 2

3 rescale your data into kilocounts. Compare the variance of the tted Gaussian t to the average number of counts. Number of Samples Counts/Time Compare your histogram to Bevington's Fig Raw Data (counts/t) Part II: The Decay of Excited Quantum States The Cs 137 (half life 33 years) emits a β particle (electron) decaying into an excited state of Ba 137. The excited Ba 137 nuclei then decay to the stable ground state of Ba 137, emitting a 662 kev γ-ray (photon). The mean life of the excited Ba 137 is a few minutes. The Cesium is chemically bound to the internal structure of the blue (or white) sources, but the accumulated Barium, which is chemically dierent, can be ushed out of the source by forcing 1 ml of saturated NaCl solution through it. Part of the Barium which has formed within the last few minutes before the source is ushed (eluted) will still be in the excited state. The instructor will show you how to elute the radioactive source. Because this process can become messy your instructor may choose to elute the sources instead of the students. It is recommended that you wear rubber gloves during this procedure. THE RULE AGAINST FOOD AND DRINK IN THE LAB MUST BE STRICTLY OBSERVED DURING THIS EPERIMENT. Deposit the used solution in the storage container after the experiment. Do not discard the solution in the sink. It is advisable to have the experiment ready to operate before you elute the radioactive source. Record the spectrum from the Cs 137 source to insure that the detection system is working properly. Remember that even when you are looking at the Cs 137 source the γ- 3

4 rays you are detecting are from the Ba 137. Also remember to move all radiation sources far away from the detector to minimize background when taking data. This experiment can be performed using the SCA, FC (frequency counter) or, most easily, using the computer as a multichannel scaler or MCS(Internal) mode. If you use the SCA then set the LLD and ULD to capture the entire photopeak and most of the Compton events. Discard the very low energy noise pulses. Remember to scan long enough ( 10 lifetimes) to determine the background count rate which must be subtracted in your analysis if you perform a simple linear analysis. Record your data in a table and use a linear least-square t to the ln (count less background) versus time in order to determine the lifetime. If you are using the computer then follow the instructions on the attached pages to use the computer as a multichannel scaler. A dwell time of 10 or 20 sec is optimal for a lifetime of a few minutes. Remember to record long enough ( 10 lifetimes) to determine the background count rate which must be subtracted in your analysis. Use PHA mode initially to set a window which includes the photoionization peak and most of the Compton signal. It is possible to transfer this data directly to Excel or other curve tting program for analysis. If you choose to use Excel, refer to the Experiment #1 handout containing instructions on transferring data from UCS 30 to Excel. NOTE: Excel has no capacity for including a background term in its exponential t. CurveFit, using the User-dened function, or other products have this capacity. You should also substract the background and perform a linear least-square t to the natural log (counts less background) versus time in order to determine the lifetime. Such a plot will enable you to assess the quality of the t and verify that you observed a single exponential decay. Students desiring extra credit, or taking the course for Honors Credit, should use a properly weighted linear least square t or properly weighted nonlinear least square t. Final Questions: 1. What is N(t) when t = τ? 2. Find a relationship between τ and the half-life T 1/2, where N(T 1/2 ) = 1 2 N Explain why the observed T 1/2 of the 662 kev γ-rays from the Cs 137 source is 33 years, even though the γ-rays you are counting come from the decay of the Ba 137, which has a T 1/2, of only a few minutes. 4. Is the time interval between when the source is eluted and when you begin your measurements critical? Why or why not? 5. What is deadtime? Is the computer channel deadtime of 1.8 µsec signicant? 4

5 Personal Computer as a Multichannel Scalar The multichannel scaling function is comparable to a series of scalars. Incoming data, independent of the pulse height, is stored in one channel for a xed period of time. An internal clock pulse steps the address register to the next channel on time basis. The data collected represents a relationship expressed as the number of counts or decays per unit time. Most of the instructions from Experiment 1 for using your PC as a Multichannel Analyzer are applicable here. Only the changes necessary for switching to multichannel scalar mode will be given here. Switching to Multichannel Scalar Mode 1. All counts between the LLD and ULD are counted in the MCS mode. The LLD should be set properly to cut o the low amplitude noise before switching from the MCA to the MCS mode. 2. To switch from multichannel analyzer mode to multichannel scalar mode, click on `Mode'. Click on `MCS(Internal)'. 3. To set the dwell time, click on `Settings'. Click on `MCS'. Chose a `Dwell Time'. A dwell time of about 10 sec is a good choice for a lifetime of several minutes. You need to have enough counts in each channel so that you have small statistical noise. You need to have the dwell time much less than the lifetime in order to resolve the exponential decay. 5

6 CREATING A HISTOGRAM USING MS-ECEL 1. Import or type your data in a column. 2. Establish the boundaries between bins in another column. 3. Click on Tools. 4. Click on Data Analysis. 5. Select histogram and click OK. 6. For input range click on your rst piece of data and drag the cursor through all of your data in the rst column. 7. Although bin range is optional, you will likely want to use the bin ranges established earlier to create your desired historgram. Select the rst bin boundary and drag the cursor through the entries in the second column. 8. Select Chart Output. 9. You should new see the chart but the bars will have gaps between them and chart will likely be too small. 10. To x the bar separation, just right click on one of the bars, click on Format Data Series, then Options, and chose a gap width of Stretch chart to the desired size. 12. Finally, you should be careful about your bin labels. MS-Excel will label a bin a with its upper limit and construct a histogram with the bar centered to the right of the label. This can be xed in the histogram table. 6