Autumn, 2016 Physics 3700 S. Durkin, R. Leonard ...durkin/phys3700/Autumn2016/P3700... · Experimental Physics Instrumentation and Data ... Data is commonly reported in the form of

Autumn, 2016 Physics 3700 S. Durkin, R. Leonard Experimental Physics Instrumentation and Data Analysis Lab

(Applied Statistics for Physicists)

LaboratoryInstructions

TableofContents

Experiment1 2 ObservingChance → Data&Distributions 2Experiment2 3 ArrivingRays,DepartingDarts → TheBinomialDistribution 3Experiment3 5 CascadingBalls → TheNormalDistribution 5Experiment4 7 SortingParticles → CurveFitting 7Experiment5 9 ResolvingEnergy → GoodnessofFit 9Experiment6 13 TimingDecay → Estimation 13

Autumn, 2016 Physics 3700 S. Durkin, R. Leonard

2

Experiment 1

Observing Chance → Data & Distributions

The field of statistics begins by describing data that vary according to, what appears to be, random chance. These simple experiments provide a start for understanding the related processes of collecting and analyzing data. Data is commonly reported in the form of histograms and compared with theoretical distributions. An important skill is using computer programs to simulate the mechanical systems.

• Histograms can be created using Scientific Data Analysis and Visualization (SciDAVis). • Programs (C++) can be created using Dev-C++ (Bloodshed Software).

Tossing a Coin Mechanical Experiment Toss a coin 100 times and record the number of heads and tails. Plot the occurrence fractions for heads and tails. Compare the recorded number of heads with the expected number. (A brief qualitative statement will suffice.)

Simulation Create a computer program to simulate the tossing of a coin. Use the program to toss the virtual coin 10!, 10!, and 10! times. Overlay the occurrence fractions on the same plot. Compare each data sample with the expected probability distribution for a coin toss.

Rolling a Die Mechanical Experiment Roll a six-sided die 100 times. Record each roll of the die, make a histogram, and plot the occurrence fractions. Compare the recorded numbers with the expected numbers.

Simulation Create a computer program to simulate the rolling of a die 10!, 10!, and 10! times. Overlay the occurrence fractions for each of the computer runs on the plot for the 10! rolls of the die. Again, compare each data sample with the expected probability distribution for a six-sided die roll.

Rolling Two Dice Mechanical Experiment Roll two six-sided dice 100 times, and plot the occurrence fractions for the sum of the two values. Plot the theoretical expectation for the probability distribution on the same plot as the measured distribution. (Derive a formula for the probability of obtaining a particular value.) Compare theory (the derived formula) and experiment. Simulation Create a program to simulate the rolling of two dice. (Start with a copy of the previous program, and modify it.) Again, use this program to roll the dice 10!, 10!, and 10! times. Plot these results on a single histogram. Overlay these results with the 10!-roll plot. Once again, compare these results with the theoretical expectations.


3

Experiment 2

Arriving Rays, Departing Darts → The Binomial Distribution

The binomial distribution is, in a fundamental sense, the progenitor of all other distributions. The Poisson distribution is a special limiting case. These experiments explore these two distributions, which are discrete as opposed to continuous.

Dice and Discrete Distributions Gather a set of a dozen dice (six-sided). Toss the set 100 times. Pick your favorite integer from 1 to 6, and, for each toss, record the number of times that it comes up. Make a histogram (normalized) that displays the fractions for the possible results (0-12). Add error bars. (Use Taylor, Section 3.2.)

• Consider a binomial distribution as a model. Plot the theoretical expectations along with the experimental results. Compare the mean and variance with theoretical expectations.

• Now consider a Poisson distribution having the same mean (theoretical) as the binomial. Plot the Poisson distribution along with the binomial expectations.

Question: How would you change the dice and the experiment, if you wanted to generate a Poisson distribution from the toss of dice? (Locate an example of such a crazy die.)

Cosmic Rays and the Poisson Distribution The rate of cosmic-ray particles passing through a Geiger counter (or your body) is governed by Poisson statistics. The Geiger counter has a wire strung inside a gas-filled chamber. The wire (anode) is held at high voltage (a few kV) with respect to the chamber (cathode). A traversing particle knocks electrons off a few of the gas molecules. The electrons accelerate toward the wire under the strong electric field resulting in an avalanche of electrons. The electronic signal produces the click sound. The signal is also sent to an electronic device, which uses the LabVIEW Geiger program to count the number of particles traversing the chamber.

• Turn on the Geiger counter. First set it to the Audio position. You should hear an occasional click indicating that the counter is working. Then set it to the On position. (When finished return the counter to the Off position.)

• Run the Geiger counter program. (Open the LabVIEW application and choose the Geiger with Timer virtual instrument program.) The Front Panel should appear.

• Run a short trial experiment by using the default settings: Count Interval = 10 seconds and Total Number of Intervals to Run = 6. (Therefore the program will take data for 1 minute.) Click the right arrow button at the top of the window to start the program. The number of events in each interval will be written out to a file (after a prompt for the file name) after the minute is over. Stop the counting by clicking on the icon that looks like a stop sign. The file can be opened in a text editor (Notepad), and the numbers in the file can be copied and pasted into SciDAVis.

• Now run the full experiment by setting the Count Interval to 15 seconds and the Total Number of Intervals to Run to 80. (Therefore the program will take data for 20 minutes).

• Make a histogram of the number of events per 15-second interval. Calculate the mean of the distribution (average number of events in a 15-second interval). Use the mean in the Poisson distribution formula to calculate the expectation probabilities for the number of events per time interval. Superimpose these results on your histogram.

Questions: How many data points have zero events, and how many have 10 or more events? What are the Poisson predictions for these numbers? Overall, is the experimental distribution consistent with the expectation from Poisson statistics?


4

Darts and the Value of π In this exercise we will determine the value of π (=3.14159…) in a strange way. Experiment (Dart Throwing) Determine π by throwing a dart 100 (and perhaps more) times. First, create a dartboard target on a regular sheet of paper. Inscribe a circle within a square. (Reason how the value of π is related to the ratio of the circle’s area to the square’s area.) Resist the temptation to be an expert dart thrower, for your tosses should be uniformly distributed throughout the square. Thus π can be determined from the probability of a dart falling within the circle (assuming that it is also in the square). Calculate the value of your π together with its uncertainty. Calculate the theoretical uncertainty as follows. (Make use of Taylor, Chapter 10.) The variance for the number in the circle (with N throws in the square) is 𝑁𝑝𝑞. Here, 𝑝 = 𝜋 4 and 𝑞 = 1− 𝑝. The standard deviation (SD) is the square root. Divide by N to get the SD per throw. Multiply by 4 to get the SD for the value of 𝜋 determined by this procedure.

Calculate the discrepancy between your value of π and the true value of π . (Express this in relative terms; that is, as a fraction: absolute discrepancy divided by true value.) Simulation (Monte Carlo Integration) Determine π by writing a computer (e.g. C++) program. One method (called Monte Carlo Integration) uses the random number generator as follows:

• Generate 2 random numbers each in the range [0,1]. Call one random number x, the other y. • Assume that (x, y) represents the coordinates of a point inside a unit square. (Note: the units used to

measure the size of the sides of the square are arbitrary.) • Assume that there is a circle inscribed in this square with radius = ½, and calculate if (x, y) falls

inside the circle. (Note: Alternatively, consider a quarter-circle with radius = 1.) • Repeat the above procedure keeping track of the number of (x, y) points generated (= dart tosses) and

the number of times the point lies within the circle. • Calculate π after all of the points have been generated.

Determine π using runs of 102, 103, 104, 105 and 106 points. For each case, calculate the relative discrepancy between your calculated value and the true value of π. Plot the absolute value of the relative discrepancies versus the number of points using logarithmic scales for each axis. As a significant comparison, plot on the same log-log graph the relative theoretical standard deviation (𝜎! 𝜋). For this comparison set, connect the five data points with an appropriate curve.

Determine π using 100 separate computer trials of 100 points each. For each trial calculate π and make a histogram of your 100 values. Calculate the average value of π and its standard deviation. Compare this with the expected theoretical standard deviation.

Questions • How does the accuracy of your value of π increase with the number of points? (For example, does

the accuracy increase linearly with the number of points or according to the square root?) • Where does the result you obtained from tossing the darts fall on the histogram? Using the histogram

as a probability distribution for obtaining π from 100 dart tosses, comment on your expertise as a (random) dart player.

• What is the connection of this dart throwing experiment to the binomial distribution?


5

Experiment 3

Cascading Balls → The Normal Distribution

Measurement errors tend to follow the normal distribution (Gauss distribution) in virtually every area of quantitative science. The central limit theorem (CLT), a general and important result from probability theory, provides an explanation for this tendency. (See TAYLOR, Section 10.5 for a succinct and somewhat implicit connection between the binomial distribution and the CLT. Note that the CLT is not explicitly named in the text.)

Central Limit Theorem Suppose that a certain quantity is the sum of many small random variables. Each of these has its own distribution law but with finite mean and variance. Then, for this random variable sum:

• the mean will be equal to the sum of the individual means, • the variance will be equal to the sum of the individual variances, and • the distribution will be normal (a Gauss distribution). Note that the CLT is valid even if the summed variables are from different probability distributions! All that is required is that the distributions have finite means and variances, and that no one term dominates the sum. Thus, since uncertainties tend to arise from a large number of sources adding together, under a wide range of circumstances a normal distribution will result. This is why we so often see the familiar bell curve in statistical results.

Experiment with Mechanical Histogram Machine (Ball Cascade) In this exercise, we will see that a normal distribution governs the landing location of stainless steel ball bearings. The apparatus (a Galton box) consists of a grid of evenly spaced pins sandwiched between two plates. When a ball is dropped into the grid, it can scatter to the left or right on the first pin encountered. The scattered ball then hits the next pin and scatters again left or right. The process continues until it reaches the bottom of the grid where there are evenly spaced slots to receive the ball. The array of slots (slightly tilted) acts like a histogram machine with each slot representing a bin in the histogram. The number of right and left scatterings determines the final landing location. There are 14 rows of staggered pins. Each scattering contributes a deviation from the center horizontal location where the ball is released. The deviation can have positive or negative sign. (Since the preceding drop might not be exactly vertical, the probabilities might not be exactly half and half.) The CLT suggests that the location should be distributed approximately normally (according to the Gauss distribution). Procedure • Insert the blocker all the way, so that no ball will escape from the slots. • Gather ~70 ml of balls in a beaker. (This corresponds to ~300 balls, enough to almost fill up the

central slot occasionally.) • Hold the funnel with the base in the notch at the top of the apparatus and pour the balls slowly into

the funnel. (Pouring too fast will cause the balls to clog up inside the funnel. • Count the number of balls in each slot to produce a histogram. • Pull out the blocker and hold the ball drain over a beaker to collect the balls. (You might need to

lightly shake the apparatus to collect all the balls, as they often jam up.)

(Any balls that happen to fall on the floor can easily be recaptured with the green magnets. Remember: Don’t lose your bearings, or you may not find your way to the end of the experiment!)


6

Histogram Statistics Compare your histogram (with error bars) to a normal distribution. First calculate the average location and standard deviation. Superimpose a normal probability distribution onto the histogram. Ensure that the Gauss distribution has the same mean, standard deviation, and normalization. Comment on this comparison.

Variation of Experimental Mean Repeat the procedure 9 more times for a total of 10 trials. Compute the mean for each trial and the standard deviation of these mean values across the 10 trials. Compare with the value expected based on the ~300 ball distribution.

Hint: The uncertainty of the single-ball landing location is estimated by the standard deviation of the multi-ball distribution of locations. The uncertainty of the mean of the multi-ball distribution is estimated by the standard deviation of the multi-trial distribution of means. The latter uncertainty is smaller than the former, and one should be able to predict how much smaller.

Simulation by Computer Use the computer to illustrate the CLT. This exercise uses the properties of a built-in random number generator that one finds in a program library. The random number generator gives us numbers uniformly distributed in the interval [0,1]. Problem (See TAYLOR, Section 5.2, and Problem 5.9.) Show, using the integral definitions of the mean and variance, that this uniform distribution function has these properties:

𝜇 =12 𝜎! =

112

Random Number Sum Write a program to add 12 random numbers (each from an identical uniform distribution).

• Calculate the expected mean and variance of the sum (using the CLT). • Generate three separate cases for the number of 12-number sums: 10!, 10! and 10!. • Group the resulting data, for each case, in 48 equal-sized bins (1 4 unit each). • Compare the three cases, with error bars included, to the expected normal distribution. Notes • Normalize the expected Gauss distribution properly. (This will involve both the standard deviation

and the bin size as well as, of course, the number of data sums.) • Choose the bin number in a single computer code statement. Think of a way to convert a general

real number into the appropriate integer bin number. (Yes, one could scan through and test each bin one-by-one. However, this would be an inefficient solution to the problem!)

• Locate the bin at the midpoint of the bin. (For example, the 1st bin, from 0 to 1 4, will plot at 1/8.) • Save this program. It will be useful again.


7

Experiment 4

Sorting Particles → Curve Fitting

Previously we detected high-energy particles with the Geiger counter. Our equipment automated their counting and sorting based on arrival time. Again, we will count and sort high-energy particles. However, now we will sort them based on their energies.

To automate this task, we will use a combination of systems. A subtask for the lab will be to understand, to a degree, these various systems. Another subtask is to perform an energy calibration. We rely on a certain response that is proportional to energy. Determining that proportional response involves the fitting of data pairs to a line. Within the study of statistics, this constitutes a prominent example from the general topic of curve fitting.

Preparation These activities should be completed before starting the lab.

• Pass the radiation safety course. (Submit the certificate via Carmen.) • Read THE ART OF EXPERIMENTAL PHYSICS (Preston and Dietz). • Read NUCLEAR SPECTROSCOPY WITH THE COMPUTER.

Equipment We will perform some elementary nuclear physics experiments using radioactive sources that emit gamma rays (or positrons). (Typical energies for gamma-ray photons range up to a few MeV.) The information transmission chain involves these steps:

• Sources of radioactivity (cobalt-60, sodium-22, cesium-137) • Detector (sodium iodide [NaI] scintillator crystal with photomultiplier and phototube base) • Amplifier • Interface to computer • Computer running LABVIEW program called MCA (Multi-Channel Analyzer).

Measurement of Spectra Measure energy spectra of nuclides of cobalt, sodium and cesium: Co-60, Na-22, and Cs-137. Each of these sources emits gamma-ray photons. These are the particles we detect and sort by energy.

Settings First, without a specific source, view the background. Ensure that the upper threshold adjustment is set to 5 V, and the lower threshold adjustment is set to 0 V. (Voltmeters are available for adjusting these thresholds.) Then, measure the Co-60 spectrum, and adjust the gain of the NaI Spectroscopy Amplifier so that the upper peak in the spectrum is at about 4 V. After this, maintain the same amplifier gain for all measurements, as we want the same channel-to-energy calibration for all sources.

Spectra Data Measure each of the three spectra separately. Accumulate enough data, until the spectrum has smoothed itself out. (The time needed will depend on the strength of the source, perhaps as much as 10 minutes.) Write each spectrum out to a data file. Then import the data to the data analysis software. Plots Include a plot of the spectrum from each of the sources (three plots total) in your lab report. Then provide an additional plot with all three spectra superimposed (with distinguishing colors). Each should be normalized the same by dividing by the total number of respective events. (Connect the points, for each spectrum, with a continuous curve rather than as a bin histogram. Error bars are unneeded.)


8

Energy Calibration There are a total of five gamma rays in these three spectra that appear as approximately symmetric peaks (two for Co-60, two for Na-22 and one for Cs-137). Calibration Data • Record the MCA voltage closest to each of the peaks. • Determine the corresponding gamma-ray energy for each peak. These energies can be found using

information in THE ART OF EXPERIMENTAL PHYSICS. This will involve some careful thinking. (See Issues below.)

Calibration Curve Calibrate the detector by making a linear least-squares fit of energy (E) versus voltage (V). That is, determine the constants A and 𝐵 in the relationship:

𝐸 = 𝐴 + 𝐵 ∙ 𝑉

• Write a program to fit the data to a line, and find the constants. (TAYLOR, Chapter 8) • Use a statistical software package (SCIDAVIS) to fit your data to a line.

Compare with the two sets of constants just determined. (They should be nearly identical.)

Issues: Nuclear and Particle Physics This entire laboratory course is devoted to understanding probability and statistics in physics. Although the course is not concerned with any specific area of physics, this experiment has a context of nuclear systems with particle emissions. On a secondary basis, we want to understand our system at an appropriate level, so as to be able to interpret the statistics of our measurements. In this regard, address the following ideas associated with the observed spectra.

Note that one of the Na-22 spectrum peaks is due to positron annihilation (e+e! → 𝛾𝛾). Here each gamma photon has an energy equal to the rest-mass energy of an electron, 0.511 MeV. However, only one of the photons goes into the NaI crystal. Where does the other photon go? Include an explanation in your report. (Hint: Consider conservation laws.)

You should see a number of other interesting features, as well. Notably, the Cs-137 spectrum includes an X-ray peak at the low-energy end of the spectrum. (The blue sources do not show this X-ray line!) Also, Cs-137 has a very obvious Compton shelf and edge with a backscatter peak. Include a brief discussion of these phenomena.


9

Experiment 5

Resolving Energy → Goodness of Fit

A particular energy peak observed in the gamma-ray spectrum from a radioactive source is, presumably, evidence of many photons having the same energy. Nevertheless, the peak appears somewhat spread in energy. This lab focuses on both describing this observed spread in energy of the peaks as well as explaining that spread. Along the way, the notion of resolution of a detector will be defined. The testing of an associated mathematical model will employ the important statistical concept of goodness of fit.

Explaining the Spread First think about an explanation, and distinguish two alternatives. The spread could be due to uncertainty that is introduced by the detector. However, consider that the source photons would already and inevitably be distributed in energy to some degree. Fundamental quantum mechanics (the Heisenberg Uncertainty Principle) dictates that uncertainties in energy and time are related. These photons were created from a nuclear system that existed for some finite time. Therefore, the energy will be accordingly uncertain. Although the logic is subtle and the detailed mathematics a bit involved, the uncertainty in energy (Δ𝐸) is related (approximately) to the nuclear system existence time (Δ𝑡) by the simple expression Δ𝐸~ℏ/Δ𝑡. (Planck’s constant ℏ, notably, is nonzero in the quantum world.)

Calculation of (Quantum Mechanical) Natural Width Look at the cesium-137 peak. Although Cs-137 lasts about 30 years, it decays into barium-137. This nuclear system then emits the photon we detect after about 2 or 3 minutes (∆t~10! s). Use the idea of quantum uncertainty to estimate the spread in this photon’s energy. (Express the value in MeV to the nearest power of ten.) [ℏ~10!!"MeV ∙ s] Compare this uncertainty in energy to the width of the observed peak from your data. For the width, use the full width at half maximum (FWHM). (See TAYLOR, p. 156, Problem 5.12.) Judge whether quantum uncertainty of the source photons adequately explains the observed peak width. This calculation does not disprove the idea that a spread in the source photons is significant. In general, both source and detector spreading might be comparable. However, we will assume the photons to be essentially mono-energetic, and effects within the detector best explain the observed spread in energy.

Describing the Spread Now think about a description, and recognize two contrasting approaches. Observation The experimentalist will measure the spread width (W) as the FWHM and relate this to the peak energy E. Thus, one can define the relative quantity called energy resolution R.

𝑅 =𝑊𝐸 with 𝑊 taken to be the FWHM

Theory A Gauss distribution seems to be a good starting point for a mathematical model of the spectrum around a peak. Remember that the normal (Gauss) distribution naturally results from the sum of a large number of contributing error effects.

Connecting Observation and Theory Relate the two measures of the width of the peak: observational (FWHM) and theoretical (𝜎). [That is, prove the relationship found in TAYLOR, Problem 5.12.]


10

Estimating Resolution Simple Estimate (directly from the definition) Refer to the gamma-ray spectral data from the previous lab experiment, in which a five-point calibration of energy versus detector voltage was determined. The useful photon energy peaks are from Co-60 (2), Na-22 (2) and Cs-137 (1). Calculate the resolution for each peak after measuring its FWHM. Technically, one should use energies to calculate energy resolution. However, for this purpose, it is sufficient to use voltage as a substitute for energy. The resolution, one should notice, is inherently dimensionless. If energy were directly proportional to voltage, then the resolution, so calculated, would be exactly the same. With a small offset in the energy-voltage calibration, the results for resolution will be only slightly different.

Development of The Theoretical Model More realistically, the region near the gamma-ray peak can be thought to consist of two components. The gamma ray itself, which we can model with a Gaussian, is the signal (s). If there were no specific radioactive source, then we would see the background (b). For the signal, model the number of events in bin i (with center 𝑥!) to be:

𝑁!! = 𝐴 ∙ exp − !!

𝑧!! 𝑧! ≡𝑥! − 𝜇𝜎

We would like to determine the fitting parameters A, 𝜇 and 𝜎 that best describe our data. However, we also have background in our spectrum. Model the number of background events in the bin (in the region of interest) as:

𝑁!! = 𝐵 + 𝐶𝑥! with a total of 𝑁! = 𝑁!! + 𝑁!!

Again, the parameters B and C are to be determined from our data. Thus, the data will be represented by the sum of signal and background functions. Concept Question Why is the linear background model a reasonable assumption in the region of interest? Curve Fitting Algorithm The problem now is to determine the five parameters (A, 𝜇, 𝜎, B, C). The basic idea is to choose these such that the probability of obtaining this particular data set is maximized (Principle of Maximum Likelihood). This reduces (with a special assumption about the uncertainties) to the technique of Least Squares. This particular problem is rather difficult, since the above function is nonlinear in some of the variables. A set of five coupled non-linear equations must be solved. The program SCIDAVIS includes just such a feature that enables the fitting of more complicated functions to data.

Improved Estimate (indirectly from a Gaussian Curve Fit) Use the relevant procedure (outlined in the appendix to the lab instructions) to find the mean and standard deviation for each peak. From these (and remembering the relation between FWHM and 𝜎):

• Calculate the energy resolution. • Plot resolution versus energy. • Fit this data using a power-law function: 𝑅 = 𝐾 ∙ 𝐸!!, with K and m as free parameters. • Explain why the value of m might be close to one-half. Hint: For this last explanation, start with the definition of resolution. Then use the idea that the energy inferred by the detector comes from counting a large number of scintillated photons all of the same energy. Consider fluctuations in the count from one gamma-ray photon to the next.


11

Simulating a Source-Detector Combination The goal here is to generate by computer simulation a histogram similar to that generated in our actual experiment. However, unlike the actual experiment, we can easily allow the detector resolution to vary. Source Specifically, we will model the cobalt-60 gamma-ray spectrum. The Co-60 decay results in two distinct gamma-ray photons having energies of 1.172 and 1.333 MeV. (Ignore other possibilities, which occur less than 1% of the time.) Detector Assume that the response of the detector is given by a Gaussian function. Adapt the Gaussian distribution generator of the earlier lab. Recall that, according to the central limit theorem, the sum of random numbers gives a good Gaussian distribution. By adding 12 uniform random numbers we generated a distribution with mean of 𝜇! = 6 and standard deviation of 𝜎! = 1. The trick is to transform to a distribution with mean of energy 𝜇! and standard deviation 𝜎!. The key to this transformation is that the relative deviation (at bin i ) is the same:

𝑧! ≡𝐸! − 𝜇!𝜎!

=𝑥! − 𝜇!𝜎!

Exercise Validate this transformation using the definition of a Gauss distribution. That is, show that if properly normalized with one variable, then the distribution will automatically be normalized with the other.

Combining Source and Detector For each Co-60 decay one must generate separate gamma-ray photons, one from the distribution with the low energy and one from the high energy. Note that the standard deviations as well as the means for the two distributions will be different. This is because the resolution is taken to be the same for each peak. (Although resolution has already been shown to be mildly energy-dependent, ignore this effect.) Generation of Histograms Use the following prescription to simulate the energy response of the detector to Co-60.

• Divide an energy window into discrete bins. (Use 25 bins of size 15 keV beginning at 1.065 MeV.) • Set the resolution. (Successively apply the cases of 2%, 6%, and 18%.) • Generate a random number (normally distributed) for each peak. • Transform to energy variables separately for each peak. • Assign the two events to energy bins (if within the window). [Use one code line to bin an event!] • Repeat for 10! simulated decays. • Plot a histogram of the energy spectrum for each of the three resolution values.

Interpretation • Which of the three histograms looks most like your actual Co-60 spectrum? • For the 6% case, explain why the peaks do not have the same height. • For the 2% and 18% cases, comment on the ability to distinguish the two peaks.


12

Appendix: Fitting a Gamma-Ray Peak Using SCIDAVIS Data Set • Copy data from the region of the peak. (Choose perhaps 15-25 points.) • Plot this peak-region data. Fitting Procedure • Use (from the Analysis menu) the Fit Wizard. • Enter the function.

A * exp ( -0.5 * ( (x-mu) / sig)^2 ) + B + C*x

• Enter the parameter names of the function.

A, B, C, mu, sig

• Click the Fit>> button.

Initialization • Enter Initial guesses for the parameters. (The best way to make these guesses is to plot the data in a

separate window along with the fitting function. Adjust the parameter values until the fit looks good by eye. Then transfer the values to the Fit Wizard.)

• Select (for the Y Error Source) Statistical (Poisson). • Click the Fit button.

Result • Observe, from a successfully convergent fit, the plotted fit function along with the data. (An

unsuccessful fit will have failed to converge. One must return to reenter initial guesses for the parameters.)

• Record the best-fit parameter values. (Note that these values, according to the premise of the maximum likelihood algorithm, maximize the probability that the model in question generated this particular set of data.)

• Record the value of chi squared per degree of freedom. • Judge the goodness of fit according to the chi-squared statistical measure. (This will involve

interpreting the corresponding probability.) Notes • All result values should be recorded with due recognition of significance of figures. Observe the

parameter uncertainties in the output. • Also observe the measure of correlation (called 𝑅!). When this value is very close to 1, the data and

model function will be seen to be very similar by eye. Nevertheless, the goodness of fit might turn out to be poor (low probability of data based on the model). The opposite results are also possible (poor correlation but strong goodness of fit). Try to explain these situations. Remember, large differences with little data might be rather insignificant, but small differences with much data could be very significant!


13

Experiment 6

Timing Decay → Estimation

The goal of this lab is to estimate the decay lifetime of a radioactive substance. Estimation involves determining not only the best value for this quantity but also the experimental uncertainty. The lifetime has already been measured many times before, so we want to compare with the standard value. The discrepancy between these two values compared with the experimental uncertainty then forces us to judge the validity of our presumed theoretical model. We must ask whether we are measuring what we think we are. If not, then we suspect that other systematic errors, rather than random errors, might exist.

Theoretical Model The arrival of cosmic rays is described well by a Poisson distribution; that is, the arrival is random but at a definite unchanging rate. The case of nuclear reactions with a consequent emission of radiation is similar, except that, typically, the amount of substance undergoing change is not resupplied. Rather, the number of radioactive nuclei progressively decreases with time. The activity (R), which we know how to measure, is proportional to the abundance (N). That constant of proportionality, having dimensions of inverse time, will effectively define our lab focus: the lifetime (𝜏). The activity is defined as the opposite of the rate of change of the abundance. (The rate of change is negative, so the activity is positive.) Putting these two definitions together and solving the resulting differential equation yields the familiar exponential decay behavior. The activity is proportional to the abundance, so both of them decline exponentially from their initial values (𝑁! and 𝑅!). In summary,

𝑅 ≡1𝜏 ∙ 𝑁 and 𝑅 ≡ −

𝑑𝑁𝑑𝑡 so

𝑁𝑁!

=𝑅𝑅!

= exp −𝑡𝜏

One could, in principle, determine the lifetime from the first relationship, but the abundance is not as easy to find. So, in practice, it is easier to observe the activity over a period of time. Then the lifetime becomes a fitting parameter for the activity-time relationship. Estimation is the procedure by which one determines such a parameter from a given measured data set.

Nuclear System Cesium to Barium We are familiar with cesium-137 as a radioactive source. It is abundant because it is a product of uranium fission in nuclear reactors (and bombs). It is not otherwise abundant in nature, because it decays within a few decades. When it decays, by a neutron transforming to a proton (beta decay), it becomes barium-137. (See the energy diagram in THE ART OF EXPERIMENTAL PHYSICS.) The Cs-137 beta decay results first in a so-called metastable state of Ba-137. This decays typically after a few minutes, and gives off the signature photon of energy 0.662 MeV. So, when cesium decays, we actually see barium decay.

Chemical Separation Process Since Ba-137 is abundant (because Cs-137 is abundant) and because its lifetime is of a convenient duration (not too long or too short), it would seem to be relatively easy to measure its lifetime. However, within a given mass of Cs-137, the disappearance of Ba-137 is matched by its ongoing generation. (This is called secular equilibrium.) In order to observe the Ba-137 decay, we must somehow separate the barium daughter from the cesium parent. We will employ a system called an isogenerator in which cesium is attached to a resin, and barium is removed from it with a particular solution. The process, formally called elution (washing) in the field of chemistry, is also referred to by the colorful phrase milking the cow. We will milk the cow to obtain barium and leave cesium behind.


14

Preparation Energy Window We can reduce the background level greatly by focusing on the very numerous events of the Ba-137 peak. (These gamma rays give up their full energy to one electron in the NaI crystal.) Therefore, we will set a narrow energy window centered on that peak energy. This can be accomplished using the Upper Level (UL) and Lower Level (LL) controls on the spectroscopy amplifier (along with a voltmeter). Input pulses from the detector that are above the voltage level set by the UL control will not send a trigger signal to the computer (via the TRG OUT connector). The same is true for input signals below the LL setting. Data Procedure • Set the energy window. Using the program MCA with Timer and a Cs-137 source, adjust the

window. To avoid making the voltage window too small: • Apply a wide-open (0-5 V) window, and measure a Cs-137 spectrum for 10 s. Note how many

events are in the peak channel. • Adjust to show only the main peak (about one standard deviation each way), and record another

spectrum for 10 s. The number of events in the peak channel should be about the same as before. If it is significantly lower, then open the window a bit. (You may have to start and stop the program several times.)

• Run a short trial experiment. In order to be ready for the actual experiment, perform the following test of the data acquisition system. • Set the Total Time of Counting (min) to 2 and Duration of Count Time (sec) to 15. • Run the program, and after two minutes it will prompt you for a file name to store the data. • Observe the file that contains eight 15-sec time bins with roughly equal event counts.

• Enter timing parameters for the full experiment. Set the program to count for a total of 30 minutes using half-minute intervals (60 event counts). (Note that these values are, strictly speaking, the integrated number of events in half-minute intervals. We will take them to represent the instantaneous activity as a function of time.)

Barium Sample • Observe a demonstration. Your instructor will prepare a barium sample. • Follow safety precautions. One must take special care with a radioactive liquid. Wear gloves, eye

protection and lab coat during the initial phase of this experiment. • Set your detector apparatus for the liquid-sample mode of operation (with vertical test tube). • Collect 8 drops of the Ba-137 eluate in a test tube. Put a cap (lightly) on the tube.

Data Acquisition • Collect data (30 minutes with 30-second intervals). • Store the data and exit the program. • Return the tube (with the liquid) to a test-tube rack.

Visualization • Import to the data analysis software. • Assign midpoint times to the time interval bins. • Plot activity versus time and observe.


15

Data Analysis Standard Result for Lifetime Our lab estimates will be compared with a standard value from previous results. The best estimates in the physics literature for the Ba-137 lifetime (as opposed to the half-life) are reported as

𝜏s = 3.682± 0.002 minutes. We will determine the lifetime of our radioactive sample in two separate ways.

Determination of Lifetime by a Linear Fit We would expect a pure exponential function to appear as a straight line, if we plotted the natural logarithm of the function instead. The inferred lifetime value is related to the slope of this line.

• Plot the natural logarithm of the activity versus time. • Restrict the data. Observe and select the portion of the curve that does indeed appear to be linear.

• Toward late times the activity will level off at a background level. • Toward early times the measured activity may also level off, because the detector cannot respond

fast enough to the true activity. (This phenomenon of dead time seems to occur when the activity is greater than about 15,000 events per 30-second interval.)

• Estimate the lifetime. • Fit the selected (natural log) data to a line (using SCIDAVIS). Find the lifetime from the slope. • Judge the goodness of fit of the line. • Report the lifetime in the form 𝜏e = 𝜇! ± 𝜎! in minutes. (Always eliminate insignificant figures.) • Compare the estimated lifetime with the standard value.

• Interpret the result. Based on the discrepancy between experimental and standard values, decide whether systematic errors seem to exist.

Notes:

• The uncertainty in the lifetime is related to the uncertainty in the slope. (Show this relationship.) • Think about how the uncertainty in the slope is calculated by the data analysis program. (Apply

TAYLOR, Section 8.6. Argue that the variation in data point uncertainties is small, and apply this assumption appropriately.)

• The systematic way of treating this situation is the method of weighted least squares. (See TAYLOR, Problems 8.9 and 8.26.)

Determination of Lifetime with a Background Correction We generally see a small background that contaminates the data. (Perhaps the elution process is imperfect and allows some cesium to follow when we milk the cow.) This background can easily be incorporated into the fitting model by adding a constant.

𝑅 𝑡 = 𝐴 ∙ exp −𝑡𝐵 + 𝐶

Use this function to perform the estimation process, and speculate about systematic errors. (We will not attempt to model the dead time phenomenon, so judiciously eliminate any aberrant early time data.)

• Fit the data, and judge the goodness of fit. • Report the estimate (with uncertainty), and compare with the standard value.


16

Simulation of Nuclear Decay (Rejection Sampling) Simulate the actual observed nuclear decay of barium. Then fit the simulated data to the model.

Algorithm • Generate a random point within a rectangle. The coordinates are found as follows:

• Generate a random time t in the range of interest (0-30 minutes). • Generate a random number y between 0 and 1.

• Accept this point if it falls within the distribution function defined by 𝑦 𝑡 = 𝑅(𝑡)/𝐴. The constants are taken as those obtained from the fitting of the actual data.

• Assign the accepted point to its corresponding time bin as with the experimental data. • Repeat until the total number of accepted events is the same as that in the experiment.

Analysis • Estimate, as before, the lifetime. Follow the fit / judge, report / compare process. (Compare, though,

with the lifetime obtained from the actual data rather than with the standard value.) • Generate 10 separate values for the lifetime by repeating for 9 additional trials. Compare this multi-

trial standard deviation with the uncertainty that was just reported.

Notes • The acceptance-rejection algorithm is a type of Monte-Carlo method. Compare with the simulation

of dart throwing in our previous experiment. In order to understand clearly, sketch a picture of the exponential decay within the randomly filled rectangle. (In this case, though, many darts get wasted!)

• We do not expect the distribution function to be a properly normalized probability distribution function. It is merely rationalized so that it begins at 1.

Documents

Autumn, 2016 Physics 3700 S. Durkin, R. Leonard ...durkin/phys3700/Autumn2016/P3700... · Experimental Physics Instrumentation and Data ... Data is commonly reported in the form of