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I. Theory and Background

II. III. Beta Ray Procedure

IV. Beta Ray Analysis

V. Error Analysis Notes

VI. Beta Ray Computer Programs

A few pointers about the Data Analysis

What does artifact mean?

If you compare the spectrum taken in this lab to ones taken elsewhere, you will find in your spectrum, an extra peak on the lower momentum side of the k-peak (see figure 2.c "artifact"). Before we automated the scan process with a computer, the current in the magnet was adjusted by hand, and it was impossible to obtain the resolution necessary to see this anomaly. But now, with the advent of the computer, intrinsic errors in the beta ray spectrometer have become apparent.

Figure 6a

To understand this, consider a source with a finite width emitting a monochromatic beam of β-rays subject to a uniform transverse magnetic field. Rays (electrons) are emitted in all directions. However, due to the Lorenz force they are confined to travel along arcs of the same radius r. Thus, each electron can travel along one of these arcs, but not the same distance. Thus a detector of width Δx "sees"

Figure 6b

an intensity distribution similar to that of Figure 6b. This implies that in the experiment, when a polychromatic beam is emitted and one "sits" at a bin and counts electrons, the count is smaller than it should be, because some electrons are falling into the neighboring bins, due to the width of the intensity distribution. In essence our raw spectrum is the result of what we should see (the true spectrum) with this distribution (the error function) smeared into it. Mathematically this is known as convolution. We would write

G = T * E

So can we mathematically deconvolute our data and find the true spectrum? Yes in some special cases, but not here because the error function is not a constant as we vary B and take our data. Also, the data are not accurate enough to warrant a detailed mathematical treatment. Instead, we smooth the curve by hand in the region where there is an extra bump on the side of the peak, and proceed with the analysis.

Figure 6c Note the "artifact" peak to the left of the k-peak at Bin # 2200; see Figure 2a

So now we can deconvolute our data and find our true Beta spectrum Figure 6D. Resultant Beta spectrum, see the K-peak at Bin \# 2200 and L-peak to the right of it.

Figure 6d Resultant Beta Spectrum, see K-peak at Bin #2200 and L-peak to the right of it

Data Analysis and Fermi-Kurie Plots

The Fermi-Kurie plot is simply an algorithm that transforms your spectrum to another view of the energy spectrum. It uses the formula [B10] in the National Bureau of Standards reference. The expectation of this experiment is to experimentally determine Q₁ and Q₂ , respectively the high and low energy 'Y' intercept points. FK (Fermi-Kurie ) transform makes it easier to determine Q₁ and Q₂ . First you must start with your full data spectrum from channel 0 to 4095 summed up and down data shifted together and then compensated, calibrated, and background-subtracted. Hit "Control+h" to see the key strokes for analyze. Analyze is a DOS program that requires, (8) eight letter names only with no spaces, for names of Files and Folders. Make a Folder in your My Documents folder for your data saving. Use (8) letter name to save your data ending in \*.DAT. The Excel files need to be opened and re-saved as "Text delimited with Tab" files ending in \*.DAT. Then analyze will recognize the files. After you have acquired your data, using the summed up and summed down files save them as the Fileup1.dat and filedn1.dat in excel as "Text delimited with Tab" files. Now close them from excel and open analyze and open the files.

Shift them together by the your calculated number, if they look okay then go on with the analysis.

If they are not perfect with respect to the hysteresis effects, then you will have to adjust the data yourself. You will need to shift each Y value by the different between the Y's at each X. In some instances you will have to average the Y's.

Then take your data and subtract background, No negative numbers allowed. Calibrate your X axis, low and high points for momenta and compensate it "Q" in analyze.

Save this data set file as a new data set in analyze(8) letters max. You will want to save the files as you go to each step. Good practice in case of computer cashes, etc. Note that you can save files in ASCII or Binary format.

Now you are ready for the Fermi-Kurie plot;

Crop out unwanted bad data points like negative numbers, etc using the "Alt+C" command in analyze with the use of markers. Fermi-Kurie "Alt+K" you should get something like the below graph.

Fit a line to this curve and get the end-point Y axis, Q2. Pick the area above the noise above the K-peak where it is linear. Now do a inverse Fermi-Kurie plot. You will get something like the following graph.

You should repeat the steps above to get the high energy component and the high energy end-point Q1.

Your result should be the high energy component as seen below.

Your combined graphs should be like the following;

Reduce your data by following the numbered steps listed below, but first read Ref. 1 for detailed explanations of how beta-ray spectra are analyzed. Also read about data analysis.

In your Fermi-Kurie analysis, use units of and for momentum and energy respectively.

As you work on the data, keep in mind what your ultimate goals are: To observe the beta decay spectrum; to model the phenomenon with a Fermi-Kurie plot that incorporates the theory of beta decay; to determine the maximum energy available in the decay process. The above Figure shows four separate phenomena. Your data are the sum of these. We would like to separate this sum into the individual curves shown above, where they are shown overlapping but not added, as they are in the experimental data. See Appendix D about the Fermi-Kurie plot.

1. From the masses of ¹³⁷Ba and ¹³⁷Cs, calculate the total energy available to the electron mass energy + electron kinetic energy + neutrino energy in the beta decay. What is the maximum kinetic energy with which the ¹³⁷Ba can recoil? Must this kinetic energy be taken into account in your experiment?
2. Combine your MYDATUP1.DAT and MYDATDN1.DAT files to reduce the effects of hysteresis and to yield a single data file which you will use in all your analyses. Should you add the two sweeps? Should you displace the two spectra by a constant amount or a variable amount? Should you shift them right or left an amount that depends on channel number? Some thought and justification are needed here. Plot your two raw spectra and your combined spectrum.
3. The momentum resolution Δp is proportional to the momentum p itself (you should derive this), and therefore, Δp is proportional to the current, since I is proportional to p. Divide the number of counts in each channel of your combined spectrum by I (or a number proportional to I). This is your compensated momentum spectrum N_c. Plot N_c vs. η.
   1. Calibrate your momentum scale in units of η by using the accepted value of the k-peak momentum (, or ), the bin in which the k-peak appears, and the bin number that corresponds to zero B-field (and hence zero momentum). Without accounting or correcting for hysteresis effects, how large an error in momentum would you have unwittingly obtained for the k-conversion electron momentum?
   2. Use the region of your spectrum that lies above the highest-momentum beta-particle to estimate the background radiation (you may assume the background is linear) and subtract it from your data. Plot the resultant spectrum.
   3. Make a Fermi-Kurie plot of your compensated spectrum. If the data were perfect, and the theory exact, then the F-K plot would be a straight line. Using a least-square-fit line, determine the end-point of the high-energy beta component. How would the end-point plot change if the neutrino did not have zero rest mass? This need not be a mathematical explanation and Appendix F, for comments about error analysis.
   4. By taking the inverse-Fermi-Kurie plot of the best-fit line obtained above you will have the contribution of the high-energy beta-component to your compensated spectrum (N_c). Subtract the higher energy component from your compensated spectrum to leave only the lower-energy component.
   5. Make a Fermi-Kurie plot of the subtracted spectrum in the vicinity of the end-point of the lower-energy beta spectrum. Determine the end-point of the low-energy component.
   6. Compare the measured line width and the expected spectrometer resolution for the conversion electron peak.
   7. Check out Error Analysis Notes and the video on Error Analysis.