PHYS272_Lab12

=Physics for Scientists and Engineers - Lab 12 PreLab =  For this assignment you will choose two possible experiments for your lab final. For each of those experiments give: = = =Physics for Scientists and Engineers - Lab 12 =
 * A title
 * A list of the independent and dependent variables
 * A list of the governing equations
 * A paragraph explaining the principles being tested.

Objectives:

 * To investigate the statistical nature of radioactive decay.
 * Measure the absorption of gamma radiation by matter.

Equipment:

 * Geiger counter and stand, test tube holder, right angle clamp and stand
 * Cs-137 gamma ray sources
 * Computer with Capstone and Graphical analysis software
 * Set of lead plates

Physical Principles:
 Radioactive decay is an inherently random process. If you record the number of counts, N, in a time, t, and repeat the experiment numerous times, you expect a Gaussian distribution of N’s centered on N ave with an uncertainty reflected in the standard deviation given by,
 * The Statistical Nature of Nuclear Decay** 

math \delta N = \sigma = sqrt{N_{\mathrm{ave}}} \ \ \ \ \ \ \ \mbox{(1)} math

If you perform the experiment once, the best estimate of the count is

math N \pm \sqrt{N} math  The estimated fractional uncertainty (relative error) then is, 

math \displaystyle \frac{\sigma}{N_{\mathrm{ave}}} = \frac{1}{\sqrt{N_{\mathrm{ave}}}} \ \ \ \ \ \ \ \mbox{(2)} math

The point being that a longer counting time will collect more data and reduce the relative error.


 * Absorption **

The intensity, I, decreases exponentially with the distance, x, that the radiation travels through a material.

math I\ =\ I_0\ e^{-k\,x}\ \ \ \ \ \ \ (3) math

where I 0 is the intensity entering the material and k is the absorption coefficient characteristic of the material.

A. Means Testing
 Place a gamma source disk with half-life measured in years flat on the table. Balance the Geiger counter directly on the disk exactly centered. Have a Data Table and Histogram open in the Displays window. The Table should display mean, standard deviation and counts. Select measurements to Geiger Counts. Set the duration of counts to 2 Hz.
 * 1) Collect exactly 200 counts in a table to generate a Gaussian distribution with the mean, standard deviation given. Increase or decrease the bin size to make a pleasing histogram.
 * 2) Repeat the experiment for 200 more counts, generating a second Gaussian distribution (maybe not too different from the first. A second histogram may be plotted on the same axis as the previous data for comparison. Again note the mean and standard deviation.
 * 3) For a third set of data, pick up the Geiger counter and then try to center it in exactly the same spot as on the previous two. Collect 200 more counts with all three experiments visualized as histograms.

Be sure to record histograms of your distributions in your lab writeup.

Compare the standard deviations with the square root of the means to verify Eq. (1). Calculate the estimated fractional uncertainties from the ratio of Eq. (2) and record as estimated percentages of uncertainty. Perform a two-sample t-test using [] to decide whether the data from experiments 1 and 2 are significantly different. What would be a reasonable null hypothesis? Web-research your choice for the parameter, α, in the t-test and what it means. Record your p value and describe its significance.

Perform the two-sample t-test to compare the means of experiments I) and III). Record p value and make a statistical inference.

B. Counting Statistics
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Place a gamma source disk with half-life measured in years flat on the table. Balance the Geiger counter directly on the disk exactly centered. Have a Data Table and Histogram open in the Displays window. The Table should display mean, standard deviation and counts. Select measurements to Geiger Counts. Begin by taking 10 counts at 2Hz (t = 0.5 seconds) and record the mean, standard deviation and number of counts. **Do not bump the source or detector during this experiment!** Adjust the sample rate to 1 Hz (t = 1 second) and repeat for 10 more counts, recording the same information. Continue this process for t = 2 seconds, 5 seconds and finally 10 seconds.
 * 1) Attempt to show graphically that the mean number of counts is proportional to the sample rate time.
 * 2) Show graphically that the standard deviation is proportional to the square root of the mean number as in Eq. (1).
 * 3) Show graphically that the relative error is inversely proportional to the square root of the mean as in Eq. (2).

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em; line-height: 1.5;">C. Gamma ray absorption by lead
Place a Cs gamma ray source on the table under the GM tube held a few inches above by the test tube clamp. Take a 1 minute reading and record the number of counts. Balance a stack of 1 – 5 lead plates on top of the source between source and detector without changing the positions of either source or detector and take additional 1 minute readings as you add sheets. (Lead is fairly toxic material, so keep the plates in the plastic wrap. Given that each plate is 0.5 cm thick, determine the distance, x, through lead as x = 0.5 N (cm).
 * Data Collection **


 * Data Analysis **
 * 1) Plot the number of counts vs. thickness of the absorber, including error bars.
 * 2) Fit the data to the appropriate function.
 * 3) Compare the fit values with those that are commonly accepted for this problem.
 * 4) Plot the same data on a linear graph ln(counts) vs x, and use the linear fit to compare to the same values.
 * 5)  Compare with the generally accepted value of absorption for lead of k = 1.19 cm -1 for 0.6616 MeV gamma rays.
 * 6) Use your data to predict what thickness of lead would be needed to stop 99% of gamma rays.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">D. CT Scan
Draw an approximately 5 cm-diameter circle on your table and divide into quarters. Place a Cs-137 source directly across the diameter of the circle from the Geiger counter –directly facing each other - and record the number of counts, I 0, for 1 minute. Consider the gamma source to be producing a radiation intensity, I 0, at the detector across the circle.

Place three different column-shaped objects (a stack of plastic dice, a stack of pennies and a lead column) each in turn at the center of the circle so that each directly absorbs gamma rays between the source and Geiger counter and record counts of I A, I B , I C for one minute each.

Calculate the fraction of the intensity transmitted and absorbed by each object, i, as,

math \displaystyle T_i = \frac{I_i}{I_0} math

And

math A_i = 1-T_i math

Place lead column in position A, copper in position B, dice stack in position C and leave position D (a column of air) as shown below. Consider your “patient” to be the bundled stacks of tissue, A, B, C, and D.

__Use the word document on LearningHub to do the next part.__

<span style="font-family: 'Times New Roman',serif; font-size: 16px;">For the four circles drawn below, adjust the transparency of each square by double-clicking on each square, choosing Shape Fill/Texture/More Textures and using the slider to adjust the transparency percentage to represent the fractional transmission.



<span style="font-family: 'Times New Roman',serif; font-size: 16px;">Place a copy of your “patient’s” transverse plane, in your report Take 1 minute counts from the six angles shown below and record the total number of counts for each angle.



Put data into the spreadsheet called //CT Scan Computations// which is available on LearningHub and compute the fractional transmission for sites A, B, C and D.

<span style="font-family: 'Times New Roman',serif; font-size: 16px;">For the four squares drawn below, adjust the transparency of each square by double-clicking on each square, choosing Shape Fill/Texture/More Textures and using the slider to adjust the transparency to represent the fractional transmission.



<span style="font-family: 'Times New Roman',serif; font-size: 16px;">Place a copy of your square, cross-sectional “image” in your report.

<span style="font-family: 'Times New Roman',serif; font-size: 16px;">Visually compare the “patient” transverse plane with the transverse plane “image”.