Experiments in Nuclear Chemistry

Physical Chemistry Lab
Chem 345 Lab Notes
Brian Mork 1997-2011
References: Overview

The purpose of this lab is to learn about very low-intensity radiation (so low that particle quantization dominates the data collection) and appropriate ways of presenting the statistical data. In addition, you'll have a chance to learn about radioactive sources and solutions, sometimes used as chemical tracers. The Geiger-Mueller counter you'll use is a pulse-counting instrument that uses gas amplification, which makes it remarkably sensitive, but whose simple construction makes it relatively inexpensive. The experiments that are designed to accomplish this purpose deal with the operating plateau of the Geiger tube, resolving-time corrections, probability density functions, half-life calculations, attenuation of nuclear radiation, and the inverse square law.

The Geiger counter consists of two electrodes with a gas at reduced pressure between the electrodes. The outer electrode is usually a cylinder, while the inner (positive) electrode is a thin wire positioned in the center of the cylinder. The voltage between these two electrodes is maintained at such a value that virtually any ionizing particle entering the Geiger tube will cause an electrical avalanche within the tube. The Geiger-Mueller tube used in this experiment is called an end-window tube because it has a thin window at one end through which the ionizing radiation enters. The Geiger counter does not differentiate between kinds of particles or energies; it tells only that a certain number of particles (betas and gammas for this experiment) entered the detector during its operation. The voltage pulse from the avalanche is typically >1 V in amplitude. These pulses are large enough that they can be counted with an external counter directly, without amplification. For the work here at USAFA, we have integral counting/timing units. In this experiment the properties of the Geiger counter will be studied and several fundamental measurements will be made.

You should have five sources available: Barium 137 (Actually, 137Cs, which continuously decays to 137Ba, which in turn is rinsed out of the dropper with a chelating EDTA solution), Strontium 90 (90Sr), Cobalt 60 (60Co), Lead 210 (210Pb) and two split disk sources of Thallium 204 (204Tl). DO NOT HANDLE THE SOURCES WITHOUT GLOVES AND TWEEZERS. If you take radioactive dirt or dust from the sources or containers with you all day, or worse, get it in your body, that's bad.

Operating Plateau for your Geiger Tube

The purpose of this experiment is to determine the voltage plateau for the Geiger tube and to establish a reasonable operating voltage for the tube. Figure 2.2 shows a counts-vs-voltage curve for a typical Geiger tube that has an operating point in the vicinity of 1000 V. The region between R1 and R2, corresponding to operating voltages, V1 and V2, is called the Geiger region. Voltages less than V2 (more negative in magnitude) cause a continuous discharge in the tube and will definitely shorten the life of the tube. If you hear a hissing, cracking, or other electrical sound, turn down the magnitude of the voltage!

Place the 210Pb source about 2 cm from the windows of the Geiger tube. Set the timer for manual time (so it won't automatically shut off), and slowly increase the voltage magnitude until the counter just begins registering counts. The point is called the starting voltage in Figure 2.2. Starting voltages are rarely greater than 900 V and can be as low as 250 V. Reset the counter, set the timer for 1-min timing intervals, and count for 1 min. Increase the voltage by 50 V or 25 V steps (choose whichever you have time for and will give the better graph) and count again for 1 min. Continue making measurements at intervals until you have enough data to plot a curve similar to that in Fig. 2.2 (but don't try to get the "swoop up" shape at voltages higher in magnitude than V2--you'll damage the tube). The region between V1 and V2 is usually less than 300 V. A sharp rise in the counting rate will be observed if you go just above V2. If this happens, the upper end of the plateau has been reached. Reduce the voltage immediately. Choose the operating point for your instrument at ~50 to 70% of the plateau range. Evaluate your Geiger tube by calculating the performance factor, as defined below. This performance factor, PF, should be less than 10.

PF=((R2-R1)/(V2-V1))*100/R1

Counting Statistics

Poisson DistributionEach measurement made for a radioactive sample is independent of all previous measurements, because radioactive decay is a random process. However, for a large number of individual measurements, the deviation of the individual count rates from what might be termed the "average count rate" behaves in a predictable manner. For example, small deviations from the average are much more likely than large deviations. In this experiment we will see that the frequency of occurrence of a particular deviation from this average can be determined with a certain degree of confidence. In addition, you'll ascertain whether your data fits a Normal or Poisson model better. You've all seen a Normal distribution ("the curve" for grades, for example). You may not have seen a Poisson distribution before, so there's an example to the right.

Place a source far enough away from the window of the GM tube so that ~20-25 counts can be obtained in a time period.  If the background count level is more than this, just use background counts.  Without moving the source, take more than 50 independent runs. The goal of the data reduction is to compare the experimental distribution to two models: Normal and Poisson, and ascertain which fits the data better.

Calculate the average count rate for your measurements using Equation #13. Because the distribution of "counted values" does not theoretically match a Normal (Gaussian) curve, the standard deviation is not calculated the normal way, built into many calculators. Instead, the standard deviation is given by Equation #15. Notice that specifying only the average or only the standard deviation completely defines a Poisson distribution, unlike a Normal distribution where they both have to be specified.

average equation
Poisson Standard Deviation
Determine RAVG and your estimate of the standard deviation.

Plot a bar graph histograms of your data and superimpose on the graph a Poisson function generated from your average, scaled to a matching horizontal scale. In Equation #17, use RAVG for mu. Note that, because of the X-factorial (x!) function, a Poisson distribution is defined only for integer x values. This is appropriate for any count-based measurement; you can't have a fractional count value.

Poisson

Gaussian
Poisson Distribution
Gaussian
Now determine your standard deviation assuming the distribution was Normal (RAVG stays the same, and the standard deviation can be calculated using the "normal" formula in your calculator). Use these new values to plot a Normal distribution (Equation #18), superimposed on the graph. Does your histogram noticeably match one type of distribution rather than the other?  Which standard deviation seems to better represent the spread in your data?  You may wish to see the MathCad template N-VS-P.MCD.

Resolving-Time Corrections for the Geiger Counter

The Geiger counter is a relatively slow electrical device. Whenever it gets a particle count, it goes "blind" for a certain amount of time afterwards. Typically this is in the microsecond range. When used for counting rates above several thousand counts/min, it is necessary to make a dead-time correction to obtain the true counting rate of particles impinging on the sensor. In order to determine this response time (TR), use the split sources provided in lab. Count the right half first, then both halves, and finally the left half. When counting only half the source, use the blank half to keep the geometry constant.

Adjust the geometry so the stronger 1/2 disk give 4000-5000 counts per minute.  Placing these sources in the the tray rightside up or upside down seems to make a big difference.  Once you decide on the placement of the sources and detector, don't change the geometry for the rest of the resolving time experiment.  Place the right half of the source in front of the window, record a 1-min count. Define this count to be R1. Place the left-half of the source along with the right-half and make a 1-min count. Define this quantity to be RT. Remove the right half and count the left half for 1 min. Define this quantity to be R2. After recording many repetitions of R1, R2, and RT, calculate the resolving time (TR) from Equation #2, below. What are the units on TR?

Equation #2, determination of Tr

Calculate an estimate for TR two ways, and comment on which is better and why:  First, calcuate TRvalues first for each set of data, then determine an average/error.  Secondly, calculate average/error in R1, R2, and RT, and then propagate the errors into TR. Because TR is used throughout the rest of the lab, be sure to get a good value. Specifically, take enough data to get less than 50% relative error in your TR value.  This may include going back and doing even more measurements of the count values after an initial run through data reduction.  If so, be sure to use the same geometry you were using before.

Once you have a TR value, any observed counting rate, R0, you take in lab can be converted to a true radiation rate, R, from  Equation #3.  Plot a graph of 100% * (R-R0)/R0 (relative error) versus hypothetical R0 values over a range of R0 values that gives approximately 1-10% error.  This plot will show you how much error you're getting rid of when you use TRIn addition to resolving-time corrections, be sure to correct for background count values in subsequent experiments.

Equation #3, use of Tr

Radioactive History Dating and Half-life Determination

Determining the time a radioactive source was made, and determining the half-life of a source use exactly the same mathematical formulas. It's just a matter of having 4 variables, and rearranging to solve for the single unknown. You'll determine the half-life of a 137Ba source, and then determine the manufacturing date of the 60Co or 90Sr source.

Note the time when you put 3 or 4 drops of the Barium solution onto a cotton swab in an observation plate, and immediately collect a 1 minute count, using a geometry that gives several thousand counts. Every other minute, again take a 1-minute count measurement. Continue doing this until the count value stops decreasing or until the noise in the count values dominates the decrease in count values. Using the math introduced below, determine what the observed half life is. Compare this to the literature value.

Determine the present activity in Curies of the Cobalt and Strontium sources. Note that your Geiger counter only samples part of the total emission from the source. You'll need to do a geometric correction to estimate the total emission in all directions (a "full sphere" correction) after doing TR and background corrections. Record the activities at the time of manufacture penned on the face of the disks. Look up literature values for the half lives of the isotopes. With these 3 pieces of information, calculate when the decaying elements were created. Compare this to the actual manufacture dates written on the sources.

For any radioactive source, it's true that the disappearance of radioactive material (atoms per second, or similar units ) is proportional to the amount of radioactive material present. Mathematically, -dR/dt = kR.  Using this information, derive Equation #16, where R is the measured rate of decay, lambda is a constant, and t is time.

Ratio is exponential decay

When determining the half life of the Barium, you simply need to plot the above equation (y=mx form) and use the slope. Specifically, plug in the variable t1/2, that you define to be the half-life, the lambda you determine from your data, and what you know the ratio (R0/R) to be, then solve for t1/2. To determine the age of the Co or Sr, simply plug in the values for R and R0, and solve for t. As always, do not forget to determine and discuss precision estimates.  For example, include confidence limits with your reported age determination.

Linear Absorption Coefficient

When gamma radiation passes through matter, it undergoes absorption primarily by Compton, photoelectric, and pair-production interactions. The intensity of the radiation is thus decreased as a function of distance in the absorbing medium. The mathematical expression for intensity, I, is given by Equation #4:

exponential decay

where Io is the original intensity of the beam, I is the intensity transmission coming through an absorber of distance, depth, or thickness x. Mu is defined as the linear absorption coefficient for the absorbing medium. Rearrange this equation to get an equation of the same form as Equation #16, above. The "half value layer" (HVL) of an absorbing medium is defined as that thickness, X1/2, which will cut the initial intensity in half; that is , I/I0 = 0.5. Because we're using rates of decay rather than intensities, this would be written R/R0 = 0.5.

Radiation is not attenuated just by thickness of material; it matters what the material is. Thickness and density contribute to the shielding value of a material. Specifically, thickness times density gives a new term called "density-thickness." With a caliper, measure the actual thickness of the plates you used. Multiply these values by the density of lead. Do you match the units and the numerical quantities of the values printed on the inside lid of the box?  Do these errors significantly affect your HVL determination?

Use a source of your choice and the box of lead shielding plates to determine the HVL for Lead. Use a source and geometry so you get an unshielded intensity of several thousand CPM.  Take data for different thicknesses from unshielded down to a shielding thickness giving you about 25% of the original count.  Plot a graph based on Equation #4 where mu can be calculated from the slope.  As you change the shielding, be sure to not change the source to detector distance.
 

Inverse Square Law

There are many similarities between ordinary light rays and gamma rays. Let us assume that we have a light source or a radiation source that emits particles at a rate R0 per unit time. It is reasonable to assume that these particles are given off in an isotropic manner, that is, equally in all directions. If we place the source in the center of a conceptual spherical shell, and measure the number of particles per second per cm2 of the spherical shell, this observed count rate, R, will be R = R0/A0, where A0 is the total area of the sphere in cm2. Putting in the geometry of a sphere into the equation yields Equation #10.

sphere correction

R is seen to vary as 1/r2, when r is the radius of the sphere. The purpose of this experiment is to verify Equation 10.

Choose an appropriate source and set the GM tube at the proper operating voltage. Count for a period of time long enough to get reasonable statistics (~4000 counts) over a range of 2-10 cm.  In order to get repeatable and stable distance measurements, set the Geiger tube in a ring stand clamp and lay the sources, in their cradle, on the ring stand base. Note that for the longer distances you can accept lower count values or raise the count time, and then later reduce all the numbers to a counts/minute basis. Which is the better method?

As in the previous tasks, correct the measured activities for dead time and background.  Determine R0 for each distance. Should these be the same? Report the average R0 for all the distances. Now, determine R0 an alternate way: plot Equation #10, and use the slope to calculate R0. How do the values compare? Which do you trust?


This page is maintained by Brian Mork.  Last modification was January 2011.  Send suggestions for changes and comments via email.