The Effects of Distance on Count
Rate of a Radioactive
Source—Laboratory Experiment






INTRODUCTION

Since its discovery by Henri Becquerel in 1896, much has been learned about nuclear radiation. Radiation travels in all directions, in straight lines, from the center of its source, like rays from the sun. As the radiation moves farther from the source, it becomes less intense. This decrease in intensity is proportional to the square of the distance between the source and the detecting device. This relationship is commonly referred to as the inverse square law of radiation. In this experiment, we will explore and attempt to calculate the relationship between the distance from a radioactive source and the intensity of the radiation emitted.

EQUIPMENT

  1. Radiation measuring devices
  2. Radioactive test sources (1.0 m Ci is sufficient)
  3. Stopwatches
  4. Metric rulers or vernier calipers
  5. Graph paper
THEORY

Inverse Square Relationship

The counts displayed by a Geiger counter depend on the number of particles that are entering the Geiger tube. Therefore, the counts depend upon, among other things, the distance of the tube from the source. A point source emitting a total of N particles, emits them in all directions. Imagine a sphere with a radius of r with the source at its center. Every particle emitted would have to pass through the surface of this sphere. Since the surface area of a sphere of radius r is A = 4p r2, we can calculate the total number of particles per unit area (P) of a sphere centered on the source simply by:

P =                              (1)
 If the distance r (from the source to the detector) is increased from r1 to r2, the area of the sphere, A also increases but N remains the same. Therefore, P decreases and:

if  P1 =   and   P2 =   then  = 

And since the counts, C detected by the Geiger tube are directly proportional to the number of particles per unit area, P we can say:

=                              (2)
 Hence, for a point source, the counts decrease as 1/r2 with the distance from the source. It is often desirable to graphically analyze experimental findings, such as the inverse square relationship. We can further analyze this relationship by transforming the hyperbolic inverse square graph into a graph of a straight line. Equation (1) can be rewritten:
P = [N/4p]                     (3)
 Taking the natural log (ln) of both sides of this equation gives the familiar slope-intercept form for the equation of a line:

ln P = ln                                                  (4)

Or

  ln P = -2 ln r + ln [N/4p]                                   (5)                 where the slope of the line is -2.
 
 

PROCEDURE

  1. Safety

    2. High doses of radiation can be damaging to living human cells. The radioactive sources with which we will be working today have very low activities and pose no health risk to us. They should, however, still be treated with care and respect.
  2. Background Count

    3. This measurement must be made with the source several meters from the Geiger counter. Do not obtain your source until this measurement has been made. Record the number of background counts on your data sheet for a time of at least 10 minutes. Calculate and record the background rate in counts per minute on the data sheet.
  3. Inverse Square Relationship
1.    Obtain a test source. Record the source type, activity, half-life (t1/2), and calibration date of your source on the data sheet.
2.    Place your source on the top shelf of the counter. Measure the distance between the detector and the top of the source using the ruler or calipers, and record this reading on your data sheet.
3.    Take a reading for at least 1-minute. Record the total counts and the duration of the measurement on the data sheet.
4.    Move your source to the next shelf down. Repeat the measurements taken in steps 2 and 3 above and record the results on your data sheet. Repeat this step until at least 5 total counts have been taken and recorded.
5.    Return the test source to your lab instructor. ANALYSIS
  1. In part B of the procedure, why is it important to measure the background radiation level with the test source several meters away?
  2. Calculate the decay constant (l ) of your source, where l = . Note that the half-life has units of years, and so l will have units of "per year". This should be converted into units of "per second".
  3. What is meant by the terms "half-life" and "decay constant" as they pertain to radioactive decay?
  4. On regular graph paper, plot a graph of counts (C) vs. distance from the source (r). What does the shape of the graph indicate?
  5. On regular graph paper, plot a graph of the natural log of counts vs. natural log of distance from the source. Compare the slope of your graph to the theoretical slope predicted in Eq. 5.
  6. According to the inverse square law, when the distance is doubled from 8 cm to 16 cm, the reading should decrease to 1/4 its initial reading. Do your results agree with the inverse square relationship? If not, give reasons and explain.

Inverse Square Law Lab
Data Sheet





BACKGROUND READING
 
Total Counts
Duration (s)
CPM
     

 

TEST SOURCE DATA
 

Source Type: ________________________

Activity: ________________________

Half-life: ________________________

Calibration: ________________________
 
Distance (m)
Total Counts
Duration (s)
CPM
Net CPM
         
         
         
         
         
         
         
         
         
         

 

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Copyright 1999, Thomas McNulty
Last updated 9 August 1999