# (1) to Verify the Inverse Square Law for Beta-Rays

Experiment 29

Purpose

(1)  To verify the inverse square law for beta-rays.

(2)  To measure the half-life for the β-decay of Ag108.

Apparatus

(a) Geiger-Muller tube, scaler with high voltage supply, timer

(b) 1 long-lived β-source (Cl 36 or Tc99)

(c) 1 short-lived β-source (Ag108)

(d) semi-log graph paper, ruler

Theory

Inverse-square law

Consider a point source of radiation emitting in all directions emanating from the source. Suppose that the total amount of radiation per unit time emitted in all directions is constant. Then on any sphere of radius R, centered at the source, the intensity (amount of radiation per unit time per unit area) will be proportional to 1/R2 (amount/4π R2). In the case of radioactivity, particles are emitted. For sources which emit a sufficiently large number of particles in all directions, we will perform an experiment which should verify the inverse-square law.

Half-Life

The radioactive decay of an extremely large number of nuclei is a random statistical process. The activity (number of decays per unit time) at any given time is proportional to the number of “not-yet-decayed nuclei” present at that instant. It can be shown that the activity decreases exponentially with time. Thus if we use semi-logarithmic graph paper to plot the activity of a source vs. the time, we should get a straight line plot. Sometimes the beginning of such a plot will not be a straight line because the source will contain several different types of nuclei decaying at different rates. The latter type of source usually consists of several short-lived components together with a long-lived one. When the short-lived components have decayed away, the long-lived component remains to yield the straight line plot.

Since the activity is proportional to the number of not-yet-decayed nuclei, when the latter

is reduced to half of their initial value, then the activity is likewise cut in half. The time it

takes for such (halving) process is called the half-life of the source. The half-life T1/2

may be read directly from the semi-log plot of activity vs. time.

Procedure Part I. Variation of Counting Rate with Inverse Square of Distance from the Source

The variation of distance between source and detector is achieved by placing the source

on the different shelves of an array. Measure the distance of each shelf to the detector.

The source used will usually be Cl36. Let N denote the number of counts per given

time-interval. Label the shelves 1 through 7 and measure the distance of each one from

the top in cm. Your data-sheet may then be arranged as follows.

SHELF
# / d
(cm) / d2
(cm2) / N / Δt
(sec) / N/ Δt
(1/sec) / N · d2
Δt
(cm2/sec)
1 / 10
2 / 20
3 / 20
4 / 30
5 / 50
6 / 70
7 / 100

The Δt column above indicates the time period during which N is to be determined.

Calculate the product Nd2. Is it constant for all shelves? Explain the discrepancies.

Δt

Procedure Part II.Half-Life of Ag108

A radioactive silver foil will now be inserted in the top shelf of the array. Measure the number of counts during 15 sec. intervals for a period of several minutes by the following procedure:

Starting at t = 0, measure N for the first 15 sec.

Reset the counter and wait until t = 30 sec.

Starting at t = 30 sec., measure N for the next 15 sec.

Reset the counter and wait until t = 60 sec., etc., repeating this counting procedure for the

next several minutes.

The number of counts obtained during each 15 sec. interval may be taken as proportional to the activity at the beginning of each interval.

Associating the number of counts N so obtained with the beginning time t of each

15 sec counting interval, plot N vs. t on a “semi-log paper”, as explained by your

instructor. Draw the “straight line of best fit” and determine the half-life T½.

Submit the lab report as directed by your instructor.

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