Experiment XI

Absorption of Gamma-Rays by Lead and Aluminum

I. Background.

The short wavelength electromagnetic radiation emitted by excited atomic nuclei is called gamma (g) radiation. The energy of g-ray photons may range from millions of electron-volts down to energies that are more typical of the x-ray region.

High energy g-rays are the most penetrating of all electromagnetic radiation and are not appreciably absorbed by even several centimeters of materials such as concrete or wood. Dense metals, particularly lead, are effective g-ray absorbers, however.

In this experiment you will measure the absorption of the 0.662 MeV g-rays emitted by 137Cs first by aluminum, then by lead.

Let I be the intensity of g radiation detected at some distance from the source. Interposing an absorbing material of thickness X between source and detector will reduce the intensity by a certain amount. One might expect that for a small increase DX in absorber thickness, the fractional change in intensity, DI/I would be proportional to DX. This situation is represented be by the Figure 11.1. I is the intensity incident on the slab of thickness DX and I' is the emergent intensity. Thus DI = I' - I and DI/I = -mDX where m is a proportionality constant called the absorption coefficient. m depends on the energy of the incident radiation and the material being used. The minus sign indicates that for increasing DX, DI is negative, i.e., I' is less than I. If the absorber is not thin, one way to determine the emergent intensity would be to divide the thickness X into many slabs of small thickness DX. The equation DI = -mIDX could then be applied successively to each thickness DX using the emergent intensity from the preceding slab as the incident intensity on the next. A second method is to integrate the above equation using calculus. This yields the result in a form that is much easier to apply. The result is:

I = Ioe-mx ...........(11.1)

where Io is the incident intensity, that is, I = Io when x = 0. I is the intensity left after the radiation traverses a thickness x of material and e = 2.718 is the base of the system of natural logarithms.

The process by which absorbing materials can reduce the intensity of g radiation are: 1) the photoelectric effect, 2) the Compton effect and 3) pair production. Only the first two effects occur in the present experiment as pair production requires g ray energies of at least 1.02 MeV.

II. Determination of the Absorption Coefficient and Half-Value Thickness of Aluminum for 0.662 MeV G-Rays.

A. Data Acquisition.

Adjust the equipment as in the preceding experiment: baseline at 64.2% and window at 4%. Place the source in the bottom slot of the holder and take 9 readings of the number of counts in a 10 second interval when no absorber is present. Place 0.25 inches of aluminum absorber over the source and take another nine readings. Repeat for absorber thickness of 0.50, 0.75, 1.25 and 1.75 inches.

Find the average number of counts for each absorber thickness and their absolute uncertainties, i.e., N ± N /3.

B. Data Analysis.

Since I = Ioe-mx is not a linear relationship, a plot of I versus X on linear graph paper is not convenient for determining the absorption coefficient m. However, if the natural logarithm of the above expression is taken, the result is

ln I = ln Io - mx. ...........(11.2)

Hence, if the exponential absorption law is obeyed, a plot of ln I versus x for your data should yield a straight line with slope -m. To see if this is the case, plot your data both by hand and by means of the computer. For the hand plot use semi-log graph paper with two cycles along the Y axis. Draw a best fit straight line through your data and find its slope, -m. To find the slope you will need to determine the ratio (ln Y2 - ln Y1)/(X2 - X1) from suitable points (X1,Y1) and (X2,Y2). Record the value you obtain for m including units on your graph and show your calculation.

For the computer plot of your data, use the computer program "Quattro Pro". Load the template "Gamma_rays.wb3" into Quattro Por and print the on-line instructions. Follow the instructions and obatin a slope for your plot. Include the number of counts for the zero thickness.

Print both the analysis and the plot. From the linear regression analysis, what is the value m? Compare this with the value for µ obtained from the hand graph. What uncertainty does your computer analysis give for the value of m?

A useful quantity to specify for an absorbing material addition to its absorption coefficient is its half-value thickness X½. This is the thickness which absorbs half of the incident g- rays. The half-value thickness may be found in two ways. Either the value for X at which the count rate drops to one-half its initial value may be read directly from the graph or X½ may be determined from m from the relation,

X½ = 1/m ln 2. ............... (11.3)

To see how the latter relation arises, note that at X = X½, the equation

I = Ioe-mx becomes I = Io/2 = Ioe-m

Thus, ln ½ = -mX½ or X½ = 1/m ln 2.

Read X½ directly from your hand graph. Then calculate X½ from your computer analysis value for m. What is the percentage difference using the two methods? Express your half-value thickness both in inches and centimeters.

III. Determination of the Half-Value Thickness and Absorption Coefficient of Lead for 0.662 MeV g-Rays.

Repeat Part II above using lead absorbers with thicknesses of 0, 0.125, 0.25, 0.50 and 0.75 inches. Compare your absorption coefficients and half-value thicknesses for lead and aluminum.

Since the absorption of g-rays is due to their interaction with the electrons in a material, it follows that denser materials are better g-ray absorbers. (Neutrons, on the other hand, are slowed by colliding with nuclei and are more effectively reduced in intensity by low Z nuclei.) The ratio of the number of electrons per unit volume in lead and aluminum is roughly in the ratio of their densities. Does this account roughly for the difference in half value thickness that you measured for the two materials? Note that close agreement is not to be expected since the inner electrons in lead are more tightly bound than those in aluminum and are more effective in reducing g-ray intensity through the photoelectric effect.

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