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 ^{137}Cs 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 = I_{o}e^{-mx} ...........(11.1)

where I_{o} is the incident intensity, that is, I = I_{o}
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 = I_{o}e^{-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* I_{o} - 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* Y_{2} - *ln* Y_{1})/(X_{2}
- X_{1}) from suitable points (X_{1},Y_{1}) and
(X_{2},Y_{2}). 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 X_{½}.
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 X_{½} may be determined
from m from the relation,

X_{½} = ^{1}/_{m}
*ln* 2. ............... (11.3)

To see how the latter relation arises, note that at X = X_{½},
the equation

I = I_{o}e^{-mx} becomes
I = I_{o}/2 = I_{o}e^{-mx½}

Thus, *ln* ½ = -mX_{½}
or X_{½} = ^{1}/_{m}
*ln* 2.

Read X_{½} directly from your hand graph. Then calculate
X_{½} 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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© 1996 Dr. H. K. Ng.

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