Phys 312 - Assignment 11 - Due 16 Apr 98

1.

Detection of plastic explosives. Plastic explosives are a favorite of terrorists due to the difficulty of detecting them. These explosives tend to contain large quantities of nitrogen. In order to detect the explosive, suppose we irradiate all luggage with thermal neutrons. The neutrons are absorbed by the nitrogen nuclei in the reaction $^{14}\mathrm{N}(n,\gamma )^{15}\mathrm{N}$. The $^{15}\mathrm{N}$ nucleus is created in an excited state and decays 14% of the time by the emission of a $\gamma$-ray of energy 10.8 MeV.

(a)

Look up the thermal neutron cross section for this process.

(b)

Suppose that the explosive ("Semtex") has a density of $\rho =1.7\ \mathrm{gm}\ \mathrm{cm}^{-3}$, and is 34% nitrogen by weight. What is the probability of a thermal neutron being captured by a nitrogen nucleus when traversing a 10 cm tube of Semtex?

(c)

What flux I₀ of slow neutrons incident on a 10 cm tube of Semtex is needed to produce 1000 gamma rays per second? [Hint: don't forget the 14% branching ratio.]

2.

Solar temperatures. Here we'll try to estimate some relevant temperatures in the sun. Assume you start with an initially diffuse cloud of hydrogen and helium atoms (initially at rest), which subsequently collapses under its gravitational attraction.

(a)

Use dimensional analysis to show that the total energy released in the gravitational collapse is about

$$E_{\odot }={\frac{GM_{\odot }^{2}}{R_{\odot }}}.$$ (1)

Calculate this energy.

(b)

How does this energy compare to the total energy that could be released by (hypothetical) chemical reactions in the sun? To the total energy that could be released by converting all the hydrogen into helium? (You can assume that the cloud has the "primitive" abundance of helium--it does not make much difference as long as you have mostly hydrogen).

(c)

Assuming that all of the energy (1) is converted into heat (a rather dubious assumption), estimate the interior temperature of the sun. Compare with the accepted value of the temperature at the center of the sun (look it up).

(d)

The intensity of solar radiation has a peak at a wavelength of 490 nm. What is the surface temperature of the sun? As you go from the surface of the sun toward to center, what is the approximate temperature gradient dT/dx? How can this temperature gradient be maintained?

3.

List three radioactive nuclides that naturally occur (in easily detectable amounts) on Earth today. What are the half -lives of these nuclides? How did they originate? At least one of them should have a half-life of less than a thousand years. Explain how is it possible to find nuclides with a half-life so much shorter than the age of the Earth.