Question

In a study, 2-MeV neutrons enter a target made of water and are gradually absorbed in the water.

Assume:

1. The neutrons don't interact with oxygen

2. Their capture cross section before they thermalize is 2.5 barns.

Calculate (for the neutrons):

a. Mean free path.

b. Linear momentum as they the target.

c. Velocity as they enter the target.

d. Calculate the neutron's lifetime in the medium (average time before they are captured).

Indicate all units.

Answer 1

Considering the assumptions given,

a) The mean free path is calculated as per the following formula:

MFP = 1/ [A*D*C/W] where

A = Avogadro number 6.023 * 10²³

D = Density of water = 1 gram/ cc

C = Capture cross section = 2.5 barns = 2.5 * 10^-24 cm²

W = molecular weight of water (H₂O) = 18

Substituting the values in the above formula, we get Mean Free Path = 0.1195457 m

c) Let's find the velocity and later find the momentum for ease of calculation;

Given, 2-MeV neutrons

1 eV = 1. 602 * 10^-19 J

converting the energy in terms of Joules, we get 3.204 * 10^-13 J

as per the conservation of energy, this energy is being converted into kinetic energy of neutrons, which gives us

3.204 * 10^-13 J = .5 * m * v² where m is the mass if neutron = 1.6749 * 10^-27 Kg and v = velocity of neutron

=> from this, we get v = 1.957 * 10⁷ m/s which is the velocity as they enter the target.

b) linear momentum = m * v

substituting the values of mass and velocity of the neutron, we get 3.2761 Kg-m/s as linear momentum as they enter the target

d) lifetime = MFP/ v

Substituting the values we got in a) and c) in the above formula, we get 6.11 ns

so, the neutrons lifetime in the medium is 6.11 nano seconds