(a) A neutron of mass mn and kinetic energy K makes a...

(a) A neutron of mass $m_{\mathrm{n}}$ and kinetic energy $K$ makes a head-on elastic collision with a stationary atom of mass $m$. Show that the fractional kinetic energy loss of the neutron is given by $$ \frac{\Delta K}{K}=\frac{4 m_{\mathrm{n}} m}{\left(m+m_{\mathrm{n}}\right)^{2}} $$ Find $\Delta K / K$ for each of the following acting as the stationary atom: (b) hydrogen, (c) deuterium, (d) carbon, and (e) lead. (f) If $K=1.00 \mathrm{MeV}$ initially, how many such head-on collisions would it take to reduce the neutron's kinetic energy to a thermal value $(0.025 \mathrm{eV})$ if the stationary atoms it collides with are deuterium, a commonly used moderator? (In actual moderators, most collisions are not head-on.)

(a) A neutron of mass $m_{\mathrm{n}}$ and kinetic energy $K$ makes a head-on elastic collision with a stationary atom of mass $m$. Show that the fractional kinetic energy loss of the neutron is given by
$$
\frac{\Delta K}{K}=\frac{4 m_{\mathrm{n}} m}{\left(m+m_{\mathrm{n}}\right)^{2}}
$$
Find $\Delta K / K$ for each of the following acting as the stationary atom:
(b) hydrogen, (c) deuterium, (d) carbon, and (e) lead. (f) If $K=1.00 \mathrm{MeV}$ initially, how many such head-on collisions would it take to reduce the neutron's kinetic energy to a thermal value $(0.025 \mathrm{eV})$ if the stationary atoms it collides with are deuterium, a commonly used moderator? (In actual moderators, most collisions are not head-on.)

Key Concepts

Elastic Collision

An elastic collision is one in which both the kinetic energy and momentum of the system are conserved. This concept is fundamental because, during the collision, no energy is lost to other forms like heat or deformation, allowing for a clear analysis of energy transfer between particles.

Conservation of Momentum and Energy

The problem utilizes the conservation laws of momentum and kinetic energy to derive the relation for the fractional kinetic energy loss. These conservation principles enable the calculation of velocities and energies post-collision by ensuring that the total momentum and energy before and after the collision remain constant.

Fractional Kinetic Energy Loss

This is a measure of the proportion of kinetic energy lost by the initially fast particle (here, the neutron) during the collision. The provided formula, (4 m? m) / (m + m?)², quantifies how the kinetic energy is partitioned between the neutron and the target atom after the collision.

Mass Ratio Effects

The ratio of the masses of the colliding particles plays a crucial role in the energy transfer efficiency. The derived expression shows that the maximum energy transfer occurs when the masses are equal, which is why light atoms like hydrogen or deuterium are more effective in slowing down neutrons compared to heavier atoms like lead.

Neutron Moderation

Neutron moderation is the process by which fast neutrons are slowed down to thermal energies using collisions with nuclei. The efficiency of this process depends on the energy loss per collision, which is determined using the fractional kinetic energy loss. In reactor physics, moderating materials (often light nuclei) are selected because they maximize energy transfer per collision, thereby reducing the number of collisions needed to achieve thermal energies.

Transcript

00:01 In this question, we have a collision between a moving neutron with stationary atom.

00:12 Then this is before the collision.

00:15 And then after the collision, we have neutron maybe moving backward.

00:24 And then the atom will move with a final velocity bf.

00:29 We want to find the fractional loss of genetic energy of the neutron, show that it is given in the...

00:37 We need to show delta k over k is equal to 4 mx times m by m plus m square.

00:57 This is what we need to show in part a.

00:59 And then we need to calculate this number for different atom, such as hydrogen, deuterium, carbon and lead, and calculate the number of head -on collisions needed to reduce the neutrons kinetic energy from one mbb to 0 .025 eb.

01:18 So, and the collision is with the deuterium.

01:29 So to do part a, we'll write down two equations based on conservation of momentum and energy.

01:49 And then for conservation of energy, we have the initial kinetic energy of the neutron is equal to the final kinetic energy of the neutron plus the final kinetic of the atom.

02:12 So we want to find delta k with k and delta k for the neutron is half mn b and i square minus half m b n f square.

02:24 And this is equal to half m b f square.

02:28 So the task is to find of express vf in terms of vni, then it's easier to do this problem.

02:40 So we have two equations, okay? equation 1 and equation 2.

02:49 So we want to eliminate vnf.

02:53 So from 1, we have vnf equals to mnvni minus mvf divide by mn.

03:02 Then we'll substitute this into two.

03:14 This is what we have.

03:16 Okay, half, mn, b, and i square is equal to half mn, mn, b, and i minus mbf divide by mn square plus half mv square...