Neutron Slowing Down: Kinematics

1. Neutron Slowing Down: Kinematics B. Rouben McMaster University Course EP 6D03 – Nuclear Reactor Analysis (Reactor Physics) 2009 Jan.-Apr.

2. Contents • We start the discussion of the energy dependence of the neutron flux by deriving the kinematics of neutron-nucleus collisions in the neutron slowing-down process. • Reference: Duderstadt & Hamilton, Section 2.I-D pp. 34-45

3. Kinematics of Collisions • Kinematics  relationship between pre- and post-collision momentum and energy in neutron-nucleus scattering. • We label the neutron n and the moderator nucleus N. • Using the scale of atomic masses, it is a very good approximation to treat the neutron as having mass 1 and the nucleus as having mass A (the moderator’s atomic mass): • mn = 1 • mN = A

4. Stationary Nuclei & Centre-of-Mass System • We also make the assumption that the moderator nuclei are at rest. This is a good approximation for very fast-moving neutrons (recall that a 1-MeV neutron travels at 13,800 km/s). • The moderator nuclei are stationary in the reactor’s frame of reference, which is by convention called the “lab system” – which we shall denote by a subscript L. • We need also to consider the “center-of-mass system” of the neutron and the nucleus – which we shall denote by a subscript CM.

5. Collision in L and CM Systems • Consider the collision in both the L and the CM systems:

6. Velocities Before Collision • In the L and CM systems, let the velocities of the neutron and nucleus respectively be • With respect to the L system, the center-of-mass system is itself moving, with velocity • The velocities in L and CM are then related by:

7. Velocities After Collision • Let the velocities and energies after the collision be denoted with a superscript apostrophe: • By conservation of momentum and energy, it is easy to show that after the collision the speeds of the neutron and of the nucleus in CM are not changed, only their direction of motion (the notation for speeds is as for velocities, but without the arrow): (use was made of Eq. (3)) • i.e., in the collision, the neutron may be scattered in CM through the angle CM: the nucleus would then be moving after the collision in the direction given by angle CM -. In L the angle of scattering of the neutron would be different, say L. Note that the angle of motion of the nucleus in L after the collision is notL -!

8. Angles of Scattering in L and CM • The velocity of the center of mass is not affected by the collision: • The diagram below relates the L and CM angles of scattering. • From the diagram

9. Relationship Between Scattering Angles • By dividing the parts of Eq. (7) we get the relation • But from the two parts of Eq. (3) we have • So that Eq. (7) becomes

10. Relationship Between Scattering Angles • From Eq. (9), we get

11. Neutron Speed & Energy After Collision • From the diagram on slide 11 we can relate the neutron speeds before and after the collision, using the cosine law: • This, together with Eqs. (2) and (5), gives: • The neutron energies in L, before and after, are in the same ratio:

12. Neutron Energy After Collision • Since the numerator in Eq. (13) cannot be larger than the denominator, we see that the neutron cannot gain energy in L on account of the collision – as we suspected, i.e. • It is convenient to rewrite Eq. (13) using the new variable • In terms of , Eq. (13) becomes

13. Neutron’s Final Energy Range • The minimum final energy of the neutron in L is obtained in a backscattering collision (L= CM = ): • Note that the neutron can lose all its energy in L only in a backscattering collision with a nucleus characterized by  = 0 (A = 1), i.e., only hydrogen (1H1) can stop a neutron in a single (head-on) collision. • In general, summarizing Eqs. (14) and (17), the neutron’s energy in L goes from EnL before the collision: • i.e., the range of the final neutron’s energy has a width of (1-)EnL.

14. Probability of Specific Energy • What is the probability of the neutron ending up with any specific energy value in that range? • To answer this question, note first that for typical moderators, i.e., those with light nuclei (e.g., A  12), scattering is isotropic in the center-of-mass system for starting neutron energies EnL 10 MeV, therefore the probability of scattering is independent of the (differential) solid angle d = d cos(CM) dCM. • And note also from Eq. (16) that • We can then conclude that the probability of attaining any value of the final neutron energy in L is also uniform in the allowed range, EnL EnL.

15. Average Value of Neutron’s Final Energy • That is, the probability of scattering to any value of the final neutron energy has the constant value (a very important conclusion) • This also means that the scattering cross section can be written • From Eq. (19), the average value of the final neutron energy in L is

16. Average Energy Loss • The average loss in energy of the neutron is therefore • Examples: • In a collision with a hydrogen nucleus (1H1,  = 0), a neutron loses on average half its energy. • In a collision with a deuterium nucleus (2H1,  = 1/9), a neutron loses on average 4/9 of its energy.

17. Neutron Lethargy • The quantity u “neutron lethargy” is a function of the neutron’s energy as it slows down, and is defined as • With that definition, the neutron lethargy increases as the neutron slows down, as is appropriate for the name “letahrgy”. • The gain in lethargy after a collision, denoted , is

18. Interactive Discussion/Exercises • 1): • 2) This formula cannot be used as is for hydrogen ( = 0). Show, by writing  = e-x and letting x, that it reduces in that case to  = 1. • 3) Note that the average gain in lethargy is constant from collision to collision. This can be used to determine the average number of collisions to thermalize a neutron (say, from 2 MeV to 1 eV).

19. Interactive Discussion/Exercises • On average, how many collisions does it take for a 2-MeV neutron to be thermalized by a hydrogen moderator to an energy of 1 eV? • Same question, but for a deuterium moderator • Same question, but for a carbon (graphite) moderator • Are the answers above sufficient to get a sense of the different total distances travelled by a neutron as it is thermalized by the different moderators, or do you need additional information, and, if so, which?

20. END