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Neutron Slowing Down: Kinematics B. Rouben McMaster University Course EP 6D03 – Nuclear Reactor Analysis (Reactor Physics) 2009 Jan.-Apr. 2009 January 1 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 2009 January 2 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 2009 January 3 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. 2009 January 4 Collision in L and CM Systems Consider the collision in both the L and the CM systems: 2009 January 5 Velocities Before Collision In the L and CM systems, let the velocities of the neutron and nucleus respectively be vnL , vnCM , vNL 0, and vNCM (1) With respect to the L system, the center-of-mass system is itself moving, with velocity vnL AvNL vnL 0 vnL vC ( 2) 1 A A 1 A 1 The velocities in L and CM are then related by: A vnCM vnL vC vnL 1 A 1 vNCM vNL vC vC vnL (3) A 1 2009 January 6 Velocities After Collision Let the velocities and energies after the collision be denoted with a superscript apostrophe: ' ' ' ' vnL , vnCM , v NL , and v NCM ( 4) 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): ' A 1 vnCM vnCM vnCM ' vnL , and vNCM vNCM ' vnL (5) A 1 A 1 (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 -! 2009 January 7 Angles of Scattering in L and CM The velocity of the center of mass is not affected by the collision: ' ' vC vnL vnCM vnL vnCM ( 6) The diagram below relates the L and CM angles of scattering. From the diagram vnL sin L vnCM sin CM ' ' and vnL cos L vC vnCM cosCM ' ' ( 7) 2009 January 8 Relationship Between Scattering Angles By dividing the parts of Eq. (7) we get the relation vnCM sin CM ' sin CM sin CM tan L ( 7) vC vnCM cosCM ' vC cosCM vC cosCM ' vnCM vnCM But from the two parts of Eq. (3) we have AvnL v 1 vnCM AvC , i.e. C (8) A 1 vnCM A So that Eq. (7) becomes sin CM tan L ( 9) 1 cosCM A 2009 January 9 Relationship Between Scattering Angles From Eq. (9), we get 1 1 cos L sec L 1 tan 2 L 1 1/ 2 1 sin CM 2 1 2 cosCM A 1 cosCM A 1/ 2 10 1 2 2 cosCM 1 A A 2009 January 10 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: vnL vnCM 2vnCM vC cos CM vC '2 '2 ' 2 (11) This, together with Eqs. (2) and (5), gives: 2 2 A 2 A 1 1 2 vnL '2 vnL 2 vnL vnL cosCM vnL A 1 A 1 A 1 A 1 A2 2 A cosCM 1 2 vnL (12) A 12 The neutron energies in L, before and after, are in the same ratio: A2 2 A cosCM 1 E ' EnL (13) nL A 12 2009 January 11 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. EnL EnL , i.e. ' EnL EnL ' max (14) It is convenient to rewrite Eq. (13) using the new variable A 1 2 (15) A 1 In terms of , Eq. (13) becomes (1 ) (1 ) cosCM E ' nL EnL (16) 2 2009 January 12 Neutron’s Final Energy Range The minimum final energy of the neutron in L is obtained in a backscattering collision (L= CM = ): E'min 1 1 (1) E EnL (17) nL nL 2 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: EnL EnL EnL ' (18 ) i.e., the range of the final neutron’s energy has a width of (1-)EnL. 2009 January 13 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) dCM. And note also from Eq. (16) that EnL (the neutron' s energy after the collision) is linear in cos(CM ). ' 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. 2009 January 14 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) P EnL EnL 1 ' , EnL EnL EnL (19 ) ' 1 EnL This also means that the scattering cross section can be written s E s EnL EnL ' , EnL EnL EnL ( 20 ) ' 1 EnL From Eq. (19), the average value of the final neutron energy in L is P EnL EnL EnLdEnL EnL EnL 'avge ' ' ' EnL ' 2 EnL 1 EnL 1 E 1 EnL E EnLdEnL ' ' nL nL 1 EnL 2 EnL EnL 2 EnL 1 2 2 EnL ( 21) 21 EnL 2 2009 January 15 Average Energy Loss The average loss in energy of the neutron is therefore 1 1 E EnL E avge nL EnL 'avge EnL EnL (22) 2 2 Thus we can write that, on each collision, the average fractional loss of energy of the neutron is : E avge 1 ( 23) E 2 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. 2009 January 16 Neutron Lethargy The quantity u “neutron lethargy” is a function of the neutron’s energy as it slows down, and is defined as E u ln 0 (24), E where E0 is defined as an upper lim it to the neutron energy , and is often taken as E0 10 MeV (25) 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 E0 E E u ln ln 0 ln nL log arithmic energy loss (26) EnL ' E nL E' nL 2009 January 17 Interactive Discussion/Exercises ' 1): U sin g the probability distribution for EnL [ Eq.(18)], and the definition of the average gain in leth arg y after a collision, show that the latter is equal to 1 ln 1 A 12 ln A 1 (27) 1 2A A 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). 2009 January 18 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? 2009 January 19 END 2009 January 20

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cross sections, neutron energy, reactor physics, slowing down, diffusion theory, neutron transport, nuclear engineering, cross section, transport theory, neutron source, target nucleus, nuclear reactions, nuclear reactor, discrete ordinates, neutron scattering

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posted: | 9/15/2010 |

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