1. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 1 NEWS Need to fix a date for the mid-term exam? The first week after vacation!

2. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 2 Chapter3 Lecture 1 THERMAL NEUTRONS

3. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 3 1.Review 2.Neutron Reactions 3.Nuclear Fission 4.Thermal Neutrons 5.Nuclear Chain Reaction 6.Neutron Diffusion 7.Critical Equation

4. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 4 Introduction Energy distribution of thermal neutrons Effective cross section for thermal neutrons. The slowing down of reactor neutrons Scattering angles in L and C.M systems Forward scattering in L system Transport mean free path and scattering cross section Average logarithmic energy decrement Slowing-down power and moderating ratio Slowing-down density Slowing-down time Resonance escape probability The effective resonance integral Lecture content:

5. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 5 4.1 Introduction Question: why it is necessary to reduce the neutron energies from 2 Mev to thermal energies? Neutrons born in a fission  2 Mev Answer: very high fission cross section for thermal neutrons as compared to that for the nonfission capture cross section. A reactor which is designed that almost all neutron fissions occur with neutrons of thermal energies Thermal Reactor This type represents the main object of interest to us in this course

6. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 6 4.1 Introduction because the existence of large number of neutrons in reactor physical processes To get some understanding of the physical processes involved statistical approach kinetic theory of gases it is necessary to employ a statistical approach very similar to that of the kinetic theory of gases.

7. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 7 4.2 Energy Distribution of Thermal neutrons not of uniform energy  Neutrons in nuclear reactors are not of uniform energy but are distributed over an energy range that extend from very slow to very fast neutrons of about 17 Mev. See Figure 6.1, Page 135  Fast neutrons are continuously being produced by fission  Slow neutrons are continuously being absorbed  leads to fission  creation of fast neutron.

8. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 8 4.2 Energy Distribution of Thermal neutrons Question: How the compensation for the steady loss of slow neutrons through absorption is made? Answer: by rapid and efficient slowing down of the fast neutrons so as to maintain the supply of slow neutrons. This process is called the moderation or thermalization moderation process is achieved by a moderating material and called moderator which is incorporated in the reactor.

9. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 9 4.2 Energy Distribution of Thermal neutrons Question: How the presence of moderator slows down the fast fission neutrons? Answer: by elastic collisions between the moderator nuclei and the neutrons until the average kinetic energy of neutrons corresponds to that of the moderator nuclei. See Figure 6.2, page 136

10. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 10 4.2 Energy Distribution of Thermal neutrons Answer: Higher scattering cross section compared to the absorption cross section. Question: what is the best moderator? This allows a rapid reach of thermal energies and avoid the nonproductive absorption.

11. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 11 4.2 Energy Distribution of Thermal neutrons  When neutrons are thermalized  their energy distribution will be approximately Maxwellian, corresponding to the temperature of the surrounding medium  For neutrons in thermal equilibium with the moderator  velocity distribution will be given by the Maxwell-Boltzmann expression: (4.1) Number of neutrons whose velocities lie between v and v+dv is given by dn=n(v)dv

12. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 12 4.2 Energy Distribution of Thermal neutrons (4.1) This distribution is shown in Figure 6.3, page 136.

13. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 13 4.2 Energy Distribution of Thermal neutrons  The energy distribution of the neutrons can be written in the form: (4.2) This distribution is shown in Figure 6.4, page137

14. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 14 4.2 Energy Distribution of Thermal neutrons  The energy distribution of the neutrons can be written in the form: (4.2)  The distribution functions n(v) and n(E) are called “ density function” Give: the number of neutrons per unit velocity interval and per unit energy interval, respectively. What means?? and Very important

15. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 15 4.2 Energy Distribution of Thermal neutrons represented by the small shaded areas in figure 6.3 and 6.4 probable velocity The velocity distribution function has a maximum for a value of the velocity v p called most probable velocity give the number of neutrons that are found within a small velocity or energy interval Equations 4.1 and 4.2

16. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 16 4.2 Energy Distribution of Thermal neutrons Question: How the most probable velocity is calculated? Answer: By differentiating (4.1), and setting this equal to zero (4.4) Can you prove this ????

17. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 17 4.2 Energy Distribution of Thermal neutrons  Energy corresponding to most probable velocity is  Neutron temperature is  The energy distribution equation (4.2) leads to the most probable energy E 0 by the same method  derivation and make it equal to zero Can you prove this?

18. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 18 4.2 Energy Distribution of Thermal neutrons Conclusion: the most probable energy E 0 is not equal to the energy corresponding to the most probable velocity E p but it is half of it In combination with The average velocity is related to v p Prove this???

19. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 19 4.2 Energy Distribution of Thermal neutrons Summary

20. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 20 4.2 Energy Distribution of Thermal neutrons The energy distribution of neutrons in a reactor is not exactly Maxwelllian, it corresponds to a temperature which is slightly higher than that of the moderator material. WHY???? Reality: Existence of a steady influx of high-energy neutrons from fission a steady absorption of the low-energy neutrons by the fissionable material + Consequence

21. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 21 4.2 Energy Distribution of Thermal neutrons The energy distribution of neutrons in a reactor is not exactly Maxwelllian, it corresponds to a temperature which is slightly higher than that of the moderator material. WHY???? Effect of raising the high-energy end of the Maxwellian distribution and depressing the low-energy portion of the distribution. Consequence See Figure 6.5, page 139

22. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 22 4.2 Energy Distribution of Thermal neutrons  The actual neutron distribution will therefore correspond to an effective temperature that is somewhat higher than that of the actual reactor material. Consequence effective rise of neutron temperature effect is called: spectrum or thermal hardening

23. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 23 4.2 Energy Distribution of Thermal neutrons  The elevation of the neutron temperature T n above that of the moderator temperature T can be calculated by the relation: valid for A<25 and Example 6.1, page 140

24. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 24 4.3 Effective Cross Section for Thermal Neutrons  We have seen that the absorption cross sections for slow neutrons are strongly dependent on the neutron energies.  In many cases, this dependence follows the 1/v law Answer: Because we have not only one speed of neutrons in the thermal region but a range. Then we have to find an average value that is the effective one Example: reaction (n, alpha) with boron is a typical example for the 1/v behaviour Question: why effective cross section مقطع فعلي??

25. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 25 4.3 Effective Cross Section for Thermal Neutrons However, Not all absorption reactions for slow neutrons follow strictly the 1/v law a correction factor is introduced  called “not 1/v” or f-factor. Ranges from 0.85 (for Gadolinium) to 1.5 (for Somarium). See Table6.1, page 140

26. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 26 4.3 Effective Cross Section for Thermal Neutrons  For more rigorous calculations, f-factors should be considered temperature dependent. Because thermal neutrons have a spread in a wide range of speed the corresponding cross section is also has spread over a wide range we use an effective cross section which is defined as:  Value of the cross section that results when the total number of absorption per second per unit volume is averaged over the neutron flux Therefore

27. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 27 4.3 Effective Cross Section for Thermal Neutrons If neutron absorption Xsec obeys the 1/v law However, the average speed is defined as the effective cross section is equal to the cross section for neutrons with speeds equal to the average speed for thermal neutron distribution

28. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 28 4.3 Effective Cross Section for Thermal Neutrons How is calculated?? conventionally taken with reference to the most probable velocity v p for the thermal neutrons Note that with a factor of = 0.886

29. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 29 4.3 Effective Cross Section for Thermal Neutrons  The thermal neutron cross sections are tabulated based on the following condition: speed=2200 meters/sec most probable speed of a Maxwellian neutron distribution at 293.6K and corresponds to a neutron energy of 0.0253 ev Question: how the effective neutron cross section at a temperature T is obtained from the tabulated value? Answer:

30. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 30 4.3 Effective Cross Section for Thermal Neutrons Example 6.2, page 142

31. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 31 4.4 The Slowing Down of Reactor Neutrons Question: the rapid slowing down and thermalization of the fast fission neutrons are an important phase in the design and operation of a nuclear reactor. Why?? Question: How neutrons slow down in a nuclear reactor? Answer: it is necessary to reduce neutron loss due to nonfission and resonance absorption and to increase the fissioning rate Answer: The most important contribution to slowing down of neutrons in a nuclear reactor are elastic collisions with the moderator nuclei. whereby a neutron transfers a portion of its kinetic energy to its collision partner.

32. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 32 4.4 The Slowing Down of Reactor Neutrons  Goal: to get an indicator of the amount of energy transferred from the neutron to the moderator nucleus in an elastic collision.? the collision process needs to be studied in more detail Hence  Collision between two particles can be described either in: 1. Laboratory system (L system)  the target nucleus (moderator) is initially at rest 2. Center of Mass system (CM system)  the center of mass of the colliding particles is at rest initially and remains throughout the collision See Figure 6.6, page 143

33. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 33 4.4 The Slowing Down of Reactor Neutrons  by definition of the CM of two particles: The total linear momentum of the colliding particles is zero in the CM system Where m is neutron mass, M is moderator nucleus mass, v c is neutron velocity and –V is the velocity of M That is why CM system is often more convenient for purpose of calculation However, L system is the frame in which all experimental measurements are made. It is important to establish relations that will permit us to relate the motion of a particle in one system to its motion in the other system.

34. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 34 4.4 The Slowing Down of Reactor Neutrons  See Figure 6.6, page 143. It is clear that the velocity of the target nucleus in the CM system is equal and opposite to the velocity of the CM in the Lab system. (4.19)  Applying the condition that the total linear momentum of the colliding particles must be zero in the CM system (4.20)

35. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 35 4.4 The Slowing Down of Reactor Neutrons  See Figure 6.7, page 144. and Combining and we can find

36. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 36 4.4 The Slowing Down of Reactor Neutrons  Conservation of energy: speeds after collision are not changed in the CM system (see Figure 6.8) they must be also collinear to give zero linear momentum The result of the collision in the CM system is a rotation of the particle system by an angle of

37. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 37 4.4 The Slowing Down of Reactor Neutrons  See Figure 6.9, page 145 the collision in the two systems Where v 1 and v 2 are the velocities of m and M, respectively, after the collision in the L system

38. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 38 4.4 The Slowing Down of Reactor Neutrons  The following geometrical relations can be deduced from the figure: (4.22) (4.23)

39. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 39 4.4 The Slowing Down of Reactor Neutrons (4.25)  By introducing the mass number A of the moderator nucleus, and the mass number 1 for the neutron, we can write with only a negligible error from (4.20) we can get and  Consider: E 0 and E 1 is the neutron energy in the L system before and after the collision respectively. by combining Equations 4.22 and 4.25 the relationship between E 0 and E 1

40. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 40 4.4 The Slowing Down of Reactor Neutrons For This type of collision causes the maximum energy transfer from neutron to the moderator nucleus i.e, largest possible energy loss for the neutron in a single collision

41. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 41 4.4 The Slowing Down of Reactor Neutrons Let’s set Maximum energy loss: The maximum fractional energy loss is

42. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 42 4.4 The Slowing Down of Reactor Neutrons Hence:(4.31) greatermoderator nucleus is the smaller Important conclusion Maximum fractional energy loss is the greater  mass of the moderator nucleus is the smaller light nuclei to be more effective moderators than heavy nuclei. See Examples 6.3 and 6.4 page147

43. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 43 4.4 The Slowing Down of Reactor Neutrons See Examples 6.3 and 6.4 page147

44. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 44 4.5 Scattering Angles in L system and CM system

45. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 45 4.5 Scattering Angles in L system and CM system neutron energy after scattering, E 1, is given by it is very useful in this form because the angular dependence of the scattering in the CM system is very simple

46. A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 46 Homework Problems: 1, 3, 4, 6, 7, 10 and 14 of Chapter 6 in Text Book, Pages 168 الى لقاء آخر, بعد قليل ان شاء الله