A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 1 CHAPTER 3 LECTURE 2 THERMAL NEUTRONS

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

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 3 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:

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Forward Scattering in the L System Neutrons show a preferential forward scattering in the L system From the scattering diagram  In the CM the scattering is spherically symmetric (isotrope). this means

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Forward Scattering in the L System  The deviation of the L system scattering from that of the CM system can be expressed in terms of the average value of See Section 6.6 in the book

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Forward Scattering in the L System Case 1 : the average forward component becomes very small, i.e. the angle is 90 and the scattering is almost isotropic Case 2 : lighter nuclei pronounced preference for the forward scattering in the L system

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Transport Mean free Path and Scattering Cross Section Question: what is the consequence of the predominance of the forward scattering for neutrons in the L system Answer: average distance traveled by the neutron before it is scattered through an angle of 90 o will be greater than the corresponding average distance for isotopic scattering the mean free path will be affected consequence this preference in traveling in the forward direction must be taken into account Hence a neutron will move a greater distance away from its origin, on the average, in a given number of collisions that would do if there were no preferred forward scattering directions because

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Transport Mean free Path and Scattering Cross Section  The increased effective mean free path for nonisotropic scattering is called the transport mean free path Where is scattering mean free path  The corresponding cross section is called transport cross section is defined as:

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Transport Mean free Path and Scattering Cross Section See Example 6.7, page 155

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Average Logarithmic Energy Decrement When considering the thermalization of fast neutrons to predict or estimate the number of collisions that are required for most of the neutrons to have their initial fission energy E 0 reduced to thermal energies E t It is important  Go back to section 6.6 in the book, examine example 6.6., page153: The average fractional energy loss per collision is

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Average Logarithmic Energy Decrement Observe also: Observe that: independent from the initial energy of neutrons Lethargy  We introduce the concept of Lethargy of the neutron and defined as: then Where E 0 here is some constant initial reference value

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Average Logarithmic Energy Decrement See Figure 6.14, page 156

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Average Logarithmic Energy Decrement Definition: The average change in log E between two successive collisions is called the average logarithmic energy (lethargy) decrement per collision, represented by Question: how to calculate ? Assumptions: probability per unit energy loss

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Average Logarithmic Energy Decrement The denominator is equal to 1

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Average Logarithmic Energy Decrement Special cases: Very important conclusion: Practically no energy is lost by a neutron in a collision with a heavy nucleus. This agrees with what was found previously that heavy nucleus are poor moderators See Example 6.8, Page158

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Average Logarithmic Energy Decrement Question : how the number of collisions n required to reduce the neutron energy from initial energy E 0 to a final thermal energy Answer: by the knowledge of the average logarithmic energy decrement per collision See Example , page 158…..very important

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-Down power and Moderating Ratio Question: what are the criteria for a substance to be a best moderator? Answer: What else? number of scattering nucleus per unit volume, N 0 solid substances are be better moderators than gases Hence there are three quantities that determine the slowing-down ability (quality) N0

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-Down power and Moderating Ratio slowing-down power (sdp) Definition: slowing-down power (sdp) Is this complete?NO Interpretation of sdp : average loss in the logarithm of the energy (lethargy of the neutron ) per unit distance of travel in the moderator To give a complete description of the suitability of a substance as moderator, we must also absorption properties specify its absorption properties for thermal neutrons A substance would not be suitable as a moderator if its absorption cross section is high

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-Down power and Moderating Ratio What is a good moderator? large sdp and a small absorption cross section ratio between the sdp over the macrscopic absorption cross section Definition of moderating ratio (mr): ratio between the sdp over the macrscopic absorption cross section Case of a moderator with a combination of elements

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-Down power and Moderating Ratio See Table 6.2, Page 161 Example 6.12, page 160

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density Definition of slowing-down density: it is the rate at which the neutron per unit volume of a moderator slow down past a particular energy E it is denoted by discontinuous slowing-down process is discontinuous neutron energy change in finite steps after each collision existence of the absorption resonances for neutron of discrete energies + not simple the variation of q with energy is not simple Because heavy nuclei moderator Special case: heavy nuclei moderator: the average energy loss per collision is small enough, then we can simplify the problem by treating the slowing down as a virtually continuous process See Figure 6.14, page 156

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density heavy nuclei moderator Special case: heavy nuclei moderator: the average energy loss per collision is small enough, then we can simplify the problem by treating the slowing down as a virtually continuous process See Figure 6.14, page 156

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density Let’s now assume the following: 1- continuous slowing-down process (heavy nuclei moderator)  continuous slowing-down model. By using this model we can obtain some information about the energy distribution of the neutrons during their passage from high energies to thermal energies. 2- Rate at which neutrons enter an energy interval between E and E+dE is equal to the rate at which neutrons leave it. See Figure 6.15, page 162

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density Let’s now assume the following: Perfect case: Rate of production of neutrons with an initial energy E 0 = constant = Q Slowing-down density q(E) will remain the same at all energies and it will be equal to Q No neutrons are lost either by escaping (leakage) from the assembly or through being absorbed by the materials of the assembly before they have reached thermal energies +

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density Simple case study: The number of neutrons per cm 3 whose energies lie between E and E+dE is. (4.58) Why negative sign?

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density. (4.60)

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density. neutron flux per unit energy is inversely proportional to the energy E  We can have this equation from previous one Physical interpretation: Scattering loss=neutron influx gain

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down density. Physical interpretation: Scattering loss=neutron influx gain scattering loss rate of neutron influx called collision density Example 6.13, page 163

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down time. Combining and the time required for the neutrons to slow down from an initial energy E 0 to a final energy E t If we can assume that and remain constant over the slowing-down range of energy The total slowing-down time T integration slowing-down time through energy interval

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Slowing-down time. Replacing See Example 6.14, page 164

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability. Previously, we considered the assumption that If the moderator has a measurable absorption cross section slowing-down density q will no longer be constant, but will depend on the neutron energies The change in q, as a neutron pass from an energy E to an energy E- is due to neutron absorption equal to the rate of neutron absorption per cm 3 =

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability.  the absorption loss is analogous to the scattering loss  slowing-down density with absorption present, q(E), will be only a fraction of its value Q in the absence of absorption is a measure of the fraction of neutrons that have escaped absorption and still survive after having been slowed down from their initial energy E 0 to an Energy E

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability. Answer: various neutron resonances which are predominant in the epithermal region of energies for the moderator and thermally nonfissionable components (for example U 238 ) of the fuel materials of the reactor Question: what are the origin of these neutron absorptions?

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability. Answer: various neutron resonances which are predominant in the epithermal region of energies for the moderator and thermally nonfissionable components (for example U 238 ) of the fuel materials of the reactor Question: what are the origin of these neutron absorptions?  The factor measures the extent to which the neutrons are successful in evading these resonance traps called the resonance escape probability  The steady-state condition within the energy interval when neutron absorption is included Scattering loss + absorption loss = influx

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability. substituting and the lhs of When this combined with

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability. The physical meaning of the lhs of equations number of neutron collisions terminating in an absorption divided by the total number of collisions, i.e., both those leading to absorption and those leading to scattering event represents the probability of a neutron capture per collision The factor is the average number of collisions corresponding to an increase in the neutron lethargy by an amount the product of the two factors represents the probability of a neutron absorption, as the neutron energy changes by an amount corresponding to a change in lethargy of

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability. By integrating between the limits of neutron energies E and E 0, we get With the integral being a summation of the partial probabilities of a neutron absorption; thus we can say that we have here the total probability per neutron of being absorbed as it is moderated from an initial energy E 0 to a final energy E If we put Since the initial slowing-down density q(E 0 ) is equal to the neutron production rate Q

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons Resonance Escape Probability. Comparingand In all practical cases the exponent is small, so that we can expand it and, neglecting higher terms than the second, obtain the result neutron absorption probability probability of the neutron escaping absorption during its passage between the energies E 0 and E is the resonance escape probability that is why

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons The effective Resonance Integral. the resonance integral called Assumption:remaining fairly constant over the range of integration and The resonance escape probability can therefore be expressed by:

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons The effective Resonance Integral. The integral in this expression is known as the effective resonance integral and the integrandthe effective absorption cross section

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons The effective Resonance Integral. since resonance absorption of the various fuels that are of interest in reactor assemblies vary in an irregular manner, the value of the resonance integral must be found empirically The number of nuclei per cubic centimeter, N 0, refers to the number of resonance absorber nuclei present; thus if all resonance absorption is ascribed to uranium, N 0 is simply the number of uranium nuclei per cm 3 Observations:

A. Dokhane, PHYS487, KSU, 2008 Chapter3- Thermal Neutrons 42 Homework Problems: 15, 16, 17 of Chapter 6 in Text Book, Pages 169 الى اللقاء بعد الاجازة ان شاء الله اجازة ممتعة!!