A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 1 NEWS Lecture1: Chapter 0 is already on my Website

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 2 Chapter 1 Neutron Reactions March 2008

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 3 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 Chapter1- Neutron Reactions 4 Slow neutron reactions Nuclear reaction cross section Neutron cross section Determination of the cross section Attenuation of neutrons Macroscopic cross section and mean free path Neutron flux and reaction rate Energy dependence of neutron cross sections Fission cross section What we will see in this chapter?:

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 5 Leibstadt NPP, Switzerland

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 6 Nuclear Reactors

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 7 Uranium 235 Neutron Fission Why neutron reactions? They are important for reactor physics since the reaction is initiated by neutrons.

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Slow neutron reactions Why slow neutron ? Nuclear reactions with slow neutrons are the most important reactions in reactor physics and nuclear engineering Consider the following general reaction: a + X  b + Y Here the projectiles a are assumed to be NEUTRONS with kinetic energy not more than few Mev. There are many case reactions: radiative capture 1- if b is gama-ray it is a (n, gama) process  radiative capture  it is important to consider it in the nuclear reactor design  responsible of neutron loss and requires protective measures of shielding to protect against gama radiation

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Slow neutron reactions There are many case reaction: scattering process 2- if a and b are identical particle, i.e. neutrons  scattering process: a. inelastic scattering: if the target nucleus is raised into an excited state by transfer of kinetic energy from the incident neutron. b. Elastic scattering: if no transfer of energy. fission process 3- if b is not an elementary particle (neutron, proton, deutron, or alpha-particle)  two nuclei of intermidiate masses  fission process

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Slow neutron reactions To be retained: the probability of the occurance of any one of the processes is strongly dependent on the neutron energy. What happen when U 235 is the target? All the three process can happen but with different probabilities. Example: bombardment with slow neutron  the relative frequency of occurence for fission, radiative capture, and scattering if around 60, 10, and 1, respectively.

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Nuclear Reaction cross section Terminology: 1) Probability of occurrence of a particular nuclear reaction = cross section of this process. Example: In previous example for probability of occurrence of fission, radiative capture and scattering we can say: the fission cross section is 6 time higher than that of the radiative capture.. 2) Probability that a given reaction will occur between one neutron and one neucleus = microscopic cross section. Example: fission cross section : nuclear collisions that lead to fission of the target nucleus.

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Nuclear Reaction cross section Unit of nuclear reaction cross section unit is Barn 1 barn = cm 2

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Neutron Cross Section  If a neutron beam is allowed to pass through a slab of target material ( see Figure 4.1, page84 ), it will emerge with reduced intensity. Why? Because: Many processes have taken place during its passage: scattering, absorption (gama-emission, fission),…

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Neutron Cross Section  Attenuation of neutron beam (loss in intensity) can be described by the total cross section : fission and radiative capture (nonfission) elastic and inelastic scattering

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions determination of Cross Section Goal : relationship between cross section from one side and measurable observables (incident beam intensity, number of interactions,…) Definitions:: Beam intensity I (flux density or simply neutron flux): number of neutrons crossing unit area perpendicular to beam in 1 Sec. n neutron density and average neutron velocity

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions determination of Cross Section Consider : Homogeneous beam of neutrons pass through thin sheet of target material of area A, thickness t, and have N 0 nuclei per cm 3 A t Total number of nuclei = Available nuclear target area = : cross section per nucleus (microscopic cross section) Probability Probability for one incoming neutron to hit a nuclear target area is equal to the ratio of this area to the total area A. Probability of interaction per neutron = = = =

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions determination of Cross Section Number of reactions per second = interaction rate Number of reactions per second = interaction rate = (probability of interaction per neutron) X (number of incident neutrons per second) Interaction rate r = r = V is the target volume. Hence Definition: Macroscopic cross section is:

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions determination of Cross Section Interaction rate per unit volume: Macroscopic cross section is rate of interactions per unit volume per unit neutron flux. Very often the target material is specified by its areal nuclear density relationship between volume density and areal density Cross section is reaction rate per nuclear density per neutron flux

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions determination of Cross Section For target material with known density,, and atomic weight, M, N 0 can be calculated N is Avogadro number

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions determination of Cross Section Example 4.1, page 88

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Attenuation of Neutrons If a neutron beam passes through matter  suffers a reduction in intensity  emerging beam will be attenuated or weakened. Why? Because of the collision processes that leads to scattering or absorption of the neutrons.

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Attenuation of Neutrons  Consider a homogeneous neutron beam (same energy and direction) passes through a slab of material of 1 cm 2 cross-section area.  Incident flux = I 0  after penetrating a distance x is I.  after a penetration of distance dx it will become I-dI  change in flux is –dI between x and x+dx

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Attenuation of Neutrons  change in flux = number of collision that have occurred in dx Hence: number of collision = reaction rate since Hence:  integration  This formula shows that corresponds to a linear absorption coefficient with dimensions of reciprocal length. Penetration through a distance equal to reduces the beam intensity by a factor of.

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Attenuation of Neutrons. Example 4.2, page 90

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Macroscopic Cross Section and Mean Free Path  Define the penetration distance mean free path Physical meaning: average distance that a neutron can travel in the material without making a collision. Beam densityneutron density: number of neutrons per cm 3 that can penetrate to a distance x without making a nuclear collision of any kind.  After penetrating a further distance dx, of the n remaining neutrons, a number dn will undergo a collision and drop out where dn is obtained by: there is a mistake in the book, in equation You should add dx By differentiation

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Macroscopic Cross Section and Mean Free Path  The combined path length of the group of neutrons that travel a distance x and then suffer a collision within the short distance between x and x+dx is xdn.  If we consider the total combined path length = It can be understood as collection of many groups dn 0 for distance x 0, dn 1 for x 1, dn 2 for x 2, ….dn i for x i ….. The average path legth is obtained by: Prove that This can be proven ??? homework ex.8 give a detailed calculation for equation 4.19, page 90-91

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Neutron Flux and Reaction Rate large number of neutronsgas   For large number of neutrons, it is physically convenient to consider them as a gas  description of movements and random motion based on molecular theory of gases.  Assymptions:  neutrons as a gas  move with a speed (all neutrons have the same speed)  mean free path  Time interval between two successive collisions for a given neutron is: Hence:  number of collisions per second by neutron  For n neutrons per cm 3  total number of collisions/cm3 sec = number of collisions per neutron/sec X number of neutrons/cm 3 this similar to that found for neutron beam traveling in a given direction

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions Macroscopic Cross Section and Mean Free Path  So far, no distinction has been made for scattering and absorption collision as they both contribute to the attenuation  HENCE: Average path length between successive collisions for neutrons moving in a medium of macroscopic cross section is equal to Attenuation : scattering + absorption  In analogy, we can define : mean free path for scatteringmean free path for absorption Note that : Prove this?