NeutronsPart 2 of 12

Navy Recruiting District DenverCDR Mike Wenke – XO

ET1 (SS) Matt Byron – Nuke CoordinatorENS Titus Reed

OC Kellan Downing25 August 2011

Microscopic Cross Section

• Cross section (σ) – “Target Area”– Controls probability of

reaction happening– Larger than geometric

cross section of nucleus– Measured in barn (b)

1b=10-24cm2

• Partial Cross Sections– Each reaction has its

own cross section– Total cross section is

sum of partial cross sections

Energy Dependence of Cross Section

• Microscopic Cross Section is dependent on:– Identity of target nucleus– Identity of incident particle – Kinetic energy of incident particle

•

Macroscopic Cross Section

• Macroscopic Cross Section (Σ)– Total nuclear target area of a material

Σ=Nσ• N=number of atoms per unit area• σ=area per atom

– Are additive (Σt=Σa+Σs)

– For mixed material the macroscopic cross section is the sum of the macroscopic cross section of each component

Mean Free Path• The mean free path– The average distance a particle travels before

colliding with anothermean free path = λ=1/Σ

– Total mean free path (λt): average distance before any type of collision

– Absorption mean free path (λa): average distance before collision that results in an absorption reaction

– Scattering mean free path (λs ): average distance before collision that results in a scattering reaction

sat 111

Attenuation• Attenuation Law– Describes the change

in the intensity of a beam of particles as it passes through a medium

– Number of particles decreases exponentially with position

– Number never becomes zero even at very large distances

xΣ0

te(x)

φ(x) = is number of particles at position xφ0 = initial number of particlesΣ = macroscopic cross sectionx = distance from material surface

Neutron Slowdown

• Prompt neutrons born in fission process have an average energy of 2 MeV delayed neutrons average 0.4 MeV

• Mechanisms– Elastic and inelastic scattering are the only processes

that removs energy without removing neutrons from the cycle.

– Inelastic scattering plays a minor role• Threshold energy is on the order of several keV• Cross section is much smaller than elastic cross section for

most nuclei

Moderator Materials

• Material Selection– The amount of energy

lost per collision• energy lost increases as

the mass of the target nucleus decreases

– Magnitude of scattering cross section• the larger the better

– Magnitude of absorption cross section• the smaller the better

• Common Moderators– Ordinary water (H2O)

– Heavy water (D2O)

– Beryllium– Graphite (carbon)

Probability Density Function For The Energy Of Scattering Neutrons

Possible final energies of scattered neutrons

αE0<E<E0

E= Final energyE0= Initial Energy

mT = mass of target

mn = neutron's mass

)m(m

)m(mα

nT

2nT

Maximum possible neutron energy loss

Δemax =E0(1-α) On average each elastic scattering event decreases energy by a factor of (1+α)/2

Quantification of Moderator Effectiveness

• Slow Down Power (ξΣs)– Measure of material’s ability to

reduce neutron energy– Does not account for absorption– ξΣs=ξ/λs

• ξ = Average logarithmic energy decrement

• Σs = Macroscopic scattering cross sections

• λs = scattering mean path

• Moderating Power– accounts for absorption

reactions– ξΣs/Σa

• Increase in temperature– Lower the peak height– Peak energy is shifted to

right– The distribution widens

Maxwell-Boltzmann Distribution

• Kinetic energy distribution that a burst of neutrons eventually have, assuming:– infinite environment– non-absorbing

• Most probable energy– E(eV)=8.61x10-5 xT(K)– Assumes no absorptions

Deviation From Maxwell-Boltzmann

• Absorption removes more neutrons from the lower energy peak– Shifts distribution to higher

energy– Lowers peak– Referred to as hardening

• Continuous production of fast neutrons:– Known as a slowdown

source– More neutrons in the

higher energy range

• Finite reactor size– Smaller effect– More high energy neutrons

escape than low energy– Known as diffusion cooling

Neutron Density and Flux

• Neutron Density– Represented by “n”– Typically units are

neutrons/cm3

– Varies with position in reactor

• Neutron Flux (φ)– Chance of neutron

reacting with a nucleus is dependent on neutron flux

– φ=nν– Thermal flux (φth) – flux

of thermal neutrons• φth=nthν• Where ν is the average

speed of the thermal neutrons

Reaction Rates

• Number of nuclear reaction of a particular type in a given amount of time

• R=φΣ– φ = proton flux– Σ = Macroscopic cross section– Typical units are Reactions/ cm3-second

• There are many different reaction rates just like there are different microscopic cross sections

Power Density

• The energy released per fission event is constant. (200 MeV for thermal fusion of 235U)

• PD=kRf=kφthΣf

– PD = power density– Rf = fission reaction rate– φth = thermal proton flux– Σth = thermal macroscopic cross section– k=εk’

• ε = fast fission factor (account for fission that occurs while protons are slowing down)

• k’= constant that contains reactor volume

Slowing Down Length

• Neutrons travel in only straight lines between collisions• Absorption stops neutron progress• Scattering changes direction of neutron• Slowing down distance is related

to crow flight distance by:

Slowing Down Length

• The mean free path length is the average length of each straight line that makes up the neutrons path

• A large slowing down distance, Ls, is associated with a large mean free path, s, and a large nuclear mass

• Large Ls means more spreading out of particles, so proper moderators must be chosen for each individual reactor based on the reactor’s desired size (large, small, etc.)

Choosing the Correct Moderator

• Scattering in the moderator dominates all scattering in the reactor

• Scattering cross section for the moderator is directly proportional to the density of the moderator

• Thus desired slowing down length can be achieved: Ls

2 = (Ls2)ref x (ref /)

Migration Length

• Measure of the straight-line distance traveled by a neutron from its birth in the fast region to its absorption in the thermal region

• Depends on the slowing down length and thermal diffusion length:

M = sqrt( Ls2 + L2)

Neutron Life Cycle

• Power generated by a reactor is proportional to the thermal neutron density, nth

• nth changes by neutron multiplication• Ratio of fission neutrons (nth) produced in two

successive fissions determines whether reactor power is constant or changing

Neutron Life Cycle

Life Cycle in Arbitrary Volume

• ELFPLThFN

Six-Factor Formula

• Ni+1 = Ni x Nf x Nth x p x f x x • Where:– Ni+1 = number of neutrons in next generation– Ni = Number of neutrons in cycle– Nf = Fast Non-Leakage Factor– Nth = Thermal Non-Leakage Factor– P = resonance escape probability– f = thermal utilization factor– = reproduction factor– = fast fission factor

Factor Definitions

• Nf = fraction of neutrons beginning each generation that do not leak out while slowing down

• P = fraction of thermalized, slowing down neutrons which do not leak out

• Nth = fraction of thermal neutrons that do not leak out of the reactor (are absorbed)

• F = of all the thermal neutrons absorbed in the reactor, the fraction that are absorbed in the fuel

• = number of fission neutrons produced per thermal neutron absorbed in the fuel

• = ratio of total fission rate (fast + thermal) to the thermal fission rate

Buckling and Leakage

• In Reactor analysis, buckling (B2) is a measure of the overall curvature of the flux (how fast the flux is changing vs. the actual flux itself)

• Infinite reactor system as buckling = 0• Large values of B2 mean a large surface area to

volume ratio of reactor, and vice versa • The further a neutron travels in slowing down or

thermal diffusion, the greater chance it will reach the core’s surface and leak out, thus losing a chance to continue the chain reaction

Flux Shapes• Neutrons crossing the reactor surface have no chance at returning

• Neutron flux at reactor boundaries is very low

•Flux is highest at center because amount of relative fuel present is high

• Flux increases as the slope increases

Flux Shapes

• Flux is greatest at the reactor’s core, where chance of leakage is low

• Reactor is surrounded by an unfueled region called a reflector

• Reflector has large scattering cross-section so some neutrons return to reactor to be thermalized