basic reactor theory and reactions- presentation 2
DESCRIPTION
NRD Denver's Nuclear DEP Meeting Lesson #2TRANSCRIPT
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