NUEN 304: Nuclear Reactor Analysis

Lecture 1 – Course Introduction

Sean Simonian
Graduate Research Assistant
Department of Nuclear Engineering
Thermal Hydraulics Verification and Validation Laboratory
 About Me
Education:
• McCallum High School – Austin, TX (2011)
• Austin Community College (ACC) – Austin, TX (2016)
• BSc, Nuclear Engineering – Texas A&M University (2018)
• PhD, Nuclear Engineering – Texas A&M University (exp.: 2026)
Work history:
• Math, Physics, and Chemistry Tutor – ACC (2 years)
• High School Physics Teacher – Manor ISD (3 years)
• Graduate Research Assistant – Texas A&M (3 years)
• CFD Student Intern – Oak Ridge National Lab (Summer 2024)
First author publications:
(1) Pros and Cons of Various Coal-to-Nuclear Projects: Case Study Analysis of the
Limestone Coal Plant in Texas, Presented at ICONE 2023 in Japan.
(2) Steady-State and Transient System Curve Prediction, With Input Uncertainty, for the
Molten Salt Research Reactor in Abilene Texas, Presented at ASTFE 2025 in DC.
(3) Verifying VERTEX-CFD Simulations of Heated Lid-Driven Cavities, VVUQ 2025.
Syllabus Overview
 What is Nuclear Reactor Analysis?
• Goal: understand how and why reactors work
• This requires knowledge of:
- Neutron-nucleus interactions
- Properties of reactor materials
Ԧ 𝐸, 𝑡, Ω)
- Neutron distributions in (𝑟,
- Computing production and loss rates
- Critical heterogeneous reactor design
- Energy conversion processes
Simplified diagram of a graphite
• Putting it all together; power reactors!!! moderated; water cooled nuclear reactor.
• 1000 MWe nuclear plant ≡ 400K homes
• This is of course very cool, but great care is needed…
 Early History of Nuclear Discovery

Henri Becquerel J.J. Thompson Earnest Rutherford O. Hahn, F. Strassman, L. Oak Ridge
Radioactivity The Electron1 The Nucleus, Proton2 Meitner, O. Frisch National Lab
(fluorescence) (cathode ray tubes) 4He + 14N →17O + p+ Nuclear Fission X-10 Reactor
(Pu, power 48’)
Marie & Pierre Curie James Chadwick Enrico Fermi et. al.
Po and Ra The Neutron3 Self-Sustaining
α n Chain Reaction Los Alamos
(U pitchblende) (Po՜Be ՜ paraffin + p+)
(Chicago Pile 1) National Lab
Albert Einstein Trinity Test
E = mc2 1939? (25 kT Pu bomb)

Time [years]
1896 1897 1898 1905 1919 1932 1938 1942 1944 1945
Optional: Want to dig deeper? Check out Ch. 2 of Dr. Adam’s Notes
 Cold, Thin Target Experiment
Consider a mono-energetic, mono-directional beam of neutrons incident on a
cold, thin target, containing a single nuclide. Let us define:
n
𝐼 = Beam intensity = Rate per unit area at which neutrons reach target 2⋅s
3
cm
𝑁 = Atom density of the target atoms/cm
𝐴 = Target-beam interaction area cm2 𝐼
Δ𝑥 = Target thickness cm
Crossing Rate Aerial Density
Collision Rate ∝ 𝐼𝐴 ⋅ (𝑁Δ𝑥)
Total Microscopic Cross-Section
Collision Rate = 𝜎𝑡 ⋅ 𝐼 𝐴 ⋅ 𝑁 Δ𝑥 𝑁, 𝐴 Δ𝑥
particles cm2 particles atoms
s 𝑎𝑡𝑜𝑚 s cm2
 Microscopic Cross Sections
The constant of proportionality, 𝜎𝑡 , is called the microscopic total cross
section. Its units are:
Area per nucleus
Caveat: nuclear interactions are governed by quantum mechanics and 𝜎𝑡 can
be many orders of magnitude larger than the cross-sectional area of the
nucleus (e.g. 135Xe), but it is still helpful to think of 𝜎𝑡 as:
The effective cross-sectional area a nucleus presents to a neutron for a rxn
Microscopic cross-sections are commonly given in units of:
“Barns” (b), 1b = 10-24 cm2
Microscopic cross-sections depend on:
1. Relative speed between the neutron and nucleus
2. The type of nuclide
 Cross-Sections for Different Reactions
The total microscopic cross-section, 𝜎𝑡 , is related to the total interaction rate
between neutrons and nuclei. It is the sum of the microscopic cross-sections for
all the different reactions a neutron can have when it interacts with a nucleus:
𝜎𝑡 ≡ 𝜎𝑎 + 𝜎𝑠

𝜎𝑡 = 𝜎𝑓 + 𝜎𝑐 + 𝜎𝑛,𝛼 + 𝜎𝑛,2𝑛 + 𝜎𝑛,3𝑛 + ⋯ + 𝜎𝑒 + 𝜎𝑖𝑛 + 𝜎𝑝 + 𝜎𝑒,𝑟

So 𝜎𝑥 , where x = a, f, s, c, …, can be thought of as the effective cross-sectional

area the nucleus presents to a neutron for a type “x” reaction. Thus:

𝜎𝑥 Can be helpful to draw

= Probability of a type "x" reaction a pie chart with a
𝜎𝑡 portion being sigma_a
and another being
sigma_s
 Different Reaction Cross-Sections
When a neutron interacts with a nucleus many types of reactions can occur. Here is a
non-exhaustive list of the reactions more commonly seen:

σe = microscopic elastic scattering cross section

σe,r = microscopic resonance elastic scattering cross section 𝜎𝑠
σp = microscopic potential scattering cross section
σin = microscopic inelastic scattering cross section
σγ or σc = microscopic radiative capture cross section (gamma(s) emitted)
σf = microscopic fission cross section
σn,2n = microscopic cross section for 2 neutrons emitted
𝜎𝑎
σn,3n = microscopic cross section for 3 neutrons emitted
σn,α = microscopic cross section for alpha particle emitted
σn,p = microscopic cross section for proton emitted

Tip: If a reaction changes the target’s nuclear structure (A or Z) → it’s absorption

 235U Microscopic Cross-Section
𝜎=𝑓 𝐸 which significantly
impacts neutron distributions,
reactor analysis and design.
Some notable features for 235U:
• 𝜎𝑓 ≫ 𝜎𝑐 (or denoted as 𝜎𝛾 )
• 𝜎𝑓 ≈ 𝜎𝑡 for thermal neutrons
• 𝜎 follows 1/v at low energies

Neutrons in thermal equilibrium:

𝑣 ≈ 2200 m/s, 𝐸𝑡ℎ = 0.025 eV,
𝜎𝑓 ≈ 600 b, and 𝜎𝑐 ≈ 100 b.
Thus, at this energy, a 1n + 235U
reaction most likely results in: Fission
 Macroscopic Cross-Sections
Notice that in our thin target collision rate expression, the microscopic cross-
section was multiplied by the atom density of the target. This product is called:

Σ𝑥 = Macroscopic Cross-Section for a type x reaction

Σ𝑥 = 𝑁𝜎𝑥 𝑁 = 𝜌𝑁𝐴 /𝑀 (Only valid for a SINGLE nuclide)

1 atoms cm2
cm cm3 atom
What is the physical meaning of Σ𝑥 ?
The expected number of type-x reactions per neutron distance traveled.
The macroscopic cross-section depends on:
Composition, Density, Temperature, Neutron Energy
 Mixtures of Nuclides
Now, lets generalize our previous results for single nuclei to consider mixtures
of multiple nuclides. Define:

𝑦 𝑦
Collision Rate with Y-nuclei = 𝐼𝐴Δ𝑥 Σ𝑡 = 𝐼𝐴Δ𝑥𝑁 𝑦 𝜎𝑡

Collision Rate with Z-nuclei = 𝐼𝐴Δ𝑥 Σ𝑡𝑧 = 𝐼𝐴Δ𝑥𝑁 𝑧 𝜎𝑡𝑧 Thin, cold target, with two
nuclides, Y and Z.

Do the two nuclides interfere with each other’s reactions with neutrons? NO
∴ The total collision rate is the sum of the y and z rates
𝑦
Mixture Collision Rate = 𝐼𝐴Δ𝑥 Σ𝑡𝑚𝑖𝑥 = 𝐼𝐴Δ𝑥(Σ𝑡 + Σ𝑡𝑧 )
Generalizing to a mixture of 𝑛 nuclides: Σ𝑡𝑚𝑖𝑥 = σ𝑛𝑖=1 Σ𝑡𝑖 = σ𝑛𝑖=1 𝑁 𝑖 𝜎𝑡𝑖
 Neutron Attenuation
Returning to our mono-energetic, mono-directional neutron beam incident on a
target that is now arbitrarily thick. Define:
𝐼(𝑥) ≡ intensity of uncollided neutrons
at a distance 𝑥 into the target 𝐼0

Let’s apply general conservation to tell us more:

Change in Q = Gains of Q – Losses of Q
Q is a conserved quantity (amount, density, …) 𝑥=0 𝑥=𝐿
Conservation of the uncollided intensity between 𝑥 and 𝑥 + Δ𝑥 tells us:
Change in Intensity = Gains of Intensity – Losses of Intensity
collision rate
𝐼 𝑥 + Δ𝑥 − 𝐼 𝑥 = 0 − 𝐼 𝑥 Σt Δ𝑥 (no gains, loss = )
area
𝐼 𝑥 + Δ𝑥 − 𝐼 𝑥
lim = −𝐼 𝑥 Σt
Δ𝑥՜0 Δ𝑥
 Neutron Attenuation (cont.)
Applying the limit definition of the derivative yields:
𝑑𝐼
= −𝐼 𝑥 Σt , I 0 = I0 𝐼0
𝑑𝑥
This is a linear, 1st order, separable ODE
1
න 𝑑𝐼 = − න Σ𝑡 𝑑𝑥
𝐼 𝑥 𝑥=0 𝑥=𝐿

ln 𝐼 𝑥 = −Σ𝑡 𝑥 + 𝐶
𝐼 𝑥 = exp 𝐶 ⋅ exp −Σ𝑡 𝑥

Apply IC: 𝐼 0 = 𝐼0 = exp 𝐶 ⋅ exp 0 ⇒ I x = I0 exp(−Σ𝑡 𝑥)

exponentially
Conclusion: uncollided neutrons are attenuated as they pass through matter.
 Mean Free Path
It is often useful to know the average path length that a neutron travels
between collisions. We give it a name: the mean free path
To compute the neutron mean free path, we need the probability that the
neutron travels some distance 𝑥 before colliding within 𝑑𝑥 about 𝑥:
𝑝 𝑥 𝑑𝑥 ≡ probability that the neutrons first collision is in 𝑑𝑥 about 𝑥
= (first collision rate in dx about x) / (incident rate)
𝐼 𝑥 Σt dx 𝐼0 exp(−Σ𝑡 𝑥)Σ𝑡 𝑑𝑥
= =
𝐼0 𝐼0

𝑝 𝑥 𝑑𝑥 = exp(−Σ𝑡 𝑥)Σ𝑡 𝑑𝑥
Mean (average) of the distance 𝑥 comes 𝑥2
‫𝑥 𝑓 𝑥׬‬ 𝑤 𝑥 𝑑𝑥 W(x) is a weight
from the definition of the average of a 𝑓ҧ = 1
𝑥2
function, it must be
positive and have
function 𝑓(𝑥) over the interval 𝑥1 , 𝑥2 ‫𝑤 𝑥׬‬ 𝑥 𝑑𝑥 a zero somewhere.
1
 Mean Free Path (Derivation)
Here, the weight function w(x) is the probability density function p(x), thus:
∞
‫׬‬0 𝑥𝑝 𝑥 𝑑𝑥
Mean Free Path = ∞
‫׬‬0 𝑝 𝑥 𝑑𝑥
∞ ∞
= ‫׬‬0 𝑥𝑝 𝑥 𝑑𝑥 = Σ𝑡 ‫׬‬0 𝑥 ⋅ exp(−Σ𝑡 𝑥)𝑑𝑥

1
Mean Free Path = 𝑥ҧ =
Σ𝑡

• Inverse proportionality: a high cross-section implies a small mean free path

• Interactions are probabilistic: statistical fluctuations about the mean value
• We can define mean free paths for various individual reactions as well:
1 1
Absorption Mean Free Path = Scattering Mean Free Path =
Σ𝑎 Σ𝑠
 Temperature Dependence
Σ𝑥 = 𝑁𝜎𝑥 , 𝜎𝑥 contains the complicated part; dependence on the relative speed
between the neutron and the nucleus. But without it, nuclear reactors would be:
Less Safe
• Nuclei in a medium wiggle (simple harmonic oscillators)
• These vibrations can be treated as isotropic
• Then, the velocity distribution is well-approximated by a:
Maxwellian distribution that only depends on the material temperature
• In such cases, dependence on 𝑣𝑟𝑒𝑙 between the neutron and nucleus simplifies:
Assume Isotropic Assume Maxwellian
𝜎(𝑣Ԧ𝑛 , 𝑉𝑎 ) 𝜎(|𝑣Ԧ𝑛 |, 𝑉) 𝜎 𝑣𝑛 , 𝑇 ≡ 𝜎(𝐸, 𝑇)

• Assume: cross-sections averaged over the Maxwellian, drop the T argument

 Doppler Broadening

Microscopic total cross-section for 233U (left) and a zoomed in view of a particular resonance centered at 𝐸0
illustrating how the resonance broadens with increasing temperature (right).
• An increase in reactor temperature results in: Doppler Broadening
• Temperature ↑, parasitic neutron capture ↑, providing: Negative Feedback
• Doppler broadening occurs away from resonances too, but not as prominently
 Neutrons Emerging from Fission
Consider the fission reaction:
1 235
0𝑛 + 92𝑈 ՜ Fission Reaction Byproducts
(fission fragments, neutrons,
betas, neutrinos)
Neutrons born from fission are:
(1) Independent of the incident neutron E
(2) Emitted isotopically
(3) Emitted with a distribution of energies

We give this energy distribution a name:

Fission spectrum for 235U and its cumulative distribution
Fission Spectrum = 𝜒(𝐸) units: [1/eV] function. Example meaning of the CDF: ~60% of neutrons are
born with E < 2 MeV, ~90% are born with E < 6 MeV.

𝜒(𝐸) describes the probability per unit energy that a neutron is born w/ energy E.
 Generalizing to Non-Beam Cases
Now, we generalize our results non-beam cases. Recall our expression for the
collision rate given a beam of intensity 𝐼, energy 𝐸, incident normal to a thin
target whose nuclei are vibrating in a Maxwellian distribution at temperature 𝑇:
𝐶𝑅 = 𝐼 ⋅ Σ𝑡 𝐸 ⋅ 𝐴Δx = 𝐼 ⋅ Σ𝑡 𝐸 ⋅ ∀
𝐶𝑅
= 𝐶𝑅𝐷 = 𝐼 ⋅ Σ𝑡 𝐸
∀
# rxns n ⋅ cm # rxns
3
=
cm ⋅ s cm3 ⋅ s n ⋅ cm

Does this interpretation of the beam intensity (total path length made by
neutrons – per unit volume – per unit time) make sense? Let’s investigate…
 Scalar Flux (Derivation)
Consider mono-energetic neutrons moving isotopically in a small volume.
Define:
Path length of a0 Number of0
3
𝑛 = neutron density [n/cm ] Total Path Length = single neutron Neutrons
𝑣 = neutron speed [cm/s]
Total Path Length = 𝑣𝑑𝑡 ⋅ (𝑛𝑑∀)
𝑑∀ = control volume [cm ]3

𝑑𝑡 = time interval [s] Path-Length Rate per Unit Volume = 𝑛𝑣

Path-Length Rate per Unit Volume = 𝜙 = 𝑛𝑣

(or path-length rate density)

Terminology: “Path-Length Rate” =

Total distance traveled by neutrons, per unit time.
 Neutron Scalar Flux
n−cm
The Neutron Scalar Flux 𝜙 𝑟,
Ԧ𝑡 represents the total distance traveled
cm3 ∙s
by neutrons of all energies and moving in all directions, per unit volume, per
unit time, at some position 𝑟,
Ԧ and some time 𝑡.

Relation to Neutron Beam Intensity 𝐼: in a mono-energetic, mono-directional

beam, all neutrons are moving in the same direction, and we have 𝜙 𝑟,
Ԧ 𝑡 = 𝐼.

However, neutrons in a nuclear reactor DO NOT have the same speed or

direction, they have a continuous distribution both in energy and direction.

Let us discuss the “hierarchy” of neutron fluxes to better understand this…

 Hierarchy of Neutron Fluxes in NUEN
𝜓 𝑟,
Ԧ 𝑡, 𝐸, Ω = Angular Flux = neutron path-
length rate density in volume, energy, and
direction. Units: n − cm
cm3 ∙ s ∙ eV ∙ sr
Integrating the angular flux over all solid angles
(over 4𝜋) eliminates the angular dependence:
න 𝜓 𝑟,
Ԧ 𝑡, 𝐸, Ω 𝑑Ω = 𝜙 𝑟,
Ԧ 𝑡, 𝐸
4𝜋

𝜙 𝑟,
Ԧ 𝑡, 𝐸 = Energy-Dependent Neutron Flux =
neutron path length rate density in volume and
Illustrating the 7-D phase space (7 independent
energy. Units: n − cm variables) needed to describe a neutron
cm3 ∙ s ∙ eV distribution in space, time, energy, and direction
𝑟Ԧ = (𝑥, 𝑦, 𝑧) (3) 𝐸 (1)
Ω = 𝜑, 𝜃 (2) 𝑡 (1).
 Hierarchy of Fluxes (continued)
Finally, integrate the energy-dependent neutron flux over all neutron energies:
∞
න 𝜙 𝑟,
Ԧ 𝑡, 𝐸 = 𝜙 𝑟,
Ԧ𝑡
0
to obtain the familiar neutron scalar flux. 𝜙 𝑟, Ԧ 𝑡 is sometimes referred to as
the “total” scalar flux (because it includes the tracks made by neutrons of all
energies, moving in all directions). In summary:
Quantity Flux Term = (Density Term)*(Speed) Governing Equation Units
n − cm
Angular Flux 𝜓 𝑟,
Ԧ 𝑡, 𝐸, Ω = 𝑛 𝑟,
Ԧ 𝑡, 𝐸, Ω ∙ 𝑣(𝐸) Neutron Transport Eq.
cm3 ∙ s ∙ eV ∙ 𝑠𝑟
Energy-Dependent n − cm
𝜙 𝑟,
Ԧ 𝑡, 𝐸 = 𝑛 𝑟,
Ԧ 𝑡, 𝐸 ∙ 𝑣(𝐸) Multi-Group Diffusion Eq.
Scalar Flux cm3 ∙ s ∙ eV
n − cm
Total Scalar Flux 𝜙 𝑟,
Ԧ 𝑡 = 𝑛 𝑟,
Ԧ 𝑡 ∙𝑣 1-Group Diffusion Eq.
cm3 ∙ 𝑠
Tip: we use the same symbol for total and energy dependent fluxes. Sorry. You
will have to know from the context and/or units which one you are working with.
 Important Points About Fluxes
n−cm
𝜙 𝑟,
Ԧ 𝑡, 𝐸 is a path-length rate density in space and in energy, units:
cm3 ∙s∙eV

Question:
How many neutrons in a nuclear reactor have exactly 1 MeV of kinetic energy?
ZERO. Continuous probability distributions can’t be evaluated at a point. We can
ask: how many neutrons are in some energy range (e.g.: 0.99 MeV to 1.01 MeV).

Question:
Can we say what the path-length rate is at some location 𝑟?Ԧ
NO. You can’t cover distance over a single point. It doesn’t make sense to ask
about path-length rate at a particular point. It does make sense to ask what the
path-length rate density is at a point.
 Reaction Rate Densities
We need fluxes to compute reaction rates.

Assume: neutrons are distributed isotropically, then the energy-dependent

neutron flux 𝜙 𝑟,
Ԧ 𝑡, 𝐸 contains the most detailed information we need.

Assume: nuclei vibrate in a Maxwellian distribution and cross-sections have

been properly averaged over the distribution of nucleus velocities. Then, the
reaction rate density 𝑅𝑅𝐷 is: Energy-dependent reaction rate density
∞
𝑅𝑅𝐷 𝑟,
Ԧ 𝑡 = න 𝜙 𝑟,
Ԧ 𝑡, 𝐸 Σ 𝐸, 𝑇 𝑑𝐸
0
# rxns n − cm # rxns
= eV
cm3 ∙ 𝑠 cm3 ∙ s ∙ eV n − cm
Physical meaning of 𝑅𝑅𝐷 𝑟, Ԧ 𝑡 : expected number of reactions (due to neutrons
of all energies), per unit volume, per unit time, at some location 𝑟,
Ԧ at time 𝑡
 Different Reactions and Reaction Rates
It should be no surprise there is a similar expression for each type of reaction:
∞
Fission Rate Density = න 𝜙 𝑟,
Ԧ 𝑡, 𝐸 Σf 𝐸, 𝑇 𝑑𝐸
0
∞
Scattering Rate Density = න 𝜙 𝑟,
Ԧ 𝑡, 𝐸 Σs 𝐸, 𝑇 𝑑𝐸
0
∞
Absorption Rate Density = න 𝜙 𝑟,
Ԧ 𝑡, 𝐸 Σa 𝐸, 𝑇 𝑑𝐸
0
To go from a reaction rate density to reaction rate 𝑅𝑅 we take the reaction
rate density and integrate over the volume of the system (∀𝑠𝑦𝑠 ):
∞
𝑅𝑅 =න න 𝜙 𝑟,
Ԧ 𝑡, 𝐸 Σ 𝐸, 𝑇 𝑑𝐸𝑑∀
∀𝑠𝑦𝑠 0
# rxns n − cm # rxns
= eV [cm3]
𝑠 cm3 ∙ 𝑠 ∙ eV n − cm
 Example 1 (Adams, pg. 80-81)
Suppose you have been analyzing a nuclear reactor and you have determined
the steady-state energy-dependent scalar flux, 𝜙 𝑟, Ԧ 𝐸 , everywhere in the
reactor and for all neutron energies. You also know all the macroscopic cross
sections everywhere in the reactor.
Question #1: How would you find the total number of neutrons in the reactor?
∞
𝑛𝑡𝑜𝑡 = ම න 𝑛 𝑟,
Ԧ 𝐸 𝑑𝐸𝑑∀
∀ 0
𝜙 𝑟,
Ԧ𝐸 𝜙 𝑟,
Ԧ𝐸
𝑛 𝑟,
Ԧ𝐸 = =
𝑣(𝐸) 2𝐸/𝑚
∞
𝜙 𝑟,
Ԧ𝐸
𝑛𝑡𝑜𝑡 = ම න 𝑑𝐸𝑑∀
∀ 0 2𝐸/𝑚
 Example 2 (Adams, pg. 81)
Question #2: Assuming each fission produces 200 MeV of energy and 190
MeV of this gets deposited in the reactor (with the other 10 MeV carried away
by leaking neutrinos, gammas, and neutrons), what is the reactor power?
Energy Energy Deposited
Power = = ∙ Fission Rate
Time Fission
𝐸𝑓 = 190 MeV / fission
∞
𝐹𝑅 = ම 𝐹𝑅𝐷 𝑑∀ , 𝐹𝑅𝐷 = න Σ𝑓 𝑟,
Ԧ 𝐸 𝜙 𝑟,
Ԧ 𝐸 𝑑𝐸
∀ 0
∞
190 MeV
∴ 𝑃= ම න Σ𝑓 𝑟,
Ԧ 𝐸 𝜙 𝑟,
Ԧ 𝐸 𝑑𝐸 𝑑∀
Fission ∀ 0
MeV MeV # fissions n − cm 3
= eV [cm ]
s fission n − cm cm3 ∙ s ∙ eV
 Example 3 (Adams, pg. 81-82)
Question #3: At what rate are neutrons produced by fission in the reactor?
n
Production Rate = ම production rate density 𝑑∀
s ∀

Production Rate Density = # neutrons per fission fission rate density

∞
= න 𝜈 𝐸 Σf 𝑟,
Ԧ 𝐸 𝜙 𝑟,
Ԧ 𝐸 𝑑𝐸
0
n n # fissions n − cm
= eV
cm3 ∙ s 3
fission n − cm cm ∙ s ∙ eV
∞
n
∴ Production Rate = ම න 𝜈 𝐸 Σf 𝑟,
Ԧ 𝐸 𝜙 𝑟,
Ԧ 𝐸 𝑑𝐸 𝑑∀
s ∀ 0
 Lecture Summary
Reactor behavior depends on neutron production and loss rates, many of which are
governed by neutron-nucleus reactions. Here are the key facts we need to know:
1. Reaction rates are volume integrals of reaction rate densities
2. Reaction rate densities are usually integrals over energy of 𝜙 𝑟,Ԧ 𝑡, 𝐸 Σ 𝐸, 𝑇
3. Fluxes are the product of a neutron density and neutron speed
4. 𝜙 𝑟,
Ԧ 𝐸, 𝑡 is a path-length rate per unit volume and per unit energy
5. Σ𝑥 (𝐸) is the expected number of type-x reactions per neutron distance traveled
# 𝑛𝑢𝑐𝑙𝑒𝑖 𝑖 𝑖
6. Σ𝑥 𝐸 = σ𝑖=1 𝑁 𝜎𝑥 (𝐸)
7. There is a different 𝜎 for each type of reaction
8. 𝜎 depends on 𝑣𝑟𝑒𝑙 between the neutron and nucleus, but if we assume nuclei
vibrate isotopically in a Maxwellian distribution, and properly average over these
molecular motions, then 𝜎 = 𝑓(𝐸, 𝑇) (neutron energy and material temperature)
9. Mean-free path for a type-x reaction is 1/Σ𝑥 𝐸 (this is history independent)
Thank you for your time! Any Questions?