FRP2

Question 1

Which are the variables that describe neutron distribution in a reactor?

Answer 1

position, time, direction of flight and energy(spectrum)
3 + 2 + 1 + 1 = 7 variables

Question 2

Why we want to know the neutron distribution in the reactor?

Answer 2

• assess the stability of the fission chain reaction
• calculate multiplication factor
• estimate neutron flux
• compute fuel burn-up (long term behaviour)
• simulate accident condition (short term behaviour)

Question 3

Why the diffusion equation is not satisfactory to describe neutron distribution?

Answer 3

It assumes high collsion frequency between neutrons (not true) and it is not locally valid near neutron sources, sinks and boundaries.

Question 4

What is the Boltzmann equation?

Answer 4

Also known as Transport equations it was developed for rarified gas but it’s appliable for neutrons. It is intrinsically non-linear but in the case of neutronsthe mutual interaction term can be neglected.
There are two main formulation: integral and integrodifferential. It can be solved both numerically and with MC methods

Question 5

What are the main assumptions of the problem?

Answer 5

• neutron mass at rest is 939 MeV
• neutron’s energy is 10 MeV, we’ll say 15 MeV to be conservative
• No relativistic effects since 15 &laquo_space;939
• neutrons are point particles described by the cross section
• even if neutrons are fermions the density is too low for Pauli’s prinicple to have a relevant effect
• Magnetic moment is neglected
• neutrons are assumed stable since half life in void is minutes order while in therma reactors it’s 10^{-3;-5} and in fast reactor is 10^{-6}
• No wave behaviour so they are described by position and velocity
• Heisenberg principle is not a problem
• Since burn-up is a long scale effect we consider timesteps in wich we consider burn-up constant
• No temperature effects on cross sections

Question 6

Why can we consider no wave behaviour for neutrons in the core?

Answer 6

pag.1

Question 7

Does heisenberg principle count for neutrons in the reactor core?

Answer 7

pag.2

Question 8

Why we do not consider scattering between neutrons?

Answer 8

pag.3

Question 9

Why we assume burn-up to be constant when solving Boltzamnn transport equation?

Answer 9

Because it would introduce non-linearity since
burn-up -> Number of fissionable nuclei -> number of fissions -> increase burn-up
So we consider steps for increasing burn-up

Question 10

Is the neutron density a good indicator for the actual neutron distribution?

Answer 10

It is a statistical quantity so it is good only if the variance is low. It’s a Markovian process so the neutron distribution is a poissonian. So the std = sqrt(n) = sqrt(10^{10}) = 10^{10}.
This is not true at start up when the neutron density is lower

Question 11

What kind of variable is the flight direction?

Answer 11

A versor \omega with modulus = 1.
It is defined in polar coordinates.
Theta is the polar angle
Phi is the azimuthal angle
pag.4

Question 12

How is the neutron velocity defined?

Answer 12

v- = v-(E,\omega) = v \omega-

Question 13

What is the neutron angluar density?

Answer 13

N(r-,\omega-,E,t)
describes the neutron population in position r- with flight direction \omega-, with energy E at time t
It’s the unknown of Boltzmann equation

Question 14

What is the neutron density?

Answer 14

n(r-,E,t)
describes the neutron population in position r- with energy E at time t.
It is obtained by integrating the neutron angular density in the angle.

Question 15

What is the neutron flux?

Answer 15

phi(r-,E,t) = n(r-,E,t) v(E)
is the product between the neutron density and the neutron velocity v(E).

Question 16

What is the Scalar angular flux?

Answer 16

Phi(r-,\omega-,E,t) = v N(r-,\omega-,E,t)
It is also called scalar flux or simply flux

Question 17

What is the Density of neutron flux?

Answer 17

Phi-(r-,E,t) = v- N(r-,\omega-,E,t)
It is also called the vectorial flux.

Question 18

During a differential time dt, how many neutrons with direction \omega- in d\omega- and with energy E in dE will cross the differential area dA?

Answer 18

pag.5

Question 19

What it the total neutron flux?

Answer 19

phi(r-,E,t) = n v
Is the integral in 4\pi di (N v = Phi)

Question 20

What is the neutron current?

Answer 20

J-(r-E,t)
is the integral of (\omega Phi(r-\omega-,E,t) ) in d\omega [4\pi]
Also known as net density current, represents the net flux of neutrons corssing a unit area in position r- at time t with energy E

Question 21

What is the average velocity vector?

Answer 21

Is the average respect to all directions of flight
<v(r-,E,t)> =INT[4pi] v-(r-,\omega-,E,t) N(r-,\omega-,E,t)/n(r-,E,t) d\omega
where N(\omega-)/n = probability of neutrons to have \omega- flight direction

This allows us to have a consistent current defintion with the elctrical analogy:

<v-> = J-(r-,E,t) / n(r-,E,t)
</v->

Question 22

Talk about the crossection.

Answer 22

It express the probability of interaction between two particles.
In ur case the interaction between neutron and nuclei.
We write with small sigma(r-,E) and is cm^-1. It’s a pdf per unit length, if multiplied by the velocity it is a pdf per unit time.
specific reaction (n,x) has cross section = sigma_x
We neglect dependency of cross section on Omega introducing the Isotropic Media hypotesis

Question 23

What is the number of reactions (n,x) during time dt in dV with neutrons of energy dE and flight directions d\omega?

Answer 23

N(r,om,E,t) sigma_x v dt dV dE dOmega
If we want to consider multiple reaction we just perform a sum on sigma_i

Question 24

What kind of reactions can neutron go through?

Answer 24

• Elastic scattering (n,n)
• Anelastic scattering (n,n)
• Fission (n,f)
• production of 2 neutrons (n, 2n)
• radiative capture (n,gamma)
  We also have independent sources like(alfa,n), spontaneous fission or cosmic rays

Question 25

What are the probability density function of appearance of a neutron?

Answer 25

Q(r-,omega-,E,t)
Is the pdf of appearance of 1 neutron in position r-, with flight direction \omega-, energy E at time t.
the rate of appeareance is
Q dV dOmega dE