What Is the Difference Between Angular Neutron Flux and Neutron Current Vector?

[badvot]

Hi,
I am sorry if my question seems a bit basic but I find it confusing to understand the differences between the angular neutron flux and the neutron current vector.
I read the definitions from multiple textbooks (Lamarsh, Stacey, Duderstadt) but my idea is that: despite the fact that the angular flux is a scalar quantity, doesn't it have the direction information built in its definition, i mean that if we are to compute phi(r,omega,E,t), this will give the number of neutrons moving with velocity in this particular omega direction which is the equivalent of what we will get if we took the dot product of the current vector by the unit vector that describe the direction omega.
i hope i have illustrated my POV clearly.
Thanks for advance.

 

[Astronuc]

despite the fact that the angular flux is a scalar quantity, doesn't it have the direction information built in its definition

Yes it does, but it is a scalar quantity dependent on the solid angle.

In neutron transport theory, one has neutron angular density, given by N(r,Ω,E,t), and it is defined as
probable (or expected) number of neutrons at the position r with direction Ω and energy E at time f, per unit volume per unit solid angle per unit energy. It is just the number of neutrons, without a direction, but it is dependent on the direction of interest.

The product of the neutron speed v and the neutron angular density is called the neutron angular flux, which is given by $\Phi$(r,Ω,E,t) = v * N(r,Ω,E,t), where v is the speed, not the velocity of the neutrons. It is also a scalar.

The net number of neutrons crossing a surface element per unit energy in unit time is called the neutron current, and it is given by

$\vec{J} (r, E, t) = v \int{\Omega N(r, \Omega, E, t)\ d\Omega}$

or expressed in terms of the neutron angular flux

$\phi (r, E, t) = \int_{4\pi} \Phi(r, \Omega, E, t)\ d\Omega$

$\vec{J} (r, E, t) = \int_{4\pi} {\Omega \Phi(r, \Omega, E, t)\ d\Omega}$

and the neutron flux and current are the zeroth and first moment of the neutron angular flux.

https://en.wikipedia.org/wiki/Moment_(mathematics)

I used the notes from the following, which expresses this information nicely. See page 412-414, or pages 4-6 in the pdf (Section 1.2, Description of neutrons)
https://www.osti.gov/etdeweb/servlets/purl/20854879

 

[rpp]

Hi,
I am sorry if my question seems a bit basic but I find it confusing to understand the differences between the angular neutron flux and the neutron current vector.
I read the definitions from multiple textbooks (Lamarsh, Stacey, Duderstadt) but my idea is that: despite the fact that the angular flux is a scalar quantity, doesn't it have the direction information built in its definition,

In nuclear engineering terminology, the "scalar flux" is usually the angular flux integrated over all angles. Technically the angular flux is also a scalar value, but it is the flux with direction "omega". I think this agrees with what you are saying, but the term "scalar" may cause some confusion in NE terminology.

i mean that if we are to compute phi(r,omega,E,t), this will give the number of neutrons moving with velocity in this particular omega direction

Slight correction, but it is the number of neutrons times the velocity with energy E.

To be more precise, $\psi(r,\Omega,E,t) \, d\Omega \, dE$ is the number of neutrons times the velocity about $dE$ and $d\Omega$.

which is the equivalent of what we will get if we took the dot product of the current vector by the unit vector that describe the direction omega.
i hope i have illustrated my POV clearly.
Thanks for advance.

One point to make is that the current vector is the "net" flow of neutrons, so it is the flux in one direction minus the flux going in the opposite direction.

If the flux is isotropic, then diffusion would be valid and it would be true that the net flow of neutrons in a certain direction is the dot product of the current vector.

However, the angular flux is often not isotropic, so you cannot make this approximation. The angular flux is usually very dependent on the angle, the flux can even be discontinuous in the angle space. Therefore, it is not as simple as taking the dot product of the current vector.