3: Flux and Current

1. 3: Flux and Current B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 2012 Sept.-Dec.

2. Contents • Concepts: • Angular Neutron Flux, Angle-Integrated Flux • Angular Neutron Current, Angle-Integrated Current • Reaction rates

3. Neutron Density • Imagine neutrons in a unit volume at a given instant t. • The neutron density n is a function of time t, position , energy E, and direction of motion  , i.e. - units are, e.g., neutrons/cm3 ( n.cm-3). • Note: We will deal at first with time-independent, or quasi-time-independent, situations, so drop the t variable for now: Then neutron density  • The direction of motion in 3 dimensions, , is also called the “solid angle”. Actually, to define any solid angle , we need only 2 quantities, e.g., 2 angles in a “polar” co-ordinate system: a polar angle and an azimuthal angle (see next slide).

4. Polar Co-Ordinate System for Solid Angle

5. Neutron Flux(es) • The most general neutron flux is the angular flux, i.e., the product of neutron density n and speed υ. • The angular flux can be thought of as a “beam” in 1 direction and energy, i.e., angular flux is analogous to beam intensity! • By summing the angular flux over angles or energies, we get: • Angle-integrated flux for 1 energy E (this can be thought of as summing neutron beams over all directions) • Total angular fluxover all energies: [integrate Eq. (1) over E] • Total angular flux or total flux over a range of energies [integrate Eq. (1) or (2) over a partial range of E] • Total flux over all energies [integrate Eq. (1) over all angles and all energies]

6. Integrating Over Solid Angle

7. Units of Neutron Flux • Fluxes fhave units of: (neutrons.cm-3*cm.s-1) = (n.cm-2.s-1) • Discussion: Prove to yourself that a flux (e.g., per energy or per direction) can be visualised as the total neutron path length traversed per unit time. • Note: Neutron flux can of course depend on time, but for now we drop the label t in most cases.

8. Neutron Current(s) • The angular neutron current is the vector quantity made from the angular flux by multiplying it by the unit vector in the direction of motion ( ): • Just as we did for flux, we can define the angle-integrated current for 1 energy: • This is a vector summation, so there is a priori no simple relationship between the angle-integrated current and the angle-integrated flux! • We could also integrated current over a range of energies.

9. Components of Current • Because the current is a vector, we can also focus on components of the vector in any direction, for instance the components Jx, Jy, Jz, of the total current J along the x, y, and z axes. • We can also focus on partial currents, i.e., partial components of the current, e.g., Jz+and Jz- would be the components of the total current J along the +z and –z directions respectively. • In that case, for example, Jz= Jz++ Jz-

10. Neutron Flux: Graphical View • An equivalent way to define the neutron flux is to visualize an arrow associated with each neutron in a unit volume. The arrow shows the direction of motion of the neutron, and its length denotes the neutron’s speed (see figure in next slide). • The sum of all the arrow lengths of given magnitude and direction is the angular flux. • The sum of all the arrow lengths of given magnitude, regardless of direction, is the angle-integrated flux. • Summing all arrows gives the total flux, integrated over angle and energy.

11. Neutron Flux Unit Volume Total flux  = sum of all arrow lengths in unit volume = total path length traversed by all neutrons in unit volume per unit time

12. Difference Between Flux & Current • Flux is a scalar quantity; current a vector quantity. • Since the “unit volume” over which flux is defined can be as small as we like, we can think of flux as the total number of particles emerging from, or, more usefully, converging to a point (or single nuclide), per unit area per unit time. We just add up arrow lengths. • On the other hand, when we add the arrows vectorially to get the current, arrows in opposite directions cancel out! • This shows how total current is not generally related to total flux.

13. Calculating Reaction Rates • Since angular flux is similar to a beam intensity, then we can calculate the reaction rate for neutrons at point r, energy E and direction of motion  as • Now, when neutrons react with nuclei, the nuclei do not “care” from which direction the neutrons came. •  In calculating reaction rates for neutrons of a given energy, we can simply work with angle-integrated fluxes: • The total reaction rate over a range of neutron energies can be obtained by integrating over energy E.

14. Neutrons Crossing a Boundary • But to calculate the number of neutrons moving from one place to another (or crossing the reactor’s surface), we do need the angular flux. • In general, to calculate the number of neutrons crossing a unit area of a given plane in direction (i.e., the current crossing the unit area at angle ), we need to take into account the angle between  and the plane, i.e., the dot product of  with the normal to the plane, just as we do when we consider the number of rays of sunlight warming a given area on the earth’s surface (see next slide).

15. Neutrons Crossing Unit Area of a Plane If we want to sum all neutrons moving across the plane from one side to the other, we need to consider the range of . For neutrons crossing the plane from below,  is between 0 and /2 ( = cos  = 0 to 1). For neutrons crossing from above to below,  is between /2 and  ( = -1 to 0).

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