3: Flux and Current

1. 3: Flux and Current B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 2014 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 basic neutron-flux quantity 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, we get: • Angle-integrated flux for 1 energy E (this can be thought of as summing neutron beams over all directions) • By doing other integrations we can also define: • Total angular flux[Eq. (1)] integrated over a range of energies • Total flux integrated 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! Can you think of an example? • 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 “colliding” with 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 the angle-integrated flux: • 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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