**The neutron flux density**,

**Ф**, is the number of neutrons crossing through some arbitrary cross-sectional unit area in

**all directions**per unit time. It is a

**scalar quantity,**and it can be calculated as the

**neutron density (n)**multiplied by

**neutron velocity (v)**.

The section on the neutron cross-section determined the probability of a neutron undergoing a specific neutron-nuclear reaction. It was determined the mean free path of neutrons in the material under specific conditions. These parameters influence the **criticality of the reactor core**. In other words, we do not know anything about the** power level** of the reactor core. If we want to know the reactor core’s **reaction rate** or **thermal power**, it is necessary to know **how many neutrons** are traveling through the material.

It is convenient to consider the** neutron density**, the number of neutrons existing in one cubic centimeter. The symbol **n** represents the neutron density with units of neutrons/cm^{3}. In reactor physics, the** neutron flux** is more likely used because it expresses better the** total path length** covered by **all neutrons**. Their velocity determines the total distance these neutrons can travel each second. The **neutron flux density** value is calculated as the **neutron density (n)** multiplied by **neutron velocity (v)**.

**Ф = n.v**

where:**Ф – neutron flux (neutrons.cm ^{-2}.s^{-1})**

**n – neutron density (neutrons.cm**

^{-3})**v – neutron velocity (cm.s**

^{-1})**The neutron flux**, the number of neutrons crossing through some arbitrary cross-sectional unit area in **all directions** per unit time, is a **scalar quantity**. Therefore it is also known as the **scalar flux**. The expression **Ф(E).dE** is the total distance traveled during one second by all neutrons with energies between E and dE located in 1 cm^{3}.

The connection to the **reaction rate**, respectively, the reactor power is obvious. Knowledge of the **neutron flux** (the **total path length** of all the neutrons in a cubic centimeter in a second) and the** macroscopic cross-sections** (the probability of having an interaction **per centimeter path length**) allows us to compute the rate of interactions (e.g.,, rate of fission reactions). **The reaction rate** (the number of interactions taking place in that cubic centimeter in one second) is then given by multiplying them together:

where:**Ф – neutron flux (neutrons.cm ^{-2}.s^{-1})**

**σ – microscopic cross-section (cm**

^{2})**N – atomic number density (atoms.cm**

^{-3})## Neutron Flux and Intensity – Examples

We have to distinguish between the** neutron flux** and the **neutron intensity**. Although both physical quantities have the** same units**, namely, neutrons.cm^{-2}.s^{-1}, their physical interpretations are different. In contrast to the neutron flux, the neutron intensity is the number of neutrons crossing through some arbitrary cross-sectional unit area** in a single direction** per unit time (a surface is perpendicular to the direction of the beam). The **neutron intensity** is a **vector quantity**.

## Neutron Flux – Uranium vs. MOX

Note that there is a difference between neutron fluxes in the uranium fueled core and the MOX fueled core. The average neutron flux in the first example, in which the neutron flux in a **uranium-loaded reactor core** was calculated, was **3.11 x 10 ^{13 } neutrons.cm^{-2}.s^{-1}**. Compared to this value, the average neutron flux in

**100% MOX fueled core**is about

**2.6 times lower**(

**1.2 x 10**), while the reaction rate remains almost the same. This fact is of importance in the reactor core design and the design of reactivity control. It is primarily caused by:

^{13 }neutrons.cm^{-2}.s^{-1}**Higher fission cross-section of**. The fission cross-section is about 750 barns in comparison with 585 barns for^{239}Pu^{235}U.**Higher energy release per one fission event**. A MOX core does not require the neutron flux of the uranium fueled core to generate the same energy.**Larger fissile loading.**The main reason is the larger fissile loading. In MOX fuels, there is a relatively high buildup of^{240}Pu and^{242}Pu. Due to the relatively lower fission-to-capture ratio, there is a higher accumulation of these isotopes, parasitic absorbers, resulting in a reactivity penalty. In general, the average**regeneration factor η**is lower forfuel than for^{239}Pu^{235}U

The relatively lower average neutron flux in MOX cores has the following consequences on reactor core design:

- Because of the lower neutron flux and the larger thermal absorption cross-section for
, reactivity worth of control rods, chemical shim (PWRs), and burnable absorbers is less with MOX fuel.^{239}Pu - The high fission cross-section of
and the lower neutron flux lead to^{239}Pu**greater power peaking**in fuel rods that are located near water gaps or when MOX fuel is loaded with uranium fuel together.

## Neutron Flux and Fuel Burnup

In a power reactor** over a relatively short period** (days or weeks), the atomic number density of the fuel atoms remains relatively constant. Therefore, the **average neutron flux remains constant** in this short period when a reactor is operated at a constant power level. On the other hand, the atomic number densities of fissile isotopes over months decrease due to the fuel burnup, and the macroscopic cross-sections decrease. This result is slow** to increase in the neutron flux** to keep the desired power level.