## Neutron Flux Density

In the section the neutron cross-section, it was 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 influences 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 **reaction rate** or **thermal power** of the reactor core, it is necessary to know **how many neutrons** are traveling through the material.

It is convenient to consider the** neutron density**, that is the number of neutrons existing in one cubic centimeter. The neutron density is represented by the symbol **n** 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**. The total distance these neutrons can travel each second is determined by their velocity and therefore 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**, which is 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})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}**. In comparison with 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 in 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**. In order to generate the same amount energy a MOX core do not require such the neutron flux as a uranium fueled core.**larger fissile loading.**The main reason is in the larger fissile loading. In MOX fuels, there is relatively high buildup of^{240}Pu and^{242}Pu. Due to the relatively lower fission-to-capture ratio, there is higher accumulation of these isotopes, which are parasitic absorbers and that results 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 is MOX cores has 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** of time (days or weeks), the atomic number density of the fuel atoms remains relatively constant. Therefore in this short period, also the **average neutron flux remains constant**, when reactor is operated at a constant power level. On the other hand, the atomic number densities of fissile isotopes over a period of months decrease due to the fuel burnup and therefore also the macroscopic cross-sections decrease. This results is slow** increase in the neutron flux** in order to keep the desired power level.