Relationship between neutron density

11 Relationship between neutron density, neutron flux and thermal power 

The neutron flux is the product of the neutron density and the neutron speed. Assuming that all the neutrons are travelling at the same speed, equal to the average speed, c , the reactor flux averaged over the complete core, Xavf neutrons/mVs will be the product  

The assumption of a constant speed, c , implies that the reactor flux will be proportional to the neutron density, n. The fission rate of the reactor per m of fuel, F, is given by the product of the neutron flux and the total cross-sectional area for fission  

Consider, for example, a PWR containing 80 te of uranium oxide enriched to 3%, with an average moderator temperature of 310 C. The effective specific volume of uranium oxide pellets will be 1.11 x 10 m /kg. The fission cross-section at a moderator temperature of 20°C/293 K, t/(293), is 582 x 10 m for U-235, while the corresponding figure for plutonium-239 is 743 x 10 m. Calculate NfOf at the start of life and at the end of life, and hence deduce the variation in the constant of proportionality, a. 

88 X 2400 = 2112kg. 1 kg-mol of U-235 has a mass of 235 kg and contains Avogadro s number of nuclei, namely 6.022 x 10 . Therefore the number of U-235 nuclei in the whole core will be  

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Hence at a moderator temperature of 310°C/S83K, = 128 X V583 = 3091 m/s. Substituting into equation (21.33) gives the relationship between neutron density, n, and average power density, at the start of core life as... 

Since the speed of the thermalized neutrons depends only on moderator temperature, the same average speed of 3091 m/s holds. Thus the relationship between neutron density and power density at the end of life is... 

The relationship between neutron density and power density at the start of life has been derived in equation... 

Once it had been shown that crystals diffract X rays, the relationship between the observed effect and the experimental conditions was put on a sound mathematical basis by Max von Laue, Paul P. Ewald and many others.X-ray diffraction by crystals represents the interference between X rays scattered by the electrons in the various atoms at various locations within the unit cell. It must, however, be stressed again that any molecule or ion can diffract X rays or neutrons. It is only when this diffraction is reinforced by the repetition of the unit cell in the crystal that diffraction by atoms is a conveniently observable effect, for example as spots of differing intensity on photographic film. Of particular interest to chemists and biochemists is the work by W. L. Bragg,who demonstrated that measurement of the diffraction patterns gives information on the distribution of electron density in the unit cell, (i.e., the arrangement of atoms within this unit cell). 

Thus the constant of proportionality has decreased by about 30% over the life of the core. More accurate relationships between thermal power density and neutron density at different stages of the reactor run may be available from the design calculations or from plant-specific reactor physics data. 

It is sometimes more convenient to describe the neutron population in terms of the neutron kinetic energy instead of the speed. In this event, the neutron density and the flux are defined per unit of energy. In order to show the relationship between the speed and the energy definitions, let us consider the function... 

One of the important problems to be considered in this chapter is the determination of the relationship between the flux and the slowing-down density for various physical situations. As an introductory treatment of this problem we carry out the following calculation, namely, compute the number of scattering collisions which occur per unit volume per unit time, using first the concept of the neutron flux and second the concept of the slowing-down density. We write the scattering-collision density in terms of the flux by means of Eq. (4.58). 

We assume a very general relationship between flux and reactivity, and then find the limitations on and the exact function of the reactivity. In order to find these limitations, we begin by establishing the diffusion equation for thermal neutrons in a nonequilibrium system. We next modify that diffusion equation for the effect of delayed neutrons and simplify. This gives us a differential equation involving flux and delayed-neutron-precursor density as space and time variables. Next, we write an equation for the net rate of formation of the delayed-neutron precursors. 

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