Uranium 239, which may be formed as the result of U-238 absorbing a neutron, is radioactive and decays by 0 emission, with a half life of 23-j minutes, to neptunium 239. This also decays by (3 emission, with half life 2.3 days, to plutonium 239, an a emitter of half life 24 000 years.

The above may be written:

238U + n 239U ——

23 t m

/3

— 239Np ——

2.3d

— 239-Pu (2.4 x 104y)

absorption of neutrons of any energy including slow neutrons of very low energies. The uranium 238 is called a fertile material because the absorption of the neutrons, which we have seen previously it most readily does in the resonance capture peaks, leads to the formation of the fissile material Pu-239.

Similarly thorium 232 is also a fertile material because neutron absorption leads, via protactinium 133 t0 the fissile material uranium 233:

&

233-Th + n — 233-Th —————

22m

— 233-Pa ——

27.4d

— 233-U (1.6 x 10-у)

Thus the fertile materials U-238 and Th-232 yield the fissile materials Pu-239 and U-233 respectively.

Considering the centre channel of core where the

radius (г) = 0 and Jo (ar) = 1, using Equation (1.12) the total heat generated from the channel inlet (-L/2) to a point (z)

Qv = Q max 1 cos bz dz

J — L. 2

= Q max/b(sin bz + sin bL/2)

The heat absorbed by the coolant in flowing along the channel from (-L/2) to (z) can be evaluated by the product of the mass flow (m) specific heat (Cp) and the temperature rise (t — t,).

For steady state conditions within the channel the rate of heat generation must equate with the rate of heat absorption by the coolant. (This is an important concept when considering methods of reactor control.)

Thus: Q max/b(sin bz + sin bL/2)

= m Cp (t-tj) (1.13)

(neglecting the conduction of heat along the fuel element)

It should be emphasised that this equation applies only to steady state conditions. For transient situa­tions within the core, i. e., when temperature levels are unstable other more complex relationships apply. The solution of equation (1ЛЗ) gives the expression:

t — tj — Д t/2 [1 + sin bz/(bL/2)]

which enables the coolant temperature distribution along the channel to be determined.

Similar expressions can be derived giving the sheath surface temperature and fuel temperature variations along the channel. Figure 1.25 shows the resulting heat flux and temperature distributions. The principal points to note from the curves (which are similar for the majority of the fuel channels) are:

• The maximum rate of coolant temperature rise is at the centre of the channel.

• The maximum sheath temperature occurs at approx­imately two-thirds of the channel length. The posi­tion and value of this temperature is particularly important in the case of magnox reactors where it is monitored usually by means of a thermocouple fixed at this position.

• The maximum fuel temperature occurs at approxi­mately three-quarters of the channel length.

The temperature gradients at any plane in the chan­nel are from the fuel to the coolant via the sheath, although the magnitude of these gradients will vary considerably along the channel giving varying rates of heat transfer.

These patterns of temperature distribution are based upon a cosine pattern of flux distribution and, al­though they give an accurate picture for the majority of channels operating at normal design conditions, any deviation from the cosine shape will produce different temperature distributions. Deviations from the cosine shape can occur for instance under ‘start­up’ conditions or with channels which are immediately adjacent to large absorbers of reactivity such as con­trol rods.

From our day-to-day experience of billiard-ball type collisions a high energy transfer collision can intui­tively be deduced to require a target nucleus of mass not too dissimilar to that of the impinging neutron.

On this basis it would be expected that hydrogen, whose single proton nucleus approximates well to the mass of neutron, would be the ‘best’ target nu­cleus and the heavier nuclei to be progressively less effective. This indeed is so. Applying Newton’s Laws of Motion to a target nucleus of mass number A it is possible to calculate the number of collisions required for a neutron to lose energy from some initial value E0, say, to a final value E. The mathe­matical expression is:

Number of scattering collisions = logn (E0/E)/£

where £, known as the logarithmic energy decrement, is given by

£ = [1 ~ (A — l)2/2A][logn (A + I)/(A — I)]

— 2/{A + 2/3)

for all but the very low mass number nuclei.

(In the case of hydrogen £ = 1.)

Thus to reduce neutron energy from 2 MeV to 0.025

eV requires

[logn (2 x 106)/(2.5 x 10"2)]£ = !8.2/£

In Table 1.6 the number of collisions required to thermalise 2 MeV neutrons are given for different nuclei. Also included are ordinary ‘light’ water FbO and ‘heavy’ water D2O where ‘D’ refers to deuterium, an isotope of hydrogen with a nucleus of one proton and one neutron. The smaller the moderator mass number the larger the value of £ or the fewer the scattering collisions to thermalise the neutrons.

These are identical with the helium nucleus and are emitted with a well-defined speed from unstable nu­clei. They are relatively heavy, being about 7000 times heavier than the orbital electrons with which they collide, and travel relatively slowly compared with 0 particles. The alpha particle carries two positive charges; orbital electrons are therefore attracted to the a particle and pulled away from their parent atom. Thus as a particles traverse a material their energy is quickly dissipated. Because they move in straight lines and lose energy continuously (in effect) in a large number of ionising collisions, a particles of a given energy have a definite range in a given material. Typically a particles travel a few centimetres in air and are stopped by a sheet of paper or the outer layers of the skin.

1.5.2 Beta particles

These are electrons originating in and emitted from unstable nuclei with energies that are found to range from zero to a definite maximum value. Being much lighter than the a particle, the 0 particles generally travel much faster and have less time in which to react with individual atoms as they pass by. Moreover, having only a single negative electric charge, the elec­tric forces are weaker than for the a particle. Thus in comparison to a particles they ionise less, lose their energy less quickly and travel further. Typically 0
particles may travel from zero to several metres in air and about a centimetre in water or living tissue.

It is appropriate at this stage to comment briefly on the emission energies of a and 0 particles. For a given a emitting radioactive isotope the a particles are all ejected with exactly the same energy. It would be expected that 0 emission would be mono-energetic also and not have the range of energies found in practice. This puzzled the early researchers; in 1931 W. Pauli postulated (the postulation was not experimen­tally confirmed until 1956) that 0 emission is accom­panied by a second particle: the neutrino with 0+ and the antineutrino with 0“. The energy of emission is shared between the two particles thus explaining the observed energies of 0 radiation: ranging from a maxi­mum value, when all the available energy is in the 0 particle, to zero, when all the energy is in the neutrino/antineutrino. The neutrino and antineutrino are particles of zero mass and zero charge and thus possessing only energy. They pass through the mass of the_earth without hindrance, perhaps one in 1010 reacting. No further comment will be made here; the reader is referred to the literature for further infor­mation on these ephemeral particles.

4.6.4 Neutron flux distribution

Evaluation of the non-leakage probability factor re­quires knowledge of the rate of diffusion and the distribution of neutrons throughout the reactor core. The neutrons tend to diffuse from regions of high flux to regions of lower flux. Thus a stable distribution is established such that the production, absorption and diffusion of neutrons into and out of each region of the reactor is in balance. Since neutrons can only escape completely from the reactor at its boundaries this leads to a lower neutron flux at the edge of the reactor rather than at the centre.

The neutron diffusion theory is similar to that developed for other diffusion controlled phenomena: temperature in a heated body, dye in a liquid solution, smoke particles in air. The mathematics of the diffu­sion theory, the derivation of the diffusion equation and application of its solution to various core geo­metries may be readily found in reactor physics text­books. Here the solution of the thermal neutron flux distribution is presented for a cylindrical core height L and radius R:

ф — Фо cos (тг L/L’) Jo (2.405 R/R’)

where L’ and R’ are the extrapolated height and ra­dius of the reactor, see Fig 1.15, and Jo is a Bessel function. The equation gives the relative flux variation; the absolute magnitude is determined by the arbitrary constant 0o-

It may be seen that the neutron flux ф does not fall to zero at the physical boundaries of the core. If this were so no neutrons would leak from the reactor as there would be no neutrons at the core boundary. The flux level at the edge of the core is such that it appears to be falling to zero at points outside the core. These points form the extrapolated boundary of the reactor.

Thus the neutron flux distribution in a cylindrical reactor, Fig 1.15, is given by:

•

Axial flux shape — cosine.

• Radial flux shape — Jo Bessel function.