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As we have seen in section 3.5, U is formed by neutron capture by Th followed by two beta decays:

The presence of protactinium imposes limits on the admissible neutron flux when using solid fuels. This limitation is due to two detrimental effects of protactinium:

1. Protactinium captures neutrons, and thus decreases the reactivity of the reactor.

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2. After a reactor stop, the Pa inventory decays to U, which leads to an increase of the reactivity and of k. This increase may lead to reactor criticality. The characteristic time for such an evolution is of the order of the half-life of 233Pa, i. e. about one month. Corrective actions could easily be taken by inserting a negative reactivity. However, the advantage that passive safety of hybrid systems represents would be lost. It is, thus, interesting to keep the system subcritical in all instances.

The evolution equations of the Th-Pa-U system read

Thus, at equilibrium,

and

«U = «Pa A = AdTh

«Th «Th aU’)’ a{j>(A + dPaa}’)

For thermal neutrons oPa = 43 barns and for fast neutrons = 1.12 barns.

The lifetime of Pa in the neutron flux is only significantly shortened if ‘ > A/oPa, i. e. ‘> 7 x 1015n/cm2/s for thermal reactors and

‘ > 2.7 x 1017 n/cm2/s for fast reactors. Except for the very high-flux molten salt reactor which has been proposed by Bowman [2], such fluxes are never reached, so that can be neglected with respect to A. Thus, at equilibrium,

It is seen that the amount of protactinium is a measure of the neutron flux. It is also proportional to the specific power of the reactor, itself proportional to the density of fission vz, with

Since the specific power is the limiting factor of reactor designs rather than the neutron flux we express the modifications to the reactivity due to Pa in terms of the number of captures in uranium. The multiplication coefficient reads

(a)

nU0U

(a) (a) (a)

nU0U + nTh°Th + nPa0Pa + P

and

For thermal reactors the ratio 0^ /oTh = 74 while /oTh = 24 for fast

reactors. It follows that, for a given decrease of kx, fast reactors allow a specific power 3 times larger than thermal reactors and, hence, three times more compact cores.

After a reactor stop, the protactinium will decay into 233U, leading to an increase of kx. Asymptotically, the final value of kx will be

Since the perturbation on the reactivity is small, we can write that the relative change with respect to the unperturbed value of kr is

In order to estimate the true reactivity excursion, the decrease of the reactivity during irradiation has to be taken into account so that the total, maximum excursion is

Akr «c(1 + a) (a) kr (a)

P^ = l0.5"-‘ + ~T"P-

For fast reactors we get

One sees that the limit on the specific power is ten times more stringent for thermal systems than for fast ones. For Akr/kr = 2 x 10~2 the correspond­ing capture densities are on the order of 3 x 1013 for fast systems and

2.5 x 1012 for thermal ones. The corresponding fluxes are then 4 x 1015 for fast reactors and 4 x 1013 for thermal reactors.

In conclusion, it appears that the protactinium effect greatly favours fast reactors if solid fuels and the Th-U cycle are to be used. This is not true for

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the U-Pu cycle where Np, which plays a role analogous to Pa, has a much shorter half-life of 2.35 days.