Half-Life & Flux: Is Decay Formula Relevant?

[Moniz_not_Ernie]

Given an initial mass of some isotope subjected to a constant neutron flux, how fast will the mass drop off? Would not the survival curve look exactly like the curve for radioactive decay? Both cases describe a starting mass subjected to a constant transformative force at a rate that depends on the size of the population. The target mass shrinks in both cases, whether the cause is internal or external. Can’t the decay formula can be used for both? Does this concept have a name?

 

[Astronuc]

Given an initial mass of some isotope subjected to a constant neutron flux, how fast will the mass drop off? Would not the survival curve look exactly like the curve for radioactive decay? Both cases describe a starting mass subjected to a constant transformative force at a rate that depends on the size of the population. The target mass shrinks in both cases, whether the cause is internal or external. Can’t the decay formula can be used for both? Does this concept have a name?

One is referring to transmutation, and like one calculates the rate of change by decay, one can calculate the rate of transmutation.

The rate of decay is given by λn(t), where λ is the decay constant and n(t) is the number of atoms (of the radionuclide) at a given time, and the change in n(t) is given by d n(t) / dt = -λ n(t), which indicates n(t) is decreasing with time due to decay.

Similarly, the rate of transmutation is proportional to σφ, where σ is the total microscopic cross-section for neutron absorption reactions and φ is the neutron flux. It's actually more complicated since σ and φ are functions of energy. The product σφ is analogous to λ, and d N(t) = -σφ N(t). For a monoenergetic source, the evaluation of σφ is pretty straightforward, but for a neutron energy spectrum, one has to integrate over the range of neutron energies.

I assume here that the target is a stable isotope. If the isotope is unstable, i.e., it is decaying, then one must consider decay and transmutation, and so, the rate of loss is proportional to (λ+σφ).

One would use the term 'depletion rate', for the loss of an isotope, or production rate for the product of transmutation. When we discuss the consumption of U-235 in a nuclear reactor, we often refer to 'depletion' rather than consumption, but it's the same thing.

 

[Moniz_not_Ernie]

I associate transmutation with a change of element, not just a new isotope of the same element. Is there a special verb for that? “The neutron plumps the 235-U nucleus if it doesn’t cause fission…”

I didn’t specify stable, so your lambda+ term will help. This is in reference to my nuclear fleet sim. I’ll start a separate thread for that.

With respect to the consumption of 235-U, is there a term for its half-life in a reactor? For instance, if the fuel rods stay in the reactor for four years, and the enrichment drops from 5% to 1% (cut in half twice), the fuel has the equivalent of a two year half-life?

 

[Astronuc]

I associate transmutation with a change of element, not just a new isotope of the same element. Is there a special verb for that? “The neutron plumps the 235-U nucleus if it doesn’t cause fission…”

Transmutation applies to changing an isotope of A to A+1, because it is a change, and often the nuclide is unstable. There are cases where an (n,γ) reaction produces a stable isotope, e.g., Xe-135 to Xe-136, or Gd-155 to Gd-156. The significance is that the A+1 isotope will have a very different cross-section than the original isotope.

With respect to the consumption of 235-U, is there a term for its half-life in a reactor? For instance, if the fuel rods stay in the reactor for four years, and the enrichment drops from 5% to 1% (cut in half twice), the fuel has the equivalent of a two year half-life?

No, not really, since in reactors, we have batches of fuel. Each cycle is designed with so many effective full power days with some margin. Most fuel operates between 3 (2 x 18-months) and 4 (2 x 24-months) years, with some fuel held over for a third cycle to 4.5 or 6 years, and rarely to a fourth cycle. During a third or fourth cycle, the fuel is most likely on the edge of the core. Some of the consumed U-235 is offset by the conversion of U-238 to Pu-239, -240, and -241. Beyond a burnup of about 30 GWd/tU, more fissions occur in the Pu than in U, which is inherent in LWRs.

 

[Moniz_not_Ernie]

I'm surprised the old fuel is placed at the rim. The neutron flux should be less out there. I would think placing fresh fuel at the edge would minimize the difference in energy output from the center out to the edge. I've heard hot spots are a problem. Doesn't putting fresh fuel at the center create a big hot spot?

 

[Astronuc]

I'm surprised the old fuel is placed at the rim. The neutron flux should be less out there. I would think placing fresh fuel at the edge would minimize the difference in energy output from the center out to the edge. I've heard hot spots are a problem. Doesn't putting fresh fuel at the center create a big hot spot?

High burnup fuel is placed on the periphery of the core, and sometimes, special absorber assemblies are placed there, in order to minimize the neutron fluence to the core baffle, core barrel and reactor pressure vessel. When plants were designed in the 1960s and 1970s, we didn't have 40 years of experience. It wasn't until 10 to 20 years down the road, the industry discovered the effects of neutron embrittlement in pressure vessel steels.

http://www.tms.org/pubs/journals/jom/0107/odette-0107.html

Fresh fuel is loaded with burnable poisons (typically gadolinia in selected UO₂ fuel rods, sometime erbia, and in most Westinghouse fuel, enriched ZrB₂). Boron is enriched in B-10, which undergoes an (n,α) reaction. The boron and gadolinia are depleted during the first cycle, and they hold down the reactivity in the fresh fuel. In some PWRs, special burnable poison assemblies contain borosilicate glass are used, and I believe in some cases, B₄C distributed in alumina is also used.