# Resonance escape probability after a change in temperature

Gold Member
Hi, I'm reading Lamarsh's book "Introduction to nuclear reactor theory" and in chapter 7 he discusses the influence of temperature on the resonance escape probability. He states that after an increase in Temperature, as a consequence of Doppler broadening, the microscopic absorption cross section of the fuel goes down and so the flux increases. If I interpreted correctly he then states that since we have "more neutrons" (the flux is higher) we have more collision and more absorption near the resonance. I don't understand this. The flux is higher, this I understand, but the cross section is smaller! So I though the effects should cancel... Why does the increased flux seem to play a bigger role than the lower value of the microscopic cross section?

Thanks
Ric

mathman
I am not familiar with the details in this situation. However if the effect of Doppler broadening is less than the the effect of increased flux, then his conclusion makes sense..

Gold Member
I am not familiar with the details in this situation. However if the effect of Doppler broadening is less than the the effect of increased flux, then his conclusion makes sense..
Ok, but I have no idea if that is true or not

Astronuc
Staff Emeritus
He states that after an increase in Temperature, as a consequence of Doppler broadening, the microscopic absorption cross section of the fuel goes down and so the flux increases
Where exactly (on what page) is one reading the discussion of resonance broadening? See Figure 2-32 in the text for Doppler broadening of the capture cross section of 238U at the 6.67 eV resonance.

Gold Member
Where exactly (on what page) is one reading the discussion of resonance broadening? See Figure 2-32 in the text for Doppler broadening of the capture cross section of 238U at the 6.67 eV resonance.
Page 229 (Paragraph 7-3) "Temperature dependence of resonance escape"

PS: In chapter 2 I have one big problem: just below Figure 2-32 Lamarsh says that the area under the curve is independent of temperature and he says to check is simply by mathematical means. What is the physical explanation though?

Astronuc
Staff Emeritus
If I interpreted correctly he then states that since we have "more neutrons" (the flux is higher) we have more collision and more absorption near the resonance.
Ok, I see the source of the confusion.

The flux is not higher or increasing, but rather the proportion of the flux near the energy of the resonance increases, so more neutrons are captured. Consider a stream along which fish swim. If one has a net of 1 m2 area, one will catch so many fish. Use a 2 m2 net, one will catch more fish, but the number of fish swimming in along the stream is not changing. Of course, if one catches more fish, there will be less fish on the other side of the net. The broadening of the resonance increases the energy range over which the neutrons would experience the resonance and be captured, even as the peak magnitude of resonance decreases.

That the area of the broadening resonance is constant is an artifact of the mathematical model/approximation.

The resonances of 238U and 240Pu are important with respect to limiting reactivity increases in light water (thermal) reactors.

dRic2
Gold Member
The broadening of the resonance increases the energy range over which the neutrons would experience the resonance and be captured, even as the peak magnitude of resonance decreases.
Yes, but (back to your analogy with fishes in a river) if you double the net you also decrease its "efficiency". When a fish gets in the net there is a probability that (even if it is inside it) it won't get captured (he escapes). If you double the area of the net you also increase the probability that if the fish gets "into" the net he might as well escape.

As I'm seeing things it is assumed that if a neutron gets in the resonance region it is automatically absorbed, but this not true.

I'll try a little different approach...

With Doppler broadening, the *peak* resonance cross section decreases, but the width of the resonance increases. The net effect is that the total number of absorption's increases, so the *average* cross section increases.

There is a "self-shielding effect" where the flux at the resonance energy decreases, but the overall net average absorption cross section still decreases.

Higher fuel temperatures --> higher resonance absorption.

Astronuc
Staff Emeritus
Yes, but (back to your analogy with fishes in a river) if you double the net you also decrease its "efficiency". When a fish gets in the net there is a probability that (even if it is inside it) it won't get captured (he escapes). If you double the area of the net you also increase the probability that if the fish gets "into" the net he might as well escape.

As I'm seeing things it is assumed that if a neutron gets in the resonance region it is automatically absorbed, but this not true.
The main point of the fish analogy is with respect to the neutron flux, which does not increase.

In a reactor, the neutron flux has an energy spectrum from about 0.001 eV to 10 MeV, or 10 orders of magnitude. At any energy there is a finite probability that the neutron will be absorbed by a fuel or non-fuel nucleus. In most cases, the reaction will be radiative capture, while in the fuel, the outcome could also be fission in addition to radiative capture. As a resonance broadens, more neutrons of different energies corresponding to the range of energies of the resonance are absorbed (interact with the nuclei associated with the resonance), whereas for a narrow resonance, most neutrons would pass by the resonance.

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I'll try a little different approach...

With Doppler broadening, the *peak* resonance cross section decreases, but the width of the resonance increases. The net effect is that the total number of absorption's increases, so the *average* cross section increases.

There is a "self-shielding effect" where the flux at the resonance energy decreases, but the overall net average absorption cross section still increases.

Higher fuel temperatures --> higher resonance absorption.

Oops - I had the wrong sign in my previous post. Corrected above. (Is there a way to edit my earlier post?)

Gold Member
Hi, sorry for my late reply. First of all, thank you very mach for all the help.

I think I'm almost there, but I didn't get the time to put it all together. I wrote some of my conclusions on a piece of paper and tomorrow I will post here what I finally figured out. Hope you can wait and check what I did.

Thanks again

Gold Member
Ok, sorry but it is pretty long to write what I concluded and you'll be bored (since you know it already). I just have one last doubt: is this true for both homogeneous and heterogeneous rector ? I think it is, but in heterogeneous reactor the effect is higher due to shielding effects

Astronuc
Staff Emeritus
Ok, sorry but it is pretty long to write what I concluded and you'll be bored (since you know it already). I just have one last doubt: is this true for both homogeneous and heterogeneous rector ? I think it is, but in heterogeneous reactor the effect is higher due to shielding effects
Resonance absorption is significant in both system. In a heterogeneous reactor, where the fuel is separated from the moderator, not only is there shielding by the resonance absorbers, but also the fuel temperature is much greater than the moderator temperature, particularly early in a reactivity transient. In LWRs, Pu-isotopes accumulate, especially on the surface of the cylindrical fuel pellets, and the resonance absorption (at 1.056 eV) by 240Pu becomes very significant.

A reference: Treatment of Resonances of Plutonium Isotopes at eV Energy Region in Tight Lattice Cell
https://www.tandfonline.com/doi/pdf/10.1080/18811248.1989.9734321

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dRic2
Ok, sorry but it is pretty long to write what I concluded and you'll be bored (since you know it already). I just have one last doubt: is this true for both homogeneous and heterogeneous rector ? I think it is, but in heterogeneous reactor the effect is higher due to shielding effects

If I understand your question correctly, are you asking if resonance absorption higher in a heterogeneous system? The answer is no, resonance absorption will be higher in a homogeneous system.

Consider the case of the first "pile" reactor in Chicago which had uranium slugs arranged in a graphite moderator. All neutrons are born in the fuel with fast energy. In thermal reactors, you try to slow the neutrons down to a lower energy so they will have a higher probability of causing a fission reaction with U-235 (the U235 fission cross section is about 500x larger for thermal neutrons than fast neutrons). If the neutron stays in the fuel, there is a very good chance it will be absorbed in the resonance regions. However, if the fast neutron escapes the fuel at high energies, slows down in the graphite, then re-enters the fuel at thermal energies, it will bypass the resonance energies. If you have a homogeneous system, the neutrons are more likely to be absorbed by the fuel resonances. Therefore, resonance absorption is *lower* in heterogeneous systems.

This process is described by the four-factor formula. I won't describe it here, there are many web pages that can describe it better than in a thread comment. I am not a big fan of the 4-factor formula, but it does describe the "neutron life cycle" and the importance of slowing down outside the fuel.

dRic2