# Quick question about neutron scattering and absorption cross sections

Does the absorption cross section to scattering cross section ratio of an isotope vary with neutron energy or stay constant?

I have heard that cross sections in general are inversely proportional to velocity (eg the fission cross section of U235 is about 1000 times higher for thermal neutrons than fast neutrons) which would suggest the answer is yes. But I have not found an explicit answer.

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Astronuc
Staff Emeritus
Does the absorption cross section to scattering cross section ratio of an isotope vary with neutron energy or stay constant?

I have heard that cross sections in general are inversely proportional to velocity (eg the fission cross section of U235 is about 1000 times higher for thermal neutrons than fast neutrons) which would suggest the answer is yes. But I have not found an explicit answer.
Some nuclides have 1/v dependency on cross-sections, and others less so.

One can find neutron-nuclide cross-sections here.
http://www.nndc.bnl.gov/sigma/index.jsp

That doesn't necessarily answer the question, it's still not clear if the ratio of absorption to scattering cross sections (xs) stays the same for all isotopes or not. It should hold true for isotopes were both cross sections follow the 1/v rule, a better way to phrase the question might be 'do the cross sections of isotopes that do not follow the 1/v rule change in the same way?' (eg if an isotope was found to have a scattering xs proportional to 1/3v would the absorption xs also follow this?)

I found a convenient table of absorption and scattering cross sections of most isotopes for thermal neutrons here:

http://www.ncnr.nist.gov/resources/n-lengths/list.html" [Broken]

I was able to put this into a spreadsheet for further analysis, unfortunately I cannot find a similar table for fast neutrons and entering each value individually would be very tedious. If I could find such a table I could answer the question pretty easily.

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QuantumPion
Gold Member
The ratio

$$\frac{\Sigma_{a}(E)}{\Sigma_{s}(E)}$$

is definitely not constant for all isotopes since they all behave quite differently with higher energy neutrons. The relationship may be constant for 1/v absorbers in the thermal range for some isotopes, but not all. For example, Carbon 12 has a constant scattering cross section over a large energy range where it is a 1/v absorber (thus the ratio would be constantly changing as a function of energy). The ratio for nuclides with resonances would obviously not be constant.

If I had to guess I would say the ratio holds when the neutron energy is below the first excitation energy of the isotope. The 1/v breaks down for the scattering xs at this point because it increases (rather than continuing to decrease with increasing energy) due to inelastic scattering cross sections summing with the elastic. Not sure whether the absorption xs continues to obey the 1/v rule or is also modified by excited states of the nucleus (resonances).

I'm researching this to compile a list of isotopes that can be used in reactor cores without absorbing too many neutrons. It's a shame helium 4 doesn't have a larger scattering xs as it is basically the only isotope that doesn't absorb neutrons. A helium cooled and moderated reactor would have some nice properties if the xs were larger.

QuantumPion
Gold Member
If I had to guess I would say the ratio holds when the neutron energy is below the first excitation energy of the isotope. The 1/v breaks down for the scattering xs at this point because it increases (rather than continuing to decrease with increasing energy) due to inelastic scattering cross sections summing with the elastic. Not sure whether the absorption xs continues to obey the 1/v rule or is also modified by excited states of the nucleus (resonances).

I'm researching this to compile a list of isotopes that can be used in reactor cores without absorbing too many neutrons. It's a shame helium 4 doesn't have a larger scattering xs as it is basically the only isotope that doesn't absorb neutrons. A helium cooled and moderated reactor would have some nice properties if the xs were larger.
Helium is not a very good moderator due to its low density and inability to form denser chemical compounds. In any case, the quantity you are after is the Moderating Ratio:
$$\frac{\xi\Sigma_{s}}{\Sigma_{a}}$$

where the average lethargy gain, $$\xi = 1-\frac{(A-1)^{2}}{2A} ln(\frac{A+1}{A-1})$$

http://www.tpub.com/content/doe/h1019v1/css/h1019v1_131.htm"

The best common moderator by far is heavy water. For some reason I thought tungsten carbide was a good one too but I can't remember and am too lazy to calculate it tonight. :zzz:

Basically, to answer your question, the only good isotopes are H, D, C, and Be.

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Lighter nuclei are more effective at neutron moderation:

Purely from the point of view of moderation you are right about heavy water, but in terms of efficiency and plant cost not necessarily as water cooled and moderated plants operate high pressure (2-300bar) (more expensive safety systems) with lower temperature differentials (lower conversion efficiency). Also high purity heavy water (99.75% in CANDU reactors) is energy intensive and expensive to isolate. In principle a molten salt reactor running high temperature with low pressure using C12 as a moderator should be safer, cheaper and more efficient than a heavy water based reactor.

BWR/PWR run dT around 2-300, advanced HTR and MSR could be 800-1000.

The UK AGR reactors use a C12 moderator with CO2 coolant at medium pressure (~50bar) and temperatures (dT ~4-500). Problem is over time the properties of the graphite change with neutron irradiation and they lose mass. I'm doing a project on the characterisation of nuclear graphites and the damage mechanisms. The project allows speculation on novel methods of preventing the damage, which is why I have interest in all isotopes, not necessarily just for moderation. For instance one idea was to use graphite powder in stacked zirconium tins, from an isotope point of view this is ok because the absorption xs of zirconium is low, but at high temperatures it strips the oxygen from CO2. It might be a possibility in future helium cooled AGRs though. Another idea was to use a ceramic coating on the graphite like aluminium oxide to reduce mass loss. Alas the absorption xs of aluminium is too high. I'm sure there is a solution though.

Astronuc
Staff Emeritus
The lighter the nucleus the better the moderating or slowing down capability. A proton is a great moderator since it has roughly the same mass as a neutron. The moderator ratio however suffers because a proton can absorb a neutron and become a deuteron.

Atomic/molecular density is another key factor. Metallic hydrogen/deuterium would be a great moderator - except for the fact it requires cryogenic temperatures, or extremely high pressures - well beyond typical structural materials used for pressure vessels.

Another consideration is the thermodynamic/power generation cycle, e.g., Brayton, Stirling, Rankine, etc. In general, the objective with nuclear energy is to produce electricity of which the most common form is AC at 50/60 Hz. Some small local systems use 400 Hz.

For a reactor design the principal design criteria are:

1. Efficient thermal generation
2. Retention of fission/activiation/transmutation products
3. Coolability - the fuel must be cooled such that technical limits are not violated
4. Controlability - the thermal generation must be controlled (shutdown if necessary) such that criteria 2 and 3 are not violated

The first criterion is the reason for using nuclear energy. The second one is the key health and safety criterion (radionuclides must be isolated from the environment/biosphere).

The third and fourth criteria follow from the second criteria. Basically the reactor/fuel must remain within technical design limits such that release of radionuclides to the environment is precluded.

I should have mentioned that the carnot cycle is the most efficient theoretical heat engine, this is never achieved in the real world power cycles but it does stand that a higher dT results in higher thermal efficiencies, for all cycles (thermodynamics 101 really). A high temperature reactor could be used in a bottoming cycle where the heat is first used in high temperature industrial process (eg for hydrogen production) and then leftover heat is used to power a turbine and generator. In this way even higher utilisation of thermal energy can be achieved than in the carnot cycle.

I'm convinced that in the future the industry will move away from water cooled and moderated designs, as all suffer from the high pressure and low temperature limitations. All other designs, gas cooled or molten metal/salt cooled use graphite as a moderator. Graphite moderated designs so far have lifetimes of 40-60 years but after this the graphite deterioration is to severe. It's not fully understood how the graphite changes:

This is for isotropic reactor grade graphite (gilsocarbon) at 450ºC. A better method of implementing graphite as a moderator needs to be devised for cost effective long lived reactors using graphite.

QuantumPion
Gold Member
I think this thread has gone completely off topic :uhh:

Perhaps I should create a new thread, but I think the original question has been answered. The 1/v rule is obeyed below the first excitation energy of the isotope.

The other information was related to my motivation for asking about cross sections.

QuantumPion