Calculate ##\xi## for Neutron Slowing Down in Moderators

[rpp]

I am looking at the average lethargy gain of a neutron when colliding with a moderator, which is often referred to with the greek letter "xi" (see for example, Duderstadt and Hamilton page 324).

There is a general equation for calculating $\xi$,

$\xi = 1 + \frac{\alpha}{1-\alpha} \ln \alpha$

And a table showing values of $\xi$ for several moderators.

I can successfully reproduce the value of $\xi$ for all of the single isotope moderators, but I cannot figure out how to calculate $\xi$ for compounds such as H2O and D2O.

For example, $\xi$ is: 1 for H, 0.12 for O, and 0.92 for H2O.

Any ideas?

 

[Astronuc]

The collision parameter, α, is defined as:

α = ( (A-1)/(A+1) )², where A = atomic mass. It works well for a nucleus, but not a molecule.

The parameter is derived from classical mechanics, 2-body kinematics for neutron collision with a stationary nucleus. See pages 39-45 in Duderstadt and Hamilton, or one of Lamarsh's introductory texts on nuclear engineering or nuclear reactor physics.

Atomic mass doesn't really apply to a molecule, although one could possibly derive an effective A based on atomic density and cross-sections. However, it would fall apart at thermal energies where the neutron energies in the eV range fall below the atomic binding energies and involve inelastic collisions which include excitation of the molecule with rotations or internal vibrations. For fast neutrons >> eV, the nuclei are relatively immobile.

 

[rpp]

Yes, it can be calculated for singe nucleus.

My question is, how is the value calculated for H2O and D2O?
Most books quote the same numbers, so I am guessing there is a common source.

 

[Astronuc]

Yes, it can be calculated for singe nucleus.

My question is, how is the value calculated for H2O and D2O?
Most books quote the same numbers, so I am guessing there is a common source.

It goes back to the 1950s. My copy of Nuclear Reactor Engineering by Samuel Glasstone and Alexander Sesonske, published by D. Van Nostrand, 1955, 1963, has the definition of the collision parameter. I suspect the calculation was done during the early 1950s and probably resides in an AEC report, or it could have been described in a journal like Nucleonics. Glasstone and Sesonske (GS) provide the background similar to the descriptions found in Lamarsh's texts, Duderstadt and Hamilton (Nuclear Reactor Analysis, Wiley, 1976), and Weston Stacey's Nuclear Reactor Physics (Wiley, 2001, 2004).

Interestingly, GS do not provide a reference to the theory development. It seems to be rather basic classical mechanics, at least, for a neutron scattering off a single atom, and they assume it's pretty straightforward.

The collision parameter is mentioned in this 1962 report -
http://web.ornl.gov/info/reports/1962/3445605700195.pdf

Another parameter of interest is the average logarithmic energy decrement per collision, ξ, which is defined by

ξ = 1 + (α ln α)/(1-α) ~ 2 / (A +2/3)

GS has an expression for the mean ξ for a compound, which I will post later.

Other texts which might have the theory are:
S. Glasstone and M. C. Edlund, "The Elements of Nuclear Reactor Theory," D. Van Nostrand Co, 1952
R. V. Meghreblian and D. K. Holmes, "Reactor Analysis," McGraw-Hill Book Co, 1960
A. M. Weinberg and E. P. Wigner, "The Physical Theory of Neutron Chain Reactors," University of Chicago Press, 1958

I don't have these books, but I did use them back in the 1970s. They were in the university library.

The collision parameter is given a cursory treatment in the text by George I. Bell and Samuel Glasstone, Nuclear Reactor Theory, Robert E. Krieger Publishing Co., 1968, 1970, 1974, 1982, which provides neutron transport theory. Chapter 7, Neutron Thermalization, discusses neutron scattering, including theory applied to graphite and water as moderators. I'll have to see if they reference some of the early theoretical work.

 

[Hologram0110]

GS has an expression for the mean ξ for a compound, which I will post later.

I was curious, so I checked my copy of GS (3rd Ed.) it is given on page 164 as eqn 3.58. It is simply the average of the logarithmic energy decrement for all isotopes in the compound weighted by their scattering cross-sections.

As Astronoc pointed out this relationship would break down for sufficiently low energy neutrons. At high energies the chemical binding energy is not significant enough to significantly change the results. However as the kinetic energy decreases this should become more and more significant. There would be significant complications scattering off a molecule because now you can introduce other types of energy into the system (like rotational and vibrational).

 

[Astronuc]

$\overline{\xi} = \frac{\Sigma_i (\sigma_{si} \xi_i)}{\Sigma_i\sigma_{si}}$

as Hologram0110 mentioned

so for H₂O
$\overline{\xi} = \frac{ 2\sigma_{sH} \xi_H + \sigma_{sO} \xi_O}{2\sigma_{sH} + \sigma_{sO} } = \frac{ 2\sigma_{sH} + \sigma_{sO} \xi_O}{2\sigma_{sH} + \sigma_{sO} }$

because $\xi_H = 1$