## DD Fusion Anisotropy Coefficients

It may be difficult to separate "theory" from "application," but let''s see if this helps facilitate the discussion.
Liam David
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### DD Fusion Anisotropy Coefficients

The attached paper contains equations/coefficients for the angular and energy distributions of neutrons generated from deuterons incident on a thin target, in the laboratory frame. The data are plotted below. I'm unable to locate the original source of these data: G.J. Csikai, CRC handbook of fast neutron generators, 1987.

https://doi.org/10.1088/1748-0221/16/10/p10001

Attachments
MCNP Dose evaluation around D-D and D-T neutron generators and shielding design.pdf
(4.13 MiB) Downloaded 17 times
Rex Allers
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### Re: DD Fusion Anisotropy Coefficients

Liam, thanks. Interesting.

It took me a few minutes to understand what I think you have given us. Here is a basic description of things I didn't grock immediately. Hope I didn't get this wrong.

The first link is the source of the paper and you already attached the pdf version in the post.

The paper includes two equations for simulating the results. The first gives a relative neutron count at theta. The second gives neutron energy. Each formula has a table of coefficents that are provided for 4 different beam energy levels.

You ran these two equations for the D-D case and generated the two angular plots in your message.

I presume the equations were derived by curve fitting the data from actual measurements. That actual data is what you say you haven't been able to find.

Two additional notes:
1) From a simple apparatus diagram in the paper, the beam direction is from 180 toward 0 on the angle scale.

2) The neutron count levels are not absolute, they are scaled relative to the value at 90 deg. Note that in the first plot all the values are 1 at 90 deg. The other values are relative to this.

Let me know if I got any of this wrong.
Rex Allers
Liam David
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### Re: DD Fusion Anisotropy Coefficients

Rex, you're completely correct. Thanks for adding the all-too-important details that I neglected. The beam comes in from the left, from 180 to 0, and the data have been normalized to the 90-degree value. The energies and spatial distribution go hand-in-hand in that the forward beam of neutrons will also have higher energies.

These distributions are for a beam impinging on a stationary, thin target, i.e. the region of neutron formation has zero thickness (order of 1-5um usually). The deuterons quickly lose energy due to the target, and you thus get fusions at energies lower than the incident beam and distributed by depth. I believe the energy range (not the depth) is what leads to the higher-order terms than (cos(theta))^2.

If you have no energy loss, the neutron distribution would be purely a (cos(theta))^2 distribution due to spin-orbit coupling (due itself to the deuteron center-of-mass and center-of-charge offset) which gives rise to p-wave scattering (see https://ir.library.ontariotechu.ca/bits ... sequence=3 and https://link.springer.com/chapter/10.10 ... -45878-1_6). For those curious, this is because in scattering theory, d(sigma)/d(Omega) = |f|^2, where the p-wave term of f is f ~ P_1(cos(theta)). Here sigma is the cross-section, Omega is the solid angle, and P_1 is the 1st Legendre polynomial P_1(x) = 2x-1. The cosine argument gets squared, which is where the distribution arises. Integrate the neutron emission for each "slice" of the target, taking into account the highly nonlinear stopping power of ions in matter (Bragg peak) and I think you'd get the coefficients in that paper. The fact that the A2 terms in table 2 are the largest in magnitude for their rows supports this idea. There's a chance I'm wrong, but it seems to be a pretty solid explanation.

I suspect these measurements were made using a D+ beam on a pre-loaded titanium or zirconium target. The beam spot size was likely on the order of a few mm to cm, as is the case for most such systems since it helps distribute the heat load and reduce target sputtering. I do not know if these data reflect that finite size, or if they have been altered to represent a point source. My guess is that the measurements were all performed in the far-field, and hence the finite size makes no difference.

A fusor does have areas that behave like thin targets, such as the cathode and endcaps. Beam-background gas fusion is technically not a thin target as the range of D+ and D2+ in D2 gas is on the order of cm, but like with the thin target, one could theoretically integrate the distribution along the appropriate paths to generate the distribution. This would be very complex, as it would require you to have an idea of the spatially-varying ion energy distribution.
Richard Hull
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### Re: DD Fusion Anisotropy Coefficients

I enjoyed the paper on neutron tube dose. I note that even with no shielding the D-D fusion neutron dosage is intrinsically zero at 50kev energies (fusor type operation which is far less efficient and far more isotropic.) It is important to note that all the tables at the end are computed for a B.O.T. tube operated at 200kev and the doses are still not frightening for D-D in short exposures at 2 meters. Of course, we all knew this.

A technician forced to operate such a system in a work related environment would need all the shielding he could get.

As systems made in these forums get smaller, with pressures and voltages higher, X-rays remain the #1 issue. Neutrons will always be secondary at normal operational distances up to 100kev. I doubt we will see a lot of operation at this level here.

Thanks for the refs. and paper.

Richard Hull
Progress may have been a good thing once, but it just went on too long. - Yogi Berra
Fusion is the energy of the future....and it always will be
Retired now...Doing only what I want and not what I should...every day is a saturday.
Chris Seyfert
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### Re: DD Fusion Anisotropy Coefficients

Liam, you may be interested in the Los Alamos report LAMS-2162 "DD and DT Neutron Source Handbook", Seagrave et al., 1957. Like you mentioned, this is primarily of relevance only for beam-on-target work.

https://permalink.lanl.gov/object/tr?wh ... t/LA-02162

I have a half-dozen or so references on this as it was relevant to my previous job, if that is of interest I can dig it out of the archives for you. Most of it is older material referenced in the above report, but I believe there were a few newer papers as well.
Liam David
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### Re: DD Fusion Anisotropy Coefficients

Thanks for the link, Chris. I haven't seen this report before, although I vaguely remember seeing it referenced in other works. I'm definitely interested in whatever else you've got. I've done a lot of searching for papers/reports like this, but especially the older stuff can be really hard to find.
Chris Seyfert
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### Re: DD Fusion Anisotropy Coefficients

Sure thing. Archive has been mined, here are the papers.
Attachments
1987 Krauss Angular DD.pdf
(1.45 MiB) Downloaded 12 times
1966 Theus Angular DD.pdf
(785.02 KiB) Downloaded 4 times
1957 Fuller Angular DD.pdf
(246.45 KiB) Downloaded 4 times
1955 Chagnon Angular DD.pdf
(488.44 KiB) Downloaded 4 times
1954 Preston Angular DD.pdf
(1.53 MiB) Downloaded 4 times
1952 Baker Angular DD.pdf
(200.35 KiB) Downloaded 5 times
1949 Hunter thesis Angular DD.pdf
(3 MiB) Downloaded 5 times
1949 Hunter Angular DD.pdf
(479.89 KiB) Downloaded 4 times