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Fusion cross sections

  1. Mar 30, 2015 #1
    Is there a book or database that lists all (or many) pairs of isotopes along with the energy at which they have the higest probability of fusion and what that probability is? Or is there a program that uses models to make a guess given a pair? Or a formula I could use to make the guess myself?

    Bonus if it gives reaction products.
  2. jcsd
  3. Mar 30, 2015 #2


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    For the actual data, you want




    (ETA: I gave the wrong url there)

    These are reaction cross section databases.

    In general, if you're looking for any nuclear data, nndc is a good place to start.

    However, if you want to estimate a fusion cross section above the barrier, there are a bunch of simple models you can use. The simplest (and naturally, most wrong, but it's good to first order) model is the classical fusion cross section, which is:

    ## \sigma_{fus} (E) = \pi r_b^2 (1-\frac{V_B}{E})##

    For ##E>V_B## and ##\sigma_{fus}=0## otherwise.

    Where ##r_b## is the distance of closest approach, and ##V_B## is the barrier height. You can find a derivation of this in any introductory text or in a bunch of places online. This leaves out a lot of key components - the centrifugal potential, relativistic effects, tunnelling (see: Classical Model), coupling to internal degrees of freedom (see: Classical Model). There are obviously very many other more sophisticated models, but you'd need to be more specific as to what you're trying to actually achieve here.
  4. Mar 30, 2015 #3
    Thanks. I'm new at this. I'm a comp sci graduate without much physics training. I looked at the websites but I'm a bit slow to figure it out. I was looking for something that would let me enter something like: hg202, ga69, 11 million electron volts
    and it would spit back something like: 20% probability.

    Can you tell me how to plug that into those fields?

    My perspective is from intertial electrostatic confinement fusion which is based on accelerating the reactants toward each other as opposed to thermal approaches like tokamak.
  5. Mar 30, 2015 #4
    Is there a website that would educate me more on the subject?
  6. Mar 30, 2015 #5


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    The trouble here is that not every beam/target combination has been measured (there are a lot of combinations!). For instance, 202Hg would be a terribly hard target to make, and I don't even know about Ga beams, but it's not something very accessible theoretically. According to the Nuclear Science References database, nothing with that beam/target combination has never been published. Also, an 11MeV beam will be far far below the barrier for that measurement, so a cross section would be rather hard to get experimentally. (A quick/very dodgy calculation gives the barrier at 212 MeV. So, the probability for fusion for this example is near enough to 0 to make no difference).

    So your best bet will be theoretical models, it seems. The closest thing to a website that does what you want will be


    It does empirical or channel coupling calculations. I've not used it in detail myself (I'm a reactions person, but not a fusion person), but it seems fairly user friendly. In the first section, you plug in the reaction details, further down are some other physics that goes into the models. For now, I suggest you take the defaults. I just did 16O + 208Pb (the standard example, both nuclei are doubly magic) and it worked fine. I then tried to do 68Ga + 202Hg, and it's still calculating. So.

    Now, of course, all these quantities are in cross sections, and it seems you want a "percentage" probability. So what you will have to do is compare that cross section to the total reaction cross section - that is, the probability of anything happening. To first order, you should compare it to elastic scattering, which you can calculate explicitly, although you should really consider transfer processes, which would involve another calculation.

    Does this help? I'll try and think of a good website, but it seems like you need some basic nuclear physics? Try Krane, Introductory Nuclear Physics. It's the classic text. I'll think of a website, but there isn't one that immediately springs to mind.
  7. Mar 30, 2015 #6


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    A reaction like Ga on Hg would use a heavy ion accelerator, and there is not much that is practical if one wants to produce energy, which is the usual goal with fusion. Certain radioisotopes may be produced for specific reasons, but it is usually done with a light projectile because one has to overcome the Coulomb repulsion, which is proportional to Zp*Zt, where Zp is the projectile atomic number and Zt is the target atomic number.

    Superheavy elements are created with special reactions, e.g., Ca on Bk, Cm or other TU element. Ca is accelerated to 0.1c.

    or Ni into Pb or Bi.

    It is doubtful that IC systems are going to be in the MeV range.
  8. Mar 30, 2015 #7


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    Astronuc has an excellent point. It didn't even occur to me, since, well, I work at a heavy ion accelerator, and my group studies heavy ion fusion. (We're on the 117 paper that he linked to, for instance. That was a good day. But we also study fusion/fission for many other reasons, none of which are related to power generation) So whenever anyone says "fusion", my mind doesn't jump to "power generation from nuclear fusion", which is what a lot of people (on PF, but also elsewhere) seem to do.

    This makes me wince whenever anyone writes

    Because, well, no, not really.

    In the case of IC studies, unless you're doing something really exotic, the cross sections tend to be well known, because they are candidates for fusion power. In which case, there will be a wealth of information in experimental results.
  9. Mar 31, 2015 #8
    Thanks! I was able to find a few PDFs that look like they will be very educational. Which of these is the optimal energy for the reaction?

    VCB(spherical) = 76.461 MeV
    RCB = 11.588 fm
    hω = 4.68 MeV
    Lcr = 61
    Ecr = 119.0 MeV
    two limit orientations:
    V1B(nose-to-nose) = 76.81 MeV
    hω1 = 2.30 MeV
    V2B(side-by-side) = 78.01 MeV
    hω2 = 2.18 MeV
    Bass barrier = 77.09 MeV
  10. Mar 31, 2015 #9
    Using http://nrv.jinr.ru/nrv/webnrv/fusion/
    I tried mercury 202 and gd 154 it said "Coulomb barrier was not found"
    Did I do something wrong or does it just not have data or . . .?

    I tried doing 2H and 6Li, but it wanted deformation numbers, but when I clicked the button there were not any numbers. What should I do?

    You guys are awesome. Thanks for answering al these questions.
  11. Mar 31, 2015 #10


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    If by "optimal energy" you mean "maximum cross-section", then none of them, really. It's not really a concept used in nuclear physics, like I've been saying. The best I can give you are a cross sections as a function of energy, and you can pick the peak you want. What you have there are a bunch of parameters for the reaction. Can you please provide a source for those numbers?

    But roughly:

    VCB(spherical) - the barrier height for an equivalent system of spherical nuclei
    RCB - touching radius
    Lcr - Critical angular momentum above which fusion will be suppressed
    Ecr - Critical energy (related to the angular momentum) above which fusion will be suppressed.

    This "critical energy" is probably the closest you'll get to the number you want. So, to get maximum fusion cross section, you'd probably want to sit a bit below that, but above the average barrier. You'd probably do pretty well around 100 MeV. But that's an estimate.

    V1B(nose-to-nose), hω1, V2B(nose-to-nose), hω2, come into something called the coupled channels model. By coupling to internal degrees of freedom (i.e. excitations of the nucleus), the colliding nuclei go into a superposition of their internal states. This means that rather than one classical barrier, you get a distribution of barriers. This results in an enhancement (~factor of 100) of the fusion cross section below the barrier. For deformed nuclei, you can get the extremes by considering the orientation of the nuclei. Nose-to-nose will result in a lower effective barrier, and side-by-side results in a higher effective barrier. The coupled channels model is one of the triumphs of nuclear reaction physics in the last few decades.

    Bass Barrier - the average barrier you get if you use the Bass model to calculate it.

    I don't want to be rude, but it seems like you are missing some really basic knowledge here. Before you go on with your project, you should really read some basic reaction theory, so you can understand the numbers you're putting in to any model. Garbage in, garbage out, and all that.
  12. Mar 31, 2015 #11


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    You were trying to study 202Hg + 154Gd? Well, for a start, fusion of that would give you Z = 144. In the superheavies, the concept of the classical barrier becomes, well, problematic. The only reason they exist is due to shell model effects. There won't be much of a barrier at all.

    For 2H and 6Li, you're at the other end of the nuclear chart - the concept of deformation is tricky to apply when you've only got 2 or 6 nucleons hanging about. In any case, since you want to know where the maximum cross-section is, whatever values you use doesn't matter too much, since you want to be above the barrier, and deformation/channel couplings are a near/below barrier issue.

    You've really got to know what you're doing!

    I just came accross this: http://link.springer.com/article/10.1007/BF01414268 I'm not at work at the moment, so don't have access, but take a look, might be useful.

    Also, I'm curious -- why you care? Above the barrier, the cross sections change pretty slowly. See for instance http://inspirehep.net/record/1102876/files/fig6.png (b).

    (ETA: See the way the models the authors use vastly underestimate the fusion cross-section below the barrier? That's those channel couplings I was talking about)

    For light nuclei, things move a bit more rapidly, but there's standard tables for reactions applicable for fusion power: http://www.kayelaby.npl.co.uk/images/p548.jpg
  13. Mar 31, 2015 #12
    Thanks! Everything you said was completely new to me. I only learned last week that fusion produces more energy for lighter elements than for heavier.

    That first article looks very interesting but too pricey for me. Those other tables were very informative.
  14. Mar 31, 2015 #13
    You asked for a source for the numbers. It was http://nrv.jinr.ru/nrv/webnrv/fusion/ but I have since closed the window. If I figure out which isotopes it was I'll let you know. Was there something interesting in there?

    You asked why I care, I was told that mercury and some other element (or isotopes of) have a suprising high cross section under the right circumstances, though I don't know if this shows up in the models. So I was curious to look. Also I have a hobby interest in in fusion especially Philo Farnsworth's stuff.

    I'll get busy with those PDFs. Thanks for all the info.
  15. Apr 3, 2015 #14
    The plasma formulary: http://www.nrl.navy.mil/ppd/sites/www.nrl.navy.mil.ppd/files/pdfs/NRL_FORMULARY_13.pdf has empirical fits for fusion reactions relevant to energy applications.

    Energy balance is key to a successful fusion reactor. The rate that heat is produced in fusion must be balanced by all sources of energy loss. Energy is created via fusion which is a nuclear processes. Energy is lost radiatively (which are atomic processes) and due to plasma conduction and convection.

    In-order to understand why high Z ( atomic number) materials like Hg aren't consider for nuclear power applications you have to look at the atomic physics. consider bremsstrahlung radiation. The rate at which a material radiatively cools via bremsstrahlung scales with the square of the ion charge. Materials, like Hg, that have large atomic numbers will rapidly cool radiatively. A fully stripped Hg ion, which has a charge 80 times larger than hydrogen, will radiatively cool roughly 6,400 times faster than hydrogen. So all other things being equal sustaining a Hg fusion reaction will be 6,400 times harder than sustaining a hydrogen (D-T) reaction.

    I admit that this is a simplification, and there are a multitude of other effects that one has to consider. They wont change the main point. Sustaining a Hg fusion reaction is significantly more challenging that sustaining a D-T reaction.
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