The "Cross Section" is used to measure or report the probability of a reaction occurring. (simple analogy to describe cross section due to Peierls ) In general, the probability of a reaction is highly dependent on the energy of the projectile (as well as what the projectile is). More on calculations using these cross sections can be found on the fusor math page. Note that the data is generally given for a "monoenergetic" distribution of the projectile, while in real life, the projectiles have a distribution of energies. Some references give data for Maxwellian distributions with a specified peak, and others give cross sections for thermalized projectiles (in the case of neutrons) with an average velocity of 2200 m/sec (Maxwellian at "room temperature").
Much research has been done over the years to experimentally determine the reaction cross sections for various reactions of interest. Since this data is useful for weapons design, a lot of it was (or still is) classified, particularly in some energy ranges. The links below for tabular data are for data extracted using the web based interface to ENDF, which can be found at: http://www.nndc.bnl.gov/exfor/endf00.htm The text forms are tab delimited, the spreadsheet files have the same data, with the addition of a VLOOKUP formula that finds the matching crosssection for a given energy.
D(d,n) Deuterium Target, deuteron projectile. (table text 13K) (excel 4.0 sheet 45K) (plot GIF 5K)
T(d,n) Tritium Target, deuteron projectile (table text 2K) (excel 4.0 sheet 10K) (plot GIF ground state 5K) (plot GIF 1st level 5K)
For many uses (planning and modelling), an analytical expression that represents the general shape of the curve is more useful. In this case, the 1998 NRL Plasma Formulary, pp 44 and 45, has a empirical approximation for these curves given as:
sigma (in barns)= (A5 + A2/((A4-A3*E)^2+1))/(E*(exp(A1/sqrt(E))-1)
E is the particle energy in keV and where the An are the "Duane coefficients" are given as:
| D(d,p) | D(d,n) | T(d,n) | 3He(d,p) | T(t,2n) | 3He(T,pn),etc. | |
| A1 |
46.097
|
47.88
|
45.95
|
89.27
|
38.39
|
123.1
|
| A2 |
372
|
482
|
50200
|
25900
|
448
|
11250
|
| A3 |
4.36E-4
|
3.08E-4
|
1.368E-2
|
3.98E-3
|
1.02E-3
|
0
|
| A4 |
1.220
|
1.177
|
1.076
|
1.297
|
2.09
|
0
|
| A5 |
0
|
0
|
409
|
647
|
0
|
0
|
However, when comparing the fit of this approximation against the data from ENDF, I found that the fit of the second reaction (D(d,n)) could be made much better at higher energies by using the coefficients in the following table (using Excel's solver to minimize the squared difference from ENDF data). I haven't researched the origin of the coefficients in the formulary, there may be some numerical problem (i.e. a high sensitivity to some small insigificant change in a parameter) or just a different data set used to derive the numbers.
| D(d,n) | Revised | Original |
| A1 | 56.962 | 47.88 |
| A2 | 482.28 | 482 |
| A3 | 5.54E-6 | 3.08E-4 |
| A4 | 0.0894 | 1.177 |
| A5 | 0.2486 | 0 |
The original references cited in the formulary are:
G.H. Miley, H. Towner and N. Ivich, Fusion Cross Section and Reactivities, Rept. COO-2218-17, University of Illinois, Urbana, IL, 1974
B.H. Duane, Fusion Cross Section Theory, Rept. BNWL-1685, Brookhaven National Laboratory, 1972
nuc/sigma.htm - revised 6 January 2000, Jim
Lux
revised 6 May 2005, put new link in for ENDF site
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