Cross sections of nuclear reactions

The probability that a nuclear reaction may occur is given by the cross section of the reaction, which is comparable with the rate constant of a bimolecular chemical reaction. Considering the general equation for a binuclear reaction 

Cross sections are defined only for certain nuclear reactions. For example, is the cross section for the (n, y) reaction of the nuclide A, and is the absorption cross section of the nuclide A in which all absorption processes of certain projectiles, in particular neutrons, in A are summarized, independently of the kind of reaction they are inducing. Another quantity that is often used is the total cross section 

If scattering processes are left out of consideration, is equal to the absorption coefficient ft for certain particles or photons. 1 is then equal to the mean free path of the particles or photons in the absorber. If absorption and scattering are taken into account, the absorption coefficient fi is given by the total macroscopic cross section 

All cross sections depend on the energy of the particles or photons. This energy dependence cr = f( ) is called the excitation function, because the energy of the incident particle or photon is transferred to the nucleus as excitation energy. The dependence of the neutron absorption cross section on the neutron energy is shown schematically in Fig. 8.6. Even neutrons with kinetic energies corresponding to liquid-air temperature Ea 0.01 eV) are able to enter a nucleus without any 

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Emanation Gaseous products of radioactive decay, in particular radon Excitation functions Cross sections of nuclear reactions as a function of the energy of the projectiles... 

Gesamtdrehimpuls- Quantenzahl, total angular momentum quantum number 4. Gesamtquerschnitt von Kernprozessen, total cross section of nuclear reactions 14. Gesamtwirkungsquerschnitt, Messung, total cross section, measurement 39-g-g-Kerne, erste angeregte Zustande, even-even nuclei, first exited states 320. 

Due to the complexity of nuclear forces, predictions from nuclear model calculations for the cross sections of nuclear reactions and their dependence on energy and detection angle are difficult, and even if it is possible, they are not precise enough for analytical purposes. Thus, one has to rely on measured data. There are many published experimental cross-section data in the basic literature on nuclear physics, but usually the same problem occurs again neither are the experimental conditions the same, nor are the precisions good enough for NRA. For depth profile measurements, the knowledge of precise resonance parameters is crucial, and often the published experimental data have to be remeasured to fulfill the requirements of the technique. 

Because the cross-sections for nuclear reaction are usually lower than the cross-sections for elastic scattering of projectiles used in RBS or in elastic recoil detection analysis (ERDA), higher currents must be used to obtain comparably high intensity in... 

Rhenium-186 is usually produced by the 185Re(n,y) reaction using a nuclear reactor. The cross section of this reaction is 1.12 0.02 x 10 26m2 (112 + 2... 

Tc is available through the /l -decay of Mo (Fig. 2.1.B), which can be obtained by irradiation of natural molybdenum or enriched Mo with thermal neutrons in a nuclear reactor. The cross section of the reaction Mo(nih,v) Mo is 0.13 barn [1.5], Molybdenum trioxide, ammonium molybdate or molybdenum metal are used as targets. This so-called (n,7)-molybdenum-99 is obtained in high nuclidic purity. However, its specific activity amounts to only a few Ci per gram. In contrast, Mo with a specific activity of more than in Ci (3.7 10 MBq) per gram is obtainable by fission of with thermal neutrons in a fission yield of 6.1 atom % [16]. Natural or -enriched uranium, in the form of metal, uranium-aluminum alloys or uranium dioxide, is used for the fission. The isolation of Mo requires many separation steps, particularly for the separation of other fission products and transuranium elements that arc also produced. 

The low cross-section of the reaction of I(n,p) I with fast neutrons (cf Table 2.3) and a low abundance of neutrons with energies higher than 9MeV, which are needed for this reaction, in the neutron spectrum of a nuclear reactor result in a detection limit which is not sufficient for iodine determination in most types of foodstuffs, even if an RNAA procedure is applied. However, this reaction, which is completely independent in relation to the reaction of I(n, ) I with thermal and epithermal neutrons may be useful for cross-checking results in analysis of foodstuff samples with higher iodine contents, using the so-called self-verification principle in NAA (Byrne and Kucera, 1997). Detection limits of various NAA modes, which were achieved in the authors laboratory are compared in Table 2.4. 

Atoms. Strictly speaking, all the atoms formed via the nuclear recoil methods are unusual species because they all possess kinetic energies far in excess of those that could be formed by conventional chemical means (5-12). On the qualitative side, such hot atoms will open various reaction channels that are not possible for their lower energy counterparts. On the quantitative side, the cross sections of various reactions for the hot atoms are normally very different from those for the thermal atoms. An additional unique feature of the atoms formed via nuclear recoil techniques is that they invariably include radioactive isotopes and, therefore, they are capable of functioning as tracers for their subsequent interactions. 

The probability that any nuclear particles will interact with stable nuclei is given in terms of the cross-section of the reaction. Its unit of measurement is the barn (1 X 10 cm ), and is indicative of the order of magnitude expressed for the cross sectional area of nuclei. 

The danger of errors in the experimental determination of the thermal neutron fluence caused by other nuclear reactions is small. Co is also produced in stainless steels by (n,p) reaction from Ni and by (n,a) reaction from Cu, with both reactions being induced by fast neutrons. Since the cross sections of both reactions are very small (5 10 cm and 4.2 10 cm respectively), no interference is to be expected in light water reactors when monitor materials of the usual purity are used. Due to the 5.28-year halflife of Co, details of the irradiation history of this monitor material can only be neglected at gross irradiation periods of up to one year for longer periods, interruptions in the operation history of the plant, such as shutdowns, load reductions etc., have to be corrected for. 

The starting nuclide (deuterium) is present in natural water with an isotope abundance of 0.015%. An additional formation of during reactor operation by the nuclear reaction H (n, y) can be neglected because of the small neutron capture cross section of this reaction. Even if one does not take into account the water exchange which takes place during plant operation, an increase in the natural concentration of only 0.002 to 0.0035% would occur during one fuel cycle such an enhancement is too small to influence production to any measurable extent. The thermal neutron capture cross section for the formation of from amounts to about 3.6 10 cm (MaxweU spectrum). In the PWR primary coolant, between 40 and 150 GBq per fuel cycle are produced by this reaction, depending on the primary coolant volume, the water volume of the active zone in the reactor core and other parameters. 

In the natural nitrogen isotope mixture " N shows an abundance of 99.64% the cross section of this reaction is about 1.81 10 cm. Because of the usually very low levels of dissolved nitrogen impurities in the eoolant, the reaction is in most cases the source of about 90% of the total C production in the plant outside the nuclear fuel. The (n, y) reaction with the C content of the coolant impurities, as well as release from the C inventories in the fuel and the cladding do not contribute to the inventory of the coolant to an extent worth mentioning. 

Activation analysis is a method that in some cases allows one to determine small amounts of nitrogen, of the order of 1 pg or less. It consists of irradiating the sample in a suitable generator of particles or rays, most frequently in a nuclear reactor, and measuring the decay of the new radioactive nuclides produced. The radioactivity A induced in such a manner can be calculated from equation (3), where N is the number of target atoms, a the cross section of the reaction, 0 the irradiation flux, t the irradiation time, and A the decay constant of the new nuclide . 

As we have seen, the vector properties of molecular collisions offer much richer information than that provided by scalar properties, such as the total cross-section of a reaction or the energy content of the reaction products. To illustrate this point, consider a simple atom-transfer reaction, which will be abstractly written as A -f BC AB -I- C. For this process, we can readily identify four vectors. These are the initial relative velocity v of the reagents (A, BC), the final relative velocity v of the products (AB, C), the initial rotational angular momentum of the reagent molecule BC, denoted by j, and the final rotational angular momentum of the product molecule AB, denoted by j. Here we have assumed, for simplicity, that no photons are emitted or absorbed in the collision process, and that electronic or nuclear spin angular momenta are non-existent or are randomly oriented and do not couple to other angular momenta present. A simple example of such a case would be the atom-transfer reaction O -F CS CO + S. 

J. W Mayer, E. Rimini. Ion Beam Handbook for Material Analysis. Academic Press, New York, 1977. Useful compilation of information which includes Q values and cross sections of many nuclear reactions for low-Z nuclei. Also has selected Y yield spectra and y-ray energies for (p, y) reactions involving low to medium-Z nuclei. 

The energy of the y-rays is indicative of the isotope present, and the intensity of the y-rays is a measure of the concentration of the isotope in the sample. The limitation of this method is that, in order to have a nuclear reaction, the repulsive Coulomb barrier has to be overcome. For incident particles of energy up to 3 MeV, the only accessible elements are the light elements with Z< 15 the cross-sections of the remaining elements become rapidly negligible. 

Fig. 4. Comparison of hydrogen content measurements. The curves show IR absorption measurements, using absorption cross sections of Brodsky el al., and Fang el al., whose absolute calibration was done by NRA, and total hydrogen measurements using 15N nuclear reaction. Note the discrepancy that arises at high hydrogen pressures, because of the presence of hydrogen that is not infrared active. (Reprinted with permission from the American Institute of Physics, Ross, R., Tsong, I.S.T., Messier, R., Lanford., W., Burman, C (1982). J. Vac. Sci. Tech. 20, 406.)...

Detailed information about specific nuclear reactions at different stages of stellar evolution and the measurement of nuclear reaction cross-sections is given (together with a good overview of the whole of astrophysics) in... 

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