In the eV~keV energy, the neutron elastic scattering cross section is usually constant (if no resonance) which is determined by the potenial scattering. But, when the neutron energy goes to cold energy region, such as 10 meV or lower, the elastic scattering cross section will increase and be proportional to 1/v. Why? Thank you.
http://www.tpub.com/doenuclearphys/nuclearphysics39.htm
For a considerable number of nuclides of moderately high (or high) mass numbers, an examination of the variation of the absorption cross section with the energy of the incident neutron reveals the existence of three regions on a curve of absorption cross section versus neutron energy. This cross section is illustrated in Figure 1. First, the cross section decreases steadily with increasing neutron energy in a low energy region, which includes the thermal range (E < 1 eV). In this region the absorption cross section, which is often high, is inversely proportional to the velocity (v). This region is frequently referred to as the "1/v region," because the absorption cross section is proportional to 1/v, which is the reciprocal of neutron velocity. Following the 1/v region, there occurs the "resonance region" in which the cross sections rise sharply to high values called "resonance peaks" for neutrons of certain energies, and then fall again. These energies are called resonance energies and are a result of the affinity of the nucleus for neutrons whose energies closely match its discrete, quantum energy levels. That is, when the binding energy of a neutron plus the kinetic energy of the neutron are exactly equal to the amount required to raise a compound nucleus from its ground state to a quantum level, resonance absorption occurs.
Dear Jan
Thank you for your answer.
But, my question is not about the absorption that is with the σ∝1/v rule in low energy region.
Actually, my question is about the elastic scattering in cold neutron region.
Attached is the cross section of H1, you might see it is constant within the region of [0.1eV, 0.01MeV], but when En is
Dear Yigang Yang,
are you sure that your plot shows indeed the elastic scattering cross section? For low energies (thermal = 25 meV), the s-wave is dominating, and thus the scattering cross section becomes independent of the energy whereas the capture cross section increases with 1/v towards smaller energies.
The capture cross section of 1H is 0.33 barns at 25 meV (thermal energy). I assume that your plot shows cross sections in fm^2; then the low-energy capture cross section should appear with 1/v and a value of 33 fm^2 at 25 meV (consistent with your figure).
Peter Mohr
Dear Peter
Actually, the plot is the total cross section of H1 with neutron. But, as the capture cross section is quite small, the total cross section approximates the elastic scattering cross section in low energy region.
I update the plot. You may see the total cross section and capture cross section there.
Yigang
Consider the neutron as a deBroglie wave. The wavelength shrinks and therefore the extension of the particle proportional to 1/Sqrt(E)=1/v. This can be related to the geometric interpretation of the cross section of pi(r_neutron + r_target). However, I agree with Peter, the plot shows the absorption cross section. Yet, the same deBroglie wavelength argument should be valid for elastic scattering. Hope this helps.
Thank you all for the fruitful discussion.
Finally, I accept the explanation given by Professor Peter Mohr (we discussed through several messages).
If the temperature of target is 0K, then the elastic scatteirng cross section would be constant and usually (approximately) can be expressed as 4*π*R^2, where R is the radius of the target. This cross section is called "potential scattering" or "shape-elastic scattering" cross section.
If the temperature of target is not 0K (for example, room temerature,293K), the elastic scattering cross section will be increased by a 1/v vector, where v is the incident neutron's speed.
Recalling cross section is (outgoing flux* r^2*solid angle) / incident flux, where the numerator is kept constant(because the speed of target nucleus is larger than the incident neutron's speed) while the denominator is proportional to v, thus the elastic cross section is proportional to 1/v.
I suspect that the plotted elastic scattering data with 1/v tails are not "zero-degree" data, as are tabulated in ENDF and similar files (in which the elastic cross section do approach a constant value as the neutron energy goes to zero), but "Doppler-broadened" data, such as can be produced from ENDF data with processing systems such as NJOY. Doppler-broadened data are intended to be used in conjunction with the density of (assumed stationary) target atoms to get a correct reaction rate (or scattering rate). Doppler broadening takes into account the effective change in the energy dependence of neutron cross sections due to the motion of the target nuclei. At neutron energies considerably below the temperature of the medium, reactions take place not due to the motion of the neutrons, but due to atoms of the medium impinging on effectively stationary neutrons. The modification of the zero-degree data that is necessary to get correct reaction rates in the above-mentioned model becomes larger and larger as the neutron energy is lowered, and that is the reason that the broadened cross section rises as the neutron energy goes to zero.
Pr Muir is correct. The 1/v behaviour is due to themal motion of the target nuclei. See ref: B. Pritychenko, S.F. Mughabghab, Neutron Thermal Cross Sections, Westcott Factors, Resonance Integrals, Maxwellian
Averaged Cross Sections and Astrophysical Reaction Rates Calculated from the
ENDF/B-VII.1, JEFF-3.1.2, JENDL-4.0, ROSFOND-2010, CENDL-3.1 and EAF-2010
Evaluated Data Libraries.
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