Introduction to Nuclear Criticality Safety

REVIEW TOPICS PART V

Continuing . . .

VARIATION OF CROSS SECTIONS WITH NEUTRON ENERGY

Cross sections for neutron interactions are in many cases not constant, but vary with neutron energy. In the low energy region of most nuclei, the capture cross section varies at 1/√E, or as 1/v since the neutron speed v is proportional to √E (E is the neutron energy). The 1/v region is illustrated in Figure 3.5 of Lamarsh & Baratta (p. 65). In the same figure, a sharp peak, or resonance, is shown in the intermediate energy range. This characteristic of cross sections can be explained by considering a neutron incident upon a nuclide   (Duderstadt & Hamilton 1976, p. 24):

 CrossSectionNeutronEnergy.png

The energy available for such reaction in the center of mass (CM) energy is:

CMenergy.png

where . . .

 

Value: 3

Match the items.

The task is to match the lettered items with the correct numbered items. Appearing below is a list of lettered items. Following that is a list of numbered items. Each numbered item is followed by a drop-down. Select the letter in the drop down that best matches the numbered item with the lettered alternatives.

a. neutron mass

b. nuclear mass

c. neutron kinetic energy in the L system

 

The actual energy of the excited level of the compound nucleus is much higher due to the additional binding energy of the added neutron Eb.   If Ec + Eb is very close to a nuclear energy level of the compound nucleus , it can be expected that the probability for compound nucleus formation will be much larger than if Ec + Eb does not "match" this energy level.   Therefore, we expect that the cross sections for such compound nuclear reactions will exhibit sharp peaks, or resonances, at those neutron energies E for which this energy matching occurs.  

Resonances occur in cross sections because nuclei can be very hungry for neutrons of certain energies, and at the same time can find other neutrons quite distasteful. This is understandable in terms of the energy levels of a nucleus illustrated below (adapted from Bennet & Thomson 1988, Figure 2.11).

Match the numbers in the chart with the correct description.

EnergyResonance.png

Energy Levels and Resonance Absorption

 

Value: 6

Match the items.

The task is to match the lettered items with the correct numbered items. Appearing below is a list of lettered items. Following that is a list of numbered items. Each numbered item is followed by a drop-down. Select the letter in the drop down that best matches the numbered item with the lettered alternatives.

a. Eb, binding energy of a neutron in

b. Ground state of the target nucleus plus a neutron at rest

c. Ec, the reduced energy of the incident neutron

d. Excited states of

e. Ground state of compound nucleus

f. Resonance for this value of Ec

 

If the energy of the neutron is such as to produce the compound nucleus at or very near one of its excited states, as shown in the figure above, the probability of the interaction taking place is very high. This corresponds to a high value of the cross section.   If, on the other hand, the energy of the incident neuron is such as to produce the compound nucleus at some energy midway between two excited levels, the probability of the interaction is very low.   This corresponds to a low value of the cross section.  

In Summary: 

When the energy is just right, the cross section will be high.   Otherwise, it will be much lower.   The high part is called a (select the correct combination below):

HighPart.png

 

Value: 2

(Binding Energy + Kinetic Energy in CM)       =

 
 
 
 

The cross section for any reaction involving a compound nucleus will exhibit resonances. The only reaction that does not involve a compound nucleus is [_____] scattering, in which the neutron scatters off of the nucleus without ever penetrating it (as previously stated). Select the correct term below.

 

Value: 1

The only reaction that does not involve a compound nucleus is [_____] scattering, in which the neutron scatters off of the nucleus without ever penetrating it (as previously stated).

 
 
 
 

Under reasonable assumptions, it is possible to derive a formula for s around a resonance.   For example, the Breit-Wigner formula (Lamarsh & Baratta, equation 3.23) gives σ as a function of E, for E at and near a single-level resonance. Characteristics of nuclei of different mass numbers are summarized in the figure below (adopted from Figure 2-1 in Duderstadt & Hamilton 1976).

NucleiMassCharacteristics.png

Based on the figure above, the following general trends can be stated about the following reactions (select the correct values for each type of reaction):

 

Value: 2

Radiative Capture: The probability of radiative capture occurring increases as A [_____] and as E [_____].

 

Select the correct term combination.

 
 
 
 

 

Value: 2

Elastic Scattering: The probability of elastic scattering occurring increases as A [_____] and as E [_____].

 
 
 
 

 

Value: 2

Inelastic Scattering: The probability of inelastic scattering occurring increases as A [_____] and as E [_____].

 
 
 
 

Exceptions to these trends exist for the "magic" nuclei with nucleons known as "magic numbers" (these were discussed previously).


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