Reaction Types, Cross-Sections

First, a short overview over the most important reactions useful for laser-induced neutron production is given.

1. (γ,n) reactions: If a γ-quantum interacts with an atomic nucleus, it can

be absorbed and its energy will be used to excite the nucleus. In the case where the excitation energy is higher than the nuclear binding energy, one or more neutrons (or protons, depending on the nuclear structure) is emitted from the core. In general, the threshold energy required for this to happen is the nuclear binding energy for the weakest bound particle, which ranges from 1.7 MeV in the case of 9_{Be over ∼ 15 MeV for very stable (even-even} paired) light nuclei like carbon to _∼ 8 MeV for heavy nuclei. This energy is not equal to the well known mean binding energy per nucleon (Fig. 3.1) especially for light elements, since some valence-nuclei might be less tightly bound. A broad resonance (the giant dipole resonance) in the cross-section is located at _≈ twice this energy. The peak cross-sections for this kind of reaction range from _∼ 2 mbarn in the case of light nuclei (deuterons, 9_{Be) to ∼ 800 mbarn for} 208_{Pb. In a laser-plasma experiment, the number} of γ-photons emitted at these high energies is relatively small due to their Boltzmann-like spectrum. Hence, (γ,n)-reactions give rise to a moderate amount of low energy neutrons. Lacking an accurate γ- or electron-, yield- and temperature-measurement, these neutrons could not be attributed to a particular source and were therefore not further treated.

Figure 3.1: Binding energy / nucleon (red line) and the threshold for the (γ,n)-reaction (dots) for a selected number of stable nuclides as a function of the atomic number Z. The scale for the threshold values is plotted at the right side. Dots arranged in a vertical column belong to a series of isotopes of one element. The threshold values are color coded separately for even-even (blue), even-odd (magenta), odd-odd (cyan) and odd-even (green) isotopes (data taken from [38]).

In analogy to these (γ,n)-reactions, also the emission of x neutrons can be triggered in (γ, xn) reactions. Since here roughly x-times the binding en- ergy has to be transferred to the nucleus, the threshold for these reactions

increases in approximately equidistant steps. This makes these reactions useful for determining the temperature of the hot γ-spectrum.

2. (p,n), exchange or stripping reactions: The excitation energy necessary

for the emission of a neutron can also be supplied by the interaction of an ion with the nucleus. The simplest projectile is a proton, which can either be captured into an energetically favorable state and release its binding energy or simply knock out a neutron from the nucleus by its momentum transfer. Heavier ions can exchange nucleons with the target and therefore reach an energetically more stable configuration, which can also lead to the freeing of neutrons. In the case of two particles in the exit channel, the neutron spectrum is monoenergetic for a given projectile energy and neutron emission angle. However, since the angle and energy spread of laser-emitted particles is large, only strongly exothermal reactions (Q Eproj.) will yield roughly monoenergetic neutrons. Which process takes place in a particular case depends on the combination of target, projectile and momentum transfer. The cross-sections for these processes are in the range of 100 mbarn up to one barn and therefore quite large.

0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 56 Fe(p,n)56Co d(d,n)3He d(p,n)2p to ta lc ro ss e ct io n [b a rn ] energy [MeV]

Figure 3.2: Neutron production cross-section for the reactions d(d,n)3_{He, d(p,n)2p and} 56_Fe(p,n)56_{Co [39].}

3. Fusion reactions: For light nuclei, the fusion reactions can be a source of

monoenergetic neutrons, since they sometimes fulfill the necessary criteria of high energy release combined with low threshold. Some of them are a special case of exchange or stripping reactions. As can be seen from Fig. 3.1, light nuclei have a low binding energy per nucleon. By fusing together, they can obtain a higher binding energy, which is equivalent to a net energy release.

Two fusion reactions are particularly important for laser plasma interaction studies:

(a) d-d fusion:

2_{H +}2_H

−→ 3He (0.8MeV) + n (2.45MeV), (3.1)

and its equivalent reaction 2_{H +}2_H

−→ 3H (1.0MeV) + p (3.02MeV). (3.2)

Also two endothermic reactions in a d-d collision shall be mentioned here, but they play only a minor role for the overall neutron production.

2_{H +}2_H

−→ n + p +2H₋2.22MeV, (3.3)

2_{H +}2_H

−→ 2n + 2p₋4.44MeV (3.4)

with a threshold of 4.44 MeV and 8.89 MeV, respectively. (b) d-T fusion:

2_{H +}3_H

−→ α(3.52MeV) + n (14.07MeV). (3.5)

The competing reaction 2_{H +}3_H

−→ 5He +γ+ 16.7MeV (3.6)

(3.7) has a very small cross-section due to the stability of the 4_{He nucleus} produced in the first case. Also in this case there exist two endothermal neutron production reactions

2_{H +}3_H

−→ n + p +3H₋2.22MeV (3.8)

2_{H +}3_H

−→ 2n +3He₋2.99MeV (3.9)

with a threshold of 5.56 MeV and 7.46 MeV, respectively.

The cross-section of the reactions 3.1,3.2,3.5 are plotted in Fig. 3.3. While the first two cross-sections are nearly equal, the tritium cross-section exceeds the others by almost two orders of magnitude. Using laser generated ions of energies below 1 MeV, by switching to d-t fusion reactions the neutron output could be boosted by the same factor.

However, due to the radioactivity of tritium, experiments using the (d,T) fusion reaction were not performed in this work. The reaction3.2 took place in the experiments too, but it was not looked at. Therefore only reaction 3.1

10-3 10-2 10-1 100 101 1x10-6 1x10-5 1x10-4 1x10-3 1x10-2 1x10-1 1x100 1x101

deuteron energy [MeV]

cr o ss -s e ct io n [b a rn s] d(d,n)3He d(d,p)T d(T,n)4He

Figure 3.3: Fusion reaction cross-section for the d-d and d-t fusion reactions (data taken from [39])