3. Methodology

### 3.1 Theoretical treatment

### 3.1.1 Neutron emission anisotropy

The well-known nuclear fusion reaction D (d,n)He<sup>3</sup> produced by a parallel monoenergetic beam of high-energy deuterons propagating along Z-axis at its interaction with an optically "thin" target (a cloud) of deuterium gas or low temperature deuterium plasma (i.e., when its thickness is much less than the mean free path of deuterons in the cloud) is characterized in the laboratory system of coordinates [20, 21] by the following formula:

$$Q = \frac{4}{3}E\_n - \frac{1}{3}E\_d - \frac{2\sqrt{2}}{3} \left(E\_d \cdot E\_n\right)^{1/2} \cos\theta \tag{2}$$

where Q is the energy released in the reaction, Ed is the energy of a fast deuteron bombarding the deuteron target, En is the energy of a neutron, and θ is the angle in relation to Z-axis that neutron is emitted to.

This equation can be resolved, and it gives for neutron energy En:

$$\sqrt{E\_n} = 0.3535 \cdot \sqrt{E\_d \cdot \cos \theta} \pm \left\{ \left( 0.125 \cdot \cos^2 \theta + 0.250 \right) \cdot E\_d + 2.475 \right\}^{1/2} \tag{3}$$

For the angle θ = 90°, this formula takes the simplest form:

$$E\_n = \frac{3}{4}Q + \frac{1}{4} \ E\_d \tag{4}$$

For the parallel beam of, e.g., 200 and 500-keV deuterons propagating along Z-axis (the angle θ = 0°) and bombarding plasma, the data on neutrons energy that escape the plasma volume (i.e., a target) at some angles are depicted in Tables 2 and 3.


#### Table 2.

Energy of neutrons produced by 200-keV deuterons with the exit angle θi.


#### Table 3.

Energy of neutrons produced by 500-keV deuterons with the exit angle θi.

Figure 10 represents an angle distribution of the effective differential crosssection σeff of the reaction D(d,n)He<sup>3</sup> in the laboratory system of coordinate. This picture is valid again for the low-intensity parallel monoenergetic beam of deuterons of energy Ed = 500 keV interacting with a "thin" target of deuterium plasma of relatively low temperature [20, 21]. Neutron stream density (fluence) is proportional to the effective differential cross-section σeff. Now, we can define the neutron anisotropy for the particular plasma device as a ratio of fluencies obtained in different directions:

$$A(\theta\_i) = \varphi(\theta\_i) / \varphi(\mathfrak{M}^\circ) \tag{5}$$

where A(θi) is the anisotropy of neutrons emitted at an angle of θ<sup>i</sup> to the direction of the beam of fast deuterons (i.e., to Z-axis of a DPF oriented from its anode), φ(θi) is the fluence of neutrons emitted at an angle of θi, φ(90o ) is the fluence of neutrons emitted at an angle of 90<sup>o</sup> , and various i are the subsequent positions where anisotropy is calculated and/or measured. From the next section, one can get that each i denotes the angle that corresponds to the measuring position location.

It gives for the data on anisotropy of neutron streams at various angles normalized to the value at 90° that produced by the beam of 500-keV deuterons at its interaction with a low-temperature deuterium plasma the following values (Table 4).

Thus, the theoretical angular distribution of neutron intensity produced in an "optically" thin deuterium gas (or low-temperature plasma) target by a

#### Figure 10.

A graph of the effective differential cross-section σeff for neutrons taking part in the reaction D(d,n)He<sup>3</sup> that is presented for the laboratory system of coordinate in dependence on the angle. The diagram is calculated for the case of the mono-energetic beam of deuterons having energy Ed = 500 keV and interacting with a so-called "thin" target of deuterium plasma of fairly low temperature.

Taxonomy of Big Nuclear Fusion Chambers Provided by Means of Nanosecond Neutron Pulses DOI: http://dx.doi.org/10.5772/intechopen.89364


Table 4.

Anisotropy coefficient on the exit angle normalized to the value at 90°.

low-intensity parallel mono-energetic beam of 100-keV deuterium ions as test particles (usually for the DPF deuteron spectrum obtained in various conditions, different authors give a figure in the range of hundreds KeV—see e.g., [8, 22]) looks similar to the eight digit (see Figure 11 plotted for the monoenergetic deuterons of 100-keV energy). In a center-of-mass system, it is symmetrical, whereas in a laboratory coordinate frame, it is slightly shifted in the direction of the beam propagation with A(θi) ≈ 2.0 and 1.5 for 0 and 180° correspondingly.

However, one has to take into account that the main part of neutrons generated in DPF is produced by gyrating deuterons [8, 22]. These particles escape pinch plasma (target) at a certain effective angle. So, it must give for the direction of Z axis the lower values of anisotropy counted for 100-keV deuterons: A(θi) ≈ 1.7 and 1.2 for 0 and 180°, respectively.

Moreover, the spectrum of fast deuterons in DPF devices is not monoenergetic: it extends till MeV range following the power law with a peak at hundred keV [8]. As it is known, the deuterons generating neutrons are captured for a certain time by self-produced magnetic fields and then fly out from the pinch under an appreciable angle as it was mentioned above. Besides, this flow of deuterons is very dense and intense (so it may be better characterized as a fast-moving cloud—high-energy and almost relativistic plasma jet) [8, 22].

All these features must result in the obtained experimental data for neutron anisotropy and spectra in smoothing of the pictures compared with the aforementioned theoretical ones. One may expect that the energy of deuterons producing neutrons in a DPF will occupy an energy range in the interval between the abovementioned values (i.e., for deuterons energy distributions with their peak energy somewhere from 100 till 500 keV).

#### 3.1.2 Time-of-flight spectral measurements

As it is well known [20, 21], the TOF technique converts a temporal behavior of the ns pulse of the neutron emission reflected in the pulse shape for the PMT + S positioned at the close vicinity to the generator into the pulse shape reproducing

Figure 11.

Theoretical angular distribution of neutron intensity produced in an "optically" thin gas target by a lowintensity parallel monoenergetic beam of 100 keV deuterium ions: in a center-of-mass system (a) and in a laboratory coordinate frame (b).

spectral characteristics of neutron radiation when the fast probes is moved to a certain distance (the speed distribution of particles transfers into the space one).

Time-of-flight of neutrons measured by means of PMT + S can be transformed into the energy distribution of the neutrons producing the neutron pulse. Their maximum may be expressed by a ratio (6) presented in books [20, 21]:

$$E\_{MeV} = (\text{72.24 } l\_m/t\_{ns})^2 \tag{6}$$

In this formula, EMeV is neutron energy in (MeV), lm is a distance in (m), and tns is a flying time interval in (ns).

To transform temporal behavior of a neutron pulse in the OT into spectral distribution of neutrons by their energy values in the OT, the distances l from the source till the PMT + S for the observation of spectra must be much longer compared with the size of the neutron pulse in space (in our case l >> 10 cm).

### 3.2 Neutron activation techniques application for the "clean-room"conditions

The first stage of the characterization experiments is the investigation of the angular characteristics of neutron yields of the PF-6 device itself by activation methods. General arrangement of the two stands with SACs and other activation detectors as well as with two stands containing PMT + Ss-related equipment is shown in Figure 9 in the positions of a "clean-room" condition (a). Note that there is a difference in heights of the positions of PMT + Ss and SACs: the PMT + Ss are situated in the plane of Z-axes of the PF-6 and PF-1000U chambers (that are directed horizontally in relation to the floor) but the SACs are placed 70 cm higher.

This taxonomy of the PF-6 device was provided in the most "clean" hall. However, it must be noted here that these conditions are not absolutely "clean." Indeed, the device itself has four capacitors filled with a castor oil (scatterers), the concrete floor and ceiling are presented, four coils of cables and four separating transformers are the elements of the PF-6 construction. All of these parts are rather bulky scatterers/absorbers. Due to these obstacles, we shall use the term "absolute" neutron yield in the subsequent text for the figures that will represent the values which are only correlate with another instruments' data in the dissimilar positions. Thus, these data are the "virtual" readings, or the "absolute" quantities with the identical but unknown standardizing coefficient.

During the experimental simulations, the neutron yield (YnTOTAL) is monitored using two SAC(s). The shots of the PF-6 device when the YnTOTAL magnitude was in the range of 10<sup>8</sup> –10<sup>9</sup> neutrons per pulse were taken into account only. Side by side with two silver activation counters (SACs), the activation detectors based on Be and Y were used. The Y neutron detectors give the data that are correlated with the neutron yield obtained from SAC quite well (see Figure 12).

The calibration measurements were produced during the successive 33 shots with SAC-1 and SAC-2. These probes were placed normally to the Z-axis of the chamber of the PF-6 device from its two opposite sides. Then keeping the position of the probe with SAC-1, the stand No. 2 was relocated along the way around the PF-6 device with seven different stops shown in Figure 13.

The procedure looks as follows. There are two cages. Each cage has a SAC placed 70 cm above the Z-axis of the PF-6 chamber (Figure 13b). The Z axis covers the axis of symmetry of the PF device. The X axis is horizontal, so the Y axis is vertical.

For any (x,y,z) point θ = arccos {z/ (x<sup>2</sup> + y<sup>2</sup> + z2 )} is the angle between vector x !, y !, z ! and the <sup>Z</sup> axis. This gives the spatial angles <sup>θ</sup>i° that differ from those

Taxonomy of Big Nuclear Fusion Chambers Provided by Means of Nanosecond Neutron Pulses DOI: http://dx.doi.org/10.5772/intechopen.89364

Figure 12.

Data of the absolute neutron yield measured during the experimental session with yttrium and silver activation counters averaged over 10–20 shots for each point of the graph.

Figure 13.

Presentation of the diagram of the immobile position of the stand No. 1 and seven successive locations of the movable stand No. 2: top view—(a) and side view—(b) during measurements of the angular distribution of "absolute" neutron yield by SACs for the PF-6 device in a "clean" situation.

"flat" angles αi° depicted in Figure 13a. From literature, one may find that neutrons produced in DPF and irradiated in all directions perpendicular to Z-axis (i.e., for the detector SAC-1 in its immobile place and for the detector SAC-2 in the position 4) has usually the energy peak near the value equal to 2.5 MeV for reacting deuteron having energy of hundred' keV [20, 21].

Step 1: At the beginning, the background (i.e., cosmic radiation) data are fixed by both SACs. It is repeated five times and the mean figures (usually it is �30–40 counts in the case) are computed for each collection of shots.

Step 2: On the assumption that for both SACs placed at 90° (i.e., perpendicular to the Z-axis of the PF-6 chamber—position No. 4 for a movable SAC-2), neutron yields are the same (with this device it was proved many times), and their individual sensitivities ratio is calculated as Yn1/Yn2(position 4) = Q1. This is a standardization coefficient Q1 for all subsequent calculations for different positions of a SAC-2 and for dissimilar neutron outputs in various collections of shots. For example, in this set of experiments, Q1 = 1.374.

Step 3: Data that were collected by the SAC-1 and by the SAC-2 in its different locations (Yn1(i) and Yn2(i)) are averaged over about 10–20 "good" shots during a single session, so the mean figures for each collection of shots Yn1(i)measured and Yn2(i)measured are calculated after deducting background values.

Step 4: At that time, the "actual" magnitudes of Yn2(i) of the SAC-2 are calculated with the detector sensitivity correction:

$$Y\_{n2(i)real} = Y\_{n2(i)measured} \times Q\_1 \tag{7}$$

Step 5: The remoteness r2i of the detector SAC-2 from the PF-6 chamber are computed for each of its positions; then a standardization procedure is produced according to the knowledge of the real distance lengths between the source and the SAC-2: rSAC-2. Subsequently by applying the r �<sup>2</sup> law, the factors for the neutron yield belonging to all locations of the detector SAC-2 are gained:

$$k\_i = \left(r\_{\text{SAC}-2i}/r\_{\text{SAC}-2(position\ 4)}\right)^{-2} \tag{8}$$

Step 6: By means of the multiplication of the above-mentioned "actual" neutron yields Yn2(i)real with these factors ki and regularizing them by the character for Yn1(i) in each collection of shots "i," a coefficient of the anisotropy A in the "clean" room is gained finally:

$$A = k\_i \left( Y\_{n2(i)real} / Y\_{n1(i)} \right) \tag{9}$$
