3. Results and Discussion

Both, α particles and scattered protons reach the CR-39 type track detector, therefore, it is necessary to tell them apart using the track diameters as a distinguishing feature. We measured α particle tracks on a CR-39 type track detector using a standard ²⁴¹Am source emitting α particles at an energy of 5.49 MeV, as shown in Figure 2(a). The track diameters of α particles are near to 10 μm, which is consistent with the results of Hicks [Reference Hicks16], who etched the CR-39 type track detector with a solution of the same concentration and temperature as we did. According to Hicks’ result, the proton track diameters are smaller than α particle track diameters. Combining our results with Hicks’, we believe that there is a great difference in track diameter between protons and α particles, from Figure 2(b) it is obvious to distinguish the tracks with different diameters. Here, the elliptical tracks of protons are caused by the large angle of backscattering. As shown in Figure 1(b), the collected particles on the CR-39 type track detector arise from nine different incident spots. Therefore, particles directed to incident spots far away will have a large incident angle, which leads to the elliptical tracks.

Figure 2:

Tracks on a CR-39 type track detector. (a) 5.49 MeV α particle tracks from a ²⁴¹Am radioactive source after 3 h of etching. (b) Experimental particle tracks after 3 h of etching. (c) Experimental particle tracks after 1 h of etching. (d) Experimental particle tracks with the 5 um Al after 3 h of etching.

Figure 3 shows the number of detected particle tracks as a function of the track diameter with different etching times. The red line in the figure represents a double-gauss-curve fitting. For both etching times of 1 hour and 3 hours, there are two Gaussian peaks. The first peak is due to protons, and the second peak is caused by α particles.

Figure 3:

Statistical histogram of particle tracks after different etching times. (a) Particle tracks after 1 h of etching. (b) Particle tracks after 3 h of etching. The red line in the figure is the double-gauss-curve fitting for the track number as a function of its diameter.

Figure 2(b) and Figure 2(d) display the tracks without Al film and with 5 μm Al film after 3 h etching, respectively. We find that the number of tracks on a CR-39 type track detector with Al film is much less than that without Al film because most of the particles are protons and low-energy protons are filtered out by Al film, which is consistent with the results shown in Figure 3.

In addition, Figure 3 shows that when the etching time is 1 h, the proton track diameter and α track diameter overlap considerably; when the etching time is 3 h, the overlap area of the two tracks is small. This is due to the fact that the etching rates of proton and α particle tracks are different. As the etching time increases, the gap in track diameter will gradually increase. Therefore, it is more convincing to distinguish the two particle species with an etching time of more than 3 h.

However, there still remains some ambiguity because some particle tracks have a large range of mutual masking after an etching time of 3 hours. Therefore, we decided to calculate the yield based on an etching time of 1 hour.

As shown in Figure 3(a), because the proton track diameter and α track diameter overlap considerably, we finally choose the intersection of two Gaussian curves to distinguish two particle species. Thus, tracks with a diameter of more than 2.7 um are considered to be due to α particles. With this assumption, the relative systematic error is 7%.

As shown in Figure 1(b), for any region (for example, the red circle on the detector), on a CR-39 type track detector, the collected particles are the contribution of nine incident proton beams. We select nine regions on a CR-39 type track detector with uniform brightness and fewer bubbles near each spot, and then the equations are constructed as follows:

(2) $\begin{array}{}\sum _{j=1}^9\frac{{\Omega }_{i,j}}{4\pi }{\rho }_{i,j}{N}_{\alpha ,j}=N_{CR39}\left(i\right),i=\mathrm{1,2},\mathrm{...},9,\end{array}$

where N _α,j is the α particle yield corresponding to the j_th incident beam, N _CR39(i) is the number of α particles detected on the i _th CR-39 type track detector region, Ω _ij is the solid angle of the j _th proton beam to the i _th CR-39 type track detector region. ρ _ij represents the ratio of α particles that can escape from the target to the total amount of α particles that are produced.

In the thick target case, the change of reaction cross-section caused by proton energy deposition inside the target must be considered. The total reaction cross-section with the thick target is as follows [Reference Taulbjerg and Sigmund17]:

(3) $\begin{array}{}\sigma =\frac{1}{3}\left(\frac{1}{n}\frac{\text{d}N\left(E\right)}{\text{d}E}\frac{\text{d}E}{\text{d}R}+\frac{\mu }{n}\frac{\cos \theta }{\cos \phi }N\left(E\right)\right),\end{array}$

where n is the atomic density of the target, N (E) is the α particle yield per proton with the proton energy E, μ is the absorption coefficient of the target nucleus, θ is the angle between the normal direction of the target plane and the direction of the incident particle, φ is the angle between the normal direction of the target plane and the detector. The factor 1/3 corrects for the fact that for each reaction, three α particles are created.

In our experiment, the proton energy is 140 keV–172 keV, and the penetration depth into the boron target is about 1 um. The energy of α particles produced by p¹¹B reaction is around 1 MeV and 4 MeV. According to the SRIM code and the α spectrum with the incident proton energy of 165 keV [Reference Becker, Rolfs and Trautvetter10], we can safely assume that almost all α particles pass through 1 um thick boron and reach the CR-39 type track detector. Thus, the second term in formula (3) can be ignored.

Figure 4(a) shows the α yield per proton for different energies of the incident proton beams, where the red line is the curve fitting the experimental data with a standard deviation of 24%. Figure 4(b) shows the cross section of p¹¹B fusion, where the present data are obtained based on the formula (3) and the error of the cross-section was 28%. We can find that the resonance appears near 156 keV, and the measured resonance cross-section is 45.6 ± 12.5 mb. The position of the peak is slightly shifted to the left from the recognized resonance energy of 163 keV [Reference Oliphant and Rutherford6], and the value of the cross-section is consistent with the previous works [Reference Anderson, Dwarakanath, Schweitze and Nero8–Reference Becker, Rolfs and Trautvetter10, Reference Munch and Uldall Fynbo18].

Figure 4:

(a) The α yield per proton. The red line is the curve fitting. (b) p¹¹B cross-section (the data are from References [Reference Anderson, Dwarakanath, Schweitze and Nero8–Reference Becker, Rolfs and Trautvetter10, Reference Munch and Uldall Fynbo18]). Here, we have corrected Becker’s value by multiplying a factor of 2/3.

The main errors are due to: (1) the fitting error of 24% in Figure 4(a); (2) the statistical error of 12% caused by particles counting on the CR-39 type track detector; and (3) the systematic error of 7%.

The higher values of the first four cross-sections result from proton backscattering. As shown in Figure 3(a), the proton track overlaps with the α particle track in a large range, so some protons will be mistaken for α particles during particle counting. In addition, the first proton beam may not be perpendicular to the target but at an angle to the target plane, which will cause large backscattering and a high particle count.

The shift of the resonance peak is mainly due to the very limited data near the resonance peak and the use of a thick target. According to Munch [Reference Munch and Uldall Fynbo18], the resonance width in the lab system of 163 keV is 5.76 keV, while the energy interval of proton beams in our experiments is 4 keV, which means that there are too few energy points measured near the resonance peak. Therefore, this causes a large error in the fitting of the peak position, thus leading to the shift of the peak.

When comparing all these cross-sections, we notice all values deviate significantly from the measurement of Becker et al. [Reference Becker, Rolfs and Trautvetter10]. We suspect that this is due to the normalization problem. Becker believes that a correction factor of 2 should be used when calculating the cross-section because two out of three α particles can be detected for one fusion reaction with detectors covering a large solid angle, while others divide the total α particle yields by a factor of 3. Considering our experiment, we have chosen several regions in the CR-39 type track detector to count the α particles. For the image shown in Figure 2, its size is 238.9 μm × 183.77 μm, and the solid angle of the image relative to the closest incident point is ΔΩ ≈ 10⁻⁴ rad. It is suggested that the α particles produced by $p^{11}B$ reaction are isotropically in the center of mass system [Reference Becker, Rolfs and Trautvetter10], if the total α number produced in reaction is N _total, then the α number that can be collected in a region with a solid angle ΔΩ should be N _total × ΔΩ/4π. Considering that one reaction will produce three α particles, therefore, a factor of 3 should be divided when calculating the cross-section. In Figure 3(b) we have corrected Becker’s value by multiplying a factor 2/3.