4.1 The Slope Method
4.1.1 Sewing by Ratio
The data-retrieval and application of the ratios in the slope-method in this work follow the steps presented in the flowchart in fig. 4.1.
Figure 4.1: Flow-chart diagram for the Slope method.
The intensity-datapoints NDj are acquired by projecting down onto the intensity-γ-ray energy plane for the 0+ g.s. and first excited 2+ state diagonals in the first genera-tion matrix as shown in Fig. 4.2 by a terminal-interactive tcl script through MaMa [27].
width along the excitation energy axis, while the integration along the γ-ray axis are per-formed over an interval whose start -and endpoint values are decided by a linear interpo-lation as a function ofEγ displayed as the diagonals in the first generation matrix in Fig.
4.2.
Figure 4.2: Illustration of the integration-bands formed from a linear interpolation in the first generation matrix for 144Nd.
This projection is done iteratively from around 4 MeV toSn where the red, or-ange and green lines indicate integration start -and end respectively. The oror-ange line rep-resents the common border of the integration bands and are half the level space difference.
The average value of the 100 keV excitation energy bins hExi are taken as the ideal value of the centroid value of theγ-ray energy Eγ. This decision was done to match the theoret-ical value whose discrepancy is due to method of extraction, namely MaMa curve-fit and detector variance. The ratio described in 4.6 also shows the value to be of less importance as the Eγi3 ratio becomes near unity both for higher energy and for the low separation of the ground state and first excited state, an example of the difference between the expected ideal case vs projection is shown for 144Nd in Fig. 4.3.
CHAPTER 4. EXTRACTING THEγSF AND NLD: THE OSLO AND SLOPE METHOD 4.1. THE SLOPE METHOD
Figure 4.3: The expectedγ-ray energy given by Eγ =Ex and Eγ =Ex−E(2+) compared to the actual centroid of the integrated number of counts for the two diagonals.
The ratio will be discussed more in the following chapter and is strongly de-pendent on the intensity. In short due to the near constant ratio of the spin-distribution over the quasi-continuum and the short energy-gap between the g.s. and first excited state which creates a ratio of theγ-ray Eγ energies near unity relative to the next itera-tion i.e. a linear trend. This becomes even more apparent for the two higherA nuclei of neodymium as the gap shortens between the two states. The γ-ray energies used for the ratio method are defined as the average excitation energy between bini and i+ 1:
Ex(i+ 1)−Ex(i)
2 =hEγ(i, i+ 1)i. (4.9)
The spin-distribution is calculated using an expect script interacting with robin [27] by looping over the same excitation bins as defined in Eq. 4.9 where the average excitation energy hExias input and retrieving the percentage of 0+ g.s. and 2+ first excited state spins with python from the distribution files written for each excitation bin. Robin offers 4 options for the spin-distribution cut-off formula described briefly in section 2.2.1. The ratio of the 2+ to 0+ g.s. distribution as a function of the average excitation energyhExi is shown in Fig. 4.4 below and will be discussed in more detail in the next chapter.
Figure 4.4: The fraction of the spin distribution that feed the final state, denoted by pD0+
and pD2+. The ratio is given for each of the spin cut-off forumla.
The preliminary γSF of γ-rays feeding the g.s. and first excited state calculated by Eq. 4.4 are shown in Fig. 4.5
Figure 4.5: The intermediate strength function ofγ-rays feeding the ground state and first excited state at 696 keV of 144Nd.
CHAPTER 4. EXTRACTING THEγSF AND NLD: THE OSLO AND SLOPE METHOD 4.1. THE SLOPE METHOD
The calculations of the normalized f0+,2+(Eγ(i)) strength values for each exci-tation bin were performed by creating a midpoint value hEγi,i+1i between the two γ-ray energiesEγi for the strength values of the 0+ and 2+ states for excitation bins i and i+ 1 respectively as shown in Fig. 4.6.
Figure 4.6: The initial point acts as a reference for all other iterations that follows. The ratio will only increase as the intensity ofγ-rays feeding the g.s. and first excited states fall with increasingEγ.
The iteration then proceeds after pulling the two values of f0+,2+(Ei+1) up by multiplying the intermediate values with the ratio R to obtain a normalized and functional form as shown in Fig. 4.7. Hence the procedural name of sewing the values together as a method to get theslope of the γSF:
f0+,2+(Ei+1)norm =R×f0+,2+(Ei+1). (4.10)
Figure 4.7: After the two initial point-pairs have been sewn together the new ratio is ap-plied for the next mean value pfEγ. The real-case analog is shown in Fig. 4.9.
The ratio increases as an exponential function as seen for 144Nd in Fig. 4.8. As the intensity decreases with excitation energy the distance between the consecutive pairs of intermediate γSF-values increases as the previous pairs are sewn up as shown in fig. 4.7.
Figure 4.8: The ratio plotted as a function of excitation energy for 144Nd.
The result can be seen by comparing the two figures of the sewn and non-sewn
CHAPTER 4. EXTRACTING THEγSF AND NLD: THE OSLO AND SLOPE METHOD 4.1. THE SLOPE METHOD
γSF pairs for 144Nd in Figs. 4.9, 4.10
Figure 4.9: The intermediateγSF of 144Nd plotted as point-pairs of the g.s. and first ex-cited state. The title states excitation energy but as explained above the ideal case of Ex=Eγ is assumed.
Figure 4.10: The final sewn slope of the 144NdγSF is obtained by applying the ratio over all bins.