Study of Some Thorium Isotopes near to the Closed Shell (82 and 126) ^†

Abstract

Using the interacting bosons model-one (IBM-1), the nuclear structure of the thorium isotopes ²²⁴Th, ²²⁶Th, and ²²⁸Th were examined in this study, which are near from closed shell 82 and 126. By acquiring this element’s energy levels and comparing them to actual values, which provide an indication of these isotopes membership in a specific determination, it is possible to determine that ^224Th and ²²⁶Th belong to transition region between SU(3) and O(6) but ²²⁸Th belong to the SU(3) limit. The ratio of the fourth to the second energy level $E_4^{+}/E_2^{+}$ with other ratios $E_6^{+}/E_2^{+}$ and $E_8^{+}/E_2^{+}$, the order of practical levels, are first exam to determine the limit that belong. Using IBM program to find theoretical energy level and compared with practical one, also the agreement between the theoretical probability of electric transitions B(E2) through IBMT program was investigated. The IBMP program was used to study the surface potential of the nucleus, which provides insight into the deformation that occurs in the nucleus and from the contour lines deviation.

1. Introduction

The development of the theories of the nucleus, there are two different models. One model is the shell model developed by Mayer and Jensen which deals with single-particle motion of the constituent nucleons in a mean-field potential. The nuclear shell model holds up well near magic nuclei. The model, however, is unable to account for nuclei’s rotations and vibrations, which are seen in areas of the nuclear chart that are distant from the magic nuclei. The other model is the working of Bohr and Mottelson together. There are three primary patterns, deformed rotation SU(3), spherical vibration U(5), and $\gamma$-soft unstable O(6), are still used as standards to assess the structural integrity of nuclei [1].

The intermediate and heavy atomic nuclei can be well described by the interacting boson model-one (IBM-1). It reproduces the majority of the low-lying states of such nuclei by adjusting a number of parameters [2].

Many nuclear collective phenomena have been successfully addressed by the interacting boson model of Arima and Iachello. The fundamental concept is that neutron and proton bosons with zero spins and 2 spin respectively, can be characterized the low energy collective degrees of freedom in nuclei [3].

2. The Model (IBM)

The shell model and the collective model were the two main pillars of the nuclear theory up to 1970. Feshbach I. and Iachello F. presented the interacting boson model (IBM) in 1973, and in 1974, Arima A. and Iachello F. developed it, a new model based on a three approach, that is group theoretical or algebraic [4].

The IBM is a model that explains how low-lying, positive parity quadrupole-collective states behave when they are present in even-even deformed nuclei for medium and heavy one. There is no distinction between protons and neutrons in the IBM-1 model. Several extensions of the original IBM-1 framework have been developed, including the IBM-2, which distinguishes between protons and neutrons, and the IBM-3 and IBM-4, which can explain light nuclei [5,6].

They pretended that the shell model indicates that the low-lying collective states of such nuclei arise from interacting nucleon pairs coupled with angular momentum L = zero or 2 (s boson and d boson, respectively), with energies(s, d), where s is frequently equal to zero.

The (IBM) is founded on the following presumptions and is based on the spherical shell model and collective model of atomic nucleus [7].

• Bosons, or pairs of fermions, are used to describe pairs of active nuclear particles or holes that are close to “closed shells”.
• The overall number of bosons affects this model (N).

Where N = Nп + Nυ

Nп = proton boson number

Nυ = neutron boson number

Since they form an anti-symmetric state, identical nucleons have higher angular momentum in even numbers.

3.

The variety of shells that are depicted in the shell model are limited to just the straightforward s-shell (L = 0) and d-shell (L = 2) [8,9].

The Hamiltonian of the IBM-1 as a multipole expansion, grouped into different boson-boson interactions as [10,11].

$$\mathrm{Ĥ}=ϵ {\widehat{n}}_d+a_0\widehat{P}.\widehat{P}+a_1\widehat{L}.\widehat{L}+a_2\widehat{\mathcal{Q}}.\widehat{\mathcal{Q}}+a_3{\widehat{T}}_3.{\widehat{T}}_3+a_4{\widehat{T}}_4.{\widehat{T}}_4$$

(1)

where: ${\widehat{n}}_d=[d^{†}.\tilde{d}]$ the number of d bosons:

• $\widehat{P}=$ 1/2 $\left(\tilde{d} . \tilde{d}\right)-$ 1/2 $\left(\tilde{s} . \tilde{s}\right)$ the pairing operator for d and s bosons;
• $\widehat{L}=\sqrt{10} {\left[d^{†}\times \widetilde{d }\right] }^{ \left(1\right)}$ the operator of angular momentum;
• $\widehat{\mathcal{Q}}$ = ${\left[d^{†}\times \tilde{s}+s^{†}\times \tilde{d}\right]}^{\left(2\right)}+$ $\chi {\left[d^{†}\times \tilde{d}\right]}^{\left(2\right)}$ the operator of quadrupole;
• ${\widehat{T}}_3={\left[d^{†}\times \tilde{d}\right] }^{\left(3\right)}$ the operator of octupole;
• ${\widehat{T}}_4={\left[d^{†}\times \tilde{d}\right]}^{\left(4\right)}$ the operator of hexapole;
• $ϵ$ = $ϵ$d − $ϵ$s the boson energy.

The IBM for axially symmetric rotors, spherical and vibrators schematically describes the analytically solvable dynamical symmetries SU(3), U(5), and O(6) [12].

Hamiltonian of SU(3) limit is:

$$\widehat{H}=a_1\widehat{L}\widehat{L}+a_2\widehat{Q}\widehat{Q}$$

(2)

and for transition limit $\mathrm{O}\left(6\right)$ $\to \mathrm{S}\mathrm{U}\left(3\right)$:

$$\widehat{H}=a_0\widehat{P}.\widehat{P}+a_1\widehat{L}.\widehat{L}+a_2\widehat{Q}.\widehat{Q}$$

(3)

3. Electric Transitions

The absolute electric transition rates serve as a rigorous test for various theories in addition to being a sensitive aspect of nuclear structure. Most of the B(E2) values that are currently known were obtained using coulomb excitation. The following is a possible representation of the electromagnetic transition rates operators’ general form [13].

$${\widehat{T}}_m^l={\alpha }_2{\delta }_{l_2}{\left[d^{†}\times \tilde{s}+s^{†}\times \tilde{d}\right]}_m^2+{\beta }_l{\left[d^{†}\times \tilde{d}\right]}_m^l+{\gamma }_0{{\delta }_l}_0{{{\delta }_m}_0\left[s^{†}\times \tilde{s}\right]}_0^0$$

(4)

where: $l$ = 0,1,2,3,4,…, m = 0,1,2,3,4,…, and ${\gamma }_0, {\alpha }_2, {\beta }_l$ reflect the particular form of the transition operator and are parameters that specify the different terms in the related operators.

The electric operators of quadrupole transition expressed as [14]:

$${\widehat{T}}_m^{\left(E2\right)}={\alpha }_2{\left[d^{†}\tilde{s}+s^{†}\tilde{d}\right]}_m^{\left(2\right)}+{\beta }_2{\left[d^{†}\tilde{d}\right]}_m^{\left(2\right)}$$

(5)

where effective charge $e_B={ \alpha }_2$ and ${\beta }_2=\chi { \alpha }_2$

The electric quadrupole reduced transition probabilities B (E2) is defined as:

$$B\left(E2,l_i\to l_f\right)=\frac{1}{2l_i+1}{\left|<l_i‖{\widehat{T}}^L‖l_f>\right|}^2$$

(6)

4. IBM Potential Surface

The potential energy surface (PES) gives the nucleus a final shape and the Hamiltonian function $PE(N,\beta ,\gamma )$ is [8,9]:

$$PE\left(N,\beta ,\gamma \right)=\raisebox{1ex}{$〈N, \beta ,\gamma \left|H\right|N,\beta ,\gamma 〉$}\!\left/ \!\raisebox{-1ex}{$⟨N,\beta ,\gamma \left|N,\beta ,\gamma ⟩\right.$}\right.$$

(7)

The energy surface has been calculated as a function of boson number (N), deformation ($\beta$) and deviation angle ($\gamma$):

$$PE\left(N,\beta ,\gamma \right)=\frac{N{\epsilon }_d}{(1+{\beta }^2)}+\frac{N(N+1)}{{\left(1+{\beta }^2\right)}^2}\left(a_1{\beta }^4+a_2{\beta }^3\cos 3\gamma +a_3{\beta }^2+a_4\right)$$

(8)

where $a_i$ are the coefficients, $\beta$ is a total deformation of the nucleus. The shape is prolate if $\gamma$ = 0, and if $\gamma$ = 60, the shape becomes oblate [12]. $\gamma$ represents the divergence from the focal symmetry and correlates with the nucleus. The equations below provided the potential energy surface for three dynamical symmetries [15,16].

$$PE(N,\beta ,\gamma )\propto \left\{\begin{array}{c}U\left(5\right): {\epsilon }_{d}N\frac{{\beta }^2}{1+{\beta }^2}\\ SU\left(3\right): kN(N-1)\frac{\frac{3}{4}{\beta }^4-\sqrt{2}{\beta }^3\cos 3\gamma +1}{{(1+{\beta }^2)}^2}\\ O\left(6\right): k^{\prime }N(N-1){(\frac{{1-\beta }^2}{1+{\beta }^2})}^2\end{array}\right.$$

(9)

where $k\propto a_2 \mathrm{and} \acute{k}\propto a_0$ in Hamiltonian.

5. Results and Discussion

The isotopes of (²²⁴Th, ²²⁶Th, ²²⁸Th) energy levels were extracted using an IBM software program and the Hamiltonian as equation 1, using estimated parameters for transition determination SU(3) and O(6) for first two isotope but pure limit SU(3)for the last one were listed in

Table 1. The parameters in the program of the Hamiltonian.

The Isotopes	N	$\mathit{E}\mathit{P}\mathit{S}$ $\mathbf{i}\mathbf{n} \mathbf{M}\mathbf{e}\mathbf{V}$	$\mathit{P}.\mathit{P}$ $\mathbf{i}\mathbf{n} \mathbf{M}\mathbf{e}\mathbf{V}$	$\mathit{L}.\mathit{L}$ $\mathbf{i}\mathbf{n} \mathbf{M}\mathbf{e}\mathbf{V}$	$\mathit{Q}.\mathit{Q}$ $\mathbf{i}\mathbf{n} \mathbf{M}\mathbf{e}\mathbf{V}$	${\mathit{T}}_3.{\mathit{T}}_3$ $\mathbf{i}\mathbf{n} \mathbf{M}\mathbf{e}\mathbf{V}$	${\mathit{T}}_4.{\mathit{T}}_4$ $\mathbf{i}\mathbf{n} \mathbf{M}\mathbf{e}\mathbf{V}$	Chi Unit Less	SO6 Unit Less
224Th	8	0.0000	0.0013	0.0082	−0.0102	0.0000	0.0000	−0.1700	1
226Th	9	0.0000	0.0013	0.0082	−0.0069	0.0000	0.0000	−0.1700	1
228Th	10	0.0000	0.0000	0.0000	0.1500	−0.5490	0.0008	−0.0242	1

with the parameters in the program in MeV (mega electron volt) unit:

There are decrease in Q.Q value in minus sign with increase of boson number N for first two isotopes, because of increase of the number of neutrons which given unstable motion in nucleus and its properties is between rotation and gamma unstable. The last one has the properties of rotation motion only and the parameters of the Hamiltonian for SU(3) limit.

The energy levels for ²²⁴Th, ²²⁶Th, and ²²⁸Th that arise from these parameters are illustrated in Figure 1, Figure 2 and Figure 3 respectively, and they are in good accord with the findings of experiments for ground band. Many energy levels are ambiguous or unknown.

Figure 1 represents the energy levels for ²²⁴Th, which have practical energy levels for the ground band only, while through the IBM program, the results for the ground band and the beta band, the ratio of the energy levels are: $E_4^{+}/E_2^{+}=2.89, E_6^{+}/E_2^{+}=5.8$ and $E_8^{+}/E_2^{+}=10.7$, so expected limit for this isotope is SU(3). ²²⁶Th isotope looks like a ²²⁴Th for energy levels as Figure 2 and the ratio of energy levels close to these value too.

The ²²⁸Th isotope, it contains practical levels for the ground band, the beta band, and the gamma band as Figure 3. The energy ratios are equal to 3.13, 6.2 and 11.3 respectively and the results from the program were compared with practical value, which denoted to SU(3) limit. The congruence in the practical and theoretical values, this gives the validity of the results, and this congruence was only achieved by adding parameter to the Hamiltonian, which is called symmetry breaking.

Table 2. Experimental data of B(E2) and its coefficients for isotopes.

The Isotopes	Boson Numbers N	$\mathit{B}(\mathit{E}2;2_1^{+}\to 0_1^{+})$ e2b2	E2SD(eb)	E2DD(eb)
224Th	8	0.77614	0.143292	−0.423866
226Th	9	1.341716	0.188401	−0.557299
228Th	10	0.026489	0.057542	−0.040279

denoted to the value for exist value for the first electric transition probability from data sheet and their coefficients in IBMT program E2SD, E2DD.

Equations (6) and (7) can be used to determine the electric transition for the isotopes which calculated from IBMT program, and the results are shown in the

Table 3. Some electric transitions B(E2) for ²²⁴Th.

${\mathit{J}}_{\mathit{i}}\to {\mathit{J}}_{\mathit{f}}$	B(E2) e2b2
	Exp.	IBM-1
$2_1^{+}\to 0_1^{+}$	0.77614	0.7643948
$2_1^{+}\to 0_2^{+}$	-	0.0703174
$3_1^{+}\to 2_1^{+}$	-	0.8552228
$3_1^{+}\to 2_2^{+}$	-	0.7218213
$3_2^{+}\to 2_1^{+}$	-	0.0016745
$3_3^{+}\to 2_{1 }^{+}$	-	0.0008739
$4_1^{+}\to 2_1^{+}$	-	1.0741140
$4_1^{+}\to 2_2^{+}$	-	0.0249889
$6_1^{+}\to 4_1^{+}$	-	1.0869880
$6_1^{+}\to 4_2^{+}$	-	0.0542134

,

Table 4. Some electric transitions B(E2) for ²²⁶Th.

${\mathit{J}}_{\mathit{i}}\to {\mathit{J}}_{\mathit{f}}$	B(E2) e2b2
	Exp.	IBM-1
$2_1^{+}\to 0_1^{+}$	1.341716	1.7095460
$2_1^{+}\to 0_2^{+}$	-	0.1313972
$3_1^{+}\to 2_1^{+}$	-	0.0161234
$3_1^{+}\to 2_2^{+}$	-	0.1522753
$3_2^{+}\to 2_1^{+}$	-	0.0024018
$4_1^{+}\to 2_1^{+}$	-	2.4107670
$4_1^{+}\to 2_2^{+}$	-	0.0330782
$6_1^{+}\to 4_1^{+}$	-	2.4773540
$6_1^{+}\to 4_2^{+}$	-	0.0778072

and

Table 5. Some electric transitions B(E2) for ²²⁸Th.

${\mathit{J}}_{\mathit{i}}\to {\mathit{J}}_{\mathit{f}}$	B(E2) e2b2
	Exp.	IBM-1
$2_1^{+}\to 0_1^{+}$	1.382405	1.4301790
$2_1^{+}\to 0_2^{+}$	-	0.0000000
$3_1^{+}\to 2_1^{+}$	-	0.0000000
$3_1^{+}\to 2_2^{+}$	-	0.0116564
$3_2^{+}\to 2_1^{+}$	-	0.0000002
$4_1^{+}\to 2_1^{+}$	-	0.0000000
$4_1^{+}\to 2_2^{+}$	-	0.0002252
$6_1^{+}\to 4_1^{+}$	-	0.0007039
$6_1^{+}\to 4_2^{+}$	-	0.0000000

for some selected electric transitions.

Table 6. The coefficients of potential energy in MeV unit.

The Isotopes	${\mathit{ϵ}}_{\mathit{s}}$	${\mathit{ϵ}}_{\mathit{d}}$	$\mathit{A}1$	$\mathit{A}2$	$\mathit{A}3$	$\mathit{A}4$
224Th	−0.0510	0.0390	0.0000	−0.0040	−0.0410	0.0000
226Th	−0.0340	0.0420	0.0000	−0.0030	−0.0280	0.0000
228Th	−0.1210	0.2430	0.0570	−0.0250	−0.0970	0.0000

has the parameters of the coefficients of potential energy surface from the Equation (7), and applying IBMP program. These coefficients such as in the program.

The symmetric shape for two sides: prolate and oblate for ²²⁴Th and contour lines for the potential energy denoted in Figure 4. There is asymmetry in both side because the deformation in the distribution of the potential in surface specially in 30 < $\gamma$ < 60 and the maximum value of potential reach to 1 MeV.

Figure 5 represent to the potential for ²²⁶Th, which clear there is asymmetry in both sides and the potential gathered in $\beta$ < 1, reach to 800 KeV, but the last Figure 6 for ²²⁸Th, refers to asymmetry in both sides and unequal distribution in contour lines with less potential energy than the previous reach to 240 MeV because there is stability in this isotope near the closed shell 126.

6. Conclusions

The estimating numerous energy levels for (²²⁴Th, ²²⁶Th, and ²²⁸Th) is close to the data sheet obtained. Uncertain and unknown levels were estimated from the results using IBM-1 program with a good agreement.

Well calculated B(E2) values and fit with experimental findings data. These electric transitions used to estimate the limit of isotope. However there are differences between them because of how these isotopes’ nuclei were deformed.

²²⁴Th and ²²⁶Th are in SU(3) and O(6) region as expected from the results, and there is deformation in the surface, but SU(3) limit for ²²⁸Th.

There is decrease in the deformation of the nuclei within increase in boson number and near the closed shell.