_{1}

^{*}

The Cranking Nilsson model is applied to calculate the single-particle energy eigenvalues and eigenfunctions of nuclei in a strongly deformed potential. Accordingly, The L. D. Energy, the Strutinsky inertia, the L. D. inertia, the volume conservation factor
, the smoothed energy, the BCS energy, the G-value and the electric quadrupole moment of the five uranium isotopes:
^{230}U,
^{232}U,
^{234}U,
^{236}U and
^{238}U are calculated as functions of the deformation parameter. Furthermore, the single-particle Schrodinger fluid is applied to calculate the rigid-body model, the cranking-model and the equilibrium-model moments of inertia of the five uranium isotopes. Moreover, the collective model is applied to calculate the rotational energies of these isotopes. The best potential and deformation parameters are also given.

As we go far from closed shells, some new very simple and systematic features start to show up for some nuclei. This is true for nuclei with mass number A in the range

The success of the description of many nuclei by means of deformed potential can be taken as an indication that by distorting a spherical potential in this manner we automatically obtain the right combination of spherical eigenfunctions that makes the corresponding Slater determinant a better approximation to the real nuclear wave function. From this point of view, the deformed potential is a definite prescription for a convenient mixing of various configurations of the spherical potential. The absolute values of the rotational energies or equivalently the moments of inertia require the knowledge of the fine details of the intrinsic nuclear structure. Consequently, the investigation of the nuclear moments of inertia is a sensitive check for the validity of the nuclear structure theories [

Theoretical investigations of Ref. [

It is well known that nearly all fully microscopic theories of nuclear rotation are based on or related in some way to the cranking model, which was introduced by Inglis [

1) In principle, it provides a fully microscopic description of the rotating nucleus. There is no introduction of redundant variables, therefore, we are able to calculate the rotational inertia parameters microscopically within this model and get a deeper insight into the dynamics of rotational motion.

2) It describes the collective angular momentum as a sum of single-particle angular momenta. Therefore, collective rotation as well as single-particle rotation, and all transitions in between such as decoupling processes, are handled on the same footing.

3) It is correct for very large angular momenta, where classical arguments apply.

A simple and widely used way to describe the change of the single-particle structure with rotation is given by the Cranked Nilsson model (CNM) [

In addition to individual nucleons changing orbits to create excited states of the nucleus as described by the shell model, there are nuclear transitions that involve many (if not all) of the nucleons. Since these nucleons are acting together, their properties are called collective, and their transitions are described by a collective model [

The study of the velocity fields for the rotational motion led to the formulation of the concept of the Schrödinger fluid [

Sadiq et al. [^{230?240}Unuclei. The energy levels, deformation systematics, E2 transition probabilities and g-factors are calculated. The calculation reproduces the observed positive parity yrast bands and B(E2) transition probabilities. The observed deformation trend of low-lying states in Uranium nuclei depends on the occupation of down-sloping components of high j orbits in the valence space. The low-lying states of yrast spectra are found to arise from 0-quasiparticle (qp) intrinsic states, whereas the high spin states possess multi-qp structure.

Doma, Kharroube, Tefiha and El-Gendy [^{170}Yb, ^{172}Yb and ^{174}Yb, hafnium: ^{176}Hf, ^{178}Hf and ^{180}Hf and tungsten: ^{182}W, ^{184}W and ^{186}W nuclei. Moreover, they have applied the single-particle Schrödinger fluid to calculate the nuclear moment of inertia of the nine mentioned nuclei by using the rigid-body model and the cranking model. Furthermore, they applied the CNM to calculate the L. D. energy, the Strutinsky inertia, the L. D. inertia, the volume conservation factor

In the present paper, we applied the concept of the single-particle Schrödinger fluid to calculate the cranking-, the rigid body- and the equilibrium-models moments of inertia for the five uranium isotopes: ^{230}U, ^{232}U, ^{234}U, ^{236}U and ^{238}U. Furthermore, we applied the CNM to calculate the L. D. energy, the Strutinsky inertia, the L. D. inertia, the volume conservation factor

parameter γ, which are assumed to vary in the ranges

The single particle Hamiltonian in the CNM assumes the form [

where

Here, the oscillator parameters

The second term in the right-hand side of Equation (2.1) is given by

In the above equations

parameter and

The non-deformed oscillator parameter

The stretched square radius

The hexadecapole potential is defined to obtain a smooth variation [

where the

and

In Equation (2.9) t refers to the stretched coordinates

The Hamiltonian

Hence, to the first order in

where

Also, direct substitution for the different quantities in the operator

The method of finding the energy eigenvalues and eigenfunctions of the Hamiltonian H, Equation (2.1), can be summarized as follows:

1) Solving the Schrödinger’s equation of the Hamiltonian (2.10)

exactly.

2) Modifying the functions

3) Using the functions obtained in step 2) to construct the complete function

4) Constructing the Hamiltonian matrix H by calculating its matrix elements with respect to the basis functions defined in step 3).

5) Diagonalizing the Hamiltonian matrix H to find the energy eigenvalues

The solutions of the equation

where

The radial wave functions

where

The last function in the right-hand side of Equation (2.17) is the associated Laguerre polynomial. Since the nucleon has spin

The classifications of the functions

Wave functions with given values of the number of quanta of excitations N, the orbital angular momentum quantum number

The functions

Accordingly, we obtain 15 wave functions, states, namely

The classifications of these states in terms of the functions

The matrix elements of the Hamiltonian

It is easy to show that the matrix elements of the operator

where

Also, the matrix elements of the spherical-harmonic operators

We define the total energy by [

where the single-particle spin contribution

The summations in (2.23) run over the occupied orbitals in a specific configuration of the nucleus. The shell energy is now calculated from

where

and

The pairing energy is an important correction that should decrease with increasing spin and becomes essentially unimportant at very high spins. To obtain an (

with g_{1}/g_{0 }≈ 1/3. Furthermore, the number of orbitals included in the pairing calculation should vary as

The total nuclear energy is now calculated by replacing the smoothed single- particle sum by the rotating-liquid-drop energy and adding the pairing correction

or

where

From the single-particle wave functions, the electric (or mass) quadrupole moment may be calculated as

where

The problem of a single quantal particle moving in a time-dependent external potential well was formulated specifically to emphasize and develop the fluid dynamical aspects of the matter flow [

The single-particle wave function

We use polar form of the wave function and isolate the explicit time dependence in

where

where

The average potential field is assumed to be in the form of anisotropic harmonic oscillator potential. The intrinsic energy of the single particle state is, then

In terms of the frequencies

Applying the time-dependent perturbation method and using the equation arising from the first-order perturbation of the wave function we can calculate the first-order time-dependent perturbation correction to the wave function explicitly as function of the number of quanta of excitations corresponding to the Cartesian coordinates and the quantity

which is a measure of the deformation of the potential.

We use the cranking-model formula for the calculation of the moment of inertia. After the inclusion of the residual pairing interactions by the quasiparticle formalism, the formula for the x-component of the moment of inertia is given in terms of the matrix elements of the single-particle angular-momentum operator corresponding to the rotation around the intrinsic x-axis, the variational parameters of the Bardeen-Cooper-Sehrieffer wave function corresponding to the single particle states and the quasiparticle energy of this state [

We now examine the cranking moment of inertia in terms of the velocity fields. Bohr and Mottelson [

the cranking moment of inertia is identically equal to the rigid moment of inertia:

We note that the cranking moment of inertia

The following expressions for the cranking-model and the rigid-body model moments of inertia,

where E is the total single particle energy, given by (3.5) and q is the ratio of the summed single particle quanta in the y-and z-directions

q is known as the anisotropy of the configuration. The total energy E and the anisotropy of the configuration q are easily calculated for a given nucleus with mass number A, number of neutrons N and number of protons Z. Accordingly, the cranking-model and the rigid-body model moments of inertia are obtained as functions of the deformation parameter β and the non-deformed oscillator parameter

The two most important developments in nuclear physics were the shell model and the collective model. The former gives the formal framework for a description of nuclei in terms of interacting neutrons and protons. The latter provides a very physical but phenomenological framework for interpreting the observed properties of nuclei. A third approach, based on variational and mean-field methods, brings these two perspectives together in terms of the so-called unified models. Together, these three approaches provide the foundations on which nuclear physics is based. They need to be understood carefully, in order to gain an understanding of the foundations of the models and their relationships to microscopic theory as given by recent developments in terms of dynamical symmetries.

On the basis of the collective model, we calculated the rotational energies by using the following formula [

where A is the reciprocal-moment of inertia of the nucleus,

We have calculated the reciprocal moments of inertia by using the cranking model and the rigid-body model of the single-particle Schrödinger fluid for the even-even deformed uranium isotopes; ^{230}U, ^{232}U, ^{234}U, ^{236}U and ^{238}U as functions of the deformation parameter β, which is allowed to vary in the range from -0.50 to 0.50 with a step equals 0.005. The equilibrium values for the moments of inertia of the five isotopes are considered as the values for which the cranking model and the rigid-body model are equal for each isotope.

In ^{230}U, ^{232}U, ^{234}U, ^{236}U and ^{238}U with respect to the deformation parameter β. Since the reciprocal values of the rigid-body moments of inertia of these isotopes are slowly varying with respect to β, we present only in ^{234}U with respect to β. It is of interest to notice that, two values of the deformation parameter β, one of which is positive and the other is negative, produced good agreement between the calculated and the experimental moments for the five isotopes.

In ^{230}U, ^{232}U, ^{234}U, ^{236}U and ^{238}U, in KeV. The values of the deformation parameter β which produced the best values of the moments of inertia in each case are also given in this table. The corresponding experimental values are given in the last column [

In

It is seen from ^{238}U. The calculated value is in good agreement with the corresponding experimental one.

Case | β | ||||
---|---|---|---|---|---|

0.175 | 8.581 | 8.662 | |||

^{230}U | −0.18 | 6.69 | 8.532 | 8.611 | 8.68 |

^{ } | 0.240 | 8.576 | 8.654 | ||

0.190 | 8.201 | 8.253 | |||

^{232}U | −0.20 | 6.69 | 8.163 | 8.202 | 8.28 |

^{ } | 0.240 | 8.189 | 8.248 | ||

0.19 | 7.190 | 7.263 | |||

^{234}U | −0.20 | 6.69 | 7.172 | 7.233 | 7.29 |

^{ } | 0.250 | 7.189 | 7.260 | ||

0.20 | 7.512 | 7.561 | |||

^{236}U | −0.21 | 6.68 | 7.422 | 7.494 | 7.57 |

^{ } | 0.250 | 7.501 | 7.555 | ||

0.19 | 7.742 | 7.801 | |||

^{238}U | −0.21 | 6.68 | 7.683 | 7.722 | 7.82 |

^{ } | 0.255 | 7.738 | 7.787 |

Case | β | γ degrees | ||||
---|---|---|---|---|---|---|

^{230}U | 0.185 | 7.0 | 8.62 | 8.68 | 13.87 | N/A |

^{232}U | 0.195 | 10.1 | 8.20 | 8.28 | 13.27 | N/A |

^{234}U | 0.195 | 8.7 | 7.21 | 7.29 | 12.61 | N/A |

^{236}U | 0.205 | 8.6 | 7.50 | 7.57 | 12.45 | N/A |

^{238}U | 0.195 | 8.5 | 7.74 | 7.82 | 13.75 | 13.90 |

In the numerical calculations of the rotational energies of the even-even deformed isotopes: ^{230}U, ^{232}U, ^{234}U, ^{236}U and ^{238}U, we have used the formula, given by Equation (4.1) by Doma and El-Gendy [

It is seen from

In ^{230}U, ^{232}U, ^{234}U, ^{236}U and ^{238}U for values of the deformation parameter β and the nonaxiality parameter γ, which produced good agreement with the corresponding experimental findings.

It is seen from

On the other hand, it is well-known that the quantity that characterizes the deviation from spherical symmetry of the electrical charge distribution in a nucleus is its quadrupole moment Q. If a nucleus is extended along the axis of symmetry, then Q is a positive quantity, but if the nucleus is flattened along the axis,

Nucleus | Case | in Kev | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

^{230}U | Calc. Exp. | 51.68 51.72 | 169.47 169.50 | 346.89 347.10 | 578.05 578.20 | 856.31 856.40 | 1175.59 1175.70 | 1531.47 1531.60 | 1921.14 1921.20 | 2320.61 - | 2757.49 - |

^{232}U | Calc. Exp. | 47.46 47.57 | 156.34 156.56 | 322.49 322.6 | 540.87 541 | 805.8 805.8 | 1111.5 1111.5 | 1453.49 1453.7 | 1828.04 1828.1 | 2232.71 2231.5 | 2665.84 2659.7 |

^{234}U | Calc. Exp. | 43.48 43.5 | 143.30 143.35 | 296.06 296.07 | 497.02 497.04 | 741.2 741.2 | 1023.8 1023.8 | 1340.75 1340.8 | 1687.16 1687.8 | 2064.41 2063 | 2465.55 2464.2 |

^{236}U | New Exp. | 45.12 45.24 | 149.40 149.48 | 309.72 309.78 | 522.22 522.24 | 782.3 782.3 | 1085.3 1085.3 | 1426.48 1426.3 | 1805.11 1800.9 | 2210.14 2203.9 | 2628.98 2631.7 |

^{238}U | Calc. Exp. | 44.78 44.91 | 148.22 148.41 | 307.11 307.21 | 517.76 517.8 | 775.7 775.7 | 1076.5 1076.5 | 1415.42 1415.3 | 1789.04 1788.2 | 2200.01 2190.7 | 2632.24 2618.7 |

Case | β | γ | L.D. energy MeV | Strutinsky inertia 1/MeV | L.D. inertia 1/MeV | smoothed energy MeV | BCS energy MeV | G-value MeV | |
---|---|---|---|---|---|---|---|---|---|

^{230}U | 0.240 | 5˚ | 4.441 | 145.42 | 113.5 | 1.0042 | 2833.4 | 3.15 | 0.089 |

^{232}U | 0.240 | 5˚ | 4.453 | 147.07 | 115.7 | 1.0053 | 3948.2 | 2.64 | 0.089 |

^{234}U | 0.250 | 5˚ | 4.464 | 145.47 | 116.9 | 1.0075 | 2766.1 | 2.11 | 0.089 |

^{236}U | 0.250 | 5˚ | 17.04 | 149.65 | 120.2 | 1.0134 | 27535 | 1.55 | 0.088 |

^{238}U | 0.255 | 5˚ | 17.12 | 111.24 | 151.6 | 1.0135 | 2729.9 | 1.47 | 0.088 |

it is negative. According to the results of the electric quadrupole moments of the five nuclei, the five uranium isotopes have prolate deformation shape.

Moreover, it is seen from the obtained results that the calculated values of the rotational energies of the five even-even deformed uranium isotopes are in good agreement with the corresponding experimental data for all values of the total spin I.

The author would like to thank the editor and reviewers for their great helpful remarks and comments.

Kharroube, K.A. (2017) Study of Even ^{230-238}U Isotopes by Using Cranked Nilsson Model, Single Particle Schrodinger Fluid and Collective Model. Open Journal of Microphysics, 7, 36-52. https://doi.org/10.4236/ojm.2017.72003