Nuclear deformation

STRUCTURAL BASIS OF THE ONSET OF NUCLEAR DEFORMATION AT NEUTRON NUMBER 90... 

According to Eq. (11), the force constant for the normal vibration Q, can be identified with the term in braces and can be negative if the second term, which is positive, exceeds the first term. If the force constant is negative, the energy should be lowered by the nuclear deformation Qi, and the second-order distortion from the symmetrical nuclear arrangement would occur spontaneously. 

If the full molecular symmetry is assumed, the ground states of the cation radical of fulvalene and the anion radical of heptafulvalene are both predicted to be of symmetry by using the semiempirical open-shell SCF MO method The lowest excited states of both radicals are of symmetry and are predicted to be very close to the ground states in the framework of the Hiickel approximation these states are degenerate in both cases (Fig. 4). Therefore, it is expected that in both these radicals the ground state interacts strongly with the lowest excited state through the nuclear deformation of symmetry ( — with the result that the initially assumed molecular... 

In order to seek the most soft nuclear deformation in an excited state, the approximation is again made of replacing the sum over excited states in Eq. (17) by a dominant term corresponding to the next higher excited state. Now, the transition density p between the nth excited state corresponding to the orbital jump and the mth... 

Nuclear deformation may result in repulsive electric forces overcoming attractive strong nuclear forces, in which case fission occurs. 

Up to this point, we have assumed that all nuclei are spherical in shape. That is not true. There are regions of large stable nuclear deformation in the chart of nuclides, that is, the rare earths (150 < A < 180) and the actinides (220 < A < 260). We shall discuss these cases in more detail later in this chapter when we discuss the electric moments of nuclei. 

A recent analysis by Thielemann et al. [THI83] of the effects of B" delayed processes on the progenitors of the Th-U-Pu chronometers showed that these processes (delayed fission in particular) did indeed significantly influence the final abundances of the chronometer progenitors. This leads to a long age for the Galaxy. In view of the importance of this result, it is useful to re-examine the calculation with a nuclear model that includes the effects of nuclear deformation on the B decay rates, fission barriers, and neutron separation energies self-consistently. 

A consistent picture of the details presented above, based on the idea that nuclear deformation is due to the residual force between valence neutrons and valence protons, nas been proposed [W0082, HEY83]. 

The information on nuclear deformation from the IS-data indicates that the steep increase, observed between N = 88 and 90, is restricted to a few elements around gadolinium (Z = 64). The effect becomes less pronounced in the lighter as well as heavier elements. In barium (Z 56) and ytterbium (Z = 70) the step vanishes completely and there is a smooth transition from spherical to strongly deformed shapes above N = 82. This behaviour indicates that the Z = 64 proton subshell closure, with its stbilizing effect for spherical shapes, plays an important role. 

Chart of nuclei. The stable isotopes are represented by black squares. The known isotopes are indicated in addition to the magic proton and neutron numbers. The smooth lines enclose regions with a nuclear deformation B 0.2. Those isotopes are marked by open bars where optical spectroscopy yielded information on nuclear ground state properties in long isotopic chains. This picture gives the status as of September 1985. 

The N = 83, 85 and 87 isotones above the neutron-shell closure at N = 82 are described essentially by the f7 shell-model state. Calculations withing the particle-rotor model of Larsson et al. [77], assuming small nuclear deformations of the core, account qualitatively for the measured nuclear moments (cf. Fig. 3). The variation in the spectroscopic quadrupole moments reflects the successive filling of the f j neutron shell. This is realized from the formula [78] ... 

Figure 16.5 shows the variation in nuclear deformation calculated for the fission barrier of 298114 Qf particular interest are the small local fluctuations at small deformation. The minimum of 8 MeV at zero deformation constrains the nucleus to a spherical shape. Spontaneous fission is a very slow process in this situation since it involves tunneling through the 8 MeV barrier. These local fluctuations in the potential energy curve in Figure 16.S result from adding corrections for shell effects to a liquid drop model. The resistance to deformation associated with closed shell nuclei produces much longer half-lives to spontaneous fission than would be expected from calculations based on a standard liquid drop model.

The second excited states in even-even nuclei. The systematics of the second excited states have been discussed recently by Nagasaki and Tamura and by Scharff-Goldhaber and Weneser. They find a very noticeable difference between the spins and energies of these states in nuclei with neutron numbers (N) above or below 90. This is the neutron number which corresponds to a rapid increase in the nuclear deformation revealed by the atomic isotope... 

In the strong coupling model the angular momentum of the unpaired nucleon processes about the axis of nuclear deformation with a constant projection Q%. The total angular momentum [JU) is compounded of the nucleon angular momentum (jfi) and the rotational angular momentum ( S), and its projection on the axis of symmetry is given by the quantum number K. In the absence of the excitation of vibrational states, rotation is developed only in directions perpendicular to the axis of symmetry and then K=Q. Associated rotational states correspond to a particular value of K ovQ) and different rotational systems are developed on the basis of states with different values of K,... 

The systematics of E2 transitions have been discussed by Sunyar, by Temmer and Heydenburg , and by Ford. The last two authors plot the square of the nuclear deformation (which is calculated from the transition probability and is proportional to it) against the neutron number (Fig. 66). The deformation rises very sharply to a flat maximum for N near 90, and then falls slowly as the closed shell at Pb is approached. The rapid rise near N = 90, it should be noted, corresponds to the rapid rate of increase in the nuclear deformation measured by the isotope shift. No corresponding analogue of the sharp rise in the isotope shift just above iV = 82, discussed by Brix and Kopfermann , has yet been detected. 

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