In this section the results of some models of the structure of low-lying states of even-even nuclei will be presented. Emphasis will be placed on predictions of the properties, and in particular, of the quadrupole moment of the first excited state. The literature is very extensive and some representative references will be given in the appropriate places. The first few subsections are devoted to the simplest phenomenological models. These yield many simple predictions which seem to be fairly accurate for some nuclei. Later subsections give very brief descriptions of some of the more complex models. It is usually true that the more complex the model, the fewer general predictions can be made. One must then rely on the availability of calculations appropriate to the particular nucleus in question to test these models.
1.2.1 The Harmonic Vibrational Model
The Hamiltonian of this system is [Bo75a, Ei75]
A 2bA
7T
•t t.+ — c,a, *a, ,
X X
2 X X X
(1.39)where TT^ are the momenta conjugate with the deformations which are the operator analogues of the deformations defined in subsection 1.1.1. The inertia parameters and stiffness parameters can either be empirically determined, or can be calculated by making further assumptions about the nucleus. The subscript X labels the multipolarity of the vibration;
X=2, the quadrupole vibration, is
the lowest order vibration.The Hamiltonian (eq. 1.39) can be cast in a form involving creation and annihilation operators. The resulting eigenstates are labelled by the number of phonons of multipolarity X, the total angular momentum J, its projection M on the z axis and whatever other quantum numbers U which are needed. I will denote the state vectors by I t>N2N 3 ... JM). The energy levels of the harmonic oscillator are
[Da68]
E = 2 (N^ + X + k) ti/c^/B^ . (1.40)
X
This equation seems to imply that the energy of the vacuum state, that is, the state with = 0 for all X, is infinite since /C^/B^ increases with increasing X in any realistic model. One might infer therefore, that there is an upper limit on the possible values of X. A shape of multipolarity X can be pictured as X protrusions from a sphere. It has been argued that each protrusion must contain a particle so that
an upper limit on X is given by the number of fundamental particles in the nucleus.
One phonon states have angular momenta J = X and two phonon states are degenerate multiplets with J
=0,2,4,.,.,2X if the phonons are
identical and J = |X1-X2 |,|X1-X2 |+1,...X 1+X2 otherwise. The values of J for a level of three identical phonons are given by Davidson [Da68]. Bohr and Mottelson [Bo75a] give general methods for determining theseJ values. The parity of the levels is even unless the state contains an odd number of odd multipolarity phonons.
In order to determine the relative energies of the levels, it is necessary to have relations between the value of and for various A. These are often derived by assuming that the vibrational motion of the nucleus can be described as the flow of an inviscid incompressible fluid of constant mass density. One then obtains [Ei75]
and
= 3ARq/4ttA (1.41)
(A-l) (A+2) FTÜ - 3Z2e 2 8tt2£ 0 (2A+1) R 0
(1.42)
where R Q is the mean radius, 0 the surface tension, Z the atomic number and A the mass of the nucleus. It follows that the first excited state is the = 6 ^ state.
It is commonly stated [e.g. Bo75a, Ku75] that, in this model, matrix elements of the electric multipole operator M(El,y) between
states |uN 2 . . . J M ) and |u'N2 ...J'M') are zero unless the rule
. N a - N - = ±6n (1.43)
holds. This rule implies that all electric multipole moments are zero. Eisenberg and Greiner [Ei75] show that this is correct to first order only. They demonstrate that, for a nucleus with a shape and density given by eqs. 1.1 and 1.2, the electric multipole operators expressed in terms of the collective coordinates 0U have the form
3eZR„ M c o n (EX'y) A+2 OL + ~ T 7 ~ Ay 4/tt
I
Ai A2v xv
2 /(2Aj+l) (2A2+1) 2A+1 X C ( A 1A2A;000)C(A1A2A;V1-V2y) a x v a x v + 0 ( a ^ ) (1.44)analogous with eq. 1.10 for the classical multipole moments of this charge distribution. The selection rule of eq. 1.43 applies only to the matrix elements of the first term of eq. 1.44. In particular, the quadrupole moment of the first excited state (the = ^ 2 state)
given to second order by [Ei75]
Q2+ = - ZeRjjMB.O,)'54 • (1.45)
There will also be non-zero values of matrix elements for transitions which do not obey eq. 1.43, but these will be small compared to those which do obey eq. 1.43 as they arise from higher order terms in eq.
1.44.
The reduced transition probability B (0\ ; i s defined by
B(gX;J. +J_) := (2 J . +1) " 1 | < J J| M (OX) || J . ) | 2 , (1.46)
i f l f l
where 0 is either E or M. For the first excited state, this model predicts
B (E2 ; 0+ 2 + ) = Z2e2R^h (B. C. ) "^ (1.47)
32TTZ 0 2 2
to lowest order in the electric quadrupole operator. The applicability of this model to real nuclei is often judged by comparing both the energy level predictions of eq. 1.40 and the prediction
B(E2; J+ + 2 + )/B(E2;2+ + 0 + ) = 2 (1.48)
(J+ being any of the "2 phonon" levels) with experimental values.
The lower part of a vibrational energy spectrum is shown in fig.
1/3 -2
1.2. The parameters Z = 80, A = 198, R 0 =1.25 A fm and 0 = 17 MeV fm were used to generate the relative spacings of the levels (see eqs.
1.41 and 1.42). This value of the surface tension 0 is consistent with the semi-empirical mass formula [Bo69]. The transitions shown in fig. 1.2 are all those for which eq. 1.43 holds.
A quantity which will be required in the analysis of the
experiments and which is not readily available from other experimental data (see ch. 4) is the sign of the product
N 2 Nj Nj tt J