The Hamiltonian for a non-spherical rigid body rotating about its centre of mass is [Ei75]
H = S ^lJ/T ,
, k k
k
(1.59)
where k numbers the principal axes of the intrinsic (body-fixed) system, L are the components of the angular momentum operator and 7
K K
are the principal moments of inertia.
The simplest case is that of axial symmetry and it is usual to take the 3 axis as the symmetry axis, so that Ij = I2 / I 3. Since the 3 axis is a symmetry axis, the angular momentum cannot have a component in that direction. The eigenstates are then identified by the total angular momentum J and its projection M on the space-fixed z axis. The energies of these eigenstates are given by
E = ^ h 2J ( J + l ) / I 1 . (1.60)
Because of the invariance of the Hamiltonian (1.59) under reflections in the intrinsic 1-2 plane, only even values of J are allowed [Bo75a].
To compute reduced transition probabilities, the eigenstates and multipole operators are written in terms of the intrinsic axes and the Euler angles which relate the intrinsic and space-fixed systems (eqs. 1.22 and 1.23). In particular, the spectroscopic quadrupole moments
E2
are related to the intrinsic quadrupole moment Q 0 by eq. 1.24 (with K = 0) and the reduced transition probabilities are given by
E2 2
B (E2; -*■ J f ) = [C(Ji2 J f ;000) Qq ] . (1.61)
If the nucleus is taken to have a Y 2 deformation only, then [Da68]
3 B 2 ß2 (1.62)
where B 2 is an inertia parameter (eq. 1.39). Using eq. 1.10, and writing 4/3 7TpQRg = Ze, one finds for J = 2
E2 6zeR^ f 2 //5 Q
Q 2
+ = - 28TT ^2l1+T
/ ¥ M---JSince there is no second 2 + state in this model, the product p 1.50 is not defined.
4
(1.63)
of eq.
The relaxation of the requirement of axial symmetry allows the angular momentum to have a projection K on the intrinsic 3 axis. However, this projection is no longer a good quantum number as the Hamiltonian does not commute with L 3. The eigenstates can be labelled by and M where J and M are the usual angular momentum quantum
numbers, and n counts the number of states of that particular J from the ground state. The eigenstates are then [Ei75]
IJ M> = 2 A (Y) I JKM> , (1.64) n , JKn
k
where |jKM) are eigenfunctions of that part of the Hamiltonian which is diagonal in L 3. The parameters A J K n (Y) depend on the asymmetry angle Y (eq. 1.14) and are tabulated in refs. Da59 and Ei75.
For Y <15°, the states of eq. 1.64 are dominated by one value of K [Da59]. In the limit of axial symmetry (i.e. Y 0) the energies of the levels are given by"[Da68]
E ft
2li
j
(
j+D
+hfi (1.65)and the moment of inertia I x about the intrinsic 1 axis approaches the value given by eq. 1.62. Also, the moment of inertia I 3 about the
intrinsic 3 axis, given by [Da68]
1 3 = 4 B 232 si n 2 Y , (1.66) approaches zero. Clearly then, the energies of the levels are
functions of Y and this can be used to determine Y for a given nucleus. For example, the energies of the two J = 2 states are [Da58]
E
4b23‘
sin 2 3Y 1
V
1- f
si" 23Y
(1.67) and that of the J = 3 state isE 9ft2
2b232 sin 2 3y . (1.68)
A table of level energies for many levels has been computed by Moore and White [Mo60] and fig. 1.3 shows some of these energies. One simple result of this model, which allows its applicability to be readily tested, is the existence of certain identities between various excitation energies. Two of these are [Ei75]
E (2 1) + E (2 2 ) = E (31) ,
4E (22) + E (22 ) = £ ( 5 ^ . (1.69)
The spectroscopic moment of the first excited state of a non- axial rigid rotor is given by [Da58]
0E+ = " Q f y cos 3y (9 - 8 s in2 3y) ^ (1.70)
which reduces to eq. 1.24 for y
expressions for reduced tran
y. Some of these are [Da58]
, = 0 (and hence K = 0) . The m od tt/3
expressions for reduced transition probabilities are also functions of
B(E2;2n ->0) = h(QQ ) 1 ± 3 - 2 sin2 3y
J 9 - 8 s in2 3y
(+ sign for n = l , - sign for n = 2 ) ,
B ( E2;22 + 2 1) = y ' ( Q ^ V Sln --
9 - 8 s i n 2 3y
(1.71)
It is concluded [Da59] that E2 transitions between levels in this model fall into three categories:
(a) Transitions between members of the "ground state band" (i.e. those levels which remain when y = 0) and transitions between members of the "anomalous band". These are strong
t r a n s itions.
(b) Transitions between members of the two bands which have the same J values. The reduced transition probabilities for these transitions are zero for y = 0 and rise to values of the order of those of group (a) for y = 30°.
y(deg)
Fig. 1.3: The dependence of the relative spacings of the lowest energy levels predicted by the asymmetric rigid rotational model on the asymmetry angle y. The levels are labelled by their angular momenta. The scale on the y axis is in units of the excitation energy of the first J = 2 state at y = 0.
(c) Other transitions which are possible under angular momentum selection rules. These have zero reduced transition
probabilities for y = 0 and y = 30° and small values in between. An exception is the transition 3* ■> 4* which behaves as if it belongs to group (b).
In this model [Is69]
p, < 0 . (1.72)
4
1.2.4 The Rotation-Vibration Model
Many nuclei have level schemes which appear to consist of rotational bands, but with relatively low-lying excited 0+ states. Such level schemes can be produced by considering vibrations about a non-spherical equilibrium. Two such models have been described [Fa65, D a 6 0 ] . Both consider harmonic quadrupole vibrations about equilibrium values 30 ,Y0 of the deformation parameters (eq. 1.14) thus
V(ß,Y)
=4 c ß ( ß - ß 0)2
+!sC (Y-Y0)z . (1.73)In one model, the equilibrium deformation is assumed to be axially symmetric (yQ =0), while the other model considers 3 vibrations only
(C = oo) .
Y
The Hamiltonian for the rotations and vibrations of a system with a non-spherical but axially symmetric equilibrium shape may be written
II = II +11 +11 ,
r v rv (1.74)
where explicit forms for each part are given by Eisenberg and Greiner [Ei75]. The first two parts are diagonal in L 3 and can be easily solved. The coupling Hamiltonian H may be treated, for low angular momenta at least, as a small perturbation [Ei75].
Eigenstates of the Hamiltonian H + H are described as |n_n KJM)
r v 1 3 Y
where n^ and n^ are the number of quadrupole phonons for
3
andy
vibrations respectively, J and M are the usual angular momentumintrinsic 3 axis. The energies of these eigenstates are given by [Ei75]
E = E r [J(J+l) - K 2 ] + E (3(n[3 +h) + E (*s|k | + 2n + 1) , (1.75)
where the characteristic energies E E 0 and E