An explanation for forking and backbending in rotational spectra

Volume 47B, number 3

PHYSICS LETTERS

12 November 1973

AN EXPLANATION FOR FORKING AND BACKBENDING IN ROTATIONAL SPECTRA * B.C. SMITH and A.B. VOLKOV

Department of Physics, McMaster University, Hamilton, Ontario, Canada Received 5 October 1973 It is shown that symmetric deformed nuclei are unstable to asymmetric deformation at sufficiently high angular momentum. Forking and backbending are natural consequences of this instability. Recently several examples of "forking" have been observed [1,2] which exhibit "backbending" [3,4] in the structure of the ground state rotational band. A striking example is the case 102pd studied by ScharfGoldhaber et al. [1 ]. It is extremely significant that a higher band of the fork represents a continuation of the variable moment of inertia (VMI) solution for the states up to the backbending region. We shall show that a proper generalization of the VMI procedure suitable to describe 3' soft nuclei, or in fact most rotational even-even nuclei, leads naturally to an explanation for both forking and backbending. It is shown that variational considerations make it probable that symmetric rotating nuclei must become unstable to asymmetric deformations at some critical angular momentum. The critical angular momentum decreases with 7 softness with J = 8 representing the smallest forking value for the traditional collective rotational Hamiltonian. Although many investigators consider the VMI model of Mariscotti et al. [5] as an ad hoc fitting procedure, the model has been shown [6-8] to follow from very general variational principles provided reasonable assumptions are made concerning the intrinsic wave function generating the rotational band. Briefly, the argument based on a collective model is (a more microscopic argument based on Hartree-Fock intrinsic states is also possible) that a Hamiltonian of the form 3

H=Ho({×i}) + ~=lHV((×i}) j2 exists where

(1)

{Xi} represents some appropriate set of

¢' Work supported in part by the National Research Council of Canada.

non-rotational coordinates, e.g./3 and 3', Jv are generators of rotations around the body fixed axes which, in the most general case, are defined as the principle axes of an asymmetric system, and Ho({Xi} ) and Hv((×i)) represent "radial" parts of the Hamiltonian. In a J = 0 (ground) state only Ho({×} ) is operative. In a variational solution the ground state wave function can be determined as the solution of

Eo ((~i)) = (~k((ai)) Ino I ~b({~i}))

(2)

subject to the conditions ~Eo ({~i})/a ~i = 0

(3)

where the set of the parameters {ori) characterizes the wave function. Eqs. (3) represent N equations if the wave function is characterized by N parameters. These parameters may include shape parameters (quadrupole moments, hexade capole moments, etc.) or in a more microscopic calculation, pairing gaps, quasi-particle mixing coefficients, etc. and consequently, the procedure used here in principle includes many of the specific microscopic models considered by other investigators [9-12]. We shall in general suppress reference to the radial coordinates {Xi). The N conditions eq. (3) determine an optimum set of parameters {~i(0)} for the ground state. The existence of a collective rotational band implies that the excited states can be written in the form (ignoring D 2 rotational symmetries for the moment) ~ J K = ~ ({°ti(J)})

DJMK(O102 03)

(4)

where ¢~i(J) = ai(O ) + Ai( J). The ,~i( J) become the variational parameters for the excited states and are assumed to vary in some appropriately continuous fashion. The energy hypersurface 193

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PHYSICS LETTERS

E o ({c@) = ( ~MJKI go I ~ J K )

(5)

has the property that N Eo({ai})=Eo({ei(0)})+

~

i
CijAiA/+O(A 3) (6)

for small A i since E o ({o~i (0)}) is the equilibrium ground state energy. Assuming small Ai, for the sake of simplicity, a variational solution for an excited state is obtained by finding 3

Ej,K({Ai(J)}=(~JKIHo +v~=1HvJ21~JK)

(7)

Ci/Ai(J)Ai(J ) i<~f=l ~

1

v=l

h2 -J~I({Ai(J)})

aEj,K

(8)

where higher order terms in Ai(J ) have been ignored and where moments of inertia are defined as ff{ai(J)~:). (9)

In the case of cylindrical symmetry (7) will reduce, for a K = 0 band, to

Ej ({Ai(J)}) = E ° ({c~i(0)})

(10)

N

+S-~Ci/Ai(J)A/(J)+½ t~2 J ( J + l ) i<.j J({Ai(J)})

'

still subject to the N conditions eq. (8), where only one moment of inertia of the form (9) is relevant. Any N - 1 of the eq. (8) plus the defining equation for the moment of inertia can in principle be used to find

Ai(J) = f i [ . J ( J ) = bi[J(J)

-

(13)

The last two equations constitute the usual VMI procedure. A generalization of the VMI procedure for an asymmetric system would be 3 Ej, K=Eo + ~ C u v [ J , ( J ) - J u ( 0 ) ] [ J v ( J ) - J v ( 0 ) ] ~,u=l J

2

J

(14)

(D~IKIJv[DMK)'

(15)

A proper calculation should actually use a trial wave function of the form

azx/(j) ( { a ; ( j ) } ) = 0,

=(~{ai(J)}lHvl

= 0.

OEj,K/O Jr(J) =O.

J 2 J (DMKIJvIDMK),

subject to the conditions that

½~2 j v 1 ( ( A i ( J ) } )

aEj/a J j

subject now to the three conditions

3 + ~7

where higher order terms in ( J ( J ) - J(0)) are neglected. The remaining unused conditional eq. (8) can be rewritten in the form

h2 + u=l ~ 2 Ju(J)

N =Eo({O~i(0)} ) +

12 November 1973

- J(0)]

(11)

J(0)] + O([ 3 ( J ) - 3(0)] 2),

where .J(0) is the value obtained in (9) when

o~i(J)

~JM = ~K CvK ~({°ti(J)})DJK

(010203)'

(16)

where the coefficients CuK satisfy the requirements of the D 2 rotation symmetry group [13] and are determined variationally, e.g., by matrix diagonalization for an asymmetric rotor Hamiltonian. The bands of interest in this work are given by the lowest solution of the matrix diagonalization for even J. Other solutions represent possible excited states. Rather than use the most general form eq. (14) we shall use a more specific model suggested by the usual form of the quadrupole collective model. Thus, we assume that the moments of inertia -Ju are proportional to sin 2 (3' - -5 2 7ru)' We further simplify the problem by assuming that the hypersurface E o ({t~i(J)}) is represented adequately by a paraboloid around the equilibrium values of {ai(J)} and that the ground state has 3' = 0, i.e., cylindrical symmetry. In this case the generalized VMI may be written, after the appropriate elimination of N - 2 of the parameters {Ai(J)},

Ej( Jj,'y)

(17)

= ~i(0).

Using this result (10) can be written as

Ej = E o +~-1 C [ J ( J ) + ~2 J ( J + l ) / 2 J ( J ) , 194

J(0)] 2

=½C(Jj-.Jo)2 + D ( . J j - J o ) T (12)

+½ E'),2 +

~i2R(,),)/ZJj

subject to the two conditions

OEj/b Jj=O, and 3Ej/OT=O.

(18, 19)

Volume 47B, number 3

PHYSICS LETTERS

,2c

5'0

IOC

12 November 1973

,

MeV-I

4'0

8C

6C 3.0

I 0"04

I 0"08 (~w) e

Ej(~)

I 0'12

J 0'16

(MeV) =

Fig. 2. Backbending plots for the two cases shown in fig. l. (MeV)

The broken lines show the higher energy "VMI" solutions (7 ~" 0 °) while the backbending part of the solid line shows the Yrast asymmetric solutions.

2-0

1.0

0.0

O*

15*

30 °

0°

15"

30 °

Fig. 1. Generalized VMI rotational energies as a function of the asymmetry parameter 7. Case A illustrates a "r "soft" nucleus and case B a relatively 7 "hard" nucleus. Minima which represent generalized VMI solutions are marked on the diagrams. The coefficients C, D and E describe the energy paraboloid in the two parameter ( J j - -To) and 7 space. The coefficients could in principle be found by an appropriate microscopic calculation [14]. R (7) is the lowest solution of the diagonalization of the operator

3 R - ~

=

j2 sin2 (7 - -~ 2 11.1.,)

in a D 2 symmetry representation. No real attempt has been made to fit data in this preliminary investigation, but the qualitative results of the analysis show remarkable agreement with experimental findings. Fig. 1 shows the energy Ej(',/) which is the solution of eqs. (17) and (18) as a function of~, for two choices of C, D and E, one of which represents a 7 "soft" case while the other represents a relatively

"hard" 7 case. The minima of E j ( 7 ) represent solutions of the generalized VMI and in both cases an asym. metric solution occurs at a critical J ( J = 8 is the lowest possibility for this model). The "VMI" solution (7 ~ 0 °) also persists. The instability of a classical rotating liquid drop to asymmetric deformation has been demonstrated by Lyttleton [15] and has been discussed by Davidson [16] in connection with the probable instability of symmetric rotating nuclei. Our analysis supports this instability which has been suggested by Mottelson [ 17] and other investigators [18, 19]. Fig. 2 shows the Jversus ~ 2 plots [3] for our solutions. The lower energy (Yrast) asymmetric band displays typical backbending behaviour while the "VMI" band persists as an almost straight line. For the particular cases we have studied the asymmetric band returns to "VMI" like behaviour for sufficiently high J. This is a consequence of our paraboloid assumption. A generalization of the energy surface along the lines of Bose and Varshni [20] and Nogami and Ross [21 ] would lead to modified curves. Another interesting feature of this analysis is the natural occurrence of near degeneracy at the beginning of the forking. This degeneracy would be hard to explain in terms of ordinary band crossing involving, for example, a two quasi-particle excitation. Since the shapes of the two intrinsic states in this model are quite different, even substantial mixing of the two states, which may occur with near degeneracy, should 195

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PHYSICS LETTERS

not lead to significant removal of the degeneracy since off-diagonal matrix elements would be small. However, fairly large mixing between the states at the fork probably occurs as evidenced by the transition between bands at the forking point.

References [1 ] G. Scharff-Goldhaber et al., Phys. Lett. 44B (1973) 416. [2] R.A. Warner et al., Phys. Rev. Lett. 31 (1973) 835. [3] A. Johnson, H. Ryde and S.A. Hjorth, Nucl. Phys. A179 (1972) 753. [4] A. Johnson and Z. Szymanski, Phys. Reports 7C (1973) 181. [51 M.A.J. Mariscotti, G. Scharff-Goldhaber and B. Buck, Phys. Rev. 178 (1969) 1864. [6 ] T.K. Das, R.M. Dreizler and A. Klein, Phys. Lett. 34B (1971) 235. [7] A.B. Volkov, Phys. Lett. 35B (1971) 299.

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[8] A.B. Volkov, Phys. Lett. 41B (1972) 1. [9] B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett.5 (1960) 511. [10] J. Krumtinde and Z. Szymanski, Phys. Lett. 36B (1971) 157. [11] R.A. Sorenson, Proc. Orsay Colloquium on Intermediate nuclei (Orsay 1971) 70. [12] F.S. Stephens and R.S. Simon, Nucl. Phys. A183 (1972) 257. [ 13 ] A.S. Davydov, Quantum mechanics (Addison-Wesley, Reading, Mass. 1965) § 46. [14] M. Baranger and K. Kumar, Nucl. Phys. 62 (1965) 113. [15 ] R.A. Lyttleton, The stability of rotating liquid masses (Cambridge University Press, Cambridge 1953). [16] J.P. Davidson, Rev. Mod. Phys. 37 (1965) 105. [17] B.R. Mottelson, The nuclear structure Symp. of the Thousand Lakes, Joutsa, Finland, 1970. [18] S.M. Abecasis and E.S. Hermandez, Nucl. Phys. A180 (1972) 485. [19] R.K. Gupta, Phys. Rev. C7 (1973) 2476. [20 ] Y.P. Varshni and S. Bose, Phys. Rev. C6 (1972) 1770. [21] C.K. Ross and Y. Nogami, Nucl. Phys. A211 (1973) 145.