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Superdeformed rotational bands in the presence of Y44 deformation
4 August 1994 PHYSICS LETTERS B EI~SEVIER
Physics Letters B 333 (1994) 294-298
Superdeformed rotational bands in the presence of Y44 deformation Ikuko Hamamoto a, Ben Mottelson b a Department of Mathematical Physics, Lund Institute of Technology, University of Lund, Lund, Sweden b NORDITA, Blegdamsvej 17, Copenhagen, Denmark and ECT*, c/o Dipartimento di Fisica, 1-38050 Povo (Trento), Italy Received 18 May 1994; revised manuscript received 13 June 1994 Editor: C. Mahaux
Abstract The observation of AI = 4 staggering in the rotational spectra of superdeformed nuclei suggests the occurrence of Y~ deformations in the nuclear shape with associated C4v point-symmetry for the rotational Hamiltonian. We have investigated the general class of Hamiltonians with such symmetry. In addition, we require the axially symmetric terms to favour rotation about an axis that is perpendicular to the long axis of nuclear shape. The AI = 4 staggering can indeed result from the tunneling between the four equivalent minima that occur in the plane perpendicular to the superdeformation symmetry axis, but the occurrence of this effect is a subtle matter depending sensitively on the axially symmetric terms in the Hamiltonian.
In the spectroscopy o f s u p e r d e f o r m e d nuclei, rotational s e q u e n c e s have been o b s e r v e d in which states w i t h A I = 4 exhibit a systematic energy displacement w i t h respect to the c o m p l e m e n t a r y s e q u e n c e o f states w i t h spins l y i n g h a l f way in b e t w e e n ( [ 1,2] ). T h e occurrence o f such a term in the rotational energy suggests that the strongly prolate spheroidal deformation has been slightly perturbed by a term with s y m m e t r y Y44 w i t h respect to the s y m m e t r y axis. In the presence o f such a term the rotational energy can acquire a fourth o r d e r t e r m proportional to (12 - I2) 2 2 reflecting the C4o s y m m e t r y o f the shape. I f such a system w e r e to rotate about the f o u r f o l d s y m m e t r i c l o n g axis (13 ~ I ) the o c c u r r e n c e o f A I = 4 structure in the rotational sequences w o u l d be an i m m e d i a t e and direct c o n s e q u e n c e o f the Y44 d e f o r m a t i o n [ 3 ]. H o w ever, the very stability o f the s u p e r d e f o r m e d shapes results f r o m the large m o m e n t o f inertia associated w i t h rotation about directions perpendicular to the 3axis (/3 < < I ) . Thus, as w e show in the present note,
the occurrence o f A I = 4 rotational structure in the sup e r d e f o r m e d yrast region is a rather subtle p h e n o m e n a w h i c h depends in an important m a n n e r on the axially s y m m e t r i c terms in the rotational H a m i l t o n i a n 1. t After the present work was completed, we received from Dr. Flibotte an abstract of a contribution by I.M. Pavlichenkov and S. Flibotte to the Conference on Physics from Large Gamma-Ray Detector Arrays to be held in Berkeley, August, 1994, in which they exhibit a fit to the experimental data of Ref. [ 1] on the basis of the formalism of Ref. [ 3 ]. This analysis involves a Hamiltonian with C4v symmetry but is based on a solution with the angular momentum aligned along a direction near to the 4-fold axis of the nuclear shape. Identifying this axis as the principal axis of the prolate superdeformed shape, we recognize this solution as one in which the angular momentum is aligned along a direction almost orthogonal to that for which the moment of inertia has its maximum value. We believe such solutions to be inappropriate for reasons given in the text. An alternative interpretation of Pavlichenkov and Flibotte's work (as suggested to us by our referee) would identify the 4-fold axis (called the 3-axis in their work) with the axis of rotation, which is assumed perpendicular to the long axis of the deformation. In this interpretation, however, their Hamiltonian is
0370-2693/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSD10370-2693 ( 9 4 ) 0 0 7 8 3 - 4
L Hamamoto, B. Mottelson/PhysicsLettersB 333 (1994) 294-298 We study the Hamiltonian
H=AI~+B1(I~
-I~)2+B2(I~+I~)
295
3 2
(1)
where we have omitted the 13-independent term proportional to I ( I + 1) that describes the regular I dependence o f the energy o f the yrast states. The first term in ( 1 ) describes the strong preference o f the leading order rotational energy for an orientation o f I perpendicular to the prolate symmetry axis; this implies that we are interested in the region o f parameter space where A is positive and large. The B2 term is the axially symmetric part o f the 4th order rotational energy; in order that the angular momentum remain perpendicular to the 3-axis we must require 2B2I 2 < A. We can, without loss o f generality, confine the parameter space to Bx > 0, provided we study both positive and negative values o f B2. It is useful to view the Hamiltonian ( 1 ) as defining an energy function on the sphere [I[ = constant; the positions of the maxima, minima, and saddle points of this function are illustrated in Fig. 1, and are seen to depend on the value o f the parameter 2 (B1 + Bz) 12/A. Quite generally there are four minima in the equatorial plane and in the yrast states the angular momentum will be localized in the neighborhoods o f these minima. This structure gives rise to quartets o f states, the individual member o f which can be labeled by the quantum number (A1, A2, B1, B2, and E ) o f the C4o point symmetry group. The fact that the intrinsic shape carries C4o symmetry implies that the C4v quantum number o f a state in the spectrum is determined by the intrinsic configuration and all the states o f a given rotational band will carry this same quantum number. (The argument here is identical to that which leads to the condition K = f~ for the rotational bands associated with an axially symmetric intrinsic shape [4] .) Thus, o f the four states associated with the minimum o f ( 1 ) only a single C4~ combination is appropriate for generating the rotational band associated with a particular intrinsic state. To illustrate the variety o f yrast structures that can result from the Hamiltonian ( 1 ), we have directly digrossly deficient since it neglects the term, AI~, (assuming the prolate axis to be the 1-axis) which describes the leading order anisotropy of the moment of inertia. In the presence of this term, the C4 symmetry is badly violated and the rotational dynamics is modified in a major way.
(a) 1 3
(b)
~
2
~
2 1
minimum • , ,, maximum •
saddlepoint ....... separatrix
o
Fig. 1. Classical phase space for C4v rotational Hamiltonian. Provided the minima of the rotational Hamiltonian lie in the equator (i.e. 2B212 < A), there are two possibilities for the arrangement of maxima, minima, and saddle points: (a) for 2 (B t+ B2 )12 > A; and (b) for 2(B1 4- B2)I 2 < A. For each of the two cases, two views of the spherical surface are shown. On the left we look from the direction of the I-axis and the positive 2- and 3-axes are labelled. On the right we look from the direction of the 3-axis and the positive 1- and 2-axes are labelled. In case (a), the energy surface exhibits two different classes of local maximum which are indicated by filled triangles and filled squares, respectively. Table 1 Hamiltonian
A
B]
B2
AI = 4 structure in yrast band
H1 //2 H3
100 100 100
1 1 I
- 1/2 0 -1
irregular regular oscillation no oscillation
agonalized this Hamiltonian for several different values of the parameters (A, B1, and B2) all o f which are chosen so that the classical motion corresponds to rotation about an axis in the equatorial plane. (see Table 1 and Fig. 2). It is seen that for some values o f the parameters the splitting o f the quartets implies a regular A I = 4 alternation in the energies o f the quantum states labelled by the quantum numbers A 1 or B1 (see H 2 ) , for other values ( H 3 ) no alternation
L Hamamoto, B. Mottelson/Physics Letters B 333 (1994) 294-298
296
I
[
I
[ •" + •
03 uJ I03 I-. 03 LLI
O
_J LL
O (.9 Z
_,.I
[
I I I I B1 ( 1 degen.) E ( 2 degen.) A1 ( 1 degen.)
nILl Z LU
5.4
The tunneling matrix element in the WKB approximation is obtained by evaluating the action
4.6
1.3
1.4
3.0
1.5
2.0
S12 = //3(~b)d~b.
2.3
•
Z~
•
ZX
/,
•
&
ZX
+
+
+
+
+
+
+
+
&
•
&
•
•
&
•
•
1.1 (-1)
8.6 (-2)
6.9 (-2)
minl
where the momentum as a function of angle, I3(~b), is obtained by solving the equation
H 2 = 100132+ (112- 122)2 6.5 (-1}
4.0 (-1)
2.7 (-1)
1.9 (-1)
1.4 (-1)
H(/3, 40 = E0 5.7 (-2)
•
z~
•
A
•
Z~
•
Zk
•
+
+
+
+
+
+
+
+
+
t,
•
t,
•
&
•
&
•
&
H 3 = 100132 + (112- 122)2 - (112+ I22)2 1.4
(1)
(o)
(-1) (-2) (.4) (-5) (-6)
1.6
1.3
8.4
5.1
2,8
Z~ +
Z~
A
A
t=
A
~.
+
+
+
+
+
+
•
•
•
•
•
Q
I
I
I
I
I
I
1.7
I
I
I
Fig. 2. Energy splittings within yrast quartets for three different C4v rotational Hamiltonians. The figure shows the relative positions of the states within the yrast quartets as a function of I; the states of the representation E are unshifted by the tunnelling, while the absolute value of the splitting between A1 and BI is given above each quartet (the number in parenthesis denotes the power of ten). The figure is appropriate for even A nuclei (i.e. integer I). For odd A, the quartets are split into two doublets and the C4v classification must be replaced by the spinor representations of the double point group C~; the splitting patterns are similar except that unshifted E states are absent and the A1, B1 states are replaced by the two-fold degenerate states E1 and E2.
occurs, while for still others (H1) the tunnelling between minima generates an erratic pattern as a function of I, sometimes shifting a given symmetry up and sometimes down without any obvious regularity. In order to understand these behaviors we examine the semi-classical evaluation of the tunnelling matrix element connecting adjacent minima of the rotational Hamiltonian. For this purpose we introduce canonically conjugate variables (/3, ~b) on the spherical surface II1 = I, I1 = ~ / ~ - q cos ¢
q sin ¢
(3)
,/
(-1) (-1) (-1) (-2) (-3) (-3) (-4) (-5)
10 12 14 16 18 20 22 24 26 SPIN (I)
12 = V / ~ -
(2)
rain2
H I = 100132 + (112. 122)2 - 0.5(112 + 122)2
m, >" L9
H=AI2+(I2-I23)2(BlCOSZ2d)+Bz)
(4)
EO = B2 I4
where E0 is the energy of the classical minimum of the Hamiltonian (2). The interesting regions of parameter space, determine two different types of behavior for the solutions to (3) and (4). a) if:
2B21z [ B1 A < 1- VBI+B2
(5)
then: I3(q~) is pure imaginary for all values of q~ in between the two minima and the resulting tunnelling amplitude exp(iSj2) is a real number without any I dependent phase. b) if:
2B212 1 >
A
¢ > 1-
B1 B1+B2
(6)
then: the angular momentum 13(~b) becomes complex (acquires a real part) while ~b is intermediate between the two minima and the tunnelling amplitude involves a factor cos(~eSl2) which can change sign as a function of I and thus give rise to staggering in the yrast energies 2. The region of the parameter space satisfying (6) is shown as a shaded area in Fig. 3. The fact that the Hamiltonian H3 defines a trajectory lying entirely in the region below the shaded area of Fig. 3 explains the absence of/-dependence in the pattern of quartet splittings for this Hamiltonian. z It should be noted that the conditions for ~}~eS12 -~ 0 ale quite different from the condition in Fig. 1 that determines the qualitative structure of the classical energy surface. This reflects the fact that the tunnelling takes place in a phase space with complex dimensions and thus reveals features of the dynamics that are invisible in the real phase space of classical mechanics.
I. Hamamoto,B. Mottelson/ PhysicsLettersB 333 (1994)294-298 3.0 3.0 2.0
J
A=90
2.0
cE 1.0 c~
I
t
•
~'~ 1.0 A=100~,~,
0.0 I
I
,,
297 i
I
I
i
I
H = AI32+ (112- Ia2)a yrast A1-band
^,
,.
M
,
~,' ,/'
' ,,,"
A ", *
t
•
0.0
-1.0 ,','~I . i ' ~ ' A=120 ~t i /
-1.0 0.0
1.0
2.0 3.0 4BlI 2 / A
4.0
5.0
For Hamiltonians o f the type o f H 2 (with B2 = 0), the real part o f the integral ( 3 ) can be simply evaluated
=
)
7(1
for I >
"
V
~
-2.0
Fig. 3. Parameter space for C4v rotational Harniltonian. In the shaded region the tunnelling matrix element between equatorial minima acquires an / dependent real phase, which is a necessary condition for obtaining AI = 4 structure. The upper limit, correponding to the condition 2B2/2 < A, is imposed in order to ensure that the rotation is about an axis lying in the equator.
~eS]~
~\ i/
A 4B1
-3.0 16
20
24
28 SPIN (I)
32
36
Fig. 4. Splittings of yrast quartet for three C4v Hamiltonians of type H2 (B2 = 0). The figure is obtained by direct diagonalization of the rotational Hamiltonian and plots only the eigenvalue of the lowest A1 state from which a smooth averaged/-dependent term has been subtracted [ 1]. The effect of the /-independent phase shift in Eq. (7) is clearly visible; for the value A = 101.5 the AI = 4 term completely vanishes.
(7)
0
for/<
The alternating sign o f the tunnelling amplitude for A I = 2 implied by ( 7 ) leads to the A I = 4 alternating order o f the A 1 and B 1 energies exhibited by the Hamiltonian H 2 in Fig. 2. The phase shift ~ ( A / B I ) 1/2 in the tunnelling amplitude ( 7 ) has been confirmed by comparison with the direct diagonalization o f the Hamiltonian (see Fig. 4 ) , and it is found that A/B], should be replaced by ( A / B ] ) - 3h2 in order to agree with the exact quantal results. The fact that the phase o f the tunnelling depends on the parameters A and BI provides a mechanism that may explain the inversion o f the A I = 4 staggering that has sometimes been observed in the experimental data. Another feature o f the Hamiltonian H 2 that should be noted is the fact that rotational bands with C4v quantum number E (the two dimensional representation) should not exhibit A I = 4 staggering (this state is unshifted by the tunnelling) while the intrinsic structures A1 and B1 should give rise to staggering with opposite sign. Concerning the Hamiltonian H1, the value o f ~eS]2 in the limit o f large I can be written "IT
~eS12 ~ 4 I x / ~ c o s ( ~ - ) .
(8)
The occurrence o f the irrational number x/2 c o s ( ~ / 8 ) in the coefficient o f ~rI in ( 8 ) explains the irregular variation in the quartet splittings o f the Hamiltonian H1. In the present note we have focused on the connection o f the A I = 4 staggering and the structure o f the rotational Hamiltonian. The values o f the Y44 deformation and the resulting BI term in the Hamiltonian is a separate issue that must be understood in terms o f the intrinsic structure o f the superdeformed states. As a first step in establishing this connection we have studied the Y44moments o f cranked high j intruder orbitals. We find that indeed these orbitals carry appreciable }~4 moments and that the lowest and the second-lowest intruder orbitals carry moments o f opposite sign. This fact may be related to the empirical observation [ 1 ] that in the A = 150 region A I = 4 staggering occurs for configurations with a single N = 7 intruder but not for configurations with two particles in N = 7 orbitals. It need hardly be emphasized that the present article does not attempt a fit to the available experimental data and thus questions such as t h e / - d e p e n d e n c e o f the A I = 4 term and the specific origin o f the observed inversions in the staggering are not addressed. Although we believe such fits could be obtained, we feel that at present we do not have a sufficiently gen-
298
L Hamamoto, B. Mottelson/Physics Letters B 333 (1994) 294-298
eral understanding of the behaviour of the spectra in the very large parameter space defined by A, B1, B2, and I to make such an adjustment of parameters. We would like to thank S. Flibotte, B. Haas and F.S. Stephens for private communications concerning the AI = 4 structures. The present work was initiated during the International Workshop on High Spins and Novel Deformation held at ECT* in Trento, Dec., 1993. I.H. would like to express her appreciation of the stimulating atmosphere provided by ECT*.
References [1] [2] [3] [4]
S. Flibotte et al., Phys. Rev. Lett. 71 (1993) 4299. B. Cederwall et al., Phys. Rev. Lett. 72 (1994) 3150. 1.M. Pavlichenkov,Phys. Reports 226 (1993) 173. A. Bohr and B. Mottelson, Nuclear Structure, Vol II, pp. 7 ff, Benjamin, New York (1975).
Physics Letters B 333 (1994) 294-298
Superdeformed rotational bands in the presence of Y44 deformation Ikuko Hamamoto a, Ben Mottelson b a Department of Mathematical Physics, Lund Institute of Technology, University of Lund, Lund, Sweden b NORDITA, Blegdamsvej 17, Copenhagen, Denmark and ECT*, c/o Dipartimento di Fisica, 1-38050 Povo (Trento), Italy Received 18 May 1994; revised manuscript received 13 June 1994 Editor: C. Mahaux
Abstract The observation of AI = 4 staggering in the rotational spectra of superdeformed nuclei suggests the occurrence of Y~ deformations in the nuclear shape with associated C4v point-symmetry for the rotational Hamiltonian. We have investigated the general class of Hamiltonians with such symmetry. In addition, we require the axially symmetric terms to favour rotation about an axis that is perpendicular to the long axis of nuclear shape. The AI = 4 staggering can indeed result from the tunneling between the four equivalent minima that occur in the plane perpendicular to the superdeformation symmetry axis, but the occurrence of this effect is a subtle matter depending sensitively on the axially symmetric terms in the Hamiltonian.
In the spectroscopy o f s u p e r d e f o r m e d nuclei, rotational s e q u e n c e s have been o b s e r v e d in which states w i t h A I = 4 exhibit a systematic energy displacement w i t h respect to the c o m p l e m e n t a r y s e q u e n c e o f states w i t h spins l y i n g h a l f way in b e t w e e n ( [ 1,2] ). T h e occurrence o f such a term in the rotational energy suggests that the strongly prolate spheroidal deformation has been slightly perturbed by a term with s y m m e t r y Y44 w i t h respect to the s y m m e t r y axis. In the presence o f such a term the rotational energy can acquire a fourth o r d e r t e r m proportional to (12 - I2) 2 2 reflecting the C4o s y m m e t r y o f the shape. I f such a system w e r e to rotate about the f o u r f o l d s y m m e t r i c l o n g axis (13 ~ I ) the o c c u r r e n c e o f A I = 4 structure in the rotational sequences w o u l d be an i m m e d i a t e and direct c o n s e q u e n c e o f the Y44 d e f o r m a t i o n [ 3 ]. H o w ever, the very stability o f the s u p e r d e f o r m e d shapes results f r o m the large m o m e n t o f inertia associated w i t h rotation about directions perpendicular to the 3axis (/3 < < I ) . Thus, as w e show in the present note,
the occurrence o f A I = 4 rotational structure in the sup e r d e f o r m e d yrast region is a rather subtle p h e n o m e n a w h i c h depends in an important m a n n e r on the axially s y m m e t r i c terms in the rotational H a m i l t o n i a n 1. t After the present work was completed, we received from Dr. Flibotte an abstract of a contribution by I.M. Pavlichenkov and S. Flibotte to the Conference on Physics from Large Gamma-Ray Detector Arrays to be held in Berkeley, August, 1994, in which they exhibit a fit to the experimental data of Ref. [ 1] on the basis of the formalism of Ref. [ 3 ]. This analysis involves a Hamiltonian with C4v symmetry but is based on a solution with the angular momentum aligned along a direction near to the 4-fold axis of the nuclear shape. Identifying this axis as the principal axis of the prolate superdeformed shape, we recognize this solution as one in which the angular momentum is aligned along a direction almost orthogonal to that for which the moment of inertia has its maximum value. We believe such solutions to be inappropriate for reasons given in the text. An alternative interpretation of Pavlichenkov and Flibotte's work (as suggested to us by our referee) would identify the 4-fold axis (called the 3-axis in their work) with the axis of rotation, which is assumed perpendicular to the long axis of the deformation. In this interpretation, however, their Hamiltonian is
0370-2693/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSD10370-2693 ( 9 4 ) 0 0 7 8 3 - 4
L Hamamoto, B. Mottelson/PhysicsLettersB 333 (1994) 294-298 We study the Hamiltonian
H=AI~+B1(I~
-I~)2+B2(I~+I~)
295
3 2
(1)
where we have omitted the 13-independent term proportional to I ( I + 1) that describes the regular I dependence o f the energy o f the yrast states. The first term in ( 1 ) describes the strong preference o f the leading order rotational energy for an orientation o f I perpendicular to the prolate symmetry axis; this implies that we are interested in the region o f parameter space where A is positive and large. The B2 term is the axially symmetric part o f the 4th order rotational energy; in order that the angular momentum remain perpendicular to the 3-axis we must require 2B2I 2 < A. We can, without loss o f generality, confine the parameter space to Bx > 0, provided we study both positive and negative values o f B2. It is useful to view the Hamiltonian ( 1 ) as defining an energy function on the sphere [I[ = constant; the positions of the maxima, minima, and saddle points of this function are illustrated in Fig. 1, and are seen to depend on the value o f the parameter 2 (B1 + Bz) 12/A. Quite generally there are four minima in the equatorial plane and in the yrast states the angular momentum will be localized in the neighborhoods o f these minima. This structure gives rise to quartets o f states, the individual member o f which can be labeled by the quantum number (A1, A2, B1, B2, and E ) o f the C4o point symmetry group. The fact that the intrinsic shape carries C4o symmetry implies that the C4v quantum number o f a state in the spectrum is determined by the intrinsic configuration and all the states o f a given rotational band will carry this same quantum number. (The argument here is identical to that which leads to the condition K = f~ for the rotational bands associated with an axially symmetric intrinsic shape [4] .) Thus, o f the four states associated with the minimum o f ( 1 ) only a single C4~ combination is appropriate for generating the rotational band associated with a particular intrinsic state. To illustrate the variety o f yrast structures that can result from the Hamiltonian ( 1 ), we have directly digrossly deficient since it neglects the term, AI~, (assuming the prolate axis to be the 1-axis) which describes the leading order anisotropy of the moment of inertia. In the presence of this term, the C4 symmetry is badly violated and the rotational dynamics is modified in a major way.
(a) 1 3
(b)
~
2
~
2 1
minimum • , ,, maximum •
saddlepoint ....... separatrix
o
Fig. 1. Classical phase space for C4v rotational Hamiltonian. Provided the minima of the rotational Hamiltonian lie in the equator (i.e. 2B212 < A), there are two possibilities for the arrangement of maxima, minima, and saddle points: (a) for 2 (B t+ B2 )12 > A; and (b) for 2(B1 4- B2)I 2 < A. For each of the two cases, two views of the spherical surface are shown. On the left we look from the direction of the I-axis and the positive 2- and 3-axes are labelled. On the right we look from the direction of the 3-axis and the positive 1- and 2-axes are labelled. In case (a), the energy surface exhibits two different classes of local maximum which are indicated by filled triangles and filled squares, respectively. Table 1 Hamiltonian
A
B]
B2
AI = 4 structure in yrast band
H1 //2 H3
100 100 100
1 1 I
- 1/2 0 -1
irregular regular oscillation no oscillation
agonalized this Hamiltonian for several different values of the parameters (A, B1, and B2) all o f which are chosen so that the classical motion corresponds to rotation about an axis in the equatorial plane. (see Table 1 and Fig. 2). It is seen that for some values o f the parameters the splitting o f the quartets implies a regular A I = 4 alternation in the energies o f the quantum states labelled by the quantum numbers A 1 or B1 (see H 2 ) , for other values ( H 3 ) no alternation
L Hamamoto, B. Mottelson/Physics Letters B 333 (1994) 294-298
296
I
[
I
[ •" + •
03 uJ I03 I-. 03 LLI
O
_J LL
O (.9 Z
_,.I
[
I I I I B1 ( 1 degen.) E ( 2 degen.) A1 ( 1 degen.)
nILl Z LU
5.4
The tunneling matrix element in the WKB approximation is obtained by evaluating the action
4.6
1.3
1.4
3.0
1.5
2.0
S12 = //3(~b)d~b.
2.3
•
Z~
•
ZX
/,
•
&
ZX
+
+
+
+
+
+
+
+
&
•
&
•
•
&
•
•
1.1 (-1)
8.6 (-2)
6.9 (-2)
minl
where the momentum as a function of angle, I3(~b), is obtained by solving the equation
H 2 = 100132+ (112- 122)2 6.5 (-1}
4.0 (-1)
2.7 (-1)
1.9 (-1)
1.4 (-1)
H(/3, 40 = E0 5.7 (-2)
•
z~
•
A
•
Z~
•
Zk
•
+
+
+
+
+
+
+
+
+
t,
•
t,
•
&
•
&
•
&
H 3 = 100132 + (112- 122)2 - (112+ I22)2 1.4
(1)
(o)
(-1) (-2) (.4) (-5) (-6)
1.6
1.3
8.4
5.1
2,8
Z~ +
Z~
A
A
t=
A
~.
+
+
+
+
+
+
•
•
•
•
•
Q
I
I
I
I
I
I
1.7
I
I
I
Fig. 2. Energy splittings within yrast quartets for three different C4v rotational Hamiltonians. The figure shows the relative positions of the states within the yrast quartets as a function of I; the states of the representation E are unshifted by the tunnelling, while the absolute value of the splitting between A1 and BI is given above each quartet (the number in parenthesis denotes the power of ten). The figure is appropriate for even A nuclei (i.e. integer I). For odd A, the quartets are split into two doublets and the C4v classification must be replaced by the spinor representations of the double point group C~; the splitting patterns are similar except that unshifted E states are absent and the A1, B1 states are replaced by the two-fold degenerate states E1 and E2.
occurs, while for still others (H1) the tunnelling between minima generates an erratic pattern as a function of I, sometimes shifting a given symmetry up and sometimes down without any obvious regularity. In order to understand these behaviors we examine the semi-classical evaluation of the tunnelling matrix element connecting adjacent minima of the rotational Hamiltonian. For this purpose we introduce canonically conjugate variables (/3, ~b) on the spherical surface II1 = I, I1 = ~ / ~ - q cos ¢
q sin ¢
(3)
,/
(-1) (-1) (-1) (-2) (-3) (-3) (-4) (-5)
10 12 14 16 18 20 22 24 26 SPIN (I)
12 = V / ~ -
(2)
rain2
H I = 100132 + (112. 122)2 - 0.5(112 + 122)2
m, >" L9
H=AI2+(I2-I23)2(BlCOSZ2d)+Bz)
(4)
EO = B2 I4
where E0 is the energy of the classical minimum of the Hamiltonian (2). The interesting regions of parameter space, determine two different types of behavior for the solutions to (3) and (4). a) if:
2B21z [ B1 A < 1- VBI+B2
(5)
then: I3(q~) is pure imaginary for all values of q~ in between the two minima and the resulting tunnelling amplitude exp(iSj2) is a real number without any I dependent phase. b) if:
2B212 1 >
A
¢ > 1-
B1 B1+B2
(6)
then: the angular momentum 13(~b) becomes complex (acquires a real part) while ~b is intermediate between the two minima and the tunnelling amplitude involves a factor cos(~eSl2) which can change sign as a function of I and thus give rise to staggering in the yrast energies 2. The region of the parameter space satisfying (6) is shown as a shaded area in Fig. 3. The fact that the Hamiltonian H3 defines a trajectory lying entirely in the region below the shaded area of Fig. 3 explains the absence of/-dependence in the pattern of quartet splittings for this Hamiltonian. z It should be noted that the conditions for ~}~eS12 -~ 0 ale quite different from the condition in Fig. 1 that determines the qualitative structure of the classical energy surface. This reflects the fact that the tunnelling takes place in a phase space with complex dimensions and thus reveals features of the dynamics that are invisible in the real phase space of classical mechanics.
I. Hamamoto,B. Mottelson/ PhysicsLettersB 333 (1994)294-298 3.0 3.0 2.0
J
A=90
2.0
cE 1.0 c~
I
t
•
~'~ 1.0 A=100~,~,
0.0 I
I
,,
297 i
I
I
i
I
H = AI32+ (112- Ia2)a yrast A1-band
^,
,.
M
,
~,' ,/'
' ,,,"
A ", *
t
•
0.0
-1.0 ,','~I . i ' ~ ' A=120 ~t i /
-1.0 0.0
1.0
2.0 3.0 4BlI 2 / A
4.0
5.0
For Hamiltonians o f the type o f H 2 (with B2 = 0), the real part o f the integral ( 3 ) can be simply evaluated
=
)
7(1
for I >
"
V
~
-2.0
Fig. 3. Parameter space for C4v rotational Harniltonian. In the shaded region the tunnelling matrix element between equatorial minima acquires an / dependent real phase, which is a necessary condition for obtaining AI = 4 structure. The upper limit, correponding to the condition 2B2/2 < A, is imposed in order to ensure that the rotation is about an axis lying in the equator.
~eS]~
~\ i/
A 4B1
-3.0 16
20
24
28 SPIN (I)
32
36
Fig. 4. Splittings of yrast quartet for three C4v Hamiltonians of type H2 (B2 = 0). The figure is obtained by direct diagonalization of the rotational Hamiltonian and plots only the eigenvalue of the lowest A1 state from which a smooth averaged/-dependent term has been subtracted [ 1]. The effect of the /-independent phase shift in Eq. (7) is clearly visible; for the value A = 101.5 the AI = 4 term completely vanishes.
(7)
0
for/<
The alternating sign o f the tunnelling amplitude for A I = 2 implied by ( 7 ) leads to the A I = 4 alternating order o f the A 1 and B 1 energies exhibited by the Hamiltonian H 2 in Fig. 2. The phase shift ~ ( A / B I ) 1/2 in the tunnelling amplitude ( 7 ) has been confirmed by comparison with the direct diagonalization o f the Hamiltonian (see Fig. 4 ) , and it is found that A/B], should be replaced by ( A / B ] ) - 3h2 in order to agree with the exact quantal results. The fact that the phase o f the tunnelling depends on the parameters A and BI provides a mechanism that may explain the inversion o f the A I = 4 staggering that has sometimes been observed in the experimental data. Another feature o f the Hamiltonian H 2 that should be noted is the fact that rotational bands with C4v quantum number E (the two dimensional representation) should not exhibit A I = 4 staggering (this state is unshifted by the tunnelling) while the intrinsic structures A1 and B1 should give rise to staggering with opposite sign. Concerning the Hamiltonian H1, the value o f ~eS]2 in the limit o f large I can be written "IT
~eS12 ~ 4 I x / ~ c o s ( ~ - ) .
(8)
The occurrence o f the irrational number x/2 c o s ( ~ / 8 ) in the coefficient o f ~rI in ( 8 ) explains the irregular variation in the quartet splittings o f the Hamiltonian H1. In the present note we have focused on the connection o f the A I = 4 staggering and the structure o f the rotational Hamiltonian. The values o f the Y44 deformation and the resulting BI term in the Hamiltonian is a separate issue that must be understood in terms o f the intrinsic structure o f the superdeformed states. As a first step in establishing this connection we have studied the Y44moments o f cranked high j intruder orbitals. We find that indeed these orbitals carry appreciable }~4 moments and that the lowest and the second-lowest intruder orbitals carry moments o f opposite sign. This fact may be related to the empirical observation [ 1 ] that in the A = 150 region A I = 4 staggering occurs for configurations with a single N = 7 intruder but not for configurations with two particles in N = 7 orbitals. It need hardly be emphasized that the present article does not attempt a fit to the available experimental data and thus questions such as t h e / - d e p e n d e n c e o f the A I = 4 term and the specific origin o f the observed inversions in the staggering are not addressed. Although we believe such fits could be obtained, we feel that at present we do not have a sufficiently gen-
298
L Hamamoto, B. Mottelson/Physics Letters B 333 (1994) 294-298
eral understanding of the behaviour of the spectra in the very large parameter space defined by A, B1, B2, and I to make such an adjustment of parameters. We would like to thank S. Flibotte, B. Haas and F.S. Stephens for private communications concerning the AI = 4 structures. The present work was initiated during the International Workshop on High Spins and Novel Deformation held at ECT* in Trento, Dec., 1993. I.H. would like to express her appreciation of the stimulating atmosphere provided by ECT*.
References [1] [2] [3] [4]
S. Flibotte et al., Phys. Rev. Lett. 71 (1993) 4299. B. Cederwall et al., Phys. Rev. Lett. 72 (1994) 3150. 1.M. Pavlichenkov,Phys. Reports 226 (1993) 173. A. Bohr and B. Mottelson, Nuclear Structure, Vol II, pp. 7 ff, Benjamin, New York (1975).