Download Quantum rotor and identical bands in deformed nuclei

Revista
Mexicana de Física 40, Suplemento
1 (199.0 12-21
Quantum rotor and identical bands in deformed nuclei*
J.P.
Department
DRAAYER
AND
c.
BAIIRI
o/ Physics and Astronomy, Louisiana State University
Baton Rouge, Louisiana USA 70803-4001
Received 16 February 1994; aeeepted 14 May 1994
AnSTRACT.
A quantum rotor with intrinsic spin, where the rotation is about tile L-axis instead
of the J-axis, is proposed for the interpretation of identieal band and spin alignment phenomena
in defonned nuelei. This seheme models a many-partiele, shell-model thcory for strongly deformed
nuelei with the total pseudo-spin of the system deeoupled frorn the rotational motion. The seheme
is applied to an analysis to the superdeformed bands in 19'Hg and 10. Hg.
RESUMEN.
Se propone un rotor cuántico con espín intrínseco, donde la rotación se hace alrededor
del eje L, en lugar del eje J, para la interpretación de bandas idénticas y los fenómenos de alineamiento de espines en núcleos deformados. Este esquema es un modelo de la teoría del modelo dp
capas de muchos cuerpos, apropiado para núcleos muy deformados con el pseudoespín total del
sistema desacoplado del movimiento de rotación. El esquema se aplica al análisis de las bandas
superdeformadas en 19'Hg y ,., Hg.
PACS:21.10.Re; 21.10.Ky; 21.60.Cs
l.
INTRODUCTION
Two extensions of classical rigid rotor dynamics are fonnd in quantum systems with nonzero intrinsic spin S [1,21. One (called the J-rotor in this paper) simply replaces the
rotational angular momentum L by the total angular momentum J, where J includes S.
This simple model, which has been thoroughly studied since the earliest days of quantum
mechanics [31 has been used to interpret diverse rotational phenomena in both physics
and chemistry. The other scheme -called the L-rotor in this paperis less well studied
because its physical relevan ce has heretofore not been fully realized. In this case, the rotation is about the L-axis rather than the J-axis; spin degrees of freedom are decoupled
from the rotational motion. The L-rotor picture emerges as a naturallimit
for strongly deformed rare-earth and actinide nuclei when a many-particle, shell-model coupling scheme
with good total pseudo-spin symmetry, which decouples from the rotational motion, is
employed [4-71. Identical bands are interpreted within this framework as pseudo-orbital
angular momentum (L) rotational sequences (L-rotor series) associated with different
pseudo-spin (8) projections .
• Supported in part by the U.S. National Science Foundation.
12
QUANTUMROTORANOIDENTICALllANOSIN OEFOR~lEONUCLEI
3
13
3.
J-rotor
1.-filfOl'
FIGURE 1. Schematic vector diagrams for L-rotor and J-rotor systems. a) In the L-rotor picture,
the system rotates about the L-axis. The rotational angular momenlum
L, intrinsic spin S, and
total angular momentum J are good quantum numbers, as well as the projection K L of L on
the intrinsic symmetry axis (3) and the projection M of J on the laboratory axis (z). The five
(KLLSJM)
independent quantum numbers are sufficient to label L-rotor states. The J vector
precesses about the L axis with various projections [( of J 00 tite intrinsic syrnmetric axis. The
alignment label D can be considered to be the projection of S along the rotation axis. b) In the
J-rotor picture, the system rotates about the J-axis. In this case total orbital angular momentum
L is not a good quantum number; there is a family of L values which generate a circle which is the
base of a circular cone with spin S as the hypotenuse and the head of the J-vector as its apex.
The J-axis is neither perpendicular to the Ks plane nor does it pass through the center of the
base of the conic section. The spin-projection K s and the projection K of J are good quantum
numbers (K L = K - K s is not independent). The J -rotor, like the L-rotor, has five good quantum
numbers (KsSKJM).
2. L-ItOTOIt MODEL
Consider the simple model of a rotor with core angular momentum L, intrinsic spin S,
and total angular momentum J = L + S (Fig. la). The hamiltonian for this system is
H =
L aa La + bL . S + cS
2,
(1)
a
when a seIf-interaction term generated by L. S is assumed and where aa is the rotational
inertial parameter around the intrinsic a-th axis (equal to 1/2Ia where la is the corresponding moment of inertia). For a fixed-spin system the S2 term is a constant and is
maintained for completeness. The second term can be replaced by the J2 operator, since
L . S = ~(J2 - L2 - S2). For the case of an axially symmetric rotor (al = a2 = a i a3)
14
J.P.
DRAAYER AND
this Hamiltonian
C.
BAHRI
reduces to
(2)
The energies of this elementary
E(L,S)} = (a -
D
L(L
system are giveu by the simple result
+ 1) + (a3 -
a)J(1
+ ~J(J + 1) +
(c -~)
5(5
+ 1),
(3)
where J(L is the eigenvalue of L3. The correspondiug eigenstates have the form hJ( LLSJ M)
where '1 is a running integer index used to distinguish multiple occurrences of a given J(LL
combination, and M is the projection of J on the laboratory-fixed z-axis. This LS-coupled
scheme differs from the eigenstates hJ(sSJ( J;'vJ) of the J-rotor which have the projections J(s of S and J( of J on the intrinsic symmetric axis of the system as good quantum
numbers (Fig. lb).
When a band label D = J - L is introduced, the E(L,S)} of Eq. (3) can be re-written
in the form
E(L,S)} = aL(L
+ 1) + bLD + (a3 -
a)J(1
+ ~D(D + 1) +
(c -
n
5(5
+ 1).
(4)
The quantity D introduced in this description (D = S, S - 1, ... , -S) indicates the (spin)
alignment of .the band relative to the reference (D = O) bando For S = 1, there are
three bands with alignments D = 1,0 (reference), and -1. When S =~, only two bands
exist and they are shifted by half-integer amounts from the reference structure; in nuelear
physics these would be for the odd-A neighbor of an even-even parent, assuming the
addition of the odd nueleon induces no change in the internal structure of the system. In
general there are 25 + 1 bands with total angular momentum J = Jm;n, Jmin + 2, ... when
J( L = O and J = Jmin, Jm;n + 1, J, ... when J( L # O. The Jmin values are rather cornplicated
functions of J(L, S, D, ami x = mod(L,2): Jm;n = D + x + 2[(5 - D + 3 - 2x)/4] for
J(L = O and Jm;" = D + [\L + rnax(O, [(S - D - 2I\L + 1)/2]) for J(L # O where [w] is the
greatest integer function. lf the coefficient b of L. S is positive (negative), the D = S band
lies highest (lowest) while the D = -S band Hes lowest (highest) for a given L value. For
the special case when b = 2a, Eq. (3) reduces to
E(L,S)} = aJ(J
+ 1) + (a3 -
a)J(1
+ (e -
a)S(S
+ 1),
(5)
which gives the energies of a J-rotor with the projection governed by J(L rather than J(.
Sincc the projection quantuffi numbcrs are Bot mcasurable, when b = 2a it is impossible
to distinguish the L-rotor and the J-rotor pietures based solely on excitation energies.
Stretehed intraband (interband) electric quadrupole (E2) transitions in deforrned nuelei
are strongly enhanced (inhibited). AH E2 transitions with tJ.D # O IIl\lSt therefore be
strongly suppressed if the L-rotor picture is to describe nuelear physics phenomena. The
QUANTUM
ROTOR ANO IDENTICAL
de-excitation energies within a band (assuming
geornetry) are given from Eq. (4) as
DANOS IN DEFORMED
!:J.L = 2 transitions
NUCLEI
and a prolate
15
rotor
!:J.E(L.D)(a, b) = E(L+2.S)J+2 _ E(L.S)J
= 2a(2L
(6)
+ 3) + 2bD,
or for the speeial b=2a case from Eq. (5),
!:J.EJ (a) = E(L+2.S)J+2 _ E(L,S)J
= 2a(21
(7)
+ 3).
A plot of intraband !:J.E(L,D)(a, b) versus L values therefore yields lines with identieal
slopes (4a) but with different intereepts (6a + 2bD). The E2 seleetion rules are obtained
from the expression
B(E2;J'
~ J) =
where Qe is the eleetrie quadrupole
2
/
+1
1("Y[{LLSJIIQelh'[{~L'S'J'W,
operator
[101 ami
= bss'V(21
("Y[(LLSJIIQelh'[{~L'S'J'}
x IV(S'
+ 1)(21' + 1)
J L2; L' J)("Y[( LLllQelh' [(~L'}.
The IV -funetion in Eq. (7) denotes a Raeah recoupling coefficient. For intraband
transitions, this equation reduces to
B(E2;J'
~ J) =
1(
25
471"
L.
-/\L
2 ~')
O
l\L
(8)
2Z
IV(S'JL2;L'J)1
2¡J(S)'Wu.
(9)
B(E2)
(10)
where Z is the atomic number and ¡J(S) the usual collective model deformation parameter
which may be spin dependent. Representative results are displayed in Fig. 2. Note that as
the value of J increases, the intraband B(E2) transition strengths for the same .I values
for the D = -1 and D = + 1 bands become equa!.
3.
PSEUDO-SPIN
REALIZATION
The L-rotor picture emerges in the context of a mauy-particle, shell-model theory whellever space-like and spin-like degrees of freedmn decouple. The many-particle,
pseudospacejspin coupling se heme for heavy deformed nuclei is an example [4-61. In this case,
replacing the normal single-particle orbital angular momentum (1) and spin (8) operators
by their pseudo-orbital (1) and pselldo-spiu (8) cOllntcrparts transforms the one-body orbitorbit (VII) and spin-orbit (VI,) interaetions into their corresponding pseudo forms VII = VII
and VI, = 4vII - VI, 171. (While a whole class of such transformations can be identified, see
16
J.P.
DRAAYER
.,.,2(iJ}2.
AND
C.
BAHRJ
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FIGURE
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QUANTUM
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(ti)
1111':.')11./1
ROTOR ANO IDENTICAL
RANOS IN OEFORMEO NUCLEI
2
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FIGURE 2. B(E2) strengths for a I( L = O scenario \vith b = 2a: a S = O L-rotor and tite
corresponding S = 1 series. Strength for L < 16h are shown in a) (previous page), while those for
L> IOh in b) (this pagel. Transition strengths less Ihan 0.00IZ2{32W.u. are suppressed fOl c1arily.
18
J.P. DRAAYERANDC. BAHRI
FIGURE 3. Various eoupling seenarios for interaeting L-rotors. a) The
JJ
seheme in whieh the
rotational angular momentum and spin oí the protons are coupled first, and then this total angular
momentum is coupled to the corresponding toatl nentran angular momentuffi. b) The LS coupling
scheme where the rotational angular momenta oí the protons and ocutraos are coupled first, and
then this total rotational angular momentum is eoupled to the complementary total spin. e) The
SU(3) strong-coupling scheme in which the eoupling of SU(3) quantum number for protons and
neutrons is done first and then (as in the LS scheme) this is followed by coupling the total rotational
angular momentum to the total intrinsic spin.
for example Refs. [8,91, only the latter special one preserves the oscillator structure and
therefore admits a many-partic1e, shell-model theory wit~ known group properties.) As a
consequence of the fact that VI, '" O, the many-partic1e S is a good quantum number in
such a theory [51. Furthermore, since this special transformation carries the harmonic oscillator hamiltonian into a pseudo-oscillator (Ho --> Ho + ñw) and the deformation inducing
quadrupole-quadrupole
interaction (Q. Q), which dominates the residual interaction, into
its pseudo (quadrupole-quadrupole)
counterpart (Q. Q) with (at most) small correction
terms, the L-rotor picture models a many-partic1e, shell-model theory with a strong deformation inducing residual interaction and good many-partic1e pseudo-spin symmetry under
the replacement L --> L and S --> S (where L is the sum of the pseudo-orbital angular
momenta of the valence partic1es and S is the corresponding many-partic1e pseudo-spin)
which couple to the total angular momentum J. Note that the totalmany-partic1e
angular
momentum J is left invariant under the pseudo-spin transformation. The L-rotor picture
models a many-partic1e pseudo-LS coupling scheme.
Valence protons and neutrons in heavy nuc1ei fill different major oscillator shells. An
appropriate scheme for simulating a many-partic1e, shell-model theory in this case is therefore two interacting L-rotors. (The superposition of these two rotors does not violate the
Pauli PrincipIe beca use they refer to <:iifferent partic1e types.) The model can assume various forms depending upon whether L~ of the protons and Lv of neutrons first couple to
their own spins [J = (L~ + S~) + (Lv + Sv) = J~ + Jvl or first couple with each other
with the spin coupling done last [J = (L~ + Lv) + (S~ + Sv) = L + SI. Since the protonneutron quadrupole-quadrupole
field favors a product configuration displaying the maximum deformation, the second scenario is preferred in nature. In the pseudo-spacejspin
QUANTUM
ROTOR ANIl IllENTICAL
BANIlS IN IlEFORMEIl
NUCLEI
19
pictnre, this is accomplished through the strong coupling of pseudo-SU(3) representations,
(5,./,.) x (.\vl'v) ~ (.\¡i); the pseudo-spins (5. and 5v) are coupled to total 5 in the usual
way (see Fig. 3).
In its simplest form the pseudo-space/spin
coupling scheme is not a complete shellmodel theory because it only takes direct account of nueleons occnpying normal parity
orbitals. Nueleon pairs distributed in the unique parity intruder orbitals (one for protons
and one for neutrons) are treated as spectators which (at most) contribute to the manypartiele dynamics in an adiabatic way. This assumption can obviously only be valid for
¡evels well below and high above the backbending regio n which signals the importance of
strong pair alignment phenomena. Silllilarly, the L-rotor picture can only be considered
applicable in a regime where pair alignlllent effects do not dominating the dynamics. The
normal and unique parity parts of the proton ami neutron spaces are then considered to
be weakly coupled with nueleons in the unique parity orbitals serving only to renormalize
the collective dynamics generated by interactions among the partieles in the unique parity
orbitals. Though this picture is an over simplification, the scheme has been shown to work
reasonably well so long as there is no pair breaking nor pair scattering of nueleon from
the nnique parity levels to the normal <lIles or vice versa [llj.
4.
SUPERDEFORMATlON
APPLlCATION
Rotational bands in several deformed nuclei have been found to have identical (within
:: 2% for normal deformed nuelei [121 and :: 1% for superdeformed nuelei [13-18]) transition energies. For example, certain superdeformed bands in 194Hg appear to be nearly identical to a superdeformed band in 192HgJI3-18). The many-partiele, pseudo-spin sC~leme
can be applied in this case by assigning S = O to the superdeformed band in 192Hg, S = O
to superdeformed
band (1) in 19411g,and 5 = 1 to superdeformed bands (2) and (3) in
19411g.The non-zero pseudo-spin is then associated with neutro n alignment and different
5 values have the possibility of distinet defonnation parameters: j3(O)['92I1g], j3(O)['94Hg,
band (1)]' and j3{l)[194Hg, band (2) and (3)1. These band assignlllents, which assumes a
manY-Jlartiele, pseudo-spin dynamics with the two additional neutrons in 19411gas compared with 19211g occupying normal parity orbitals, are different from earlier analyses
made within the context of a single- partiele picture, Refs. [19-211. Regarding this difference it is indeed unfortunate that the available data, which are limited to the transition
energies only, provide no guidance as to whether the extra neutrons are expected to lie in
the normal or unique parity orbitals. Assigning the extra neutrons to the normal parity
part of the space is consistent with the fact that the experimental valnes for the transition
energies within bands (2) and (3) are almost identical, differing from one another only by
a " shift in the total angular momentull1. The deforll1ation parameters 13(5) can be determined by fitting to the measured B(E2) transition strengths, or derived microscopically
using the appropriate pseudo-SU(3) coufiguration. This letter follows the assignll1ent for
final total angular 1I10mentum JI given in the Ref. [161.
Thc £-rotor picture predicts tluce V-hanus fOf an S = 1 configuratiollj
howcvcr, sOlne
of the bands may not be separately distinguishable. For examJlle, if b = 2a the transition
energies of the b = + 1 band match those of the b = -1 band identically. Since the
20
J.P. DRAAYERANDC. BAIIRI
Superdeformed bands far Hg
5=1
II9-!Hg(IJ
D=O
5=0
200.0
)000
400,0
500,0
600.0
700.0
800.0
Ey
FIGURE4. Transitian energies far an L-rotar system with D-band assignments. The L-rotar parameters are a = 4.5 keY, b = 9.0 keY, and k = 92 keY and 74 keY far S = Oand S = 1, respectively.
The tap band (heavy lines) far each pair gives the experimental results far superdefarmed bands
in the identified Hg isatapes while the battam ane (light lines) is the L-rotar descriptian.
B(E2} strengths are alsa equal far large J values (see Fig. 2), the transitian strength far
states belonging to the b = :1:1 pair appear to be twice as strang as for the b = O band,
a result that appears to be in agreement with the experimento
Equation (5) was fit to each of the four superdeformed Hg bands for L values in the
range L = 10-38h correspanding ta E, = 255-735 keY where the complete experimental
results are available. The bands have nearly the same inertial parameter a = 4.5 keY. The
L. 5 self-interactian strength far 194Hg can be deduced fram the energy shift af band (2)
campared ta the reference band (3), and is b = 9.0 keY. Since the kinematic (/(1) = L/w)
and dynamic (/(2) = (d2 E/d2 L)-I) maments af inertia are nat precisely equal and the
transitian energies nat equally spaced (as appased ta the case af rigid ratars), ane cannat
deduce unambiguausly the intercept af the f!,.E versus L curves. Hawever, if a canstant k
replaces the 6a term in Eqn. (5), this canstant k turos out to be the same as far the same
S value (see Fig. 4). Within the framewark af this madel, the integer alignment in the Hg
isatapes occurs as a cansequence af the b = 2a conditian.
5.
CONCLUSIONS
An L-ratar picture is impartant when strangly defarmed canfiguratians are favared and
the caupling between spatial and spin degrees af freedam is weak. The madel gives rise ta
L(L + 1) ratatianal sequences assaciated with each af the (25 + 1) spin arientatians, and
this allows an identificatian af a spin alignment band label D = J - L. The L. 5 caupling
determines any deviatian fram the reference band (D = O) -which can be significant for
QUANTUM
ROTOR
ANO
IDENTICAL
nANOS
IN OEFOR~IEO
NUCLEl
21
low L values but is negligible for high L values- and its strength can be extracted from
shifts in the excitation spectra. A value b = 2a for the L . 5 strength leads to integer
alignment and produces J(J + 1) rotational sequences.
\Ve have shown that the appearance of identical superdeformed bands in 194Hg is consistent with an [-rotor picture which emerges naturally within the context of many-partiele
theory with good total pseudo-spin sy_mmetr~ and its pseudo-5U(3) and pseudo-symplectic
extensions. The occurrence of only 5 + 1 (5 +
numbers of D bands for integer (halfinteger) pseudo-spin nuelei, inslead of the expected 25 + 1 distinct bands, happens when
b = 2a. For members of lhe same pseudo-spin multiplel, this band degeneracy doubles the
B(E2) transition strengths of DolO bands with respect to those of the reference (D = O)
bando Measurements of B(E2) rates for heavy deformed nuelei are crucial for proving
or disproving the model. Deviations from this simple picture, such as differences in the
kinematic and dynamic moments of inertia, are expected to teach us additional physics
concerning, for example, the alignment of partieles in the unique parity intruder levels.
t)
ACKNOWLEDGMENTS
Comments and suggestions from R.V. Janssens are gratefully acknowledged. The authors
also acknowledge helpful discussions with H. Naqvi and J. Escher as well as comments on
the manuscript from D. Troltenier and A. 13l0khin.
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