Revista Mexicana de Física 40, Suplemento 1 (199.0 12-21

Quantum rotor and identical bands in deformed nuclei*

J.P. DRAAYER AND c. BAIIRI

Department o/ Physics and Astronomy, Louisiana State UniversityBaton 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 insteadof the J-axis, is proposed for the interpretation of identieal band and spin alignment phenomenain defonned nuelei. This seheme models a many-partiele, shell-model thcory for strongly deformednuelei with the total pseudo-spin of the system deeoupled frorn the rotational motion. The sehemeis 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 alrededordel eje L, en lugar del eje J, para la interpretación de bandas idénticas y los fenómenos de alinea-miento de espines en núcleos deformados. Este esquema es un modelo de la teoría del modelo dpcapas de muchos cuerpos, apropiado para núcleos muy deformados con el pseudoespín total delsistema desacoplado del movimiento de rotación. El esquema se aplica al análisis de las bandassuperdeformadas 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 non-zero intrinsic spin S [1,21. One (called the J-rotor in this paper) simply replaces therotational 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 quantummechanics [31 has been used to interpret diverse rotational phenomena in both physicsand chemistry. The other scheme -called the L-rotor in this paper- is less well studiedbecause its physical relevan ce has heretofore not been fully realized. In this case, the ro-tation is about the L-axis rather than the J-axis; spin degrees of freedom are decoupledfrom the rotational motion. The L-rotor picture emerges as a naturallimit for strongly de-formed rare-earth and actinide nuclei when a many-particle, shell-model coupling schemewith good total pseudo-spin symmetry, which decouples from the rotational motion, isemployed [4-71. Identical bands are interpreted within this framework as pseudo-orbitalangular momentum (L) rotational sequences (L-rotor series) associated with differentpseudo-spin (8) projections .

• Supported in part by the U.S. National Science Foundation.

12

QUANTUMROTORANOIDENTICALllANOSIN OEFOR~lEONUCLEI 13

1.-filfOl'

3

J-rotor

3.

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, andtotal angular momentum J are good quantum numbers, as well as the projection K L of L onthe 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 vectorprecesses about the L axis with various projections [( of J 00 tite intrinsic syrnmetric axis. Thealignment label D can be considered to be the projection of S along the rotation axis. b) In theJ-rotor picture, the system rotates about the J-axis. In this case total orbital angular momentumL is not a good quantum number; there is a family of L values which generate a circle which is thebase 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 thebase of the conic section. The spin-projection K s and the projection K of J are good quantumnumbers (K L = K - K s is not independent). The J -rotor, like the L-rotor, has five good quantumnumbers (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 + cS2,a

(1)

when a seIf-interaction term generated by L. S is assumed and where aa is the rotationalinertial parameter around the intrinsic a-th axis (equal to 1/2Ia where la is the corre-sponding moment of inertia). For a fixed-spin system the S2 term is a constant and ismaintained for completeness. The second term can be replaced by the J2 operator, sinceL . S = ~(J2 - L2 - S2). For the case of an axially symmetric rotor (al = a2 = a i a3)

14 J.P. DRAAYER AND C. BAHRI

this Hamiltonian reduces to

(2)

The energies of this elementary system are giveu by the simple result

E(L,S)} = (a - D L(L + 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(LLcombination, and M is the projection of J on the laboratory-fixed z-axis. This LS-coupledscheme differs from the eigenstates hJ(sSJ( J;'vJ) of the J-rotor which have the projec-tions J(s of S and J( of J on the intrinsic symmetric axis of the system as good quantumnumbers (Fig. lb).

When a band label D = J - L is introduced, the E(L,S)} of Eq. (3) can be re-writtenin 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 arethree bands with alignments D = 1,0 (reference), and -1. When S =~, only two bandsexist and they are shifted by half-integer amounts from the reference structure; in nuelearphysics these would be for the odd-A neighbor of an even-even parent, assuming theaddition of the odd nueleon induces no change in the internal structure of the system. Ingeneral there are 25 + 1 bands with total angular momentum J = Jm;n, Jmin +2, ... whenJ( L = Oand J = Jmin, Jm;n+ 1, J, ... when J( L # O. The Jmin values are rather cornplicatedfunctions of J(L, S, D, ami x = mod(L,2): Jm;n = D + x + 2[(5 - D + 3 - 2x)/4] forJ(L = O and Jm;" = D + [\L + rnax(O, [(S - D - 2I\L + 1)/2]) for J(L # O where [w] is thegreatest integer function. lf the coefficient b of L. S is positive (negative), the D = S bandlies highest (lowest) while the D = -S band Hes lowest (highest) for a given L value. Forthe 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 impossibleto distinguish the L-rotor and the J-rotor pietures based solely on excitation energies.

Stretehed intraband (interband) electric quadrupole (E2) transitions in deforrned nueleiare strongly enhanced (inhibited). AH E2 transitions with tJ.D # O IIl\lSt therefore bestrongly suppressed if the L-rotor picture is to describe nuelear physics phenomena. The

QUANTUM ROTOR ANO IDENTICAL DANOS IN DEFORMED NUCLEI 15

de-excitation energies within a band (assuming !:J.L = 2 transitions and a prolate rotorgeornetry) are given from Eq. (4) as

!:J.E(L.D)(a, b) = E(L+2.S)J+2 _ E(L.S)J

= 2a(2L + 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 + 3).

(6)

(7)

A plot of intraband !:J.E(L,D)(a, b) versus L values therefore yields lines with identiealslopes (4a) but with different intereepts (6a + 2bD). The E2 seleetion rules are obtainedfrom the expression

B(E2;J' ~ J) = / 1("Y[{LLSJIIQelh'[{~L'S'J'W,2 + 1

where Qe is the eleetrie quadrupole operator [101 ami

("Y[(LLSJIIQelh'[{~L'S'J'} = bss'V(21 + 1)(21' + 1)

x IV(S' J L2; L' J)("Y[( LLllQelh' [(~L'}.

(8)

(9)

The IV -funetion in Eq. (7) denotes a Raeah recoupling coefficient. For intraband B(E2)transitions, this equation reduces to

B(E2;J' ~ J) =251( L. 2 ~') IV(S'JL2;L'J)12Z2¡J(S)'Wu. (10)471" -/\L O l\L

where Z is the atomic number and ¡J(S) the usual collective model deformation parameterwhich may be spin dependent. Representative results are displayed in Fig. 2. Note that asthe value of J increases, the intraband B(E2) transition strengths for the same .I valuesfor the D = -1 and D = +1bands become equa!.

3. PSEUDO-SPIN REALIZATION

The L-rotor picture emerges in the context of a mauy-particle, shell-model theory whell-ever space-like and spin-like degrees of freedmn decouple. The many-particle, pseudo-spacejspin coupling seheme for heavy deformed nuclei is an example [4-61. In this case,replacing the normal single-particle orbital angular momentum (1) and spin (8) operatorsby their pseudo-orbital (1) and pselldo-spiu (8) cOllntcrparts transforms the one-body orbit-orbit (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 AND C. BAHRJ

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QUANTUM ROTOR ANO IDENTICAL RANOS IN OEFORMEO NUCLEI 17

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FIGURE 2. B(E2) strengths for a I( L = O scenario \vith b = 2a: a S = O L-rotor and titecorresponding S = 1 series. Strength for L < 16h are shown in a) (previous page), while those forL> 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 J J seheme in whieh therotational angular momentum and spin oí the protons are coupled first, and then this total angularmomentum is coupled to the corresponding toatl nentran angular momentuffi. b) The LS couplingscheme where the rotational angular momenta oí the protons and ocutraos are coupled first, andthen this total rotational angular momentum is eoupled to the complementary total spin. e) TheSU(3) strong-coupling scheme in which the eoupling of SU(3) quantum number for protons andneutrons is done first and then (as in the LS scheme) this is followed by coupling the total rotationalangular momentum to the total intrinsic spin.

for example Refs. [8,91, only the latter special one preserves the oscillator structure andtherefore admits a many-partic1e, shell-model theory wit~ known group properties.) As aconsequence of the fact that VI, '" O, the many-partic1e S is a good quantum number insuch a theory [51. Furthermore, since this special transformation carries the harmonic oscil-lator hamiltonian into a pseudo-oscillator (Ho --> Ho + ñw) and the deformation inducingquadrupole-quadrupole interaction (Q. Q), which dominates the residual interaction, intoits pseudo (quadrupole-quadrupole) counterpart (Q. Q) with (at most) small correctionterms, the L-rotor picture models a many-partic1e, shell-model theory with a strong defor-mation inducing residual interaction and good many-partic1e pseudo-spin symmetry underthe replacement L --> L and S --> S (where L is the sum of the pseudo-orbital angularmomenta 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 angularmomentum J is left invariant under the pseudo-spin transformation. The L-rotor picturemodels a many-partic1e pseudo-LS coupling scheme.

Valence protons and neutrons in heavy nuc1ei fill different major oscillator shells. Anappropriate scheme for simulating a many-partic1e, shell-model theory in this case is there-fore two interacting L-rotors. (The superposition of these two rotors does not violate thePauli PrincipIe beca use they refer to <:iifferent partic1e types.) The model can assume var-ious forms depending upon whether L~ of the protons and Lv of neutrons first couple totheir own spins [J = (L~ + S~) + (Lv + Sv) = J~ + Jvl or first couple with each otherwith the spin coupling done last [J = (L~ + Lv) + (S~ + Sv) = L + SI. Since the proton-neutron quadrupole-quadrupole field favors a product configuration displaying the max-imum 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 usualway (see Fig. 3).In its simplest form the pseudo-space/spin coupling scheme is not a complete shell-

model theory because it only takes direct account of nueleons occnpying normal parityorbitals. Nueleon pairs distributed in the unique parity intruder orbitals (one for protonsand one for neutrons) are treated as spectators which (at most) contribute to the many-partiele 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 ofstrong pair alignment phenomena. Silllilarly, the L-rotor picture can only be consideredapplicable in a regime where pair alignlllent effects do not dominating the dynamics. Thenormal and unique parity parts of the proton ami neutron spaces are then considered tobe weakly coupled with nueleons in the unique parity orbitals serving only to renormalizethe collective dynamics generated by interactions among the partieles in the unique parityorbitals. Though this picture is an over simplification, the scheme has been shown to workreasonably well so long as there is no pair breaking nor pair scattering of nueleon fromthe nnique parity levels to the normal <lIlesor 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]) transi-tion energies. For example, certain superdeformed bands in 194Hg appear to be nearly iden-tical to a superdeformed band in 192HgJI3-18). The many-partiele, pseudo-spin sC~lemecan be applied in this case by assigning S = O to the superdeformed band in 192Hg, S = Oto superdeformed band (1) in 19411g,and 5 = 1 to superdeformed bands (2) and (3) in19411g.The non-zero pseudo-spin is then associated with neutro n alignment and different5 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 amanY-Jlartiele, pseudo-spin dynamics with the two additional neutrons in 19411gas com-pared with 19211goccupying normal parity orbitals, are different from earlier analysesmade within the context of a single- partiele picture, Refs. [19-211. Regarding this differ-ence it is indeed unfortunate that the available data, which are limited to the transitionenergies only, provide no guidance as to whether the extra neutrons are expected to lie inthe normal or unique parity orbitals. Assigning the extra neutrons to the normal paritypart of the space is consistent with the fact that the experimental valnes for the transitionenergies within bands (2) and (3) are almost identical, differing from one another only bya " shift in the total angular momentull1. The deforll1ation parameters 13(5) can be deter-mined by fitting to the measured B(E2) transition strengths, or derived microscopicallyusing the appropriate pseudo-SU(3) coufiguration. This letter follows the assignll1ent forfinal total angular 1I10mentum JI given in the Ref. [161.

Thc £-rotor picture predicts tluce V-hanus fOf an S = 1 configuratiollj howcvcr, sOlneof the bands may not be separately distinguishable. For examJlle, if b = 2a the transitionenergies 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 param-eters 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 bandsin 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 farstates 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 therange L = 10-38h correspanding ta E, = 255-735 keY where the complete experimentalresults are available. The bands have nearly the same inertial parameter a = 4.5 keY. TheL. 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 thetransitian energies nat equally spaced (as appased ta the case af rigid ratars), ane cannatdeduce unambiguausly the intercept af the f!,.E versus L curves. Hawever, if a canstant kreplaces the 6a term in Eqn. (5), this canstant k turos out to be the same as far the sameS value (see Fig. 4). Within the framewark af this madel, the integer alignment in the Hgisatapes occurs as a cansequence af the b = 2a conditian.

5. CONCLUSIONS

An L-ratar picture is impartant when strangly defarmed canfiguratians are favared andthe caupling between spatial and spin degrees af freedam is weak. The madel gives rise taL(L + 1) ratatianal sequences assaciated with each af the (25 + 1) spin arientatians, andthis allows an identificatian af a spin alignment band label D = J - L. The L. 5 cauplingdetermines 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 fromshifts in the excitation spectra. A value b = 2a for the L . 5 strength leads to integeralignment and produces J(J + 1) rotational sequences.\Ve have shown that the appearance of identical superdeformed bands in 194Hg is consis-

tent with an [-rotor picture which emerges naturally within the context of many-partieletheory with good total pseudo-spin sy_mmetr~ and its pseudo-5U(3) and pseudo-symplecticextensions. The occurrence of only 5 + 1 (5 + t) numbers of D bands for integer (half-integer) pseudo-spin nuelei, inslead of the expected 25 + 1 distinct bands, happens whenb = 2a. For members of lhe same pseudo-spin multiplel, this band degeneracy doubles theB(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 provingor disproving the model. Deviations from this simple picture, such as differences in thekinematic and dynamic moments of inertia, are expected to teach us additional physicsconcerning, for example, the alignment of partieles in the unique parity intruder levels.

ACKNOWLEDGMENTS

Comments and suggestions from R.V. Janssens are gratefully acknowledged. The authorsalso acknowledge helpful discussions with H. Naqvi and J. Escher as well as comments onthe manuscript from D. Troltenier and A. 13l0khin.

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R1582.