scissors and twist modes) in heavy deformed nuclei e56

scissors and twist modes) in heavy deformed nuclei e56

REVISTA MEXICANA DE FÍSICA 46 SUPLEMENTO
l. 71-76
SEPTIEMBRE 2000
MI strengths (scissors and twist modes) in heavy deformed nuclei e56-160Cd)
J.P. Draayer and G. Popa
Departmem
LOllisian{l Srate University
Batol1 Rouge, LA 70803-4001, U.SA
of PhysiC!i Qlld ASlrollomy.
J.G. Hirsch
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico
ApartadoposraI70-543,
04510 México, D.F, Mexico
C. Vargas
Departamento de Física, Centro de /lll'estigaci Ofl y de Estudios Avanzados de/Instituto
Apartado postall4-l40,
07000 México, D.F, Mexico
Politecflico Nacional
Recibido el 28 de febrero de 2(X}();aceptado el 5 de mnyo de 2000
The Elliott SUD) Model. extended for hcavy nuclei using pscudo-spin. is lISCU lo study low-lying magnctic dipole excitations in dcforroce! lluclei. Thc Hamiltollian includcs single-parliclc encrgics as well as quadrupolc-quadrupole and pairing ¡nteraction terms. Sinec
the quadrupole-quadrupole
interaction dominates, a strong SU(3)~dictated truncalion 01' the model spnce is used. With interaetion strcngths
lixed hy syslematics. good agreemenl with experimental results is found for the excitation and f\11 transition spectra of the well-deformed
evel1-cven 15ú~160Gd ¡sotopes.
Kl')'lI'ords: SU(3) Model; pseudo-spin;
magnetic dipole excitations: 156-160Gd
El modelo SU(3) de Elliot. extendido usando el pseudo-spit¡ para núcleos pesados, es empIcado para es!udiar excitaciones magnéticas
dipolares de haja energía en núcleos deformados. El hamiltoniano incluye energías de partícula independiente e interacciones de apareamiento
y cuadrupolo-cuadrupolo. Dado la dominación de la interacción euadrupolar se emplea un e"pacio fuertemente lmncado basado en la simelria
SU(3). Usando intensidades de las interacciones basadas en la sistemática se ohtiene un huen acuerdo con los datos experimentales para las
energías de excitación y las transiciones M 1 de los isótopos deformados par-par 15(i-160Gd.
Dl's('riprorcs:
Modelo SU(3); pseudo-spill;
excitaciones magnéticas dipolarcs:
160Gd
156-
PAes: 21.60.Fw: 23.20.-g: 27.70.+q
I. Introduction
In reeent years considerable experimental [1-5] and theoretieal [6-9J cffort has been foeused on the study 01' low-Iying
enhanccd MI transition strengths (2-4 single-particle units)
round in rare earth and actinide nuclei. A simple geometrical
inlerprctation of this phenomena is that al' a collective magl1etic dipole excitatian resulting from relative rotational ascillations of the proton and neutron distrihutions against each
othcr, a scissors-Iike actian that ¡ed to the christening al' this
excitation as the "scissors" moJe [10]. It is also clear, howevcr, that non-collective features in the nuclear interaction
are necessary for a complete interpretation of the experimcntal results. In particular. the fragmentation uf the MI transition strcngth among severallevels that are closely packed and
clustercd aroul1J a few strong transition peaks [11J cannot he
rcproduced hy a model that incIudes only colleclÍve degrees
of freedom. Early on (see Ref. 12 and referenccs therein),
models hased 011 more l11icroscopic principIes. while still CI11ploying the hasic concept ol' the rclativc rotalianal mol ion of
lhe proton and neutron distrihutions. wcrc proposed for cxplaining the origin of this collective phenomena. In this artiele, the pseudo SU(3) shell model is used to deserihe Ihe
l1on-collcctive as well as the underlying callective nature of
low-Iying MI transitions in strongly deformcd nuclei.
The pseudo SU(3) model is a many-partide shell-model
thcory that takes full aJvantage of pseudo-spin symmetry [13,141. which in heavy Iluclci is manifested in the near
degeneraey ofthe orhila pairs ((I-I)j=I+I/2,
(1+ 1)]=I-l/2J.
Since Ihe tic algehra 01' the pseudo oscillator is the same as
for the norlllal oscillalor, the pseudo SU(3) symmetry can
he lIsed to partition the full space into distinct subspaces.
Since its introduction in the late sixties [15, 16], the pseudo
SU(3) model had been applied to various propert;cs 01'heavy
deformed nuele; [17-19]. hut these have been Iimiled to
schematic nucleon-nuclcon interactions hecause of technical diflieuItics related to the ealeulation 01'SU(3) matrix elements 01' Illore general interactions. Howcver, a cade is now
availahle [201 thal removes this limitations and allows for
the introduclion nI' interactions, likc pairing which is important rOl'un adcquatc description 01' experimental results. into
pseudo SU(3) model ealeulations [21],
Thc pseudo SU(J) model can be used lO give a
microscopic shcll-modcl interpretation oí' the "scissors"
mode [22, 23J. As noted ahoye, a geometrical interpretation
of this phenomena in terms oí' a scissors-like relative motion
72
J,P. DRAAYER. G. POPA, lG. HIRSCH. AND C. VARGAS
o •.lhe proran and nCUlron distrihutions--ealled
lhe Two Rolor t\lodcl (TRM)-pararnClcrizcs
Ihe mOlian in Icrms of an
angle H llctwecn lhe principal axcs ofaxially
syrnrnclric proIon and neutron distrihutions.
shcll-Illodcl schcl1lcs-hoson
and an intrinsic
action,
parl lhal describes
rn
H¡n!
as wcll as fCfmion, howcvcr, (his scissors aClion is associ¡¡Ict! with lhe rclativc molion nI' "valcncc" protons ami neutrons only. nol lhe cntirc mass dislrihution.
Tllat such all intcrprctatioll is a corrcct (lile can he secn from lhe raer Iha1
only in this case do lhe cncrgctics (1 + state al 2-3 McV)
within lhe shcll-modcl pit:(ure une discovers Ihal "{\Vis'" modes are possihle for triaxial
match ohservatíans.
In adJition,
has a very simple interrretalion
pseudo SU(3) mode!.
wilhin lhe framcwork
Hin!
The starting roint for a gcolllclrical
inlcrpretation
of lhe
scissors motle within lhe fralllework 01' lhe pseudo SU(3)
shell l1lotlcl is the well.knowll
relation 01' the SU(3) sYIllIl1l'lry group to the sYlllllletry grour 01' lhe tri axial rolor,
/lot(:\) [29.:10], As has heen shown 123,31). a similar relalion holtls for the case 01' two coupled quanturn rotors, one
represenling prolons and the olher nClItrons. Based on a cm.
rcspontlcllcc
oct\\'een (he Casimir operators el ano C:I 01'
SU(J) antl invariants of Ihe rolor group.
2
Tr (O' (e, 1>,,1>,,)]
Tr(o'(e,1>rro1>,,)]
.¡
= 32 C
2
+ 2.
J
I.;Ul ~I =
•. ,1 ),.\-(.,.,+ .\,,
/3(/' + 1)
2.\ + l' + 3 '
This correspondcnce
SU(3) Hami1tonian
also suggests
dC'.!,
(5)
by a two-
+ 1)(.\" + l)e2
+ (/', + 1)("" + 1)1>2]+ E~.
=
where
Eb
""'0 (110 + ~)
is a constant
w" are dcfined
+ 1''''0
(110 + ~)
amI lhe oscillalor
+ E:,.
frequencies
(6)
Wo antl
hy
Wo
=
w. =
)12,.d(.\,
+ 1)(.\" + 1),
)12,'d(/,
+ 1)(/"
+ 1).
(7)
Here Ao and Ao• respectively,
are [he relalive angular momenta aboul an axis perpendicular
to and in lhe symrnc[ry
plane of the proton.neulron
system.
Thc structure 01' the intrinsic Hamillonian
aIlows for
<In interpretation
of the coupled SU(3) irreps (Arr, Itrr) and
(A", JI,,) for protons ami neutrons, respectively. in lerms 01'
oscillator funclions. According to lhe Littelwood
rules 132]
for coupling Young diagrams, lhe allowed producl configurations can he expressetl in malhematicallcfms
wilh the hclp
01"three quanlUm nllmhers (11I.1.k) [31 J:
(.\,/')=
EB(.\, + .\,,-211I+1./,,+
I'v - 2/+
lII)k.
(8)
where the parametcrs
l antl 1// are defined in a flxed range
given by the vallles nI' the initial SU(3) representations,
In
3
= :¡C",
]
+ 1'" + 3 (.\ + l' + 1) ,
this fonnulation
(A, JI) tum out to he independent 01' k which
serves to dislinguish belween Illultiple occurrences
01' equiv.
alent (.\,1') irreps in the tensor produel. The numher 01" k
values is equal lo the maximum
ouler multiplicity,
Prnax
(p
1,2, .... PlIlax)' The l and m lahels can he idenlified 123) with excitation t)uanla al' the lwo dimensional
os-
=
eiliator given hy Eq. (6)
(9)
(2)
(hat one should rewrile the
This corrcsponds (o 1\\'0 t1istinct lypcs 01' 1 + lllo1ion, lhe scissors and twis! Illodes, antl their realization
in lerms 01' lhe
pseudo SU(3) mode!.
The leading
product.
(3)
in lerrns 01' a rotational
(./'.! -
dC2
-
(I)
.'"1 1 flllS
-----
Lv)2
inler-
111.1.1 •.
Ihe irrcdu<:ihle fcprescnlalion
(irrcp) labels (A, Ji) 01' lhe total
SU(3) can he mapped onto the eolieetive shape variahles (f!
amI I ) of lhe joint rolor syslelll 130].
" h
¡I- = -;:- (
-
= c(l~ + I~) + :ld[(.\,
01' the
2. ShclI-l1Jodel gcol1Jctry
('(Lrr
in such a way lllal l/ilJ! can he approximated
dimensional,
anisotropic oscillalor, Explicitly,
distrihulions hecausc rotations hy cPrr and óv ahout the :-axes
nI' Ihe prolon and neulron distributions
emerge as addilional
dcgrces 01' frcedom. (In atlJilion to Ref, 21, see Refs. 2427). As rOl" lhe scissors modc, it is the rclative differcncc in
these rotations Ihat gives rise lo an MI cxcitation. This !lCW
Illode, which together wilh the scissors construclion
determines lhe overall slructure 01' the M 1 transition spcclfllm,
=
=
the proton.neutron
part.
(4)
irrep (11I./)
(.\.1')
=
(O. O). 01" the SU(3) tensor
= (.\, + A,,,
1',
+ 1"')'
(10)
is associatcd wilh a rninimulll in lhe relalive angular tlisplaceIIIclHs uf lhe prolon and nClllron Subsyslems, geIlerating maximllm deform<llion rOl" lile composilc system. AII otller vallles
Re¡'. M('x. I-'í.\. "¡(, SI (20()O) 71-76
MI STRENGTHS
(SCISSORS AND TWIST MODES) IN HEAVY DEFORMED
for m and 1 provide larger expeelalion values for Ihe angular
variahles B ami Ó. rcspeclivcly. which, in turn. are rclated lo
internal cxcitation cncrgics. Assuming prolatc shapcs fOf lhe
parent distrihutions, A<7 > ¡tu. cach phonon added 10Ihe scissors mode produces a largcr ¡ncrease in cnergy than a phonoo
added lo Ihe Iwisting molion. This refleels lhe softness 01'Ihe
(wist dcgrce of frecdom relative lo lhe stiffcr scissors modc.
Thc conliguration
(A, 11)
=
=
A gcneralization ofthis picture is possihle fOftriaxial distrihutions. In this casc, with eithcr Jl1f or JLv are non-7.cro, J
secanJ scissors statc appcars which is given by
=
(Arr +A" -1,
Ilrr +,1" -1)
e
::l.
~ o.,
+
'1
+
O
::o
;r;
0.6
F";Il-:IIH'I,tOlti'>I'
0.4
O.,
O
2.00 1
=
(Arr + Av + 1, Ilrr + 11"" - 2).
FIGURE l. The experimenlal ~11 transilion strenglh spectrum of
160Gd (1J. Note how the transilions are c1usteredaround local cen.
Iroids and that sorne of the c1uslersappcar to be more fragrnented
then others.
3. Shell.rnodel ealculations
To investigate the effect of syrnmetry hreaking terms in
the interactian on the fragmentation 01' the geomclrical
modes [i, llJ, a generalization 01'lhe Hamillonian (3) introduced ahove was used [22J,
HpSU(3)
=-
( U2
+ Drr
The most general situation is when JI1f and Jlv are both
nonzcro. that ¡s. when both the proton and neutron distrihutions are triaxial. Thcn two more 1+ statcs can be identifleJ,
one heing Ihe (111.1)
(0,1) eonliguration wilh
=
73
1.2
N.
( 12)
or (m, 1) = (1, 1J. In lerms 01'an underlying geomelrical pielure, this structure is a superposition af a <p twisting motion
on top uf the lowest scissors conflguration. The encrgy separation 01' Ihe (m,l) ~ (1,0) and (1,1) modes is a resull 01'
different intrinsic l1lotions.
(A. 11)
(156-160Gd)
(1 1)
is lhe tlrst scissors-likc configuration. 11is always pan of lhe
tensor product dccOInposilion and contains a)1I
1+ statc
that is the handhead 01' a l\
1 hand. lis SU(3) parameters are (m, 1) = (1, O), and therefore eorresponds lo a single
cxcitation of Ihe () Illode. This can he intcrprClcd as lhe shcllmodel equivalen! 01' Ihe scissors mode 01' lhe TRM. Note,
however, tha! in conlrast with lhe TRM, hcrc the relative angular displJcements are associateJ with the JeformeJ valencc
nucleon dislributions only.
(A, 11)
NUCLEI
( 13)
It differs fmm the others not only through its intrinsic energy hUI also heeause it helongs lo a f{ = O hand. The
fourlh state, (111,1, kJ = (1,1,2) differs from lhe seeond,
(m, 1, k)
(1, 1, 1), only in its value oflhe ouler mulliplieity
paramcter (J. In sUlTlmary.a pure pseudo SU(3) picture givcs
rise to a maximum 01"four ] + states: scissors, Iwist. and a
Jouhly degcncralc scissors-plus-twisl moJe.
=
Thc experimental rcsults [11. Fig. l. suggest a much
largcr number oí' 1+ statcs wilh non-zero M l transition probabilitics to the 0+ ground state. This can he undcrslood in
terrns 01"the fragmentation 01'Ihe pure symmetry states under
the intlucnce of SU(3) hrcaking residual intcractions. which
is the topic of Ihe following section.
+ (ls,VIIl)C2 + U;iC3 + bh]'2 + eJ ,
L Ii. + Dv L If.
- GrrHj, - GvHf"
(14)
l.
i?..
Ir..
whcre the single-particle angular momcnta
and
are
one-body terms and the proton and neutro n pairing terms Hp
and H'íJ are two-body interactions. Also, a parameter a.1Im
was included to allow an energy shift for SU(3) irreps with
eilher A or 1I odd as Ihese heloog lo differenl irreps (Bo,
n = 1,2,3, ralhe, lhan A) 01'the inlrinsie Vierergruppe (D,)
symmetry [33J.
To select an appropriate set of SU(3) hasis functions, one
f1rstdetennines the proton and neutron occupancies hy filling
pair-wise from below the single-particle Icvcls of the gencr.
alized Nilsson Hamillonian [341,
"o
=
"w +
el .s
+ Dl2
-mw
2 ,
['2
r f310
'Y,
']
+ SIIl
fñ (Y2 +Y
2)'
v2
-
(15)
for values 01' (J and 'Y that give lhe lowest total energy 01'
the combined prolon and neutron systcms. Dne then determines the number of valencc-space nuclcons in the normal
and unique parity Icvels, the Intter bcing intruder states that
are pushed down inlo the valcnee spaee from ¡he nex! higher
shell hy the slrong spin.orhit interaclion. An overall simplifying assumption made in mosl pseudo SU(3) model ealeulalions is that the relevaot dynamies can he deserihcd hy
laking ¡nto account the nucleons in the normal parity sector
only [35]; the nuclcons in intrudcr statcs (unique parity sector) are assumed lo follow in an adiabatic manner the motion
of the nucleons in normal parity sector with their effect repre-
HI'I'. Mex. f'ís. 46 SI (2000) 71-76
74
lP. DRAAYER. G. POPA, J.G HIRSCH, AND C. VARGAS
1' __
"
~: r~.-~:
6'_
;-
!
12
2'
1'
1'--"
6'
2" o'
,
,-
O-
S'-_M'
O
;;-
"
~
6'__
6'
4' __
4'
2'--2"
0'--0'
o
"
~
M'-_R'
~
b' __
6'
4' __
4'
01>
"'
o
6'
2'
2" o'
,
~: ~:
~.-~:
1.2
e;
6'_
e
~:====~:
~
•
"
o
,-
O'
6' __
6'
4' __
4'
2" __
2"
hp
Th~"ry
0'--0'
Up thy
o
e)(p thy
e)(p thy
200
eJ{p thy
2:.0
1,IlJ
Enl."rg)' IMeV]
FlCiUR E 2. Encrgy anJ MI tmnsilion spcClra for the even-cvcn 156-16°(Jd ¡SOIOpcS In l. Experimental c'(cilatioo cnergics and E2 transition
slrcngths wcrc lIscd as input in a fitting routinc todetermine paramctcrs oflhc Ilamiltoniall fm cach syslcm. These werc then used to calculatc
the thcorctical spcctrum and corresponding :.11 transilion slrengths.
TAHI.E 11. TOlal B(t\ll) transition strcnglhs [Jt~] from experi.
mcnl [1,281 and calculalion (22). Experimental and Iheoretical val.
lICSfor H(E2.0i -+ 2i) transition in f?b2 are also givcn.
TA BLE l. Dcformation and occupation numbers fOf the Gd isolOpcs
lIscd in this study. Thcsc pararncters are also used in a determinalion ofthc SU(3) oasis slalcs.
NudcllS
,
¡¡
¡,~Ú(Jd
(UO
00
15HGd
(UI
(¡OS,S4'
1fioGd
0.29
0
0
:::; '"'( :::; (jO
:::; '"'( ~
nA
UN
8
6
6
4
8
6
6
6
8
6
8
6
nÑ
4
0
l/A
Nucleus
[,,'(.]
B(E2.0t
-; 27)
(e'b']
Exp.
Thcory
Exp.
Theory
¡~fi(jd
3.40
2.91
4.66
4.79
1~,KGd
02
3.02
5.02
5.23
~.29
5.19
5.()()
¡hoGd
sentcd through a rcparametcrization of the thcory. f-or Ihe nU
r1ci investigalcd here. the occupalion numbcrs and lhe corrcsponding dcformations (3 and l' are given in Table I.
In the ReL 22 aH SU(3) hasis siates with Cf 2: Cf",,,,
were selecteJ with Cfi, set so thal all proton (o = rr) antl
neulron (O: = ti) irrcp~~nlyingbelow approximately 6 l\.1cV
'\o'ereincluded in the analysis. Thcn all possiblc couplings 01'
Ihese prolOn antl neutron SU(3) irreps, again within 6 McV of
¡he preJicted grounJ state, were taken lo give coupleJ SU(3)
irreps that form basis slalcs of the model spacc. Also. only
slales with J ~ 8 and 5' = O were considcred.
The paramcters 1'orthe Harniltonian given in Eq. (14) ano
Ihe clTcctive chargcs C1f = 1 + qeff ano el-' = qeff useo in lhe
£(2) Iransition operalor,
¿ IJ(MI)
4.21
4
T¡\I(E)
"
= A Ir'
¿
'"
'"¿Cu1u .'(')V
1 I2Af (Tu ('))
1 •
wcrc detcrmincd through a fitting procedurc Ihat incluoco as
input all known levels wilh J S S up through 2 MeV in energy amI sclcctcJ B(E2) transition strcngths. This proccdurc
gavc. in general. good agrccment bet\Vccn the experimental
and thcorcticalnumhers (Fig. 2 and Tahle 11).
Rcccnt calculations \Verc oone lIsiog a rctlncd vcrsion
of the pseudo SU(3) formalism. The rcalistic Hamiltonian
thal was uscd in thcse calculations too k singlc-particlc cncrgics and the quatlrupolc-quadrupolc anJ Illonopole pairing
strengths from systcmalics.
( 16)
Rev. Mex. ,..,:,.. 46 SI (2000) 71-76
+ (/f".,
\./ + hf2
., + asym C':1 + (13C3.
(17)
MI STRENOTHS
MODES) IN HEAVY DEFORMED NUCLEI (156-1600d)
(SCISSORS ANDTWIST
~.~
75
1. 5
S'
2
-
8'
"-,.
7'
"
Il'-_s'
5':=5'
6'-_6'
,'-,.
"-,.
1
"-,. ,.-,.
4'-_4'
•
0'-0'
6'_-1>-
,._,.
o
,'-l'
0'-0'
-0.5
Exp
Th Exp
Th Exp
Th
Exp
o
Th
4.00
FIGURE 3. Thc experimental and theoretical graund-balld cncrgy speclra and the corrcsponding MI transilion spectrum for lMGd. The
Ihcorcticnl resuhs were calculated wilh singlc-parlicle cncrgics. quadrupole-quadrupolc. and pairing ¡nreraetion strengths frem systematics.
Thc othcr (rOlor) pararnctcrs were fil 10 the experimental encrgics below 2 McV.
TABLE 111. D(MI) transition slrengths [tt~] in (he purc syrnrnc.
try [¡mil 01' the pseudo SU(3) model. Thc strong coupleó pseudo
SU(3) irrep (..\, p)gs Cm the graund Mate is givcn w¡lh its proton
and neulron sub-irreps and the irrcps associated w¡lh the 1 + states.
(A', ,1') 1 +. In addition. cach transition is labeled as a scissors (s) or
(wiSl(l) or combination mode.
N"cle"s (.\,,1") (.\",1'") (.\,1')9' (.\'.1")1+ 8(1\11) mode
l'.'H58Gd
H,oGd
(10,4)
(10,4)
(18,0)
(18,4)
(28,4)
(26.5)
1.91
(28.8)
(27.3)
(26.9)
1.61
1.77
(27.7)1
(27.7)'
(29.6)
1.82 s + f
0.083 t + .,
0.56
f
.<
s
+f
s
The olher parameters of the Hamiltonian (17), namely, a,
{J, (/.3, anll aS'ylll' were determined by fitting the low-energy
speclra to lhe experimental numbers. This refined model was
applicd to 156Gd. As with lhe earlier ca1culations, the lowenergy spectra was described wel!. In addition, lhe distrioUlion of B(M 1) transition strengths was determined 10 he
cJoser lo the experimental values lhan in lhe previous calculalion (Fig. 3).
In all cases Ihe M I lransilion operalm [19J
T,: = J3/4/IN
¿{!I~L~ +!I~S~},
(18)
with orbit q-faclors
9~ ~ 1,
!J~ ~ 0,
( 19)
complex transition speclrum which is in much betler agreemenl with experimental results. One finds a number oflransilions thal are close lo the observed ones. Also, the centroid of
the experimental and theoretical MI lransition slrength distribUlion are usually founu lo lie al approximately Ihe same encrgy, so good overall agreement wilh experiment is obtained.
The lolal MI slrenglh. whieh fm Ihe full Hamillonian is
a hillower Ihen for ils pure SU(3) limit due lo destruelive inlerferenee associaled wilh Ihe l11ixing (see Tables 11 and IlI).
shows reasonahle agrcement wilh the experimenlal results,
underestimating them slightly for lhe Gd isolopes. A possihle reason for lhis discrepancy is missing spin I admixtures
in lhe wavefunctions which are known to add MI slrength to
Ihe syslem [3GJ.
4. Conclusion
The pseudo SU(3) model uffers a mieroseopic shell-model
inlerpre(ation oí"the "scissors" mode. In uddilion. il reveals
an addilional "twist" degree 01' freedom that corrcsponds lo
allowcd relalive angular motion of the proton and/or neutron
suh-distrihutions when (hey are lriaxial. Each MI modes is
assoeialed wilh a well-defined SU(3) irrep.ln Ihe pure SU(3)
symrnctry limit oí" the theory lhere are up to four non-zero
MI transitions from cxcited 1+ stales lo Ihe ground slate.
The summcd strenglh is in good agreement wilh experiment.
By adding SU(3) syml11eIrybreaking one-body and Iwo-body
pairing intcraclions to lhe Hamiltonian. it is possibIe lo descrihe the expcrimentally ohserved fragmenlation 01'the MI
sIrenglh.
(i.e., with no cffective q-factors) was used lo determine Iran-
silion slrcnglhs hetween the 0+ ground state and 1+ states.
In Ihe pure SU(3) Iimil of the lheory, there are at mosl I"our
allowcd MI transitions. These, logelher with lhcir classification as the scissors mocle. twisl mode, or a comhination are
listed wilb tbe corresponJing SU(3) irreps in Tahle 111.
I3ecause 01' the SU(3) syrnmetry hreaking terrns
(H;,;") and H;Y, lhis simple piclure gives way to a more
lL ....
Acknowledgments
Supponed by Ihe U.S. National Seienee Foundalion Ihrough
a U.S.-Mcxico Coopcrativc Rescarch grant (lNT-9500474). a
regular granl (PHY-9970769). and a Cooperative Agreemenl
(EPS-9720652) Ihal ineludes malehing frol11 Ihe Louisiana
Board of Regenls Suppml Fund; and by Conaeyl (Mexieo).
Rel'. Mex. Fís. 46 SI (2000) 71-76
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