Chiral vibrations in the A=135 region

0.24 MB·English

by  Daniel Almehed

Chiral vibrations in the A=135 region
Daniel Almehed∗
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
Stefan Frauendorf†
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA and
Institut fu¨r Stralenphysik, Forschungszentrum Dresden-Rossendorf, 01314 Dresden, Germany
(Dated: February 1, 2008)
Chiral vibrations are studied with the RPA plus self-consistent tilted axis cranking formalism in
theA=135region. Inthismethodchiralvibrationsappearasaprecursortothestaticchiralregime.
The properties of the RPA phonons are discussed and compared to experimental data. We discuss
thelimits thechiral region and thetransition to thenon harmonic regime.
7
0 PACSnumbers: 21.60.-n,24.30.Cz,21.30.x
0
2
I. INTRODUCTION molecules the tunneling time is so long that they stay in
p
one of the enantiomers and the level splitting is unmea-
e
S Chirality is an important symmetry in many physical surable. However, for CH3NHF the tunneling is rapid.
6 systems. Inmoleculesitisastaticpropertyofthegeome- The splitting is in the order of 100 meV, i.e. the tun-
neling frequency is in the order of 3000 GHz, which is
trythatappearsinmoleculeswithmorethanfourdiffer-
somewhat larger than the average rotational frequency
] ent atoms, which can have left-handed and right-handed
h ofthemoleculeatroomtemperature. Fortheknownnu-
enantiomers. In these cases the left- and right-handed
t
- geometry is connected by reflection in a plane. In par- clearcases,the left-rightconversionisalwaysrapid. The
l observed energy distance between the chiral partners is
c ticle physics chirality is a dynamical feature of mass-less
u in the order of 100 keV [8], except in very narrow spin
particles,whichindicatesorientationoftheintrinsicspin
n regions, where the bands cross. This is comparable with
relative to the linear momentum. In a rotating nucleus
[ the rotational frequency in the observed spin range.
it arises as a combination of a dynamical property, the

1 angularmomentum,withastaticproperty,thereflection TACisamicroscopicmeanfieldmethodthathasbeen
v symmetric triaxial shape of the nucleus. If the angular showntoverywelldescribetheenergyandtheintra-band
9 momentum vector lays outside any of the three princi- transition rates of the lower of the two chiral partner
6 pal planes, there are a left-handed and a right-handed bands, see e.g. [9]. However, it gives either one achiral
9
arrangement, which are connected by the time reversal self-consistent solution or two degenerate chiral ones. It
0
. operation. Thiskindofchiralityismanifestedasapairof cannotdescribetheleft-rightmode,whichisawellknown
9 (almost)degeneraterotationalbands withthe same par- deficiencyofthemeanfieldsolutionwhenitbreaksspon-
0
ity[1]. Pairsofsuchbandshavebeenseenexperimentally taneously a symmetry (cf. e.g. [1, 10]). In a finite sys-
7
0 in the mass regions A=105, A=135 and A=190 and the- tem, as the nucleus, the symmetry breaking develops in
: oretically described using mean field tilted axis cranking a gradual way. First a precursor appears as a slow vi-
v (TAC) models [2, 3], two-particle-rotor models [4, 5] or bration around the symmetric configuration. Becoming
i
X extensions of the IBA model [6, 7]. increasinglyanharmonic,the vibrationchangesinto tun-
neling between two asymmetric configurations, which is
r Chirality appears both in molecules and nuclei as a
a progressivelyinhibited.
spontaneouslybrokensymmetry: ThemanybodyHamil-
In order to describe the splitting between the two
tonian describing these systems is invariant with respect
bands one has to go beyond the mean field approxi-
to the chiral operation (space inversion or time rever-
mation. So far this has only be done in the frame-
sal, respectively), which has the consequence that the
work of two-particle-core coupling models [4, 5, 6, 7].
exact eigenfunctions are achiral. However, there exist
Thesestudies showadevelopmentfromchiralvibrations
verygoodapproximatesolutions,whicharechiral,where
toward tunneling between static left- and right-handed
theleft-handedandright-handedconfigurationshavethe
configurations with increasing angular momentum. Al-
same energy. The exacteigenfunctions are oddand even
though the quantal nature the core-particle models ac-
superpositionsofthesechiralsolutions. Theenergysplit-
counts well for this aspect of symmetry breaking, they
ting between these states is given by the matrix element
are basedon severalassumptions the validity of whichis
between the two chiral configurations, which is the in-
notassured. Examplesare: Rigidshapeandirrotational-
verse time for tunneling from one to the other. In most
flow moments of inertia for a triaxial rotor core [4, 5],
which are not consistent with microscopic cranking cal-
culations [7], the application of the IBA core, and the
∗Electronicaddress: [email protected] IBAFF coupling scheme [6, 7]. A microscopic treatment
†Electronicaddress: [email protected] starting from the TAC mean field seems important for
2
a better understanding of nuclear chirality. Obviously
thetransitionfromchiralvibrationstochiralrotationin- TABLE I: Quadrupoleforce strength κ0.
volves large amplitude collective motion, a treatment of κ0 [MeV]
which goes beyond the scope of this paper. However the 130Cs 0.0367
regime of chiral vibration can be treated by the random 132La 0.0359
phase approximation(RPA) approachas long as the en- 134Pr 0.0348
ergy splitting between the bands is large enough for the 136Pm 0.0355
assumption of a harmonic vibration remaining reason- 138Eu 0.0363
able. As the energy splitting decreases the anharmonic 140Tb 0.0357
130La 0.0372
effects will become more important until the transition
134La 0.0337
point where the RPA energy becomes zero and the TAC
mean field becomes chiral. The presentpaper is devoted
to a study of the spin range before the transition point
wheretheRPAapproximationwillallowusinvestigating RPA calculations we need an interaction Hamiltonian.
the chiral vibrations and provide insight into the nature Hence, we choose using the self-consistent TAC with the
of nuclear chirality. QQ-force and a constant pair gap (PQTAC [15]) in two
In the static chiral picture on would expect the two major N-shells (N =4 and N =5):
low up
bands to have very similar electromagnetic transition
rates. Recent experiment have shown different intra- 2 5 κ
band transition rates in the two chiral partner bands in H′ = h0+ 2NQ(mN)Q(mN)(−)m+1
134Pr[7,11]while the bandsin128Cs[12]and135Nd[13] mX=−2NX=4
have similar transition rates. The different transition ∆ P++P ω J (1)
ratesin134Prhavebeen interpretedasbeing due tocou- − τ τ τ − ·
τX=n,p (cid:0) (cid:1)
pling of shape degrees of freedom to the orientation de-
gree of freedom [7] or even that the bands are not chiral where h is the spherical Woods-Saxon energy [2] and
0
partner bands at all [11]. Q(N)arethedimensionlessquadrupoleoperatorsforeach
m
Chiral pairs of rotational bands have mainly been
N-shell with an N-dependent force strength [16][24]:
suggested for nuclei with odd proton and odd neutron
number. In the A=135 region, they are build on the 1/3
N B 2Z(N)
πh νh−1 configuration. Candidate bands have κ =κ low− . (2)
11/2 ⊗ 11/2 N 0 N B (cid:18) A (cid:19)
been seen in the odd-odd nuclei 126−132Cs, 130−134La, −
132−134Pr, 136Pm and 138−140Eu. In this work we fo- ThevalueoftheparameterB ischosentoapproximately
cus on the N=75 isotone chain and the Z=57 isotope reproduce the Strutinsky potential energy surface [15].
chainwhichrepresentthe centralpartofthis region. We Setting the parameter B = 0.5 gives a similar prolate
have performed calculations in the framework of tilted oblate mass difference and γ-softness as the SCTAC cal-
axis cranking (TAC) and random phase approximation culations in this mass region. In other mass regions a
(RPA) for the twin bands in the N=75 isotones and the different value of B may be needed. The parameters
Z=57 isotopes built on the {πh11/2,νh11/2} configura- κ0, listed in table I, are adjusted to reproduce the mo-
tion. Examples for chiral partner bands exist also in ment of inertia of the SCTAC calculations. The an-
odd-even nuclei [13, 14]. The case of 135Nd has been gular velocity in the cranking term, ω J, is defined
studied in [13] by means of the method presented in this as ω = ω sinϑcosϕ, ω = ω s·inϑsinϕ, and
1 2
paper. In principle, chiral partner bands should also ex- ω = ω cosϑ, where ϑ and ϕ are the tilt angles [2].
3
istineven-evennucleibuildontwo-particletwo-holelike The monopole pair operatorsare denoted by P , and we
τ
configurations. For these and even more complex con- use a proton and neutron pairing gap, ∆ , that is 80%
n,p
figurations our method can be directly applied. We de- of the odd-even mass difference in this mass region.
scribe the formalism in section II. The results for the
In the TAC calculations for the A=135 region, chiral
energy, amplitudes and transition rates are presented in
solutions appear as a transient phenomenon where the
section III. An analysis of the structure of the phonons
angularmomentum vectormovesfromthe principalaxis
isgiveninsectionIV,whereitwillbedemonstratedthat
planespannedbythe1-(short)and3-(long)axes(ϕ=0)
formostofthecasesthechiralcharacterprevails,i.e. the
to the principal plane spanned by the 2- (intermediate)
coupling to the shape degrees of freedom is weak. and3-axes(ϕ=90◦)[2]. TheRPAmethoddescribesthe
regime when the TAC angular momentum lays in one of
theprincipalplanes. Theexcitedchiralpartnerbandcan
II. FORMALISM bedescribedasachiralvibration,whichisaperiodicmo-
tion of the orientationof the deformed potential relative
Most earlier TAC calculations for chiral bands have to angular momentum vector, i. e. a mode in the ϑ and
used the Strutinsky renormalization for calculating the ϕ degrees of freedom centered about their TAC equilib-
energies (SCTAC [15]). Since we intend to carry out riumvalues. Therearetwocriticalfrequencies. Thefirst
3
is associated with the transition of the TAC mean field
solution into the chiral regime at ϕ = 0, and the sec-
80
ondis associatedwith the transitionout ofchiralregime
at ϕ = 90◦. These transitions of the mean field are are
accompanied by an instabilities of the RPA, where the
energy of the lowest RPA phonon becomes zero. 70
WhentheTACsolutionhas0< ϕ <90◦(staticchiral
| |
regime) our present approach gives zero energy splitting
60
between the two chiral configurations with ϕ = ϕ.
±| | theta
However, higher order terms in our Hamiltonian would
give rise to tunneling between the left-handed and the
right-handed solution which causes an energy splitting 50
between the two bands in this region too. Figure 1
40
0 0
-0.05 ϕ=0 γ fixed -0.05
ϕ minimised γ minimised
30
-0.1 -0.1
0 20 40 60 80
V] V]
Me-0.15 -0.15Me phi
E’ [ -0.2 -0.2 E’ [ FIG. 2: (Color online) Potential energy surface in the rotat-
ing frame for 134Pr at ω = 0.4 MeV/~ as a function of ϑ
-0.25 -0.25 andϕwithconstantdeformation. Theequipotentiallinesare
separated by 25 keV.Darkercolor represent lower energy.
-0.3 -0.3
-30 -20 -10 0 10 20 0 20 40 60 80
γ ϕ
where h is the diagonal mean field Hamiltonian
mf
FIG. 1: Potential energy in the rotating frame for 134Pr at
ω = 0.4 MeV/~. The two solid curves are calculated with hmf =Emf + ekα†kαk (4)
γ or ϕ fixed while the dotted curve is calculated with a full Xk
minimization.
in terms of the self-consistent mean field quasi particle
operatorsα ,andH istheresidualinteraction. Wein-
k res
shows the TAC energy in the rotating frame, spanned troducethequasi-bosonapproximationb† =α†α†,where
µ i j
by the three principal axes of the triaxial density distri- the b† are treated as exact bosons and µ i > j. The
bution, as a function of the tilt angle ϕ and the triaxial µ ≡
HamiltonianH isrewritteninRPAorderbyonlykeeping
deformation parameter γ. The calculation is done for
terms up to second order in the boson operators [10],
the πh ,νh configuration in 134Pr at ω = 0.4
11/2 11/2
{ }
MeV. The energy surface is very flat especially in the ϕ 1
H =E + A b†b + B b†b† +h.c. ,
direction. Whenweallowforthefullminimizationinthe RPA mf µν µ ν 2 µν µ ν
Xµ,ν Xµ,ν (cid:0) (cid:1)
orientationdegrees of freedom the surface becomes sym-
(5)
metric in γ, because the change of sign is equivalent
± where matrices A and B are hermitian and determined
with a reorientation of the axes. It is therefore enough
from the residual interaction [10]. The matrix elements
to consider γ >0 only. Figure 2 shows the potential en-
for the QQ-interactionare
ergy surface in the rotating frame (total routhian) as a
function of ϑ and ϕ. One can see the very soft nature of 2
the potential in the ϕ direction. A = δ E + κ qmqm∗, (6)
µν − µν µ µ ν
The RPA calculates the harmonic excitations around mX=−2
the mean field minimum. It will describe the system 2
as long as we are in the chiral vibrational regime well Bµν = κ qµmqνm∗, (7)
before the transition to static chirality. The transition mX=−2
pointcorrespondstowheretheRPAsolutiongoestozero
where qm are the quadrupole matrix elements in quasi-
energy and the tilted mean field acquires a nonzero ϕ. µ
boson, approximation [10]
After solving the mean field problem the Hamiltonian
can be written as Qm = qmb† +qm∗b , (8)
µ µ µ µ
Xµ
H =h +H (3) qm = b Qm , (9)
mf res µ h µ i
4
andE =e +e arethetwoquasi-particleenergies. (We 0.8
µ i j
0.7 Experiment Experiment
suppressedthesummationovertheshellnumberN.) We 1st RPA 1st RPA
solve the RPA equations V] 0.6 2nd RPA 2nd RPA
Me 0.5 E2qp E2qp
hHRPA,Oλ†i=ERPAOλ†, (10) E’-E’ [210000....2431 Z=55 130Cs Z=57 132La
usingthestrengthfunctionmethodof[10,17]. TheRPA
0
eigenmode operators Oλ† are 0.7 Experiment Experiment
1st RPA 1st RPA
V] 0.6 2nd RPA 2nd RPA
where the RPA Oamλ†p=litXuµdeXsµλXbµλ†µ−anYdµλYbµµλ,are obtained(1b1y) E’-E’ [Me2100000.....24531 Z=5E92q p134Pr Z=6E12q p136Pm
solvingthestandardsetoflinearequationsresultingfrom
0
Eq. (10) together with the normalization condition 0.7 Experimet 1st RPA
1st RPA 2nd RPA
SincewhOeλu,sOeλa†′sie=paXraµblXeµfλoXrcµλe′,−thYisµλsYetµλo′f=linδλeaλ′r.equat(io1n2s) E’-E’ [MeV]21000000......246531 Z=632En2 qd1p 3R8PEAu Z=6E52q p140Tb
is strongly simplified [17].
0
TherearetworotationalspurioussolutionsintheRPA 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5
h_ω [MeV] h_ω [MeV]
spectrum. OneatzeroenergyinducedbytheJ operator
z
andoneatthe rotationalfrequencyω inducedby theJ
+ FIG.3: (Coloronline)Energydifferenceintherotatingframe
operator. Numerically the spurious solutions decouple
between the two chiral partner bands in the N=75 isotone
fromthephysicalRPAsolutionsinastablemannerifthe
chain. RPA results are compared with experimental data [5,
mean field problem is solved accurately enough. In the 7,9,18,19]. WealsoplotthesecondRPAphononenergyand
discussion below we will only refer to the physical RPA thetwo-quasi particle energy, E2qp. The experimental values
solutions, which are in all cases well decoupled from the showthedifferencebetweentheexperimentalrouthiansofthe
spuriousones. The RPAphononenergygivesthe energy two bandsat grid points of ω calculated by interpolation.
splitting between the zero-phonon lower band and the
excitedone-phononbandata givenrotationalfrequency
ω. From the RPA amplitudes we derive the inter-band larger N we approach the shell closing where the defor-
transitionratesusingthemethodof[17]. SinceRPAdoes mation disappears. Smaller N leads to disappearance of
not give any contribution to the intra-band transition the triaxiality. LargerZapproachesthe protondripline,
rates, we use the TAC results [15] for those. whereverylittle highspindataareavailable. Inthelow-
Z part of the A=135 region the TAC energy minimum
sits at ϕ = 0 at low spin. As we move towards larger Z
III. RESULTS theminimummovesoverto ϕ=90◦. Withincreasingω,
ϕ = 90◦ is preferred, because the intermediate axis has
The results for the TAC and RPA calculations for the the largest moment of inertia [7].
N=75isotonesandtheZ=57isotopesispresentedbelow. In the lighter nuclides the energy of the lowest RPA
Wesolvetheself-consistentTACmeanfieldproblemplus phonon is substantially smaller than the mean field two-
RPAoftheHamiltonianinEq.(1)fortheπh11/2⊗νh−111/2 quasiparticle energy and the energy of the second RPA
configuration. phonon. Inthe heavier nuclides the difference is smaller.
For the lowest phonon being a well developed collective
excitation it must have a substantially smaller energy
A. Energies than the next excitations. When the lowest phonon sits
inaregionwhereotherexcitationsarelocatedthecollec-
We present the results of the TAC+RPA calculations tive strength will become fragmented, and the observed
in Fig. 3 and 4, which show the energy of the lowest bands will be much like two-quasiparticle excitations.
RPA phonon and the experimental energy splitting be- Figure 3 shows the results for the N=75 isotones. The
tween the chiral bands, which are identified with each RPA results show a clear trend which reflects proper-
other. Forreferencewealsoplotthe energyofthelowest ties the TAC mean field solutions. In the lighter sys-
two-quasiparticle excitation as well as the energy of the tems, where the TAC solutions has ϕ = 0, the phonon
second collective RPA phonon. The nuclei studied here energy decreases with spin, reaching zero at the critical
representthechiralregionforodd-oddnucleiintheN=75 frequency for the onsetofstatic chirality(ϕ>0). In the
isotonechainandtheZ=57isotopechain. ForlowerZor heavier systems, where the TAC solution has ϕ = 90◦
5
already at the band head, the phonon energy increases with the flat potential in ϕ direction in Fig. 2 and our
with increasing angular momentum. 136Pm is the limit- analysis of the composition of the RPA phonon wave
ing case where we have a chiral vibration around ϕ = 0 function discussed in section IV below. In 134Pr the two
and decreasing phonon energy at low spin, then a static band cross over in experiment. In 130Cs and 135Nd (not
chiral regime 0 < ϕ < 90◦, and finally a second chiral shown here, see [13]) one sees in experiment an avoided
vibrationaroundϕ=90◦ and increasing phononenergy. crossing with some interaction. In all three cases, the
Comparing with experiment, 130Cs, 134Pr, 138Eu have crossing frequency correlates with the frequency where
the right trend and magnitude of the energy splitting. the phonon energy goes to zero. At the moment, it re-
132La,136Pmhaveamuchmoreconstantenergysplitting mains unclear why these nuclei are different from 132La
in the experiment than given by the RPA calculations. and 136Pr with a nearly frequency-independent energy
difference between the bands. To understand this phe-
0.8 nomenon further it is clearly necessary to take into ac-
0.7 Experiment count large amplitude anharmonic effects, which will be
1st RPA
V] 0.6 2nd RPA done in a forthcoming publication [22].
Me 0.5 E2qp
E’-E’ [21000...243 B. Transition rates
130
N=73 La
0.1
Inaddition to the energysplitting, the RPA alsogives
0.8
0.7 Experiment thetransitionratesbetweenthetwochiralpartnerbands.
1st RPA In Fig. 5-10 we show the calculated B(E2) and B(M1)
MeV] 00..65 2En2qdp RPA inter-band rates. The intra-band values calculated by
E’ [100..43 mapeparnosaochfTthAeCya[1r5e]tahreesaalmsoeifnocrltuhdeedtw.oWbiathnidns.thTehperiensteenrt-
E’-20.2 band reduced transition probabilities are given by
134
N=77 La
0.1
0 B(E2,I I m)= 1 e r2Ylab 0 2 m= 2,...,2
0 0.1 0.2 0.3 0.4 0.5 → − τ 2m −
h_ω [MeV] (cid:10) (cid:12) (cid:12) (cid:11) (13)
(cid:12) (cid:12)
and
FIG.4: (Coloronline)Energydifferenceintherotatingframe
B(M1,I I m)= 1 µlab 0 2 m= 1,0,1, (14)
betweenthetwochiralpartnerbandsinZ=57isotopes. RPA → − 1m −
results are compared with experimental data [20, 21]. We (cid:10) (cid:12) (cid:12) (cid:11)
also plot the second RPA phonon energy and the two-quasi where 0 is the RPA gro(cid:12)und(cid:12)state and 1 is the first
| i | i
particle energy,E2qp. excitedRPA state. The chargeseτ are for 1 protonsand
0 neutrons. The spin parts of the magnetic operator are
attenuated by the factor of 0.7. We include terms up to
Figure 4 show the results for the Z=57 isotopes. The
linearRPAorderinthequadrupole,r2Y andmagnetic
magnitudeoftheenergysplittingisreproducedbutthere 2m
µ transition operators. The transition operators are
is a problem with the angular frequency trend in 132La. m
transformed from the intrinsic frame into the lab frame
In experiment 132La looks similar to 130La while in the
by means of the relation
calculations it looks more like 134La.
OurRPAcalculationsgivethecorrectenergysplitting Tlab = Dl (ϕ,ϑ π/2,0)T (15)
away from the region where the RPA energy approaches lm νm − lν
Xν
zero,whichiswherewecanexpecttheRPAtoworkwell.
However,spindependenceisnotalwayswellreproduced. wherethe π/2rotationbringstheintrinsicsystemfrom
−
The RPA phonon energy tends to approach zero faster x-axis to the z-axis representation.
andmore oftenthan inexperiment, whichindicates that Theinter-bandM1transitionsaremuchstrongerinthe
anharmonic effects are needed to understand the data lighter N=75 isotopes where the B(M1) values are typi-
in this regime. The RPA phonon energy is generally in- cally around1/3ofthe intra-bandvalues. In the heavier
creasingordecreasingwithrotationalfrequency. Theex- N=75 isotopes as well as in 134La the inter-band B(M1)
perimental energy splitting, on the other hand, does not are weak. For the E2 transitions in the lighter N=75
change much with ω in some cases, which seem to ap- isotopesthedominantinter-bandtransitionisI I 2,
→ −
pear predominantly in the frequency regions where the except approaching the critical frequency the transition
TAC gives static chirality, 0 < ϕ < 90◦, i.e. TAC+RPA I I 1 increases in strength. In the heavier N=75
→ ±
predicts zero splitting. The constant energy splitting in isotopes B(E2,I I + 1) becomes the largest inter-
→
these regions should be attributed to the tunneling be- band value. However, the I I +1 or I +2 will often
→
tween the left- and right-handed solutions. Its relatively not be seen in experiment due to their low or negative
large value of about 300keV indicates strong mixing be- energy. The ratios of the different transition proba-
tween the two chiral configurations, which is consistent bilities vary substantially over the studied region. The
6
0.12 1.2 0.12
22B(E2) [eb]000...0000.4681 BBBBB(((((EEEEE22222,,,,, IIIII----->>>>>IIIII-+-+)12o12u))t))oooouuuutttt 130Cs BBB(((MMM111,,, III--->>>III-+)1o1u)t)oouutt 130Cs 0001...4682µB(M1) [] 22B(E2) [eb]000...0000.4681 BBBBB(((((EEEEE22222,,,,, IIIII----->>>>>IIIII-+-+)12o12u))t))oooouuuutttt136Pm BBB(((MMM111,,, III--->>>III-+)1o1u)t)oouutt136Pm 0001...4682µB(M1) []
0.02 0.2 0.02 0.2
0 0 0 0
22B(E2) [eb]000...0000.4681 BBBBB(((((EEEEE22222,,,,, IIIII----->>>>>IIIII-+-+)12o12u))t))oooouuuutttt 132La BBB(((MMM111,,, III--->>>III)-+o1u1t))oouutt132La 0001...4682µB(M1) [] 22B(E2) [eb]000...0000.4681 BBBBB(((((EEEEE22222,,,,, IIIII----->>>>>IIIII)-+-+o12u12t))))oooouuuutttt138Eu BBB(((MMM111,,, III--->>>III-+)1o1u)t)oouut1t38Eu 0001...4682µB(M1) []
0.02 0.2 0.02 0.2
0 0 0 0
22B(E2) [eb]000...0000.4681 BBBBB(((((EEEEE22222,,,,, IIIII----->>>>>IIIII-+-+)12o12u))t))oooouuuutttt 134Pr BBB(((MMM111,,, III--->>>III-+)1o1u)t)oouutt 134Pr 0001...4682µB(M1) [] 22B(E2) [eb]000...0000.4681 BBBBB(((((EEEEE22222,,,,, IIIII----->>>>>IIIII-+-+)12o12u))t))oooouuuutttt140Tb BBB(((MMM111,,, III--->>>III-+)1o1u)t)oouutt140Tb 0001...4682µB(M1) []
0.02 0.2 0.02 0.2
0 0 0 0
0.15 0.2 0.25 0.3 0.35 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
h_ω [MeV] h_ω [MeV] h_ω [MeV] h_ω [MeV]
FIG. 5: The RPA inter-band transition rates in 130Cs, 132La FIG.7: TheRPAinter-bandtransitionratesin136Pm,138Eu
and 134Pr. and 140Tb.
1.2 2
1.2 2
22B(E2) [eb]000...4681 BB((EE22,, II-->>II--21))iinn 130Cs B(M1, I->I-1)in 130Cs 11.52µB(M1) [] 22B(E2) [eb]000...4681 BB((EE22,, II-->>II--21))iinn 136Pm B(M1, I->I-1)in 136Pm 011..552µB(M1) []
0.5 0.2
0.2
0 0
01 B(E2, I->I-2) 132La 132La 0 1 138Eu 138Eu 1.5
22B(E2) [eb]000...468 B(E2, I->I-1)iinn B(M1, I->I-1>in 11.52µB(M1) [] 22B(E2) [eb]000...468 B(E2, I->I-2)in B(M1, I->I-1)in 01.52µB(M1) []
0.5 0.2 B(E2, I->I-1)in
0.2
0 0
0 0
1 B(E2, I->I-2) 134Pr 134Pr 1 140Tb 140Tb 1.5
22B(E2) [eb]000...468 B(E2, I->I-1)iinn B(M1, I->I-1)in 11.52µB(M1) [] 22B(E2) [eb]000...468 B(E2, I->I-2)in B(M1, I->I-1)in 01.52µB(M1) []
0.5 0.2 B(E2, I->I-1)in
0.2
0 0
0 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.15 0.2 0.25 0.3 0.35 0.15 0.2 0.25 0.3 0.35 0.4 h_ω [MeV] h_ω [MeV]
h_ω [MeV] h_ω [MeV]
FanIGd.1364:PTr.he TAC intra-band transition rates in 130Cs, 132La aFnIGd.1480:TTbh.eTACintra-bandtransitionratesin136Pm,138Eu
IV. SHAPE AND ORIENTATION AMPLITUDES
B(E2,I I 1) and B(M1,I I 1) are typically
→ ± → ±
the strongest.
TheenergyoftheRPAphononandthetransitionrates
There are experimentaldata for the inter-bandtransi- donotgivedirectinformationaboutthe degreetheRPA
tion rates in 134Pr [23]. Our calculated rates are some- phonon is dominated by orientation or shape fluctua-
what larger than what is seen in experiment. There are tions. The potential energy surfaces as Fig. 1 and 2 give
alsodataavailablefortheratioofB(M1 )/B(M1 )in some hint ofwhichare the importantdegreesof freedom
out in
136Pm[19],butunfortunatelynotinthe frequencyrange but they do not provide conclusive information, because
where we have RPA results. the missing collective mass parameter is as important as
7
0.12 1.2 wheretheoverallsignisambiguousduetothefreedomof
0.1 BB((EE22,, II-->>II-+11))out 130La 130La 1 the phaseofthe wave-function,butthe relativesignsbe-
22B(E2) [eb]000...000468 BBB(((EEE222,,, III--->>>III-+)2o2u)t)ooouuuttt BBB(((MMM111,,, III--->>>III-+)1o1u)t)oouutt 000...4682µB(M1) [] tfiowfxeewedhnifcrthahmeofediiisffsegraievtnteatncchboeymdtphtooenttewhneotsTsaeAlrfeC-croesnoleslvuisattineotnn.t,Tmthheeeanbshofiadepylde-
0.02 0.2 quadrupole moments q = Q and q = q = Q .
0 0 2 −2 2
h i h i
0 0 Its orientation in the laboratory frame is given by the
0.1 BB((EE22,, II-->>II-+11))out 134La 134La 1 three Euler angles ψ, ϑ, ϕ. We decompose the ∆Qm
22B(E2) [eb]000...000468 BBB(((EEE222,,, III--->>>III-+)2o2u)t)ooouuuttt BBB(((MMM111,,, III--->>>III-+)1o1u)t)oouutt 000...4682µB(M1) [] iibnnotdoteyra-mfixpseaodrtffcrthahamantegiewssigtoehfnψecro,antϑset,danϕbt,yqamand,rewionhrtiiocehnataipstaieorxtnptorhefaststheides
0.02 0.2 generated by changes of the shape, which are expressed
0 0 in terms ofchangesof the deformationparametersε and
0.15 0.2 0.25 0.3 0.35 0.15 0.2 0.25 0.3 0.35 0.4
h_ω [MeV] h_ω [MeV] γ. By means of the Wigner functions Dm2m′(ψ,ϑ,ϕ),
we express the five components of the quadrupole op-
FIG. 9: The RPA inter-band transition rates in 130La and erators Q in the laboratory system in terms of the
134La. m
two intrinsic quadrupole moments Q = Qεcosγ and
0
Q =Q =Qεsinγ/√2, i.e.
1.2 2 2 −2
22B(E2) [eb]000...4681 BB((EE22,, II-->>II--21))iinn 130La B(M1, I->I-1)in 130La 011..552µB(M1) [] wtiohQnemrep(aψQr,aϑmi,sϕeat,eεcr,soγ,n)sat=nadnDtt,m2hεe0,Qthγ0r+eaer(cid:0)eDEtum2hle2er+staaDnngm2dle−asr2,d(cid:1)ϕQd,e2ϑf,o,r(ma1n9ad)-
0.2
ψ specify the orientation of the body-fixed coordinates
0 02 in the laboratory frame. We derive linear expressions
1 B(E2, I->I-2) 134La 134La for ∆Q in terms of the fluctuation in Euler angles and
B(E2, I->I-1)in 1.5 m
22B(E2) [eb]000...468 in B(M1, I->I-1)in 1 2µB(M1) [] shape paramete∂rsQγmand ε
0.5 ∆Qm = ∆αi, αi ψ,ϑ,ϕ,γ,ε . (20)
0.2 ∂α ∈{ }
Xi i
0 0
0.15 0.2 0.25 0.3 0.35 0.15 0.2 0.25 0.3 0.35 0.4
h_ω [MeV] h_ω [MeV] Inverting this system of linear equation, we express the
fluctuation of Euler angles and of the shape parameters
FIG. 10: The TAC intra-band transition rates in 130La and in terms of ∆Q and mean field expectation values,
134La. m
∆Q
1
∆ψ = , (21)
2q
the curvature of the potential. In order to analyze the 2
∆Q
nature of the RPA phonons we calculated the fluctua- ∆ϑ = −1 , (22)
tionsofthefivequadrupoleoperators,whicharethende- q2+√2q0
composedintofluctuationsoftheEuleranglesandshape ∆Q
−2
∆ϕ = , (23)
fluctuations. q √2q
2 0
We define the fluctuation of the quadrupole operator −
∆ε √3q q ∆Q + √3q +q ∆Q
QmgeneratedbytheRPAphononasthedifferenceofthe = 2− 0 0 2 0 2, (24)
dispersionsintheone-phononstate 1 =O† 0 andzero- ε (cid:0) (cid:1) 2(q02+(cid:0)q22) (cid:1)
| i 1| i
phononstate|0i,whereO1†createsthelowestphonon,i.e. ∆γ = √3q2+q0 ∆Q0− √3q2−q0 ∆Q2. (25)
(cid:0) (cid:1) 2(q2+(cid:0)q2) (cid:1)
∆Q2 = (16) 0 2
m
1 (Q 1 Q 1 )2 1 0 (Q 0 Q 0 )2 0 . The results are shown in Fig. 11 and 12. One can
m m m m
−h | | i − −h | | i clearly see that the lowest phonon in this region is dom-
(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11)
Usin(cid:12)g that in RPA(cid:12) approxi(cid:12)mation 1 Q 1 (cid:12) = inated by orientation fluctuations especially in the ϕ di-
m
h | | i
0 Q 0 =q we find rection, which is consistent with the TAC potential en-
m m
h | | i
ergy surface in Fig. 2. The amplitude of ∆ϕ is typically
∆Q2 =2 Q ,O† 2. (17) 30-40◦. The fluctuations in ϕ diverge at the point where
m m 1 the RPA energy goes to zero, i.e. at the transition to
h i
aplanar chiral rotation. The fluctuations in ϑ are 10-
We define, ∆Qm, for m=−2,...,2 as 20◦ while the fluctuations in ψ are half that. In allthese
cases,theshapefluctuationsaresmallinamplitudecom-
∆Qm =√2 Qm,O1† (18) pared to the orientation fluctuations. The fluctuations
h i
8
70 70
60 60
50 50
ϕ 40 ϕ 40
∆ 30 ∆ 30
20 20
10 10
0 0
60 130Cs 136Pm 60 130La
50 132La 139Eu 40 132La
∆θ 234000 134Pr 140Tb ∆θ 20 134La
10 0
0
-20
60 60
50
40 40
ψ
∆ 30 ψ
20 ∆ 20
10
0
0
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 -20
h_ω [MeV] h_ω [MeV] 0.1 0.h_2ω [MeV0].3 0.4
FIG.11: (Coloronline)ThefluctuationofEuleranglesinthe
FIG.13: (Coloronline)ThefluctuationofEuleranglesinthe
RPA phonons in the N=75 isotone chain. Cs, La and Pr in
RPAphonons in the Z=57 isotones.
theleft panels and Pm, Eu and Tb in the right panels.
80
60
0 4 136Pm 00 40
∆ε/ε*10 02 113302CLas 113480ETub ∆ε/ε*1--2420000 113302LLaa
134Pr -60 134La
-2
10
6
4 5
γ 2 ∆γ 0
∆
0
-2 -5
-4 -10
0.1 0.2 0.3 0.4
0.1 0.2 h_ω 0[M.3eV]0.4 0.1 0.2 h_ω 0[M.3eV]0.4 0.5 h_ω [MeV]
FIG. 12: (Color online) The fluctuation of shape parameters FIG. 14: (Color online) The fluctuation of shape parameters
in the RPA phonons in the N=75 isotone chain. Cs, La and in the RPAphonons in theZ=57 isotones.
Pr in theleft panels and Pm, Eu and Tb in theright panels.
134La can no longer be thought of as a chiral vibration
even though it has large components of orientation fluc-
in ε are few percent of the equilibrium values and the
fluctuations ∆γ < 6o for all cases but 134La (cf. next tuations in the wavefunction. The largejump in several
of the 134La curves is due to the level crossing between
paragraph). This is in contrastto Ref. [7], which, taking
the first and second RPA solution seen in Fig. 4.
into account the shape degrees of freedom in the frame-
workofthetheIBAFFtwo-particle-coreapproach,found
a strong coupling between the shape and orientation de-
greesoffreedom. Inourmicroscopicapproach,thelowest V. CONCLUSIONS
phonon is a rather clean chiral vibration.
Figures 13 and 14 show how the structure of the low- We have used the RPA method to study the energy
est RPA phonon develops when moving away from the splittingbetweenchiralpartnerbands. TheRPAphonon
N=75 chain. In 130La it is dominated by orientation energygivesthecorrectsizeoftheenergysplittingatlow
fluctuations but not in the ϕ direction as for the N=75 spin in most cases but the trend with increasing angular
cases. This nuclide also has relatively strong γ fluctua- momentum is not always reproduced. The RPA calcula-
tions especially at large ω. In 134La with N=77 we are tionsalsogivetheinter-bandtransitionrates. Astheen-
approaching the N=82 shell closure. This leads to a re- ergy splitting between the chiral partner bands becomes
duction in the deformation and increased amplitudes of small the anharmonic effects are expected to get more
the shape fluctuations in the RPA phonon. The case of and more important.
9
By analyzing the RPA amplitudes we found that near ingtomoreneutron-richnucleistructureofthefirstRPA
Z=57 and N=75 the lowest phonon is a collective chiral phononchangestoacomplexmixtureofshapeandorien-
vibration. It is dominated by large amplitude fluctua- tation degrees of freedom, reflecting the approach of the
tions in the orientation angle ϕ. The amplitudes of the shell closure at N=82. In the more neutron deficient nu-
shape oscillationsare found to be muchsmallerthan the cleitheεdeformationisincreasingandtheγdeformation
ones of the orientation oscillations. The large amplitude is decreasing which reduces the collectivity of the lowest
natureoftheϕ-fluctuationsindicatethatforafullunder- RPAsolutionsthat takesontwo-quasiparticlecharacter.
standingofthe chiralbandsone hastotakeintoaccount Thecalculationoftheinter-bandtransitionratesshow
effects thatgobeyondthe harmonicRPAapproximation that the ratios of the different ∆I components of the
in the orientation degrees of freedom. Anharmonicities transitions vary relatively fast with N and Z.
become dominant where the energy of the first RPA so-
lution goes to zero and the fluctuation amplitude in ϕ
reaches 90◦. Studies of the consequences of anharmonic-
ities for the nature of the band crossing and the source
Acknowledgment
of the different in band transitionrates seenin 134Prare
to be published in the near future [22].
The quite pure chiral nature of the RPA solution This work was supported by DOE grant DE-FG02-
seems to be localized around the N=75 isotones. Mov- 95ER4093and the ANL-ND Nuclear Theory Institute.
[1] S.Frauendorf, Rev. Mod. Phys. 73, 463 (2001) (2007)
[2] V.I. Dimitrov, S. Frauendorf and F. D¨onau, Phys. Rev. [14] S. Zhu et al.,Phys. Rev.Lett. 91, 132501 (2003)
Lett 84, 5732 (2000) [15] S. Frauendorf, Nucl. Phys.A677, 115 (2000)
[3] P.Olbratowski,J.Dobaczewski,J.DudekandW.Ploci- [16] M. Barranger and K. Kumar, Nuc. Phys. A110, 490
ennik,Phys.Rev. Lett. 93, 052501 (2004) (1968)
[4] S. Frauendorf and J. Meng, Nucl. Phys. A617, 131 [17] J. Kvasil and R. Nazmitdinov, Sov. J. Part. Nucl. 17,
(1997) 265 (1986)
[5] K.Starosta et al.,Phys. Rev.C 65, 044328 (2002) [18] T. Koike, K. Starosta, C.J. Chiara, D.B. Fossan, and
[6] S.Brant, D.VretenarandA.Ventura,Phys.Rev.C 69, D.R. LaFosse, Phys. Rev.C 67, 044319 (2003)
017304 (2004) [19] D.J. Hartley et al.,Phys.Rev.C 64, 031304(R) (2001)
[7] D.Tonev et al.,Phys. Rev.Lett. 96, 052501 (2006) [20] T. Koike, K. Starosta, C.J. Chiara, D.B. Fossan, and
[8] K.Starosta et al.,Phys. Rev.Lett 86, 971 (2001) D.R. LaFosse, Phys. Rev.C 63, 061304 (2001)
[9] A.A.Hecht et al.,Phys. Rev.C 63, 051302 (2001) [21] R.A. Bark et al.,Nucl. Phys.A 691, 577 (2001)
[10] P.RingandP.Schuck,TheNuclearMany-BodyProblem, [22] D. Almehed and S.Frauendorf To be published
(Springer,New York,1980) [23] D. Tonev et al.,Submittedto Phys. Rev.C
[11] C.M. Petrache, G.B. Hagemann, I. Hamamoto and [24] Ref.[16]usesaN-dependentquadrupoleoperatorinstead
K.Starosta, Phys. Rev.Lett. 96, 112502 (2006) ofN-dependentκbuttwowaysofwritingthisfactor are
[12] E. Grodner et al.,Phys. Rev.Lett. 97, 172501 (2006) equivalent.
[13] S. Mukhopadhyay et al., Submitted to Phys. Rev. Lett.