Description of electromagnetic and favored $\alpha$-transitions in heavy odd-mass nuclei

0.54 MB·

by  A. Dumitrescu

Description of electromagnetic and favored α-transitions in heavy odd-mass nuclei
A. Dumitrescu1,4 and D.S. Delion1,2,3
1 ”Horia Hulubei” National Institute of Physics and Nuclear Engineering,
407 Atomi¸stilor, POB MG-6, Bucharest-Ma˘gurele, RO-077125, Romaˆnia
2 Academy of Romanian Scientists, 54 Splaiul Independen¸tei, Bucharest, RO-050094, Romaˆnia
3 Department of Biophysics, Bioterra University, 81 Gˆarlei str., Bucharest, RO-013724, Romaˆnia
4 Department of Physics, University of Bucharest, 405 Atomi¸stilor,
POB MG-11, Bucharest-Ma˘gurele, RO-077125, Romaˆnia
Wedescribeelectromagneticandfavoredα-transitionstorotationalbandsinodd-massnucleibuilt
uponasingleparticlestatewithangularmomentumprojectionΩ6= 1 intheregion88≤Z ≤98. We
2
usetheparticlecoupledtoaneven-evencoreapproachdescribedbytheCoherentStateModel(CSM)
6 andthecoupledchannelsmethodtoestimatepartialα-decaywidths. Wereproducetheenergylevels
1 of the rotational band where favored α-transitions occur for 26 nuclei and predict B(E2) values
0 for electromagnetic transitions to the bandhead using a deformation parameter and a Hamiltonian
2 strength parameterfor each nucleus, togetherwith an effectivecollective chargedependinglinearly
onthedeformation parameter. Whereexperimentaldataisavailable, thecontributionofthesingle
n
particle effective charge tothe total B(E2) valueis calculated. The Hamiltonian describing the α-
a
nucleusinteractioncontainstwoterms,asphericallysymmetricpotentialgivenbythedouble-folding
J
of the M3Y nucleon-nucleon interaction plus a repulsive core simulating the Pauli principle and a
3
quadrupole-quadrupole (QQ) interaction. The α-decaying state is identified as a narrow outgoing
1
resonanceinthispotential. Theintensityofthetransitiontothefirstexcitedstateisreproducedby

the QQ coupling strength. It depends linearly both on the nuclear deformation and the square of
]
h thereduced width for the decay to the bandhead,respectively. Predicted intensities for transitions
t tohigherexcitedstatesareinareasonableagreementwithexperimentaldata. Thisformalismoffers
-
l a unified description of energy levels, electromagnetic and favored α-transitions for known heavy
c odd-mass α-emitters.
u
n PACSnumbers: 21.60.Gx,23.60.+e,24.10.Eq
[

1
I. INTRODUCTION mostly in the 84 < Z < 88 region. Several unfavored
v
0 transitions are treated in this paper and predictions are
6 A brief overview of the α-emission process in even- made for the properties of the g.s. g.s. α-trasition in
→
1 odd-mass superheavy nuclei. The unfavored g.s. g.s.
even nuclei is helpful for the understanding of the more
3 α-decay in odd-mass nuclei in the region 64 Z →112
complex situation in odd-mass emitters.
0 isalsotreatedinRef. [9],withthe purposeof≤inves≤tigat-
. In the case of transitions to excited states, the single-
1 ing the effect of the difference in the spin and parity of
particle levels around the Fermi surface play the domi-
0 thegroundstatesontheα-particleanddaughternucleus
nant role and the corresponding decay widths are very
6 preformation probability. The calculations are done in
1 sensitive to the structure of the daughter nucleus [1, 2].
the framework of the extended cluster model, with the
: An important problem is the study of the α-daughter
v Wentzel-Kramers-Brillouinpenetrability andassaultfre-
interaction. One of the most popular approaches is the
i quency, together with aninteractionpotential computed
X doublefoldingprocedure[3]. Thismethodhasbeenused
on the basis of the Skyrme SLy4 interaction.
r togetherwiththecoupledchannelsapproachandarepul-
a sive core simulating the Pauliprinciple in order to study In the current paper, we expand the method previ-
the α-decay fine structure in transitional and rotational ously used in Ref. [5] for the even-even case by allow-
even-even nuclei [4]. For a thorough study of the struc- ing the coupling of an odd-particle to a core described
tureandα-emissionspectruminvibrational,transitional by a coherent function. We study the energy levels and
and rotational even-even nuclei, see Ref. [5]. electromagnetic transition rates of this nucleus and then
Several calculations for the fine structure of the emis- couple an α-particle to it in order to describe the emis-
sion spectrum have already been made in the case of sion spectrum for the case of favored transitions. Our
odd-mass α-emitters. For example, in Ref. [7] a multi- method is to employ an (I,l) coupling procedure in the
channelclustermodeltogetherwiththecoupledchannels laboratory frame, between the angular momentum I of
equation is used to calculate branching ratios to excited the daughternucleusandthe orbitalangularmomentum
states for favored transitions in heavy emitters, in the l oftheα-particle,similartoNilsson’soriginal(I,j)cou-
region 93< Z < 102. In Ref. [8], a microscopic method plingmethodforthedescriptionofnuclearspectrainthe
is employed with a Skyrme SLy4 effective interaction. intrinsic frame [6], where j is the angular momentum of
Starting from the Hartree-Fock-Bogoliubov vacuum and theoddparticle. Weshowthatusingasmallbasishaving
quasiparticleexcitations,theα-particleformationampli- a single value for l in each channel, we can use a QQ α-
tude is calculated for the α-decay to various channels daughterinteractiontogeneratesimultaneouslyresonant
2
states of even or odd parity at the same reaction energy with Jˆ=√2J +1 and (0)(d) given by
IJ
andQQcouplingstrength. The partialdecaywidths ob-
tainedthis wayarein goodagreementwiththe available
1
experimental data. IJ(0)(d)= PJ(x)ed2P2(x)dx, (5)
Z0
in terms of the Legendre polynomial .
II. THEORETICAL BACKGROUND J
P
For an odd-mass nucleus, the state of total angular
momentum I and projection M is given by projecting
In this section we present the theoretical tools re-
out the product between the coherent state (1) and the
quired for the calculation of energy levels and electro-
singleparticlestateψ ,wherej isashorthandnotation
magnetic transition rates for odd-mass nuclei, as well as jm
for all of the quantum numbers of the state, that is
thecoupled-channelsmethodthatisappliedtothestudy
of the fine structure of the α-emission spectrum.
Φ =PI [ψ φ ]. (6)
IM M0 j g
II.1. Nucleon coupled to a coherent state core A straighforwardcalculation leads to the following re-
sult
A description of the surface dynamics of a deformed
even-evennucleuswasproposedforthefirsttimeinRefs. ΦIM = XIJj ϕ(Jg)⊗ψjm IM, (7)
[10, 11] by using a coherent state of quadrupole bosons. XJ h i
Ageneralizationto alltypes oflow-energycollectivemo-
with normalization coefficients XJj given by
tion was proposed in Refs. [12, 13] and was extensively I
developed in Refs. [14, 15] as the coherent state model
(CSM). A review paper on this topic is available in Ref. −1
[16],aswellasinthetextbook[17]. Here,wewillpresent XJj = NJ(g) hjJ;Ω0|IΩi , (8)
inaconcisemannerthemainideasofthemodel,andthen I (cid:16) (cid:17)
−2
extend them to the coupling of an additional nucleon to (g) ( jJ′Ω0IΩ )2
the even-even core. The final goal is to describe a rota- sJ′ NJ′ h | i
(cid:16) (cid:17)
tionalbandbuiltuponagivensingle-particlestateofthe P
odd nucleon. wherethe bra-ketproductdenotesaClebsch-Gordanco-
We begin by assuming that the intrinsic state of an efficient and Ω is the fixed z-projection of the single-
axiallydeformedeven-evennucleusisgivenbyacoherent particle angular momentum j. More details on this pro-
superposition of quadrupole bosons b† with µ=0 cedure can be consulted in Ref. [18].
2µ The states built upon the bandhead I = j = Ω that
follow the sequence I = Ω,Ω+1,Ω+2,... constitute a
φg =ed(b†20−b20) 0 , (1) rotational band. In the Nilsson model, these states are
| i | i labeledbythe setΩπ[Nn Λ], whereπ isthe parity, N is
z
where 0 is the phonon vacuum and the quantity d is the principal quantum number, n the number of nodes
| i z
called deformation parameter [14]. of the radial wavefunction in the z direction and Λ the
The physical states that define the ground band are projection of the single-particle orbital angular momen-
obtained by angular momentum projection tum. The last three numbers act only as labels, as the
good quantum numbers are only Ω and π.
ThesimplestHamiltonianthatcandescribesucharo-
ϕ(g) = (g)PJ φ . (2)
| JMi NJ M0| gi tational structure consists of two terms [18]:
PJ is the projection operator which has the integral
M0
representation
H =A b† b A r2 b†+˜b Y . (9)
1 2· 2− 2 2 2 · 2
(cid:16) (cid:17)
PJ = 2J +1 dω J (ω)R(ω), (3) where by dot we denoted the scalar product. A1 is a
MK 8π2 DMK strengthparameterrequiredtofitexperimentaldataand
r Z A is the strength of the particle-core QQ interaction.
2
with ω the set of three Euler angles, DMJ K(ω) a Wigner For a given ladder operator al, we have
function and R(ω) the rotation operator.
(g) is the norm of the projected state, given by the
forNmJula a˜lµ =(−)µal−µ. (10)
Forthe descriptionoftherotationalbandtheonlyrel-
NJ(g) = Jˆ2IJ(0)(d) −21 ed22, (4) ecvoarenttepramraimsectoemrmisoAn1. dInusetetaodthoef sfaoclvtinthgatthteheeigpeanrvtaiclluee-
h i
3
3.5 3.5 II.2. Electromagnetic transitions
3 (a) Ω=3/2 3 (b) Ω=5/2
2.5 2.5
The B(E2) values of electric quadrupole transitions
2 2
EI EI follow from both collective and single particle contribu-
1.5 1.5
1 1 tions
0.5 0.5
0 0
0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 1
d d B(E2;I2 →I1)=[Iˆ hI1||q0cQc2||I2i+ (14)
3.5 3.5 2
3 (c) Ω=7/2 3 (d) Ω=9/2 1
+ I qspQsp I ]2,
2.5 2.5 Iˆ h 1|| 0 2 || 2i
2 2 2
EI 1.5 EI 1.5 where qc and qsp are effective charges.
1 1 0 0
Thecollectivequadrupoletransitionoperatorhasboth
0.5 0.5
harmonic and anharmonic contributions
0 0
0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4
FIG. 1. Normalidzed energy levels EI as functiodn of defor- Qc2µ =b†2µ+˜b2µ+aq b†2⊗b†2 2µ+(b2⊗b2)2µ , (15)
mation d, for different values of the single particle angular (cid:20)(cid:16) (cid:17) (cid:21)
momentum projection Ω. with a the anharmonic strength. Its reduced matrix
q
elements on the states of the core are
q d
problem by a full diagonalization procedure, a simpler ϕ(g) qcQc ϕ(g) = eff J 2;00J 0 (16)
approach, involving the analytical expression for the di- h J1 || 0 2|| J2 i Jˆ2NJ(1g)NJ(2g)h 1 | 2 i×
agonal matrix elements of the Hamiltonian (9) in the 2 2
Jˆ2 (g) +Jˆ2 (g) ,
basis of Eq. (7) suffices: × 1 NJ1 2 NJ2
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
with q given by a linear formula in d
eff
3
IM H IM =A d2f d N + (11)
1 jΩI
h | | i − 2 × 2
(cid:18) (cid:19) q =qc 1 a d . (17)
j2;Ω0jΩ j2;10j1 , eff 0 −r7 q !
×h | ih 2 | 2i
The single particle quadrupole transition operatorhas
with fjΩI given by the occupation number representation
1
hIj;Ω−Ω|J0i2IJ(1)(d) Qs2pµ = ˆ2hj1||r2Y2||j2i c†j1c˜j2 2µ. (18)
fjΩI = PJ hIj;Ω−Ω|J0i2IJ(0)(d), (12) Explicit expresXjs1ijo2ns for the matr(cid:16)ix elem(cid:17)ents of these
J
operators over the states of the odd-mass nucleus follow
P
from the above results and the use of standard angular
in terms of the function
momentum algebra. For our particular case of fixed j,
the final formulas are
d
(1)(x)= (0)(x). (13)
IJ dxIJ I qcQc I = XJ1jXJ2jIˆIˆ ( )j−I1 (19)
h 1|| 0 2|| 2i I1 I2 1 2 − ×
The shape of such a spectrum is dependent both on JX1J2
the deformationparameterandonthe valueofΩ, ascan (I J I J ;j2) ϕ(g) qcQc ϕ(g) ,
be seen in Fig. 1. ×W 1 1 2 2 h J1 || 0 2|| J2 i
I qspQsp I = XJ1jXJ1jIˆIˆ ( )j+I2 (20)
Whilethisapproachisadequateforthepurposeofthis h 1|| 0 2 || 2i I1 I2 1 2 − ×
paper, if a greater precision in the description of the nu- XJ1
clear energy spectrum is required, then more terms can (I jI j;J 2) j qspr2Y j ,
×W 1 2 1 h || 0 2|| i
be added to the Hamiltonian (9), as shown in Ref. [18].
with a Racah coefficient.
Letusalsomentionthatthedevelopmentpresentedhere W
All reduced matrix elements are defined in the usual
and expanded upon in Ref. [18] is appropriate for any
convention
rotational band built upon an angular momentum pro-
jection Ω = 1. The special case Ω = 1 requires a modi-
6 2 2 1
fication of the formalism and will be treated in a future lmT l′m′ = l′m′;λµlm l T l′ . (21)
paper. h | λµ| i ˆlh | ih || λ|| i
4
II.3. α-emission in the coupled channels approach
4A
Thedecayphenomenonofinterestconnectstheground µ=m D , (28)
N
4+A
state of the parent nucleus of angular momentum I to D
P
anexcited state ofangularmomentum I of the daughter
and anα-particle ofangular momentum l, in sucha way a term describing the motion of the daughter HD b†2
that the total angular momentum IP is conserved: and an α-daughter interaction with monopole (cid:16)an(cid:17)d
quadrupole components
P(I ) D(I)+α(l). (22)
P
→
V b†,R =V (R)+V b†,R . (29)
An importantremarkis thatboththe initialstate ofthe 2 0 2 2
parentand the final state of the daughterare built upon (cid:16) (cid:17) (cid:16) (cid:17)
the same single particle orbital j. This is known as a A detailedstudy ofthis potentialcanbe found in Ref.
favored α-transition, due to the fact that it usually has [4]. There it is shown that the monopole component has
a large branching ratio. The situation where the initial a pocket-like shape
and final single particle orbitals are different is known
as an unfavored α-transition. For the favored case, the
transition from the ground state to the bandhead built V (R)=v V¯ (R),R>R (30)
0 a 0 m
atop the j orbital in the daughter nucleus generally has
=a(R R )2 v ,R<R ,
thehighestdecaywidth,andtransitionsonexcitedstates − min − 0 m
of the band form the fine structure of the spectrum.
obtained through the matching of a harmonic oscilla-
The total wavefunction of the α-daughter system can
tor to the nuclear plus Coulomb potential V¯ obtained
be assumed to be separable in radial and angular parts 0
by the method of the double folding procedure of the
and expanded in the angular momentum basis
M3Y particle-particle interaction with Reid soft core
parametrisation(Refs. [19–21]and the book [1] for com-
Ψ b†,R = fIl(R) b†,ω , (23) putational details).
2 R ZIl 2 The number v acts as a quenching factor of the nu-
a
(cid:16) (cid:17) XIl (cid:16) (cid:17) clear force. v = 1 implies an α-particle existing with
a
where the angularcomponents are givenby the coupling certainty, and a value v <1 is required in order to sim-
a
to goodangularmomentum betweena wavefunction for ulate the formation of the α-particle on the nuclear sur-
the odd-massdaughternucleusandasphericalharmonic face. Since branching ratios tend to have a weak depen-
for the α-particle dence on this parameter [4], it can be adjusted in order
to reproduce the total decay width Γ [22]. Another pos-
sibility is to leave the interaction potential unquenched
ZIl b†2,ω = ΦIM b†2 ⊗Ylm(ω) IPMP . (24) and to consider the spectroscopic factor
(cid:16) (cid:17) h (cid:16) (cid:17) i
Here, R = (R,ω) is the relative vector between the two
Γ
fragments. Each pair of angular momentum values de- S = expt , (31)
fines a decay channel Γtheor
as a measure of the particle formation probability, as in
(I,l)=c. (25) Ref. [23].
The repulsive core on the second line of equation (30)
ThefunctionΨmustsatisfythestationarySchr¨odinger simulates the Pauli principle, namely the fact that the
equation α-particle can exist only on the nuclear surface. Its pa-
rameters can be adjusted so that the first resonance in
the potential corresponds to the experimental Q-value.
HΨ b†,R =Q Ψ b†,R , (26)
2 α 2 ThematchingradiusRm andthe pointRmin atwhich
(cid:16) (cid:17) (cid:16) (cid:17) theoscillatorpotentialattainsthelowestvaluearefound
with Q the Q-value ofthe decayprocess. The Hamilto-
α throughthemethodofRef. [4],whichrequirestheequal-
nian
itybetweentheexternalattractionandinternalrepulsion
together with their derivatives. This makes the total
~ interaction continuous and dependent on the repulsive
H =−2µ∇2R+HD b†2 +V b†2,R (27) strength a and potential depth v0. In our study, a has
(cid:16) (cid:17) (cid:16) (cid:17) a fixed value of 50 MeV for all nuclei and v0 is fitted in
features a kinetic operator depending on the reduced each case through the experimental Q-value.
mass µ of the system The second term of Eq. (29) is the QQ interaction
5
II.4. Resonant states
V b†,R = C (R R )dV0(R) (32) The measured α-decay widths are by many orders
2 2 − 0 − min dR × of magnitude smaller than the Q-value. Thus, an α-
(cid:16) (cid:17) √5[Qc Y (ω)] , decaying state is almost a bound state, this being the
× 2⊗ 2 0 main reason way the stationary approach is a very good
approximationof the emissionprocess. The state canbe
with C serving as an α-nucleus coupling strength.
0 identified with a narrow resonant solution of the system
The angular functions entering the expansion of Eq.
of equations (33), containing only outgoing components.
(23) are orthonormal. Using this, one obtains in a stan-
In order to solve this system of equations we first define
dardwaythe systemofcoupleddifferentialequationsfor
the internal and external fundamental solutions which
radial components
satisfy the boundary conditions
d2fdI1ρl21(R) = AI1l1;I2l2(R)fI2l2(R), (33) RIl,L(R) R−→→0δIl,LεIl , (38)
I1 XI2l2 HI(+l,L)(R) ≡GIl,L(R)+iFIl,L(R)R−→→∞
with the coupling matrix having the expression δ H(+)(κ R) δ [G (κ R)+iF(κ R)] ,
Il,L l I ≡ Il,L l I l I
where ε are arbitrary small numbers. Here, the chan-
Il
l (l +1) V (R) nel indexes label the component and L the solution,
AI1l1;I2l2(R)= 1 ρ12 + Q 0 E −1 δI1l1;I2l2 + GIl(κIR) and FIl(κIR) are the standard irregular and
(cid:20) I1 α− I1 (cid:21) regular spherical Coulomb functions, depending on the
1
+ V b†,R , (34) momentum κI in the channel c, defined by Eq. (35).
Qα−EI1hZI1l1| 2(cid:16) 2 (cid:17)|ZI2l2i Each component of the solution is built as a super-
position of N independent fundamental solutions. We
in terms of the reduced radius imposethematchingconditionsatsomeradiusR1 inside
the barrier and obtain
f (R )= (R )M = (+)(R )N (39)
ρI =κIR, κI = 2µ(Q~α2−EI). (35) Il 1 XL RIl,L 1 Il,L XL HIl,L 1 Il,L
r dfIl(R1) = dRIl,L(R1)M = dHI(+l,L)(R1)N ,
Notice that κI has the same value for all the channels dR dR Il,L dR Il,L
of fixed I, so the supplementary l-index can be omitted XL XL
both for the wave number and reduced radius. where N are called scattering amplitudes. One thus
Il,L
The coupling term of the matrix is found by the same arrives at the following secular equation
methods as in the previous sections to be
(R ) (+)(R ) (R ) (R )
R 1 H 1 R 1 G 1 =0 . (40)
hZI1l1|V2 b†2,R |ZI2l2i= The fir(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)sdtRdc(RoRn1d)itdioHn(+diR)s(Rfu1l)fi(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ll≈ed(cid:12)(cid:12)(cid:12)(cid:12)dfoRdr(RRth1)e cdoGmd(RRp1l)e(cid:12)(cid:12)(cid:12)(cid:12)xenergiesof
XJ1jX(cid:16)J2j ϕ(cid:17)(g) Qc ϕ(g) l Y l Iˆ2IˆIˆˆj the resonant states. They practically coincide with the
I1 I2 h J1 || 2|| J2ih1|| 2|| 2i P 1 2 real scattering resonant states, due to the fact that the
JX1J2 imaginary parts of energies are much smaller than the
J I j
1 1 corresponding real parts, which implies vanishing regu-
( )I2−IP+l2 (I l I l ;I 2) J I j (36)
× − W 1 1 2 2 P  2 2  lar Coulomb functions FIl inside the barrier. The roots
2 2 0
  of the equation (40) do not depend upon the matching
radius R , because both internal and external solutions
1
 
where the curly brackets denote a 9j-symbol. Since the satisfy the same Schr¨odinger equation. The unknown
reduced matrix element between the states of the core is coefficients M and N are obtained from the nor-
Il,L Il,L
alinearfunctionofthe deformation[15],onecanexpress malization of the wave function in the internal region
this linearity in terms of an effective α-nucleus coupling
strength having a different anharmonic parameter aα R2 f (R)2dR=1 , (41)
Il
| |
Il ZR0
X
where R is the external turning point.
2 2
C =C0 1 aαd . (37) From the continuity equation, the total decay width
−r7 ! can be written as a sum of partial widths
6
800
Ω=3/2
Ω=5/2
Ω=7/2
Γ= ΓIl = ~vI lim fIl(R) 2 = (42) 700 Ω=9/2
R→∞| |
Il Il
X X
= ~v N 2, 600
I Il
| |
Il
X 500
V)
withvI thecentre-of-massvelocityatinfinityinthegiven A (ke1 400
channel
300
~κ
vI = I. (43) 200
µ
100
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
d
III. NUMERICAL APPLICATION FIG. 2. Hamiltonian strength parameter A1 versus deforma-
tion d for rotational bands built atop different values of the
odd nucleon angular momentum projection Ω.
All the experimental data with which we have tested
the model has been provided by the ENSDF data set
maintained by BNL [24]. In this paper we have studied
fraovtoartieodntarlabnasnitdioinnswinhi2c6hotdhde-pmaarsesntαd-eemcaitytserissbwuhieltreattohpe 222...468 III===ΩΩΩ+++123 (a) Ω=3/2 222...468 (b) Ω=5/2
2.2 I=Ω+4 2.2
asingleparticleorbitalofangularmomentumprojection EI 2 I=Ω+5 EI 2
Ω6= 12. Additionally, this band must be described in the E/I+1 11..68 E/I+1 11..68
formalismofanoddnucleoncoupledtogoodangularmo- 1.4 1.4
1.2 1.2
mentumwithaCSMcore. Thedeformationparameterd 1 1
wasobtainedbyfitting availableenergylevels relativeto 0.8 0.5 1 1.5 2 2.5 3 3.5 4 0.8 0.5 1 1.5 2 2.5 3 3.5 4
thebandhead. Anumberofabout4levelsisrequiredfor d d
2.8 2.8
the determination of a reliable deformation. As can be 2.6 (c) Ω=7/2 2.6 (d) Ω=9/2
seenfromFig. 1,there exists adeformationrangewhere 2.4 2.4
2.2 2.2
a large shift of the parameter’s value has little impact EI 2 EI 2
on the energy levels. Because of this, when fewer energy E/I+1 11..68 E/I+1 11..68
levels are available, the fit becomes unreliable. In these 1.4 1.4
1.2 1.2
circumstances we have determined the deformation pa- 1 1
0.8 0.8
rameterbystudying the systematicsofenergylevelsand 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4
d d
deformations for the neighboring nuclei with good ex- FIG.3. Experimentalenergylevelratios EI+1 asafunction
perimental data. A quadratic trend is observed in the EI
of thedeformation parameter d together with thetheoretical
dependence of the Hamiltonianstregth parameterA1 on curves,separatelyforeachvalueofthesingleparticleangular
the deformation,asevidenced inFig. 2, where we assign momentum projection Ω.
the nuclei with separate symbols for each value of Ω .
The fitting formula is
A1(d)=55.583d2−119.283d+150.409 (44) B E2;9+ 5+ =170 30 W.u.. (45)
σ =68.410, 2 → 2
(cid:18) (cid:19)
Usingthesystematicsforthecollectiveeffectivechargeqc
agreeing qualitatively with the similar treatment made 0
asfunctionofdestablishedinRef. [5]ourmodelpredicts
for the ground bands of even-even nuclei in Ref. [5].
a value
The agreement between the ratio of experimental en-
ergy levels assigned to the deformation parameter d and
the theoretical ratio EI+1 is shown in Fig. 3, with sepa- 9+ 5+
EI B E2; =117.8 W.u.. (46)
rate panels for different values of Ω.
2 → 2
Onthetopicofelectromagnetictransitions,onenotices (cid:18) (cid:19)
asurprisinglackofmeasuredB(E2)valuesforodd-mass The difference up to the experimental value can be ob-
α-emitters. Only one such value can be found in the tained by tweaking the value of the single particle ef-
database,forthetransition 9+ 5+ inthegroundband fective charge q0sp, which in this case must be equal to
of Th229. It is given by 2 → 2 q0sp = 7.004 (W.u.)21. Due to the lack of experimental
7
data, a systematics of single particle effective charges 1
Il=5/2 0
cannot currently be made, but we present predictions Il=7/2 2
Il=9/2 2
for B(E2;Ω+2 Ω) values based on the systematics Il=11/2 4
Il=5/2 1
of the collective e→ffective charge from Ref. [5], together IIll==79//22 13
0.5 Il=11/2 3
with results concerning energy levels in Table I.
To study α-transitions, we make use of the so-called
decay intensities
fIl 0
Γ
Υ =log Ω0, (47)
Il 10 Γ
Il
-0.5
and we will employ the notation Υ , i = 1,2,3 to re-
i
fer to decay intensities for the transitions to the first,
second and third excited state respectively in any rota-
-1
tional band of bandhead angular momentum projection 5 6 7 8 9 10 11 12 13 14 15
R (fm)
Ω 6= 21. Notice that, in principle, each intensity Υi is FIG. 4. Solutions to the system (33) for the favored decay
given by the sum processU233 →Th229+α4. Solid linesrepresent radial func-
92 92 2
tions of even orbital angular momentum l while dashed lines
represent radial functions of odd l. The sets of functions of
Υ = Υ , (48) fixedparityareobtainedsimultaneouslyforthesamereaction
i Il
energy and QQ coupling strength.
l
X
whereI isfixedbytheangularmomentumofthedaugh-
ternucleusinthatparticularstateandl followsfromthe
triangleruleforthecouplingtototalangularmomentum of even and odd resonances for each α-decay process of
I . However, it is sufficient to consider only one l-value energy Q , in an attempt to fit experimental data. One
P α
for each state. This is due to the fact that the standard will thus obtain a total of eight radial functions in the
penetrability P through the Coulomb barrier, defined solution,four in eachresonance,as can be seenin Fig. 4
Il
by the factorization for the decay process
Γ =2P (R)γ2 (R), (49) U233 Th229+α4. (50)
Il Il Il 92 → 92 2
decreases by one order of magnitude for each increas- Wehaveobservedthatfor23decayprocessesoutofthe
ing value of l. Therefore, one would expect to be able total of 26 studied, C can be tweaked in order to match
to make a reasonable prediction of the fine structure of theexperimentalvalueofΥ foradecaywidthwithl=0
1
the α-emissionspectrumusingabasisofjustfourstates, corresponding to the α-transition to the bandhead and
one state for the bandhead and an additional state for the first decay width having l = 2 obtained in the even
eachexcited energy level. In the cases where experimen- resonance corresponding to the α-transition to the first
tal data concerning the energy of the last state was not excited state. Simultaneously, the ratio between decay
available, we used the CSM core + particle prediction widths corresponding to the same l =0 for the decay to
provided by the fit of the lower energies. thebandheadandthefirstvalueofl=3forthedecayto
It turns out however that the basis suggested above the second excited state obtained in the odd resonance
needs to be enlarged, due tot the fact that the parity of yielded a very good estimate of Υ , while the ratio be-
2
a resonance is fixed by whether the l-values involvedare tween decay widths correspondingto l =0 for the band-
even or odd. Since the interaction (29) conserves parity, head decay and l = 4 for the decay to the third excited
one must construct separate resonances of fixed even or statefoundintheevenresonancehavegivenareasonable
oddparity. Theevenonefollowsthesequenceofminimal value for Υ . One of the exceptions is the decay to the
3
l-values in each channel as l = 0,2,2,4, while the odd daughter nucleus Am241, where the available data con-
95
one follows the sequence l = 1,1,3,3. Thus, each basis cerning Υ ,i = 1,2 suggests a doublet structure in the
i
offourstateshavingagivenparityconstructsaseparate emissionspectrumthat canbe reproducedby employing
resonantsolutionofthesystem(33). Itisimportantthat thesamel=0widthforthebandheadtransitionandthe
both resonances are found at the same reaction energy two decay widths with l = 2 obtained in the even reso-
Q and same QQ coupling stregth C. It is possible to nance. TheotherexceptionconcernsthetwoAcisotopes
α
achieve this for the potential of Eq. (30) by adjusting in our data set. In these cases, the decay width of angu-
the depth v so that both resonances generated at the lar momentum l = 0 and the first l = 2 width obtained
0
same C match in terms of the reaction energy. Using intheevenresonancecanbe usedtoreproducethevalue
this, one can then tweak the effective coupling strength of Υ , situation in which the l=0 width and the second
2
C of Eq. (37) to simultaneously generate different sets width of angular momentum l = 1 in the odd resonance
8
0.14 0.14
0.12 0.12
0.1 0.1
C 0.08 C 0.08
0.06
0.06
0.04
0.04
0.02
0.02 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.4 2.6 2.8 3 3d.2 3.4 3.6 3.8 4 100γ2Ω0
FIG. 5. Effective α-nucleus coupling strength C versus de- FIG. 6. Effective α-nucleus coupling strength C versus the
formation parameter d. reduced width γ2 for α-transitions to thebandhead.
Ω0
2.5 3
(which corresponds to the transition to the first excited exp
pred
state) will reproduce Υ reasonably. 2 penet 2.5
1
When plotted against the deformation parameter, the 1.5 2
Υ1 Υ2
valuesofC obtainedfromtheabovefitfollowthepredic- 1 (a) 1.5 (b)
tionofEq. (37)byexhibitingalineartrendwithrespect 0.5 1
to d, as seen in Fig. 5. The parameters of the linear fit
0 0.5
are 0 5 10 15 20 25 30 0 5 10 15 20 25 30
n n
5
C = 0.088, a =0.971, σ =0.023. (51) 4.5
0 α
− 4
3.5
Thiscouplingstrengthcanbeinterpretedasameasure 3
ofα-clustering. Toseethis,weusethereducedwidthγ2 Υ3 2.5 (c)
Ω0 2
introducedinEq. (49). ItturnsoutthatC showsalinear 1.5
correlationwithγ2 withapositiveslope,ascanbe seen 1
Ω0 0.5
in Fig. 6. The parameters are given by 0 5 10 15 20 25 30
n
FIG. 7. Intensities of the favored α-transitions Υi to the
first three excited states in rotational bands as function of
C =10.096γ2 +0.037 (52) the index number n in the first column of Tables I and II.
Ω
Open circles denote the experimental data, filled circles are
σ =0.021.
the values predicted by the coupled channels method with a
particle+CSMcorestructuremodelanddarktrianglesshow
In Fig. 7 we present in separate panels the values of
thebarrier penetration estimates.
theintensitiesΥ , i=1,2,3obtainedbythemethodpre-
i
sentedabove,versustheindexnumbernfoundinthefirst
columnofTablesIandII.Withopencirclesweshowex-
perimentaldata andwith filledcircles we givethe values partialspectroscopicfactorsforeachchannelandthelog-
predictedbythecoupledchannelsmethodwithaparticle arithm of the hindrance factor as
+CSMcorestructuremodel. Darktrianglespresentthe
S
crudest barrier penetration calculation, where the inten- log HF =log Ω0 =Υexp Υtheor. (54)
sities follow from the ratios of penetrabilities computed 10 Il SIl Il − Il
at the same values of l as in the coupled channels ap-
This quantity shows the importance of the extra-
proach
clustering in the decay process to excited states that is
notconsideredwithinourmodel. InFig. 8wehaveplot-
P ted these logarithms versus the neutron number. It is
Υ =log Ω0. (53)
i 10 P clearlyshownthatcouplinganα-particletothedaughter
Il
nucleus with the required strength needed to reproduce
All emission data is presented in Table II. one intensity (usually Υ , with the exception of Ac iso-
1
As we mentioned, the spectroscopic factor defined by topes where Υ is reproduced) allows one to predict the
2
Eq. (31) accounts for clustering effects. One can define valuesofthe otherintensities within afactorusually less
9
0.35 0.5 0
0.175 0.25
F1 F2
logH10 0 logH10 0 -0.5
-0.175 (a) -0.25 (b)
-1
-0.35 -0.5
5 10 15 20 25 30 5 10 15 20 25 30 S
1.2 N-126 N-126 log10
-1.5
0.6
F3
logH10 0 -2
-0.6 (c)
-1.2 -2.5
5 10 15 20 25 30 5 10 15 20 25 30
N-126 N-126
FIG. 8. Logarithm of the hindrance factor HFi versus FIG. 10. Logarithm of the spectroscopic factor S versus
neutron number N −126, separately for each excited state neutron numberN −126.
i=1,2,3.
for heavier nuclei than what is observed experimentally.
4.5
Υ
Υ1
Υ2
4 3
IV. CONCLUSIONS
3.5
We analyzed the available experimental data for fa-
3
voredα-transitions to rotationalbands built upon a sin-
Υi 2.5 gle particle angular momentum projection Ω = 1. The
6 2
nuclear structure was modeled as an odd-nucleon cou-
2
pledtoa coherentstate even-evencore,the energylevels
of each band being fitted through the use of a deforma-
1.5
tionparameterdandHamiltonianstrengthparameterA
1
1 that is related to the deformation through a quadratic
dependence. B(E2) values can be predicted using the
0.5
20 40 60 80 100 120 140 160 180 200 220 systematics of the collective effective charge as function
Ei (keV) of deformation established in Ref. [5]. In the absence
FIG.9. Υi valuesversusexcitationenergyEi relativetothe of experimental data that allows the study of the sin-
bandhead in each case.
gle particle effective charge contribution, it is expected
thatthesepredictedvaluesaresmallerthanwhatwillbe
observed in reality.
The fine structure of the α-emission spectrum was
than 3.
studied using the coupled channels method, through a
We note that the universal decay law treated in Refs.
QQ interaction tweaked by a coupling strength that be-
[23] and [25] is once again manifested in the dependence
haves linearly with respect to the deformation parame-
of the decay intensities on excitation energies. In Fig.
ter and reduced width for the g.s. Ω transition. The
9 we have represented all of the Υ values as function
i →
predictedvaluesoftheintensitiesareinreasonableagree-
of the corresponding excitation energy E relative to the
i
mentwithexperimentaldata,usuallywithinafactorless
bandhead for each collective structure analyzed in this
than 3. With additional developments in the structure
paper. We observe a universal linear correlation with
part, it is expected that the model will be useful for the
parameters
study of the case Ω= 1 as well.
2
Υ =0.017E +0.169, σ =0.316. (55)
i i
ACKNOWLEDGMENTS
As a final remark, the logarithm of the spectroscopic
factor of Eq. (31) can be represented as a function of This work was supported by the grants of the Roma-
neutron number, like in Fig. 10. This quantity shows nian Ministry of Education and Research, CNCS UE-
a decreasing trend with the neutron number, meaning FISCDI, PN-II-ID-PCE-2011-3-0092, PN-09370102 and
that the unquenched potential predicts shorter half-lives by the strategic grant POSDRU/159/1.5/S/137750.
10
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