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Nuclear deformation and radii in heavy nuclei
Takashi Nakatsukasa
Center for Computational Sciences, University of Tsukuba
ELPH"%�1- ��467 �"%2@ELPH, Tohoku University, 2019.3.20-21
Collaborators:Shuichiro Ebata (Tokyo Int. Tech.)Kouhei Washiyama (Kyushu Univ.)
#��'2019.1.25�.
Content of the talk
• Focus: Nuclear deformation
• Shell effect
– Origin of deformation
– Origin of sphericity
– “Islands of inversion” in n-rich isotopes
• Mysteries associated with excited 0+ states
– Need multi-facet data
Shape transition
spherical deformed
Increasing neutrons
Phonon excitation
Rotational (NG mode)
EI ≈I(I +1)
2ℑEI ≈ωvibI
Sudden changes in R4/2 signify changes in structure, usually from spherical to deformed structure
Onset of deformation à
Sph.
Def.
+
+=2
42/4 EER
R. Casten
R. Casten
+
+=2
42/4 EER
Experimental evidences, �
Z
�� � N
Single-particle and collective motions
• Period of nucleonic Fermi motion
/F~ R/vF ~ 10-22 s– Collision time
/c >> /F
• Low-energy collective motion
– /coll >> /F
Symmetry-breaking mechanism
• Unified model=Bohr-Mottelson>
• Adiabatic approx.
H = −B∂α2 2+V (α)+Hsp (ξ )+Hcoup (α,ξ )
τα >> τξ
Heff (α) = −B2∂α + φ0 ∂αφ0( )
2+V (α)+ε0 (α)+Φ0 (α)
Avoid degeneracy
Induced by couplingbetween collective & single-particle motions
Hamiltonian
a x x a= + +( ) ( ) ( ) ( )H H H H , , 1coll sp int
where Hcoll is the collective Hamiltonian to describe the low-energy shape vibrations. Hsp corresponds to the single-particle (shell model) Hamiltonian at the spherical shape,
x x x a= + =( ) ( ) ( )H T V , 0sp kin . Tkin is the kinetic energyterm and the nuclear self-consistency requires the potential
x a( )V , to vary with respect to the shape α. The interactionbetween the collective and single-particle motions, given bythe third term x a( )H ,int , is indispensable to take into accountthis important property of nuclear potential.
2.2. Symmetry breaking mechanism
The coupling term in equation (1) could lead the nucleus todeformation. This is associated with the SBS mechanism. Toelucidate the idea, let us adopt a simple adiabatic (Born–Oppenheimer) approximation. First, we solve the eigenvalueproblem for the Schrödinger equation for the variables ξ witha fixed value of α
a f a a f añ = ñ�( ) ∣ ( ) ( ) ∣ ( ) ( )H , 2n n ndef
where a aº +( ) ( )H H Hdef sp int . This gives the adiabaticcollective Hamiltonian, a a a= + �( ) ( ) ( )( )H Hn
nad coll , for eachintrinsic eigenstate f x a x f aº á ñ( ) ∣ ( );n n . ( )H n
ad is an effectiveHamiltonian for the collective variables α. The total wavefunction is given by a product of the intrinsic and thecollective parts [15], a x y a f x aY =( ) ( ) ( )( ), ;n
nn .
There are two possible mechanisms of the SBS in theunified model to realize the deformed ground state with a ¹ 0.When a� ( )0 strongly favors the deformation, even if a( )Hcollhas the potential minimum at a = 0, the adiabatic potential in
a( )( )Had0 may have a deformed minimum. Apparently, this
mechanism requires the deformation-driving nature ofx a( )H ,int , which we call ‘coupling-driven mechanism’. On the
other hand, there is another mechanism which can deform thenucleus even if the spherical shape (a = 0) is favored by theadiabatic ground state a� ( )0 . This is due to the additionalcoupling caused by the kinetic term of H ;coll a =( )Tkin
- ¶a( )B1 2 2 . We adopt units of =� 1 throughout the presentarticle. Roughly speaking, the SBS takes place when the levelspacing in a� ( )n is smaller than the additional coupling. It isanalogous to the Jahn-Teller effect in the molecular physics[16], which we call ‘degeneracy-driven mechanism’.
2.3. Degeneracy-driven SBS; diagonal approximation
In order to understand the degeneracy-driven mechanism, wemake the argument simpler, neglecting the off-diagonal ele-ments, f a a f aá ñ( )∣ ( )∣ ( )Tn kin 0 ( ¹n 0). Integrating the intrin-sic (single-particle) degrees ξ, the effective Hamiltonian forthe collective variable α is obtained as
a f a f a a a a= á ñ = + + F�( ) ( ) ∣ ∣ ( ) ( ) ( ) ( )( )
( ) ( )H H H ,3
eff0
0 0 coll0
0 0
where ( )Hcoll0 is identical to Hcoll except that its kinetic energy is
modified into a f f= - ¶ + á ¶ ña a( ) ( ) ( ∣ )( )T B1 2kin0
0 02. This is
equivalent to introduction of a ‘vector’ potential [17],a f fº á ¶ ña( ) ∣A i 0 0 . If the coordinate α is one-dimensional,
the ‘vector’ potential a( )A can be eliminated by a gaugetransformation, ò a a( )( )Aexp i d . However, the following‘scalar’ potential remains
å
å
a f f f f
f f f f
f a a f aa a
F = ᶠ- ñá ¶ ñ
= ᶠñá ¶ ñ
=á ¶ ñ
-
a a
a a
a
¹
¹ � �
( ) ∣( ∣ ∣) ∣
∣ ∣
( ) ∣( ( )) ∣ ( )( ) ( )
( )
B
B
BH
12
1
12
12
. 4
nn n
n
n
n
0 0 0 0 0
00 0
0
def 0
0
2
From equation (4), it is apparent that aF ( )0 is positive andbecomes large where the adiabatic ground state is approxi-mately degenerate in energy, »� �n0 ( ¹n 0). When thespherical ground state (a = 0) shows degeneracy, it could besignificantly unfavored by aF ( )0 . The system tends to avoidthe degenerate ground state, which leads to the SBS withnuclear deformation.
We would like to emphasize again that the couplingbetween the collective (shape) degrees of freedom α and theintrinsic (single-particle) motion ξ is essential to produce thenuclear deformation. This is apparent for the coupling-drivenmechanism, and is also true for the degeneracy-driven case. Ifthe coupling term x a( )H ,int is absent, the adiabatic statesf x( )n are independent of α, thus, produce no gauge potentials,
a a= F =( ) ( )A 00 . We also note here that the presentargument on the degeneracy-driven (Jahn-Teller) mechanismexplains why the instability of a spherical state occurs, but notwhat kind of deformation takes place. This will be discussedin sections 3.3 and 6.2.
2.4. Field coupling
The oscillation of the variable α correspond to the shapevibration. Thus, it can be quantized to a boson operator. Inorder to describe the vibrational motion associated with α, weintroduce a boson space with the n-phonon state ñ∣n . When αis small, we may linearize the coupling term in equation (1)with respect to α as
x a ka x= -( ) ( ) ( )H F, , 5int
where κ is a coupling constant which depends on thenormalization of α and F. If the operator F is given, thenormalization of α is usually chosen as follows. The action ofthe one-body operator F on the ground state (a Slaterdeterminant) produces many one-particle-one-hole states;F ñ = å F ñáF F ñ∣ ∣ ∣ ∣F Fph ph ph0 0 . This is identified with the
operation of α in the collective (boson) space:
å aáF F ñ = á = = ñ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( )F n n1 0 . 6ph
ph 02 2
The coupling constant κ can be also determined by this self-consistency. See chapter 6 of BM2 for details of the fieldcoupling techniques.
If the matrix elements of F are identical to those of α asin equation (6), the field coupling(5) can be interpreted as an
3
Phys. Scr. 91 (2016) 073008 Invited Comment
Heavy nuclei favor deformation?
Nakatsukasa et al. Phys. Scr. 91 073008 (2016)
Hamiltonian
a x x a= + +( ) ( ) ( ) ( )H H H H , , 1coll sp int
where Hcoll is the collective Hamiltonian to describe the low-energy shape vibrations. Hsp corresponds to the single-particle (shell model) Hamiltonian at the spherical shape,
x x x a= + =( ) ( ) ( )H T V , 0sp kin . Tkin is the kinetic energyterm and the nuclear self-consistency requires the potential
x a( )V , to vary with respect to the shape α. The interactionbetween the collective and single-particle motions, given bythe third term x a( )H ,int , is indispensable to take into accountthis important property of nuclear potential.
2.2. Symmetry breaking mechanism
The coupling term in equation (1) could lead the nucleus todeformation. This is associated with the SBS mechanism. Toelucidate the idea, let us adopt a simple adiabatic (Born–Oppenheimer) approximation. First, we solve the eigenvalueproblem for the Schrödinger equation for the variables ξ witha fixed value of α
a f a a f añ = ñ�( ) ∣ ( ) ( ) ∣ ( ) ( )H , 2n n ndef
where a aº +( ) ( )H H Hdef sp int . This gives the adiabaticcollective Hamiltonian, a a a= + �( ) ( ) ( )( )H Hn
nad coll , for eachintrinsic eigenstate f x a x f aº á ñ( ) ∣ ( );n n . ( )H n
ad is an effectiveHamiltonian for the collective variables α. The total wavefunction is given by a product of the intrinsic and thecollective parts [15], a x y a f x aY =( ) ( ) ( )( ), ;n
nn .
There are two possible mechanisms of the SBS in theunified model to realize the deformed ground state with a ¹ 0.When a� ( )0 strongly favors the deformation, even if a( )Hcollhas the potential minimum at a = 0, the adiabatic potential in
a( )( )Had0 may have a deformed minimum. Apparently, this
mechanism requires the deformation-driving nature ofx a( )H ,int , which we call ‘coupling-driven mechanism’. On the
other hand, there is another mechanism which can deform thenucleus even if the spherical shape (a = 0) is favored by theadiabatic ground state a� ( )0 . This is due to the additionalcoupling caused by the kinetic term of H ;coll a =( )Tkin
- ¶a( )B1 2 2 . We adopt units of =� 1 throughout the presentarticle. Roughly speaking, the SBS takes place when the levelspacing in a� ( )n is smaller than the additional coupling. It isanalogous to the Jahn-Teller effect in the molecular physics[16], which we call ‘degeneracy-driven mechanism’.
2.3. Degeneracy-driven SBS; diagonal approximation
In order to understand the degeneracy-driven mechanism, wemake the argument simpler, neglecting the off-diagonal ele-ments, f a a f aá ñ( )∣ ( )∣ ( )Tn kin 0 ( ¹n 0). Integrating the intrin-sic (single-particle) degrees ξ, the effective Hamiltonian forthe collective variable α is obtained as
a f a f a a a a= á ñ = + + F�( ) ( ) ∣ ∣ ( ) ( ) ( ) ( )( )
( ) ( )H H H ,3
eff0
0 0 coll0
0 0
where ( )Hcoll0 is identical to Hcoll except that its kinetic energy is
modified into a f f= - ¶ + á ¶ ña a( ) ( ) ( ∣ )( )T B1 2kin0
0 02. This is
equivalent to introduction of a ‘vector’ potential [17],a f fº á ¶ ña( ) ∣A i 0 0 . If the coordinate α is one-dimensional,
the ‘vector’ potential a( )A can be eliminated by a gaugetransformation, ò a a( )( )Aexp i d . However, the following‘scalar’ potential remains
å
å
a f f f f
f f f f
f a a f aa a
F = ᶠ- ñá ¶ ñ
= ᶠñá ¶ ñ
=á ¶ ñ
-
a a
a a
a
¹
¹ � �
( ) ∣( ∣ ∣) ∣
∣ ∣
( ) ∣( ( )) ∣ ( )( ) ( )
( )
B
B
BH
12
1
12
12
. 4
nn n
n
n
n
0 0 0 0 0
00 0
0
def 0
0
2
From equation (4), it is apparent that aF ( )0 is positive andbecomes large where the adiabatic ground state is approxi-mately degenerate in energy, »� �n0 ( ¹n 0). When thespherical ground state (a = 0) shows degeneracy, it could besignificantly unfavored by aF ( )0 . The system tends to avoidthe degenerate ground state, which leads to the SBS withnuclear deformation.
We would like to emphasize again that the couplingbetween the collective (shape) degrees of freedom α and theintrinsic (single-particle) motion ξ is essential to produce thenuclear deformation. This is apparent for the coupling-drivenmechanism, and is also true for the degeneracy-driven case. Ifthe coupling term x a( )H ,int is absent, the adiabatic statesf x( )n are independent of α, thus, produce no gauge potentials,
a a= F =( ) ( )A 00 . We also note here that the presentargument on the degeneracy-driven (Jahn-Teller) mechanismexplains why the instability of a spherical state occurs, but notwhat kind of deformation takes place. This will be discussedin sections 3.3 and 6.2.
2.4. Field coupling
The oscillation of the variable α correspond to the shapevibration. Thus, it can be quantized to a boson operator. Inorder to describe the vibrational motion associated with α, weintroduce a boson space with the n-phonon state ñ∣n . When αis small, we may linearize the coupling term in equation (1)with respect to α as
x a ka x= -( ) ( ) ( )H F, , 5int
where κ is a coupling constant which depends on thenormalization of α and F. If the operator F is given, thenormalization of α is usually chosen as follows. The action ofthe one-body operator F on the ground state (a Slaterdeterminant) produces many one-particle-one-hole states;F ñ = å F ñáF F ñ∣ ∣ ∣ ∣F Fph ph ph0 0 . This is identified with the
operation of α in the collective (boson) space:
å aáF F ñ = á = = ñ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( )F n n1 0 . 6ph
ph 02 2
The coupling constant κ can be also determined by this self-consistency. See chapter 6 of BM2 for details of the fieldcoupling techniques.
If the matrix elements of F are identical to those of α asin equation (6), the field coupling(5) can be interpreted as an
3
Phys. Scr. 91 (2016) 073008 Invited Comment
!"#$ % = !'( + !*+,((%)
Finite-size effect (1): SSB in finite time
τ SSB ~ ℑ !
Symmetry restoration and its time scale
Gap in the NG modes; “massive”
ΔE ~ ℑ−1
EI = E0 +I(I +1)2ℑ
ψ(t) = e−iHt cI II∑ = e−iE0t cI exp −iI(I +1)t 2ℑ[ ] I
I∑
Rotational spectra
Deformed state
ℑ : Moment of inertia
τ SSB >> τ F ≈10−22 s
Pairing case
Symmetry restoration in a “rotating” frame
EN = EN0+µ(N − N0 )+
(N − N0 )2
2ℑ
ψ(t) = e−iH 't cN NN∑ = e−iE0t cN exp −i(N − N0 )
2 t 2ℑ⎡⎣ ⎤⎦ NN∑
Rotational spectra
Hamiltonian in the frame rotating with a constant angular frequency, μ H ' = H −µ(N − N0 )
τ SSB ~ ℑ !
When the moment of inertia is large enough, the symmetry breaking is realized.
τ SSB >> τ F
Finite size effect (2): Shell effect
svvrMrV lsll!"!
" ⋅++= 222
21)( ω
Shell structure/effect
→ Magic numbers (closed shell):
(N,Z)=2, 8, 20, 28, 50, 82, 126
Traditionally, we have been thinking it as
Shell effect → Symmetry Breaking
Alternatively, we may think it as
Shell effect → Symmetry restoring
324 REvIEws QP MQDERN PHYsIcs ~ APRIL 1972
lhLd(9CYUJ
LLI
LLJ
OI—lX
LLI
MOREZ'N
FERMlENERGY
LESS
FIG. II-1. Qualitative il-lustration of the connectionbetween the level density atthe Fermi energy and thebinding energy of the nucleus.
BOUND NUCLEUS
level density is smaller, because then the nucleonsoccupy deeper and more bound states. Conversely, thenucleus is less bound if the level density is increasednear the Fermi energy.This effect of the variation of the single-particle level
density in the vicinity of the Fermi energy on thenuclear binding energy is, in fact, a particular exampleof the general rule that, in quantal systems, degeneracyleads to reduced stability. This is so because even aninfinitely small perturbation of a degenerate systemproduces a finite response in the system due to re-arrangement ofmany close states. In nuclei, the bunchingof levels expresses, of course, only an approximatedegeneracy and, therefore, it requires a finite thoughsmall perturbation to reveal this feature of the de-generacy. In all other respects, the situation is analogousto the one met in some problems of molecular and solidstate physics; see, e.g., the Jahn —Teller rule in thetheory of molecules (Landau and Lifshitz, 1959).The nuclear ground state, as well as any other
relatively stable state, should thus correspond to thelowest possible degeneracy or, in other words, thelowest density of states near the Fermi energy. Fromthis, a new definition of a "magic" nucleus (or, moregenerally, of a shell closure) follows: it is the one, whichis the least degenerate, i.e., which has the lowest densityof intrinsic states at the Fermi level, among its neigh-bors. It should be noted that neither this definition northe concept of shells introduced above involves anyassumption concerning the nuclear shape. As the shelldistribution is also a function of the nuclear shape, andpronounced shells appear in deformed nuclei, one alsohas to generalize somewhat the concept of "magicity".Instead of being connected only to definite nucleonnumbers, "magicity" should be characterized by boththe nucleon number aed some characteristic deforma-tion of the nuclear shape at which the shell closuresoccur. A magic nucleus need not be spherical and, inaddition to the familiar magic numbers of nucleonsfor spherical nuclei, one can also speak of magicdeformed nuclei connected to other nucleon numbers.In the same way as, e.g., the presence of the familiarshell closures for Z=82, N=126 in a spherical shapenucleus is responsible for the increased stability ofspherical shapes in nuclei around lead-208, the deformed
WXMX oC~i
MWOCKXp, '
DEFORMATION ~
FIG II 2 Qualitativepicture of the distribu-tion of single-particlestates in the deformednucleus. The low-density regions (shellclosures) are indicatedby circles.
shape shell closures appearing for some other nucleonnumbers (e.g., X=100, 152) are responsible for theincreased stability of distorted shapes (correspondingto quadrupole distortions with Pm equal to 0.2 or 0.3)for the rare-earth and actinide nuclei. Confirmationhereof can be found in some empirica, l data. Here oneshould mention an observation by W. D. Myers andW. Swiatecki who in their analysis of nuclear massesfound systematic deviations in the middle of the rare-earth region and, independently of the theoreticalresults, found it compelling to introduce some kind ofmagicity also for deformed nuclei. This observation ledthem to improve their phenomenological shell correc-tions in a direction consistent with the calculations(Myers and Swiatecki, 1966a).In order to illustrate our point of view, we present in
Fig. II-2 a schematic fingerprint of single-particle leveldistributions in a deformed nucleus, where the magicshell closures as the regions of a locally low level densityare indicated by circles. In real nuclei, these regionsexpose an anomalous stability depending on specificshapes and nucleon numbers.As we shall explain later, the shell distribution is
expected to change appreciably with even a relativelysmall variation of the nuclear shape of the order of
20 j~ and, consequently, a nucleus can havemore than one shape where the condition of the lowestdegeneracy is locally fulfilled. Thus, one can also speak ofmagic nuclei in connection with a second shell closure ina strongly deformed nucleus. An important example isfound at N =146, where the nucleus becomes magic fora strongly distorted shape. A pronounced potential welldevelops in the nuclei around this nucleon number at adeformation characterized by the ratio of the twonuclear axes equa, l to 1.8—2.0.In fission, the nucleus may be caught into this other
well and stay there for a relatively long time. In this
Nuclear deformation
• “Traditional” understanding
– Bulk (liquid drop) part favors spherical shape
– Quantum (shell) effect makes the nucleus deformed
• Strutinsky method
– Shell correction: A1/3
– Surface energy: A2/3
• Favoring spherical shape in a heavy region?
324 REvIEws QP MQDERN PHYsIcs ~ APRIL 1972
lhLd(9CYUJ
LLI
LLJ
OI—lX
LLI
MOREZ'N
FERMlENERGY
LESS
FIG. II-1. Qualitative il-lustration of the connectionbetween the level density atthe Fermi energy and thebinding energy of the nucleus.
BOUND NUCLEUS
level density is smaller, because then the nucleonsoccupy deeper and more bound states. Conversely, thenucleus is less bound if the level density is increasednear the Fermi energy.This effect of the variation of the single-particle level
density in the vicinity of the Fermi energy on thenuclear binding energy is, in fact, a particular exampleof the general rule that, in quantal systems, degeneracyleads to reduced stability. This is so because even aninfinitely small perturbation of a degenerate systemproduces a finite response in the system due to re-arrangement ofmany close states. In nuclei, the bunchingof levels expresses, of course, only an approximatedegeneracy and, therefore, it requires a finite thoughsmall perturbation to reveal this feature of the de-generacy. In all other respects, the situation is analogousto the one met in some problems of molecular and solidstate physics; see, e.g., the Jahn —Teller rule in thetheory of molecules (Landau and Lifshitz, 1959).The nuclear ground state, as well as any other
relatively stable state, should thus correspond to thelowest possible degeneracy or, in other words, thelowest density of states near the Fermi energy. Fromthis, a new definition of a "magic" nucleus (or, moregenerally, of a shell closure) follows: it is the one, whichis the least degenerate, i.e., which has the lowest densityof intrinsic states at the Fermi level, among its neigh-bors. It should be noted that neither this definition northe concept of shells introduced above involves anyassumption concerning the nuclear shape. As the shelldistribution is also a function of the nuclear shape, andpronounced shells appear in deformed nuclei, one alsohas to generalize somewhat the concept of "magicity".Instead of being connected only to definite nucleonnumbers, "magicity" should be characterized by boththe nucleon number aed some characteristic deforma-tion of the nuclear shape at which the shell closuresoccur. A magic nucleus need not be spherical and, inaddition to the familiar magic numbers of nucleonsfor spherical nuclei, one can also speak of magicdeformed nuclei connected to other nucleon numbers.In the same way as, e.g., the presence of the familiarshell closures for Z=82, N=126 in a spherical shapenucleus is responsible for the increased stability ofspherical shapes in nuclei around lead-208, the deformed
WXMX oC~i
MWOCKXp, '
DEFORMATION ~
FIG II 2 Qualitativepicture of the distribu-tion of single-particlestates in the deformednucleus. The low-density regions (shellclosures) are indicatedby circles.
shape shell closures appearing for some other nucleonnumbers (e.g., X=100, 152) are responsible for theincreased stability of distorted shapes (correspondingto quadrupole distortions with Pm equal to 0.2 or 0.3)for the rare-earth and actinide nuclei. Confirmationhereof can be found in some empirica, l data. Here oneshould mention an observation by W. D. Myers andW. Swiatecki who in their analysis of nuclear massesfound systematic deviations in the middle of the rare-earth region and, independently of the theoreticalresults, found it compelling to introduce some kind ofmagicity also for deformed nuclei. This observation ledthem to improve their phenomenological shell correc-tions in a direction consistent with the calculations(Myers and Swiatecki, 1966a).In order to illustrate our point of view, we present in
Fig. II-2 a schematic fingerprint of single-particle leveldistributions in a deformed nucleus, where the magicshell closures as the regions of a locally low level densityare indicated by circles. In real nuclei, these regionsexpose an anomalous stability depending on specificshapes and nucleon numbers.As we shall explain later, the shell distribution is
expected to change appreciably with even a relativelysmall variation of the nuclear shape of the order of
20 j~ and, consequently, a nucleus can havemore than one shape where the condition of the lowestdegeneracy is locally fulfilled. Thus, one can also speak ofmagic nuclei in connection with a second shell closure ina strongly deformed nucleus. An important example isfound at N =146, where the nucleus becomes magic fora strongly distorted shape. A pronounced potential welldevelops in the nuclei around this nucleon number at adeformation characterized by the ratio of the twonuclear axes equa, l to 1.8—2.0.In fission, the nucleus may be caught into this other
well and stay there for a relatively long time. In this
Experimental signature of deformation
�� � N
, �
Z
+
+=2
42/4 EER
�����3���+�(&
• 8:;9<���+�
• Kohn-Sham-BdG (HFB) �$�
[ ])();(),(),(),(),(),( ttTtstjtJttE qqqqqqq κτρ!!!"
spin-currentkinetic
pair density
ℎ − 1 Δ−Δ∗ − ℎ − 1 ∗
4565
= 754565
Theoretical predictionEnergy density functional calculation
82
SkM* EDF
http://massexplorer.frib.msu.edu
Deformation map
�� � N
Nuclear deformation
• “Opposite” way of thinking
– Nucleus is genetically deformed.
– Shell effect makes the nucleus spherical.
82
Neutron-driven “deformed island/peninsula”
• “IoI” is common in very n-rich isotopes
Z=40 isotopes
SkM*
SLy4
Neutron-driven “deformed island/peninsula”
• “IoI” is common in very n-rich isotopes
Z=40 isotopes
SkM*
SLy4
Density distributions
9 = 56
9 = 58
9 = 60
9 = 62
E1 strength in Zr isotopes
N = 60 − 72Deformed region(prolate)
N = 40 − 58Spherical
N > 82
N = 76 − 92Spherical
N = 74oblate
7 [ MeV ]
Zr isotopes? @
A(7)
[ BC fm
C]
0
0.05
0.1
0.15
0.2
0.25
0.3
0 5x106 1x107 1.5x107 2x107 2.5x107 3x107
E [eV]
94Zr(γ, *)Berman et al. 1967
Neutron-driven “deformed island/peninsula”
• “IoI” is common in very n-rich isotopes
Z=50 isotopes
SkM*
SLy4
UNEDF1
• “IoI” is common in very n-rich isotopes
SkM*
SLy4
UNEDF1
Z=50 isotopes
SkM*
SLy4
UNEDF1
Neutron-driven “deformed island/peninsula”
• “IoI” is common in very n-rich isotopes
Z=82 isotopes
−0.4
0.4
0
0.2
−0.2
8C
Neutron-driven “deformed island/peninsula”
“Genetic” deformation
group), incorporating 2p -2h neutron excitations across theN ¼ 20 shell, have been carried out and show that the energyof these configurations could indeed drop and even cross theregular 0ℏ!, N ¼ 20 closed-shell configuration, resulting ina different structure appearing at low excitation energy(Caurier et al., 1998). This is illustrated in Fig. 2 for theMg nuclei in which the monopole gap, which is related tothe gap in the single-particle spectrum appearing between thefilled sd shell and the unfilled fp shell, is shown.
Figure 2 shows that a minimum in energy is reached atN ¼ 20, increasing when adding neutrons to the fp shell-model orbitals. Next, the correlation energy !ðEÞcorr, definedas
!ðEÞcorr ¼ hð2p -2h Þ; 0þ j H j ð2p -2h Þ; 0þ i% hð0p -0h Þ; 0þ j H j ð0p -0h Þ; 0þ i; (7)
is shown and gives the extra binding energy within thetruncated spaces, as a result of creating a 2p -2h neutronexcitation relative to a closed N ¼ 20 core. It is immediatelyclear that the intruder configuration, in particular, at and nearto N ¼ 20, corresponds to a more correlated state comparedto the 0ℏ! states. Thus, low-lying 2p -2h intruder configura-tions are favored only at and near to the N ¼ 20 neutron shellclosure.
Similar shell-model calculations have been performedat the N ¼ 28 shell closure, incorporating 2p -2h neutron
excitations from the 1f7=2 into the higher-lying pf orbitals.Results are obtained from the Z ¼ 28 doubly closed shell56Ni nucleus, down to the Z ¼ 12, 14, neutron-rich, Mg andSi nuclei (Caurier, Nowacki, and Poves, 2004).
The results of the N ¼ 20 and N ¼ 28 regions can becombined and are shown in Fig. 3, in which the energy ofthe lowest 0þ 2p -2h intruder configuration is given relative tothe 0ℏ! reference energy. It is clear that at N ¼ 20 a zone of‘‘inversion’’ appears in which the intruder configuration be-comes the lowest-lying state. Here we point out that the largedrop and ‘‘flat’’ behavior in energy of the 0þ2 intruder statesfits well with the schematic analysis of intruder 0þ statesdescribed in Sec. II.A.3.a, and more, in particular, in Fig. 4. InFig. 4, one observes a rather flat variation of the intruderexcitation energy as a function of N (or Z).
For the N ¼ 28 isotones, on the other hand, the stability of48Ca inhibits the formation of such an inversion but allows forlow-lying intruder 0þ states in 52Cr. However, moving to theneutron-rich N ¼ 28 nuclei, it shows that for 40Mg and 42Si,an inversion appears, mainly because of the large correlationenergy and the almost constant monopole energy for Z & 20.Mixing is needed to obtain a more realistic description in theN ¼ 28 region, which requires large-scale shell-model cal-culations with, for Z & 20, a valence space that consists ofthe full sd shell for the protons and the pf shell for neutrons.For Z > 20, the full fp shell for both protons and neutronshas been used (Caurier, Nowacki, and Poves, 2004).
The first set of conclusions following from the above shell-model studies is the fact that, if one enters a region of nucleiwith a number of valence protons and a closed-neutron shellor subshell, one has to consider the balance between, on oneside, the tendency to stabilize nuclei in a spherical shape for0ℏ! configurations, and, on the other side, the deformation-driving tendency when allowing the closed shells to bebroken with the subsequent formation of 2p -2h , etc.configurations.
One can deduce an approximate expression for the corre-lation energy which varies as nval!np -h with nval the number
of valence nucleons outside of the closed shells and np -h , thenumber of particle-hole pairs excited across the closed shell.Such a dependence results when the residual proton-neutron
18 1820 2022 2224 2426 2628 28N N
10
20
30
0 0
E)
VeM(
m
10987654
321
0p-0h2p-2h
)Ve
M(E
c
X X
2p1/21f5/2
1f7/2
1d3/22s1/21d5/2
2p3/2
20
FIG. 2 (color online). Schematic view of the 2p -2h neutron ex-citations from the sd shell into the fp shell (upper panel). In thelower part, the energy gap which equals the difference of themonopole energy for the normal and 2p -2h intruder 0þ configura-tion (left panel) and the correlation energy in the normal and 2p -2hintruder 0þ configuration [as derived from Eq. (7)] (right panel) aregiven for the Mg (Z ¼ 12) nuclei. From Caurier et al., 1998.
FIG. 3 (color online). Relative position of the lowest normal andlowest neutron 2p -2h intruder 0þ states, resulting from diagonaliz-ing in the separate subspaces. From Caurier et al., 1998, andCaurier, Nowacki, and Poves, 2004.
Kris Heyde and John L. Wood: Shape coexistence in atomic nuclei 1471
Rev. Mod. Phys., Vol. 83, No. 4, October–December 2011
Heyde, Wood, RMP 83 (2011)
Nakatsukasa, et al. Phys. Scr. 91 073008 (2016)
Excited 0+ states
• Unsolved mysteries in (stable/near-stable) nuclei
• Radii & transition form factor may be a possible solution??
Shell gap or emergence of 0#I*J
• Opening of the shell gap
• Emergence of 0#I*J
subshell occupancies, such as presented in Fig. 37 for the Geisotopes. [We note that transition densities deduced frominelastic electron scattering also can reveal important infor-mation about differences in the structure of 0þ configurations
(Bazantay et al., 1985).] It is evident that the concept of asingle pairing condensate (single vacuum) upon which allcollectivity is built is probably never realized in nuclei.
A useful overall view of excited 0þ states in nuclei ispresented in Table VII which shows the lowest known casesacross the mass surface. In particular, many of these cases liein regions of established shape coexistence and, indeed, havebeen identified as coexisting structures.
4. Where to look
Shape coexistence at low energy has now emerged in awidely spread number of mass regions as the result of avariety of dominant structural factors. We suggest criteriafor further searches below, but we caution that the historicalrecord has been rather full of surprises.
The overriding factor that appears to be needed for theappearance of shape coexistence at low energy is a competi-tion between an energy gap and a residual interaction thatlowers the energy of configurations involving promotion ofnucleons across the gap.
The occurrence of the ‘‘gap and interaction’’ mechanismmost commonly can arise in singly closed shell regions nearmidshell (for the other kind of nucleon) and V!;". The Z ¼ 50and 82 regions near N ¼ 66 and 104, respectively, are clearmanifestations of this. But this rule needs an exception for theN ¼ 50 and 82 regions near Z ¼ 39 and 66 where low-energyshape coexistence is not observed: The suppression effect ofsubshell gaps at Z ¼ 40 and 64 appears to be the answer. Themost valuable data in support of this idea are those at Z# 40,N # 60 as manifested in Figs. 27 and 28 which show thatnuclei at and close to double-subshell gaps can exhibit shapecoexistence via the suppression of ground-state collectivity.Thus, shape coexistence in the N ¼ 50 and 82 regions needs
8+ 8+ 8+ 8+ 8+
6+ 6+ 6+6+
0+6+
4+
2+
0+
0+
0+
0+0+
0+
2+ 2+2+ 2+
4+ 4+ 4+ 4+
131ns 190ns3589 2760 2644 2531 2428
2612 2498 2424 22822520
3448
3077
2186
1761
0
0
00
0
1510 14311415
1395
2283 2187 20992082
71 s 2.1 s 480ns
9040 50
Zr 9242 50
Mo 9444 50
Ru 9646 50
Pd 9848 50
Cd
FIG. 50. A similar pattern, as the one in Fig. 49 (the Ni isotopes)for the N ¼ 50 isotones, also involving the same shell-modelorbitals. The data are from Nuclear Data Sheets.
TABLE VI. Mixing strength (in units of keV) used in the description of energy, decay, and transferreaction properties of coexisting structures.
Isotope Vmix Quantities fitted Reference
72Kr 310 E Becker et al., 1999; Korten, 2001; Bouchez et al., 200374Kr 340 E Becker et al., 1999; Korten, 2001; Bouchez et al., 200376Kr 250 E Becker et al., 1999; Korten, 2001; Bouchez et al., 200378Kr 200 E Becker et al., 1999; Korten, 2001; Bouchez et al., 200398Sr 67 E, BðE2Þ, #2ðE0Þ Mach et al., 1989
34 BðE2Þ, #2ðE0Þ Wu, Hua, and Cline, 2003100Zr 115 E, BðE2Þ, #2ðE0Þ Mach et al., 1989
88 BðE2Þ, #2ðE0Þ Wu, Hua, and Cline, 200398Mo 326 BðM1Þ Rusev et al., 2005100Mo 321 BðM1Þ Rusev et al., 2005112;114Cd 297 $ðt; pÞ O’Donnell, Kotwal, and Fortune, 1988152Sm 310 #2ðE0Þ Kulp et al., 2007176Pt 180 E Dracoulis et al., 1986178Pt 210 E Dracoulis et al., 1986180Pt 220 E Dracoulis et al., 1986182Pt 230 E Dracoulis et al., 1986184Pt 240 E Dracoulis et al., 1986186Pt 220 E Dracoulis et al., 1986188Pt 400 E Dracoulis et al., 1986192Pb 52 BðE2Þ, #2ðE0Þ Van Duppen, Huyse, and Wood, 1990194Pb 51 BðE2Þ, #2ðE0Þ Van Duppen, Huyse, and Wood, 1990
Kris Heyde and John L. Wood: Shape coexistence in atomic nuclei 1507
Rev. Mod. Phys., Vol. 83, No. 4, October–December 2011
Heyde, Wood, RMP 83 (2011)
Puzzles: Multi-phonon statesTextbook example of phonon excitataion: 112Cd
J. Phys. G: Nucl. Part. Phys. 37 (2010) 064028 P E Garrett and J L Wood
0
2
20 4
320 4 6
1-phonon
2-phonon
3-phonon
2
3
Figure 1. Levels expected in the harmonic quadrupole vibrator up to the three-phonon state,labelled by their spins and parities and with energies in ‘oscillator’ units, hω.
0 0
3 2066
2 618
0 1433 4 1416
2 2121 4 2082
0 1871
6 2168
2 1312
1-phonon
2-phonon}
}3-phonon
Figure 2. Partial level scheme displaying the levels in 112Cd that have traditionally been assignedas phonon states, arranged for easy comparison with figure 1. Energies are given in keV.
candidates for such a degree of freedom, based on the observation of excited states with therequisite spins and parities, made nuclei in this region text-book cases [4, 5] for this remarkablysimple mode of nuclear collective excitation. The example of 112Cd, organized so that theexcited states match those in figure 1, is shown in figure 2.
A much less well-recognized set of properties of the harmonic quadrupole vibrator is itselectromagnetic properties, i.e. its electric quadrupole or E2 moments and E2 transition rates.Primarily, this is because these properties are extremely difficult to measure. In consequence,the almost complete lack of information on E2 properties for the Cd isotopes led to the strongbelief, based on excitation energy information alone, that the Cd isotopes are the text-bookexample of harmonic quadrupole vibrations in nuclei.
Advances in techniques for precisely measuring E2 moments and E2 transition rateshave dramatically changed in the last 20 years. This has come about through the possibilityof carrying out multi-step Coulomb excitation and picosecond to femtosecond lifetimemeasurements of excited states and through the capability of accurately measuring branchingratios of low-energy high-lying transitions. The latter capability is crucial because the intensityof the interesting ‘collective’ transitions are attenuated by E5
γ and this results in the mostimportant transitions being extremely difficult to observe.
The expected pattern of E2 properties for a harmonic quadrupole vibrator is shownin figure 3. All diagonal E2 matrix elements are zero because of the simple selectionrule that the E2 operator is a linear combination of the raising and lowering operators forvibrations and so directly connects any given state with phonon number N only to states withphonon number N + 1 or N − 1, i.e. the E2 operator has diagonal matrix elements equalto zero. The transition matrix elements reflect their dependence on the phonon number, e.g.B(E2;N = 2 → N = 1) = 2B(E2;N = 1 → N = 0). Transitions involving stateswith N > 2 also have factors which depend on coefficients of fractional parentage, but fora given state the sum of the decay branches equals the simple phonon number scaling, e.g.!
B(E2; 3+, N = 3 → 2+, 4+, N = 2) = 3B(E2;N = 1 → N = 0).
2
J. Phys. G: Nucl. Part. Phys. 37 (2010) 064028 P E Garrett and J L Wood
0
2
20 4
320 4 63-phonon
1
22 2
3 3
7_5 4_
7
36__35 15__
7 6_7
10__7
11__7
2-phonon
1-phonon
Figure 3. B(E2) values, relative to the B(E2; 2+1 → 0+
1) value, expected for a quadrupoleharmonic vibrator.
0 0
2 1312
3 2066
0 1433 41416
0 1871
2 2121 6216842082
2 618
1
0.49(8)0.00033(3) 1.98(17)
100<9.5
0.89(27)<0.014
<0.07 2.05(56)
0.83(27)
2.3(7)0.9(3)
______
Figure 4. Systematics of B(E2) values for the even–even Cd isotopes 110−116Cd. Transitions arelabelled with their B(E2) values in W.u. with 1σ uncertainties on the last digit in the parentheses;the listing of two numbers reflects asymmetric uncertainties with +1σ and −1σ , respectively.Values without uncertainties are relative B(E2) values, or upper limits. Dashed arrows indicateunobserved transitions where upper limits have been established. Of particular note are the greatlydisparate values in 114Cd between results of Coulomb excitation and lifetime measurements forsome levels, the most serious of which occurs for the 1842 keV 2+ level; the values obtained fromCoulomb excitation are listed above the transitions and are up to a factor of ≈35 greater than thosederived from the most stringent lifetime limits.
Figure 4 shows the low-lying excited states in 112Cd grouped on the basis of B(E2)
values [6, 7]. Evidently, the pattern does not match that shown in figure 3. In particular, atthe two-phonon level, the B(E2) value for decay from the 2+ state is a factor of 4 weakerthan expected, and that of the 0+ state is extraordinarily small. At the three-phonon level, the0+ state has no observed decay to the two-phonon 2+ state [7, 8], and the 2+ member has anenhanced decay to the 0+ member of the two-phonon triplet only. This disagreement is inmarked contrast to the similarities between figure 1 and figure 2. Indeed, this result came asa complete surprise [6, 9]. Further, the disagreement is not peculiar to 112Cd: this observedpattern of B(E2) values persists from 110Cd to 116Cd [6–17], as shown in figure 5.
Historically, a natural response to any observed deviation from the harmonic quadrupolevibrator has been to invoke ‘anharmonicities’: there is a very large literature covering this, butit is more useful here to look further at data. For 114Cd there are multi-step Coulomb excitation
3
J. Phys. G: Nucl. Part. Phys. 37 (2010) 064028 P E Garrett and J L Wood
0
2
20 4
320 4 63-phonon
1
22 2
3 3
7_5 4_
7
36__35 15__
7 6_7
10__7
11__7
2-phonon
1-phonon
Figure 3. B(E2) values, relative to the B(E2; 2+1 → 0+
1) value, expected for a quadrupoleharmonic vibrator.
0 0
2 1312
3 2066
0 1433 41416
0 1871
2 2121 6216842082
2 618
1
0.49(8)0.00033(3) 1.98(17)
100<9.5
0.89(27)<0.014
<0.07 2.05(56)
0.83(27)
2.3(7)0.9(3)
______
Figure 4. Systematics of B(E2) values for the even–even Cd isotopes 110−116Cd. Transitions arelabelled with their B(E2) values in W.u. with 1σ uncertainties on the last digit in the parentheses;the listing of two numbers reflects asymmetric uncertainties with +1σ and −1σ , respectively.Values without uncertainties are relative B(E2) values, or upper limits. Dashed arrows indicateunobserved transitions where upper limits have been established. Of particular note are the greatlydisparate values in 114Cd between results of Coulomb excitation and lifetime measurements forsome levels, the most serious of which occurs for the 1842 keV 2+ level; the values obtained fromCoulomb excitation are listed above the transitions and are up to a factor of ≈35 greater than thosederived from the most stringent lifetime limits.
Figure 4 shows the low-lying excited states in 112Cd grouped on the basis of B(E2)
values [6, 7]. Evidently, the pattern does not match that shown in figure 3. In particular, atthe two-phonon level, the B(E2) value for decay from the 2+ state is a factor of 4 weakerthan expected, and that of the 0+ state is extraordinarily small. At the three-phonon level, the0+ state has no observed decay to the two-phonon 2+ state [7, 8], and the 2+ member has anenhanced decay to the 0+ member of the two-phonon triplet only. This disagreement is inmarked contrast to the similarities between figure 1 and figure 2. Indeed, this result came asa complete surprise [6, 9]. Further, the disagreement is not peculiar to 112Cd: this observedpattern of B(E2) values persists from 110Cd to 116Cd [6–17], as shown in figure 5.
Historically, a natural response to any observed deviation from the harmonic quadrupolevibrator has been to invoke ‘anharmonicities’: there is a very large literature covering this, butit is more useful here to look further at data. For 114Cd there are multi-step Coulomb excitation
3
Ideal phonon excitation spectra
Garrett, Wood, JPG 37, 064028 (2010)
B(E2) strengths are inconsistent with multi-phonon interpretation?
Inconsistent with the interpretation of beta-vib.?V(β)
ββ-vib.
Beta vibration
R6 Topical Review
Figure 1. Ratios of B(E2) values for the decay of the 2+0+
2level to the ground state band. The
B(E2; 2+0+
2→ 4+
gsb) values, predicted to be the largest from the Alaga rules, have been taken as
the reference. The broken horizontal lines are the Alaga rule values. Only those data for whichall three transitions have been observed are plotted. The B(E2; 2+
0+2
→ 2+gsb) values plotted have
been computed assuming a pure E2 transition (the mixing ratios δ are not known in most cases).Therefore, they can be considered as upper limits (BNL data).
rates; with an opposite sign for M2 compared with M1, the B(E2) value for spin increasingtransitions will be larger than the B(E2) value for spin-decreasing transitions [30]. Therefore,the data presented in figure 1 may be used to argue that, in most cases, the sign of the mixingmatrix element for β-to-ground band transitions is the same as that for γ -to-ground bandtransitions [31]. B(E2) ratios that are above the Alaga expectations suggest a remeasurementof the branching ratio may be in order, or for the case of the 2+
0+2→ 2+
gsb transition, the presenceof significant M1 intensity. Such large M1 components would be surprising, however, if thedecay were from a β-vibrational state where one expects collective E2 transitions.
Higgs amplitude mode
Garrett, JPG 27, R1 (2001)
N
Consistent with the beta-vibration picture
Coriolis coupling effect
B(E2; Ii → I f ) =M1 +M2 I f (I f +1)− Ii (Ii +1)⎡⎣ ⎤⎦
Shimizu and Nakatsukasa, Nucl. Phys. A611 (1996) 22
M1, M2 can be calculated with the cranking approach at the limit of ωrot = 0.
0
50
100
150
200
250
84 86 88 90 92 94 96 98
B(E2
,0+ 2-
>2+ 1)
(W.u
.)
N
Gd LO+NLOLO
0
5
10
15
20
25
84 86 88 90 92 94 96 98
B(E2
,0+ 1-
>2+ 2)
(W.u
.)
N
Gd LO+NLOLO
Matsuzaki and Ueno, PTEP 2016 (2016) 043D03Nakatsukasa et al., Phys. Scr. 91, 073008 (2016)
1– 2173
8– 2888
Oct0
1– 9633– 1041
5– 1222
7– 1506
9– 1879
11– 2327
13– 2833
1– 1681
3– 1779
5– 1976
7– 2177
9– 2507
11– 2905
13– 3392
g
2+ 122
4+ 366
6+ 707
8+ 1125
10+ 1609
12+ 2149
14+ 2736
0+ 0
β
2+ 811
4+ 1023
6+ 1311
8+ 1666
10+ 2058
12+ 2526
14+ 3292
0+ 685
i
2+ 1293
4+ 1613
6+ 2004
0+ 1083
2+ 1944
0+ 1755
γ
2+ 1086
4+ 1371
6+ 1728
8+ 2140
10+ 2662
9+ 2376
11+ 2832
3+ 1234
7+ 1946
5+ 1559
Γ
4+ 1757
6+ 2040
8+ 2392
9+ 2587
7+ 2205
5+ 1891
2+ 1769
4+ 2052
6+ 2417
9+ 3018
3+ 1907
7+ 2623
5+ 2237
4+ 2402
Oct1
1– 1511
6– 19307– 2004
8– 2201
9– 2290
10– 2510
11– 2640
5– 17644– 1683
3– 1579
2– 1530
152Sm collective rotational bandsby Sharpey-Schafer
Oct0
1– 9633– 1041
5– 1222
7– 1506
9– 1879
11– 2327
13– 2833
g
2+ 122
4+ 366
6+ 707
8+ 1125
10+ 1609
12+ 2149
14+ 2736
0+ 0
i
2+ 1293
4+ 1613
6+ 2004
0+ 1083
γ
2+ 1086
4+ 1371
6+ 1728
8+ 2140
10+ 2662
9+ 2376
11+ 2832
3+ 1234
7+ 1946
5+ 1559
Γ
4+ 1757
6+ 2040
8+ 2392
9+ 2587
7+ 2205
5+ 1891
Oct1
1– 1511
6– 19307– 2004
8– 2201
9– 2290
10– 2510
11– 2640
5– 17644– 1683
3– 1579
2– 1530
1– 717
3– 738
5– 755
7– 670
9– 628
11– 578
13– 558
2+ 689
4+ 656
6+ 604
8+ 541
10+ 467
12+ 377
14+ 556
0+ 685
2+ 672
0+ 652 2+ 683
4+ 680
6+ 689
9+ 642
3+ 673
7+ 677
5+ 677
4+ 645
1– 662
8– 687
Coexisting (nearly) identical bands in 152Sm
Oct0βiβ γβ Γβ Oct1ββ
Summary
• Deformed nuclei
– SSB in finite time
– All the heavy nuclei are deformed in both (either) the ground and (or) excited states.
• Spherical nuclei
– Ground states: Rarer in heavier nuclei
– Excited states: Even rarer, could be none
• Unsolved mysteries associated with exc. 0+
– Shell gap or low-lying exc. 0+
– Multi-phonon, beta-vib, pair isomer?
