Possible Analog States of the Hoyle State in Heavier Nuclei

 
NATIONAL RESEARCH CENTRE "KURCHATOV INSTITUTE
 
   
Possible analogs of the Hoyle state
in heavier 4N nuclei
 
* Through the Hoyle state, elements
heavier than 
12
C are formed in the
Universe
* Hoyle's state is not described within the
shell model.
* The structure of the state is unknown
A prototype of the α-condensate state?
*
 
Increased radius is predicted within
many theoretical models
 
 
2
7.65 MeV 0
+
2
 state in 
12
C (Hoyle state)
 
* Stll actual 
K.C.W. Li et al. Phys. Rev. C 105, 024308  (2022) | T. Otsuka et al., Nature Comm. 13, 2234 (2022)
 
Our result: 2.9 fm; 60% probability of
configuration with 3 
α
-clusters on the
lowest s-orbit
Calculations from first principles: 2.8 fm
(difference between radii of the g.s. and
the Hoyle state – 0.36 fm) 
T. Otsuka et al.,
Nature Comm. 13, 2234 (2022)
 
 
 
R
rms
 of the Hoyle state
 
D
i
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f
r
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r
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I
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+
 
 
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+
)
 
R
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M
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1
7
,
 
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.
 
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C
 
8
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4
6
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3
 
(
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0
9
)
.
Method – Modified diffraction model
 
3
 
4
                 
Analogs of the Hoyle state (
12
C, 0
+
2
) in 
16
O
 
We determined directly the rms radii of 
16
O in a number of states with excitation energies
up to 15.1 MeV applying the MDM.
No significant radius enhancement in any states was observed.
We did not confirm the existence of a dilute state with super-large radius associated with the
15.1-MeV 0
+
6
 state, which was predicted by the αBEC model. 
The rms radius of 
16
O in this state
was found similar to the radius of 
16
O in the ground state.
From this point of view, the 0
+
6
 state cannot be considered as an analog of the Hoyle state in 
12
C.
 
αBEC
Y. Funaki et al.,
PRL 101, 082502 (2008).
 
MDM 
A. A. Ogloblin et al.,
Phys. Rev. C 94, 051602(R) (2016)
Overview of current situation with 
20
Ne
 
Experimental works
 
 
 
 
 
 
 
 
 
 
 
 
 
AMD calculations: 
Y. Kanada-En’yo and K. Ogata, Phys. Rev. C 101, 064308 (2020)
In the structure calculation of 
20
Ne with AMD, 
16
O + α cluster structures were obtained in the parity-doublet
K
π
 = 0
+
1
 and K
π
 = 0
1
 bands, and the 
12
C +2α-like structure was obtained in the K
π
 = 2
 band. The AMD
calculation reproduced the experimental B(E2) of in-band transitions. It also described the experimental
form factors of the 0
+
1
 , 2
+
1
 , and 4
+
1
 states
R. Bijker and F. Iachello 
Nuclear Physics A 1006, 122077 (2021).
Available experimental data in 
20
Ne 
 
can be described in terms of the bi-pyramidal cluster configuration
suggested by Brink in 1970.
 
All observed bands with band-heads up to an excitation energy of about 12 MeV
can be accounted for in terms of the vibrationless ground state band, of the nine vibrational modes expected
on the basis of D
3h
 symmetry (3 singly degenerate and 3 doubly degenerate), and of six of the double
vibrational
 
bands expected on the basis of D
3h
 symmetry. Strong evidence for a quasimolecular structure of
20
Ne with D3h symmetry.
 
5
S. Adachi 
et al., Physics Letters B 819, 136411 (2021)
conducted the coincidence measurement of alpha particles
inelastically scattered from 
20
Ne at 0◦ in order to search for the
5α condensed state in 
20
Ne. Comparing the measured excitation
energy spectra and decay branching ratio with the statistical
decay-model calculation, found that the newly observed states
 
at
E
x
 = 23.6, 21.8, and 21.2 MeV in 
20
Ne are strongly coupled to the
0
+
6
 state in 
16
O.  These states are the candidates for the 5α
condensed state in 
20
Ne?
 
Theoretical works
 
J. Zhang, W.D.M. Rae, Nucl. Phys. A 564, 252
(1993).
 
6
Overview of current situation with 
20
Ne
 
Theoretical works
 
 
 
 
 
 
 
 
 
 
 
 
 
framework based on energy density functionals: 
P. Marević et al.,
Phys. Rev. C 97, 024334 (2018)
The structure of the lowest positive- and negative-parity bands of
20
Ne has been analyzed using a beyond mean-field approach based
on relativistic energy density functionals both for the ground-state
band as well as for the excited K
π
 = 0
±
 bands. A good agreement
with experimental results for the energies of the lowest positive-
parity states, as well as with available data on low-energy negative-
parity states. In particular, the spectroscopic properties of 
20
Ne
have been calculated at a level of accuracy comparable to those
obtained using more specific models, such as antisymmetrized
molecular dynamics.
ab initio calculations: 
A.C. Dreyfuss et al., Phys. Rev. C 102, 044608
(2020).
The formalism was demonstrated in a study of the 
16
O + 𝛼 cluster
structure of 
20
Ne, through inspection of the relative motion wave
functions of the clusters within the 
20
Ne ground state and the 1
resonance (ab initio SA-NCSM calculations). The importance of
correlations in developing cluster structures was shown.
 
g.s. 0
+
 – 1.64 MeV 2
+
 – 4.25 MeV 4
+
 
7
Band K
π
 = 0
+
1
 
𝑅
rms
 (g.s.)= 2.87 ± 0.03 fm
 
𝑅
rms
(1.63) = 3.0 ± 0.2 fm
 
AMD: Y. Kanada-En’yo and K. Ogata, Phys. Rev.  C 101, 064308 (2020).
 
16
O + 
𝛼 
cluster structure
Normal radius
 
4.97 MeV 2
5.62 MeV 3
–7.00 MeV 4
–8.45 MeV 5
 
8
Band K
π
 = 2
-
 
𝑅
rms
 (g.s.)= 2.9 ± 0.1 fm
 
AMD: Y. Kanada-En’yo and K. Ogata, Phys. Rev.  C 101, 064308 (2020).
 
12
C + 2
𝛼 
cluster structure
Normal radius
 
5.79 MeV 1
 – 7.17 MeV 3
 – 10.26 MeV 5
 
9
Band K
π
 = 0
-
1
 
AMD: Y. Kanada-En’yo and K. Ogata, Phys. Rev.  C 101, 064308 (2020).
 
16
O + 
𝛼 
cluster structure
10% radius increase
 
𝑅
rms
(5.79) = 3.4 ± 0.2 fm
 
𝑅
rms
(7.17) = 3.4 ± 0.2 fm
 
6.73 MeV 0
+
 
– 7.42 MeV 2
+
 – 9.03 MeV 4
+
 
10
Band K
π
 = 0
+
2
 
𝑅
rms
 (g.s.)= 2.87 ± 0.03 fm
 
𝑅
rms
(6.73) 
 3.6 fm
 
 
Perspectives
: we need to check other high-lying
0
+
 states, new experimental data are needed
 
Current results are published: 
A.S. Demyanova
et al., arXiv:2206.09475 (2022)
 
We obtained 25% radius increase for this state
comparing to the g.s. radius.
Practically the same increase we observed for
the Hoyle state in 
12
C.
The 6.73 MeV state is located 2 MeV above
the 𝛼-emission threshold, so our result can
be the argument for the possible alpha-
condensate structure.
 
11
Next goal – 
24
Mg
 
Currently no radius enhancement was observed for low-lying excited of 
24
Mg
Perspectives
: we need to check other high-lying states, especially 0
+
Our colleagues last results: 
Dong-Xi Wang et al., Chin. Phys. C (2022), to be published
: A
number of resonances up to 30 MeV excitation in 
24
Mg with the 
α
+
20
Ne cluster configuration
were observed, most of the states are decaying to 
20
Ne in the g.s. and two first excited states
(experiment 
16
O(
12
C,
24
Mg->
α
+
20
Ne)
α
).
 
Preliminary results for low-lying states
 
The rms radii of 
16
O in a number of states with
excitation energies up to 15.1 MeV were
determined. No significant radius enhancement in
any states was observed.
The 𝑟𝑚𝑠 radii of 
20
Ne in a number of states with
excitation energies up to 7 MeV were determined.
20% radius enhancement was obtained for the K
𝜋
 =
0
1
 band members. Moreover, our estimates of the
radius of the 0
+
2
 state, the head of the K
𝜋
 = 0
+
2
 band,
showed 25% radius increase. Obtained result can
speak in favor of possible 𝛼-condensate structure
of the 0
+
2
 state
Preliminary results for the 
24
Mg - no significant radius
enhancement in any states was observed, analysis is
in process
Next goal – 
28
Si
Conclusions
 
12
Slide Note

For measuring the radii of excited states we proposed a method based on the analysis of the inelastic diffraction scattering. We applied the model to studying the α- and 3He-scattering on 12C. The analysis allowed determining the enhanced radii of some excited states in 12C including the famous Hoyle state (0+, E* = 7.65 MeV). Besides the diffraction method the inelastic nuclear rainbow scattering also can be used for measuring the radii of unstable states. The enhancement of the radius of the excited state is detected by the shift of the inelastic rainbow minimum relatively the elastic one. The application of this approach to the inelastic scattering leading to excitation of the Hoyle state gave results consistent with those obtained by the diffraction method. Considerable attention has been drawn to the studies of α-cluster states in 12C, especially the second 0+ state, at Ex= 7.65MeV (so called the Hoyle state).

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Research conducted at the National Research Centre, Kurchatov Institute, explores possible analogs of the Hoyle state in heavier 4N nuclei, focusing on the 7.65 MeV 0+2 state in 12C (Hoyle state). The study reveals insights into the structure and characteristics of the Hoyle state, crucial for understanding the formation of elements heavier than 12C in the Universe. The research utilizes a modified diffraction model to determine diffraction radii and investigates analog states in 16O. Experimental works on 20Ne further shed light on condensed states associated with specific excitation energies, contributing to the broader understanding of nuclear physics.


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  1. NATIONAL RESEARCH CENTRE "KURCHATOV INSTITUTE POSSIBLE ANALOGS OF THE HOYLE STATE IN HEAVIER 4N NUCLEI

  2. 7.65 MeV 0+2state in 12C (Hoyle state) Rrmsof the Hoyle state * Through the Hoyle state, elements heavier than 12C are formed in the Universe * Hoyle's state is not described within the shell model. * The structure of the state is unknown A prototype of the -condensate state? * Increased radius is predicted within many theoretical models Our result: 2.9 fm; 60% probability of configuration with 3 -clusters on the lowest s-orbit Calculations from first principles: 2.8 fm (difference between radii of the g.s. and the Hoyle state 0.36 fm) T. Otsuka et al., Nature Comm. 13, 2234 (2022) 2 * Stll actual K.C.W. Li et al. Phys. Rev. C 105, 024308 (2022) | T. Otsuka et al., Nature Comm. 13, 2234 (2022)

  3. Method Modified diffraction model Diffraction radii are determined from minima (maxima) positions in d /d Inelastic scattering (0+ 0+) Elastic scattering 2 d d 1( ) x ( ) ~J x el d d 2 0 (0 0) ~ ( ) x J = x qR dif Rdifis the only parameter of the model. As it has the dimension of length it can be directly connected with a real (rms) nuclear radius Model: Rrms(exc.st) = Rrms(gr.st) + [Rdif(in) - Rdif(el)] The method was tested in the analysis of the radius of Hoyle state in 12C A.S. Demyanova et al., Int. J. Mod. Phys. E17, 2118 (2008). A.N. Danilov et al., Phys. Rev. C 80, 054603 (2009). 3

  4. Analogs of the Hoyle state (12C, 0+2) in 16O MDM A. A. Ogloblin et al., Phys. Rev. C 94, 051602(R) (2016) BEC Y. Funaki et al., PRL 101, 082502 (2008). We determined directly the rms radii of 16O in a number of states with excitation energies up to 15.1 MeV applying the MDM. No significant radius enhancement in any states was observed. We did not confirm the existence of a dilute state with super-large radius associated with the 15.1-MeV 0+6state, which was predicted by the BEC model. The rms radius of 16O in this state was found similar to the radius of 16O in the ground state. From this point of view, the 0+6state cannot be considered as an analog of the Hoyle state in 12C. 4

  5. Overview of current situation with 20Ne Experimental works S. Adachi et al., Physics Letters B 819, 136411 (2021) conducted the coincidence measurement of alpha particles inelastically scattered from 20Ne at 0 in order to search for the 5 condensed state in 20Ne. Comparing the measured excitation energy spectra and decay branching ratio with the statistical decay-model calculation, found that the newly observed states at Ex= 23.6, 21.8, and 21.2 MeV in 20Ne are strongly coupled to the 0+6state in 16O. These states are the candidates for the 5 condensed state in 20Ne? Theoretical works J. Zhang, W.D.M. Rae, Nucl. Phys. A 564, 252 (1993). AMD calculations: Y. Kanada-En yo and K. Ogata, Phys. Rev. C 101, 064308 (2020) In the structure calculation of 20Ne with AMD, 16O + cluster structures were obtained in the parity-doublet K = 0+1and K = 0 1bands, and the 12C +2 -like structure was obtained in the K = 2 band. The AMD calculation reproduced the experimental B(E2) of in-band transitions. It also described the experimental form factors of the 0+1, 2+1, and 4+1states R. Bijker and F. Iachello Nuclear Physics A 1006, 122077 (2021). Available experimental data in 20Ne can be described in terms of the bi-pyramidal cluster configuration suggested by Brink in 1970. All observed bands with band-heads up to an excitation energy of about 12 MeV can be accounted for in terms of the vibrationless ground state band, of the nine vibrational modes expected on the basis of D3hsymmetry (3 singly degenerate and 3 doubly degenerate), and of six of the double vibrational bands expected on the basis of D3hsymmetry. Strong evidence for a quasimolecular structure of 20Ne with D3h symmetry. 5

  6. Overview of current situation with 20Ne Theoretical works framework based on energy density functionals: P. Marevi et al., Phys. Rev. C 97, 024334 (2018) The structure of the lowest positive- and negative-parity bands of 20Ne has been analyzed using a beyond mean-field approach based on relativistic energy density functionals both for the ground-state band as well as for the excited K = 0 bands. A good agreement with experimental results for the energies of the lowest positive- parity states, as well as with available data on low-energy negative- parity states. In particular, the spectroscopic properties of 20Ne have been calculated at a level of accuracy comparable to those obtained using more specific models, such as antisymmetrized molecular dynamics. ab initio calculations: A.C. Dreyfuss et al., Phys. Rev. C 102, 044608 (2020). The formalism was demonstrated in a study of the 16O + ? cluster structure of 20Ne, through inspection of the relative motion wave functions of the clusters within the 20Ne ground state and the 1 resonance (ab initio SA-NCSM calculations). The importance of correlations in developing cluster structures was shown. 6

  7. Band K= 0+1 g.s. 0+ 1.64 MeV 2+ 4.25 MeV 4+ 16O + ? ? cluster structure Normal radius ? ?rms(g.s.)= 2.87 0.03 fm ? ?rms(1.63) = 3.0 0.2 fm AMD: Y. Kanada-En yo and K. Ogata, Phys. Rev. C 101, 064308 (2020). 7

  8. Band K= 2- 4.97 MeV 2 5.62 MeV 3 7.00 MeV 4 8.45 MeV 5 ? ?rms(g.s.)= 2.9 0.1 fm 12C + 2? ? cluster structure Normal radius AMD: Y. Kanada-En yo and K. Ogata, Phys. Rev. C 101, 064308 (2020). 8

  9. Band K= 0-1 5.79 MeV 1 7.17 MeV 3 10.26 MeV 5 16O + ? ? cluster structure 10% radius increase ?rms(5.79) = 3.4 0.2 fm ?rms(7.17) = 3.4 0.2 fm AMD: Y. Kanada-En yo and K. Ogata, Phys. Rev. C 101, 064308 (2020). 9

  10. Band K= 0+2 6.73 MeV 0+ 7.42 MeV 2+ 9.03 MeV 4+ ? ?rms(g.s.)= 2.87 0.03 fm ? ?rms(6.73) 3.6 fm We obtained 25% radius increase for this state comparing to the g.s. radius. Practically the same increase we observed for the Hoyle state in 12C. The 6.73 MeV state is located 2 MeV above the ?-emission threshold, so our result can be the argument for the possible alpha-condensate structure. Perspectives: we need to check other high-lying 0+states, new experimental data are needed Current results are published: A.S. Demyanova et al., arXiv:2206.09475 (2022) 10

  11. Next goal 24Mg Preliminary results for low-lying states Currently no radius enhancement was observed for low-lying excited of 24Mg Perspectives: we need to check other high-lying states, especially 0+ Our colleagues last results: Dong-Xi Wang et al., Chin. Phys. C (2022), to be published: A number of resonances up to 30 MeV excitation in 24Mg with the +20Ne cluster configuration were observed, most of the states are decaying to 20Ne in the g.s. and two first excited states (experiment 16O(12C,24Mg-> +20Ne) ). 11

  12. Conclusions The rms radii of excitation energies up to 15.1 MeV were determined. No significant radius enhancement in any states was observed. The ??? radii of20Ne in a number of states with excitation energies up to 7 MeV were determined. 20% radius enhancement was obtained for the K?= 0 1band members. Moreover, our estimates of the radius of the 0+2state, the head of the K?= 0+2band, showed 25% radius increase. Obtained result can speak in favor of possible ?-condensate structure of the 0+2state Preliminary results for the24Mg - no significant radius enhancement in any states was observed, analysis is in process Next goal 28Si 16O in a number of states with 12

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