In-Medium Cluster Binding Energies and Mott Points in Low Density

Nuclear Matter

K. HagelSSNHIC 2014Trento, Italy8-Apr-2014

Clustering and Medium Effects in

Low Density Nuclear Matter

Outline

• Experimental Setup• Clusterization and observables in low

density nuclear matter.• Clusterization of alpha conjugate

nuclei• Summary

3

Cyclotron Institute, Texas A & M University

Beam Energy: 47 MeV/uReactions:40Ar + 112,124Sn

35 MeV/u40Ca + 181Ta, Ca, C

35 MeV/u28Si + 28Si

35 MeV/u40Ca + 181Ta, Ca, C

35 MeV/u28Si + 28Si

4

• 14 Concentric Rings• 3.6-167 degrees• Silicon Coverage• Neutron Ball

S. Wuenschel et al., Nucl. Instrum. Methods. A604, 578–583 (2009).

Beam Energy: 47 MeV/uReactions: 40Ar + 112,124SnNIMROD

beam

Low Density Nuclear Matter• Systems studied

– 47 MeV/u 40Ar + 112,124Sn– 35 MeV/u 40Ca + 181Ta (preliminary data)

• Use NIMROD as a violence filter– Take 30% most violent collisions

• Use spectra from 40o ring– Most of yield from intermediate velocity

source• Coalescence analysis to extract

densities and temperatures– Equilibrium constants– Mott points– Symmetry energy

Coalescence Parameters

𝒅𝟑𝑵 (𝒁 ,𝑵 ,𝑬𝑨 )𝒅 𝑬𝑨𝒅 𝜴 =𝑹𝒏𝒑

𝑵 𝑨−𝟏

𝑵 !𝒁 ! { 𝟒𝝅𝟑 𝑷𝟎

𝟑

[𝟐𝒎𝟑 (𝑬−𝑬𝒄 ) ]𝟏/𝟐 }𝑨−𝟏

[ 𝒅𝟑𝑵 (𝟏 ,𝟎 ,𝑬 )𝒅 𝑬𝒅𝜴 ]

𝑨

𝒅𝟑𝑵 (𝒁 ,𝑵 )𝒅𝒑𝟑 =𝑹𝒏𝒑

𝑵 𝑨𝟑 (𝟐 𝒔+𝟏 )𝒆 (𝑬𝟎 /𝑻 )

𝟐𝑨 (𝒉𝟑

𝑽 )𝑨−𝟏[ 𝒅𝟑𝑵 (𝟏 ,𝟎)

𝒅𝒑𝟑 ]𝑨

𝑉=[( 𝑍 !𝑁 ! 𝐴32𝐴 ) (2𝑠+1 )𝑒 (𝐸 0/𝑇 )]1

𝐴−1 3h3

4𝜋 𝑃03

𝒅𝟑𝑵 (𝒁 ,𝑵 )𝒅𝒑𝟑 ∝𝑹𝒏𝒑

𝑵 𝒇 (𝑷 𝟎)[ 𝒅𝟑𝑵 (𝟏 ,𝟎)𝒅𝒑𝟑 ]

𝑨

PRC 72 (2005) 024603

vsurf, cm/ns

t avg,

fm/c

Temperatures and Densities• Recall vsurf vs time calculation• System starts hot• As it cools, it expands

47 MeV/u 40Ar + 112Sn

Equilibrium constants from α-particles model predictions

• Many tests of EOS are done using mass fractions and various calculations include various different competing species.

• If any relevant species are not included, mass fractions are not accurate.

• Equilibrium constants should be independent of proton fraction and choice of competing species.

• Models converge at lowest densities, but are significantly below data

• Lattimer & Swesty with K=180, 220 show best agreement with data

• QSM with p-dependent in-medium binding energy shifts PRL 108 (2012) 172701.

𝐾 𝑐 ( 𝐴 ,𝑍 )= 𝜌(𝐴 ,𝑍)𝜌𝑝𝑍 𝜌𝑛

(𝐴−𝑍 )

Density dependent binding energies• From Albergo, recall that• Invert to calculate binding energies• Entropy mixing term 𝑙𝑛 [𝐾 𝑐 /𝐶 (𝑇 )]=𝐵

𝑇 −𝑍𝑙𝑛( 𝑍𝐴 )−𝑁𝑙𝑛 (𝑁𝐴 )

𝑲 𝒄(𝑨 ,𝒁 )=𝐂 (𝐓 )𝒆(𝑩(𝑨 ,𝒁 )𝑻 )

Δ 𝐹=𝑇 (𝑍𝑙𝑛( 𝑍𝐴 )+𝑁𝑙𝑛(𝑁𝐴 ))

PRL 108 (2012) 062702

Symmetry energy

• Symmetry Free Energy– T is changing as ρ increases– Isotherms of QS calculation that includes in-medium modifications to cluster

binding energies• Entropy calculation (QS approach)• Symmetry energy (Esym = Fsym + T S∙ sym)

– quasiparticle mean-field approach (RMF without clusters) does not agree with the data

S. Typel et al., Phys. Rev. C 81, 015803 (2010).

PRC 85, 064618 (2012).

Alpha clustering in nuclei• Ikeda diagram (K. Ikeda,

N. Takigawa, and H. Horiuchi, Prog. Theor. Phys. Suppl. Extra Number, 464, 1968.)

• Clusterization of low density nuclear matter in collisions of alpha conjugate nuclei

• Role of clusterization in dynamics and disassembly.

Estimated limit N = 10α for self-conjugate nuclei(Yamada PRC 69, 024309)

40Ca + 40Ca

28Si + 40Ca

40Ca + 28Si 28Si + 28Si40Ca + 12C 28Si + 12C40Ca + 180Ta

28Si + 180Ta

Data Taken

10, 25, 35 MeV/u

Alpha-like multiplicities

• Large number of events with significant alpha conjugate mass for all systems

Vparallel vs Amax

• Observe mostly PLF near beam velocity for low E*• More neck (4-7 cm/ns) emission of α-like fragments with

increasing E*

𝐸∗=∑𝑖=1

𝑀

𝐾 𝑐𝑝 (𝑖 )+𝑀𝑛 ⟨𝐾 𝑛⟩−𝑄

• Heavy partner is near beam velocity

• alphas originate from neck emission

Origin of alpha conjugate clusters

Source Frame study of Origin of clusters

Origin of alpha conjugate clusters(continued)

Source Frame Origin of clusters(continued)

Summary• Clusterization in low density nuclear matter

– In medium effects important to describe data– Equilibrium constants

• EOS Implications– Density dependence of Mott points– Symmetry Free energy -> Symmetry Energy

• Clusterization of alpha conjugate nuclei– Large production of α-like nuclei

• Ca + Ca• Ca + Ta• Ca + C

– Neck emission of alphas important

Outlook and near future• Low density nuclear matter

– We have a set of 35 MeV/u 40Ca+181Ta and 28Si+181Ta

• Disassembly of alpha conjugate nuclei– Analysis on presented systems

continues– Have Si + C, Si + Ta (almost calibrated)

and Ca + Si

Collaborators

J. B. Natowitz, K. Schmidt, K. Hagel, R. Wada, S. Wuenschel, E. J. Kim, M. Barbui, G. Giuliani, L. Qin, S. Shlomo, A. Bonasera, G. Röpke, S. Typel, Z. Chen, M. Huang, J. Wang, H. Zheng, S. Kowalski, M. R. D. Rodrigues, D. Fabris, M. Lunardon, S. Moretto, G. Nebbia, S. Pesente, V. Rizzi, G. Viesti, M. Cinausero, G. Prete, T. Keutgen, Y. El Masri, Z. Majka, and Y. G. Ma