Constraining the properties of dense matter

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Constraining the properties of dense matter
William Lynch, Michigan State University

• What is the EOS ?
•  1. Theoretical approaches ?
• 2. ExampleT0 with Skyrme ?
•  3. Present status ?
• a) symmetric matter
• b) asymmetric matter and symmetry term.
• 4. Astrophysical relevance ?
• B. Summary of first lecture ?
• C. What observables are sensitive to the EOS and
  at what densities?
• 1. Binding energies ?
• 2. Radii of neutron and proton matter in nuclei
  ?
• 3. Giant resonances ?
• 4. Particle flow and particle production
  symmetric EOS ?
• 5. Particle flow and particle production
  symmetry energy ?
• D Summary ?
•  

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Theoretical Approaches

• Variational and Bruekner model calculations with
  realistic two-body nucleon-nucleon interactions
  (see Akmal et al., PRC 58, 1804 (1998) and refs
  therein.)
• Variational minimizes ltHgt with elaborate grounds
  state wavefunction that includes nucleon-nucleon
  correlations.
• Incorporate three-body interactions.
• Some are "fundamental"
• Others model relativistic effects.
• Relativistic mean field calculations using
  relativistic effective interactions, (see
  Lalasissis et al., PRC 55, 540 (1997), Peter Ring
  lectures)
• Well defined transformations under Lorentz boosts
• Parameterization can be adjusted to incorporate
  new data.
• Skyrme parameterizations (Vautherin and Brink,
  PRC 5, 626 (1972).)
• Requires transformation to local rest frame
• Computationally straightforward - example

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Example Skyrme interaction.
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• Hint use the expressions for the differential
  increases in potential energy per unit volume
  above and do a parametric integration over ? from
  zero to one.

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Brown, Phys. Rev. Lett. 85, 5296 (2001)
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• The density dependence of symmetry energy is
  largely unconstrained.

• Unlike symmetric matter, the potential energy of
  neutron matter is repulsive.

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Constraints on symmetric and asymmetric matter EOS
E/A (?, ?) E/A (?,0) ?2?S(?) ? (?n-
?p)/ (?n ?p) (N-Z)/A?1
Danielewicz et al., Science 298,1592 (2002).
Danielewicz et al., Science 298,1592 (2002).

• Constraints come mainly from collective flow
  measurements.
• Know pressure is zero at ??0.
• Results from variational calculations and
  Relativistic mean field theory with density
  dependent couplings lie within the allowed
  boundaries.

• Neutron matter EOS also includes the poorly
  constrained pressure from the symmetry energy.
• The uncertainty from the symmetry energy is
  larger than that from the symmetric matter EOS.

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Type II supernova (collapse of 20 solar mass
star)

• Supernovae scenario (Bethe Reference)
• Nuclei H?He?C?...?Si?Fe
• Fe stable, Fe shell cools and the star collapses
• Matter compresses to ?gt4?s and then expands
• Relevant densities and matter properties
• Compressed matter inside shock radius ?0lt?lt10?0,
  ??0.40.9
• What densities are achieved?
• What is the stored energy in the shock?
• What is the neutrino emission from the
  proto-neutron star?
• Clustered matter outside shock radius mixed
  phase of nucleons and nuclear drops - nuclei
  ?lt?0, ??0.30.5
• How much energy is dissipated in vaporizing the
  drops during the explosion?
• What is the nature of the matter that interacts
  and traps the neutrinos?
• What are the seed nuclei that are present at the
  beginning of r-process which makes roughly half
  of the elements?

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Neutron Stars
Neutron Star Structure Pethick and Ravenhall,
Ann. Rev. Nucl. Part. Sci. 45, 429 (1995)

• Neutron Star stability against gravitational
  collapse
• Stellar density profile
• Internal structure occurrence of various phases.
• Observational consequences ?
• Stellar masses, radii and moments of inertia.
• Cooling rates of proto-neutron stars
• Cooling rates for X-ray bursters.

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Some examples
Neutron star radii
Cooling of proto-neutron stars
2
1.5
Lattimer , Ap. J., 550, 426 (2001).
0.5
0

• Neutrino signal from collapse.
• Feasibility of URCA processes for proto-neutron
  star cooling if fp gt 0.1. This occurs if S(?) is
  strongly density dependent.
• pe- ? n? n ? pe-

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• These equations of state differ only in their
  density dependent symmetry terms.
• Clear sensitivity to the density dependence of
  the symmetry terms

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Summary of last lecture

• The EOS describes the macroscopic response of
  nuclear matter and finite nuclei.
• It can be calculated by various techniques.
  Skyrme parameterizations are a relatively easy
  and flexible way to do so. .
• The high density behavior and the behavior at
  large isospin asymmetries of the EOS are not well
  constrained.
• The behavior at large isospin asymmetries is
  described by the symmetry energy.
• The symmetry energy has a profound influence on
  neutron star properties stellar radii, maximum
  masses, cooling of proto-neutron stars, phases in
  the stellar interior, etc.

?(?,0,?) ?(?,0,0) d2?S(?) d (?n- ?p)/
(?n ?p) (N-Z)/A
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Binding energies as probes of the EOS
BA,Z av1-b1((N-Z)/A)²A - as1-b2((N-Z)/A)²A2
/3 - ac Z²/A1/3 dA,ZA-1/2 CdZ²/A,

• Fits of the liquid drop binding energy formula
  experimental masses can provide values for av,
  as, ac, b1, b2, Cd and ?A,Z.
• Relationship to EOS
• av ?(?s,0,0) avb1S(?s)
• as and asb2 provide information about the density
  dependence of ?(?s,0,0) and S(?s) at
  subsaturation densities ? ? 1/2?s . (See
  Danielewicz, Nucl. Phys. A 727 (2003) 233.)
• The various parameters are correlated. Coulomb
  and symmetry energy terms are strongly
  correlated. Shell effects make masses differ from
  LDM.
• Measurement techniques
• Penning traps ?qB/m
• Time of flight TOFdistance/v B?mv/q
• Transfer reactions A(b,c)D
  Q(mAmb-mc-mD)c2
• Mass compilations exist e.g. Audi et al,.,NPA
  595, (1995) 409.

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Neutron and proton matter radii

• A simple approximation to the density profile is
  a Fermi function ?(r)?0/(1exp(r-R)/a).
• For stable nuclei, Rp has been measured by
  electron scattering to about 0.02 fm accuracy.
• (see G. Fricke et al., At. Data Nucl. Data Tables
  60, 177 (1995).)

208Pb
?(r) (fm-3)
r (fm)

• Neutron radii can be measured by hadronic
  scattering, which is more model dependent and
  less accurate (?Rn ? 0.2 fm) because the
  interaction is mainly on the surface.
• a ? 0.5 0.6 fm for stable spherical nuclei, but
  near the neutron dripline, an can be much larger.
• Strong interaction radius for 11Li is about the
  same as that for 208Pb.

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Comparison of Rn and Rp

• The asymmetry in the nuclear surface can be
  larger when S(?) is strongly density dependent
  because S(?) vanishes.more rapidly at low density
  when S(?) is stiff.
• Stiff symmetry energy ? larger neutron skins.
  (See Danielewicz lecture.)
• Measurements of 208Pb using parity violating
  electron scattering are expected to provide
  strong constraints on ltrn2gt1/2- ltrp2gt1/2 and on
  S(?) for ?lt ?s. Uncertainties are of order 0.06
  fm. (see Horowitz et al., 63, 025501(2001).)
• The upper figure shows how the predicted neutron
  skins depend on Psym??2dS(?)/d ?
• Analyses of ltrn2gt1/2- ltrp2gt1/2 for Na isotopes
  have placed some constraints on S(?) for ?lt ?s,
  (see Danielewicz, NPA 727, 203 (2003).?

Brown, Phys. Rev. Lett. 85, 5296 (2001)
softer
stiffer
at ?0.1 fm
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Radii of Na isotopes
Suzuki, et al., PRL 75, 3241 (1995)

• ? ltrp2gt1/2 0.1 fm

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• Proton radii are determined by measuring atomic
  transitions in Na, which has a 3s g.s. orbit.
• Neutron radii increase faster than Rr0A1/3,
  reflecting the thickness of neutron skin, e.g.
  RMF calculation.

• The relationship between cross-section and Na
  interaction radius is
• Getting the actual neutron radius is model
  dependent.

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Giant resonances

• Imagine a macroscopic, i.e. classical excitation
  of the matter in the nucleus.
• e.g. Isoscaler Giant Monopole (GMR) resonance
• GMR provides information about the curvature of
  ?(?,0,0) about minimum.
• Inelastic ?? particle scattering e.g. 90Zr(?,??
  )90Zr can excite the GMR. (see Youngblood et
  al., PRL 92, 691 (1999).)
• Peak is strongest at 0?

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Giant resonances 2

• HW 3 Assume that we can approximate a nucleus as
  having a sharp surface at radius R and ignore the
  surface, Coulomb and symmetry energy
  contributions to the nuclear energy.
• In the adiabatic approximation show that
• Show that
• Show that
• In practice there are surface, Coulomb and
  symmetry energy corrections to the GMR energy.
  (see Harakeh and van der Woude, Giant
  Resonances Oxford Science...)
• Leptodermous expansion

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Giant Resonances 3

• Isovector Giant Dipole Resonance neutrons and
  protons oscillate against each other. The
  restoring force is the surface energy of the
  nucleus.
• Danielewicz has shown that EGDR depends on the
  surface symmetry energy but not on the volume
  symmetry energy. (Danielewicz, NP A 727 (2003)
  233.)

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Probes of the symmetric matter EOS

• Nuclear collisions are the only way to make
  variations in nuclear density under
  experimentally controlled conditions and obtain
  information about the EOS.
• Theoretical tool transport theory
• Example Boltzmann-Uehling-Uhlenbeck eq. (Bertsch
  Phys. Rep. 160, 189 (1988).)
• Describes the time evolution of the Wigner
  transform of the one-body density matrix
  (quantum analogue to classical phase space
  distribution)
• classically, f ( the number of
  nucleons/d3rd3p at ) .
• Semiclassical time dependent Thomas-Fermi
  theory
• Each nucleon is represented by 1000 test
  particles that propogate classically under the
  influence of the mean field U and subject to
  collisions due to the residual interaction. The
  mean field is self consistent, at each time step,
  one
• propogates nucleons, etc. subject to the mean
  field and collisions, and
• recalculates the mean field potential according
  to the new positions.

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Constraining the EOS at high densities by
laboratory collisions
AuAu collisions E/A 1 GeV)
pressure contours
density contours

• Two observable consequences of the high pressures
  that are formed
• Nucleons deflected sideways in the reaction
  plane.
• Nucleons are squeezed out above and below the
  reaction plane. .

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Procedure to study high pressures

• Measure collisions
• Simulate collisions with BUU or other transport
  theory
• Identify observables that are sensitive to EOS
  (see Danielewicz et al., Science 298,1592 (2002).
  for flow observables)
• Directed transverse flow (in-plane)
• Elliptical flow out of plane, e.g.
  squeeze-out
• Kaon production. (Schmah, PRC C 71, 064907
  (2005))
• Analyze data and model calculations to measured
  and calculated observable assuming some specific
  forms of the mean field potentials for neutrons
  and protons. At some energies, produced
  particles, like pions, etc. must be calculated as
  well.
• Find the mean field(s) that describes the data.
  If more than one mean field describes the data,
  resolve the ambiguity with additional data.
• Constrain the effective masses and in-medium
  cross sections by additional data.
• Use the mean field potentials to apply the EOS
  information to other contexts like neutron stars,
  etc.

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Directed transverse flow
Partlan, PRL 75, 2100 (1995).
target
AuAu collisions EOS TPC data
Ebeam/A
projectile

• Event has elliptical shape in momentum space.
• The long axis lies in the reaction plane,
  perpendicular to the total angular momentum.
• Analysis procedure
• Find the reaction plane
• Determine ltpx(y)gt in this plane
• note

y/ybeam (in C.M)

• The data display the s shape characteristic of
  directed transverse flow.
• The TPC has in-efficiencies at y/ybeamlt -0.2.
• Slope is
  determined at 0.2lty/ybeamlt0.3

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Determination of symmetric matter EOS from
nucleus-nucleus collisions
Danielewicz et al., Science 298,1592 (2002).
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• The boundaries represent the range of pressures
  obtained for the mean fields that reproduce the
  data.
• They also reflect the uncertainties from the
  effective masses in in-medium cross sections.

• The curves labeled by Knm represent calculations
  with parameterized Skyrme mean fields
• They are adjusted to find the pressure that
  replicates the observed transverse flow.

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Probes of the symmetry energy
?(?,0,?) ?(?,0,0) d2?S(?) d (?n- ?p)/
(?n ?p) (N-Z)/A

• Common features of some of these studies
• Vary isospin of detected particle
• Sign in Uasy is opposite for n vs. p.
• Shape is influenced by stiffness.
• Vary isospin asymmetry ? of reaction.
• Low densities (?lt?0)
• Isospin diffusion
• Neutron/proton spectra and flows
• Neutron, proton radii, E1 collective modes.
• High densities (??2?0) ??
• Neutron/proton spectra and flows ?
• ?? vs. ?- production ??

?0.3
Uasy (MeV)
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Constraining the density dependence of the
symmetry energyObservable Isospin diffusion in
peripheral collisions

• In a reference frame where the matter is
  stationary
• D? the isospin diffusion coef.
• Two effect contribute to diffusion
• Random walk
• Potential (EOS) driven flows
• D? governs the relative flow of neutrons and
  protons
• D? decreases with ?np
• D? increases with Sint(?)

softer
Shi et al, C 68, 064604 (2003)
stiffer

• R is the ratio between the diffusion coefficient
  with a symmetry potential and without a symmetry
  potential.

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Probe Isospin diffusion in peripheral collisions

• Vary isospin driving forces by changing the
  isospin of projectile and target.
• Probe the asymmetry ?(N-Z)/(NZ) of the
  projectile spectator after the collision.
• The asymmetry of the spectator can change due to
  diffusion, but it also can changed due to
  pre-equilibrium emission.
• The use of the isospin transport ratio Ri(?)
  isolates the diffusion effects
• Useful limits for Ri for 124Sn112Sn collisions
• Ri 1 no diffusion
• Ri ?0 Isospin equilibrium

mixed 124Sn112Sn n-rich 124Sn124Sn p-rich
112Sn112Sn
P
?
N
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Sensitivity to symmetry energy
Stronger density dependence

• The asymmetry of the spectators can change due to
  diffusion, but it also can changed due to
  pre-equilibrium emission.
• The use of the isospin transport ratio Ri(?)
  isolates the diffusion effects

Weaker density dependence
Lijun Shi, thesis
Tsang et al., PRL92(2004)
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Probing the asymmetry of the Spectators

• The the shift can be compactly described by the
  isoscaling parameters ? and ? obtained by taking
  ratios of the isotopic distributions

• The main effect of changing the asymmetry of the
  projectile spectator remnant is to shift the
  isotopic distributions of the products of its
  decay

Liu et al., (2006)
Tsang et. al.,PRL 92, 062701 (2004)
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Determining ?Ri(?)

• Statistical theory provides

• Consider the isoscaling ratio Ri(X), where X ??
  or ?
• When X depends linearly on ?2
• By direct substitution
• true for known production models
• linear dependences confirmed by data.

?
?
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Probing the asymmetry of the Spectators

• The the shift can be compactly described by the
  isoscaling parameters ? and ? obtained by taking
  ratios of the isotopic distributions

• The main effect of changing the asymmetry of the
  projectile spectator remnant is to shift the
  isotopic distributions of the products of its
  decay

Liu et al., (2006)
Tsang et. al.,PRL 92, 062701 (2004)
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Constraints from Isospin Diffusion Data
M.B. Tsang et. al.,PRL 92, 062701 (2004) L.W.
Chen, C.M. Ko, and B.A. Li,PRL 94, 032701
(2005) C.J. Horowitz and J. Piekarewicz,PRL 86,
5647 (2001) B.A. Li and A.W. Steiner,nucl-th/0511
064
124Sn112Sn data
C
B
Approximate representation of the various
asymmetry terms used in BUU calcuations Esym(?)
32(?/?0)? (?n - ?p) /(?n ?p)2 g 0.5, 1.0,
1.6 (for cases A, B, C)
A
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• Interpretation requires assumptions about isospin
  dependence of in-medium cross sections and
  effective masses

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Final Summary

• The EOS describes the macroscopic response of
  nuclear matter and finite nuclei.
• It can be calculated by various techniques.
  Skyrme parameterizations are relatively easy.
• The high density behavior and the behavior at
  large isospin asymmetries of the EOS are not well
  constrained.
• The behavior at large isospin asymmetries is
  described by the symmetry energy.
• It influences many nuclear physics quantities
  binding energies, neutron skin thicknesses,
  isovector giant resonances, isospin diffusion,
  etc. Measurements of these quantities can
  constrain the symmetry energy.
• The symmetry energy has a profound influence on
  neutron star properties stellar radii, maximum
  masses, cooling of proto-neutron stars, phases in
  the stellar interior, etc.
• Constraints on the symmetry energy and on the EOS
  will be improved by planned experiments. Some of
  the best ideas have not yet been discovered.

?(?,0,?) ?(?,0,0) d2?S(?) d (?n- ?p)/
(?n ?p) (N-Z)/A
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Influence of production mechanism on isoscaling
parameters
Primary Before decay of excited fragments,
Final after decay of excited fragments

• Statistical theory
• Final isoscaling parameters are often similar to
  those of the primary distribution
• Both depend linearly on ?
• ? R(?)R(?)

• Dynamical theories
• Final isoscaling parameters are often smaller
  than those of primary distribution
• Both depend linearly on ?
• ? R(?)R(?)
• Doesn't matter which one is correct.

final
R
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Test of linearity using central collisions

• Data analyzed in well-mixed region at
  70???cm?110?.

• Linearity is demonstrated for ?, ? and
  ln(Y(7Li)/Y(7Be))??-?

Liu et al., (2006)
Liu et al., (2006)
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Asymmetry term studies at ??2?0

• Densities of ??2?0 can be achieved at E/A??400
  MeV.
• Provides information about direct URCA cooling in
  proto-neutron stars, stability and phase
  transitions of dense neutron star interior.
• S(?) influences diffusion of neutrons from dense
  overlap region at b0.
• Diffusion is reduced, neutron-rich dense region
  is formed for soft S(?).

• Densities of ??2?0 can be achieved at E/A??400
  MeV.
• Provides information about direct URCA cooling in
  proto-neutron stars, stability and phase
  transitions of dense neutron star interior.

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First observable pion production
Yong et al., Phys. Rev. C 73, 034603 (2006)

• The enhanced neutron abundance at high density
  for the soft asymmetry term (x0) leads to a
  stronger emission of negative pions for the soft
  asymmetry term (x0) than for the stiff one
  (x-1).
• In delta resonance model, Y(??-)/Y(??)?(?n,/?p)2
• ?- /?? means Y(??-)/Y(??)
• Coulomb interaction has a strong effect on the
  pion spectra
• Coulomb repels ?? and attracts ??-.

soft
stiff

• The density dependence of the asymmetry term
  changes ratio by about 15 for neutron rich
  system.
• How does one reduce sensitivity to systematic
  errors?

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Double ratio pion production

• Double ratio involves comparison between neutron
  rich 132Sn124Sn and neutron deficient
  112Sn112Sn reactions.
• Double ratio maximizes sensitivity to asymmetry
  term.
• Largely removes sensitivity to difference between
  ?- and ? acceptances.

Yong et al., Phys. Rev. C 73, 034603 (2006)
soft
stiff
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Independent observable n/p spectra

• Neutrons are repelled and protons are attracted
  by the asymmetry term (in neutron rich matter).
• The Coulomb interaction has somewhat the opposite
  effect.
• Sensitivity can be maximized by constructing a
  double ratio
• Removes sensitivity to calibration and efficiency
  problems

Li et al., arXivnucl-th/0510016 (2005)
stiff
soft
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Alternate observable n-p differential transverse
flow

• Transverse directed flow is usually obtained by
  plotting the mean transverse momentum ltpxgt vs.
  the rapidity y.
• The neutron-proton differential flow is defined
  here to be
• Sensitivity to acceptance effects might be
  minimized by constructing the difference

Li et al., arXivnucl-th/0504069 (2005)
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Constraints on momentum dependence of mean fields
and in-medium cross sections
Li et al., Phys. Rev. C 69, 011603(R) (2004)

• Li et al., Phys. Rev. C 71, 054603 (2005)

40Ca100Zn E/A200 MeV

• We need calculations of the corresponding double
  ratios.
• Not clear that we have a good way to distinguish
  momentum and density dependencies.

• Important to control the number of n-p
  collisions, p-p and n-n collisions
• compare 37Ca112Sn to 37Ca124Sn
• compare 52Ca112Sn to 52Ca124Sn.

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