Lecture 4: Cooking in the cosmic kitchen: Applications to astrophysics

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Lecture 4 Cooking in the cosmic kitchen
Applications to astrophysics
Instead of detailing the astrophysical context
which you will see very nicely from other
lectures I will concentrate on shell model
issues what can you trust and what is uncertain?
Example 1 Neutrino capture cross-section on 37Cl
(solar neutrino detector) Example 2 Electron
capture on 54Fe in pre-supernova core Example 3
Inverse ?-decay (?-capture) on 56Fe during
core-collapse SN Example 4 Nonresonant capture
7Be(p,?)8B -- solar fusion cross-section
bonus round binding energies/drip lines,
level densities
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Weak interactions and configuration mixing
The interacting shell model is the method of
choice for weak interactions (?-decay,
?-capture, e--capture) Why?
Ans Pauli blocking/ unblocking
example
these orbits are filled
Now consider neutrino and anti-neutrino capture
Hartree-Fock configuration
OK!
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NO! requires too much energy!!
NO! Pauli blocked!
But, if ? c
...
Some weak processes (usually p? n ) blocked
because neutron orbits already occupied. But
configuration mixing, even a little, can unblock
by creating holes for the new neutron to go
into. thus Weak processes sensitive to
configuration mixing
OK!
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Example 1The Chlorine solar-neutrino experiment
3/2
5.15
37Cl ??37Ar e-
Sun
1.61
5/2
1.42
1/2
3/2
3/2
0.814
37Ar
37Cl
37Cl
Q-value of reaction 814 keV solar ?s have
energies up to 15 MeV can determine
cross-sections for lower transitions directly by
?-decay
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37Cl ? cross-section
Transitions are relatively low energy, so
long-wavelength approximation applies Most
transitions are Gamow-Teller one-body transition
operator ??- (no radial dependence dont
worry about HO, WS, HF single-particle wfns)
1p1/2 0d5/2 1p3/2 0d7/2
Shell-model space full sd-shell ( 0s1/2 0d3/2
0d5/2 above inert 16O core) 9 valence protons,
12 valence neutrons, each in 12 valence states so
in this simplest approximation, neutrons actually
are inert as well. number of
M-scheme many-body states 37
0d3/2 1s1/2 0d5/2
active space inert core
0p1/2 0p3/2
0s1/2
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37Cl ? cross-section
Two issues interaction and model space
There is a well-tested shell-model interaction
for full sd-shell calculations, the
Brown-Wildenthal USD interaction 3 s.p.energies
and 63 tbmes. Based upon realistic G-matrix
but subsequently fitted to 440 sd-nuclei levels
If one sticks to sd configurations, neutrons are
effectively inert. What about intruder states,
excitations of neutrons from sd to pf shell? The
problem here interactions is much more poorly
constrained.
As it turns out, cross-section for solar
neutrinos dominated by low-lying states, so
these uncertainties affect the cross-section by
less than 4 ?solar 1.08 ? 10-42 cm2
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Where do shell model interactionscome from,
anyway?
Two sources Phenomenological these
follow well-known behavior Q(r1) Q(r2) induces
deformation also arises naturally from
multipole expansion of generic V(r1, r2)
enhanced in nuclei delta-force ?(r1-r2)
short-range of nuclear force pairing odd-even
binding energy, pairing gaps (also arises
directly from multipole expansion of
delta-force) Realistic effective
interactions derived from NN potentials integra
te out nasty short-range behavior by G-matrices
(ladder diagrams), Feshbach effective
interaction theory , Lee-Suzuki theory, etc.
Until recently, still had to make
phenomenological adjustments to get good
agreement with experiment. Today purely ab
initio effective interactions give good (not
great) agreement need three-body corrections.
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Quenching of gA in nuclei
For free neutrons, gA/gV 1.26. For shell-model
calculations of Gamow-Teller transitions, best
fit to experiment by gA/gV ? 1.
This is either an in-medium effect
(delta-isobar excitations)
or one needs effective
transition operators the same way one
needs effective interactions due to truncation of
many-body space. Impossible to calculate for
empirical interactions such as USD. In
principle, possible to compute in conjunction
with G-matrix, Feshbach, Lee-Suzuki, (but almost
never done). For E2 gamma-decays similar need
effective charge (enhanced) This is known to be
a case of needing effective transition
operator due to truncation of the model space.
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Example 2 Electron capture on 54Fe in stellar
core
In the last day of the life of a massive star,
electron capture on Fe-region nuclei determine
the lepton fraction Ye
When the Fe iron reaches the Chandrasekhar mass,
it collapses. The Chandrasekhar mass and
subsequent evolution of the supernova depends
critically on Ye .
Fe,Ni
C, O, Mg,Si
54Fe(e-, ?)54Mn, etc.
H,He
pre-supernova massive star not to scale
0
4
1
2
can be thermally populated in stellar core T
?1.5 MeV
2
0
3
54Fe
54Mn
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pf-Shell calculations for Fe
Valence particles in pf-shell 1p1/2 1p3/2 0f5/2
0f7/2 above 40Ca inert core 6 valence protons,
8 valence neutrons half a billion basis
states! Can truncate the many-body basis, say
to 1p3/2 0f7/2 or to just 2 or 4 particles out
of 0f7/2 into 1p1/2 1p3/2 0f5/2...
unfortunately no natural way to truncate, and
results depend sensitively upon model space, by
as much as 10-100 ?!
from Langanke Martinez-Pinedo, 2002
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Example 3 Neutrino-capture on 56Fe during core
collapse Supernova
As the Fe core reaches the Chandrasekhar mass and
collapses, the gravitational binding energy is
released as ??s, with E? 15-30 MeV.
?
56Fe(?,e-)56Co is a reaction of interest
?
Not only does this make the model space large,
these so-called forbidden transitions (1- etc.)
are sensitive to the shape of radial wfns,
e.g. HO vs. WS vs. HF ...
6 valence protons, 10 valence neutrons in pf
shell, 108 basis states but E? is large enough
to excite opposite parity states from sd-shell
into pf-shell or from pf-shell
up into gsd-shell
Computed using hybrid approach shell-model for
GT, contiruum RPA for forbidden Agreement with
experiment with neutrinos from muon decay ? 2.4
? 10-40 cm-2
These issues also relevant to parity violation,
detection of WIMPs
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Example 4 Protron capture on 7Be
Solar fusion reactions 7Bep ? 8B ? 8B
?8Be ? ?
detected by 37Cl experiment
The cross-section for 7Be(p,?) 8B is critical to
interpreting 37Cl experiment measured in lab
down to KEp 110 keV...but in Suns core, most
captures occur at KEp 20 keV.
Can the shell model help us? After all, for such
a light nucleus we ought to be able to easily
generate a very large model space...
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Asymptotic wavefunctions
Want to compute ? 8B j1(qr) Y1 7Be p ?
electric dipole (E1) operator
unbound proton in continuum
Here, q is very small. But, unfortunately, when
we do the integral r is not restricted to lt R
nuclear radius, because the proton comes from
infinitely far away!!
The integral depends on the asymptotic form of
the unbound proton wavefunction, and one must
integrate out to hundreds of fm!!
Can model as a single proton being captured by a
mean-field potential (cf. Jennings,
Karataglidis, and Shoppa, 1998)
This gives the energy dependence, but not the
absolute normalization.
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Spectroscopic factors
Ideally, would like to use fully microscopic
theory and indeed, calculations are under way
using GFMC but difficult to modeling
long-distance and short-distance simultaneously.
Approximate solution use one-body potential to
get one-body asympotic behavior normalized by
shell-model spectroscopic factor
This can be easily computed in the shell model
however, once again, consistency is the main
problem.
destroys a particle that occupies state n
Ive oversimplified for detailed discussion see
Escher, Jennings, and Sherif (2001)
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Other applications...?
Binding energies/ driplines needed for
r-process nucleosynthesis Shell model gives very
good agreement with binding energies, but better
at interpolation than extrapolation. (This is
also true of the methods of choice for binding
energies Hartree-Fock and semiclassical liquid
drop models. But the latter are computationally
cheaper). Can be useful for specific instances
when calibrated by nearby masses. Level
densities that is, number of excited states
per MeV of excitation energy. Needed for
statistical models (Hauser-Feshbach) of neutron
capture in nucleosynthesis. Typically use crude
phenomenology or combinatorial arguments. Shell
model would be ideal, except that one needs all
states, which eliminates usually approach of
Lanczos ? few low-lying states. Can use Monte
Carlo evaluation of path integrals (but limited
choice of interactions) or spectral distribution
methods.
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The future of the Shell Model
-- More powerful computers larger model space
(from 106 to 109 in 1 decade) -- New techniques
in shell-model framework (e.g. Monte Carlo
shell model)even larger basis sizes -- New
exact calculations for light nuclei
complement and benchmark shell-model
calculations -- Better understanding of
fundamental NN interaction and effective
interactions improving foundation of shell
model calculations -- Useful for nuclear
astrophysics, method of choice for weak
interactions