10,914 research outputs found
Exploration of Resonant Continuum and Giant Resonance in the Relativistic Approach
Single-particle resonant-states in the continuum are determined by solving
scattering states of the Dirac equation with proper asymptotic conditions in
the relativistic mean field theory (RMF). The regular and irregular solutions
of the Dirac equation at a large radius where the nuclear potentials vanish are
relativistic Coulomb wave functions, which are calculated numerically.
Energies, widths and wave functions of single-particle resonance states in the
continuum for ^{120}Sn are studied in the RMF with the parameter set of NL3.
The isoscalar giant octupole resonance of ^{120}Sn is investigated in a fully
consistent relativistic random phase approximation. Comparing the results with
including full continuum states and only those single-particle resonances we
find that the contributions from those resonant-states dominate in the nuclear
giant resonant processes.Comment: 16 pages, 2 figure
Effect of Resonant Continuum on Pairing Correlations in the Relativistic Approach
A proper treatment of the resonant continuum is to take account of not only
the energy of the resonant state, but also its width. The effect of the
resonant states on pairing correlations is presented based on the relativistic
mean field theory plus Bardeen-Cooper-Schrieffer(BCS) approximation with a
constant pairing strength. The study is performed in an effective Lagrangian
with the parameter set NL3 for neutron rich even-even Ni isotopes. The results
show that the contribution of the proper treatment of the resonant continuum to
pairing correlations for those nuclei close to neutron drip line is important.
The pairing gaps, Fermi energies, pairing correlation energies, and binding
energies are considerably affected with a proper consideration of the width of
resonant states. The problem of an unphysical particle gas, which may appear in
the calculation of the traditional mean field plus BCS method for nuclei in the
vicinity of drip line could be well overcome when the pairing correlation is
performed by using the resonant states instead of the discretized states in the
continuum.Comment: 19 pages, 8 Postscript figur
The Gamow-Teller Resonance in Finite Nuclei in the Relativistic Random Phase Approximation
Gamow-Teller(GT) resonances in finite nuclei are studied in a fully
consistent relativistic random phase approximation (RPA) framework. A
relativistic form of the Landau-Migdal contact interaction in the spin-isospin
channel is adopted. This choice ensures that the GT excitation energy in
nuclear matter is correctly reproduced in the non-relativistic limit. The GT
response functions of doubly magic nuclei Ca, Zr and Pb
are calculated using the parameter set NL3 and =0.6 . It is found that
effects related to Dirac sea states account for a reduction of 6-7 % in the GT
sum rule.Comment: 9 pages, 1 figur
Unraveling Projection Heads in Contrastive Learning: Insights from Expansion and Shrinkage
We investigate the role of projection heads, also known as projectors, within
the encoder-projector framework (e.g., SimCLR) used in contrastive learning. We
aim to demystify the observed phenomenon where representations learned before
projectors outperform those learned after -- measured using the downstream
linear classification accuracy, even when the projectors themselves are linear.
In this paper, we make two significant contributions towards this aim.
Firstly, through empirical and theoretical analysis, we identify two crucial
effects -- expansion and shrinkage -- induced by the contrastive loss on the
projectors. In essence, contrastive loss either expands or shrinks the signal
direction in the representations learned by an encoder, depending on factors
such as the augmentation strength, the temperature used in contrastive loss,
etc. Secondly, drawing inspiration from the expansion and shrinkage phenomenon,
we propose a family of linear transformations to accurately model the
projector's behavior. This enables us to precisely characterize the downstream
linear classification accuracy in the high-dimensional asymptotic limit. Our
findings reveal that linear projectors operating in the shrinkage (or
expansion) regime hinder (or improve) the downstream classification accuracy.
This provides the first theoretical explanation as to why (linear) projectors
impact the downstream performance of learned representations. Our theoretical
findings are further corroborated by extensive experiments on both synthetic
data and real image data
Axially deformed relativistic Hartree Bogoliubov with separable pairing force
A separable form of pairing interaction in the channel has been
introduced and successfully applied in the description of both static and
dynamic properties of superfluid nuclei. By adjusting the parameters to
reproduce the pairing properties of the Gogny force in nuclear matter, this
separable pairing force is successful in depicting the pairing properties of
ground states and vibrational excitations of spherical nuclei on almost the
same footing as the original Gogny force. In this article, we extend these
investigations for Relativistic Hartree Bogoliubov theory in deformed nuclei
with axial symmetry (RHBZ) using the same separable pairing interaction. In
order to preserve translational invariance we construct one- and
two-dimensional Talmi-Moshinsky brackets for the cylindrical harmonic
oscillator basis. We show that the matrix elements of this force can then be
expanded in a series of separable terms. The convergence of this expansion is
investigated for various deformations. We observe a relatively fast
convergence. This allows for a considerable reduction in computing time as
compared to RHBZ-calculations with the full Gogny force in the pairing channel.
As an example we solve the RHBZ equations with this separable pairing force for
the ground states of the chain of Sm-isotopes. Good agreement with the
experimental data as well as with other theoretical results is achieved.Comment: 8 pages, 5 figures. accepted by Phys. Rev.
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