Photo:
Paul Greenlees

**FIDIPRO team against the background of the
famous Jyväskylä footbridge and frozen lake (January 6, 2009).**

From the left Jacek Dobaczewski, Francesco Raimondi, Alessandro Pastore, Gillis Carlsson, Pekka Toivanen, Kazuhito Mizuyama, Rayner Rodriguez-Guzman, and Jussi Toivanen. (Marcin Borucki, absent from the photo)

The
FIDIPRO project at the University of Jyväskylä

The project is jointly funded by the
Academy of Finland and University of Jyväskylä within the Finland Distinguished
Professor Programme (FIDIPRO). One of the 2006 FIFIPRO grants has been
attributed to Jacek Dobaczewski of University of Warsaw, who is now leading the
FIDIPRO team at the Physics Department of University of Jyväskylä (JYFL).
Currently the team is composed of 8 researchers and students working at JYFL,
while the project leader shares his time between stays in Finland and Poland.

The project [1] will last for five years
(2007-2011) and has for its objectives advanced studies in theoretical nuclear
structure physics, in strong synergy with experimental studies performed at
JYFL and with other worldwide initiatives, in order to investigate properties
of exotic nuclei. The project will develop and implement new theoretical
methods in this domain of physics as well as train young theorists within the
M.Sc. and Ph.D. educational programmes at the University of Jyväskylä. The
project is strongly linked with other similar activities in the world, like for
example UNEDF in the United States [2].

The nuclear science community is now
very much aware of the fact that theory efforts in this domain of physics are
grossly underfunded in Europe. In a long run, this may create quite inadequate
research conditions at existing, built, and planned large-scale European
experimental facilities. Important step towards improving upon this situation
has been made in Finland; the country, which distinguishes itself on the
European level not only by a high level of science funding in general, but also
by novel ideas in allocating these funds. The extent and scope of the FIDIPRO
project at JYFL is much larger than anything that happen in the European
nuclear theory for quite some number of years – an entirely new and focused
initiative was created in conjunction with a very successful experimental
research centre.

The main idea behind the FIDIPRO
project is to bridge the gap between the fundamental, *ab initio *studies
in nuclear structure physics and experiment. Recently, these basic research
directions in nuclear theory have achieved important progress in deriving
nuclear properties from first principles and in constructing very sophisticated
theoretical methods. However, a similar progress in describing heavy nuclei, especially
those far from stability, is still not available. To make it happen, the method
of choice is the energy density functional (EDF) approach, which, on the one
hand, is rooted in fundamental nuclear properties, and, on the other hand, must
be adjusted to data in a proper and professional way. The main focus of the
FIDIPRO project is on increasing the precision and accuracy of describing
exotic nuclei.

Nuclear shell structure is a
centrepiece of describing detailed nuclear properties. The quest for a
spectroscopic-quality EDFs begins, therefore, with attempts to improve the
description of single-particle energies. In 2008, the FIDIPRO team published a
study [3] showing that single-particle energies in doubly
magic nuclei depend almost linearly on the coupling constants of the nuclear
EDF. Therefore, they can be very well characterized by the linear-regression
coefficients, which were calculated for the coupling constants of the standard
Skyrme functionals, see Fig 1. By using the regression coefficients, we were
able to perform a one-step adjustment of the coupling constants to experimental
data and show that the current parameterisations of the local EDF does not
allow for obtaining rms deviations below 1.1 MeV. This result clearly points to
a necessity of extending standard parameterisations beyond the current form.

**Figure 1. **Regression coefficients
illustrating the rate of change of centroid energy (upper panels) and
spin-orbit splitting (lower panels) of the 1d_{5/2 }orbital (relative
to the 1s_{1/2 }energy) with the EDF coupling constants listed in the
abscissa. From Ref. [3].

Regression analysis methods,
relevant for a reliable determination of good nuclear EDF parameter sets for
nuclear mass fits were also studied within the FIDIPRO project [4]. In particular, a simple model for nuclear binding
energies and its regression analysis were used to study the validity and errors
of the model's parameters and uncertainties of predicted nuclear masses. This
work has won recognition in the new on-line APS journal physics.aps.org that
aims at *spotlighting exceptional research* [5].

Another work, completed in 2008,
achieved a construction of nuclear EDFs in terms of derivatives of densities up
to sixth, next-to-next-to-next-to-leading order (N^{3}LO). A
phenomenological functional built in this way conforms to the ideas of the
density matrix expansion and is rooted in the expansions characteristic to
effective theories. It builds on the standard functionals related to the
contact and Skyrme forces, which constitute the zero-order (LO) and
second-order (NLO) expansions, respectively. At N^{3}LO, the full
functional with density-independent coupling constants, and with the isospin
degree of freedom taken into account, contains 376 terms, while the functionals
restricted by the Galilean and gauge symmetries contain 100 and 42 terms,
respectively, see Fig. 2. For functionals additionally restricted by the
spherical, space-inversion, and time-reversal symmetries, the corresponding
numbers of terms are equal to 100, 60, and 22, respectively.

**Figure 2. **Numbers of terms in the
EDF with density dependent and density independent coupling constants, Eqs.
(28) and (30) of Ref. [6], respectively, plotted in logarithmic scale as a
function of the order in derivatives. From Ref. [6].

Related to the single-particle
energies, we currently study the role of density-dependent coupling constants
in the nuclear EDF. The density dependence is systematically introduced into
all the coupling constants. Preliminary results show that the single-particle
energies cannot be too much corrected in this way, but some improvement can be
obtained in the description of bulk nuclear properties.

A large part of our activity now aims at creation of an efficient Quasiparticle Random Phase Approximation (QRPA) computer program, designed for deformed nuclei. Our goal is to create a tool that would allow for a rapid determination of giant-resonance and beta-decay properties for nuclei across the mass chart. To solve the large-dimensional QRPA equations for deformed nuclei, a variant of Lanczos diagonalization method is adapted for QRPA. Two calculation strategies will be used: either a few lowest RPA phonons can be accurately calculated by using restarted Lanczos method, or strength functions can be calculated by using Lanczos moments method. First, a spherical code working on the harmonic-oscillator basis will be constructed for testing and benchmarking purposes.

As a follow-up of Refs. [3] and [6], we now build a numerical code to solve the
self-consistent equations with all the N^{3}LO terms taken into
account. The code will be used to adjust new coupling constants to experimental
data. Our main focus is at improving the description of single-particle
energies. Within the same framework, we also study properties of nuclear matter
and conservation of equation of continuity.

Within the FIDIPRO project two
lecture courses were delivered at JYFL, in 2007 and 2008. They covered the
subject of modern EDF methods in nuclear structure physics and were attended by
MSc. and PhD. students of the University of Jyväskylä. At present two of the
FIDIPRO team members are the PhD. students and one is an ERASMUS MSc. student.

Within our dissemination activities
we organised three topical workshops. On October 25–27, 2007, we organized at
JYFL the **First FIDIPRO-JSPS Workshop on Energy Density Functionals in Nuclei**.
The workshop was devoted to modern approaches and methods based on using the
EDF methods in nuclear physics, with particular emphasis on first-principle
derivations, new parameterisations, and applications including those going
beyond the mean-field limit. We hosted over 50 participants, of whom 15 came
from Japan, 5 from the United States, and 20 from all over the Europe.

Last year, on October 9–10, 2008 we
organised the **FIDIPRO-UNEDF collaboration meeting on nuclear
energy-density-functional methods**. We had 10 visitors from the United
States, Poland, Belgium, and France and about the same number of the JYFL
participants. Instead of having formal talks, only specific discussion points
were briefly introduced by selected participants and then covered in general
open discussions. The main goal of the meeting was to review current and future
projects, distribute tasks, set priorities, and define sequences of steps in
developing the codes. This year, we continue our series by organising on April
20–24, 2009 the **Arctic FIDIPRO-EFES Workshop: Future Prospects of Nuclear
Structure Physics**.

The FIDIPRO project has specific
tasks to fulfil within a well-defined frame of time and funding. It constitutes
a concerted effort of a quite substantial workforce to achieve specific goals.
Although results of scientific research cannot be guaranteed or predicted with
certainty, we expect that improvements in describing nuclear global properties,
like masses and beta-decay rates, can be achieved. These are very much in
demand by experimental groups striving to perform crucial measurements in
exotic nuclei, currently intensely investigated. All codes developed within the
FIDIPRO project will be rapidly published and made available to other
researchers.

- More extensive and complete
information is available on the project web page: http://www.jyu.fi/accelerator/fidipro/.
- G.F. Bertsch, Nuclear Physics
News
**17**(2007) 31 - M. Kortelainen, J. Dobaczewski, K.
Mizuyama, and J. Toivanen, Phys. Rev. C
**77**(2008) 064307. - J. Toivanen, J. Dobaczewski, M.
Kortelainen, and K. Mizuyama, Phys. Rev. C
**78**(2008) 034306. - http://physics.aps.org/synopsis-for/10.1103/PhysRevC.78.034306
- B.G. Carlsson, J. Dobaczewski,
and M. Kortelainen, Phys. Rev. C
**78**(2008) 044326.

Jacek Dobaczewski

University of Warsaw

University of Jyväskylä