1. Anatoli Afanasjev Mississippi State University (MSU), USA.
Single-particle degrees of freedom in covariant density functional theory: the successes and challenges.
Anatoli Afanasjev
Mississippi State University (MSU), USA.
Motivation and introduction
Impact of time-odd mean fields on single-particle states
Comparison with experiment (at and beyond men field)
Competition of particle-vibration coupling and tensor force
Impact on nuclear extremes (neutron-drip line and superheavy nuclei)
6. Conclusions

2. For future ECT* workshops:
please, provide proper breakfast for nuclear physicists

3. 1. Introduction

4. Covariant density functional theory (CDFT)
The nucleons interact via the exchange of effective mesons 
 effective Lagrangian
Long-range
attractive
scalar field
Short-range
repulsive vector
field
Isovector
field
- meson fields
Mean
field
Eigenfunctions

5. Densities
Single-particle energies
Klein-Gordon equations for bosons
CDFT is the only nuclear structure theory with a potential of its connection to the low energy domain of QCD, the basic theory of the strong
interaction !!!.

6. Non-relativistic Schrodinger
Approximations in the modelling of nuclear many-body problem.
General approach (DFT level)
Non-relativistic Schrodinger
equation:
Skyrme and Gogny DFT
Relativistic Dirac equation:
covariant DFT (CDFT)
Range of interaction
Zero range - point coupling models
in CDFT (no mesons)
- Skyrme DFT
Finite range - meson exchange models
in CDFT
- Gogny DFT
Effective density dependence
CDFT : explicit (DD-ME2, DD-PC1)
- non-linear (through the powers of mesons) (NL1, NL3*)
Skyrme and Gogny DFTs: different prescriptions for density dependence

7. 2. Impact of time-odd mean fields on the single-particle states

8. Nuclear magnetism (NM)=Time-odd (TO) mean fields
Pairing is neglected
in the calculations
Single-particle states
are characterized by
signature
Abbreviations:
NM – nuclear magnetism is included
WNM – nuclear magnetism is neglected

9. the r-meson is very small in the majority of the cases
Dominance of the w-meson in time-odd mean fields
isoscalar
isovector
The contribution of
the r-meson is very small
in the majority of the cases
Weak dependence on
the parametrization
BT states: time-odd mean fields
are defined with ~15% accuracy
A.A., PRC 78, (08)

10. odd-mass nuclei: parametrization dependence
Impact of time-odd mean fields on binding energies of
odd-mass nuclei: parametrization dependence
W. Satula, Proceedings of Nuclear Structure'98 conference, Gatlinburg,
Tenn., USA
ENM-EWNM [MeV]
Skyrme
CDFT
Ar-isotopes
Additional binding due to NM (always
attractive) only weakly depends on the
covariant EDF.
In Skyrme DFT calculations the time-odd
mean fields can be both attractive and repulsive (sign even depends on mass region)

11. 3-A. Different aspects of the single-particle motion

12. of two rotational bands
Relative (effective)
alignments
of two rotational bands
depends both on the alignment
properties of single-particle
orbital(s) by which the two bands
differ and on the polarization
effects induced by the particles
in these orbitals
The impact of the particle(s) on
kinematic and dynamic moments
of inertia is well reproduced
AA, G.A. Lalazissis, P. Ring,
Nucl. Phys. A 634 (1998) 395.
AA and P.Ring, Phys. Scripta,
T88, 10 (2000)

13. Impact of particle(s) on charge quadrupole moments:
example of SD bands in the A~150 mass region
Experimental and calculated relative charge quadrupole moments
DQ0=Q0(Band)-Q0(152Dy(1)) of the 149Gd(1), 151Tb(1) and 151Dy(1).
The detailed structure of the configurations of these bands relative to the
doubly magic 152Dy core is given in column 2.
AA, G.A. Lalazissis, P. Ring, Nucl. Phys. A 634 (1998) 395.
This impact is rather well reproduced in non-relativistic and relativistic DFT
Satula et al, PRL 88, 5182 (1996)
M.Matev et al, PRC 76, (2007)
R.W. Laird et al, PRL 88, (2002)
On the contrary, the DQ0 quantity is not uniquely defined in phenomenological
models based on the Nilsson or Woods-Saxon potentials

14. Effective mass of nucleon at the Fermi surface and
single-particle states (their energies and wave functions).
Phenomenological potentials (Woods-Saxon, Nilsson) :
m*(kF)/m=1.0 since they are fitted to experimental single-particle energies.
However,
The coupling of vibrations to nucleons moving in levels lying close to the Fermi energy of deformed nuclei is found to lead to a number of effects:
(i) shifts of the single-particle levels of the order of 0.5 MeV
towards the Fermi energy and thus to an increase of the
level density (as a consequence m*(kF)/m ~ 1.4),
(ii) fragmentation of the single-particle strength (spectroscopic
factors S< 1)
Non-rotating deformed nuclei: F. A. Gareev, S. P. Ivanova, V. G. Soloviev, S.I. Fedotov, Single—particle energies and wave functions of the Saxon—Woods potentials
and nonrotational states of odd-mass nuclei in the region 150 < A < 190,
Physics of Elementary Particles and Atomic Nuclei, v.4, issue 2 (1973) 357
(see table next page)
Rotating deformed nuclei: P. Donati et al, PRL 84, 4317 (2000)

15. Phenomenological potentials (Woods-Saxon, Nilsson) : continuation
165Ho
Skyrme DFT : m*(kF)/m =
Inclusion of single-particle states into the fit of the force always leads to m*(kF)/m~1.0
Gogny DFT: m*(kF)/m = (D1S)
Covariant DFT: mL*(kF)/m = (NL1) to 0.66 (NL3*)
Landau
effective
mass
M. Jaminon et al,
PRC 40 (1989) 354

16. 3-B. How do we extract the energies of the “single-particle” states:
“absolute” versus “relative” energies

17. How single-particle spectra are obtained in experiment
Core
Binding energy
Core - nucleon
Core + nucleon
Include polarization effects due to
1. deformation (def) and
2. time-odd (TO) mean fields as well as
3. energy corrections due to particle-vibration coupling (PVC)

18. is calculated fully self-
For each configuration (blocked solution) the binding energy Ei
is calculated fully self-
consistently (including also
time-odd mean fields)

19. 3-C. Review of the accuracy of the description of experimental data

20. particle-vibration coupling + TO, TE polarization effects
E. Litvinova, AA, PRC 84, (2011)
NL3*
parametrization
particle-vibration coupling
+ TO, TE polarization effects

21. Statistical distribution of deviations of the energies of
one-quasiparticle states from experiment
The description of deformed states
at DFT level is better than spherical
ones by a factor 2-3 (and by a
factor ~1 (neutron) and ~2 (proton)
as compared with spherical PVC
calculations)
Triaxial CRHB; fully self-consistent
blocking, time-odd mean fields
included, NL3*, Gogny D1S pairing,
AA and S.Shawaqfeh,
PLB 706 (2011) 177
Two sources of deviations:
Low effective mass (stretching of the energy scale)
Wrong relative energies of the states
Similar problems in
Skyrme and Gogny DFT

22. Deformed one-quasiparticle states: covariant and
non-relativistic DFT description versus experiment
J.Dobaczewski, AA, M.Bender, L.M.Robledo, Yue Shi,
NPA 944 (2015) 388

23. AA and O.Abdurazakov,
PRC 88, (2013)
AA, Phys. Scr. 89 (2014)
054001

24. proton or neutron pairing (b) Alignment properties of blocked
Increase of J(1) in
odd-proton nucleus
as compared with
even-even 240Pu
is due to blocking
which includes:
Decrease of
proton or neutron pairing
(b) Alignment
properties of blocked
proton or neutron state

25. 3-D. Role of particle-vibration coupling: some important results.

26. The impact of particle-vibration coupling on spin-orbit splittings.
The inclusion of particle-vibration coupling decreases the accuracy of the description of spin-orbit
splittings.
E. Litvinova, AA, PRC 84,
(2011)
The absolute deviations per doublet are 0.34 MeV [0.50 MeV], 0.23 MeV [0.56 MeV] and 0.26 MeV [0.45 MeV] in the mean field (“def+TO”) [particle-vibration coupling (“def+TO+PVC”)] calculations in 56Ni, 132Sn and 208Pb, respectively.

27. The impact of particle-vibration coupling on pseudospin doublets.
Exp (protons)
Exp (neutrons)
def+TO
def+TO+PVC
PVC substantially improves
the description of splitting
energies in pseudospin
doublets as compared
with mean field calculations.
Observed similarity of the
splitting energies of proton
and neutron pseudospin
doublets with the same
single-particle structure in
medium and heavy mass
nuclei can only be
reproduced when the
particle-vibration coupling
is taken into account.
E. Litvinova, AA, PRC 84, (2011)

28. Spectroscopic factors
The absolute values of
experimental spectroscopic
factors are characterized by
large ambiguities and
depend
strongly on the reaction
employed in experiment
and the reaction model
used in the analysis

29. Competion of quasi-particle vibration coupling with tensor force
4. Single-particle degrees of freedom:
Competion of quasi-particle vibration coupling with tensor force
AA, E.Litvinova, PRC 92, (2015)

30. Tensor force
Schematic picture of the expectation values of the tensor operator S12 when the spins are either aligned with (prolate configuration) or perpendicular to (oblate configuration) the relative distance vector . The function f(r) is negative, favoring a prolate shape for the deuteron.
Deuteron: S12=-1 less binding, unbound
S12=+2, more binding, assumes prolate configuration

31. Tensor interaction in Skyrme DFT
Recent extensive review on
effective tensor interaction –
H.Sagawa and G. Colo,
PPNP 76, 76 (2014).
Sb (Z=51) isotopes
e(ph11/2) – e(pg7/2) [MeV]
Strongest “evidence” for effective tensor interaction from the energy splitting of spherical states
Skyrme DFT - G. Colo et al,
PLB 646 (2007) 227
e(ni13/2) – e(nh9/2)
N=83 isotones

32. Other examples: CDFT and Gogny DFT
Relativistic Hartree-Fock -
pion tensor coupling
G. A. Lalazissis et al, PRC 80,
(2009)
Gogny D1S
GT2 = D1S + plus tensor force
- T. Otsuka et al,
PRL 97, (2006)

33. The excitation energies of the lowest 2+ and 3- collective states
Sn nuclei in Skyrme QRPA
B.G. Carlsson et al, PRC 86,
(2012)
Relativistic QRPA
Neutron number N

34. Impact of quasiparticle-vibration coupling on the spectra
AA, E.Litvinova, PRC 92, (2015)
NL3* covariant energy density functional

35. Relativistic quasiparticle-vibration coupling calculations:
(1) the NL3* functional and (2) no tensor interaction
Our analysis clearly indicates that both QVC and tensor interaction act
in the same direction and reduce the discrepancies between theory and experiment for the splittings of interest. As a consequence of this
competition, the effective tensor force has to be weaker as compared
with earlier estimates.

36. Fragmentation of the single-particle strength
J. P. Schiffer et al, PRL 92,
(2004) – the states
of interest are single-particle
ones (S=1)
J. Mitchell, PhD thesis,
University of Manchester,
(2012) – strong fragmentation of the single-particle strength
(cannot be accounted at
the DFT level)
M. Conjeaud et al, NPA 117,
449 (1968) and O. Sorlin
Prog. Part. Nucl. Phys. 61,
602 (2008) also support low
S~0.5 for ph11/2 state in
mid-shell Sb isotopes
B.P.Kay et al, PRC 84, (2011)
PLB 658, 216 (2008)

37. Example of generic
problems of many
functionals:
Deformed shell gaps at
N=152 and Z=100
Towards spectroscopic quality DFT:
Improvement of the functionals
at the DFT level
Accounting of (quasi)particle-
vibration coupling
3. Inclusion of tensor interaction
(not clear at this point)

38. 4-A. Impact on nuclear extremes (neutron-drip line)
AA, S. Agbemava, D. Ray and P. Ring, PLB 726, 680 (2013)
S. Agbemava, AA, D. Ray and P. Ring, PRC 89, (2014)
AA, S.E. Agbemava, D. Ray, P. Ring ,PRC 91, (2015)

39. How to minimize the uncertainties in theoretical predictions?
AA, S. Agbemava, D. Ray and P. Ring, PLB 726, 680 (2013)
S. Agbemava, AA, D. Ray and P. Ring, PRC 89, (2014)
J.Erler et al et al, Nature 486 (2012) 509
How to minimize the uncertainties in theoretical predictions?

40. Single-particle states as the sources of uncertainties
AA, S.E. Agbemava, D. Ray, P. Ring
PLB 726 (2013) 680
DD-ME2
DD-ME2
Rn (Z=86)
NL3* drip line-diamonds

41. The shell structure still survives in neutron-rich nuclei

42. The SHE and two-neutron drip line – the common
source of uncertainties (single-particle states)
DD-ME2
Rn (Z=86)
NL3* drip diamonds
Predictions of two-neutron drip
line for DD-Med and DD-PC1 are
closer to DD-ME2 than to NL3*

43. Neutron single-particle energies for the ground-state
Two-neutron drip lines: the impact of uncertainties in
single-particle energies
Neutron single-particle
energies for the
ground-state
configurations of the
Rn isotopes calculated
at their equilibrium
deformations as a
function of neutron
number N. Note that the
transition to deformation
removes the 2j + 1
degeneracy of the
spherical orbitals.
AA, S.E. Agbemava, D. Ray, P. Ring ,PRC 91, (2015)
calculational scheme “B”

44. Reducing uncertainties for two-neutron drip line
Known nuclei
Two-neutron
drip line nuclei
Thin lines – 10 CEDF, thick – 4 CEDF
Solid–10 CEDF, dashed– 4 CEDF

45. Theoretical uncertainties are most pronounced for
transitional nuclei (due to soft potential energy surfaces) and in
the regions of transition between prolate and oblate shapes.
Details depend of the description of single-particle states

46. 4. An extreme of superheavy nuclei
Detailed results in
S. Agbemava et al, PRC 92, (2015)
Covariant density functional theory: Reexamining the
structure of superheavy nuclei

47. Theoretical uncertainties in the prediction of the
sizes of shell gaps.
Thin lines – all 10 CEDF’s,
thick – 4 CEDF
(NL3*,DD-ME2,
DD-MEd,DD-PC1)
Mass dependence of single-particle level density (~A1/3) is taken into account

49. DD-PC1: Experimental Z=116, 118 nuclei are oblate PC-PK1: Experimental
Z=118 nucleus
is spherical
Other experimental
SHE are prolate
Open circles –
experimentally
observed nuclei

50. Potential
energy
surfaces
in axially
symmetric
RHB
calculations
with separable
pairing

51. The source of oblate shapes – the low density of s-p states

53. Why I never ventured for two-quasiparticle states?

54. 2-quasi-particle states:
how they are important?
Exp.
Theory
Setup: each theoretical single-particle
level deviates
from experimental one by 0.5 MeV a a
Energies of 2-quasiparticle states can either
be close to experiment or to deviate from
experiment by ~ 1 MeV
Single-particle energies b c b c
However, even this picture is considerably
simplified, see next slide

55. 2-quasi-particle states:
Continuation 1
The 1-qp (odd nuclei) and 2-qp (even-even and odd-odd) spectra
will be stretched as compared with experiment in self-consistent
approaches based on low-effective mass of nucleon
(see A.V.Afanasjev et al, PRC 67 (2003) for 1-qp spectra)
2. Time-odd effects have to be included. There is considerable
uncertainty in their definition in Skyrme type models
3. There are the cases of the mixing of 2-neutron qp and 2-proton qp states.
This type of mixture have to be included in the calculations or only the
cases where such mixing is not important are considered.
4. The admixtures of vibrational states to some 2-qp states
cannot be neglected

56. 2-quasi-particle states:
Continuation 2: Gallagher-Moszkowski splitting
Intrinsic states of 2
protons in 2-qp proton state
- the projections of the single-quasiparticle
angular momentum on the symmetry axis
due to residual p-p (or n-n)
interaction of unpaired particles
typically several hundreds keV’s but could reach 0.5 MeV
in rare earth region (A.Goel et al, PRC 45, 221 (1992))
Cannot be treated in the 1D-cranking models since
no projections K+, K- on the symmetry axis are considered
no residual interaction of unpaired nucleons is included
UNRESOLVED PUZZLE: the parameters of the residual interaction of
unpaired nucleons are completely different in even-even and odd-odd nuclei

57. Conclusions: Different nuclear phenomena are well described in the
CDFT framework at and beyond the mean field level.
2. At present stage, we can also estimate theoretical uncertainties
and their propagation beyond the known region of nuclei
3. Many theoretical uncertainties emerge from inaccuracies in
the description of the single-particle states. At present, this
is a real bottleneck of the DFT models (both relativistic and
non-relativitic ones).
4. Note that different phenomena/observables or regions
of nuclear chart are differently affected by the uncertainties
in the single-particle energies. For example, the
descriptive/predictive power for rotational systems is not
strongly affected by them.

59. Spectroscopic factors

60. to the single-particle energies
The danger of the fit of the model parameters on the DFT level
to the single-particle energies
M. Zalewski et al,
PRC 80, (2009) and
PRC 77, (2008).
Skyrme DFT
A direct fit of the isoscalar
spin-orbit (SO) and both
isoscalar and isovector
tensor coupling constants
to the f5/2-f7/2 SO splittings
in 40Ca, 56Ni, and 48Ca nuclei
requires a drastic reduction
of the isoscalar SO strength
and strong attractive tensor
coupling constants.
The PVC changes SO splitting in the f5/2-f7/2 doublet by 1.5 MeV

61. Systematic theoretical uncertainties are defined by the spread
(the difference between maximum and minimum values
of physical observable obtained with employed set of CEDF’s).
NL3*, DD-ME2, DD-MEd, DD-PC1 [ also PC-PK1 in superheavy nuclei ]

62. Theoretical errors/uncertainties in description/predictions
of physical observables
Systematic errors/uncertainties apply to a set of the functionals
- related to the choice of the basic
physics behind the functional
Statistical errors/uncertainties - apply only to a given functional
- due to choice of experimental data
and the selection of adopted errors
in the fitting protocol (the latter
is to a degree subjective especially
for the quantities of different
dimensions)
S. Brandt, Data analysis. Statistical and Computational Methods for Scientists
and Engineers. (Springer International Publishing, Switzerland, 2014).
2. J. Dobaczewski et al, J. Phys. G 41 (2014)

63. These values are comparable or somewhat better than the
ones obtained in systematic Hartree–Fock+BCS calculations of deformed
nuclei with SIII, SkM* and SLy5 Skyrme forces and FRDM
calculations employing phenomenological folded-Yukawa potential
[L. Bonneau et al, PRC 76 (2007) ], which show the agreement with experiment at about 40% level.

64. DFT calculations suggest that band 1 in 158Er is observed up to spin around 75
AA, Yue Shi, W.Nazarewicz,
PRC 86, (R) (2012)

65. The impact of particle-vibration coupling on spin-orbit splittings:
the comparison with Skyrme PVC for neutron subsystem of 208Pb
The change of spin-orbit splitting
induced by PVC
Covariant
PVC
Skyrme
PVC
doublet
3d5/2-3d3/2
+0.14
+0.15
3p3/2-3p1/2
-0.16
-0.15
2g9/2-2g7/2
+0.32
-0.1
2f7/2-2f5/2
-1.0
-0.6
2i13/2-1i11/2
-0.75
Skyrme PVC
G.Colo et al, PRC 82, (2010)

66. to the single-particle energies
The danger of the fit of the model parameters on the DFT level
to the single-particle energies
M. Zalewski et al,
PRC 80, (2009) and
PRC 77, (2008).
Skyrme DFT
A direct fit of the isoscalar
spin-orbit (SO) and both
isoscalar and isovector
tensor coupling constants
to the f5/2-f7/2 SO splittings
in 40Ca, 56Ni, and 48Ca nuclei
requires a drastic reduction
of the isoscalar SO strength
and strong attractive tensor
coupling constants.
The PVC changes SO splitting in the f5/2-f7/2 doublet by 1.5 MeV