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Probing effective NN interaction
at band termination
W. Satuła
IFT Univ. of Warsaw
in collaboration with
R.A. Wyss, H. Zduńczuk, M. Zalewski, M. Kosmulski, G. Stoicheva, D. Dean, W. Nazarewicz,
H. Sagawa (+exp), A. Bhagwat, J.Meng ...
Outline:
• towards effective superfluid local density approximation (SLDA)
- general remarks
• pairing: volume-, mixed- or surface-type
- selectivity/resolution of nuclear data
• odd-even staggering (OES) in high-spin isomers – new opportunities to
study:
- blocking of pair correlations
- single-particle, time-odd, and residual p-n effects
• termination in N~Z and N=Z nuclei in A~50 mass-region
M. Zalewski/saturday
- fine tuning of particle-hole field
H. Zduńczuk/poster
- fine tuning of shell-model interaction
• 73Kr – dynamical manifestation of T=0 pn-pairing at high spins?
Skyrme-force as a particular realization of effective ph interaction
Fourier
Long-range part of NN interaction
(must be treated exactly!!!)
local
correcting
potential
infinite number
of equivalent
effective theories
Skyrme interaction:
10(11)
parameters
LEDF:
Y | H | Y
Slater determinat
(number conserving)
20 parameters
lim da
a
Gogny:
0
density-dependent
spin-orbit
Fitting the Skyrme force parameters or the nuclear LEDF
Saturation point of symmetric infinite nuclear matter:
- saturation density ( ~0.16fm-3)
- energy per nucleon (-16 +0.2MeV)
- incompresibility modulus (210 +20MeV)
- isoscalar effective mass (???)
Asymmetric infinite nuclear matter and the isovector properties:
- symmetry energy ( 30+ 2MeV)
- isovector effective mass (GDR sum-rule enhancement)
- neutron-matter EOS (Wiringa, Friedmann-Pandharipande)
Finite nuclei [masses,radii,sp levels]:
- surface properties (semi-infinite nuclear matter)
- realistic mean level-density (masses)
entire ZOO of parameterizations !!!
TOWARDS EFFECTIVE (local) PAIRING THEORY
(I) Low-momentum transfer expansion in particle-particle channel:
v(k,k’) = g + g2(k-k’)2 + g4(k-k’)4 +.....
S. Hilaire et al. PLB531 (2002) 61
...in r-space:
D1S Gogny
exp.
th.
Dn (MeV)
1.5
1.0
Gaussian Dirac-delta model
leads to Gogny pairing
0.5
• offers excellent agreement with data
0
• no cut-off is needed
However, the so called coherence
length i.e. spatial extension of
the nucleonic Cooper pair x:
x
0
(200MeVfm)2
= dx =
20
60
1.4fm-1
(hc)2kF
(mc2)D
1000MeV
40
>>
1MeV
1
kF
80
100
120 140
three-point filter
2.0
neutron number
interparticle distance
In this context the use of finite range pairing
force can be viewed as rather unnecessary
complication.
TOWARDS EFFECTIVE (local) PAIRING THEORY
(II) resolution Gogny versus local (DDDI) pp interaction:
DDDI:
E. Garrido et al. PRC60, 064312 (1999)
PRC63, 037304 (2001)
dots represent the
Gogny gap
Cut-off!!!
(otherwise divergent!)
TOWARDS EFFECTIVE (local) PAIRING THEORY
(IIIa) in-medium effects towards the SLDA approach:
[Superfluid Local Density Approximation]
Major obstacle in constructing SLDA is:
we use cut-off!!!
anomalous density
(pairing tensor)
(usual, but not
at all satisfactory
solution)
ultraviolet
divergence
In particular, in infinite homogenous system (example):
regular
isolate and regularize
divergent term
„regularize” means in practice simply
„remove divergent part”
(relate to scattering amplitude; use dimensional regularization;
introduce counter-terms [regulators] with explicit cut-off)
;
LOCAL EFFECTIVE PAIRING THEORY
(IIIb) Bulgac-Yu SLDA approach:
A.Bulgac, Y.Yu, PRL88, 042504 (2002)
A.Bulgac, PRC65, 051305(R) (2002)
local HFB
In fact, for sufficiently
large Ec gap is cutoff
independent
40MeV
45MeV
50MeV
35MeV
110Sn
30MeV
Ec=20MeV
Formally, gap depends on
both the effective (running)
coupling constant and on
QP cut-off energy
interaction distance
Ec~pc2/2mr & ropc~h
Ec~h2/mro2 ~ 40MeV
J.Dobaczewski & W.Nazarewicz
Prog. of Theor. Phys. Suppl. No. 146 (2002)
b2
surface
mixed
volume
0.25
0.20
0.15
0.10
Dn
1.5
0.6
0.4
0.2
0
20 40 60 80
1.0
N,Z
40
0.5
14
20
82
40
60
50Cr
SLy4
0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
1.0
50
28
Eexp-Eth [MeV]
Dn
dD (MeV)
0.05
2.0
Dn,p (MeV)
Pairing/resolution (1)
spherical HFB
126
80 100 120 N,Z
1.5
Dn [MeV]
2.0
Pairing/resolution (2)
Hilaire et al. PLB531 (2002) 61
Vo(1-r(r)/ri)
0.20
EXP
b2
0.15
0.10
0.05
46Ti
SLy4
Eexp-Eth [MeV]
5.0
4.5
4.0
3.5
3.0
2.5
surface
mixed
volume
1.0
1.5
Dn [MeV]
2.0
M. Kosmulski – licentiate thesis
High-spin isomers - new opportunities to study pairing correlations
Odd-Even Binding Energy Effect in the High-Spin Isomers:
Are Pairing Correlations Reduced in Excited States?
A. Odahara, Y. Gono, T. Fukuchi, Y. Wakabayashi,
H. Sagawa, WS, W. Nazarewicz, PRC72, 061303(R), (2005)
Stretched configurations:
N=83
DE ~~ 8.5MeV ~~ const.
Hence, the OES:
is similar in GS and HSI!!!!
no blocking???
We expect:
Dracoulis et al. PLB419, 7 (1998)
0.8
0.4
full
TE
0
p-Fermi
energy
HF SLy4/HSI
HF SkO/HSI
EXP {GS
HSI
61 62 63 64 65 66
Z
• sp contribution to OES
• time-odd terms (within
self-consistent models)
esp
0.5
0.0
-0.5
-1.0
-1.5
-2.0
d3/2
s1/2
h11/2
64
hole in
[402]5/2
fixed
occup.
DIPM
data
Spherical
sp spectrum
WS
Z
SLy4
61 63
SkO
D(Z) [MeV]
3.0
2.5
2.0
1.5
1.0
0.5
0
Isomerism of the same type!
Oblate shapes at HSI (-0.2)
Nearly spherical GS
HF-SLy4
GS
{
D(Z) [MeV]
Are Pairing Correlations Reduced in Excited States? (II)
d5/2
Strutinsky calculations with pairing:
2.0 OES/DPES
1.5
0.25
0.20
0.15
61 63 65 67
Z
9
8
DPES
DPES+dpn
D(Z) [MeV]
10
|dpn| [MeV]
DEHSI [MeV]
Excitation energy
GAP/OES
EXP
DIPM
dpn
1.0
2.0 Dproj
Dexp(Z)
1.5
1.0
DIPM
DPES
143 144 145 146 147 148 149 150
Nd Pm Sm Eu Gd Tb Dy Ho
+10%
+15%
HSI
GS
61 62 63 64 65 66
Z
Enhanced pairing is needed (see also Xu et al. PRC60,051301 (1999))
Blocking is too strong!!!
This study reveals that many effects can contribute
to OES in particular:
pairing
single-particle effects
time-odd effects (nuclear magnetism)
residual pn interaction (odd-odd)
RMF
...and the question is...
Is blocking under control?
Time-odd
fields
Rutz et al. PLB468 (1999) 1
p-h
Consider the energy difference between stretched
(terminating) configurations in A~50 mass region
-1 n+1
n
DE = E( d3/2 f7/2
) - E( f7/2 I )
Imax
max
• the best examples of almost unperturbed sp motion
• uniquely defined (in N=Z)
• config. mixing beyond mean-field is expected to be mariginal
(in particular all pairs are broken)
• shape-polarization effects included already at the level of the SHF
• time-odd mean-fields (badly known) can be tested
these are ideal for fine tune particle-hole interaction!!!!
f7/2
20
d3/2
energy scale
(bulk properties)
spin-orbit
dominates!!!
~ 0 light
~ ½ heavy
nuclei
Mean-field versus Shell-Model
„isospin symmetry restoration” in N=Z nuclei original HF
result for
pph excitation
T=1
nph
dET
pph
centroid
dET
T=0
Can be evaluated from
mirror-symmetric nuclei e.g.
40Ca
from 40K and
42Sc
etc.
1.5
1.0
dET [MeV]
Isospin symmtery
„restoration” in
N=Z nuclei:
DEth-DEexp [MeV]
G.Stoitcheva, WS, W.Nazarewicz, D.J.Dean,
M.Zalewski, H.Zduńczuk, PRC73, 061304(R) (2006)
1.8
DZ
1.4
SkO
1.0
40
42
44
A
0.5
0
SM
SkO
-0.5
42Ca
43Sc
44Ca
45Sc
44Sc
46Ti
45Ti
47V
42
46V
40Ca Sc44
Shifted by 480keV
T=0 pn pairing???
reduced s-o
Ti
N=Z
HFB calculations including T=0 and T=1 pairing
J. Terasaki, R. Wyss, and P.H. Heenen PLB437, 1 (1998)
48Cr
-1
d3/2
g9/2
isoscalar
pairing
•
•
•
•
Skyrme interaction in p-h
DDDI in p-p channel
fully self-consistent theory
no spherical symmetry
no T=0 at
low spins
Non-collective (oblate)
rotation
T=1 collapses
4
4
[nf7/2
pf7/2
]16
Collective (prolate)
rotation
+
(termination)
data
BFZ mechanism: Isovector dependence of shell-model
matrix elements
DE=1/2b[T(T+1)-Tp(Tp+1)-Th(Th+1)]
particle
contribution
Tp=T + 1/2
single-hole
contribution: Th=1/2
N = Z (Tp=1/2):
DE=-3b/4
N = Z (Tp=T-1/2):
DE=b(T-1/2) /2
DESM-DEEXP [MeV]
„... It is a grave error to assume that
the p-h intraction is independent of
isotopic spin...”
... isotopic dependence of p-h interaction
can be approximated by a monopole potential vT~bt1.t2
modified SM
SM
0.4
0.2
0
-0.2
-0.4
The SM overestimates b
by ~700keV!!!
Vphph(JT=0)+3d
Vphph(JT=1)-d
d=175keV
42Ca
Single j-shell J-T SM phenomenology:
43Sc
44Ca
45Sc
44Sc
45Ti
(2j+3)VT=1,n=2-(2j+1)VT=0,n=2-2VT=1,J=0
bsym =
2(2j+1)
Bansal & French, Phys.Lett. 11, 145 (1964)
Zamick, Phys. Lett. 19, 580 (1965)
46Ti
47V
42
46V
40Ca Sc44
VT=0+3d
VT=1-d
Ti
N=Z
bsym- 4d
N. Zeldes, Handbook of Nuclear Properties, Clarendon Press, Oxford, 1996, p.13
Low-spin particle-hole intruder states
SM versus modified-SM
modified SM
SM
3-
ESM-EEXP [MeV]
0.8
3/2+
3/2+
0.6
3-
0.4
3-
3-
0.2
3- 3-
2-
3-
0
3/2+
3-
3-
40Ca
42Sc
44Sc
44Ca 43Sc
3/2+
3/2+
3/2+
42Ca
3/2+ 3- 3
3
44Ti
45Sc
46Ti
45Ti
47V
46V
Concluding remarks
Consistent superfluid local density approximation
is just behind our doors!
Volume-, mixed- or surface-like local pairing
- selective nuclear data exist but must be systematically
identified (and understood) thrughout the nuclear chart
OES in high-spin isomeric states:
-new opportunities to study blocking, TO terms,
residual-pn, and mean-field (sp-splitting) effects
Termination in N~Z, A~50 nuclei:
- excellent laboratory for fine-tuning of
ph MF interaction and SM interaction
73Kr
– possible fingerprint of enhanced T=0 pairing
(1)
73Kr
- a fingerprint of T=0 pairing?
R.Wyss, P.J. Davis, WS, R. Wadsworth
Conventional TRS calculations involving only T=1 pairing:
Ix
3qp (-,-)
(+,+)
30
2.0
73Kr
(-,-)
25
1.5
20
1.0
3qp
5qp
15
0.5
10
0.0
1qp
-0.5
0.5
1.0
g
40
fp
1.5
0.5
hw [MeV]
1qp
5
1.0
1.5
0.5
1.0
1.5
hw [MeV]
|1qp> = a+n(fp)|0>
|3qp> = a+ng a+pg a+p(fp)|0>
<1qp|E2|3qp> ~ 0 (one-body operator)
Kelsall et al., Phys. Rev. C65 044331 (2005)
Ew [MeV]
2.5
73Kr
negative parity
negative parity
73Kr:
positive parity
D [MeV]
(2) 73Kr a fingerprint of T=0 pairing?
What makes the 1qp and 3qp configurations alike?
Scattering of a T=0 np pair
TRS involving T=0 and T=1
73
in Kr
pairing
n(fp)
p(fp)
Dn
1.0 D
ng9/2
pg9/2
p
0.5
DT=0
n(fp)
p(fp)
0
ng9/2
pg9/2
73Kr
30
n(fp)(-)
vacuum
n(fp)
ng9/2
n(fp)
ng9/2
ng9/2(+)
p(fp)
pg9/2
p(fp)
pg9/2
pg9/2 p(fp)(-)
3qp configuration
Ix
1qp configuration
25
20
theory
15
10
exp
5
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
hw [MeV]
Ew [MeV]
(3)
73Kr
a fingerprint of T=0 pairing?
2.0
Conventional TRS calculations involving only T=1 pairing
in neighbouring nuclei:
negative parity
positive parity
all bands
Ix 75
(-,+)
Rb
75Rb
3qp
30
1.5
25
1.0
20
1qp
0.5
15
0.0
10
-0.5
5
0.5
1.0
hw [MeV]
1.5
(+,+)
3qp
1qp
0.5
1.0
1.5
0.5
hw [MeV]
1.0
1.5
Excellent agreement was obtained in:
Tz=1 : 74Kr,76Rb, D. Rudolph et al. Phys. Rev. C56, 98 (1997)
Tz=1/2: 75Rb, C. Gross et al. Phys. Rev. C56, R591 (1997)
Tz=1/2: 79Y, S.D. Paul et al. Phys. Rev. C58, R3037 (1998)