PAIRING PROPERTIES OF SUPERHEAVY NUCLEI A. Staszczak, J. Dobaczewski and W. Nazarewicz
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9/28/2006 19:01 (00) PAIRING PROPERTIES PAIRING PROPERTIES OF SUPERHEAVY NUCLEI OF SUPERHEAVY NUCLEI A. Staszczak, J. Dobaczewski and W. Nazarewicz A. Staszczak, J. Dobaczewski and W. Nazarewicz (KFT UMCS) (IFT UW) (ORNL & UT) (KFT UMCS) (IFT UW) (ORNL & UT)
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PAIRING PROPERTIES OF SUPERHEAVY NUCLEI A. Staszczak, J. Dobaczewski and W. Nazarewicz (KFT UMCS) (IFT UW) (ORNL & UT). The constrained HF procedure The constraints act as the external fields capable to deform the nucleus in different ways - PowerPoint PPT Presentation
Transcript of PAIRING PROPERTIES OF SUPERHEAVY NUCLEI A. Staszczak, J. Dobaczewski and W. Nazarewicz
9/28/2006 19:01 (00)
PAIRING PROPERTIESPAIRING PROPERTIESOF SUPERHEAVY NUCLEIOF SUPERHEAVY NUCLEI
A. Staszczak, J. Dobaczewski and W. NazarewiczA. Staszczak, J. Dobaczewski and W. Nazarewicz
(KFT UMCS) (IFT UW) (ORNL & UT)(KFT UMCS) (IFT UW) (ORNL & UT)
9/28/2006 19:01 (00)
The constrained HF procedure
The constraints act as the external fields capable to deform the nucleus in different ways
The collective coordinates can be defined in a natural way by measuring the deformations generated by the various constraints
The constrained mean field theory defines the deformed states(BCS- or HFB-type) that solve the variational equation:
with the constraint conditions
ˆˆ ˆ{ } { } , { } { } , { } { } .j jq N q N q Z q Z q Q q q
{ }q
212
ˆˆ ˆ ˆ{ } { } 0N Z j j jjq H N Z c Q q
quadratic multipole constraints
The multipole constraints prescribe different kinds of deformation characterized by the set of parameters
1 2{ } { , ,..., }.Nq q q q
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is the many-body nuclear (non-relativistic) Hamiltonian H
2
2( )
1
ˆˆ 1ˆ ˆ2 2 2
A Aii effiij
i i j
ppH V
m Am
center-of-mass “projection” term(in the VAP technique),
to eliminate spurious mode associated with the broken translational symmetry
nuclear effective interaction term(Skyrme, Gogne type forces)
To describe the fission process most “important” are the low-multipolarity mass moments, i.e.,
2 *20 20
3 *30 30
4 *40 40
ˆ ( ) 16 5 ( , ),
ˆ ( ) 4 7 ( , ),
ˆ ( ) 4 9 ( , ).
i i ii
i i ii
i i ii
Q r r Y
Q r r Y
Q r r Y
“nuclear stretching”
“reflection-asymmetry”
“necking”
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-50 0 50 100 150 200 250 300 350
-1804
-1802
-1800
-1798
-1796
-1794
-1792
-1790
-1788
-1786
-1784
EIII
Tot
al E
nerg
y [M
eV]
Q20
[b]
680 (14) 816 (15) 969 (16) 1000 1140(17) 1330(18) 1540(19) 1771(20)
reflection-symmetric path 240Pu
EA E
IIE
B
Size of the basis
1140 s. p. states of deformed 3D h. o.
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Seniority pairing:Seniority pairing:J. Dudek, et al., J. Phys. G6 (1980)447.
/ /
/
18.95 0.078 , 88 17.90 0.176 , 88
19.3 0.084 , 88 13.3 0.217 , 88
n p
n p
n p
n p
N Z N ZZ Z
A AG G
N Z N ZZ Z
A A
fG G
1 2
-31 2 0
0
12
/0
/1 2
2 , 0.16 fm , 1,
0, delta interaction (DI)
1, density-depende
" "
" "
"
nt delta interaction (DDDI)
, (MIX)
, 1n p n p
volume pair
r r
r r
ing
surface pairing
mixe
V r r V
"d pairing
Zero-range pairing force:Zero-range pairing force:R. R. Chasman, Phys. Rev. C 14 (1976)1935.G. F. Bertsch and H. Esbensen, Ann. Phys. (N.Y.) 209 (1991)327.
/n pf/
0n pV
In the paring (BCS) window n/pN (or Z) s. p. states are taken, and parameters are chosen to reproduce n/p for 252Fm.
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The spectral (average) gap:
/
/
// , : or s.p. states
n p
n p
k k k n pkn p
k kk
v uN Z
v u
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SLy4
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SLy4
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SLy4
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-50 0 50 100 150 200 250 300-2056
-2052
-2048
-2044
-2040
-2036
-2032
Etot Q40
MIX DDDI DI G
Quadrupole Moment Q20
[b]
Tot
al E
nerg
y E
tot [M
eV] 288Rf
184
0
10
20
30
40
50
60
70
80
90
Hexadecapole M
oment Q
40 [b2] -50 0 50 100 150 200 250 300
-2080
-2070
-2060
-2050
-2040
-2030
Quadrupole Moment Q20
[b]
288Rf184T
ota
l En
erg
y E
tot [M
eV
]
0.0
0.5
1.0
1.5
2.0
Pa
iring
Ga
p
n/p [Me
V]
-50 0 50 100 150 200 250 300-2116
-2112
-2108
-2104
-2100
-2096
-2092
Etot Q40
MIX DDDI DI G
298114184
Quadrupole Moment Q20
[b]
Tot
al E
nerg
y E
tot [M
eV]
0
10
20
30
40
50
60
70
80
90 Hexadecapole M
oment Q
40 [b2]
-50 0 50 100 150 200 250 300-2140
-2130
-2120
-2110
-2100
-2090
298114184
Quadrupole Moment Q20
[b]
Tot
al E
nerg
y E
tot [M
eV]
0.0
0.5
1.0
1.5
2.0
Pairing G
ap n/p [M
eV]
-50 0 50 100 150 200 250 300
-2140
-2136
-2132
-2128
-2124
-2120 310126184
Etot Q40
MIX DDDI DI G
Quadrupole Moment Q20
[b]
Tot
al E
nerg
y E
tot [M
eV]
0
10
20
30
40
50
60
70
80
90
Hexadecapole M
oment Q
40 [b2] -50 0 50 100 150 200 250 300
-2160
-2150
-2140
-2130
-2120 310126184
Quadrupole Moment Q20
[b]
Tot
al E
nerg
y E
tot [M
eV]
0.0
0.5
1.0
1.5
2.0
Pairing G
ap n/p [M
eV]
SLy4
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Quadrupole Moment Q20 [b]
Total Energy Etot [MeV]
Hexadecapole M
oment Q
40 [b2]
-50 0 50 100 150 200 250 300
-2068
-2064
-2060
-2056
-2052
-2048 290Sg184
MIX0102030405060708090
-50 0 50 100 150 200 250 300-2056
-2052
-2048
-2044
-2040
-2036
-2032288Rf
184
MIX0102030405060708090
-50 0 50 100 150 200 250 300
-2084
-2080
-2076
-2072
-2068
-2064 292Hs184
MIX
0102030405060708090
-50 0 50 100 150 200 250 300
-2096
-2092
-2088
-2084
-2080
-2076 294Ds184
MIX
0102030405060708090
-50 0 50 100 150 200 250 300-2108
-2104
-2100
-2096
-2092
-2088
-2084296112
184
MIX
0102030405060708090
-50 0 50 100 150 200 250 300-2116
-2112
-2108
-2104
-2100
-2096
-2092
298114184
0102030405060708090
MIX
-50 0 50 100 150 200 250 300-2124
-2120
-2116
-2112
-2108
-2104
-2100300116
184
MIX
0102030405060708090
-50 0 50 100 150 200 250 300
-2128
-2124
-2120
-2116
-2112
-2108 302118184
MIX
0102030405060708090
-50 0 50 100 150 200 250 300
-2132
-2128
-2124
-2120
-2116
-2112 304120184
MIX
0102030405060708090
-50 0 50 100 150 200 250 300-2136
-2132
-2128
-2124
-2120
-2116
-2112306122
184
MIX0102030405060708090
-50 0 50 100 150 200 250 300
-2136
-2132
-2128
-2124
-2120
-2116 308124184
MIX
0102030405060708090
-50 0 50 100 150 200 250 300
-2136
-2132
-2128
-2124
-2120
-2116 310126184
MIX
0102030405060708090
Etot
Q40
Etot
Q40
MIX
SLy4
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Etot
n
p
Etot
n
pQuadrupole Moment Q20 [b]
Total Energy Etot [MeV]
Pairing G
ap n/p [M
eV]
-50 0 50 100 150 200 250 300-2160
-2150
-2140
-2130
-2120
-2110
310126184
MIX0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2160
-2150
-2140
-2130
-2120
-2110
308124184
0.0
0.5
1.0
1.5
2.0
MIX
-50 0 50 100 150 200 250 300-2160
-2150
-2140
-2130
-2120
-2110
306122184
0.0
0.5
1.0
1.5
2.0
MIX
-50 0 50 100 150 200 250 300-2150
-2140
-2130
-2120
-2110
-2100
302118184
MIX0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2150
-2140
-2130
-2120
-2110
-2100
0.0
0.5
1.0
1.5
2.0
MIX
300116184
-50 0 50 100 150 200 250 300-2140
-2130
-2120
-2110
-2100
-2090
MIX
298114184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2130
-2120
-2110
-2100
-2090
-2080
296112184
MIX
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2120
-2110
-2100
-2090
-2080
-2070
294Ds184
MIX
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2110
-2100
-2090
-2080
-2070
-2060
292Hs184
MIX
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2090
-2080
-2070
-2060
-2050
-2040
290Sg184
MIX
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2080
-2070
-2060
-2050
-2040
-2030288Rf
184
MIX
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2160
-2150
-2140
-2130
-2120
-2110
304120184
MIX
0.0
0.5
1.0
1.5
2.0
MIX
SLy4
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SummarySummary
We compared “volume”, “surface” and “mixed” zero-range nuclear pairing forces with monopole pairing.
Applying “mixed” -interaction we have investigated total binding energies (barriers) and spectral pairing gaps along fission paths of even-even superheavy nuclei with N = 184.
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KONIEC
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P. Möller, J. R. Nix, and K.-L. Kratz Atomic Data’97
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Quadrupole Moment Q20 [b]
Total Energy Etot [MeV]
Hexadecapole M
oment Q
40 [b2]
-50 0 50 100 150 200 250 300-2056
-2052
-2048
-2044
-2040
-2036
-2032288Rf
184
Etot
Q40
0102030405060708090
-50 0 50 100 150 200 250 300-2072
-2068
-2064
-2060
-2056
-2052
-2048290Sg
184
Etot
Q40
0102030405060708090
-50 0 50 100 150 200 250 300
-2084
-2080
-2076
-2072
-2068
-2064
Etot
Q40
292Hs184
0102030405060708090
-50 0 50 100 150 200 250 300
-2096
-2092
-2088
-2084
-2080
-2076
Etot
Q40
294Ds184
0102030405060708090
-50 0 50 100 150 200 250 300-2116
-2112
-2108
-2104
-2100
-2096
-2092
298114184
Etot
Q40
0102030405060708090
-50 0 50 100 150 200 250 300
-2136
-2132
-2128
-2124
-2120
-2116
Etot
Q40
0102030405060708090
306122184
-50 0 50 100 150 200 250 300-2108
-2104
-2100
-2096
-2092
-2088
-2084
Etot Q
40
296112184
0102030405060708090
-50 0 50 100 150 200 250 300
-2132
-2128
-2124
-2120
-2116
-2112 304120184
Etot
Q40
0102030405060708090
-50 0 50 100 150 200 250 300-2140
-2136
-2132
-2128
-2124
-2120
-2116
Etot
Q40
0102030405060708090
308124184
-50 0 50 100 150 200 250 300-2124
-2120
-2116
-2112
-2108
-2104
-2100
300116184
Etot
Q40
0102030405060708090
-50 0 50 100 150 200 250 300-2140
-2136
-2132
-2128
-2124
-2120
-2116310126
184
Etot
Q40
0102030405060708090
-50 0 50 100 150 200 250 300
-2128
-2124
-2120
-2116
-2112
-2108 302118184
Etot
Q40
0102030405060708090
G
SLy4
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Quadrupole Moment Q20 [b]
Total Energy Etot [MeV]
Pairing G
ap [MeV
]
-50 0 50 100 150 200 250 300
-2150
-2140
-2130
-2120
-2110 302118184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300
-2160
-2150
-2140
-2130
-2120
310126184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300
-2120
-2110
-2100
-2090
-2080
294Ds184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2110
-2100
-2090
-2080
-2070
-2060
292Hs184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2150
-2140
-2130
-2120
-2110
-2100300116
184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300
-2160
-2150
-2140
-2130
-2120
308124184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300
-2160
-2150
-2140
-2130
-2120
306122184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2160
-2150
-2140
-2130
-2120
-2110304120
184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300
-2130
-2120
-2110
-2100
-2090 296112184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2140
-2130
-2120
-2110
-2100
-2090298114
184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300
-2090
-2080
-2070
-2060
-2050
290Sg184
0.0
0.5
1.0
1.5
2.0
-50 0 50 100 150 200 250 300-2080
-2070
-2060
-2050
-2040
-2030
288Rf184
0.0
0.5
1.0
1.5
2.0
Etot
n
p
Etot
n
p
G
SLy4
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