Transcript Document

J. P. Draayer , K. D. Sviratcheva, C. Bahri and A. I. Georgieva
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Models
 Sp(4) model (sp(4)~so(5)) for a description of pp+nn+pn
isovector (isospin T=1) pairing correlations in light and
medium mass nuclei following Helmers’s approach:
Outline
“quasi-spin” approach of Helmers
U (2(2 j  1)) ( j n )  Sp (2 j  1) ( j  )  SO (5) (    ,t ,n ,b ,T )
(b )
n
j )
(a )
(a)
 U (2 j  1) (T )  Sp (2 j  1) (  ,t )  SO ( 3) ( J )
vs. conventional seniority scheme of Racah and Flowers

(b )
(a )
U (2(2 j  1)) ( j n )  U (2 j  1) (T )  Sp (2 j  1) (  ,t )  SO ( 3) ( J )
 Spq(4) model for a description of non-linear effects in quantum
mechanical nuclear systems: q-deformation takes into account
 additional interactions between nucleons [such as higher-order
many-body correlations] while preserving fundamental laws.
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Nuclei with Mass 32<A<100
Nuclei with Mass 32 <A<100 N=Z nuclei
82
126
protons (Z) 50
Stellar
20
rapid proton
capture path
82
28
50
20
28
XXIII DGMTP, China 2005
Microscopic
description of
pairinggoverned 0+
states in
even-A nuclei
neutrons
(N)On the Physical Significance of q
Deformation in Many-body Physics
Identical-Particle Pairing
su2
 = –1/2
n
1
†
A     ' 
2 1   , ' 
jp
j
   1
j m
†
j  j1 m   j
jp
N 
p
 =1/2
†
c j ,m , c j , m , '  ( A   )
†
{c j ',m ', ' ,c j ,m , }   j ', j m ',m  ',
j
 c
†
j ,m ,
c
j  j1 m   j

J=0 pair of identical
fermions


su(2)
Orbit j
Dimension
2=2j+1
XXIII DGMTP, China 2005
†
On the Physical Significance of q
Deformation in Many-body Physics
j ,m ,
Identical-Particle plus Proton-Neutron Pairing
sp4
 = –1/2
n
†
A     ' 
1
2 1   , ' 

j
  1
 =1/2
Isospin:
T 
1
2
j m
j  j1 m   j
jp
N 
j  j1 m   j
†
j
 1 / 2
  c
†
j ,m ,
c
c
†
j ,m ,
c
1
j ,m ,
2
1
2
T0 
2

sp(4)
Orbit j
Dimension
2=2j+1
XXIII DGMTP, China 2005
j ,m ,
N 1  N  1
Add J=0 np pairs tonn and pp pairs

†
j  j1 m   j    1 / 2
j
 
†
c j ,m , c j , m , '  ( A   )
jp
Number of particles:
p

jp
On the Physical Significance of q
Deformation in Many-body Physics
Identical-Particle plus Proton-Neutron Pairing
… and Many-body Interactions
 = –1/2
n
†
A     ' 
jp
j
   1
2 1   , ' 
 =1/2
Isospin:
T 
1
2
j m

†
j ,m ,
jp
j
j  j1 m   j
N 
Number of particles:
p

spq4
1
jp

 
†
j , m , '
 ( A  )
†
 1 / 2
c
†
j ,m ,
c
j ,m ,
j  j1 m   j    1 / 2
j
 

j  j1 m   j
†
j ,m ,
1

j ,m ,
2
T0 
1
2
N 1  N  1
2

J=0 pairs gain higher-order interactions
spq(4)


Orbit j
Dimension
2=2j+1
XXIII DGMTP, China 2005
{
jm 
,

j1 m 1
1
}
q
 
1
q

N 2
2
 jm  , j
On the Physical Significance of q
Deformation in Many-body Physics
1 m 1 1
spq(4) Algebra
Spq41 algebra
jp
†
A     ' 
   1
2 1   , ' 
spq(4)
j
j  j1 m   j
Isospin:
1
T 
2
1
sp(4)
{
,
jm 

{ jm  , 
[N  , 
†
jm 
†
j1 m 1 1
†
j ,m , '


XXIII DGMTP, China 2005
jp
} q 1  q

]   '

 
j  j1 m   j

†
j ,m ,
1
jp
j

j ,m ,
2

†
j , m , '
 ( A  )
†
 1 / 2
c
†
j ,m ,
c
j ,m ,
1
2
T0 
N 1  N  1
2
Observables remain
nondeformed
2
†
j ,m , '
†
j ,m ,
j  j1 m   j    1 / 2
j
N 2 
}  0,    1

 
N 
Number of particles:
q
j m
{
(† )
jm 
(† )
,  j1 m 1 1 }  0
[ N  ,  j ,m ' ,  ' ]      '
j, m ' ,  '
On the Physical Significance of q
Deformation in Many-body Physics
Spq(4) Model: Important Features
Spq4 model
Spq(4) model does not violate physical laws fundamental to
a nuclear system: q-deformation accounts for many-body
interactions without affecting the concepts of the standard
quantum mechanics [no space-time geometry deformation]!
Conserves:
same physics, but more of it
Angular momentum J [and its algebra], hence Hq
transforms as a scalar under 3-dimensional rotations in real
coordinate space;
Total number of particles N and third projection of isospin
T0.
Makes possible the analytical modeling of a set of manybody interactions, which are in general important yet rather
complicated to handle.
q-Deformation is a transformation to quasi-particles that
keeps their number fixed and allows them to interact via manybody forces.
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Spq(4)  SUq(2) Spq(4)
U(q)(1) SUq(2)  U(1)
Isospin symmetry
suq
T(2)
suq0(2)

T
 T0 
[T , T ]  2

 2 

 N  2 
[ A , A 0 ]  2


4  
†
0
 N 1   
†
[ A  1 , A ]   2


2 
N
2

[O

(0)
,O
( )
]  O
XXIII DGMTP, China 2005
( )
†
 , A 0 , A 0
p-p
n-n

[2]  [2]
1
2
N 1  
2
O(±) raising (lowering) operator

O(0) third projection operator

T0
p-n

Proton-neutron pairs
4
For all:
N
T 0 , T  , T
Like-particle pairs
suq±(2)

u(q)(1)
N
1
†
, A  1, A
On the Physical Significance of q
Deformation in Many-body Physics
1
Sp (4)  SU (2)  USp(4)
(1)  SU(2)  U(1)
Isospin symmetry
suT(2)
[T , T ]  2
2
 X k

q
q
q q
k
 kX
k
Proton-neutron pairs
su0(2)

†
0
[ A , A0 ]  2

N  2

4
†
1
[A ,A ]  2
N 1  
2
q 1

 X
[O

(0)
,O
( )
]  O
XXIII DGMTP, China 2005
( )
T0
p-n

N
†
 , A 0 , A 0
p-p
n-n
N 1  
2
For all:
N
T 0 , T  , T
2
Like-particle pairs
su
 ±(2)
T
T0
kX
u (1)
O(±) raising (lowering) operator

O(0) third projection operator
N
1
†
, A  1, A
On the Physical Significance of q
Deformation in Many-body Physics
1
Spq(4)  SUq(2)  Uq(1): Casimir Operators
C2
C2(suq(2))
Isospin symmetry
suq
T(2)
0
su
q (2)
T
 1 
2  T  1 T  1 1
2  2
2
T 0 , T  , T
p-n
Proton-neutron pairs
 1 2  2 n1  n 1 
2  

2 
2
 1
2  2 n 
 n 1  
1
 1

2

 1
2
2
1   n 0    n 0


1
1 
1
  2


2
   
 

n+1(–1): number of proton (neutron) pairs;
 of pn pairs.
n0: number
XXIII DGMTP, China 2005


2
†
 , A 0 , A 0
p-p
n-n
Like-particle pairs
±
su
q (2)
N
[2]  [2]

N 1  
1
2
2
†
, A  1, A
4
On the Physical Significance of q
Deformation in Many-body Physics
1
Diagonal Second-order Operator O2(spO2
q (4))
of spq(4)
O2(spq (4)) =

k  0, 1,T
k
C 2 ( suq
k(2)


2
2 
N  1    
)  1 2  N 1   
   
 

1 
1  2

2    2


 
[2]  [2]
 1   1;  T  1 .

2
4



N+1(–1): number of protons (neutrons)
XXIII DGMTP, China 2005
1
On the Physical Significance of q
Deformation in Many-body Physics
Diagonal Second-order Operator
2(spq (4))
O2 ofOspq(4),
gammas
2
1 
 0
 
q
1
q

1
2
2 X k
2
n+1(–1): number of proton (neutron) pairs;
n0: number of pn pairs.

XXIII DGMTP,
China 2005

2 X k
 X k
q
kX
q
 kX
On the Physical Significance of q
Deformation in Many-body Physics
Diagonal Second-order Operator O2(spO2
q (4))
ofspq(4)
O2(spq (4)) =

k  0, 1,T
k
C 2 ( suq
k(2)


2
2 
N  1    
)  1 2  N 1   
   
 

1 
1  2

2    2


 
 1   1;  T  1 .


[2]  [2]

4
•Diagonal in the q-deformed basis set
•Reduces to the Casimir invariant of sp(4) in the
nondeformed limit

•Zeroth-order approximation of O2 commutes
with all the
q-deformed generators
•Gives direct relation between the expectation values of
the second-order products of the operators that build O2
•Result can be used to provide for an exact solution of a qdeformed model Hamiltonian
N+1(–1): number of protons (neutrons)
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
1
2
The Model(s)
Sp(4) Dynamical Symmetry
Hcl
H  eNˆ

†
†
†
G A A  A A
 A A
0 0
1 1
1 1


E
2
T

2 
 FA
†
A
0 0

Nˆ

2

 C

2
Nˆ
4




†
n
n0
†
( A1 )
n 1
0
basis states

Isovector (isospin 1)
J=0 pairing interaction

 Nˆ ( Nˆ  1)

4  
2
E
†
( A1 ) 1 ( A 0 )
Diagonal isoscalar
(isospin 0) pn force
Symmetry term

 2
D T
 0

 Symmetry
 breaking
4 
Nˆ
Describe pairing-governed isobaric analog 0+ states (IAS)
Include ground states for all even-even and only some (N~Z)
odd-odd nuclei
On the Physical Significance of q
Deformation in Many-body Physics
XXIII DGMTP, China 2005
The Model(s)
Hq eq Nˆ
G

Spq(4) Dynamical Symmetry
Hq
†
†
†
A A  A A
 A A
0
0
1

1
1 1
q

q  1

Eq
2
 2
T

†
F A A 
q 0 0



2 
 N



Hcl
1 N  N
2 C q

 

2 
 1 
2
2
XXIII DGMTP, China 2005


2

 1
2

2
1 
Dq
T  1



  0

2
On the Physical Significance of q
Deformation in Many-body Physics
The Model(s)
Hq eq Nˆ
G


Spq(4) Dynamical Symmetry
Hq
†
†
†
A A  A A
 A A
0
0
1

1
1 1
q
Eq
2
 2
T

†
F A A 
q 0 0




2 
has the Spq(4) SUq(2)
dynamical symmetry
contains the original
q  1symmetry
Sp(4) dynamical
 N



1 N  N
2 C q

 

2 
 1 
2
2
= Hcl + 3-b
+4-b+5-b…
XXIII DGMTP, China 2005


2

 1
2

2
1 
Dq
T  1



  0

2
{eq,Gq,Eq,Fq,Cq,Dq}= {e,G,E,F,C,D}
On the Physical Significance of q
Deformation in Many-body Physics
If:
From a Nondeformed Perspective
Hq=Hcl+m-b
H  eNˆ

†
†
†
G A A  A A
 A A
0 0
1 1
1 1


2
T

2 
E
 FA
Sp(q)(4) Dynamical Symmetry
†
A
0 0

Nˆ

2

 C

2
Nˆ
4

q=1
oneand
twobody
terms



 Nˆ ( Nˆ  1)

4  
2
E
 many-body terms
XXIII DGMTP, China 2005

X  

 2
D T
 0
q q
X
qq



4 
Nˆ
X
1
X 1
2
 X (1  
q=e
2

6
On the Physical Significance of q
Deformation in Many-body Physics
)
From a Nondeformed Perspective
 2C
C
N

2  2
N

2
96 
{
2
(3)

 c 1 c 2 c 3 c 3 c 2 c 1
†
 1 2
†
†
N(N-1)

16 
 24   5 V
2
(2)
†
F A A 
q 0 0
Cq = C
1 N  N
2 C q

 

2 
 1 
2
XXIII DGMTP, China 2005
 2 
(1)
 6  8 V
V
(4 )


V
(3)
(2)
V

(4 )
}  ...
 c 1 c 2 c 3 c 4 c 4 c 3 c 2 c 1
†
 1 2
3

q=e

2 

 6V
V
Illustrative Example

2

 1
2
†
†
†
3

2
1 
Dq
T  1



  0

2
On the Physical Significance of q
Deformation in Many-body Physics
From a Nondeformed Perspective
 2C
C

2
96 
{
2

2 

16 
2
 6V
V
(3)

 c 1 c 2 c 3 c 3 c 2 c 1
†
 1 2
†
 24   5 V
(2)

(1)
 6  8 V
†
3
q=e
N(N-1)
N

2  2
N

Illustrative Example
V
may not
be
negligible
(4 )

V
(3)
(2)
V

(4 )
}  ...
 c 1 c 2 c 3 c 4 c 4 c 3 c 2 c 1
†
 1 2
†
†
†
3
E.g.,the energy contribution of the four-body
interaction can reach a magnitude of several
MeV in nuclei in the upper fp +1g9/2 shell.
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Parameters
Parameters
16S16
20Ca20 28Ni28
2
pp+nn+pn G/ 0.702
pairing
F/ 0.007
N ( N  1) C
0.815
2
T0 D
0.127
symmetry
E/2 -1.409
energy
9.01
e
s.p. energy S
1.72

min
0.496

4
11
0.453 0.296
0.072 0.056
0.473 0.190
0.149 -0.307
-1.120 -0.489

9.36 9.57
16.1 300.3
0.732 1.79

ND total number of nuclei;
np number of fit parameters
ND
 E 0 ,i
Sp ( 4 )
 
S min
ND  np

 E 0,i
exp
i 1
ND  np
We fix the values of the parameters so that Hcl remains
unchanged when deformation is introduced – this is because
Hcl reproduces reasonably well the overall behavior common
for all the nuclei in a shell such as…
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics

2
Lowest Isobaric Analog 0+ State (Binding) Energy
Agree well with
experiment ()
BE
400
E0 (MeV)
40Ca
EC0 (MeV)
A=100
300
Semiempirical
estimate(*)
200
100
1f7/2
T0
Coulomb correction
J. Retamosa, E. Caurier, F. Nowacki and A.
Poves, Phys. Rev. C 55, 1266 (1997).
Energy spectra of pairinggoverned 0+ IAS of 319 nuclei
and only 6 parameters
56Ni
-10
-5
5
10
A=56
-100
1f5/2 2p1/2 2p3/2 1g9/2
A=78
(*)P. Moller, J. R. Nix and K.-L. Kratz, At. Data Nucl. Data Tables 66, 131 (1997).
XXIII DGMTP, China 2005
On -200
the Physical Significance of q
Deformation in Many-body Physics
T0
Energy Spectra of Isobaric Analog 0+ States
1d3/2
18Ar18
Without varying
parameters
32
S
core
30
25
20
15
10
5
15
10
5
th exp
22
22
23
23
24Cr22
E0 (MeV)
Very good
agreement with
experiment
Higher
0
Ti
V
1f7/2
22Ti24
23V25
24Cr24
40Ca
core
22Ti26
24Cr26
30
25
20
15
10
5
30
25
20
15
10
5
26Fe22
th exp
XXIII DGMTP, China 2005
25Mn23
th
26Fe24
th exp
25Mn25
th
26Fe26
th exp
On the Physical Significance of q
Deformation in Many-body Physics
Detailed Structure: Two-Proton Drip Line
2p-DripLine
S2p =E0(Z)–E0(Z–2) (MeV)
N=Z
30
neutron number
20
10
0
56Ni
-10
core
-20
1f5/2 2p1/2 2p3/2 1g9/2
33
38
43
Zr36
Sr34
Kr32
Se30
Ge28
Ga29
Br33
As31
XXIII DGMTP, China 2005
Rb35
48
Y37
Z
Zr38
Comparison of
Sp(4) model to
other models:
50
Cd46
Pd44
28
Ru42
Mo40
Nb41
Y39
Ag47
Rh45
Tc43
1
nd
nd
 S 2 p ,i  S 2 p ,i  ,MeV
Sp ( 4 )
#
2
i 1
#E. Ormand, Phys. Rev. C 55, 2407 (1997).
#P. Moller, J. R. Nix and K.-L. Kratz, At. Data
Nucl. Data Tables 66, 131 (1997).
#B. A. Brown, 
R. R. C. Clement, H. Schatz and
A. Volya, Phys. Rev. C 65, 045802 (2002).
0.32
0.78
0.43
On the Physical Significance of q
Deformation in Many-body Physics
Detailed Structure: N=Z Irregularities
56Ni
core
Interaction between
FiniteEnergyDifference
E
E
the last proton and the
Non-pairing likeparticle
interaction
2
0
Z2
0
ZN
last neutron
MeV
MeV
Z
XXIII DGMTP, China 2005
2
N
On the Physical Significance of q
Deformation in Many-body Physics
Detailed Structure: Staggering Behavior
40Ca
1f7/2
FED: Ca
core
E0
n
n: valence number of particles
XXIII DGMTP, China 2005
Agree well with
experiment
2E0
 2n
On the Physical Significance of q
Deformation in Many-body Physics
Detailed Structure: Staggering Behavior 1f5/22p1/22p3/21g9/2
56Ni
core
20 i =-6
16 i =-5
12
8
20 i =-4
i =-3
16
12
8
20 ii =-2
=-1
16
12
8
60
E0/n (MeV)
theory
experiment
FED: Ni
A=64
A=66
A=76
A=78
A=88
A=90
A
70
80
90
-10
-5
0
T0
5
10
2E0/nT0 (MeV)
10
5
0
-5
-10
A
60
70
80
XXIII DGMTP, China 2005
90
100 -10
T0
-5On the Physical
0
5
10of q
Significance
Deformation in Many-body Physics
~
Isovector Pairing Gap, 
Significant pp,nn and pn Interplay
40Ca
Pairing gaps I
(MeV)
core
symmetry
term
pairing
T0
T0
T0
Z1{E Z1,N1+ E Z1,N1 – 2E Z,N} (MeV)
˜      2

pp
nn
pn
odd-odd, ZN
CORE
Z1,N
1
XXIII DGMTP, China 2005
+
CORE
CORE
Z1,N – 2 Z1,N
On the Physical Significance of q
1Deformation
1 Physics
in Many-body
Pairing Gaps
40Ca
1f7/2
Pairing gaps II
(MeV)
core
Like-particle
pairing gap
56Ni
core
XXIII DGMTP, China 2005
1f5/22p1/22p3/21g9/2
On the Physical Significance of q
Deformation in Many-body Physics
From a Global Scale to a Local Scale
Global to local
QuickTime™ and a
DV/DVCPRO - NTSC decompressor
are needed to see this picture.
http://svs.gsfc.nasa.gov/
XXIII DGMTP, China 2005
The new feature is
an extension of the
theory to
include
nonlinear
local
deviations
from the
pairing
solution as
realized
through qdeformation of the
sp(4) algebra.
On the Physical Significance of q
Deformation in Many-body Physics
Physical Significance of q
The q-deformation adds
to the theory, which
describes quite well the
overall nuclear
behavior, a mean-field
correction along with
2-, 3-, and many-body
interactions of a local
character that can be
responsible for residual
single-particle and
many-body effects.
q: Physical
significance
Local non-linearity
Spq(4)
Sp(4)
Global behavior
H=
Hcl
Many-body terms
two-body interaction (q = 1)
quantum two-body interaction (q ≠ 1)
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Physical Significance of q
Interaction strength parameters
q: Physical
significance
R
in H remain fixed;
R1
cl
q-deformation does not influence the non-deformed two-body
interaction
H=
Hcl
many-body terms
two-body interaction (q = 1)
quantum two-body interaction (q ≠ 1)
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Physical Significance of q
‘q’
Interaction strength parameters
q: Physical
significance
R
in H remain fixed;
cl
q-deformation does not influence the non-deformed two-body
interaction
H=
Hcl
many-body terms
two-body interaction (q = 1)
quantum two-body interaction (q ≠ 1)
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Physical Significance of q
‘q’
Interaction strength parameters
q: Physical
significance
R
in H remain fixed;
cl
q can vary within different
nuclei to reflect experimental
energies.
The fundamental properties of q
cannot be “mocked up” by allowing the strengths of the
nondeformed interaction to absorb its effect.
H=
Hcl
many-body terms
two-body interaction (q = 1)
quantum two-body interaction (q ≠ 1)
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Solutions for q (Near-closed Shell Nuclei)
40Ca
core
Solutions
for
q
E –E
(MeV)
q
0
0,exp
48
22Ti
5
4
Near-closed
Shell Nuclei
3
52
27Co
2
1
-2
-1
-1
E0 nondeformed
energy predicted by
Sp(4), q=1
XXIII DGMTP, China 2005
1
2
q=e
Solutions for the
deformation
parameter:
detects possible
presence of local
effects

Deviation within an
individual nucleus of E0 from
the experimental value E0,exp
On the Physical Significance of q
Deformation in Many-body Physics
Solutions for q (Midshell Nuclei)
40Ca
core
q=e
Solutions
for
q
E –E
(MeV)
q
0
Midshell
Nuclei
0,exp
48
22Ti
5
4
3
52
27Co
2
1
-2
-1
-1
E0 nondeformed
energy predicted by
Sp(4), q=1
XXIII DGMTP, China 2005
1
2

No solution: theoretical
prediction closest to the
experiment occurs at the
nondeformed point, q=1.
On the Physical Significance of q
Deformation in Many-body Physics
Smooth Dependence of q on Nuclear Characteristics
q=e
Smooth

Ni of q
Smooth
behavior
behavior
56
q-deformation
as prescribed
by the Spq(4)
model is not
random in
character but
rather
fundamentally
related to the
very nature of
the nuclear
interaction.
core
1.0
0.8
Zn
0.6
0.4
0.2
Peaks at N=Z [with
significant values of q]
1.0
0.8
Ge
0.6
0.4
0.2
Many-body nature of
the interaction is most
important away from
mid shell
1.0
Se
0.8
0.6
0.4
0.2
5
XXIII DGMTP, China 2005
10
15
20
N–1
On the Physical Significance of q
Deformation in Many-body Physics
Functional Dependence of q on Model Quantum Numberss q=e

1.0
0.8
Based on theDependence
discrete solutions found
q: Functional
within each nucleus, q is assigned a
functional
dependence on N and T0:
Zn
0.6
0.4
0.2
 (N, T0)=
1.0
0.8
2
 N
 N

  1 
 1
  2  2 ( N  2 ) e
Ge
2
2

0.6
0.4
  4  ( N  2 ) T0
0.2
1.0
N
1,
 ( x )  
0,
0.4

0.2
5
XXIII DGMTP, China 2005
10
15
1
2
Se
0.8
0.6
1  T 0 
 

2  3 / 2 
20
x0
x0
N–1
On the Physical Significance of q

Deformation
in Many-body Physics
Functional Dependence of q on Model Quantum Numberss q=e

1.0
0.8
q: Functional
Dependence
 = –2.13,  = 0.37,  = 3.07,  = 0.15
1
2
3
4
Zn
0.6
0.4
0.2
f5/2pg9/2
 (N, T0)=
Fit to even-even
1.0
nuclei in 1f7/2 and
0.8
1f5/22p1/22p3/21g0.69/2
2
 N
 N

  1 
 1
  2  2 ( N  2 ) e
Ge
2
2

0.4
  4  ( N  2 ) T0
0.2
1.0
N
1f5/22p1/2
2p3/21g9/2
0.4

0.2
5
XXIII DGMTP, China 2005
10
15
1
2
Se
0.8
0.6
1  T 0 
 

2  3 / 2 
20
N–1
q=1
q≠1
SOS 271.63 130.21
1.79
1.28

On the Physical Significance of q
Deformation in Many-body Physics
Comparison to Nondeformed Energies
q: uniformly
superior
1f
E –E
(MeV)
(q)
0
0,exp
q=e
40Ca
core
7/2
One reason
may be that
the qdeformed
fermions,
unlike usual
quasi1f5/22p1/2
particles,
The model with the local q improves 2 1g
q≠1
indeed obey the energy prediction compared to the 3/2 9/2 q=1
SOS 271.63 130.21
nondeformed global model and
the
1.79
1.28

fundamental reproduces more closely the
experiment.
laws.
On the Physical Significance of q
Deformation in Many-body Physics
XXIII DGMTP, China 2005
q-Parameter and Many-body Interactions
q=e
q in mid-mass nuclei
Determined statistically,
1, 2, 3, and 4 provide
an estimate for the
overall significance of
q-deformation within a
shell.
1
0.75
0.5
0.25
0
0
 (N, T0)
56Ni
T0
core
20
15
5
10
N+1
XXIII DGMTP, China 2005
N
N–1
15
10
5
N–1
20 0
On the Physical Significance of q
Deformation in Many-body Physics
q-Parameter and Many-body Interactions
I.
q=e
q in mid-mass nuclei
Significant many-body
interactions are detected
away from mid-shell
and tend to peak at
even-even N=Z nuclei
where strong pairing
correlations
are expected. 1
0.75
0.5
The pair formation favors the 0.25
0
non-negligible higher-order
0
interactions between the pair
constituents that are detected
via the Spq(4) model.
XXIII DGMTP, China 2005
 (N, T0)
N
56Ni
T0
core
20
15
5
10
N+1
N–1
15
10
5
N–1
20 0
On the Physical Significance of q
Deformation in Many-body Physics
q-Parameter and Many-body Interactions
II.
q in mid-mass nuclei
q=e
Around mid-shell the
deformation adds little
improvement to the q=1  (N, T ) N
0
56Ni
theory. For these nuclei
core
T0
the many-body
interactions as
prescribed
by Spq(4)
1
0.75
are
20
0.5
negligible
0.25
15
and the model 0
10
is not sufficient 0
N–1
5
to describe other
10
5 N–1
15
types of local effects
N+1
20 0
that may be present.
On the Physical Significance of q
Deformation in Many-body Physics
XXIII DGMTP, China 2005
Conclusion
Conclusion I
Microscopic description
of pairing-governed 0+
states in nuclei
include 0+ ground states
for all even-even and
some odd-odd nuclei
with nuclear masses
with protons and
32<A<164
neutrons occupying
the same major shell even-A: even-even
odd-odd
• two-body interactions
• many-body interactions
• common nuclear properties
• local non-linearity
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Conclusion
• Obeys standard quantum
mechanics
• Conserves angular
momentum J and number
of particles N
• two-body interactions
• many-body interactions
• common nuclear properties
• local non-linearity
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Conclusion
 Reasonable prediction of ground and excited pairing-governed
isobaric analog 0+ state energies
 Reproduction of global trends and smaller fine features
in nuclear dynamics
• Two-proton drip line
• N=Z irregularities
• Pairing gaps
• Staggering behavior
 Isobaric analog 0+ states possess a simple Sp(4) dynamical
symmetry
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics
Conclusion
 Introduction of higher-order many-body interactions: q
 Physical significance of q-deformation: in the very
nature of nuclear interaction
• Decoupling of q-deformation from the two-body
interaction: strengths of the nondeformed interaction
cannot absorb the effect of the deformation.
• Local non-linear effects (within individual nucleus)
• Uniformly superior q≠1 results to those in the
nondeformed limit
• Smooth dependence on nuclear characteristics
• Non-negligible higher-order many-body interactions
(q) in regions of dominant pairing correlations
XXIII DGMTP, China 2005
On the Physical Significance of q
Deformation in Many-body Physics