### Spectroscopy at the Particle Threshold

```Spectroscopy at the Particle
Threshold
H. Lenske
1
Agenda:
• Pairing in the continuum
• Nuclear Polarizability and Spectral Functions
• Continuum spectroscopy and Fano-Resonances
• Summary
2
Pairing in the Continuum:
Quasiparticle Resonances
3
Extended HFB Theory as Coupled Channels
Problem: The Gorkov-Equations
   
  
H 







E






 ( H   )   
 
  
 ~ u(jq) (r) | (s) jm ;  ~ v(jq) (r) | (s) jm
Mean-Field Hamiltonian (q  p,n):
Pairing- Field & Density (q  p, n) :
2
 q  1 VSE q
H 
 2  U()
2m
2 j  1 (q)
q (r)  
| v n j (r) |2
4
n j
2
2 j  1 (q)
( q )*
 q (r )  
u n j ( r ) v n j ( r )
4
n j
Spectrum of the Gorkov Equation:
Extended HFB Theory: Pairing Self-Energies
• Energy Shifts and Widths
• Spectral Functions for particles and holes
Pairing in Infinite Nuclear Matter
(k F )  
d 3q
 2 
0
V
(
k
,
q
)
u(q;
k
)
v(q;
k
)
~
N(k
)
V
(k F , k F ) ( k F ) I  k F ,  
SE
F
F
F
F
SE
3
G (k F )  N(k F )VSE0 (k F , k F )  I  k F ,   
(k F ) ~ e F (k F ) e
1
G (k F )
3(1G ( k F ))
G ( kF )
7
Pairing in Infinite Nuclear Matter
Free Space SE (S=0,T=1) Interaction:
(Bonn-B Potential)
Pairing is a LOW DENSITY Phenomenon
Pairing Gap  and Anomal Density 
Pairing Correlations in Nuclear Matter
in Symmetric Nuclear Matter
Pairing Gap 
Anomal Density 
Pairing-Field in a Nucleus
RA
RA
11Li
: Continuum HFB Spectral Functions
Neutron Spectrum :
 Dissolution of Shell Structures!
11Li
: Continuum
HFB g.s. Densities
g.s. Densities
g.s. Densities  r2 :
for q  p, n : (q)  v(jq) (r) | (s) jm 
 q (r )  
j
q
2 j 1
(q)
2
de
|
v
(
e
,
r
)
|
j
4 
Neutron Spectral Functions in 9Li(3/2-):
Pairing in the Continuum
S. Orrigo, H.L., PLB 677 (2009)
14
Pairing Resonances in Dripline Nuclei
9Li+n
10Li
S. Orrigo, H.L., PLB 677 (2009) & ISOLDE newsletter Spring 2010, p.5 15
Continuum Spectroscopy at REX-ISOLDE:
d(9Li,10Li)[email protected]
10Li=9Li+n
S. Orrigo, H.L., PLB 677 (2009) & ISOLDE newsletter Spring 2010, p.5
Data: H. Jeppesen et al., REX-ISOLDE Collaboration, NPA 738 (2004) 511 & NPA 748 (2005) 374.
16
New experimental results (Dec. 2013):
10Li continuum spectroscopy at TRIUMF




1 3 3 5 
 , 
2 2 2 2 
S. Orrigo, M. Cavallo, F. Capppuzzello et al.
17
Spectral Structures by
Dynamical Polarization
18
Beyond the Mean-Field:
Short-range Correlations in Nuclear Matter
Momentum Distribution
n(p) = N(kF)  a(w, p) dw
PLB483 (2000) 324
NPA723 (2003) 544
NPA (2005)in print
Nuclear Dynamics…
20
QRPA Response in
10Be
Eth [MeV]
Eexp [MeV]
2+
3.220
3.368
1-
6.423
5.960
0+
6.513
6.179
2-
6.446
6.263
3-
7.372
7.371
J c
4-
(9.270)
1+
7.122
3+
7.159
0-
7.374
DCP Neutron Spectral Distributions in
[0+ × 1/2+]: 0.79
[2+ × 5/2+]: 0.18
[0+ × 1/2-]: 0.58
[2+ × 3/2-]: 0.28
11Be
Spectral Distributions in Carbon Isotopes
…normalized to sum rule
E1 Dipole
Polarizability of C-Isotopes:
HFB+QRPA results
Multipole polarizabilties
coefficients by sum rules:
S n ( )   E
n
a
2
a T 0
S 1 ( )
P 
S 0 ( )
Longitudinal Momentum Distributions: 17,19C → 16,18C + n
Carbon Target, Elab 900 AMeV
•Binding: Correlation Dynamics
17C
•17C(5/2+,g.s.)
• Sn(the.)=715keV
• C2S(g.s.) = 0.41
G(the.): 132 MeV/c
G(exp.): 143 ± 5 MeV/c
s(-1n,the.): 124 mb
s(-1n,exp.): 129± 22 mb
•Binding: Correlation Dynamics
•19C(1/2+,g.s.)
19C
• Sn(the.)=263keV
• C2S(g.s.) = 0.40
G(the.): 69 MeV/c
G(exp.): 68 ± 3 MeV/c
s(-1n,the.): 192 mb
s(-1n,exp.): 233± 51 mb
Dynamical Core Polarization:
•HFB g.s.:
„3-body renormalized“ GMatrix
• ph-Interactions:
Fermi Liquid Theory
Hole
Spectrum
Particle
Spectrum
Fano Resonances
Interactions of Closed and Open
Channels:
Fano Resonances
27
The Spectral Situation encountered in Atoms,
• A closed channel E* is embedded into a continuum of open channels
• E* interacts via V(r) with open channels given by scattering states
• E* Interacts via V(r) with closed channels, e.g. of (simple) bound states
 Bound
State Embedded into the Continuum - BSEC
28
Examples:
•
Atoms: self-ionizing states of multi-electron configuration
•
Nuclei: Multi-particle-hole states above threshold
•
Mesons: Confined qq-configurations embedded into the
continuum of meson-meson scattering states, e.g. (1232),
(770), Y‘‘(3770)…
•
Baryons: Confined qqq-configurations embedded into the
continuum of meson-nucleon scattering states, e.g. (1232),
N*(1440), L(1405)…
29
Visualizing Quantum Interference in Microscopic Systems:
Asymmetric Fano-Line Shapes of Resonances
Y E   aii( d )    d  bk k(c)
i
k
30
Historically:
The famous Silverman-Lassettre data
He(e,e‘)He*(1P) @ 500eV
Note: q must be negative – q=-1.84
31
Fano-Resonances in Nuclei
32
Hamiltonian and Wave function
 H11b

H  0
 V31

0
H 22s
V32
V13 

V23 
H 33x 
The coupled equations (core nucleus integrated out):

b
n
   zn   nJ V13 j ' c z j ' c  0 

c
  s.p. motion w.r.t. the g.s.
s
 j    z   nJ V23 j ' c z j 'c  0
c


j'

 (  E JC ) z j ' c   j ' c V31 n zn   d  ' j ' c V32  ' z '  0
n
Multi-channel Fano wave function:
33
Extension to Several Open Channels
• n=2 open channels
• n=2 energetically degenerate solutions with outgoing flux
G1   VE
2
; G 2   WE  G   G i
2
i
   arctan
G(E )  
P

G( E )
E  E  G ( E )
 dE '
G( E ')
E ' E
34
Solution 1: fully mixed
1
P
z( E , E ') 
sin    ( E  E ') cos 
 EE'
G1
G2
sin 
a1 
; b1E ' 
z ( E , E ') ; c1E ' 
z ( E , E ')
G
G
G
Solution 2: continuum mixed
a2  0 ; b2 E '
G1
G2

 ( E  E ') ; c 2 E '  
 ( E  E ')
G
G
„Dark States“
 Resonance superimposed on a smoothly varying
background!
35
Multi-channel Coupling
36
Resonance Scenarios in Nuclear Physics
The Fano-Wave Function:
37
Reaction Matrix Elements and Formation
Cross Section
M    Y E , 
  | T |  

1
| T |   
 | T |   sin  
 cot  
*  E ,


 VE

|
T
|



E
,




The (single channel) Fano-Formula:
| q  cot  |
s  =s 
~ M 
2
1  cot 
 | T |  

q=
 E , | T |  
2
s
1
s  ~
 | T |  
*  E ,
 VE
2
2
s
38
Correlation Dynamics in an Open Quantum System:
d-wave Fano-Resonances in 15C
G~60…140keV
Sonja Orrigo, H.L., Phys.Lett. B633 (2006)
39
DD-Dynamics at Threshold
Channel Coupling and the Line Shape of Y(3770)
D  D  threshold
3739.2 MeV
D0 D0 threshold
3729.7 MeV
q=-2.1±0.6
X(3900) ??
3.65
 (2s / 3686)
Xu Cao, H. L., PRL, submitted
40
Summary
•
Dynamics at the particle threshold
•
Pairing at the dripline/in the continuum
•
Nuclear polarizabilities
•
Fano resonances in atomic nuclei
•
Tools for continuum spectroscopy
•
Universality of quantum interference
…with contributions by
Sonja Orrigo (Valencia) and Xu Cao (Giessen/Lanzhou)
41
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