Sum rule and Giant Resonances
Sum rule and giant resonances
Giant resonances are typical collective modes of excitation at high
energy, and exhaust major portion of the sum rule
Isoscalar (T=0)
Monopole (0+)
Quadrupole (2+)
Octupole (3−)
Isovector (T=1)
Monopole (0+)
Dipole (1 −)
Quadrupole (2+)
Gamow-Teller (1+)
Spin-monopole (1+)
Spin-dipole (0 −, 1 −, 2 −)


 f λ (r )Yλ µ




 τ i f λ (r )Yλ µ



 τ iσ j f 0 ( r )


τ i f 0 ( r )( σ × Y1 ) Iµ
Sum rule
m1 =
∑
En n F 0
∑
Enp n F 0
2
⇒
(Energy - weighted) Sum Rule
n
mp =
2
n
Odd-p moments can be expressed by the ground-state expectation value.
m1 =
∑
n
mp =
∑
n
If Fˆ =
∑
1
En n F 0 =
0 [ F , [ H , F ]] 0
2
2
1
p
En n F 0 =
0 [[ F , H ], [ H , [ H , F ]]] 0
2
2

f (ri ) and the Hamiltonian does not have momentum dependence,
i
A
1
m1 =
0 ∑ (∇ f ) i2 0
2m i = 1
2
1 2  ∂ 2
m3 =  
E (η )
2
2 m ∂η
η =0
, E (η ) = 0 e
−η G
ηG
He
m
0 , G = − [H , F ]
2
Physical meaning of sum rule
Nucles under an impulse external field
A

V (t ) = Fδ (t ) = δ (t )∑ f ( ri )
i= 1


Nucleons i change its momentum by ∆ pi = − ∇ i f (ri )
The nucleon at t<0 has zero velocity expectation value, then this field

creates the velocity field:  
∇ f (r )
v (r ) = −
Energy transferred to nucleon i is
m
∆εi =
 2
∇ i f ( ri )
2m
Thus, the energy weighted sum rule has the following physical meaning.
∑
En n F 0
n
2
A
1
=
0 ∑ (∇ f ) i2 0
2m i = 1
The energy absorbed by
nucleus:
(Exc. energy) x (probability)
The energy transferred to nucleons
in the nucleus
Meaing of m3
2
1 2  ∂ 2
m3 =  
E (η )
2
2 m ∂η
η =0
E (η ) = 0 e
−η G
ηG
He
m
0 , G = − [H , F ]
2

If f (r ) = r λ Yλ µ (rˆ) and the Hamiltonian does not have momentum
dependence,
G= −
ηG
The state e
(
)
m
1
[ H , F ] = ∑ ∇ r λ Yλ µ (rˆ) i ⋅ ∇
2
2 i
0
i
 
λ
introduces the velocity field v (r ) ~ ∇ r Yλ µ (rˆ)
Its energy curvature with respect to the “deformation” is directly
related to m3.
E1 sum rule
“E1 recoil charge”
Isovector dipole operator
A
Fˆ = e∑ (τ z ) i ( zi − Rz ) =
i= 1
N
∑
i= 1
Ze
zi −
A
N+ Z
Ne
zi
∑
i= N + 1 A
Assuming the ground state has I=0,
NZ
m1 =
(1 + κ )
2 Am
The interaction usually contains the isospin-dependent terms.
κ > 0
This includes effects of meson exchange current.
κ = 0 ⇒ Thomas-Reiche-Kuhn (TRK) sum rule
Isoscalar giant resonance
Fˆ =
∑

f ( r ) = r λ Yλ 0 (rˆ)

f (ri )
i
The previous formulae lead to
2
2
A
1 λ (2λ + 1)
1
2
∂


m1 =
0 ∑ ri 2 λ − 2 0 , m3 =  
E (η )
2
2m
4π
2 m ∂η
i= 1
η =0
E (η ) = 0 e
−η G
ηG
He
(
1 A
0 , G = ∑ (∇ r λ Yλ 0 ) ⋅ ∇
2 i= 1
)
i
For instance, for the quadrupole operator,
2
1 2  5
m3 =  
⋅8 T
2  m  16π
<T >: Kinetic energy expectation value
Giant quadrupole resonance energy
ω
2
2+
4T
m3
~
=
≈ 2ω 0 ,
2
m1 mA r
( T
≈ mω 02 A r 2
2
)
Fermi Liquid Properties
1 ∂2
m3 ∝
E (η ) ∝ T
2
2 ∂η
η =0
The density change of the ISGQR is a surface type, however, the
restoring force for ISGQR originates from the kinetic energy.
pz

P( p)
pz
py
px
py
px
The vibration leads to a deformation in the momentum distribution
This is different from low-lying surface vibrations and different
from the classical (incompressible) liquid model.
Four-current sum rule
∑
n
Suzuki, Rowe, NPA286 (1977) 307.
Suzuki, Prog. Theor. Phys. 64 (1980) 1627.
 


 
i
0 j (r ) n n ρ (r ' ) 0 = −
ρ 0 (r )∇ δ (r − r ' )
2m
 

∇
⋅
j
(
r
)
=
i
[
ρ
(
r
), H ]
Using the continuity equation

 
and a property of the density operator ∫ ρ (r ) f (r )dr =
A
∑
i= 1
f (ri )
we can obtain the energy-weighted sum rule for the density operator
∑
n



1
En 0 ρ ( r ) n n F 0 = −
∇ ⋅ ρ 0 (r )∇ f (r )
2m
The normal m1 sum rule can be easily derived from this formula.
Taking the photoexcitation operator
∑
n

f ( r ) = r λ Yλ 0 (rˆ)

λ λ − 1 dρ 0
En 0 ρ ( r ) n n F 0 = −
r
Yλ µ (rˆ)
2m
dr

dρ
0 ρ (r ) n ~ r λ − 1 0 Yλ µ (rˆ) Transition density of the Tassie
dr
model for the giant resonance
Time-Dependent Density
Functional Theory (TDDFT)
Basic ideas of the unified
(collective) model
• Nucleons are independently moving in a
potential that slowly changes.
– Collective motion induces oscillation/rotation of the
potential.
– The fluctuation of the potential changes the nucleonic
single-particle motion.
Consistent with the idea of
Time-Dependent Mean-Field Theory
or
Time-Dependent Density-Functional Theory
Time-dependent density-functional
theory (TDDFT)
• Basic theorem of DFT (Hohenberg-Kohn)
• Basic theorem of TDDFT (Runge-Gross)
• Perturbative regime: Linear response and
random-phase approximation
– Matrix formulation
– Green’s function method
– Real-time method
– Finite amplitude method
• Non-perturbative regime
– Theories of large-amplitude collective motion
Density Functional Theory
• Quantum Mechanics
– Many-body wave functions;


Ψ (r1 , , rN )
• Density Functional Theory
– Density clouds;

F [ ρ (r )]
The many-particle system can be described by a functional of
density distribution in the three-dimensional space.
Hohenberg-Kohn Theorem (1)
Hohenberg & Kohn (1964)
The first theorem
Density ρ(r) determines v(r) ,
except for arbitrary choice of zero point.

A system with a one-body potential v ( r )
Hv = H +
∑

v( ri )
HvΨ
v
gs
= E gsv Ψ
v
gs
i
=
∑
i
2
pi
+
2m
∑
i< j
 
w(ri , r j ) +
∑

v( ri )
i


gs
(
)
v
r
↔
Ψ
↔
ρ
(r
Existence of one-to-one mapping：　v v )
Strictly speaking, one-to-one or one-to-none
v-representative

① v( r ) ↔ Ψ
v
gs
Here , we assume the non-degenerate g.s.
Reductio ad absurdum: Assuming


Ψ gsv
different　and　produces
v( r )
v' ( r )
the same ground state　
v
v
v
( H + V )Ψ gs = E gs Ψ gs
V = ∑ v( ri )
i
− ) ( H + V ' )Ψ
v
gs
= E Ψ
v'
gs
V '=
v
gs
∑

v' ( ri )
i
(V − V ' )Ψ
v
gs
= ( E gsv − E gsv ' )Ψ
v
gs
V and V’ are identical except for constant. → Contradiction
② Ψ
v
gs
↔ ρv
Again, reductio ad absurdum
v
assuming different states Ψ gs , Ψ
ρv
same density
E gsv = Ψ
v
gs
< Ψ
v'
gs
H+V Ψ
v'
gs
( H + V )Ψ
v
gs
= E gsv Ψ
v
gs
( H + V ' )Ψ
v'
gs
= E gsv ' Ψ
v'
gs


(
)
(
v
r
,
v
'
r
with ) produces the
v
gs
H + V Ψ gsv '
 


v
v'
E gs < E gs + ∫ dr [ v( r ) − v' ( r ) ] ρ v ( r )
H + V = H + V '+ (V − V ' )
 

v
v
Ψ gs V Ψ gs = ∫ dr v( r ) ρ v ( r )
Replacing V ↔ V’
 


E gsv ' < E gsv + ∫ dr [ v' ( r ) − v( r ) ] ρ v ( r )
∴ E gsv + E gsv ' < E gsv + E gsv ' Contradiction !
ρ v v-representative.
Here, we assume that the density　is
For degenerate case, we can prove one-to-one


v( r ) ↔ ρ v (r )
Hohenberg-Kohn Theorem (2)
The second theorem
There is a energy density functional and the variational principle
determines energy and density of the ground state.
Any physical quantity must be a functional of density.

v( r ) ↔ Ψ
↔ ρv

Many-body wave function Ψ [ ρ ( r ) ] is a functional of densityρ(r).　From theorem (1)
Ev [ ρ ] ≡ Ψ [ ρ ] H + V Ψ [ ρ
Energy functional for
external potential v (r)
Ev [ ρ v ] = Evgs < Ev [ ρ
Ev [ ρ ] = FHK [ ρ ] +
∫
v
gs
]
  
ρ ( r ) v ( r ) dr
]
Variational principle holds for vrepresentative density
FHK [ ρ ] : v-independent universal functional
The following variation leads to all the ground-state properties.
{
δ F[ ρ ] +
∫
  
ρ ( r ) v ( r ) dr − µ
(∫
)}
 
ρ ( r ) dr − N = 0
In principle, any physical quantity of the ground state should be a
functional of density.
Variation with respect to many-body wave functions
↓
Variation with respect to one-body density
↓
Physical quantity


Ψ (r1 ,  , rN )

ρ (r )

A[ ρ (r )] = Ψ [ ρ ] Aˆ Ψ [ ρ ]
v-representative→ N-representative
Levy (1979, 1982)
The “N-representative density” means that it has a corresponding
many-body wave function.
Ritz’ Variational Principle
Min Ψ ( r1 ,.., rN ) H Ψ ( r1 ,.., rN ) ⇒ Ψ
HΨ
gs
( r1 ,.., rN ) =
E gs Ψ
gs
( r1 ,.., rN )
gs
Decomposed into two steps
 Min Ψ H Ψ 
Min Ψ H Ψ = Min



ρ ( r )  Ψ → ρ ( r )

F [ ρ ( r ) ] ≡ Min Ψ H Ψ
Ψ → ρ (r)
( r1 ,.., rN )
One-to-one Correspondence
External potential
Minimum-energy state
Ψ
Ground state

V (r )
Ψ
V
Density

ρ (r )
v-representative
density

ρ V (r )
Time-dependent “HK” theorem
Runge & Gross (1984)
One-to-one mapping between time-dependent density ρ(r,t)
and time-dependent potential v(r,t)
except for a constant shift of the potential
Condition for the external potential:
Possibility of the Taylor expansion around finite time t0
∞
1
k
v(r, t ) = ∑
vk (r )( t − t0 )
k = 0 k!
The initial state is arbitrary.
This condition allows an impulse potential, but forbids adiabatic switch-on.
∂
i
Ψ (t ) = H (t ) Ψ (t )
∂t
Schrödinger equation:
Current density follows the equation
[
]
∂
i j(r, t ) = Ψ (t ) ˆj(r ), H (t ) Ψ (t )
∂t
(1)
Different potentials, v(r,t) , v’(r,t), make time evolution from the same
vk (r ) − vk' (r ) ≠ c for ∃ k
initial state into Ψ(t)、Ψ’(t)
 ∂ 
 
 ∂t
k+ 1
{
j(r, t ) − j' (r, t )} t = t = − ρ (r, t0 )∇ wk (r )
 ∂ 
wk (r ) =  
 ∂t
∴
0
k
{ v(r, t ) − v' (r, t )} t = t
j(r, t ) ≠ j' (r, t )
0
Continuity eq.
= vk (r ) − v'k (r ) ≠ c
ρ (r, t ) ≠ ρ ' (r, t )
at t > t0
Problem 1: Two external potentials are different, when their expansion
∞
1
k
v(r, t ) = ∑
vk (r )( t − t0 )
k = 0 k!
has different coefficients at the zero-th order
v0 (r ) − v0' (r ) ≠ c
Using eq. (1), show
∂
{ j(r, t ) − j' (r, t )} t = t0 = − ρ (r, t0 )∇ w0 (r )
∂t
w0 (r ) = { v(r, t ) − v' (r, t )} t = t = v0 (r ) − v'0 (r ) ≠ c
0
Next, if
v0 (r ) − v0' (r ) = c
then, show
 ∂ 
 
 ∂t
2
{
, but
v1 (r ) − v1' (r ) ≠ c ,
j(r, t ) − j' (r, t )}
= − ρ (r, t0 )∇ w1 (r )
t = t0
Problem 2: Using the continuity equation and the following equation
 ∂ 
 
 ∂t
k+ 1
{
j(r, t ) − j' (r, t )} t = t = − ρ (r, t0 )∇ wk (r )
0
 ∂ 
wk (r ) =  
 ∂t
prove that
 ∂ 
 
 ∂t
k+ 2
k
{ v(r, t ) − v' (r, t )} t = t
{ ρ (r, t ) −
ρ ' (r, t )}
0
= vk (r ) − v'k (r ) ≠ c
= ∇ ⋅ { ρ (r, t0 )∇ wk (r )}
t = t0
Then, show that the right-hand side cannot vanish identically, with
∇ wk (r ) ≠ 0
One-to-one Correspondence
External potential

V (r , t )
Time-dependent state
starting from the initial state
Ψ (t0 )
Time-dependent
density
TD state
v-representative
density
Ψ (t )

ρ V (r,t)
V (t )
The universal density functional exists, and the variational
principle determines the time evolution.
From the first theorem, we have ρ(r,t) ↔Ψ(t). Thus, the variation of the
following function determines ρ(r,t) .
∂
S [ ρ ] = ∫ dt Ψ [ ρ ](t ) i − H (t ) Ψ [ ρ ](t )
t0
∂t
t1
~
S [ ρ ] = S [ ρ ] − ∫ dt ∫ drρ (r, t )v(r, t )
t1
t0
The universal functional
~
S [ρ ]
is determined.
v-representative density is assumed.
TD Kohn-Sham Scheme
Real interacting system
TD state

V (r , t )
TD density
Ψ (t)
V

ρ (r,t)
Virtual non-interacting system

Vs (r , t )
TD state
Ψ (t)
TD density
S

ρ (r,t)
Time-dependent KS theory
Assuming non-interacting v-representability

ρ (r,t) =
Time-dependent Kohn-Sham (TDKS) equation
N
∑
i= 1
 2
φ i (r ,t)
 2 2

 −
∇ + vs [ ρ ](r, t )  φ i (r, t )
 2m

δ S [ρ ]
vs [ ρ ](r, t ) =
δ ρ (r, t )
t1
∂
S [ ρ ] ≡ S [ ρ ] − ∫ Φ D [ ρ ](t ) i − T Φ D [ ρ ](t )
t0
∂t
∂
i φ i (r, t ) =
∂t
Solving the TDKS equation, in principle, we can obtain the exact time
evolution of many-body systems.
The functional depends on ρ(r,t） and the initial state Ψ0 .
Time-dependent quantities
→ Information on excited states
Ψ (0) =
∑
cn Φ
⇒
n
Ψ (t ) =
n
∑
cn e − iEnt Φ
n
n
Energy projection
1
2π
∫
∞
−∞
Ψ (t ) e iEt dt =
∑
cn Φ
n
δ ( E − En )
n
Finite time period →
Finite energy resolution
T ~1 Γ
1
2π
∫
∞
−∞
iEt − Γ t 2
Ψ (t ) e e
dt =
∑
n
cn
Γ 2
Φ
2
2
π ( E − E n ) + ( Γ 2)
n
TDHF(TDDFT) calculation in 3D real space
H. Flocard, S.E. Koonin, M.S. Weiss, Phys. Rev. 17(1978)1682.
Small-amplitude limit
(Random-phase approximation)
One-body operator under a TD external potential
i
∂
ρ (t ) = [ hKS [ ρ (t )] + Vext (t ), ρ (t )]
∂t
Assuming that the external potential is weak,
ρ (t ) = ρ 0 + δ ρ (t )
h(t ) = h0 + δ h(t ) = h0 +
∂
i δ ρ (t ) = [ h0 , δ ρ (t )] + [δ h(t ) + Vext (t ), ρ 0 ]
∂t
δh
δρ
⋅ δ ρ (t )
ρ0
Let us take the external field with a fixed frequency ω,
Vext (t ) = Vext (ω )e − iω t + Vext+ (ω )e + iω t
The density and residual field also oscillate with ω,
δ ρ (t ) = δ ρ (ω )e − iω t + δ ρ + (ω )e + iω t
δ h(t ) = δ h(ω )e − iω t + δ h + (ω )e + iω t
The linear response (RPA) equation
ω δ ρ (ω ) = [ h0 , δ ρ (ω )] + [δ h(ω ) + Vext (ω ), ρ 0 ]
Note that all the quantities, except for ρ0 and h0, are non-hermitian.
A
∑ ( δ ψ (t )
δ ρ (ω ) = ∑ ( X (ω )
δ ρ (t ) =
i
i= 1
A
i= 1
i
φ i + φ i δ ψ i (t ) )
φ i + φ i Yi (ω ) )
This leads to the following equations for X and Y:
ω X i (ω ) = ( h0 − ε i ) X i (ω ) + Qˆ {δ h(ω ) + Vext (ω )} φ i
ω Yi (ω ) = − Yi (ω ) ( h0 − ε i ) − φ i {δ h(ω ) + Vext (ω )} Qˆ
Qˆ =
A
∑ (1 −
i= 1
φi φi
)
These are often called “RPA equations” in nuclear physics.
X and Y are called “forward” and “backward” amplitudes.
If we start from the TDHF with a “density-independent” Hamiltonian
(not from the energy functional), then, there is other ways to formulate
the RPA. (see TextBooks)
Matrix formulation
ω X i (ω ) = ( h0 − ε i ) X i (ω ) + Qˆ {δ h(ω ) + Vext (ω )} φ i
ω Yi (ω ) = − Yi (ω ) ( h0 − ε i ) − φ i {δ h(ω ) + Vext (ω )} Qˆ
(1)
Qˆ =
A
∑ (1 −
i= 1
φi φi
If we expand the X and Y in particle orbitals:
X i (ω ) =
∑
m> A
φ m X mi (ω ) ,
Yi (ω ) =
∑
m> A
φ m Ymi* (ω )
Taking overlaps of Eq.(1) with particle orbitals
 A
  *
 B
B
 (Vext ) mi 
 1 0    X mi (ω ) 







−
ω
=
−

*



A 
 0 − 1   Ymi (ω ) 
 (Vext ) im 
Ami ,nj = (ε m − ε )δ mnδ ij + φ m
Bmi ,nj = φ m
∂h
∂ ρ jn
∂h
∂ ρ nj
φi
ρ0
φi
ρ0
In many cases, setting Vext=0 and solve the normal modes of excitations:
→ Diagonalization of the matrix
)
Lowest negative-parity states
(SGII functional)
cal
exp
Green’s function method
ω δ ρ (ω ) = [ h0 , δ ρ (ω )] + [δ h(ω ) + Vext (ω ), ρ 0 ]
(2)
Multiply Eq.(2) with φ i φ i from the right and from the left:
δ ρ (ω ) φ i φ i = ( ω + ε i − h0 ) [Vscf , ρ 0 ] φ i φ i
(3-1)
φ i φ i δ ρ (ω ) = φ i φ i [Vscf , ρ 0 ]( ω − ε i + h0 )
(3-2)
−1
−1
Vscf (ω ) ≡ Vext (ω ) + δ h(ω )
Sum up with respect to occupied orbitals i, then, add (3-1) and (3-2), using
δ ρ (ω ) = { ρ 0 , δ ρ (ω )}
the orthonormalization condition for KS orbitals (ρ2=ρ):
δ ρ (ω ) =
∑ {G (ε
0
i
+ ω )Vscf φ i φ i + φ i φ i Vscf G0 (ε i − ω )}
i
G0 ( E ) ≡ ( E − h0 )
−1
If the Vscf is local, we can rewrite this as follows:
δ ρ (r; ω ) =
∑ ∫ dr{G (r, r ' ; ε
0
i
}
+ ω )Vscf (r ' )φ i (r ' )φ i* (r ) + φ i (r )φ i* (r ' )Vscf (r ' )G0 (r ' , r : ε i − ω )
i
=
∫ drΠ
0
(r, r ' ; ω )Vscf (r ' ; ω )
where the independent-particle response function is defined by
Π 0 (r , r ' ; ω ) ≡
∑ ∫ dr{φ
i
i
}
(r )G0* (r, r ': ε i − ω * )φ i* (r ' ) + φ i* (r )G0 (r, r ' ; ε i + ω )φ i (r ' )
Green’s function method (cont.)
An advantage of the Green’s function method is that we can treat the
continuum exactly.
Shlomo and Bertsch, NPA243 (1975) 507.
ω → ω + iη
Π 0 (r , r ' ; ω + iη ) =
∑ ∫ dr{ φ
i
}
(r )G0( + )* (r, r ': ε i − ω )φ i* (r ' ) + φ i* (r )G0( + ) (r , r ' ; ε i + ω )φ i (r ' )
i
G0( ± ) ( E ) ≡ ( E ± iη − h0 )
−1
In case h0 is spherical, the Green’s function can be easily obtained by the
partial-wave expansion:
1
ul (r< )vl( + ) (r> )
*
ˆ
G ( E ) = 2m ∑
Y
(
r
)
Y
(rˆ' )
lm
lm
(+ )
rr ' lm W [ul , vl ]
(+ )
0
In case h0 is deformed, we can construct the Green’s function by using the
following identity:
T.N. and Yabana, JCP114 (2001) 2550; PRC71 (2005) 024301.
(± )
(± )
(± )
(± )
Gdef
( E ) = Gsph
( E ) + Gsph
( E )( hdef − hsph )Gdef
(E)
Real-time method
In the RPA calculations (matrix formulation & Green’s function method),
the most tedious part is the calculation of the residual induced fields:
δ h(ω ) =
δh
δρ
⋅ δ ρ (ω )
ρ0
In the original time-dependent equations, this effect is included in the
self-consistent potential:
h[ ρ (t )] = h0 + δ h(t ) ,
δ h(t ) =
δh
δρ
⋅ δ ρ (t )
ρ0
Therefore, in principle, the RPA can be achieved by solving the TD
Kohn-Sham equations, starting from the ground state with a weak
perturbation.
∂
i φ i (r , t ) =
∂t
 2 2

 −
∇ + vs [ ρ ](r, t ) + Vext (t )  φ i (r, t )
 2m

Skyrme TDDFT in real space
Time-dependent Hartree-Fock equation
(
− iη~ ( r )
)

∂
i ψ i (rσ τ , t ) = hSk [ ρ ,τ , j, s, J ](t ) + Vext (t ) ψ i (rσ τ , t )
∂t
3D space is discretized in lattice
n = 1,Mt
Single-particle orbital: ϕ i (r, t ) = {ϕ i (rk , t n )}k = 1,Mr ,
i = 1,  , N
N: Number of particles
y [ fm ]
Mr: Number of mesh points
Mt: Number of time slices
Spatial mesh size is about 1 fm.
Time step is about 0.2 fm/c
Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301
X [ fm ]
Calculation of time evolution
Time evolution is calculated by the finite-order Taylor
expansion
t+ ∆ t

ψ i (t + ∆ t ) = exp − i ∫ h(t ' )dt ' ψ i (t )
t


≈
∑
( − i∆ t h(t +
n
n!
∆ t 2) )
n
ψ i (t )
Violation of the unitarity is negligible if the time step is
small enough:
∆ t ε max < < 1
ε max
The maximum (single-particle) eigenenergy in the model space
Real-time calculation of
response functions
1. Weak instantaneous external
perturbation
Ψ (t ) Fˆ Ψ (t )
Vext (t ) = η Fˆδ (t )
3. Calculate time evolution of
Ψ (t ) Fˆ Ψ (t )
5. Fourier transform to energy domain
dB (ω ; Fˆ )
1
= −
Im ∫ Ψ (t ) Fˆ Ψ (t ) e iω t dt
dω
πη
dB (ω ; Fˆ )
dω
ω [ MeV ]
Real-time dynamics of electrons
in photoabsorption of molecules
1. External perturbation　t=0
Vext ( r, t ) = − ε riδ (t ), i = x, y , z
2. Time evolution of dipole moment
di ( t ) ∝
∫ r ρ (r, t )
i
E
at t=0
Ethylene molecule
Comparison with measurement (linear optical absorption)
TDDFT accurately describe optical absorption
Dynamical screening effect is significant
PZ+LB94
i



∂
ψ i (r , t ) = h[n(r , t )]ψ i (r , t )
∂t
with
Dynamical screening
without
i



∂
ψ i (r , t ) = h[n0 (r )]ψ i (r , t )
∂t
TDDFT
Exp
Without dynamical screening
(frozen Hamiltonian)
T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114(2001)2550.
－－
－
－
－
－
－
Eext ( t )
++
+
+
+
++
Eind ( t )
Photoabsorption cross section in C3H6 isomer molecules
Nakatsukasa & Yabana, Chem. Phys. Lett. 374 (2003) 613.
TDLDA cal with LB94 in 3D real space
•
33401 lattice points (r < 6 Å)
•
Isomer effects can be understood in terms of symmetry and antiscreening effects on bound-to-continuum excitations.
Cross section [ Mb ]
•
Photon energy [ eV ]
Neutrons
16
O
δ ρ n (t ) = ρ n (t ) − ( ρ 0 ) n
Time-dep. transition density
δρ> 0
δρ< 0
δ ρ p (t ) = ρ p (t ) − ( ρ 0 ) p
Protons
16
O
18
O
Prolate
10
20
30
Ex [ MeV ]
40
26
Mg
24
Triaxial
Prolate
10
20
30
Ex [ MeV ]
Mg
40
10
20
30
Ex [ MeV ]
40
Si
Si
28
30
Oblate
Oblate
10
20
30
Ex [ MeV ]
10
20
30
Ex [ MeV ]
40
40
Ca
44
Prolate
40
48
Ca
10
20
Ca
30
Ex [ MeV ]
10
10
20
30
Ex [ MeV ]
40
20
30
Ex [ MeV ]
40
Finite Amplitude Method
T.N., Inakura, Yabana, PRC76 (2007) 024318.
A method to avoid the explicit calculation of the residual fields (interactions)
ω X i (ω ) = ( h0 − ε i ) X i (ω ) + Qˆ {δ h(ω ) + Vext (ω )} φ i
ω Yi (ω ) = − Yi (ω ) ( h0 − ε i ) − φ i {δ h(ω ) + Vext (ω )} Qˆ
(1)
Residual fields can be estimated by the finite difference method:
δ h(ω ) =
ψ
i
1
( h[ ψ ' , ψ
η
]− h )
= φ i + η X i (ω ) ,
0
ψ
'
i
= φ i + η Yi (ω )
Starting from initial amplitudes X(0)and Y(0), one can use an iterative
method to solve eq. (1).
Programming of the RPA code becomes very much trivial, because we
only need calculation of the single-particle potential, with different bras
and kets.
Fully self-consistent calculation
of E1 strength distribution
Inakura, T.N., Yabana, in preparation
Z
SkM*
Rbox= 15 fm
Γ = 1 MeV
N
Large Amplitude Collective Motion
Beyond the small-amplitude
approximation
• In the small-amplitude limit, the normal
modes are obtained by diagonalizing the
RPA matrix.
→ “Quantization” is on hand.
• Large amplitude collective motion
– Real-time approach to non-linear response
– Adiabatic TDHF
– Self-consistent collective coordinate method
Real-time approach to non-linear
response
• In principle, non-linear response can be
studied with the real-time method.
– Accuracy
– Applicability
TDHF(TDDFT) calculation in 3D real space
H. Flocard, S.E. Koonin, M.S. Weiss, Phys. Rev. 17(1978)1682.
Ionization by Laser
Electrons in a strong electric field
Laser field, E ～ Electric field of ions binding electrons
Laser frequency ω ～ HOMO-LUMO gap
Re-scattering process: A new probe for electronic structure
Wave packet split by the laser field
Re-accelerated toward the origin
Scattered by the remaining part of itself
eE(t)z
γ«1
z
Keldysh parameter
γ = τ tunnelω laser
Higher-order harmonic generation
ψ (t) = α φ + β e

ik ⋅ r − iE k t / 
d (t ) = ψ ( t ) z ψ ( t ) ≈ α * β φ z e
I (ω ) ∝
∫
dte iω t d A ( t )

ik ⋅ r − iE k t / 
+ cc
eE(t)z
2
Ne
z
HOMO orbital in N2
(Molecular tomography)
Ab inito calculation
N2 molecule
2x1014W/cm2, 800nm laser
Calculated by Yabana
50fs
ψ (t) = α φ + β e

ik ⋅ r − iEk t / 
d (t ) = ψ ( t ) z ψ ( t ) ≈ α β φ z e
*
I (ω ) ∝
∫ dte
iω t
dA(t)
2

ik ⋅ r − iEk t / 
+ cc
|dA(ω ) | 2
Comparison with experiments
Calculation
10
Experiment
8
10
7
10
6
10
5
10
4
10
3
15
N2 0 deg.
N2 45 deg.
N2 90 deg.
Ar
20
25
30
35
Harmonic order
40
45
θ
Large amplitude collective motion
(LACM) in nuclei
• Fission
• Decay of superdeformed band
• Shape-coexistence phenomena
Φ = c1
+ c2
Adiabatic theories of LACM
• Baranger-Veneroni, 1972-1978
ρ (t ) = e iχ (t ) ρ 0 e − iχ (t )
•Expansion with respect to χ
• Villars, 1975-1977
• Eq. for the collective subspace
(zero-th and first-order w.r.t. momenta)
∂V
Q (q ) Φ (q ) = 0
∂q
∂
δ Φ (q ) [ H , Q (q )] + iM (q ) − 1
Φ (q ) = 0
∂q
δ Φ (q ) H −
•Non-uniqueness problem
“Validity condition”
(Goeke-Reinhard, 1978-)
Goeke, Reinhard, Rowe, NPA359 (1981)
408
Approaches to Non-uniqueness Problem
(1) Yamamura-Kuriyama-Iida, 1984
Requirement of “analyticity”
(ex) Moya de Guerra-Villars, 1978)
(2) Rowe, Mukhejee-Pal, 1981
Therefore, in principle, we can
determine a unique collective path
Requirement of “Point transf.” and
in the ATDHF. The higher-order in p
equations up to O(p2)
can be systematically treated.
In practice, it is only applicable to
simple models.
There is no systematic way to go
beyond the second order in p.
In practice, the method is
applicable to realistic models as
well.
Non-adiabatic theories of LACM
• Rowe-Bassermann, Marumori,
Holzwarth-Yukawa, 1974• Local Harmonic Approach (LHA)
• Curvature problem
• Correspondence between, Q,P
↔ Infinitesimal generator, is not
guaranteed.
δ Φ (q ) H −
∂V
Q (q ) Φ (q) = 0
∂q
δ Φ (q ) [ H , Q (q )] + iM (q ) − 1 P (q ) Φ (q ) = 0
δ Φ (q ) [ H , P (q )] − iC (q )Q (q ) Φ (q ) = 0
• Marumori et al, 1980• Self-consistent collective
coordinate (SCC) method
• The problems of LHA are
solved.
• The SCC equation is solved by
the expansion with respect to
(q,p).
δ Φ ( q, p ) H −
∂H
∂H
Q−
P Φ (q) = 0
∂q
∂p
H ≡ Φ ( q, p ) H Φ ( q, p )
• “Adiabatic” approx. → LACM
(Matsuo, TN, Matsuyanagi,
2000)
TDHF(B) → Classical Hamilton’s form
Blaizot, Ripka, “Quantum Theory of Finite Systems” (1986)
Yamamura, Kuriyama, Prog. Theor. Phys. Suppl. 93 (1987)
The TDHF(B) equation can be described by the classical form.
For instance, using the Thouless form
1

z = exp z µ ν aµ+ aν+  Φ
2

0
The TDHF(B) equation becomes in a form
∂H
+
(
1
+
z
z)
∂ z+
∂H
iz + = − 2(1 + z + z )
(1 + zz + )
∂z
iz = 2(1 + zz + )
The Holstein-Primakoff-type mapping
leads to
∂H
∂β +
∂H
iβ + = − 2
∂β
iβ = 2
β µ ν = ( ξ + iπ
+
H ( z, z ) =
[
zH z
z z
β µ ν = z (1 + z + z )1/ 2
)µν
2
]
µν
∂H
ξµ ν =
∂ π µν
iπ µ ν = −
∂H
∂ ξ µν
TDHF(B) →Small amplitude limit
Small fluctuation around the HF(B) state
ξ ,π H ξ ,π = Φ
0
HΦ
2qp index α = ( µ ν )
 A
1
1
+ +
− TrA +
a a , aa  *
2
2
B
(
0
Aα β = Φ 0 [(aa )α , [ H , (a + a + ) β ]] Φ
0
B   aa

* + +
A  a a
)



, Bα β = Φ 0 [(aa )α , [ H , (aa ) β ]] Φ
0
α
Rewriting the last term in terms of variables (ξ , π α )
 A B  κ 
1
+



ξ , π H ξ , π = ERPA + (κ * , κ )  *
,
κ
≡
β
(
1
−
β
β )≈ β
*
*
2
 B A  κ 
1
1
( A + B )α β ξ α ξ β + ( A − B )α β π α π β
2
2
α
µ
Linear point transformation (ξ , π α ) → (q , pµ ) leads to
1
= ERPA + ∑ pµ2 + ω n2 (q µ ) 2
2 n
= ERPA +
[
]
qµ =
δ
µν
1
ωµ
∑ (X
α
µ
+ Yµ
)
α ξ
α
,
pµ =
ν
∂ qµ
αβ ∂q
2
=
(
A
−
B
)
,
ω
δ
µ
α
β
∂ξ
∂ξ
µν
ω
µ
∑ (X
α
µ
− Yµ
)
α
πα
∂ξ α
∂ξ β
=
( A + B )α β
µ
∂q
∂ qν
Decoupled classical motion within
the point transformation
Klein, Walet, DoDang, Ann. Phys. 208 (1991) 90
Expanding the classical Hamiltonian w.r.t. momentum up to 2nd order
1
H (ξ , π ) = Bα β π α π
2
β
+ V (ξ ) , B
∂ 2H
≡
∂π α∂π
αβ
(ξ , π ) → (q, p)
Point transformation
q µ = f µ (ξ ) , ξ
α
= g α (q)
β π =0
∂ gα ∂ ξ α
g ≡
=
,
∂ qµ
∂ qµ
α
,µ
pµ = g ,αµ π α , π α = f ,αµ pµ
∂f µ
∂ qµ
f ≡
=
∂ξ α ∂ξ α
µ
,α
Point transformation conserves the quadratic form in
momenta.
1
H ( q, p ) =
2
B µ ν pµ pν + V (q) , B µ ν ≡ f ,αµ Bα β f ,νβ
γβ
β
Metric tensor: Bα β : defined by Bα γ B = δ α
Shift-up and down of
α
µ
β
indexes:
Chain rules: g ,µ f , β = δ α ,
V ,α ≡ Bα β V, β
f ,αµ g ,αν = δ ν µ
(=canonical variable cond.)
Assuming that there is a decoupled path (1-dim. collective submanifold)
q1 : Collective coordinate, q n : Non - collective coord.
Decoupling condition: q = pn = 0 ⇒ q = p n = 0
n
n
qn
q1 = f 1 (ξ )
(1) V,n = 0
(1) V,α = V,1 f ,α1
(2) B n1 = 0
(2) Bα β f ,1β = B 11 g ,α1
(3) B,11
n = 0
11 1
(3) B,11
α = B,1 f ,α
1
1
1
1
Decoupling condition (1) ↔ HF(B) with the constraint q = Φ (q ) Qˆ (q ) Φ (q )

∂ V 1

δ  V (ξ ) − 1 q  = δ { H (ξ , π = 0) − λ q1 (ξ )} = δ Φ (q1 ) H − λ Qˆ (q1 ) Φ (q1 ) = 0
∂q


Using the decoupling conditions (2) and (3), we may construct the
constraint operator Q(q). More precisely speaking, we can determine the
2qp parts of Q(q).
µ
Differentiating the chain relation V,α = V, µ f ,α
V,α β = V, µ ν f ,αµ f ,νβ + V, µ f ,αµ β
The last term indicates that the second derivative of the potential is not
covariant. This can be rewritten in a covariant derivative
V;α β ≡ V,α β − Γ αγ β V,γ , V;µ ν ≡ V, µ ν − Γ µρν V, ρ
V;α β = V;µ ν f ,αµ f ,νβ
Here, two different definitions of the metric tensor are possible:
(i) Riemannian type
Mass tensor as the metric tensor
Γ αγ β ≡
1 γδ
B ( Bδ α , β + Bδ β ,α − Bα β ,δ
2
)
Affine connection
(ii) Symplectic type
Metric tensor
Γ αγ β ≡
Kα β =
∑
µ
f ,αµ f , βµ , K α β =
1 γδ
K ( Kδ α , β + K δ β ,α − Kα β ,δ ) = g ,γµ f ,αµ β
2
∑
µ
g ,αµ g ,βµ
Kα β = Bα β
With this metric, the decoupled space is assumed to be “flat”.
A certain combination of the decoupling conditions (1-3) leads to the
following Local Harmonic Equation (LHE) (with metric tensor Kij):
,β
;α
f =ω f
1
,β
V
2
1
,α
V;α, β ≡ B β γ V;α γ , V;α β ≡ V,α β − Γ αγ β V,γ
(i) Riemannian LHE
The condition (3) is equivalent to that the decoupled
collective path is geodesic with metric tensor of Bα β
δ
∫
B11 (q1 )dq1 = 0 ⇒ f ,α1β − Γ αγ β f ,γ1 + Γ 111 f ,α1 f ,1β = 0
Then, using the condition (2), we can derive the LHE above.
ω 2 = V;11 = B 11 (V,11 − Γ 111 V,1 )
(ii) Symplectic LHE
Without the condition (3), we can derive the LHE.
ω 2 = V;11 = B 11V,11
1
Either neglect, or determine by a certain condition, the curvature f ,α β
Riemannian LHE vs Symplectic LHE
Symplectic LHE is (almost) identical to the “adiabatic” approximation of the
Self-consistent Collective Coordinate (SCC) Method
Original formulation: Matsuo, TN, Matsuyanagi, Prog. Theor. Phys. 103 (2000) 959
Gauge-invariant formulation: Hinohara et al, PTP 117 (2007) 451
We believe that the Symplectic LHE (ASCC) is superior to the Riemannian
LHE in the following reasons:
• Extension to lift the restriction to the point transformation can be
consistently achieved.
• Both formalisms coincide with the RPA at equilibrium. However, in case
of superconducting nuclei, the “extended” symplectic LHE naturally
becomes identical to the QRPA.
• Nambu-Goldstone modes are automatically separated from the
decoupled collective variables, as zero-energy solutions.
Separation of Nambu-Goldstone modes
Extended “point” transformation
1
q = f (ξ ) + f (1) µ α β π α π
2
pµ = g ,αµ π α + O (π 3 )
µ
µ
β
+ O (π )
4
ξ
α
πα
1 (1)α µ ν
g
pµ pν + O ( p 4 )
2
= f ,αµ pµ + O ( p 3 )
= g α (q ) +
This extension leads to the modification of mass parameter, but the other
~
formulation is kept invariant.
B α β ≡ Bα β − V f (1) µ α β
,µ
(1) Symmetry operator S = momentum
ps = g ,αs π α + O (π 3 )
{ ps , H }PB = 0
TN, Walet, DoDang, PRC61 (1999) 014302
g ,αsV,α = 0 ⇒
V;α β g ,αs = 0
(2) Symmetry operator S = coordinate
q s = f s (ξ ) +
1 (1) sα β
f
π α π β + O (π 4 )
2
~
B α β f , βs = Bα β f , βs − V,µ f (1) µ α β f ,βs
= Bα β f , βs − V, µ f (1) sα β f , βµ
= Bα β f , βs − V, β f (1) sα β = 0
({q µ , qν }PB = 0)
({q s , H }PB = 0)
Collective path and re-quantization
Solve the constrained MF eq. and LHE to obtain self-consistent solutions
Symplectic LHE
Adiabatic SCC
(CMF) V,α = V,1 f ,α1
(CMF) δ φ (q) Hˆ − (∂ V ∂ q)Qˆ (q) φ (q) = 0
(LHE) V;α, β f ,1β = ω 2 f ,α1
(LHE) δ φ (q) Hˆ (q), iQˆ (q) − B(q) Pˆ (q) φ (q ) = 0
(
V;α, β ≡ B β γ V,α λ − V,1 f ,α1 γ
)
δ
[
]
φ (q) [ Hˆ − (∂ V ∂ q)Qˆ (q), Pˆ (q) i ] − C (q )Qˆ (q)
1
[
[
−
Hˆ , (∂ V ∂ q)Qˆ (q)] , Qˆ (q)] φ (q) = 0
2 B(q )
We obtain a series of “Slater determinants”, as the solutions.
φ (q1 ) , φ (q2 ) , φ (q3 ) , 
GCM
B (q1 ) , B (q2 ) , B (q3 ) ,
V (q1 ) ,V (q2 ) ,V (q3 ) ,
H ( q, p ) =
q1 = f 1 (ξ )
“Collective Hamiltonian”
 ∂ 
 ∂ 
1
1
 B (q) 
 + V (q)
B (q) p 2 + V (q) ⇒
B (q) 
2
2
 i∂ q 
 i∂ q 
Applications to simple models
Applications to O(4) models
Model Hamiltonian
Monopole+ “Quadrupole” pairing + “Quadrupole” int.
P0 ≡
σ
jm
∑
(
)
(
)
1
1
1
G0 P0+ P0 + P0 P0+ − G2 P2+ P2 + P2 P2+ − χ Q 2
2
2
2
∑ c j − mc jm , P2 ≡ ∑ ∑ σ jm c j − mc jm , Q ≡ ∑ d j ∑ σ
H = h0 −
j m> 0
j m> 0
 1 m < Ω j /2
= 
− 1 m > Ω j /2
Parameters
ε 1 = 0, ε 2 = 1.0, ε 3 = 3.5
d1 = 2.0, d 2 = 1.0, d 3 = 1.0
Ω 1 = 14, Ω 2 = 10, Ω 3 = 4
j
jm
c +jm c jm
m
ε3
ε2
ε1

2Ω j = 2 j + 1
Hinohara, TN, Matsuo, Matsuyanagi, PTP115 (2006) 567
Potential
Mass: M=1/B
Larger G0
Effects of time-odd MF
D (q ) ≡ φ (q ) Q φ (q )
Cranking Mass
Exact
Adiabatic
SCC
CHB with
Mcranking
Time-odd effects are neglected in the cranking mass !
Curvature
effects
Qˆ ( q) =
∑
µν
QµAν ( aµ+ aν+ + h.c.) + ∑ QµBν aµ+ aν
µν
f ,α1 β
In this model, requiring
the gauge invariance, we
can determine them.
The curvature effects are
weak.
Neglected calc
Full calc
Model of protons
and neutrons
T.N. & Walet, PRC58 (1998) 3397
Gp=0.3
Neutrons
Gp=10
Protons
Upper orbital has a larger
quadrupole moment
CHB+Mcr LHE
Exact
Adiabatic vs Diabatic Dynamics
Review: Nazarewicz, NPA557 (1993) 489c
The problem has been discussed since the paper by Hill and Wheeler (1953)
The pairing interaction plays a key role for configuration changes at level
crossings.
ε
ε
t ic
a
b
ia
d
ε
β
β
ε
adia
b
β
“Specialization energy”
atic
β
Spontaneous fission
life time is much larger
for odd nuclei.
Applications to more realistic models:
Separable-force model
Calculations carried out by Dr Nobuo Hinohara (YITP, Japan)
Shape coexist ence in N~Z~40 r egion
Se
68
neut r on single par t icle ener gy
Kr
72
50
42
40
36
34
34
38
28
20
Z,N = 34,36 (oblat e magic number s)
Z,N = 38 (pr olat e magic number )
Fischer et al . Phys.Rev.C67 (2003)064318.
Bouchez et al. Phys.Rev.Lett.90(2003) 082502.
 oblat e-pr olat e shape coexist ence
 oblat e gr ound st at e
 shape coexist ence/mixing
Skyr me-HFB: Yamagami et al . Nucl.Phys.A693 (2001)579.
Mi cr os copi c t heor y t o des cr i be s hape coexi s t ence
 Lar ge-Scale Shell Model Calculat ion
 Dimension becomes too large for medium-heavy nuclei
(1013 dim for 80Zr, 40Ca core)
Too har d t o per for m !
 68Se: Kaneko et al. Phys.Rev.C70 (2004)051301.
Model Space: 56Ni core, fpg-shell 1.6 x 108 dim
 GCM
 72Kr: Bender et al. Phys. Rev. C74 (2006) 024312.
 Skyrme interaction
 Generator Coordinate: axial symmetric deformation
The t r iaxial defor mat ion is ignor ed.
 Adiabat ic TDHF
Adiabatic Self-consistent Collective Coordinate Method
68Se, 72Kr: Kobayasi et al. (Prog.Theor.Phys.112(2004), 113(2005))
 Almehed et al. (Phys. Lett. B604 (2004)163.)
Impor t ance of t r iaxial defor mat ion is discussed
Pair ing + Quadr upole Model (68Se, 72Kr )
Microscopic Hamiltonian
SP energy + Pairing (Monopole, Quadrupole) + Quadrupole interaction
Model Space
two major shells (Nsh=3,4) (40Ca core)
Parameters
sp energy: Modified Oscillator
interaction strength
monopole pairing and quadrupole int. strength:
adjusted to the pairing gaps and deformations of Skyrme-HFB
(Yamagami et al. NPA693(2001) )
quadrupole pairing strength G2:
 G2 = 0
 G2 = (G2)self　(self-consistent value) Sakamoto and Kishimoto PLB245 (1990) 321.
(G2)self restores the Galilean invariance in RPA order,
which was broken by the monopole pairing.
Collect ive pat h in 68Se
G2=0: Kobayasi et al., PTP113(2005), 129.
Collective potential
Moving-frame QRPA frequency
Collective mass
Moment of Inertia
β
γ
collective path
 Triaxial deformation connects two local minima
 Enhancement of the collective mass and MoI by the quadrupole pairing
Due to the contribution from the time-odd component
Prog.Theor.Phys.115(2006)567.
Collect ive pat h in 72Kr
G2=0: Kobayasi et al., PTP113(2005), 129.
Collective potential
Collective mass
q=0
Moment of Inertia
prolate
oblate
Moving-frame QRPA frequency
γ
β
β
β
γ
γ
γ
β
 Bifurcation of the path
 Triaxial degrees of freedom: important
 Enhancement of the collective mass and MoI
by the quadrupole pairing
Basic Scheme of ASCC Met hod (2)
3rd Step: Requantize the collective Hamiltonian.
Collective wave function
Collective Hamiltonian
vibrational wavefunctions
K: 3-axis component of angular momentum
3-axis: quantization axis, symmetry axis(γ=0o)
boundary conditions for collective wave functions
 periodic boundary condition at γ=0°and 60°for 68Se
Kumar and Baranger Nucl. Phys. A92 (1967) 608.
 box boundary condition for 72Kr
4th Step: Calculate EM transitions
E2 transitions, spectroscopic quadrupole moments …
Ener gy spect r a of 68Se
 two rotational bands
( ) …B(E2) e2 fm4
effective charge: epol = 0.904
 02+ state
 quadrupole pairing reduces ex.energy EXP：Fischer et al., Phys.Rev.C67 (2003) 064318.
Collect ive Wavefunct ions of 68Se
G2 = 0
G2 = G2self
localization
 I = 0: oblate and prolate shapes are strongly mixed via a triaxial degree of freedom
 ground band: mixing of different K states、excited band: K=0 dominant
　oblate-prolate mixing: strong in 0+ states, reduced as angular momentum increases
Ener gy Spect r a of 72Kr
effective charge is adjusted to this value
 two rotational bands
2
4
(
)
…B(E2)
e
fm
 small inter-band B(E2): shape mixing rather weak
EXP：Fischer et al., Phys.Rev.C67 (2003) 064318, Bouchez, et al., Phys.Rev.Lett.90 (2003) 082502.
Gade, et al., Phys.Rev.Lett.95 (2005) 022502, 96 (2006) 189901
Collect ive wavefunct ions of 72Kr
G2 = 0
G2 = G2self
oblate
 01+ state: well localized around oblate shape
 02+ state: weak oblate-prolate shape mixing
 other states: well defined shape character
prolate
oblate
prolate
Spect r os copi c quadr upol e moment
Fission
Staszczak et al.
•
Optimal path to fission
•
Diabatic vs Adiabatic dynamics
•
Collective mass parameters
Self-consistent, self-determined,
microscopic description of
nuclear fission
Summary
• Liquid-drop, shell, unified models, cranking model
• Nuclear structure at high spin and large
deformation
• Sum-rule approaches to giant resonances
• Basic theorem for the Time-dependent densityfunctional theory (TDDFT)
• Linearized TDDFT (RPA) and elementary modes
of nuclear excitation
• Theories of large-amplitude collective motion
• Anharmonic vibrations, shape coexistence
phenomena