Shell Model
A Unified View of Nuclear Structure
Frederic Nowacki1
18th STFC UK Postgraduate Summer School
Lancaster, August 24th-September 5th-2015
1Strasbourg-Madrid Shell-Model collaboration
Bibliography
Basic ideas and concepts in nuclear physicsan introductory approachHeyde K.IOP Publishing 1994
Shell model applications in nuclear spectroscopyBrussaard P.J., Glaudemans P.W.M.North-Holland 1977
The nuclear shell modelHeyde K.Springer-Verlag 1994
The nuclear shell modelA. Poves and F. NowackiLecture Notes in Physics 581 (2001) 70ff
The shell model as a unified view of nuclear structureE. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, A. P. ZukerRev. Mod. Phys. 77, 427 (2005)
Shell structure evolution and effective in-medium NN interactionN. SmirnovaEcole Joliot-Curie 2009
Outline
Lecture 1: Introduction, basic notions, shell model codes
and calculations
Lecture 2: Lanczos structure functions, Effective
Interactions
Lecture 3: Shell model applications to nuclear
spectroscopy
Reminder of Shell Model Problem
CORE
Define a valence space
Derive an effective interaction
HΨ = EΨ → HeffΨeff = EΨeff
Build and diagonalize the
Hamiltonian matrix.
In principle, all the spectroscopic properties are described
simultaneously (Rotational band AND β decay half-life).
Reminder of basis representation
Reminder of basis representation:
We represent a Slater determinant by a machine word, where
each state is a bit (0 empty 1 occupied)
1/2 3/2 1/2- 1/2 -1/2 -3/2 1/2 3/2 1/2- 1/2 -1/2 -3/2
0 0 1 1 1 1 1 1 1 10 0
12 11 10 9 8 7 6 5 4 3 2 1
Mn
i=
Mp
0p1/2 0p3/2 0p1/2 0p3/2
≡ a†10a
†9a
†8a
†7 b
†4b
†3b
†2b
†1 |0〉
After diagonalization, the eigenstates of the system are linear
combinations of Slater Determinants of the basis:
|Ψα〉 =∑
i
ci |Φi〉 with |Φi〉 =∏
k=nljmτ
a†k |0〉 = a
†k1...a
†kA|0〉
Diagonalisation with Lanczos algorithm
The Lanczos algorithm consist in the construction of an
orthonormal basis by orthogonalization of the states Hn|1〉,obtained by the repeated action of the hamiltonian H, on a
basis state |1〉 called pivot. From this procedure results a
tridiagonal matrix. In the first step we write:
H|1〉 = E11|1〉 + E12|2〉
where E11 is just 〈1|H|1〉 = 〈H〉,the mean value of H.
Lanczos Algorithm
E12 is obtained by normalization :
E12|2〉 = H|1〉 - E11|1〉 = (H − E11)|1〉
In the second step:
H|2〉 = E21|1〉 + E22|2〉 + E23|3〉
The hermiticity of H implies E21 = E12
E22 is just 〈2|H|2〉and E23 is obtained by normalization :
E23|3〉 = (H− E22) |2〉 - E21|1〉
Lanczos Algorithm
At rank N, the following relations hold:
H|N〉 = ENN−1|N − 1〉 + ENN|N〉 + ENN+1|N + 1〉
Lanczos Algorithm
At rank N, the following relations hold:
H|N〉 = ENN−1|N − 1〉 + ENN|N〉 + ENN+1|N + 1〉
EN−1N = ENN−1, ENN = 〈N|H|N〉
Lanczos Algorithm
At rank N, the following relations hold:
H|N〉 = ENN−1|N − 1〉 + ENN|N〉 + ENN+1|N + 1〉
EN−1N = ENN−1, ENN = 〈N|H|N〉
and ENN+1|N + 1〉 = (H − ENN)|N〉 - ENN−1|N − 1〉
Lanczos Algorithm
It is explicit that we have built a tridiagonal matrix
〈I|H|J〉 = 〈J|H|I〉 = 0 if |I − J| > 1
E11 E12 0 0 0 0
E12 E22 E23 0 0 0
0 E32 E33 E34 0 0
0 0 E43 E44 E45 0
Lanczos convergence
RANDOM STARTING VECTOR
3
-6.345165 11.335118 29.120687
6
-21.344259 -7.802025 4.637278 16.927858 29.308309
9
-30.092574 -19.653950 -9.343311 0.467972 10.265731
12
-32.722076 -24.462806 -17.104890 -9.353111 -1.628857
15
-32.930624 -26.709841 -22.335011 -15.957805 -9.401645
18
-32.952147 -28.028244 -24.233122 -19.625844 -14.772679
21
-32.953570 -28.413699 -25.350732 -22.676041 -18.180356
24
-32.953655 -28.537584 -26.244093 -23.883982 -20.534298
27
-32.953658 -28.559930 -26.542899 -24.362551 -22.197866
30
-32.953658 -28.563001 -26.646165 -24.887184 -23.559799
33
-32.953658 -28.564277 -26.912739 -26.199181 -24.299165
36
-32.953658 -28.564535 -27.102898 -26.382496 -24.409357
39
-32.953658 -28.564567 -27.148522 -26.416873 -24.529055
42
-32.953658 -28.564570 -27.156735 -26.425250 -24.724078
45
-32.953658 -28.564570 -27.158085 -26.427319 -24.910915
48
-32.953658 -28.564570 -27.158371 -26.428021 -25.107898
Lanczos convergence
RANDOM STARTING VECTOR
3
-6.345165 11.335118 29.120687
6
-21.344259 -7.802025 4.637278 16.927858 29.308309
9
-30.092574 -19.653950 -9.343311 0.467972 10.265731
12
-32.722076 -24.462806 -17.104890 -9.353111 -1.628857
15
-32.930624 -26.709841 -22.335011 -15.957805 -9.401645
18
-32.952147 -28.028244 -24.233122 -19.625844 -14.772679
21
-32.953570 -28.413699 -25.350732 -22.676041 -18.180356
24
-32.953655 -28.537584 -26.244093 -23.883982 -20.534298
27
-32.953658 -28.559930 -26.542899 -24.362551 -22.197866
30
-32.953658 -28.563001 -26.646165 -24.887184 -23.559799
33
-32.953658 -28.564277 -26.912739 -26.199181 -24.299165
36
-32.953658 -28.564535 -27.102898 -26.382496 -24.409357
39
-32.953658 -28.564567 -27.148522 -26.416873 -24.529055
42
-32.953658 -28.564570 -27.156735 -26.425250 -24.724078
45
-32.953658 -28.564570 -27.158085 -26.427319 -24.910915
48
-32.953658 -28.564570 -27.158371 -26.428021 -25.107898
Lanczos convergence
RANDOM STARTING VECTOR
3
-6.345165 11.335118 29.120687
6
-21.344259 -7.802025 4.637278 16.927858 29.308309
9
-30.092574 -19.653950 -9.343311 0.467972 10.265731
12
-32.722076 -24.462806 -17.104890 -9.353111 -1.628857
15
-32.930624 -26.709841 -22.335011 -15.957805 -9.401645
18
-32.952147 -28.028244 -24.233122 -19.625844 -14.772679
21
-32.953570 -28.413699 -25.350732 -22.676041 -18.180356
24
-32.953655 -28.537584 -26.244093 -23.883982 -20.534298
27
-32.953658 -28.559930 -26.542899 -24.362551 -22.197866
30
-32.953658 -28.563001 -26.646165 -24.887184 -23.559799
33
-32.953658 -28.564277 -26.912739 -26.199181 -24.299165
36
-32.953658 -28.564535 -27.102898 -26.382496 -24.409357
39
-32.953658 -28.564567 -27.148522 -26.416873 -24.529055
42
-32.953658 -28.564570 -27.156735 -26.425250 -24.724078
45
-32.953658 -28.564570 -27.158085 -26.427319 -24.910915
48
-32.953658 -28.564570 -27.158371 -26.428021 -25.107898
Lanczos convergence
RANDOM STARTING VECTOR
3
-6.345165 11.335118 29.120687
6
-21.344259 -7.802025 4.637278 16.927858 29.308309
9
-30.092574 -19.653950 -9.343311 0.467972 10.265731
12
-32.722076 -24.462806 -17.104890 -9.353111 -1.628857
15
-32.930624 -26.709841 -22.335011 -15.957805 -9.401645
18
-32.952147 -28.028244 -24.233122 -19.625844 -14.772679
21
-32.953570 -28.413699 -25.350732 -22.676041 -18.180356
24
-32.953655 -28.537584 -26.244093 -23.883982 -20.534298
27
-32.953658 -28.559930 -26.542899 -24.362551 -22.197866
30
-32.953658 -28.563001 -26.646165 -24.887184 -23.559799
33
-32.953658 -28.564277 -26.912739 -26.199181 -24.299165
36
-32.953658 -28.564535 -27.102898 -26.382496 -24.409357
39
-32.953658 -28.564567 -27.148522 -26.416873 -24.529055
42
-32.953658 -28.564570 -27.156735 -26.425250 -24.724078
45
-32.953658 -28.564570 -27.158085 -26.427319 -24.910915
48
-32.953658 -28.564570 -27.158371 -26.428021 -25.107898
Lanczos convergence
48Cr
Dim (t=2) = 6.105
Dim (full space) = 2.106
Ca40
f7/2 f7/2
Π ν
f5/2p1/2p3/2
f5/2p1/2p3/2
STARTING VECTOR :EIGENVECTOR OF A SMALLER SPACE
ITER= 1 DIA= -31.105920 NONDIA= 4.642871
3
-32.578285 -21.260843 5.090417
6
-32.929531 -27.208522 -16.116780 -1.200061 14.816894
9
-32.952149 -28.024347 -22.702052 -13.782511 -3.514506
12
-32.953553 -28.345536 -25.965169 -20.636169 -12.806719
15
-32.953655 -28.528301 -26.951521 -22.532438 -18.004439
Lecture 2
Lanczos structure functions,Effective Interactions
Computation of transition operators
Given a one-body transition operator O(r), how do we compute
〈Ψf |O|Ψi〉
here Ψi and Ψf are many-body wave functions obtained from
shell-model diagonalization
One body operators:
O =A∑
i=1
o(r(i)) −→ O =∑
i ,j
〈i |o|j〉a†i aj
We need to know
the value of our one body operator between single particlewave functions 〈i|o|j〉
the one body density matrix elements 〈Ψf |a†i aj |Ψi〉
Computation of transition operators
for 〈i |o|j〉, one needs (eventually) to know the radial part of
the wave function: usually harmonic oscillator, sometimes
wood-saxon.
〈i |o|j〉 =
∫
d3r φ∗i oφj
for the one body density matrix elements, same procedure
as for the hamiltonian:
a†5a2|001011〉 = |011001〉
now we know the procedure to compute :
EL gamma decay rLYL0
ML gamma decay rL−1YL0
β decay :
Fermi decay : τ±Gamow-Teller decay: στ±
Spectroscopic factors a†i,ai
Lanczos Structure Function
We can be interested in the transition matrix elements
〈Ψf |O|Ψi〉 for many final states |Ψf 〉
example: β decay half-life calculation
Determine initial state |Ψi〉
Determine all posible final states |Ψf 〉
Compute matrix elements 〈Ψf |O|Ψi〉
λf =ln 2
Kf (Z ,W f
0)[Bf (F ) + Bf (GT )]
Determine total decay rate:
λ =ln 2
T1/2
=∑
f
λf
Lanczos Strength Function
Let O be an operator acting on some initial state |Φini 〉, we obtain the state
O|Φini 〉 whose norm is the sum rule of the operator O in the initial state:
S = |O|Φini〉| =√
〈Φini |O2|Φini〉
Depending on the nature of the operator O,the state O|Φini 〉 belongs to the same nucleus (if O is a e.m transition operator) or
to another (Gamow-Teller, nucleon transfer, a†
j /aj , ββ, ...)
If the operator O does not commute with H, O|Φini 〉 is not necessarily an
eigenvector of the system BUT it can be developped in energy eigenstates:
O|Φini〉 =∑
f
S(Ef )|Ef 〉 and 〈Φini |O2|Φini〉 =
∑
f
S2(Ef )
with S(Ef ) = 〈Ef |O|Φini〉 being the strength function (or structure
function)
Lanczos Structure Function
If we carry on the Lanczos procedure
using |O〉 = O|Φini〉 as initial pivot.
then H is again diagonalized to obtain the eigenvalues |Ef 〉
U is the unitary matrix that diagonalizes H and gives the
expression of the eigenvectors in terms of the Lanczos vectors:
U =
|E1 |E2〉 |E3〉 ... |EN〉|O〉|2〉|3〉::
|N〉
S(Ef ) = U(1, f ) = 〈Ef |O〉 = 〈Ef |O|Φini〉
How good is the Strength function S obtained at iteration N
compared to the exact one S?
Lanczos Structure Function
Any distribution can be characterized by the moments of the
distribution.
E = 〈O|H|O〉 =∑
f
Ef S2(Ef )
mn = 〈O|(H − E)n|O〉 =∑
f
(Ef − E)nS2(Ef )
Gaussian distribution characterized by two
moments (E , σ2 = m2)
g(E) = 1
σ√
2πexp(− (E−E)2
2σ2 )
Eg(
E)
2σ
Lanczos Structure Function
Lanczos method provides a natural way of determining a finite
number of momenta.
Initial vector |1〉 = |O〉√〈O|O〉
E12|2〉 = (H − E11)|1〉E23|3〉 = (H − E22)|2〉 − E12|1〉. . .ENN+1|N + 1〉 = (H − ENN)|N〉
−EN−1N |N − 1〉
whereENN = 〈N|H |N〉, ENN+1 = EN+1N
Each Lanczos iteration gives informa-tion about two new moments of the dis-tribution.
E11 = 〈1|H |1〉 = E
E212 = 〈O|(H − E11)
2|O〉 = m2
E22 =m3
m2
+ E
E223 =
m4
m2
−m2
3
m22
− m2
Diagonalizing Lanczos matrix after N iterations gives an approximation to
the distribution with the same lowest 2N moments.
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
Evolution of Strength Distribution
GT Strength on 48Sc
Evolution of Strength Distribution
GT Strength on 48Sc
48Ca(p,n)48Sc Strength Function
48Ca(p,n)48Sc Strength Function
Quenching of GT operator in the pf -shell
Nucleus Uncorrelated Correlated Expt.
Unquenched Q = 0.74
51V 5.15 2.42 1.33 1.2 ± 0.154Fe 10.19 5.98 3.27 3.3 ± 0.555Mn 7.96 3.64 1.99 1.7 ± 0.256Fe 9.44 4.38 2.40 2.8 ± 0.358Ni 11.9 7.24 3.97 3.8 ± 0.459Co 8.52 3.98 2.18 1.9 ± 0.162Ni 7.83 3.65 2.00 2.5 ± 0.1
Quenching of GT strength in the pf -shell
Quenching of GT strength in the pf -shell
Quenching of M1 operator in the pf -shell
KB3 interaction
Neumann-Cosel et al.
Phys. Lett. B433 1 (1998)
Quenching of M1 operator in the pf -shell
GXPF1 interaction
−4 −2 0 2 4 6 8 10µfree
(µN)
−4
−2
0
2
4
6
8
10
µexp (
µ N)
−4 −2 0 2 4 6 8 10µeff
(µN)
−4
−2
0
2
4
6
8
10
µexp (
µ N)
M. Homma, T. Otsuka, B. A. Brown, T. Mizusaki
Phys. ReV. C69, 034335 (2004)
Quenching of M1 operator in the pf -shell
0
0.04
0.08
6 8 10 12 140.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
6 8 10 12 14Excitation Energy (MeV)
0.0
0.5
1.0
Orbital
Spin
Shell−ModelTotal
Expt.
B(M
1) (
µ N2)
52Cr
K. Langanke, G. Martinez-Pinedo,
P. Von Neumann-Cosel, and A. Richter
Phys. Rev. Lett. 93, (2004) 202501
KB3G interaction
Quenching of GT operator in the pf -shellIf we write
|i〉 = α|0~ω〉+∑
n 6=0
βn|n~ω〉,
|f 〉 = α′|0~ω〉+∑
n 6=0
β′n|n~ω〉
then
〈f ‖ T ‖ i〉2 =
αα′ T0 +∑
n 6=0
βnβ′n Tn
2
,
n 6= 0 contributions negligible
α ≈ α′
projection of the physical wavefunction in the
0~ω space is Q ≈ α2
transition quenched by Q2
GT and M1 strengths in 12C
0 50 100 150 200E (MeV)
0
0.02
0.04
0.06
0.08
GT
Str
engt
h
0
0.02
0.04
0.06
0.08
GT
Str
engt
h
4~ω
6~ω
/10
/10
67% in 0~ω states
78% in 0~ω states
0 50 100 150 200E (MeV)
0
0.1
0.2
B(M
1) (
µ N2 )
0
0.1
0.2
0.3
B(M
1) (
µ N2 )
4~ω
6~ω
/10
/10
64% in 0~ω states
75% in 0~ω states
Correlations in nuclei
V. R. Pandharipande, I. Sick and P. K. A. deWitt
Huberts, Rev. mod. Phys. 69 (1997) 981
SCGF microscopic description
Lecture 2
Effective Interactions
Symmetries of the hamiltonian
The information related to the two body interaction is fully
contained in the matrix elements:
〈(j1j2)JTMTz
|V |(j3j4)JTMTz
〉
Symmetries of the hamiltonian have the following
consequences:
at fixed J (T) values, the matrix elements corresponding to
all possible values of M (Tz) are equal
matrix elements between different J (T) values are
vanishing
The effective interaction
Different approaches are possible to determine the effective
interaction, or equivalently the set of matrix elements:
〈(j1j2)JT |VJTj1j2j3j4
|(j3j4)JT 〉
EMPIRICAL
E icalc. =
p∑
k=1
c(i)k vk
and minimize the function
Q2 =N∑
i=1
(E icalc. − E i
exp.)2 i. e.
∂Q2
∂vr= 0 = ∂
∂vr
(N∑
i=1
((p∑
k=1
c(i)k vk )− E i
exp.)2
)
for r=1, ... p
The effective interaction
N∑
i=1
((
p∑
k=1
c(i)k
vk )− E iexp.))c
(i)r = 0
from thereN∑
i=1
((p∑
k=1
c(i)k vk )− E i
exp.) =N∑
i=1
c(i)r E i
exp.
i.e. a set of p linear equations with p parameters with the
hypothesis that the wave functions do not change (c(i)k fixed)
Effective interactions
choose
initial
parameters
calculate
hamiltonian
matrix
calculate
eigenvectors
eigenvalues
no
yesconv.? end
parameters
of
variation
new
interaction
parameters
new
linear
system
Realistic Interactions
Free nucleon-nucleon interaction:
N N
V(1,2)
• Spin state S=0,1
• Charge state pp,pn,nn (isospin T)
• Spatial state L=even,odd
δ
• Deuteron properties
=⇒ Realistic potentials : Hamada-Johnston, Paris, CD-Bonn,
Argonne, Idaho-A
v(NN) = vEM (NN) + vπ(NN) + vR(NN)
Failure of realistic potentials in many-body
calculations
NN potentails with hard core are not suited for low energy nuclear physicscalculations. Before using in medium, the realistic potential is ”softened”(G-matrix, Vlowk , SRG). Effective interactions for SM take into account the coreparticles and scattering to outer space by perturbation (MBPT).
0
2.5
5
8 14 16
Ee
xc(
2+)
(Me
V)
Neutron number
O, Z=8
THE EXP 0
2.5
5
20 28 32
Neutron number
Ca, Z=20
THE EXP
Realistic effective interactions
Exp. KB KLS Bonn A Bonn B Bonn C
2+1 excitation energy
44Ca 1.16 1.45 1.43 1.31 1.25 1.2646Ca 1.35 1.45 1.42 1.26 1.22 1.2348Ca 3.83 1.80 1.60 1.23 1.30 1.4150Ca 1.03 1.41 1.35 1.27 1.10 1.17
〈(f 72)8|ΨGS〉 0.468 0.381 0.214 0.345 0.437
56Ni model space (f 72p 3
2)16
56Ni 2.70 0.39 0.31 0.43 0.42 0.42
〈(f 72)16|ΨGS〉 0.04 0.015 0.018 0.011 0.019
〈np3/2〉 4.5 5.2 5.7 5.2 5.0
Realistic effective interactions
Exp. KB KLS Bonn A Bonn B Bonn C
2+1 excitation energy
44Ca 1.16 1.45 1.43 1.31 1.25 1.2646Ca 1.35 1.45 1.42 1.26 1.22 1.2348Ca 3.83 1.80 1.60 1.23 1.30 1.4150Ca 1.03 1.41 1.35 1.27 1.10 1.17
〈(f 72)8|ΨGS〉 0.468 0.381 0.214 0.345 0.437
56Ni model space (f 72p 3
2)16
56Ni 2.70 0.39 0.31 0.43 0.42 0.42
〈(f 72)16|ΨGS〉 0.04 0.015 0.018 0.011 0.019
〈np3/2〉 4.5 5.2 5.7 5.2 5.0
The realistic interactions do not reproduce the shell closure
N or Z=28back
Modern nuclear forces: N3LO
We use pions and nucleons asdegrees of freedom.
The effective Lagrangian isclassified using a systematicexpansion based on a powercounting in terms of (Q/Λχ)ν ,where ν is called chiral order andΛχ is the hard scale (∼700MeV)
ν=0 is called leading order, ν > 1are called next-to-ν − 1 leadingorders.
Note hierarchy of nuclear forces.
Coupling constants (LEC)adjusted to phase shifts anddeuteron properties.
E. Epelbaum et al., Rev. Mod. Phys. 81 (2009) 1773.
N3LO and no-core SM calculations
0
4
8
NN+NNN Exp NN
3 + 3
+1
+
1 +
0 +; 1
0 +; 1
1 +
1 +
2 +
2 +
3 +
3 +
2 +; 1
2 +; 1
2 +
2 +
4 +
4 +
2 +; 1
2 +; 1
hΩ=14
0
4
8
12
16
NN+NNN Exp NN
3/2-
3/2-
1/2-
1/2-
5/2-
5/2-
3/2-
3/2-
7/2-
7/2-
5/2-
5/2-
5/2-
5/2-
1/2-; 3/2
1/2-; 3/2
0
4
8
12
16
NN+NNN Exp NN
0 +
0 +
2 +
2 +
1 +
1 +
4 +
4 +
1 +; 1
1 +; 1
2 +; 1
2 +; 1
0 +; 1
0 +; 1
0
4
8
12
16
NN+NNN Exp NN
1/2- 1/2
-
3/2-
3/2-
5/2-
5/2-
1/2-
1/2-
3/2-
3/2-
7/2-
7/2-
3/2-; 3/2
3/2-; 3/2
10B
11B
12C 13
C
Excitation energies (in MeV) in light nuclei in NOCSM with chiral EFT interactions.NN force at N3LO level, 3N force from N2LO.P. Navratil et al., Phys. Rev. Lett. 99 (2007) 042501.
Effective 2-body HamiltoniansImportant features
/ Effective 2-body Hamiltonians derived from realistic
potentials fail to reproduce correctly the spectroscopy of
many-body systems and bulk properties of nuclei-no right
binding, no double shell closures...
, The matrix elements depend very little on the potential used
(Argonne, Paris, Bonn, N3LO...) and the method of
regularization (G-matrix, Vlowk , ...). The link to phase shifts is
then nearly model independent.
,/ NN potentials are nowadays ”perfect”. We need a NNN
force.
What we can do ?
we have solved the problem of s.p. energies replacing
them by the experimental ones
we can treat all the matrix elements as free parameters
and fit them to the many-body data (e.g. USD, USDb
interactions)
or we can proceed in a more general manner: understand
the physics behind the set of the matrix elements and
change only some of them
Outline
Separation of the effective Hamiltonian: monopole and
multipole
3-body forces: corrections to the monopole Hamiltonian
Some useful definitions
Separation of the effective HamiltonianMonopole and multipole
From the work of M. Dufour and A. Zuker (PRC 54 1996 1641)
Separation theorem:
Any effective interaction can be split in two parts:
H = Hmonopole + Hmultipole
Hmonopole: spherical mean-field
Zresponsible for the global saturation properties and for the
evolution of the spherical single particle levels.
Hmultipole: correlator
Zpairing, quadrupole, octupole...
Important property:
〈CS ± 1|H|CS ± 1〉 = 〈CS ± 1|Hmonopole|CS ± 1〉
Effective HamiltonianMonopole and Multipole Hamiltonians
V =∑
JT
V JTijkl
[
(a+i a+
j )JT (ak al)
JT]00
In order to express the number of particles operators
ni = a+i
ai ∝ (a+i
ai)0,
particle-hole recoupling :
V =∑
λτ
ωλτikjl
[
(a+i
ak )λτ (a+
jal)
λτ]00
ωλτikjl ∝
∑
JT
V JTijkl
i k λ
j l λ
J J 0
12
12 τ
12
12
τ
T T 0
Effective HamiltonianMonopole and Multipole Hamiltonians
Hmonopole corresponds only to the terms λτ=00 and 01 which
implies that i = j and k = l and writes as
Hmonopole =∑
i
niǫi +∑
i≤j
ni .nj Vij
Hmultipole corresponds to all other combinations of λτ
Effective HamiltonianMultipole Hamiltonian
Hmultipole can be written in two representations, particle-particle
Hmultipole =∑
ik<jlΓ
WΓijkl [(a
†i a
†j )
Γ(akal)Γ]0,
where Γ = JT or in particle-hole
Hmultipole =∑
ik<jlΓ
√
(2γ + 1)
√
(1 + δij)(1 + δkl)
4ωγ
ikjl [(a†i ak)
γ(a†j al)
γ ]0,
where γ = λτThe W and ω matrix elements are related by a Racah transformation:
ωγikjl =
∑
Γ
(−)j+k−γ−Γ
i j Γl k γ
WΓijkl(2J + 1)(2T + 1),
WΓijkl =
∑
γ
(−)j+k−γ−Γ
i j Γl k γ
ωγikjl(2λ+ 1)(2τ + 1).
Effective HamiltonianMultipole Hamiltonian
In the preceding expressions we can replace pairs of indices by a single one
ij ≡ x , kl ≡ y , ik ≡ a et jl ≡ b, and diagonalise the matrices W Γxy and
f γab = ωγab
√
(1 + δij)(1 + δkl)/4, by unitary transformations UΓxk , u
γak :
U−1
WU = E =⇒ WΓxy =
∑
k
UΓxk U
Γyk E
Γk
u−1
fu = e =⇒ fγab =
∑
k
uγaku
γbke
γk ,
then
Hmultipole =∑
k,Γ
EΓk
∑
x
UΓxk Z
+xΓ ·∑
y
UΓyk ZyΓ,
Hmultipole =∑
k,γ
eγk
(∑
a
uγakS
γa
∑
b
uγbkS
γb
)0
[γ]1/2,
that we call representations E and e.
Effective HamiltonianMultipole Hamiltonian
E-eigenvalue density for the KLS interaction in the pf+sdg major shells ~ω = 9. Eacheigenvalue has multiplicity [Γ].
Effective HamiltonianMultipole Hamiltonian
e-eigenvalue density for the KLS interaction in the pf+sdg major shells ~ω = 9. Eacheigenvalue has multiplicity [γ].
0
5000
-4 -3 -2 -1 0 1 2 3 4 5
num
ber
of s
tate
s
e-eigenvalue density
2 0+4 0+
1 0+ 3 0-
1 1+
1 0-
Energy (Mev)
Effective HamiltonianMultipole Hamiltonian
E-eigenvalue density for the KLS interaction in the pf+sdg major shells ~ω = 9, afterremoval of the five largest multipole contributions. Each eigenvalue has multiplicity [Γ].
0
2000
-10 -8 -6 -4 -2 0 2 4
num
ber
of s
tate
s
1 0+ 0 1+ 2 0+
Energy (Mev)
E-eigenvalue densityepsilon=2.0
Multipole Hamiltonian
Hmultipole can be written in two representations, particle-particleand particle-hole. Both can be brought into a diagonal form.When this is done, it comes out that only a few terms arecoherent, and those are the simplest ones:
L = 0 isovector and isoscalar pairing
Elliott’s quadrupole
~σ~τ · ~σ~τ
Octupole and hexadecapole terms of the type rλYλ · rλYλ
Besides, they are universal (all the realistic interactions give
similar values) and scale simply with the mass number
Interaction particle-particle particle-holeJT = 01 JT = 10 λτ = 20 λτ = 40 λτ = 11
KB3 -4.75 -4.46 -2.79 -1.39 +2.46FPD6 -5.06 -5.08 -3.11 -1.67 +3.17
GOGNY -4.07 -5.74 -3.23 -1.77 +2.46
Multipole Hamiltonian
γ eγ
1 eγ
2 eγ
1+2 M 〈u1|M〉 〈u2|M〉 〈u1|u′1〉 〈u2|u
′2〉 α
2
11 1.77 2.01 3.90 στ .992 .994 .999 1.000 .94
20 -1.97 -2.14 -3.88 r2Y2 .996 .997 1.000 1.000 .9510 -1.02 -0.97 -1.96 σ .880 .863 .997 .994 1.04
21 -0.75 -0.85 -1.60 r2Y2τ .991 .998 .999 .997 .94
40 -1.12 -1.24 -2.11 r4Y4
Γ EΓ1 EΓ
2 EΓ1+2 P 〈U1|P〉 〈U2|P〉 〈U1|U
′1〉 〈U2|U
′2〉 α
2
01 -2.95 -2.65 -5.51 P01 .992 .998 1.000 .994 1.04810 -4.59 -4.78 -10.12 P10 .928 .910 .998 .997 .991
Effective HamiltonianMonopole Hamiltonian
The evolution of effective spherical single particle energies with
the number of particles in the valence space can be extracted
from Hmonopole. In the case of identical particles the expresion
is:
ǫj(n) = ǫj(n = 1) +∑
i
V 1ij ni
The monopole Hamiltonian Hmonopole also governs the relative
position of the various T-values in the same nucleus, via the
terms:
bij Ti · Tj
Even small defects in the centroids can produce large changes
in the relative position of the different configurations due to the
appearance of quadratic terms involving the number of
particles in the different orbits
Effective HamiltonianClosed shell vs 4p4h states in 56Ni
ECS = 16 ǫf + 16∗152 Vff
2p3/22p1/21f5/2Π ν
1f7/2
r
f
E4p4h = 12 ǫf + 4 ǫr
66Vff + 48Vfr + 6Vrr
2p3/22p1/21f5/2Π ν
1f7/2f
r
∆ E = 4(ǫf − ǫr ) + 48(Vff − Vfr ) + 6(Vff − Vrr )
Effective HamiltonianMonopole Hamiltonian
table
Effective HamiltonianMonopole Hamiltonian: quadratic effects
Two shells (i and j ) system:
Hmonopole = E0 + niǫi + njǫj +ni(ni − 1)
2Vii +
nj(nj − 1)
2Vjj + ninjVij
It is possible to rewrite this equation and separate a global term H0
(depending only on the total number of particles n = ni + nj ) from a linear
term H1 and a quadratic term H2 in ni et nj to get:
Hmonopole = E0 + n ǫ0 +n(n − 1)
2W0
︸ ︷︷ ︸
+ Γij [ǫ1 + (n − 1)W1]︸ ︷︷ ︸
+ Γ(2)ij W2︸ ︷︷ ︸
= H0 + H1 + H2
with
Γij =Djni − Dinj
Di + Dj
Γ(2)ij =
Di Dj
2
(2ninj
Di Dj
− ni(nj − 1)
Di(Dj − 1)− ni(nj − 1)
Dj(Di − 1)
)
Effective HamiltonianMonopole Hamiltonian: quadratic effects
νΠ
1p3/2
1p1/2
1d5/22s1/2
1d3/2
νΠ
1p3/2
1p1/2
1d5/22s1/2
1d3/2
0 1 2 3 40
5
10
15
20
25
30
35
40
45
lineairequadratique
kp-kh excitation energies for 16O
3-body forcescorrections to the monopole Hamiltonian
The 2-body potentials are now ’perfect’. Exact calculations are
possible for light systems. The bad spectroscopy has to be then
related with the lack of 3-body forces.
Let’s add 3-body term to the monopole Hamiltonian
Hmonopole =∑
i
eini
︸ ︷︷ ︸
+∑
i≤j
aijnij +∑
i≤j
bijTij
︸ ︷︷ ︸
+∑
ijk
aijknijk +∑
ijk
bijkTijk
︸ ︷︷ ︸
,
1-body 2-body 3-body
where
nijk = ninjnk , Tijk = TiTjTk , etc.
3-body forces I
One should remember that the 2-body force contributes to the
1-body piece of the effective interaction, when we make
summation over orbits in the core:
∑
c
aicninc = ni
∑
c
aicnc ≡ niei
Similarly, the 3-body force contributes to 1-body and 2-body
terms.
∑
c
aijcninjnc = ninj
∑
c
aijcnc ≡ ninjaij
∑
cc′aic′cnincncc′ = ni
∑
cc′aicc′ncnc′ ≡ niei
3-body forces II
ZDifferent studies (no-core, coupled-cluster) suggest that the
residual 3-body force is much smaller than 1-and 2-body parts
of the 3-body force. As a first step one should look to the
contributions of the 3-body terms to 1 and 2-body pieces.
Contributions from
3-body force to 4He
binding energy (from
Hagen et al. Phys.
Rev. C 76, 034302,
2007)
Shell gapDefinition
The shell gap ∆ is defined as the difference of binding energies
(positively defined)
∆ = [BE(N)− BE(N − 1)]− [BE(N + 1)− BE(N)]
= 2BE(N)− BE(N + 1)− BE(N − 1)
ZThe uncorrelated shell gap is therefore equal to the difference
of the corresponding ESPE.
Computing session
This afternoon:
you will learn what a Hamiltonian file contains
you will check yourself how calculated spectra differ
depending on the interaction used (tuned, not tuned)
you will do some mathematics on the paper, please bring
calculators!
you will learn the usage of the option 52 to
calculate/change monopoles
you will ”repare” yourself a bad interaction
you will calculate ESPE
Further reading
E. Caurier et al., The shell model as a unified view of the
nuclear structure, Rev. Mod. Phys. 77 (2005) 427.
M. Dufour, A.P. Zuker, The realistic collective nuclear
Hamiltonian, Phys. Rev. C52 (1996) 1641.
A.P. Zuker, Separation of the monopole contribution to the
nuclear Hamiltonian, nucl-th/9505012.
A.P. Zuker, Three body monopole corrections to the
realistic interactions, Phys. Rev. Lett.90 (2003) 042502.
A. Schwenk, A.P. Zuker, Shell model phenomenology of
low momentum interactions, Phys. Rev. C74 (2007)
061302.
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