Post on 13-Mar-2020
Quantum Monte Carlo Studies ofthe Structure of Light Nuclei
Robert B. Wiringa
Physics Division, Argonne National Laboratory
WORK WITH
Joe Carlson, Los Alamos
Ken Nollett, Argonne
Muslema Pervin, Argonne
Steve Pieper, Argonne
Rocco Schiavilla, Jefferson Lab & Old Dominion
Work not possible without extensive computer resources:
Argonne Laboratory Computing Resource Center (Jazz)Argonne Math. & Comp. Science Division (BlueGene/L)
NERSC IBM SP’s (Seaborg, Bassi)
Work supported by U.S. Department
of Energy, Office of Nuclear PhysicsPhysics Division
GOAL OF ab-initio L IGHT-NUCLEI CALCULATIONS
We seek to understand nuclei as collections of interacting nucleons by reliably solving the
many-nucleon Schrodinger equation for realistic Hamiltonians of the form
H =X
i
Ki +X
i<j
vij +X
i<j<k
Vijk + ...
Using quantum Monte Carlo methods we want to compute
• Binding energies, excitation spectra, relative stability
• Densities, moments, transition amplitudes, cluster-cluster overlaps
• Low-energyNA & AA scattering, astrophysical reactions
With accurate calculations we can rigorously test a given Hamiltonian.
At present our methods are limited to light (A ≤ 12) nuclei
and local potentials with weak quadratic-momentum dependence.
ARGONNE V18
Ki = − ~2
4[( 1
mp+ 1
mn) + ( 1
mp− 1
mn)τzi]∇
2i
vij = vγij + vπ
ij + vIij + vS
ij =P
p vp(rij)Opij
vγij : pp, pn & nn electromagnetic terms
vπij ∼ [Yπ(rij)σi · σj + Tπ(rij)Sij ] ⊗ τ i · τj
vIij =
P
p IpT 2
π (rij)Opij
vSij =
P
p[P p +Qpr + Rpr2]W (r)Opij
Opij = [1, σi · σj , Sij ,L · S,L2,L2(σi · σj), (L · S)2]
+ [1, σi · σj , Sij ,L · S,L2,L2(σi · σj), (L · S)2] ⊗ τi · τj
+ [1, σi · σj , Sij ,L · S] ⊗ Tij
+ [1, σi · σj , Sij ,L · S] ⊗ (τiz + τjz)
Sij = 3σi · rijσj · rij − σi · σj Tij = 3τizτjz − τi · τj
Fits Nijmegen PWA93 data base of 1787pp & 2514np observables forElab ≤ 350 MeV
with χ2/datum = 1.1 plusnn scattering length and2H binding energy
THREE-NUCLEON POTENTIALS
Urbana IX (UIX)
Vijk = V 2πPijk + V R
ijk
Illinois 2 (IL2)
Vijk = V 2πPijk + V 2πS
ijk + V 3π∆Rijk + V R
ijk
THE MANY-BODY PROBLEM
Need to solve
HΨ(~r1, ~r2, · · · , ~rA; s1, s2, · · · , sA; t1, t2, · · · , tA) = EΨ(~r1, ~r2, · · · , ~rA; s1, s2, · · · , sA; t1, t2, · · · ,
si are nucleon spins:± 12
ti are nucleon isospins (proton or neutron):± 12
2A ד
AZ
”
complex coupled2nd order differential equations in3A dimensions
(number of isospin states can be reduced)
12C: 270,336 coupled equations in 36 dimensions
Coupling is strong:
• 〈vtensor〉 is ∼ 60% of total 〈vij〉
• 〈vtensor〉 = 0 if no tensor correlations
-1 0 1
-1
0
1
fem
tom
eter
MJ = 0
femtometer-1 0 1
-1
0
1
MJ = 1
VARIATIONAL MONTE CARLO
Minimize expectation value ofH
EV =〈ΨV |H|ΨV 〉
〈ΨV |ΨV 〉≥ E0
Trial function (s-shell nuclei)
|ΨV 〉 =
2
41 +X
i<j<k
UT NIijk
3
5
"
SY
i<j
(1 + Uij)
# "
Y
i<j
fc(rij)
#
|ΦA(JMTT3)〉
|Φd(1100)〉 = A| ↑ p ↑ n〉 ; |Φα(0000)〉 = A| ↑ p ↓ p ↑ n ↓ n〉
Uij =X
p=2,6
up(rij)Opij ; UT NI
ijk = −ǫVijk(rij , rjk, rki)
Functionsfc(rij) andup(rij) are obtained numerically from solution of coupled differential
equations containingvij .
Correlation functions
0 1 2 3 4 5 6 7 8r (fm)
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
fc,utτ (2H)
fc,utτ (3H)
fc,utτ (4He)
fsp,utτ (6Li)
φp (6Li)
fc
10 utτ
φp
fsp
[f c(4He)]
3
[f c(3H)]
2
Trial function (p-shell nuclei)
⇒ A
8
<
:
Y
i<j≤4
fss(rij)X
LS[n]
“
βLS[n]
Y
k≤4<l≤A
fsp(rkl)Y
4<l<m≤A
fpp(rlm)
|Φα(0000)1234Y
4<l≤A
φLS[n]p (Rαl)
˘
[Yml1 (Ωαl)]LML
⊗ [χl(12ms)]SMS
¯
JM[νl(
12t3)]T T3
〉
9
=
;
Permutation symmetry
A [n] L (T, S)
6 [2] 0, 2 (1, 0), (0, 1)
[11] 1 (1, 1), (0, 0)
7 [3] 1, 3 (1/2, 1/2)
[21] 1, 2 (3/2, 1/2), (1/2, 3/2), (1/2, 1/2)
[111] 0 (3/2, 3/2), (1/2, 1/2)
8 [4] 0, 2, 4 (0, 0)
[31] 1, 2, 3 (1, 1), (1, 0), (0, 1)
[22] 0, 2 (2, 0), (1, 1), (0, 2), (0, 0)
[211] 1 (2, 1), (1, 2), (1, 1), (1, 0), (0, 1)
Diagonalization
in βLS[n] basis to produce energy spectraE(Jπx ) and orthogonal excited statesΨV (Jπ
x )
Expectation values
ΨV (R) represented by vector with2A × (AZ) spin-isospin components for each space
configurationR = (r1, r2, ..., rA); Expectation values are given by summation over samples
drawn from probability distributionW (R) = |ΨP (R)|2:
〈ΨV |O|ΨV 〉
〈ΨV |ΨV 〉=
X Ψ†V (R)OΨV (R)
W (R)/
X Ψ†V (R)ΨV (R)
W (R)
Ψ†Ψ is a dot product andΨ†OΨ a sparse matrix operation.
Scaling of calculation
A P NS × NTQ
(×8Be)
4He 4 6 16×2 0.001
6Li 6 15 64×5 0.036
8Be 8 28 256×14 1.10B 10 45 1024×42 24.12C 12 66 4096×132 530.
GREEN’ S FUNCTION MONTE CARLO
Projects out lowest energy state from variational trial function
Ψ(τ) = exp[−(H − E0)τ ]ΨV =X
n
exp[−(En − E0)τ ]anψn
Ψ(τ → ∞) = a0ψ0
Evaluation ofΨ(τ) done stochastically in small time steps∆τ
Ψ(Rn, τ) =
Z
G(Rn,Rn−1) · · ·G(R1,R0)ΨV (R0)dRn−1 · · · dR0
using the short-time propagator accurate to order(∆τ)3
Gαβ(R,R′) = eE0δτG0(R,R′)〈α|
"
SY
i<j
gij(rij , r′ij)
g0,ij(rij , r′ij)
#
|β〉
where the free many-body propagator is
G0(R,R′) = 〈R|e−Kτ |R′〉 =
»r
m
2π~2τ
–3A
exp
»
−(R − R′)2
2~2τ/m
–
andg0,ij andgij are the free and exact two-body propagators
gij(rij , r′ij) = 〈rij |e
−Hijτ |r′ij〉
Mixed estimates
〈O(τ)〉Mixed =〈ΨV |O|Ψ(τ)〉
〈ΨV |Ψ(τ)〉; 〈O(τ)〉 ≈ 〈O(τ)〉Mixed + [〈O(τ)〉Mixed − 〈O〉V ]
〈H(τ)〉Mixed =〈Ψ(τ/2)|H|Ψ(τ/2)〉
〈Ψ(τ/2)|Ψ(τ/2)〉≥ E0
Propagator cannot containp2, L2, or (L · S)2 operators:
Gβα(R′,R) has onlyv′8〈v18 − v′8〉 computed perturbatively with extrapolation (small for AV18)
Reliable in Faddeev (3H), hyperspherical harmonic & Yakubovsky (4He) comparisons
Fermion sign problem limits maximumτ :
Gβα(R′,R) brings in lower-energy boson solution
〈ΨT |H|Ψ(τ)〉 projects back fermion solution
Exponentially growing statistical errors
Constrained-path propagation, removes steps that have
Ψ†(τ,R)Ψ(R) = 0
Possible systematic errors reduced by10−20 unconstrained steps before evaluating observables.
GFMC propagation of three states in6Li
0 0.5 1 1.5 2 2.5
-32
-30
-28
-26
-24
τ (MeV-1)
E(τ
) (
MeV
)
2+
3+
gs
6Li
α+d
GFMC FOR SECOND EXCITED STATES OF SAMEJπ
TheΨT are constructed by non-orthogonal basis diagonalization in p-shell wave functions.
Example:7Li(5/2-) has 4 symmetry possibilities:2F[43] , 4P[421] ,4D[421] , 2D[421]
〈ΨT (2nd 52
−)|ΨT (1st 5
2
−)〉 = 0 , but〈ΨGFMC(2
nd 52
−)|ΨT (1st 5
2
−)〉 need not be zero.
Will e−(H−E0)τΨT (2nd 52
−) → ΨGFMC(1
st 52
−) ?
Can use〈ΨGFMC(i)|H|ΨGFMC(j)〉 and〈ΨGFMC(i)|ΨGFMC(j)〉 to rediagonalize
0 0.2 0.4 0.6 0.8 1-35
-30
-25
-20
-15
τ (MeV-1)
E(τ
) (
MeV
)
4th 5/2-
3rd 5/2-
2nd 5/2-
1st 5/2-
7Li(5/2-)AV18 + Illinois-2
GFMC Propagation
0 0.2 0.4 0.6 0.8 1
-0.05
-0.03
-0.01
0.01
0.03
τ (MeV-1)
⟨Ψi|Ψ
1⟩
⟨Ψ2|Ψ1⟩⟨Ψ3|Ψ1⟩⟨Ψ4|Ψ1⟩
Binding energy results forA=3,4
Hamiltonian Method 3H 3He 4He
Argonnev′
8VMC* 25.44(2)
(no EM) GFMC1 25.93(2)
FY2 25.94(5)
HH3 25.90(1)
SVM4 25.92
EIHH5 25.944(10)
CRCGV6 25.90
NCSM7 25.80(20)
Argonnev18 VMC* 7.50(1) 6.77(1) 23.70(2)
GFMC1 7.61(1) 6.89(1) 24.07(4)
F/FY2 7.623 6.924 24.28
PHH/HH3 7.623 6.925 24.18
Argonnev18 VMC* 8.31(1) 7.56(1) 27.72(2)
+ Urbana IX GFMC1 8.46(1) 7.70(1) 28.33(2)
F/FY2 8.478 7.760 28.50
PHH/CHH3 8.480 7.749 28.46
Experiment 8.482 7.718 28.296
* Arriaga, Pandharipande, Wiringa 1 Carlson, Pieper, Wiringa 2 Kamada, Nogga, Glockle 3 Viviani, Kievsky, Rosati
4 Varga, Suzuki 5 Barnea, Leideman, Orlandini 6 Hiyama, Kamimura 7 Navratil, Barrett
-70
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
Ene
rgy
(MeV
)
α+d
α+t
2α 2α+n9Be+n
6Li+α
ΨT
GFMC
0+
4He 1+
3+2+
6Li3/2−1/2−
7/2−5/2−
7Li
0+
2+
4+
1+3+
8Be3/2−
5/2−1/2−7/2−9/2−
9Be0+2+
3+
4+
1+
10Be3+1+2+
4+
10B
α+d
α+t
2α 2α+n9Be+n
6Li+αArgonne v18 + IL2
VMC Trial &GFMC Calculations
-60
-55
-50
-45
-40
-35
-30
-25
-20E
nerg
y (M
eV)
α+2n
α+d
α+t
6He+2n
7Li+n
2α
AV18
UIXIL2 Exp
0+
4He
1+
0+
2+
6He 1+
3+2+1+
6Li
5/2−
5/2−
3/2−1/2−
7/2−5/2−5/2−
7/2−
3/2−
7Li
1+
0+
2+
8He
2+1+
0+
3+1+
4+
8Li
0+
2+
4+
2+1+3+
8Be
α+2n
α+d
α+t
6He+2n
7Li+n
2α
Argonne v18without & with various Vijk
GFMC Calculations
-65
-60
-55
-50
-45
-40
-35
-30E
nerg
y (M
eV)
8Li+n
2α+n9Be+n
6Li+α
AV18 UIX IL2 Exp3/2−
3/2−
1/2−
?5/2−
7/2−
9Li
3/2−
5/2−1/2−
9Be
0+
2+
0+
3,2+
10Be3+1+
10B
3+
1+
8Li+n
2α+n9Be+n
6Li+α
Argonne v18without & with VijkGFMC Calculations
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0E
nerg
y (M
eV)
AV18
IL2 Exp
1+
2H 1/2+
3H
0+
4He 1+3+2+
6Li 3/2−1/2−7/2−5/2−
7Li 2+1+
0+
3+
4+
8Li0+2+
4+
1+3+
8Be
3/2−1/2−5/2−7/2−
9Li
9/2−
5/2+
7/2+
3/2+
1/2+
3/2−5/2−1/2−
7/2−
9Be
3+1+
0+2+
4+
0+2+,3+
10Be3+1+2+4+
10B
3+
1+
2+4+
0+
12C
Argonne v18With Illinois-2
GFMC Calculations
-5
0
5
10
15
20
25E
xcita
tion
ener
gy (
MeV
)
AV18 +IL2 Exp
1+
0+
2+
2+0+
6He
1+
1+
3+
2+
1+
6Li
5/2−
5/2−
3/2−1/2−
7/2−
5/2−5/2−
7/2−
3/2−1/2−
7Li
2+
2+
2+1+
0+
3+1+
4+
3+
1+
8Li
1+
0+
2+
4+
2+1+3+
4+
0+
2+
3+2+
8Be
3/2−
3/2−
1/2−
?
5/2−
7/2−
9Li
9/2−3/2−
5/2+
7/2+
3/2+
1/2+
3/2+
3/2−1/2+5/2−1/2−5/2+3/2+
7/2−
3/2−
5/2−
9/2+
5/2+7/2+
9Be
0+
3+
2+
1+
0+
2+1−
2−
2+
4+
0+
2+,3+
10Be
3+1+
2+
2−
4+
1+
3+
1+2+
3+
10B
3+
1+
2+
2−
4+
1+
3+
1+
2+
Argonne v18With Illinois-2
GFMC Calculations 4 March 2006
NEW ILLINOIS POTENTIALS – PROGRESSREPORTI
• Illinois 1–5 parameters determined in 2000.
– Fits made toA ≤ 8 only
– Preliminary nuclear matter calculations at Urbana (Morales, Pandharipande, Ravenhall)
suggested at most IL2 is viable
– Improved GFMC results in worse8He agreement
• Started new fitting up toA = 10
• Michele Viviani (Pisa) finds sign error in one piece ofAσ in V 3πijk
– Formula was published correctly, but incorrectly programmed
– Increased attraction for all nuclei
• New fit made with correctedAσ : IL7
– parameters weaker than for IL2 because of increased attraction
– better quality reproduction of energies than IL2
– so far have not found any significant difference in other observables
• Nuclear and neutron matter are probably still too soft
-70
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20E
nerg
y (M
eV)
IL2IL7
Exp
0+
4He
1+
0+2+
2+0+
6He
1+
1+
3+
2+1+
6Li
3/2−
3/2−
1/2+
5/2+
1/2−
3/2−
5/2−
(5/2)−
7He
5/2−5/2−
3/2−1/2−
7/2−5/2−5/2−
7/2−
3/2−1/2−
7Li
1+0+2+
0+
2+
8He 2+
2+
2+1+
0+
3+1+
4+
3+
1+
8Li
1+
0+
2+
4+
2+1+
3+4+
0+
2+
3+2+
8Be
1/2−
9He
3/2−
3/2−1/2−
?5/2−
7/2−
9Li
9/2−3/2−
5/2+
7/2+
3/2+
1/2+
3/2+
3/2−1/2+5/2−1/2−5/2+3/2+
7/2−
3/2−
5/2−
9/2+
5/2+7/2+
9Be
0+
10He
2+
0+
(1-,2-)1+
10Li0+
3+
2+
1+
0+2+1−
2−
2+
4+
0+
3,2+
10Be 3+1+
2+
2−
4+
1+
3+
1+2+
3+
10B
Argonne v18With Illinois Vijk
GFMC Calculations22 October 2007
GFMC FOR SCATTERING STATES
GFMC treats nuclei as particle-stable system – should be good for energies of narrow resonances
Need better treatment for locations and widths of wide states and for capture reactions
METHOD
• Pick a logarithmic derivative,χ, at some large boundary radius (RB ≈ 9 fm)• GFMC propagation, using method of images to preserveχ atR, findsE(RB, χ)
• Phase shift,δ(E), is function ofRB , χ,E• Repeat for a number ofχ until δ(E) is mapped out
Example for5He(12
−)
• “Bound-state” boundarycondition does notgive stable energy;Decaying to n+4Hethreshold
• Scattering boundarycondition producesstable energy.
0 0.1 0.2 0.3 0.4-27
-26
-25
-24
-23
τ (MeV-1)
E(τ
) (
MeV
)
"bound state"
Log-deriv = -0.168 fm-1
4He + n – PARTIAL-WAVE CROSS SECTIONS
• Hale phase shifts fromR-matrix
analysis up toJ = 92
of data
• Tilted error bars from
δ(RB , χ, E ± ∆E)
• AV18+IL2 was not fit to5He,
good prediction of32
−& 1
2
−
resonances (both locations
and widths)
• 4He+n scattering length also
well reproduced
Nollett, et al., Phys.Rev.Lett.99, 022502 (2007)
0 1 2 3 4 50
1
2
3
4
5
6
7
Ec.m. (MeV)
σ LJ
(b)
12
+
12
-
32
-
R-Matrix
Pole location
RMS RADII OF HELIUM ISOTOPES
Recent measurement of6He charge radius at Argonne and8He at GANIL• Single4He or6He or8He atoms trapped• Isotope shift of an atomic transition measured• Small〈r2〉1/2 dependence of shift extracted using precise atomic calculations• 6He half-life only 0.807s;8He half-life only 0.119s
GFMC radius strongly dependent on propagated separation energy8He charge radius smaller than for6He
0.8 0.9 1 1.1 1.2 1.31.86
1.90
1.94
1.98
2.02
2.06
Esep′ = H′(4He) - H′(6He) (MeV)
rms
r p (f
m)
AV8’ modified
SSCC V8’ modified
IL2, other
IL2, VR’=1.254
IL6, VR’, ES=1.84
IL6, VR’, ES=1.48
IL6, 3 trm, RWS=4.0
IL6, 3 trm, ES=1.4
6He
2 2.2 2.4 2.6 2.8 3 3.21.78
1.80
1.82
1.84
1.86
1.88
Esep′ = H′(6He) - H′(8He) (MeV)
rms
r p (f
m)
AV8’ modified
SSCC V8’ modified
IL6, VR’ varied
IL6, VR’ varied
IL6, 3=trm, l3bc=6
IL6, 3=trm, l3bc=5
Experiment
8He
4,6,8HE DENSITIES
• 4He central density twice that of nuclear matter!• Neutrons drag4He center of mass around – spread out density
– results in charge radius of6He> 4He (2.08 fm vs 1.66 fm)• 6He & 8He have large neutron halos due to weak binding of neutrons• Neutron halo of6He more diffuse than that of8He – smallerEsep
0 0.5 1 1.5 2 2.5 30.00
0.05
0.10
0.15
r (fm)
Den
sity
(fm
-3)
4He6He - Proton6He - Neutron8He - Proton8He - Neutron
0 2 4 6 8 10 1210-5
10-4
10-3
10-2
10-1
r (fm)
Den
sity
× r2
(f
m-1
)
4He6He - Proton6He - Neutron8He - Proton8He - Neutron
TWO-NUCLEON DENSITIES
ρpp(r) =X
i<j
〈Ψ|δ(r − |ri − rj |)1 + τi
2
1 + τj
2|Ψ〉
0 0.5 1 1.5 2 2.5 30.000
0.005
0.010
0.015
0.020
r12 (fm)
ρ pp
(fm
-3)
4He6He - Proton8He - Proton
pair rms radii
rpp rnp rnn4He 2.41 2.35 2.416He 2.51 3.69 4.408He 2.52 3.58 4.37
TWO-NUCLEON KNOCKOUT – (e, e′pN)
• Recent (still being analyzed) JLab expt. for12C(e, e′pN)
• Measured back to backpp andnp pairs• Pairs with relative momentum 2–3 fm−1 show 10–20× np enhancement (preliminary).
0 1 2 3 4 5q (fm
-1)
10-1
100
101
102
103
104
105
ρ NN(q
,Q=
0) (
fm6 )
AV18/UIX 4He
AV4’4He
pp np
0 1 2 3 4 5q (fm
-1)
10-1
100
101
102
103
104
105
106
ρ NN(q
,Q=
0) (
fm6 )
3He
4He
6Li
8Be
• VMC calculations for3He,4He, and8Be show this effect
• Effect disappears when tensor correlations are turned off
• Shows importance of tensor correlations to> 2 fm−1.
GFMC FOR OFF-DIAGONAL MATRIX ELEMENTS
We can generalize the “mixed” estimates of expectation values for off-diagonal matrix elements
〈Ψf (τ)|O|Ψi(τ)〉 ≈ 〈O(τ)〉Mi+ 〈O(τ)〉Mf
− 〈O〉V ,
where
〈O〉V =〈Ψf
T |O|ΨiT 〉
q
〈ΨfT |Ψf
T 〉q
〈ΨiT |Ψi
T 〉,
〈O(τ)〉Mi=
〈ΨfT |O|Ψi(τ)〉
〈ΨiT |Ψi(τ)〉
v
u
u
t
〈ΨiT |Ψi
T 〉
〈ΨfT |Ψf
T 〉,
〈O(τ)〉Mf=
〈Ψf (τ)|O|ΨiT 〉
〈Ψf (τ)|ΨfT 〉
v
u
u
t
〈ΨfT |Ψf
T 〉
〈ΨiT |Ψi
T 〉,
Electromagnetic Transitions ofA = 6, 7 Nuclei – Widths in eV
JPi → JP
f Transition VMC GFMC Expt6Li(3+) →6Li(1+) E2 (10−4) 3.86 4.68(5) 4.40(34)6Li(0+) →6Li(1+) M1 (100) 7.10 6.86(2) 8.19(17)7Li( 1
2
−) →7Li( 3
2
−) E2 (10−7) 2.61 3.24(7) 3.30(20)
7Li( 12
−) →7Li( 3
2
−) M1 (10−3) 4.74 4.58(3) 6.30(31)
7Li( 72
−) →7Li( 3
2
−) E2 (10−2) 1.29 1.74(2) 1.50(20)
7Be( 12
−) →7Be( 3
2
−) E2 (10−7) 4.24 6.00(7) —
7Be( 12
−) →7Be( 3
2
−) M1 (10−3) 2.69 2.62(1) 3.43(45)
Weak Transitions ofA = 6, 7 Nuclei – log(ft)
JPi → JP
f Transition VMC GFMC Expt6He(0+) →6Li(1+) GT 2.901 2.916 2.9107Be( 3
2
−) →7Li( 3
2
−) F & GT 3.288 3.302 3.32
7Be( 32
−) →7Li( 1
2
−) GT 3.523 3.542 3.55
7Li( 12
−) / 7Li( 3
2
−) F & GT 10.38% 10.25% 10.44%
Isospin-mixing in8Be
Experimental energies of 2+ states
Ea = 16.626(3) MeV
Eb = 16.922(3) MeV
and 2α decay widths:
Γa = 108.1(5) keV
Γb = 74.0(4) keV
Assume isospin mixing of 2+;1 and 2+;0*
states due to isovector interactionH01:
Ψa = αΨ0 + βΨ1
Ψb = βΨ0 − αΨ1
α2 + β2 = 1
Decay throughT = 0 component only
Γa/Γb = α2/β2
α = 0.7705(15)
β = 0.6375(19)
-60
-56
-52
-48
-44
-40
-36
Ene
rgy
(MeV
)
8Li
2+;1
3P[431]
7Li+n
1+;1
3+;13
D[431]
8Be
α+α0
+;01
S[44]
2+;01
D[44]
4+;01
G[44]
2+;0+1
1+;1
1+;0
3+;0+1
8B
2+;1
7Be+p
1+;1
3P[431]
Ea,b =H00 + H11
2±
s
„
H00 − H11
2
«2
+ (H01)2
H00 = 16.746(2) MeV
H11 = 16.802(2) MeV
H01 = −145(3) keV
F. C. Barker [Nucl.Phys.83, 418 (1966)] estimated the Coulomb matrix element
connecting the 2+;1 and 2+;0* states asHC01 = −67 keV
The 1+;1 and 1+;0 and the 3+;1 and 3+;0 levels also mix, but decay by nucleon emission. Barker assigns
values of:
α1 = 0.24 ; β1 = 0.97 ; H01 = −120(1) keV
α3 = 0.41 ; β3 = 0.91 ; H01 = −62(15) keV
Isospin-mixing matrix elements in keV
H01 KCSB V CSB Vγ (Coul) (Mag)
2+;1⇔2+;0* VMC −107(2) −2.5(2) −23.8(4) −80.8(12) −69.4(11) −11.4(1)
GFMC −115(3) −3.1(2) −21.3(6) −90.3(26) −78.3(25) −12.0(2)
Barker −145(3) −67
1+;1⇔1+;0 VMC −70(1) −1.8(1) −17.5(3) −50.4(9) −50.6(9) 0.2(1)
GFMC −102(4) −2.9(2) −18.2(6) −80.3(30) −79.5(30) −0.8(2)
Barker −120(1) −54
3+;1⇔3+;0 VMC −67(1) −1.4(1) −12.7(3) −52.0(6) −41.0(6) −12.0(2)
GFMC −90(3) −2.5(2) −14.8(6) −73.1(21) −60.9(21) −12.2(2)
Barker −62(15) −32
2+;1⇔2+;0 VMC −13(1) −0.2(1) −2.4(1) −10.4(3) −6.1(2) −4.3(1)
GFMC −6(2) −0.4(2) −1.3(4) −4.4(12)
CONCLUSIONS
We have made much progress in calculating light nuclei
• 1 − 2% calculations ofA = 6 – 12 nuclear energies are possible• Illinois Vijk give average binding-energy errors< 0.7 MeV for A = 3 − 12
– Vijk required for overallP -shell energies
– Also required for spin-orbit splittings and several levelorderings• Charge radii are in good agreement with experiment• GFMC for off-diagonal matrix elements in progress• GFMC for scattering states has been initiated• VMC calculations of single- and multi-nucleon momentum distributions
and there is still much to do
• Lots of scattering states and reactions to be done– n+3H, p+α, n+6He, n+8He, n+9Li, α+α, etc.– astrophysical reactions:3He+α →7Be, p+7Be→8B, n+(α+α) →9Be, etc.
– All big-bang nucleosynthesis, solar, & somer-process seeding reactions should be accessible
• Calculations of– overlaps, spectroscopic factors, asymptotic normalization coefficients– electromagnetic and weak transitions inA ≥ 8 nuclei– meson-exchange current contributions to moments and transitions– more unnatural-parity &2~ω excited states
• 12C including 2nd 0+ (Hoyle) state