Diamagnetic EDMs and Nuclear Structure J. Engel University of North - - PowerPoint PPT Presentation
Diamagnetic EDMs and Nuclear Structure J. Engel University of North Carolina February 15, 2013 One Way Things Get EDMs Starting at fundamental level and working up: N Underlying fundamental g theory generates three T -violating NN
University of North Carolina February 15, 2013
Starting at fundamental level and working up: Underlying fundamental theory generates three T-violating πNN vertices:
New physics
Starting at fundamental level and working up: Underlying fundamental theory generates three T-violating πNN vertices: Then neutron gets EDM, e.g., from chiral-PT diagrams like this:
New physics
Nucleus can get one from nucleon EDM or T-violating NN interaction:
π ¯ g γ
Nucleus can get one from nucleon EDM or T-violating NN interaction:
π ¯ g γ
VPT ∝
g0τ1 · τ2 − ¯ g1 2 (τz
1 + τz 1) + ¯
g2 (3τz
1τz 2 − τ1 · τ2)
−¯ g1 2 (τz
1 − τz 2) (σ1 + σ2)
mπ|r1 − r2|
Nucleus can get one from nucleon EDM or T-violating NN interaction:
π ¯ g γ
VPT ∝
g0τ1 · τ2 − ¯ g1 2 (τz
1 + τz 1) + ¯
g2 (3τz
1τz 2 − τ1 · τ2)
−¯ g1 2 (τz
1 − τz 2) (σ1 + σ2)
mπ|r1 − r2| Finally, atom gets one from nucleus. Electronic shielding makes the relevant nuclear object the “Schiff moment” S ≈
p r2 pzp + . . . rather than the dipole moment Dz .
Nucleus can get one from nucleon EDM or T-violating NN interaction:
π ¯ g γ
VPT ∝
g0τ1 · τ2 − ¯ g1 2 (τz
1 + τz 1) + ¯
g2 (3τz
1τz 2 − τ1 · τ2)
−¯ g1 2 (τz
1 − τz 2) (σ1 + σ2)
mπ|r1 − r2| Finally, atom gets one from nucleus. Electronic shielding makes the relevant nuclear object the “Schiff moment” S ≈
p r2 pzp + . . . rather than the dipole moment Dz .
Job of nuclear theory: calculate dependence of S on the ¯ g’s.
Theorem (Schiff)
The nuclear dipole moment causes the atomic electrons to rearrange themselves so that they develop a dipole moment opposite that of the
the electrons’ dipole moment exactly cancels the nuclear moment, so that the net atomic dipole moment vanishes.
Consider atom with nonrelativistic constituents (with dipole moments
k
dk and a Hamiltonian H =
p2
k
2mk +
V ( rk) −
Ek
Consider atom with nonrelativistic constituents (with dipole moments
k
dk and a Hamiltonian H =
p2
k
2mk +
V ( rk) −
Ek = H0 +
k(1/ek)
dk · ∇V ( rk) K.E. + Coulomb dipole perturbation
Consider atom with nonrelativistic constituents (with dipole moments
k
dk and a Hamiltonian H =
p2
k
2mk +
V ( rk) −
Ek = H0 +
k(1/ek)
dk · ∇V ( rk) = H0 + i
(1/ek)
pk, H0
dipole perturbation
The perturbing Hamiltonian Hd = i
(1/ek)
pk, H0
The perturbing Hamiltonian Hd = i
(1/ek)
pk, H0
|˜ = |0 +
|m m| Hd |0 E0 − Em = |0 +
|m m| i
k(1/ek)
dk · pk |0 (E0 − Em) E0 − Em =
(1/ek) dk · pk
The induced dipole moment d′ is
= ˜ 0|
ej rj |˜
The induced dipole moment d′ is
= ˜ 0|
ej rj |˜ = 0|
k(1/ek)
dk · pk
j ej
rj
k(1/ek)
dk · pk
= i 0|
rj,
k(1/ek)
dk · pk
= − 0|
= −
= − d
The induced dipole moment d′ is
= ˜ 0|
ej rj |˜ = 0|
k(1/ek)
dk · pk
j ej
rj
k(1/ek)
dk · pk
= i 0|
rj,
k(1/ek)
dk · pk
= − 0|
= −
= − d So the net EDM is zero!
Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM d. Post-screening nucleus-electron interaction proportional to Schiff moment:
ep
p − 5
3R2
ch
Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM d. Post-screening nucleus-electron interaction proportional to Schiff moment:
ep
p − 5
3R2
ch
If, as you’d expect, S ≈ R2
N
D, then d is down from D by O
N/R2 A
Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM d. Post-screening nucleus-electron interaction proportional to Schiff moment:
ep
p − 5
3R2
ch
If, as you’d expect, S ≈ R2
N
D, then d is down from D by O
N/R2 A
Ughh! Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10Z 2 ≈ 105.
Th nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM d. Post-screening nucleus-electron interaction proportional to Schiff moment:
ep
p − 5
3R2
ch
If, as you’d expect, S ≈ R2
N
D, then d is down from D by O
N/R2 A
Ughh! Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10Z 2 ≈ 105. Overall suppression of D is only about 10−3.
S ∝ Z 2, so experiments are in heavy nuclei but can’t solve Schr¨
scheme, then account for omitted physics by modifying operators.
S ∝ Z 2, so experiments are in heavy nuclei but can’t solve Schr¨
scheme, then account for omitted physics by modifying operators.
Paradigm: Density functional Theory
H¨
calculation with appropriate effective interaction (density functional).
S ∝ Z 2, so experiments are in heavy nuclei but can’t solve Schr¨
scheme, then account for omitted physics by modifying operators.
Paradigm: Density functional Theory
H¨
calculation with appropriate effective interaction (density functional). Nuclear version: Mean-field theory with density-dependent interactions (called Skyrme interactions) built from delta functions and deriviatives of delta functions plus whatever corrections one can manage, e.g. projection of deformed wave functions onto states with good angular momentum mixing of several mean fields . . .
S ∝ Z 2, so experiments are in heavy nuclei but can’t solve Schr¨
scheme, then account for omitted physics by modifying operators.
Paradigm: Density functional Theory
H¨
calculation with appropriate effective interaction (density functional). Nuclear version: Mean-field theory with density-dependent interactions (called Skyrme interactions) built from delta functions and deriviatives of delta functions plus whatever corrections one can manage, e.g. projection of deformed wave functions onto states with good angular momentum mixing of several mean fields . . . Density functional still obtained largely through phenomenology.
HFB, 2-D lattice, SLy4 + volume pairing
Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (2005)
β
Nuclear ground state deformations (2-D HFB)
Ref: Dobaczewski, Stoitsov & Nazarewicz (2004) arXiv:nucl-th/0404077
Need to calculate S = Sz =
0| VPT |m m| Sz |0 E0 − Ei + c.c. where H = Hstrong + VPT.
Need to calculate S = Sz =
0| VPT |m m| Sz |0 E0 − Ei + c.c. where H = Hstrong + VPT. Hstrong represented either by Skyrme density functional or by simpler effective interaction, treated non-self-consistently. VPT either included nonperturbatively or via explicit sum over intermediate states. Nucleus either forced artificially to be spherical or allowed to deform.
explicit corrections in RPA, which sums over intermediate states.
explicit corrections in RPA, which sums over intermediate states. SzHg ≡ a0 g ¯ g0 + a1 g ¯ g1 + a2 g ¯ g2 (e fm3) a0 a1 a2 SkM⋆ 0.009 0.070 0.022 SkP 0.002 0.065 0.011 SIII 0.010 0.057 0.025 SLy4 0.003 0.090 0.013 SkO′ 0.010 0.074 0.018 Dmitriev & Senkov RPA 0.0004 0.055 0.009
explicit corrections in RPA, which sums over intermediate states. SzHg ≡ a0 g ¯ g0 + a1 g ¯ g1 + a2 g ¯ g2 (e fm3) a0 a1 a2 SkM⋆ 0.009 0.070 0.022 SkP 0.002 0.065 0.011 SIII 0.010 0.057 0.025 SLy4 0.003 0.090 0.013 SkO′ 0.010 0.074 0.018 Dmitriev & Senkov RPA 0.0004 0.055 0.009 Range of variation here doesn’t look too bad. But these calculations are not the end of the story.
If deformed state has good intr. Jz = K, averaging over angles gives: |J, M = 2J + 1 8π2
MK(Ω)ˆ
R(Ω) |ΨK dΩ
If deformed state has good intr. Jz = K, averaging over angles gives: |J, M = 2J + 1 8π2
MK(Ω)ˆ
R(Ω) |ΨK dΩ Matrix elements; J, M| ˆ Si |J′, M′ ∝
j
dΩ dΩ′ × (some D-functions) × ΨK| ˆ R−1(Ω′) ˆ Sj ˆ R(Ω) |ΨK
rigid defm.
− − − − − − →
Ω≈Ω′
(Geometric factor) × ΨK|ˆ Sz|ΨK
Sintr.
If deformed state has good intr. Jz = K, averaging over angles gives: |J, M = 2J + 1 8π2
MK(Ω)ˆ
R(Ω) |ΨK dΩ Matrix elements; J, M| ˆ Si |J′, M′ ∝
j
dΩ dΩ′ × (some D-functions) × ΨK| ˆ R−1(Ω′) ˆ Sj ˆ R(Ω) |ΨK
rigid defm.
− − − − − − →
Ω≈Ω′
(Geometric factor) × ΨK|ˆ Sz|ΨK
Sintr.
For expectation value in J = 1
2 state:
S = ˆ SzJ= 1
2 ,M= 1 2 =
⇒
Sintr. spherical nucleus
1 3 ˆ
Sintr. rigidly deformed nucleus Exact answer somewhere in between.
Deformation actually small and soft — perhaps worst case scenario for mean-field. But in odd nuclei, that’s the limit of current
plus an economy of approach. Otherwise more or less equivalent.
1Has some “issues”: doen’t get ground sate spin correct, limited for now to
axially-symmetric minima, which are sometimes a little unstable, true minimum probably not axially symmetric . . .
Deformation actually small and soft — perhaps worst case scenario for mean-field. But in odd nuclei, that’s the limit of current
plus an economy of approach. Otherwise more or less equivalent.
1 2 3 4 5 r⊥ (fm) 1 2 3 4 5 z (fm)
2 4 6 δ ρp (arb.)
Induced change in density distribution indicates delicate Schiff moment.
1Has some “issues”: doen’t get ground sate spin correct, limited for now to
axially-symmetric minima, which are sometimes a little unstable, true minimum probably not axially symmetric . . .
Like before, use a number of Skyrme functionals: Egs β Eexc. a0 a1 a2 SLy4 HF
0.97 0.013
0.022 SIII HF
0.012 0.005 0.016 SV HF
0.68 0.009
0.016
Like before, use a number of Skyrme functionals: Egs β Eexc. a0 a1 a2 SLy4 HF
0.97 0.013
0.022 SIII HF
0.012 0.005 0.016 SV HF
0.68 0.009
0.016 SLy4 HFB
0.83 0.013
0.024 SkM* HFB
0.82 0.041
0.069
Like before, use a number of Skyrme functionals: Egs β Eexc. a0 a1 a2 SLy4 HF
0.97 0.013
0.022 SIII HF
0.012 0.005 0.016 SV HF
0.68 0.009
0.016 SLy4 HFB
0.83 0.013
0.024 SkM* HFB
0.82 0.041
0.069
QRPA — — — 0.010 0.074 0.018
Authors of these papers need to revisit/recheck their results. Improve treatment further:
Variation after projection Triaxial deformation
Authors of these papers need to revisit/recheck their results. Improve treatment further:
Variation after projection Triaxial deformation
Ultimate goal: mixing of many mean fields (aka “generator coordinates”)
Here we treat always VPT as explicit perturbation: S =
0| Sz |m m| VPT |0 E0 − Em + c.c. where |0 is unperturbed ground state.
Calculated 225Ra density
Here we treat always VPT as explicit perturbation: S =
0| Sz |m m| VPT |0 E0 − Em + c.c. where |0 is unperturbed ground state.
Calculated 225Ra density
Ground state has nearly-degenerate partner |¯ 0 with same opposite parity and same intrinsic structure, so: S − → 0| Sz |¯ 0 ¯ 0| VPT |0 E0 − E¯ + c.c. ∝
E0 − E¯
Here we treat always VPT as explicit perturbation: S =
0| Sz |m m| VPT |0 E0 − Em + c.c. where |0 is unperturbed ground state.
Calculated 225Ra density
Ground state has nearly-degenerate partner |¯ 0 with same opposite parity and same intrinsic structure, so: S − → 0| Sz |¯ 0 ¯ 0| VPT |0 E0 − E¯ + c.c. ∝
E0 − E¯ S is large because Sintr. is collective and E0 − E¯
0 is small.
When intrinsic state | is asymmetric, it breaks parity.
When intrinsic state | is asymmetric, it breaks parity. In the same way we get good J, we average over orientations to get states with good parity: |± = 1 √ 2
± |
When intrinsic state | is asymmetric, it breaks parity. In the same way we get good J, we average over orientations to get states with good parity: |± = 1 √ 2
± |
and |¯ 0 = |−, we get S ≈ 0| Sz |¯ 0 ¯ 0| VPT |0 E0 − E¯ + c.c. And in the rigid-deformation limit 0| ˆ O|¯ 0 ∝ | ˆ O| = ˆ Ointr. again like angular momentum.
225Ra Results
Hartree-Fock calculation with our favorite interaction SkO’ gives SRa = −1.5 g ¯ g0 + 6.0 g ¯ g1 − 4.0 g ¯ g2 (e fm3) Larger by over 100 than in 199Hg!
225Ra Results
Hartree-Fock calculation with our favorite interaction SkO’ gives SRa = −1.5 g ¯ g0 + 6.0 g ¯ g1 − 4.0 g ¯ g2 (e fm3) Larger by over 100 than in 199Hg! Variation a factor of 2 or 3.
Judgment in upcoming review article (based on spread in reasonable calculations):
Nucl. Best value Range a0 a1 a2 a0 a1 a2
199Hg
0.01 0.01 0.02 0.005 – 0.02
0.01 – 0.03
129Xe
225Ra
6.0
4 — 20
Uncertainties pretty large, particularly for g1 in 199Hg (range includes zero). How can we reduce them?
Improving the many-body theory to handle soft deformation, though probably necessary, is tough. But can also try to optimize density functional.
Improving the many-body theory to handle soft deformation, though probably necessary, is tough. But can also try to optimize density functional.
6 12 18 24 30 36 42
Energy (MeV)
6 12 18 24 30 36
10
−3 Strength (fm 6/MeV)
SkP SkO’ SIII EX2 EX1
Isoscalar dipole operator contains r2z just like Schiff
functionals reproduce measured distributions, e.g. in 208Pb.
VPT probes spin density; functional should have good spin response. Can adjust relevant terms in, e.g. SkO’, to Gamow-Teller resonance energies and strengths.
Here there have been important recent developments.
0.2 0.3 0.4 2.0 2.5 3.0 3.5 Octupole moment Q30 [(10 fm)3] 0.2 0.3 0.4 HF BCS Schiff moment [(10 fm)3]
SKM* SKO' SLy4 UDF0 SKXc SIII SKM* SKO' SLy4 SKXc SIII UDF0
225Ra
SkO’
L 229Pa 225Ra 223Rn
∆ ∆ ∆ ∆N=0.6–0.9 ∆ ∆ ∆ ∆P=0.6–0.9
moment, which will be extracted from measurements of E3 transitions.
Here there have been important recent developments.
0.2 0.3 0.4 2.0 2.5 3.0 3.5 Octupole moment Q30 [(10 fm)3] 0.2 0.3 0.4 HF BCS Schiff moment [(10 fm)3]
SKM* SKO' SLy4 UDF0 SKXc SIII SKM* SKO' SLy4 SKXc SIII UDF0
225Ra
SkO’
L 229Pa 225Ra 223Rn
∆ ∆ ∆ ∆N=0.6–0.9 ∆ ∆ ∆ ∆P=0.6–0.9
moment, which will be extracted from measurements of E3 transitions.
!"
./ 0/ 1/ 2/ 3! "! 4!
!0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !" !" !" !" !"
This is 224Ra; transitions in 225Ra will be measured soon.
Thanks for your kind attention.