t violation, edms, and nuclear structureuser.physics.unc.edu/~engelj/talks/dipole-darmstadt.pdf ·...

T Violation, EDMs, and Nuclear StructureJ. Engel

University of North CarolinaJanuary 23, 2014

Outline1. Why T is different from other symmetries2. Connection between EDMs and T

3. What to calculate to make contact with experiment4. Shielding and Schiff theorem5. Calculations in 199Hg6. Importance of octupole deformation7. Calculations in 225Ra8. Using data to improve calculations

The T Operator in QM is Different.

Not linear:T [x , p]T−1 = −[x , p]so i is odd under T .Has no eigenstates in the conventional sense:

T |a〉 = |a〉 −→ T (α |a〉) = α∗T |a〉 = α∗ |a〉 6= α |a〉

for α complex.Typical physical states |J,M〉 not even close toeigenstates of T .As a result, T violation doesn’t show upas “mixing of states with opposite T .”

The T Operator in QM is Different. Not linear:

T [x , p]T−1 = −[x , p]so i is odd under T .Has no eigenstates in the conventional sense:T |a〉 = |a〉 −→ T (α |a〉) = α∗T |a〉 = α∗ |a〉 6= α |a〉

for α complex.Typical physical states |J,M〉 not even close toeigenstates of T .

As a result, T violation doesn’t show upas “mixing of states with opposite T .”

The T Operator in QM is Different. Not linear:

T [x , p]T−1 = −[x , p]so i is odd under T .Has no eigenstates in the conventional sense:T |a〉 = |a〉 −→ T (α |a〉) = α∗T |a〉 = α∗ |a〉 6= α |a〉

for α complex.Typical physical states |J,M〉 not even close toeigenstates of T .As a result, T violation doesn’t show upas “mixing of states with opposite T .”

Is T Violated in the Real World?

Yup!Violation is seen in decay of K-mesons (direct) andB-mesons (through CP violation).And we strongly believe that T (≡ CP) violation played animportant role in the early universe, causing excess ofmatter over antimatter.

What is the Source of T -Violation?K and B phenomena almost certainly due to a phase in the3 × 3 CKM matrix, which converts (d , s, b) to “weakeigenstates” that couple to W and Z .

But this violation is too weak to cause “baryogenesis”, whichmust arise outside the standard model, e.g. throughsupersymmetryheavy neutrinosHiggs sector . . .To confuse things more, there’s the “strong CP problem.”We need to see T -violation outside mesonic systems tounderstand its sources. EDM’s are not sensitive to CKM Tviolation, but are to other sources. They’re already puttingextreme pressure on supersymmetry.

What is the Source of T -Violation?K and B phenomena almost certainly due to a phase in the3 × 3 CKM matrix, which converts (d , s, b) to “weakeigenstates” that couple to W and Z .But this violation is too weak to cause “baryogenesis”, whichmust arise outside the standard model, e.g. throughsupersymmetryheavy neutrinosHiggs sector . . .

To confuse things more, there’s the “strong CP problem.”We need to see T -violation outside mesonic systems tounderstand its sources. EDM’s are not sensitive to CKM Tviolation, but are to other sources. They’re already puttingextreme pressure on supersymmetry.

What is the Source of T -Violation?K and B phenomena almost certainly due to a phase in the3 × 3 CKM matrix, which converts (d , s, b) to “weakeigenstates” that couple to W and Z .But this violation is too weak to cause “baryogenesis”, whichmust arise outside the standard model, e.g. throughsupersymmetryheavy neutrinosHiggs sector . . .To confuse things more, there’s the “strong CP problem.”

We need to see T -violation outside mesonic systems tounderstand its sources. EDM’s are not sensitive to CKM Tviolation, but are to other sources. They’re already puttingextreme pressure on supersymmetry.

What is the Source of T -Violation?K and B phenomena almost certainly due to a phase in the3 × 3 CKM matrix, which converts (d , s, b) to “weakeigenstates” that couple to W and Z .But this violation is too weak to cause “baryogenesis”, whichmust arise outside the standard model, e.g. throughsupersymmetryheavy neutrinosHiggs sector . . .To confuse things more, there’s the “strong CP problem.”We need to see T -violation outside mesonic systems tounderstand its sources. EDM’s are not sensitive to CKM Tviolation, but are to other sources. They’re already puttingextreme pressure on supersymmetry.

Connection Between EDMs and T ViolationConsider nondegenerate ground state |g.s. : J,M〉. Symmetryunder rotations Ry (π) for vector operator like ~d ≡∑i ei~ri ,〈g.s. : J,M| dz |g.s. : J,M〉

= −〈g.s. : J,−M| dz |g.s. : J,−M〉 .

R−1R R−1RT takes M to −M , like Ry (π). But ~d is odd under Ry (π) andeven under T , so for T conserved〈g.s. : J,M| dz |g.s. : J,M〉 = + 〈g.s. : J,−M| dz |g.s. : J,−M〉 .

T−1T T−1T

Together with the first equation, this implies〈dz〉 = 0 .

If T is violated, argument fails because T takes |g : JM〉 tostates with J,−M , but different-energy.

Of course T =CP .

Connection Between EDMs and T ViolationConsider nondegenerate ground state |g.s. : J,M〉. Symmetryunder rotations Ry (π) for vector operator like ~d ≡∑i ei~ri ,〈g.s. : J,M| dz |g.s. : J,M〉 = −〈g.s. : J,−M| dz |g.s. : J,−M〉 .

R−1R R−1R

T takes M to −M , like Ry (π). But ~d is odd under Ry (π) andeven under T , so for T conserved〈g.s. : J,M| dz |g.s. : J,M〉 = + 〈g.s. : J,−M| dz |g.s. : J,−M〉 .

T−1T T−1T

Together with the first equation, this implies〈dz〉 = 0 .

If T is violated, argument fails because T takes |g : JM〉 tostates with J,−M , but different-energy.

Of course T =CP .

EDMs Sensitive to New PhysicsIn standard model only one phase. Diagrams cancel to highorder, e.g.:

+ ...i sinδf fW

γ

f fW

γ

f’f’ f’ f’−i sin +δ

SUSY has many phases. Low-order diagrams uncanceled, e.g.:

e iθf f

γ

f~

Thus, EDMs are insensitive to standard-model CP butsensitive to extra-standard-model CP .

There are EDM Experiments on Neutrons, Atoms . . .Basic principle:

B E BE

f f’

H = −~µ · ~B − ~d · ~E

and there is a change in precession frequency (linear in E )when ~E is flipped.

There are EDM Experiments on Neutrons, Atoms . . .Basic principle:

B E BE

f f’

H = −~µ · ~B − ~d · ~Eand there is a change in precession frequency (linear in E )when ~E is flipped.

One Way Things Get EDMsStarting at fundamental level and working up:

Underlying fundamental theorygenerates three T -violating πNNvertices:

Then neutron gets EDM, e.g.,from chiral-PT diagrams like this:

N

?π

g

n p n

π−

γ

g g

New physics

How Diamagnetic Atoms Get EDMsNucleus can get one from nucleonEDM or T -violating NN interaction: π

g

γ

VPT ∝[g0τ1 · τ2 −

g1

2(τz1 + τz1 ) + g2 (3τz1τz2 − τ1 · τ2)] (σ1 − σ2)

− g1

2(τz1 − τz2 ) (σ1 + σ2) ·(∇1 −∇2)exp (−mπ |r1 − r2|)

mπ |r1 − r2|+ contact term

Finally, atom gets one from nucleus. Electronic shielding makesrelevant nuclear object the “Schiff moment” 〈S〉 ≈ 〈∑p r2p zp + . . .〉.

Job of nuclear theory: calculate dependence of 〈S〉 onthe g’s (and on the contact term and nucleon EDM).

How Diamagnetic Atoms Get EDMsNucleus can get one from nucleonEDM or T -violating NN interaction: π

g

γ

VPT ∝[g0τ1 · τ2 −

g1

2(τz1 + τz1 ) + g2 (3τz1τz2 − τ1 · τ2)] (σ1 − σ2)

− g1

2(τz1 − τz2 ) (σ1 + σ2) ·(∇1 −∇2)exp (−mπ |r1 − r2|)

mπ |r1 − r2|+ contact termFinally, atom gets one from nucleus. Electronic shielding makesrelevant nuclear object the “Schiff moment” 〈S〉 ≈ 〈∑p r

2p zp + . . .〉.

Job of nuclear theory: calculate dependence of 〈S〉 onthe g’s (and on the contact term and nucleon EDM).

How Does Shielding Work?Theorem (Schiff)The nuclear dipole moment causes the atomic electrons torearrange themselves so that they develop a dipole momentopposite that of the nucleus. In the limit of nonrelativisticelectrons and a point nucleus the electrons’ dipole momentexactly cancels the nuclear moment, so that the net atomicdipole moment vanishes.

How Does Shielding Work?ProofConsider atom with non-relativistic constituents (with dipolemoments ~dk ) held together by electrostatic forces. The atomhas a “bare” edm ~d ≡∑

k~dk and a Hamiltonian

H = ∑k

p2k2mk

+∑k

V (~rk ) −∑k

~dk · ~Ek

= H0 +∑k (1/ek )~dk · ~∇V (~rk )= H0 + i

∑k

(1/ek ) [~dk · ~pk ,H0

]K.E. + Coulomb dipole perturbation

How Does Shielding Work?The perturbing HamiltonianHd = i

∑k

(1/ek ) [~dk · ~pk ,H0

]shifts the ground state |0〉 to|0〉 = |0〉+∑

m

|m〉〈m|Hd |0〉E0 − Em

= |0〉+∑m

|m〉〈m| i∑

k (1/ek )~dk · ~pk |0〉 (E0 − Em)E0 − Em

= (1 + i∑k

(1/ek )~dk · ~pk) |0〉

How Does Shielding Work?The induced dipole moment ~d ′ is~d ′ = 〈0|

∑j

ej~rj |0〉

= 〈0|(1− i

∑k (1/ek )~dk · ~pk) (∑

j ej~rj)

×(1 + i

∑k (1/ek )~dk · ~pk) |0〉

= i 〈0|[∑

j ej~rj ,∑

k (1/ek )~dk · ~pk] |0〉= − 〈0|

∑k

~dk |0〉 = −∑k

~dk

= − ~d

So the net EDM is zero!

All is Not Lost, Though. . .The nucleus has finite size. Shielding is not complete, andnuclear T violation can still induce atomic EDM DA.Post-screening nucleus-electron interaction proportional toSchiff moment:〈S〉 ≡

⟨∑p

ep

(r2p −

5

3〈R2ch〉

)zp

⟩+ . . .

If, as you’d expect, 〈S〉 ≈ R2Nuc 〈DNuc〉, then DA is down from〈DNuc〉 by

O(R2Nuc/R2

A

)≈ 10−8 ,

Ughh! Fortunately the large nuclear charge and relativisticwave functions offset this factor by 10Z 2 ≈ 105.Overall suppression of DA is only about 10−3.

All is Not Lost, Though. . .The nucleus has finite size. Shielding is not complete, andnuclear T violation can still induce atomic EDM DA.Post-screening nucleus-electron interaction proportional toSchiff moment:〈S〉 ≡

⟨∑p

ep

(r2p −

5

3〈R2ch〉

)zp

⟩+ . . .If, as you’d expect, 〈S〉 ≈ R2Nuc 〈DNuc〉, then DA is down from〈DNuc〉 by

O(R2Nuc/R2

A

)≈ 10−8 ,

Ughh! Fortunately the large nuclear charge and relativisticwave functions offset this factor by 10Z 2 ≈ 105.Overall suppression of DA is only about 10−3.

Theory for Heavy Nuclei〈S〉 ∝ Z 2, so experiments are in heavy nucleibutcan’t solve Schrodinger eq’n for A > 40. Usually approximate,then account for omitted physics by modifying operators.

Paradigm: Density functional TheoryHohenberg-Kohn-Sham: Can get exact density from Hartreecalculation with effective interaction (density functional).Nuclear version: Mean-field theory with density-dependentinteractions (named after Skyrme) built from delta functionsand deriviatives of delta functions, plus corrections, e.g.:projection of deformed wave functions onto states withgood angular momentummixing of several mean fields. . . Density functional still largely phenomenological.

Theory for Heavy Nuclei〈S〉 ∝ Z 2, so experiments are in heavy nucleibutcan’t solve Schrodinger eq’n for A > 40. Usually approximate,then account for omitted physics by modifying operators.Paradigm: Density functional TheoryHohenberg-Kohn-Sham: Can get exact density from Hartreecalculation with effective interaction (density functional).

Nuclear version: Mean-field theory with density-dependentinteractions (named after Skyrme) built from delta functionsand deriviatives of delta functions, plus corrections, e.g.:projection of deformed wave functions onto states withgood angular momentummixing of several mean fields. . . Density functional still largely phenomenological.

Theory for Heavy Nuclei〈S〉 ∝ Z 2, so experiments are in heavy nucleibutcan’t solve Schrodinger eq’n for A > 40. Usually approximate,then account for omitted physics by modifying operators.Paradigm: Density functional TheoryHohenberg-Kohn-Sham: Can get exact density from Hartreecalculation with effective interaction (density functional).Nuclear version: Mean-field theory with density-dependentinteractions (named after Skyrme) built from delta functionsand deriviatives of delta functions, plus corrections, e.g.:projection of deformed wave functions onto states withgood angular momentummixing of several mean fields. . . Density functional still largely phenomenological.

Nuclear Deformation

Deformed Skyrme Mean-Field Theory

"#"$#%&! '()*+,!-.+,/0*+,1!'/23+,.4)5! F!

Zr-102: normal density and pairing density

HFB, 2-D lattice, SLy4 + volume pairing Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (2005)

G=HI!β"JKLM&N76! +OKI!β"

JKLM&N7"J8L!1!PNQN!G@/2R!+5!/)N1!9AS;N!T+<N!U!J"&&$L!

Applied EverywhereNuclear ground state deformations (2-D HFB)

Ref: Dobaczewski, Stoitsov & Nazarewicz (2004) arXiv:nucl-th/0404077

"#"$#%&! %V!'()*+,!-.+,/0*+,1!'/23+,.4)5!

Varieties of Recent Schiff-Moment CalculationsNeed to calculate dependence of

〈S〉 ≈∑m

〈0|S |m〉〈m|VPT |0〉E0 − Em

+ c .c .

on gi for H = Hstrong + VPT .Hstrong represented either by Skyrme density functional orby simpler effective interaction, treated on top of separatemean field.VPT either included nonperturbatively or via explicit sumover intermediate states.Nucleus either forced artificially to be spherical orallowed to deform.

199Hg: The Atom with the Best Experimental LimitOur First Attempt: Spherical HFB + Core Polarization1. Skyrme-HFB in 198Hg, with several Skyrme interactions.2. Perturbation theory for interaction between last neutronand core.First order in W ≡ VPT since it is very weak.RPA order in Skyrme interaction.

Lowest-order diagramsfor Schiff moment

a

a

S

Wa

a

S

W

Building QRPADiagrams like theseare summed. . .

yielding this.We also includethese.

These we evaluatebut find negligible:

a

a

a

a

· · ·

a

a

c

a

a

a

a

a

a

This is all fine and dandy — gives sensible results— but we really want a one-step calculation thatincludes deformation.

Building QRPADiagrams like theseare summed. . .yielding this.

We also includethese.


a

a

a

a

· · ·

a

a

c

a

a

Sz02

W 31

a

a

i

Sz20

V13

W 11

a

a

iSz02

V31

W 11

BA

a

a

a

a

a

a


Building QRPADiagrams like theseare summed. . .yielding this.We also includethese.


a

a

a

a

· · ·

a

a

c

a

a

Sz02

W 31

a

a

i

Sz20

V13

W 11

a

a

iSz02

V31

W 11

A B

a

a

a

a

a

a


Building QRPADiagrams like theseare summed. . .yielding this.We also includethese.


a

a

a

a

· · ·

a

a

c

a

a

Sz02

W 31

a

a

i

Sz20

V13

W 11

a

a

iSz02

V31

W 11

A Ba

a

a

a

a

a


Deformation and Angular-Momentum RestorationIf deformed state |ΨK 〉 has good intrinsic Jz = K , average overangles gives:|J,M〉 = 2J + 1

8π2

∫DJ∗MK (Ω)R(Ω) |ΨK 〉 dΩ

Matrix elements (with more detailed notation):〈J,M|Sm |J ′,M ′〉 ∝

∫ ∫ ∑n

dΩ dΩ′ × (some D-functions)× 〈ΨK |R−1(Ω′)Sn R(Ω) |ΨK 〉

rigid defm.−−−−−−→Ω≈Ω′ (Geometric factor)× 〈ΨK |Sz |ΨK 〉︸︷︷︸〈S〉intr.For expectation value in J = 1

2 state:〈S〉 = 〈Sz〉J= 1

2 ,M= 12

=⇒ 〈S〉intr. spherical nucleus13 〈S〉intr. rigidly deformed nucleus

Exact answer somewhere in between.


8π2



∫ ∫ ∑n


rigid defm.−−−−−−→Ω≈Ω′ (Geometric factor)× 〈ΨK |Sz |ΨK 〉︸︷︷︸〈S〉intr.

For expectation value in J = 12 state:

〈S〉 = 〈Sz〉J= 12 ,M= 1

2=⇒

〈S〉intr. spherical nucleus13 〈S〉intr. rigidly deformed nucleus



8π2



∫ ∫ ∑n


rigid defm.−−−−−−→Ω≈Ω′ (Geometric factor)× 〈ΨK |Sz |ΨK 〉︸︷︷︸〈S〉intr.For expectation value in J = 1

2 state:〈S〉 = 〈Sz〉J= 1

2 ,M= 12

=⇒ 〈S〉intr. spherical nucleus13 〈S〉intr. rigidly deformed nucleus


Deformed Mean-Field Calculation Directly in 199HgDeformation actually small and soft — perhaps worst casescenario for mean-field. But in odd nuclei, that’s the limit ofcurrent technology1. VPT included nonperturbatively andcalculation done in one step. Except for deformation should bemore or less equivalent to RPA.

0 1 2 3 4 5r⊥ (fm) 0 1

2 3

4 5

z (fm)-4

-2

0

2

4

6

δ ρ p

(ar

b.) Oscillating PT -odddensity distributionindicates delicateSchiff moment.

1Has some “issues”: doen’t get ground-state spin correct, limited for now to axially-symmetric minima, which are sometimes a little unstable, true minimum probably notaxially symmetric . . .

Results〈S〉Hg ≡ a0 gg0 + a1 gg1 + a2 gg2 (e fm3)

Use a number of Skyrme functionals:Egs β Eexc. a0 a1 a2

SLy4 HF -1561.42 -0.13 0.97 0.013 -0.006 0.022SIII HF -1562.63 -0.11 0 0.012 0.005 0.016SV HF -1556.43 -0.11 0.68 0.009 -0.0001 0.016SLy4 HFB -1560.21 -0.10 0.83 0.013 -0.006 0.024SkM* HFB -1564.03 0 0.82 0.041 -0.027 0.069RPA Ave. QRPA — — — 0.010 0.074 0.018

Hmm. . .

What to Do About DiscrepancyAuthors of these papers need to revisit/recheck theirresults.Improve treatment further:Variation after projectionTriaxial deformation

Ultimate goal: mixing of many mean fields, aka “generatorcoordinates”Still a ways off because of difficulties marrying generatorcoordinates to density functionals.

Schiff Moment with Octupole DeformationHere we treat always VPT as explicitperturbation:〈S〉 =∑

m

〈0|S |m〉〈m|VPT |0〉E0 − Em

+ c .c .

where |0〉 is unperturbed ground state. Calculated 225Ra densityGround state has nearly-degenerate partner |0〉 with sameopposite parity and same intrinsic structure, so:〈S〉 −→ 〈0|S |0〉〈0|VPT |0〉

E0 − E0

+ c .c . ∝ 〈S〉intr. 〈VPT 〉intr.E0 − E0

Why is this? See next slide.〈S〉 is large because 〈S〉intr. is collective and E0 − E0 is small.


m

〈0|S |m〉〈m|VPT |0〉E0 − Em

+ c .c .


E0 − E0


Why is this? See next slide.

〈S〉 is large because 〈S〉intr. is collective and E0 − E0 is small.


m

〈0|S |m〉〈m|VPT |0〉E0 − Em

+ c .c .


E0 − E0


Why is this? See next slide.〈S〉 is large because 〈S〉intr. is collective and E0 − E0 is small.

A Bit More Detail on Parity DoubletsWhen intrinsic state | 〉 is asymmetric, it breaks parity.In the same way we get good J , we average over orientationsto get states with good parity:|±〉 = 1√

2

(| 〉 ± | 〉

)

These are nearly degenerate if deformation is rigid. So with|0〉 = |+〉 and |0〉 = |−〉, we get

〈S〉 ≈ 〈0|Sz |0〉〈0|VPT |0〉E0 − E0

+ c.c.

And in the rigid-deformation limit〈0|O|0〉 ∝ 〈 |O| 〉= 〈O〉intr.

like angular momentum.

A Bit More Detail on Parity DoubletsWhen intrinsic state | 〉 is asymmetric, it breaks parity.In the same way we get good J , we average over orientationsto get states with good parity:|±〉 = 1√

2

(| 〉 ± | 〉

)These are nearly degenerate if deformation is rigid. So with|0〉 = |+〉 and |0〉 = |−〉, we get

〈S〉 ≈ 〈0|Sz |0〉〈0|VPT |0〉E0 − E0

+ c .c .

And in the rigid-deformation limit〈0|O|0〉 ∝ 〈 |O| 〉= 〈O〉intr.

like angular momentum.

Spectrum of 225Ra350

9/2+ 321

300-

7l24 Ia27 g& -- -4

250 - 243 (13/2+)L (7/2+) 236

(~~2~)~

3/2_----_ 5L2+

200

i

5l2+ 379

150- 312 + 149

720 5f2--

x2+ fli K=3!2 bands

tOo-- 912i 'O"

712-A

50 i/2- 55 3f2i

42

5i2t25 312- 3l

O- l/2+- 0

K I TIP bands i

Fig. 5. Proposed grcxxping of the low-lying states OF 2zSRa into rotation& bands. T’ke two members of tke f? = $- band have been reported in a study of the ‘%?r decay 2oj; they are not observed in the

present study.

of the favored K * = z* band. (We have chosen to show in fig. 4 the M 1 multipolarity for the 134 keV y so that this apparent con%& in the data will not be overlooked by the reader.)

Definitive I” assignments for the remaining levels above 236 keV are difficult to make fram the available data, although the y-ray multipolarities and o-transition hindrance factors provide at least some insight. Again, the low value (23) of the hindrance factor of the rw-transition to the 394.7 keV Ievel is quite interesting, but no definite conclusion can be drawn regarding the I” assignment of this fevei.

Parity doublet|0〉

|0〉

225Ra ResultsHartree-Fock calculation with our favorite interaction SkO’gives

〈S〉Ra = −1.5 gg0 + 6.0 gg1 − 4.0 gg2 (e fm3)Larger by over 100 than in 199Hg!

Variation a factor of 2 or 3.

Current “Assessment” of UncertaintiesJudgment in recent review article (based on spread inreasonable calculations):

Nucl. Best value Rangea0 a1 a2 a0 a1 a2

199Hg 0.01 ±0.02 0.02 0.005 – 0.05 -0.03 – 0.09 0.01 – 0.06129Xe -0.008 -0.006 -0.009 -0.005 – -0.05 -0.003 – -0.05 -0.005 – -0.1225Ra -1.5 6.0 -4.0 -1 – -6 4 — 24 -3 – -15

Uncertainties pretty large, particularly for g1 in 199Hg (rangeincludes zero). How can we reduce them?

Grounding the Calculations: HgImproving many-body theory to handle soft deformation,though probably necessary, is tough. But can also try tooptimize density functional.

0 6 12 18 24 30 36 42Energy (MeV)

0

6

12

18

24

30

36

10−

3 S

tren

gth

(fm

6 /MeV

)

SkPSkO’SIII

EX2

EX1

Isoscalar dipole operatorcontains r2z just like Schiffoperator. Can see how wellfunctionals reproduce measureddistributions, e.g. in 208Pb.

More on Grounding Hg Calculation

VPT probes spin density;functional should have goodspin response. Can adjustrelevant terms in, e.g. SkO’,to Gamow-Teller resonanceenergies and strengths.

Grounding the Calculations: RaImportant new developments here.

0.2

0.3

0.4

2.0 2.5 3.0 3.5Octupole moment Q 30 [(10 fm) 3]

0.2

0.3

0.4HFBCS

Sch

iff m

omen

t [(1

0 fm

)3 ]

SKM*

SKO'SLy4

UDF0SKXc

SIII SKM*

SKO'SLy4

SKXc

SIIIUDF0 225Ra

SkO’L

229Pa

225Ra

223Rn

∆∆∆∆ N=0.6–0.9

∆∆∆∆P=0.6–0.9

〈S〉intr. correlated with octupolemoment, which will be extractedfrom measured E3 transitions.

Coulomb Excitation

5

Projectile (Z1,A1)

Target (Z2,A2)

b$

v

Sommerfeld parameter:

!"

#$%&'()*+,+-

./

0/

1/

2/

3!

"!

4!

!0

!0

!0

!0

!0

!0

!0

!0!0

!0!0

!"

!"

!"

!"

!"

1

“Safe” Coulex:

Reduced matrix elements: < 0+||E3||3 >224RaGaffney et al., Nature

Transitions in 225Ra to bemeasured soon.

Grounding the Calculations: RaImportant new developments here.

0.2

0.3

0.4

2.0 2.5 3.0 3.5Octupole moment Q 30 [(10 fm) 3]

0.2

0.3

0.4HFBCS

Sch

iff m

omen

t [(1

0 fm

)3 ]

SKM*

SKO'SLy4

UDF0SKXc

SIII SKM*

SKO'SLy4

SKXc

SIIIUDF0 225Ra

SkO’L

229Pa

225Ra

223Rn

∆∆∆∆ N=0.6–0.9

∆∆∆∆P=0.6–0.9

〈S〉intr. correlated with octupolemoment, which will be extractedfrom measured E3 transitions.

Coulomb Excitation

5

Projectile (Z1,A1)

Target (Z2,A2)

b$

v

Sommerfeld parameter:

!"

#$%&'()*+,+-

./

0/

1/

2/

3!

"!

4!

!0

!0

!0

!0

!0

!0

!0

!0!0

!0!0

!"

!"

!"

!"

!"

1

“Safe” Coulex:

Reduced matrix elements: < 0+||E3||3 >224RaGaffney et al., NatureTransitions in 225Ra to bemeasured soon.

In Sum. . .Calculations have become sophisticated, but we still have a lotof work to do. Octupole-deformed nuclei more under controlthan 199Hg.

THE END

Thanks for your kind attention.

t violation, edms, and nuclear structureuser.physics.unc.edu/~engelj/talks/dipole-darmstadt.pdf ·...

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