Theoretical study of ﬁssion barriers in odd-A nuclei using ...irfu.cea.fr/dphn/Espace_Theorie/May2006/talks/Perez.pdf · S. Pérez and L.M. Robledo Theoretical study of ﬁssion

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IntroductionTheoretical Framework

Justification of this approximationCalculation of Fission Barriers

Theoretical study of fission barriersin odd-A nuclei using the Gogny force

S. Pérez and L.M. Robledo

Departamento de Física TeóricaUniversidad Autónoma de Madrid

Saclay, 9th May 2006

Workshop on the Theories of Fission and Related Phenomena

S. Pérez and L.M. Robledo Theoretical study of fission barriers in odd-A nuclei using the Gogny force

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Outline

1 IntroductionMotivation

2 Theoretical FrameworkMean Field

3 Justification of this approximationFT-HFB Theory

4 Calculation of Fission BarriersDetails of the calculationNumerical Results


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Motivation

Outline






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Motivation

Introduction

Aims_

We are interested in having a microscopic description of fissionbecause we want to understand the phenomenon by itself.

_

Such description would be very helpful to make predictions forneutron rich nuclei, essential for stellar nucleosynthesiscalculations, as well as for superheavies.


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Motivation

Introduction

Aims_

We are interested in having a microscopic description of fissionbecause we want to understand the phenomenon by itself.

_

Such description would be very helpful to make predictions forneutron rich nuclei, essential for stellar nucleosynthesiscalculations, as well as for superheavies.


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Motivation

Introduction

Aims

In general, we would like to have a microscopic description offission in odd-A nuclei using a mean field theory with realisticinteractions. The traditional procedure (blocked HFB) implies thebreaking of time-reversal symmetry, which means a highcomputational cost , specially for the study of fission processes.

The alternative is to use Equal Filling Approximation , whichkeeps the time-reversal symmetry (and also axiallity).


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Motivation

Introduction

Aims




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Motivation

Introduction

Aims




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Motivation

Introduction

Aims




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Mean Field

Outline






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Mean Field

HFB Theory

The Hartree-Fock-Bogoliubov theory has been used.The quasi-particle creation and annihilation operators are given by themost general linear transformation from the particle operators:(

αk

α†k

)= ∑l

(U† V †

V T UT

)kl

(cl

c+l

)The matrices U and V will be the variational parameters whichminimise the HFB energy:

〈H〉= Tr(tρ)+1

2Tr(Γρ)− 1

2Tr(∆κ∗)

ρij = 〈c+j ci〉= (V ∗V T )ij ; κij = 〈cj ci〉= (V ∗UT )ij

Γij = ∑kl

vikjl ρlk ; ∆ij =1

2 ∑kl

vijkl κkl


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Mean Field

Trial Ground State

The ground state of even-even nuclei is represented as the QPvacuum:

|φ〉= ∏k>o

αk αk |−〉 ⇒ ρ = V ∗V T andκ = V ∗UT

However, the ground state of odd nuclei can be characterised bymeans of:

|φ1qp〉= α+i |φ〉 ⇒

ρkk ′ =(V ∗V T

)kk ′ +

{Uki U∗k ′ i −V ∗ki Vk ′ i

}κkk ′ =

(V ∗UT

)kk ′ +

{Uki V ∗k ′ i −V ∗ki Uk ′ i

}Drawback

With this ansatz the time-reversal symmetry is broken.


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Mean Field

Time-reversal invariant description

To restore the time reversal symmetry, an intuitive solution comes interms of the Equal Filling Approximation (EFA):

ρefakk ′ =

(V ∗V T

)kk ′ +

1

2

{UkiU

∗k ′ i −V ∗kiVk ′ i +UkiU

∗k ′ i −V ∗kiVk ′ i

}κefa

kk ′ =(V ∗UT

)kk ′ +

1

2

{UkiV

∗k ′ i −V ∗kiUk ′ i +UkiV

∗k ′ i −V ∗kiUk ′ i

}Without any justification, and using it in the same way as in theordinary HFB theory, the energy is computed as:

Eefa = Tr(tρefa)+1

2Tr(Γefaρefa)− 1

2Tr(∆efaκefa∗)

the U and V matrices are computed by solving the HFB equation(t +Γefa ∆efa

−∆∗efa −t∗−Γ∗efa

)(Uk

Vk

)=

(Uk

Vk

)Ek

again without any foundationS. Pérez and L.M. Robledo Theoretical study of fission barriers in odd-A nuclei using the Gogny force

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FT-HFB Theory

Outline






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FT-HFB Theory

HFB Theory

Writing the density matrix and the pairing tensor in term of matrices:

R efa =(

ρefa κefa

−κefa∗ 1−ρefa∗

)=

(U V ∗

V U∗

)(f 00 1− f

)(U+ V+

V T UT

)

where

fk ={

120

k = i or k = iotherwise

which reminds us the Finite-Temperature HFB Theory!


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FT-HFB Theory

Expectation value of the Hamiltonian

From the self-consistent FT-HFB theory, the energy could be expandedthrough:

E = Tr(DH) =1

Z ∑{n}〈φ|αk1 , · · · ,αkN Hα+

k1, · · · ,α+

kN|φ〉e−β(Ek1 +···+EkN )

Both expressions give the same results:

Eefa =1

4

{〈φ|H |φ〉+ 〈φ|αiHα+

i |φ〉+ 〈φ|αiHα+i|φ〉+ 〈φ|αiαiHα+

iα+

i |φ〉}

=

= Tr(tρefa)+1

2Tr(Γefaρefa)− 1

2Tr(∆efaκefa∗)

Therefore, we have, again, a theory based on the variational principle on theEFA energy given above.


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FT-HFB Theory

Gradient method

Since the FT-HFB equations are nonlinear, they can be solved byiteration.The use of iterative diagonalization generates some problems aboutthe convergence and has complications that arise in cases with one ormore constraints. These facts drive us to employ the gradient method.The gradient method can be used because the EFA equations comefrom the variational principle on the EFA energy.This method is based on the Thouless parametrisation of the mostgeneral HFB wave function:

|φ1(Z )〉= N exp[ ∑k<k ′

Zkk ′α+k α+

k ′ ] |φ0〉


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FT-HFB Theory

Advantages of having a justification

Thanks to this justification, we can extend the use of the EFA in moreadvanced frameworks, such as the time-dependent HFB theory,projection methods, etc.Specifically, the adiabatic approximation of the TD-HFB theory hasbeen used to evaluate the collective masses involved in the CollectiveHamiltonian.


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Details of the calculationNumerical Results

Outline






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Effective Interaction

The Gogny density dependent effective force with the D1Sparametrisation has been used:

V12 =2

∑i=1

(Wi +Bi Pσ−Hi Pτ−Mi PσPτ) e− (~r1−~r2)2

µ2i

+ i WLS(←−−−−−∇1−∇2)×δ(~r1−~r2)(

−−−−−→∇1−∇2) · (~σ1 +~σ2) (1)

+ t0 (1+ x0Pσ)δ(~r1−~r2)[

ρ(~r1 +~r2

2)]γ

+VCoul


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Approximations and Basis

We have used the Slater approximation to compute the Coulombexchange term:

ECE =−3

4e2

(3

π

)1/3 Z[ρp(−→r )]3/4d3−→r

The QP operators have been expanded into an axially symmetricharmonic oscillator basis including 17 shells.In order to have an accurate fission driving coordinate, we have usedthe Q20 degree of freedom.The three lowest QP states for each Jz value(Jz = 1

2 , 32 , 5

2 , 72 , 9

2 and 112 ) have been selected to be blocked.


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Half-life formula

The spontaneous fission half life has been calculated using thetraditional WKB approximation:

Tsf (s) = 2.8610−21(1+e2S

)where S is the action

S =Z b

adQ20

√2B (Q20)(V (Q20)−E0)

where B is the collective mass computed in the framework of theadiabatic TDHFB.In the collective potential energy V (Q20) the rotational energycorrection has been taken into account.


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Projection techniques

In order to better describe the odd-A nuclei we can implement theparity and number parity projection, and check when it is important toapply them.

Parity Projection

Since the reflexion symmetry is broken when octupole deformationappears, we will check how and when parity projection is important.

-

Number Parity Projection

Since we are describing an odd-A nuclei, only odd quasi-particlesstates should be considered. Therefore, in order to keep only onequasi-particle states, the number parity projection must be applied.

EefaPRO =

1

2

{〈φ|αi Hα+

i |φ〉+ 〈φ|αi Hα+i|φ〉

}S. Pérez and L.M. Robledo Theoretical study of fission barriers in odd-A nuclei using the Gogny force

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Parity Projection

Interesting examples on which to apply parity projection should haveoctupole deformation in the ground state. We have chosen the 229Radium isotope to make this study.

Projection After Variation drawbacks:

In order to be close to the Variation After Projection method, we shallstudy the Projection After Variation solution in terms of the octupolardegree of freedom, even if the mean field ground state doesn’t showoctupolarity.


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Parity Projection

0 1000 2000 3000 4000 5000 6000Q3 (fm3)

-1738

-1737

-1736

-1735

-1734

-1733

-1732

-1731

E (M

eV)

01030507

09

11

13

229Ra

Therefore we can extract from these calculations that this projection isimportant when the octupolarity is in a certain range, β∼ 0.01−0.1 ,not more not less.


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We have used the 235U fission barrier to see the effects of thisprojection.

235U

0 5000 10000 15000Q20 (b)

-1776

-1772

-1768

-1764

E (M

eV)

1/2 3/2 5/2

7/2 9/2 11/2


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0 5000 10000 15000-1780

-1775

-1770

-1765

-1760

E H

FB (M

eV)

235U 1st jz=1/2

0 5000 10000 15000Q_20 (fm²)

0

0.05

0.1

0.15

0.2

Epr

oj. -

E_

EFA

(MeV

)

It is easy to check that the modification is very small and of the orderof 100keV. Therefore we can state that the Equal Filling Approximationis a very good way to describe odd-A nuclei, since to number paritycorrection is very small.


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Odd-A fission calculations

We have applied the EFA method to the calculation of two fissionbarriers, one for a nucleus blocking in the proton channel, that is23593 Np142 , and other blocking in the neutron channel , 235

92 U143 .We have restricted to a mean field like study (EFA) as we know thatParity and Number Parity projections produce very tiny changes on theenergies of the configurations relevant to fission


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Outline






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The Selection of the even nuclei

We have two possibilities to select the even-even neighbour nucleus ofthe 235U , let’s say 234U and 236UIn order to check whether this selection is important or not, we havecomputed the the PES for 235U blocking the first state with jz = 1

2 andusing the quasi-particle states of both even neighbour nuclei 234U and236U.

235U

0 20 40 60 80 100 120 140Q20 (b)

-1778

-1776

-1774

-1772

-1770

-1768

-1766

-1764

E (M

eV)

1/2


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234U and 236U

235U

0 10 20 30 40 50 60 70 80 90 100110120130140150Q20 (b)

0

2

4

6

8

10

12

14

E (M

eV)

1/2

It is easy to see that the results are exactly the same, so we canconclude that it doesn’t matter which even neighbour nucleus isselected.


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Comparison

An average calulation has been perfomed, just to have a reference tocompare with.

0 10 20 30 40 50 60 70 80 90 100 110 120Q20 (b)

-1778

-1776

-1774

-1772

-1770

-1768

-1766

-1764

-1762

-1760

E (M

eV)

234U235Np AVE.235U AVE.


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235U

Energy

235U

0 20 40 60 80 100 120 140Q20 (b)

-1784

-1780

-1776

-1772

-1768

-1764

-1760

E (M

eV)

1/2 3/2 5/2

7/2 9/2 11/2


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235U

Ground state spectrumCalculated Measured

0

0.2

0.4

0.6

0.8

1

1.2

E (M

eV)

1/2 +

7/2 -

5/2 +

5/2 -

5/2 +

0

0.2

0.4

0.6

0.8

E (M

eV)

7/2 -

1/2 +5/2 +

5/2 +3/2 +

7/2 +

5/2 -3/2 - 1/2 -

1/2 - 1/2 +


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235U

Calculated

Ground state spectrum Fission Isomer spectrum

0

0.2

0.4

0.6

0.8

1

1.2

E (M

eV)

1/2 +

7/2 -

5/2 +

5/2 -

5/2 +

0

0.2

0.4

0.6

0.8

1

1.2

E (M

eV)

5/2 +

11/2 +

9/2 -1/2 +

3/2 -


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235U

Pairing energy

235U

0 10 20 30 40 50 60 70 80 90 100110120Q20 (b)

02468

101214161820

-Epp

(MeV

) 1/2 3/2 5/2

7/2 9/2 11/2

prot.neut.

EFAAVE


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235U

Collective masses

235U

0 20 40 60 80 100Q20 (b)

0

4

8

12

16

20

24

BA

TDH

F A

4/3 (b

-2)

1/2 3/2 5/2

7/2 9/2 11/2


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235U Spontaneous fission half-life

The spontaneous fission half-lifes are some orders of magnitudelargers in odd-A nuclei than in even-A nuclei.This is because of :1) The specialisation energy:

0 10 20 30 40 50 60 70 80 90 100 110 120Q20 (b)

-1778

-1776

-1774

-1772

-1770

-1768

-1766

-1764

-1762

-1760

E (M

eV)

234U235Np AVE.235U AVE.


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235U Spontaneous fission half-life

2)The larger inertia B(Q20) due to the smaller pairing energy in odd-Anuclei.

235U

0 10 20 30 40 50 60 70 80 90 100110120Q20 (b)

02468

101214161820

-Epp

(MeV

) 1/2prot.neut.

EFAAVE 235U

0 20 40 60 80 100Q20 (b)

0

4

8

12

16

20

24

BA

TDH

F A

4/3 (b

-2)

1/2

Measured Calculated234U 4.471023s 1.91019s235U 3.241026s 5.71037s236U 7.651023s 3.341023s


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235Np

Energy

235Np

0 20 40 60 80 100 120 140Q20 (b)

-1785

-1780

-1775

-1770

-1765

-1760

E (M

eV)

1/2 3/2 5/2

7/2 9/2 11/2


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235Np

Ground state spectrumCalculated Measured

0

0.2

0.4

0.6

0.8

1

E (M

eV)

5/2 -

5/2 +

1/2 +3/2 -3/2 +

0

0.2

0.4

0.6

0.8

1

E (M

eV)

5/2 +5/2 -

1/2 -

3/2 -

7/2 -


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235Np

Calculated

Ground state spectrum Fission Isomer spectrum

0

0.2

0.4

0.6

0.8

1

E (M

eV)

5/2 -

5/2 +

1/2 +3/2 -3/2 +

0

0.2

0.4

0.6

0.8

1

E (M

eV)

7/2 +9/2 -3/2 +3/2 -1/2 +3/2 -

5/2 -

1/2 -

1/2 -


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235Np

Pairing energy

235Np

0 10 20 30 40 50 60 70 80 90 100110120Q20 (b)

02468

101214161820

-Epp

(MeV

) 1/2 3/2 5/2

7/2 9/2 11/2

prot.neut.

EFAAVE


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235Np

Collective masses

235Np

0 20 40 60 80 100 120 140Q20 (b)

0

4

8

12

16

20

24

BA

TDH

F A

4/3 (b

-2)

1/2 3/2 5/2

7/2 9/2 11/2


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237Plutonium Fission Study

We have calculated, in the same way as for 235Np and 235U the fissionbarrier of the 237Pu and 237Am

0

0.2

0.4

0.6

0.8

1

E (M

eV)

7/2 +

237Pu

experimental g.s.

1/2 +

237Pu

experimental g.s.

5/2 +

237Pu

experimental g.s.

3/2 +

237Pu

experimental g.s.

5/2 +

237Pu

experimental g.s.

7/2 +

237Pu

experimental g.s.

1/2 -

237Pu

experimental g.s.

5/2 -

237Pu

experimental g.s.

1/2 +

237Pu

experimental g.s.

0

0.2

0.4

0.6

0.8

1

E (M

eV)

1/2 +

7/2 +

237Pu

ground state

5/2 +

3/2 -

237Pu

ground state


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237Plutonium Fission Study

0

0.2

0.4

0.6

0.8

1

E (M

eV)

1/2 +

7/2 +

237Pu

ground state

5/2 +

3/2 -

237Pu

ground state

0

0.2

0.4

0.6

0.8

1

E (M

eV)

5/2

11/2

237Pu

fission isomer

9/2

3/2

237Pu

fission isomer

The experimental excitation energy of the fission isomer is 2.6 MeV.Our result for this value is 3.82 MeV.


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237Americium Fission Study

The experimental ground state of 237Am is 52−

0

0.2

0.4

0.6

0.8

1

E (M

eV)

5/2 +

3/2 -

237Am

ground state

0

0.2

0.4

0.6

0.8

1

E (M

eV)

1/2 7/2

237Am

fission isomer

3/2

9/2

237Am

fission isomer

5/2

Our result for the excitation energy of the fission isomer is 2.12 MeV.S. Pérez and L.M. Robledo Theoretical study of fission barriers in odd-A nuclei using the Gogny force

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Summary and Projects

We have justified the EFA in terms of the FT-HFB.

Based on this justification, techniques beyond mean field can beused.

The small effect of the Number Parity Projection shows that EFAis a good approximation for odd-A nuclei.

Important results have been obtain when Parity projections hasbeen applied, such in the Radium isotopes.

A study of the fission barrier of 235U and 235Np has been carriedout. Results regarding the spin, parity and energy of the groundand shape isomeric states are very satisfactory.

These promising results make us be optimistic about the scope ofthe EFA.


Documents

Theoretical study of ﬁssion barriers in odd-A nuclei using ...irfu.cea.fr/dphn/Espace_Theorie/May2006/talks/Perez.pdf · S. Pérez and L.M. Robledo Theoretical study of ﬁssion