Theoretical study of fission barriers in odd-a nuclei using the Gogny force

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Theoretical study of fission barriers in odd-a nuclei using the Gogny force S. Pérez and L.M. Robledo Departamento de Física Teórica Universidad Autónoma de Madrid Saclay, 9th May 2006 Workshop on the Theories of Fission and Related Phenomena

Outline 1 Motivation 2 Mean Field 3 FT-HFB Theory 4

Outline Motivation 1 Motivation 2 Mean Field 3 FT-HFB Theory 4

Motivation Aims _ We are interested in having a microscopic description of fission because we want to understand the phenomenon by itself. _ Such description would be very helpful to make predictions for neutron rich nuclei, essential for stellar nucleosynthesis calculations, as well as for superheavies.

Motivation Aims _ We are interested in having a microscopic description of fission because we want to understand the phenomenon by itself. _ Such description would be very helpful to make predictions for neutron rich nuclei, essential for stellar nucleosynthesis calculations, as well as for superheavies.

Motivation Aims In general, we would like to have a microscopic description of fission in odd-a nuclei using a mean field theory with realistic interactions. The traditional procedure (blocked HFB) implies the breaking of time-reversal symmetry, which means a high computational cost, specially for the study of fission processes. The alternative is to use Equal Filling Approximation, which keeps the time-reversal symmetry (and also axiallity).

Motivation Aims In general, we would like to have a microscopic description of fission in odd-a nuclei using a mean field theory with realistic interactions. The traditional procedure (blocked HFB) implies the breaking of time-reversal symmetry, which means a high computational cost, specially for the study of fission processes. The alternative is to use Equal Filling Approximation, which keeps the time-reversal symmetry (and also axiallity).

Motivation Aims In general, we would like to have a microscopic description of fission in odd-a nuclei using a mean field theory with realistic interactions. The traditional procedure (blocked HFB) implies the breaking of time-reversal symmetry, which means a high computational cost, specially for the study of fission processes. The alternative is to use Equal Filling Approximation, which keeps the time-reversal symmetry (and also axiallity).

Motivation Aims In general, we would like to have a microscopic description of fission in odd-a nuclei using a mean field theory with realistic interactions. The traditional procedure (blocked HFB) implies the breaking of time-reversal symmetry, which means a high computational cost, specially for the study of fission processes. The alternative is to use Equal Filling Approximation, which keeps the time-reversal symmetry (and also axiallity).

Outline Mean Field 1 Motivation 2 Mean Field 3 FT-HFB Theory 4

HFB Theory Mean Field The Hartree-Fock-Bogoliubov theory has been used. The quasi-particle creation and annihilation operators are given by the most general linear transformation from the particle operators: ( ) ( αk U V = ) ( ) cl l α k The matrices U and V will be the variational parameters which minimise the HFB energy: V T U T H = Tr(tρ) + 1 2 Tr(Γρ) 1 2 Tr( κ ) kl c + l ρ ij = c + j c i = (V V T ) ij ; κ ij = c j c i = (V U T ) ij Γ ij = v ikjl ρ lk ; ij = 1 kl 2 kl v ijkl κ kl

Trial Ground State Mean Field The ground state of even-even nuclei is represented as the QP vacuum: φ = α k α k ρ = V V T andκ = V U T k>o However, the ground state of odd nuclei can be characterised by means of: φ 1qp = α + i φ ρ kk = ( V ) V T kk + { U ki U k i V } ki V k i κ kk = ( V ) U T kk + { U ki V k i V } ki U k i Drawback With this ansatz the time-reversal symmetry is broken.

Mean Field Time-reversal invariant description To restore the time reversal symmetry, an intuitive solution comes in terms of the Equal Filling Approximation (EFA): ρ efa kk = ( V ) V T kk + 1 { Uki U k 2 i V ki V k i + U ki U k i V } ki V k i κ efa kk = ( V ) U T kk + 1 { Uki V k 2 i V ki U k i + U ki V k i V } ki U k i Without any justification, and using it in the same way as in the ordinary HFB theory, the energy is computed as: E efa = Tr(tρ efa ) + 1 2 Tr(Γefa ρ efa ) 1 2 Tr( efa κ efa ) the U and V matrices are computed by solving the HFB equation ( )( ) ( ) t + Γ efa efa Uk Uk t Γ efa = E k efa again without any foundation V k V k

Outline FT-HFB Theory 1 Motivation 2 Mean Field 3 FT-HFB Theory 4

HFB Theory FT-HFB Theory Writing the density matrix and the pairing tensor in term of matrices: ( ρ R efa efa κ = efa κ efa 1 ρ efa ) ( U V = V U )( f 0 0 1 f )( U + V + V T U T ) where f k = { 1 2 0 k = i or k = ī otherwise which reminds us the Finite-Temperature HFB Theory!

FT-HFB Theory Expectation value of the Hamiltonian From the self-consistent FT-HFB theory, the energy could be expanded through: E = Tr(DĤ) = 1 φ α k1,,α kn Hα + k Z 1,,α + k N φ e β(e k 1 + +E kn ) {n} Both expressions give the same results: E efa = 1 4 { φ H φ + φ α i Hα + i φ + φ α i Hα + i φ + φ α i α i Hα + ī α + i } φ = = Tr(tρ efa ) + 1 2 Tr(Γefa ρ efa ) 1 2 Tr( efa κ efa ) Therefore, we have, again, a theory based on the variational principle on the EFA energy given above.

Gradient method FT-HFB Theory Since the FT-HFB equations are nonlinear, they can be solved by iteration. The use of iterative diagonalization generates some problems about the convergence and has complications that arise in cases with one or more constraints. These facts drive us to employ the gradient method. The gradient method can be used because the EFA equations come from the variational principle on the EFA energy. This method is based on the Thouless parametrisation of the most general HFB wave function: φ 1 (Z) = N exp[ k<k Z kk α + k α+ k ] φ 0

FT-HFB Theory Advantages of having a justification Thanks to this justification, we can extend the use of the EFA in more advanced frameworks, such as the time-dependent HFB theory, projection methods, etc. Specifically, the adiabatic approximation of the TD-HFB theory has been used to evaluate the collective masses involved in the Collective Hamiltonian.

Outline 1 Motivation 2 Mean Field 3 FT-HFB Theory 4

Effective Interaction The Gogny density dependent effective force with the D1S parametrisation has been used: V 12 = 2 (W i + B i ˆPσ H i ˆPτ M i ˆPσ ˆPτ ) e ( r1 r2)2 µ 2 i i=1 + i W LS ( 1 2 ) δ( r 1 r 2 )( 1 2 ) ( σ 1 + σ 2 ) (1) [ + t 0 (1 + x 0 ˆPσ )δ( r 1 r 2 ) ρ( r γ 1 + r 2 )] + V Coul 2

Approximations and Basis We have used the Slater approximation to compute the Coulomb exchange term: E CE = 3 4 e2 ( 3 π ) 1/3 Z [ρ p ( r )] 3/4 d 3 r The QP operators have been expanded into an axially symmetric harmonic oscillator basis including 17 shells. In order to have an accurate fission driving coordinate, we have used the Q 20 degree of freedom. The three lowest QP states for each J z value (J z = 1 2, 3 2, 5 2, 7 2, 9 2 and 11 2 ) have been selected to be blocked.

Half-life formula The spontaneous fission half life has been calculated using the traditional WKB approximation: where S is the action T sf (s) = 2.8610 21 ( 1 + e 2S) S = Z b a dq 20 2B (Q 20 )(V (Q 20 ) E 0 ) where B is the collective mass computed in the framework of the adiabatic TDHFB. In the collective potential energy V(Q 20 ) the rotational energy correction has been taken into account.

Projection techniques In order to better describe the odd-a nuclei we can implement the parity and number parity projection, and check when it is important to apply them. Parity Projection Since the reflexion symmetry is broken when octupole deformation appears, we will check how and when parity projection is important. - Number Parity Projection Since we are describing an odd-a nuclei, only odd quasi-particles states should be considered. Therefore, in order to keep only one quasi-particle states, the number parity projection must be applied. EPRO efa = 1 { φ α i Hα + i 2 φ + φ α i Hα + i } φ

Parity Projection Interesting examples on which to apply parity projection should have octupole deformation in the ground state. We have chosen the 229 Radium isotope to make this study. Projection After Variation drawbacks: In order to be close to the Variation After Projection method, we shall study the Projection After Variation solution in terms of the octupolar degree of freedom, even if the mean field ground state doesn t show octupolarity.

Parity Projection -1731-1732 13 11-1733 E (MeV) -1734-1735 09 01 03 05 07-1736 -1737 229 Ra -1738 0 1000 2000 3000 4000 5000 6000 Q 3 (fm 3 ) Therefore we can extract from these calculations that this projection is important when the octupolarity is in a certain range, β 0.01 0.1, not more not less.

Number Parity Projection We have used the 235 U fission barrier to see the effects of this projection. 7/2 9/2 11/2-1764 235 U -1768 E (MeV) -1772-1776 1/2 3/2 5/2 0 5000 10000 15000 Q 20 (b)

Number Parity Projection 235U 1st jz=1/2-1760 E HFB (MeV) -1765-1770 -1775-1780 0 5000 10000 15000 0.2 Eproj. - E_ EFA (MeV) 0.15 0.1 0.05 0 0 5000 10000 15000 Q_20 (fm²) It is easy to check that the modification is very small and of the order of 100keV. Therefore we can state that the Equal Filling Approximation is a very good way to describe odd-a nuclei, since to number parity correction is very small.

Odd-A fission calculations We have applied the EFA method to the calculation of two fission barriers, one for a nucleus blocking in the proton channel, that is 235 93 Np 142, and other blocking in the neutron channel, 235 92 U 143. We have restricted to a mean field like study (EFA) as we know that Parity and Number Parity projections produce very tiny changes on the energies of the configurations relevant to fission

Outline 1 Motivation 2 Mean Field 3 FT-HFB Theory 4

The Selection of the even nuclei We have two possibilities to select the even-even neighbour nucleus of the 235 U, let s say 234 U and 236 U In order to check whether this selection is important or not, we have computed the the PES for 235 U blocking the first state with j z = 1 2 and using the quasi-particle states of both even neighbour nuclei 234 U and 236 U. -1764-1766 235 U -1768 E (MeV) -1770-1772 -1774-1776 -1778 1/2 0 20 40 60 80 100 120 140 Q 20 (b)

234 U and 236 U 14 12 235 U 10 E (MeV) 8 6 4 2 0 1/2 0 10 20 30 40 50 60 70 80 90 100110120130140150 Q 20 (b) It is easy to see that the results are exactly the same, so we can conclude that it doesn t matter which even neighbour nucleus is selected.

Comparison An average calulation has been perfomed, just to have a reference to compare with. -1760-1762 -1764 234 U 235 Np AVE. 235 U AVE. -1766 E (MeV) -1768-1770 -1772-1774 -1776-1778 0 10 20 30 40 50 60 70 80 90 100 110 120 Q 20 (b)

235 U Energy E (MeV) -1760-1764 -1768-1772 -1776-1780 -1784 235 U 0 20 40 60 80 100 120 140 Q 20 (b) 7/2 9/2 11/2 1/2 3/2 5/2

235 U Ground state spectrum Calculated 1.2 Measured 0.8 1/2-1/2 + E (MeV) 1 0.8 0.6 0.4 0.2 5/2-7/2-5/2 + 5/2 + E (MeV) 0.6 0.4 0.2 3/2-7/2 + 5/2 + 1/2 + 1/2-5/2 3/2 + 5/2 + 0 1/2 + 0 7/2 -

235 U Calculated Ground state spectrum 1.2 Fission Isomer spectrum 1.2 E (MeV) 1 0.8 0.6 0.4 5/2-7/2-5/2 + 5/2 + E (MeV) 1 0.8 0.6 0.4 1/2 + 11/2 + 3/2-9/2-0.2 0.2 0 1/2 + 0 5/2 +

235 U Pairing energy 7/2 9/2 11/2 -E pp (MeV) 20 18 16 14 neut. prot. 12 10 8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100110120 Q 20 (b) AVE 235 U EFA 1/2 3/2 5/2

235 U Collective masses 7/2 9/2 11/2 B ATDHF A 4/3 (b -2 ) 24 20 16 12 8 4 235 U 1/2 3/2 5/2 0 0 20 40 60 80 100 Q 20 (b)

235 U Spontaneous fission half-life The spontaneous fission half-lifes are some orders of magnitude largers in odd-a nuclei than in even-a nuclei. This is because of : 1) The specialisation energy: -1760-1762 234 U 235 Np AVE. -1764 235 U AVE. -1766 E (MeV) -1768-1770 -1772-1774 -1776-1778 0 10 20 30 40 50 60 70 80 90 100 110 120 Q 20 (b)

235 U Spontaneous fission half-life 2)The larger inertia B(Q 20 ) due to the smaller pairing energy in odd-a nuclei. -E pp (MeV) 20 18 16 14 12 10 8 6 4 neut. prot. AVE 235 U EFA 1/2 2 0 0 10 20 30 40 50 60 70 80 90 100110120 Q 20 (b) B ATDHF A 4/3 (b -2 ) 24 20 16 12 8 4 0 235 U 1/2 0 20 40 60 80 100 Q 20 (b) Measured Calculated 234 U 4.4710 23 s 1.910 19 s 235 U 3.2410 26 s 5.710 37 s 236 U 7.6510 23 s 3.3410 23 s

235 Np Energy 7/2 9/2 11/2-1760 -1765 235 Np E (MeV) -1770-1775 -1780-1785 0 20 40 60 80 100 120 140 Q 20 (b) 1/2 3/2 5/2

235 Np Ground state spectrum Calculated 1 3/2-3/2 + 1/2 Measured 1 7/2-0.8 0.8 E (MeV) 0.6 0.4 5/2 + E (MeV) 0.6 0.4 3/2-1/2-0.2 0.2 0 5/2-0 5/2-5/2 +

235 Np Calculated Ground state spectrum 1 0.8 3/2-3/2 + 1/2 Fission Isomer spectrum 1 0.8 1/2-1/2 - E (MeV) 0.6 0.4 5/2 + E (MeV) 0.6 0.4 0.2 0 5/2-0.2 0 3/2-3/2-9/2-5/2-1/2 + 3/2 + 7/2 +

235 Np Pairing energy 7/2 9/2 11/2 -E pp (MeV) 20 18 16 14 neut. prot. 12 10 8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100110120 Q 20 (b) AVE 235 Np EFA 1/2 3/2 5/2

235 Np Collective masses 7/2 9/2 11/2 B ATDHF A 4/3 (b -2 ) 24 20 16 12 8 4 235 Np 1/2 3/2 5/2 0 0 20 40 60 80 100 120 140 Q 20 (b)

237 Plutonium Fission Study We have calculated, in the same way as for 235 Np and 235 U the fission barrier of the 237 Pu and 237 Am 237 Pu experimental g.s. 237 Pu ground state 1 1 3/2-0.8 1/2 + 0.8 5/2 + E (MeV) 0.6 0.4 5/2-7/2 + 3/2 + 1/2-5/2 + 5/2 + E (MeV) 0.6 0.4 7/2 + 0.2 1/2 + 0.2 0 7/2 + 0 1/2 +

237 Plutonium Fission Study 237 Pu ground state 237 Pu fission isomer 1 3/2-1 3/2 0.8 5/2 + 0.8 9/2 E (MeV) 0.6 0.4 7/2 + E (MeV) 0.6 0.4 0.2 0.2 11/2 0 1/2 + 0 5/2 The experimental excitation energy of the fission isomer is 2.6 MeV. Our result for this value is 3.82 MeV.

237 Americium Fission Study The experimental ground state of 237 Am is 5 2 1 1 0.8 237 Am ground state 0.8 237 Am fission isomer E (MeV) 0.6 0.4 0.2 3/2 - E (MeV) 0.6 0.4 0.2 9/2 5/2 0 5/2 + 0 7/2 1/2 3/2 Our result for the excitation energy of the fission isomer is 2.12 MeV.

Summary and Projects We have justified the EFA in terms of the FT-HFB. Based on this justification, techniques beyond mean field can be used. The small effect of the Number Parity Projection shows that EFA is a good approximation for odd-a nuclei. Important results have been obtain when Parity projections has been applied, such in the Radium isotopes. A study of the fission barrier of 235 U and 235 Np has been carried out. Results regarding the spin, parity and energy of the ground and shape isomeric states are very satisfactory. These promising results make us be optimistic about the scope of the EFA.

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