1

Continuum Hartree-Fock Random Phase Approximation Description of Isovector Giant Dipole Resonance in

28O, 60Ca and 80Zr.

Emilian NicaTexas A&M University

Advisor: Dr.Shalom Shlomo

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PlanI. Introduction

a. Overviewb. Collective Modesc. Isovector Giant Dipole Resonanced. Collective Motion Models

II. Theorya. Overviewb. Hartree-Fock Mean-Field Approximationc. Nucleon-nucleon Interactiond. Random Phase Approximatione. Nuclear Response

III. Results and Discussion

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Part I

Introduction

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Overview General Remarks

The present discussion deals with the study of the Isovector Giant Dipole Resonance (IVGDR) in symmetric (80Zr) and neutron-rich (28O, 60Ca) nuclei.

The goal of our calculations was to determine whether the small peaks in the RPA response function were indeed due to low-energy resonance effects or to particle emission threshold energies.

We used a Hartree-Fock (HF) Continuum Random Phase Approximation (CRPA) to determine the nuclear response of the three nuclei under study.

The highly accurate HF-CRPA was applied to account for particle excitation into the continuum and to exclude the most common sources of error.

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Collective Modes General Description

There are several collective modes in nuclei. A few are illustrated below.

In the isoscalar case, the neutron and proton ensembles oscillate in phase (i.e. dashed line). For the isovector, the neutrons and protons oscillate out of phase.

Monopole Dipole Quadrupole

Isoscalar

(T=0)

Isovector

(T=1)

L=0 L=1 L=2

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Isovector Giant Dipole Resonance Brief History

The Isovector Giant Dipole Resonance (IVGDR) was the first collective motion seen in nuclei. It was discovered in 1947 by G.C. Baldwin and G.S. Klaiber.

This was done by studying the photon absorption cross section for different nuclei. The giant resonance corresponds to the prominent peak in the graph below.

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Liquid Drop Model In the Liquid Drop model the IVGDR is described as the oscillation of

the neutron liquid against the proton liquid with a restoring force related to the neutron-proton interaction.

The neutron density is given by

and the proton density is given by

In the expressions above ρ0n and ρ0p are the neutron and proton saturation densities respectively , 2ε is the small amplitude of the oscillation and is the oscillating frequency.

Microscopic Description In the microscopic treatment of collective motion the nuclear wave

function is described as a linear combination of particle-hole excitations. We used the Hartree-Fock (HF) based Continuum Random Phase

Approximation (CRPA) to calculate the response function of the IVGDR in symmetric (80Zr) and neutron-rich nuclei (28O and 60Ca).

),cos()(),( 0 trA

Zt on

nn

rr

).cos()(),( 0 trA

Nt op

pp

rr

Collective Motion Models

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Part II

Theory

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Overview Microscopic Model

The microscopic model involves the use of HF-CRPA calculations to describe the collective motion in nuclei. In essence, its use involves the collective effect of single-particle excitations on the structure of the nucleus.

The HF-CRPA calculations model the collective motion through changes in the ground-state nuclear wave function induced by particle-hole excitations.

The collective aspect of the motion is accounted for by considering all possible single-particle excitations and the overall effect these have on the nuclear properties.

Our calculations were based on a Green’s function HF-CRPA. This allows for solutions in coordinate space as opposed to configuration space and for a simplification of the procedure.

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Hartree-Fock Mean-Field Approximation General Remarks

The Hartree-Fock Mean-Field Approximation is a method used to solve the nuclear many-body problem.

The Hartree-Fock calculations allow us to determine the nuclear ground-state wave function and energy.

The Hartree-Fock method and results provide a basis (i.e. wave function , Hamiltonian and single-particle energies) for the introduction of particle-hole excitations in the RPA.

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Hartree-Fock Mean-Field Approximation Hamiltonian Operator

The total Hamiltonian of the many-body nuclear system can be written as a sum of the single-particle kinetic energies and two-body interactions (potentials)

V is the short-range, nucleon-nucleon interaction and it is written here in general form. The potential V can be chosen a specific form (i.e. Skyrme interaction).

Mean-Field Approximation The many-body Schrodinger equation is difficult to solve. We

will use a mean-field approximation to considerably simplify our models. In a mean-field approximation each particle moves independently of other nucleons in a central potential U representing the interaction of a nucleon with all the other nucleons.

.)(1

A

ji

A

ii ijVTH

EH

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Slater Determinant Because of the antisymmetrization of the overall nuclear wave function

Φ0 in the mean-field approximation, it can be written as a determinant known as the Slater determinant

The arguments 1, 2…, A above denote the coordinates of the particles (position and spin) while λ are the single-particle wave functions.

By interchanging any two rows the sign of the wave function will change (i.e. antisymmetry).

)(...)2()1(

.

.

.

)(...)2()1(

)(...)2()1(

!

1),...,1(

222

111

0

A

A

A

AA

AAA

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Variational Principle By applying the variational principle to the expectation value of the

Hamiltonian and normalizing through the use of Lagrange multipliers

we obtained after some calculations the HF equations

The HF equations can be solved in an iterative way by “guessing” an initial set of wave functions λ, solving the equations above for new single-particle wave functions and repeating the process until convergence (i.e. ελ converge).

),1()1()2()2()12()2()1()2()2()12()2()1( **1

dVdVT

.,...2,1 A

0),...,1(),...,1( *00

*0 AHA

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Nucleon-nucleon interaction In our calculation we used a simplified Skyrme type effective nucleon-

nucleon interaction given by

where the parameters t0, t3 and α are obtained by fitting the HF results to experimental data.

In our calculations we have used the values of:

t0=-1600 MeV fm3

t3=12500 MeV fm4

α=1/3

),(26

1)( 30 ji

jijiij ttV rr

rrrr

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Random Phase Approximation Microscopic Treatment

In the microscopic treatment of collective vibrations within the RPA method the new wave function is a linear combination of all possible single-particle excitations.

In our treatment of the IVGDR we will use a Green’s function formalism instead of a “standard” RPA procedure (i.e. configuration space).

Linear Response Theory The bare particle-hole Green’s function is defined as

In the preceding equation, h is the occupied single-particle wave function with the corresponding single-particle energy εh, H0 is the HF Hamiltonian and is the excitation energy of the nucleus.

).'(11

)(),',(00

*)0( rrrr hh hh

h HHG

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The RPA Green’s function is obtained from

The term Vph in the expression above stands for the particle-hole interaction and it is directly linked to the RPA procedure since it represents the additional interaction introduced by single-particle excitations.

The formalism must account for the possibility of the excitation of a particle beyond the binding energy threshold (i.e. into the continuum).

.)1( 1)0()0( GVGG phRPA

Random Phase Approximation

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Continuum Effects Because of the possibility of particle excitation into the continuum the

single-particle Green’s function is obtained from

ulj is the regular solution to the HF Hamiltonian and vlj is the irregular solution (i.e. numerical solution starting from a low r to r and vice versa). The r and r stand for lesser and greater of r and r’, respectively, while W is the Wronskian

The boundary condition for the irregular solution at r is:

For positive energies E, this describes an outgoing wave asymptotically.

We note that in our calculations we have used a very small value for the smearing parameter : /2=0.01 MeV.

.dr

duv

dr

dvuW

./)()(21

),',(2

0

Wrvrum

EHEg ljljlj

rr

.2

exp~)(2

r

mEirv

Random Phase Approximation

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Nuclear Response General Remarks

We define a scattering operator F such as

The nuclear response is given by the Strength Function S(E):

where and denote the ground and excited states, respectively.

In the Green’s function formalism

Another useful value is the transition density. In our case it can be obtained using the relation

'.),,'(Im1

)'()(

),( rrrrr dEGfEES

EEt

,)(0)(2

nnEEnFES

.)(Im1

)( fGfTrES

.)(i

ifF r

0 n

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Part III

Results and Discussion

Orbits Proton Neutron Proton Neutron Proton Neutron0s1/2 -26.824 -26.824 -29.919 -29.919 -30.792 -30.7920p3/2 -17.882 -17.882 -23.776 -23.776 -25.569 -25.5690p1/2 -17.882 -17.882 -23.776 -23.776 -25.569 -25.569

----------0d5/2 -7.614 -7.614 -16.350 -16.350 -19.139 -19.1390d3/2 -7.614 -7.614 -16.350 -16.350 -19.139 -19.1391s1/2 -5.225 -5.225 -13.591 -13.591 -16.438 -16.438

---------- ----------0f7/2 -7.952 -7.952 -11.714 -11.7140f5/2 -7.952 -7.952 -11.714 -11.7141p3/2 -4.674 -4.674 -7.998 -7.9981p1/2 -4.674 -4.674 -7.998 -7.998

---------- ---------- ----------0g9/2 -3.536 -3.5360g7/2 -3.536 -3.5361d5/2 -0.255 -0.2551d3/2 -0.255 -0.2552s1/2 -0.255 -0.255

28O 60Ca 80Zr

HF Single Particle energies (MeV)

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Results

Figure 1.The IVGDR free and RPA response functions for 60Ca (in arbitrary units).

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Results

Figure 2. Similar to Figure 1 for 28O.

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Figure 3. Similar to Figure 1 for the symmetric nucleus 80Zr.

Results

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Discussion The figures show (solid line) and (dashed line) of

the free (top) and RPA (bottom) of the response functions for 28O, 60Ca and 80Zr.

For a resonance has a peak at a resonance energy E=ER, and decreases with E, going through zero at ER.

Example 60Ca:(i) Threshold energies: Sn=4.674 MeV, Sp=13.591 MeV.

(ii) The sharp peaks at 8.398, 8.917 and 11.676 MeV are due to proton bound to bound particle-hole excitations π0d -> π0f, π1s -> π1p

and π0d -> π1p, respectively.

(iii) The neutron particle-hole excitation are all to the continuum. Note the threshold effect of enhancement in above 4.674 MeV and at around 6 MeV.

(iv) In the RPA excitation strength ( ) we observe the collective IVGDR above 11 MeV with threshold enhancement at low excitation energies.

fGfIm

fGfIm

fGfIm

fGfIm

fGfRe

fGfRe

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SummaryWe have carried out the HF-based-Continuum RPA calculation of the IVGDR response function for the symmetric nucleus 80Zr and the neutron-rich nuclei 28O and 60Ca.

We have demonstrated the important threshold effect of enhancement in the IVGDR excitation strength at low excitation energies in neutron-rich nuclei associated with loosely bound orbits.

In 80Zr, due to the large single-particle separation energies, the corresponding threshold enhancements in the excitation strength are negligible.

In common RPA calculations with discretized continuum these strength enhancements incorrectly appear as low-lying excited states, also termed soft dipole modes.

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Acknowledgments

Work done at:

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Work supported by:

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Hartree-Fock Mean-Field Approximation Antisymmetry

The Pauli exclusion principle states that no two identical fermions can occupy the same state. A consequence of this is that interchanging the positions of 2 particles will change the sign of the overall wave function.

Let Ψab(r12) be a wavefunction of two particles with r12 the distance between the particles and a , b the occupied orbits. The Pauli exclusion principle implies that Ψab(r12) → 0 as r12 → 0.

If Ψab(r12)= Ψa(r1) Ψb (r2) it is not necessary that the above condition is respected. A better expression would be

, which vanishes when r1=r2.

)()()()()( 122112 rrrrr babaab

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Random Phase Approximation General Remarks:

The Random Phase Approximation (RPA) is a method to describe excitations of ground state particles. For a single particle excitation the RPA shifts a particle from its ground state into a higher energy state. A schematic illustration of this is shown below.

In a Slater Determinant formalism this corresponds to the annihilation of one state (i.e. a row) and the creation of a new state in place of the previous one. The energy of the system is the eigenvalue corresponding to the new nuclear wave function.