Elsevier

Physics Reports

Volume 393, Issue 1, March 2004, Pages 1-86
Physics Reports

Extended theory of finite Fermi systems: collective vibrations in closed shell nuclei

https://doi.org/10.1016/j.physrep.2003.11.001Get rights and content

Abstract

We review an extension of Migdal's Theory of Finite Fermi Systems which has been developed and applied to collective vibrations in closed shell nuclei in the past 10 years. This microscopic approach is based on a consistent use of the Green function method. Here one considers in a consistent way more complex 1p1h⊗phonon configurations beyond the RPA correlations. Moreover, these configurations are not only included in the excited states but also explicitly in the ground states of nuclei. The method has been applied to the calculation of the strength distribution and transition densities of giant electric and magnetic resonances in stable and unstable magic nuclei. Using these microscopic transition densities, cross sections for inelastic electron and alpha scattering have been calculated and compared with the available experimental data. The method also allows one to extract in a consistent way the magnitude of the strength of the various multipoles in the energy regions in which several multipoles overlap. We compare the microscopic transition densities, the strength distributions and the various multipole strengths with their values extracted phenomenologically.

Introduction

The atomic nucleus is a complicated quantum mechanical many-body system with a rich excitation spectrum. The experimental investigations of nuclei during the past 50 years have provided us with an overwhelming amount of excellent data which have to be understood in the framework of a quantum mechanical many-body theory. Moreover, we expect new data from the coming radioactive beam facilities for atomic nuclei far from the stability line, which will give us information about, for example, nuclei with a large neutron-to-proton ratio. Some of the features of the nuclei, such as their level fluctuations, can be described in terms of random matrix theory, which is now interpreted as the manifestation of chaotic motion. This stochastic nature of the levels is, however, only one aspect of the spectrum of the atomic nucleus which shows, on the other hand, many regular features, such as single-particle structure and collective modes. This regular behavior is a consequence of the Pauli principle and the special structure of the nucleon–nucleon interaction that leads to the concept of a mean field. As a consequence of the mean field, one is able to define quasiparticles in the sense of Landau and to apply Landau's theory of Fermi liquids [1] to the nuclear many body problem, as has been done by Migdal in his theory of finite Fermi systems (TFFS) [2].

In this article we focus on the question of how nuclei respond to a weak external field. This corresponds in lowest order to linear response and Migdal's equations look formally very similar to it. As we will show in the next section, however, the range of validity is much larger than one would expect from the conventional derivation of the linear response equations. Migdal's equations are derived within the many body Green functions (GF) theory. Using this powerful formulation of the many-body problem one is able to obtain equations which in principle are “exact”. This is due to a renormalization procedure similar to the one used in quantum field theory. In the original version of the TFFS one considers explicitly only the propagation of particle–hole pairs in the nuclear medium. All the other configurations, such as the two particle-two holes ones, are renormalized into an effective two-body interaction and effective operators. The final equations have the same form as the corresponding equations of the conventional linear response theory, or random phase approximation (RPA).

The important observation by Migdal was that the effective interaction and effective operators so defined depend only weakly on the mass number and the energy. Therefore, as in Landau's original theory, these effective quantities can be parameterized and the corresponding parameters should be the same for all nuclei except for the lightest ones. Moreover, these parameters have been chosen to be density-dependent, as the interaction inside and outside of nuclei may be quite different. These assumptions have been shown to be correct in numerous applications and they were and still are the most important features of TFFS. One can hope that the same parameters are also applicable to nuclei far from stability where we shall next apply the theory.

It is well known that pairing correlations are important in non-magic nuclei. In this case on has to define quasiparticles analogously to the BCS theory, which include the pairing gap, and in the linear response equation one has to consider as well the change of the pairing gaps [2], [3] in a consistent way. The equations of the extended theory look formally identical to the quasiparticle RPA (QRPA) [4].

The standard TFFS is not a self-consistent theory in the sense that one starts with an effective interaction from which one obtains simultaneously the mean-field quantities such as single-particle energies, single-particle wave functions and the particle–hole interaction for the linear response. Instead, one has independent parameters for the single-particle model and the particle–hole interaction. Of the various extensions of Migdal's original theory that have been made over the years, the first were directed at taking self-consistency effects into account [5], [6].

It has been shown that the TFFS can also be applied to strongly deformed nuclei [8]. In that case the appropriate quasiparticles can be deduced from the deformed shell model of Nilsson. The extension of TFFS to a second order response theory has been developed [9] and the formalism for the calculation of electric and magnetic moments of excited states in even–even nuclei, transitions between excited states, as well as isomer shifts of rotational states in deformed nuclei, can be found in [10], [11].

In addition, the Δ-isobar has been included in the space of quasiparticles [12] and, in the effective interaction, the effect of the one pion exchange, which turned out to be crucial for the existence of pion condensation and the properties of spin modes [7], [13], [14].

As mentioned earlier, both the linear response theory and (effectively) the TFFS are based on the RPA, which is restricted to a configuration space which includes one particle–one hole (1p1h) configurations only. If applied to collective modes in nuclei it gives values of, for example, the centroid energies and total strengths of giant resonances in good agreement with the data. The widths of the resonances and their fine structure, however, are not. These can only be obtained if one includes in the conventional RPA and its related approaches the coupling of the 1p1h excitations to more complex configurations. Therefore, if one wants to apply TFFS to a realistic theoretical interpretation of giant resonance experiments one has to extend the formalism to include explicitly in the theory more complex configurations than 1p1h pairs.

The main aim of this article is therefore to discuss such extensions of Migdal's theory and their application to giant resonances in nuclei. The essential feature—that which determines the necessity and the title of our approach—is of course the explicit taking of complex configurations into account. After a short review of the standard TFFS we discuss in a qualitative way possible extensions of the theory. Here we will give physical arguments for why the coupling of the most collective low-energy phonons to the single-particle and single-hole propagators is the most important one and which higher configurations are necessary for a quantitative description of the widths of giant resonances.

In Section 2 we derive the basic equations of the extended TFFS (ETFFS) for closed shell nuclei. As in the original TFFS, the GF method is used, and for convenience we restrict ourselves to doubly closed shell nuclei, where pairing correlations can be neglected. As we shall see, there exist several stages of sophistication of the extension, and we will discuss these in some detail. As a first approximation we include the complex configurations in the excited states only. This already gives rise to the fragmentation of the multipole strength, but the total strength remains unchanged. In the next step we include also the more complex configurations in the ground state of the nuclei. This gives a further fragmentation but, in addition, gives changes in the magnitudes of the electric and magnetic transition strengths, which turns out to be important for a quantitative comparison of the theoretical results with the data. The final formulas of the extended theory are given in r-space, which is especially appropriate for including effects of the single-particle continuum.

In Section 3 we apply the various versions of our theory to giant multipole resonances (GMR) in medium and heavy mass nuclei. The comparison with experiment demonstrates the power of our new theoretical framework and, simultaneously, the importance of the different steps of our approach.

In Section 4 the calculations of transition densities within our approach are reviewed. With this information one is able to derive cross sections that can be compared directly with electron and alpha scattering experiments. One major difference between the microscopically calculated transition densities and the phenomenological ones is an energy dependence of the microscopic ones. In this connection we discuss also the problems that arise in the analysis of giant resonances in nuclei in which the resonances are very broad and where the various multipolarities overlap.

Finally, we summarize our review and discuss possible extensions that may be important for a theoretical understanding of nuclei far from the stability line.

Migdal [2] has applied Landau's theory of interacting Fermi systems to atomic nuclei. Here one has first to deal with two kinds of fermions—protons and neutrons—and second with a relatively small number of particles. In the past 30 years Migdal's theory has been applied successfully by many groups to various nuclear structure problems. (See, for example, the reviews [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].)

Landau's theory deals with infinite systems of interacting fermions, such as liquid 3He or nuclear matter. If there were no interaction, the system would simply be a collection of independent particles, each characterized by its spin and a wave number k. Landau's basic assumption is that the interacting system can be obtained from the non-interacting one by an adiabatic switching on the interaction. In particular, there should be a one-to-one correspondence between the single-particle states of non-interacting systems and the so-called quasiparticle states in interacting systems. These quasiparticles behave in the correlated system like real particles in a non-interacting system. They obey Fermi–Dirac statistics and occupy, like the non-interacting particles, corresponding quasiparticle states up to the Fermi energy. In order to define such quasiparticles Landau considers the total energy of an interacting system as a functional of the occupation function n(k) of the quasiparticle states k. The quasiparticle energies are then given as first functional derivatives of the total energy with respect to n(k) and the interaction between the quasiparticles is defined as its second functional derivative. Using this approach, one can calculate properties of the excited system of a Fermi liquid, such as the zero sound mode in 3He. In infinite systems a quasiparticle differs from a real particle essentially by its mass, because both can be described by a plane wave. The effective mass of the quasiparticle is, in general, momentum–dependent and is deduced from experiments. The renormalized quasiparticle interaction depends on spin and momenta and is expanded at the Fermi surface in terms of Legendre polynomials with free parameters—the well-known Landau parameters—that are also determined from experiments.

Migdal extended these ideas to finite Fermi systems and applied his TFFS to atomic nuclei. Here the quasiparticles are the single-particle states of the nuclear shell model, which can be obtained experimentally from the neighboring odd-mass nuclei of closed shell nuclei. The quasiparticle interaction here is defined in the same way but it is isospin dependent. As in the infinite system, one expands the interaction at the Fermi surface in terms of Legendre polynomials and the parameters of the expansion, the famous Landau–Migdal parameters, are considered universal. They are also determined from experiment.

The TFFS is for that reason a semi-phenomenological microscopic theory. It is a microscopic theory because all its fundamental equations are derived rigorously from first principles, however it also contains phenomenological aspects such as the above-mentioned quasiparticle energies and the quasiparticle interaction. All the parameters are well-defined microscopically and in principle could be calculated starting from the bare nucleon–nucleon interaction. Such calculations, however, are very involved and in actual calculations one has to make severe approximations so that we cannot expect to obtain full quantitative agreement with the phenomenological parameters. Nevertheless the calculated interaction parameters are in surprisingly good agreement with the phenomenological ones [17].

As mentioned above, the energy E of an interacting system may be considered to be a functional of the occupation functions n(k) of the quasiparticles E=E(n(k)). If one excites the system, one basically changes the occupation functions by an amount δn(k). The corresponding change of the energy isδE=kϵ0(k)δn(k)+k,kfω(k,k′)δn(k)δn(k′)=kϵ0(k)+kfω(k,k′)δn(k′)δn(k)=kϵ(k)δn(k),where the ϵ0(k) are the equilibrium energies of the quasiparticles.

The quasiparticle energies ϵ(k) and the interaction between a quasiparticle and a quasi-hole fω(k,k′) defined in this way are the first and second derivatives of the energy functional with respect to the occupation functions:ϵ(k)=δEδn(k),fω(k,k′)=δ2Eδn(k)δn(k′).

In nuclear matter we have an explicit spin and isospin dependence:fω(k,k′)=F(k,k′)+F′(k,k′)τ·τ′+[G(k,k′)+G′(k,k′)τ·τ′]σ·σ.

In addition to the central interaction one has in principle also to consider tensor and spin–orbit forces which, however, have been neglected in most of the actual calculations. If one denotes the momenta before the collision by k1=k and k2=k then, by translational invariance, the momenta after the collision are given by k3=k+q and k4=k′−q, where q is the momentum transfer. In the third article of Ref. [1] Landau showed that the main contribution (singularity) to the full scattering amplitude should come from particle–hole pairs with small q. He also renormalized the integral equation for this amplitude in such a way that it contained a microscopic analog of the function fω(k,k′) and an integration over these particle–hole pairs within a small region near the Fermi surface. In this case the interaction depends only on the angle θ between k and k. This suggests an expansion in Legendre polynomials, e.g.:F(k,k′)=lFlPl(cosθ).

The constants Fl are the famous Landau–Migdal parameters. One introduces dimensionless parameters by definingFl=C0fl,where C0 is the inverse density of states at the Fermi surface:C0=dndϵ−1ϵ=ϵF.Likewise, one may also expand the other components of interaction (1.3) and define the parameters fl′,gl,gl for the various terms.

Some of the Landau parameters can be related to bulk properties of the nucleus, such as the compression modulus K,K=kF2d2(E/A)dkF2=62kF22m(1+2f0),the symmetry energy β,β=132kF22m(1+2f0′)and the effective mass mm/m=1+23f1.

The various parameters have to be determined from experiments and can then be used to predict other experimental quantities. In particular, these parameters enter into the equation for δn(k,ω), the solution of which determines small amplitude excitation of Bose type in Fermi systems. In nuclei these excitations correspond to the giant multipole resonances that we are going to investigate.

If one restricts the expansion in Eq. (1.4) to the lowest order, l=0, then the interaction Eq. (1.3) is a constant in momentum space, which corresponds in r-space to a delta function in (rr′). The next order, l=1, is the derivative of a delta function, which introduces a momentum dependence into the particle–hole (ph) interaction. In the application of TFFS to nuclei nearly all calculations have been performed with only the l=0 component of the interaction, which corresponds in r-space to the following form of the Landau–Migdal interaction:F(r,r′)=Fph(r,r′)=C0[f(r)+f′(r)τ1·τ2+(g+g′τ1·τ2)σ1·σ2]δ(rr′).

In finite nuclei, one has to introduce density-dependent parameters because it is obvious that the interaction inside a nucleus is different from the interaction in the outer region of the nucleus. The most often used ansatz isf(r)=fex+(fin−fex0(r)/ρ0(0)and similarly for the other parameters. Here ρ0(r) is the density distribution in the ground state of the nucleus under consideration and fin and fex are the parameters inside and outside of the nucleus. In actual calculations it turned out that g and g′ depend only weakly on the density, so that one uses the same parameters inside and outside. This density dependence of the Landau–Migdal interaction is the basic reason for its success and universal applicability.

Most of the results discussed so far were obtained within the phenomenological version of the theory of Fermi liquids. In the third article of Ref. [1] Landau gave a microscopic justification of his original phenomenological theory using the GF technique. There he derived in a fully microscopic way the basic equations of his theory. He showed, in particular, that the scattering amplitude of two quasiparticles is connected with the response function R (and two-particle GF) in the ph channel and that the quantity fω(k,k′) is connected with the forward scattering amplitude of two quasiparticles at the Fermi surface.

The starting point of the TFFS are the equations for the energies and transition amplitudes of excited states in even–even nuclei and the equations for moments and transitions in odd mass nuclei. As in Landau's theory, these equations can be also derived using the GF technique.

The one-particle GF is defined byG12(t1,t2)=−i〈A0|Ta1(t1)a+2(t2)|A0〉and the two-particle GF byK1234(t1,t2;t3,t4)=〈A0|Ta1(t1)a2(t2)a+3(t3)a+4(t4)|A0〉.Here a and a+ denote time-dependent annihilation and creation operators, |A0〉 is the exact eigenfunction of the ground state of an A-particle system and T is the time-ordering operator. In the following we use the single-particle states ϕλ(r,s) of a nuclear shell model as basis states; therefore the subscripts 1...4 stand for a set of single-particle quantum numbers.

We also need the response function R, which is defined asR(12,34)=K(23,14)−G(21)G(34),where the indices denote space and time coordinates.

The calculation of the excitation spectra of even–even nuclei and their transition probabilities can often be reduced to the calculation of the strength function, which describes the distribution of the transition strength in a nucleus induced by an external field V0(ω)S(ω)=n≠0|〈An|V0|A0〉|2δ(ω−ωn),where ωn=EnE0 is the excitation energy, while |An〉 and |A0〉 refer to the excited and ground states of a nucleus with mass number A, respectively. As the response function R(ω) in the energy representation has the spectral expansionR12,34(ω)=−∞dϵiR12,34(ω,ϵ)=nφn0∗21φn043ω+ωniδφn012φn0∗34ω−ωn+iδ,whereφn012=〈An|a1+a2|A0〉is the transition amplitude between the ground state and the excited state n, the strength function S(ω) is completely determined by the response function R(ω):S(ω)=1πlimΔ→+0Im1234V0∗21R12,34(ω+iΔ)V043.

This response function is defined by a Bethe–Salpeter equation in the ph channel [2]:R12,34(ω,ϵ)=−G31(ϵ+ω)G24(ϵ)+5678G51(ϵ+ω)G26(ϵ)−∞dϵ′iU56,78(ω,ϵ,ϵ′)R78,34(ω,ϵ′),which is shown graphically in Fig. 1.1.

In Eq. (1.19) G is the exact one-particle GF, and U is the irreducible amplitude in the ph channel, which is the unrenormalized ph interaction. (For details see Ref. [18].) The one-particle GF G and the response R are related by a system of nonlinear equations [2], [19]. In particular, this system of equations includes the relationU(12,34)=iδΣ(3,4)δG(1,2),where Σ is the so-called mass operator related to the one-particle GF by the Dyson equationG12(ϵ)=G012(ϵ)+34G013(ϵ)Σ34(ϵ)G42(ϵ)and G012(ϵ)=((ϵ−p2/2m)−1)12 is the one-particle GF of a free particle.

Here we will briefly describe the derivation of the basic equations of the standard TFFS in such a way as to clarify the explicit inclusion of complex configurations to be discussed in Section 2. For a more detailed derivation see [2], [15].

As mentioned before, the GF G and the response function R are determined self-consistently by a system of non-linear equations. This is in principle an exact formulation of the (non-relativistic) A-particle problem, but is of little use for practical applications.

In order to arrive at solvable equations, one applies Landau's quasiparticle concept and his renormalization procedure, which he developed for the microscopic theory of Fermi liquids [1]. For the nuclear many-body problem, it was done by Migdal [2] in his TFFS.

As a first step, one splits the one-particle GF into a quasiparticle pole part which is diagonal in the shell model basis and a remainder,G12(ϵ)=δ12a11−n1ϵ−ϵ1+iδ+n1ϵ−ϵ1iδ+Gr12(ϵ).Here a1 denotes the single-particle strength of the shell model pole, ϵ1 is the single-particle energy, n is the quasiparticle occupation number (1 or 0) and Gr(ϵ) is that part of the exact one-particle GF that remains when the shell model pole has been removed. It is assumed in Migdal's TFFS that Gr(ϵ) is a smooth function of ϵ in the vicinity of ϵF. The dominance of the first term in Eq. (1.22) with respect to the ϵ-dependence of the one-particle GF is one of the basic assumptions on which the TFFS rests. The validity of this assumption is crucial for the reliability of the results of all TFFS calculations. As we will see in the following, there might be additional pole terms in the expansion of Eq. (1.22) that originate from more complex configurations and which have to be considered in addition to the simple shell model poles. This extension of the theory will be the main topic of this review.

The main goal in the following is to obtain an equation for the response function that can be solved in practice because R(ω) contains all the information we need. In Eq. (1.19) the product of two one-particle Green functions enters. Here it is important to realize that with the quasiparticle ansatz in Eq. (1.22) this product can be separated at low transferred energies into a part A, which depends strongly on the energy ω and has a δ-function maximum with respect to ϵ for ϵ1 approaching ϵ2 (that is, near ϵF) and a weakly energy-dependent part B,−G(ϵ+ω)G(ϵ)=A(ϵ,ω)+B(ϵ,ω).A is given byA12,34(ϵ,ω)=2πia1a2δ13δ24n2−n1ϵ1−ϵ2−ωδϵ−ϵ122,whereas B, which contains all the rest, does not give rise to a pronounced ω-dependence.

We now insert Eq. (1.24) into the equation for the response function and integrate over ϵ to obtain in compact notationR(ω)=(A+B)−(A+B)UR(ω),where A is the shell-model ph-propagator:A12,34(ω)=−∞dϵiA12,34(ϵ,ω)=a1a2δ13δ24n2−n1ϵ1−ϵ2−ω.

With the help of Landau's renormalization procedure one can rewrite Eq. (1.25) in such a way that only the known A appears explicitly in the equation, whereas the unknown B changes U into the renormalized ph interaction F and gives rise to effective charges for the external fields. We introduce a renormalized response function R̃(ω), which is connected with the original response function R(ω) by the relationR=eqR̃eq+Beq,whereeq=1−FBand the renormalized ph interaction F satisfies the integral equationF=U−UBF.

A detailed investigation of these equations shows that F depends smoothly on the energy variables. For that reason the energy dependence is neglected when F is parameterized. The equation for R̃(ω) is given byR̃=A−AFR̃and the explicit form isR̃12,34(ω)=n2−n1ϵ1−ϵ2−ωδ13δ24n2−n1ϵ1−ϵ2−ω56F12,56R̃56,34.This is the basic equation of the TFFS.

Due to the renormalization procedure only the 1p1h configurations appear explicitly in the final equation, whereas all the more complex configurations give rise to renormalized quantities: the ph-interaction F and the effective charges eq. These quantities are not calculated within the theoretical framework but are parameterized in the way that was discussed before for the ph-interaction. Due to conservation laws the electric proton and neutron operators have eqp=1, eqn=0, respectively, and only spin-dependent operators have to be renormalized. In all practical cases the second term on the right side of Eq. (1.27) does not contribute, so it is sufficient to solve Eq. (1.31). Here all quantities are known. The single-particle energies are given by the nuclear shell model or taken, as far as possible, from experiment and the parameters of the interaction and the effective operators have been deduced from experimental data.

Eq. (1.31) has exactly the same form as the conventional RPA equation that, however, has been derived using approximations from the outset. In the present derivation, , are still exact. For that reason one is able to obtain relations (the Ward identities) between the effective operators and the effective interaction. In cases where conservation laws exist these relations determine the effective operators completely. In addition, the present derivation shows more clearly the range of validity of the theory, which naturally also applies to the conventional RPA equation. Moreover, the GF formalism provides a natural basis for an extension of the theory, as we will see in the next section.

For practical reasons one solves not Eq. (1.31), but the related equation for the change of the density matrix ρ12(ω) in the external field V0(ω), which is defined asρ12(ω)=−34R̃12,34(ω)eqV043(ω).The equation for ρ12 follows from Eq. (1.31) and has the formρ12(ω)=−34A12,34eqV0433456A12,34F34,56ρ56(ω).The expression for the strength function is then given byS(ω,Δ)=−1πIm12eqV0∗21ρ12(ω+iΔ),where Δ is a (finite) smearing parameter which simulates a finite experimental resolution and at the same time phenomenologically can include configurations not dealt with explicitly in the approach under consideration.

, are the main equations that are used in the calculations within the TFFS.

Most of the calculation that we will present here have been performed in r-space and not in the configuration space of a shell model basis. For the cases of RPA and TFFS the method was suggested in [20], [21] and was applied by us to the ETFFS [78]. The main reason for this choice is that the r-space representation is much more appropriate for the treatment of the single-particle continuum, as first pointed out in [20], [21]. Therefore we give here some relevant equations in the coordinate representation.

Eq. (1.33) has the formρ(r,ω)=−∫A(r,r′,ω)eqV0(r′,ω)d3r′−∫A(r,r1,ω)F(r1,r2)ρ(r2,ω)d3r1d3r2.The ph propagator A, given byA(r,r′,ω)=12n2−n1ϵ1−ϵ2−ωϕ1(r2(r′)ϕ2(r1(r′),can be rewritten asA(r,r′,ω)=−1n1ϕ1(r1(r′)[G(r′,r1+ω)+G(r′,r1−ω)]using the formula for the one-particle GFG(r,r′;ϵ)=2ϕ2(r2(r′)ϵ−ϵ2,where ϕ2(r) are the single-particle wave functions calculated in a mean-field potential.

The summation in Eq. (1.37) is over states below the Fermi surface, i.e. the single-particle continuum is already contained in Eq. (1.38). On the other hand, the coordinate part of this GF Glj=glj/rr′ can be expressed in closed form in terms of the regular y(1)lj and irregular y(2)lj solutions of the one-dimensional Schrödinger equation asglj(r,r′;ϵ)=2m2y(1)lj(r<;ϵ)y(2)lj(r>;ϵ)/Wlj(ϵ),where r< and r> denote the lesser and the greater of r and r′, respectively and W is the Wronskian of the two solutions. The irregular solution y(2)lj is determined by the boundary conditions at ∞; e.g., for neutronsy(2)lj(r→∞)∼exp(−kr)for negative energies ϵ<0 andy(2)lj(r→∞)∼expikr−πl2ljfor positive energies ϵ>0, where k=2m|ϵ|/ℏ and δlj is the scattering phase for the mean nuclear potential considered.

Thus, the functions ylj(1,2) are calculated numerically if the mean potential is known. For ϵ<0 the functions glj have no imaginary part; that is, the 1p1h states have automatically no width if the smearing parameter Δ=0.

Inclusion of the single-particle continuum makes it possible to obtain a physical envelope of the resonance without using a smearing parameter, that is, to obtain directly the escape width Γ.

In the complex configuration problem under consideration, using the representation of the single-particle wave functions (λ-representation) gives matrices of a very large rank especially, in the case of treatment of the ground state correlations caused by complex configurations. Using the coordinate representation affords a big numerical advantage in this problem because the rank of matrices is determined not by the number of configurations but by the number of mesh points used in solving the corresponding integral equation.

In order to distinguish GMR from other collective excitations, one can define them as follows:

  • (1)

    The form and the width Γ of the resonance depend rather weakly on A; as a rule, the dependence ΓA−2/3 is used.

  • (2)

    The resonance mean energy E also depends weakly on A; usually one uses EA−1/3.

  • (3)

    The resonance width is small compared with its excitation energy.

  • (4)

    The resonance exhausts a large fraction of its energy weighted sum rule (EWSR)—usually more than 50 percent.

The last of these is the most quantitative characteristic of the resonance and justifies its name “giant”.

In 1971–72 in inelastic electron [22], [23] and proton [24] scattering, giant multipole resonances (GMR) were detected that were different from the well-known isovector electric dipole resonance. That was the starting point of a period, sometimes called a renaissance of giant resonance physics, of very rapid and intensive development. A large amount of experimental data on the GMR in stable nuclei—principally their energies, total strengths, widths and resonance envelopes—has been accumulated and discussed since the mid-1970s. Currently there exists experimental information on more than 20 different types of GMR that were detected in a large number of nuclei in a broad range of excitation energies. A detailed review, with experimental results up to the end of the eighties, is presented in Ref. [25]. Further information can be found in the proceedings of the last four international conferences devoted to giant resonances [26], [27], [28], [29], and in a recent monograph [30] that gives an excellent review of the present experimental situation. Conventional theoretical methods such as the RPA and QRPA, and their comparison with the data, have been the subject of many review articles and books (e.g., [30], [65], [25], [88] and [67,90,12,89,66,15,16]). For nuclei with A>40, the experimental situation is essentially settled. There is no longer any major controversy over the centroid energies and the total strengths of GMR, and the theoretical interpretation of these data within the framework of the conventional theoretical methods is also clear. These methods, however, do not allow description of the widths and the fine structure of the resonances, nor do they offer any possibility to analyze complex spectra with overlapping resonances. This is the main subject of our review and will be presented in the next sections.

In the following we illustrate and briefly discuss typical results that have been obtained within the continuum TFFS (CTFFS), which is, as mentioned before, formally identical to the continuum RPA (CRPA). We show in Fig. 1.2, Fig. 1.3, Fig. 1.4 and Table 1.1 the hadron strength functions for the E2 isoscalar (IS) and isovector (IV) resonances in 208Pb and the E2 IS resonance in 40Ca and some of their integral characteristics. The calculations have been performed in the coordinate representation within the CTFFS, i.e., using , , . In order to simulate the finite experimental resolution, we introduced a smearing parameter Δ with a value of 250keV. As the TFFF is not self-consistent, as discuss in the beginning, we have to determine the parameters of the effective interaction, , , from experiment by fitting some specific theoretical results to experimental data. In our calculations we always used, with the exception of fex, the following Landau–Migdal interaction parameters, which were adjusted previously to various experimental quantities [101], [102] (as we use in the following the lowest order interaction only, we drop the index zero for the parameters):fin=−0.002,fex′=2.30,fin′=0.76,g=0.05,g′=0.96,C0=300MeVfm3.

For the nuclear density ρ0(r) in the interpolation formula (1.11) we chose the theoretical ground state density distribution of the corresponding nucleus,ρ0(r)=ϵi⩽ϵF1(2ji+1)R2i(r),where Ri(r) are the single-particle radial wave functions of the particular Woods–Saxon potential used. For other details of the calculations, including the definitions used in Table 1.1, see Section 3.1.3.

For the parameter fex we have used the values fex=−1.9 and −2.2 for 208Pb and 40Ca, respectively. These parameters were adjusted to reproduce the energies of the first excited 21+ level in 208Pb and 31 level in 40Ca. We will see in 3 Application to giant resonances, 4 Microscopic transition densities and the calculations of cross sections that the same parameters (1.42) can also be used if more complex configurations are considered explicitly. Even the parameter fex changes only slightly in the latter case. We have used these parameters in all our calculations for stable and unstable closed shell nuclei from 16O to 208Pb, where we investigated many different types of GMR. In our opinion, the reasonable agreement with experiment we obtained confirms the assumed universality [2] of the parameters of the Landau–Migdal interaction, Eq. (1.10).

In Fig. 1.2, Fig. 1.3, Fig. 1.4 the CTFFS results are given together with the results of the calculations without taking the effective interaction into account (“free response”). As the parameters fin,fex (fin,fex) for the isoscalar (isovector) resonance are negative (positive), the IS (IV) resonances are shifted to lower (higher) energies, when the interaction is included, compared with the free response. For the latter we have the shell model estimate for 208PbE≃2×41A−1/3=13.8MeV while, according to Table 1.1, Ēis=8.1MeV, Ēiv=18.2MeV or, if one uses another definition for Ē=E2,0 (see Section 3.1.3), Ēis=10.2MeV, Ēiv=19.3MeV. In both cases—with and without interaction—reasonable depletion values (90–100 percent for large energy intervals) of the corresponding EWSR have been obtained (see Table 1.1). The depletion is in satisfactory agreement with the corresponding experimental values of the EWSR. (See 3 Application to giant resonances, 4 Microscopic transition densities and the calculations of cross sections and, in particular, Table 3.3, Table 3.7.) In Table 1.1 we give also the experimental values of the mean energies. It should be noted, however, that the experimental data were obtained, as a rule, for intervals which are much smaller than those given in Table 1.1. Moreover, the experimental mean energies are determined in different ways. Therefore the comparison with experiment should be made more carefully, and this will be done in Section 3, where we will also discuss the width problem.

We have obtained a relatively good description of the mean energies and total strengths of the GMR under consideration. These are typical results of the CRPA.

In Fig. 1.2, Fig. 1.3, Fig. 1.4 we have chosen the smearing parameter noticeably larger than the experimental resolution in order to simulate at least some of the decay width not included in the present approach. Nevertheless, one cannot see in Fig. 1.2, Fig. 1.3, Fig. 1.4 any resemblance to an observed resonance because the smallest width among the three resonances under discussion is the one for the E2 IS resonance in 208Pb, which has an experimental width of 3.1±0.3MeV [35], [40]. In other words, the widths of resonances, which are among the most important characteristics, are not reproduced within the CRPA. Even in medium–mass nuclei, where the role of the continuum (escape width) is much larger, the theoretical widths are still in disagreement with the experimental ones. The reason is well known: the one particle–one hole configurations describe only the escape widths, which—in general—are only a small fraction of the total widths. For a realistic description one has to include the spreading widths, which are due to the coupling of the one particle–one hole configurations to more complicated configurations. One possible solution to this problem will be discussed next.

We have seen that the Landau–Migdal theory is based an a microscopic many-body theory with, however, important elements taken from experiment. For that reason it is quite natural that an extension of that successful theoretical frame work is also based on experimental facts. It is well known from nuclear spectroscopy that in odd-mass nuclei that differ from a magic nucleus by one nucleon (or hole), the coupling of the low-lying phonons of the even nucleus to the odd particle (or hole) plays an important role. It gives rise to a strong fragmentation of the corresponding single-particle (hole) strength over a range of the phonon energy. This observation gives us the possibility to include in the standard TFFS the coupling of the 1p1h states to more complex configurations. We will show that with this coupling to the low–lying phonons one includes those 2p2h configurations that give rise to the strongest fragmentation of the giant resonances, which—together with the coupling to the single-particle continuum—makes it possible to calculate quantitatively the strength distribution of the GMR.

It is clear now that GMR are a universal property of nuclei. The investigations of GMR are not only important for a detailed understanding of the structure of nuclei, but they are also an important tool for a better understanding of nuclear reaction mechanisms involved in the excitation of the different types of these resonances. Moreover, we obtain from the investigation of GMR additional information on the Landau parameters. The most important of these is f0in, which is connected with the compressibility of nuclear matter and is therefore of crucial importance in astro-physics. The isoscalar electric monopole resonance (breathing mode), on the other hand, is closely related to this parameter and therefore one would need to know this resonance in nuclei far from stability in order to obtain the dependence of the compression modulus on the number of protons and neutrons. The magnetic resonances are related to the parameters g and g′ of the spin- and isospin-dependent parts of the forces. The latter is related to pions in nuclei and is of special interest in connection with the possibility of pion condensation. Phenomenologically, GMR inform us about the nuclear shape (splitting of the E1, E2 and E0 resonances in deformed nuclei), and about volume, surface and other kinds of vibrations.

The understanding of the widths of GMR is obviously connected with the damping of small amplitude vibrations in finite systems, as we shall soon discuss. Thus the general problem of how energy from highly ordered excitations is dissipated in nuclei, including the question of transition from order to chaos, can be clarified through GMR studies [31].

Many ideas from GMR physics have been used in other applications, such as the recent investigations of metallic clusters [49] and the fullerene molecules [50]. See also Ref. [30, Chapter 11].

Necessity of inclusion of complex configurations and single-particle continuum. As discussed in Section 1.2.5, the standard continuum TFFS or the continuum RPA in closed shell nuclei are able to describe only two integral characteristics of GMR: their mean energies and total strengths. The quantity that is of equal importance, the strength distribution of the GMR (i.e., their widths) cannot be reproduced within this approximation. The reason for this failure has been already been indicated above: the coupling of the 1p1h configurations to more complex configurations, which gives rise to the spreading width, has so far been neglected. The escape width, which is included in the present continuum approaches, represents in the giant resonance region only the lesser part of the total width. There are several reasons why one would like do describe theoretically the widths of the GMR quantitatively:

  • 1.

    the intellectual challenge to develop a microscopic theory that gives a quantitative explanation for the collective motion in strongly interacting finite Fermi systems;

  • 2.

    the new insight into the fine structure of the GMR due to the rapid improvement of the experimental resolution to ΔE<10keV, which needs to be understood (see, for example, Ref. [51]);

  • 3.

    the need for microscopically derived strength distributions that quantitatively reproduce the date on resonances in the medium mass region, where the various multipole resonances overlap in energy. The conventional analyses with phenomenological transition densities are no longer applicable because they introduce strong uncertainties, as we shall discuss in the following sections.

The inclusion of the single–particle continuum gives the physical envelope for processes at excitation energies higher than the nucleon separation energy. For giant resonances this gives the escape width Γ↑, the magnitude of which depends significantly on the mass number, the excitation energy, the multipolarity, etc., so that it is not justified, especially in the calculations of such delicate properties as fine structure and decay characteristics, to simulate the role of the continuum by a constant smearing parameter. The exact microscopic treatment of the continuum is therefore crucial for a realistic theory of giant resonances.

In addition to this, a realistic microscopic theory of collective motion in nuclei has also to consider more complex configurations than those included in the RPA. There are new data, e.g. [144], [148], [145], and the largely unsatisfactory explanation of the older results concerning the low-lying structures in cross sections in a wide excitation energy range around the nucleon binding energy [53], [54], [55], [56], [57], [111], see also Section 3.2.5. An extended theory will also have implications for the interpretation of experimental data obtained with modern germanium detectors and gamma spectrometers such as EUROBALL cluster, EUROBALL and others [58], [59]. Unprecedentedly high resolution and high efficiency of detecting gamma rays with energies up to 20MeV have already given new and very precise information, not only on deformed nuclei, but also on low-lying levels in odd and even–even spherical nuclei. In fact, these detectors give direct information about the low-lying complex configurations containing phonons [58], [59], [60], [61], [62], which may be seen again in the fine structure of the giant resonances. At last, in order to explain the large amount of (5 lines) available data on the decay properties of GMR's gained in experiments with coincidences of secondary particles [52], [30] it is also necessary to take complex configurations into account (see, for example, [74], [76]).

It is clear that in the immediate future the number of such data will increase rapidly and that these results require improved microscopic approaches for their interpretation.

Ground state correlations caused by complex configurations. The ground state correlations (GSC) problem has a long history. (See, for example, references in [63], [68] and also the article [69].) It started with the Hartree–Fock approximation (HF), where the effects of the Pauli principle was included in the calculation of the ground state of fermion systems. Some specific ground state correlations are taken into account if one calculates excited states within the RPA. The most important consequence of these ground state correlations is that the energy-weighted sum rule for the transition probabilities is conserved. This is not so in the Tamm–Dankoff approximation, which starts from the uncorrelated HF ground state. During the past ten years, in connection with the development of the extended TFFS, where configurations beyond the 1p1h states are included in the excited states, the question arose as to how far one has also to consider the same configurations in the ground state. [68], [84], [85]. There is a fundamental difference between the effects of GSC in the RPA and their effects in models where more complex configurations are included explicitly. The RPA GSC do not lead to the appearance of new transitions compared with the TDA, but only shift the energies and redistribute the transition strengths.

The GSC induced by the more complex configurations, on the other hand, lead not only to a redistribution of strength but also to new transitions, which give rise to a change of the EWSR [96], [85]. Thus these GSC are at least as important and physically interesting as the RPA GSC. Actually, their consequences are much richer and far-reaching. The present approach is an extension of the previously developed Extended Second RPA (see, for example [96], [89]), in which uncorrelated 2p2h GSC have been considered.

We will see in the following that these effects play a noticeable—sometimes decisive—role in the theoretical description of the experimental data. The most striking example obtained within the GF approach is the explanation [93], [95] of the observed M1 excitations in 40Ca and 16O with energies of about 10 and 16MeV, respectively, solely as a result of ground state correlations.

There is increasing interest in the structure of nuclei far from the stability line. The study of these exotic nuclei, is of importance not only in itself [42], [43], [51], [114], [115], but also for its relevance to astrophysics [42], [43], [44], [45], [51]. The ETFFS we are discussing here may play an important role in the analysis of the experiments done at the proposed radioactive beam facilities. As mentioned earlier, one of the crucial quantities one wants to know is the breathing mode in nuclei with very different numbers of protons and neutrons. This will give us the compression modulus as a function of the proton and neutron number, which is needed for the extrapolation to nuclear matter. We may suppose that in nuclei far from stability, even with closed shells, the high-lying spectra may be as complicated as in the medium mass nuclei, where the various multipole resonances overlap and a microscopic theory is necessary for the analysis in order to obtain reliable nuclear structure information.

If we extrapolate our present knowledge of unstable nuclei to nuclei far away from the stability line we may expect two characteristic features: (i) there will be very low-lying collective states and (ii) the nucleon separation energy may also be relatively low. For these reasons a realistic theory has to treat the continuum in an exact way, and the phonon coupling is not only important for the analysis of GMR but also for a quantitative understanding of the low-lying spectrum. In connection with the application of the present approach, one has to investigate the extent to which the Landau–Migdal parameters are dependent on the numbers of protons and neutrons. Our extensive experience indicates that this dependence may be quite weak, so we can at least use the present parameters as a good starting point. The second important input into the theory concerns the single-particle spectrum and the single-particle wave functions. Here one may use self-consistent approaches, for example, those that are based on density functionals, in order to obtain a reliable quasiparticle basis. Investigations in these directions are in progress.

It should be emphasized that only a reliable inclusion of the single-particle continuum can make it possible to do calculations for nuclei with the separation energy near zero. This is important for understanding drip-line nuclei and for astrophysical studies. For neutron-reach nuclei with the separation energy near zero, this is of prime interest because of the absence of the Coulomb barrier. The CRPA calculations in 28O have shown that the strength distributions of the E2 [46] and isovector E1 [47] resonances are very different from those for 16O; the resonances are more spread out, shifted down and have a noticeable low-lying strengths. The effect of complex configurations is also noticeable, at least for the isoscalar E2 resonance in 28O [48]. It should be pointed out, however, that except for Ref. [48] and the calculations we shall present in Section 3.3, there exists almost no theoretical information about the role of complex configurations in unstable nuclei [30].

We can summarize our discussion so far by asserting that a microscopic theory that is able to describe quantitatively the structure of collective excitations in nuclei and which is based on the mean-field approximation has to include four major effects:

  • 1.

    the 1p1h RPA, which creates collectivity out of the uncorrelated ph states, as a starting point;

  • 2.

    complex configurations beyond the 1p1h states, which give rise to a fragmentation of the collective states derived from the RPA;

  • 3.

    the single-particle continuum;

  • 4.

    ground state correlations induced by the complex configurations under consideration.

In addition, one should not use separable forces, because one then must use different forces for each mulipolarity, which strongly reduces the predictive power of the theory. Indeed one needs an interaction that is universal for the whole periodic table, or at least that changes only very little with the mass number, and which should be adjusted to quantities other than those that one is going to calculate. As we shall see, the GF approach that we are going to discuss in that what follows allows the inclusion of all these effects simultaneously.

It is obvious that, compared with the simple 1p1h configuration problem, the present task is much more difficult—both theoretically and numerically. In addition we shall develop and apply various stages of sophistication of our theory to the nuclear structure problem in order to clarify the different effects. At present there exist several other approaches that have considered some of the effects mentioned above. These are reviewed in [41], [89] (“pure” 2p2h configurations) and [65], [66], [90] (configurations with phonons).

In the past, microscopic theories of GMR have been developed using two different approaches: RPA + continuum on the one hand side [20], [21] and RPA + complex configurations on the other [64], [65], [113], [91], [92], [93], [94], [95], [98]. As we have seen, however, both extensions of the RPA are need to explain the data. The first of these can now be considered solved, and there exist several numerical methods for it. One, which was mentioned in Section 1.2.4, uses the GF method. There one considers the one-particle continuum exactly (for a contact interaction) by transforming the RPA equation into the coordinate representation. Other methods for solving this problem have also been developed that even admit the use of nonlocal forces [71], [70]. As for the problem of including complex configurations, the most advanced approach is the quasiparticle–phonon model for magic and non-magic nuclei by Soloviev and his co-workers [64], [65]. These authors, however, used separable forces in order to reduce the numerical difficulties of the problem, and they leave out the single-particle continuum. In addition, the ground state correlations are included only partially, that is, mainly on the RPA or QRPA level.

The microscopic theory for GMR that satisfies the requirements mentioned above turned out to be quite difficult to formulate and especially to realize numerically if one uses non-separable forces, as shall will do.

There have been some successful developments in this direction in the past ten years. The first attempts of this kind, which simultaneously consider RPA configurations, the single-particle continuum (escape width Γ↑) and more complex (2p2h [72], [73] or 1p1h⊗phonon [75], [74]) configurations (spreading width Γ↓) using non-separable forces were made in [72], [73], [74], [75] for some closed shell nuclei. These authors investigated various types of GMR using, of course, different approximations and methods.

The model developed in Ref. [74] used only 1p1h⊗phonon configurations and it considered only a particle–phonon interaction. It is based on the Bohr–Mottelson model for the strength function of the phonons. This model was also successfully used [76] to calculate partial branching coefficients of the proton decay of the isobar-analog and Gamow–Teller resonances in 208Bi. The papers [74], [76] and [72] were the first articles in which all three (that is, the above-mentioned items 1, 2 and 3) microscopic mechanisms of GMR formation were used to explain such delicate properties as the decay characteristics of the GMR. It was also shown in [74], [76] that the complex 1p1h⊗phonon configurations noticeably improve the description of the decay characteristics. The advantage of this method [74], [76] is a self-consistency (on the RPA level), that is, the phonons that are used in the extended theory have been obtained in RPA using the calculated interaction. In this development, however, ground state correlations due to complex configurations have been ignored.

The most extensive investigations of GMR, which include the above-mentioned effects, were performed within our ETFFS approach, where calculations for stable and unstable closed shell nuclei have been made. The theory is based on the consistent use of the GF method [78], [79], [80], [81], [82], [83], [84], [85], [86], [87]. The ETFFS simultaneously takes into account 1p1h, complex 1p1h⊗phonon configurations, the single-particle continuum and ground state correlations both of the RPA type and of those caused by the complex configurations under consideration. In addition, in its final equations it includes explicitly both the effective particle–hole interaction and the quasiparticle–phonon interaction.

A theoretical approach that takes into account the 2p2h configuration space including the full 2p2h interaction is numerically hardly solvable if one also uses a realistically large configuration space. For that reason the main approximation in ETFFS concerns the selection of the 2p2h configurations. In our approach, guided by experimental observations, we include the most important correlations in the 2p2h space by coupling phonons (correlated 1p1h states) to a one-particle and one-hole state. With this procedure we obtain effectively 1p⊗phonon and 1h⊗phonon configurations. If we then couple an additional hole and particle, respectively, to the previous configurations we obtain 1p1h⊗phonon configurations where part of the 2p2h interaction is included. As one can see from the applications, these configurations are indeed the most important ones for the understanding of the spreading width of GMR. The configurations with phonons also nicely explain a part of the low-lying spectrum in the neighboring odd mass nuclei. Configurations with phonons are used in many theoretical approaches [88], [65], [66].

There is, however, an additional fact that greatly simplifies the problem, and that is the existence of a small parameter for closed shell nuclei [88]:α=〈1||g||2〉2(2j1+1)ωs2<1,where 〈1||g||2〉 is a reduced matrix element of the amplitude for low-lying phonon creation with the energy ωs and 1 represents the set of single-particle quantum numbers n1, l1 and j1 for spherical nuclei. Henceforth, when we refer to the g2 approximation, it will be understood that the dimensionless α is small. Using this small parameter affords following advantages:

  • 1.

    We obtain a general principle for selecting terms: as α is small, we may restrict ourselves to 1p1h⊗phonon configurations, which correspond to second order in g (two-phonon configurations correspond to terms of order g4). Because we use the g2 approximation in the propagators of our integral equations, our approach is not the usual perturbative theory in g2.

  • 2.

    For the widths of GMR the most important contribution comes from low-lying phonons, which give rise to a strong energy dependence in the energy range of the high lying collective (1p1h) RPA solutions. Therefore we may confine ourselves to the most collective low-lying phonons, which are restricted in number. The effects of the other phonons are effectively already included through the phenomenological parameters of our approach.

  • 3.

    The restriction to a small number of collective phonons noticeably reduces the numerical difficulties. This is especially important for the present approach, in which non-separable forces and the GSC induced by complex configurations are considered.

  • 4.

    As some of the 1p1h⊗phonon configurations are treated Landau–Migdal parameters that are determined within the 1p1h approximation may change. As we restrict ourselves to the g2 approximation and the collective low-lying phonons, this effect in the actual calculations is found to be small.

The ETFFS approach is, like the original TFFS, a semi-microscopic theory. As our approach is based on the TFFS, we actually do not need additional experimental input beyond that used already in the TFFS. We must, however, “correct” some of the parameters in order to avoid double counting. Such corrections can be performed fully consistently within our approach. The most important corrections refer to the single-particle energies which are taken—as far as possible—from experiment, or else from a shell model potential that is carefully adjusted to the corresponding closed shell nucleus. The single-particle wave functions are also taken from that model. These quantities contain contributions from the same phonons that enter the complex configurations under consideration. In order to avoid double counting due to these phonons, the single-particle model has to be “refined” from this mixing. The procedure for this will be described in Section 3.1.2. The complex configurations that we treat explicitly in our extended approach are also included implicitly in the force parameters of the original TFFS approach. Therefore, in principle, one has also to correct the Landau–Migdal parameters.

There exists so far no self-consistent theoretical approach that includes all the effects discussed above. In such an approach one would start with an effective two-body interaction, a density functional or an effective Lagrangian, that would allow to determine the single-particle energies and wave functions and the ph interaction. As within such a procedure the phonon effects are not included, our extended theory would be the natural formalism in which to do it. In our extended version of the TFFS we did not include the so-called tadpole graphs with the low-lying phonons under consideration that have been used in the self-consistent version of the TFFS [6]. Their contribution is contained effectively in our “refined” mean field.

Section snippets

General description of collective excitations, including the particle–hole and quasiparticle–phonon interaction

As mentioned before, the original TFFS allows to calculate only the centroid energies and total transition strength of giant resonances because the approach is restricted to 1p1h configurations. In order to describe more detailed nuclear structure properties one has to include higher configurations. Here we describe the derivation of the main ETFFS equations. These equations contain both the quasiparticle–phonon interaction and the effective ph interaction in a general form.

Scheme of the calculations

In the previous section the basic equation (2.57) of the ETFFS for the case of closed shell nuclei has been formulated in coordinate space for the change of the density matrix in an external field V0(r,ω) with the energy ω. The strength function (Eq. (2.59)) gives the energy distribution of the excitation strength under consideration:S(ω,Δ)=dB(EL)dω=−1πImΠ(ω+iΔ),whereΠ(ω+iΔ)=∫d3r[ẽqV0(r)]ρ(r,ω+iΔ)is the polarizability propagator and Δ is a smearing parameter. By using this we take into

Comparison of microscopic and phenomenological transition densities

The transition densities ρtrL, which are necessary to describe the nuclear structure in calculations of cross sections, are simply connected with our density matrix ρL(r,E+iη) determined in Eq. (2.57):ρtrL(r,ΔE)=1πΣB(EL)ImEminEmaxdEρL(r,E+iη),where ΣB(EL) is the B(EL) value summed over the interval ΔE.

The isovector E1 transition densities calculated in our approach for the large energy interval were obtained in Ref. [85]. There is no significant difference between the continuum RPA and our

Conclusion

In this review we presented a new microscopic many-body theory for the structure of closed shell nuclei. The extended theory of finite Fermi systems (ETFFS) is based on the Landau–Migdal theory of finite Fermi systems (TFFS) and includes in a consistent way configurations beyond the 1p1h level. A large part of this review is concerned with the application of this new approach to giant resonances in closed shell nuclei. As in the standard TFFS [2], we formulate the theory within the framework of

For further reading

G.J. O'Keefe, M.N. Thompson, Y.I. Assafiri, R.E. Pywell, K. Shoda, Nucl. Phys. A 469 (1987) 239.

Acknowledgements

We thank our colleagues S.T. Belyaev, P.F. Bortignon, M. Harakeh, V.A. Khodel, S. Krewald, P. von Neumann-Cosel, Nguyen Van Giai, A. Richter, P. Ring, E.E. Saperstein, A.I. Vdovin, J. Wambach, D.H. Youngblood, V. Zelevinsky for fruitful discussions. We also thank V.I. Tselyaev for collaboration and stimulative discussions. The authors are greatly indebted to J. Durso for his careful reading of the manuscript. J.S. thanks Tony Thomas for many useful discussions and the hospitality he enjoyed in

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