Excitation of the electric pygmy dipole resonance by inelastic electron scattering

TL;DR

This study investigates the excitation mechanisms of the electric pygmy dipole resonance (PDR) in $^{140}$Ce via inelastic electron scattering, revealing its predominantly transversal nature and potential to provide new insights into nuclear structure.

Contribution

It introduces detailed calculations of PDR excitation in electron scattering using QRPA and QPM models, highlighting the reaction's sensitivity to transversal nuclear currents and its complementarity to other methods.

Findings

PDR excitation is mainly transversal at backward angles.

Electron scattering cross sections for PDR are much smaller than for the first 2+ state.

At extreme backward angles, PDR can dominate the scattering cross section.

Abstract

To complete earlier studies of the properties of the electric pygmy dipole resonance (PDR) obtained in various nuclear reactions, the excitation of the 1$^-$ states in $^{140}$Ce by $(e,e')$ scattering for momentum transfers $q=0.1-1.2$~fm$^{-1}$ is calculated within the plane-wave and distorted-wave Born approximations. The excited states of the nucleus are described within the Quasiparticle Random Phase Approximation (QRPA), but also within the Quasiparticle-Phonon Model (QPM) by accounting for the coupling to complex configurations. It is demonstrated that the excitation mechanism of the PDR states in $(e,e')$ reactions is predominantly of transversal nature for scattering angles $\theta_e \approx 90^o-180^o$. Being thus mediated by the convection and spin nuclear currents, the $(e,e')$ like the $(\gamma,\gamma')$ reaction, may provide additional information to the one obtained from Coulomb- and hadronic excitations of the PDR in $(p,p')$, $(\alpha,\alpha')$, and heavy-ion scattering reactions. The calculations predict that the $(e,e')$ cross sections for the strongest individual PDR states are in general about three orders of magnitude smaller as compared to the one of the lowest $2^+_1$ state for the studied kinematics, but that they may become dominant at extreme backward angles.

Equations

$${\left(\frac{d\sigma }{d\Omega }\right)}_{\lambda }\propto \left\{V_L({\theta }_e){\left|F_{\lambda }^C(q)\right|}^2+V_T({\theta }_e){\left|F_{\lambda }^E(q)\right|}^2\right\}$$

$$F_{\lambda }^C(q)\propto {\int }_0^{\infty }{\rho }_{\lambda }(r)j_{\lambda }(qr)r^{2}dr$$

$$F_{\lambda }^E(q)$$

$$F_{\lambda }^E(q)\approx \frac{E_x}{q}\sqrt{\frac{\lambda +1}{\lambda }}{F}_{\lambda }^C(q)$$

$$\mathrm{R}({\theta }_e)=\frac{V_L({\theta }_e)}{\frac{E_x}{q}\sqrt{\frac{\lambda +1}{\lambda }}{V}_T({\theta }_e)}$$

$${\left(\frac{d\sigma }{d\Omega }\right)}_{\lambda }\propto \sum _{m_s,\mu }{\left|A(\lambda \mu m_s)\right|}^2$$

$$|{\Psi }_{\lambda \mu }^{\nu }⟩$$

$$[Q_{{\lambda }_1{\mu }_1{i}_1}^{+}{Q}_{{\lambda }_2{\mu }_2{i}_2}^{+}{]}_{\lambda \mu }=\sum _{{\mu }_1\mu 2}⟨{\lambda }_1{\mu }_1{\lambda }_2{\mu }_2|\lambda \mu ⟩Q_{{\lambda }_1{\mu }_1{i}_1}^{+}{Q}_{{\lambda }_2{\mu }_2{i}_2}^{+}.$$

$${\left(\frac{d\sigma }{d\Omega }\right)}_{\lambda \nu }\propto \sum _{m_s,\mu }{\left|\sum _i{R}_i(\lambda \nu )A_i(\lambda \mu m_s)\right|}^2$$

$$S(B(E1),E_x)\propto \sum _{\nu }\frac{B_{\nu }(E1)}{(E_x-E_{\nu }{)}^2+(\Gamma /2{)}^2}$$

Full text

Dedicated to Pier Francesco Bortignon

1 Introduction

A group of low-lying $1^{-}$ states in neutron-rich heavy nuclei below the particle emission threshold is often referred to as the Pygmy Dipole Resonance (PDR). The excitation probability of the PDR by photons is about two orders of magnitude smaller as compared to the Giant Dipole Resonance (GDR). Nonetheless the high selectivity of the electromagnetic interaction to the excitation of dipole states already allowed to observe the PDR in experiments with tagged photons as a bump of unresolved states with a width of about 2-3 MeV Lasz1 ; Lasz2 . Later, nuclear resonance fluorescence (NRF) experiments with high resolution in Darmstadt He97 and Gent Go98 and follow-up studies during the last 20 years also at the ELBE accelerator of the Helmholtz-Zentrum Dresden-Rossendorf ELBE and at the High Intensity $\gamma$-ray Source (HI$\gamma$S) operated by Triangle University Nuclear Laboratory (TUNL) TUNL , identified the PDR fine structure, i.e. hundreds of 1- states were observed in spherical nuclei at the PDR excitation energy.

Recently, other probes were used to investigate the PDR properties: $1^{-}$ states which form the PDR, were studied in $(\alpha,\alpha^{\prime}\gamma)$ aa , $(p,p^{\prime})$ pp , (17O,17O${}^{\prime}\gamma$) 17O , and $(p,p^{\prime}\gamma)$ Ce-last reactions, in which the detection of the $\gamma$ decay photon in coincidence with the scattered particle was used to select the corresponding excitation of a $1^{-}$ state. For example, the spectrum of the $1^{-}$ states in 208Pb obtained in the $(p,p^{\prime})$ reaction at very small scattering angles ($\theta_{\rm lab}<1^{o}$, where the excitation process is purely determined by the Coulomb interaction between projectile and target pp ) resembles closely the NRF spectrum Rye . At the same time, Coulomb- and strong ($NN$) interactions between projectile and nucleus play an important role in the excitation of the PDR states in the other reactions mentioned above. As a result of the different sensitivity of various reactions, some $1^{-}$ states are observed in NRF spectra but not in reactions with hadronic probes and vice versa. Also, the relative excitation strengths of different individual states deviate appreciably. Contrary to the GDR states where the $E1$-strength is concentrated merely in a single collective level called 1p-1h doorway state and a spreading over many states of 2p-2h, 3p-3h, $\ldots$ character (see, e.g., BBB ), the PDR is characterized by probably a few doorway states. We will return to this point in some detail below.

For a detailed account of the present status of studies of the PDR properties we refer to a recent review article rew .

In the present work we consider the possibility of using electrons as a projectile to supply further information on the properties of the $1^{-}$ states belonging to the PDR. As it will become clear from results presented below, the prerequisite for an experimental verification of them are the availability of (i) low energy electron beams and (ii) high-resolution and large acceptance magnetic spectrometers. Both conditions are, e.g., fulfilled at the S-DALINAC (Superconducting Darmstadt Electron Linear Accelerator) and its spectrometers LINTOTT and Q-CLAM i . Some selective excitations of isoscalar and isovector electric dipole transitions below the electric giant resonance region were, e.g., investigated in 12C, 16O, 40Ca and 208Pb ii ; iii ; iv ; v . Furthermore, some benchmark high-resolution $(e,e^{\prime}x)$ experiments with $x=p,n,\alpha$ and the decay of the Giant Dipole Resonance (GDR) in the doubly magic nuclei 40Ca and 48Ca were also performed at the S-DALINAC vi ; vii ; viii ; viv ; vv but in general, the information on the observation of detailed strength distributions of 1- states in the $(e,e^{\prime})$ reactions is very sparse.

Concerning the physical origin of low energy electric dipole strength and its particular distribution we note in passing that there exists at present still no clear picture about the relevant excitation mechanism. Recent self-consistent Random-Phase Approximation (RPA) calculations with various finite-range forces in 16O and 40Ca Papa1 and also 48Ca Papa2 have shown that, e.g., nuclear surface vibrations might mix with skin modes and thus influence the pygmy dipole strength. It is stated clearly there that an electroexcitation experiment of the $(e,e^{\prime})$ type could eventually help to“improve the different models aspiring to describe reliably the low-energy dipole strength of nuclei” Papa2 . This point has also been independently emphasised in Rei . To provide some estimates for the feasibility of $(e,e^{\prime})$ experiments is the main purpose of this article.

The cross section for the excitation of natural parity states in $(e,e^{\prime})$ reactions has a longitudinal and transversal component. It is expected that the longitudinal or Coulomb term gives rise to a distribution of electric dipole strength over energy quite similar to the one seen in NRF experiments, at least at small momentum transfer. However, the transversal part is mediated by nuclear currents, and thus provides an alternative mechanism to excite the same set of PDR states in addition to the Coulomb and $NN$ excitations.

It is thus important

– to investigate at which kinematics the transversal mechanism dominates over the longitudinal one in the excitation of the PDR states, and to compare it to the behaviour of the excitation of the collective GDR and

– to provide realistic estimates of the $(e,e^{\prime})$ cross section for the excitation of the PDR levels.

Electrons with incident kinetic energies from 30 to 120 MeV will be considered. Such energies can be provided by the S-DALINAC in Darmstadt, where the detector system allows for measurements in a wide range of scattering angles, including backward scattered electrons close to 180o which can be detected with high angular resolution i ; Lut .

The calculations have been performed for 140Ce, a semi-magic nucleus in which the PDR has already been studied in $(\gamma,\gamma^{\prime})$, $(p,p^{\prime})$, and $(\alpha,\alpha^{\prime})$ reactions Ce-last .

2 Plane-wave Born approximation

The theory of inelastic scattering of electrons on nuclei is well developed and may be found in textbooks (see, e.g., Ube ). The plane-wave Born approximation (PWBA) is usually sufficient for simple estimates. In the PWBA, the differential $(e,e^{\prime})$ cross section for excitation of a natural parity state of multipolarity $\lambda$ can be written as HB83

[TABLE]

where $V_{L}(\theta_{e})$ and $V_{T}(\theta_{e})$ are the longitudinal and transversal kinematic factors, respectively, and $\lambda$ denotes the multipolarity of the transition. Nuclear structure information on the excited state enters via the charge transition density $\rho_{\lambda}(r)$ into the Coulomb form factor

[TABLE]

and via the transition current densities $J_{\lambda,\lambda\pm 1}(r)$ into the electric form factor

[TABLE]

where $q$ denotes the three-momentum transfer and $j_{\lambda}(qr)$ is the spherical Bessel function. Any interference between Coulomb and electric form factors is neglected in the PWBA.

At small $q$-values, Siegert’s theorem Sie may be applied, resulting in

[TABLE]

where we have used relativistic units ($\hbar=c=1$). When combining Eqs. (1) and (4), the quantity

[TABLE]

indicates whether the longitudinal or the transversal contribution dominates in the nuclear excitation process. This quantity is shown in Fig. 1 by a solid line together with the kinematic factors $V_{L}(\theta_{e})$ and $V_{T}(\theta_{e})$. The calculation was performed for a hypothetical $1^{-}$ state with an excitation energy of $E_{x}=8$ MeV excited by $E_{e}=70$ MeV electrons. Notice that the excitation via the Coulomb form factor (${\rm R}(\theta_{e})>1$) dominates in a wide range of scattering angles $\theta_{e}$, except for very forward ($\theta_{e}=0^{o}-8^{o}$) and backward ($147^{o}-180^{o}$) angles.

When the momentum transfer $q$ is small, it is also possible to perform a Taylor expansion of the Bessel function $j_{\lambda}(qr)$ in Eq. (2). Keeping only the first term, the square of the Coulomb form factor is closely related to the reduced transition probability $B(E\lambda)$ of the excited state, $|F^{C}_{\lambda}(q)|^{2}\propto q^{2\lambda}~{}B(E\lambda)$.

These simple estimates lead to the expectation that the distribution of $E1$ strength of states in the region of the PDR in $(e,e^{\prime})$ experiments at small $q$-values is rather similar to the one in $(\gamma,\gamma^{\prime})$ measurements. Indeed, at fixed kinematics, the $(\gamma,\gamma^{\prime})$ excitation cross section is strictly proportional to $B(E\lambda)$ (see, e.g., Go94 ; wf ). Some deviations are possible only at very large scattering angles of electrons.

The nuclear structure information on the states which form the PDR is contained in transition charge and current densities, which enter into Eqs. (2) and (3). They were calculated for 140Ce within the quasiparticle-phonon model Solo ; LoIu . The model employs a nuclear Hamiltonian which includes the mean field for protons and neutrons (a phenomenological Woods-Saxon potential is usually used), monopole pairing, and residual interactions in a separable multipole form. Excitations of even-even nuclei are treated as quasi-bosons (phonons), the excitation energies and internal fermion structure of which are obtained by solving equations of motion of the quasiparticle random phase approximation (QRPA). This yields the eigenenergies and wavefunctions the one-phonon states.

The distribution of the $B(E1)$ strength over the one-phonon $1^{-}$ states in 140Ce in the PDR energy region is shown in Fig. 2 (top-left). The states with the largest $B(E1)$ values are marked with an asterix. They will be discussed in more detail below. Notice that their $B(E1)$ values are almost two orders of magnitude smaller as compared to the one-phonon states which form the GDR in Fig. 2 (top-right).

Transition charge density, $\rho_{1}(r)$, and current densities, $J_{1,0}(r)$ and $J_{1,2}(r)$, of some selected one-phonon $1^{-}$ states are presented in Fig. 3 with their excitation energies indicated in the right-top corner of each panel. The ones belonging to the PDR (GDR) are shown in the left (right) column. The following effective charges for protons (Z) and neutrons (N) have been used: $e_{Z(N)}={\rm N(-Z)/A}$ for the $B(E1)$ values and charge densities, and the effective $g$-factors: $g_{l}^{Z(N)}=e_{Z(N)}$ for the convection current, and $g_{s}^{\rm eff}=0.8~{}g_{s}^{\rm free}$ for the magnetization current.

The charge transition densities of the $1^{-}$ states, which form the GDR, have a strong surface peaking, typical for collective vibrations. Protons and neutrons oscillate out of phase and due to different signs of the effective charges, they add constructively. The interference of 1p1h components in the wave function of one-phonon $1^{-}$ states from the PDR energy region has a destructive nature Pon . Their charge transition densities are peaking in the interior of the nucleus where their main 1p1h component is dominating. Accordingly, the position of minima and maxima varies from state to state.

The PWBA differential $(e,e^{\prime})$ cross sections for the excitation of three selected one-phonon $1^{-}$ states, which belong to the PDR (GDR) are presented in the left (right) part of Fig. 4 as a function of the angle of the scattered electrons. They are calculated for an incident energy of 70 MeV. The contribution of the longitudinal and transversal components is shown separately by dashed and dotted lines, respectively. Our conclusion about the longitudinal and the transversal contributions based on Eq. (5) and Fig.1 remains valid for the GDR states. But for the PDR states the transversal component determines the excitation in a wide range of angles from $90^{o}$ to $180^{o}$. For all three PDR states the differential cross section has a very similar shape with deep minima between $40^{o}$ and $50^{o}$ and an almost flat behaviour for scattering angles $\theta_{e}>70^{o}$.

3 Distorted-wave Born approximation

Within the distorted-wave Born approximation (DWBA), one solves the Dirac equation for the incoming and outgoing electrons in the Coulomb field of the nucleus in terms of partial waves. Schematically, the differential cross section has the form:

[TABLE]

where $m_{s}$ and $\mu$ are the projections of the spin $\vec{s}$ of the incoming electron and of the angular momentum transfer $\vec{\lambda}$, respectively. The expression for the transition amplitudes $A(\lambda\mu m)$ may, e.g., be found in Ube ; Tuan . An essential detail is that $A=A_{C}+A_{E}$, where both Coulomb ($A_{C}$) and electric ($A_{E}$) amplitudes are calculated by folding the nuclear charge and current transition densities, respectively, with the partial waves of the incoming and outgoing electron and with the propagator of the virtual photon. This implies that the DWBA accounts for the interference between two mechanisms for the excitation of natural parity states, (longitudinal) Coulomb and (transversal) electric.

One of the DWBA problems is a poor convergence of the radial integrals, particularly for dipole excitations at backmost scattering angles. We employ here the complex-plane rotation method developed in VF70 ; we1 , to overcome this problem. This allows us to cover all scattering angles from $0^{o}$ to $180^{o}$. The computation time is sped up by a multiple convergence acceleration in the sum over the final-state partial waves YRW54 . However, such an acceleration is not possible for angles $\theta_{e}\lesssim 10^{\circ}$.

DWBA calculations have been performed for all one-phonon $1^{-}$ states of 140Ce with an excitation energy below 20 MeV (42 states in total). We have considered incident energies from 30 to 120 MeV and scattering angles from 40o to 180o. Special attention is paid to incident energies of 40, 70, and 110 MeV. They correspond roughly to the energies which may be achieved with the present set-up at the S-DALINAC after the beam passes the linac once, twice, and four times. The main results of the calculations are presented in Figs. 5-8.

With Figs. 5 and 6 we continue the discussion on the role of the longitudinal and transversal mechanisms of excitation which was started in connection with Figs. 1 and 4. One state at 6.51 MeV belonging to the PDR (left column) and one at 15.50 MeV belonging to the GDR (right column) are considered in each of the figures. The dependence of the differential cross sections on scattering angle is presented in Fig. 5 for the incident energies of 40, 70, and 110 MeV. The dependence on incident energy is shown in Fig. 6 for the scattering angles $60^{o}$, $120^{o}$, and $175^{o}$. The DWBA cross sections are plotted by solid thick lines. By artificially setting $J_{1,L}(r)=0$ or $\rho_{1}(r)=0$ one obtains the excitation of the states by pure Coulomb (dashed line) or electric (dotted line) mechanisms, respectively.

The analysis of the results in Figs. 5 and 6 yields conclusions similar to the ones drawn from the PWBA predictions in the previous section: the electric term in the excitation of the GDR plays the most important role only for very backward scattering, while for the PDR it may determine the cross section already at $90^{o}$. However, for some kinematics the interference between the Coulomb and electric parts may be extremely important (see, e.g., right-top panel of Fig. 5). In our examples this interference has often a destructive nature.

The DWBA results in Figs. 5 and 6 are also compared to the PWBA predictions (thin solid lines). Basically, the DWBA leads to a smoothing of the sharp structures in PWBA due to the folding procedure. Although in some cases (e.g., right-center panel of Fig. 5) the agreement between the two approximations is rather good, in other cases they may disagree by an order of magnitude or more with each other.

Cross sections for the states marked with an asterix in Fig. 2 are plotted as a function of scattering angle (Fig. 7) and as a function of bombarding energy (Fig. 8). The selected states have the largest $B(E1)$ values in the PDR (left column) and GDR (right column) energy regions. The cross sections for other one-phonon $1^{-}$ states look rather similar except at the largest $q$-values in the studied kinematical range.

Low incident energies and small or modestly large scattering angles provide the biggest cross sections for the excitation of the PDR in inelastic electron scattering experiments. How large they are, can be seen from a comparison with the excitation cross sections for the $2^{+}_{1}$ and $3^{-}_{1}$ states (with excitation energies 1.596 MeV and 2.464 MeV, respectively) presented in the left panels of Figs. 7 and 8 by solid and dotted lines. One notices that the one-phonon $1^{-}$ PDR states are about two orders of magnitude weaker excited in $(e,e^{\prime})$ reactions than the $2^{+}_{1}$ state in a wide range of kinematics under consideration. An exception is the backmost scattering. The main reason is that for the excitation of the $2^{+}_{1}$ and $3^{-}_{1}$ states the transversal part plays a marginal role even at $\theta_{e}\to 180^{o}$. It is sizeable, however, for the PDR states. But when changing the scattering angle from $40^{o}$ to $180^{o}$, the PDR excitation cross sections drop by one to three orders of magnitude, depending on the incident electron energy.

4 Fine structure of the PDR

In order to discuss the absolute values of the $(e,e^{\prime})$ excitation cross sections for the $1^{-}$ states in the PDR energy region, which one expects to measure in an experiment, it is necessary to account for the fact that the one-phonon $1^{-}$ states, discussed in the previous sections, are embedded in more complex two-, three-, etc. phonon states. The excitation of the latter from the ground state is very weak as compared to the excitation of the one-phonon states, but their density increases rapidly with excitation energy. The interaction between the one-phonon and more complex states leads to a fragmentation of the strength carried by the one-phonon excitations into components from many states with more complex wave functions. In other words, we are dealing with the decay of the doorway one-phonon states owing to the interaction with more complex background states in the spirit of Ref. BBB .

In the QPM this decay is implemented by describing excited states with a wave function which contains one-phonon (first term), two-phonon (second term), and higher components

[TABLE]

where $Q^{+}_{\lambda\mu i}$ is the creation operator of a phonon with multipolarity $\lambda$ and its projection $\mu$, and where $i=1,2,3\ldots$ is the ordered number of the one-phonon states for a given $\lambda$. The phonon operators act on $|\Psi_{g.s.}\rangle$ which is the wave function of the ground state of even-even nuclei, identified with the phonon vacuum. Multiphonon configurations are built up of phonons of different multipolarities $(\lambda_{1},\mu_{1})$, $(\lambda_{2},\mu_{2})$, coupled to the same $(\lambda,\mu)$ as the one-phonon term

[TABLE]

The eigenenergies of the states described by the wave functions (7), as well as the coefficients $R_{i}(\lambda\nu)$ and $P^{\lambda_{2}i_{2}}_{\lambda_{1}i_{1}}(\lambda\nu)$, are obtained by the diagonalization of the model Hamiltonian on the set of these wave functions. Since the model Hamiltonian is already prediagonalized on the QRPA level, one-phonon configurations do not interact with each other, but they mix in the wave function (7) due to their interaction with the same set of complex configurations.

The transition densities of the states (7) have the form of (7) where phonon operators are replaced by transition densities of one-, two-, etc. configurations. Neglecting the transition densities of the complex configurations, the cross section for excitation of the $\nu$-th ($\nu=1,2,3,\ldots$) state (7) in $(e,e^{\prime})$ reactions can be written as:

[TABLE]

where $A_{i}$ are the transition amplitudes for the $i$-th one-phonon state.

The first QPM calculation with the wave function (7) for the PDR states was performed for 140Ce in late 90-ies (see Fig. 2 in Ref. He97 ) and compared to the results of one of the first NRF experiment in which the fine structure of the PDR was observed. The model Hamiltonian was diagonalized in the basis of interactive one-, and a limited number of two-, and three-phonon configurations. The basis of complex configurations was extended later in wf : two- and three-phonon configurations were built up from the phonons with multipolarities from 1*±* to 9*±* and were cut above 8.5 MeV. All 42 one-phonon 1- configurations (discussed in the previous sections) were included in order to account for the GDR contribution at low excitation energies. The diagonalization yields 1157 $1^{-}$ states $\nu$ below 8.5 MeV. We will use this set of states in the discussion of the PDR below.

The fragmentation process of the $B(E1)$ strength of the doorway one-phonon $1^{-}$ states in the PDR energy region is demonstrated in the left part of Fig. 2. To guide the eye, we also present in Fig. 2 (bottom) the strength function

[TABLE]

of the distribution where $E_{\nu}$ are the eigenenergies of the states (7) and $B_{\nu}(E1)$ are their reduced transition probabilities. The strength functions here are calculated with an artificial width $\Gamma=0.1$ MeV and presented in arbitrary units.

The strongest states described by Eq. (7) in Fig. 2 (bottom) have $B(E1)$ values which are almost one order of magnitude smaller than the doorway ones in Fig. 2 (top). For the predictive power of the present set of the QPM wave functions we refer to Fig. 2 in Ce-last . It combines information on excitation of the individual PDR levels in 140Ce as observed in $(\gamma,\gamma^{\prime})$, $(p,p^{\prime})$ and $(\alpha,\alpha^{\prime})$ reactions in comparison with the calculation of the corresponding reaction cross sections performed with this set. Although it is not possible to establish a one-to-one correspondence between experiment and theory, a comparison of the calculations for single excitations in three different reactions with the experimental results on an absolute scale shows good agreement Ce-last . Also, calculations and experimental NRF data are found in good agreement concerning the degree of fragmentation and on the integrated strength, if the sensitivity limit of the experiments is taken into account wf . All together, it leads us to expect that employing the same set of wave functions in the calculation of the $(e,e^{\prime})$ cross sections will provide realistic values for the excitation of the strongest levels in the experiment.

The results of the DWBA calculations with the QPM wave functions Eq. (7) are displayed in Fig. 9 for an incident energy of 70 MeV and scattering angles $60^{o}$, $120^{o}$, and $180^{o}$. The strength functions in Fig. 9 are defined similar to Eq. (9) with the replacement of the $B_{\nu}(E1)$ quantities by the corresponding $(e,e^{\prime})$ cross sections $(d\sigma/d\Omega)_{\nu},$ and are presented in arbitrary units which are different for different panels. As in the case of the $B(E1)$ quantities, the largest cross sections of an individual $\nu$-th state in Fig. 9 (left) are about one order of magnitude smaller than the cross sections of the doorway one-phonon states for the same kinematics.

To discuss the relative $(e,e^{\prime})$ cross sections in the excitation of the PDR and GDR, an additional diagonalization of the QPM Hamiltonian has been performed by extending the basis of two-phonon configurations up to 19 MeV. No three-phonon configurations have been accounted for in this calculation. The fine structure of the GDR strength is shown in Fig. 2 (right bottom) and Fig. 9 for the $B(E1)$ values and $(e,e^{\prime})$ cross sections, respectively.

Figure 9 (left) demonstrates that, depending on the kinematics, the shape of the PDR excitation in inelastic electron scattering may vary dramatically and deviate from the distribution of the $B(E1)$ values which present the $q=0$ limit. For some kinematics the summed cross sections of all PDR states from the $(e,e^{\prime})$ reaction are even larger as compared to the summed ones for the GDR states. But the absolute values of the cross sections are small under such kinematical conditions.

5 Conclusion

The excitation of the $1^{-}$ states in 140Ce by inelastically scattered electrons with incident energies from 30 to 120 MeV is investigated. The scattering angle is varied from $40^{o}$ to $180^{o}$. This kinematical range covers a momentum transfer $q$ from 0.1 to 1.2 fm*-1*. We consider $1^{-}$ states which belong to the PDR and GDR. Their structure is described within the one-phonon QRPA, and by accounting for the coupling to complex configurations within the quasiparticle-phonon model.

It is demonstrated that Coulomb scattering is the dominant excitation mechanism for the GDR states in an $(e,e^{\prime})$ reaction in a wide range of scattering angles, except for the very backward scattering. On the contrary, the PDR states are predominantly excited by transverse electric scattering mediated by the nuclear current for scattering angles in a large angular region from $90^{o}$ to $180^{o}$. Also, the interference between the longitudinal and transversal components plays an important role for them. The latter effect is a distinctive feature of the DWBA calculations, while it is neglected in the PWBA.

The calculations show that the fine structure of the PDR in $(e,e^{\prime})$ reactions may change substantially, depending on the kinematics, especially at large scattering angles. We predict that the $(e,e^{\prime})$ excitation cross sections of the strongest individual $1^{-}$ states are about three orders of magnitude lower than the respective cross section for the $2^{+}_{1}$ state, except for very large scattering angles where the significant transversal contributions to the cross section for the PDR dominates. However, the absolute values of the cross section are rather small.

In this context we finally note that in earlier search for $M1$ and $M2$ giant resonances in 140Ce at the DALINAC the measured high-resolution spectra – $\Delta E$ varied between 28 and 48 keV (FWHM) – at backward angles showed no sign for excited $1^{-}$ states between excitation energies from 7.5 to 10 MeV Meu . With the improved electron beams from the S-DALINAC and its high-resolution spectrometers there is now, however, found hope to detect them.

We thank Peter von Neumann-Cosel for discussions concerning the topic of the present studies. This work was supported by the Deutsche Forschungsgemeinschaft under Contract No. SFB-1245.

We have dedicated this article to our late colleague and friend Pier Francesco Bortignon with whom we have discussed the physics of nuclear excitations for many years.