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Giant dipole resonance in 4d photo absorption of atomic barium
Volume 46A, number 2
PHYSICS LETTERS
GIANT DIPOLE RESONANCE
IN 4d PHOTOABSORPTION
3 December 1973
OF ATOMIC BARIUM
G. WENDIN CECAM*, Batiment 506, 91-Campus d’orsay, France
Received 25 July 1973 We calculate the photoabsorption cross section for the 4d1° and 5p6 shells of atomic barium using the random phase approximation (RPAE) includinn relaxation. For both shells the oscillator strength goes mainly into a collective vibration of the entire shell. In this letter we present some new theoretical results for the photoabsorption cross section of atomic barium which further support the idea that collective resonances do occur in atoms. Recent experiments [l] show that the continuum cross section for the 4dlo shell in the rare-earth metals sharply peaks 15-20 eV above the NIV v threshold, the width of the peak being lo-20 eV.‘These peaks are also seen in electron energy loss spectra [2]. In two recent papers [3,4] we have demonstrated that the corresponding broad peak in xenon (-35 eV) can be interpreted in terms of a strongly damped collective resonance involving coherent motion of all electrons in the 4d1° shell. It is then highly tempting to interpret also the peaks in the rare-earth spectra as manifestations of a fairly well defined collective resonance, analogous to the giant dipole resonances in nuclear gamma ray absorption spectra. Such a peak should be essentially an atomic feature but solid state effects could be important for an accurate determination ofthe position, width and structure of the peak. From the present calculation we conclude that a collective giant resonance in the 4d1° shell is a common feature of all the elements from xenon to praseodymium and we feel that the same is true for several of the following elements in the rare-earth series. In the case of barium we have also made a preliminary investigation of the 5p6 cross section. Again we find a collective resonance but this time slightly below the OIIa threshold (fig. lb). Due to the continuum background one should then see very pronounced structure in the O,Ifl threshold region around 20 eV, similar to what is observed for the 3p absorption
* Present and permanent address: Institute of Theoretical Physics, Fack, S-402 20 Giiteborg 5, Sweden.
around the MIIJn threshold in the transition metals. We would like to suggest that in the beginning of the transition series the observed peaks mainly are of atomic, collective origin although the 3d-bandwidth may be quite important for an accurate numerical description. Our calculation is based on a diagrammatic analysis of the atomic polarizability, as discussed in detail in refs. [3,4] for the case of xenon. The idea is to choose a zeroth-order approximation that describes single particle-hole pair excitations in a frozen environment and let the perturbation expansion explicitly account forall effects of dynamical electronelectron interaction, such as polarization, core relaxation, shake up and shake off, multiply excited resonance etc. Collective resonances would then show up in the theory as qualitatively new poles, not predicted in zeroth order, arising from coupling “many” electrons. This definition of zeroth-order approximation leads us to use Hartree-Fock wave functions for occupied (hole) states and to calculate excited (particle) states with the HF non-local potential built up from frozen ground state orbitals minus one (the so called VN-’ ion potential) [3,4] . The zerothorder result for the (4d; nf)l P photoabsorption cross section (-i(4dlzlnf)12) is shown in fig. la and reflects the fact that the excited 4f-orbital has become localized in the same region as the 4d-orbital while all other f-states are far out., ((4di4f) - -0.80) side the core. We have found two major effects of dynamical interaction that strongly influence this static zeroth-order result: (i) Correlations between density fluctuations in the 4d1° shell. A virtual particle-hole pair excitation polarizes the medium (mainly the 4d1° shell) and becomes itself modified by the induced field. In a strong coupling case perturbation theory does not 119
Volume
46A, number
2
PHYSICS
(al /
Present theory
4d-4f --I
lb)
-j
Zeroth - order: i
0
1.0
20
3 .o W Photon energy
(rydl
Fig. 1. Continuum photoabsorption cross section for atomic barium. Present theory to he compared with experiment: Random phase approximation with exchange (RPAE) including relaxation of the 4d-hole; --~ shows the effect of omitting ground state correlations. The short vertical solid and dashed lines mark the position of the collective resonance. The length of the zeroth order 4d-4f line only illustrates the strength and has no numerical significance. ~.-.~ is the zeroth order continuum and -- ~~ shows the effect of including relaxation. Hatched areas indicate the remaining part of the discrete spectrum. (a) 4d’O shell, (4d; nf)‘P channel. (b) 5p6 shell, (5~; nd)‘P channel. The results are preliminary but should illustrate the basic features. Note that the position of the “true” 0~1~ threshold has been guessed, not calculated.
suffice and one has to take the polarization expansion (bubble diagrams) to infinite order. This leads to the so called random phase approcimation with exchange (RPAE), having the power to describe collective oscillations of a system. (ii) Core relaxation in response to the individual charges of the hole and the particle. Point (i) above
120
LETTERS
3 December
1973
describes the response of the system to the dipole field of the particle-hole fluctuation but “assume” that the fluctuation looks neutral. This is far from true for an atomic system. especially when an electron is excited to high lying states or to the continuum. The system then effectively responds to a full core hole by relaxation, shake up and shake off. The relaxation shift of a 4d-hole in atomic barium is estimated [S] to be -0.45 ryd and the relativistic shift -0.30 ryd so that in this calculation we take the energy of the 4d-hole to be 7.25 ryd (- -8.00 ryd in HF)fov excitation of 5f and higher. The 4d 4f excitation is special in many ways because it carries nearly ail the of the zeroth-order spectral strength and looks almost neutral. In the present calculation we assume that there is no relaxation shift associated with the 4d -4f transition and also no relativistic shift. In [3,4] we formulated the RPAE method in terms of a reaction matrix accounting for the (4d; nf)lP intrachannel interaction including ground state correlations, and we have used the same set of equations in the present calculation. It turns out that the intrachannel interaction couples the 4d--4f excitations so strongly together that the energy of the normal mode solution is pushed far up into the continuum. As a consequence our calculated photoabsorption cross section for the (4d; nolP channel (fig. la) shows a giant dipole resonance in the continuum above the NW v threshold and the photon energy primarily goes’into a vibrational mode of the 4d1° shell.
Reference: [ 1) T.M. Zimkina and S.A. Gribovskii.
[2]
[3] [4] [S]
J. Physique (France) 32 (1971) C4-282; R. Haensel, P. Rabe and B. Sonntag, Solid State Commun. 8 (1970) 1845. P. Trebbia and C. Colliex. Phys. Stat. Solidi (b) 58 Nr 2 (1973) to be published; P. Trebbia, th’ese Docteur 3” cycle. Orsay, France (1973). G. Wendin, J. Phys. BS (1972) 110. G. Wendin, J. Phys. B6 (1973) 42. G. Wendin, Phys. Lett. j to be published.
PHYSICS LETTERS
GIANT DIPOLE RESONANCE
IN 4d PHOTOABSORPTION
3 December 1973
OF ATOMIC BARIUM
G. WENDIN CECAM*, Batiment 506, 91-Campus d’orsay, France
Received 25 July 1973 We calculate the photoabsorption cross section for the 4d1° and 5p6 shells of atomic barium using the random phase approximation (RPAE) includinn relaxation. For both shells the oscillator strength goes mainly into a collective vibration of the entire shell. In this letter we present some new theoretical results for the photoabsorption cross section of atomic barium which further support the idea that collective resonances do occur in atoms. Recent experiments [l] show that the continuum cross section for the 4dlo shell in the rare-earth metals sharply peaks 15-20 eV above the NIV v threshold, the width of the peak being lo-20 eV.‘These peaks are also seen in electron energy loss spectra [2]. In two recent papers [3,4] we have demonstrated that the corresponding broad peak in xenon (-35 eV) can be interpreted in terms of a strongly damped collective resonance involving coherent motion of all electrons in the 4d1° shell. It is then highly tempting to interpret also the peaks in the rare-earth spectra as manifestations of a fairly well defined collective resonance, analogous to the giant dipole resonances in nuclear gamma ray absorption spectra. Such a peak should be essentially an atomic feature but solid state effects could be important for an accurate determination ofthe position, width and structure of the peak. From the present calculation we conclude that a collective giant resonance in the 4d1° shell is a common feature of all the elements from xenon to praseodymium and we feel that the same is true for several of the following elements in the rare-earth series. In the case of barium we have also made a preliminary investigation of the 5p6 cross section. Again we find a collective resonance but this time slightly below the OIIa threshold (fig. lb). Due to the continuum background one should then see very pronounced structure in the O,Ifl threshold region around 20 eV, similar to what is observed for the 3p absorption
* Present and permanent address: Institute of Theoretical Physics, Fack, S-402 20 Giiteborg 5, Sweden.
around the MIIJn threshold in the transition metals. We would like to suggest that in the beginning of the transition series the observed peaks mainly are of atomic, collective origin although the 3d-bandwidth may be quite important for an accurate numerical description. Our calculation is based on a diagrammatic analysis of the atomic polarizability, as discussed in detail in refs. [3,4] for the case of xenon. The idea is to choose a zeroth-order approximation that describes single particle-hole pair excitations in a frozen environment and let the perturbation expansion explicitly account forall effects of dynamical electronelectron interaction, such as polarization, core relaxation, shake up and shake off, multiply excited resonance etc. Collective resonances would then show up in the theory as qualitatively new poles, not predicted in zeroth order, arising from coupling “many” electrons. This definition of zeroth-order approximation leads us to use Hartree-Fock wave functions for occupied (hole) states and to calculate excited (particle) states with the HF non-local potential built up from frozen ground state orbitals minus one (the so called VN-’ ion potential) [3,4] . The zerothorder result for the (4d; nf)l P photoabsorption cross section (-i(4dlzlnf)12) is shown in fig. la and reflects the fact that the excited 4f-orbital has become localized in the same region as the 4d-orbital while all other f-states are far out., ((4di4f) - -0.80) side the core. We have found two major effects of dynamical interaction that strongly influence this static zeroth-order result: (i) Correlations between density fluctuations in the 4d1° shell. A virtual particle-hole pair excitation polarizes the medium (mainly the 4d1° shell) and becomes itself modified by the induced field. In a strong coupling case perturbation theory does not 119
Volume
46A, number
2
PHYSICS
(al /
Present theory
4d-4f --I
lb)
-j
Zeroth - order: i
0
1.0
20
3 .o W Photon energy
(rydl
Fig. 1. Continuum photoabsorption cross section for atomic barium. Present theory to he compared with experiment: Random phase approximation with exchange (RPAE) including relaxation of the 4d-hole; --~ shows the effect of omitting ground state correlations. The short vertical solid and dashed lines mark the position of the collective resonance. The length of the zeroth order 4d-4f line only illustrates the strength and has no numerical significance. ~.-.~ is the zeroth order continuum and -- ~~ shows the effect of including relaxation. Hatched areas indicate the remaining part of the discrete spectrum. (a) 4d’O shell, (4d; nf)‘P channel. (b) 5p6 shell, (5~; nd)‘P channel. The results are preliminary but should illustrate the basic features. Note that the position of the “true” 0~1~ threshold has been guessed, not calculated.
suffice and one has to take the polarization expansion (bubble diagrams) to infinite order. This leads to the so called random phase approcimation with exchange (RPAE), having the power to describe collective oscillations of a system. (ii) Core relaxation in response to the individual charges of the hole and the particle. Point (i) above
120
LETTERS
3 December
1973
describes the response of the system to the dipole field of the particle-hole fluctuation but “assume” that the fluctuation looks neutral. This is far from true for an atomic system. especially when an electron is excited to high lying states or to the continuum. The system then effectively responds to a full core hole by relaxation, shake up and shake off. The relaxation shift of a 4d-hole in atomic barium is estimated [S] to be -0.45 ryd and the relativistic shift -0.30 ryd so that in this calculation we take the energy of the 4d-hole to be 7.25 ryd (- -8.00 ryd in HF)fov excitation of 5f and higher. The 4d 4f excitation is special in many ways because it carries nearly ail the of the zeroth-order spectral strength and looks almost neutral. In the present calculation we assume that there is no relaxation shift associated with the 4d -4f transition and also no relativistic shift. In [3,4] we formulated the RPAE method in terms of a reaction matrix accounting for the (4d; nf)lP intrachannel interaction including ground state correlations, and we have used the same set of equations in the present calculation. It turns out that the intrachannel interaction couples the 4d--4f excitations so strongly together that the energy of the normal mode solution is pushed far up into the continuum. As a consequence our calculated photoabsorption cross section for the (4d; nolP channel (fig. la) shows a giant dipole resonance in the continuum above the NW v threshold and the photon energy primarily goes’into a vibrational mode of the 4d1° shell.
Reference: [ 1) T.M. Zimkina and S.A. Gribovskii.
[2]
[3] [4] [S]
J. Physique (France) 32 (1971) C4-282; R. Haensel, P. Rabe and B. Sonntag, Solid State Commun. 8 (1970) 1845. P. Trebbia and C. Colliex. Phys. Stat. Solidi (b) 58 Nr 2 (1973) to be published; P. Trebbia, th’ese Docteur 3” cycle. Orsay, France (1973). G. Wendin, J. Phys. BS (1972) 110. G. Wendin, J. Phys. B6 (1973) 42. G. Wendin, Phys. Lett. j to be published.