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Relativistic calculations for highly correlated atomic and highly charged ionic systems
ARTICLE IN PRESS
Radiation Physics and Chemistry 76 (2007) 404–411 www.elsevier.com/locate/radphyschem
Relativistic calculations for highly correlated atomic and highly charged ionic systems Fumihiro Koikea,, Stephan Fritzscheb a
Physics Department, School of Medicine, Kitasato University 1-15-1 Kitasato, 228-8555, Japan b Universitaet Kassel, Heinrich-Plett-Strasse, 40, 34132, Kassel, Germany Received 31 August 2005; accepted 2 October 2005
Abstract The effect of electron correlations in many electron atoms or many electron atomic ions are discussed extensively based on the MultiConfiguration Dirac Fock (MCDF) calculations. In a precision atomic physics the relativistic treatment of the system is indispensable. The correlation effects and the relativistic effects are no more additive if one want to treat the many electron systems accurately. In the excited states, the single electron orbitals are modified in accordance with the vacancies near the atomic center. The electron correlations may be evaluated from the nonorthogonality of the single electron orbitals. Several examples have been given. The electronic configurations with the same total parities may interact each other even in the cases the constituent single electron orbitals have opposite parities. Such the configuration interactions may provide us with characteristic interference structures in the optical emission or absorption spectra. The anormally of the extreme ultra-violet optical emission spectra of highly charged tin ions has been illustrated as an example. r 2006 Elsevier Ltd. All rights reserved. PACS: 32.80.t; 32.80.Fb Keywords: Atomic structure; Atomic transition; Optical emission; EUV; Relativistic theory; MCDF; Configuration interaction; Electron correlation
1. Introduction In recent years, the spectroscopic measurement in atomic physics experiments gives us quite a sophisticated set of data that requires an accurate theoretical treatment. Also in the field of plasma physics, requirement for detailed and sophisticated description of the plasma is becoming noticeable in the study of confinement nuclear fusion as well as the other plasma applications. Because a plasma model is based on the Corresponding author.
E-mail address: [email protected] (F. Koike).
atomic models that may facilitate appropriately the corresponding plasma simulations, a precision plasma model requires precision atomic data. For plasmas containing heavy elements, we must take into account the effects arising from such complex species. In Fig. 1, we illustrate the situation schematically. To obtain theoretical values with reasonable precision, we find that a fully relativistic treatment for the problem is indispensable. In fact, the value b ¼ v=c, which is the ratio of the orbital electron velocity v and the speed of light c, is about 14=1370:1 in silicon 1 s orbitals, for example. The relativistic correction could be a few percent of the total atomic state energy and also of
0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.10.044
ARTICLE IN PRESS F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
405
Multi-Configuration and Non-Local Treatment
Dirac Equation and more
Electron Correlation
Schroedinger Equation
Precise Atomic Structure Calculation
Relativistic Treatment
Local State-Independent Potentials
Fig. 1. A schematic illustration for the steps towards precision atomic structure calculations. Because the effects of electron correlations and of the relativistic behavior of atomic electrons are not additive, both the effects must be evaluated simultaneously.
the electronic transition energy if one considers the deep inner-shell excitations; the relativistic effects cannot be canceled out by subtraction of the lower state energies from the upper state ones. Large spin–orbit term splittings may also appear. The electron correlation energy is in general a few percent of the total atomic electron energy. Consideration of the electron correlations between various electronic configurations is also indispensable if we want to determine the atomic transition energies within the accuracy of several electron volts or less. And, furthermore, an important point is that we cannot, in general, discriminate between the relativistic effects and the correlation effects out of the actual atomic transition energies or excitation strengths, and sometimes the relativistic shift of the orbital energies could even enhance or suppress the electron correlation effects. In this sense, we should point out that both the relativistic and correlation effects must be treated simultaneously. In this report, we, firstly, in Section 2, discuss the characteristic properties of Dirac Hartree Fock or multiconfigutration Dirac Hartree Fock methods, which are a group of methods based on variational principles for the set of single electron orbitals. It is pointed out that the concept of a single electron orbital would have to be modified if one would optimize the atomic states individually. We briefly review extensive efforts to carry out the atomic structure calculations by a number of authors. Secondly, the electron correlations in highly excited atomic or ionic states including the deep inner shell excitations and/or multiple excitations will be reviewed. We illustrate a couple of numerical examples with the corresponding experimental data. The electron
correlation effects is important even in highly ionized atoms. In heavy elements, a pair of electrons may come close to provide us with large correlation energies. The energy of correlations of atomic electrons can be considered as typically of the order of a few electron volts. We must take into account the electron correlation effects, when the required accuracy of the atomic transition energies falls in this range. The orbital contractions in the atomic excitation may play a key role, which will be shown in Section 3. Thirdly, in Section 4, we discuss the electron correlations due to the inter-(sub)shell mixing. In general, the electronic configurations with the same total parity may interact each other even in the cases the constituent single electron orbitals have opposite parities; a configuration state function recovers its total parity after a two-electron replacement. As an example of such configuration interactions, we illustrate an anomaly in the extreme ultra-violet optical emissions of highly charged tin ions Sn12þ . Finally, in Section 5, we give concluding remarks. 2. Characteristics of the Dirac Hartree Fock methods As for the description of the Hartree Fock method, we may refer to the formalism of Lipkin (1973). We consider an N-electron wavefunction under the independent particle model. We take ayki as a creation operator of the single electron orbital. The N-electron wave function may be represented by jUi ¼
N Y i¼1
ayki j0i,
(1)
ARTICLE IN PRESS F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
406
where j0i represents the vacuum and the anti-symmetrization of the electron orbitals will be made if necessary. In the Hartree Fock method we apply a variational condition dhUjHjUi ¼ hUjHjdUi ¼ 0
(2)
under an appropriate ortho-normality constraint for wavefunctions. We take dyi as a small variation of ayki . Then, the variation jdUi may be represented as jdUi jU þ dUi jUi ¼
N X
ð1ÞP dyj
j¼1
N Y
ayki j0i,
i¼1ðiajÞ
(3) where P in ð1ÞP is the number of permutations that is required to place the operator dyj on top of the product. Substituting Eqs. (1) and (3) into Eq. (2), we obtain hUjHjdUi ¼
N X
ð1ÞP h0jakj jHjdyj j0i ¼ 0.
(4)
j¼1
From this equation, we see that the (D)HF method diagonalizes H with respect to the single electron displacements on the basis of independent particle view point. We also find that the atomic states that diagonalize the multi-electron displacements cannot be represented by one unique configuration state function (CSF) jUi. And, further on, we may recognize that an atomic state jUi may be optimized for each atomic state independently of the other states. The basis sets fayki g may differ between different atomic states. And, furthermore, we may note that we can obtain excited states by choosing the basis sets fayki g as appropriate; there is no restriction for the atomic state jUi to remain in the ground state in the framework of the (D)HF method. In the multiconfiguration Dirac Hartree Fock method, we define the atomic state function (ASF) in terms of configuration state functions (CSFs). We define an ASF jWm i by CSFs jUn i as jWm i
X
jUn icnm
with jUn i
n
N Y i¼1
ay ðnÞ j0i. ki
(5)
We optimize jWm i applying the variational condition that * + X d Un cnm jHjWm n
*
¼
X
+ dfUn g cnm jHjWm
n
* þ
X
+ Un dfcnm gjHjWm
¼0
ð6Þ
n
under an appropriate orthonormality constraint for the orbital basis sets, CSFs and ASF. We note here that the
ASF jWm i may be optimized individually with respect to the index m, and, therefore, orbital basis sets are not necessarily orthogonal between the sets for different m. For example, a 1s orbital in the first set jð1sÞ1 i may be a linear combination of the second set jðnsÞ2 i; we may here write as X X jð1sÞ1 i ¼ jðnsÞ2 ihðnsÞ2 jð1sÞ1 i ¼ jðnsÞ2 isn . (7) n
n
The states described by the other set {ai ð2Þ} may be represented by a state with multiple excitations on the basis of fai ð1Þg; the effect of electron correlations may be evaluated by introducing a separate optimization for individual states. However, in the calculation of atomic states, we should note that the symmetry quantum numbers are not the subject of the optimization and therefore the basis sets are always orthonormal with respect to the symmetry quantum numbers. In the relativistic calculations, the electronic Hamiltonian of the system is chosen as in the following: X X 1 H¼ hDC ðiÞ þ r i ioj ij ! X 1 ðai ri Þðaj rj Þ þ ai aj þ , ð8Þ 2rij r2ij ioj where hDC ðiÞ is the single electron Dirac Hamiltonian, and the last term is the Breit correction term. The symmetries of the ASFs are specified by their parity, total angular momentum, and its magnetic component. And they are expanded in terms of the configuration state functions (CSFs) that represent virtual single, or multiple excitations from occupied to unoccupied (sub)shells. In these couple of decades, extensive efforts have been made for computer codes to calculate the atomic structures on the basis of multiconfiguration relativistic treatment. Grant and coworkers (Dyall et al., 1989; Parpia et al., 1996) have developed a set of programs called GRASP (General Purpose Relativistic Atomic Structure Program). As an extension of GRASP, Fritzsche and his coworkers have developed a set of codes (Fritzsche et al., 2002a,b; Fritzsche, 2001; Fritzsche et al., 2000; Fritzsche and Anton, 2000; Fritzsche et al., 1999; Fritzsche and Grant, 1997; Fritzsche and Froese Fischer, 1997; Fritzsche and Grant, 1995) that calculate the atomic transition and other properties, which is called RATIP (Relativistic Atomic Transitions and Ionization Properties). Kagawa and his coworkers have developed a set of relativistic configuration interaction codes (Kagawa, 1975; Kagawa et al., 1991). Another set of multiconfiguration Dirac Fock programs has been developed by Kim and his coworkers (Mohr and Kim, 1992). And, Klapisch and his coworkers have developed a set of programs that is called HULLAC with local approximation of the exchange integrals (Klapisch et al., 1997).
ARTICLE IN PRESS F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
processes. They observed a pair of 1s1 2s1 ð3 SÞ 2p6 3s1 3p1 , and 1s1 2s1 ð1 S Þ2p6 3s1 3p1 double electron excitations at 924.78 eV and at about 935 eV of photon energies. Those two peaks are about 10 eV apart from each other. This value is somewhat large compared to the usual understanding of the singlet-doublet separation in neon 1s1 2s1 configurations. To find out the origin of this largeness, we have calculated the electron density distribution of the Ne2þ 1s1 2s1 2p6 configuration using both the MCDF wavefunctions of neutral neon and the doubly charged neon atomic ion. Fig. 3 shows the charge density plots from those two sets of wavefunctions. We see that in the actual Ne2þ 1s1 2s1 2p6 ions, the electronic charges are relaxed to the atomic center. It may be concluded that this contraction of the electronic wavefunctions due to the 1s2s double hole creation is the cause of the enlargement of the singlet–triplet separation in Ne 1s1 2s1 ð3;1 S Þ2p6 3s1 3p1 states.
In the rest of the present report, we introduce a couple of calculations on the remarkable phenomena relating to the electronic correlations of atoms using GRASP family codes.
3. Importance of the state dependent atomic orbital wavefunctions 3.1. The 4d–4f photoexcitation of Xeþ It is well known that there is a strong concentration of the oscillator strength in the 4d photoexcitations of xenon atoms in the photon energies around 100 eV, which is called the giant resonance. Sano et al. (1996) made a measurement on the photoionization of 4d-electrons in singly charged Xe ions. And a number of subsequent measurements have been carried out by Koizumi et al. (1997, 1996). They found a double hump structure in the photoion spectra of Xeþ and analyzed them by means of the MCDF calculation. They found that the 4f orbital suffers a strong collapse when the system has a simultaneous 5p to 6p shake up excitation. We illustrate the usual and collapsed wavefunctions in Fig. 2. Due to this 4f orbital collapse, the 4f orbital could have a strong interaction with 4d orbital which realize the double hump structure in the observed spectra. And we also note here that this type of collapsed orbital may not be expanded in terms of the nf orbital wavefunctions without collapse; the optimization of the collapsed orbital is compulsory if we want to gain an insight into the Xeþ 4d 9 4f 1 5s2 5p4 6p1 shake up excitations.
3.3. Li 3s orbital contraction triggered by K, L electron escape by photoabsorption In the last decade, there have been an extensive study of the lithium hollow atomic state excitations (Azuma et al., 1995). Azuma et al. (1997) have observed for the first time the K, L double shell hollow atomic state in lithium atoms. Fig. 4 shows their experimental spectrum with the results of an elaborate MCDF calculation. We can see that the oscillaor strength of the three electron excitations is still large to make the states experimentally observable. The largeness of the oscillator strength originates from the 3s and 3p orbital contraction in the hollow atomic states. In Fig. 5, we illustrate the change of the charge distributions from the lithium ground state to the lithium 333 resonance states. We can
3.2. Ne 1s and 2s resonant double photoexcitation Recently, Oura et al. (2004) have made an experiment on neon 1s and 2s resonant double photoexcitation
Xe + 4 d 94 f 1 5s2 5 p5 6 p0 6p
0.5
5f 6f
7f
0.0
Radial Wavefunction
4f
4d – 4 f +
-0.5 4d
6p
4f
0.5
5p – 6 p
Collapse
-1.0
Xe + 4 d 94 f 1 5s2 5 p4 6 p1
0.0 7p
-0.5 5p
-1.0
4d
0
5
407
10 Radial Distance (a0 )
15
20
Fig. 2. The collapse of the 4f orbital wavefunction due to the 5p–6p shake up excitation in Xeþ 4d photoionizations.
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WithOUT Relaxation After the Inner-Shell Hole Creation
Charge Density (1 /a0)
8
6 With Relaxation After the InnerShell Hole Creation
4 Contraction of electron orbitals
2
0
0
0.5
1 1.5 2 Radial Distance (a0 )
2.5
3
Fig. 3. The charge density plots from two sets of wavefunctions. We see that in the actual Ne2þ 1s1 2s1 2p6 ions, the electronic charges are relaxed to the atomic center.
× 10- 4 2 s 2 2p
3s 23p + 3p3
2
2s2p3s+2 s 23p
Cross Section (kb)
Oscillator Strength
3
1
15.5 15.0 14.5 14.0 13.5 13.0 175.3 12.5 174.5 175.0 175.5 176.0 Photon Energy (eV)
eV
233 175.6 eV
0 140
150
160
170
180
Photon Energy (eV) Fig. 4. Calculated oscillator strength distribution for Li hollow atomic excitations. The inset is the experimental photoion spectra around 175 eV of photon energy. The theoretical resonance peaks are convoluted with small artificial natural width and instrumental width.
see that the range of the charge density in 333 resonance is rather short compared with the ones in 223 or 233 resonance; this may give a fairly large overlap of the 333 state configuration with the ground state configuration.
4. Electron correlations due to the inter-(sub)-shell mixing In these decades, the 13.5 nm range extreme ultraviolet (EUV) light emissions of many electron atomic ions have become of a strong interest in relation to the short wavelength light source of the semiconductor lithography technologies. One of the best candidates for
such an EUV light source is considered to be of the intra N shell (n ¼ 4 shell) transitions or inter N–O shell transitions of tin (Sn) or xenon (Xe) multi-charged atomic ions. Extensive efforts to produce an accurate atomic data for the use of radiation transport calculations have been carried out by a number of authors (Sasaki et al., 2004; Koike et al., 2005; Churilov, 2004; O’Sullivan and Faukner, 1994; Svendsen and O’Sullivan, 1994). Because the range of the wavelength required by the side of the lithography optics instrumentation is very narrow, the required accuracy of the atomic structure calculation is also high accordingly. We have to evaluate, on one hand, the intra N shell electron correlations precisely for excited states as well as the
ARTICLE IN PRESS F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
409
3
Charge Density (1 /a0)
Ground State
222 Resonance
2
223 Resonance
233 Resonance
1 333 Resonance
0 0
2
4
6
8
10
12
Radial Distance (a0) Fig. 5. Charge density distributions of Li hollow atomic states.
Configuration Mixing
4f 0
4f1
81 eV
4 d3
4 d1
72 eV
4 p5
4 p6
47 eV
4 s2
4s2
E = 81 – 72 = 9 eV ~ 4% of 4 d binding energy 253 eV Fig. 6. The scheme of configuration interactions in Sn12þ 4d–4f excited states. An excited state odd parity electronic configuration 4s2 4p6 4d 1 4f 1 can be generated by replacing two even parity 4d electron orbitals by two odd parity electron orbitals 4p and 4f from a configuration 4s2 4p5 4d 3 4f 0 . If the unperturbed energies of these two configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 are not very different in values, strong interactions between these two configurations may be expected.
ground states. On the other hand, we have to notice that, in this type of ions, especially of the Sn ions, a peculiar behavior in the emission spectra has been observed by O’Sullivan and Faukner (1994). They have pointed out that the narrowing and the shift take place in the 4f –4d EUV light emission spectra. They have explained these phenomena as due to the interactions between 4p6 4d w1 4f 1 and 4p5 4d wþ1 4f 0 configurations, where w is an integer that runs from 1 to 9. This is another type of the electron correlations which are different from the ones discussed in the previous section, and we may note that those correlations may also be interesting from the view point of basic atomic physics. To gain a further insight into the effects, we have carried out careful MCDF calculations for 4d w (w ¼ 1–9) atomic ions with atomic number Z ¼ 48–56, using GRASP and RATIP family computer codes. At first,
we chose the system Sn12þ as a candidate for detailed investigation of the electron correlation effects, since the ground state of Sn12þ has 4d 2 partially filled open subshell as its outermost subshell, and the EUV light emission falls near to the range of the wavelength around 13.5 nm. In Fig. 6, we illustrate the scheme of configuration interactions that we took into consideration. An excited state odd parity electronic configuration 4s2 4p6 4d 1 4f 1 can be generated by replacing two even parity 4d electron orbitals with two odd parity electron orbitals 4p and 4f from a configuration 4s2 4p5 4d 3 4f 0 . If the unperturbed energies of these two configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 are not very different in values, we may expect strong interactions between these two configurations. In the result of an elaborate MCDF calculation, we found that in the case of Sn12þ , the energy differences between the neighboring single
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electron orbitals are quite close as indicated in Fig. 6; the difference of the energy separations between 4f and 4d, and 4d and 4p has turned to be 9 eV, which is only 4% of the 4d electron orbital energy. And, furthermore, we have found that the major peak positions of the 4s, 4p, 4d, and 4f radial wavefunctions almost coincide. The configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 have good reasons to suffer strong mixing as of the effect of configuration interactions. The optical 4p–4d and 4f –4d transitions cannot be discriminated from the others, in other words, the transitions take place coherently at the same time, providing us with quite a peculiar EUV emission spectrum. In Fig. 7, we have plotted the oscillator strength distribution for 4d–4f band transitions of Sn12þ ions. Each spectrum gives a simple sum over the oscillator strengths for transitions of all the possible combinations of the members of multiplets in the excited and ground states. The calculated spectrum has been smoothed out by convoluting a natural width function of the artificial width 0.05 nm. The dashed curve represents the spectrum in which the coherent transitions of 4d–4f and 4p–4d has been taken into account. The dotted curve represents the distribution of only the 4d–4f transitions, and the solid curve represents the distribution of only the 4p–4d transitions. We can see that a constructive interference between the 4d–4f and 4p–4d transitions is taking place at a shorter wavelength region, and also that a destructive interference is taking place at a longer wavelength region. We
can confirm the enhancement of the optical transition due to the interference effects in a wavelength region at 13 eV. We have, further, explored if any similar effects can be found in the atomic ions with different atomic numbers and charge states. We finally realized that those interference phenomena are quite common to the ions in this range; a similar effect has been observed also in Xe ions, and further on, in almost all the atomic ions with Z ¼ 48–56.
5. Discussion We have reported a couple of characteristic features of the electron correlation effects based on the multiconfiguration Dirac Fock type calculations using the atomic codes from GRASP and RATIP. In these calculations, the atomic states are optimized individually to the specific energy levels. The single electron wavefunctions with the same principal quantum numbers may differ between the atomic energy levels; the concept of the independent particle wavefunctions are restricted in the framework of the MCDF theory. However, if we trace the change in the nature of a single electron orbital with the same set of symmetry index in the atomic excitation and de-excitation processes, the concept of the single particle model will be recovered by introducing the primitive idea of the contraction or stretching of the orbital. And we can
4 d - 4 f & 4 p - 4 d Transitions of Sn12+ Ions 4d - 4f + 4p - 4d
Intensities (arb. units)
Interference considered
4 d - 4 f only
4 p - 4 d only
Wavelength (nm)
10
12
14 Constructive
16
18
20
Destructive
Interference Fig. 7. The oscillator strength distribution for 4d–4f band transitions of Sn12þ ions. Each spectrum gives a simple sum over the oscillator strengths for transitions of all the possible combinations of the members of multiplets in the excited and ground states. The calculated spectrum has been smoothed out by convoluting a natural width function of the artificial width 0.05 nm. The dashed curve represents the spectrum in which the coherent transitions of 4d–4f and 4p–4d has been taken into account. The dotted curve represents the distribution of only the 4d–4f transitions, and the solid curve represents the distribution of only the 4p–4d transitions.
ARTICLE IN PRESS F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
organize such a picture on good theoretical foundations. The key to distinguishing the single electron quantum numbers are the number of nodes in the orbital wavefunction. The MCDF method may give a good base for those pictures. The parity of the multiple electron system may be recovered by the replacement of the two single electron orbitals of opposite parities with respect to the parity of the original occupied orbitals. In the atomic ions with 4d partially filled shell, the orbitals 4s, 4p, 4d, and 4f may sometimes be almost equidistant in the single electron orbital energy. In these cases, the 4p–4d excitation and 4d–4f excitation may have almost the same energies as far as the difference of the single electron orbital energy concerned. In the framework of the MCDF calculations, these two excitations cannot be discriminated between each other with respect to the total parity and the total angular momentum of the ionic systems; they are mixed through the configuration interactions, in other words, they undergo serious interference in the course of excitation or de-excitations. We have introduced the modification of the photoemission spectra in tin atomic ions. These types of phenomena induced by the interference between the configurations with two opposite parities of the single electron orbitals are found to be quite common to the systems of highly charged ions with the atomic number from 48 to 56.
Acknowledgments This work is partly supported by the Leading Project for advanced semiconductor technology of MEXT Japan. This work is partly supported by Grant in Aid of scientific Research from JSPS Japan. One of the authors (F. Koike) would like to express his thanks to Professor Nishihara of the Institute of Laser Engineering in Osaka University for his valuable suggestions and his continuous encouragement.
References Azuma, Y., Hasegawa, S., Koike, F., Kutluk, G., Nagata, T., Shigemasa, E., Yagishita, A., Sellin, I.A., 1995. Phys. Rev. Lett. 74, 3768–3771.
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Azuma, Y., et al., 1997. Phys. Rev. Lett. 79, 2419. Churilov, S.S., 2004. Phys. Scr. 70, 126. Dyall, K.G., et al., 1989. Comput. Phys. Commun. 55, 425. Fritzsche, S., 2001. Comput. Phys. Commun. 141, 163–174. Fritzsche, S., Anton, J., 2000. Comput. Phys. Commun. 124, 354–357. Fritzsche, S., Froese Fischer, C., 1997. Comput. Phys. Commun. 99, 223–234. Fritzsche, S., Grant, I.P., 1995. Comput. Phys. Commun. 92, 111–126. Fritzsche, S., Grant, I.P., 1997. Comput. Phys. Commun. 103, 277–286. Fritzsche, S., Koike, F., Sienkiewicz, J.E., Vaeck, N., 1999. Phys. Scr. T 80, 479–481. Fritzsche, S., Froese Fischer, C., Dong, C.Z., 2000. Comput. Phys. Commun. 124, 340–353. Fritzsche, S., Froese Fischer, C., Gaigalas, G., 2002a. Comput. Phys. Commun. 148, 103–123. Fritzsche, S., Fricke, B., Gaigalas, G., Jacob, T., Tomaselli, M., 2002b. Comput. Phys. Commun. 147, 612–616. Kagawa, T., 1975. Phys. Rev. A 12, 2245; Kagawa, T., 1980. Phys. Rev. A 22, 2340. Kagawa, T., Honda, Y., Kiyokawa, S., 1991. Phys. Rev. A 44, 7092–7107. Klapisch, M., et al., 1997. J. Quant. Spectrosc. Radiat. Transfer 58, 687. Koike, F., Azuma, Y., Hasegawa, S., Nagata, T., Nishihara, K., 2005. J. Electron Spectrosc. Relat. Phenom. 144–147, 1227–1228. Koizumi, T., Awaya, Y., Gonno, M., Itoh, Y., Kimura, M., Kojima, T.M., Kravis, S., Oura, M., Sano, M., Sekioka, T., Watanabe, N., Yamaoka, H., Koike, F., 1996. J. Electron Spectrosc. Relat. Phenom. 79, 289–292. Koizumi, T., Awaya, Y., Fujino, A., Itoh, Y., Kitajima, M., Kojima, T.M., Oura, M., Okuma, R., Sano, M., Sekioka, T., Watanabe, N., Koike, F., 1997. Phys. Scr. T 73, 131–132. Lipkin, H.J., 1973. Quantum Mechanics—New Approaches to Selected Topics. North Holland, Amsterdam, p. 267. Mohr, P.J., Kim, Y.K., 1992. Phys. Rev. A 45, 2727. O’Sullivan, G., Faukner, R., 1994. Opt. Eng. 33, 3978. Oura, M., et al., 2004. Phys. Rev. A 70, 062502. Parpia, F.A., FroeseFischer, C., Grant, I.P., 1996. Comput. Phys. Commun. 94, 249. Sano, M., Itoh, Y., Koizumi, T., Kojima, T.M., Kravis, S.D., Oura, M., Sekioka, T., Watanabe, N., Awaya, Y., Koike, F., 1996. J. Phys. B 29, 5305–5313. Sasaki, A., et al., 2004. Appl. Phys. Lett. 85, 5857. Svendsen, W., O’Sullivan, G., 1994. Phys. Rev. A 50, 3710.
Radiation Physics and Chemistry 76 (2007) 404–411 www.elsevier.com/locate/radphyschem
Relativistic calculations for highly correlated atomic and highly charged ionic systems Fumihiro Koikea,, Stephan Fritzscheb a
Physics Department, School of Medicine, Kitasato University 1-15-1 Kitasato, 228-8555, Japan b Universitaet Kassel, Heinrich-Plett-Strasse, 40, 34132, Kassel, Germany Received 31 August 2005; accepted 2 October 2005
Abstract The effect of electron correlations in many electron atoms or many electron atomic ions are discussed extensively based on the MultiConfiguration Dirac Fock (MCDF) calculations. In a precision atomic physics the relativistic treatment of the system is indispensable. The correlation effects and the relativistic effects are no more additive if one want to treat the many electron systems accurately. In the excited states, the single electron orbitals are modified in accordance with the vacancies near the atomic center. The electron correlations may be evaluated from the nonorthogonality of the single electron orbitals. Several examples have been given. The electronic configurations with the same total parities may interact each other even in the cases the constituent single electron orbitals have opposite parities. Such the configuration interactions may provide us with characteristic interference structures in the optical emission or absorption spectra. The anormally of the extreme ultra-violet optical emission spectra of highly charged tin ions has been illustrated as an example. r 2006 Elsevier Ltd. All rights reserved. PACS: 32.80.t; 32.80.Fb Keywords: Atomic structure; Atomic transition; Optical emission; EUV; Relativistic theory; MCDF; Configuration interaction; Electron correlation
1. Introduction In recent years, the spectroscopic measurement in atomic physics experiments gives us quite a sophisticated set of data that requires an accurate theoretical treatment. Also in the field of plasma physics, requirement for detailed and sophisticated description of the plasma is becoming noticeable in the study of confinement nuclear fusion as well as the other plasma applications. Because a plasma model is based on the Corresponding author.
E-mail address: [email protected] (F. Koike).
atomic models that may facilitate appropriately the corresponding plasma simulations, a precision plasma model requires precision atomic data. For plasmas containing heavy elements, we must take into account the effects arising from such complex species. In Fig. 1, we illustrate the situation schematically. To obtain theoretical values with reasonable precision, we find that a fully relativistic treatment for the problem is indispensable. In fact, the value b ¼ v=c, which is the ratio of the orbital electron velocity v and the speed of light c, is about 14=1370:1 in silicon 1 s orbitals, for example. The relativistic correction could be a few percent of the total atomic state energy and also of
0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.10.044
ARTICLE IN PRESS F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
405
Multi-Configuration and Non-Local Treatment
Dirac Equation and more
Electron Correlation
Schroedinger Equation
Precise Atomic Structure Calculation
Relativistic Treatment
Local State-Independent Potentials
Fig. 1. A schematic illustration for the steps towards precision atomic structure calculations. Because the effects of electron correlations and of the relativistic behavior of atomic electrons are not additive, both the effects must be evaluated simultaneously.
the electronic transition energy if one considers the deep inner-shell excitations; the relativistic effects cannot be canceled out by subtraction of the lower state energies from the upper state ones. Large spin–orbit term splittings may also appear. The electron correlation energy is in general a few percent of the total atomic electron energy. Consideration of the electron correlations between various electronic configurations is also indispensable if we want to determine the atomic transition energies within the accuracy of several electron volts or less. And, furthermore, an important point is that we cannot, in general, discriminate between the relativistic effects and the correlation effects out of the actual atomic transition energies or excitation strengths, and sometimes the relativistic shift of the orbital energies could even enhance or suppress the electron correlation effects. In this sense, we should point out that both the relativistic and correlation effects must be treated simultaneously. In this report, we, firstly, in Section 2, discuss the characteristic properties of Dirac Hartree Fock or multiconfigutration Dirac Hartree Fock methods, which are a group of methods based on variational principles for the set of single electron orbitals. It is pointed out that the concept of a single electron orbital would have to be modified if one would optimize the atomic states individually. We briefly review extensive efforts to carry out the atomic structure calculations by a number of authors. Secondly, the electron correlations in highly excited atomic or ionic states including the deep inner shell excitations and/or multiple excitations will be reviewed. We illustrate a couple of numerical examples with the corresponding experimental data. The electron
correlation effects is important even in highly ionized atoms. In heavy elements, a pair of electrons may come close to provide us with large correlation energies. The energy of correlations of atomic electrons can be considered as typically of the order of a few electron volts. We must take into account the electron correlation effects, when the required accuracy of the atomic transition energies falls in this range. The orbital contractions in the atomic excitation may play a key role, which will be shown in Section 3. Thirdly, in Section 4, we discuss the electron correlations due to the inter-(sub)shell mixing. In general, the electronic configurations with the same total parity may interact each other even in the cases the constituent single electron orbitals have opposite parities; a configuration state function recovers its total parity after a two-electron replacement. As an example of such configuration interactions, we illustrate an anomaly in the extreme ultra-violet optical emissions of highly charged tin ions Sn12þ . Finally, in Section 5, we give concluding remarks. 2. Characteristics of the Dirac Hartree Fock methods As for the description of the Hartree Fock method, we may refer to the formalism of Lipkin (1973). We consider an N-electron wavefunction under the independent particle model. We take ayki as a creation operator of the single electron orbital. The N-electron wave function may be represented by jUi ¼
N Y i¼1
ayki j0i,
(1)
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406
where j0i represents the vacuum and the anti-symmetrization of the electron orbitals will be made if necessary. In the Hartree Fock method we apply a variational condition dhUjHjUi ¼ hUjHjdUi ¼ 0
(2)
under an appropriate ortho-normality constraint for wavefunctions. We take dyi as a small variation of ayki . Then, the variation jdUi may be represented as jdUi jU þ dUi jUi ¼
N X
ð1ÞP dyj
j¼1
N Y
ayki j0i,
i¼1ðiajÞ
(3) where P in ð1ÞP is the number of permutations that is required to place the operator dyj on top of the product. Substituting Eqs. (1) and (3) into Eq. (2), we obtain hUjHjdUi ¼
N X
ð1ÞP h0jakj jHjdyj j0i ¼ 0.
(4)
j¼1
From this equation, we see that the (D)HF method diagonalizes H with respect to the single electron displacements on the basis of independent particle view point. We also find that the atomic states that diagonalize the multi-electron displacements cannot be represented by one unique configuration state function (CSF) jUi. And, further on, we may recognize that an atomic state jUi may be optimized for each atomic state independently of the other states. The basis sets fayki g may differ between different atomic states. And, furthermore, we may note that we can obtain excited states by choosing the basis sets fayki g as appropriate; there is no restriction for the atomic state jUi to remain in the ground state in the framework of the (D)HF method. In the multiconfiguration Dirac Hartree Fock method, we define the atomic state function (ASF) in terms of configuration state functions (CSFs). We define an ASF jWm i by CSFs jUn i as jWm i
X
jUn icnm
with jUn i
n
N Y i¼1
ay ðnÞ j0i. ki
(5)
We optimize jWm i applying the variational condition that * + X d Un cnm jHjWm n
*
¼
X
+ dfUn g cnm jHjWm
n
* þ
X
+ Un dfcnm gjHjWm
¼0
ð6Þ
n
under an appropriate orthonormality constraint for the orbital basis sets, CSFs and ASF. We note here that the
ASF jWm i may be optimized individually with respect to the index m, and, therefore, orbital basis sets are not necessarily orthogonal between the sets for different m. For example, a 1s orbital in the first set jð1sÞ1 i may be a linear combination of the second set jðnsÞ2 i; we may here write as X X jð1sÞ1 i ¼ jðnsÞ2 ihðnsÞ2 jð1sÞ1 i ¼ jðnsÞ2 isn . (7) n
n
The states described by the other set {ai ð2Þ} may be represented by a state with multiple excitations on the basis of fai ð1Þg; the effect of electron correlations may be evaluated by introducing a separate optimization for individual states. However, in the calculation of atomic states, we should note that the symmetry quantum numbers are not the subject of the optimization and therefore the basis sets are always orthonormal with respect to the symmetry quantum numbers. In the relativistic calculations, the electronic Hamiltonian of the system is chosen as in the following: X X 1 H¼ hDC ðiÞ þ r i ioj ij ! X 1 ðai ri Þðaj rj Þ þ ai aj þ , ð8Þ 2rij r2ij ioj where hDC ðiÞ is the single electron Dirac Hamiltonian, and the last term is the Breit correction term. The symmetries of the ASFs are specified by their parity, total angular momentum, and its magnetic component. And they are expanded in terms of the configuration state functions (CSFs) that represent virtual single, or multiple excitations from occupied to unoccupied (sub)shells. In these couple of decades, extensive efforts have been made for computer codes to calculate the atomic structures on the basis of multiconfiguration relativistic treatment. Grant and coworkers (Dyall et al., 1989; Parpia et al., 1996) have developed a set of programs called GRASP (General Purpose Relativistic Atomic Structure Program). As an extension of GRASP, Fritzsche and his coworkers have developed a set of codes (Fritzsche et al., 2002a,b; Fritzsche, 2001; Fritzsche et al., 2000; Fritzsche and Anton, 2000; Fritzsche et al., 1999; Fritzsche and Grant, 1997; Fritzsche and Froese Fischer, 1997; Fritzsche and Grant, 1995) that calculate the atomic transition and other properties, which is called RATIP (Relativistic Atomic Transitions and Ionization Properties). Kagawa and his coworkers have developed a set of relativistic configuration interaction codes (Kagawa, 1975; Kagawa et al., 1991). Another set of multiconfiguration Dirac Fock programs has been developed by Kim and his coworkers (Mohr and Kim, 1992). And, Klapisch and his coworkers have developed a set of programs that is called HULLAC with local approximation of the exchange integrals (Klapisch et al., 1997).
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processes. They observed a pair of 1s1 2s1 ð3 SÞ 2p6 3s1 3p1 , and 1s1 2s1 ð1 S Þ2p6 3s1 3p1 double electron excitations at 924.78 eV and at about 935 eV of photon energies. Those two peaks are about 10 eV apart from each other. This value is somewhat large compared to the usual understanding of the singlet-doublet separation in neon 1s1 2s1 configurations. To find out the origin of this largeness, we have calculated the electron density distribution of the Ne2þ 1s1 2s1 2p6 configuration using both the MCDF wavefunctions of neutral neon and the doubly charged neon atomic ion. Fig. 3 shows the charge density plots from those two sets of wavefunctions. We see that in the actual Ne2þ 1s1 2s1 2p6 ions, the electronic charges are relaxed to the atomic center. It may be concluded that this contraction of the electronic wavefunctions due to the 1s2s double hole creation is the cause of the enlargement of the singlet–triplet separation in Ne 1s1 2s1 ð3;1 S Þ2p6 3s1 3p1 states.
In the rest of the present report, we introduce a couple of calculations on the remarkable phenomena relating to the electronic correlations of atoms using GRASP family codes.
3. Importance of the state dependent atomic orbital wavefunctions 3.1. The 4d–4f photoexcitation of Xeþ It is well known that there is a strong concentration of the oscillator strength in the 4d photoexcitations of xenon atoms in the photon energies around 100 eV, which is called the giant resonance. Sano et al. (1996) made a measurement on the photoionization of 4d-electrons in singly charged Xe ions. And a number of subsequent measurements have been carried out by Koizumi et al. (1997, 1996). They found a double hump structure in the photoion spectra of Xeþ and analyzed them by means of the MCDF calculation. They found that the 4f orbital suffers a strong collapse when the system has a simultaneous 5p to 6p shake up excitation. We illustrate the usual and collapsed wavefunctions in Fig. 2. Due to this 4f orbital collapse, the 4f orbital could have a strong interaction with 4d orbital which realize the double hump structure in the observed spectra. And we also note here that this type of collapsed orbital may not be expanded in terms of the nf orbital wavefunctions without collapse; the optimization of the collapsed orbital is compulsory if we want to gain an insight into the Xeþ 4d 9 4f 1 5s2 5p4 6p1 shake up excitations.
3.3. Li 3s orbital contraction triggered by K, L electron escape by photoabsorption In the last decade, there have been an extensive study of the lithium hollow atomic state excitations (Azuma et al., 1995). Azuma et al. (1997) have observed for the first time the K, L double shell hollow atomic state in lithium atoms. Fig. 4 shows their experimental spectrum with the results of an elaborate MCDF calculation. We can see that the oscillaor strength of the three electron excitations is still large to make the states experimentally observable. The largeness of the oscillator strength originates from the 3s and 3p orbital contraction in the hollow atomic states. In Fig. 5, we illustrate the change of the charge distributions from the lithium ground state to the lithium 333 resonance states. We can
3.2. Ne 1s and 2s resonant double photoexcitation Recently, Oura et al. (2004) have made an experiment on neon 1s and 2s resonant double photoexcitation
Xe + 4 d 94 f 1 5s2 5 p5 6 p0 6p
0.5
5f 6f
7f
0.0
Radial Wavefunction
4f
4d – 4 f +
-0.5 4d
6p
4f
0.5
5p – 6 p
Collapse
-1.0
Xe + 4 d 94 f 1 5s2 5 p4 6 p1
0.0 7p
-0.5 5p
-1.0
4d
0
5
407
10 Radial Distance (a0 )
15
20
Fig. 2. The collapse of the 4f orbital wavefunction due to the 5p–6p shake up excitation in Xeþ 4d photoionizations.
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WithOUT Relaxation After the Inner-Shell Hole Creation
Charge Density (1 /a0)
8
6 With Relaxation After the InnerShell Hole Creation
4 Contraction of electron orbitals
2
0
0
0.5
1 1.5 2 Radial Distance (a0 )
2.5
3
Fig. 3. The charge density plots from two sets of wavefunctions. We see that in the actual Ne2þ 1s1 2s1 2p6 ions, the electronic charges are relaxed to the atomic center.
× 10- 4 2 s 2 2p
3s 23p + 3p3
2
2s2p3s+2 s 23p
Cross Section (kb)
Oscillator Strength
3
1
15.5 15.0 14.5 14.0 13.5 13.0 175.3 12.5 174.5 175.0 175.5 176.0 Photon Energy (eV)
eV
233 175.6 eV
0 140
150
160
170
180
Photon Energy (eV) Fig. 4. Calculated oscillator strength distribution for Li hollow atomic excitations. The inset is the experimental photoion spectra around 175 eV of photon energy. The theoretical resonance peaks are convoluted with small artificial natural width and instrumental width.
see that the range of the charge density in 333 resonance is rather short compared with the ones in 223 or 233 resonance; this may give a fairly large overlap of the 333 state configuration with the ground state configuration.
4. Electron correlations due to the inter-(sub)-shell mixing In these decades, the 13.5 nm range extreme ultraviolet (EUV) light emissions of many electron atomic ions have become of a strong interest in relation to the short wavelength light source of the semiconductor lithography technologies. One of the best candidates for
such an EUV light source is considered to be of the intra N shell (n ¼ 4 shell) transitions or inter N–O shell transitions of tin (Sn) or xenon (Xe) multi-charged atomic ions. Extensive efforts to produce an accurate atomic data for the use of radiation transport calculations have been carried out by a number of authors (Sasaki et al., 2004; Koike et al., 2005; Churilov, 2004; O’Sullivan and Faukner, 1994; Svendsen and O’Sullivan, 1994). Because the range of the wavelength required by the side of the lithography optics instrumentation is very narrow, the required accuracy of the atomic structure calculation is also high accordingly. We have to evaluate, on one hand, the intra N shell electron correlations precisely for excited states as well as the
ARTICLE IN PRESS F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
409
3
Charge Density (1 /a0)
Ground State
222 Resonance
2
223 Resonance
233 Resonance
1 333 Resonance
0 0
2
4
6
8
10
12
Radial Distance (a0) Fig. 5. Charge density distributions of Li hollow atomic states.
Configuration Mixing
4f 0
4f1
81 eV
4 d3
4 d1
72 eV
4 p5
4 p6
47 eV
4 s2
4s2
E = 81 – 72 = 9 eV ~ 4% of 4 d binding energy 253 eV Fig. 6. The scheme of configuration interactions in Sn12þ 4d–4f excited states. An excited state odd parity electronic configuration 4s2 4p6 4d 1 4f 1 can be generated by replacing two even parity 4d electron orbitals by two odd parity electron orbitals 4p and 4f from a configuration 4s2 4p5 4d 3 4f 0 . If the unperturbed energies of these two configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 are not very different in values, strong interactions between these two configurations may be expected.
ground states. On the other hand, we have to notice that, in this type of ions, especially of the Sn ions, a peculiar behavior in the emission spectra has been observed by O’Sullivan and Faukner (1994). They have pointed out that the narrowing and the shift take place in the 4f –4d EUV light emission spectra. They have explained these phenomena as due to the interactions between 4p6 4d w1 4f 1 and 4p5 4d wþ1 4f 0 configurations, where w is an integer that runs from 1 to 9. This is another type of the electron correlations which are different from the ones discussed in the previous section, and we may note that those correlations may also be interesting from the view point of basic atomic physics. To gain a further insight into the effects, we have carried out careful MCDF calculations for 4d w (w ¼ 1–9) atomic ions with atomic number Z ¼ 48–56, using GRASP and RATIP family computer codes. At first,
we chose the system Sn12þ as a candidate for detailed investigation of the electron correlation effects, since the ground state of Sn12þ has 4d 2 partially filled open subshell as its outermost subshell, and the EUV light emission falls near to the range of the wavelength around 13.5 nm. In Fig. 6, we illustrate the scheme of configuration interactions that we took into consideration. An excited state odd parity electronic configuration 4s2 4p6 4d 1 4f 1 can be generated by replacing two even parity 4d electron orbitals with two odd parity electron orbitals 4p and 4f from a configuration 4s2 4p5 4d 3 4f 0 . If the unperturbed energies of these two configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 are not very different in values, we may expect strong interactions between these two configurations. In the result of an elaborate MCDF calculation, we found that in the case of Sn12þ , the energy differences between the neighboring single
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electron orbitals are quite close as indicated in Fig. 6; the difference of the energy separations between 4f and 4d, and 4d and 4p has turned to be 9 eV, which is only 4% of the 4d electron orbital energy. And, furthermore, we have found that the major peak positions of the 4s, 4p, 4d, and 4f radial wavefunctions almost coincide. The configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 have good reasons to suffer strong mixing as of the effect of configuration interactions. The optical 4p–4d and 4f –4d transitions cannot be discriminated from the others, in other words, the transitions take place coherently at the same time, providing us with quite a peculiar EUV emission spectrum. In Fig. 7, we have plotted the oscillator strength distribution for 4d–4f band transitions of Sn12þ ions. Each spectrum gives a simple sum over the oscillator strengths for transitions of all the possible combinations of the members of multiplets in the excited and ground states. The calculated spectrum has been smoothed out by convoluting a natural width function of the artificial width 0.05 nm. The dashed curve represents the spectrum in which the coherent transitions of 4d–4f and 4p–4d has been taken into account. The dotted curve represents the distribution of only the 4d–4f transitions, and the solid curve represents the distribution of only the 4p–4d transitions. We can see that a constructive interference between the 4d–4f and 4p–4d transitions is taking place at a shorter wavelength region, and also that a destructive interference is taking place at a longer wavelength region. We
can confirm the enhancement of the optical transition due to the interference effects in a wavelength region at 13 eV. We have, further, explored if any similar effects can be found in the atomic ions with different atomic numbers and charge states. We finally realized that those interference phenomena are quite common to the ions in this range; a similar effect has been observed also in Xe ions, and further on, in almost all the atomic ions with Z ¼ 48–56.
5. Discussion We have reported a couple of characteristic features of the electron correlation effects based on the multiconfiguration Dirac Fock type calculations using the atomic codes from GRASP and RATIP. In these calculations, the atomic states are optimized individually to the specific energy levels. The single electron wavefunctions with the same principal quantum numbers may differ between the atomic energy levels; the concept of the independent particle wavefunctions are restricted in the framework of the MCDF theory. However, if we trace the change in the nature of a single electron orbital with the same set of symmetry index in the atomic excitation and de-excitation processes, the concept of the single particle model will be recovered by introducing the primitive idea of the contraction or stretching of the orbital. And we can
4 d - 4 f & 4 p - 4 d Transitions of Sn12+ Ions 4d - 4f + 4p - 4d
Intensities (arb. units)
Interference considered
4 d - 4 f only
4 p - 4 d only
Wavelength (nm)
10
12
14 Constructive
16
18
20
Destructive
Interference Fig. 7. The oscillator strength distribution for 4d–4f band transitions of Sn12þ ions. Each spectrum gives a simple sum over the oscillator strengths for transitions of all the possible combinations of the members of multiplets in the excited and ground states. The calculated spectrum has been smoothed out by convoluting a natural width function of the artificial width 0.05 nm. The dashed curve represents the spectrum in which the coherent transitions of 4d–4f and 4p–4d has been taken into account. The dotted curve represents the distribution of only the 4d–4f transitions, and the solid curve represents the distribution of only the 4p–4d transitions.
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organize such a picture on good theoretical foundations. The key to distinguishing the single electron quantum numbers are the number of nodes in the orbital wavefunction. The MCDF method may give a good base for those pictures. The parity of the multiple electron system may be recovered by the replacement of the two single electron orbitals of opposite parities with respect to the parity of the original occupied orbitals. In the atomic ions with 4d partially filled shell, the orbitals 4s, 4p, 4d, and 4f may sometimes be almost equidistant in the single electron orbital energy. In these cases, the 4p–4d excitation and 4d–4f excitation may have almost the same energies as far as the difference of the single electron orbital energy concerned. In the framework of the MCDF calculations, these two excitations cannot be discriminated between each other with respect to the total parity and the total angular momentum of the ionic systems; they are mixed through the configuration interactions, in other words, they undergo serious interference in the course of excitation or de-excitations. We have introduced the modification of the photoemission spectra in tin atomic ions. These types of phenomena induced by the interference between the configurations with two opposite parities of the single electron orbitals are found to be quite common to the systems of highly charged ions with the atomic number from 48 to 56.
Acknowledgments This work is partly supported by the Leading Project for advanced semiconductor technology of MEXT Japan. This work is partly supported by Grant in Aid of scientific Research from JSPS Japan. One of the authors (F. Koike) would like to express his thanks to Professor Nishihara of the Institute of Laser Engineering in Osaka University for his valuable suggestions and his continuous encouragement.
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