12968
J. Phys. Chem. 1994, 98, 12968-12972
Photoinduced Bifurcation between Steady and Oscillatory States in the Fe(CN)64--Hz0~ Reaction in a Flow Reactor Yoshihito Mori, Gyula Rgbai,' and Ichiro Hanazaki" Institute for Molecular Science, Myodaiji, Okuzaki 444, Japan Received: July 20, 1994; In Final Fonn: August 26, 1994@
The photoinduced bifurcation structure in the Fe(CN)$--H202-H2S04 system in a continuous-flow stirred tank reactor has been determined. The photo-induced transition between the low-pH steady state and the oscillatory state shows a hysteresis, similarly to the dark reaction reported previously. The action spectrum for the photoinhibition of oscillations suggests that the primary light absorber in the visible region is Fe(CN)a3-, similarly to the photoinduction of oscillations reported previously, while in the UV region Fe(CN)a4- is also active. The primary photochemical processes are shown to be the following: F ~ ( C N ) G ~ - H2O Fe(CN)5H203CN- and Fe(CN)63- H20 !3' Fe(CN)5H2O2- CN-. The state diagram spanned by the irradiation light power and the flow rate as external control parameters is presented, on the basis of which the mechanism of oscillations is discussed qualitatively.
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1. Introduction
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Among the photosensitive reactions (1) and (2) given above, (1) is widely accepted in the literature. There is, however, a Hexacyanoferrate(II) and hexacyanoferrate(II1) complexes in controversly over the light sensitivity of Fe(CN)63-. According aqueous solution have been studied extensively on their to Asperger, reaction 2 is not accelerated by light at all, and photochemi~try~-~ and reactivity with hydrogen p e r ~ x i d e . ~ - ' ~ addition of Fe(CN)63- to the Fe(CN)64- solution reduces the The Fe(CN)64--H202 reaction is known to exhibit lightrate of reaction 1, because it acts as an internal light filter.13 sensitive, large-amplitude pH oscillations in a CSTR (continuThis is in obvious contradiction with other reports2 and with ous-flow stirred tank reactor).8 Small-amplitude pH oscillations the experimental results of the photoinduction of oscillations are also observed in a batch reactor when the reaction takes found in the reaction between F ~ ( C N ) Gand ~ - H202.I2 Results place in a slightly alkaline acetonitrile-water mixture." of batch experiments on the photoresponse of pure diluted It has been shown'O that, in the dark, the present system in a solutions of Fe(CN)a4and Fe(CN)a3will also be presented CSTR shows either a low-pH steady state (SSL) at higher flow in this paper to clarify this point. rates or a high-pH steady state (SSH) at lower flow rates. At Finally, the dynamical behavior of the system in a CSTR as medium flow rates, an oscillatory state (OSC) exists with a a function of the light power and the flow rate will be bistable region (BIS) between OSC and SSL. summarized in the form of a state diagram. On these bases, it Photoirradiation of the CSTR system has been reported to is intended to establish the reaction mechanism which would alter the amplitude of oscillations and to cause a bifurcation account for all the observed photoresponses as well as the between steady and oscillatory states at a critical value of the dynamical behavior in the dark. light intensity. Action spectrum for the photoinduction of oscillations suggests12 that the primary light absorber in the 2. Experimental Section visible region is Fe(CN)a3-, which is produced in the reactor by the oxidation of Fe(CN),j4-. Both Fe(CN)63- and Fe(CN)64Materials. Reagent grade K$e(CN)s3H20, K3Fe(CN)6, HZare found to be active for the ultraviolet illumination. These so4 (Katayama), and 30% stabilizer-free H202 solution (Mitresults suggest that the primary photochemical processes in the subishi) were used. [H202] was determined by permanganocase of the photoinduction of oscillations are reactions 1 and metric titration.14 Water was distilled and filtered before use. 2: Stock solutions of F ~ ( C N ) Gand ~ - Fe(CN)e3- were bubbled with nitrogen to purge dissolved 0 2 and C02 and were kept from Fe(CN);H,O Fe(CN),H203CN(1) light at 25.0 f 0.3 "C in a thermostat (Eyela, NlT-110) to maintain the equilibrium of the reversible aquation of Fe(CN)64and Fe(CN)63- to the monoaquacyanoferrate(IUII1) complexes. Fe(CN);H 2 0 =%Fe(CN),H202CN(2) Light Source. Monochromatic light obtained with a monochromator and a 500-W Xe lamp (Shimadzu, RF-502) was used In the Fe(CN)64--H202 system, not only photoinduction but to illuminate the reaction mixture. The spectral width of the also photoinhibition of oscillationscan be observed under certain monochromatic light was 12 nm for the construction of circumstances. The purpose of this paper is to establish the bifurcation diagrams and 10 nm for the determination of action bifurcation structure of the photoinhibition to complement the spectrum. The light intensity was varied by either a variable previous results for the photoinduction of oscillations12and to density filter (Sigma 278(2)U) or a set of fixed ND filters. The clarify whether the same or different photochemical processes variable filter was driven by a stepping motor (Oriental, are responsible for the two phenomena. For this purpose, we UPD534M-A). The light intensity was measured by a calibrated present here new results of measurements on the bifurcation photodiode (Hamamatsu, S-1723-05) at the incident window structure and the action spectrum for the photoinhibition of of the reactor. oscillations. CSTR Experiments. A double-jacket acxyl resin reactor with Abstract published in Advance ACS Absrracfs, November 15, 1994. a volume of 11.5 mL was used. It was equipped with a pH
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0022-365419412098-12968$04.50/0
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J. Phys. Chem., Vol. 98, No. 49, 1994 12969
Photoinduced Bifurcation
2
0.5
\
n" Ot
b
5 L 0
5000
10000
Time / s
Figure 1. Result of continuous scanning of the illumination light power (PO).(a) The linear scanning profile of PO at 420 nm. (b) Response of the system pH. Experimental conditions are [Fe(CN)b4-]0 = 3.30 mM, [H202]0 = 50.0 mM, [H+]o= 1.80 mM, ko = 2.46 x s-I, and temperature = 25.0 f 0.1 "C. Arrows represent the corresponding critical light powers (Pt', P y ) .
electrode (Horiba, 6366), a magnetic stirrer, a quartz window for illumination, three inlet tubes, and an ovefflow hole. &Fe(CN)6 (9.9 mM), H202 (150 mM), and H2S04 (2.7 mM) solutions were fed separately, at the same flow rates, into the CSTR with a peristaltic pump (Eyela, MP-32). In this work, the normalized flow rate (b)was used throughout, which is the sum of the three input flow rates divided by the volume of the reactor. The reaction mixture was stirred vigorously. The pH was continuously measured and the data were collected and stored on a computer (NEC, PC-9801UX) through an AD board (Contec, AD12-16F(98)). All the experiments were performed in a dark box to avoid disturbing effects of room light. The temperature of the reactor was kept at 25.0 f 0.1 "C by using a thermostat (Eyela, "IT-1300). Batch Experiments. A 1.5 x 1.5 x 5.0 cm3 quartz cell surrounded with a plastic jacket for temperature-regulated water circulation was used. The plastic wall of the jacket had a quartz window for irradiation. The reactor was equipped with a pH electrode (Horiba, 6366) and a magnetic stirrer bar. A continuous N2 stream was maintained over the solution to keep it from C02 and 0 2 .
3. Results and Discussion Bifurcation Diagram. Figure 1 shows the light-power dependence of pH of the reaction mixture for the illumination at 420 nm, where the light power is continuously scanned with a variable ND filter. The b value is chosen so that the system is at SSL (pH a 4.8) in the dark. As the illumination light power (PO)increases, the steady-state pH value increases monotonically. The system bifurcates from SSL to OSC discontinuously at a critical value (P:'), beyond which it keeps oscillating. The oscillation amplitudes decrease on increasing POfurther, and oscillations are completely inhibited at another critical value (PF),where the system bifurcates to SSH. When PO is decreased down from a high value corresponding to SSH, bifurcation takes place from SSH to OSC at P F . No hysteresis is observed at this bifurcation point. Further decrease of PO brings the system from OSC into SSL at Pg3, which is much lower than P:'. Hence there is clearly a hysteresis at this bifurcation point. The existence of hysteresis is confirmed further in a separate experiment shown in Figure 2, where PO
0
5000
10000
Time 1 s
Figure 2. Confirmation of hysteresis in the neighborhood of the subcritical bifurcation point. Experimental conditions are the same as those in Figure 1. In this case, PO is decreased from a higher value down through the critical point and then increased back to the initial value. Ptl and P y stand for the two critical values of PO,corresponding to the hysteresis.
Po/ mW
Figure 3. Bifurcation diagram for the Fe(CN)64--H202-H2S04 system constructed on the basis of Figures 1 and 2.
is decreased from OSC to SSL and then increased back to OSC. With decreasing PO,oscillations cease at PO = PF,while with increasing PO,they start at PO = P:'. The basic question that should be addressed in this context is whether the observed hysteresis is due to the slow response of the system to the scanned light power or it is a genuine chemical phenomenon. One should keep in mind that the scanning rate applied here is very slow, and the system should have enough time to respond. Nevertheless additional experiments were performed with fixed light power at several points around P:' and PF. The existence of the light-assisted chemical hysteresis was confirmed by these experiments. A bifurcation diagram is shown in Figure 3. The curves between PF and Pt2 represent the minimum and maximum pH values of oscillations. The arrows show the points where the system starts to oscillate (at P:') for increasing light power and where oscillations cease (at PF) for decreasing light power. It is clearly seen that the photoinduced transition between SSL and OSC has a bistable region with a hysteresis and the initiation and termination of oscillations occurs abruptly. This bifurcation may therefore be assigned to the subcritical Hopf bifurcation. On the other hand, the transition between OSC and SSH shows no hysteresis and the oscillation amplitude seems to decrease gradually to zero while PO increases to
12970 J. Phys. Chem., Vol. 98, No. 49, 1994 81
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Mori et al.
I
2.0
t
1.5
$
1.0
d 3
0" 0.5
t
0.0
Po""
u 0.1
5~
c
0.2
Po I mW
Figure 4. Bifurcation diagram for the Fe(CN)64--H~0~-H~S04 system. Experimental conditions are the same as those in Figure 3, except for ko = 1.65 x s-'. Ps4 is the critical light power. 0.8
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1
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1
1
1
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1
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1
Figure 6. Wavelength dependence of the relative cross section for the photoinhibition of oscillations in the Fe(CN)64--H20~-H~S04 system. Measurements are made at the bifurcation point between SSH and OSC for each wavelength. Open circles, the relative cross section (uR);curve (a), absorption spectrum of the reaction mixture; curve (b), absorption spectrum of the hexacyanofenate(III) solution; curve (c), absorption spectrum of the hexacyanoferrate(JI) solution. The concentrations for curves (b) and (c) are chosen so that the sum of optical densities for (b) and (c) gives curve (a).
to examine if the same primary process is responsible for the photoinhibition of oscillations. Figure 6 shows the action spectrum at the bifurcation point corresponding to the photoinhibition of oscillations at the OSC SSH transition. The relative cross section (OR)is defined by the following equation:
-
3 0.4 -
. a" E
15
0.2-
SSL
u 1
Wavelength / nm
1.5
2
2.5
Figure 5. State diagram spanned by the illumination light power (PO) and the flow rate (h). Closed circles represent the bifurcation points measured in the present work. Open triangles, open circles, closed rectangles, and closed triangles represent a high-pH steady state (SSH), an oscillatory state (OSC), a low-pH steady state (SSL), and a bistable state between SSL and OSC, respectively, in the dark.'O
P y , where a transition from OSC to SSH takes place, suggesting that it is the supercritical Hopf bifurcation. Figure 4 shows a bifurcation diagram for a different flow rate, where the system oscillates in darkness. As PO is increased, the amplitude decreases gradually and oscillations cease at P y , which corresponds to the supercritical Hopf bifurcation. A state diagram is shown in Figure 5 in the plane spanned by POand b. The diagram includes points obtained previously in the dark experiments.'O If POis increased from SSL to OSC while lq, is kept constant, oscillations start at PO beyond the border of BL2. In order for the system to stay in SSL while PO is increased further, k0 has to be increased as indicated by arrows at point (a). This means that increasing PO has qualitatively the same effect as decreasing h;namely, the effect of increasing light intensity can be compensated by increasing b. Similarly, decreasing POat (b) or (c) has the same effect as increasing b. On the basis of these observations, we can conclude for both of the photoinhibition and photoinduction of oscillations that the system can show the same dynamical behavior under illumination as in darkness only if the higher rates of chemical reactions under illumination are compensated with increased flow rate. In our previous paper,12 we identified the primary light absorber at the bifurcation point corresponding to the photoinduction of oscillationsfor SSL (BIS) OSC. It is now required
-
uR= ( / ~ D / P E ) [~ exp(-2.303~)1-'
(3)
where hv is the photon energy, PE is the critical light power, and D is the optical density of the reaction mixture at the bifurcation point. Peaks at 420 and 300 nm and a shoulder at 320 nm in the action spectrum coincide with the absorption spectrum of F~(CN)G~-. This suggests that Fe(CN)63- could be the primary light absorber for the bifurcation in both visible and ultraviolet regions. The spectrum of Fe(CN)64- also shows a broad shoulder at 300-340 nm. Although the absorptivity Of is much smaller than that of Fe(CN)63-, the former is known to be kinetically more active. Therefore the contribution of Fe(CN)64- to the light sensitivity can be significant in the UV region. In the case of photoinduction of oscillations (SSL(B1S) OSC),12we have performed the prereactor illumination experiments, where either pure Fe(CN)64- or Fe(CN)63- solution is illuminated in a prereactor before they are introduced into the CSTR. Both Fe(CN)64- and Fe(CN)a3- were found to be effective in the photoinduction of oscillations. The same may apply to the case of photoinhibition of oscillations. To clarify further this point, we studied the photoresponse of pure F~(CN),S~and Fe(CN)63- in a batch reactor. Batch Experiments. When reactions 1 and 2 take place in an unbuffered neutral aqueous solution, the pH rises because the protonation of the released CN- occurs below pH 9.5. We have confirmed this in a batch reactor as shown in Figure 7. This result also supports the effects of illumination in a prereactor mentioned above for the photoinduction of oscillations; namely, Fe(CN)sH203- or Fe(CN)5H2O2- produced upon illumination in the prereactor flows into the main CSTR, causing oscillations. GAsp6r and B e ~ kmeasured ~ ? ~ the pH change in the irradiated 0.1 M Fe(CN)64- solution. We have confirmed their results and found that the pH approaches a limiting value (pH 9.5) under continuous illumination when [Fe(CN)64-] is above 0.06 M. However, at lower concentrations, which apply to the oscillatory system, an interesting overshoot on the pH-
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Photoinduced Bifurcation 9.5
J. Phys. Chem., Vol. 98, No. 49, 1994 12971 for the photoresponse of the system. The aquapentacyanoferrate(J.I)/(III) pair has been proposed to be the key species in this oscillatory reaction as they are involved in a crucial autocatalytic redox cycle (7)-(13):
1
Fe(CN)5H,02- i- Fe(CN);Fe(CN),H203- i- Fe(CN);-
-
Fe(CN),H2O3- iH202
i -
Fe(CN)5H202-
65
50
H,O,
tlme/mln 150
Figure 7. Effect of illumination in a batch reactor. Illumination at 360 nm. PO = 1.58 mW, T = 25 "C. The light was switched on and off at the arrows. Solid line, 10-mL solution of 3.3 mM Fe(CN)64-;
Fe(CN)5H,03-
Fe(CN)5H,02-
+ H+
(4)
However, its response is slower and does not exhibit any overshoot-undershoot phenomenon. Furthermore, the pH change upon switching on and off the light is not completely reversible. The original pH measured in darkness can never be reached again, and the solution looses its light sensitivity under prolonged (several hours) irradiation with light power given in Figure 7. This can be the reason for the contradictory reports on the light sensitivity of Fe(CN)a3-. Mechanistic Considerations. The overshoot-undershoot behavior observed in a batch reactor when the Fe(CN)a4solution is illuminated suggests that there must be, in addition to the photoaquation (reaction l), another composite reaction that affects the pH. It is obvious that any further aquation of the Fe(CN)sH2O3- complex, formed in reaction 1, cannot account for the pH decrease, because the released CN- increases rather than decreases the pH. It has, however, been known that aquapentacyanoferrate(II) tends to dimerize in a slow reaction. (5)
2Fe(CN),H,03- t Fe2(CN),,(H,0),6-
The dimer may be involved in an acid dissociation equilibrium (6). After the rapid increase of pH by (l), (6) would slowly decrease it as the dimer accumulates under illumination. When the light is turned off, the reverse of both reactions (1) and (6) takes place to give rise to the undershoot of pH. Fez(CN),o(H,0),6- t Fez(CN),o(H,0)(OH)7-
+ H+
(6)
In the following, we shall examine the reaction scheme proposed previously8Jo~12 and try to establish a unified view
+ H,O
+ HO,' + H+
Fe(CN):-
Fe(CN),0H3-
HO,'
Fe(CN),H,O*-
dashed line, 3.3 mM Fe(CN),j3-.
time curve was observed under continuous illumination (solid line in Figure 7). First, the pH rises rapidly then decreases slowly to a photostationary state. After switching off the light the pH goes through a minimum and returns to its original dark value, indicating that the photoinduced reaction is completely reversible in this case. This behavior was observed between 0.002 and 0.06 M complex concentrations, using 360-nm UV light with different PO. We found that Fe(CN)a3- also responds to irradiation with an increase in pH as shown in Figure 7 (dashed line). The aquapentacyanoferrate(II1) complex is involved in acid-base equilibrium (4) with a pK4 = 8.4 at 25 "C, which limits the rise in pH at around pH 8.5.
+ OH'
+ OH' + OH-
Fe(CN);Fe(CN);-
(7)
(8) (9)
+ 20H'
+ OH' -Fe(CN);- + OH-I-H,O Fe(CN),H203- + CN+ H,O Fe(CN)5H202- + CNf
(10) (1 1) (12)
(13)
where (12) and (13) are the "dark" versions of (1) and (2), respectively. The concentrations of aquapentacyano complexes are obviously important in this cycle. The autocatalytic formation of OW accelerates steps @)-(lo) when [Fe(CN)5H203-] is relatively high and terminates when this concentration decreases dramatically because of its consumption by (10). After the autocatalytic cycle has come to an end because of the lack of Fe(CN)gH@, reaction 11 eliminates OH', and Fe(CN)$1203can accumulate again through (12) and (13) and the interconversion reaction (7), creating favorable conditions for a new autocatalytic cycle. First we consider the dark reaction on the basis of this scheme. When the reaction is run in a CSTR at relatively high l ~ the , system stays in SSL (low pH, high [H+] steady state), since there is not enough time for the reaction to build up the autocatalytic cycle; production of aquapentacyano complexes by (12) and (13) is slow in the dark, and the steady concentration of Fe(CN)5H203- is below the threshold for the autocatalysis to occur. When ko is too low, supply of [H+] cannot compete with its consumption by (l), (2), (8), and (1 1). Therefore [H+] e 0 and autocatalytic process (10) cannot occur. The system stays in SSH (high pH, low [H+] steady state). Oscillations occur for moderate k~ values. When ko is increased in SSH, the supply of [H+] overcomes it exceeds the its consumption, and at a certain value of ko threshold for the autocatalysis. However, at this stage, the supply of [H'] is still not sufficient so that the upper limit of [H+] for each oscillation is low, which limits the oscillation amplitude. As ko increases, this limit increases with increase of amplitude, while the lower (high pH) limit remains nearly constant ([H+] e 0). This accounts for the supercritical nature of the bifurcation. On the other hand, when ko is decreased in SSL, [Fe(CN)5HzO3-] increases because of the decreasing rate of its flow-out. At a critical value g, it reaches the threshold for the autocatalysis to start. In contrast to the bifurcation at the system contains sufficient [H+] at the critical point so that oscillations start with a large amplitude, giving a subcritical nature to this bifurcation. Existence of hysteresis may be understood as follows; when is decreased in SSL, [Fe(CN)5H2O3-] is determined by the competition of its flow-out and the production due to (12) or
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12972 J. Phys. Chem., Vol. 98, No. 49, 1994
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(13) (11). When is increased in the OSC region, the sustained oscillations enhance the production of Fe(CN)5H202through the reverse reaction of (7). At the peak of an oscillation where Fe(CN)sH203- is consumed by the autocatalytic step, Fe(CN)sH2O2- serves as a reservoir to regenerate Fe(CN)5H203through (7) in addition to its generation by (12), making the rate of production of Fe(CN)5H203- higher than that in the case of decreasing k~from SSL. This means [Fe(CN)5H2O3-] is still above the threshold at ko = kk; a higher value of k~ = (>koH) is required to stop oscillations. Now let us consider the photoinduced bifurcations. In the case of Figure 3, the system is in SSL in the dark, where [Fe(CN)5H2O3-] is low. Illumination enhances its production through (1) or (2) (7) to bring the system to subcritical bifurcation. This brings about the same effect as decreasing k~ in the dark reaction. A further increase of PO increases [Fe(CN)sH203-], which enhances the consumption of [H+] through (8) and (10). The oscillation amplitude decreases gradually and finally the system reaches the supercritical bifurcation point where [H+] FZ 0. Existence of hysteresis may also be understood consistently with the dark reaction; when POis increased in SSL, [Fe(CN)5H203-] is low and [Fe(CN)5H202-] is also low. It is required to increase PO to PE' to reach the threshold. When POis decreased in OSC, the sustained oscillations may produce additional Fe(CN)5H2O2- by (lo), which can be a source of regeneration of Fe(CN)5H203-. Therefore, higher [Fe(CN)5H203-] is expected when PO is decreased from OSC to SSL,which makes the threshold power lower than P?. The dimerization (6) of aquapentacyanoferrate(II), which is thought to take place under continuous illumination, may also be important in the oscillatory mechanism, especially in explaining the hysteresis behavior, because the dimer can also act as a source of regeneration of [Fe(CN)5H2O3-]. However, no sufficient experimental data are available yet to discuss the possible contribution of reaction 6.
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4. Conclusion In this paper, we have determined the bifurcation structure for photoinhibition of oscillations in the Fe(CN)a4--H&-
H2SO4 system in a CSTR, which complements our previous report on the photoinduction of oscillations.8 On the basis of these results, a state diagram has been established for this system, taking the light power and the flow rate as control parameters. Action spectrum for the photoinhibition of oscillations shows that Fe(CN)a3- is the primary light absorber, while Fe(CN)64- may also be active in the ultraviolet region. On the basis of this and the reaction scheme proposed previously for the photoinduction case,8 it has been shown that the photoresponse of the system can be understood in a unified manner for both photoinduction and -inhibition as well as the bifurcation in the dark reaction.'O Although the scheme is qualitative at present, it seems to account for the observed behavior of the system reasonably. We are trying to make a simulation calculation on the basis of this scheme to c o n f i i that the model is also useful in a quantitative treatment.17
Acknowledgment. We thank Mr. Okazaki, a student of the Graduate University for Advanced Studies, for his technical assistance in the data processing and continuous scanning measurements. References and Notes (1) On leave from Institute of physical Chemistry, Kossuth Lajos University, H-4010 Debrecen, Hungary. (2) Adamson, A. W.; Waltz, W. L.; Zinato, E.; Watts, Z. D.; Fleischauer, P. D.; Lindholm, R. D. Chem. Rev. 1968,68,541. (3) GASP&, V.; Beck, M. T. Magy. K2m. Folydirat 1982,88, 433. (4) GASP&, V.; Beck, M. T. Polyhedron 1983,2, 387. (5) Barb. W. G.: Baxendale.. J. H.: K. R. Trans. . George. - .P.: .Harerave. Far&y SO~.' 1951,47,591. (6) Brav. D. G.: Thomuson. R. C. Inora. Chem. 1994,33, 905. (7) Rabai, G.;Kustin, K.;Epstein, I. R.2. Am. Chem. Soc. 1989,111, 3870. (8) Rabai, G.;Kustin, K.; Epstein, I. R. J. Am. Chem. Soc. 1989,111, 8271. (9) Mori, Y.;Srivastava, P. K.; Hanazaki,I. Chem. Lerf. 1991, 669. (10) Mori, Y.; Hanazaki, I. J. Phys. Chem. 1993,97,7375. (11) Rabai, G. J. Chem. SOC., Chem. Commun. 1991,1083. (12) Mori, Y.; Hanazaki, 1. J . Phys. Chem. 1992,96,9083. (13) Asperger, S. Faraday Trans. 1952,48, 617. (14)Treadwell, F. P. Analytical Chemistry, 8th ed.; Hall, W. T., translator; John Wiley & Sons: New York, 1935; Vol. 2, p 570. (15) Hanazaki, I. J . Phys. Chem. 1992,96,5652. (16)Davis, G.; Garafalo, A. R. Inorg. Chem. 1976, 15, 1101. (17) Mbai, G.; Mori, Y.; Hanazaki,I. Manuscript in preparation.
