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Laser study of no-phonon lines in the inhomogeneously broadened spectra via photochemical hole burning
Journal oil un~ines~eji~e 31 & 32(1984) 744 749
744
“lot tlt .lloIlan 3 Amsterdam
LASER STUDY OF NO PHONON LINES IN THE INHOMOGENEOUSLY BROADENED SPECTRA VIA PHOTOCHEMICAL HOLE BURNING Karl K. Rebane Institute of Physics of the Estonian SSR Academy of Sciences, 202400 Tartu, 142 Riia Str. , USSR When inhomogerieous broadening is eliminated, the no-phonon lines in spectra (absorption, luminescence, hot luminescence, light scattering, excitation of luminescence, hole burning) actually become the optical analogs of the MUss bauer y resonance lines of very high resolution and sensitivity. Photochemi cal hole-burning is an effective method to eliminate the inhomogeneous broadening and to perform high resolution frequency-selective photochemistry. Photochemical hole burning by cw-lasers provides high resolution spectro scopy of molecules and new possibilities of high capacity data storage; by picosecond pulses storage of their temporal structure, tine and-space domain holography, photochemically accumulated photon echo phenomena. 1. INTRODUCTION The no-phonon line, especially the purely electronic no phonon line (PEL),ir the optical spectra of impurities in olid matrices is an interesting and useful phenomenon in the luminescence, spectroscopy, physics and chemistry of molecules and solids. PEL5 obtain very narrow homogeneous linewidths and high peak intensities at helium temperature’. It holds not only for impurity ions or atoms in single crystals, but also for large nolecules in different solid 3.The presence of PEL matrices (polycrystals, glasses, polymer films) as well~ (the optical analog of the Mbssbauer line”46’1) in the spectra provides new possibilities for high resolution spectroscopy and frequency selective photo chemistry (see e.g. 2, 3, 6, 8 and references therein). Because of their very snall 1inuwidth~ thu nu—ph.rriun linus, eper..ially th~PELs, ate ux~r’umely suns; tive to inhomogeneities in the matrice structure. The inhomogeneous broadening (iB) is tremendous in comparison with the low temperature homogeneous line widths 1(T). The latter approach electronic lifetime determined width at T—O, i.e. (O)~10 3:iO ~ cm1 for allowed non quenced purely electronic transitions and even less for the forbidden ones’. iB is the reason why the PELs and vibro— nic no-phonon lines often do not show up in conventional spectra at all. Even in ab sorption and luminescence spectra with well pronounced vi broni c structure measured under conventional broad line excitation (e.g. the Shpolskii spectra9’1) the line widths of PELs are because of iB about 1cm’, i.e. 1O3 1O~times broader than the corresponding homogeneous linewidths U. The enormous lB of PELs is a serious 0022 23I3/84/S03.000 Elsevier SLience Publishers By. (North Holland Physics Publishing Division)
K.K. Rebane / Laser stud3’ ofno-phonon lines
745
disadvantage in comparison with the Mössbauer spectroscopy and this is the very reason why for a quite long time the extremely small linewidths and high peak intensities of homogeneous PEL5 were considered to present purely theoretical interest. The last decade of development in laser spectroscopy has provided two effective and relatively simple* methods to get rid of the iB: the site-selec1 and the photochemical hole burning in tive excitation of fluorescence’°” spectra’2”3 (more precisely — in the inhomogeneous distribution function of the PELs of the impurity molecules2’7). The photochemical hole burning (PHB) provides new technological applications such as narrow-line optical filters and information storage (sme 7, 8 and refern~nces therein). Sn it appears that in these aspects the inhoniogeneous broad bands present a very useful feature of the impurity spectra of solids. 2. PHOTOCHEMICALLY ACCUMULATED STIMULATED PHOTON ECHO There are several review papers on the rapidly growing field of studies and applications of PHB 14,15,3,7,8,16 In full accordance with the basic principles of theoretical physics the spectra are the responses of the matter, radiation and detectors to the stimula applied for a certain measurement. Spectra depend crucially on the way of excitation, on the properties of the spectral device, on the choice of the timewindow to detect them (on time-dependent optical and MUssbauer spectra see 7 and references therein). The best example of the possibilities to create spectra using the sources of coherent radiation is NMR studies. Lasers give us the possibility to create a great variety of optical spectra and PHB may be considered as one of the methods to deal with inhomogeneous spectra. Various developments of this method are possible. A method based on picosecond laser excitation has recently been developed1119. In conventional frequency—spectra (PHB) techniques sharp single holes are burned in inhomogeneously broadened impurity absorption bands using near-monochromatic radiation of CW lasers. Later afterwards, within their lifetime, the holes are studied to get data on the homogeneous spectra. In the time-domain approach PHB is carried out by sequences of picosecond pulses having the spectral width about 3—5 Ps, i.e. much larger than the homogeneous widths of no-phonon lines. In the case of one pulse the intensities of its Fourier-components are fixed by P1-tB, the phases of components are lost. In the case of two—pulse excitation one can see that if T relaxation time and t
-
2t (T2 is the phase delay between the two pulses) the summary intensity has
There are variations of these two and several much more sophisticated non-linear methods. *
746
A A. Rchain’
0onon (tort
Lan r turlr o/ ito p
a contribution from the interference. This interference leads to a sinusoidal modulation of the spectral intensity of PHB and of the sample transparency spectrum.
Passing a third probing pulse through the sample results in delayed
echo pulses stimulated from the burned in persistent ground state population spectral grating.
50
40
______________
042 crnT1
~
10
~O5
00
;J~26Ds36oA
WAVE NUMBER
FIGURE 1 Transparency spectrum of the sample (prepared by polymerization of porphyrazirte styrol solution) after 20 mU/cm2 dose irradiation with picosecond laser beam passed through a 12 mm basis Fabry-Perot etalone. Insert the envelope of the hole measured with a rough 1 A resolution; main frame - fine structure of the hole, measured by scanning a 0.075 cm 1 (FWHM) dye laser. This phenomenon, called by the authors~19 the photochemically accumulated stimulated photon echo (PASPE), related to three—pulse stimulated photon echoes20 displays several special features. First, while the stimulated photon echo can occur only within the relaxation time limits of light-induced transient frequency-space population gratings, the lifetime of PASPE is determined by the very long lifetime of PHB photoproducts. This allows the spectral gratings in PASPE to be accumulated to a very high contrast, which in turn results in very high relative intensity echo signals (up to about 50 per cent). Secondly, due to the accumulation effect, PASPE experiments can be performed well under modest linear excitation and in first approximation a simple theory
K.K. Rebane /Laser study of no-phonon lines
747
relying on linear filtration theory can be estab21. lished In experimental investigations18 of PASPE a synchronously pumped pico— second laser was applied ~
:
x20
.
to generate echo signals
•
in porphyr-type low-ternperature impurity systems. Spectral gratings were first burned by repeated weak picosecond multipulse
• •
z
HPB and echoes were after-
I
/
0
100
probing beam through the
~TIME -‘ -~ ,-
wards detected by passing a still weaker picosecond
.~.
(psi
-‘-.-.
.-
---
--
sample and focussing it on the entrance slit of a synchroscan streak camera system. Both spectral and temporal responses of the
FIGURE 2 Streak image of a single picosecond probe pulse after travelling through the sample prepared as described in Fig. 1. The passed through probe pulse (upper left) is followed by two stimulated echo pulses. For time reference a fragment of the PHB pulse sequence is also presented.
sample were measured (Fig. 1 and 2 From the dependence of the echo si nals intensit g on the delay between the picosecond HB pulses phase
relaxation time can be estimated. In the case of 2.5x104 M octaethylporphin in 1.8 K polystyrol matrix T 2 was about 300 ps (Fig. 3). Now let us suppose the PHB sequence to comprise a signal pulse with an arbitrary temporal and spatial amplitude and phase distribution and a plane wave reference pulse which is short enough to be considered a a-pulse compared to the signal. It was demonstrated in 21 that if T2>>t exhaustive temporal and spatial information about the signal can be stored in the sample by PHB and later reproduced by PASPE. Again within the limits of weak excitation a linear approach is valid and the formalism of ordinary holographic image storage can well be extended 22.to the field of photosensitive materials with very high spectral resolution
A A. Rehane / La~r studt’ 01 no—pltonon lutes
748
FIGURE 3 Streak images showing the PHB sequence (a) and resulting PASPE after (c) 0.5 2 (b) signals 1 5 mU/cm2 mU/cm and 2.5 mU/cm2 (d) exposures. The sample was prepared by polymerization of octaethyl porphin 2.5M styrol solution.
0
100
200
300 ps
3. CONCLUDING REMARKS 1. Photochernical hole burning in spectroscopy has an essential advantage in comparison with the site-selective spectroscopy: the hole burning and its study may be done with the same laser, no additional spectral device is needed 22 2. By taking optically thick samples (and losing in transparency of holes) we can create holes narrower than the linewidth of the hole burning laser. We can approach the natural linewidth of the purely electronic line. 3. To study the shapes of these sharp holes (narrower than the narrowest laser linewidth in our experiment) we can apply Doppler shift measurements in a way quite similar to MUssbauer methods and no laser is needed to study the holeshapes. For 0.01 cm1 holewidths the velocities about tens of msec’ are needed23. 4. In some tasks fast photoburning and detection may be decisive. In our laboratory evidence has been obtained that the holewidths for molecules in glassy matrices deorease with decreasing holeburning+detection time. ACKNOWLEDGEMENT The author is grateful to P.M. Saari and A.K. Rebane for many stimulatino discussinos.
K.K. Rebane / Laser studt of no-phonon lines
749
REFERENCES 1) K.K. Rebane, Impurity Spectra of Solids (Plenum Press, New York, 1970). 2) K.K. Rebane, Zh. Prikladnoi Spektroskopii 37 (1982) 906 (in Russian). 3) R.I. Personov, in: Spectroscopy and Excitation Dynamic on Condensed Molecular Systems, eds. V.M. Agranovich and R.M. Hochstrasser (North Holland, Amsterdam, 1983) p. 55. 4) E.D. Trifonov, Doklady Akademii Nauk SSSR 147 (1962) 826 (in Russian). K.K. Rebane and V.V. Hizhnyakov, Optika i Spektroskopija 14 (1963) 491. 5) E.F. Gross, B.S. Razbirin and S.A. Permogorov, Doklady Akademii Nauk SSSR 154 (1964) 1306 (in Russian). 6) RH. Silsbee and D.B. Fitchen, Rev. Mod. Phys. 36 (1964) 433. 7) K.K. Rebane, Proc. Int. Conf. Lasers’82 (New Orleans, Lousiana, Dec. 13-17, 1982; 1983) p. 340. 8) A. Szabo, Proc. Int. Conf. Lasers 80, New Orleans, Lousiana, Dec. 15-19, 1980, ed. C. Collins (STS Press, McLean, 1981) p. 374. 9) E.V. Shpolskii, Uspekhi Fiz. Nauk 71 (1960 215; 77 (1962) 32; 80 (1963) 255 (in Russian). 10) A. Szabo, Phys. Rev. Lett. 25 (1970) 924. 11) R. Personov, E. Al shits and L. Bykovskaya, Opt. Commun. 6 (1972) 169. 12) A.A. Gorokhovski, R.K. Kaarli and L.A. Rebane, UETP Lett. 20 (1974) 216. 13) B.M. Kharlamov, R.I. Personov and L.A. Bykovskaya, Opt. Commun. 12 (1974) 191. 14) L.A. Rebane, Zh. Prikladnoi Spektroskopii 34 (1981) 1023 (in Russian). 15) L.A. Rebane, A.A. Gorokhovski and U.V. Kikas, Appl. Phys. B29 (1982) 235. 16) U. Friedrich and D. Haarer, Angewandte Chemie 23 (1984) 113. 17) A.K. Rebane, R.K. Kaarli and P.M. Saari, Optika i Spektroskopija 55 (1983) 405 (in Russian); U. Mol. Structure 114 (1984) 343. 18) A.K. Rebane, R.K. Kaarli, P.M. Saari, A. Anijalg and K. Timpmann, Optics Commun. 47 (1983) 173. 19) A.K. Rebane and R.K. Kaarli, Chem. Phys. Lett. 101 (1983) 317. 20) W.H. Hesselink and D.A. Wiersma, in: Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, eds. V.M. Agranovich and R.M. Hochstrasser (North Holland Publishing Co., 1983). 21) P.M. Saari and A.K. Rebane, Eesti NSV Tead. Akad. Toimet., FUUs. Matem. in print (in Russian). 22) S. Voelker, R.M. Macfarlane and J.H. van derWaals, Chem. Phys. Lett. 53(1978)8. 23) K.K. Rebane and V.V. Palm, Optika
Spvktroskupija 57 (1984) 381 ( ri Russ an).
744
“lot tlt .lloIlan 3 Amsterdam
LASER STUDY OF NO PHONON LINES IN THE INHOMOGENEOUSLY BROADENED SPECTRA VIA PHOTOCHEMICAL HOLE BURNING Karl K. Rebane Institute of Physics of the Estonian SSR Academy of Sciences, 202400 Tartu, 142 Riia Str. , USSR When inhomogerieous broadening is eliminated, the no-phonon lines in spectra (absorption, luminescence, hot luminescence, light scattering, excitation of luminescence, hole burning) actually become the optical analogs of the MUss bauer y resonance lines of very high resolution and sensitivity. Photochemi cal hole-burning is an effective method to eliminate the inhomogeneous broadening and to perform high resolution frequency-selective photochemistry. Photochemical hole burning by cw-lasers provides high resolution spectro scopy of molecules and new possibilities of high capacity data storage; by picosecond pulses storage of their temporal structure, tine and-space domain holography, photochemically accumulated photon echo phenomena. 1. INTRODUCTION The no-phonon line, especially the purely electronic no phonon line (PEL),ir the optical spectra of impurities in olid matrices is an interesting and useful phenomenon in the luminescence, spectroscopy, physics and chemistry of molecules and solids. PEL5 obtain very narrow homogeneous linewidths and high peak intensities at helium temperature’. It holds not only for impurity ions or atoms in single crystals, but also for large nolecules in different solid 3.The presence of PEL matrices (polycrystals, glasses, polymer films) as well~ (the optical analog of the Mbssbauer line”46’1) in the spectra provides new possibilities for high resolution spectroscopy and frequency selective photo chemistry (see e.g. 2, 3, 6, 8 and references therein). Because of their very snall 1inuwidth~ thu nu—ph.rriun linus, eper..ially th~PELs, ate ux~r’umely suns; tive to inhomogeneities in the matrice structure. The inhomogeneous broadening (iB) is tremendous in comparison with the low temperature homogeneous line widths 1(T). The latter approach electronic lifetime determined width at T—O, i.e. (O)~10 3:iO ~ cm1 for allowed non quenced purely electronic transitions and even less for the forbidden ones’. iB is the reason why the PELs and vibro— nic no-phonon lines often do not show up in conventional spectra at all. Even in ab sorption and luminescence spectra with well pronounced vi broni c structure measured under conventional broad line excitation (e.g. the Shpolskii spectra9’1) the line widths of PELs are because of iB about 1cm’, i.e. 1O3 1O~times broader than the corresponding homogeneous linewidths U. The enormous lB of PELs is a serious 0022 23I3/84/S03.000 Elsevier SLience Publishers By. (North Holland Physics Publishing Division)
K.K. Rebane / Laser stud3’ ofno-phonon lines
745
disadvantage in comparison with the Mössbauer spectroscopy and this is the very reason why for a quite long time the extremely small linewidths and high peak intensities of homogeneous PEL5 were considered to present purely theoretical interest. The last decade of development in laser spectroscopy has provided two effective and relatively simple* methods to get rid of the iB: the site-selec1 and the photochemical hole burning in tive excitation of fluorescence’°” spectra’2”3 (more precisely — in the inhomogeneous distribution function of the PELs of the impurity molecules2’7). The photochemical hole burning (PHB) provides new technological applications such as narrow-line optical filters and information storage (sme 7, 8 and refern~nces therein). Sn it appears that in these aspects the inhoniogeneous broad bands present a very useful feature of the impurity spectra of solids. 2. PHOTOCHEMICALLY ACCUMULATED STIMULATED PHOTON ECHO There are several review papers on the rapidly growing field of studies and applications of PHB 14,15,3,7,8,16 In full accordance with the basic principles of theoretical physics the spectra are the responses of the matter, radiation and detectors to the stimula applied for a certain measurement. Spectra depend crucially on the way of excitation, on the properties of the spectral device, on the choice of the timewindow to detect them (on time-dependent optical and MUssbauer spectra see 7 and references therein). The best example of the possibilities to create spectra using the sources of coherent radiation is NMR studies. Lasers give us the possibility to create a great variety of optical spectra and PHB may be considered as one of the methods to deal with inhomogeneous spectra. Various developments of this method are possible. A method based on picosecond laser excitation has recently been developed1119. In conventional frequency—spectra (PHB) techniques sharp single holes are burned in inhomogeneously broadened impurity absorption bands using near-monochromatic radiation of CW lasers. Later afterwards, within their lifetime, the holes are studied to get data on the homogeneous spectra. In the time-domain approach PHB is carried out by sequences of picosecond pulses having the spectral width about 3—5 Ps, i.e. much larger than the homogeneous widths of no-phonon lines. In the case of one pulse the intensities of its Fourier-components are fixed by P1-tB, the phases of components are lost. In the case of two—pulse excitation one can see that if T relaxation time and t
-
2t (T2 is the phase delay between the two pulses) the summary intensity has
There are variations of these two and several much more sophisticated non-linear methods. *
746
A A. Rchain’
0onon (tort
Lan r turlr o/ ito p
a contribution from the interference. This interference leads to a sinusoidal modulation of the spectral intensity of PHB and of the sample transparency spectrum.
Passing a third probing pulse through the sample results in delayed
echo pulses stimulated from the burned in persistent ground state population spectral grating.
50
40
______________
042 crnT1
~
10
~O5
00
;J~26Ds36oA
WAVE NUMBER
FIGURE 1 Transparency spectrum of the sample (prepared by polymerization of porphyrazirte styrol solution) after 20 mU/cm2 dose irradiation with picosecond laser beam passed through a 12 mm basis Fabry-Perot etalone. Insert the envelope of the hole measured with a rough 1 A resolution; main frame - fine structure of the hole, measured by scanning a 0.075 cm 1 (FWHM) dye laser. This phenomenon, called by the authors~19 the photochemically accumulated stimulated photon echo (PASPE), related to three—pulse stimulated photon echoes20 displays several special features. First, while the stimulated photon echo can occur only within the relaxation time limits of light-induced transient frequency-space population gratings, the lifetime of PASPE is determined by the very long lifetime of PHB photoproducts. This allows the spectral gratings in PASPE to be accumulated to a very high contrast, which in turn results in very high relative intensity echo signals (up to about 50 per cent). Secondly, due to the accumulation effect, PASPE experiments can be performed well under modest linear excitation and in first approximation a simple theory
K.K. Rebane /Laser study of no-phonon lines
747
relying on linear filtration theory can be estab21. lished In experimental investigations18 of PASPE a synchronously pumped pico— second laser was applied ~
:
x20
.
to generate echo signals
•
in porphyr-type low-ternperature impurity systems. Spectral gratings were first burned by repeated weak picosecond multipulse
• •
z
HPB and echoes were after-
I
/
0
100
probing beam through the
~TIME -‘ -~ ,-
wards detected by passing a still weaker picosecond
.~.
(psi
-‘-.-.
.-
---
--
sample and focussing it on the entrance slit of a synchroscan streak camera system. Both spectral and temporal responses of the
FIGURE 2 Streak image of a single picosecond probe pulse after travelling through the sample prepared as described in Fig. 1. The passed through probe pulse (upper left) is followed by two stimulated echo pulses. For time reference a fragment of the PHB pulse sequence is also presented.
sample were measured (Fig. 1 and 2 From the dependence of the echo si nals intensit g on the delay between the picosecond HB pulses phase
relaxation time can be estimated. In the case of 2.5x104 M octaethylporphin in 1.8 K polystyrol matrix T 2 was about 300 ps (Fig. 3). Now let us suppose the PHB sequence to comprise a signal pulse with an arbitrary temporal and spatial amplitude and phase distribution and a plane wave reference pulse which is short enough to be considered a a-pulse compared to the signal. It was demonstrated in 21 that if T2>>t exhaustive temporal and spatial information about the signal can be stored in the sample by PHB and later reproduced by PASPE. Again within the limits of weak excitation a linear approach is valid and the formalism of ordinary holographic image storage can well be extended 22.to the field of photosensitive materials with very high spectral resolution
A A. Rehane / La~r studt’ 01 no—pltonon lutes
748
FIGURE 3 Streak images showing the PHB sequence (a) and resulting PASPE after (c) 0.5 2 (b) signals 1 5 mU/cm2 mU/cm and 2.5 mU/cm2 (d) exposures. The sample was prepared by polymerization of octaethyl porphin 2.5M styrol solution.
0
100
200
300 ps
3. CONCLUDING REMARKS 1. Photochernical hole burning in spectroscopy has an essential advantage in comparison with the site-selective spectroscopy: the hole burning and its study may be done with the same laser, no additional spectral device is needed 22 2. By taking optically thick samples (and losing in transparency of holes) we can create holes narrower than the linewidth of the hole burning laser. We can approach the natural linewidth of the purely electronic line. 3. To study the shapes of these sharp holes (narrower than the narrowest laser linewidth in our experiment) we can apply Doppler shift measurements in a way quite similar to MUssbauer methods and no laser is needed to study the holeshapes. For 0.01 cm1 holewidths the velocities about tens of msec’ are needed23. 4. In some tasks fast photoburning and detection may be decisive. In our laboratory evidence has been obtained that the holewidths for molecules in glassy matrices deorease with decreasing holeburning+detection time. ACKNOWLEDGEMENT The author is grateful to P.M. Saari and A.K. Rebane for many stimulatino discussinos.
K.K. Rebane / Laser studt of no-phonon lines
749
REFERENCES 1) K.K. Rebane, Impurity Spectra of Solids (Plenum Press, New York, 1970). 2) K.K. Rebane, Zh. Prikladnoi Spektroskopii 37 (1982) 906 (in Russian). 3) R.I. Personov, in: Spectroscopy and Excitation Dynamic on Condensed Molecular Systems, eds. V.M. Agranovich and R.M. Hochstrasser (North Holland, Amsterdam, 1983) p. 55. 4) E.D. Trifonov, Doklady Akademii Nauk SSSR 147 (1962) 826 (in Russian). K.K. Rebane and V.V. Hizhnyakov, Optika i Spektroskopija 14 (1963) 491. 5) E.F. Gross, B.S. Razbirin and S.A. Permogorov, Doklady Akademii Nauk SSSR 154 (1964) 1306 (in Russian). 6) RH. Silsbee and D.B. Fitchen, Rev. Mod. Phys. 36 (1964) 433. 7) K.K. Rebane, Proc. Int. Conf. Lasers’82 (New Orleans, Lousiana, Dec. 13-17, 1982; 1983) p. 340. 8) A. Szabo, Proc. Int. Conf. Lasers 80, New Orleans, Lousiana, Dec. 15-19, 1980, ed. C. Collins (STS Press, McLean, 1981) p. 374. 9) E.V. Shpolskii, Uspekhi Fiz. Nauk 71 (1960 215; 77 (1962) 32; 80 (1963) 255 (in Russian). 10) A. Szabo, Phys. Rev. Lett. 25 (1970) 924. 11) R. Personov, E. Al shits and L. Bykovskaya, Opt. Commun. 6 (1972) 169. 12) A.A. Gorokhovski, R.K. Kaarli and L.A. Rebane, UETP Lett. 20 (1974) 216. 13) B.M. Kharlamov, R.I. Personov and L.A. Bykovskaya, Opt. Commun. 12 (1974) 191. 14) L.A. Rebane, Zh. Prikladnoi Spektroskopii 34 (1981) 1023 (in Russian). 15) L.A. Rebane, A.A. Gorokhovski and U.V. Kikas, Appl. Phys. B29 (1982) 235. 16) U. Friedrich and D. Haarer, Angewandte Chemie 23 (1984) 113. 17) A.K. Rebane, R.K. Kaarli and P.M. Saari, Optika i Spektroskopija 55 (1983) 405 (in Russian); U. Mol. Structure 114 (1984) 343. 18) A.K. Rebane, R.K. Kaarli, P.M. Saari, A. Anijalg and K. Timpmann, Optics Commun. 47 (1983) 173. 19) A.K. Rebane and R.K. Kaarli, Chem. Phys. Lett. 101 (1983) 317. 20) W.H. Hesselink and D.A. Wiersma, in: Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, eds. V.M. Agranovich and R.M. Hochstrasser (North Holland Publishing Co., 1983). 21) P.M. Saari and A.K. Rebane, Eesti NSV Tead. Akad. Toimet., FUUs. Matem. in print (in Russian). 22) S. Voelker, R.M. Macfarlane and J.H. van derWaals, Chem. Phys. Lett. 53(1978)8. 23) K.K. Rebane and V.V. Palm, Optika
Spvktroskupija 57 (1984) 381 ( ri Russ an).