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Dead layer effects in the ultraviolet reflectance of excitons in solid rare gases
Solid State Communications, Voi.26, pp.425-428. Q Pergamon Press Ltd. 1978. Printed in Great Britain
0038-1098/78/O515-0425
$02.00/0
DEAD LAYER EFFECTS IN THE ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES W. Andreoni Laboratoire de Physique Appliqu~e, EPF-Lausanne, Switzerland M. De Crescenzi Laboratoire d'Optique des Solides, Universit@ Paris VI, Paris, France and Istituto di Fisica, Universit~ di Roma, Italy and E. Tosatti GNSM-CNR, Istituto di Fisica Teorica, Universit~ di Trieste and International Centre for Theoretical Physics, 34100 Trieste, Italy
(Received 14 March 1978 by F. B~sani) We propose an explanation of the apparent doubling of exciton peaks of solid Kr and Ar seen in reflectivity in terms of spatial dispersion and of a surface dead layer for bulk excitons. Speculative arguments as to the origin of the dead layer are presented, that include the role of surface excitons, as well as a possible thin superficial liquid film. Reflectivfty experiments performed over the recent years by the DESY group on thick samples of solid rare gases have revealed the occurrence of what appear to be sharp splittings of some of the n = 1 exciton lines into doubletsl,2, 3. This is particularly evident in Arand Kr as shown i; Figs. la and ib, which reproduce the original data of Haensel et al.1,2. Of eachdoublet of peaks, the
Kr
A
/
0 C
Ar 0
¢-
(9
}
0
0 0 tY
o, •
|
12.
"-)---
e
,
i I
13.
10.
E (ev)
Fig. la : Reflectance spectrum of solid Ar in arbitrary units (Haensel et al., R e f . I).
I
11.
I
12, E(ev)
Fig. ib : Reflectance spectrum of solid Kr in arbitrary units (Haensel et al., Ref. 2).
lowest is, as expected, very close in energy to the transverse exciton energies known independently from absorption 4,5. The higher energy peak, marked by an arrow in Fig. 1 is however
not expected, and has not received an adequate explanation so far. It should be noted that the relative heights of the two peaks are not uniquely defined; they do instead depend criti425
426
ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES
cally on sample preparation 3 and other experimental conditions 6. Search for this exciton splitting in transmission data in the literature leads to the intriguing conclusion that the higher energy peak is usually absent 4,5, except for weaker structures recently detected 7 , and tentatively reported to be very dependent upon the incidence angle 8 . Energy loss spectra also show an unsplit exciton peak in all cases 9 . The first possibility to consider is that of a true exciton splitting, caused by some symnetry lowering. In cubic synmetry, each of these exciton is Fir, and splitting into a doublet would require an axial distortion, of which there is no other evidence. One could conceivably overcome this by considering a final (dynamical Jahn-Teller) state consisting of a linear superposition of the star of distorted states that are equivalent in the cell I0. This distorted state has overall cubic symmetry and a split exciton, and given a long enough lifetime, ought to be realised in all cubic semiconductors and insulators. However, a Jahn-Teller splitting should be equally visible in reflection, absorption and energy loss, which is not the case. Furthermore, different investigations yield very close values for the doublet splitting (~ 0 and 0.15 eV in Ar; ~0.12 eV and 0.08 eV in Kr). The strong temperature dependence typical of a Jahn-Teller splitting I0 is completely absent. Therefore, this hypothesis is not
IEL
V
Ar
R
j/
.75
m=6.mo I- =0.2mev 6b~2.7
.50
---I=O
I =50,&
/
.25
// ET
!
1P_1
Fig. 2
I
123
1215 E(ev)
: Calculated reflectivities with (solid line) and without (dashed line) the dead layer (parameters appropriate to a Ar exciton).
Voi.26, No.7
tenable and a completely different mechanism is called for to account for the observed facts. In the following, we suggest II that the observed reflectivity exciton splitting is only apparent, and could be ascribed to the wellknown Hopfield-Thomas (HT) interference phenomenon~ 2 if a thin surface "dead layer" is supposed to exist where bulk-frequency excitons do not propagate. A sharp extra reflection peak is known to develop, due to the existence of a dead layer, coupled to the finite total exciton mass (spatial dispersion) in many semiconductors, notably CdS 12,13, ZnTe 12, InP 14, G a ~ 15, ZnO .16, and PbI217. Physically, the phenomenon originates from the fact that if the excitons propagate with a kinematical mass M, which is not infinite, then two separate polariton modes couple to the light beam at the surface rather than only one. The energy branching into the two modes is determined in a rather involved way by the so-called "additional boundary conditions ''12,18. At the longitudinal exciton energy, the group velocity of one of the modes vanishes, and the resulting singularity in the density of states reflects back on the other coupled channels 19. This may produce a violent interference and a reflection peak if a thin dielectric film, such as a dead layer, is present on the surface. In practice, the extra peak may or may not show up, depending on several facts : I) there must be sizable spatial dispersion; 2) there must be a surface exciton-free layer, of thickness typically no less than ~0.i~ ; 3) the dumping must not be too big; 4) the oscillator strength must be reasonably high. In a rare gas, conditions 3) and 4) are certainly met. Condition I) is also fulfilled in t2~at the rare gas excitons are kr~m to p r o p a g a t e ; thus their mass, even though large of order one to several electron masses, is finite. A dead layer is however a new feature for a rare gas which must be invoked here for the first time; we now show that the case for its existence is indeed a very strong one. We have calculated exciton reflectivities of Ar and Kr using the HT model, with a suitable exciton mass M, lifetime I/F, and a dead layer whose thickness £ is taken as a parameter. Fig. 2 shows that indeed as £ increases an extra reflectivity peak appears at approximately the longitudinal exciton frequency ~L" The shape and magnitude of the new peak are sensitive to the choice of lifetime, to the background dielectric constant Eo, and of course to £, but are rather insensitive to the precise value of the mass. The dead layer thicknesses £ necessary to yield a strong extra reflection peak comparable to that observed are of the order of 30~ in Kr and 50 ~ in Ar. These are large values, for exan~le they are about one order of magnitude larger than the exciton radii. We point out now several facts that are quite naturally explained by our model. First, the extra peak is predicted in reflectivity, but not in absorption and in electron energy loss, which is experimentally correct. We stress that the finding of a corresponding peak in transmission is not an argument against this model; on
Voi.26, No.7
ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES
the contrary, the transmission is related to the absorption coefficient via the reflectivity of bo.th, f r o n t and back s u r f a c e s , and such s t r u c t u r e s are expected, and i n p r i n c i p l e c a l c u l a b l e . Second, we n o t i c e the very near coincidence of the e x t r a r e f l e c t i o n peak e n e r g i e s (12.50 eV i n Ar, 10.37 eV and 10.97 eV i n Kr) with l o n g i t u d i n a l e x c i t o n e n e r g i e s measured by e l e c t r o n loss (12.50 eV i n Arg, 10.55 eV and 11.20 eV i n Kr9. For Kr there appears to be a c o n s i s t e n t energy s h i f t of the whole spectrum of ~ - 0 . 2 0 eV to be s u b s t r a c t e d before comparison with o p t i c a l s p e c t r a . Once t h i s i s done, the s p e c t r a agree very w e l l ) . These experimental f i n d i n g s are i n s t r i k i n g agreement with the f a c t t h a t , i n the HT model, the e x t r a peak i s always very near ~L, independent of a l l parameters. Third, the apparent r e f l e c t i o n peak energy s p l i t t i n g s are p r e d i c t e d to be always proport i o n a l to the e x c i t o n o s c i l l a t o r s t r e n g t h s . In Ar 1, the lowest energy e x c i t o n a t 12.10 eV i s weak, and appears to be u n s p l i t , while the s t r o n g e r higher e x c i t o n i s s p l i t by ~0.15 eV ( i n good agreement with a c a l c u l a t e d L-T s p l i t t i n g of 0.19 eV21. In Kr2, the apparent s p l i t t i n g s are 0.12 eV (j = 3/2) and 0.08 eV (j = 1 / 2 ) , and an o s c i l l a t o r s t r e n g t h of approximately 3:2 i s indeed observed 22. All these facts make in our view the case extremely strong in favour of the dead layer model. The question that remains open is what the possible nature of the dead layer might be. As already seen, a minimum value of £, provided by the exciton radius 12, is far too small in this case. Deep surface irregularities, crystal growth by islands and pools, etc., seem ruled out by the observation of sharp interference patterns 2 3.
A possible explanation could be connected with the recent observation that surface excitons exist both in Ar 7 and K~ 23, several hundreds of b~eV's below the corresponding bulk excitons. There is thus a layer, of still unknown depth, where the exciton, that is essentially an atomic-like excitation weakly modified by overlap of the electronic wavefunctions, has a very different energy from the deep bulk. The exciton amplitude at the bulk frequency must therefore fall to zero continuously before reaching the surface. If we assume that the center of mass exciton wavefunction takes on in this situation a behaviour like, say, sin qz/%, where q is an exciton wavelength, then this simulates an effective dead layer which is an appreciable fraction of We finally note that both thermal and quantum fluctuations must be strongly enhanced at a free rare gas surface over their bulk magnitudes. As a consequence, a thin liquid film might subsist on these surfaces, down to very low temperatures. Due to rarefaction, the exciton in the liquid has lower energy than in the solid, closer to the free atom excitation. A liquid layer and its in~nediate neighbourhoods would then constitute an effective dead layer for excitons at the bulk frequency. In conclusion, a dead layer model is proposed that accounts for the extra peak structure observed in the reflectivity of solid Kr and Ar in considerable detail. Speculations on possible reasons for the presence of a bulk exciton depletion layer at the surface are advanced that could be readily tested experimentally.
REFERENCES
i. 2. 3. 4. 5. 6. 7.
8. 9. I0. ii.
12. 13. 14.
427
HAENSEL R., KEITEL G., KOCH E.E., SKIBOWSKI M. and SCHREIBER P., Phys. Rev. Lett. 23, 1160 (1969). HAENSEL R., KEITEL G., KOCH E.E., SKIBOWSKI M. and SCHREIBER P., Opt. Comm. 2, $9 (1970). For a complete review see SONNTAG B., in "Rare Gas Solids", ~d. M.K. Klein and J.A. Venables, Acad. Press N.Y. 1976), Vol. II, p. 1020. SCHARBER S.R. and WEBBER S.E., J. Chem. Phys. 35, 3977 (1971). BALDINI g., Phys. Rev. 128, 1562 (1962). ASAF U. and STEINBERGER I']T., Proc. of the Vth Int. Conf. on Vacut~n Ultraviolet Radiation Physics, Montpellier 1977 (Ed. M.C. Castex, M. Powey and N. Powey) Vol. I, p. 205. SAILE V., private communication; SAILE V., PhD thesis (unpublished) 1976; SAILE V., SKIBOWSKI M., S T E I N ~ W., GURTLER P., KOCH E.E. and KOZEVNIKOV A., Phys. Rev. Lett. 37, 305 (1976); SAILE V., S T E I N ~ W. and KOCH E.E., Proc. of the Vth Int. Conf. on VacuL~n Ultraviolet Radiation Physics, Montpellier 1977 (Ed. M.C. Castex, M. Powey and N. Powey) Vol. I, p. 199. SAILE V. and KOCH E.E., private co~unication. BOSTANJOGLO O. and SCI~qIDT L., Phys. Lett. 22, 130 (1966); S C ~ I D T L., Phys. Lett. 36A, 87 (1971); SCHMIDT L., PhD thesis (unpublished): data quoted in ref. 3. See e.g. ENGINAN R., "The Jahn-Teller Effect in Molecules and Crystals" (Wiley-lnterscience J.B. Birks, ed., London 1972). ANDREONI W., DE CRESCENZI M. and TOSATrl E., Proc. of the Vth Int. Conf. on Vacuum Ultraviolet Radiation Physics, Montpellier 1977 (Ed. M.C. Castex, M. Powey and N. Powey) Vol. I, p. 220. HOPFIELD J.J. and THOMAS D.G., Phys. Rev. 132, 563 (1963). EVANGELISTI F., FROVA A. and PATELLA F., Phys. Rev. BI0, 4253 (1974). EVANGELISTI F., FISCHBACH J.V. and FROVA A., Phys. Rev. B9, 1516 (1974).
428
ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES
Voi.26, No.7
15. SELL D.D., STOKOWSKI S.E., DINGLE R. and DI LORENZO J.V., Phys. Rev. B7, 4568 (1973). 16. LAGOIS J. and HUMMER K., Phys. Stat. Sol. (b) 72, 393 (1975). 17. HARBEKE G. and TOSATrI E., Nuovo Cimento 22B, 8-7 (1974); GROSMANN M., BIELLMANN J. and NIKITINE S., in "Springer Tracts in Modern Physics", Vol. 73, 242 (1973). 18. PEKAR S.I., Sov. Phys. JETP 34, 813 (1955). 19. HOPFIELD J.J., "Physics of Semiconductors", Proc. of the VIII Int. Conf., Kyoto (1966), p. 77. 20. ZIMMERER G., in Course on Synchrotron Radiation Research, (ed. A.N. ~,hancini and I.F. Quercia, 1976), Vol. I, p. 435. 21. ANDREONI W., ALTARELLI M., and BASSANI F., Phys. Rev. BII, 2352 (1975). 22. The difference of the oscillator strengh ratios in the two cases is incidentally well understood on the basis of a different interplay of the spin-orbit interaction of the hole states and of the electron-hole exchange). 23. SAILE V. and KOCH E.E., private cormmmication.
0038-1098/78/O515-0425
$02.00/0
DEAD LAYER EFFECTS IN THE ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES W. Andreoni Laboratoire de Physique Appliqu~e, EPF-Lausanne, Switzerland M. De Crescenzi Laboratoire d'Optique des Solides, Universit@ Paris VI, Paris, France and Istituto di Fisica, Universit~ di Roma, Italy and E. Tosatti GNSM-CNR, Istituto di Fisica Teorica, Universit~ di Trieste and International Centre for Theoretical Physics, 34100 Trieste, Italy
(Received 14 March 1978 by F. B~sani) We propose an explanation of the apparent doubling of exciton peaks of solid Kr and Ar seen in reflectivity in terms of spatial dispersion and of a surface dead layer for bulk excitons. Speculative arguments as to the origin of the dead layer are presented, that include the role of surface excitons, as well as a possible thin superficial liquid film. Reflectivfty experiments performed over the recent years by the DESY group on thick samples of solid rare gases have revealed the occurrence of what appear to be sharp splittings of some of the n = 1 exciton lines into doubletsl,2, 3. This is particularly evident in Arand Kr as shown i; Figs. la and ib, which reproduce the original data of Haensel et al.1,2. Of eachdoublet of peaks, the
Kr
A
/
0 C
Ar 0
¢-
(9
}
0
0 0 tY
o, •
|
12.
"-)---
e
,
i I
13.
10.
E (ev)
Fig. la : Reflectance spectrum of solid Ar in arbitrary units (Haensel et al., R e f . I).
I
11.
I
12, E(ev)
Fig. ib : Reflectance spectrum of solid Kr in arbitrary units (Haensel et al., Ref. 2).
lowest is, as expected, very close in energy to the transverse exciton energies known independently from absorption 4,5. The higher energy peak, marked by an arrow in Fig. 1 is however
not expected, and has not received an adequate explanation so far. It should be noted that the relative heights of the two peaks are not uniquely defined; they do instead depend criti425
426
ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES
cally on sample preparation 3 and other experimental conditions 6. Search for this exciton splitting in transmission data in the literature leads to the intriguing conclusion that the higher energy peak is usually absent 4,5, except for weaker structures recently detected 7 , and tentatively reported to be very dependent upon the incidence angle 8 . Energy loss spectra also show an unsplit exciton peak in all cases 9 . The first possibility to consider is that of a true exciton splitting, caused by some symnetry lowering. In cubic synmetry, each of these exciton is Fir, and splitting into a doublet would require an axial distortion, of which there is no other evidence. One could conceivably overcome this by considering a final (dynamical Jahn-Teller) state consisting of a linear superposition of the star of distorted states that are equivalent in the cell I0. This distorted state has overall cubic symmetry and a split exciton, and given a long enough lifetime, ought to be realised in all cubic semiconductors and insulators. However, a Jahn-Teller splitting should be equally visible in reflection, absorption and energy loss, which is not the case. Furthermore, different investigations yield very close values for the doublet splitting (~ 0 and 0.15 eV in Ar; ~0.12 eV and 0.08 eV in Kr). The strong temperature dependence typical of a Jahn-Teller splitting I0 is completely absent. Therefore, this hypothesis is not
IEL
V
Ar
R
j/
.75
m=6.mo I- =0.2mev 6b~2.7
.50
---I=O
I =50,&
/
.25
// ET
!
1P_1
Fig. 2
I
123
1215 E(ev)
: Calculated reflectivities with (solid line) and without (dashed line) the dead layer (parameters appropriate to a Ar exciton).
Voi.26, No.7
tenable and a completely different mechanism is called for to account for the observed facts. In the following, we suggest II that the observed reflectivity exciton splitting is only apparent, and could be ascribed to the wellknown Hopfield-Thomas (HT) interference phenomenon~ 2 if a thin surface "dead layer" is supposed to exist where bulk-frequency excitons do not propagate. A sharp extra reflection peak is known to develop, due to the existence of a dead layer, coupled to the finite total exciton mass (spatial dispersion) in many semiconductors, notably CdS 12,13, ZnTe 12, InP 14, G a ~ 15, ZnO .16, and PbI217. Physically, the phenomenon originates from the fact that if the excitons propagate with a kinematical mass M, which is not infinite, then two separate polariton modes couple to the light beam at the surface rather than only one. The energy branching into the two modes is determined in a rather involved way by the so-called "additional boundary conditions ''12,18. At the longitudinal exciton energy, the group velocity of one of the modes vanishes, and the resulting singularity in the density of states reflects back on the other coupled channels 19. This may produce a violent interference and a reflection peak if a thin dielectric film, such as a dead layer, is present on the surface. In practice, the extra peak may or may not show up, depending on several facts : I) there must be sizable spatial dispersion; 2) there must be a surface exciton-free layer, of thickness typically no less than ~0.i~ ; 3) the dumping must not be too big; 4) the oscillator strength must be reasonably high. In a rare gas, conditions 3) and 4) are certainly met. Condition I) is also fulfilled in t2~at the rare gas excitons are kr~m to p r o p a g a t e ; thus their mass, even though large of order one to several electron masses, is finite. A dead layer is however a new feature for a rare gas which must be invoked here for the first time; we now show that the case for its existence is indeed a very strong one. We have calculated exciton reflectivities of Ar and Kr using the HT model, with a suitable exciton mass M, lifetime I/F, and a dead layer whose thickness £ is taken as a parameter. Fig. 2 shows that indeed as £ increases an extra reflectivity peak appears at approximately the longitudinal exciton frequency ~L" The shape and magnitude of the new peak are sensitive to the choice of lifetime, to the background dielectric constant Eo, and of course to £, but are rather insensitive to the precise value of the mass. The dead layer thicknesses £ necessary to yield a strong extra reflection peak comparable to that observed are of the order of 30~ in Kr and 50 ~ in Ar. These are large values, for exan~le they are about one order of magnitude larger than the exciton radii. We point out now several facts that are quite naturally explained by our model. First, the extra peak is predicted in reflectivity, but not in absorption and in electron energy loss, which is experimentally correct. We stress that the finding of a corresponding peak in transmission is not an argument against this model; on
Voi.26, No.7
ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES
the contrary, the transmission is related to the absorption coefficient via the reflectivity of bo.th, f r o n t and back s u r f a c e s , and such s t r u c t u r e s are expected, and i n p r i n c i p l e c a l c u l a b l e . Second, we n o t i c e the very near coincidence of the e x t r a r e f l e c t i o n peak e n e r g i e s (12.50 eV i n Ar, 10.37 eV and 10.97 eV i n Kr) with l o n g i t u d i n a l e x c i t o n e n e r g i e s measured by e l e c t r o n loss (12.50 eV i n Arg, 10.55 eV and 11.20 eV i n Kr9. For Kr there appears to be a c o n s i s t e n t energy s h i f t of the whole spectrum of ~ - 0 . 2 0 eV to be s u b s t r a c t e d before comparison with o p t i c a l s p e c t r a . Once t h i s i s done, the s p e c t r a agree very w e l l ) . These experimental f i n d i n g s are i n s t r i k i n g agreement with the f a c t t h a t , i n the HT model, the e x t r a peak i s always very near ~L, independent of a l l parameters. Third, the apparent r e f l e c t i o n peak energy s p l i t t i n g s are p r e d i c t e d to be always proport i o n a l to the e x c i t o n o s c i l l a t o r s t r e n g t h s . In Ar 1, the lowest energy e x c i t o n a t 12.10 eV i s weak, and appears to be u n s p l i t , while the s t r o n g e r higher e x c i t o n i s s p l i t by ~0.15 eV ( i n good agreement with a c a l c u l a t e d L-T s p l i t t i n g of 0.19 eV21. In Kr2, the apparent s p l i t t i n g s are 0.12 eV (j = 3/2) and 0.08 eV (j = 1 / 2 ) , and an o s c i l l a t o r s t r e n g t h of approximately 3:2 i s indeed observed 22. All these facts make in our view the case extremely strong in favour of the dead layer model. The question that remains open is what the possible nature of the dead layer might be. As already seen, a minimum value of £, provided by the exciton radius 12, is far too small in this case. Deep surface irregularities, crystal growth by islands and pools, etc., seem ruled out by the observation of sharp interference patterns 2 3.
A possible explanation could be connected with the recent observation that surface excitons exist both in Ar 7 and K~ 23, several hundreds of b~eV's below the corresponding bulk excitons. There is thus a layer, of still unknown depth, where the exciton, that is essentially an atomic-like excitation weakly modified by overlap of the electronic wavefunctions, has a very different energy from the deep bulk. The exciton amplitude at the bulk frequency must therefore fall to zero continuously before reaching the surface. If we assume that the center of mass exciton wavefunction takes on in this situation a behaviour like, say, sin qz/%, where q is an exciton wavelength, then this simulates an effective dead layer which is an appreciable fraction of We finally note that both thermal and quantum fluctuations must be strongly enhanced at a free rare gas surface over their bulk magnitudes. As a consequence, a thin liquid film might subsist on these surfaces, down to very low temperatures. Due to rarefaction, the exciton in the liquid has lower energy than in the solid, closer to the free atom excitation. A liquid layer and its in~nediate neighbourhoods would then constitute an effective dead layer for excitons at the bulk frequency. In conclusion, a dead layer model is proposed that accounts for the extra peak structure observed in the reflectivity of solid Kr and Ar in considerable detail. Speculations on possible reasons for the presence of a bulk exciton depletion layer at the surface are advanced that could be readily tested experimentally.
REFERENCES
i. 2. 3. 4. 5. 6. 7.
8. 9. I0. ii.
12. 13. 14.
427
HAENSEL R., KEITEL G., KOCH E.E., SKIBOWSKI M. and SCHREIBER P., Phys. Rev. Lett. 23, 1160 (1969). HAENSEL R., KEITEL G., KOCH E.E., SKIBOWSKI M. and SCHREIBER P., Opt. Comm. 2, $9 (1970). For a complete review see SONNTAG B., in "Rare Gas Solids", ~d. M.K. Klein and J.A. Venables, Acad. Press N.Y. 1976), Vol. II, p. 1020. SCHARBER S.R. and WEBBER S.E., J. Chem. Phys. 35, 3977 (1971). BALDINI g., Phys. Rev. 128, 1562 (1962). ASAF U. and STEINBERGER I']T., Proc. of the Vth Int. Conf. on Vacut~n Ultraviolet Radiation Physics, Montpellier 1977 (Ed. M.C. Castex, M. Powey and N. Powey) Vol. I, p. 205. SAILE V., private communication; SAILE V., PhD thesis (unpublished) 1976; SAILE V., SKIBOWSKI M., S T E I N ~ W., GURTLER P., KOCH E.E. and KOZEVNIKOV A., Phys. Rev. Lett. 37, 305 (1976); SAILE V., S T E I N ~ W. and KOCH E.E., Proc. of the Vth Int. Conf. on VacuL~n Ultraviolet Radiation Physics, Montpellier 1977 (Ed. M.C. Castex, M. Powey and N. Powey) Vol. I, p. 199. SAILE V. and KOCH E.E., private co~unication. BOSTANJOGLO O. and SCI~qIDT L., Phys. Lett. 22, 130 (1966); S C ~ I D T L., Phys. Lett. 36A, 87 (1971); SCHMIDT L., PhD thesis (unpublished): data quoted in ref. 3. See e.g. ENGINAN R., "The Jahn-Teller Effect in Molecules and Crystals" (Wiley-lnterscience J.B. Birks, ed., London 1972). ANDREONI W., DE CRESCENZI M. and TOSATrl E., Proc. of the Vth Int. Conf. on Vacuum Ultraviolet Radiation Physics, Montpellier 1977 (Ed. M.C. Castex, M. Powey and N. Powey) Vol. I, p. 220. HOPFIELD J.J. and THOMAS D.G., Phys. Rev. 132, 563 (1963). EVANGELISTI F., FROVA A. and PATELLA F., Phys. Rev. BI0, 4253 (1974). EVANGELISTI F., FISCHBACH J.V. and FROVA A., Phys. Rev. B9, 1516 (1974).
428
ULTRAVIOLET REFLECTANCE OF EXCITONS IN SOLID RARE GASES
Voi.26, No.7
15. SELL D.D., STOKOWSKI S.E., DINGLE R. and DI LORENZO J.V., Phys. Rev. B7, 4568 (1973). 16. LAGOIS J. and HUMMER K., Phys. Stat. Sol. (b) 72, 393 (1975). 17. HARBEKE G. and TOSATrI E., Nuovo Cimento 22B, 8-7 (1974); GROSMANN M., BIELLMANN J. and NIKITINE S., in "Springer Tracts in Modern Physics", Vol. 73, 242 (1973). 18. PEKAR S.I., Sov. Phys. JETP 34, 813 (1955). 19. HOPFIELD J.J., "Physics of Semiconductors", Proc. of the VIII Int. Conf., Kyoto (1966), p. 77. 20. ZIMMERER G., in Course on Synchrotron Radiation Research, (ed. A.N. ~,hancini and I.F. Quercia, 1976), Vol. I, p. 435. 21. ANDREONI W., ALTARELLI M., and BASSANI F., Phys. Rev. BII, 2352 (1975). 22. The difference of the oscillator strengh ratios in the two cases is incidentally well understood on the basis of a different interplay of the spin-orbit interaction of the hole states and of the electron-hole exchange). 23. SAILE V. and KOCH E.E., private cormmmication.