Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption
Results of investigations of spectral characteristics in the fundamental absorption range for the glass-like alloys HgSe - GeSe₂ are represented. To explain the phenomenon of anomaly growth of the static disorder, the model of deforming tensions is discussed. The hypothesis concerning a sharp change...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2002
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| Цитувати: | Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption / V.V. Bozhko, V.V. Halyan, O.V. Parasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 2. — С. 163-169. — Бібліогр.: 17 назв. — англ. |
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irk-123456789-1211622017-06-14T03:06:30Z Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption Bozhko, V.V. Halyan, V.V. Parasyuk, O.V. Results of investigations of spectral characteristics in the fundamental absorption range for the glass-like alloys HgSe - GeSe₂ are represented. To explain the phenomenon of anomaly growth of the static disorder, the model of deforming tensions is discussed. The hypothesis concerning a sharp change of physical-and-chemical properties for the transition over the double eutectic point on the stable phase diagram of the HgSe - GeSe₂ system with a changing glass-creating matrix is suggested. 2002 Article Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption / V.V. Bozhko, V.V. Halyan, O.V. Parasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 2. — С. 163-169. — Бібліогр.: 17 назв. — англ. 1560-8034 PACS: 78.20.Bh, 78.40 http://dspace.nbuv.gov.ua/handle/123456789/121162 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Results of investigations of spectral characteristics in the fundamental absorption range for the glass-like alloys HgSe - GeSe₂ are represented. To explain the phenomenon of anomaly growth of the static disorder, the model of deforming tensions is discussed. The hypothesis concerning a sharp change of physical-and-chemical properties for the transition over the double eutectic point on the stable phase diagram of the HgSe - GeSe₂ system with a changing glass-creating matrix is suggested. |
| format |
Article |
| author |
Bozhko, V.V. Halyan, V.V. Parasyuk, O.V. |
| spellingShingle |
Bozhko, V.V. Halyan, V.V. Parasyuk, O.V. Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Bozhko, V.V. Halyan, V.V. Parasyuk, O.V. |
| author_sort |
Bozhko, V.V. |
| title |
Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption |
| title_short |
Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption |
| title_full |
Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption |
| title_fullStr |
Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption |
| title_full_unstemmed |
Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption |
| title_sort |
urbach’s edge of glassy hgse-gese₂ alloys: static disorder and temperature dependence of optical absorption |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121162 |
| citation_txt |
Urbach’s edge of glassy HgSe-GeSe₂ alloys: static disorder and temperature dependence of optical absorption / V.V. Bozhko, V.V. Halyan, O.V. Parasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 2. — С. 163-169. — Бібліогр.: 17 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT bozhkovv urbachsedgeofglassyhgsegese2alloysstaticdisorderandtemperaturedependenceofopticalabsorption AT halyanvv urbachsedgeofglassyhgsegese2alloysstaticdisorderandtemperaturedependenceofopticalabsorption AT parasyukov urbachsedgeofglassyhgsegese2alloysstaticdisorderandtemperaturedependenceofopticalabsorption |
| first_indexed |
2025-07-08T19:18:44Z |
| last_indexed |
2025-07-08T19:18:44Z |
| _version_ |
1837107615797084160 |
| fulltext |
163© 2002, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2002. V. 5, N 2. P. 163-169.
PACS: 78.20.Bh, 78.40
Urbach�s edge of glassy HgSe-GeSe2 alloys: static disorder
and temperature dependence of optical absorption
V.V. Bozhko1), V.V. Halyan1) , O.V. Parasyuk2)
1)Department of Solid State Physics, Volyn State University, 13 prospect Voli, 43009 Lutsk, Ukraine
E-mail: galyan@lab.univer.lutsk.ua
2)Department of Inorganic and Physical Chemistry, Volyn State University, 13 Voli av., 43009 Lutsk, Ukraine
Abstract. Results of investigations of spectral characteristics in the fundamental absorption
range for the glass-like alloys HgSe � GeSe2 are represented. To explain the phenomenon of
anomaly growth of the static disorder, the model of deforming tensions is discussed. The
hypothesis concerning a sharp change of physical-and-chemical properties for the transition
over the double eutectic point on the stable phase diagram of the HgSe � GeSe2 system with a
changing glass-creating matrix is suggested.
Keywords: fundamental absorption, static disorder, glass-creating matrix.
Paper received 06.03.02; revised manuscript received 17.05.02; accepted for publication 25.06.02.
1. Introduction
The research of new materials is one of the basic direc-
tion in physics of amorphous semiconductors.
Multicomponent chalcogenide glasses are very promis-
ing, first of all because of switching properties and the
optical storage ones. Represented system of similar to
glass alloys was researched for the first time. Significant
factors for determination of optical, photoelectric and
radiation properties of glass-like materials are the state
and form of fundamental absorption range. Investiga-
tion of the optical absorption spectra in a wide range of
temperatures and for large region of glass creation is a
powerful tool to ascertain the electronic structure of these
semiconductors and to predict physical properties of
glasses.
The optical spectra of glasses of the HgSe � GeSe2
system within the range of fundamental absorption are
represented. Characteristic parameters of Urbach�s rule
are determined. Experimental data are analysed on the
basis of the generalized Urbach�s rule and the principle
of equivalence between static and dynamic parts of a gen-
eral structural disorder [3,4,5].
2. Materials, glass preparation and methods of
investigation
The synthesis of glasses was carried out in the evacuated
(0.1 Pa) silica ampoules, upper parts of which were
alloys:thermostated by the asbestos tape for decreasing
losses from the gas phase. The high purity elements (Ge
99.9999 wt.%, Se 99.997 wt.%) and previously synthe-
sised HgSe (Hg 99.999 wt.%) were used for preparation
of the charges. The sample was heated up to the maximal
temperature 1270 K, then were kept for 10 hours. Subse-
quently, the ampoules were quenched in 25% solution of
NaCl. Glassy phase in the alloys was controlled using
X-ray diffraction (DRON-3M diffractometer) and micro-
structure analysis (MMU-3 microscope). The alloys
(HgSe)õ(GeSe2)1-õ were obtained in the interval 0≤õ≤0.4
with step x=0.1 and in the interval 0.42≤õ≤0.6 with step
x=0.02. For optical measurements the samples in the form
of parallel-sided plates polished down to 0.12-0.20 mm
thickness were prepared. The quality of surfaces was
checked by MII-4 interferometer.
The absorption coefficient measurements were real-
ized using the spectrometer IKS-12 with PbS
164 SQO, 5(2), 2002
V.V. Bozhko et al.: Urbach�s edge of glassy HgSe-GeSe
2
alloys...
10
100
1000
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
0
1 0
2 0
3 0
4 0
4 2
4 4
4 6
4 8
5 0
5 2
5 4
Fig. 1. Spectra of the optical absorption edge for glasses of the HgSe � GeSe2 system at 290K (the numbers imply mol.% HgSe)
10
100
1000
0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5
0
1 0
2 0
3 0
4 0
4 2
4 4
4 6
4 8
5 0
5 2
5 4
Fig. 2. Specta of the optical absorption edge for glasses of the HgSe � GeSe2 system at 77K (the numbers imply mol.% HgSe)
V.V. Bozhko et al.: Urbach�s edge of glassy HgSe-GeSe
2
alloys...
165SQO, 5(2), 2002
photodetector for the samples (42-54 mol.%) HgSe � (58-
46 mol.%) GeSe2 and DMR-4 spectrometer with silicon
photodetector for the samples (0-40 mol.%) HgSe � (100-
60 mol.%) GeSe2. In both above mentioned cases, the
usual circuit of synchronized detection was used. The
measurements were carried out at T=290 K and T=77 K
using vacuum cryostat in the spectral ranges:
1) 550 � 2500 nm for PbS photodetector;
2) 450 � 1100 nm for silicon photodetector.
For glassy alloys containing 100 mol.% GeSe2, 70
mol.% GeSe2, 54mol.% GeSe2, 50mol.% GeSe2,
48mol.% GeSe2 the temperature dependencies of the lo-
cation of optical absorption edge were evaluated (Fig. 3)
using the above mentioned spectrometers and measure-
ment procedures.
3. Results
The absorption factor a dependencies from photon en-
ergy hn at 290 K and 77 K are represented in Fig. 1,2,
respectively. The ordinate axis is represented in the loga-
rithmic scale.
When the content of HgSe increases up to 54 mol.%
the shift of the straight lines describing the spectral de-
pendencies in the lower energy range is observed. Ac-
cording to the X-ray data the alloys from the interval
(56-60) mol.% HgSe are crystalline, therefore the results
of its investigation are not considered in present paper.
The alloy containing 42 mol.% HgSe is characterized by
the most flatness of the approximating line. With the in-
creasing HgSe content more than 42 mol.%, the absorp-
tion edge are still being shifted to the low-energy direc-
tion, and decreasing Urbach�s exponential tail is ob-
served. According to the results of spectral distribution
the magnitude of Eo is evaluated which represents the
width of exponential edge (or its steepness [2,4]).
Ε
0
= α
ν
ln∂
∂h
For all alloys the Å0 parameter value was estimated in
the range α=80�300cm�1.
Two samples containing 100, 70 mol.% GeSe2 and
several samples with the considerable content of HgSe
(46, 50, 52 mol.%) were selected from the alloys of the
HgSe � GeSe2 system. In Fig. 3 we can see the graphs of
the temperature as dependencies of optical absorption
edges for the above mentioned samples. The hν(T) func-
tion shows the linear dependence.
Non-linear sections from the low-temperature side
occurring in the phosphate or silicate glasses [5] are not
registered.
In consequence of the linearity of hn(T) the tempera-
ture coefficient for the variation of the absorption edge
location can be estimated (Table 1).
Fig. 3. Temperature dependence of the absorption edge (the numbers imply mol.% HgSe)
166 SQO, 5(2), 2002
V.V. Bozhko et al.: Urbach�s edge of glassy HgSe-GeSe
2
alloys...
dT
hd )( νβ =
According to Table 1 and Fig. 4, the temperature co-
efficient β changes the sign from �+� to �-� at the transi-
tion from the glasses with small content of HgSe to the
alloys where the content of HgSe exceeds 42 mol.%.
Using the spectral distribution of the absorption fac-
tor, we can evaluate the energy gap width Eg (values
Eg and hν were determined as energy for absorption
factor α = 100 cm-1). In Fig. 4 the composition depend-
ence of Åg for glassy alloys at Ò=290, 77Ê is given.
As we can see the decrease of the energy gap occurs
for the 42 mol.% HgSe � 58 mol.% GeSe2. Here Åg for
glasses containing less than 42 mol.% HgSe at T=77 K
is larger than that at the room temperature. At the same
time, for the alloys containing 42 and more mol.% HgSe,
the energy gap is larger for the alloys examined at the
liquid-nitrogen temperature. It is well agreed with the
experiment where the change of the β sign from �+� for
the glasses having small content of HgSe to ��� for the
alloys with considerable content of this component is
observed.
4. Theory and discussion
Obtained exponential dependencies (Fig. 1,2) would be
expressed by Urbach�s rule [1,4,8,9]. In general case the
spectral-temperature dependence for absorption edge is
evaluated from the expression
( ) exp,, 0ανα =XTh
( )( )
( )XTE
TEh g
,0
−ν
(1)
where α0 is a constant,
hν � photon energy,
Eg(T) � temperature function of the optical gap width,
E0(T, X) � function of the overall disorder present in sys-
tem.
Å0 is a quantitative characteristic of disorder for glassy
alloys and represents contributions of the dynamic (ther-
mal) and static disorder components [3]:
Å0(Ò, Õ) = Ê(u2
ò + u2
õ) (2)
<u2>ò � root-mean-square thermal shifting of atoms;
Fig. 4. The optical ionization energy determined
Mol.%HgSe 0 30 46 50 52
β
K
eV410−× 6.164 6.918 � 4.347 � 3.1 � 3.311
Table 1. Temperature coefficients of the absorption edges
V.V. Bozhko et al.: Urbach�s edge of glassy HgSe-GeSe
2
alloys...
167SQO, 5(2), 2002
<u2>õ � root-mean-square static shifting of atoms caused;
K � deformation potential constant.
Actually, in amorphous structures the dynamic and
static components of overall structural disorder occur.
But regarding the particular systems one of these compo-
nents is dominant. For glassy materials, the base of struc-
tural disordering is static disorder (possible exception is
SiO2 [4]):
<u2>õ >> <u2>ò
Then the function
E0(T, X) = E0(X) = K <u2>x ≡E0 (3)
does not depend on temperature now.
For those structures the so-called glassy modifica-
tion of Urbach�s rule is realised (1)
( )
+=
00
exp,
T
T
E
h
Th g
νανα ; β
0
0
E
T = (4)
In [4] the equality (3) is easily evaluated analytically
from (1) using linear dependence Eg(T) [10,11]
( ) ( ) TETE gg β−= 0 (5)
Where Åg(0) � gap energy at zero temperature;
dT
dEg=β � thermal coefficient of gap energy.
The proportion (2) is equivalence of static and dy-
namic types of disorder. If the edge shift occurs without
variation of the slope angle (strictly speaking, this case is
realised for most of the HgSe � GeSe2 glasses) then the
temperature dependence hν(T) completely coincides with
the temperature variations of Eg(T) ( Fig.3).
( ) ( ) ( ) TTEEThh gg βνν =−=− )(00 (6)
It has to be noted that the equations (4), (5), (6) is
correct only for the temperature range where the func-
tional dependence of the absorption edge hν(T) is linear.
In glassy systems below the some fictitious temperature
Ò < Ò′ and above the glass-transition temperature Ò > Òg
the non-linear temperature dependence of the absorption
edge is observed. For Ò > Òg the irreversible structural
transitions of the glass-creating matrix occur [12,13]. At
Ò < Ò′ for the description of Åg(T) or hν(T) we should
consider the phonon contribution [14,15] been prevalent
in the low-temperature range. Discussion about the hν(T)
behaviour at Òg<T<T′ oversteps the limits of the present
paper and will not be considered in the further analysis
of the experimental data for the HgSe � GeSe2 glasses.
Fig. 5. Dependence of static disorder on the HgSe content
168 SQO, 5(2), 2002
V.V. Bozhko et al.: Urbach�s edge of glassy HgSe-GeSe
2
alloys...
The glassy modification (4) of the generalized Urbach�s
rule is characteristic of the HgSe � GeSe2 alloys (see Fig.
5). It follows from (3) as far as Å0 is not dependent on
temperature function yet.
The exceptions are the 42 mol.% HgSe and 54 mol.%
HgSe glasses. In the last case Å0 decreases when tem-
perature puts down. The same regularity is exhibited in
some silicate glasses [4] and also for As2Te3, As2Se3 [1].
But the most typical it is for the crystal media whose ab-
sorption edge is expressed by (1). In such materials static
lattice disturbance is small comparing with the ampli-
tude of atom heat oscillations, then
〈u2〉ò >> 〈u2〉x
In the framework of the one-phonon approximation
[16], the mean square shifting of atoms is proportional to
kT.
Å0(Ò, X) = E0(T) = K 〈u2〉ò = KγkT
(7)
γ has the dimensionality inverted to Ê, [γ] = À2/åV.
From the equation (1) using (5), (7) we can obtain
�crystalline� modification of Urbach�s rule
( ) ( )
−
=
kT
Eh
K
h g
êr
01
exp
ν
γ
ανα (8)
Where
kKêr γ
βαα exp0= .
In the HgSe � GeSe2 alloys the �crystalline� modifi-
cation of Urbach�s rule appears only in the � boundary�
sample where the glassy state is existed. As it has been
shown previously, the alloys containing more than 54
mol.% HgSe are the crystalline ones. If we follow for the
Å0 variation with increasing of the HgSe content (Fig.
5), we should note that in the interval (0-40) mol.% HgSe
the value of Å0 is nearly constant. But for the ranges (44-
54) mol.% HgSe the Å0 downtrend is seen evidently. This
means that the structure ordering of glassy alloys gradu-
ally carries out with the next transition of the �glassy�
for crystalline Urbach�s rule. As a consequence, the crys-
tallization of medium occurs (56 � 60 mol% HgSe). On
the assumption of the above mentioned we can do the
conclusion that the alloy is as if �prepares� for the crys-
tallization.
In another sample with 42 mol.% HgSe, the functional
temperature dependence of Å0 is also observed (Fig 5).
It is emerged an interesting question: is it also the con-
sequence of the dominant influence of the dynamic dis-
ordering? Is it necessary to use the �crystalline� modi-
fication of Urbach�s rule (8) used for the 54 mol.%
HgSe sample? As we can see from Fig. 5, the magni-
tude of the Å0 parameter at 77 K much exceeds the mag-
nitude at 290 K. It follows from this that the condition (7)
can not be applied.
Å0(Ò, X) ≠ ÊγkT
Actually, it is an evident fact that the mean-square
shifting 〈u2〉ò at the low temperature can not exceed the
heat shifting of atoms at high temperature.
The anomalous Å0 increasing can be explained on
the basis of the next model. Under the influence of pho-
tons, electrons from valence band occupy the energy lev-
els in the tail of conduction band. In this connection the
overflow of the energy levels occurs so at the transition to
the next energy level electron does not occupy the higher
energy state En+1 (En+1>En) but jumps for the same level
En. From the energy viewpoint, these transitions are un-
profitable in local scope. But in the range of the whole
glassy medium it is quite admissible. The overflow of the
energy levels results in the existence of large deforma-
tion tension and pressure on atoms by glass-creating net-
work. As a consequence, the quite large disordering of
atoms is observed for the 24 mol.% HgSe alloy (see Fig.
5). At high temperatures, due to the heat energy kT the
considerable part of the charges moves from the local-
ized states to the delocalized ones above the transition
point Åñ [1]. Besides, the charges located in the condition
band over Åñ (for electrons) can form Debye screening
[19]. These factors promote decreasing of the tensions in
amorphous medium. Therefore, the Å0 decreasing occurs
at 290 K comparing with the liquid-nitrogen tempera-
ture.
The oneness of the 42 mol.% HgSe alloy was con-
firmed in [20]. In the glassy GeSe2 there must be the same
polyhedrons (the [GeSe4]-4 tetrahedral) as in crystalline
GeSe2. But in the glassy alloy the tetrahedrons form the
continuous structure network disordered in space. When
we deal with the complex glass the new basic structures
exist (BS). According to the HgSe � GeSe2 phase dia-
gram [21] the composition of 42 mol.% HgSe � 58 mol.%
GeSe2 corresponds to the binary eutectic point. In
aftereutectic alloys the GeSe2 BS are primary formed, in
eutectic alloys the GeSe2 and Hg2GeSe4 BS are formed
simultaneously, and for the undereutectic ones the pri-
mary Hg2GeSe4 and secondary GeSe2 BS are formed.
The represented experimental data show the change
of physical properties for the transition from glass with
small content HgSe to samples with large content of this
component. For transitions though point of binary eu-
tectic the band gap (Fig. 4) is changed. Thermal coeffi-
cient β changes its sign, too. In aftereutectic alloys in
comparison to undereutectic ones the value of static dis-
order E0 is decreased (Fig. 5). The hypothesis of possible
cause of its phenomena is discussed. According to one of
the basic cause of structural change, the similar to glass
alloys have structural-phase transition in the point of bi-
nary eutectic, by another words, the change of germa-
nium glass creating matrix (BS � GeSe2) by the mercury-
germanium one (BS � Hg2GeSe4).
V.V. Bozhko et al.: Urbach�s edge of glassy HgSe-GeSe
2
alloys...
169SQO, 5(2), 2002
5. Conclusions
The optical distributions of the absorption factor of the
alloys in the system HgSe � GeSe2 were researched. Ab-
sorption boundary for these glasses is explained accord-
ing to Urbach�s rule. Difference between similar to glass
and crystal modifications of Urbach�s rule is shown and
represented for the researched system of alloys. Crystal
modification is realized only in the sample with 54 mol%
HgSe, which find of border the creation of glass, though
its structure is not crystalline. The change of physical
properties of these alloys for transition over the point of
double eutectic (42 mol% HgSe) is fixed. It corroborates
the results [16,17] where the change structural unit for-
mation order in under-eutectic and after-eutectic alloys
was observed, too. The hypothesis about changing the
glass-creating matrixes in the point of double eutectic is
discussed. Anomaly growth of the static disorder E0 is
explained using the model of deformation tensions.
References
1. G. D. Cody, T. Tiedje, B. Abeles, B. Broks, Y. Goldstein,
Phys. Rev. Lett. 47, p.1480 (1981)
2. I. Vanshtein, A. Zatsepin, V. Kortov, Phys. Chem. of Glass. 25
(1999) 85.
3. I. Vanshtein, A. Zatsepin, V. Kortov, Yu. Schapova,
Phys.Tverdogo. 42 (2000) 224.
4. Abe S., Toyozava Yu. Density of the electron states and
absorption edge in amorphous semiconductors In: Amorphous
semiconductors and devices based on it. Metallurgy, Mos-
cow, 1986, 376 p.
5. N.F. Mott, E.A. Devis, Electron processes in non-crystalline
materials, Mir, Moscow, 1982, 652 p.
6. G.D. Cody. J. Non-Cryst. Solids 141, 3 (1992).
7. Ya. Klyava, Fizika Tverdogo Tela. 27 p. 1350 (1985).
8. T. Skettrup, Phys. Rev. B 18 (1978) 2622.
9. Y.P. Varshni. Physica. 34 (1967) 149.
10. J.R. Zakis, H. Fritzsche. Phys. Stat. Sol. B 64 p. 123 (1974).
11. A. Andreyev, B. Kolomiyets, T. Mazets, A. Manuk�an, S.
Pavlov, Fizika Tverdogo Tela 18 p. 53 (1976).
12. H.Y. Fan. Phys. Rev. 82 (1951) 900.
13. H. Fan. Photon-electron interaction in crystals in the absence
of the applied fields, Mir, Moscow, 1969. 127 p.
14. J.A. Reissland, The physics of phonons, Mir, Moscow, 1975.
367 p.
15. A. Abrikosov. Introduction to the normal metal theory,
Nauka, Moscow, 1972. 288 p.
16. O.V. Parasyuk. Thesis, Lviv, 1996, in Ukrainian.
17. S. Motrya, Yu. Voroshilov, M. Potoriy et al. Ukr. Chem. J. 52
p. 807 (1986).
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