Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses
Revised in this work is the correlation equation Q₁ = 2.3×π/D in the FSDPrelated void-based model for As₂S(Se)₃ chalcogenide glasses between the first sharp diffraction peak (FSDP) position, Q₁, and nanovoid diameter, D, are modified to be presented in the form of Q₁ = 1.75×π/D, taking into ac...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2013
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irk-123456789-1176822017-05-27T03:05:48Z Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses Kavetskyy, T.S. Revised in this work is the correlation equation Q₁ = 2.3×π/D in the FSDPrelated void-based model for As₂S(Se)₃ chalcogenide glasses between the first sharp diffraction peak (FSDP) position, Q₁, and nanovoid diameter, D, are modified to be presented in the form of Q₁ = 1.75×π/D, taking into account a newly deduced formula for positron lifetime, τ₂, versus void radius, R. It is valid for those molecular substrates for R up to 5 Å and when no orthopositronium (o-Ps) or very small fractions of o-Ps components are observed in positron annihilation lifetime spectroscopy. 2013 Article Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses / T.S. Kavetskyy // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 2. — С. 136-139. — Бібліогр.: 26 назв. — англ. 1560-8034 PACS 61.43.Fs http://dspace.nbuv.gov.ua/handle/123456789/117682 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Revised in this work is the correlation equation Q₁ = 2.3×π/D in the FSDPrelated
void-based model for As₂S(Se)₃ chalcogenide glasses between the first sharp
diffraction peak (FSDP) position, Q₁, and nanovoid diameter, D, are modified to be
presented in the form of Q₁ = 1.75×π/D, taking into account a newly deduced formula for
positron lifetime, τ₂, versus void radius, R. It is valid for those molecular substrates for R
up to 5 Å and when no orthopositronium (o-Ps) or very small fractions of o-Ps
components are observed in positron annihilation lifetime spectroscopy. |
| format |
Article |
| author |
Kavetskyy, T.S. |
| spellingShingle |
Kavetskyy, T.S. Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Kavetskyy, T.S. |
| author_sort |
Kavetskyy, T.S. |
| title |
Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses |
| title_short |
Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses |
| title_full |
Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses |
| title_fullStr |
Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses |
| title_full_unstemmed |
Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses |
| title_sort |
modified correlation equation in the fsdp-related void-based model for as₂s(se)₃ chalcogenide glasses |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117682 |
| citation_txt |
Modified correlation equation in the FSDP-related void-based model for As₂S(Se)₃ chalcogenide glasses / T.S. Kavetskyy // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 2. — С. 136-139. — Бібліогр.: 26 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT kavetskyyts modifiedcorrelationequationinthefsdprelatedvoidbasedmodelforas2sse3chalcogenideglasses |
| first_indexed |
2025-07-08T12:36:59Z |
| last_indexed |
2025-07-08T12:36:59Z |
| _version_ |
1837082336500383744 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 136-139.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
136
PACS 61.43.Fs
Modified correlation equation in the FSDP-related void-based model
for As2S(Se)3 chalcogenide glasses
T.S. Kavetskyy
Ivan Franko Drohobych State Pedagogical University, Solid-State Microelectronics Laboratory,
24, I. Franko str., 82100 Drohobych, Ukraine
Abstract. Revised in this work is the correlation equation Q1 = 2.3×π/D in the FSDP-
related void-based model for As2S(Se)3 chalcogenide glasses between the first sharp
diffraction peak (FSDP) position, Q1, and nanovoid diameter, D, are modified to be
presented in the form of Q1 = 1.75×π/D, taking into account a newly deduced formula for
positron lifetime, 2, versus void radius, R. It is valid for those molecular substrates for R
up to 5 Å and when no orthopositronium (o-Ps) or very small fractions of o-Ps
components are observed in positron annihilation lifetime spectroscopy.
Keywords: chalcogenide glass, first sharp diffraction peak, positron annihilation, voids.
Manuscript received 01.06.12; revised version received 17.01.13; accepted for
publication 19.03.13; published online 25.06.13.
Recently, nanoscale void-species structure has been
studied [1, 2] in the As2Se3 glass using combination of
X-ray diffraction in respect to the first sharp diffraction
peak (FSDP-related XRD), treated within a void-based
model [3-5], and positron annihilation lifetime
spectroscopy (PALS), analyzed within the two-state
positron trapping model [6-8]. Assuming that the same
nanovoids are responsible for both FSDP and PALS data
in terms of Jensen et al. [9], the analytical correlation
relationship between the FSDP position, Q1, and
nanovoid diameter, D, has been presented as [1, 2]
Q1 = 2.3 /D. (1)
In the further work [10], it was shown using the
examples of both g-As2Se3 and g-As2S3 (g-glassy) that
the analytically found Eq. (1) seems to be similar to the
relationship Q1 = 2.5×π/r, reported by Gaskell [11]
and/or Ehrenfest’s formula 1.23 = Q1 r/2π reported by
Rachek [12], where r is a characteristic distance that
corresponds to the inter-atomic correlations detected
from the pair distribution function, G(r).
The Monte-Carlo simulation results for g-As2Se3
[13, 14] and g-As2S3 [15] have also been applied to
confirm the validity of Eq. (1) for these chalcogenides.
Basically, the Eq. (1) comes from the Elliott
interpretation [3-5] of nature of the anomalous first sharp
diffraction peak (FSDP) as a chemical-order prepeak in
the concentration-concentration structure factor, arising
from the clustering of interstitial nanovoids around
cation-based structural units, tested for AX2-type glassy-
like materials (e.g. GeS(Se)2), and resulting in the
empirical relationship between the position of the FSDP,
Q1, and atom-void separation like to diameter of
nanovoids, D:
Q1 = 3/2D. (2)
If Eq. (2) is presented through a coefficient k,
depending on the type of glass structure, the Elliott
formula may be written as
Q1 = k /D, (3)
where k = 1.5 for tetravalent AX2-type glasses.
To find a correlation equation in the case of
pyramidal type As2S(Se)3 glasses, the Jensen et al. [9]
regularity between the positron lifetime, (in ns), and
vacancy volume, V (in Å 3), has been applied [1, 2] in the
form of expression:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 136-139.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
137
Table. The values of positron lifetime 2 as typical for chalcogenide glasses as well as the void radius R and void volume V
calculated from Eqs. (4) to (6).
2 (ns) 0.35 0.36 0.37 0.38 0.39 0.40 0.41
Jensen’s et al. approach, Eq.(4)
R (Å) 2.73 2.80 2.88 2.95 3.02 3.08 3.15
V (Å3) 85 92 100 108 115 123 131
Liao’s et al. approach, Eq.(5)
R (Å) 2.26 2.40 2.50 2.60 2.71 2.81 2.91
V (Å3) 48 58 65 74 83 93 103
Liao’s et al. approach, Eq.(6)
R (Å) 2.05 2.16 2.28 2.40 2.51 2.63 2.75
V (Å3) 36 42 50 58 66 76 87
0.240 + 0.0013 V. (4)
Recently, Liao et al. [16] reported a newly
modified positron lifetime-free volume correlation
equation for determination of free volumes up to the
mean radius R = 5 Å in polymeric systems, where no
orthopositronium (o-Ps) (3 = 1…4 ns) or very small
fractions of o-Ps components are observed in PALS,
which is based on the infinitive potential spherical model
(originally proposed by Tao [17]) and presented as
2 = 0.260 [1 (R/(R + 3.823)) + 1/2
sin(2R/(R + 3.823))]1. (5)
For simplicity, it was shown in [16] that R2
correlation for R < 5 Å could be approximated and
empirically fitted by a linear equation as
2 = 0.174(1 + 0.494R), (6)
where 2 and R are expressed in the units of nanoseconds
and angstroms, respectively.
As chalcogenide glasses in their chemical nature
are covalent disordered inorganic polymers [18], and, in
respect to positron annihilation, they should be
considered as inorganic polymers without o-Ps
component or with very small fraction of o-Ps
component in PALS (see, for example, [8, 19, 20]), the
Liao et al. correlation equations (5) and/or (6) are valid
in the first approximation for these materials and can be
used obviously for As2S(Se)3 alloys.
The values of void radius R or void volume V (V =
4/3R3) estimated using positron lifetime-free volume
correlation equations are given in Table. As seen, the
difference between values of V calculated from Eqs. (5)
and (6) is 15 Å3, which is rather small to be affected on
the average size of voids, whereas the void volume V
calculated from Eq. (4) is larger by 30 to 50 Å3.
Liao et al. [16] noted that Eq.(5) should be applied
for 2 0.35 to 0.40 ns, while for the shortest
2 < 0.26 ns this equation is not applicable, while the
linear Eq. (6) works better. It is noticeable that the
correlation coefficients (r2 = 0.9391) for Eq. (5) and
(r2 = 0.9268) for Eq. (6) are very similar [16], and, thus,
it is suggested that the linear equation could be also used
in more general cases for 2 0.35 to 0.40 ns when
various polymeric materials (organic and inorganic) are
examined. Taking into account that Eq. (5) is deduced
by Liao et al. [16] using the example of organic
polymers having preferential hydrocarbon environment,
while chalcogenide glasses are inorganic polymers, the
linear Eq. (6) is applied in the present work.
The values of the open-volume defect-related
positron lifetime component 2 are known to be 0.37 ns
for g-As2Se3 [21] and 0.36 ns for g-As2S3 [22].
According to Eq. (6), the values of void radius R are
2.28 Å (D = 4.6 Å) for g-As2Se3 and 2.16 Å (D = 4.3 Å)
for g-As2S3. The values of the FSDP position Q1 are
known to be 1.25 Å 1 for g-As2Se3 and 1.26 Å 1 for g-
As2S3 (see, for example [1, 2, 10, 23] and references
therein). Thus, corresponding to Eq. (3), the values of
the coefficient k are obtained to be 1.8 for g-As2Se3 and
1.7 for g-As2S3. In average, for pyramidal-type
As2S(Se)3 glasses, the value of the coefficient k is 1.75
and modified correlation equation in the FSDP-related
void-based model for these alloys can be written as
Q1 = 1.75× π/D. (7)
Therefore, using the newly modified positron
lifetime-free volume correlation equation, Eq. (6), it is
found that the established value of k = 1.75 for
pyramidal type As2S(Se)3 glasses is very close to the
above mentioned Elliott’s value of k = 1.5 for tetravalent
AX2-type glasses. On the basis of this similarity in k
values, it is suggested that the nature of the FSDP for
pyramidal-type As2S(Se)3 glasses and tetravalent AX2-
type glasses (e.g., A = Ge, X = S, Se) is the same or
very similar. In other words, in terms of the FSDP-
related void-based model [3-5], the cation-centered
nanovoids with the same or very similar size are
responsible for the FSDP occurrence in the both
pyramidal and tetrahedral type chalcogenide glasses.
This finding gives a new insight on the explanation of
experimental observation that the FSDP position in the
mixed pyramidal-tetrahedral type chalcogenide glasses,
such as As(Sb)2S3-GeS2 and As(Sb)2S3-Ge2S3, is
determined exclusively by concentration of Ge cations,
while As and Sb cations do not affect the FSDP
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 136-139.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
138
occurrence [23, 24]. The most probably it is caused by
the similar size of cation-centered nanovoids in the
pyramidal and tetrahedral type chalcogenide glasses.
From this viewpoint, the previously used Jensen
et al. [9] theoretical calculations of positron annihilation
lifetime versus open volume in application to
chalcogenide glasses should be modified (see Eq. (4)).
One of the reasons for this required modification can be
a suggestion based on the two-component electron-
positron density functional theoretical calculations
[25, 26] that a symmetric fourfold tetravacancies formed
during the clustering process have the positron lifetime
similar to that of divacancies, and fourfold trivacancies
have the positron lifetime similar to that of
monovacancies. This suggestion shows the possible
confusion in the interpretation of positron annihilation
lifetime data and explains why the conclusion made
using Eq. (4) [23] that the size of nanovoids changes
from three atomic vacancies for pyramidal type
chalcogenide glasses to di- and/or monovacancies for
tetrahedral type and pyramidal-tetrahedral type
chalcogenide glasses should be reconsidered.
Finally, on the basis of the modified correlation
equation, Eq. (7), reported here, it is easy to calculate
that the size of nanovoids in the pyramidal type
As2S(Se)3 glasses with average value of radius
R 2.22 Å (or volume V 46 Å3) corresponds to the
size of di- and/or monovacancies similarly to the size of
nanovoids in the tetrahedral type GeS(Se)2 glasses [23]
and mixture of pyramidal and tetrahedral structural units
in chalcogenide glassy network should not practically
affect the size of nanovoids responsible for the FSDP
occurrence within the FSDP-related void-based model
[3-5], as it is indeed experimentally observed [23, 24].
This agreement between calculation and experimental
results may be also considered for validity of linear
Eq. (6) to be applied for 2 0.35 to 0.40 ns in the case
of inorganic polymers like to chalcogenide glasses.
The author would like to thank Prof. Volodymyr
Tsmots (Drohobych Ivan Franko State Pedagogical
University) for stimulating discussions. This work is
supported in part by the State Fund for Fundamental
Researches of Ukraine (project #F40.2/019).
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 136-139.
PACS 61.43.Fs
Modified correlation equation in the FSDP-related void-based model for As2S(Se)3 chalcogenide glasses
T.S. Kavetskyy
Ivan Franko Drohobych State Pedagogical University, Solid-State Microelectronics Laboratory,
24, I. Franko str., 82100 Drohobych, Ukraine
Abstract. Revised in this work is the correlation equation Q1 = 2.3×π/D in the FSDP-related void-based model for As2S(Se)3 chalcogenide glasses between the first sharp diffraction peak (FSDP) position, Q1, and nanovoid diameter, D, are modified to be presented in the form of Q1 = 1.75×π/D, taking into account a newly deduced formula for positron lifetime, (2, versus void radius, R. It is valid for those molecular substrates for R up to 5 Å and when no orthopositronium (o-Ps) or very small fractions of o-Ps components are observed in positron annihilation lifetime spectroscopy.
Keywords: chalcogenide glass, first sharp diffraction peak, positron annihilation, voids.
Manuscript received 01.06.12; revised version received 17.01.13; accepted for publication 19.03.13; published online 25.06.13.
Recently, nanoscale void-species structure has been studied [1, 2] in the As2Se3 glass using combination of X-ray diffraction in respect to the first sharp diffraction peak (FSDP-related XRD), treated within a void-based model [3-5], and positron annihilation lifetime spectroscopy (PALS), analyzed within the two-state positron trapping model [6-8]. Assuming that the same nanovoids are responsible for both FSDP and PALS data in terms of Jensen et al. [9], the analytical correlation relationship between the FSDP position, Q1, and nanovoid diameter, D, has been presented as [1, 2]
Q1 = 2.3 ( (/D.
(1)
In the further work [10], it was shown using the examples of both g-As2Se3 and g-As2S3 (g-glassy) that the analytically found Eq. (1) seems to be similar to the relationship Q1 = 2.5×π/r, reported by Gaskell [11] and/or Ehrenfest’s formula 1.23 = Q1 r/2π reported by Rachek [12], where r is a characteristic distance that corresponds to the inter-atomic correlations detected from the pair distribution function, G(r).
The Monte-Carlo simulation results for g-As2Se3 [13, 14] and g-As2S3 [15] have also been applied to confirm the validity of Eq. (1) for these chalcogenides.
Basically, the Eq. (1) comes from the Elliott interpretation [3-5] of nature of the anomalous first sharp diffraction peak (FSDP) as a chemical-order prepeak in the concentration-concentration structure factor, arising from the clustering of interstitial nanovoids around cation-based structural units, tested for AX2-type glassy-like materials (e.g. GeS(Se)2), and resulting in the empirical relationship between the position of the FSDP, Q1, and atom-void separation like to diameter of nanovoids, D:
Q1 = 3(/2D.
(2)
If Eq. (2) is presented through a coefficient k, depending on the type of glass structure, the Elliott formula may be written as
Q1 = k ( (/D,
(3)
where k = 1.5 for tetravalent AX2-type glasses.
To find a correlation equation in the case of pyramidal type As2S(Se)3 glasses, the Jensen et al. [9] regularity between the positron lifetime, ( (in ns), and vacancy volume, V (in Å3), has been applied [1, 2] in the form of expression:
( ( 0.240 + 0.0013 ( V.
(4)
Recently, Liao et al. [16] reported a newly modified positron lifetime-free volume correlation equation for determination of free volumes up to the mean radius R = 5 Å in polymeric systems, where no orthopositronium (o-Ps) ((3 = 1…4 ns) or very small fractions of o-Ps components are observed in PALS, which is based on the infinitive potential spherical model (originally proposed by Tao [17]) and presented as
(2 = 0.260 ( [1 ( (R/(R + 3.823)) + 1/2( (
( sin(2(R/(R + 3.823))](1.
(5)
For simplicity, it was shown in [16] that
R
-
t
2
correlation for R < 5 Å could be approximated and empirically fitted by a linear equation as
(2 = 0.174(1 + 0.494R),
(6)
where (2 and R are expressed in the units of nanoseconds and angstroms, respectively.
As chalcogenide glasses in their chemical nature are covalent disordered inorganic polymers [18], and, in respect to positron annihilation, they should be considered as inorganic polymers without o-Ps component or with very small fraction of o-Ps component in PALS (see, for example, [8, 19, 20]), the Liao et al. correlation equations (5) and/or (6) are valid in the first approximation for these materials and can be used obviously for As2S(Se)3 alloys.
The values of void radius R or void volume V (V = 4/3((R3) estimated using positron lifetime-free volume correlation equations are given in Table. As seen, the difference between values of V calculated from Eqs. (5) and (6) is (15 Å3, which is rather small to be affected on the average size of voids, whereas the void volume V calculated from Eq. (4) is larger by 30 to 50 Å3.
Liao et al. [16] noted that Eq.(5) should be applied for (2 ( 0.35 to 0.40 ns, while for the shortest (2 < 0.26 ns this equation is not applicable, while the linear Eq. (6) works better. It is noticeable that the correlation coefficients (r2 = 0.9391) for Eq. (5) and (r2 = 0.9268) for Eq. (6) are very similar [16], and, thus, it is suggested that the linear equation could be also used in more general cases for (2 ( 0.35 to 0.40 ns when various polymeric materials (organic and inorganic) are examined. Taking into account that Eq. (5) is deduced by Liao et al. [16] using the example of organic polymers having preferential hydrocarbon environment, while chalcogenide glasses are inorganic polymers, the linear Eq. (6) is applied in the present work.
The values of the open-volume defect-related positron lifetime component (2 are known to be 0.37 ns for g-As2Se3 [21] and 0.36 ns for g-As2S3 [22]. According to Eq. (6), the values of void radius R are 2.28 Å (D = 4.6 Å) for g-As2Se3 and 2.16 Å (D = 4.3 Å) for g-As2S3. The values of the FSDP position Q1 are known to be 1.25 Å
1
-
for g-As2Se3 and 1.26 Å
1
-
for g-As2S3 (see, for example [1, 2, 10, 23] and references therein). Thus, corresponding to Eq. (3), the values of the coefficient k are obtained to be 1.8 for g-As2Se3 and 1.7 for g-As2S3. In average, for pyramidal-type As2S(Se)3 glasses, the value of the coefficient k is 1.75 and modified correlation equation in the FSDP-related void-based model for these alloys can be written as
Q1 = 1.75× π/D.
(7)
Therefore, using the newly modified positron lifetime-free volume correlation equation, Eq. (6), it is found that the established value of k = 1.75 for pyramidal type As2S(Se)3 glasses is very close to the above mentioned Elliott’s value of k = 1.5 for tetravalent AX2-type glasses. On the basis of this similarity in k values, it is suggested that the nature of the FSDP for pyramidal-type As2S(Se)3 glasses and tetravalent AX2-type glasses (e.g., A = Ge, X = S, Se) is the same or very similar. In other words, in terms of the FSDP-related void-based model [3-5], the cation-centered nanovoids with the same or very similar size are responsible for the FSDP occurrence in the both pyramidal and tetrahedral type chalcogenide glasses. This finding gives a new insight on the explanation of experimental observation that the FSDP position in the mixed pyramidal-tetrahedral type chalcogenide glasses, such as As(Sb)2S3-GeS2 and As(Sb)2S3-Ge2S3, is determined exclusively by concentration of Ge cations, while As and Sb cations do not affect the FSDP occurrence [23, 24]. The most probably it is caused by the similar size of cation-centered nanovoids in the pyramidal and tetrahedral type chalcogenide glasses.
From this viewpoint, the previously used Jensen et al. [9] theoretical calculations of positron annihilation lifetime versus open volume in application to chalcogenide glasses should be modified (see Eq. (4)). One of the reasons for this required modification can be a suggestion based on the two-component electron-positron density functional theoretical calculations [25, 26] that a symmetric fourfold tetravacancies formed during the clustering process have the positron lifetime similar to that of divacancies, and fourfold trivacancies have the positron lifetime similar to that of monovacancies. This suggestion shows the possible confusion in the interpretation of positron annihilation lifetime data and explains why the conclusion made using Eq. (4) [23] that the size of nanovoids changes from three atomic vacancies for pyramidal type chalcogenide glasses to di- and/or monovacancies for tetrahedral type and pyramidal-tetrahedral type chalcogenide glasses should be reconsidered.
Finally, on the basis of the modified correlation equation, Eq. (7), reported here, it is easy to calculate that the size of nanovoids in the pyramidal type As2S(Se)3 glasses with average value of radius R ( 2.22 Å (or volume V ( 46 Å3) corresponds to the size of di- and/or monovacancies similarly to the size of nanovoids in the tetrahedral type GeS(Se)2 glasses [23] and mixture of pyramidal and tetrahedral structural units in chalcogenide glassy network should not practically affect the size of nanovoids responsible for the FSDP occurrence within the FSDP-related void-based model [3-5], as it is indeed experimentally observed [23, 24]. This agreement between calculation and experimental results may be also considered for validity of linear Eq. (6) to be applied for (2 ( 0.35 to 0.40 ns in the case of inorganic polymers like to chalcogenide glasses.
The author would like to thank Prof. Volodymyr Tsmots (Drohobych Ivan Franko State Pedagogical University) for stimulating discussions. This work is supported in part by the State Fund for Fundamental Researches of Ukraine (project #F40.2/019).
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Table. The values of positron lifetime (2 as typical for chalcogenide glasses as well as the void radius R and void volume V calculated from Eqs. (4) to (6).
(2 (ns)�
0.35�
0.36�
0.37�
0.38�
0.39�
0.40�
0.41�
�
Jensen’s et al. approach, Eq.(4)�
�
R (Å)�
2.73�
2.80�
2.88�
2.95�
3.02�
3.08�
3.15�
�
V (Å3)�
85�
92�
100�
108�
115�
123�
131�
�
Liao’s et al. approach, Eq.(5)�
�
R (Å)�
2.26�
2.40�
2.50�
2.60�
2.71�
2.81�
2.91�
�
V (Å3)�
48�
58�
65�
74�
83�
93�
103�
�
Liao’s et al. approach, Eq.(6)�
�
R (Å)�
2.05�
2.16�
2.28�
2.40�
2.51�
2.63�
2.75�
�
V (Å3)�
36�
42�
50�
58�
66�
76�
87�
�
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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