A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited
We use the density functional method to examine the properties of the nonuniform (two-phase) fluid of twolevel atoms, a part of which is excited. From the analysis of the equation of state of a gas of two-level atoms, a part of which is excited, the following density functional of the grand thermo...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited / O. Derzhko, V. Myhal // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 703–708. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1214502017-06-15T03:03:40Z A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited Derzhko, O. Myhal, V. We use the density functional method to examine the properties of the nonuniform (two-phase) fluid of twolevel atoms, a part of which is excited. From the analysis of the equation of state of a gas of two-level atoms, a part of which is excited, the following density functional of the grand thermodynamical potential emerges width (here ΩCS[ρ(r)] is the Carnahan-Starling term, σ is the atom radius, v = 4/3πσ³, c₁ is the concentration of excited atoms, c₀ + c₁ = 1, E₁ − E₀ is the excitation energy and a is the dimensionless parameter which characterizes the atom). We use this expres Ми використовуємо метод функцiоналу густини для дослiдження властивостей неоднорiдного (двофазного) плину дворiвневих атомiв, частина з яких збуджена. На основi аналiзу рiвняння стану газу дворiвневих атомiв, частина з яких збуджена, виникає наступний функцiонал густини великого термодинамiчного потенцiалу, (тут ΩCS[ρ(r)] – доданок Карнагана-Старлiнга, σ – радiус атома, v = 4/3πσ³, c₁ – концентрацiя збуджених атомiв, c₀ + c₁ = 1, E₁ − E₀ – енергiя збудження i a – безрозмiрний параметр, який характеризує атом). Ми використовуємо цей вираз для обчислення нуклеацiйного бар’єру для фазового переходу пари в рiдину за наявностi збуджених атомiв. 2006 Article A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited / O. Derzhko, V. Myhal // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 703–708. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 64.70.Fx, 82.65.Dp, 62.60.Nh, 64.60.Qb DOI:10.5488/CMP.9.4.703 http://dspace.nbuv.gov.ua/handle/123456789/121450 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
We use the density functional method to examine the properties of the nonuniform (two-phase) fluid of twolevel
atoms, a part of which is excited. From the analysis of the equation of state of a gas of two-level atoms,
a part of which is excited, the following density functional of the grand thermodynamical potential emerges width (here ΩCS[ρ(r)] is the Carnahan-Starling term, σ is the atom radius, v = 4/3πσ³, c₁ is the concentration of
excited atoms, c₀ + c₁ = 1, E₁ − E₀ is the excitation energy and a is the dimensionless parameter which
characterizes the atom). We use this expres |
| format |
Article |
| author |
Derzhko, O. Myhal, V. |
| spellingShingle |
Derzhko, O. Myhal, V. A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited Condensed Matter Physics |
| author_facet |
Derzhko, O. Myhal, V. |
| author_sort |
Derzhko, O. |
| title |
A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited |
| title_short |
A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited |
| title_full |
A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited |
| title_fullStr |
A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited |
| title_full_unstemmed |
A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited |
| title_sort |
microscopic theory of photonucleation: density functional approach to the properties of a fluid of two-level atoms, a part of which is excited |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121450 |
| citation_txt |
A microscopic theory of photonucleation: Density functional approach to the properties of a fluid of two-level atoms, a part of which is excited / O. Derzhko, V. Myhal // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 703–708. — Бібліогр.: 24 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT derzhkoo amicroscopictheoryofphotonucleationdensityfunctionalapproachtothepropertiesofafluidoftwolevelatomsapartofwhichisexcited AT myhalv amicroscopictheoryofphotonucleationdensityfunctionalapproachtothepropertiesofafluidoftwolevelatomsapartofwhichisexcited AT derzhkoo microscopictheoryofphotonucleationdensityfunctionalapproachtothepropertiesofafluidoftwolevelatomsapartofwhichisexcited AT myhalv microscopictheoryofphotonucleationdensityfunctionalapproachtothepropertiesofafluidoftwolevelatomsapartofwhichisexcited |
| first_indexed |
2025-07-08T19:55:28Z |
| last_indexed |
2025-07-08T19:55:28Z |
| _version_ |
1837109901098221568 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 4(48), pp. 703–708
A microscopic theory of photonucleation: Density
functional approach to the properties of a fluid of
two-level atoms, a part of which is excited ∗
O.Derzhko1, V.Myhal2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 The Ivan Franko National University of L’viv, Department for Theoretical Physics,
12 Drahomanov Street, 79005 L’viv, Ukraine
Received August 26, 2005
We use the density functional method to examine the properties of the nonuniform (two-phase) fluid of two-
level atoms, a part of which is excited. From the analysis of the equation of state of a gas of two-level atoms,
a part of which is excited, the following density functional of the grand thermodynamical potential emerges
Ω[ρ(r)] = ΩCS[ρ(r)] −
6σ3a(c1, T )
π
�
|r1−r2|>2σ
dr1dr2
ρ(r1)ρ(r2)
|r1 − r2|6
with
a(c1, T ) =
1
32
a2v(E1 − E0) �c0 − c1 + 2c0c1
E1 − E0
kT �
(here ΩCS[ρ(r)] is the Carnahan-Starling term, σ is the atom radius, v = 4/3πσ3, c1 is the concentration of
excited atoms, c0 + c1 = 1, E1 − E0 is the excitation energy and a is the dimensionless parameter which
characterizes the atom). We use this expression to calculate the nucleation barrier for vapor-to-liquid phase
transition in the presence of excited atoms.
Key words: photonucleation, nucleation barrier, density functional approach
PACS: 64.70.Fx, 82.65.Dp, 62.60.Nh, 64.60.Qb
The studies of equilibrium properties of a gas of identical atoms, a part of which is in an excited
electronic state, have attracted attention for the last forty years [1–10]. Such atoms may appear due
to the electromagnetic irradiation with the frequency, which corresponds to the excitation energy
of the atom. Since the life-time of the excited state is essentially larger than the time required
for establishing the equilibrium over translational degrees of freedom, the system should exhibit
equilibrium properties at a given (nonequilibrium) concentration of excited atoms. Moreover, owing
to new effective long-range interatomic interactions – the resonance dipole-dipole interactions – one
may expect essential changes of various equilibrium characteristics due to the presence of excited
atoms. The analysis performed within the framework of the cluster expansion method confirmed
these expectations (see [7,8] and references therein). On the other hand, there are only a few
papers which may be related to experimental observations of theoretically investigated features
of the gas with excited atoms. We should mention here the papers on the effect of irradiation on
the condensation of iodine and anthracene vapor [11,12] and on the photonucleation [13–21], in
particular, in vapors of mercury and cesium. The latter studies reported the quantitative results
of resonance irradiation effect on the nucleation rate. The main conclusion of these studies is as
follows: the resonance irradiation of the nucleation zone leads to a sharp increase of the nucleation
rate. The experimental results, to our best knowledge, have not been explained so far. One may
∗The paper submitted to the Proceedings of the conference “Statistical physics 2005: Modern problems and new
applications” (August 28–30, 2005, Lviv, Ukraine).
c© O.Derzhko, V.Myhal 703
O.Derzhko, V.Myhal
try to interpret these data basing on the theory of nucleation in a supersaturated vapor which
contains excited atoms. Such atoms may appear as a result of resonance irradiation.
In what follows we present preliminary results about the nucleation phenomena in a fluid
of two-level atoms, a part of which is excited (see also [22]). In our study we use the density
functional approach developed by D.W. Oxtoby with coworkers [23]. This method permits to
obtain the nucleation rate basing on the first principles. The nucleation rate J is connected with
the nucleation barrier A, J = J0 exp
(
− A
kT
)
. The nucleation barrier can be calculated within the
framework of the classical nucleation theory which relies on the capillarity approximation [23]
Acl
kT
=
16π
3
( γ
kT
)3 1
ρ2
l ln2 s
. (1)
Here γ is the surface tension of the vapor-liquid interface, ρl is the density of liquid, s = p/p0 is
the supersaturation, p is the actual pressure of the supersaturated vapor and p0 is the equilibrium
pressure. To compute the nucleation rate according to equation (1) one has at first to construct the
vapor-liquid phase diagram determining the equilibrium pressure p0 and the liquid density ρl at
the temperature T and then to compute the surface tension γ at this temperature. This calculation
can be done within the density functional approach considering the planar vapor-liquid interface.
An alternative approach to the calculation of the nucleation barrier which does not use the key
assumption in the classical nucleation theory – the capillarity approximation – was suggested by
D.W. Oxtoby [23]. According to this scheme one has to consider a metastable vapor in a spherical
vessel of the radius R, assume the appearance of a spherical liquid droplet in the center of the
vessel and analyse the density profile and the value of the grand thermodynamical potential of such
a two-phase fluid with a spherical vapor-liquid interface. The value of the grand thermodynamical
potential of such a metastable fluid Ω(T, µ, V ) permits to calculate the nucleation barrier via the
equation
A = Ω(T, µ, V ) −
(
−p
4
3
πR3
)
. (2)
In our study we use both schemes in calculating the vapor-to-liquid nucleation barrier in a system
of two-level atoms, a part of which is excited.
To perform the theoretical analysis of the vapor-to-liquid nucleation in the presence of the
excited atoms we need an appropriate density functional of the grand thermodynamical potential.
We assume the grand thermodynamical potential to be a functional of the density with the form
Ω[ρ(r)] = kT
∫
dr1ρ(r1)
(
ln
(
Λ3ρ(r1)
)
+
−1 + 6vρ(r1) − 4v2ρ2(r1)
(1 − vρ(r1))
2
)
−
6σ3a(c1, T )
π
∫
|r1−r2|>2σ
dr1dr2
ρ(r1)ρ(r2)
|r1 − r2|
6
− µ
∫
dr1ρ(r1), (3)
where
a(c1, T ) =
a2
32
v(E1 − E0)
(
1 − 2c1 + 2(1 − c1)c1
E1 − E0
kT
)
. (4)
Here Λ is thermal de Broglie wavelength of the atom, v = 4/3 ·πσ3, σ is the radius of the atom, c1
is the concentration of the excited atoms, a = d2σ−3(E1 − E0)
−1 is the dimensionless parameter
which characterizes the two-level atom (in what follows we set a = 1), E1 − E0 is the excitation
energy, d is the value of the transitional electrical dipole moment between the ground and excited
states. The equilibrium density minimizes the grand thermodynamical potential Ω[ρ(r)], i.e. it is
the solution of the following integral equation
kT ln
(
Λ3ρ(r1)
)
+ kT
8vρ(r1) − 9v2ρ2(r1) + 3v3ρ3(r1)
(1 − vρ(r1))
3
−
12σ3a(c1, T )
π
∫
|r1−r2|>2σ
dr2
ρ(r2)
|r1 − r2|
6
− µ = 0. (5)
704
A microscopic theory of photonucleation: Density functional approach
Substituting the equilibrium density into equation (3) one obtains the value of the grand ther-
modynamical potential of the system Ω(T, µ, V ). The adopted density functional of the grand
thermodynamical potential (3) is consistent with the virial state equation obtained earlier [1]. It
takes into account the short-range interaction within the Carnahan-Starling local approximation
(the first term in the r.h.s. of equation (3)) neglecting the difference of the atom radii in the ground
and excited states. Moreover, it takes into account the long-range interactions, in particular, the
resonance dipole-dipole interactions, within the mean-field approximation (the second term in the
r.h.s. of equation (3)). Note, that the coefficient a(c1, T ) (4) depends on temperature only when
c1 deviates from zero. More sophisticated density functionals are available but they have not been
employed in the present study.
It is convenient to introduce the dimensionless units of energy, temperature, chemical potential,
length, volume, density, pressure, surface tension etc renormalizing these quantities as follows:
E → E
E1−E0
, T → kT
E1−E0
, µ → µ
E1−E0
, r → r
σ
, V → V
v
, ρ → vρ, p → pv
E1−E0
γ → γσ2
E1−E0
etc,
respectively. As a result equations (3), (4), (5) become
Ω[ρ(r)] =
3T
4π
∫
dr1ρ(r1)
(
ln
(
Λ3
v
ρ(r1)
)
+
−1 + 6ρ(r1) − 4ρ2(r1)
(1 − ρ(r1))
2
)
−
9α(c1, T )
2π2
∫
|r1−r2|>2
dr1dr2
ρ(r1)ρ(r2)
|r1 − r2|
6
−
3µ
4π
∫
dr1ρ(r1), (6)
α(c1, T ) =
a2
32
(
1 − 2c1 +
2(1 − c1)c1
T
)
, (7)
T ln
(
Λ3
v
ρ(r1)
)
+ T
8ρ(r1) − 9ρ2(r1) + 3ρ3(r1)
(1 − ρ(r1))
3
−
12α(c1, T )
π
∫
|r1−r2|>2
dr2
ρ(r2)
|r1 − r2|
6
− µ = 0. (8)
Moreover, we assume for concreteness in equations (4) and (7) a = 1. We also set without loss of
generality Λ3/v = 1.
We start with the phase diagram of the system. For this purpose we assume the constancy
of the density, ρ(r) = ρ, that immediately yields instead of equations (6) and (8) the following
expressions
Ω(ρ) = TρV
(
ln ρ +
−1 + 6ρ − 4ρ2
(1 − ρ)
2
)
− α(c1, T )ρ2V − µρV (9)
and
T ln ρ + T
8ρ − 9ρ2 + 3ρ3
(1 − ρ)
3
− 2α(c1, T )ρ − µ = 0. (10)
Solving equation (10) with respect to ρ and substituting this density into equation (9) one gets
the value of the grand thermodynamical potential Ω(T, µ, V ). One can also eliminate, using equa-
tion (10), the chemical potential µ from equation (9) thus getting the equation of state
−
Ω(T, ρ, V )
TV
=
p
T
= ρ
1 + ρ + ρ2 − ρ3
(1 − ρ)
3
−
a(c1, T )
T
ρ2. (11)
Equation (11) agrees with the virial state equation of a gas of two-level atoms, a part of which
is excited, obtained earlier [1] (see also [7,8]). Equation (10) may have more than one solution
which yield the same value of the grand thermodynamical potential Ω(T, µ, V ). Indeed, for a given
temperature T let us fix the value of the grand thermodynamical potential −Ω/V = p and solve
equation (11) with respect to ρ. At high temperatures (above the critical temperature Tc) one finds
only one solution ρ which corresponds to a certain value of the chemical potential µ in equation (10).
705
O.Derzhko, V.Myhal
At low temperatures (below the critical temperature Tc) one finds several solutions ρ with the
corresponding values of chemical potential µ which follow from equation (10). Varying the value
of the grand thermodynamical potential −Ω/V = p one finds such two densities ρv and ρl > ρv
which yield the same value of µ. The quantities T , p = p0, ρv, ρl, µ = µ0 correspond to the points
on the phase diagram where the two phases, liquid and vapor, coexist (see figure 1). In figure 1
.
.
.
.
.
p
1
r
_
.
.
.
.
.
.
.
p
1
r
_
Figure 1. The isotherms p vs ρ−1 at T = 0.6Tc(0) (left panel) and T = 0.8Tc(0) (right panel).
Bold curves correspond to c1 = 0, thin curves correspond to c1 = 0.00006 (= 0.006%). The left
(right) endpoint of the horizontal part of the isotherm (corresponding to p0) gives the liquid
density ρl (the vapor density ρv).
we display the isotherms which correspond to the temperatures T = 0.6Tc(0) ≈ 0.00176866 (left
panel) and T = 0.8Tc(0) ≈ 0.00235822 (right panel) (here Tc(0) denotes the critical temperature
Tc without excited atoms, i.e. when c1 = 0) for two concentrations of excited atoms, c1 = 0
(bold curves) and c1 = 0.00006 (= 0.006%) (thin curves). Considering at first the case c1 = 0 at
T = 0.8Tc(0) we find that the equilibrium values of the pressure, the chemical potential, the liquid
density, and the vapor density are p0 ≈ 0.00004118, µ0 ≈ −0.00996108, ρl ≈ 0.30719568, and
ρv ≈ 0.02172324, respectively. Assume further that in a system the concentration of excited atoms
becomes c1 = 0.00006. For such a fluid the equilibrium values of the pressure is p0 ≈ 0.00003179 and
the vapor with the pressure ≈ 0.00004118 becomes metastable with the value of supersaturation
parameter s ≈ 1.29559738. Moreover, the equilibrium values of the chemical potential, the liquid
density, and the vapor density of the fluid with c1 = 0.00006 at T = 0.8Tc(0) are µ0 ≈ −0.01049448,
ρl ≈ 0.32745048, and ρv ≈ 0.01596817, respectively.
To calculate the vapor-to-liquid nucleation barrier according to equation (1) one has to find the
surface tension γ. Analyzing the density profile for a planar vapor-liquid interface (for this purpose
we consider a two-phase system in a cylinder of the radius R and the height L) at T = 0.8Tc(0) and
c1 = 0.00006 and estimating Ω(T, µ0, V ) we find according to the relation γπR2 = Ω(T, µ0, V ) −
(−p0πR
2L) the value of the surface tension γ = 0.00051195. As a result one immediately gets the
value of the vapor-to-liquid nucleation barrier A/T ≈ 68.3294 (see figure 2, dash-dotted curve 3).
Obviously, since A/T becomes now finite (and decreases as c1 increases) the nucleation of liquid
from vapor becomes now possible.
On the other hand, we can calculate the nucleation barrier based on the equation (2). First we
estimate the Thompson radius r? = 2γ(kTρl ln s)−1 at T = 0.8Tc(0) and c1 = 0.00006. We obtain
r?/2 ≈ 6.614283.
706
A microscopic theory of photonucleation: Density functional approach
A
__
T
Figure 2. The dependence of the vapor-to-liquid
nucleation barrier on the concentration of ex-
cited atoms c1 at two temperatures T = 0.6Tc(0)
(curves 1 and 2) and T = 0.8Tc(0) (curves 3 and
4). The curves 1 and 3 were obtained using equa-
tion (1), the curves 2 and 4 were obtained using
equation (2).
We note that r? is rather small which may
be a reason to go beyond the classical nucle-
ation theory since the capillarity approxima-
tion cannot be justified for such small droplets.
Next we calculate the chemical potential for the
supersaturated vapor with excited atoms ac-
cording to equation (10) with ρ ≈ 0.02227178
(this value of density follows from equation (11)
for c1 = 0.00006, T = 0.8Tc(0) and p ≈
0.00004118), µ = −0.01049448 + 0.00049852,
and analyse the density profile of a spherical
droplet in the supersaturated vapor seeking for
a “stable” value of the grand thermodynami-
cal potential which plays the role of Ω(T, µ, V )
in equation (2) (for details see [23]). We find
A/T ≈ 67.12 that agrees with the value ob-
tained within the framework of the classical
nucleation theory. The described calculations
have to be repeated for other values of concen-
tration c1. Moreover, we perform such calcula-
tions for several values of temperature. Some of
our findings are collected in figure 2.
The main conclusion which can be read
off from figure 2 is as follows: the vapor af-
ter the appearance of excited atoms becomes
metastable with s > 1 and the nucleation bar-
rier for vapor-to-liquid phase transition becomes essentially diminished. This outcome agrees with
a naive expectation that the long-range resonance dipole-dipole interactions should act in favor
of liquid formation in vapor. Although the present consideration permits to obtain the nucleation
rate which can be measured experimentally much more work is required to compare theory and
experiment. Firstly, we have to analyse in detail the results for nucleation rates [15,16] obtained
using the upward thermal diffusion cloud chamber setup [24]. Secondly, we should bare in mind
that the fluids whose photonucleation has been studied have more complicated particle structure
and interparticle interactions. The comparison with experiment can therefore be only qualitative
at present, and in this respect our results are consistent with the data reported in [15,16].
One of the authors (O.D.) is grateful to the DAAD for the support of his visit to Philipps-
Universität Marburg in the autumn of 1995. He wishes to thank Dr. Hermann Uchtmann for kind
hospitality and many stimulating conversations.
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Мiкроскопiчна теорiя фотонуклеацiї: метод функцiоналу
густини для дослiдження властивостей плину дворiвневих
атомiв, частина з яких збуджена
О.Держко1, В.Мигаль2
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1
2 Львiвський нацiональний унiверситет iм. I.Франка, факультет теоретичної фiзики,
79005 Львiв, вул. Драгоманова 12, Україна
Отримано 26 серпня 2005 р.
Ми використовуємо метод функцiоналу густини для дослiдження властивостей неоднорiдного (дво-
фазного) плину дворiвневих атомiв, частина з яких збуджена. На основi аналiзу рiвняння стану газу
дворiвневих атомiв, частина з яких збуджена, виникає наступний функцiонал густини великого тер-
модинамiчного потенцiалу
Ω[ρ(r)] = ΩCS[ρ(r)] −
6σ3a(c1, T )
π
�
|r1−r2|>2σ
dr1dr2
ρ(r1)ρ(r2)
|r1 − r2|6
з
a(c1, T ) =
1
32
a2v(E1 − E0) �c0 − c1 + 2c0c1
E1 − E0
kT �
(тут ΩCS[ρ(r)] – доданок Карнагана-Старлiнга, σ – радiус атома, v = 4/3πσ3, c1 – концентрацiя збу-
джених атомiв, c0 + c1 = 1, E1 − E0 – енергiя збудження i a – безрозмiрний параметр, який хара-
ктеризує атом). Ми використовуємо цей вираз для обчислення нуклеацiйного бар’єру для фазового
переходу пари в рiдину за наявностi збуджених атомiв.
Ключовi слова: фотонуклеацiя, нуклеацiйний бар’єр, метод функцiоналу густини
PACS: 64.70.Fx, 82.65.Dp, 62.60.Nh, 64.60.Qb
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