Генерація звуку металевими нанокластерами в діелектричній матриці
Побудовано теорiю фотоакустичного ефекту, зумовленого дiєю лазерного опромiнення на металевi нанокластери, iнкорпорованi в дiелектричну матрицю. Поглинута кластерами енергiя поширюється у виглядi тепла в дiелектричнiй матрицi i генерує в нiй згiдно з термодеформацiйним механiзмом звуковi хвилi. У ро...
Gespeichert in:
| Datum: | 2010 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Ukrainian |
| Veröffentlicht: |
Відділення фізики і астрономії НАН України
2010
|
| Schlagworte: | |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/13436 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Генерація звуку металевими нанокластерами в діелектричній матриці / П.М. Томчук, М.І. Григорчук, Д.В. Бутенко // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 443-452. — Бібліогр.: 16 назв. — укр. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-13436 |
|---|---|
| record_format |
dspace |
| spelling |
Томчук, П.М. Григорчук, М.І. Бутенко, Д.В. 2010-11-08T17:29:55Z 2010-11-08T17:29:55Z 2010 Генерація звуку металевими нанокластерами в діелектричній матриці / П.М. Томчук, М.І. Григорчук, Д.В. Бутенко // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 443-452. — Бібліогр.: 16 назв. — укр. 2071-0194 PACS 78.67.Bf; 68.49.Jk; 73.63.-b; 75.75.+a https://nasplib.isofts.kiev.ua/handle/123456789/13436 534 Побудовано теорiю фотоакустичного ефекту, зумовленого дiєю лазерного опромiнення на металевi нанокластери, iнкорпорованi в дiелектричну матрицю. Поглинута кластерами енергiя поширюється у виглядi тепла в дiелектричнiй матрицi i генерує в нiй згiдно з термодеформацiйним механiзмом звуковi хвилi. У роботi отримано формули для акустичного сигналу i виявлено високу чутливiсть амплiтуди звукової хвилi до форми металевих кластерiв, а також таких параметрiв лазерного випромiнювання, як частота, поляризацiя, iнтенсивнiсть. Детально дослiджено поведiнку амплiтуди звукових коливань в областi збудження поверхневих плазмонiв. Знайдено, що ця амплiтуда при поглинаннi свiтла дискретною металевою плiвкою (системою кластерiв у матрицi) в областi плазмонних резонансiв може на кiлька порядкiв перевищувати вiдповiдну амплiтуду при поглинаннi суцiльною металевою плiвкою. Построена теория фотоакустического эффекта, обусловленного действием лазерного излучения на металлические нанокластеры, инкорпорированные в диэлектрическую матрицу. Поглощенная кластерами энергия распространяется в виде тепла в диэлектрической матрице и генерирует в ней в соответствии с термодеформационным механизмом звуковые волны. В работе получены формулы для акустического сигнала и обнаружено высокую чувствительность амплитуды звуковой волны к форме металлических кластеров, а также таким параметрам лазерного излучения, как частота, поляризация, интенсивность. Детально исследовано поведение амплитуды звуковых колебаний в области возбуждения поверхностных плазмонов. Найдено, что эта амплитуда при поглощении света дискретной металлической пленкой (системой кластеров в матрице) в области плазмонных резонансов может на несколько порядков превышать соответствующую амплитуду при поглощении сплошной металлической пленкой. We develop the theory of the photo-acoustical effect caused by a laser action on metal nanoclusters embedded in a dielectric matrix. The energy absorbed by clusters propagates through the dielectric matrix and generates sound waves in it by the thermodeformation mechanism. The formulas for an acoustical signal are derived, and the high sensitivity of the sound wave amplitude to the shape of metal clusters, as well to such parameters of a laser irradiation as the frequency, polarization, and intensity, is revealed. The behavior of the amplitude of sound vibrations in a region of the absorption of surface plasmons is studied in detail. It is found that this amplitude at the light absorption by a discrete metal film (a system of clusters in the matrix) can exceed the corresponding amplitude for the absorption by a continuous metal film in the region of plasmon resonances by several orders of magnitude. Роботу виконано за часткової фiнансової пiдтримки НАН України (проект ВЦ/138). uk Відділення фізики і астрономії НАН України Наносистеми Генерація звуку металевими нанокластерами в діелектричній матриці Генерация звука металлическими нанокластерами в диэлектрической матрице Generation of Sound by Metal Nanoclusters in a Dielectric Matrix Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Генерація звуку металевими нанокластерами в діелектричній матриці |
| spellingShingle |
Генерація звуку металевими нанокластерами в діелектричній матриці Томчук, П.М. Григорчук, М.І. Бутенко, Д.В. Наносистеми |
| title_short |
Генерація звуку металевими нанокластерами в діелектричній матриці |
| title_full |
Генерація звуку металевими нанокластерами в діелектричній матриці |
| title_fullStr |
Генерація звуку металевими нанокластерами в діелектричній матриці |
| title_full_unstemmed |
Генерація звуку металевими нанокластерами в діелектричній матриці |
| title_sort |
генерація звуку металевими нанокластерами в діелектричній матриці |
| author |
Томчук, П.М. Григорчук, М.І. Бутенко, Д.В. |
| author_facet |
Томчук, П.М. Григорчук, М.І. Бутенко, Д.В. |
| topic |
Наносистеми |
| topic_facet |
Наносистеми |
| publishDate |
2010 |
| language |
Ukrainian |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Генерация звука металлическими нанокластерами в диэлектрической матрице Generation of Sound by Metal Nanoclusters in a Dielectric Matrix |
| description |
Побудовано теорiю фотоакустичного ефекту, зумовленого дiєю лазерного опромiнення на металевi нанокластери, iнкорпорованi в дiелектричну матрицю. Поглинута кластерами енергiя поширюється у виглядi тепла в дiелектричнiй матрицi i генерує в нiй згiдно з термодеформацiйним механiзмом звуковi хвилi. У роботi отримано формули для акустичного сигналу i виявлено високу чутливiсть амплiтуди звукової хвилi до форми металевих кластерiв, а також таких параметрiв лазерного випромiнювання, як частота, поляризацiя, iнтенсивнiсть. Детально дослiджено поведiнку амплiтуди звукових коливань в областi збудження поверхневих плазмонiв. Знайдено, що ця амплiтуда при поглинаннi свiтла дискретною металевою плiвкою (системою кластерiв у матрицi) в областi плазмонних резонансiв може на кiлька порядкiв перевищувати вiдповiдну амплiтуду при поглинаннi суцiльною металевою плiвкою.
Построена теория фотоакустического эффекта, обусловленного действием лазерного излучения на металлические нанокластеры, инкорпорированные в диэлектрическую матрицу. Поглощенная кластерами энергия распространяется в виде тепла в диэлектрической матрице и генерирует в ней в соответствии с термодеформационным механизмом звуковые волны. В работе получены формулы для акустического сигнала и обнаружено высокую чувствительность амплитуды звуковой волны к форме металлических кластеров, а также таким параметрам лазерного излучения, как частота, поляризация, интенсивность. Детально исследовано поведение амплитуды звуковых колебаний в области возбуждения поверхностных плазмонов. Найдено, что эта амплитуда при поглощении света дискретной металлической пленкой (системой кластеров в матрице) в области плазмонных резонансов может на несколько порядков превышать соответствующую амплитуду при поглощении сплошной металлической пленкой.
We develop the theory of the photo-acoustical effect caused by a laser action on metal nanoclusters embedded in a dielectric matrix. The energy absorbed by clusters propagates through the dielectric matrix and generates sound waves in it by the thermodeformation mechanism. The formulas for an acoustical signal are derived, and the high sensitivity of the sound wave amplitude to the shape of metal clusters, as well to such parameters of a laser irradiation as the frequency, polarization, and intensity, is revealed. The behavior of the amplitude of sound vibrations in a region of the absorption of surface plasmons is studied in detail. It is found that this amplitude at the light absorption by a discrete metal film (a system of clusters in the matrix) can exceed the corresponding amplitude for the absorption by a continuous metal film in the region of plasmon resonances by several orders of magnitude.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13436 |
| citation_txt |
Генерація звуку металевими нанокластерами в діелектричній матриці / П.М. Томчук, М.І. Григорчук, Д.В. Бутенко // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 443-452. — Бібліогр.: 16 назв. — укр. |
| work_keys_str_mv |
AT tomčukpm generacíâzvukumetaleviminanoklasteramivdíelektričníimatricí AT grigorčukmí generacíâzvukumetaleviminanoklasteramivdíelektričníimatricí AT butenkodv generacíâzvukumetaleviminanoklasteramivdíelektričníimatricí AT tomčukpm generaciâzvukametalličeskiminanoklasteramivdiélektričeskoimatrice AT grigorčukmí generaciâzvukametalličeskiminanoklasteramivdiélektričeskoimatrice AT butenkodv generaciâzvukametalličeskiminanoklasteramivdiélektričeskoimatrice AT tomčukpm generationofsoundbymetalnanoclustersinadielectricmatrix AT grigorčukmí generationofsoundbymetalnanoclustersinadielectricmatrix AT butenkodv generationofsoundbymetalnanoclustersinadielectricmatrix |
| first_indexed |
2025-11-26T22:54:09Z |
| last_indexed |
2025-11-26T22:54:09Z |
| _version_ |
1850779178648993792 |
| fulltext |
NANOSYSTEMS
440 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
GENERATION OF SOUND BY METAL NANOCLUSTERS
IN A DIELECTRIC MATRIX
P.M. TOMCHUK,1 N.I. GRIGORCHUK,2 D.V. BUTENKO1
1Institute of Physics, Nat. Acad. of Sci. of Ukraine
(46, Nauky Ave., Kyiv 03680, Ukraine; e-mail: ptomchuk@ iop. kiev. ua )
2Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: ngrigor@ bitp. kiev. ua )
PACS 78.67.Bf; 68.49.Jk;
73.63.-b; 75.75.+a
c©2010
We develop the theory of the photo-acoustical effect caused by a
laser action on metal nanoclusters embedded in a dielectric ma-
trix. The energy absorbed by clusters propagates through the
dielectric matrix and generates sound waves in it by the thermod-
eformation mechanism. The formulas for an acoustical signal are
derived, and the high sensitivity of the sound wave amplitude to
the shape of metal clusters, as well to such parameters of a laser ir-
radiation as the frequency, polarization, and intensity, is revealed.
The behavior of the amplitude of sound vibrations in a region of
the absorption of surface plasmons is studied in detail. It is found
that this amplitude at the light absorption by a discrete metal film
(a system of clusters in the matrix) can exceed the corresponding
amplitude for the absorption by a continuous metal film in the
region of plasmon resonances by several orders of magnitude.
1. Introduction
At the irradiation of a metal nanocluster by a laser-
generated light beam, hot electrons appear in the clus-
ter. The presence of hot electrons causes the addi-
tional pressure of the electron gas on the cluster sur-
face and induces a heat flow from the cluster to the
environment. If a metal nanocluster (MN) or a sys-
tem of such nanoclusters is positioned into a dielec-
tric matrix, then both above-indicated factors, the ad-
ditional pressure and the heat flow, can generate sound
in the matrix. This occurs in the case where a laser-
generated light beam is nonstationary (for example, a
short laser pulse or a stationary laser-generated light
beam modulated with a low frequency). Such optoa-
coustic effects in a system of MNs positioned into a
transparent matrix (or on its surface) were studied in
[1, 2]. In particular, a thermodeformation mechanism
of sound generation in a matrix by the modulated heat
flow caused by the energy transfer from hot electrons
of the cluster to the dielectric matrix was considered.
Work [1] presented, for the first time, the mechanism
of sound generation by a modulated electron pressure
which appears in MN due to a change of the electron
temperature. A change of the pressure induces oscilla-
tions of the MN surface, and the oscillations of the sur-
face, in turn, generates sound in the matrix. Later on,
the same mechanism of sound generation was proposed
in [3].
In addition to acoustic oscillations of the matrix, ra-
dial oscillations in MN itself are of significant interest,
since a periodic change of the MN radius in the course
of time leads to oscillations in the relaxation dynamics
of the electron temperature (see, e.g., [4]).
In [5], a theory of the absorption of the energy of elec-
tromagnetic waves depending on the shape and size of
small metal particles was constructed. The high sensi-
tivity of the absorption to the shape of a particle and to
the polarization of a wave was established.
In the present work, we intend to study optoacous-
tic effects related to identical MNs with the spheroidal
shape. We assume that such MNs have the same orien-
tation. This can be attained, for example, if MNs are
positioned in a liquid crystal. The absorption of such
a system by MNs depends on the laser radiation po-
larization and will manifest itself in the dependence of
the acoustic signal amplitude in the matrix on the light
polarization. These effects will be considered in what
follows.
GENERATION OF SOUND BY METAL NANOCLUSTERS
We note else that similar optoacoustic effects can oc-
cur also in the case where the absorbing objects are semi-
conducting clusters incorporated into transparent solu-
tions [6].
2. Statement of the Problem
The generation and propagation of long-wave acoustic
oscillations in a dielectric matrix are described by the
equation of motion (see, e.g., [7])
ρ
∂2
∂t2
ui =
∑
j
∂
∂xj
σij , (1)
In (1), ui(r, t) is a component of the displacement vector,
ρ is the mass density of the matrix, σij are components of
the stress tensor, t is the time, and xj are components of
the coordinate vector r. With regard for the temperature
dependence, the components σij have the form [7]
σij = K
{∑
α
uαα − α (Tl − T0)
}
δij+
+2µ
{
uij −
1
3
δij
∑
α
uαα
}
, (2)
where K and µ are, respectively, the moduli of uniform
compression and shear, and α is the constant of thermal
expansion. In addition, the strain tensor in the case of
small deformations can be written as
uij =
1
2
(
∂
∂xi
uj +
∂
∂xj
ui
)
. (3)
In (2), Tl is the temperature of the lattice at the given
point, and T0 is some given temperature at a remote
distance from MN, at which no deformation is present.
If we write the displacement vector u in the form of a
sum of the vectors of longitudinal uL and transverse uT
displacements,
u = uL + uT , ∇× uL = 0, ∇uT = 0, (4)
then relations (1)–(3) yield
∇2uL −
1
s2L
∂2
∂t2
uL =
3Kα
3K + 4µ
∇Tl. (5)
In (5),
sL =
(
3K + 4µ
3ρ
)1/2
(6)
is the velocity of longitudinal sound. In a similar way
for transverse acoustic waves, we obtain
∇2uT −
1
s2T
∂2
∂t2
uT = 0, sT =
√
µ
ρ
, (7)
where sT is the velocity of transverse sound. As is seen,
the temperature gradient does not generate transverse
acoustic oscillations in a medium which is described only
by two moduli K and µ.
Let us introduce a scalar potential Ψ via
uL = ∇Ψ, (8)
Then Eq. (5) yields the following equation for it:
∇2Ψ(r, t)− 1
s2L
∂2
∂t2
Ψ(r, t) =
3Kα
3K + 4µ
[Tl(r, t)− T0] . (9)
A partial solution of this inhomogeneous equation has
the form of a retarded potential [8]
Ψ(r, t) =
1
4π
3Kα
3K + 4µ
∫
δTl(r, t− |r− r′|/sL)
|r− r′|
dr′, (10)
where |r− r′| is the distance from the observation point,
at which we seek a value of the potential, to the volume
element dV ′ = dr′. In (10), we introduced the notation
δTl(r′, t) = Tl(r′, t)− T0. (11)
As for the solution of the homogeneous equation corre-
sponding to (9), this solution describes, as was shown in
[1], the sound generation by oscillations of the MN sur-
face. Such a mechanism of sound generation becomes
significant [1] under the action of short, but powerful
laser pulses. Further, we will consider a situation where
MN undergoes the action of a continuous laser-generated
light beam, whose energy per unit volume of MN is mod-
ulated by acoustic oscillations with a low frequency ωac:
I = I0(1 + cosωact). (12)
In the situation where the light intensity varies smoothly,
we can neglect the sound generation caused by oscilla-
tions of the MN surface.
3. Heat Flows
In order to move further, we need to determine the dis-
tribution of temperatures T (r, t) which determines, ac-
cording to (9), an acoustic signal.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 441
P.M. TOMCHUK, N.I. GRIGORCHUK, D.V. BUTENKO
First, we consider a single MN which has the shape
of an ellipsoid of rotation and is located in the dielec-
tric matrix. Let such an MN be irradiated by a laser-
generated light beam.
In the general case, the balance equations used for
the determination of the electron temperature Te of a
cluster and the temperature of the metal lattice Tm can
be written as a system of two differential equations (see,
e.g., [9])Ce(Te)
∂
∂tTe = div(Ke∇Te)− g(Te − Tl) +Q,
Cl
∂
∂tTl = div(Kl∇Tl)− g(Te − Tl),
(13)
where Ce, Cl, and Ke, Kl are, respectively, the specific
heat capacities and the coefficients of heat conduction
of electrons and the lattice of MN, g is the constant of
electron-phonon energy exchange, the term g(Te − Tl)
characterizes the energy transferred by electrons to the
lattice per unit time, and Q is the energy absorbed by
electrons in unit volume of MN per unit time.
In addition to Eqs. (13) which describe heat processes
in MN, it is necessary to write else the equation for the
temperature of the dielectric matrix surrounding MN.
This equation has the form
Cm
∂
∂t
Tm = div (Km∇Tm) , (14)
Cm and Tm are, respectively, the heat capacity and the
coefficient of heat conduction of the matrix surround-
ing MN. Since it is assumed that the electrons do not
leave MN (for a dielectric), only phonons transfer heat
to the matrix. Therefore, the solutions of Eq. (14) and
the second equation in (13) and the corresponding heat
flows must be “sewed” on the boundary of MN. To this
end, we could write the corresponding boundary condi-
tions for these equations. But we choose another way
instead of the solution of (13) and (14) and the men-
tioned procedure of “sewing.” For small metal islands
under consideration, their sizes are less than the free
path of an electron, and the temperature distributions
Te and Tl over coordinates inside an island are not signif-
icant. Therefore, we can restict ourselves by the solution
of only one equation,
Cm
∂
∂t
Tm = div (Km∇Tm) +G(r, t) (15)
with the function
G(r, t) =
{
g(Te − Tl), r ∈ V,
0, r /∈ V, (16)
which does not depend on coordinates in the volume V
of MN and is equal to zero outside it. Equation (15)
describes the nonuniform heat conduction of an inho-
mogeneous isotropic body. It is obvious that the above-
described approach is exact in the case where the quanti-
ties Cl and Kl coincide, respectively, with Cm and Km.
But if such a coincidence is absent, this approach de-
scribes the process only approximately. However, this
approach is quite proper for small MNs with the almost
homogeneous temperature distributions Te and Tl inside
them. In what follows, we consider that Km is indepen-
dent of the coordinates. Then the general solution of
(15) has the form (see, e.g., [10, 11])
Tm(r, t)− T0 =
κm
π3/2Km
t∫
−∞
dt′×
×
∫
V
G(r′, t′)
[4κm(t− t′)]3/2
exp
[
− |r− r′|2
4κm(t− t′)
]
dr′, (17)
where κm/Km = Cm. The integration in (17) over
t′ is executed from −∞, because we consider that the
source G(t), which sets the initial conditions for the ho-
mogeneous equation, is switched-on at the time moment
t′ = −∞, when
Tm(r, t)|t=−∞ = T0.
To determine the explicit formula for G(r, t), we use
Eq. (13). We recall that Tl = Tm at r ∈ V in the case
where Eq. (15) is valid. We assume again that the free
path of an electron is larger than the size of MN. Then
we can omit the gradient in (13) (electron temperature is
invariable for the whole MN). For such sizes of a cluster,
Eq. (13) yields
g(Te − Tl) = Q− Ce(Te)
∂
∂t
Te. (18)
For a stationary laser-generated light beam, the last
term on the right-hand side of (18) vanishes. But, in our
case, the laser beam intensity is modulated, according to
(12), with a low (acoustic) frequency ωac. Therefore, we
get
∂
∂t
Te ∼ ωacTe. (19)
Since the heat capacity of the electron gas Ce is small
(due to its degeneration) as compared with the heat ca-
pacity of the lattice, and the frequency ωac is low, we
easily prove the inequality
Q� ωacCe(Te)Te. (20)
442 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
GENERATION OF SOUND BY METAL NANOCLUSTERS
We consider that inequality (20) is valid in our case.
Then, according to (16) and (18)–(20), we obtain
G = g(Te − T ) = Q(r, t). (21)
The energy absorbed by a cluster per unit time can be
written in the form
V Q = cSabI, (22)
where Sab is the absorption cross-section of the cluster.
In view of (12) and (22), we can write
Q(r, t) = Q0(r)℘(t), (23)
where
℘(t) =
c
V
SabI0(1 + cosωact), (24)
Q0(r) =
{
1, r ∈ V,
0, r /∈ V.
By substituting formula (23) in (17), we obtain
T (r, t) = T0 +
κm
π3/2Km
∞∫
0
dτ
℘(t− τ)
(4κmτ)3/2
×
×
∫
V
exp
[
−|r− r′|2
4κmτ
]
dr′. (25)
After the Fourier transformation, relation (25) becomes
T (r, ω) =
κm
π3/2Km
℘(ω)
∞∫
0
dτ
eiωτ
(4κmτ)3/2
×
×
∫
V
exp
[
−|r− r′|2
4κmτ
]
dr′. (26)
In (26), the integral over τ can be calculated with the
use of the formula [12]
∞∫
0
dx
x3/2
e−q/xeibx =
√
π
q
exp
(
−2
√
−ibq
)
. (27)
As a result, we get
T (r, ω) =
℘(ω)
4πKm
×
×
∫
V
dr′
|r− r′|
exp
[
−(1− i)
√
ω
2κm
|r− r′|
]
, (28)
where we used the relation
√
−i =
√
2
2 (1− i).
We now consider nanoclusters with the shape of an
ellipsoid of rotation with curvature radii R‖ (along) and
R⊥ (normally to the rotation axis) and with volume V =
4π
3 R‖R
2
⊥. At distances from a cluster greater than its
sizes, i.e.,
r � max
{
R‖, R⊥
}
, (29)
we can approximately write
|r− r′| ≈ r − r′ cosϑ, (30)
where ϑ is the angle between r and r′. Integral (28) can
be calculated in this approximation. But let us assume
that, in addition to (29), the inequality
r′
√
ω
2κm
≤
√
ω
2κm
max
{
R‖, R⊥
}
< 1, (31)
holds for low frequencies ω ∼ ωac. Then formula (28)
takes a very simple form:
T (r, ω) =
V
4π|r|
℘(ω)
Km
exp
[
−(1− i)
√
ω
2κm
|r|
]
. (32)
It is seen from (32) and (24) that, at far (as compared
with the sizes of MN) distances, the temperature gener-
ated in a dielectric matrix depends on the shape of MN
only through the absorption cross-section Sab. But Sab,
as we showed in [5], depends on the shape of MN quite
significantly.
4. Sound Generation
Having determined the distribution of temperatures in
the dielectric matrix, we now turn to relation (10) which
describes, according to (9), longitudinal acoustic oscilla-
tions. In (10), we carry out the Fourier transformation:
Ψ(r, ω) =
1
4π
3Kα
3K + 4µ
∫
dr′
|r− r′|
T (r′, ω)eiω|r−r′|/sL .
(33)
By substituting T (r′, ω) from (32) to the last formula,
we obtain
Ψ(r, ω) =
V
(4π)2
℘(ω)
Km
3Kα
3K + 4µ
×
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 443
P.M. TOMCHUK, N.I. GRIGORCHUK, D.V. BUTENKO
Fig. 1. Scheme of the arrangement of a point of observation and
a plane of metal spheroidal clusters
×
∫
dr′
|r− r′||r′|
e−(1−i)
√
ω/(2κm)|r′|eiω|r−r′|/sL . (34)
As is seen from the integrand of (34), the actual region
of integration over r′ is determined by the relation
|r′| ≤
√
2κm/ω ≡ |r0|. (35)
Therefore, at distances r > r′ from MN, we can set
|r− r′| ≈ r − r′ cosϑ′, where ϑ′ is the angle between
r and r′. In this approximation, integral (34) is easily
calculated, and we obtain
Ψ(r, ω) =
V
4π
ακm
Km
K
ρs2L
℘(ω)
ω
s2L + iκmω
s2L + (κmω/sL)2
eiω|r|/sL
|r|
.
(36)
According to (24), the Fourier-component ℘(ω) reads
℘(ω) = 2πc
SabI0
V
{
δ(ω) +
1
2
[δ(ω − ωac) + δ(ω + ωac)]
}
.
(37)
By substituting relation (37) in (36) and by performing
the inverse Fourier transformation, we have
Ψ(r, t) =
c
4π
I0
ωac
ακm
Km
K
ρs2L
Sab√
1 + (κmωac/s2L)2
×
×cos[ωac(t− |r|/sL)− δ]
|r|
, (38)
where the phase δ is determined by the relation
δ = κmωac/s
2
L.
In (38), we omit the first term with δ(ω), because it is
not related to the sound generation. It is seen that MN
with asymmetric shape generates a spherical acoustic
wave at a remote distance (as compared with the sizes
of MN). The asymmetry of MN absorbing a laser radi-
ation affects (and very significantly) only the acoustic
wave amplitude. This effect, i.e. the influence of the
shape of MN on the acoustic effect, is especially clearly
manifested in the case where we deal with a system of
MNs of the same shape, size, and orientation. For a
system of MNs in the dielectric matrix, formula (38) is
replaced by
Ψ(r, t) =
c
4π
K
ρs2L
I0
ωac
α√
1 + (κmωac/s2L)2
×
×
∑
j
κm
Km
S
(j)
ab
cos [ωac(t− |r− rj |/sL)− δ]
|r− rj |
. (39)
Relation (39) is written in the general case where the
absorption cross-sections of different MNs S(j)
ab and their
coefficients of heat conduction and thermal diffusivity
are different.
5. System of Identical Nanoclusters
We now consider a system of MNs of the same shape and
size which are positioned on a single plane in the matrix
(see Fig. 1).
At a distance remote as compared with that between
MNs, the shape of an acoustic signal depends weakly
on the specific arrangement of MNs in the plane, but it
strongly depend on their orientation.
In what follows, we will consider identical MNs of the
spheroidal shape. Therefore, we assume, in order to sim-
plify calculations, that the centers-of-mass of MNs are
positioned on a square lattice (with a lattice constant
a ), and their rotation axes are parallel to one another.
Under such an assumption, values of the vector rj in for-
mula (39) can be written in the form rj = (nja,mja, 0),
where nj and mj are integers. Then the sum over j
in (39) in the polar coordinate system can be approxi-
mately replaced by the integral∑
j
cos [ωac (t− |r− rj |/sL)− δ]
|r− rj |
≈
≈ 2π
a2
∞∫
0
cos
[
ωac
(
t−
√
z2 + ζ2/sL
)
− δ
]
√
z2 + ζ2
ζdζ =
444 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
GENERATION OF SOUND BY METAL NANOCLUSTERS
=
2π
a2
∞∫
z
cos
[
ωac
(
t− ξ
sL
)
− δ
]
dξ =
=
2π
a2
sL
ωac
sin
[
ωac
(
t− z
sL
)
− δ
]
. (40)
Here, we took the relation |r− rj | =
√
z2 + ζ2 into ac-
count and made change of the variables ξ =
√
z2 + ζ2.
To avoid a possible misunderstanding, we note that in-
tegral (40) has no exact value on the upper limit. This
uncertainty can be eliminated, by considering the effect
of damping of sound at remote distances from the source
(this can be formally realized, by adding an imaginary
term to the sound velocity sl). By substituting relation
(40) in (39) and by introducing the surface density of
MNs N0 = 1/a2, we obtain
Ψ(r, t) =
1
2
KI0
ρs2L
ακm
Km
csL
ω2
ac
N0Sab√
1 + (κmωac/s2L)2
×
× sin
[
ωac
(
t− z
sL
)
− δ
]
. (41)
Formula (41) is similar to that for an acoustic wave in the
case of a continuous film which absorbs light [11]. But
the light absorption which is set by the quantity SabN0
in (41) is anisotropic and depends on the frequency of
a laser beam in a complicated way. Respectively, the
amplitude of the z-th component of the vector of a sinu-
soidal displacement, by (40) and (8), reads
AzL = −1
2
c
ωac
ακm
Km
KI0
ρs2L
N0Sab√
1 + (κmωac/s2L)2
. (42)
It follows from relation (42) that the acoustic waves with
lower frequencies ωac cause greater displacement ampli-
tudes.
Earlier [5, Eq. (83)], we obtained the formula for the
energy absorbed by MN of the spheroidal shape per unit
time under its irradiation by a monochromatic electro-
magnetic wave with frequency ω:
W ≡ V Q =
V
2
3∑
j=1
σjj(εmω2/gj)2|E(0)
j |2
(ω2 − ω2
j )2 + (4πLjσjj/gj)2ω2
. (43)
In (42), σjj is the corresponding diagonal component of
the tensor of high-frequency conduction, εm is the per-
mittivity of the matrix, and Lj is the factor of depolar-
ization. In addition,
ω2
j =
Lj
gj
ω2
pl (44)
is the square of the frequency of a plasmon resonance,
ωpl is the frequency of plasma oscillations of electrons,
and
gj = εm + Lj(1− εm). (45)
For media with εm = 1, we obviously have gj = 1. In
(43), E(0)
j is the j-th component of the amplitude of an
electromagnetic wave which was described in [5] as
E(r, t) = E(0)ei(kr−ωt), (46)
where k is the wave vector. Formula (43) is written in the
general case for a three-axis ellipsoid. In this case, MN is
characterized by three plasmon resonances at frequencies
ωj(j = 1, 2, 3).
In order to study the role of an anisotropy of MN on
the process of sound generation, we consider the simplest
case below. Namely, let MN have the spheroidal shape.
By z, we denote the rotation axis. Then we have σxx =
σyy ≡ σ⊥; σzz = σ‖; Lx = Ly ≡ L⊥; Lz = L‖; gx = gy ≡
g⊥; gz = g‖. Крiм того, ωx = ωy ≡ ω⊥; ωz = ω‖. The
graphic dependence of these frequencies on the degree
of oblateness or elongation of MN positioned in a glass
matrix with εm = 7 is shown in Fig. 2. The formulas
for the factors of depolarization L‖ and L⊥ in (44) can
be found, e.g., in [5].
Using the above-presented notation and taking the re-
lations
W = Sab
( c
8π
√
εm|E(0)|2
)
, (47)
and (43) into account, we obtain the following formula
for the absorption cross-section:
Sab = 4πV
ε
3/2
m
c
ω4
{
(σ‖/g2
‖) cos2 θ
(ω2 − ω2
‖)
2 + (4πL‖σ‖/g‖)2ω2
+
+
(σ⊥/g2
⊥) sin2 θ
(ω2 − ω2
⊥)2 + (4πL⊥σ⊥/g⊥)2ω2
}
, (48)
Here, V is the volume of MN, θ is the angle between
the rotation axis of the spheroid and the unit vector of
the polarization of an electromagnetic wave, and compo-
nents of the tensor of conduction at frequencies ω � ν
(ν is the frequency of electron collisions) are
σ( ‖⊥ )(ω) =
9
32π
(ωpl
ω
)2 υF
R⊥
(
η(es)
ρ(es)
)
, (49)
where the functions η(es) and ρ(es) dependent of the
spheroid eccentricity es can be found in the explicit an-
alytic form, for example, in [5]. For MN of the spherical
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 445
P.M. TOMCHUK, N.I. GRIGORCHUK, D.V. BUTENKO
0 1 2 3 4 5
0
1
3
5
7
R ⊥ /R||
ω | |
ω⊥ω
|
| ,
ω
⊥
(1
0
s
)
-1
1
5
Fig. 2. Plasmon frequencies ω‖ and ω⊥ of dipole oscillations
of electrons, respectively, along (continuous curve) and across
(dashed curve) the rotation axis of a spheroid versus the degree of
oblateness or elongation of a gold spheroidal nanocluster
shape, η(0) = ρ(0) ≡ 2/3. The formula similar to (48)
also describes well the energy absorption by a metal par-
ticle in the case where the width of incident laser pulses
is large [13].
Formula (48) determines both the frequency and the
polarization dependence of the absorption. The factors
of depolarization L‖ and L⊥ define the dependence of
the absorption on the shape of MN and, according to
(44), determine also the positions of plasma resonances.
For a spherical MN, L‖ = L⊥ ≡ 1/3.
6. Discussion of Results
Using (43), we consider that the main contribution to the
absorption cross-section is given by plasma resonances.
The half-width of these resonances for the polarizations
along and across the rotation axis of the spheroid is given
by the formula
γ(‖⊥)(ω) = 2πL( ‖⊥ )σ( ‖⊥ )(ω). (50)
This half-width is a significant physical characteristic,
because it reflects the type of interactions in the system.
Above we obtained formula (41) for a scalar potential
which defines an acoustic displacement according to (8).
In this case, we considered that the laser radiation is ab-
sorbed by a system of identical spheroidal MNs, whose
rotation axes are parallel to one another and lie in the
same plane. The heat flow from these MNs generates
sound in the dielectric matrix. If a continuous metal
film absorbing the laser radiation would be on the sur-
face of the dielectric matrix instead of a system of MNs,
we would obtain a formula for the scalar potential Ψ
similar to (41). But the former will include the quantity
corresponding to the absorption of a continuous film in-
stead of the product N0Sab which determines the share
of a laser-generated light beam absorbed by MN. By
definition, the share of the energy absorbed by a contin-
uous metal film under condition of the light transmission
tending to zero is equal to
η ≈ I − IR
I
= 1−R, (51)
where IR is the intensity of a reflected laser beam, and
R is the share of the energy of a reflected radiation.
Formula (51) requires also the assumption that the film
thickness exceeds the skin depth. Otherwise, the share
of the absorbed energy would be higher.
As is known, at the normal incidence of light incoming
from vacuum, we have
R =
(n− 1)2 + κ2
(n+ 1)2 + κ2
, (52)
where n and κ are defined by the relation√
ε(ω) = n+ iκ, (53)
and the permittivity of a metal takes the form
ε(ω) = ε′(ω) + iε′′(ω) ' 1−
(ωpl
ω
)2
+ i
ν
ω
(ωpl
ω
)2
. (54)
In (54), ν is the frequency of electron-phonon collisions.
In this case, it is assumed that the frequency ω belongs
to the interval
ν < ω < ωpl. (55)
Let us use formulas (51)–(54), inequality (55), and the
assumption that1 κ � n. Then, for a continuous metal
film, we obtain
η ≈ 2
ν
ωpl
. (56)
The evaluation for a film fabricated, for example, from
gold gives η ≈ 0.006.
Thus, for a continuous metal film on the surface of a
dielectric matrix, we would obtain relation (41) for Ψ,
in which 2ν/ωpl would stand instead of N0Sab. In view
of the above discussion, it is expedient to normalize the
amplitude of acoustic oscillations (42) by an analogous
amplitude characteristic of a continuous metal film Afilm.
The ratio of these amplitudes is as follows:
A =
AzL
Afilm
=
ωpl
2ν
N0Sab. (57)
1 For example, n = 0.26 and κ = 2.16 for gold [14].
446 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
GENERATION OF SOUND BY METAL NANOCLUSTERS
Fig. 3. Amplitude of sound oscillations of the matrix versus the de-
gree of oblateness or elongation of a spheroidal MN with a volume
equal to the volume of a sphere with the radius R = 3
√
R2
⊥R‖ =
200 Å at the frequencies of the plasmon resonance Ω−0.1Ω (dashed
curve), Ω (continuous curve), and Ω + 0.1Ω (dash-dotted curve)
and at the incidence angle of a laser beam θ = π/4
In Fig. 3, we present the dependence of the ratio
of sound amplitudes (57) at θ = π/4 on the degree of
oblateness or elongation of a spheroidal MN at the plas-
mon resonance frequency ω = ωpl/
√
3 ≡ Ω characteristic
of a particle of the spherical shape and at two other fre-
quencies which are not much higher or lower than Ω.
Here and below, the calculations of (57) are carried out
with the use of formulas (42) and (48) for a gold particle
in the glass matrix (εm = 7) with the following values of
parameters: ν0◦C ' 3.39× 1013 s−1 [14] ne ' 5.9× 1022
cm−3 [15], a = 2000 Å. The other parameters were cal-
culated by the formulas
ωpl =
√
4πnee2/m, υF =
2π~
m
(
ne
3
8π
)1/3
, (58)
where e and m are, respectively, the charge and mass of
an electron, and ne is the concentration of electrons.
Curve 1 corresponds to the plasmon resonance which
arise in a spherical MN positioned in the medium with
the permittivity εm. For materials of the matrix with
lower permittivities, the resonance shifts to the side
lower ratios R⊥R‖ and approaches R⊥/R‖ = 1 as εm →
1. The most intense sound signal is observed, obviously,
at the plasmon frequencies. We have already established
in out previous studies that the shape of MN is closely
related to frequencies, at which it absorbs in the reso-
nance manner [16]. The frequencies ω < Ω (curve 2) and
ω > Ω (curve 3) correspond, respectively, to plasmon os-
cillations of electrons across and along the spheroid axis
(see Fig. 2). Comparing curves 1–3, we obtain that, for
Fig. 4. Amplitude of acoustic oscillations of the matrix versus
the frequency of a laser wave for various angles θ of its incidence
on an oblate MN (R⊥/R‖ = 1.5): π/4 (continuous curve); π/3
(dashed curve), and π/6 (dash-dotted curve). The volume of a
spheroidal MN corresponds to the volume of a sphere with the
radius R = 200 Å. The distance between MNs a = 2000 Å
more and more oblate MNs, the characteristic resonance
frequencies shift to the short-wavelength side of the spec-
trum. In this case, the half-width of the resonance curve
increases proportionally to the ratio R⊥/R‖.
If we choose MN of a certain shape (oblate or elon-
gated) and vary the carrier frequency of a laser, then
formulas (42) and (48) yield that two sound waves (dou-
blet) appear in MN of the oblate spheroidal shape in
correspondence to two plasmon resonances observed in
MN of such a shape (Fig. 4). The relative height of
peaks in the doublet can be controlled by varying the
incidence angle of an electromagnetic wave.
First, we take the incidence angle of the electromag-
netic wave relative to the rotation axis of the spheroid
to be equal to θ = π/4. That is, this angle is such
that plasmon (dipole) oscillations of electrons across and
along the rotation axis of the spheroid can be excited to
the same extent. In this case, we observe (curve 1) that
the less intense maximum located at lower frequencies
corresponds to the plasmon resonance which arises in a
spheroidal MN at the frequency ω⊥ across the rotation
axis of the spheroid, and the more intense maximum
at higher frequencies corresponds to the plasmon res-
onance at the frequency ω‖ along the rotation axis of
the spheroid. For MN of the elongated shape, on the
contrary, the less intense maximum would be at higher
frequencies and would correspond to the frequency ω⊥.
The different intensities of the peaks is caused by the
corresponding frequency dependence of the factors of de-
polarization which are present in (48). As the incidence
angle increases from π/4 to π/3 (curve 2), the intensity
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 447
P.M. TOMCHUK, N.I. GRIGORCHUK, D.V. BUTENKO
of the sound amplitude peak increases at the frequency
ω⊥ and decreases at the frequency ω‖. On the contrary,
the incidence angle decreases from π/4 to π/6 (curve 3),
the intensity of the sound amplitude peak decreases at
the frequency ω⊥ and increases at the frequency ω‖. It
is obvious that, at the angle θ = 00, the peak at the fre-
quency ω⊥ disappears in correspondence with (48), and
the intensity of the peak at the frequency ω‖ becomes
maximum. Otherwise, at an angle of π/2, the peak at
the frequency ω‖ disappears, and the peak at the fre-
quency ω⊥ remains maximum.
By concluding, it is worth noting that, at the above-
chosen sizes of MNs and distances between them, all
MNs cover only 3% of the area, in which the centers-of-
mass of MNs are positioned. In this case, the amplitude
of sound waves generated by MNs exceeds the amplitude
of waves which can arise under the same conditions in a
continuous metal film on the surface of the dielectric ma-
trix by several orders (in our case, by three orders). Such
significant difference between optoacoustic properties of
a discrete (island) metal film and a continuous metal film
is related to the fact that the light absorption by dis-
crete films reaches a maximum in the frequency range of
plasmon resonances, whereas this frequency range cor-
responds to the region of the almost complete reflection
of light for a continuous metal film.
7. Conclusions
We have developed the theory of acoustooptic phenom-
ena for metal nanoclusters incorporated in a dielectric
matrix which allows one to determine the amplitude of
acoustic oscillations of the matrix at various polariza-
tions of the incident electromagnetic wave. The obtained
analytic formulas allow one to evaluate the heat flows
between MNs and the matrix, to determine the tem-
perature of the matrix at any time moment and at any
distance from MN, and to find the dependence of the am-
plitude of an acoustic signal on elastic constants of the
medium, intensity of a laser beam, and the cross-section
of absorption by MN.
The case where the frequency of a laser beam is close
to plasmon frequencies of a spheroidal MN is studied in
detail. We have obtained the dependence of the ampli-
tude of acoustic oscillations of the matrix on the degree
of oblateness or elongation of a metal spheroidal cluster
on the frequencies of a plasmon resonance.
As a function of the frequency of a laser beam, the am-
plitude of an acoustic wave in a spheroidal MN has two
maxima with different intensities, as distinct from spher-
ical MNs, where a single maximum is observed. This is
caused by resonances which arise at the excitation by a
laser at the frequencies of oscillations of a plasmon along
and across the rotation axis of a spheroid. By the dis-
tance between the doublet peaks, we can estimate the
degree of oblateness or elongation of MN. The intensity
of the doublet peaks can be controlled by the variation
of the incidence angle of a laser beam relative to the
rotation axis of a spheroid.
We have revealed a significant difference of optoacous-
tic properties of discrete and continuous metal films on
the surface of a transparent dielectric matrix in the re-
gion of plasmon resonances.
The work was executed under the partial financial sup-
port of the NASU (project VTs/138).
1. P.M. Tomchuk, Ukr. Fiz. Zh. 38, 1174 (1993).
2. I.V. Blonsky, E.A. Elyseev, and P.M. Tomchuk, Ukr. Fiz.
Zh. 45, 1110 (2000).
3. M. Perner, S. Gresillon, J. Marz, G. von Plessen, J. Feld-
mann et al., Phys. Rev. Lett. 85, 792 (2000).
4. G.V. Hartland, J. Chem. Phys. 57, 403 (2006).
5. P.M. Tomchuk and N.I. Grigorchuk, Phys. Rev. B 73,
155423 (2006).
6. I.V. Blonsky, M.S. Brodin, Yu.P. Piryatinskii, G.M.
Tel’biz, V.A. Tkhorik, P.M. Tomchuk, and A.G. Filin,
Zh. Teor. Eksp. Fiz. 107, 1685 (1995).
7. L.D. Landau and E.M. Lifshitz, Theory of Elasticity
(Pergamon Press, London, 1970).
8. N.S. Koshlyakov, E.B. Gliner, and M.M. Smirnov, Basic
Differential Equations of Mathematical Physics (GIFML,
Moscow, 1962) (in Russian).
9. R.D. Fedorovich, A.G. Naumovets, and P.M. Tomchuk,
Phys. Rep. 328, 73 (2000).
10. Yu K. Danilenko, A.A. Manenkov, and V.S. Nechitailo,
Trudy Fiz. Inst. im. P.N. Lebedeva 101, 31 (1978).
11. L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Perg-
amon Press, Oxford, 1975).
12. A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, In-
tegrals and Series. Elementary Functions (Gordon and
Breach, New York, 1986).
13. N.I. Grigorchuk and P.M. Tomchuk, Fiz. Nizk. Temp. 34,
576 (2008).
14. N.I. Grigorchuk and P.M. Tomchuk, Phys. Rev. B 80,
155456 (2009).
15. Ch. Kittel, Introduction to Solid State Physics (Wiley,
New York, 1974).
16. P.M. Tomchuk and N.I. Grigorchuk, Ukr. Fiz. Zh. 52,
889 (2007).
Received 04.07.09.
Translated from Ukrainian by V.V. Kukhtin
448 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
GENERATION OF SOUND BY METAL NANOCLUSTERS
ГЕНЕРАЦIЯ ЗВУКУ МЕТАЛЕВИМИ НАНОКЛАСТЕРАМИ
В ДIЕЛЕКТРИЧНIЙ МАТРИЦI
П.М. Томчук, М.I. Григорчук, Д.В. Бутенко
Р е з ю м е
Побудовано теорiю фотоакустичного ефекту, зумовленого дiєю
лазерного опромiнення на металевi нанокластери, iнкорпорова-
нi в дiелектричну матрицю. Поглинута кластерами енергiя по-
ширюється у виглядi тепла в дiелектричнiй матрицi i генерує в
нiй згiдно з термодеформацiйним механiзмом звуковi хвилi. У
роботi отримано формули для акустичного сигналу i виявлено
високу чутливiсть амплiтуди звукової хвилi до форми мета-
левих кластерiв, а також таких параметрiв лазерного випро-
мiнювання, як частота, поляризацiя, iнтенсивнiсть. Детально
дослiджено поведiнку амплiтуди звукових коливань в областi
збудження поверхневих плазмонiв. Знайдено, що ця амплiту-
да при поглинаннi свiтла дискретною металевою плiвкою (си-
стемою кластерiв у матрицi) в областi плазмонних резонансiв
може на кiлька порядкiв перевищувати вiдповiдну амплiтуду
при поглинаннi суцiльною металевою плiвкою.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 449
|