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J. Phys. Chem. A 2010, 114, 5068–5075
A Microscopic-Macroscopic Analysis for Mixed Energy Transfer Schemes in Doped Amorphous Solids F. Ferraro*,†,‡ and C. Z. Hadad*,† Grupo de Quı´mica-Fı´sica Teo´rica, Instituto de Quı´mica, UniVersidad de Antioquia, Calle 67 No. 53108, bloque 2 oficina 337, Apartado Ae´reo 1226, Medellı´n, Colombia and Grupo de Quı´mica Cua´ntica RelatiVista, Facultad de Recursos Naturales, Departamento de Ciencias Quı´micas, UniVersidad Andre´s Bello, Repu´blica 275, Santiago de Chile, Chile ReceiVed: October 6, 2009; ReVised Manuscript ReceiVed: February 9, 2010
In this study we propose a methodology to treat mixed and more complex energy transfer schemes to reach the time-dependent intensities of luminescence in doped amorphous materials. We start outlining the differential equations for the time variation of microscopic probabilities of being in the relevant states and then transforming them into equations for the variation of the relevant macroscopic states populations and solve those equations. By this method, statistical approaches to the initially excited and up-converted states transient populations for up-conversion processes in the presence of cross relaxation in lanthanide-monodoped amorphous solid are calculated. The resultant formulations produce plots that show correct general tendencies and are coherent with what would be expected for systems exhibiting both mechanisms, hence they could be used in the fitting of experimental curves to calculate some important parameters. The new solution method is more convenient that the classical analysis because it permits the introduction of more realistic time dependent functions for the interacting optical centers and allows showing the time dependence of the macroscopic energy transfer rates in the equations for the dynamics of the involved populations. We apply our method to the particular case of mixing of up-conversion and cross relaxation phenomena; however, because of its general characteristics, we suggest it could be applied to other mixings or more complex schemes. I. Introduction There are multiple paths by which energy transfer processes between optical centers in doped materials can happen. At the microscopic level, energy transfer processes take place by means of electric or magnetic multipole interactions, by exchange interactions, or by emission and subsequent absorption of photons.1–3 They also can occur via different kinds of kinetic mechanisms: energy back-transfer, energy migration, cross relaxation, frequency upconversion, among others.2–7 Both the nature of the host lattice and the nature of the optical centers are determinant in the preferred kinetic mechanism by which this process occurs. However, there are several systems in which two or more mechanisms are present simultaneously.8 Fo¨rster, while studying the energy transfer between single molecules in frozen solutions,9 proposed a mathematical formalism for electric dipole interactions between optical centers and produced a statistical treatment to explain the observed macroscopic behavior. This model was generalized by Inokuti-Hirayama for multipole interactions.10 Later, Dexter developed mathematical treatments for the energy transfer microscopic rate in inorganic materials.11 From a technological point of view, an important mechanism is frequency up-conversion by energy transfer.12–16 In this mechanism electromagnetic radiation of shorter wavelength with respect to the radiation of the excitation source is obtained. This energy conversion allows applications such as display devices (because it is possible to get visible light from longer wavelength sources),17–19 biosensors (for example, determination of trace amounts of avidin based on up-conversion fluorescence by resonant energy transfer generated by a donor-acceptor pair20), up-conversion lasers (where * To whom correspondence should be addressed. E-mail: f.ferraro@ uandresbello.edu (F.G.F.),
[email protected] (C.Z.H.). † Instituto de Quı´mica, Universidad de Antioquia. ‡ Universidad Andres Bello.
it is possible to reach the population inversion through energy transfer between excited states),21–23 and other devices.24 Because of the intrinsic complexity of the up-conversion mechanism and the usual presence of other competing processes associated with it, the conventional treatment to explain the experimental luminescence curves is conducted with direct equations for the variation of the relevant populations in the involved states. However, this procedure does not account for the statistical effects on the time-resolved luminescence normally observed when some energy transfer process is present in solid materials.25 In previous works,25,26 we proposed a statistical model for timeresolved up-converted luminescence that corresponds to an extension of the Inokuti-Hirayama model and in which the upconversion mechanism is the only kinetic mechanism present. Statistical treatments for mixed schemes of energy migration or for diffusion in presence of cross relaxation and/or backtransfer are the continuous-time-random-walk model, the coherent-potential approximation, and the Gochanour-Andersen-Fayer method.27–31 Sergeyev et al.32 developed a model for the mixing between upconversion and migration of energy. The model proposed is mainly for steady state luminiscence, but has some limitations in the case of pulsed excitation.25,26 Additionally, some numerical techniques for mixed schemes consist in Monte Carlo simulations.33–35 A special case of mixed energy transfer scheme is the upconversion process in the presence of cross relaxation. Some particular examples of systems where this mix is present are Pr3+ single doped and Pr3+/Yb3+ codoped nanocrystalline Gd3Ga5O12,36 and Tm3+-doped silica fiber lasers.8 In this paper we propose an analytical treatment for the mixing of kinetic mechanisms in the context of time-resolved luminescence. In particular, the development is applied to amorphous materials monodoped with trivalent lanthanides, mainly for f-f energy transfer transitions, and for those systems where a combination between upconversion and cross relaxation is the predominant mechanism. In
10.1021/jp909569v 2010 American Chemical Society Published on Web 03/29/2010
Mixed Energy Transfer Schemes in Doped Amorphous Solids
J. Phys. Chem. A, Vol. 114, No. 15, 2010 5069 environment. This is because, inter alia, they contain the resonance integral,11,12,25,26 where the differences between energy levels are considered. Effective angular factors can be expressed in terms of spectroscopic and electronic parameters.12,26 Alternatively, Ωs and Ωs′ can representing parameters to evaluate by fitting resultant equations to experimental luminescent curves. Differential equations for the temporary evolution of the microscopic probabilities of being in the relevant states, F(1) k (t), (3) F(2) j (t), and Fj (t), for generic ions, j and k, are NV
dF(1) k (t) ) -γ1F(1) k (t) + dt dF(2) j (t) dt
∑ P(rkj)Fj(2)(t)F(0)k (t)
(2)
k*j*i
NV
) -γ2F(2) j (t) - 2
∑ W(r ) · F
-
(2) (2) j (t)Fi (t)
ij
i*j*k
NV
Figure 1. Diagram of relevant energy levels and transitions for the interacting optical centers. Initially the “j” and “i” centers are both in the intermediate state “2” and the “k” center is in the ground state, “0”. The excited “j” generic center could return to the less energetic states by radiative or nonradiative paths or could transfer its energy to or receive the energy from the neighboring “i” center (W(rij) in the figure). The “j” center also has the possibility to transfer its energy to the “k” ground state center (P(rkj) in the figure).
many systems where migration is important, the lack of energy migration is valid at low active optical center concentration (which means low pump power or low optical center concentrations). For the sake of simplicity, this condition of low active optical center concentration is another assumption of this model.
∑ P(r )F kj
dF(3) j (t) dt
Figure 1 shows a simplified transition diagram for the mixing between the energy transfer up-conversion and cross relaxation processes. It should be noted that even though the amorphous condition changes the energy levels from an optical center to another by local differences of electronic environment, we plot the energy levels as being equivalent for both optical centers. This is because they are of the same nature and we will treat the differences in average way in the energy transfer microscopic rates. Interacting optical centers are initially found in the intermediate excited state, labeled (2). The state is reached after pulsed selective excitation of intensity Iexc, short enough not to give rise to significant populations in other excited states reached by energy transfer within the excitation time. From there, optical centers have the possibility to radiatively and/or nonradiatively decay with rate γ2 ) γ2r + γ2nr. Optical centers also have the possibility to transfer their energy to other optical centers that are either in the intermediate state (2) or in the ground state (0). In the first case the result is optical centers in a second excited state of higher energy than the intermediate state (up-conversion process). In the second case optical centers appear in an excited state of lower energy than the intermediate state (cross relaxation process). Microscopic rates of up-conversion, W(rij), and cross relaxation, P(rkj), are given by the conventional approximations10–12,25,26
W(rij) )
Ωs rijs
and P(rkj) )
Ωs′ rkjs
(1)
where s ) 6, 8, and 10, for dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interaction, respectively. The terms Ωs and Ω′s are effective angular factors in this model.11,12,25,26 They are representative of any interacting optical pair in the solid, i.e., they are averages among the whole interacting pairs. This assumption ensures remediating, in part, the fact that an amorphous condition changes the energy levels (and transition probabilities) from an optical center to another by local differences of electronic
(3)
NV
) -γ3F(3) j (t) +
∑ W(r )F ij
(2) (2) j (t)Fi (t)
(4)
i*j*k
They fulfill Fz(0)(t) + Fz(1)(t) + Fz(2)(t) + Fz(3)(t) ) 1, z ≡ j,k, or i ≡ 1,2,...,NV, where NV is the total number of optical centers in a given volume, V, of the solid; γ1 ) γ1r + γ1nr and γ3 ) γ3r + γ3nr are the radiative and/or nonradiative decay rates of the first and third excited states, respectively. The partial solution of eq 3 is F(2) j (t)
II. Model
(2) (0) j (t)Fk (t)
k*j*i
)
F(2) j (0)
{
NV
exp -γ2t - 2
∑ W(r ) ∫ F t
ij
0
(2) i (t)
dt -
}
(5)
1, centers initially excited 0, centers initially in the ground state
(6)
i)1
NV
∑ P(r ) ∫ F t
kj
k)1
0
(0) k (t)dt
where
{
Fj(2)(0) )
For the i optical centers, that are in direct interaction with the j center, we could consider the following approximations:
Fi(2)(t) ≈ Fi(2)(0) ⇒ T(t) ≡
∫0t 1 dt ) t
(7)
and, -γ2t
∫0t e-γ t dt ) 1 -γe2
Fi(2)(t) ≈ Fi(2)(0) e-γ2t ⇒ T(t) ≡
2
(8) Equation 7 is valid at short times, and eq 8 is valid in the case of low active optical center concentrations. Because the ground state population is larger than the population in any other state, it is a good approximation to take
∫0t 1 dt ) t
(0) F(0) k (t) ≈ Fk (0) ⇒
(9)
By taking into account eqs 6, 8, and 9, we can rewriting eq 5 as NV(2)(0)
Fj(2)(t)
)
Fj(2)(0)e-γ2t
∏e
-2W(rij)T(t)
i)1
NV(2)(0)
∏ e-P(r )t kj
k)1
(10)
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This form suggests using the Fo¨rster-like procedure9,10 to calculate the intermediate state average probability. The procedure of Fo¨rster considers a representative donor of average behavior throughout the solid. This donor is likely to be affected by simultaneous independent processes. Of those, the distance-dependent processes, i.e., the energy transfer processes, contribute with
∫V exp(-tW(r))F(r) dV
(11)
to the decay of the representative donor by each interacting acceptor, where t is the time (or a function of time, T(t), in our case), W(r) is the energy transfer microscopic rate, F(r) is the probability distribution of the donor-acceptor distance, r, in the volume V.10 In our case we have three possible independent processes that could affect to the donor dynamic; intraoptical center decay, upconversion energy transfer, and cross relaxation energy transfer; and we have N(2) V (0) interacting optical centers (acceptors). Therefore, the intermediate state average probability is9,25,26
F¯ (2)(t) ) F¯ (2)(0)exp{-γ2t} ×
{
lim
NV(2)(0)f∞
[
R ∫r)R V
]
4πr2 -2W(r)T(t) dr e m V
NV(2)(0)
RVf∞
}{
lim
NV(0)(0)f∞
[
R ∫r)R V
]
4πr2 -P(r)t dr e m V
RVf∞
NV(0)(0)
}
(12)
where V ) is the volume of the solid, and (4πr )/(V) is the probability distribution of the donor-acceptor distance, r, in the volume V in a random collection of points. In the context of Fo¨rster-like models,9,10,25,26 the ideal random collection of points is the approximation for the possible dopant positions in amorphous solids of homogeneous distribution (without clustering or segregation) and certain concentration. The structural differences between different hosts is taken into consideration in this model by means of the minimal approach distance between two ions in the lattice Rm.25 Taking into account all the above considerations and a minimal approach distance between two ions in the lattice Rm, we can solve averages in eq 12 using the modified Inokuti-Hirayama method,25 (4πRV3 )/3
2
{
}
4π (2) NF¯ (0)Γ[1 - 3 / s](2ΩsT(t))3/s 3 × exp{-V(Rm)NF¯ (2)(0)(e-2W(Rm)T(t) - Γ[1 - 3 / s, 2W(Rm)T(t)](2W(Rm)T(t))3/s)} 4π × exp - NF¯ (0)(0)Γ[1 - 3 / s](Ωs′t)3/s 3 × exp{-V(Rm)NF¯ (0)(0)(e-P(Rm)t - Γ[1 - 3 / s, P(Rm)t](P(Rm)t)3/s)}
F¯ (2)(t) ) F¯ (2)(0)e-γ2texp -
{
}
(13)
3 s s where V(Rm) ) 4πRm /3, W(Rm) ) Ωs/Rm , and P(Rm) ) Ωs′/Rm correspond to the volume and the energy transfer microscopic rates for up-conversion and cross relaxation at Rm, respectively; Γ[a] is the gamma function, and Γ[a,x] is the incomplete gamma function. The macroscopic temporal population is found by the equation, N(2)(t) ) NFj(2)(t). To facilitate the next section analysis, we will consider dipole-dipole interactions (s ) 6) and we will not take into consideration the Rm factor. By means of eqs 7 and 9, the averaged intermediate state probability is
√t
F¯ (2)(t) ) F¯ (2)(0)e-γ2te-f
(14)
where f ) a + b and a ) (4Fj (0)Nπ Ω6)/(3), b )(4Fj (0)Nπ Ω6′)/(3). With the approximations eqs 8 and 9, Fj(2)(t) becomes (2)
3
(0)
3
F¯ (2)(t) ) F¯ (2)(0)e-γ2te-a
1-e-γ2t -b√t γ2 e
(15)
Now, we will change the previous equations for the time variation of microscopic probabilities to equations for the population dynamics. This path is more appropriate to solve complex energy transfer processes, because its partial results are simple enough to introduce the time-dependent functions for the interacting optical centers and permits the visualization of the underlying physics involved and its relationship to the classic macroscopic direct rate equations for the problem analysis. For example, it will permit an explicit understanding of the time dependency of the macroscopic energy transfer rates in the equations for the dynamics of the involved populations. Using the relationship,
NV
∑ Fj(est)(t) j)1
V
)
NV(est)(t) ) N(est)(t) ) V
N
∑ Fj(est)(t)
(16)
j)1
on eqs 2, 3 and 4, we obtain the macroscopic differential equations,
dN(1)(t) ) -γ1N(1)(t) + PCRTE(t)N(2)(t) dt
(17)
dN(2)(t) ) -γ2N(2)(t) - 2WUPTE(t)N(2)(t) - PCRTE(t)N(2)(t) dt
(18)
dN(3)(t) ) -γ3N(3)(t) + WUPTE(t)N(2)(t) dt
(19)
where N(0)(t) + N(1)(t) + N(2)(t) + N(3)(t) ) N, and N ) NV/V, is the optical density (optical centers concentration). The terms WUPTE(t) and PCRTE(t), are the macroscopic energy transfer rates of up-conversion and cross relaxation, respectively, defined by:
Mixed Energy Transfer Schemes in Doped Amorphous Solids NV(2)(t) NV(2)(t)
∑ ∑ W(rij)Fj(2)(t)Fi(2)(t)
1
WUPTE(t) )
NV(2)(t)
j)1
(20)
i*j
NV(2)(t) NV(0)(t)
∑ ∑ P(rkj)Fj(2)(t)F(0)k (t)
1
PCRTE(t) )
J. Phys. Chem. A, Vol. 114, No. 15, 2010 5071
NV(2)(t) j)1
(21)
k*j
By inserting eqs 5 and 16 into eqs 20 and 21, we obtain
WUPTE(t) )
PCRTE(t) )
(
(
ΝV
ΝV
ΝV
ΝV
)
∑ ∑ W(rij)Fj(2)(0) exp{-γ2t - 2 ∑ W(rij) ∫0 Fi(2)(t) dt - ∑ P(rkj) ∫0 F(0)k (t) dt}Fi(2)(t) /
1 NV j)1
ΝV
(
i*j
t
k
i
ΝV
∑
1 F(2)(0) exp{-γ2t - 2 NV j)1 j
ΝV
t
ΝV
ΝV
ΝV
t
i
t
k
ΝV
)
∑ ∑ P(rkj)Fj(2)(0) exp{-γ2t - 2 ∑ W(rij) ∫0 Fi(2)(t) dt - ∑ P(rkj) ∫0 F(0)k (t) dt}F(0)k (t) /
1 NV j)1
(
k*j
t
i
t
k
ΝV
∑
1 F(2)(0) exp{-γ2t - 2 NV j)1 j
)
(22)
)
(23)
∑ W(rij) ∫0 Fi(2)(t) dt - ∑ P(rkj) ∫0 F(0)k (t) dt}
ΝV
ΝV
∑
W(rij)
i
∫0t Fi(2)(t) dt - ∑ P(rkj) ∫0t F(0)k (t) dt} k
In both terms, the numerator and the denominator have been multiplied by (1/NV). The denominator in eqs 22 and 23 corresponds to the intermediate state probability average.25 The temporal derivative of the logarithm of this average is,
{
d 1 d ln(F¯ (2)(t)) ) ln dt dt NV
ΝV
∑
ΝV
Fj(2)(0) exp{-γ2 - 2
j
∑
}
ΝV
W(rij)
i
∫0t Fi(2)(t) dt - ∑ P(rkj) ∫0t F(0)k (t) dt} k
d ln(F¯ (2)(t)) ) -γ2 dt
(
(
NV
NV
ΝV
(
i*j
i
ΝV
(24)
)
∑ ∑ W(rij)Fj(2)(0) exp{-γ2t - 2 ∑ W(rij) ∫0 Fi(2)(t) dt - ∑ P(rkj) ∫0 F(0)k (t) dt}Fi(2)(t) /
2 NV j)1
NV
NV
∑
1 F(2)(0) exp{-γ2t - 2 NV j)1 j
∑
∫0t Fi(2)(t) dt - ∑ P(rkj) ∫0t F(0)k (t) dt} k
ΝV
(
k*j
∑
W(rij)
i
∫
ΝV
F(2)(t) dt 0 i
NV
∑
W(rij)
i
)
t
k
ΝV
∑
-
∑ P(rkj) ∫0 F(0)k (t) dt}F(0)k (t) /
t
1 F(2)(0) exp{-γ2t - 2 NV j)1 j
∫
ΝV
t
F(2)(t) dt 0 i
Replacing eqs 22 and 23 in eq 25 and reordering, we obtain
WUPTE(t) ) -
)
ΝV
W(rij)
i
P(rkj)Fj(2)(0) exp{-γ2t - 2
t
k
ΝV
NV
∑∑
1 NV j)1
t
{
)
∑ P(rkj) ∫0 F(0)k (t) dt} t
k
}
1 d ln(F¯ (2)(t)) + γ2 + PCRTE(t) 2 dt
(25)
(26)
From eq 19 and given the initial condition, N(3)(0) ) 0, the partial solution for the time dependent up-converted population is
N(3)(t) ) e-γ3t
∫0t eγ tWUPTE(t)N(2)(t) dt ) Ne-γ t ∫0t eγ tWUPTE(t)F¯ (2)(t) dt 3
3
By replacing eq 26 into eq 27 and carrying out the integration we obtain
{
(27)
3
}
t d ln(F¯ (2)(t)) N N(3)(t) ) - e-γ3t 0 eγ3t + γ2 + PCRTE(t) F¯ (2)(t) dt 2 dt t t t dF¯ (2)(t) N ) - e-γ3t γ2 0 eγ3tF¯ (2)(t) dt + 0 eγ3t dt + 0 eγ3tPCRTEF¯ (2)(t) dt 2 dt t t N N(3)(t) ) {e-γ3tF¯ (2)(0) - F¯ (2)(t) + (γ3 - γ2)e-γ3t 0 eγ3tF¯ (2)(t) dt - e-γ3t 0 eγ3tF¯ (2)(t)PCRTE(t) dt} 2
{
∫
∫
∫
∫
∫
where
〈∑ 〉 NV
PCRTE(t) )
P(rkj) ≈
k*j*i
∫
2F¯ (0)(0)N√π3Ω6′ 3√t
}
(28)
(29)
is an approximation to the time dependent cross relaxation macroscopic velocity calculated without the minimal approaching radius correction (see Appendix).
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Figure 2. Semilogarithmic plots for the temporal intermediate state population (from eq 15) with (UCCR, dashed-dotted curve) and without (UC, dotted curve) cross relaxation process. The solid straight line corresponds to the case of simple exponential decay luminescence, DS. The parameters employed are indicated in the insert, where F means Fj(2)(0).
Figure 3. Temporal up-converted state population (from eq 28). The solid curve corresponds to the case of pure up-conversion process (UC). When cross relaxation is added, the temporal up-converted population decreases (dashed-dotted curve).
Figure 4. Temporal up-converted population (from eq 28) for different optical densities, Ni (i ) 1, 2, 3) and for some typical parameters. The solid curve corresponds to concentration N1, the dashed curve to concentration N2, and the dashed-dotted curve to concentration N3.
Mixed Energy Transfer Schemes in Doped Amorphous Solids
J. Phys. Chem. A, Vol. 114, No. 15, 2010 5073
Figure 5. Temporal up-converted population (from eq 28) for different intensities of external excitation (through the initially excited optical centers fractions; F ≡ Fj(2)(0)) and for some typical parameters. The solid curve corresponds to F1, the dashed curve to F2 and the dashed-dotted curve to F3.
Figure 6. Temporal up-converted population (from eq 28) for different cross relaxation strengths (through the angular factor, ΩCR) and for some typical parameters.
III. Results and Discussion Figure 2 shows the effect of adding the cross relaxation process to the up-conversion process on the temporal intermediate state population (from eq 15) and for some typical parameters. In the semilogarithmic scale the simple exponential decay luminescence is the solid straight line. When the up-conversion process is added the typical simple exponential decay loss of the intermediate state luminescence appears (dotted curve in Figure 2). With the additional presence of cross relaxation, a more pronounced deviation from simple exponential decay loss is generated (dasheddotted curve in Figure 2). This is because when cross relaxation is present, it increases the total population of acceptor optical centers with the addition of a large amount of optical centers in the ground state. Figure 3 shows that the up-converted temporal population (From eq 28 and eq 8) diminishes when the cross relaxation competitive mechanism is added (pure up-conversion, UC), in solid curve, in comparison to the mixing between cross relaxation and up-conversion processes, in the dashed-dotted curve.
Figures 4 and 5 show the dependence of the up-converted population on the optical density, N, and on the intensity of external excitation (through the initially excited optical centers fractions; F ≡ Fj(2)(0), in Figure 5). It can be seen that the maximum of the temporal up-converted population increases with an increment of both quantities. The plot shows the correct tendency because the microscopic energy transfer interaction leading to up-conversion processes increases with an increment of the active intermediate state interacting optical centers, even when the cross relaxation process also increases. Finally, and for the sake of completitud in checking the results, Figures 6 and 7 show the increase in the maximum of the up-converted temporal population when the cross relaxation strength (through the angular factor, ΩCR) and the intermediate state radiative/nonradiative rate, γ2, decline. Both tendencies are close to what is expected. Therefore, the above graphical analysis shows us that the model is physically coherent, so, it could be used in the fitting of experimental curves to obtain information about the different parameters involved in mixed processes. More general and important is the realization that
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Figure 7. Temporal up-converted population (from eq 28) for different intermediate state radiative/nonradiative rates γ2, and for some typical parameters.
our methodology, which previously substitutes the equations for the variation of microscopic probabilities into equations for the variation of the relevant states populations (eqs 2-4 into eqs 17-19) is convenient to reach the up-converted population (eq 28) in cases of mixed energy transfer mechanisms. Another advantage of conversion to equations for populations is that it is easy to understand the time dependence of macroscopic energy transfer rates (WUPTE(t) and PCRTE(t) in eqs 17-19) in time-dependent resolved experiments. Hence, this kind of treatment could be helpful to avoid mistakes in analysis that begins with the direct formulation of equations for the variation of the implicated populations. Therefore, the proposed path of Section II constitutes a general methodology for posterior studies of similar complexity. IV. Conclusions The proposed analytical approach that turns previous equations for the time variation of microscopic probabilities into equations for the populations dynamics is appropriate to solve complex energy-transfer processes. It allows the introduction of realistic time dependent functions for the interacting optical centers and for explicit understanding of the time dependency of the macroscopic energy transfer rates in the equations for the dynamics of the involved populations. Even though this path was presented for the particular case of the mixture of up-conversion and cross relaxation phenomena in monodoped amorphous solids, by its general scope, we suggest it could be applied to other mixed or more complex schemes. The application to other mixed schemes will be published elsewhere. The graphical analysis for the resultant formalism show that our equations are physically coherent in the sense of reproducing the expected trends. Therefore, they could be used in the fitting of experimental curves to obtain information about the different parameters involved in mixed energy transfer processes.
state when there is only the cross relaxation process, are eqs B1 and B2, respectively
PCRTE(t) )
(
(
{
NV(2) NV(0)
∑ ∑ P(rkj)Fj(2)(0) exp
1 NV j)1 NV(0)
∑ P(rkj) ∫0 F(0)k (t) dt k
NV(2)
∑
t
{ {
1 F(2)(0) exp -γ2t NV j)1 j
-γ2t -
} )
k
NV(0)
F(0) k (t) /
})
(B1)
}}
(B2)
∑ P(rkj) ∫0 F(0)k (t) dt t
k
{
NV(2)
∑
d 1 d F(2)(0) exp -γ2t ln(F¯ (2)(t)) ) ln (2) dt dt N j)1 j V NV(0)
∑ P(rkj) ∫0 F(0)k (t) dt
(
k
1 ) -γ2 NV NV(0)
∑ P(r ) ∫ F t
kj
k
0
(0) k (t)
t
NV(2) NV(0)
∑ ∑ P(r )F kj
} )( j)1
(2) j (0)
k
dt Fk(0)(t) /
1 NV
NV(2)
∑F
{ {
exp -γ2t -
(2) j (0)exp
j)1
NV(0)
-γ2t -
∑ P(r ) ∫ F t
kj
0
k
(0) k (t)
dt
})
The second term of eq B2 corresponds to eq B1, hence after replacing and reordering we obtain
{
PCRTE(t) ) - γ2 +
}
d ln(F¯ (2)(t)) dt
(B3)
Appendix: Approximation to the Cross-Relaxation Energy Transfer Macroscopic Rate According to ref 25, an expression for the isolated cross relaxation energy transfer macroscopic rate (temporal average among optical centers), and an expression for the temporal derivative of the logarithm of the averaged probability of being in the intermediate
From eq 13 an approximated expression for the averaged probability of being in the intermediate state when there is only the cross relaxation process and without the minimal approaching radius correction is
Mixed Energy Transfer Schemes in Doped Amorphous Solids
{
F¯ (2)(t) ) F¯ (2)(0) e-γ2t exp -
}
3 4π (0) NF¯ (0)Γ[1 - 3/s](Ωs′t) /s 3 (B4)
By replacing this last expression into eq B3 we obtain
PCRTE(t) )
3 3 4π (0) F¯ (0)NΓ[1 - 3/s](Ωs′) /s (t)3-s/s 3 s
(B5) When s ) 6 we obtain eq 29. Acknowledgment. We are thankful to Universidad de Antioquia, Centro de Costo 3514, and to professor Albeiro Restrepo Cossio for the revision of the manuscript of this article. References and Notes (1) Jenkins, R. D.; Andrews, D. L. J. Phys. Chem. A 1998, 102, 10834. (2) Di Bartolo, B. Energy Transfer Processes in Condensed Matter; Di Bartolo, B., Ed.; NATO Advanced Study Institutes Series; Plenum Press: New York, 1984. (3) Malta, O. L. J. Non-Cryst. Solids. 2008, 354, 4770. (4) Ryan, T. G.; Jackson, S. D. Opt. Comm. 2007, 273, 159. (5) Luxbacher, T.; Fritzer, H.; Flint, C. J. Lumin. 1997, 71, 177. (6) Campbell, M.; Flint, C. D. Spectrochim. Acta Part A 1998, 54, 1583. (7) Fernandez, B. R.; Adam, J. L.; Mendioroz, A.; Arriandiaga, M. A. J. Non-Cryst. Solids 1999, 29, 256. (8) Jackson, S. D. Opt. Comm 2004, 203, 197. (9) Forster, T. Naturforsch, Z., (1949). 4, 321. (10) Inokuty, M.; Hirayama, F. J. Chem. Phys. 1965, 43, 1978. (11) Dexter, D. L. J. Chem. Phys. 1953, 21, 83. (12) Miyakawa, T.; Dexter, D. L. Phys. ReV. B 1970, 1, 70.
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