ISSN 0021-3640, JETP Letters, 2008, Vol. 88, No. 1, pp. 12–18. © Pleiades Publishing, Ltd., 2008. Original Russian Text © K.K. Pukhov, T.T. Basiev, Yu.V. Orlovskii, 2008, published in Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, 2008, Vol. 88, No. 1, pp. 14–20.
Spontaneous Emission in Dielectric Nanoparticles K. K. Pukhov, T. T. Basiev, and Yu. V. Orlovskii Prokhorov General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia e-mail:
[email protected] Received March 11, 2008; in final form, June 3, 2008
An analytical expression is obtained for the radiative-decay rate of an excited optical center in an ellipsoidal dielectric nanoparticle (with sizes much less than the wavelength) surrounded by a dielectric medium. It is found that the ratio of the decay rate Anano of an excited optical center in the nanoparticle to the decay rate Abulk of an excited optical center in the bulk sample is independent of the local-field correction and, therefore, of the adopted local-field model. Moreover, the expression implies that the ratio Anano/Abulk for oblate and prolate ellipsoids depends strongly on the orientation of the dipole moment of the transition with respect to the ellipsoid axes. In the case of spherical nanoparticles, a formula relating the decay rate Anano and the dielectric parameters of the nanocomposite and the volumetric content c of these particles in the nanocomposite is derived. This formula reduces to a known expression for spherical nanoparticles in the limit c 1, while the ratio Anano/Abulk approaches unity as c tends to unity. The analysis shows that the approach used in a number of papers {H. P. Christensen, D. R. Gabbe, and H. P. Jenssen, Phys. Rev. B 25, 1467 (1982); R. S. Meltzer, S. P. Feofilov, B. Tissue, and H. B. Yuan, Phys. Rev. B 60, R14012 (1999); R. I. Zakharchenya, A. A. Kaplyanskii, A. B. Kulinkin, et al., Fiz. Tverd. Tela 45, 2104 (2003) [Phys. Solid State 45, 2209 (2003)]; G. Manoj Kumar, D. Narayana Rao, and G. S. Agarwal, Phys. Rev. Lett. 91, 203903 (2003); Chang-Kui Duan, Michael F. Reid, and Zhongqing Wang, Phys. Lett. A 343, 474 (2005); K. Dolgaleva, R. W. Boyd, and P. W. Milonni, J. Opt. Soc. Am. B 24, 516 (2007)}, for which the formula for Anano is derived merely by substituting the bulk refractive index by the effective refractive index of the nanocomposite must be revised, because the resulting ratio Anano/Abulk turns out to depend on the local-field model. The formulas for the emission and absorption cross sections σnano for nanoparticles are derived. It is shown that the ratios σnano/σbulk and Anano/Abulk are not equal in general, which can be used to improve the lasing parameters. The experimentally determined and theoretically evaluated decay times of metastable states of dopant rare-earth ions in crystalline YAG and Y2O3 nanoparticles are compared with the corresponding values for bulk crystals of the same structure. PACS numbers: 78.55.-m, 78.67.Bf DOI: 10.1134/S0021364008130043
INTRODUCTION
sample. The existence of spontaneous emission was postulated in 1917 by Einstein in his quantum theory of the interaction between the equilibrium radiation and matter [9]. It was shown in this paper that the statistical equilibrium between matter and radiation can only be achieved if spontaneous emission exists together with the stimulated emission and absorption. The quantummechanical expression for the Einstein coefficient A equal to the probability of spontaneous emission from a two-level atom in a vacuum has been obtained by Dirac in [10]. In 1946, Purcell [11] showed that the spontaneous-emission probability can drastically increase if the radiating dipole is placed in a cavity (see also [12, 13] and references therein). The inverse phenomenon, i.e., the inhibition of the spontaneous emission, can take place in three-dimensional periodic dielectric structures [14]. Progress in this field is so rapid that the number of citations of [14] approaches 5000. Variations in the probability of the spontaneous emission from optical centers near the planar interface of the dielectrics have been the subject of active studies since the 1970s [15–22]. Modifications of the spontane-
Interest in studying the optical characteristics of nanosized materials, including theoretical and experimental studies of the spontaneous lifetimes of optical centers (OCs) in nanosized samples has grown considerably at present [1–8]. The difference in the OC spontaneous lifetime in nanosized samples in comparison with the corresponding value in bulk materials is of great interest not only for the fundamental physics, but also for practical applications. For example, the increased lifetime of a metastable level in a lasing medium makes it possible, by increasing the pumppulse duration several times, to reduce the power and cost of the diode laser-pump source and superluminescence losses while keeping the output-radiation energy and power intact. The adequate theoretical interpretation of the experimental results is of primary importance at the current stage of investigations. It is of great interest to derive a formula describing the spontaneous decay rate of an excitation in a nanosized object and reveal its differences from the corresponding expression for the bulk 12
SPONTANEOUS EMISSION IN DIELECTRIC NANOPARTICLES
ous emission from OCs located in the vicinity of a metal mirror were also analyzed [23]. Chew [24, 25] considered the modification of the spontaneous emission from an optical center inside and outside the dielectric sphere by modeling the optical center with an oscillating dipole. His analytical results were confirmed later on in [26] devoted to the spontaneous emission from a two-level atom inside a dielectric sphere. The problem of the spontaneous emission from an atom near a prolate spheroid was considered in [27]. The problem of the spontaneous emission from an atom in the vicinity of a triaxial nanosized ellipsoid was analyzed in a recent paper [28]. (Of course, this short overview is far from being comprehensive.) In the present paper, the spontaneous radiative decay rate of smallsized optical centers (dopant ions of transition elements) in a crystalline spherical or ellipsoidal nanoparticle with linear sizes much less than the wavelength (2R, a, b, c λ) is analyzed theoretically, and the results are compared with the experimental data for rare-earth ions in crystalline YAG and Y2O3 nanoparticles suspended in a dielectric medium. The approach applied to solve the problem is drastically different from that used in [1, 3–5, 7, 8]. THEORY Spontaneous Radiative Lifetime in Nanocrystals The rate Abulk of spontaneous electric–dipole radiative decay of a small-radius optical center in a bulk crystal is given by the known formula [29, 30] 2 1 loc A bulk = --------- = n cr ( E /E ) A 0 = n cr f L A 0 , τ bulk
A nano = n eff f L ( n eff ) A 0 .
(2)
This gives rise to a certain ambiguity in the interpretation of the experimental data related to the choice in the expression for the correction factor fL(n) (this problem was discussed in [8]). However, in our viewpoint, the most important fact is the unjustified derivation of Eq. (2) from Eq. (1). Consider this issue in more detail. If the sample is a nanocomposite medium (hereafter, JETP LETTERS
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nanocomposite), then it is expedient to average the electric field over volumes much larger than the typical irregularity volume and introduce the “effective” 2 dielectric constant εeff = n eff in accordance with the definition D = εeffE [32], where D and E are the electric induction and field averaged in the above-specified sense. Thus, replacement of the first factor in Eq. (1) ncr by neff in Eq. (2) is well justified since, firstly, the photon state density becomes n eff ρvac(ω) instead of 3
n cr ρvac(ω) and, secondly, the interaction of the macroscopic field E with the dipole, proportional to 1/ncr for the bulk crystal, is replaced by the interaction in the nanocomposite, which is proportional to 1/neff. However, the replacement of fL(ncr) by fL(neff) does not take into account the difference of the macroscopic field E from both the mean macroscopic field 〈Emed〉 in the medium, where the nanocrystals are suspended and the mean macroscopic field 〈Ecr〉 at the OC location in a nanocrystal. (Hereafter, medium refers to the medium surrounding the nanocrystals.) Correspondingly, the local-field correction given by the local-to-macroscopic field ratio at the OC location is now equal to (Eloc/〈Ecr〉)2, but not to (Eloc/E)2. Therefore, Eq. (1) for nanocrystals in a nanocomposite is reduced to 3
2
A nano = n eff ( E /E ) A 0 . loc
(3)
Obvious transformations of Eq. (3) yield 〈 E cr〉 A nano = n eff ----------E
(1)
where ncr is the crystal refractive index, A0 is the decay rate in a vacuum or in a medium with ncr = 1, and Eloc and E are the micro- and macroscopic electric fields at the optical-center location, respectively. Within the framework of any known local-field models, the ratio (Eloc/E)2 = fL called the local-field correction, which is unity in a vacuum, is a function of the refractive index ncr (see [31] for a detailed review of all local-field models). In other words, (Eloc/E)2 = fL(ncr) and fL(1) = 1. In all of the papers available to us, including [1, 3–5, 7, 8] cited above, Eq. (1) is reduced to an expression for Anano by substituting ncr with neff and instead of fL(ncr) with fL(neff) according to the adopted local-field model:
13
= n eff
〈 E cr〉 ----------E
2
2
loc
2
E ------------ A 0 〈 E cr〉
(4)
f L ( n cr ) A 0 = n eff f N f L ( n cr ) A 0 ,
where fN = (〈Ecr〉/E)2 is the correction factor describing the difference in the macroscopic field 〈Ecr〉 at the OC location from the macroscopic field E in the nanocomposite. The arguments in favor of the difference in fN from unity are clearly outlined in [33] on the example of the spontaneous radiative electron-hole recombination in double heterostructures of III–V semiconductors. Without repetition of these arguments, we merely note the difference in Eq. (4) from the corresponding formula in [33] (Eq. (5) in [33]). The main difference consists in the appearance of the factor fL in Eq. (4). In the case of the process considered in [33], this factor is evidently in unity. Moreover, the analysis in [33] is restricted to the case of spherical nanoparticles. In what follows, we turn again to the comparison of our results and the results of [33]. Thus, Eq. (4) with allowance for Eq. (1) is written as n eff -f A . A nano = -----n cr N bulk
(5)
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An important consequence of Eq. (5) is the independence of the ratio Anano/Abulk on the local-field model. The problem of the theoretical evaluation of the ratio Anano/Abulk is now reduced to the problem of determining the correction factor fN and, of course, neff. First of all, we consider the most important case of spherical nanocrystals. Spherical Nanocrystals Let us consider the case of spherical nanocrystals of radius R obeying the conditions aL 2R λ or, more exactly, 2kR 1, where k = 2π/λ, λ = λ0/neff, λ0 is the wavelength of the spontaneous emission in a vacuum, and aL is the lattice constant. If the nanoparticles are not 3
crystalline, then a L is equal to the volume of the formula unit of the initial substance. The field Ecr inside the dielectric sphere in an external field Emed is [32] E cr = [ 3/ ( ε + 2 ) ]E med ,
(6)
where ε = εcr/εmed = n cr / n med , where nmed is the refractive index of the medium. Since Ecr and Emed are parallel, the macroscopic fields E, 〈Emed〉, and 〈Ecr〉 obey the relation E = (1 – c)〈Emed〉 + c〈Ecr〉, where c is the filling factor, i.e., the volumetric ration of the nanocrystals in the medium. Then, 2
spher
fN
2
= ( 〈 E cr〉 /E ) = { 3/ [ 2 + ε – c ( ε – 1 ) ] } 2
2
(7)
and Eq. (5) for spherical nanocrystals takes the form 2 n eff 3 spher - ------------------------------------- A bulk . A nano = -----n cr 2 + ε – c ( ε – 1 )
(8)
(For definiteness, we speak of nanocrystals, but all of the results in this paper are equally valid for dielectric nanoparticles with the refractive index ncr.) The corresponding formula for neff obtained with allowance for the relations D = εeffE and E = (1 – c)〈Emed〉 + c〈Ecr〉 have the form 3cβ 2 ε eff = n eff = ε med 1 + --------------- , 1 – cβ
(9)
where β = (ε – 1)/(ε + 2). Equation (9) is merely the well-known Maxwell Garnett formula [34, 35]. Note that the rigorous solutions given in [32] for the cases |ε – 1| 1 and c 1 agree with the corresponding limits of Eq. (9). The decay rates Anano corresponding to the limiting concentrations c are given below. Equations (8) and (9) for c 1 obviously yield Anano Abulk. If c 0, then Eqs. (8) and (9) yield n med 3 2 spher ----------- . A nano / A bulk = --------n cr 2 + ε
(10)
This formula gives the ratio Anano/Abulk valid for an arbitrary functional form of fL(ncr). Equation (10) is in good agreement with the results obtained in [33] for the ratio spher A nano /Abulk in the particular case fL(ncr) = 1, as well as with the analytical result of [25] obtained without allowance for the local-field effect, as well. (In [25], the radiation of an electric dipole inside a sphere filled with a continuous dielectric medium is considered.) It should be emphasized again that the result given by spher Eq. (10) for the ratio A nano /Abulk is equivalent to the results obtained in [25, 33] obtained for the particular case fL = 1, since, according to Eq. (10), the ratio spher
A nano /Abulk does not depend on the local-field correction fL. Thus, Eq. (10) has a wider validity range and we generalize the results obtained in [25, 33] to the case where the local-field correction fL cannot be ignored. It should be noted that, as well as Maxwell Garnett formula (9), Eq. (8) is derived based on Eqs. (6) and (7), which are only valid for a uniform external field, and without allowance for the spatial fluctuations of the local electric field in the nanocomposite. (Note that this constrains filling factor c. Another constraint on the filling factor is imposed by the fact that the retardation effect is neglected.) Hence, Eq. (8) is a “matching” formula both yielding the correct result for c 1 and conforming to the results of [25] and [33] for c 0 (see the discussion above). It is not yet clear whether this formula is applicable for the intermediate concentrations, since the experimental data are scarce and the experimental determination of the concentration, the particle form, and the degree of agglomeration for nanocrystals is difficult. Ellipsoidal Nanocrystals The mathematical complication encountered in the analysis of ellipsoidal nanocrystals consists in the fact that the field Ecr inside the dielectric ellipsoid in the external electric field Emed is not parallel to Emed [33]. Hereafter, the analysis is restricted to the case c 0 (in this case, neff nmed and Emed E). The components of the field Ecr in the major axes a, b, and c of the ellipsoid are related to the components of the field E as [32] ( cr )
Eα
= E α / [ 1 + ( ε – 1 )N α ] ≡ g α E α ,
(11)
where Nα are the depolarization coefficients (Na + Nb + Nc = 1). Thus, the macroscopic field in ellipsoidal nanocrystals (nanoparticles) is anisotropic. This should be taken into account in the calculations of Anano. In the case of a two-level optical center in an isotropic field, the averaging of |E(cr)e(k, σ)d|2 over the orientations of polarization vectors e(k, σ) of the macroscopic field E(cr) acting on an optical center with the dipole moment d of the JETP LETTERS
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∑
2
transition yields const α d α /3 (here, dα are the components of d). Such averaging in an anisotropic field 2 leads to the expression const α g α |dα |2/3. As a result,
∑
ell
A nano / A bulk
n med = --------n cr
∑
α = a, b, c
∑
2 γα -------------------------------- , 1 + ( ε – 1 )N α
(12)
d are the direction cosines of where γα = dα/ α α the dipole moment of the transition in the major axes a, b, and c of the ellipsoid. (In the case of sphere, Na = Nb = Nc = 1/3, so that Eq. (12) reduces to Eq. (10).) In other words, the ratio Anano/Abulk is now dependent on the dipole orientation with respect to the major axes of the ellipsoid. This dependence is the most pronounced for the prolate and oblate ellipsoids. Note that Eq. (12) and Eq. (10) following from it are valid if the effect of the optical-center field on the polarization of the nanoparticle containing this center is negligible. The anisotropy factor K equal to the ratio of the tranell spher sition probabilities A nano and A nano in an ellipsoid and sphere, respectively, is ell
2
A nano ε+2 K = ----------- = ----------spher 3 A nano
2
∑
α = a, b, c
2 γα -------------------------------- , (13) 1 + ( ε – 1 )N α
so that the radiative decay time in nonspherical nanoparticles can vary in a wide range compared to the spherical nanoparticles. A more detailed analysis is beyond the scope of this paper and will be given elsewhere. Here, we merely note that the known Judd– Offelt parameters Ωk [36] for the radiative decay of rare-earth ions in ellipsoidal nanocrystals are functions of the depolarization factors Nα and the orientation of the crystallographic axes relative to the major axes of the ellipsoid. Integral Cross Sections for Emission and Absorption in Nanocrystals In addition to the lifetime, the integral cross sections for the emission and absorption are the important characteristics of the lasing materials. The expression for the integral cross section for electric dipole radiation in the i j band has the form [29] em σ bulk ( i
j ) = W bulk ( i
2 2 j )/ [ 8πcn cr ν ],
(14)
j) is the spontaneous decay probabilwhere Wbulk(i ity for the i j channel and ν is the average energy of the (i j) transition in cm–1, and c is the speed of light. To derive the integral cross section for the electric dipole emission in the i j band of the nanocrystals, j) by one should evidently replace Wbulk(i Wnano(i j) and ncr by neff in Eq. (14). As a result, JETP LETTERS
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σ nano ( i
j ) = W nano ( i
em
15
j )/ [ 8πcn eff ν ]. 2
2
(15)
Substitution of the expression W nano ( i
n eff - f W (i j ) = -----n cr N bulk
j ),
(16)
derived in the same way as Eq. (5), we find n cr em em -f σ . σ nano = -----n eff N bulk
(17)
The same formula holds for the integral cross section for the absorption as well. Hence, for any integral electric dipole cross section, we have n cr -f σ . σ nano = -----n eff N bulk
(18)
In the case of spherical nanoparticles, the substitution spher of Eq. (7) for f N into Eq. (18) yields 2 n cr 3 - ------------------------------------- σ bulk . σ nano = -----n eff 2 + ε – c ( ε – 1 )
(19)
Note that the factor ncr/neff in Eqs. (17)–(19) is inverse with respect to the factor neff/ncr on the right-hand sides of Eqs. (5), (8), and (16). As a result, n cr 2 - [ A nano / A bulk ] σ nano /σ bulk = -----n eff n cr 2 - [ τ bulk /τ nano ], = -----n eff
(20)
where τbulk = 1/Abulk and τnano = 1/Anano. It should be noted that, e.g., for ncr = 1.82 (YAG), an increase in the radiative decay time in a nanoparticle by a factor of 5 in comparison with the bulk crystal gives rise to a decrease in the corresponding cross section for emission by only 42%. A slight decrease in the absorption cross section of the pump and radiation cannot significantly worsen the lasing-medium characteristics, while the fivefold increase in the lifetime for the same pump power makes it possible to increase the accumulated inversion in the laser generator by a factor of 5, which drastically simplifies the lasing system and significantly reduces its cost. COMPARISON OF THE THEORY WITH THE EXPERIMENTAL RESULTS In the remaining part of the paper, the formula n cr 2 + ε 2 ----------- , τ nano /τ bulk = --------n med 3
(21)
valid for identical spherical nanoparticles with the size 2R λ for c 0 is compared with the experimental data. This is done using the plots of τnano/τbulk for acti-
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Ratio τnano/τbulk of the decay times in the nanocrystal and bulk crystal for the metastable 4F3/2 level of the Nd3+ ion in the (open circles) crystalline YAG [8] and (closed triangles) Y2O3 [39] matrices, (closed circles) 2F5/2 level of the Yb3+ ion in the crystalline Y2O3 matrix, and (closed squares) 5D0 level of the Eu3+ ion [3] in the crystalline Y2O3, measured at a late stage of the fluorescence decay, as functions of nmed. The theoretical plots τnano/τbulk for the crystalline YAG matrices (ncr = 1.82, the dashed curve) and Y2O3 (ncr = 1.84, the solid curve) are obtained using Eq. (21), which is valid at the limit c 0 (neff = nmed). The dash–dotted curves are plotted for Y2O3 (ncr = 1.84) using Eqs. (8) and (9) with c = (upper curve) 0.2 and (lower curve) 0.4.
vated nanoparticles of yttrium aluminum garnet (YAG) and yttrium oxide (Y2O3) (see figure). In the case of morphologically identical nonspherical nanoparticles (i.e., the particles of the same nonspherical shape), Eq. (12) should be used instead of Eq. (21), which yields the following formula for the decay-time ratio: n cr τ nano /τ bulk = --------1/ n med
∑
α = a, b, c
γα -------------------------------1 + ( ε – 1 )N α
2
. (22)
According to Eq. (22), as nmed becomes closer to the nanoparticle refractive index ncr, the nanoparticle-morphology effect on the radiative relaxation rate of a rareearth ion becomes weaker. This is demonstrated very well by the measurements [8] of the radiative relaxation rate of the 4F3/2 level of Nd3+ ions in YAG: (0.9 at % Nd3+) nanoparticles with the diameter 2R = 20 nm for the small filling factor (c = 1.1 × 10–3) of these particles in various immersion liquids (open circles in the figure). The circle sizes correspond to the measurement error. It is seen that the experimental points are in good agreement with the theoretical curve if nmed= 1.55– 1.85. Deviation of the experimental data from the theory becomes pronounced for nmed = 1.5 and decreases
with a decreasing nmed, which can be related to the nonspherical shape of the nanoparticles. It should be noted that the nonradiative relaxation due to the concentration Nd–Nd quenching [37] cannot be neglected even at the late stage of the fluorescence decay kinetics for a Nd3+ concentration of 0.9% by atoms. In this case, the fluorescence decay rate is determined by the sum of two rates, namely, the radiative decay rate and the cross-relaxation quenching rate. If the latter process is neglected, then the radiative decay rate is overestimated and, thus, τnano is underestimated. This can also contribute to the discrepancy between the experimental and theoretical dependences of τnano/τbulk(neff). To minimize the effect of the intercenter quenching of the excitation on the fluorescence kinetics, the lifetime in the YAG:Nd crystal (τbulk = 255 µs [38]) was determined for the Nd3+ concentration much lower than for the nanoparticles studied in [8]. The same behavior is observed for the 4F3/2 level of the Nd3+ ions in Y2O3: 0.8% Nd3+ nanoparticles with the diameter 2R = 100–300 nm (closed triangles in the figure, the rates determined at the late stage of the fluorescence decay kinetics [39]). In this case, however, the significant filling factor c of the nanoparticles in the medium JETP LETTERS
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which could not be neglected was an additional factor leading to the decrease in τnano/τbulk [see Eqs. (8) and (9) valid for spherical nanoparticles and arbitrary c]. In addition, the experiment on the dynamical light scattering revealed that particles with diameters ranging from 700 nm to 2 µm were present in the Y2O3: 0.8% Nd3+ sample due to the agglomeration. The radiative decay rate in such particles is close to the decay rate in the bulk crystal, while the measured fluorescence-decay kinetics should have a nonexponential time profile which is actually observed in [39]. In comparison with the Nd3+ ion, the smaller deviation from the theoretical curve is observed for the 2F5/2 level of the Yb3+ ion in Y2O3: 0.3% Yb3+ nanoparticles with the diameter 2R = 100–300 nm (closed circles in the figure), which is related to the weaker effect of fluorescence quenching of the Yb3+ ions on their fluorescence decay rate in the late stage of kinetics. The best agreement with the theoretical curve is observed for the fluorescence decay time of the 5D0 level of the Eu3+ ion in Y2O3: 0.1% Eu3+ nanoparticles with the diameter 2R = 7–10 nm (closed squares in the figure) [3] already for nmed = 1.3. The discrepancy for n 1 is probably related to the large filling factor of the nanoparticles and their nonspherical shape. To illustrate the behavior of τnano/τbulk = f(nmed) for c > 0, the curves calculated for Y2O3 nanoparticles (ncr = 1.84) with allowance for Eqs. (8) and (9) are plotted in the figure by the upper (c = 0.4) and lower (c = 0.2) dash–dot curves. Thus, the analysis of the experimental results and theoretical expressions shows that radiative characteristics of nanoparticles are strongly different from the corresponding characteristics of a bulk crystal. The nanoparticle characteristics can be controlled by varying their morphology, size, and volumetric content in a suspension, as well as the refractive index of the surrounding medium. As a result, new lasing and fluorescent media with advanced parameters can be created. The results obtained above can also be used for the development of the fluorescence-kinetic method for monitoring the nanoparticle size and the degree of their agglomeration in the synthesis of lasing optical ceramics, controlling the lifetime of nanophosphor fluorescence, and observing nanoagglomerates in organic and biological structures. We are grateful to V.V. Osiko for a discussion of the results and interest in this work; P.P. Fedorov, E.A. Tkachenko, and S.V. Kuznetsov for the samples of Y2O3:Nd3+ and Y2O3:Yb3+ nanoparticles provided for the experiments; and O.K. Alimov for measuring the fluorescence kinetics of these nanoparticles. This work was supported in part by the Russian Foundation for Basic Research (project no. 08-02-01058-a) and the U.S. Civilian Research and Development Foundation (grant no. RUP2-1517-MO-06). JETP LETTERS
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Translated by A. Serber
JETP LETTERS
Vol. 88
No. 1
2008
