18010
J. Phys. Chem. B 2005, 109, 18010-18017
Theoretical Calculations of Refractive Indices and Optical Effects in Spectra of Nitric Acid and Nitric Acid Monohydrate Crystals D. Ferna´ ndez-Torre,† R. Escribano,*,† V. J. Herrero,† B. Mate´ ,† M. A. Moreno,† and I. K. Ortega†,‡ Instituto de Estructura de la Materia (CSIC), Serrano 123, 28006 Madrid, Spain, and Departamento de Quı´mica Fı´sica y Analı´tica, UniVersidad de Jae´ n, Pasaje Las Lagunillas, 23071 Jae´ n, Spain ReceiVed: April 7, 2005; In Final Form: July 14, 2005
The theoretical infrared refractive indices of two systems related to atmospheric research, nitric acid (NA) and nitric acid monohydrate (NAM) crystals, have been computed using a methodology based on first-principles. The effects of lack of coherence in the infrared beam in RAIR and transmission spectra have also been treated using a model based on classical optics. The optical constants of NA crystals are presented for the first time; the results on NAM are compared to empirical values previously published with good general agreement. With the optical constants of NA, polarized reflection-absorption infrared spectra are predicted and compared to experimental spectra recorded also for the first time, for a set of varying film thickness. The global agreement is satisfactory. The effects of a number of experimental factors in transmission spectra of NAM are assessed, in an attempt to explain observed differences among experimental spectra. It is concluded that the spectral disparities are probably due to differences in the nature of the samples.
Introduction The interest in nitric acid as one of the minor components of the atmosphere has favored a large number of studies on this species, both in gas phase1 and in solid state,2 in this case either as amorphous or crystalline pure species, and also in several phases that are produced as a result of its hydration.3 Nitric acid trihydrate (NAT) is one of the main constituents of polar stratospheric clouds,4,5 in whose composition the precise role of nitric acid hydrates is still a matter of open debate. Although nitric acid (NA) and nitric acid monohydrate (NAM), due to their high HNO3 concentration, have probably no relevance in the stratosphere, they constitute very good model substances in this class for comparison between theory and experiment. Infrared spectroscopy is specially suitable for the study of these species, and a precise knowledge of their optical constants in the IR range is consequently of high importance. The most comprehensive contribution in this field is a paper by Toon et al.,6 in which they measured transmission spectra of samples of several thickness of amorphous and crystalline nitric acid hydrates. By application of the Kramers-Kronig technique, they obtained empirical optical constants for these species. An alternative procedure is that employed by Richwine et al.,7 who determined the IR complex refractive indices of NAT from spectra of aerosol samples using a Mie scattering model. The determination of optical constants from first principles has never been carried out for molecular crystals of atmospheric relevance, as far as we know. Although in general the theoretical results may not reach the accuracy of empirically determined constants, they are interesting from various points of view. The theoretical predictions yield the individual components of the real and imaginary parts of the refractive index along a set of Cartesian axes. A quick glance at these components reveals
possible privileged directions, along which the optical indices take larger values. The same can be deduced for planes, containing or not the Cartesian axes, or for any other directions. This information, extrapolated to the prediction of the infrared spectrum of the solid, may help to evaluate if an observed feature is due to orientation effects of the crystals in the sample, which is particularly relevant for reflection-absorption infrared (RAIR) spectroscopy and also for extinction spectroscopy in aerosol studies.8 We present in this investigation a theoretical study of optical effects of special interest in the spectroscopy of solids, which have not been discussed previously in detail in the literature. First we deal with the theory required to calculate the refractive index for anisotropic media, by considering the components along the axes or planes of the crystal. Next we discuss the effects of lack of coherence in reflection-absorption or transmission experiments, following the scheme initially employed by Toon et al.,6 with an extension for angles of incidence other than the normal. All these theoretical effects are applied to a study of the refractive index of nitric acid monohydrate, with comparison with the empirically determined values of Toon et al., and also of nitric acid crystals, for which there are no experimental values available. After that, we present a comparison of experimental RAIR spectra of NA of varying thickness between ca. 430 and 1700 nm, and the corresponding to theoretical predictions with various degrees of coherence. It is illustrative to observe the broadening of spectral features with increasing noncoherence and thickness of the crystal. Finally, simulations of transmission spectra of NAM under different conditions of sample thickness, degree of coherence, and incidence angle are presented and compared to already published spectra. Experimental Section
* Corresponding author. E-mail:
[email protected]. † Instituto de Estructura de la Materia (CSIC). ‡ Universidad de Jae ´ n.
The experimental system used in the present work is described elsewhere9 and only the relevant details will be given here. The
10.1021/jp0517899 CCC: $30.25 © 2005 American Chemical Society Published on Web 09/03/2005
Theoretical Calculations of Refractive Indices
J. Phys. Chem. B, Vol. 109, No. 38, 2005 18011
experiment was conducted in a ultrahigh vacuum cylindrical chamber evacuated by a turbomolecular pump to a base pressure lower than 3 × 10-8 mbar and provided with a liquid nitrogen Dewar in contact with the deposition substrate. After the Dewar was filled, the base pressure was in the 10-9 mbar range in the present experiments. The cold deposition surface is made of polished aluminum. The surface temperature could be regulated between 85 and 323 K with an accuracy of 1 K. For the deposition of nitric acid the substrate was held at 150 K, and the chamber was filled with a homogeneous pressure of nitric acid of ∼1.0 × 10-4 mbar. The film thickness could be approximately controlled (uncertainty estimated to be (10%) by monitoring the interference fringes of a He-Ne laser incident on the growing film at near normal incidence.10 Once the nitric acid films have the desired thickness, the sample is annealed at 160 K during 15 min to ensure crystallization. The substrate is then cooled again to 150 K, and at this temperature RAIR spectra are recorded with a FTIR spectrometer Bruker IFS66. The IR radiation was focused on the sample with a KBr lens. The incidence angle was 75°, and the reflected IR light was focused with a curved mirror onto a mercury cadmium telluride (MCT) detector cooled with liquid nitrogen. A polarizer SPECAC KRS-5 placed before the focusing lens was used to select the incident radiation with an electric field vector perpendicular (S) or parallel (P) to the plane of incidence. Each spectrum was obtained from the addition of 512 scans, recorded at 8 cm-1 apodized resolution. Theory Refractive Index from First Principles Calculations. Our system consists of a molecular crystal of unit cell volume Ω0, with vibrational eigenvalues at wavenumbers νm. The dielectric tensor of this system in the infrared range can be calculated as11,12
ij(ν) ) ij(∞) +
4π Ω0
∑ m
1
Sm,ij
4π2 νm2 - (ν + iΓ)2
(1)
In this equation i and j are indices along Cartesian axes, i is the imaginary unit, so that ij(ν) is a complex quantity; ij(∞) is the dielectric tensor for high wavenumbers which, as we are mainly dealing with the infrared, can be taken as the corresponding value at visible or ultraviolet frequencies; ν is the wavenumber; Γ is a damping factor; and, finally, Sm,ij is related to the Born effective charges, Z*τ,ij,12 and the normal displacements at the equilibrium configuration, Um,τ,i, as
Sm,ij ) [
∑τ ∑k Z*τ,ikUm,τ,k][∑ ∑ Z*τ′,jk′Um,τ′,k′] τ′ k′
(2)
The normal displacements appearing in (2) are normalized to satisfy
∑τ ∑k MτUm,τ,kUn,τ,k ) δmn
(3)
where Mτ is the mass of the atom labeled τ. We have adopted in this treatment the Gaussian system of units.13 Thus, when the unit cell volume is given in a03 (a0 ) bohr), Born charges in au (au ) e, elementary charge), normal displacements in mu-1/2 (mu ) atomic mass unit), and the wavenumber in cm-1, the second term of the right-hand side
of eq 1 must be multiplied by
1 e2 ≈ 2.622 × 107 1011 mua03 If we have a perfectly ordered crystal, which constitutes an anisotropic medium, and we illuminate it with a linearly polarized beam of light, the effective dielectric constant will be12
qˆ (ν) )
∑i ∑j qˆ iij(ν)qˆ j
(4)
qˆ being the unitary vector along the electric field E direction. Equation 4 adopts a simpler form if we are interested in the effective dielectric constant along the Cartesian axes
eff,i(ν) ) ii(ν)
(5)
Any other direction can be obtained as a combination of these three. On the other hand, if the incident light is unpolarized, the electric field vector will be contained in a plane perpendicular to the propagation direction. For instance, if the plane of vibration of E is the xy Cartesian plane, the average of eq 4 over all possible orientations on that plane should be evaluated as
eff,x,y ) 1 2π
∫φ)0
φ)2π
(
)( )
xx xy xz cos φ dφ(cos φ sin φ 0 ) yx yy yz sin φ zx zy zz 0
where φ defines the orientation of E in the xy plane. For any one of the Cartesian planes
1 eff,i,j(ν) ) [ii(ν) + jj(ν)] 2
(6)
Finally, if the crystal is randomly oriented, there is no difference in having polarized or unpolarized light, because the direction between the crystallographic axes and the electric field will be random anyway. In this case we have an isotropic material, because the effective dielectric constant has the same value for every incidence direction and polarization, with
1 1 eff,iso(ν) ) [xx(ν) + yy(ν) + zz(ν)] ) tr(ij) 3 3
(7)
From the dielectric constant we can infer the complex refractive index N ˆ , whose real and imaginary parts are the optical constants n and k, N ˆ ) n + ik. For a nonmagnetic crystal, the refractive index is simply the square root of the dielectric constant
N ˆ (ν) ) x(ν)
(8)
Once the refractive index is known, we can simulate many kinds of infrared spectra by means of classical optics models. In this work we deal with the simulation of RAIR and transmittance spectra of isotropic samples. Coherence Effects in RAIR Spectra. The theoretical simulation of RAIR spectra, based on classical optics, is well described elsewhere.10,14 We present here an extension to include the effect of a partial lack of coherence in the beam of light, following the lines of the treatment of ref 6. This effect appears when the phase of the light is not well described by a single value, and it is necessary to use a distribution function to
18012 J. Phys. Chem. B, Vol. 109, No. 38, 2005
Ferna´ndez-Torre et al.
Figure 1. Reflection in a typical RAIR setup: light comes from air (N0), enters a thin dielectric film (N1), and is reflected by a metallic substrate (N2).
represent it. This can have its origin, for instance, in a lack of planarity or in some degree of roughness at the sample interfaces. Figure 1 reproduces a typical RAIR experiment. In the following, we represent by N0, N1, and N2 the refractive indices of the media along which the light propagates, air (or vacuum), a film of thickness d1 of the material under study, and the metal substrate, respectively. We use the indices u,V ) 0, 1, 2 as associated with air, ice, and metal, respectively. The coefficients ruV and tuV indicate the Fresnel reflection and transmission coefficients, giving the fraction of the incident light reflected and transmitted at the UV interface, respectively, and RuVw is the total reflected fraction of the light intensity. We can approximate the reflectance of a RAIR sample by a weighted average between the reflection produced by a totally coherent interference, R012c, and the reflectance associated with a totally incoherent interference, R012nc
R012(P) ) cR012c(P) + (1 - c)R012nc(P)
(9)
where c is the degree of coherence, which can take values between 0 and 1, and P the polarization of the light, either parallel (P) or perpendicular (S) to the plane of incidence. The coherent reflectance can be calculated as described in ref 10. Note that the sign criterion for the complex index of refraction used in that paper is N ˆ ) n - ik, whereas here we have followed N ˆ ) n + ik. In practice, the formulas are the same, replacing the phase δ by -δ.
R012c(P) ) |r012c(P)|2 r012c(P) )
(10)
r01(P) + r12(P)eiδ 1 + r01(P)r12(P)eiδ
where
δ)
4π d N 2 - N02sen2Θ0 λ 1x 1
The noncoherent reflectance, or reflectance without interference, can be written as15
R012nc(P) ) R01(P) +
T01(P)T10(P)R12(P)e-2Im(δ) 1 - R01(P)R12(P)e-2Im(δ)
TuV(P) ) 1 - RuV(P)
are presented in the appropriate section below. They have been evaluated by reference to the metal reflection R02 and expressed in absorbance units. For polarized light, this corresponds to
AUR(P) ) -log[R012(P)/R02(P)]
(13)
with P ) P or S, as indicated above, and for unpolarized light
AUR ) -log[(R012(s) + R012(p))/(R02(s) + R02(p))]
(14)
Coherence Effects in Transmission Spectra. If surface reflections are ignored, which is a good approximation for thin films such as the ices we are interested in, transmission can be approximated by the Lambert-Beer law. In the following we present an extension of the theory to take surface reflections into account. The initial model is again that developed by Toon et al.6 for normal incidence, to which we have added terms to deal with any incidence angle other than normal. Our final purpose is to investigate deviations from the Lambert-Beer law that can arise when varying the film thickness and for incidence angles slightly different from 0°. These deviations might explain part of the differences found among experimental transmission measurements recorded in different laboratories on apparently the same ice material. Our model assumes a sampling geometry used in many transmission experiments, with thin sample films formed on both sides of a thick substrate, as depicted in Figure 2. In the following, subindices 1 and 2 refer to the ice films, 0 to air and S to the substrate. N1 is a complex magnitude, whereas N0 and NS are real. We are interested in the total transmission of the sample normalized by the transmission of the substrate alone, in absorbance units:
AUT(P) ) -log[T(P)/TS(P)]
(15)
The substrate transmission can be calculated as
(11)
where TuV and RuV are, respectively, the transmittance and reflectance between the corresponding interfaces, defined as
RuV(P) ) |ruV(P)|2;
Figure 2. Trajectory of the light across the sample for an incidence angle different from 0°.
(12)
The calculated RAIR spectra, given by the R012 term of eq 9,
TS(P) )
T0S(P)TS0(P) 1 - RS0(P)2
(16)
Equation 16 has been obtained under the assumption that coherence is not maintained as the beam of light traverses the substrate, which is a good approximation for thick substrates with ds of the order of hundreds of micrometers or thicker, since such values overcome the coherence length of the radiation.
Theoretical Calculations of Refractive Indices
J. Phys. Chem. B, Vol. 109, No. 38, 2005 18013
On the other hand, the total transmission of the three-layer system can contain a contribution of coherent and noncoherent interference, because transmission occurs through the two thin films and the substrate. Whereas the interference among transmitted rays in the substrate is noncoherent, the transmitted light through the two thin films may give rise to coherent interferences. We can express this situation by inclusion of two transmission fractions, as we did in the simulation of RAIR spectra.
T(P) ) cTc(P) + (c - 1)Tnc(P)
(17)
The total coherent transmission in eq 17 corresponds to the coherent transmission on both thin films 1 and 2:
T (P) )
c c Tf1 (P)Tf2 (P)
c
(18)
c c 1-R ˜ f1 (P)Rf2 (P)
c c where Tf1 , Tcf2, and Rf2 are the composite transmission and reflection factors for films 1 and 2 when the light travels from c is the reflection factor for top to bottom in Figure 2, and R ˜ f1 light traveling from bottom to top, indicated with a tilde. These composite transmission and reflection factors for films 1 and 2 can be written as15
c c (P)Tf2 (P) Tf1
|
)
t01(P)t1S(P)eiδ1
1 + r01(P)r1S(P)e
i(2δ1-φ1)
|
R ˜ f1(P) ) -
Rf2(P) )
|| 2
|
tS2(P)t20(P)eiδ2
1 + rS2(P)r20(P)e
i(2δ2-φ2)
r1S(P) + r01(P)ei(2δ1-φ1) 1 + r1S(P)r01(P)ei(2δ1-φ1)
rS2(P) + r20(P)e
|
Figure 3. Projections of the calculated refractive index for the NAM crystal along the x, y, and z Cartesian axes, and full refractive index N ˆ ) n + ik: top, imaginary part; bottom, real part. Each curve has been offset along the ordinate axis for clarity.
where the composite transmission factors take the form
Tf1(P)Tf2(P) )
2
(19)
|
2
T01(P)T1S(P)e-2Im(δ1)
1 - R01(P)R1S(P)e-2Im(2δ1-φ1) 1 - RS2(P)R20(P)e-2Im(2δ2-φ2) (22) R ˜ f1(P) ) R1S(P) +
|
i(2δ2-φ2) 2
1 + rS2(P)r20(P)ei(2δ2-φ2)
Rf2(P) ) RS2(P) +
with phases
x
4π dN 2δ1 - φ1 ) λ 1 1 2π δ1 ) λ
1-
N12
(20)
x
1-
N02sen2Θ0
x
N1
N02sen2Θ0 N12
Finally, the transmission corresponding to the incoherent interference in eq 17 is
Tnc(P) )
1 - RS2(P)R20(P)e-2Im(2δ2-φ2)
nc nc Tf1 (P)Tf2 (P) nc nc 1-R ˜ f1 (P)Rf2 (P)
2Im[δ1 + δ2] log(e)
(23)
Results
2
d2N1 1-
TS2(P)T2S(P)R20(P)e-2Im(2δ2-φ2)
Note that the Lambert-Beer expression does not depend on the polarization of the incident beam, even for incidence angles different from 0°.
N12
x
2πΠ λ
1 - R1S(P)R01(P)e-2Im(2δ1-φ1)
AUTLambert-Beer )
N02sen2Θ0
4π 2δ2 - φ2 ) dN λ 2 1
T˜ S1(P)T1S(P)R01(P)e-2Im(2δ1-φ1)
For the system represented in Figure 2, the Lambert-Beer law, in absorbance units, takes the following form when surface reflections are ignored
d1N1 1-
δ2 )
N02sen2Θ0
TS2(P)T20(P)e-2Im(δ2)
(21)
Refractive Indices. Nitric Acid Monohydrate. In a previous work16 the Born charges and normal displacements associated with the crystal of NAM were calculated. We have used here those results for the calculation of the dielectric tensor and the effective index of refraction, as indicated in eqs 1-8. The value of (∞) ) 1.542 has been taken from the literature17 and, as in ref 16, we have chosen two different values for the damping factor Γ, of 20 and 140 cm-1 for frequencies lower and higher than 2000 cm-1, respectively, to account for the larger anharmonicity of higher frequency vibrations where H-bonding is predominant. We display in Figure 3 the results for the real and imaginary part of the refractive index of NAM. Top to bottom traces in
18014 J. Phys. Chem. B, Vol. 109, No. 38, 2005 the figure represent the projections along the x, y, and z Cartesian axes, corresponding to the a, b, and c crystallographic axis of the orthorhombic NAM crystal, respectively, and the full refractive index, calculated under the isotropic assumption in an ordered crystal. The results shown in Figure 3 represent a theoretical prediction of the effective refractive indices that would be measured if a perfectly ordered NAM crystal was illuminated with polarized radiation along each of the Cartesian axes. The real part of the refractive index conveys the origin of phase changes in the beam, which can give rise to interference effects depending on the geometry of the problem under study, whereas the imaginary part is responsible for attenuation or absorption of the beam of light. Inspection of the upper part of Figure 3 reveals that the most important absorption of light for the infrared range is predicted for the electric field polarized along the x and y Cartesian axes, i.e., along the a and b crystallographic directions. This is not surprising if we consider the geometry of the molecules in the unit cell of NAM:18 they are disposed forming layers approximately along a and b and perpendicular to c. The molecular modes that produce greater absorptions would correspond to dipole moment changes approximately within the molecular plane, namely, O-H stretching modes in the 2000-3000 cm-1 region, and asymmetric bending modes around 1500 cm-1, also contained in the ab molecular plane. On the other hand, symmetric bending modes, like the umbrella bending in H3O+, produce a dipole change along the perpendicular direction, that is the c axis, and correspond to the peak calculated at approximately 1200 cm-1 in the z trace of the figure.19 It is also remarkable that vibrations of apparently similar types, like the above-mentioned O-H stretching modes, appear at different wavenumbers in the x and y polarization. These vibrations are composed of blends of symmetric and antisymmetric modes, in- and out-of-phase for the 4 H3O+ units in the unit cell, and therefore respond differently to excitation along the x or y axes. It is only through theoretical analysis like the present one that this different behavior can be revealed. A final consideration can be made with respect to previous literature spectra. Theoretical absorption spectra can be evaluated by multiplication of the imaginary part of the effective index by the corresponding wavenumber.20 If we compare the results obtained in this way (not presented here) for all four k traces of Figure 3 with absorption experimental spectra from the literature,21 it is possible to conclude that most probably the crystalline thin films on which the spectra were recorded were composed of disordered monocrystalline domains, because the better match with our calculations is found for the trace that corresponds to the full refractive index, that is, when all three crystalline axes are combined together in the isotropic solid model. For the NAM crystal, Toon et al.6 obtained empirical values of the real and imaginary part of the refractive index, by application of the Kramers-Kronig technique, using a skillful iterative method. Their results can serve as a reference to evaluate the quality of the theoretical values determined in the present work. Figure 4 reproduces the empirical values of Toon et al., together with our determination for the isotropic crystal. The general agreement is satisfactory, with very good reproduction of the higher frequency peaks, both in frequency and in numerical value, and even in the band shape, and a blueshift of ca. 100 cm-1 of the theoretical values in the 10001500 cm-1 region. The numerical values of the indices at ∼750 cm-1 are underestimated in our calculation, although the
Ferna´ndez-Torre et al.
Figure 4. Calculated effective isotropic refractive index of the NAM crystal compared with the experimental result of Toon et al.:6 top, imaginary part; bottom, real part.
Figure 5. Projections of the calculated refractive index for the NA crystal along the x, y, and z Cartesian axes, and full refractive index N ˆ ) n + ik: top, imaginary part; bottom, real part. Each curve has been offset along the ordinate axis for clarity.
agreement in frequency is excellent. These results are relevant for the discussion presented below. Nitric Acid. In a similar manner as for the NAM crystal, we have calculated the refractive index of a pure nitric acid (NA) crystal. Born charges and atomic displacements were taken from Ferna´ndez-Torre et al.,16 (∞) ) 1.492 was taken from ref 22, and we chose two different values for the damping factor Γ, of 20 and 60 cm-1, for frequencies lower and higher than 2000 cm-1, respectively, the high-frequency value being smaller in this case than that for NAM, due to the absence of H bonding in NA. In Figure 5 we plot the results for the real and imaginary part of the refractive index of NA, projected along the Cartesian axes x, y, and z, coincident with the orthorhombic crystal axes a, b, and c, respectively, and also the full refractive index, calculated under the isotropy assumption. The projections along the axes behave differently now than in the case of NAM. There is an axis whose contribution is
Theoretical Calculations of Refractive Indices
Figure 6. Comparison between experimental (top) and simulated RAIR spectra of nitric acid crystals of thickness d < 1 µm, for degree of coherence values of 1, 0.8, and 0: (left) d ) 430 nm, (right) d ) 860 nm; (black trace) P polarization, (gray trace) S polarization.
clearly dominant, namely, the y (or b) axis. This is also consistent with the geometry of the molecules within the solid,2 where most of the NO3 groups and OH bonds lie mainly along the b axis, and their corresponding stretching modes would induce the largest change in the electric dipole moment. The main contribution of the x and z axes is apparent only as a change in the relative intensity of the NO2 asymmetric and symmetric stretching bands, at 1300 and 1700 cm-1, respectively.2 The intensity of the latter is comparatively larger in the full index trace than in the curve for the y axis projection, revealing a nonnegligible distribution of electric charge displacement in the xz plane. As for NAM, we can predict the transmission spectrum by taking the calculated imaginary isotropic refractive index times the wavenumber (not shown here). This allows a comparison with experimental absorption spectra,21,23 but in this case it is not possible to draw any conclusions about the isotropy or orientation of the sample used in the experiments, because, at the level of approximation implied in our results, the matching between the experiment and the calculated spectrum for any plane containing the b-axis is just as good as that between the experiment and the isotropic result. RAIR Spectra. We have recorded RAIR spectra of pure nitric acid crystals of varying thickness, using polarized incident light in the plane of incidence (P polarization) and in a perpendicular plane (S polarization). They provide the basis for an assessment of the quality of the theoretical refractive indices, using the model described above. We have calculated NA spectra, also for both polarizations, using the theoretical refractive index of the preceding section. To compare with the observations, all predictions are made assuming an incidence angle of 75°, as employed in the experimental setup. The calculations have been carried out for the same thickness used in the experiments, with values of approximately 430, 860, 1280, and 1710 nm. Figures 6 and 7 present a comparison between experimental and calculated nitric acid RAIR spectra for thickness values 1 µm, respectively. In all panels we show
J. Phys. Chem. B, Vol. 109, No. 38, 2005 18015
Figure 7. Comparison between experimental (top) and simulated RAIR spectra of nitric acid crystals of thickness d between 1 and 2 µm, for degree of coherence values of 1, 0.8, and 0: (left) d ) 1280 nm; (right) d ) 1710 nm. (black trace) P polarization; (gray trace) S polarization.
simulations for c ) 1, i.e., for a totally coherent interference, for c ) 0.8, i.e., with a 20% of noncoherent interference, and for c ) 0, to illustrate the limit of noncoherent interference. The best match in Figure 6 takes place for c ) 1, although the calculations are fairly similar for c ) 1 and c ) 0.8, those for the S polarization being closer to the experiment for the totally coherent trace. As a general rule, the better reproduced features are the peaks at ∼ 3000 and 1700 cm-1, whereas the intensities of the peaks in the 1450 cm-1 manifold in the thinnest film, and those ca. 750 cm-1 for both films, are not well matched. The band calculated near 3400 cm-1 seems to be shifted from the stronger counterpart peaking at ∼3100 cm-1, which is in closer agreement with the experiment. All these circumstances were already found in the previous calculations for the nitric acid crystal2,16 from where the refined geometrical structure and Born charges and atomic displacements have been taken for the present prediction, and seem therefore to be inherent to the data used here. An interesting goal of the present simulation is the ability to reproduce the relative intensity of the S and P polarizations, and this is well accomplished in this investigation. In Figure 7, the P polarization trace is similarly well reproduced for c ) 0.8 and c ) 1 for the 1280 nm film, but it would probably require lower c values for the thickest crystal. The reproducibility of the observed contour is in general acceptable, although some structure observed in the experimental trace in a low-frequency shoulder of the 3000 cm-1 band is not predicted in the calculations. On the other hand, the inclusion of a certain lack of coherence is critical for the S polarization, where the intensity of the peaks in the 1300-1700 cm-1 region of the 1280 nm sample would otherwise grow dramatically, due to destructive infrared interferences,24 which were not very important for films with d < 1 µm. This behavior is similar to that observed by Toon et al.6 in transmission spectra of thin films of nitric acid hydrates, where they showed that the degree
18016 J. Phys. Chem. B, Vol. 109, No. 38, 2005
Figure 8. Simulations of transmission spectra of NAM crystals grown on AgCl and Si substrates: thick black trace, Lambert-Beer; thin black trace, coherence coefficient c ) 0; gray trace, c ) 1. Thickness values are d1 ) d2 ) 1000, 600, 400, and 200 nm, top to bottom.
of coherence decreases as the films grow thicker. The comparison of the spectra displayed along corresponding panels in Figures 6 and 7 shows that the lack of coherence does indeed increase with the thickness of the samples. Transmission Spectra Simulations. We deal in this section with spectra of nitric acid monohydrate crystals. There are a number of experimental publications showing spectra of NAM recorded under different conditions.21,23,25,6 Such spectra show remarkable differences among them, affecting the band shapes and widths, and even presenting some frequency shifts in some cases. There may be two possible reasons to explain these differences,16 related either to optical effects arising from the diverse experimental setups or to variations in the intrinsic nature of the samples. We explore in this section a number of optical effects which might have some influence on the spectra. First we simulate transmission spectra for two different substrates, in normal incidence experiments, and study the variations induced by lack of coherence and changing film thickness. We have selected for the simulations two substrates, Si and AgCl, which are common choices for transmission setups. Next, we deal with the case of a nonnull angle of incidence. It will be seen that none of these possibilities can explain the variations appearing in experimental spectra. We therefore conclude that they must be due to the nature of the crystalline samples prepared by the different authors. Normal Incidence. In most transmission setups, films are formed on either side of a cold substrate by homogeneously filling the sample compartment with gas, which results on both films having basically the same width, d1 ) d2. The size of the crystals varies usually between a few nanometers and a few micrometers. We present in Figure 8 the results of our simulations for a silver chloride substrate and a Si substrate, for increasing thickness values, with d1 ) d2 ) 200, 400, 600, and 1000 nm from bottom to top. The traces correspond to simulations both when surface reflection effects are ignored, that is, when the Lambert-Beer law is obeyed, and when
Ferna´ndez-Torre et al. reflections are taken into account, for limiting coherence coefficients of c ) 0 and 1. Si has a refractive index of ∼3.4 and AgCl of ∼2.26 Both substrates are transparent for the wavenumber range shown in the figures. The difference in the refractive index will cause different Fresnel transmission and reflection coefficients and, more importantly, diverse phase differences. Optical distortions are expected to be smaller for the second material, whose refractive index is closer to those of NAM and air, and this is what is found in the simulations. The general trend observed in Figure 8, from bottom to top, is that spectra which include reflection effects, both for c ) 0 and c ) 1, approach the Lambert-Beer prediction as film thickness grows. In the theoretical equations (18) to (23) given above, absorption is the dominant part for thick crystals. The figure shows also that for a AgCl substrate the agreement between all traces is fairly close even for thin films of 400 nm, whereas for a Si substrate a comparable concordance is only found for thickness values of 1000 nm. An alternative experimental setup directs the gas jet to one side of the substrate, giving rise to a thicker film on that side, and a thinner one on the opposite face. We carried out sets of simulations under varying conditions of d1 ) d2 and d1 . d2 for the least favorable case of the Si substrate. The result was that a good reproduction of the Lambert-Beer law was achieved for thickness values of the order of d1 ) d2 ∼ 2 µm, in the first case, and d1 > 4 µm . d2 ) 100 nm, in the second. The conclusion is that the total thickness, d1 + d2, is the relevant factor for a good agreement with the expectation of the Lambert-Beer law. Incidence at an Angle Θ * 0. If the incident radiation is not normal to the plane of the crystal, or if the surface of the crystal is not planar or parallel to the substrate, optical effects appear in transmission spectra. These effects can be calculated both for the Lambert-Beer trace and for simulations which include reflections at the interfaces according to the equations given above. Figure 9 shows such calculations for angles varying from 10° to 60°, for a simulated setup consisting of a Si substrate and two adjacent films of thickness d1 ) d2 ) 200 nm, with a coherence factor of c ) 1. For this model, the Lambert-Beer spectrum and that due to polarized radiation are different for the lowest angles, even for Θ ) 0 (shown on the bottom right trace of Figure 8). For an angle of 10° the S and P traces are close to each other. As the angle increases, the S polarization prediction becomes more and more distorted, while the P polarization gradually gets closer to the Lambert-Beer spectrum. For the largest angles shown, the optical effects distort all traces, which present a derivative-like contour at the edge of the strongest absorption, near 1500 cm-1. The LambertBeer spectrum is of course not affected by the polarization of the incident radiation. Relation to Experimental Spectra. As mentioned above, literature experimental transmission spectra of NAM show in some cases important disparities. Figure 5 of ref 16, which reproduces NAM spectra of refs 21 and 23, presents a good example of this. The corresponding samples have been prepared by different methods, like homogeneous filling of the cell or directional jet flow onto the substrate, and have been grown on diverse substrates at various deposition temperatures. The present study shows examples of how some of these factors affect the spectra, but we have found that the dissimilarities observed in the experimental spectra cannot be explained by the different conditions considered here, which include lack of planarity, or noncoherent interferences. Therefore, we must
Theoretical Calculations of Refractive Indices
J. Phys. Chem. B, Vol. 109, No. 38, 2005 18017 We have explored the effects of a number of experimental factors on transmission IR spectra of NAM. These include loss of coherence due to partial reflection, the use of two different substrates, and the nonnormal incidence of the radiation on the sample. These effects cannot explain the disparities observed in the available literature spectra. Acknowledgment. This research has been carried out with financial support from the Spanish “Ministerio de Educacio´n, Ciencia y Deporte”, project FIS2004-00456. The present investigation has been performed within the frame of a “Unidad Asociada” between the IEM (CSIC) and the Department of Analytical and Physical Chemistry of the University of Jae´n. References and Notes
Figure 9. Simulations of transmission NAM spectra for varying angle of incidence: thick black trace, Lambert-Beer; thin black trace, S polarization (coherence factor c ) 1); gray trace, P polarization (coherence factor c ) 1). Substrate: Si; thickness, d1 ) d2 ) 200 nm.
conclude that such spectral differences are probably due to the different nature of the samples being used, in the sense that they might contain amorphous regions, or be contaminated by other hydrates, or have an irregular or defective crystalline structure. Summary and Conclusions We present here a formalism for the derivation of refractive indices using Born effective charges and normal displacements from density functional calculations on dielectric crystals. These optical indices can be used to compare theoretical results with spectra obtained by different experimental techniques and using different geometrical arrangements. We describe also a treatment of the effects of lack of coherence in the light beam, associated with nonideality in the sample. The method is applied to crystals of nitric acid and nitric acid monohydrate. As far as we know, the present theoretical values are the only available data on the optical constants of NA. As for NAM, we have obtained a semiquantitative agreement with the empirical indices of Toon et al.6 We have recorded polarized RAIR spectra of nitric acid crystals, which had never been published before, within our knowledge. Comparison of the experimental and calculated spectra shows good global agreement. The spectra are best accounted for by assuming a high degree of coherence, although an increasing loss of coherence is observed with growing film thickness.
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