1. Introduction

Reflectance spectra of particulate materials for studies related to the mineral composition of planetary surfaces have been measured extensively [1 –6]. Unlike transmittance measurements, reflectance spectra are influenced by both the real ($n$) and imaginary parts ($k$) of the refractive index as well as particle size and morphology. While the optical constant, $n$, determines the phase velocity of the light propagating through the homogeneous medium, the imaginary constant, $k$, takes into account the absorption losses and is proportional to the absorption coefficient. Early studies [1 –6] showed that reflectance and emission spectra can vary as a function of particle size for a variety of rocks and minerals, as two different scattering mechanisms can lead to either maxima or minima in the same spectra: surface scattering, which results from rays that have reflected from the surface without penetration, usually leads to upward-going peaks in the reflectance spectrum. Conversely, volume scattering is due to the rays that have penetrated into the interior of the sample where the photons are either scattered or absorbed and, generally, lead to downward-going peaks when on an absorption feature. Both troughs and peaks can thus be observed in the infrared spectral region because volume and surface scattering can occur. As these two processes compete, changes in spectral contour can be encountered as a function of particle size and morphology. The scattering process that dominates is a function of both the magnitude of $n$ and $k$ (both of which vary with wavelength) as well as the particle size and morphology.

Accurate and calibrated directional-hemispherical reflectance spectra of solids [7,8] are often used to develop libraries of representative spectra for sample identification for remote sensing and in situ measurements. Successful identification of particulate samples, however, requires representative reference spectra that include the particle size of interest. For example, the NASA/JPL ASTER Spectral Library Database [9], which was developed to provide a comprehensive database of rocks and minerals, has recently been updated to include three particle size fractions. Most of these studies, however, have only examined geological materials and have not included other solids that should show similar effects. As reflectance spectroscopy is now finding more uses outside of planetary sensing [10] (e.g., for homeland security screening [11] and environmental sensing [12,13]), we consider other types of solids and the effects that particle size may have on their spectra.

In this paper, we describe results of quantitative directional-hemispherical reflectance measurements, both total and diffuse, for several compounds of varying particle size distributions using a Fourier transform infrared spectrometer equipped with an integrating sphere and operating in the 1.3–16.6 μm spectral range for several inorganic and organic compounds of varying particle size distributions. The spectral range was selected based on the natural domains of the optical components as well as the known specificity and sensitivity of the infrared. The particle size is confirmed with optical microscopy measurements to provide a better estimate than the sieve screen sizes alone. Most of the selected chemicals exhibit a wide range of spectral behavior because of a combination of strong and weak absorption bands, and all are strongly dependent on the mean particle size of the sample. We have studied particle size effects for dozens of samples; this paper discusses such effects on the spectra of four representative species: ammonium sulfate, calcium carbonate, sodium sulfate, and lactose. These materials were specifically selected, as they represent a good cross section of organic and inorganic compounds with associated spectral profiles as well as being materials likely to be encountered in typical field or Earth-science studies.

2. Experimental Methods

A. Experimental

Spectra were recorded at $4.0{\mathrm{cm}}^{-1}$ resolution using either a Bruker Optics IR Cube or an IFS 66/S Fourier transform infrared spectrometer (FTIR); each system was equipped with a two-port, 75 mm diameter, gold-coated integrating sphere, the A562 from Bruker. A beam-steering mirror in the sphere directs the incoming light from the FTIR spectrometer to either the top or bottom sample port or to another reference point on the front wall of the sphere [7]. With the mirror pointing down toward the sample port, the angle between the incident light on the sample surface and the surface normal is 14.8°. The integrating sphere has a purge gas connection and is under constant dry nitrogen purge. Via use of a specular exclusion port, two sets of spectra are recorded for each sample: the diffuse-only reflectance spectra and the total (or hemispherical) reflectance, which represents the sum of the specular and diffuse components. In this paper, however, only the total directional-hemispherical spectra (i.e., specular + diffuse rays) are reported. The hemispherical spectral reflectance is determined by ratioing the sample spectrum, which is obtained by pointing the mirror inside the sphere toward the sample in the bottom port, to a reference spectrum, which is obtained by pointing the mirror toward a reference point in the sphere. For the data in this paper, the reference position for the total spectra was the beam pointing at the diffuse gold dome at the top of the sphere, which has the same curvature as the sphere. For each spectrum, 2048 interferograms were averaged, and the sample is maintained in the bottom port for the sample and reference spectra to maintain a constant sphere albedo [14]. Standards were measured daily prior to the acquisition of sample spectra to ensure there were no major deviations in instrument performance. Prior work using reflectance standards has demonstrated that the systematic error for the total reflectance measurements is estimated to be on the order of $\pm 3\%$ [7]. Infrared transmission spectra of ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$ in KBr were recorded using a previously described instrument [15,16].

To facilitate comparison between the present infrared data versus those from a known system, we also recorded spectra in the visible and near-infrared (VNIR) regions. The VNIR system was selected such that the data produced by that spectrometer at the red end of its spectral domain overlap the data at the blue end of the Bruker/A562 spectra. For the VNIR, the spectra were recorded using an Agilent Cary 5000 dispersive spectrometer equipped with a PTFE-lined DRA-2500 sphere from Labsphere, which has been described [8]. The DRA 2500 is 150 mm in diameter and has two detectors (PbS diode/R928 photomultiplier) that provide coverage from 300 to 2300 nm. Mostly linear, the data beyond 2000 nm tend to exhibit spectral noise and more baseline drift. Spectra are calibrated using a NIST SRM 2036 wavelength standard [17] as well as periodically calibrated for the reflectance values using a series of calibrated Spectralon standards. The DRA 2500 is a comparison sphere with sample and reference ports, both of which have an 8° angle of incidence for the beam, and a specular subtraction port for sample diffuse-only measurements. The sample port is 38 mm, and samples are generally housed in custom holders having a quartz first-surface window; data are corrected for the effects of the cover window used to contain the samples [8]. Although the IR and VNIR spheres are of different design, as are the two spectrometers (dispersive versus FT), we have found that the IR and VNIR systems provide excellent agreement, both on the wavelength and reflectance axes, over the 1600–2000 nm domain.

The samples used in this research were purchased from various vendors—${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$: Aldrich 99.999%; ${\mathrm{CaCO}}_3$: Acros Organics 97%; ${\mathrm{Na}}_2{\mathrm{SO}}_4$: Sigma-Aldrich 99%; Lactose: Fluka Analytical, purity not specified. The samples were taken directly from the bottle and ground to various particle sizes using a mortar and pestle. An ATM sonic sifter separator is used to separate the ground samples into various size fractions typically using the following sieve diameters: 45, 90, 180, 250, and 500 μm, yielding the following size fractions: 0–45, 45–90, 90–180, 180–250, 250–500, and greater than 500 μm; other sieves can be used to generate additional size fractions, as was done for two of the samples discussed in this paper (${\mathrm{CaCO}}_3$ and ${\mathrm{Na}}_2{\mathrm{SO}}_4$). For IR reflectance measurements, the sieved samples were scooped into a Delrin sample cup, and the excess sample was removed by using the flat end of a spatula blade to smooth the surface; for the VNIR reflectance measurements, the samples are encased into a custom nondispersible holder with a quartz window.

A Keyence VHX-1000 digital microscope with 16-bit resolution was used to provide photomicrographs of the various samples [18] and to directly measure the particle size classification (see Fig. 1). A portion of the sample is spread out onto a glass slide, so that the particles are well separated for analysis. The microscope software differentiates the brightness and colors in the image and extracts the bright objects to produce a binary image. The software assumes all adjacent bright points are part of the same object and then calculates the area for each of these objects. The area (A) is used to calculate the mean particle diameter (d) by assuming that the particles are spherical and using the relationship $d={(4*A/\pi )}^{1/2}$. Although the assumption of spherical particles is not perfectly valid, the analysis provides a reasonable estimate of the mean particle size. For all of the samples, the calculated mean diameter generally showed good agreement with some slight overestimation of the size fractions, as determined from the sieve sizes. Average particle sizes and standard deviations ($1\sigma$) for the four sieved samples, as measured using optical microscopy, are shown in Tables 1 through 4. Because of the diffraction of visible light, the optical microscopy system will not resolve submicrometer particles [19], so that, for the smallest size fractions, the mean particle sizes may be skewed toward larger values.

Table 1. Average ${({\mathsf{NH}}_{\mathsf{4}})}_{\mathsf{2}}{\mathsf{SO}}_{\mathsf{4}}$ Particle Size Via Optical Microscopy

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Table 2. Average ${\mathsf{CaCO}}_{\mathsf{3}}$ Particle Size Via Optical Microscopy

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Fig. 1. Photomicrograph images of ${\mathrm{NaNO}}_3$ [see Ref. 18] for the first three particle size fractions: 0–45, 45–90, and 90–180 μm. Red scale bar at lower right corner is 500 μm for all images.

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3. Results and Analysis

A. Ammonium Sulfate

Ammonium sulfate, ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$, was ground and sieved to provide six sample size fractions. Table 1 shows the calculated mean diameters measured by optical microscopy along with the standard deviations ($1\sigma$) compared to the size fractions determined from the sieve sizes.

Figure 2 displays the VNIR hemispherical reflectance spectra from 11,000 to $5000{\mathrm{cm}}^{-1}$ for all six grain sizes along with the corresponding IR measurements in the 6500–$4000{\mathrm{cm}}^{-1}$ domain. The agreement between the two measurements in the overlap region is excellent, further corroborating the ordinate values of the infrared technique. The most obvious trend seen in the spectra of Fig. 2 is that the overall reflectivity or albedo increases as the mean particle size is reduced. This trend was observed with all of the samples that were measured [18,20] and has been noted in the literature by other researchers [1 –6,21 –23]. In the NIR, the observed spectral features are absorption bands attributed to either overtones or combinations of fundamental vibrational modes. Such bands are generally weak and appear as troughs in the reflectance spectra, typically not saturated [24]. In such cases, volume scattering dominates both on and off the absorption features, so that, as the particle size becomes smaller, the incident radiation sees a greater fractional surface area (i.e., a less porous surface) and passes through less of the absorbing medium; thus, (1) the overall reflectivity increases and (2) the bands generally decrease in spectral contrast with decreasing particle size. These two effects are clearly observed for the two downward-going peaks at 9400 and $7850{\mathrm{cm}}^{-1}$ in Fig. 2, labeled A and B, respectively. The trough at $6360{\mathrm{cm}}^{-1}$ (labeled C in Fig. 2), however, shows a different trend in which the spectral contrast actually increases slightly as the particle size gets smaller. This trend differs from peaks A and B in Fig. 2 because of a larger absorption coefficient such that absorption in the band center remains nearly saturated for most of the larger size fractions, while reflectivity in the wings of the absorption feature increases. We emphasize that bands such as A and B are relatively weak absorbers in the VNIR and that, with increasing particle size, the increase in spectral contrast directly leads to greater specificity and detection for the larger particle sizes. This trend, however, is reversed once the particles reach a critical size, so that absorption in the band center is no longer saturated for the smallest size fraction (i.e., 0–45 μm), the particles become optically thin, and the spectral contrast begins to decrease.

 

Fig. 2. Hemispherical reflectance spectra for ground and sieved samples of ammonium sulfate for all six grain sizes from $11000-5000{\mathrm{cm}}^{-1}$ for the VNIR data and from $6500-4000{\mathrm{cm}}^{-1}$ for the IR data. Corresponding size fractions for the VNIR and IR are plotted in the same color.

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Similar to the VNIR data, Fig. 3 shows the hemispherical infrared reflectance data over all six grain sizes. In this spectral region, the absorption cross sections are larger than the VNIR so that other trends are observed, and the reflectance is generally lower because of increased absorption. In the IR, both surface and volume scattering contribute, so that both minima and maxima are observed in the reflectance spectra, particularly below ca. $3500{\mathrm{cm}}^{-1}$.

 

Fig. 3. Hemispherical (total) IR reflectance spectra from $6000-1500{\mathrm{cm}}^{-1}$ for ground and sieved samples of ammonium sulfate for all six grain sizes.

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The minima at 4905 and $4678{\mathrm{cm}}^{-1}$ show that the spectral contrast increases as the particle size is reduced, which is opposite from most of the VNIR bands in Fig. 2 (i.e., peaks labeled A and B). This increase in spectral contrast indicates that absorption in the band center is greater because of larger absorption coefficient, so that, as the particle size is reduced, absorption in the band center remains saturated or nearly saturated but absorption in the wings does not, similar to the band at $6360{\mathrm{cm}}^{-1}$ in Fig. 2 (labeled C), except the spectral contrast continues to increase even for the smallest size fraction.

Ammonium sulfate has an anion (${\mathrm{SO}}_4^{2-}$) and cation (${\mathrm{NH}}_4^{+}$), each of which is highly symmetric and has very strong IR absorption bands. The free ${\mathrm{NH}}_4^{+}$ and ${\mathrm{SO}}_4^{2-}$ ions have tetrahedral (${\mathrm{T}}_d$) geometry, so that ${\nu }_1$ and ${\nu }_2$ are IR forbidden transitions. The other two fundamental molecular vibrations, ${\nu }_3$ and ${\nu }_4$, represent the antisymmetric stretching and bending modes, respectively, both of which are infrared active. In the crystalline state, the symmetry is lowered, so that additional vibrational frequencies can be observed from removal of degeneracies present in the normal modes for the free ions. In this paper, for simplicity’s sake, we discuss the vibrational modes associated with the free ions.

The positions and intensities of the IR absorption bands for crystalline ammonium sulfate have been measured by Miller and Wilkins [25]. The two strongest bands in the absorption spectrum are centered at $1105{\mathrm{cm}}^{-1}$ and $1410{\mathrm{cm}}^{-1}$; these bands can be assigned to ${\nu }_3({\mathrm{SO}}_4^{2-})$ and ${\nu }_4({\mathrm{NH}}_4^{+})$, respectively. These two strong molecular vibrations give rise to upward-going bands, which are peaked at 1106 and $1417{\mathrm{cm}}^{-1}$, respectively. Both peaks are known as reststrahlen bands. Reststrahlen features arise when the absorption coefficient $k$ is large so that Fresnel reflection is high, and most of the incident radiation is reflected from the first surface; thus, a peak is observed in the reflectance spectrum. Photons that do penetrate the surface are absorbed because of the large absorption coefficient, so the rays that do backscatter are dominated by surface scattering [24].

Interestingly, the reststrahlen peaks seen in Fig. 4 appear quite invariant with particle size. The difference is minor and only slightly larger than the measurement error (typically 3%), and there is no clear trend with particle size. Studies on quartz and fused silica, however, have shown different results. These works have shown that, for smaller particle sizes, the intensity of the reststrahlen band decreases as the particle size is reduced [1 –6,26]. Indeed, for ${\mathrm{SiO}}_2$, for example, the reststrahlen bands can be greatly diminished for fine particle sizes, that is, less than 5 μm [3]. Salisbury and Wald [3] report that, typically, 70–80% of the particles in their 0–75 μm size fractions are smaller than 5 μm in diameter. For most of the samples we have measured, however, the reststrahlen bands are relatively invariant with particle size for the size ranges used in this study. The one exception is sodium sulfate, which is presented in Section 3C; for this compound, a greater number of size fractions were prepared, and the intensities of the reststrahlen bands do show some variance with particle size. As shown in Table 1, however, our particle sizes for the smallest size fraction do not generally show an abundance of particles less than 5 μm; this results in a larger mean diameter for the smallest size fractions than those earlier studies; this effect, however, could also result from the limitations of using optical microscopy to quantify the particle size because this technique is not able to discern submicrometer particles.

 

Fig. 4. Hemispherical (total) IR reflectance spectra from $1600-600{\mathrm{cm}}^{-1}$ for ground and sieved samples of ammonium sulfate for all six grain sizes.

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Along with the reststrahlen bands, certain minima are consistently observed at a higher frequency adjacent to the reststrahlen bands; in Fig. 4, minima are observed at 1490 and $1188{\mathrm{cm}}^{-1}$. These features, which denote a sharp drop in reflectivity and occur on the high-frequency side of the reststrahlen band, have been observed in the reflectance spectra of powdered minerals and are referred to as Christiansen features [2 –6,21,27] and always appear on the shorter wavelength (blue) edge of the reststrahlen band. Traditionally, researchers [2,24] have attributed this minimum in reflectance to those wavelengths where the refractive index of the material undergoes a rapid change such that it approaches the refractive index of the surrounding medium, i.e., the Christiansen effect [28]. Because there is little scattering or absorption, the radiation is able to pass through the sample relatively easily, resulting in a minimum in reflectance, often near zero. As the minimum does not usually appear at exactly the Christiansen frequency (i.e., where $n=1$), Hapke [27] has attributed this feature to the region in the IR spectrum where the particle albedo changes from primarily volume scattering to surface scattering from the increase in the absorption coefficient; therefore, minimal scattering results and a minimum in reflectance is observed prior to the reststrahlen band where surface scattering dominates. The exact position of the minimum has been suggested by Salisbury and Walter [2] as a means to differentiate classes of (extraterrestrial) minerals. Because this feature is present adjacent to a reststrahlen band, materials with multiple strong absorption bands can have more than one Christiansen feature.

On the low-frequency side for each of the reststrahlen bands, an upward-going peak occurs that increases in reflectivity for the smaller size fractions. These features, which are peaked at 1300 and $843{\mathrm{cm}}^{-1}$ for ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$, are referred to as transparency peaks [2 –6] and occur in spectral regions adjacent to the reststrahlen band where the absorption coefficients are weak and volume scattering dominates. Transparency features find their origin in a process that is primarily volume scattering because the features do not actually result from absorption. Instead, minimal absorption by the particles allows more photons to survive passage through the grains and be backscattered. The effect becomes more pronounced as the grain size becomes smaller due to the increased number of particles, which yields a greater number of reflections. Thus, a broad reflectance maximum is observed, particularly for fine grain sizes.

In the ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$ crystalline absorption spectrum [25], two strong, sharp features are observed near 3055 and $3165{\mathrm{cm}}^{-1}$ and are attributed to ${\nu }_3({\mathrm{NH}}_4^{+})$. Two very weak maxima also are observed at 3184 and $2996{\mathrm{cm}}^{-1}$ in our IR reflectance spectra in Fig. 3; these peaks are assigned as reststrahlen bands. These reststrahlen features are more clearly observed in the reflectance spectra for ammonium chloride, as reported by Su et al. [20]. A nearly zero intensity feature is observed at $3330{\mathrm{cm}}^{-1}$ corresponding to the Christiansen feature. The transparency feature, on the other hand, is easily identified by the broad peak centered at $2360{\mathrm{cm}}^{-1}$, which becomes clearly evident for the two smallest particle size fractions. The other fundamental molecular vibration, ${\nu }_4({\mathrm{SO}}_4^{2-})$, is observed in Fig. 4 at $615{\mathrm{cm}}^{-1}$ near the cutoff for the detector and is also a reststrahlen band.

B. Calcium Carbonate

Calcium carbonate (${\mathrm{CaCO}}_3$) was ground and sieved to provide seven sample size fractions, which are shown in Table 2, along with the measured mean diameter as measured by optical microscopy. An additional sieve is used to provide a size fraction with a smaller mean diameter and narrower distribution for the finest particles. For this sample, microscopic images showed fine particles below 5 μm for all of the size fractions; because the microscope cannot accurately measure fine particles below 5 μm, these very fine particles are excluded from the calculation used in Table 2. The microscopic images also showed fine particles clinging to the coarse particles, which have been shown to increase the volume scattering component [3].

Table 3. Average ${\mathsf{Na}}_{\mathsf{2}}{\mathsf{SO}}_{\mathsf{4}}$ Particle Size Via Optical Microscopy

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The total and diffuse spectra were recorded for all seven size fractions in the VNIR and IR. Figure 5 shows an expanded plot of the hemispherical (total) reflectance spectra for all seven grain sizes of ground and sieved samples of calcium carbonate from $3000-1000{\mathrm{cm}}^{-1}$.

 

Fig. 5. Hemispherical IR reflectance spectra from $3000-1000{\mathrm{cm}}^{-1}$ for all seven grain sizes of ground and sieved ${\mathrm{CaCO}}_3$.

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The transmission spectrum for calcite, the most stable polymorph of calcium carbonate, has been measured by various researchers [29,30]. The IR reflectance [3,29] and emission spectra [1,23,31] for calcite have also been reported using different grain size fractions than the ones used in this study. The free carbonate ion (${\mathrm{CO}}_3^{2-}$) has trigonal planar geometry and thus belongs to the $D_{3h}$ point group. The fully symmetric C–O stretch is ${\nu }_1$ and, because it belongs to the $a_{1g}$ irreducible representation, is only Raman active. The strongest absorption band corresponds to the asymmetric stretch ${\nu }_3$ and is located at $1420{\mathrm{cm}}^{-1}$ in transmission [30]. The other two fundamental molecular vibrations correspond to ${\nu }_2({\mathrm{CO}}_3^{2-})$, the out-of-plane bending mode, and ${\nu }_4({\mathrm{CO}}_3^{2-})$, the planar deformation, at 877 and $713{\mathrm{cm}}^{-1}$, respectively [30].

For ${\mathrm{CaCO}}_3$ reflectance, we again observe that, for wavenumbers $\gtrsim 1800{\mathrm{cm}}^{-1}$, volume scattering dominates, and there is a strong inverse correlation with particle size. In those regions free of absorption bands, the albedo of the finest sieve particle spectra is three to four times greater than that of the larger particle sieve aliquot. In the longwave region shown in Fig. 5, the strongest absorption feature, ${\nu }_3({\mathrm{CO}}_3^{2-})$, occurs as a reststrahlen peak with a maximum at $1541{\mathrm{cm}}^{-1}$. The same band was observed in the transmission spectrum by Andersen and Brečević [30] at $1420{\mathrm{cm}}^{-1}$. This phenomenon for the reststrahlen bands in reflectance spectra to be offset to shorter wavelengths from the transmission peak has been observed for some of the other species as well [18] and has been attributed by Hunt and Vincent [1] to the dependence of reflectance spectra on the refractive index and the absorption coefficient, whereas transmission features are due solely to the absorption coefficient. This upward-going reflectance feature is accompanied by a Christiansen minimum near $1620{\mathrm{cm}}^{-1}$ and a transparency feature peaked at $1220{\mathrm{cm}}^{-1}$. A separate overtone ($2{\nu }_2$) is also seen at $1797{\mathrm{cm}}^{-1}$ in which the spectral contrast increases with particle size resulting (for the fine size fractions) in a deep trough adjacent to volume scattering.

An expanded plot from $1000-600{\mathrm{cm}}^{-1}$ is shown in Fig. 6. The ${\nu }_2({\mathrm{CO}}_3^{2-})$ band appears as a peak maximum at $886{\mathrm{cm}}^{-1}$ for the larger grain sizes but undergoes a contrast reversal for smaller grain sizes, as volume scattering starts to dominate. For particle sizes between 180 and 250 μm, the peak becomes asymmetric, as a trough begins to appear around $872{\mathrm{cm}}^{-1}$. For particle sizes below ca. 45 μm, the peak maximum has completely disappeared from the spectrum and is replaced by a deep trough, which is consistent with earlier nonquantitative measurements with a different set of size fractions and lower SNR [1,3,23,29].

 

Fig. 6. Hemispherical (total) IR reflectance spectra from $1000-600{\mathrm{cm}}^{-1}$ for ground and sieved samples of calcium carbonate for five grain sizes. Each spectrum (except for the purple trace) is vertically offset by 5% increments for clarity.

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This inversion from a maximum to minimum as a function of particle size is similar to the effect seen for the ${\nu }_2({\mathrm{NO}}_3^{-})$ band of ${\mathrm{NaNO}}_3$ [18]. Figure 6 clearly demonstrates the importance of grain size considerations in LWIR reflectance spectroscopy—certain pronounced maxima become pronounced minima, even though adjacent bands do not undergo any such reversals.

The absorption band at $714{\mathrm{cm}}^{-1}$ corresponds to ${\nu }_4({\mathrm{CO}}_3^{2-})$ and shows a similar inversion as ${\nu }_3$, except that surface scattering is not as efficient for the largest particle sizes; thus, a weak derivative-shaped feature appears for the largest grain sizes because both volume and surface scattering contribute to the reflectance. The spectral feature appears as a trough for most of the smaller particle sizes, as volume scattering starts to dominate.

C. Sodium Sulfate

Anhydrous sodium sulfate (${\mathrm{Na}}_2{\mathrm{SO}}_4$) was ground and sieved to obtain 13 size fractions, as shown in Table 3. A greater number of sieves were used to provide sample size fractions with a narrower distribution, thus facilitating quantification of particle size effects. We observe that the mean size measured by microscopy is slightly greater than the size fraction in most cases. A similar effect was observed by Moersch and Christensen [21] and was attributed to particles not being spherical but rather elongated, allowing some particles to pass through the sieves in a lengthwise direction. Alternatively, some particles may have clumped after sieving. Spectra were therefore remeasured five months later for the same size fractions (samples placed open in a dry box), and no significant spectral changes were observed. Other grinding experiments also indicated that the anhydrous form appeared to be more stable versus the decahydrate form of ${\mathrm{Na}}_2{\mathrm{SO}}_4$.

Table 4. Average Lactose Particle Size Via Optical Microscopy

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The total and diffuse VNIR and IR spectra were recorded for all 13 size fractions. Figure 7 shows a plot of the total IR reflectance spectra for just three of the ground and sieved samples of sodium sulfate from $6000-1400{\mathrm{cm}}^{-1}$.

 

Fig. 7. Hemispherical (total) IR reflectance spectra for ground and sieved samples of sodium sulfate for three grain sizes from $6000-1400{\mathrm{cm}}^{-1}$. The other 10 particle sizes are omitted for clarity, but similar trends are observed.

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Similar to ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$, reststrahlen features are observed for the IR-allowed stretching and bending vibrations, ${\nu }_3({\mathrm{SO}}_4^{2-})$ and ${\nu }_4({\mathrm{SO}}_4^{2-})$, respectively. An expanded plot from $1600-600{\mathrm{cm}}^{-1}$ is shown in Fig. 8, which clearly shows an upward-going doublet with peaks that occur at 1182 and $1135{\mathrm{cm}}^{-1}$. These two peaks are reststrahlen features; both are assigned to the ${\nu }_3({\mathrm{SO}}_4^{2-})$ band. Unlike ammonium sulfate in Section 3A, the symmetry is lowered for crystalline ${\mathrm{Na}}_2{\mathrm{SO}}_4$, so that degeneracies for this triply degenerate ($t_2$) vibrational mode are removed [32]. In addition, a very weak downward-going peak is observed at $976{\mathrm{cm}}^{-1}$, which can be assigned to ${\nu }_1({\mathrm{SO}}_4^{2-})$, which is IR forbidden if the ${\mathrm{SO}}_4^{2-}$ were to retain $T_d$ geometry [32].

 

Fig. 8. Hemispherical (total) IR reflectance spectra from $1600-600{\mathrm{cm}}^{-1}$ for ground and sieved samples of sodium sulfate for all 13 grain sizes.

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The Christiansen minimum appears at ca. $1270{\mathrm{cm}}^{-1}$ with a transparency band peaking at $844{\mathrm{cm}}^{-1}$. Interestingly, for ${\mathrm{Na}}_2{\mathrm{SO}}_4$, the intensity of the reststrahlen band does, in fact, decrease with particle size but only as the particle diameter approaches $\lesssim 150\mathrm{\mu m}$. For the smallest size fraction, which has a mean diameter of $26\pm 18\mathrm{\mu m}$, the reststrahlen band peaks at 16%; the reflectance has increased to 49% for the largest grain sizes ($>150\mathrm{\mu m}$). This difference is larger than the measurement uncertainty, suggesting that, at least for these narrower size fractions, a change in intensity of the ${\mathrm{Na}}_2{\mathrm{SO}}_4$ reststrahlen band is observed and is similar to what has been reported for the small size fractions of ${\mathrm{SiO}}_2$ [3,21].

The other fundamental molecular vibration ${\nu }_4({\mathrm{SO}}_4^{2-})$ also appears as a doubly peaked reststrahlen band (mode split due to lower symmetry), which is observed at 619 and $646{\mathrm{cm}}^{-1}$, as is shown in Fig. 9 on an expanded plot. For this feature, the intensity of the reststrahlen band also changes with particle size, again by a factor of ~3. The reflectance for the smallest size fraction (0–45 μm) is 4.5% and increases to 13.5% for the largest size fraction ($>500\mathrm{\mu m}$). However, we also note that the present system typically demonstrates a 1%–2% $R$ residual signal, even for substances whose reflectance is nominally zero.

 

Fig. 9. Hemispherical (total) IR reflectance spectra from $800-600{\mathrm{cm}}^{-1}$ for ground and sieved samples of sodium sulfate for all 13 grain sizes.

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D. Lactose

Our studies have also included organic compounds whose spectra tend to be more congested than simple inorganics. As an example, lactose (${\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}$) was ground and sieved to provide the five size fractions shown in Table 4, along with the mean diameters as measured by microscopy.

The total and diffuse VNIR and IR spectra were recorded for all five size fractions for ground and sieved samples of lactose. Figure 10 shows the hemispherical (total) IR reflectance spectra for all five grain sizes. Similar to the inorganic salts, the reflectivity increases as the particle size gets smaller. A smaller difference in reflectivity, however, is observed for the largest size fractions (i.e., 180–250 and 250–500 μm); this effect is most likely from less differentiation between the particle size ranges: that is, greater overlap between the two size fractions as reflected by the mean diameters and standard deviations presented in Table 4.

 

Fig. 10. Hemispherical (total) IR reflectance spectra for ground and sieved samples of lactose for all five grain sizes from $7000-600{\mathrm{cm}}^{-1}$.

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Lactose, which has two six-membered rings linked by an oxygen atom, is a large molecule with ${\mathrm{C}}_1$ symmetry and 129 normal vibrational modes; thus, the crystalline absorption spectrum is quite congested [33,34], and the total IR reflectance spectra shows multiple spectral features, as seen in Fig. 10. Strong absorption features around $3700-3000{\mathrm{cm}}^{-1}$ correspond to the O–H stretching vibrations. In the IR reflectance spectrum of Fig. 10, deep troughs are observed near 3400 and $1125{\mathrm{cm}}^{-1}$ in which the spectral contrast increases as the particle size gets smaller, indicating the absorptions remain saturated in the band centers in this region of the spectrum.

An expanded plot from $1400-600{\mathrm{cm}}^{-1}$ is shown in Fig. 11. Upon closer inspection from $1200-1000{\mathrm{cm}}^{-1}$, which is the region depicted in the gray box in Fig. 11, a few upward-going peaks are observed, indicating surface scattering behavior. These peaks appear at 1096, 1071, and $1034{\mathrm{cm}}^{-1}$ and overlap with prominent bands in the absorption spectrum [33,34]. These features, which most likely correspond to C–C and C–O stretching, are reststrahlen bands and occur in regions where the absorption coefficient is high. As observed for the inorganic compounds with similar particle size distributions, the intensities of the reststrahlen bands show little variance with particle size.

 

Fig. 11. Hemispherical (total) IR reflectance spectra for ground and sieved samples of lactose for all five grain sizes from $1400-600{\mathrm{cm}}^{-1}$. Gray box shows the small upward-going peaks assigned as reststrahlen features.

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Any associated Christiansen features, as observed for all of the inorganic compounds, however, are not clearly evident in Fig. 11. This apparent lack of associated Christiansen minima most likely results from the spectral congestion from the increased overlap and contributions from neighboring spectral features. An increase in reflectance is observed for the finest grain size from 1000 to $750{\mathrm{cm}}^{-1}$, which could correspond to a region of transparency; this region, however, overlaps with absorption features that show up as downward-going peaks in the reflectance spectrum. These troughs show an increase in spectral contrast for the finest grain size; thus, absorption remains nearly saturated in the band centers. That the Christiansen minima are completely obfuscated by other vibrational modes precludes using such minima as identifiers of chemical composition, as was shown possible for (simpler) mineral species [2].

4. Discussion

Others have experimentally studied the reflectance/emissivity of solid powders as a function of their physical parameters including, and in particular, their particle size. These include the works of authors such as Salisbury and co-workers [2,3], Mustard and co-workers [4,22], Moersch and Christensen [21], Lane and co-workers [23,31], Hunt and co-workers [1,35], Le Bras and Erard [5], and Conel [36] as well as having been modeled by authors such as Hapke and co-workers [37 –39]. However, almost all of these studies have focused on minerals for planetary studies, such as silicates and carbonates. We chose not to focus on silica or quartz for several reasons: while quartz is of interest and its optical constants have been measured, its birefringence is a complicating factor, and the small number of vibrational modes for ${\mathrm{SiO}}_2$ means it is not typical of most molecules. It also tends to form finer particles than do many species such as those measured here. Instead, we have investigated dozens of other compounds in our studies and selected four representative species. All of the samples were sieved to provide at least five particle size fractions that were quantified using optical microscopy. The works by others typically involved size distributions that were limited in number (five or fewer) and were only loosely quantified using the size bins that were obtained through the sieving process. Moreover, many of those studies focused only on very small particles, approximately 200 μm or less.

Other than the work of Moersch and Christensen [21], as well as Lane and co-workers [23,31], the present work is one of the few studies to employ large numbers of validated particle sizes; for example, 13 different size fractions in the case of anhydrous ${\mathrm{Na}}_2{\mathrm{SO}}_4$. We also have confirmatory measurement of the mean particle size for each bin via microscopy, as seen in Tables 1 through 4. Finally, along with the work of Salisbury and co-workers [2,40] and Hanssen et al. [14], we believe that the present data represent some of the few that extensively calibrate the $y$ axis for reflectance in terms of using standards of high, low, and intermediate reflectivity with differing degrees of specular and diffuse components [41]. The methods for the calibration of the A562 integrating sphere have been reported by Blake et al. [7] and have further been validated by VNIR measurements of the same materials. The present data, thus, allow a greater degree of quantitative investigation of reflectance versus particle size.

Typically, much of the short- and midwave infrared is dominated by volume scattering, and there is a strong inverse correlation with particle size. In these domains, most absorption features are weak overtone or combination bands resulting in downward-going peaks. Across the longwave infrared, however, the reflectivity is typically much lower, and the strongest fundamental vibrational modes generally manifest themselves as reststrahlen bands with upward-going peaks in the reflectance spectrum with adjacent Christiansen minima on the high-energy sides. Transparency peaks are also observed adjacent to the reststrahlen bands but appear on the low-energy side of the reststrahlen peaks; the reflectance of these upward-going peaks increases as the particle size is reduced. We note that, for the dozens of species in our study, the reststrahlen bands were fairly invariant with particle size, as shown in Figs. 4 and 5. The one exception is ${\mathrm{Na}}_2{\mathrm{SO}}_4$, as shown in Figs. 8 and 9, in which the reststrahlen band begins to wane in amplitude for the smallest size fractions ($<150\mathrm{\mu m}$). For intermediate and large size particles (i.e., $\ge 150\mathrm{\mu m}$), the peak of the reflectance band is of nearly constant amplitude, but the bands do show a diminishing reflectivity for the finest particle sizes, as seen in, for example, quartz and carbonates [3,29]. This exception is most likely from the narrower size bins, which prevent surface scattering effects from being washed out with particle size.

The presence of such disparate features can be understood by knowledge of particle morphology combined with the optical constants, $n$ and $k$. Moersch and Christensen [21] have analyzed the features by dividing into four basic classes based on the values of $n$ and $k$. Class 1 is where $k$ is large ($k>2$) and represents the reststrahlen bands; class 2 occurs where both $k$ and $n$ are moderate ($0.5<k<1$ and $n\sim 2$); class 3 represents the transparency bands where $k$ is small ($k<0.1$) and $n$ is moderate ($n\sim 2$), and class 4 is where $k$ is small ($k<0.1$) and $n$ is near unity and represents the Christiansen minima. For anhydrous ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$, the optical constants have been determined by Toon et al. [42]. These values are plotted in Fig. 12 (middle traces) along with the reflectance spectra for three of our particle size fractions of ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$ (bottom traces) as well as the transmission spectrum of ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$ in a KBr pellet (top trace).

 

Fig. 12. Hemispherical IR reflectance spectra (bottom plot) for ground and sieved samples of ammonium sulfate for three grain sizes from $3500-600{\mathrm{cm}}^{-1}$: 0–45 μm (black trace); 45–90 μm (red trace); and $>500\mathrm{\mu m}$ (purple trace). The middle frame shows the calculated $n$ (blue trace) and $k$ (green trace) values [42]. The black dotted vertical lines represent the locations of the reststrahlen bands in the reflectance spectrum. Top frame shows transmission spectrum after subtracting spectrum of pure KBr.

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The reflectance spectra for ${({\mathrm{NH}}_4)}_2{\mathrm{SO}}_4$ show multiple reststrahlen bands caused by the presence of an anion and cation with strong fundamental modes. As expected, the reststrahlen bands occur in regions where $k$ is large and $n\sim 1$ and increases sharply with decreasing frequency [43]. The height of the reststrahlen feature primarily depends on the magnitude of $k$ and is greater for the peaks at $1106{\mathrm{cm}}^{-1}$ and $615{\mathrm{cm}}^{-1}$ because of a larger $k$ ($\ge 2$). In comparison, although a broad, strong absorption feature is observed in the transmission spectrum around $3125{\mathrm{cm}}^{-1}$, the reststrahlen peaks are diminished in the reflectance spectra because their $k$ values are smaller (ca. 0.3 whereas for the three LWIR bands $1<k<2.5$). That is to say, for the two absorption bands near $3000{\mathrm{cm}}^{-1}$, we observe that the reststrahlen features are much weaker than their LWIR counterparts. Their absorption peaks are still very strong (top traces) because the absorption coefficient depends not only on $k$ but is inversely proportional to the wavelength $(\alpha =4\pi k/\lambda )$. Alternatively construed, Fig. 12 demonstrates how, for bands of comparable absorption strengths in the MWIR and LWIR, the reststrahlen amplitudes will tend to be larger in the LWIR.

Figure 12 also shows that the reflectance Christiansen features (i.e., class 4) occur where $n$ is near unity and $k$ is small but increasing with wavelength; on the other hand, the transparency bands (i.e., class 3) occur where $k$ is very small ($k<0.1$) and $n$ is moderate ($n\sim 1.5$). Height of the transparency bands depends on the magnitude of $k$ and is greater for the transparency peaks at ca. 2360 and $846{\mathrm{cm}}^{-1}$ because $k$ is nearly zero, so that very little absorption occurs and most of the light is reflected. For the transparency peak at ca. $1298{\mathrm{cm}}^{-1}$, $k$ is slightly larger ($k\approx 0.07$) due to a nearby spectral feature, so that more absorption occurs, causing the reflectance to decrease.

While most of the reststrahlen bands in our work tended to be fairly invariant with particle size, there was one exception in ${\mathrm{Na}}_2{\mathrm{SO}}_4$, which had a greater number of particle size fractions, especially for smaller particles. This invariance for the reststrahlen band differs from planetary studies on powdered minerals such as silicates, which have shown that the spectral amplitude of the reststrahlen band actually reduces as the particle size decreases [1 –6,21,26,29,35]. For those studies, very fine particles (i.e., less than 5 μm) were suggested to cause this effect. We note that, for our study, the sample preparation does not appear to result in an abundance of fine particles below 5 μm where such effects tend to appear; this difference may be from clumping or hydration during the sieving process because the inorganic salts are often hygroscopic. This effect could also result from our preparation of the ground samples, which involves using a mortar and pestle. The finest particles tend to be $>10\mathrm{\mu m}$ (as seen by optical microscopy), so this effect on the reststrahlen band may be diminished for the grain sizes used in this study.

There was one exception—the microscopic images for ${\mathrm{CaCO}}_3$ did show fine particles below 5 μm for all of the size fractions; these particles, however, are excluded from the calculation of the mean diameters presented in Table 2. The reststrahlen peak in Fig. 5 is invariant with particle size, even for the smallest size fraction, and shows a reflectivity of ca. 23%. Other researchers have measured the infrared reflectance spectrum for calcite [3,29] as well as the emission spectra [1,23] using different particle size fractions. For example, Salisbury and Wald [3] measured the spectra for two size fractions: 0–75 μm and 75–250 μm (this sample was washed with alcohol to remove clinging fines). They measured a reflectivity of $\sim 40\%$ for the larger size fraction and a diminished reflectivity of 16% for the smaller size fraction. The work by Lane [23] used a greater number of size fractions and showed little variance in the emissivity for most of the particle size fractions (710–1000, 355–500, 180–250, and 90–125 μm). For these intermediate and large size fractions, the emissivity was around 55% (i.e., 45% R), whereas the emissivity for the finest grain size (0–63 μm) was around 85% (i.e., 15% R). Research by Salisbury and Wald [3] also showed that the spectral amplitude for the reststrahlen peak decreased by ca. 50% for an unwashed sample of quartz compared to a washed sample for the 75–250 μm size fraction. They proposed that clinging fines caused the spectral shape of the spectrum to be dominated by the fine particles. As already discussed, microscopic images for calcium carbonate do show fines smaller than 5 μm clinging to the coarser particles. The clinging fine particles may, thus, affect the spectral amplitude for the reststrahlen bands of ${\mathrm{CaCO}}_3$, but these do not appear to be the reason for the other solids used in this study.

Careful inspection of the longwave region of Figs. 5 and 9 also shows that the adjacent Christiansen minima on the blue edge begins to increase, rising significantly above 0% $R$ (i.e., greater than our experimental error). However, because this phenomenon is not observed for every Christiansen minimum, it may result from adjacent transparency bands or absorption features that overlap the Christiansen minimum, particularly at small particles sizes ($<\sim 50\mathrm{\mu m}$) where the reflectivity of these features is greater. For example, the minimum for ${\mathrm{CaCO}}_3$ rises to 6.3% for the 11 μm particles at $1620{\mathrm{cm}}^{-1}$ and for ${\mathrm{Na}}_2{\mathrm{SO}}_4$ to 2.9% for the 26 μm particles at $656{\mathrm{cm}}^{-1}$. We also note that, even with careful alignment and a perfect “light trap” with 0% $R$, typical performance of our system yields a consistent but residual signal of $\sim 1.5$ to 2% $R$ baseline.

As shown in Figs. 8 and 9, the spectral contrast of the reststrahlen bands for sodium sulfate did vary as a function of particle size, unlike most of the other species that we have studied. The sample preparation for sodium sulfate involved a greater number of sieves (12 sieves versus a typical five). It may be that the intermediate and larger size fractions had narrower size bins and thus less contamination from the smaller particles. This difference may lead to a greater differentiation in the size of the reststrahlen feature but needs to be studied in more detail to determine if this is the reason.

In all cases, prominent reststrahlen features only tend to arise when $n$ and $k$ are strong, as typically found in salts with polyatomic anions or cations that have a high degree of symmetry with strong absorption coefficients; for example, ${\mathrm{SO}}_4^{2-}$, ${\mathrm{NO}}_3^{-}$, ${\mathrm{NH}}_4^{+}$, etc. Lactose, on the other hand, lacks symmetry but is a large molecule with an abundance of vibrational modes. Figure 11 appears to indicate weak reststrahlen bands, but they do not seem to display the typical features of such bands, including associated Christiansen minima. The scenario observed with lactose is the one more likely observed with a typical compound; namely, weaker reststrahlen features and multiple modes in the LWIR obfuscating any Christiansen minima as well as transparency bands. Such effects would preclude, for example, analysis of particle size via depth of the Christiansen minimum, as has been shown useful for planetary surfaces composed almost exclusively of carbonates and silicates [2]. We expect similar observations for spectroscopies of medium to large molecules, which have heretofore not been widely studied. However, there have been some studies on plants for remote sensing applications [44 –47] and early studies on powdered organic solids for sample identification [26,48]. One study probing broad leaf plants [44] did observe weak reststrahlen bands for cellulose and other organic compounds in regions with strong absorption bands. Olinger and Griffiths [49,50] used a diffuse-reflectance device to show significant reststrahlen-band specular effects as a function of particle size for durum wheat and some of its components such as sucrose and cellulose. As expected, some of the specular component was lost with sufficient dilution in a salt such as KCl. There have otherwise been few studies measuring the reflectance spectra of an organic compound as a function of particle size.

The data garnered in our studies (for organic and inorganic materials) allow for more quantitative study of the reflectance versus size relation—an empirical approach is presented here as a starting point in which we have excluded strong absorption bands, reststrahlen, or Christiansen features. That is, we temporarily ignore features with large $k$ or where $n\sim 1$, which require different consideration [21,43], and select only Moersch and Christensen classes 2 and 3. In such cases of modest to no absorption, volume scattering dominates the reflectance process [24]. Upon analysis of plotting $R$ versus particle diameter ($d$), it becomes clear that, in the theoretical limit of a very large number of particle size bins, the curve will be sigmoidal in nature, as any sample’s reflectivity is constrained to be $0<R<1$. To avoid regions where $R$ can approach 1 for the smallest grain sizes, we have selected sundry wavelengths where the reflectivity values were not saturated near high albedo. A typical plot of $R$ versus $d$ is seen in Fig. 13(a), which shows fits of an inverse relationship to the reflectance data for the seven different wavenumbers, as indicated in the legend.

 

Fig. 13. Plots of (a) the reflectivities versus particle diameter for all ground and sieved samples of sodium sulfate and (b) the hemispherical IR reflectance spectra. Dotted lines in the bottom plot correspond to the regions used in the top plot. Fits (lines) to the experimental data are also shown in the top plot.

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The fit of $R$ versus $d$ shows good agreement to the form,

Empirically, we have performed similar fits on six different molecules, both on and off weak absorption features. In analyses of each case, we used between three and 10 wavelengths (typically five or six), similar to those seen in Fig. 13(b). In most cases, the same conclusion was reached, namely, that either Eq. (1) or (2) adequately fits the data, to within statistical uncertainty. Experimental uncertainty must be borne in mind for the fits, both in terms of the abscissa and the ordinate values. Uncertainties on the $y$ axis range from 3% to 5%, but the $x$ axis values (mean particle diameter) also have substantial uncertainty values for each datum, as seen in Tables 1 through 4.

Starting rather from a physical model, however, the relationship between particle size and single-scattering albedo $\omega$ has been approximated by Hapke and Nelson [51], namely,

 

Fig. 14. Plot of the single-scattering albedo versus mean diameter for ${\mathrm{Na}}_2{\mathrm{SO}}_4$.

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However, we note that, while most of the data for ${\mathrm{Na}}_2{\mathrm{SO}}_4$ and a few other species did fit nicely to this model, the model did not fit at all well to the results of other species such as ${\mathrm{NaNO}}_3$ and ${\mathrm{CaCO}}_3$. As suggested by Johnson et al. [52] and Hapke and Nelson [51], there may be limitations of the model to particles whose mean particle size is significantly larger than the wavelength of the probing light. That is to say, for geologic materials, the model may only be applicable to particles tens of microns or more for the VNIR and perhaps hundreds of microns or more for LWIR wavelengths. There may be limitations in our data sets as well, such as reliably obtaining and confirming particle sizes smaller than $\sim 5\mathrm{\mu m}$ using optical microscopy.

5. Conclusions

These studies provide quantitative hemispherical reflectance spectra of multiple species with not only separated but quantified (via microscopy) particle sizes. As noted by other researchers, in those domains where volume scattering dominates with no or only weak absorption features, the albedo can drop significantly with increasing particle size, in some cases by a factor of 4 or more. Moreover, certain features even show “contrast reversal” with particle size, where positive-going features become negative-going for the same material. The reststrahlen bands also displayed characteristics very different from those wavelengths dominated by volume scattering. In the present work, prominent reststrahlen band features were most clearly manifested in species with high symmetry ions ($T_d$, $D_{4h}$, $D_{3h}$ etc.), ions such as ${({\mathrm{CO}}_3)}^{2-}$, ${({\mathrm{SO}}_4)}^{2-}$, ${\mathrm{NH}}_4^{+}$, etc. For the larger particle sizes ($\gtrsim 150\mathrm{\mu m}$), the reflectance values of such reststrahlen bands tended to remain invariant. But similar to what others have found for the fine particles of ${\mathrm{SiO}}_2$, the reflectance values can drop appreciably when narrow size bins are used for sample preparation: that is, as for sodium sulfate in this study. For reflectance of common materials and chemicals of low or no symmetry, the infrared reflectance spectra tend to be far more congested, of lower general reflectance, and with few appreciable reststrahlen features (i.e., approximating gray- or blackbody behavior for larger particle sizes), but possibly exhibiting some stronger features for the smaller particle sizes.

Further study is warranted to understand these complex behaviors, as it is clear that mean particle size has enormous influence on the measured directional–hemispherical reflectance spectra of bulk materials; successful identification requires sufficient and representative reflectance data to include the various particle sizes that may be encountered. The ability to model such behavior would be most beneficial. The current status is that no one model satisfactorily predicts the reflectance behavior across the entire spectrum, across different values of $n$ and $k$, and across all possible geometries (e.g., spherical versus spheroid versus jagged). The fundamental optical constants ($n$ and $k$) are needed in almost all cases to understand the reflectance/morphological properties at a more fundamental level, and work in this vein continues.