Improved Mode Assignment for Molecular Crystals Through

Oct 8, 2012 - We report the anisotropic terahertz response of oxalic acid and sucrose crystals in the 0.2–3.0 THz range using terahertz time domain ...
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Improved Mode Assignment for Molecular Crystals Through Anisotropic Terahertz Spectroscopy Rohit Singh,† Deepu Koshy George,† Jason B. Benedict,‡ Timothy M. Korter,§ and Andrea G. Markelz*,† †

Department of Physics, ‡Department of Chemistry, University at Buffalo, Buffalo, New York, United States § Department of Chemistry, Syracuse University, Syracuse NY USA ABSTRACT: We report the anisotropic terahertz response of oxalic acid and sucrose crystals in the 0.2−3.0 THz range using terahertz time domain spectroscopy on large, single crystals. We compare the observed anisotropic response with the response calculated using solid-state density functional theory and find good agreement in the orientation dependence and relative intensities of the crystal phonons. It was found that oxalic dihydrate can be reversibly converted to anhydrous by controlled relative humidity. In addition, oxalic acid was found to have a large birefringence with Δn = 0.3, suggesting the material may be useful for terahertz polarization manipulation. Sucrose has a smaller birefringence of Δn = 0.05, similar to that of x-cut quartz. The anisotropic measurements provide both mode separation and symmetry determination to more readily achieve mode assignment for the more complex sucrose spectrum. waveplate.8 The retardance of a waveplate is given by δβ = 2πνdΔn, where ν is the frequency in cm−1, d is the thickness of the waveplate, and Δn is the birefringence of the waveplate at the measurement frequency. Few materials have been shown to have a strong birefringence at terahertz frequencies. Here, we report that oxalic acid has a large birefringence with Δn ∼ 0.3 and could be useful as a waveplate material in the terahertz range. Sucrose, on the other hand, has a birefringence of only Δn = 0.05, similar to x-cut quartz9 typically used for terahertz polarimetry. Both oxalic acid and sucrose have highly anisotropic resonant absorptions and could be used as standards for polarimetry measurements. Anisotropic response from molecular crystals arises from the regular orientation of the molecules within the crystal and the strong coupling between the intramolecular vibrations to the intermolecular phonons. As a result, one might expect enhanced absorption for light polarization along the dipole derivative of the internal vibrational mode. To compare molecular crystal anisotropy measurements with density functional theory (DFT) calculations, it is preferable to begin with systems that are not overly complex and can readily be grown to ∼1 cm dimensions, larger than the diffraction-limited spot size for terahertz measurements. Both oxalic acid and sucrose fulfill these criteria. Figure 1a and b shows the monoclinic crystal structures for oxalic anhydrous and dihydrate respectively.10,11 As can be seen in Figure 1b, dihydrate oxalic acid has two additional water molecules associated with each oxalic acid molecule. In Figure 1c, we show the most common crystal morphology for large oxalic

1. INTRODUCTION Terahertz (THz) vibrational resonances of molecular crystals can be used to determine polymorphism in pharmaceuticals and the fundamental coupling between intramolecular and intermolecular motions.1 A number of groups have made significant progress at identifying and understanding the coupling of internal motions with long-range crystal motions in the terahertz frequency range.1−6 Even in the case of simple crystals, however, calculated vibrations in the terahertz range do not always agree in frequency or intensity with the measured resonances, interfering with mode assignment. In addition, as systems become more complex, modes begin to overlap further complicating assignment. To address these issues, one can use the anisotropy of the optical response of an aligned system to achieve mode separation and distinguish different resonances.7 The symmetry of calculated modes is well-defined, allowing one to more confidently assign modes. Here, we compare ab initio calculations of the anisotropic vibrational spectra of oxalic acid and sucrose crystals to anisotropic terahertz time domain spectroscopy (THz TDS) measurements. The comparison with anisotropic measurements significantly reduces the ambiguity in mode assignment, even for the somewhat complex sucrose spectrum. Beyond examining the use of anisotropic measurements for mode assignment, there is a real need for well characterized birefringent materials at terahertz frequencies for polarization manipulation and calibration. Polarization control can be used for modulation of sources for high sensitivity and fast imaging systems. Polarization modulation is also necessary for the determination of anisotropic response, such as the Faraday angle or Kerr angle. Jenkins et al. have demonstrated a high sensitivity technique for determining the real and imaginary Faraday angle at terahertz frequencies using a rotating © 2012 American Chemical Society

Received: July 24, 2012 Revised: October 2, 2012 Published: October 8, 2012 10359

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have been assigned with the some success using DFT.14,15 This agreement of the absorbance measurements with DFT calculations for nonoriented samples motivates using sucrose and oxalic acid to study anisotropic response. Although the anisotropic absorbance of a single sucrose crystal has been measured earlier,16,17 there has not been a report of the anisotropy of its index nor a comparison to the anisotropy predicted by DFT.

2. METHODS For the anisotropy characterization, large oxalic acid and sucrose crystals were made using the seeded growth method. Sucrose and oxalic acid dihydrate powder were purchased from Sigma Aldrich. Sucrose was dissolved in pure deionized (DI) water to saturation at 23 °C. Small sucrose seeds were first grown by drying the saturated solution. Seeds were then hung in a beaker filled with clean saturated solution and were left for several days at room temperature. Oxalic acid crystal growth followed a procedure similar to that found in the literature.18 Seeds were formed from drying a saturated solution of oxalic acid in DI water. For large dihydrate crystal growth, a saturated solution was made using a solvent of 75% acetone and 25% DI water. The seeds were hung in the saturated solution at 35 °C. For both sucrose and oxalic acid, crystals as large as 15 mm × 12 mm × 7 mm were obtained, and habits were sufficiently clear that one can readily identify the crystal axes. The grown crystals were polished by hand along a specific crystal face (a, b, or c) to have a large area and are sufficiently thin so that the absorbance is within the dynamic range of the terahertz system. The polished thicknesses were between 300 and 1000 μm. The polished crystals were mounted on aluminum apertures centered in a rotation stage. Transmission measurements were made every 15°, with 0° corresponding to the terahertz electric field perpendicular to the crystal b-axis. All measurements were made at room temperature. Single crystal sucrose X-ray measurements were performed for crystallographic indexing. The polished plates were mounted to glass capillaries using rubber cement. Data were collected at room temperature using a Mo Bruker microfocus turbo rotating anode source (λ = 0.7107 Å) that was focused and monochromatized with Helios multilayer optics. Given the high brilliance of the X-ray source and the large size of the crystal plates, data were collected at the edge of the plates to avoid over exposure of the APEXII CCD detector. Three ω scans (20°/scan, 0.5°/frame) were used to determine the unit cell and orientation matrix for each sample. The index of the polished face was then unambiguously determined using the face indexing routine of the APEXII software suite. Terahertz time domain spectroscopy is described elsewhere.19 It is necessary to purge the terahertz spectroscopy system with dry nitrogen gas to remove background absorption from atmospheric water. We have found that oxalic acid dihydrate crystals convert to anhydrous when placed in a dry nitrogen environment overnight. This is shown in Figure 3 which shows the measured absorption coefficient of an oxalic acid dihydrate crystal that remains at 0° orientation but is exposed to different environments over time. Initially, the characteristic peak at 1.38 THz for the dihydrate was present, as shown by the red curve. The green curve shows the same sample with the same orientation, but after 12 h exposure to the dry nitrogen atmosphere. The 1.38 THz absorbance is absent, and a new absorbance at 1.95 THz is present, indicating the

Figure 1. Crystal structures for (a) anhydrous and (b) dihydrate oxalic acid along the b axis. The red, black, and gray in the images are oxygen, carbon, and hydrogen, respectively. (c) The morphology for a large oxalic acid crystal that allows one to readily identify the crystal orientation with c = (001), p = (110), p′ = (110̅ ), r = (101), and r′ = (101̅).

acid crystals and the relationship of the facets to the crystal axes, allowing one to easily determine crystal orientation of the as-grown crystals. Oxalic acid dihydrate forms monoclinic crystals. We find that we can convert to anhydrous and then back to dihydrate through humidity control. Since the crystal remains intact during this conversion, we have assumed that the anhydrous oxalic acid is also in the monoclinic form, as shown in Figure 1a; however, we note that anhydrous can also form orthorhombic crystals. Figure 2a shows the unit cell structure

Figure 2. (a) Unit cell in a sucrose crystal looking down the a-axis. The red, black, and gray in the images are oxygen, carbon, and hydrogen, respectively. (b) The morphology for a large sucrose crystal along with the crystal axes directions relative to the large crystal planes: a = (100), p′ = (110), c ∼ c′ = (001). (d) The a-face lies in the bc plane and is not perpendicular to that a axis. (d) The c-face lies in the ab plane and is not perpendicular to the c-axis.

for crystal sucrose along the a axis, and Figure 2b shows the common sucrose crystal morphology along with the facet relationships to the crystal axes.12 All crystal structures were rendered using Mercury 2.0 from the Cambridge Structural Database.13 Previously pressed polycrystalline powder samples of oxalic acid and sucrose have been measured using terahertz time domain spectroscopy (THz TDS),2,14 and the absorbances 10360

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od.31,32 Unfortunately, these calculations were unable to provide physically reasonable values for the refractive indices and, therefore, were not pursued beyond the preliminary stages.

3. RESULTS The measurements for oxalic acid dihydrate are shown in Figure 4 for a c-face 660-μm-thick crystal. In Figure 4b shows

Figure 3. Conversion of dihydrate oxalic acid to anhydrous and back to dihydrate through relative humidity control.

sample was anhydrous. The blue curve shows the same sample at the same orientation, now after the sample has been exposed to 100% relative humidity (r.h.) for 4 days. The 1.38 THz dihydrate absorbance is again present; however the conversion does not appear complete because the 1.95 THz absorbance is still present. In addition, there is a background present that increases with frequency for the mixed phase sample. This background may arise from amorphous patches in the sample20 or scattering from microcracks in the sample that are visible after the hydration cycling. To prevent conversion from dihydrate to anhydrous during the terahertz measurements, the rotation stage included a humidity controlled sample cell. The humidity was controlled by circulating air from a Licor Dew Point generator. Humidity can be fixed anywhere from 0 to 100% r.h. It was found that for humidity >10% r.h., there was no conversion to the anhydrous phase. Computational analyses of the terahertz spectra of these crystalline samples were performed using solid-state DFT. The periodic boundary conditions implemented in the solid-state DFT simulations enable the vibrational frequencies and intensities of both the intramolecular and intermolecular motions found in the crystals to be simultaneously calculated. This approach enables the complete assignment of the lowfrequency terahertz spectra of molecular solids. The simulations in this work were performed using the CRYSTAL09 software package21,22 utilizing the B3LYP23,24 hybrid density functional in combination with the atomcentered Gaussian-type 6-31G(d,p) basis set.25 The total energy convergence criteria were ΔE < 10−8 Hartree for geometry optimizations and ΔE < 10−10 Hartree for frequency calculations. All structural optimizations were performed with unit cell dimensions, and symmetries were constrained to those reported experimentally for room temperature crystals of oxalic acid dihydrate26 and sucrose.27 Normal mode frequencies and infrared intensities were then calculated for the optimized structures. The frequency of each normal mode was calculated within the harmonic approximation by numerical differentiation of the analytical gradient of the potential energy with respect to atomic position. The infrared intensities for each normal mode were calculated from the dipole moment derivatives (dμ/dQ) determined using the Berry phase technique of calculating Born charges as polarization differences between equilibrium and distorted geometries.28−30 Simulations of theoretical refractive indices were also attempted by calculation of the dielectric tensor on the basis of evaluation of the static polarizability through the coupled perturbed Kohn−Sham (CPKS) meth-

Figure 4. The (a) truncated terahertz waveform (full waveform of 42 ps), (b) absorbance spectrum, and (c) refractive indexes for a 660-μmthick c-face dihydrate oxalic acid crystal at room temperature, where 0° corresponds to the polarization aligned along the a axis. The color code follows Figure 5b.

the absorption coefficient, and Figure 4c shows the refractive index as a function of crystal orientation. For all plots, there is a smooth increasing absorbance with frequency. Because the dihydrate crystals are of good quality and optically transparent, this background cannot be due to scattering; rather, it is likely either absorbance from water or a tail of a stronger absorbance at higher frequency. For the electric field parallel to the a-axis, there is a sharp resonance at 1.38 THz. As the crystal is rotated, there is an apparent broad resonance at 1.00 THz, and the 1.38 THz resonance is diminished and a 1.87 THz resonance appears. The strong resonances at 1.38 THz and 1.87 THz are consistent with the results by the Korter group14 for randomly oriented pressed powders. Then as the crystal is rotated for E∥b, only the 1.87 THz absorbance is present. As will be discussed, the measured absorbances and their orientation 10361

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dependence are in agreement with DFT calculations, except for the 1.0 THz broad resonance. Measurements on additional crystals show that this feature is, in fact, an artifact from the strong birefringence of oxalic acid. In Figure 4a, we see that the waveform changes significantly as the crystal is rotated. In particular, if we look at the waveform at 45°, we see a small peak near 37 ps, and immediately following, a large peak at 36.5 ps. This is due to detecting the net field transmitted by the crystal, which is the superposition of the incident field component along [100] with index 2.1 and the field component along [010] with index 1.8. The Fourier transform of this superposition gives rise to a low frequency dip in the transmitted spectrum and an apparent absorbance. For different crystal thicknesses, this feature will move in frequency, since the phase delay between the field along the [100] and the [010] will change. The effect can be removed entirely by using a sufficiently thick crystal and cutting off the waveform; however, it may not be as readily removed and identified for crystalline powder samples. As observed in the experimental terahertz spectrum of oxalic acid dihydrate, the DFT simulation produced two vibrational modes in the region below 3.0 THz. The first feature was calculated at 1.58 THz with ac polarization (10% a, 90% c), and the second was predicted at 1.93 THz with pure b polarization. As illustrated in Figure 1c, for the monoclinic oxalic acid dihydrate crystals, the c-face cut crystal will have a projection of the c crystal axis lying along the a axis; thus, the 1.58 THz mode is accessible in the c-face measurements. These results, in terms of both frequency and polarization, are in very good agreement with the experimental findings. In Figure 5, we show the terahertz measurement as a function of crystal rotation for an a-face, 200-μm-thick sucrose crystal. Figure 5a, b, and c show the transmitted terahertz waveform, the absorption coefficient, and the refractive index, respectively. The absorption coefficient spectra for different orientations are offset for each curve by 60 cm−1. In Figure 6, we show the terahertz measurement as a function of crystal rotation for a c-face, 270-μm-thick sucrose crystal. Figure 6a, b, and c shows the transmitted terahertz waveform, the absorption coefficient, and the refractive index, respectively. One can clearly see the anisotropic absorption for the crystals as the position is changed from 0 to 90°. Additional measurements show that all resonances are periodic with rotations of 180°. The absorption coefficient spectra for different orientations are offset for each curve by 80 cm−1. As illustrated in Figure 2, for the monoclinic sucrose crystals the a (c) face cut crystals will have a projection of the c (a) crystal axis lying along the a (c) axis; thus, all modes should be accessible for a single-face measurement. We fit the spectra at different rotations to Lorentzian absorbances and have listed these in Table 1, along with their amplitudes for the c-face measurements. Fits for 0° and 90° rotations for the c-face data shown in Figure 6b demonstrate that the measured spectra are well reproduced by the fits. Looking at the waveform in Figure 6a, we see a slight phase shift with increasing orientation rotation, with the refractive index in Figure 6c reflecting this. The index along the a axis is n = 1.80 and along the b axis is n = 1.85, which is a birefringence similar to x-cut quartz. Because of the strength, density, and line width of the absorbances in the region of 1.8−2.5 THz at room temperature, not all anomalous dispersion features are easily discernible.

Figure 5. The (a) truncated terahertz waveform (full waveform of 42 ps), (b) absorbance spectra, and (c) refractive indices for a 200-μmthick a-face sucrose crystal at room temperature, where 0° corresponds to the polarization aligned along the c axis. The color code follows part b. The y-offset for each curve for part b is 60 cm−1.

The terahertz spectrum of sucrose is more difficult to unambiguously assign, since there are a greater number of spectral features that are overlapping, as compared with the sparse oxalic acid dihydrate spectrum. The DFT simulations yield 12 infrared-active vibrations below 3.0 THz. The calculated frequencies agree well with the observed spectrum, but perhaps even more importantly, the infrared transition polarizations are consistent with the experimental observation that the majority of the intensity (∼60%) originates from bpolarized transitions. A list of the predicted vibrational frequencies, intensities, and polarizations are provided in Table 1. The utility of performing anisotropy measurements alongside the DFT calculations is evident in the assignment of the measured peaks. Nearly all assignments follow the frequency ordering; however, while the measured peaks at 1.88 and 1.98 THz would be assigned to 1.939 and 2.060 THz, respectively, calculated peaks due to the proximity of the frequencies, the measured and calculated orientation dependencies for these two resonances indicate that the 1.88 THz measured resonance must correspond to the 2.060 THz calculated phonon, and the 1.98 THz measured resonance must correspond to 1.940 THz calculated phonon. The 10362

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anisotropic measurements provide an additional evaluation of the accuracy of the calculations and indicate where further refinement is needed.

4. DISCUSSION Anisotropy measurements of single crystal sucrose were previously reported;16 however, our data are not in agreement with those previous measurements. Kroll et al. first measured spectra for the electric field aligned along specific crystal axes and as a function of rotation from those axes for a-cut and c-cut crystals at 5 K. From the temperature-dependent measurements of Jepsen et al., we can correlate our room temperature peaks at 1.46 and 1.74 THz with the 5 K peaks identified by Kroll et al. at 1.37 and 1.7 THz. In the earlier measurements, an orientation independent peak at 1.9 THz was reported; however, we find all absorbances to be orientation-dependent. We note that the room temperature calculations predict modes at 2.668 and 2.714 THz, with measured peaks at 2.57 THz aligned along the b axis and at 2.58 THz aligned along the a axis. These closely spaced modes with different orientation dependencies are difficult to distinguish from an orientationindependent peak. The greatest difference with the earlier measurements is the report that the 1.37 THz absorption was maximized for E∥c for an a-cut crystal and at a maximum for E∥a for a c-cut crystal. We find for the corresponding room temperature feature at 1.47 THz is maximum for E∥b for both crystal cuts. Finally, in the earlier measurements, the 1.7 THz resonance was strongest for E∥b, whereas we find it is maximum for E∥a in the c-face data. It is likely the difference between the measurements is from a difference in the actual crystal orientations. For the results reported here, we verified the crystal orientation by single crystal X-ray diffraction measurements. Comparing our measurements with calculations, we find that despite taking some care to ensure the crystal parameters correspond to room temperature values, nearly all calculated frequencies are blue-shifted relative to the measured absorbances, with the notable exception of the calculated lines 1.931 and 2.060 THz. The quality of our data is not sufficient to determine the frequency shift for the 2.91 THz peak. The calculated orientation dependence is in excellent agreement with the observed resonances. Although for the

Figure 6. The (a) truncated terahertz waveform (full waveform of 42 ps), (b) absorbance spectra, and (c) refractive indices for a 270-μmthick c-face sucrose crystal at room temperature where 0° corresponds to the polarization aligned along the a axis. The color code follows part b. The y-offset for each curve for part b is 80 cm−1. Also shown by the ▲ points are representative curve fits for E∥a and E∥b, from which frequencies and amplitudes are determined for Table 1. The straight, thin, black (b polarized) and red (a/c polarized) lines show the predicted frequencies and intensities from the DFT calculations.

Table 1. The Measured Absorbance Peak Frequencies and Amplitudes As Determined by Curve Fitting to c Face Data, along with the Calculated Frequencies, Intensities, and Orientation Dependence Determined from DFT Calculations polarization

a

experimental frequency (THz)

calculated (THz)

1.46 1.66 1.80 1.88 1.98 2.26 2.26 2.57 2.58 2.73 2.91

1.577 1.761 1.872 2.060 1.931 2.369 2.396 2.668 2.714 2.796 2.849 2.997

c-face fit amplitude (cm−1) 71 44 156 238 142 55 3 158 216 103 101

∥b ∥a ∥b ∥b ∥aa ∥a ∥b ∥b ∥a ∥b ∥a

IR intensity (km/mol)

%a

%b

%c

1.86 0.004 11.33 0.75 5.89 2.30 0.002 1.29 2.88 0.01 4.19 6.63

0.0 69.8 0.0 0.0 2.3 93.0 0.0 0.0 9.5 0.0 48.6 0.0

100.0 0.0 100.0 100.0 0.0 0.0 100.0 100.0 0.0 100.0 0.0 100.0

0.0 30.2 0.0 0.0 97.7 7.0 0.0 0.0 90.5 0.0 51.4 0.0

Maximum amplitude at 30° from a axis. 10363

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most part, the calculated relative intensities are consistent with the peak absorbance variation in the data, the absolute variation between adjacent peaks is often significantly less than predicted. In particular, the calculated lines 1.761, 2.396, and 2.796 are predicted to have somewhat smaller intensities; however, they are all readily detected (note the calculated intensities for these lines are scaled by ×100 in Figure 6b). The only significant discrepancy between the measured and predicted relative intensities is for the b∥ calculated peaks 1.872 and 2.060 THz. The calculation predicts the 2.060 THz resonance with a somewhat smaller intensity relative to the 1.872 THz resonance; however, the measurement finds the high-frequency absorbance is stronger.

REFERENCES

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5. CONCLUSION We find a large anisotropic terahertz absorption in crystals of sucrose and oxalic acid and utilize this anisotropy to improve spectral assignment. The two low-frequency modes of oxalic acid dihydrate at 1.38 THz and 1.87 THz are strongly anisotropic, and we find a large birefringence of Δn = 0.3 over the 0.2−3.0 THz range. The observed frequencies and anisotropy are in good agreement with DFT calculations. The strong birefringence can give rise to an apparent resonant absorption at lower frequencies when using THz TDS for a microcrystalline sample complicating spectral analyses. The results demonstrate that particular care needs to be taken when assigning vibrational modes in highly birefringent crystals because the birefringence will give rise to misleading absorption artifacts in the transmitted terahertz pulse. Although the large crystal growth of oxalic acid is more challenging than sucrose, the strong anisotropic narrow resonances at room temperature can be readily used for polarimetry calibration at terahertz frequencies. In addition, the giant birefringence for single crystal oxalic acid dihydrate may be useful for wave plates in this frequency range. Large anisotropic narrow resonances are also observed for sucrose, but the overlap of these resonances at room temperature may limit their usefulness as polarimetry standards. Sucrose has a slight birefringence of Δn ∼ 0.05, similar to that of x-cut quartz. The anisotropic response of sucrose enabled straightforward comparison with the calculated spectrum, by both separating overlapping modes and unambiguously identifying the mode symmetry. The sucrose measurements demonstrate that anisotropic measurements of single crystals provide sufficient feature separation that mode identification can be readily achieved. This type of mode separation of a complex overlapping absorption bands becomes increasingly important for measurements of more complicated systems, such as polypeptides and other biomolecules.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: amarkelz@buffalo.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation MRI2 Grant DBI2959989. 10364

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