Optics and IR-Spectroscopy of Polydomain Materials

Optics and IR-Spectroscopy of Polydomain Materials

Habilitationsschrift

vorgelegt am 07. Juli 2006

der Chemisch-Geowissenschaftlichen Fakultät der Friedrich-Schiller-Universität Jena

von

Dr. rer. nat. Thomas Mayerhöfer

aus Parsberg/Opf.

Gutachter:

1.

Prof. Dr. J. Popp

2.

Prof. Dr. H. Dunken

3.

Prof. Dr. D. B. Tanner

Erteilung der Lehrbefähigung am …

Table of contents

- III -

1. Introduction..............................................................................................3

2. Generalized 4×4 Matrix Formalism ...........................................................6 2.1

%H U U HPDQ ¶ s formalism: Maxwell equations and constitutive relations ...................... 7

2.2

%H U U HPDQ ¶ V I RU PDO L V P &DO FXO D W L RQ RI W KH U HI U D F W L YH L Q GL F H V DQ G W K H SRO D U L ] DW L RQ directions .................................................................................................................. 10

2.3

> O. Therefore, it seems to be strongly handicapped for practical purposes. In contrast to this restriction, the simulated and measured spectra in Fig. 5.4 correspond well over the whole Reststrahlen range between 100 cm-1 ( O 100 Pm ) to 1100 cm-1 ( O 9.1Pm ). This is obviously a contradiction, since the average crystallite diameter is about 10 Pm and therefore lies well below O. In this context it must be stated that according to ref. [3], Doll et al. have not examined the crystallite size of their sample, therefore their criterion d >> Ofor optically large crystallites is adopted presumably from the geometrical optics limit. In a follow-up paper, Doll et al.57 determined the crystallite size of different samples with optically large crystallites, including a sample similar or equivalent to that of ref. [3], to range from 5 10 Pm , which is in good agreement to the crystallite sizes in the polycrystalline Fresnoite sample Ba0K0-100G. How are these seemingly inconsistent results explainable? Basically, it seems as if averaged optical constants can only be used for crystallite sizes below O/10. Above about O/10 the anisotropy of the optical properties of the crystallites has to be taken into account. Another important problem, which deserves examination, is the question whether it is allowed to use single crystal data without modification taking into account the limited size of the microcrystals. It is well-known from the literature that the oscillator parameters (cf. section A.5) need to be modified when switching from macroscopic crystals to their microscopic counterparts due to effects related to phonon confinement (cf., e.g., ref. [58] and references therein) and anharmonicity43 (multi-phonon resonances). These effects should gain importance especially in the crystallite size region below about 100 nm, and manifest itself on the one hand in a softening of the T ĺ U XO H I RU W K H SK RQ RQ -wavevector (conservation of momentum of the photon) and on the other hand in a change of the damping constant (the

5. Experimental Verification of ARTT, ARIT and UAOPT

- 53 -

inverse lifetime of the phonon). The first effect leads to an often asymmetric band broadening resulting from the phonon dispersion curve and will not be considered in the following, since the dispersion curves for phonons of Fresnoite are not known (usually, dispersion curves are determined experimentally from neutron scattering experiments). Moreover, the comparison between experimental and simulated spectra shows that this effect needs not to be considered in this work with the exception of chapter 6. The second effect leads also to a broadening of the bands and can be simply introduced by modifying the damping constants of the single crystal by a common factor f(d) > 1 according to

Jij, pc d f d Jij , where

(5.12)

Jij represents the damping constant of the ith oscillator in the single crystal and Jij, pc

its counterpart in the respective polycrystalline material. Since f is a function of the crystallite diameter d, so too is

Jij, pc .

360 400 440 360 400 440 360 400 440 360 400 440 0.75

g =1

g = 1.1

g = 1.2

g = 1.3

reflectance (20°, s)

0.50 0.25 0.00 0.10 0.05 0.00 560 580 600 560 580 600 560 580 600 560 580 600 Simulation -1 wavenumber / cm Measurement Fig. 5.5:

Measured reflectance spectrum of randomly-oriented polycrystalline Fresnoite (BaK0-100G, incidence angle 20°) and spectra simulated with modified single crystal data (all damping constants multiplied by f, 'M 'T 1° ).

- 54 -


This common factor was determined to be about 1.3 in case of randomly oriented Fresnoite with optically small crystallites (d § QP V H H V H FW L RQ5.2.5). Since f is assumed to be indirectly proportional to the crystallite size d, the common factor should be smaller in case of randomly oriented Fresnoite with optically large crystallites. To determine f for the sample BaK0-100G, spectra were simulated applying different common factors on the damping constants of the single crystal. The results were compared with the measured spectrum. Besides a lowered intensity and a larger halfwidth of the reflectance peaks for f > 1, two spectral ranges (320 ±480 cm-1 and 540 ±620 cm-1 ) can be found, where peak-shapes were characteristically altered by the enlargement of f. The results of the simulations are illustrated in Fig. 5.5.

0.75 0.60

s-polarized

measured simulated

0.30 0.15 0.00 measured simulated

0.60

p-polarized

reflectance (20°)

0.45

0.45 0.30 0.15 0.00 100

300

500

700

900

1100

1300

-1

wavenumber / cm Fig. 5.6:

Measured reflectance spectra of randomly-oriented polycrystalline Fresnoite (BaK0100G, incidence angle 20°) and simulated spectra with f = 1.1 ( 'M 'T 1q). Lower part: p-polarization; upper part: s-polarization.

From the comparison between the simulated spectra and the measured spectrum it is obvious that for f = 1.1 the best resemblance between the peak shapes results for both of the peaks. In addition, f = 1.2 and f = 1.3 lead locally to a lower reflectance for the simulated spectrum compared to the measured spectrum in the second range (540 ±620 cm-1 ).


- 55 -

The spectra simulated with f = 1.1 for the whole spectral range of interest are depicted in Fig. 5.6 together with the measured spectra. The good resemblance justifies the use of a common factor f in case of polycrystalline materials with large crystallites. Nevertheless, since f is comparably small in the large crystallite limit, it will be usually omitted in the following ARTT and UAOPT simulations. If it is employed it will be stated explicitly. All in all, the results presented in this section prove 'RO O ¶ V D SSU RD FK 2EYL RX V O \it captures all important effects, consisting in band shifts, changes of the band shapes and the relative intensities. The next section is dedicated to an effect, which could neither be realized by Doll nor by Frech, due to the missing ability of their formalisms to calculate cross-polarization terms. 5.1.5

Non-zero cross-polarization

Theoretical derivation and consequences It is important to realize that the application of ARTT (eqn. (3.3)) implies non-zero crosspolarization terms for a polycrystalline medium consisting of optically large and anisotropic crystallites (domains). Assuming random orientation, it follows from R ps : t0, Rsp : t0

(5.13)

Tps : t0, Tsp : t0

and from

R

1 1 Rs R p Rss Rsp R pp R ps 2 2

(5.14)

together with eqn. (3.3) that

R ps

N (3)

³R : d :!0 ps

:( 3)

Rsp

N (3)

³R : d :!0 sp

:( 3)

Tps

N (3)

³Tps : d :!0

.

(5.15)

:( 3)

Tsp

N (3)

³T : d :!0 sp

:( 3)

Non-zero cross-polarization terms do always exist except for cubic crystal symmetry or certain special orientations of a crystallite or a domain. In particular, the equality signs in eqn. (5.13) are valid only for certain orientations of the crystallites in dependence of their symmetry:

- 56 -

5. Experimental Verification of ARTT, ARIT and UAOPT Optically uniaxial materials (tetragonal, trigonal, hexagonal): The optical axis must be either confined within the plane of incidence or has to be oriented perpendicular to it (cf. section 8.3.2).

-

Orthorhombic symmetry: All three dielectric axes must be oriented either parallel or perpendicular to the plane of incidence.

-

Monoclinic symmetry: The monoclinic b-axis (the dielectric axis the orientation of which is not dependent on frequency) has to be oriented parallel to the interface and either parallel or perpendicular to the plane of incidence.

-

Triclinic symmetry: The equality signs are never applicable since polarization conversion generally occurs.

Therefore, the incoherent averaging of the reflectance or the transmittance always leads to a non-zero cross-polarization as long as not only the special orientations mentioned above are involved in the average. In particular, eqn. (5.15) predicts that non-zero cross-polarization terms must exist despite of random orientation. This result is equivalent to the following statement: Polycrystalline (polydomain) materials can never be characterized by a scalar dielectric function or an index of refraction function if the anisotropy of the domains can be resolved by the probing light beam. Unfortunately, the resolution limit could not be determined so far. From the experimental results presented in this work it can be tracked down to be about 1/10 of the wavelength. This means that in the infrared spectral range at 1000 cm-1 domains can be considered as being optically small if they are smaller than about 1 Pm. In the visible range however, at a wavelength of e.g. 500 nm, every domain larger than about 50 nm must be considered as RSW L F DO O \O D U JH 6L QF H L Q W K H O L W H U DW X U H W K H W HU P³ RSW L F DO L V RW U RSL F´ L V DO ZD\ V X V H G D V EHL QJ H TXL YDO HQW W R ³ D PDW H U L DO F DQ EH FK DU D F W H U L ] H G E\ D V FDO D U GL HO H F W U L F I XQF W L RQ´ W KL V I L QGL QJ L V RI fundamental importance and has far-reaching consequences. To name just a few, Mie theory,1 e.g., can be applied only for isotropic spheres. In the context of the statement above, this means, that it is useful for polycrystalline spheres only if their domains are small compared with the wavelengths. Obviously, this is quite a harsh restriction on its applicability. Another example is the EMA (cf. Appendix, section A.6) and its application to polycrystalline materials. Usually it is used only in the so-called quasi-static limit, which means that the crystallites must be small compared with the wavelength, but it was tried to enhance the range of applicability (see, e.g., ref. [60]). Any enhancement, however, must be doomed to failure as long as the concept of an averaged dielectric function is not given up. An extended discussion with emphasis on the consequences of the interpretation of spectra is given in chapter 9.


- 57 -

Experimental verification Since the cross-polarization terms are usually comparably small, their detection is strongly depending on the quality of the polarizers and the detector. While this poses typically no problem in the visible spectral range, it is experimentally more challenging in the MIR- and especially in the FIR-spectral range. In contrast to measurements with one polarizer, the reference measurement for experiments employing crossed polarizers is carried out with parallel polarizers in a way that the polarizer before the sample is oriented just as for the crossed polarizer experiment:

Rij sample, corrected

I R ,ij sample I R ,ij aperture , I R ,ii mirror I R ,ii aperture

i, j

s, p .

(5.16)

Here, the IR are the absolute reflected intensities. The correction for the aperture is of less importance. The form of the correction scheme is a consequence of eqn. (5.14). I R ,i sample I R ,ii sample I R ,ij sample : I R ,i mirror Ri sample, corrected

sample I R ,i mirror I R ,ii mirror I R ,ii sample I R ,ij o I R ,i mirror I R ,i mirror

sample I sample I R ,ij Ri sample, corrected R ,ii I R ,ii mirror I R ,ii mirror

(5.17)

Rii sample, corrected Rij sample, corrected

Here, the correction for the aperture is omitted. The equality between IR,i(mirror) and IR,ii(mirror) holds, of course, only for ideal, non-absorbing polarizers. In practice, it is therefore necessary to use IR,ii(mirror) for experiments carried out with analyzer and polarizer. In the far- infrared spectra were recorded on a Bruker IFS 113V with a liquid helium cooled bolometer acting as detector. Unfortunately, the used wire-grid polarizers were of comparably low quality as can be seen from the strong mode leakage (IR,sp (mirror) / IR,ss (mirror) = 0 for ideal polarizers) in the spectrum depicted in Fig. 5.7, lower part. Therefore an additional correction was applied to correct for mode leakage according to

Rsp sample, corrected

I R , sp sample I R , s mirror sample I R , sp . I R , ss mirror I R , s mirror I R , ss mirror

(5.18)


reflectance (7°, sp)

- 58 -

0.04

K0100G corrected for the mode leakage K0100G KRS-5 polarizers

0.02

0.00

Mode leakage

0.3 Rsp(mirror)/Rss(mirror) 0.2 0.1 0.0 0

100

200

300

400

500

600

-1


Lower panel: mode leakage of the wire-grid polarizers. Upper panel: comparison between the cross-polarized reflectance spectrum of BaK0100G corrected for mode leakage and the uncorrected cross-polarized reflectance spectrum recorded using KRS-5 polarizers.

0.045 Simulation

reflectance (7°)

Rsp

0.030

0.015

0.000 20

120

220

320

420


520

620

-1

Comparison of the measured and simulated cross-polarized reflectance spectra of polycrystalline Fresnoite consisting of large crystallites (sample BaK0-100G) in the FIR.


- 59 -

0.045 Simulation Rs p

reflectance (20°)

R ps

0.030

0.015

0.000 400

600

800

1000


1200

1400

-1

A comparison of the measured and simulated cross-polarized reflectance spectra of polycrystalline Fresnoite consisting of large crystallites (sample BaK0-100G) in the MIR.

The comparison between the so corrected cross-polarization spectrum and a cross-polarization spectrum obtained with KRS-5 polarizers without correction for mode leakage in the upper part of Fig. 5.7 fully justifies eqn. (5.18), at least between 200 and 650 cm-1 . In the range below 200 cm-1 , however, the mode leakage of the wire-grid polarizers was comparably low. The MIR-spectra were generally not corrected for mode leakage. This was not necessary as is obvious from the near-zero intensity in the spectral ranges between about 600 to 800 cm-1 and above 1100 cm-1 . The total measured intensities are usually rather low due to the use of two polarizers, especially below 600 cm-1 in the MIR using the Bruker IFS 66. Therefore the resemblance between measured and simulated spectra is comparably poor in this spectral range as demonstrated in Fig. 5.9. With this exception, positions and shapes of the spectral features between 20 cm-1 and 1100 cm-1 have been simulated rather satisfactorily by the forward calculation based on eqn. (5.15), keeping in mind that the absolute peak intensities are not well reproduced due to the existence of holes on the surface of the samples as already detailed in the preceding section. Since the cross-polarization is dependent on the optical anisotropy of the single crystal, it is not surprising that there is no distinct peak at 860 cm-1 , because around this peak Ha |Hc holds (cf. single crystal data, Appendix, section A.5).

- 60 -


To summarize, the occurrence of non- zero cross-polarization terms for randomly oriented polycrystalline materials with large crystallites has been not only undoubtedly proved, it was also shown that it is possible to calculate these terms quantitatively. This is of special importance as will be revealed in section 5.1.7. It must be emphasized that the existence of non- zercross-polarization terms is not necessarily connected with linear dichroism. Also in regions were the principal indices of refraction are real numbers, a distinction between large and small crystallites is possible, provided, of course, that they differ. Unfortunately, these differences have to be relatively large to cause, e.g., a cross-polarization above the detection limit in a reflectance measurement if conventional instrumentation is utilized. For Fresnoite, the difference amounts to 'n |0.14 at 1500 cm-1 leading to a calculated cross-polarization Rsp

R ps

4 u105 (assuming that ka =

kc = 0, where ki is the imaginary part of the principal refractive index belonging to axis i), while measurements of the cross-polarization of the polycrystalline Fresnoite sample Ba0k (d §300 nm) show an average value of about 6×10-4 in the spectral range from 1450 ±1550 cm1

. This indicates an error of the same order, since the cross-polarization of such a sample

should equal zero. 5.1.6

Approximations

In spite of the usually fast convergence of the result of eqn. (3.3), the time needed for the numerical evaluation is still far away from enabling, e.g., a DA of the spectra of randomly oriented polydomain materials with large domains in due time. Therefore, some approximations are derived in the following, which will be compared with those known from the literature. Instead of using approximations, it would also save time to measure reflectance spectra at normal incidence. Unfortunately corresponding accessories GRQ¶ Wseem to be available (such accessories would require the use of a beam splitter with all the corresponding disadvantages), but for low angle of incidence (D < 10°) it may be allowed to use calculations under the assumption of normal incidence as a first approximation. As shown in section 2.8.1, normal incidence simplifies the 4×4 matrix formalism considerably. Additional simplifications apply for uniaxial domains as shown in section 2.8.2. Derivation To derive a first approximate formula, an averaged Euler angle M is used instead of averaging Runiaxial MT , over this angle:

5. Experimental Verification of ARTT, ARIT and UAOPT S

R

uniaxial

|

1 2 R D, M S³ 0

S

1 2 S , T d T R D, M 4 S³ 0

- 61 -

, T d T

S 4

(5.19)

This approximation is closely related to eqn. (5.10), since cos 2 M sin 2 M 1 . Moreover, 2 eqn. (5.19) turns into eqn. (5.11) for polarized incident radiation, if D 0 . A further approximation results, if the angle Tis averaged directly, too:

R

uniaxial

1 | R D, M 2

, T

S 4

12 R D, M

S 4

, T

S 4

S 4

(5.20)

Both approximations are compared to the numerical solution of eqn. (5.1) in Fig. 5.10.

0.8

Simulation (eqn. (5.1)) Approximation I Approximation II

0.6 0.4

Rs

reflectance (20°)

0.2 0.0 Rp

0.6 0.4 0.2 0.0 0.08 0.06 0.04 0.02 0.00 100

Rsp=Rps

300

500

700

900

1100

1300

-1

wavenumber / cm

Fig. 5.10: Simulated spectra of randomly-oriented polycrystalline Fresnoite according to eqn. (5.1) and the approximate formulae (5.19) (approximation I) and (5.20) (approximation II). Apparently, the approximate spectra calculated according to eqn. (5.19) resemble the spectra resulting from eqn. (3.3) very well except in the range between 1020 and 1080 cm-1 . The approximation based on eqn. (5.20) is obviously less satisfactory. Consequently, its use should be restricted to small anisotropy. Eqn. (5.20) is able to explain why the spectra of certain crystal faces resemble the spectra of respective powders/polycrystalline bulk samples.61 The resemblance is clearly of less quality with respect to the cross-polarization spectra (lower part of Fig. 5.10), where eqn. (5.19) is at least able to predict the relative

- 62 -


intensities and the peak positions correctly, but overestimates the overall intensities by a factor of about two. From eqn. (5.19) it is possible to derive an even better approximation, which is able to predict the correct cross-polarization to a high degree. Besides, this approximation is not restricted to optically uniaxial crystallites: SSS

1 R | 3³ R J, M ³ 2S 0 ³ 0 0

, T\ , R J ,M

S 8

, T\ , d d d JT\

5S 8

(5.21)

Note that this relation is exact at D= 0. For uniaxial crystal symmetry it simplifies to S

1 R | ³ R M 2S0

, T R M

S 8

, T d T,

5S 8

(5.22)

which is equivalent to eqn. (5.11) taking into account that

1 2R M x, T 1 2 R M x S4 , T R M

. , T

S 4

(5.23)

The performance of the approximation according to eqn. (5.22) is compared to that of the numerical solution of eqn. (5.1) in Fig. 5.11.

0.8

Rs

0.6 0.4 reflectance (20°)

0.2 0.0 Rp

0.6 0.4 0.2 0.0 0.04 0.03 0.02 0.01 0.00 100

Simulation (eqn. (5.1)) Approximation (eqn. (5.22))

300

500

700

900

Rsp=Rps

1100

1300

wavenumber / cm-1 Fig. 5.11: Simulated spectra of randomly-oriented polycrystalline Fresnoite according to eqn. (5.1) ( 'M 'T 1q) and the approximate formula (5.22) ( 'T 1q). The superiority of this approximation over eqs. (5.19) and (5.20) is obvious.


- 63 -

Approximations known from the literature It is interesting to note that the approximation according to eqn. (5.20) is very similar to a result of LDT (see section A.7). The basic assumption of this theory is usually that the sample under consideration is of uniaxial symmetry. Additionally, we will assume in the following that the preferred axis of the sample is oriented parallel to the electrical field vector of the incident radiation (which is, for s-polarized light, the X-axis if the plane of incidence equals the Y-Z-plane). Then, the following three orientation distributions are equivalent according to LDT: 1. a, b and c axes are perfectly aligned with the X-axis, with equal probability. 2. a, b and c axes are at the magic angle to the X-axis ( cos 2 Xa

cos 2 Xb cos 2 Xc

cos 2 K

1

3

, Xa, Xb, Xc are the , K arccos 1 3 |54.74q

direction cosines). 3. Random orientation. Note that the failure of LDT with regard to liquid or solid samples is immediately obvious from the fact that it neglects the direction of the incoming wave relative to the sample interface. Nevertheless, if the TO-LO-splitting can be neglected as for most of the organic materials, LDT might be a good first order approximation, but its breakdown is usually almost complete for inorganic materials. E.g., from the first orientation distribution a zero crosspolarization results for optically uniaxial and orthorhombic materials, which is a proof of its general incorrectness. Nonetheless, this model was also applied in the literature to model the spectra of randomly oriented materials with large domains in the following or in equivalent forms:5,62

R

uniaxial

| 1 f Ra f Rc

(5.24)

Here, f is a volume factor in equivalence to EMA (see section A.6) and should equal 1/3. Therefore, eqn. (5.24) is related to orientation distribution 1. On the other hand, it is easy to show that eqn. (5.24) also represents a possible magic angle arrangement of a single crystal, given by the Euler angles M arcsin 1 3, T 90q, if the angle of incidence D 0q , since then R

Ra cos 2 MRc Tsin 2 Mholds (cf. eqn. (5.10)).

For orientation distribution 2 it is important to realize that there exists an infinite number of equivalent arrangements as long as the wave propagation direction and the assumption of uniaxial symmetry along X is not taken into account. For example, assume that the orientation of a single crystal relative to the laboratory axes is given by the Euler angles

- 64 -


M 45qT , 90qarccos 1 3

arcsin

1 3 ( a x, b y, c z , in this work, the Euler

angles are defined with respect to the normal of the sample interface and not relative to the electrical field vector, which is the usual definition if LDT is applied). For this orientation, all transition moments are aligned at the magic angle relative to the electrical field vector of the incident radiation, if perpendicularly polarized radiation is applied (s-polarized light must be used because the polarization direction does not depend on the angle of incidence and is always parallel to X). Therefore, this orientation may be seen, under the constraints given above, as a magic angle-like arrangement, and, hence, as a possible approximation according to LDT. However, for both magic angle- like arrangements known from the literature the preferred axis is not the X-axis, since in each case simply a special orientation of a single crystal is considered (the preferred axis of a uniaxial crystal is its c-axis, which is in both cases not aligned parallel to the X-axis). As a consequence, inside the crystals in general two waves exist with two different orientations of the electrical field vector. Being strict, this prevents LDT from being applicable.

Improved magic angle based model An orientation distribution, which is a real magic angle arrangement and also conforms to LDT, can be constructed under the assumption that all possible magic angle orientations of a single crystal are taken into account. This is equivalent to an orientation distribution, where the axes of the crystallites lie on a cone around X with a cone angle G arccos 1 3 . To describe such an orientation distribution it is of advantage to use an EOR with fixed axes given by the rotation matrix A X ,Y

A X [AY W. Employing s-polarized light and an initial orientation

of the crystallites where their optical axes are oriented parallel to the Z-axis, they are first rotated around the Y-axis by an angle W arcsin 1 3 . Then, they are rotated around X. This keeps the angle between the polarization direction and the transition moments constantly at the magic angle. As a result, the reflectance with incident s-polarized light is averaged according to: S

Rs

magic angle

1 Rs W arcsin 1 3, [ d [. S³ 0

(5.25)

In Fig. 5.12 the spectra calculated according to the magic angle-arrangement (eqn. (5.25)) and the two magic angle- like-arrangements are compared to the spectrum computed from eqn.


- 65 -

(5.1) for 'M 'T 1q. Though the approximation according to eqn. (5.25) provides the best result of all magic angle-related approximations, it can be observed that all three approximations are neither able to predict peak shapes nor relative intensities satisfactorily, and, generally, the same is true regarding peak positions.

0.75

Rs

0.30 0.25

0.50

0.20 0.15

reflectance (20°)

0.10

0.25

0.05 0.00

0.00 0.20

Rsp

1080

ARTT cos2K= 1/3, [= S /4 2 cos K= 1/3 2/3 Ra + 1/3 Rc

0.15 0.10 0.05 0.00 100

1040

300

500

700

900

1100

1300

1500

-1

wavenumber / cm

Fig. 5.12: Simulated reflectance spectra of polycrystalline Fresnoite (incidence angle 20°, perpendicularly polarized radiation, continuous line: ARTT (eqn. (5.1), 'M 'T 1q), red line: magic angle- like arrangement, green line: magic angle arrangement according to eqn. (5.25), blue line: EMA-like theory (eqn. (5.24))). The reason for the failure of the approximation according to eqn. (5.25) is that the electric field inside the sample is not oriented parallel to the preferred axis in general, despite the assumed uniaxial alignment along X, because all crystallites are by assumption larger than the resolution limit of light. Therefore, inside every crystallite two different directions of polarization exist (except for [ 0 , cf. also sections 5.1.5 and 8.3). Especially the failure of the approximation according to eqn. (5.24) has far-reaching consequences, since it definitely shows that the resulting spectrum of an orientation distribution can not be calculated from a linear combination of the principal spectra. The latter is a result important for LDT and builds the basis for the definition of the so-called Dichroic Ratio DXY,17

- 66 -


DXY

UX , UY

U

T , R, A, A ,

(5.26)

where T, R and A are the transmittance, the reflectance and the absorbance (1-R-T), respectively and A is the absorption (- log (R+T)). Since eqn. (5.24) fails for polycrystalline materials in general, the Dichroic Ratio is, strictly speaking, a meaningless quantity. In summary, orientation distribution 1 and 2 cannot be equal to random orientation. Hence, their consideration can not be useful for the interpretation and modeling of spectra of polycrystalline materials consisting of anisotropic crystals with non-negligible TO-LOsplitting. In addition, it must be stated that orientation distribution 3 (random orientation) is, of course, not conformable to the assumption of a uniaxial aligned sample and the conservation of linear polarization inside the sample (even uniaxial alignment and an orientation of the preferred axis parallel or perpendicular to the polarization direction is not sufficient for the conservation of linear polarization, as will be shown in section 8.3). It is therefore obvious right from the start that orientation distribution 3 can not be equivalent to orientation distributions 1 and 2 for large domains.

5.1.7

The weight sin T

The orientational average of an orientation dependent property F assuming uniaxial symmetry of the domains can be found in two different forms in the literature:

F

F

2 SS

1 FMT , d Md T, ³ 2S2 ³ 0 0

(5.27)

2 SS

1 FMT , d Msin Td T. ³ 4S³ 0 0

(5.28)

The difference between the two equations is obviously the use of the weighting factor sin Tin eqn. (5.28). As an alternative to the use of the weighting factor, also non-equally spaced steps can be employed in the numerical evaluation of eqn. (5.27).49 The idea and justification of introducing weighting factors and/or non-H TX DO O \ V SD F H G V W H SV L V W R REW DL Q D ³ V H W RI F U \V W DO O L W H RU L HQ W DW L RQV XQL I RU PO \ GL V W U L EXW H G RY HU W KH XQL W V SK HU H «´ Weighting factors shall ensure that each orientation contributes equally to the average.49 Especially for the interpretation of NMR powder spectra, a huge number of different methods have been developed due to the lack of an analytical solution to the problem of uniform distribution:48-56 The justification of the application of the weighting factor sin Tis at the same time the best argument against it, since


- 67 -

this weighting factor is not able to remedy the high density of crystallite orientations at the poles.49 All more sophisticated methods like REPULSION49 or Gaussian Spherical Quadrature48 are based on this fundamental flaw. Alternatively to symmetry considerations, the use of the weighting factor sin Tin eqn. (5.28) may also be inspired by the derivation of the (classical) partition function of a rigid rotator zrot (see also the well known Langevin-theory!) given by: 1 h2

zrot 1 h2

2S

2 SSff

ª H rot º »d Md TdpMdpT ¼

exp « ³ ³ ³³ ¬ kT 0 0 ff

f

2 2 ª pT º Sf ª º pT d M³ exp « exp « dp d T. »dpT³ 2 » M ³ ³ ¬ 2 IkT ¼ 0 f ¬ 2 IkT sin T¼ 0 f

2SIkT h2

(5.29)

2 SS

sin Td Md T ³ ³ 0 0

Here, however, it must be emphasized that the factor sin T is introduced by the spin Hamiltonian Hrot ( H rot

2 2 ) and the proper substitution of the exponential 1 2 I pT pM sin 2 T

2 function containing pM sin 2 T to solve the integral and not by considerations on the

symmetry of the system or the attempt to obtain a uniform distribution of orientations on a unit sphere (the latter is a result of eqn. (5.29), not a condition!). Therefore, since we are interested in the infrared optical properties of solids, eqn. (5.29) can hardly be engaged for the problem at hand, but may be applicable, whenever rotation plays a role. A slightly different argument, presented for the introduction of the weighting factor used especially in combination with the interpretation of X-ray texture measurements, is that the volume fraction oriented at Tis a function of sin T.63 Orientational averaging has a broad spectrum of application; it is used in computations of light scattering for randomly oriented non-spherical particles,64,65 in the ESR and NMRspectroscopy of solids, XIII,48-56 for the interpretation of infrared spectra of randomly or partly oriented polymers and for the calculation of depolarization factors in Raman-spectroscopy, 17,66-68

to name just a few. Assuming that the result may be applicable to other properties, it is

therefore of vital interest to investigate if the correct form of averaging for the optical properties is given by eqn. (5.27) or by eqn. (5.28). In the following we will do this for randomly oriented polycrystalline Fresnoite with optically large crystallites on an experimental basis (some theoretical considerations with regard to the weighting factor can be found in the Appendix, section A.10). First of all, Fig. 5.13 gives a XIII

It can not be excluded that in case of NM R- and ESR-powder averag ing eqn. (5.28) L V ³ F RU U H F W ´as a consequence of eqn. (5.29).

- 68 -


comparison of the spectra of differently prepared samples of polycrystalline Fresnoite with large crystallites. The samples Ba0K0-100G and Ba0K50-50G were prepared by Spark Plasma Sintering (SPS, see Appendix A.9.2) from Fresnoite powders. The sample Ba0K5050G consisted originally of a mixture of 50% large ( d § Pm) and 50% small ( d § nm) crystallites. The preparation conditions, however, led to a growth of the small crystallites so that they ended up being optically large. The samples Ba0K0-100G and Ba0kG were prepared solely from powder consisting of large crystallites ( d § Pm). The last sample, Ba1k ( d

§ Pm) was prepared by the crystallization of a glass with the composition

Ba2 TiSi2 O8 + 1 SiO2 . The additional SiO2 slowed down the nucleation process and the growing of the crystals. These conditions favor the development of large crystallites as is obvious from Fig. 5.13 (further information about sample preparation and additional sample properties can be found in the Appendix A.9.2).

0.6

SPS (Ba0K0-100G) Ba0kG

0.2 0.0

SPS (Ba0K50-50G) Ba1k

0.4

G Si-O-Si SiO2

0.2 0.0 100

300

500 700 900 -1 wavenumber / cm

Q Si-O-Si as SiO2

1100

p-polarized

reflectance (20°)

Q Si-O-Si as SiO2

s-polarized

G Si-O-Si SiO2

0.4

1300

Fig. 5.13: Measured reflectance spectra of different polycrystalline Fresnoite samples (incidence angle 20°). All of the spectra depicted in Fig. 5.13 show similar relative band intensities, band shapes and band positions (of course with the exceptions of the bands around 500 and 1080 cm-1 in the spectra belonging to Ba1k, since these bands interfere with the bands of the additional SiO2 .


- 69 -

A further exception might be the band with the highest wavenumber in case of the sample Ba0kG, but the pole figure of this sample does not indicate any preferred orientation, so this effect might be due to comparably high volume fraction of voids.). Besides, for all spectra the inequality Rs > Rp holds. The overall lower intensity of the samples Ba1k and Ba0kG is due to the existence of a second phase in these samples (SiO2 (Ba1k) and air (Ba0kG), respectively). We can therefore conclude that the orientation of the crystallites, whatever their nature may be, is similar in all samples. Even in the absence of additional information from X-ray diffraction and X-ray goniometry, it would therefore be improbable that the samples possess any preferred orientation, taking additionally into consideration the similarity between the Rp and Rs-spectra.

0.75

0.50

s-polarized

simulated (sinT dMdT ) simulated (dMdT )

0.50 0.25

measured u 1.22

0.00

reflectance (50°)

870

measured

0.50 0.25

300

500

700

900

1070

0.15 0.10 0.05

1060

0.00 100

970

p-polarized

reflectance (20°)

0.25

1100

1080

1300

1500

-1

wavenumber / cm

Fig. 5.14: Measured reflectance spectra of randomly-oriented polycrystalline Fresnoite (Ba0K0-100G) and simulated spectra ( 'M 'T 1q). Lower part: p-polarization; upper part: s-polarization (1st Inset: measured spectrum multiplied by 1.22. 2nd Inset: parallel polarized radiation, incidence angle 50°). Due to the similarity of the samples, the findings with regard to orientation and the use of the weighting factor sin T that are obtained below for the sample Ba0K0-100G, apply in the same way to all other samples. In the following, the averaging schemes according to d: = d Md T and d: = sin Td Md Twill be denoted simply as averaging scheme I and II, respectively.

- 70 -


Fig. 5.14 shows the comparison between measured (sample Ba0K0-100G) and simulated reflectance spectra according to averaging scheme I and II for s- and p-polarization. In contrast to scheme I it is obvious that scheme II shows a notable worse correspondence with regard to the relative intensities. The most striking differences are the following:

Averaging scheme I -

Averaging scheme II

The peak at 122 cm-1 is noticeably

-

Almost equal intensities of the peaks located at 122 cm-1 and 162 cm-1 .

more intense than the peak at 162 cm-1 . -

The intensities of the peaks at 234

-

cm-1 and 273 cm-1 are almost the

The peak at 234 cm-1 is noticeably more intense than that at 273 cm-1 .

same. -

The band between 550 and 600 cm-1

-

shows a maximum at 581 cm-1 and a

The peaks at 581 cm-1 and at 594 cm-1 possess comparable intensities.

shoulder at 594 cm-1 . -

If

the

measured

spectrum

is

-

The peaks at 933 and 986 cm-1 show

multiplied by 1.22 (correction for the

a clearly lower intensity, whereas the

holes) the measured and simulated

band at 1065 cm-1 shows a higher

spectrum are very similar (cf. Fig.

intensity compared to the multiplied

5.14, 1st inset).

experimental spectrum.

As already stated, the lower overall intensity for the measured spectra is due to losses caused by the inferior surface quality compared to the single crystal (apparently, the SPS-sample possesses a higher surface quality compared to that of the sample Ba0kG). In addition to a lower resemblance with regard to the relative peak intensities, the introduction of the weighting factors sin Talso leads to shifts of the peak maxima: While the peaks belonging to vibrational transitions having their transition moments perpendicular to the optical axes (E ll a ± modes) of the corresponding single crystal, are shifted to higher wavenumbers, the peaks of the E ll c ±modes are moved to lower wavenumbers (a and c are the crystallographic axes, which are oriented perpendicular and parallel to the optical axis, respectively). The positions of the measured reflectance maxima for both samples and the calculated positions for both averaging schemes are compared in Table 5.1. Again, averaging scheme I is preferred by the comparison, but the differences between both averaging schemes are smaller than the spectral resolution (4 cm-1 , interpolated: 2cm-1 ), therefore this result cannot be seen as a proof.


0.75

0.25 0.00 0.50

measured simulated (dMdT ) simulated (sinT dMdT )

pp-polarized

reflectance (20°)

ss-polarized

0.50


- 71 -

0.25 0.00 450

625

800 -1 wavenumber / cm

975

1150

Fig. 5.15: Experimental reflectance spectra (parallel polarizers) of randomly-oriented polycrystalline Fresnoite (Ba0K0-100G, lower part: p-polarized incident radiation, upper part: s-polarized incident radiation, analyzer parallel to the polarizer) and simulated spectra ( 'M 'T 1q).

0.06

0.02 0.00 0.04


sp-polarized

reflectance (20°)

ps-polarized

0.04


0.02 0.00 450

625


975

1150

Fig. 5.16: Experimental cross-polarization reflectance spectra of randomly-oriented polycrystalline Fresnoite (Ba0K0-100G, lower part: p-polarized incident radiation, upper part: s-polarized incident radiation, analyzer perpendicular to the polarizer) and simulated spectra ( 'M 'T 1q).

- 72 -


In this context, it is remarkable to observe, that even for a relatively small angle of incidence of 20°, the results of the calculations for some of the peak positions differ for s- and ppolarization. Here, the high wavenumber E ll c ±mode of the single crystal located at 1025 cm-1 (TO-position, cf. Appendix, Table A.5.1) is of special interest: Calculations show a decisive difference (larger than the spectral resolution) of the peak positions calculated according to the different averaging schemes for this mode if p-polarized light and large angle of incidence (D are employed. The peak maximum of the high wavenumber E ll c ± mode can be found according to the measured spectrum at 1080 cm-1 (inset, Fig. 5.14), while the simulations predict 1079 cm-1 (averaging scheme I) and 1074 cm-1 (averaging scheme II).

Table 5.1:

Measured and calculated reflectance peak positions (D = 20°, in cm-1 ) of modes in the range between 800 cm-1 and 1100 cm-1 .

Orientation of the transition moment M

Experimental values

p-pol.

s-pol.

p-pol.

s-pol.

Calculated values d: = d: = d Md T sin Td Md T p-pol. s-pol. p-pol. s-pol.

0 ll a 0 ll a 0 ll c

930 982 1067

934 984 1065

931 984 1067

931 984 1067

931 985 1066

Ba0kG

Ba0K0-100G

931 985 1067

933 986 1065

933 986 1065

The differences between the different averaging schemes are even more pronounced if twopolarizers-experiments are employed. This can be deduced from the comparisons of the corresponding experimental and simulated spectra, which are depicted in Fig. 5.15 and Fig. 5.16, respectively. In case of the terms Rss and Rpp , it is obvious that the relative intensities of the measured spectra are again properly reflected in the simulated spectra based on averaging scheme I ( d: = d Md T). In contrast, notable deviations exist if d: is assumed to equal sin Td Md T (averaging scheme II). This is also illustrated in Table 5.2, where the ratios Ri(simulated)/Ri(measured) (i = ss, pp) of several peak maximae for the different averaging schemes are presented. High- wavenumber peaks were chosen for this comparison, because these peaks are well-separated from each other. Besides, the signal-to-noise-ratio is favorable in this spectral range. The peaks at 860 cm-1 are excluded from the discussion, since the magnitudes of the transition moments belonging to these peaks are nearly the same (the Fresnoite single crystal is almost optically isotropic in the spectral region around 860 cm-1 , cf. Appendix, section A.5).

5. Experimental Verification of ARTT, ARIT and UAOPT Table 5.2:

- 73 -

Comparison of the ratios Ri (simulated)/Ri(measured) at the peak maximae for the different average schemes ( d: = d Md Tand d: = sin Td Md T).

Orientation of the transition moment M 0 ll a 0 ll a 0 ll c

Peak-wavenumber / cmí 931 984 1067

Ri(simulated)/Ri(measured) d: = d Md T d: = sin Td Md T i=pp i=ss i=pp i=ss 1.22 1.24 1.07 1.12 1.24 1.27 1.08 1.14 1.19 1.28 1.75 1.81

Evidently, the ratios Ri (simulated)/Ri(measured) can be seen as constant, within experimental errors, and independent of the orientation of the transition moment and the mode symmetry for averaging scheme I (recall that modes with transition moments 0 ll a are of E-symmetry, whereas 0 ll c- modes belong to symmetry class A1 in case of Fresnoite). At the contrary, averaging scheme II clearly favors the modes of A1 -symmetry on expense of the E- modes. Additionally, the ratios Ri(simulated)/Ri(measured) are approximately constant only within the modes of one symmetry species. From an analysis of the experimental and simulated cross-polarization spectra Rsp and Rps, it is obvious that the resemblance of the computed and the measured relative peak intensities is quite satisfactorily in case of both averaging schemes. The same is true for the peak positions, the differences of which are again smaller than the resolution limit. However, the ratios Ri(simulated)/Ri(measured) are strongly dependent on the chosen averaging scheme, being about 1.4 for averaging scheme I, which is similar to the ratios evaluated above for i = ss, pp within experimental errors. In contrast, for averaging scheme II a fundamentally different ratio of about 2 was determined. The possibly most striking proof for the correctness of averaging scheme I is based on a comparison between simulated and measured polycrystalline Fresnoite reflectance spectra at comparably high angle of incidence. Under these conditions, averaging scheme I leads to a situation where Rp > Rs in the spectral range between 1068 and 1096 cm-1 (this will be discussed in more details in sections 7.4 and 7.5), whereas such a range does not exist if averaging scheme II is employed. This is illustrated in Fig. 5.17. The comparison given in Fig. 5.17 further shows that the differences between the two averaging schemes are somewhat leveled out at high angle of incidence and are not longer as decisive as at comparably low angle of incidence with the important exception of the range around the highest wavenumber peak.

- 74 -


1.00

d:= sinT dMdT

0.75

Rs Rp

0.50

reflectance (70°)

0.25 0.00

d:= dMdT

0.75 0.50 0.25 0.00

Experiment

0.75 0.50 0.25 0.00 100

300

500

700

900

1100

1300

-1

wavenumber / cm

Fig. 5.17: Experimental (Ba0K0-100G) and simulated spectra of randomly-oriented polycrystalline Fresnoite ( 'M 'T 1q, dielectric tensor function based on a KKA of the single crystal (see Appendix, section A.5)) at D= 70°. All in all, it has been shown in this section that the averaging scheme, which employs the weighting factor sin T is not applicable in case of polycrystalline Fresnoite with large crystallites. From the two possible explanations, namely that either the weighting factor sin T must be rejected or that in all investigated polycrystalline samples the crystallites are oriented with a weighting factor (sin T )-1 , so that the weighting factor sin Twould be compensated, the first explanation seems to be more realistic. Otherwise it would be necessary to define two different kinds of random orientation: One, where for an arbitrary plane all optical axes parallel to this plane are randomly oriented (averaging scheme I) and an additional one, where this is not the case, but which is of spherical symmetry instead (averaging scheme II, for more details see section 8.3.1 and the Appendix, section A.10). Unfortunately, the differences are less well-pronounced in the spectra of other uniaxial materials with known dielectric tensor functions (like, e.g., quartz (SiO2 ) or D-Al2 O3 ). Other techniques based on X-ray diffraction like pole figures, rocking curves and T -2Tscans (Rietveld refinement) seem to suffer from mutual inconsistencies63 and seem to be of poor sensitivity for weak textures in the proximity of random orientation. Additionally, these


- 75 -

techniques suffer from a continuous change of the reference frame relative to a coordinate system fixed inside the sample (cf. Appendix, section A.10).

5.2

ARIT

5.2.1

Convergence of the numerical evaluation of the Integrals

For the application of ARIT to uniaxial materials the initial orientation of the crystallographic c-axes was set parallel to the X-axis and the EOR E(z,x,z) was used. As a result, one index of refraction is independent of orientation and the second depends on the angles Tand \: n1 n2

H a HH a c

H cos 2 Tcos 2 \sin 2 T Hcsin 2\sin 2T a

0.7

' \, ' T 30° 15° 10° 5° 3° 2° 1° 0.5 ° 0.25°

0.6 reflectance (20°, s)

(5.30)

0.5 0.4 0.3

0.2

0.1

0.2 0.0 1000

0.1 0.0 100

300

500

700

900

1100

1050

1300

1500

wavenumber / cm-1 Fig. 5.18: Simulated reflectance spectra (based on single crystal data) for different steps '\ and 'T( '\ 'T) according to eqn. (5.30). Since the values of the integrals in eqn. (3.8) are evaluated numerically, it is important to analyze the convergence of the approximate solutions with decreasing '\ and 'Tbefore comparing simulated and measured data as this was done in section 5.1.3 for ARTT. A

- 76 -


corresponding comparison of simulated spectra, employing different steps '\( '\ 'T), is shown in Fig. 5.18. Similarly to ARTT, the solutions tend to converge very quickly. With exception of the range between 1000 cm-1 and 1100 cm-1 , the numerical solution calculated with 16 different orientations ( '\ 'T 30q ) of the single crystal seems to be already adequate. This is an important difference to powder averaging known from, e.g., NMR or EPR, where a minimum of several thousands of different orientations needs to be taken into consideration in order to obtain a satisfactory result.48,50 However, since the computational effort in case of ARIT is comparatively low (spectral resolution 1 cm-1 , 1 GHz Athlon: 70 orientations s-1 ), the following simulated spectra have been calculated with '\ 'T 1q.

5.2.2

Comparison between simulated and measured spectra

Fig. 5.19 shows the comparison of the experimental spectrum and the simulated spectra according to ARIT and EMA (cf. Appendix, section A.6). In case of uniaxial crystallite symmetry the EMA-equation (eqn. (A.6.2)) is simplified to 2

3

Ha H 1 Hc H 3 0 Ha 2 H Hc 2 H

(5.31)

with the solutions

H ,

1

4

H r H a

a

Ha 8Hc ,

(5.32)

where the solution H has no physical relevance. The reflectance or transmittance can then be calculated by the Fresnel formulae as usual.2 Both simulated spectra are based on data for Ha and Hc obtained by DA of the single crysta l (first set, Appendix, section A.5). The correspondence between simulated and measured spectrum is quite satisfactory over wide spectral ranges, especially between 500 cm-1 and 900 cm-1 and above 1100 cm-1 . In the latter range a good correspondence between experiment and simulation is mainly an indication of a surface quality of the polycrystalline sample comparable to that of the single crystal, since above 1100 cm-1 (and also between 600 cm-1 and 900 cm-1 ) Ha |Hc holds (cf. Appendix, section A.5) and, therefore, the birefringence of the single crystal is negligible (the surface quality of the sample was assessed by SFMmeasurements, cf. Fig. A.9.2).


- 77 -

0.7 measured EMA ARIT


0.6 0.5 0.4 0.3 0.2 0.1 0.0 100

300


1100

1300

Fig. 5.19: Measured reflectance spectrum of polycrystalline Fresnoite (incidence angle 20°, perpendicularly polarized radiation, sample Ba0k) and simulated spectra based on EMA and ARIT. Below 175 cm-1 and in the range between 1000 cm-1 and 1100 cm-1 , EMA allows a more accurate prediction of the experimental spectrum, whereas this theory tends to overestimate peak intensities noticeably in all other ranges, which have not been discussed so far. The deviations between measured and simulated reflectance (according to ARIT) in the range between 980 cm-1 and 1100 cm-1 may be based on errors of the single crystal data (in particular Hc), which become especially obvious in this range (see Appendix, section A.5 in combination with section 7.5). Further, it may be possible that the assumption of a continuous distribution of orientations of the crystallites within dimensions of O10 is no longer legitimate in this spectral range.

5.2.3

Critical annotations

The relative similarity between the results of EMA and ARIT, which is obvious from the comparison shown in Fig. 5.19, tends to obscure the fundamental differences between both theories. In contrast to EMA, ARIT neglects possibly important interactions between the domains, while in case of EMA the contributions of non-principal orientations are ignored. The ranges of applicability of both theories, however, are similar. Nevertheless, in case of

- 78 -


EMA, it is important to realize that this theory has its origin in electrostatics. Its application in optics is therefore severely limited as it is based on two conditions, which have been formulated, e.g., by Bohren and Huffman.1 Accordingly, the externally applied electrical field has to be uniform inside all domains and field changes have to affect all domains at the same time. Both conditions give a limit for the maximum size of the crystallites. Mathematically, these can be formulated approximately according to S n + k) a/O 1 can be found inside the square grid (Fig. 3.2b) has not been discussed so far. The theoretical spectra resulting from a treatment of this case are compared to those of the other two limiting cases in Fig. 5.24. In contrast to the spectra depicted in Fig. 5.24b and Fig. 5.24c, which have been calculated assuming an evenly spaced distribution of the angles M \ and T, the orientations to compute the spectrum shown in Fig. 5.24b were chosen in a random manner.


0.75

- 85 -

d >> O /10

c)

0.50

0.00 0.50

k=2 k=3 k = 10

b)

s-polarized


0.25 d O /10

0.25 0.00

d

O/10) is to be compared to the glass spectrum, the resulting conclusion would be that the structural units of glass and polycrystalline sample must be different due to the blue-shift of the peak maxima in the spectrum of this polycrystalline sample (notably especially for the highest wavenumber peak around 1050 cm-1 ). Since both the glass as well as the polycrystalline sample consisting of small crystallites do not depolarize polarized light (i.e., Rsp = Rps = 0), the spectrum of the latter would be the right one to compare with the glass spectrum. Hence, Fresnoite glass should consist of structural

- 94 -

6. Medium range order in glasses

units similar to those of crystalline Fresnoite. This conclusion has already been drawn from the comparison of Raman-spectra.M1,74

0.6

polycrystalline sample (d < O /10) glass


0.4 0.2 0.0

polycrystalline sample (d > O /10) glass

0.4 0.2 0.0 100

Fig. 6.1:

300


1100

1300

Upper part: experimental IRR-spectra of polycrystalline Fresnoite with optically small crystallites (d < O/10) and of Fresnoite glass. Lower part: experimental IRR-spectra of polycrystalline Fresnoite with optically large crystallites (d > O/10) and of Fresnoite glass.

It must be stated, however, that this conclusion is not completely correct, since the Ti-atom, which is (almost exclusively, cf. ref. [M1]) fivefold coordinated in Fresnoite, exists also in fourfold- and sixfold-coordination in notably amounts in the glass according to XPS and Xray-absorption- measurements.75,76

6.2

Modeling glass spectra considering medium range order

As already mentioned, conventional DA (section 9 and section A.8) can only be applied for the modeling of the optical properties if the ordered regions of the material under consideration are of cubic symmetry

H

n 2 . Cubic symmetry is normally seen as a

synonym for optical isotropy. However, the dielectric tensor function is not necessarily


- 95 -

reduced directly to a scalar for randomly oriented polycrystalline materials as presumed by EMA. Alternatively, it can be assumed with the same legitimacy, that a plane wave

E exp ª i k r - Zt º ¬ ¼experiences an averaged complex index of refraction n on reflection and refraction at a boundary between air and a material, which consist of randomly oriented ordered regions, the latter being small compared to the resolution limit. This averaged index of refraction can be calculated by ARIT (eqn. (3.8)). Since a dielectric function Hav can be assigned to the averaged index of refraction according to

Hav

2

n ,

(6.1)

the reduction of the dielectric tensor to a scalar is carried out indirectly. Consequently, Hav cannot be represented by a dispersion relation according to, e.g., eqs. (A.8.1) - (A.8.5) in contrast to the principal dielectric functions of a crystal or a domain. Note that this is also not possible if EMA is applied, since the principal dielectric functions are combined in a nonlinear way (cf. chapter 9)! Of what value is eqn. (3.8) for glasses? An important consequence of applying ARIT is that the spectrum of a polycrystalline sample can be simulated from single crystal data by varying all damping constants with a common factor (cf. section 5.2.5). Therefore, it is tempting to assume that the respective glass spectra can be generated by multiplying all damping constants of the related single crystal with a constant factor f. In contrast to Efimov,77 we think that higher damping constants for glasses are physically reasonable, since damping constants are proportional to the inverse phonon life time and therefore proportional to the inverse of the dimension of the ordered regions (cf. also refs [78] and [79]). Therefore, the small dimensions of the structural units of MRO in glasses may be one reason for the broadening of the peaks in glass spectra. In addition, effects due to disorder have to be taken into consideration. This can be done by using the convolution model (cf. Appendix, section A.8, eqn. (A.8.3)) instead of the Lorentz model (eqn. (A.8.1)) to generate the principal dielectric functions of the domains. Eqs. (3.8) and (6.1) are fully compatible with the generally accepted continuous random network theory, which describes the structure of glasses. In this theory regions of MRO are assumed, which are closely related to the structure of crystals with the same composition.73 These regions are randomly oriented in exactly the same way as the crystallites in polycrystalline samples. Therefore it should be necessary to calculate an orientational average of the optical properties of the, in general, anisotropic regions of MRO in order to model the

- 96 -


optical properties of amorphous solids. Unfortunately, the dielectric tensors of the regions of MRO are not available. As a first approximation, varied single crystal data can be used instead as this is done in case of polycrystalline materials.

0.6

simulated (f = 1.25) measured


0.4 0.2 0.0

f=1 f=3 f=5 f=7 f=9

0.4 0.2 0.0 100

300

500

700

900

1100

1300

wavenumber / cm-1 Fig. 6.2:

Upper part: simulated (according to ARIT with a factor f = 1.25), and measured spectrum of polycrystalline Fresnoite (sample Ba0k, d O /10) R = R Rs sp ps Rp

0.00 0 0.003

15

30

1.0 0.8 0.6 0.4 0.2

(b)

0.002 0.001 0.000 32

45

33

34

0.04 0.03 0.02 0.01 0.00 35 40

60

75

90

(c)

41

42

43

44

angle of incidence (°) Fig. 7.5:

(a): Calculated angular dependence of the perpendicular and parallel polarized reflectance, Rs and Rp , from an isotropic semiinfinite medium consisting of either optically small (d > O/10) ordered domains into a highly refractive incidence medium ( ninc 3 , na = 2.02 and nc = 1.96). (b): Close- up view of the calculated angular dependence of the parallel polarized reflectance, Rp , and the cross-polarized reflectance Rsp = Rps for an isotropic medium consisting of optically small (d > O/10) ordered regions near DBrew and DMin, respectively. (c): Close- up view of the calculated angular dependence of the reflectance near the onsets of total reflection.

From normal incidence up to an angle of incidence D § GL I I HU HQ F H V EHW ZH HQ W K H reflectance values provided by the different theories can be detected only in close-up views. Again, as in the previous section, where the incidence medium was vacuum (n = 1), Rp does not show a zero but a minimum at a certain angle of incidence DMin for large domain media. 7KL V PL QL PXPL V V KL I W H G D ZD\ I U RP %U H ZV W H U ¶ V DQJO H ZK H U H Rp is identically zero for an optically homogenous and isotropic medium. Furthermore, the value of Rp (large domains) at

7. The dependence of the reflectance on the angle of incidence

- 113 -

the minimum is larger than the individual cross-polarization terms Rsp and Rps alone due to the use of an incoherent averaging scheme. For small angles of incidence, Ri (large domains, i = s,p) is lower than the corresponding Ri (small domains), whereas this relation is inverted at larger D. Therefore Ri (large domains) must be equal to Ri (small domains) at a certain D. For D > 38° the difference between Rp (large domains) and Rp (small domains) increases considerably, in contrast to the difference between Rs (large domains) and Rs (small domains), which remains comparably small. At about D = 40.7° the cross-polarization terms show a marked increase. This increase is coincident with an enlargement of the difference between Rs (large domains) and Rs (small domains). Since the onset of total reflectionXVIII is shifted to a larger angle of incidence for the sample consisting of large domains, Rs (large domains) and Rs (small domains) as well as Rp (large domains) and Rp (small domains) are again equal in pairs at a certain Din the range between the marked increase of the cross-polarization terms and the onset of total reflection. The cross-polarization terms themselves show a maximum close to the onset of total reflection and are slowly converging to zero afterwards. Overall, we notice that the cross-polarization reaches comparably high values near the onset of total reflection even for a fairly small optical anisotropy of the domains. Therefore, an experimental verification of our theoretical predictions should be straightforward. The second simulation was carried out with the same set of principal indices of refraction (na,2 = 2.2 and nc,2 = 1.6) for the domains as was used in the previous section. Following the approximation given by eqn. (5.36), the result of the averaged index of refraction n = 2 as assumed, while the superior approximation according to ARIT (eqn. (3.8)) predicts

n

=

1.96 for the second set. The difference between both results is still comparably small but one order of magnitude larger than that for the set used for the first simulation. The corresponding simulations are shown in Fig. 7.6, where we again compare the results for two virtual semiinfinite exit media consisting of either large or small domains. For the small domain media, only minor differences are noticeable between the reflectance-curves based on the first (Fig. 7.5) and the second set (Fig. 7.6) of optical constants. We find a small shift of the Brewster angle (1st set: 33.8°, 2nd set: 33.3°) and of the onset of total reflection (1st set: 41.8°, 2nd set: 40.8°), but the shapes of the graphs are essentially the same. The latter statement is not true if we compare the simulated reflectances for the large domain media. From Fig. 7.6, it is obvious that there is a considerable shift of the onset of total reflection from 42.4° (1st set) to

XVIII

The angle of incidence belonging to the onset of total reflection is called the crit ical angle Dcr and is given

by Dcr

sin 1 n ninc for a s mall do main mediu m.22,99

- 114 -


47.2° (2nd set), which is about five times larger than the corresponding shift in the small domain limit. Rp still shows a minimum, but it is clearly less well-pronounced compared to that exhibited by the simulation belonging to the first set of optical constants shown in Fig. 7.5. The most striking difference, however, is that there exists a range where Rp exceeds Rs, which is an unparalleled behavior for optically isotropic media. This range extends from 33.8° up to the onset of total reflection at 47.2°. We know from further simulations (not shown), that its width depends on the difference between the principal indices of refraction na and nc and that it disappears at na § D nd nc § I RU ninc = 3 (not shown).

1.00

(a)

0.75

Small Domain Size Rs Rp Large Domain Size Rsp = Rps Rs Rp

0.043

0.50

0.041

0.25

reflectance

0.045

0 1 2 3 4 5

0.00 0 0.04

15

30

(b)

45

60

75

90

1.0 (c) 0.8 0.6

0.02

0.00 25

Fig. 7.6:

30

0.15 0.10 0.05 0.00 35 45 46 angle of incidence (°)

47

48

49

(a): Calculated angular dependence of the perpendicular and parallel polarized reflectance, Rs and Rp , from an isotropic medium consisting of optically small (d > O/10) ordered domains into a highly refractive incidence medium ( ninc 3 , na = 2.2 and nc = 1.6). (b): Close- up view of the calculated angular dependence of the parallel polarized reflectance, Rp and the cross-polarized reflectance Rsp = Rps. (c): Close- up view of the calculated angular dependence of the reflectance (large domain case) near the onset of total reflection.


- 115 -

In order to understand the existence of a range where Rp exceeds Rs, we performed several simulations of the reflectance in dependence of the orientation from a single ordered domain with the principal indices of refraction of the 2nd set (na,2 = 2.2 and nc,2 = 1.6). We decided to simulate first the angular dependence of the reflectance for principal orientations of the ordered domain. Assumed that the plane of incidence is the Y-Z-plane and the interface between incidence and second medium is the X-Y-plane (see Fig. 7.1 or inset of Fig. 7.7) these are the orientations were the optical axis is oriented parallel to the Z-axis (Fig. 7.7a, M,T = 0°), the Y-axis (Fig. 7.7b, M = 0°, T= 90°) and the X-axis (Fig. 7.7c, M,T= 90°), respectively.

1.00 (a)

D cr(c,2)

0.75 0.50

Rs Rp

0.25

reflectance

1.00 (b)

D cr(a,2) M= 0° T= 0°

0.00

D cr(a,2)

0.75 0.50

M= 0° T= 90°

0.25 0.00

0

15 30 45 60 75 90

1.00 (c)

0.50

D cr(a,2) M= 90° T= 90°

0.25 0.00

15 30 45 60 75 90

1.00 (d)

D cr(c,2)

0.75

0

D cr(a,2)

0.75

Incoherent Average

D cr(c,2)

0.50 0.25 0.00

0

15 30 45 60 75 90

0

15 30 45 60 75 90


Angular dependence of the reflectance from an anisotropic medium for the three different principal orientations and the incoherent averages of Rs and Rp . (a) M= 0°, T= 0° (optical axis parallel to the Z-axis). (b) M= 0°, T= 90° (optical axis parallel to the Y-axis). (c) M= 0°, T= 90° (optical axis parallel to the X-axis). (d) Incoherent average ( 13 R( a ),i 13 R( b ),i 13 R( c ),i , i s, p ).

- 116 -


For these orientations, the cross-polarization terms are zero and the perpendicular polarized reflectance Rs is merely depending on one of the principal dielectric constants, in contrast to Rp , which is influenced by two. All in all, we only observe two different onsets of total reflection, one at Dcr(c,2) = 32.2° and a second at Dcr(a,2) = 47.2°. These belong to the principal indices of refraction na,2 (Dcr(a,2)) and nc,2 (Dcr(c,2)). It is obvious from Fig. 7.7a and Fig. 7.7b, and also well-known in general that for an anisotropic medium Rp can exceed Rs at certain Dranges. However, the incoherent average of the reflectances according to

R

1 1 1 R MT , 0q , 90q R M 0q, T 90q R MT 3 3 3

(7.1)

has no such range, but Rp nearly reaches Rs at Dcr(c,2), indicating that Rp ascends steeper than

reflectance

Rs in the vicinity of this particular angle of incidence.

1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00

(a) M= 45° T= 45°

D cr(a,2)

D cr(c,2)

(b) M= 45° T= -45°

(c) Incoherent Average Rs Rp 0

Rsp Rps 15

30

45

85 90


Angular dependence of the reflectance from an anisotropic medium for two different non-principal orientations : :(M T . Also shown is the corresponding incoherent average (a) Mҏ = 45°,T= 45°. (b) M= 45°,T= - 45°. (c) Incoherent average ( 12 R( a ),i 12 R(b ),i , i s, p, sp, ps ).


- 117 -

In Fig. 7.8, we show simulations of the reflectance for two non-principal orientations (M= 45°, T= 45°), (M= 45°, T= - 45°) and the average reflectance R according to

R

1 1 R M S , T S R M S , T S , 4 4 2 4 4 2

(7.2)

which is a clearly better approximation for R in the large domain case than the average according to eqn. (7.1) (cf. section 5.1.6).This is obvious from the fact that the average of the reflectance of the principal orientation does not account for the cross-polarization terms, since they are zero for the principal orientations as mentioned above. The approximation according to eqn. (7.2), however, tends to overestimate these terms by a factor between 2 and 3, but gives qualitatively a correct reproduction. Analyzing Fig. 7.8c, the cross-polarization shows a distinct increase at about Dcr(c,2) and a maximum near Dcr(a,2) When compared to ARTT (eqn. (3.3), Fig. 7.6), eqn. (7.2) is not able to predict the shape of the curves of Rs and

Rp

correctly, but in contrast to eqn. (7.1) (Fig. 7.7), there exists a range in Fig. 7.8, where Rp > Rs. The onset of total reflection is for all three kinds of reflectance averages (eqs. (3.3), (7.1) and (7.2)) at Dcr(a,2) = 47.2°. Therefore, we conclude that in case of randomly oriented media with large domains the onset of total reflection is shifted to a higher angle of incidence compared to random media consisting of small domains. In particular, the onset of total reflection is determined by the largest of the principal indices of refraction of the anisotropic domains. It is equal to that of a homogenous and isotropic medium with this index of refraction. We suggest naming the corresponding effect suppressed total reflection.

7.4 7.4.1

The influence of absorption Index of refraction of the incidence medium smaller than the principal indices of refraction of the polydomain medium

In the following we will assume the uniaxial domains to be absorptive ( n

n ik , with the

real part n and the imaginary part k of the complex index of refraction n). To study the influence of absorption, we will employ a set of optical constants based on that used in section 7.2 (na = 2.2, nc = 1.6 and ninc = 1). The ki will be varied in a way that the variance is the same for ka and kc, i.e. na /nc = ka /kc (the anisotropy is kept constant). The results of the simulations are shown in Fig. 7.9 (one polarizer experiments) and Fig. 7.10 ('Ri, crosspolarized reflectances).

- 118 -


1.00

ki = 0 Small Domain Size (d > O /10) Rs Rp

0.75 0.50 0.25 0.00

ki = 1/2 ni

0.75

reflectance

0.50 0.25 0.00

ki = ni

0.75 0.50 0.25 0.00

ki = 2 ni

0.75 0.50 0.25 0.00 0

15

30

45

60

75

90


Angular dependence of the reflectances Rs, Rp from an isotropic semiinfinite medium consisting of either optically small (d > O/10) uniaxial domains (na = 2.2 and nc = 1.6) for several principal absorption indices (ki = 0, ½ ni, ni, 2 ni, i = a, c). The incidence medium is characterized by an index of refraction ninc 1 .

From Fig. 7.9 it is obvious that the minimum of Rp is shifted to higher angles of incidence with increasing ki. In addition, %U HZV W H U ¶ V DQJO H F H D V HV W R H[L V W HY HQ I RU W K H V PDO O GRPDL Q case (Rp DW DMin). Besides, we notice that the already small differences between the angular dependencies of the reflectance in the small and large domain limit for ki = 0 are further decreased with increasing principal absorption indices ki (Fig. 7.10). With regard to


- 119 -

the cross-polarization terms of the reflectance, we see that the increase of the principal absorption indices leads to a shift of the maximum to lower Dand to a drop of the crosspolarization terms, especially at higher D. Therefore, while the occurrence of absorption is the reason for indices of refraction larger than unity and for the existence of birefringence, it simultaneously reduces the optical domain size effects.

Rps = Rsp

0.0020 0.0015 0.0010 0.0005 0.0000

'Rp

0.01 0.00

ki = 0 ki = 1/2 ni

-0.01

'Rs

-0.02

ki = ni ki = 2 ni

0.010

'Ri = Ri(d >> O /10) - Ri(d Rs holds and we see that Rp displays only a minimum but no formal zero. The latter is always the case in the large domain limit if the domains are optically anisotropic as already stated. However, compared with the simulations presented in the preceding section (Fig. 7.9, upper panel) where ninc < na , nc, the differences between the small and the large domain case are much more evident. Again, these differences are reduced if we allow the index of refraction to be complex and increase the principal indices of absorption ki. If the ki values are comparable to those of the ni, the differences between the large and the small domain case become very small, even if the ratio na /nc, and therefore the anisotropy, is kept constant. This applies also to the crosspolarization terms, which are in such a case of the order of 1×10-3 and therefore almost negligible like in the case presented in the preceding section. For organic materials, however, typically exhibiting low oscillator strengths and thus comparably low ki values in the infrared,99 considerable misinterpretations may result, if domain size effects are neglected. In

this context

it

may be

worth pointing out that the well-known relation

Rs2 D 45q R p D 45q 1 (cf., e.g., ref. [99]) only holds for isotropic media, which can be characterized properly by a scalar dielectric or index of refraction function. Deviations from this relation may easily lead to the assumption of, e.g., preferential orientation of the domains, whereas the occurrence of large domains might be also responsible for the invalidity of the above given relation between Rp and Rs at D= 45°, an angle, which is one of the most commonly applied angles in ATR elements.

reflectance


- 121 -

1.00 Small Domain Size Large Domain Size 0.75 Rs Rs 0.50 Rp Rp 0.25 ki = 0 Rps= Rsp 0.00 1.00 0.75 0.50 0.25 ki = 0.03 ni 0.00 1.00 0.75 0.50 0.25 ki = 0.09 ni 0.00 1.00 0.75 0.50 0.25 ki = 0.27 ni 0.00 1.00 0.75 0.50 k = 0.81 n i i 0.25 0.00 0 15 30 45 60 75 90 angle of incidence (°)

Fig. 7.11: Angular dependence of the reflectances Rs, Rp and the cross-polarization reflectance terms Rsp and Rps (Rsp = Rps) of an isotropic semiinfinite medium consisting of optically large uniaxial domains (na = 2.2 and nc = 1.6) for several principal absorption indices (ki = 0, ½ ni, ni, 2 ni, i = a, c) and comparison with Rs and Rp of the corresponding small domain medium. The incidence medium is characterized by an index of refraction higher than any of the principal indices na and nc of the polydomain medium ( ninc 3 ). 7.4.3

Index of refraction of the incidence medium lies between the values of the principal indices of refraction of the polydomain medium

For the case of a small domain medium it is usually not seen as a special case if the index of refraction of the incidence medium lies between the values of the principal indices of

- 122 -


refraction of the polydomain medium, since its averaged index of refraction is still either smaller or larger than that of the incidence medium. These cases were already discussed in the preceding sections (with the exception of the trivial case, when the second medium has the same index of refraction as the incidence medium and, as a result, no reflection occurs). For the large domain limit, however, the conditions given above lead to a situation, which deserves special attention, as will be shown in the following. Again, we presume the same set of optical constants for the domains as in the preceding sections, na = 2.2 and nc = 1.6, but now we choose the refractive index of the incidence medium ninc = 2. Since the averaged index of refraction (small domains) does not depend on the incidence medium, its value is still given by

n = 1.96 according to ARIT. As a

consequence, the small domain medium shows total reflection at Dcr = 78.52° and, while the difference between Rp and Rs is very small, Rs is, of course, always larger than Rp (with the trivial exceptions at D= 0° and for DDcr) as shown in the top panel of Fig. 7.12. Investigating the angular dependence of the reflectance for the large domain medium, we first note that total reflection cannot occur, since na > ninc and Dcr

sin 1 n largest ninc is a complex

number. Indeed, it can be shown that total reflection is possible only for p-polarized light and domains having their optical axis oriented perpendicular to the surface. Additionally, this particular orientation results in R p !Rs if Dz0 and DDcr . Since there is a broad variety of other orientations, which also lead to R p !Rs , at least over a limited range of D, we can expect to find an angular range, where R p !Rs for the randomly oriented large domain medium. This is confirmed by the results of our simulations, which are shown in the top panel of Fig. 7.12, where we also present the simulated cross-polarization terms, which possess remarkable values in a range starting in the vicinity of D sin 1 . n lowest n inc 53.1q Allowing the domains to be absorptive while keeping the anisotropy constant, we once more notice that the domain size related effects are reduced by increasing absorption. However, at ki = 0.09 ni, Rp is still slightly larger than Rs. Therefore it might be relatively easy to find a system, which allows an experimental verification of the theoretical finding that Rp can exceed Rs in case of isotropic systems. On the other hand, increasing the ki values again lead to decreasing differences of the domain size related effects.

reflectance


- 123 -

d > O /10 Rs Rs Rp Rp Rps= Rsp ki = 0

1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00

ki = 0.03 ni

ki = 0.09 ni

ki = 0.27 ni

ki = 0.81 ni 0

15

30 45 60 angle of incidence (°)

75

90

Fig. 7.12: Angular dependence of the reflectances Rs, Rp and the cross-polarization reflectance terms Rsp and Rps (Rsp = Rps) of an isotropic semiinfinite medium consisting of optically large uniaxial domains (na = 2.2 and nc = 1.6) for several principal absorption indices (ki = 0, ½ ni, ni, 2 ni, i = a, c) and comparison with Rs and Rp of the corresponding small domain medium. The incidence medium is characterized by an index of refraction, which lies between the principal indices na and nc of the polydomain medium ( ninc 2 ). 7.4.4

Simulation of the frequency-dependent specular reflectance from a model system for a small and a high angle of incidence

The simulations of the dependence of the reflectance from a large domain medium presented in this chapter are by no means comprehensive, since we limited our simulations to optically

- 124 -


uniaxial domains with the additional restrictions that na > nc and that the ratio na nc may be constant. In this section we will drop these two limitations and show that all three cases, which were discussed in the preceding sections, can occur in principle in a usual reflectance experiment with vacuum or air as incidence medium. To this end we assume a polydomain medium (e.g., a polycrystalline material) with optically uniaxial domains and two strong oscillators in the Reststrahlen region. The transition moment of the first oscillator is presumed to be oriented perpendicular to the optical axis. Therefore, this oscillator contributes to the  na2 . The transition moment of the second oscillator is principal dielectric function Ha Q Q

 nc2 . The assumed to be oriented parallel to the optical axis and to belong to Hc Q Q

 can be described  and Hc frequency dependence of the principal dielectric functions Ha Q Q by the classical damped harmonic oscillator model:

Si 2 , 2 2    (Q Q ) i Q J i i

 Hf,i Hi Q

i

(7.3)

a, c

 As in the preceding sections, Si represents the oscillator strengths, Ji the damping constant, Q i

the oscillator position and Hf,i the dielectric background.

Table 7.1:

Parameters used to compute the principal dielectric functions Ha and Hc. Si (cm-1 )

Ji (cm-1 )

-1  Q i (cm )

Hf,i

H a

1000

10

900

3.1

Hc

1000

10

1000

2.9

In our simulations we used the parameters given in Table 7.1. The real and imaginary parts of the complex indices of refraction ni

n i ik i are displayed in Fig. 7.13 as functions of the

wavenumber. Taking into account that the ambient medium is assumed to be vacuum or air (ninc § ZH F DQ V X EGL YL GH W K H V SH FW U DO U DQJ H L QW R I L YH SD U W V ZKL FK D U H SU HV HQ W H G L QTable 7.2: (i) Up to 914 cm-1 , the real parts of the principal refractive indices are both larger than ninc. (ii) between 915 cm-1 and 1014 cm-1 the inequalities na < ninc < nc hold, while in the third range from (iii) 1015 cm-1 to 1134 cm-1 both na and nc are smaller than ninc. In the subsequent range, (iv) which extends to 1236 cm-1 , only na exceeds ninc, before (v) both na and nc are again larger than ninc.


5

 n  k

ki

0

ka kc

5 0

 n  k

5

Rps, Rsp

0

D= 20° D= 70°

0.08

Rsp=Rps

1.0

R(D= 70°)

na nc

0.5

0.0

R(D= 20°)

ni

10

d > O /10 Rs Rp

0.5

0.04 0.00 800

1000 1200 -1 wavenumber / cm

- 125 -

1400

0.0 800


1400

Fig. 7.13: Comparison of the infrared reflectance from polydomain media consisting of optically uniaxial domains (either large or small). The principal dielectric functions are generated assuming a classical damped harmonic oscillator model employing the oscillator parameters shown in Table 7.1. Left column: wavenumber dependence of the principal indices of refraction ni and indices of absorption ki of the domains, wavenumber dependence of the averaged index of refraction n and averaged index of absorption k (small domains); cross-polarization terms of the reflectance Rsp and Rps (Rsp = Rps) for two different angles of incidence D= 20° and D= 70° (large domains). Right column: comparison of the reflectance for s- and p-polarized incident radiation and two different angles of incidence D= 20° and 70°. Table 7.2:

Comparison of na , nc and ninc in the spectral range from 800 ±1400 cm-1 .

I (cm-1 )

II (cm-1 )

III (cm-1 )

IV (cm-1 )

V (cm-1 )

800 ±914

914 ±1014

1014 ±1134

1134 ±1236

1236 ±1400

ninc < na , nc

na < ninc < nc

na , nc < ninc

nc < ninc < na

ninc < na , nc

The dispersion of the real part of the averaged index of refraction n , as displayed in Fig. 7.13, demonstrates that between 1001 cm-1 and 1190 cm-1 an ATR experiment is performed if DDc, on the condition that the hypothetic polydomain medium consists of small domains. In

- 126 -


case of the large domain medium, the corresponding range is identical to range III and therefore smaller. With respect to the angular dependence of the cross-polarization terms, which are relevant only for the large domain medium, we find that the cross-polarized reflectance is higher at D= 20° up to 1093 cm-1 compared with its values at D= 70°, while above 1093 cm-1 this behavior is reversed. Rs and Rp display similar band shapes and fulfill the inequality Rs > Rp over the whole spectral range for both types of polydomain media at D = 20°. This inequality is, of course, also fulfilled at D= 70° in the small domain limit, whereas Rp > Rs holds for the large domain medium between 1130 cm-1 and 1216 cm-1 . The latter range lies nearly completely in the region where nc < ninc < na holds (denoted range IV in Table 7.2) and in which the absorption is already low compared to range III. The magnitude of the absorption presumably prevents Rp from being larger than Rs in the latter range. The size effect is clearly visible if we compare the Rs- and Rp -spectra belonging to the two types of random polydomain media. The spectral differences due to the different averaging procedures (ARIT and ARTT), which are hardly noticeable for wavenumbers below the maximum of ka (< 903 cm-1 ), begin to increase drastically after this maximum. At D= 20°, the curves of Rs and Rp converge again between 1030 cm-1 and 1070 cm-1 , before the differences once more increase until the cross-polarization terms drop down significantly at 1200 cm-1 . At D= 70° the domain size effect also starts at approximately 903 cm-1 , but the similarity between 1030 cm-1 and 1070 cm-1 is much less pronounced. The cross-polarization terms decrease at this angle of incidence at 1250 cm-1 , being again connected with a marked reduction of the domain size effect.

7.5

Experime ntal ve rification

In this section, once more Fresnoite will be used as the test material to verify the findings of the preceding sections. As already stated, Fresnoite can be characterized by two principal complex refractive indices na

n a i k a and nc

n c i k c . The dependence of the refractive

indices from the wavenumber can be evaluated either by a KKA or a DA from the principal reflectance spectra recorded from an a-c-cut of the single crystal (see Appendix, section A.5). These spectra are depicted in the range between 950 and 1150 cm-1 in Fig. 7.14 together with the calculated spectra based on KKA and DA. The mean square error of the KKA of Fresnoite is about 1/10 ( E a ) and 1/70 ( E c ) compared to that resulting from DA in the spectral


- 127 -

range of interest (for DA, however, only one oscillator was employed for each band in the


reflectance spectra).

1.00 a) 0.75 0.50 0.25 0.00 0.008

E ll a KKA E ll c KKA

'reflectance = exp. - calc.

E ll a E ll c

0.000 -0.008 1.00 b) 0.75 0.50 0.25 0.00 0.04

E ll a DA E ll c DA

'reflectance = exp. - calc.

E ll a E ll c

0.00 -0.04 950

1000


1150

Fig. 7.14: a) Comparison between experimental spectra of the Fresnoite single crystal (angle of incidence D= 20°, perpendicular polarized incident radiation) and spectra calculated based on the principal dielectric functions obtained by KKA. b) Comparison between experimental spectra of the Fresnoite single crystal (angle of incidence D= 20°, perpendicular polarized incident radiation) and spectra calculated based on the principal dielectric functions obtained by DA. The resulting principal indices of refraction functions are shown in Fig. 7.15. Equivalently to the preceding section, we subdivide the graphs into several ranges, which are characterized by the values of the real parts of the principal refractive indices in comparison with the index of refraction ninc of the incidence medium (air: ninc § . As a result of this subdivision, six parts are obtained, which are presented in Table 7.3: From the visible spectral range down to 1110 cm-1 (I), the real parts of the principal refractive indices are both larger than ninc. In the second range (II), between 1110 cm-1 and 1063 cm-1 the inequalities nc < ninc < na hold, while in the third range (III), which extends from 1063 cm-1 to 1034 cm-1 , both na as well as nc are smaller than ninc. In the subsequent range (IV),

- 128 -


which lies between 1034 cm-1 and 969 cm-1 , only nc exceeds ninc. In part V, which ranges from 969 cm-1 down to 957 cm-1 , a point is located between 961 cm-1 and 962 cm-1 , where both complex indices of refraction are equal and therefore the optical properties of Fresnoite should resemble those of a material with cubic crystal symmetry. While in range V both real parts are larger than unity, na is again smaller than ninc below 957 cm-1 in range VI. Table 7.3:

Comparison of the principal indices of refraction of Fresnoite, na and nc and ninc in the spectral range from 1150 ±950 cm-1 .

I (cm-1 )

II (cm-1 )

III (cm-1 )

IV (cm-1 )

V (cm-1 )

VI (cm-1 )

1150 ±1110

1110 ±1063

1063 ±1034

1034 ±969

969 ±957

957 ±950

ninc < nc, na

nc < ninc < na

na , nc < ninc

na < ninc < nc

ninc < nc, na

na < ninc < nc

The dispersion of the average index of refraction

n

as determined from a KKA of a

reflectance spectrum of randomly oriented Fresnoite with small crystallites, which is displayed in Fig. 7.16, demonstrates that in the two ranges between 1030 cm-1 and 1090 cm-1 and also between 976 cm-1 and 996 cm-1 an ATR experiment is performed if DDcr. The situation is different in case of a sample with large crystallites. Here the former range is identical to range III and therefore narrower (1034 cm-1 and 1063 cm-1 ), while the second does not exist, since for a large domain medium, total reflection is suppressed in this range. (see section 7.4.3). With respect to the angular dependence of the cross-polarization terms, we find that at D= 70° the cross-polarized reflectance is higher down to 1020 cm-1 compared with its values at D= 20°. At D= 20°, Rs and Rp show similar band shapes and fulfill the condition that Rs > Rp over the whole spectral range for both types of polycrystalline samples. This inequality is always fulfilled at D= 70° only for the sample consisting of small crystallites, whereas Rp > Rs holds for the large domain sample between 1069 cm-1 and 1096 cm-1 . Note that the latter range lies completely in the region where nc < ninc < na (range II in Table 7.3) and in which absorption is already comparatively low, in contrast to range III. In this range absorption prevents Rp from being larger than Rs. Hence, especially at higher angles of incidence, care must be taken not to misinterpret the occurrence of spectral ranges where Rp > Rs, e.g., by assuming preferred orientation.


4

VI V IV

II

III

I

n

3

- 129 -

na nc

2 1 0

k

3

ka kc

na= nc ka= kc

2 1 0 950

1000


1100

1150

Fig. 7.15: Real and imaginary parts of the principal indices of refraction of the Fresnoite single crystal obtained by the KKA (see Fig. 7.14).

 n  k

0.5 0.0 0.04

Rsp

Rsp

D= 20° D= 70°

0.02 0.00 950


R(D= 70°)

1.0

R(D= 20°)

 n  k

1.0

1150

0.5

0.0 0.4 0.2 0.0 950

/10 d > O Rs Rs Rp Rp


1150

Fig. 7.16: Experimental reflectance (D = 20°/70°) of randomly oriented polycrystalline Fresnoite with either optically large (sample Ba0K0-100G) or small crystallites (sample Ba0k). Left column: wavenumber dependence of the averaged index of refraction n and averaged index of absorption k (small domain sample, derived from a KKA of Rs (D = 20°)); cross-polarization terms of the reflectance Rsp for two different angles of incidence D= 20° and D= 70° (large domain sample). Right column: comparison of the reflectance for s- and p-polarized incident radiation and two different angles of incidence D= 20° and 70°.

- 130 -


Comparing the spectra of the two different types of samples, spectral differences cannot be noticed down to about 1110 cm-1 (1100 cm-1 at D= 20°) before the high- wavenumber onset of notable absorption, while they increase drastically afterwards. Around 965 cm-1 both samples show nearly the same reflectance, which could be deduced from the fact that na § nc in this

D= 20°

reflectance

D= 70°

range (vide supra).

Experiment 0.9 0.6 0.3 0.0 0.06 0.03 0.00 950 1050 0.6 0.4 0.2 0.0 0.04

Rs

Rp

Rsp

Rps

0.02 0.00 950

Simulation KKA Simulation DA 0.9 0.9 0.6 0.6 0.3 0.3 0.0 0.0 Rsp = Rps 0.06 0.06 0.03 0.03 Rsp = Rps 0.00 0.00 1150 950 1050 1150 950 1050 1150 0.6 0.4 0.2 0.0 0.04 0.02

1050

0.6 0.4 0.2 0.0 0.04 Rsp = Rps

0.02

0.00 0.00 1150 950 1050 1150 950 wavenumber / cm-1

Rsp = Rps 1050

1150

Fig. 7.17: Comparison of the experimental reflectance spectra of randomly oriented polycrystalline Fresnoite (large domains, first column from the left, sample Ba0K0-100G) with the simulated spectra based on single crystal data obtained by KKA (centered column) and DA (right column) at D= 70° (first row) and D= 20° (second row). Fig. 7.17 presents a comparison between the experimental spectra of the large crystallite sample with those simulated by ARTT based on the two different sets of optical constants as evaluated by KKA and DA. While at D= 20° the differences between the two sets have only a negligible influence on the quality of the simulated spectra, the inferiority of the set obtained by DA is obvious from the comparison at D = 70°, especially with regard to the crosspolarization terms Rsp and Rps. These terms are apparently of special sensitivity to errors concerning the optical constants determined from the single crystal. This example shows that a comparison between experimental spectra of respective polycrystalline materials with large


- 131 -

crystallites and spectra simulated by ARTT can be used to assess the quality of optical constants of single crystals (another example will be presented in section 8.2). The superiority of the optical constants derived by KKA is, however, not as complete as suggested by the mere comparison of the experimental and the calculated spectra (see Fig. 7.14). This became obvious from a closer inspection of the imaginary parts of the principal dielectric functions at wavenumbers high compared to the Reststrahlen region, where these parts should be approaching zero. In case of the data derived by KKA, we find values of the imaginary parts, which are on the average about ten times higher than those which result from DA. This is obviously a shortcoming of KKA, which can explain the non-perfect resemblance of the measured spectra and those calculated from KKA data.

reflectance (D= 70°)

a) 0.9 0.6 0.3 0.0 0.06 0.03 0.00 950 c) 0.9 0.6 0.3 0.0 0.06 0.03 0.00 950

Experimental

Rsp = Rps

1050

b) 0.9 Rs 0.6 Rp 0.3 0.0 0.06 0.03 0.00 1150 950

Simulation (J ) sc

Rsp = Rps 1050

1150

Simulation (J 2) sc d) 0.9 0.6 0.3 0.0 0.06 0.03 Rsp = Rps 0.00 1150 950 1050 1150 -1 wavenumber / cm

Simulation (J 1.5) sc

Rsp = Rps 1050

Fig. 7.18: Comparison of the experimental reflectance spectra of randomly oriented polycrystalline Fresnoite (large domains, sample Ba0K0-100G, D= 70°, a)) with the simulated spectra based on single crystal data obtained by DA with the original damping constants (b)) and with modified damping constants (c)) all damping constants multiplied by 1.5; d) all damping constants multiplied by 2. From this resemblance between measured and simulated spectra we nevertheless draw the conclusion that ARTT is valid at higher angles of incidence as well as at near normal

- 132 -


incidence. Besides the quantitative prediction of cross-polarized reflectance terms Rsp and Rps, ARTT is also able to predict band shapes, positions and relative intensities of bands quite satisfactorily. With regard to absolute intensities, the errors are larger due to the lower surface quality of the polycrystalline samples compared to that of the single crystal, the optical constants of which were used in the simulation. Another factor,

which possibly influences the absolute reflectance intensities of

polycrystalline materials, is the fact that damping constants are always higher in polycrystalline materials compared to those in the corresponding single crystalline materials. This factor, however, is of less importance than usually suspected (cf. sections 5.1.4 and 5.2.5). In case of the sample Ba0K0-100G, we found that the damping constants in the polycrystalline material differ by a factor f 1.1 from those in the respective single crystal. (cf. section 5.1.4). Nevertheless, we cannot exclude that this factor is larger in case of other materials. To assess the influence of f, we performed some model calculations with modified single crystal data using factors up to 2. The results of these calculations are presented in Fig. 7.18. Besides an, albeit slight, alteration of the peak shapes, the intensity of the crosspolarization terms is markedly reduced. Additionally, with increasing f, the range where Rp > Rs narrows and eventually vanishes at f > 2. This is another possible explanation, besides the assumption of preferred orientation, why this effect has not been described in the literature before.

7.6

Concluding re marks

In the preceding sections, we have discussed the influence of absorption on the angular dependence of the reflectance from a randomly oriented polydomain medium consisting of either small or large domains compared with the wavelength. Even under the assumption of constant anisotropy, we found that the presence of absorption levels out spectral differences between randomly oriented small and large domain media. As a consequence, it is obvious that for materials with strong absorption not only the distinction between external and internal reflection is lifted, but also the spectral differences based on the domain size are reduced. and k  and since the corresponding However, due to the asymmetric shape of n Q Q

maxima are displaced from each other, the size effect can still be noticeable in the spectra. Weakly absorbing isotropic polydomain materials with large domains on the other hand can show unexpected behavior especially in ATR spectra. Special care has to be taken in this case to avoid misinterpretations of experimental results.

8. Applications of ARTT and ARIT

- 133 -

8. Applications of ARTT and ARIT Formel-Kapitel (nächstes) Abschnitt 1 8.1

Introduction

Besides their use to simulate and understand spectra of randomly oriented polycrystalline samples, ARTT and ARIT have a much broader spectrum of possible applications, two of which will be discussed in more detail in the following sections. One application has already been touched on in section 7.5, namely their use to verify optical constants determined from single crystals. Beyond the outstanding sensibility of the cross-polarization terms to small errors of optical constants determined by DA or KKA, ARTT (and in this case also ARIT or EMA) can simply be used to check whether the optical constants derived from low-symmetry single crystals (i.e., monoclinic and triclinic crystals) or from unfavorable crystal faces, which are not parallel or perpendicular to symmetry axes or planes of the unit cell, are in error. A comparison between simulated and measured spectra of randomly oriented polycrystalline samples readily reveals the usefulness of single crystal data. Another important application of ARTT is the quantitative determination of orientation of oriented materials. Approximate theories like LDT impose strict limits on the possibility of determining orientation quantitatively, since quite different orientation distributions can give the same IR- and UV/Vis-spectra (see section A.7). A more sophisticated treatment with ARTT (provided, of course, that the region of order are not small compared to the wavelength) immediately reveals, e.g., that many of the numerous different magic angle arrangements can easily be distinguished by their different cross-polarization spectra. Based on ARTT it also follows that uniaxially oriented materials consisting of optically large domains do not possess a real optical axis unless their orientation is exactly the same as in the corresponding single crystal. Otherwise, linear polarized light becomes depolarized when incident on such a sample, even when the polarization direction is parallel or perpendicular to the preferred axis.

8.2

Verification of optical constants determined from low symmetry single crystals XIX

As is known, the real and the imaginary part of the dielectric function tensor of crystals of monoclinic or triclinic symmetry cannot be diagonalized at the same time by applying a rotation transformation.69 Reflectance spectra obtained with linearly polarized light depend therefore basically on all tensor components for a triclinic crystal. This is due to the fact that XIX

This section is based on ref. [M 16].

- 134 -


the dispersion of the dielectric function tensor depends on the orientation of the individual transition moments. In the infrared, the dielectric function tensor can be represented by100-102  İ Q

§ Hf, xx ¨ Hf, yx ¨ ¨ Hf, zx ©

Hf, xy Hf, xz · ¸ Hf, yy Hf, yz ¸ Hf, zy Hf, zz ¸ ¹

. 2 2 2 § · sin 4 cos ) sin 4 sin ) cos ) sin 4 cos 4 cos ) i i i i i i i i N Si 2 ¨ 2 ¸ 2 2 sin 4i sin)i cos)i sin 4i sin )i sin4i cos4i sin)i ¸ ¦ ¨ 2 2   Q Q Q i J i 1 i i ¨ ¸ cos 2 4i ©sin4i cos4i cos)i sin4i cos4i sin)i ¹ (8.1)

Obviously, eqn. (8.1) represents an extension of the conventional damped harmonic oscillator model (A.5): The parameters Hf,ij denote the components of the dielectric tensor at optical frequencies and the orientation of the transition moment of the ith oscillator is specified in terms of the polar and the azimuthal angle 4i and )i , respectively. Eqn. (8.1) has been applied to a number of monoclinic and triclinic materials like e.g. blue vitriol (CuSO 4 ·5H2 O),101 gypsum (CaSO4 ·2H2 O),103 Bi2 O3104 and CuO.102,105,106 Especially the IR-spectra of CuO have attracted strong interest in connection with the occurrence of superconductivity in a number of cuprates. First, the exploration of this monoclinic material suffered from the lack of single crystals large enough to be measured from the (001) face, the normal of which is oriented parallel to the C2 -axis. Therefore spectra from the ( 110 )-plane were measured instead in early works.105,106 The use of this plane lifts a significant advantage of monoclinic crystals compared to those of triclinic symmetry, namely the possibility to obtain spectra of pure transverse modes. For mixed TO ±LO modes considerable difficulties in obtaining correct oscillator parameters arise, since, e.g., band positions are depending not only on the respective oscillator positions, but also on the oscillator strengths and the directions of the transition moments relative to the surface of the sample.107 In such cases, the spectra of polycrystalline materials allow gaining access to an independent source of information and the comparison between measured and simulated spectra based on the single crystal data offer an easy way to verify these data. In this section we compare the single crystal data provided by refs. [102,106] as given in Table 8.1. Based on these data we have first calculated reflectance spectra, which allow a plain comparison between the two different sets. These spectra are shown in Fig. 8.1. Obviously, there are strong differences between the different sets. This seems to be puzzling at first, since the achieved accuracies of the fits of the corresponding measured spectra in refs.


- 135 -

[102,106@ D U H RI V L PL O D U TX DO L W \ 1RW H W K D W .X] ¶ PHQ NR HW DO GHI L QH G W K Hcrystallographic aand b-axis as reference frame,102 while Homes et al. used the c-axis instead, since one of the surfaces of their crystal was parallel to this axis.106 As the angle between a- and c- axis amounts to 99.54°, the difference between the direction perpendicular to the a-b-plane and the c-axis is only 9.54°. This is the explanation of the comparably high similarity between both E Aa,b ±spectra shown in Fig. 8.1. Concerning the E  a- and the E  b-spectra there are strong deviations. Obviously, the deviations increase with increasing oscillator strength as a consequence of the additional dependence of the band position from oscillator strength and direction of the transition moment for mixed TO-LO modes. Table 8.1:

Ref.

Parameters of CuO from DA according to refs. [102,106]. All values with the -1 exception of the )i (°) and İ (Homes) + 9.5° f are given in cm . Note that I i = 90° + Ti .X] ¶ PHQ NR 4i due to different conventions with regard to the coordinate system fixed inside the crystal. The three modes with )i 4i 90qare Au -modes, the other three modes are of Bu -symmetry. Conventions

Homes et al.106

z c, y b

.X] ¶ PHQ NR 102 et al.

x a, y b

-1  Q i (cm )

166.5 323.7 450 147.6 516 566 160.5 321.5 408.7 144.9 469.6 522.8

Ji (cm-1 ) Si (cm-1 ) 2.6 3.3 40 1.8 19 16 1.5 4.3 44.6 1.3 10.6 7.0

151 107 1442 141 857 443 166.7 169.5 805.7 123.0 908.0 957.2

)i (°)

4i (°)

90 90 90 0 0 0 90 90 90 0 0 0

90 90 90 37 266 139 90 90 90 37.0 59.5 147.6

İ f

0 · §6.7 0 ¨ ¸ ¨0 6.7 0 ¸ ¨0 0 7.4 ¸ © ¹

§7.3 0 0.8 · ¨ ¸ ¨ 0 5.9 0 ¸ ¨0.8 0 6.8 ¸ © ¹

The comparably strong differences between the different data sets of optical constants for CuO motivated us to investigate if it is possible to use the spectra of polycrystalline samples to verify single crystal data. Such a method would be of special value if the data were determined from measurements on unfavorable crystal faces as in refs. [105,106] or from measurements of triclinic crystals (for triclinic crystals favorable crystal faces GRQ¶ W H[ist in general as mentioned earlier). Measurements of reflectance spectra of polycrystalline CuO were presented e.g. in refs. [106,108,109]. The spectra taken from refs. [106] and [108] are presented in Fig. 8.2. These

- 136 -


1.00

E b

0.75 0.50

reflectance (0°)

0.25 0.00 0.75

E Aa,b (a-c-plane, c 9.5°)

H [102] H [106]

0.50 0.25 0.00

E a

0.75 0.50 0.25 0.00 100

300

500

700

900

wavenumber / cm-1 Fig. 8.1:

Calculated reflectance spectra of single crystalline CuO based on the dielectric tensor functions from refs. [102,106] (angle of incidence D= 0°, linear polarized light, intrinsic coordinate system of the crystal: x  a, y  b, z Aa,b).

reflectance (7°, unpolarized)

0.6 Experiment [108] Experiment [106] 0.4

0.2

0.0 100 Fig. 8.2:

200


600

700

Comparison between the experimental reflectance spectra of polycrystalline CuO according to refs. [106] and [108].


- 137 -

spectra show major deviations with regard to band positions, band shapes and relative intensities above 350 cm-1 . In the range between 170 and 350 cm-1 both spectra are similar except for total intensity, which might be due to a difference with regard to surface quality. Below 170 cm-1 both spectra show two bands, which are recognizably shifted to higher wavenumbers in the spectrum from ref. [108]. As source of the difference we suspect the occurrence of the optical crystallite size effect. Unfortunately, the crystallite sizes of the samples were not determined, but, as will be shown later, this assumption is fully compatible with the results from our simulations.

0.9

H [102] H

d @


0.6 0.3 0.0

H H [106] > @

0.6 0.3 0.0 100

Fig. 8.3:

300


700

900

Simulated reflectance spectra of randomly oriented polycrystalline CuO with crystallite-diameters d O /10 0.4


0.50

0.2

0.25 0.00

Experiment [106] Simulation H[102] Simulation H[106] d O/10). Lower panel: ARIT (d Rsp , Rpp >> Rps (valid for most spectral ranges). 2. Rij MT , Rij MT , Rij 180qM , T 3.

if i

j.

. Ri MT , M |Ri 45q , T

The averaging under point three is carried out only with respect to M, Tremains fixed. As already pointed out, this relation is exact at D= 0. Since the differences between the different simulated Ri-spectra are much smaller than those between the simulations and the experimental spectra, it is obvious that one polarizer experiments are not very useful to unveil orientation in a quantitative manner, even for nonnormal incidence.


- 151 -

Experiment M= 0°/90°, T= 77° M= 22.5°/112.5°/ 202.5°/292.5°, T= 77° M= 45°/135°, T= 77° M= 0° - 180°, T= 77° 0.50

s-polarized

0.25

reflectance (20°)

0.00

Fig. 8.9:

p-polarized

0.25 0.00 sp-polarized 0.10 0.05 0.00 ps-polarized 0.10 0.05 0.00 100 300

500 700 900 wavenumber / cm-1

1100

1300

Simulated and measured reflectance spectra of an oriented Fresnoite glass ceramics (Ba2 TiSi2 O8 + 0.75 SiO2 , angle of incidence D= 20°, black lines: experimental spectra, blue lines: (M= 0°/90°, T q orange lines: (M= 45°/135°, T q ѽ U H G O L QH V M= 22.5°/112.5°/202.5°/292.5°, T q green lines: (M= 0° - 180°, T q ). 1. Panel: s-polarized incident radiation. 2. Panel: p-polarized incident radiation. 3. Panel: s-polarized incident radiation, analyzer parallel to the plane of incidence. 4. Panel: p-polarized incident radiation, analyzer perpendicular to the plane of incidence.

On the other hand, the cross-polarization terms Rsp and Rps depend notably stronger on the nature of the individual orientation distribution. The cross-polarization terms belonging to the continuous distribution of M and the average of Rij in case of the four orientations (M = 22.5°/112.5°/202.5°/292.5°, T = 77°) are an obvious exception, since their graphic representations also agree within line thickness. Though these two orientation distributions can be distinguished simply by an additional measurement after a rotation of the sample

- 152 -


around the Z-axis by an angle ) (cf. section 8.3.2), it can not be excluded that there exist other discrete distributions consisting of more than four orientations, which also give similar cross-polarization spectra and may be not distinguishable by this procedure. E.g., a discrete orientation distribution consisting of different orientations with T= 77° and equally spaced steps between M= 0° and M= 180°, cannot be distinguished if the steps are smaller than a certain limit depending on the anisotropy of the structural units.Therefore the necessary value for the angle ) may be too small to generate detectable spectral changes. This point, however, should not limit the practical applicability of the method severely, since such discrete orientation distributions are not commonly found in real samples. Overall, if we compare the simulated spectra based on the continuous distribution of M-values with the experimental spectra we find a good agreement referring to peak position, peak shape and relative intensities. The differences with regard to the absolute intensities can be understood in terms of the reasons given in the preliminary remarks above (influence of the Pt-wire and the surface cracks). An important point to note, however, is that the simulations and also, within experimental errors, the measured cross-polarization spectra Rsp and Rps are equal, which is a consequence of the rotational symmetry of the sample. According to the results of the preceding chapter it is therefore impossible to distinguish between the orientations T and ±T based on measurements in the centre of the disk. Additional measurements with the aperture displaced from the centre are therefore obligatory. These also allow ruling out the non-continuous orientation distributions. At its edge the sample usually possesses a higher surface quality compared to that in the neighborhood of its centre. Thus, the correspondence between simulated and experimental Rsand Rp -spectra is much better with respect to the absolute intensities as can be seen in Fig. 8.10. The high degree of alignment at the edge is obvious from the small range of M-values necessary to bring simulation and experiment in good accordance. Since we choose the plane of incidence to be oriented perpendicular to the radius of the disk and, therefore, nearly perpendicularly to the c-axes in the area probed by the light beam (cf. Fig. 8.10), only a comparably weak cross-polarization can be measured, which is already strongly influenced by spectral noise below 1000 cm-1 . Nevertheless, the difference between a single orientation (M = 89°, T= 74°), which is equivalent to the locally preferred axis, and the orientation distribution used for the simulations (M= 80° - 98°, T= 74°) is still detectable employing cross-polarization spectra.

reflectance (20°)


Experiment M= 89°, T= 74° M= 80° - 98°, T= 74° 0.75 s-polarized 0.50 0.25 0.00 p-polarized 0.50 0.25 0.00 sp-polarized 0.010 0.005 0.000 ps-polarized 0.010 0.005 0.000 100 300 500 700 900 -1 wavenumber / cm

- 153 -

1100

1300

Fig. 8.10: Simulated and measured reflectance spectra of an oriented Fresnoite ceramics (Ba2 TiSi2 O8 + 0.75 SiO2 , angle of incidence D= 20°, black lines: experimental spectra, red lines: M= 89°, T q green lines: M= 80 - 98°, T q ). 1. Panel: s-polarized incident radiation. 2. Panel: p-polarized incident radiation. 3. Panel: s-polarized incident radiation, analyzer parallel to the plane of incidence. 4. Panel: p-polarized incident radiation, analyzer perpendicular to the plane of incidence. Note that for the second simulation we had to choose an orientation distribution with an average Msmaller than 90° to account for the slight differences between Rps and Rsp . These slight differences allow us to conclude that Thas a positive sign in agreement with the pole figures presented in ref. [113]. As already pointed out at the end of the preceding section, it is of advantage to rotate the sample around the Z-axis of the reference frame to increase these differences (recall that the reference frame is not fixed within the sample. The Z-axis in

reflectance (20°)

- 154 -


0.6 0.4 0.2 0.0 0.4 0.2 0.0

Experiment M= 74°, T= 74° M= 65 - 83°, T= 74° s-polarized

p-polarized

sp-polarized 0.03 0.00 ps-polarized 0.03 0.00 400

700 1000 wavenumber / cm-1

1300

Fig. 8.11: Simulated and measured reflectance spectra of an oriented Fresnoite ceramics at the edge of the sample (Ba2 TiSi2 O8 + 0.75 SiO2 , angle of incidence D = 20°, continuous lines: experimental spectra, red lines: M= 74°, T q , green lines: M= 65 - 83°, T q ). Same position as in Fig. 8.10, but rotated by an angle ) = -15°. 1. Panel: s-polarized incident radiation. 2. Panel: p-polarized incident radiation. 3. Panel: s-polarized incident radiation, analyzer parallel to the plane of incidence. 4. Panel: p-polarized incident radiation, analyzer perpendicular to the plane of incidence. particular is represented by the normal of the sample surface through the centre of the aperture). This is illustrated in Fig. 8.11, where spectra recorded after a rotation of the sample by ) = -15° are depicted. While the Rs- and Rp -spectra have not changed significantly, the overall intensity of the cross-polarization terms has nearly quintupled and the differences between Rps and Rsp are now obvious. This is, however, on expense of the relative difference of the cross-polarization terms between one single orientation and an orientation distribution as already pointed out in the preceding section. Due to the sharpness of the orientation distribution at the edge of the disk it is no longer possible to distinguish it from a single orientation based on the comparison between measured and simulated spectra presented in


- 155 -

Fig. 8.11, since the differences between experiment and the individual simulations are larger than those between them. Note that the spectra presented in Fig. 8.9 and Fig. 8.10 prove the existence of non- zero cross-polarization even for light polarized perpendicularly to (in case of the spectra presented in Fig. 8.10 locally) preferred axes despite being (locally) uniaxially oriented. A basic assumption underlying common treatments of the optical properties of uniaxial samples like LDT, which consist in the assumption that uniaxially oriented materials do not depolarize light if the polarization vector is oriented parallel or perpendicular to the high symmetry axis, is therefore also experimentally found to be not correct in general. 8.3.4

Concluding remarks

We have demonstrated that it is possible to distinguish different kinds of orientation distributions by IR-reflectance spectroscopy if the oriented regions or the structural units are large compared to the resolution limit of light. This possibility is based on the dependence of the peak positions on the relative orientations of the transition moments and the direction of propagation of the light waves. The peak shifts are functions of the oscillator strengths and therefore of the TO-LO splitting, respectively. As a consequence for weak oscillators as usually present in organic materials, it may be sufficient in a first approximation to consider only the relative orientation of the transition moments and the polarization directions, with the important exception of internal reflection experiments (ATR, cf. section 7.4).M13 On the other hand, our approach is not limited to the infrared spectral range and to the presence of absorption bands. Hence it is valid, e.g., in the UV/Vis-spectral range as well. In this range, the condition of structural elements being large compared to the resolution limit is usually quite easy to fulfill. We have limited our simulations and experiments to materials with optically uniaxial regions of order. However, the theoretical formalism used in this chapter is also readily applicable to lower symmetries of the structural elements. The most important finding is probably the fact that only in perfectly single crystal- like oriented samples the preferred axis is a real optical axis. In all other cases this is axis nothing more than a main axis, since depolarization occurs even if the incoming light is polarized along or perpendicular the main axis. As a further proof of this result, Fig. 8.12 shows UV/Vis-cross-polarization spectra of a uniaxially stretched poly(ethylene terephthalate)-film with a draw ratio of 2, aligned in such a way that the polarization vector was oriented either parallel or perpendicular to the draw direction. Obviously, an orientation with a completely vanishing depolarization GRHV Q ¶ W H[L V W

- 156 -


(interestingly, the cross-polarization is not symmetric with respect to the preferred axis. It becomes symmetric, however, after increasing the draw ratio). Also it is important to point out that the cross-polarization does not decrease with the wavelength following, e.g., a 1/O4 rule. Therefore scattering cannot be the only (and also not the main!) reason for the existence of depolarization.

Transmittance (crossed polars)

0.4 parallel +1° -1° 0.2

0.0 empty parallel perpendicular 0.2

0.0 400

500

600

700

800

wavelength / nm Fig. 8.12: Transmittance cross-polarization spectra of a uniaxially stretched PET film (draw ratio = 2). The spectrum without sample (empty) is shown for comparison. Lower panel: the draw direction was oriented either parallel or perpendicular to the polarization vector. Upper panel: spectra with draw direction oriented parallel to the polarization vector and after a slight rotation of 1° degree (rotation axis normal to the surface).

9. Consequences for the interpretation of spectra of polydomain materials

- 157 -

9. Consequences for the interpretation of spectra of polydomain materials Formel-Kapitel (nächstes) Abschnitt 1 The goal of optical and IR-optical spectroscopy applied to solids is usually to obtain information about their near- and middle-range structure. It is well-known that it is preferable in case of crystalline compounds to use well-ordered single crystals to that end whenever one is available. Unfortunately, this is not always the case. As a consequence, polycrystalline samples have to be used quite often instead. In these samples, the crystallites are typically randomly oriented. As a result, usually no knowledge about the directions of the transition moments is available a priori. This is a serious problem, especially if the structural units are of uniaxial crystal symmetry (tetragonal, trigonal and hexagonal), since then the contributions of doubly degenerated modes (e.g., the E- modes in case of Fresnoite) are different from those of modes polarized along the main axis. It is therefore impossible to determine to which of the principal dielectric functions a transition belongs without additional knowledge. Of course, this difficulty also exists for materials of lower symmetry of the domains (orthogonal, monoclinic and triclinic), but with the important difference that in these cases every mode contributes with the same statistical weight. Despite these obstacles, quite often DA is applied to the spectra of polycrystalline materials (or glasses) with the seeming advantage of not only obtaining the dielectric function, but also the oscillator parameters. These parameters are believed to be directly related to the structure of the unit cells. This is definitely correct if spectra of single crystals are investigated in a proper way, but usually not for polydomain materials. One reason for the failure of DA with regard to the spectra of randomly oriented polycrystalline materials has been repeatedly stated in this work: The assumption of the equivalence of optical isotropy and the possibility to characterize a material with a scalar dielectric function. Due to the existence of the crystallite size effect and the need to average over the reflection or transmission in dependence of the orientation if the domains cannot be considered as optically small, it is commonly impossible to use the concept of an averaged dielectric function or index of refraction over the whole spectral range from the UV down to the FIR. This finding does not only spoil the application of DA to polycrystalline materials in many cases, but also generally that of KKA, since, being strict, the latter has to be carried out over the whole spectral range. Consequently, the domains would have to be optically small independent of frequency, which is obviously never the case.

- 158 -


To demonstrate the failure of DA in case of randomly oriented materials with large domains we have carried out such an analysis for the sample Ba0K-100G applying the classical damped harmonic oscillator model,  H Q

Si 2 Hf ¦ 2 2    Q ) i Q J i 1 (Q i i N

(9.1)

where the angle brackets indicate the orientational average. The spectrum calculated on the basis of the parameters evaluated by DA is depicted together with the measured spectra in the range 800 ±1100 cm-1 in Fig. 9.1 (note that DA usually underlies a best- fit procedure in contrast to the forward calculations based on ARTT and ARIT/EMA shown in the preceding chapters!).

0.6

Experiment Simulation


0.4

0.2

0.0

Experiment Simulation

0.4

0.2

0.0 800

900

1000

1100

-1


Comparison of the experimental and simulated reflectance spectra of polycrystalline Fresnoite. The simulated spectra are based on a best-fit procedure applying the Fresnel equations for the reflectance (assumption: cubic crystal symmetry) in combination with eqn. (9.1).

From the comparison it is obvious that the band shapes of the experimental spectra cannot be resembled well by eqn. (9.1) in combination with the Fresnel equations2 as could be expected. Since the bands of the polycrystalline material are blue-shifted with the exception of the band at 860 cm-1 , where single crystalline Fresnoite is nearly isotropic (i.e., H H a § c), the application

9. Consequences for the interpretation of spectra of polydomain materials Table 9.1:

- 159 -

Comparison of single crystal data with data obtained from applying DA to the spectra of the polycrystalline samples Ba0k and Ba0K0-100G according to eqn. (9.1). All values in cm-1 . Single crystal

Ba0k

Ba0K0-100G

Orientation of the transition moment M M ll a/c

 Q i

Si

J i

 Q i

Si

J i

 Q i

Si

860.4/857.9

378.4/323.8

11.4/11.5

859.9

333.7

13.388

860.5

325.1

10.6

M ll a

902.1

554.3

10.4

902.9

366.5

15.9

911.2

428.0

13.3

M ll c

952.8

297.6

12.0

925.8

327.7

43.3

970.5

281.2

19.3

M ll c

1024.6

503.9

9.5

1026.9 266.1

49.4

1054.2

206.2

33.7

J i

 of DA results in oscillator positions Q , which are higher than those in the single crystal. This i

blue-shift depends on the oscillator strength and the symmetry of the mode (see Table 9.1). Besides the oscillator positions also the damping constants and the oscillator strengths are in error, the errors being again larger the higher the oscillator strengths are in the corresponding single crystal. Even if a polydomain material can indeed be characterized by a scalar dielectric function  , it is nevertheless not possible to model this dielectric function with eqn. (9.1) or other H Q

oscillator models (cf. Appendix, section A.9), because of the underlying assumption of a linear combination of the principal dielectric functions. Keeping in mind that the crystallites themselves are optically anisotropic and that the oscillators can be distinguished according to the orientation of their transition moments relative to the crystal¶ Vaxes, eqn. (9.1) can be split according to:

 H Q

Na

Nb Nc S ia 2 S ib 2 S ic 2 ¦ ¦ 2 2 2 2 2 2          (Q Q ) i Q J Q ) i Q J Q ) i Q J i 1 (Q i 1 (Q ia ia ib ib ic ic

Hf ¦ i 1

Ha Hb Hc

.(9.2)

3 3Sij2 Hj Hf, j ¦ 2 2 ; j a, b, c; N a N b N c  ) i Q  Q J i 1 (Q ij ij Nj

For uniaxial materials ( Ha Hb zHc ), it follows that

N

- 160 -


 H Q

Na

Nc S ia 2 S ic 2 Hf ¦ 2 2 ¦ 2 2  ) i Q     Q J Q ) i Q J i 1 (Q i 1 (Q ia ia ic ic

Ha Hc

.

2 Nj

2 Sij2 ; j a, c; N a N c 2 2    (Q Q ) i Q J ij ij

Hj Hf, j ¦ i 1

(9.3)

N

Therefore, DA applied to polycrystalline samples with non-cubic crystal structure is based on the assumption of an arithmetic average of the principal dielectric functions. Moreover, if the symmetry of the crystal structure is uniaxial, equal weighing of Ha and Hc in eqn. (9.1) leads to additional errors. These errors are in the main: -

Overestimation of damping constants and

-

incorrect determination of particular oscillator frequencies.

The overestimation of damping constants is linked directly to the overestimation of reflectance

due

to

the

use

of eqs.

(5.33)

( H 1 3 ) and Ha Hb Hc

(5.34)

( H 2 3 Ha 1 3 Hc ), since the first overestimation compensates for the second in the fitting procedure of DA. In addition, errors are introduced with regard to the oscillator strengths.

0.8


0.7

Experiment ¢ H ² H + H a c

0.6 0.5 0.4 0.3 0.2 0.1 0.0 800

Fig. 9.2:


1100

Comparison of the experimental and simulated (linear combination of the principal dielectric functions) reflectance spectra of polycrystalline Fresnoite consisting of small crystallites.


- 161 -

The incorrect evaluation of particular oscillator frequencies (e.g. Fresnoite: E ll c mode located at 953 cm-1 , Table 9.1) is correlated with the fact that the existence of an oscillator is not necessarily connected with the existence of a local maximum near its resonance frequency in the spectrum of a polycrystalline sample (cf. Fig. 9.2). Besides, due to orientational averaging the band shapes are different from those of damped harmonic oscillators (see section 6.3). As a consequence, DA of polycrystalline samples can only provide relevant results concerning oscillator parameters, if the differences between the principal dielectric constants are negligibly small.

- 162 -

10. Summary

10.Summary In this work the optical properties of polydomain media were investigated to gain a deeper understanding how the spectra of such materials are influenced by their domain structure and to assess how these spectra can be employed in order to obtain information about the structure of these materials. It is known from the literature, but not yet common knowledge, that the scale of heterogeneity of a polydomain medium in comparison with the wavelength is the key feature with regard to its optical properties at this wavelength provided that the domains are optically anisotropic. If the domain size is larger than about one tenth of the wavelength, then light is able to resolve the anisotropic nature of the domains. This assumption was proved in this work by IRmicroscopy on polycrystalline Fresnoite: The spectra showed a clear dependence on the position of the light beam for a sample having crystallites with an average crystallite size of about 10 Pm. In contrast, a second sample, with an average crystallite size of about 300 nm, showed no dependence of the spectra from the position of the beam. The macroscopic reflectance from the former sample can be simulated according to 'RO O ¶ V PRGHOby averaging the reflectance for all possible orientations. Before applying this model to polycrystalline Fresnoite, it was generalized within this work to arbitrary crystal symmetry and angle of incidence by using an exact 4×4 matrix formalism originally developed by Berreman and in a different but equivalent form by Yeh. 7KL V ZRU N¶ Vcontribution to the 4×4 matrix formalism consists mainly in providing solutions for the Eigenvalues of the wavevectors and the Eigenpolarizations of the waves in general bianisotropic media making the formalism fully analytical. Besides, the occurrence of singularities in the formalism was investigated and proper treatments for their avoidance were suggested. Additionally, the formalism was generalized in a way that both formalisms, that of Berreman and that of Yeh, can be arbitrarily exchanged for materials with dielectric anisotropy only. Besides the employment of an exact 4×4 matrix formalism, it was also necessary for the J HQ HU DO L ] D W L RQ RI 'RO O ¶ V D SSU RD FKto develop a new kind of orientation representation called Symmetric Euler Orientation Representation. This necessity arose from the need to simulate spectra of materials with domains having orthorhombic or lower symmetry, since in this case the application of conventional Euler orientation representations suffers from problems connected with the locations of the rotation axes, which are used in these representations. As a consequence of these problems, the results of the orientational averaging procedures (including that for optically small crystallites) depend on the initial position of the intrinsic

10. Summary

- 163 -

coordinate system with respect to the reference frame. At least for those Euler orientation representations, which are based on the use of two rotation axes, it is possible to remove this dependence partly if the orientation dependent property to be averaged is weighted with the factor sin Tand if an infinite number of orientations is employed. The latter is usually impossible. The model of Doll et al. was further extended to allow the treatment of reflectance from and that of transmittance (Average Reflectance and Transmittance Theory, ARTT) through samples consisting of both small and large domains compared to the wavelength (Unified Average Optical Properties Theory, UAOPT). The correctness of these generalizations were proved by the comparison of simulated spectra based on single crystal data and experimental spectra of randomly oriented, polycrystalline Fresnoite, which resulted in all cases in an excellent agreement between both kinds of spectra. By ARTT, in combination with the application of the 4×4 matrix formalism, it was also possible to show theoretically that large domain materials should always possess non- zero cross-polarization terms, even for random orientation, as long as the individual domains are optically anisotropic. Therefore, the reflectance or transmittance of light from or through such kind of materials always leads to a depolarization of this light. This statement is, e.g., also correct for uniaxial oriented materials even if the incident light is polarized parallel or perpendicular to the main axis except for perfect orientation. It is therefore impossible and highly erroneous to attribute scalar dielectric functions to randomly oriented materials with large domains or to describe oriented materials with a dielectric tensor function unless it is perfectly oriented or consists of small domains. This automatically excludes the possibility of a useful analysis of the spectra by conventional DA or KKA. The correctness of these theoretical findings was proved experimentally by the investigation of the cross-polarization spectra of Fresnoite ceramics with large crystallites. In all cases, ARTT was able to quantitatively predict the spectra. The experiments on oriented Fresnoite ceramics also showed that the occurrence of non- zero cross-polarization terms enable the determination of orientation in oriented materials to a much higher degree as anticipated, e.g., by the well-known and generally accepted LDT. The strangeness of or the missing familiarity with the optics of materials consisting of large and randomly oriented domains justified an extended investigation of their optical properties, especially with regard to the dependence of the reflectance from the angle of incidence. Important findings of this study are, e.g., W K DW %U HZV W H U ¶ V DQJO H GRH V Q RW H[L V W I RU W KL V NL Q G RI material and that the reflectance of parallel polarized incident light can exceed that of

- 164 -

10. Summary

perpendicular polarized light at certain angle of incidence ranges despite random orientation (the correctness of the latter finding was proved experimentally within this work). Besides, it was shown theoretically that total reflection is suppressed if the incidence medium has a refractive index, which lies between the principal indices of refraction of the domains, even if the averaged index of refraction of the domains is much smaller than that of the incidence medium. 7K H F RQY HQ W L RQ DO L GH D RI RSW L F DO L V RW U RS\ L Q W K H V HQ V H RI ³ K DYL QJ D V F DO DU GL HO H FW U L F I XQ F W L RQ´ as imparted by the textbooks can be applied only if the anisotropic nature of the domains cannot be resolved by the probing light. As already stated, this is usually the case if the domains are small compared to its wavelength. It is then common to apply effective medium theories to calculate the scalar dielectric function by an appropriate averaging of the principal dielectric functions taking into account possible interactions between the crystallites. Since such functions do exist only for orthorhombic or higher crystal symmetry, a formalism was developed in this work, which uses the Eigenvalues of the complex dielectric tensor function instead. Despite the fact that the Eigenvectors are different for the real- and the imaginary part in general, it was shown, using monoclinic CuO as an example, that both parts still obey the Kramers-Kronig relations. The results of EMA are very similar to that of a theory with completely different origin and basic assumptions, which averages the indices of refraction of the domains in dependence of their orientation in a way similar to the reflectances or transmittances in case of ARTT. This theory, called Average Refractive Index Theory (ARIT), does not restrict itself to the use of principal values but also adds indices of refractions for non-principal orientations to the average. Therefore, ARIT can readily explain the presumed existence of mixed TO-LO modes in polycrystalline materials as well as in glasses. A further contrast to EMA is that ARIT neglects any possible occurring interactions between the crystallites in accordance with ARTT. An application to polycrystalline Fresnoite with small crystallites in the infrared range shows that ARIT performs also well in the vicinity and at the Eigenfrequencies of oscillators where EMA usually shows weaknesses. The remaining differences between simulated and measured spectra are fairly small and can be eliminated by increasing all damping constants by a common factor. In case of polycrystalline Fresnoite with an average crystallite size of 300 nm this common factor was found to be about 1.3. If this factor is increased further, spectra result, which show strong similarities to those provided by amorphous or glassy materials. Since in such materials also some kind of domains exist, albeit of much smaller size compared to those in polycrystalline materials and without clear borders, it was concluded that orientational averages also plays an outstanding

10. Summary

- 165 -

role in explaining their spectra. The comparison between in such a manner simulated and measured spectra of Fresnoite, Sr- and Ge-Fresnoite glasses shows a less satisfying agreement compared to that for polycrystalline materials. This is probably caused by the existence of additional structural units of the atoms like, e.g., four-, five- and sixfold coordinated titanium in these glasses, while titanium exists nearly exclusively in fivefold coordination in the crystalline species. In case of Enstatite glass (Mg2 Si2 O6 ), where less structural moieties can be assumed to occur, the comparison between simulated and measured spectrum is much more encouraging. As already stated, conventional DA must not be applied in case of materials with large domains. However, even if small domain materials can be characterized by an averaged dielectric function or index of refraction function, it is nevertheless impossible to apply conventional DA to derive meaningful structural information by the determination of oscillator positions etc. This is a consequence of the assumption of a linear combination of the principal dielectric functions, which underlies the use of conventional DA when it is applied to the spectra of polydomain systems. Such linear combinations, however, are proved to be incorrect easily by comparing simulated and experimental spectra as demonstrated in this work. Besides, neither EMA nor ARIT support such an assumption. As a résumé it can be stated that the optics of polydomain materials and, as a consequence, the interpretation of their spectra is much more complex than often assumed. Therefore, the spectroscopy of single crystals seems to be inevitable if the primary aim is to obtain information about the molecular structure or the structure on the level of the unit cell. Nevertheless, the spectra of polycrystalline materials can be important sources of independent information, e.g. to prove that data obtained from single crystals are correct as demonstrated for CuO in this work.

- 166 -

10. Zusammenfassung

Zusammenfassung Im Rahmen dieser Arbeit wurden die optischen Eigenschaften von polydomänen Materialien untersucht

um einerseits ein

tieferes

Verständnis bezüglich des Einflusses

der

Domänenstruktur auf ihre Spektren zu erlangen und um andererseits beurteilen zu können, inwieweit sich aus diesen Spektren Informationen über die Struktur dieser Materialien gewinnen lassen. Aus der Literatur ist zwar prinzipiell bekannt, dass die Größe der geordneten Bereiche eines polydomänen Materials einen entscheidenden Einfluss auf seine optischen Eigenschaften besitzt, wenn die Domänen optisch anisotrop sind, das Wissen um diesen Größeneffekt ist jedoch nicht sehr verbreitet. Die kritische Domänengröße liegt etwa bei einem Zehntel der Wellenlänge. Sind die Domänen größer, kann Licht die anisotrope Natur der Domänen auflösen, wie Messungen mit Hilfe der IR-Mikrospektroskopie an polykristallinem Fresnoit aufzeigten: Für Proben mit zufällig orientierten, großen Kristalliten (Kristallitdurchmesser etwa 10 Pm) besaßen die Spektren eine deutliche Abhängigkeit von der Position des Lichtstrahls auf der Probe. Im Gegensatz dazu zeigten Spektren einer zweiten Probe mit zufällig orientierten, kleinen Kristalliten (Kristallitdurchmesser etwa 300 nm) keinerlei Ortsabhängigkeit. Die makroskopische Reflektanz der ersten Probe kann unter Einsatz von Dolls Modell erfolgreich modelliert werden, indem man die Reflexion über alle möglichen Orientierungen mittelt. Bevor dieses Modell auf polykristallinen Fresnoit angewendet wurde, wurde es im Rahmen dieser Arbeit auf beliebige Kristallsymmetrie und Einfallswinkel erweitert. Dies geschah durch die Anwendung eines exakten 4×4 Matrix Formalismus, welcher im Original von Berreman und, in abgewandelter, aber äquivalenter Form von Yeh entwickelt wurde. Für die ursprünglichen Formalismen wurden in dieser Arbeit explizite Lösungen für die Eigenwerte der Wellenzahlvektoren und die Eigenpolarisationen der Wellen in generellen bianisotropen Medien abgeleitet, weshalb prinzipiell auf numerische Lösungen verzichtet werden kann. Weiterhin wurde das Auftreten von Singularitäten in den Formalismen untersucht und geeignete Abwandlungen zu ihrer Vermeidung erarbeitet. Zudem wurde ein einheitlicher Formalismus entwickelt, der es im Falle des ausschließlichen Vorliegens von dielektrischer Anisotropie erlaubt, in beliebiger Weise die Formalismen von Berreman und Yeh gegeneinander auszutauschen. Neben der Anwendung eines exakten 4×4 Matrix Formalismus war es zur Verallgemeinerung von Dolls Modell weiterhin notwendig eine neue Form von Orientierungsrepräsentation namens Ä Symmetric Euler orientation representation³zu entwickeln. Diese Notwendigkeit

10. Zusammenfassung

- 167 -

basiert auf der Erfordernis auch die Spektren von Materialien mit orthorhombischen oder noch niedriger symmetrischen Domänen modellieren zu können. In diesen Fällen verhindert die in den konventionellen Euler Orientierungsrepräsentationen ungünstige Anordnung der Drehachsen den sinnvollen Einsatz dieser Repräsentationen. Als Folge der ungünstigen Anordnung sind die Resultate der Orientierungsmittelungen von der Definition der intrinsischen Koordinatensysteme,

bzw.

von

ihrer anfänglichen Orientierung

zum

Laborkoordinatensystem abhängig. Diese Abhängigkeit kann zumindest für die auf zwei Drehachsen beruhenden Eulerschen Orientierungsrepräsentationen teilweise beseitigt werden, indem die von der Orientierung abhängige Eigenschaft in der Mittelung mit dem Faktor sin T gewichtet und eine unendliche Anzahl von Orientierung berücksichtigt wird. Letzteres ist im Allgemeinen nicht möglich. Das von Doll et al. stammende Modell wurde weiterhin so verallgemeinert, dass die Berechnung der Reflexion und Transmission (Average Reflectance and Transmittance Theory, ARTT) im Fall solcher Proben möglich wurde, welche sowohl aus großen als auch aus kleinen Domänen bestehen (Unified Average Optical Properties Theory, UAOPT). Die Korrektheit dieser Generalisierungen wurde mit Hilfe des Vergleichs von auf der Basis von Einkristalldaten simulierten und gemessenen Spektren entsprechender, polykristalliner Fresnoit-Proben überprüft. Mit Hilfe der ARTT, in Kombination mit der Anwendung eines 4×4 Matrix Formalismus, konnte weiterhin gezeigt werden, dass Materialien mit großen, optisch anisotropen Domänen grundsätzlich von Null verschiedene Kreuzpolarisationsterme aufweisen sollten. Dies gilt insbesondere auch dann, wenn die Domänen zufällig orientiert sind. Deshalb führen Reflexion oder Transmission im Fall solcher Medien grundsätzlich zur Depolarisation von Licht. Diese Aussage bleibt auch dann korrekt, wenn z.B. Materialien mit uniaxialen Domänen eine makroskopische Vorzugsrichtung besitzen und das einfallende Licht parallel oder senkrecht zu dieser Vorzugsrichtung polarisiert ist, solange die Ausrichtung der Domänen nicht perfekt ist. Aus diesem Grund ist es sowohl unmöglich als auch falsch, zufällig orientierten Materialien mit großen Domänen einen skalaren dielektrischen Tensor zuzuschreiben oder orientierte Materialien mit einem dielektrischen Tensor zu charakterisieren, solange diese nicht perfekt orientiert oder klein sind. Dies schließt automatisch auch die Möglichkeit aus, die Spektren solcher Materialien mit konventioneller DA oder KKA sinnvoll zu analysieren. Die Korrektheit dieser theoretisch abgeleiteten Resultate wurde experimentell durch die Aufnahme von Kreuzpolarisationsspektren verschiedener Fresnoit-Keramiken mit großen Kristalliten überprüft. In allen Fällen konnten die Spektren quantitativ mit ARTT simuliert werden.

- 168 -

10. Zusammenfassung

Experimente, die an orientierten Fresnoit-Keramiken durchgeführt wurden, zeigten außerdem, dass das Auftreten von von Null verschiedenen Kreuzpolarisationstermen die Bestimmung der Orientierung von orientierten Materialien zu einem weitaus höheren Grad zulässt, als sich das auf der Basis der wohlbekannten und gemeinhin akzeptierten LDT ableiten ließ. Die scheinbare Seltsamkeit der bzw. die fehlende Vertrautheit mit den optischen Eigenschaften von Materialien mit großen und zufällig orientierten Domänen rechtfertigte eine ausgedehntere Untersuchung dieser Eigenschaften, speziell im Hinblick auf die Abhängigkeit der Reflexion vom Einfallswinkel. Wichtige Erkenntnisse dieser Studien sind, dass ein Brewster Winkel für diese Materialien grundsätzlich nicht existiert und dass die Reflexion von parallel zur Einfallsebene polarisierten Lichts diejenige von senkrecht polarisierten Licht in gewissen Bereichen des Einfallswinkels trotz zufälliger Orientierung der Domänen übersteigen kann (die Korrektheit letzterer Erkenntnis wurde im Rahmen dieser Arbeit experimentell nachgewiesen). Des Weiteren wurde theoretisch gezeigt, dass Totalreflexion dann nicht auftritt, wenn das Einfallsmedium einen Brechungsindex aufweist, welcher zwischen den prinzipiellen Brechungsindizes der Domänen liegt, selbst wenn der gemittelte Brechungsindex der Domänen kleiner ist als der des Einfallsmediums. *U XQ GV lW ] O L FK L V W O DXW GHQ /HKU E FK H U Q GHU 2SW L NGH U %HJU L I I Ä RSW L V FK H , V RW U RSL H³ gleichbedeutend mit der Möglichkeit ein Material mit einer skalaren dielektrischen Funktion zu charakterisieren. Wie bereits erwähnt, ist diese Aussage bei polydomänen Materialien nur dann richtig, wenn die anisotrope Natur der Domänen nicht vom einfallenden Licht aufgelöst werden kann, was grundsätzlich für Domänen gelten sollte, die klein gegenüber der Wellenlänge sind. In diesem Fall werden im Allgemeinen Effektiv-Medien Theorien eingesetzt um die skalare dielektrische Funktion mit Hilfe einer geeigneten Mittelung unter Berücksichtigung von möglichen Wechselwirkungen zwischen den Kristalliten aus den prinzipiellen dielektrischen Funktionen

zu berechnen.

Da diese jedoch

nur

für

orthorhombische oder höhere Kristallsymmetrien existieren, wurde in dieser Arbeit ein Formalismus entwickelt, der an Stelle der prinzipiellen dielektrischen Funktionen die Eigenwerte der komplexen dielektrischen Tensorfunktion benutzt. Trotz der Tatsache, dass die Eigenvektoren für die Real- und Imaginärteile generell verschieden sind, wurde am Beispiel des monoklinen CuO gezeigt, dass auch diese Funktionen den Kramers-KronigRelationen gehorchen. Die auf Basis der EMA modellierten Spektren sind denen einer im Rahmen dieser Arbeit entwickelten Theorie mit komplett unterschiedlichem theoretischen Ursprung und unterschiedlichen Voraussetzungen sehr ähnlich. Diese Theorie namens Ä Average Refractive Index Theory³(ARIT) mittelt über die Brechungsindices der Domänen

10. Zusammenfassung

- 169 -

in Abhängigkeit ihrer Orientierung in Analogie zu den Reflektanzen und Transmittanzen im Falle der ARTT. Im Rahmen der ARIT werden nicht nur die Brechungsindizes für prinzipielle Orientierungen

in

die

Mittelung

miteinbezogen,

sondern

auch

nicht-prinzipielle

Orientierungen berücksichtigt. Aus diesem Grund kann die ARIT zwanglos die bislang nur angenommene, aber nicht bewiesene Existenz von gemischten TO-LO Moden in polykristallinen Materialien wie auch in Gläsern belegen. Ein weiterer Unterschied zur EMA ist die Vernachlässigung jeglicher Wechselwirkung zwischen den Kristalliten in Analogie zur ARTT. Die Anwendung der ARIT auf polykristallinen Fresnoit mit kleinen Kristalliten im infraroten Spektralbereich zeigt, dass die ARIT auch in Bereichen, wo die EMA generell Schwächen zeigt, nämlich in der Nähe der Eigenfrequenzen der Oszillatoren, zu sehr guten Resultaten führt. Die verbleibenden Unterschiede zwischen simulierten und experimentellen Spektren sind

vergleichsweise gering und

lassen sich durch Multiplikation aller

Dämpfungskonstanten mit einem gemeinsamen Faktor größer eins eliminieren. Konkret ergibt sich für diesen Faktor ein Wert von ungefähr 1.3 für die Probe polykristallinen Fresnoits mit einem mittleren Kristallitdurchmesser von 300 nm. Wird dieser Faktor in den Modellierungen weiter vergrößert, so resultieren schließlich Spektren, die starke Ähnlichkeiten mit denen von amorphen oder glasigen Materialien aufweisen. Da auch in solchen Materialien Domänen existieren, wenn auch mit im Vergleich zu polykristallinen Materialien wesentlich kleineren Durchmessern und ohne klare Begrenzungen, kann daraus geschlossen werden, dass die Orientierungsmittelung auch eine wichtige Rolle in der Entstehung der Spektren von Gläsern und amorphen Materialien spielt. Der Vergleich zwischen auf diese Weise simulierten und experimentellen Spektren von Fresnoit, Sr-Fresnoit und Ge-Fresnoit Gläsern zeigt eine im Vergleich zu polykristallinen Materialien

weniger

zufrieden stellende

Übereinstimmung.

Ursächlich

hierfür

ist

wahrscheinlich die Existenz von zusätzlichen strukturellen Einheiten wie z.B. vier-, fünf- und sechsfach koordiniertes Titan in den Fresnoit-Gläsern, wohingegen Titan in den kristallinen Fresnoit-Spezies praktisch ausschließlich fünffach koordiniert auftritt. Im Fall von EnstatitGlas (Mg2 Si2 O6 ), in dem angenommen werden kann, dass weniger verschiedene strukturelle Einheiten auftreten, ist der Vergleich zwischen den auf Einkristalldaten basierenden Vorwärtsrechnungen und den experimentellen Spektren deutlich ermutigender. Wie bereits erwähnt, darf die konventionelle DA nicht für Materialien mit großen Domänen angewendet werden. Weiterhin lässt sich aber auch zeigen, dass selbst wenn ein Material mit einer

gemittelten

dielektrischen

Funktion

oder

einer

Funktion

des

gemittelten

Brechungsindexes charakterisiert werden kann, die Interpretation der Ergebnisse der DA

- 170 -

10. Zusammenfassung

hinsichtlich Oszillatorposition oder ±Stärke nicht ohne Weiteres sinnvoll ist. Dies folgt aus der Annahme einer linearen Kombination der prinzipiellen dielektrischen Funktionen die der konventionellen DA zugrunde liegt. Es ließ sich jedoch auf einfache Weise über den Vergleich von simulierten und experimentellen Spektren ableiten, dass solche linearen Kombinationen fundamental inkorrekt sind. Zusätzlich ist anzumerken, dass die Annahme einer linearen Kombination der prinzipiellen dielektrischen Funktionen weder durch die EMA noch durch die ARIT gestützt wird. Als Résumé lässt sich feststellen, dass die optischen Eigenschaften polydomäner Materialien generell vielschichtiger sind, und folglich die Interpretation ihrer Spektren wesentlich komplexer ist als dies generell unterstellt wird. Zur Spektroskopie von Einkristallen scheint es damit keine Alternative zu geben, falls Informationen über die molekulare Struktur oder die Struktur der Einheitszelle gewonnen werden sollen. Nichtsdestotrotz können Spektren polykristalliner Materialien eine wichtige Quelle unabhängiger Informationen darstellen, z.B. um die Qualität von Einkristalldaten zu überprüfen, wie das am Beispiel des CuO in dieser Arbeit demonstriert wurde.

Appendix

- 171 -

Appendix Formel-Kapitel 1 Abschnitt 1 A.1 Explicit forms of the Eigenvectors < and p. A.1.1 Explicit form of < Explicit form of eqn. (2.20).

Q i i

for which the reflectance has been determined experimentally.

Consequently, the number of terms P of the second sum depends on the spectral resolution. The result of a calculation according to eqn. (A.8.5) is not distinguishable from that of eqn. (A.8.4), as long as all damping constants J and all standard deviations Vi are greater than i unity. In practice, this is not a harsh restriction, since for amorphous solids the J and the Vi i are expected to be large. On the other hand, the use of eqn. (A.8.5) reduces the computational time effort to about 1/40 compared to that necessary for the use of eqn. (A.8.4) in mathematicaT M programs.126

- 192 -

Appendix

0.45

meas. cm-fit


0.30 0.15 0.00 meas. Lm-fit

0.30 0.15 0.00 100

Fig. A.8.1:

300

500 700 900 -1 wavenumber /cm

1100

1300

Upper part: experimental IRR-spectrum of Fresnoite glass and best theoretical fit (10 oscillators) according to eqs. (A.8.3) and (A.8.5) (convolution model). Lower part: experimental IRR-spectrum of Fresnoite glass and best theoretical fit (10 oscillators) according to eqn. (A.8.1) (Lorentz model). The oscillator positions provided by both fits are indicated by dashed vertical lines.

It must be emphasized that eqs. (A.8.1) - (A.8.5) are based on the inherent idea of a dispersion relation suitable for a material consisting of ordered parts of cubic symmetry, where H n 2 holds. Therefore, mixed TO-LO modes, excited by plane waves, should be forbidden in case of glasses, as for cubic crystals. Eqs. (A.8.1) and (A.8.3) were used in order to model the measured spectrum of Fresnoite glass by DA based on a least square fit of the reflectance. The results of both fits are compared to the measured spectrum in Fig. A.8.1. The quality of the fits is nearly equivalent. An obvious exception is the range from 1080 ±1180 cm-1 , where the modeling according to the convolution model (eqn. (A.8.3)) is clearly superior compared to that provided by the damped harmonic oscillator model (eqn. (A.8.1)). This illustrates that the convolution model allows extracting the optical constants of glassy or amorphous materials in general with higher accuracy.77

Appendix

- 193 -

However, there remains the question, if the oscillator parameters provided by the fit according to the convolution model are indeed more realistic compared to those resulting from other models. To examine this point, all parameters were allowed to vary freely in the least square fit of the IRR-spectra of Fresnoite glass using the convolution model. The results are shown in Table A.8.1 (set 1). Apparently, the values of the standard deviations Vi are on average higher for modes with higher wavenumber. XXV The standard deviation takes on appreciable values for some modes on expense of the damping constants, which are in these cases unrealistically small. Table A.8.1:

Two sets of oscillator-parameters (in cm-1 ) for glassy Ba2 TiSiO8 with equivalent standard deviations of the least squares fit using the convolution model (eqs. (A.8.3) and (A.8.5)).

Set 1 i

1

2

3

4

5

6

7

8

9

10

 Q i

126.6

146.5

368.0

474.4

558.9

677.7

853.3

856.4

905.1

978.1

Si

127.1

286.3

520.7

220.6

242.0

218.5

370.18

261.9

168.5

460.6

J i

23.2

89.8

163.6

16.3

116.5

0.03

0.0001

0.01

0.1

7.5

Vi

2.3

0.3

36.8

29.8

6.5

69.0

95.2

34.1

21.8

57.7

Hf 3.16 Set 2 i

1

2

3

4

5

6

7

8

9

10

 Q i

126.7

146.8

367.3

474.3

562.8

727.0

844.2

860.8

905.6

974.3

Si

128.0

285.9

514.9

220.2

236.9

369.4

240.8

236.6

181.12

483.0

J i

23.3

90.0

162.4

16.5

116.8

141.5

26.7

0.001

0.0001

0.0001

Vi

2.3

0.3

36.2

29.9

6.8

58.6

34.8

34.7

22.52

59.6

Hf 3.17

XXV

Note that due to the comparably s mall halfwidth of the band below 200 cm-1 (cf. Fig. A.8.1) the use of the same Vfor all bands as suggested in ref. [125] is impract ical.

- 194 -

Appendix

Following Efimov, a typical value for the standard deviation Vi for oxide glasses is 35 cm-1 .77 As a consequence, 68.3 % of the oscillators should be found within a wavenumber- interval with a range of 70 cm-1 and additional 31.4 %, if the range of the interval is extended to 3 Vor 210 cm-1 , respectively. We think that the justification of the assumption of such a broad variance is highly questionable, at least for stretching vibrations, since the variations of bond lengths in glasses are relatively modest compared to the variation of bond angles.80 Besides, we found, that there exists a multitude of fits of comparable quality. One example with exactly the same average deviation between measured and simulated spectrum is also given in Table A.8.1 (set 2). In this second set, oscillator 6, e.g., is shifted 50 cm-1 to higher wavenumbers compared to its position according to set 1. To avoid such ambiguities we tried to fit the spectrum with fewer oscillators, however, satisfactory resemblance between measured and calculated spectrum could not be achieved in this way. On closer inspection of the oscillator positions provided by the convolution model (eqn. (A.8.3)), we found that these positions are shifted notably towards lower wavenumbers compared to those evaluated applying the Lorentz model (eqn. (A.8.1), see Fig. A.8.1). To examine this finding we have calculated hypothetical IRR-spectra assuming only one -1 -1  oscillator ( Q /cm-1 = 10, H TO /cm = 1000, S /cm = 500, J f

2 ) by increasing the standard

deviation Vstepwise. The results are shown in Fig. A.8.2. It can be easily observed that an increasing standard deviation leads to a notable shift of the peak maximum to higher wavenumbers. On the other hand, using eqn. (A.8.1) for DA, the broadening of peaks can only be achieved by increasing damping constants. This leads, in contrast to increasing V, to slight shifts of the peak maxima to lower wavenumbers. As a consequence, oscillator positions evaluated from a given spectrum applying the convolution model depend also on the values found for the standard deviations. Hence, large values of a particular Vi mean that the related oscillator position in a glass is strongly red-shifted compared to that in a corresponding crystal if the positions of the band maxima are  comparable in both spectra. On the other hand, if the Q are assumed to be the same in the i

glass and the crystal, then the bands in the glass spectrum would be blue-shifted, which is not observed experimentally. Besides, more realistic assumptions on the standard deviation lead to only slight changes of the peak form. Therefore, the convolution model alone is unable to explain the broadening of the peaks satisfactorily in our opinion (see Fig. A.8.2). In summary, great care must be taken when using the convolution model for quantitative IR spectroscopy to avoid significant

Appendix

- 195 -

misinterpretations and ambiguities, especially if accurate oscillator positions are to be evaluated.


0.8

0.6

Lorentzian J 2 J 3 J 4 J 5

Lorentzian V= 5 V= 10 V= 20 V= 30

0.4

0.2

0.0 850

Fig. A.8.2:

1000


1000

1150

Left part: variance of the IRR-spectrum assuming one model oscillator according to eqs. (A.8.3) and (A.8.5) with varying standard deviations V -1 -1  (Q /cm-1 = 10, H 2 , Lorentzian: Vo 0 ). TO /cm = 1000, S /cm =500, J f Right part: variance of the IRR-spectrum assuming one model oscillator -1  according to eqn. (A.8.1) with varying damping constants ( Q = 1000, TO /cm -1 -1 2 ). S /cm =500, basic damping constant J/cm = 10, H f

A.9 Used samples and experimental setup

A.9.1 Fresnoite single crystal The Fresnoite single crystal employed in this work was grown from a stoichiometric melt by the Czochralski technique at the Institute of Crystal Growth (Forschungsverbund Berlin e.V.). Laue orientation measurements revealed the orientatiRQ RI W K H F U \ V W DO ¶ V D[ H V ZL W KL Q DQ H U U RU RI about 1°. To determine the single crystal data an a-c-cut was used.

- 196 -

Appendix

A.9.2 Polycrystalline Fresnoite samples Polycrystalline Fresnoite samples were prepared by melting a stoichiometric composition of raw materials (BaCO3 (p. a. Merck), TiO2 (Merck, optipur), SiO2 (Quarzmehl A, Schott)) at 1500 C°. The melt was homogenized by stirring for one hour and was soaking for an additional hour. To finally obtain a polycrystalline sample with an average crystallite size of about 300 nm, the melt was first splat cooled, resulting in a clear glass. To induce crystallization, the glass was heated with a heating rate of 10 K/min, held for 1 h at 1000 °C, and then cooled down at a rate of 10 K/min. Alternatively, if the melt was not splat cooled, but poured into a graphite mould instead, polycrystalline material with an average crystallite size of about 10 Pm resulted. Unfortunately, oriented crystallization at the surface of the material was observed. Therefore, the material was milled and two different treatments were applied afterwards to obtain a randomly oriented and compact material. One possibility

is to

press the powder

isostatically and to sinter the pressed powder at 1400 °C for 10 h. A disadvantage of this method is that the volume ratio is comparably small and the so-prepared samples possess clearly visible holes, resulting in a comparably poor surface quality even after polishing. F i g. A .9 .1 :

Preferable to the latter approach is to employ Spark Plasma Sintering (SPS) to compact the powder. SPS is a relatively recent sintering technique. It is similar to conventional hot pressing in a way that the precursors are loaded in a die and pressure is applied during sintering.

Fig. A.9.1: Schematic drawing of the SPS-apparatus.

Instead of using an external heating source, however, a pulsed direct current causes internal heating by passing through the electrically conducting pressure die and, in appropriate cases, also through the sample. Therefore, the die as well acts as a heating source and the sample is heated from both outside and inside. Besides, the setting up of an external electric field plays an important role in enhancing mass-transportation. A unique feature of the SPS process is thus to enable reaction and consolidation to occur at relatively low temperatures in a very rapid manner.127,128 A schematic drawing of a SPS-apparatus is presented in Fig. A.9.1. The densification of the large-grained Fresnoite powder was carried out in vacuum in a Spark Plasma Sintering apparatus Dr. Sinter 2050 (Sumitomo Coal Mining Co. Ltd., Japan). The

Appendix

- 197 -

densification temperature was set to 1270 °C and monitored with an optical pyrometer focused on the surface of the pressure die; upon the application of a uniaxial pressure of 50 MPa, the powder was initially heated to 600 °C, and from this temperature to the final densification temperature, a heating rate of 100 °C/min was applied. The SPS-process was also used to compact mixed powders consisting of both large and small crystallites (the small crystallites powder was obtained by milling the ceramics prepared by sintering the glass). This could only be achieved successfully for a 3:1 mixture (75 mass-% small and 25 mass-% large crystallites). For larger mass-percentages of the large crystallites the need for comparably high temperatures during the SPS-process led to a growth of the small crystallites, so that the resulting samples consisted exclusively of large crystallites after the compaction. The great advantage of the SPS-process is the strong densification of the powders. According to density measurements, the density of the SPS-compacted Fresnoite samples equals that of the single crystal within the error range. Alternatively, Fresnoite-ceramics with large crystallites or with mixtures of large and small crystallites (to be more precise, with a broad distribution of crystallite sizes) can also be prepared from glasses with the compositions Ba2 TiSi2 O8 + x SiO2 .M1 Here, the addition of SiO2 slows down the nucleation and the growth rate of the Fresnoite crystallites. An overview of the different polycrystalline Fresnoite samples used in this work can be found in Table A.9.1. All ceramics were gradually polished with 6, 3, and 1 Pm grain size diamond paste until an optically reflecting surface was achieved, which was promoted by the comparably low hardness of Fresnoite.129 Since the surface quality is a critical parameter in case of reflection measurements, Scanning Force Microscopy (SFM) was employed to assess it. The results are depicted in Fig. A.9.2, with the exception of the samples Ba0kG (surface roughness too strong for the method) and Ba0.75 (comparable to Ba0k). A)

Fig. A.9.2:

B)

C)

SFM-pictures of polycrystalline Fresnoite. A) Ba0k, B) Ba0K75-25G, C) Ba0K0-100G.

- 198 Table A.9.1:

Appendix Overview of the different polycrystalline Fresnoite samples used in this work.

Name

Composition

Preparation method

Ba0k

Ba2 TiSi2 O8

Crystallized from the glass

Ba0kG

Ba2 TiSi2 O8

Pressed and sintered powder

Average crystallite size § QP § Pm Initial Composition:

Ba0K50-50G

Ba2 TiSi2 O8

SPS

§ nm § Pm Composition:

Ba0K75-25G

Ba2 TiSi2 O8

SPS

§ QP § Pm

Ba0K0-100G

Ba2 TiSi2 O8

SPS

§ Pm

Ba1k

Ba2 TiSi2 O8 + 1 SiO2 Crystallized from the glass

§ Pm

Fig. A.9.3:

2T -scan of the sample Ba0k. The red lines indicate the position and the relative intensities of the peaks according to the literature.

Appendix

- 199 -

Fig. A.9.4:

Pole figure of the sample Ba0k. Areas with equal intensity share the same color. The most intense peak (211) according to the 2 T -scan was used (cf. Fig. A.9.3).

Fig. A.9.5:

Pole figure of the sample Ba0kG. Areas with equal intensity share the same color. The most intense peak (211) according to the 2 T -scan was used (cf. Fig. A.9.3).

- 200 -

Appendix

By X-ray diffraction (Cu-KD, 2 T « W K H V DPSO HV ZHU H SU RY H G W R EH V L QJO H -phased. An exception is, of course, the sample Ba1k with SiO2 -excess, which crystallized as tridymite. Fig. A.9.3 shows the 2 T -scan of the sample Ba0k as an example. The relative intensities of the peaks proved random orientation. X-Ray texture goniometry (pole figures) was also used to verify random orientation of the polycrystalline samples. Examples of typical pole figures are shown in Fig. A.9.4 and Fig. A.9.5 for the samples Ba0k and Ba0Gk. The non-complete uniformness of the intensity is a consequence of the non-perfect correction of the intensity, which is needed to account, e.g., for the beam divergence. All samples where inspected further by Scanning Electron microscopy. A selection of SEM-pictures is shown in Fig. A.9.6.

A)

B)

C)

D)

Fig. A.9.6:

SEM-pictures of polycrystalline Fresnoite. A) Ba0k, B) Ba0kG, C) Ba0K750G, D) Ba0K0-100G.

Appendix

- 201 -

A.9.3 Oriented Fresnoite glass ceramics (Ba2 TiSi2 O8 + 0.75 SiO2 ) Oriented Fresnoite glass ceramics were prepared by electro-chemically

induced

nucleation.113,M1,M3 This method involves the preparation of a supercooled melt, consisting of Fresnoite and additional SiO2 , in a platinum crucible in the middle of which a platinum wire is placed (Fig. A.9.7). After feeding a voltage of 1.2 V in a way that the platinum wire acts a cathode, the reduction of Ti4+ to Ti3+ induces

Fig. A.9.7: Preparation of Fresnoite glass ceramics by electrochemically induced nucleation.

nucleation and crystallization of the glass ceramics, which starts growing around the cathode. The resulting glass ceramics is highly oriented due to an anisotropic growth rate, which is a maximum parallel to the crystallographic c-axis of Fresnoite. F i g. A .9 .7

A.9.4 Measuring instruments and measurement conditions Most of the IR-reflectance measurements were performed on a FTIR-spectrometer Bruker IFS 66 in the spectral ranges 400 ±6000 cm-1 (MIR) and 100 ±450 cm-1 (FIR) at room temperature. Two different reflection accessory were used, namely an attachment from =HL V V Q RO RQJ HU F RPPH U FL DO O \ DY DL O D EO H ZL W K D I L [ H G DQJO H RI L Q FL GHQ F H RI DQ G D second attachment from Harrick for all other angle oI L QFL GHQ F HV 6H DJXO O Y DU L D EO Hangle reflectance accessory). Usually, polarized radiation was employed generated by KRS-5 polarizers in the MIR and PE-grid polarizers in the FIR. A gold mirror served as reference for the reflection measurements. Typically, the area of inspection was limited by circular apertures of 3-7 mm diameter as appropriate. Some reflectance spectra were recorded on a Bruker IFS 113 V, covering the frequency range between 22 and 6000 cm-1 (using a liquid He-cooled Bolometer as detector in the FIR) and employing an aluminum mirror as reference. The IR- microspectrometry was performed in the range between 600 ±6000 cm-1 with a %U X NH U L QI U DU H G PL F U RV F RSH $ ZKL FK ZD V DW W D FKH G W R W KH , )6 L Q U HI O H F W DQ F H PRGH (average angle of incidence: 17 °). Equation Section 6

- 202 -

Appendix

A.10 The sin TparadoxFormelabschnitt 10 The

problem discussed

below

is

of

immediate and practical relevance for both randomly and partly oriented materials as demonstrated in sections 5.1.7 and 8.3. F i g. A . 1 0. 1:

In orientational averaging two problems are frequently encountered. The first problem consists in averaging a vectorial quantity, which is confined in two dimensions. A prominent example is the polarization vector E of a naturally polarized light wave, which is always oriented perpendicular to the traveling direction in a homogenous and non-anisotropic medium. In general, the probability for a property F to have an

Fig. A.10.1: Illustration of two dimensional averaging according to eqn. (A.10.1).

orientation between - and -d - must obviously be proportional to the corresponding circumference of the segment of a circle, and therefore equal to d- for a circle with radius unity (Fig.A.10.1). As a consequence, the average F is given by:

F

2S

2S

0

0

Fd -. d - ³ ³

(A.10.1)

Depending on the symmetry of the problem it might be possible to decrease the integration range. F i g. A . 1 0. 2:

Another quite often occurring problem is the equivalent averaging in three dimensions. In correspondence

to

the two-dimensional

averaging, it is usually assumed that the probability for a certain orientation (M,T ), e.g., of an optical axis z in a randomly oriented

polycrystalline

material,

is

proportional to the solid angle sin TdMdT (cf. Fig.A.10.2). Accordingly, the average value of a property calculated by

FMT , is then

Fig. A.10.2: Illustration of three dimensional averaging according to eqn. (A.10.2).

Appendix

- 203 -

2 SS

F

FMT , sin Td Md T ³ ³ 0 0

2 SS

.

(A.10.2)

sin Td Md T ³ ³ 0 0

Eqn. (A.10.2) seems to be a generalization of eqn. (A.10.1). Therefore it should be possible to regain the latter from the former. To that end, we cut a disk from the sphere through the equator (the X-Y-plane) with coordinates MT and, indeed, eqn. (A.10.1) is regained. , 90q Another possibility, however, is to cut through the poles, e.g., M 0q , T(the X-Z-plane). The resulting equation is given by S

F

FM ³ 0

0q , T sin TT d . S sin TT d ³

(A.10.3)

0

In eqn. (A.10.3), the orientations are weighted with the factor sin Tobviously contradicting the assumption above that the probability is proportional to d-. The same is true for any other cut through the poles. This result may be illustrated further by the following example: Say, a randomly oriented polycrystalline sample with optically uniaxial crystallites is investigated. According to eqn. (A.10.2), one finds that for an arbitrary coordinate system all optical axes oriented parallel to the X-Y-plane are randomly oriented within this plane. The result is different for all planes perpendicular to the X-Y-plane, i.e., for the X-Z-plane. If the optical axes were oriented parallel to these planes, one would find that the probability of having an orientation ( M 0q , T) is increasing with increasing angle between the optical axis and the Z-axis by the factor sin T . However, since we can choose the coordinate system freely by assumption, we can also use a new coordinate system X¶ Y¶ Z¶ , which is related to the old by (X¶Z, Y¶X, Z¶Y). Now if eqn. (A.10.2) was right, it would result that all optical axes parallel to the X¶ Y¶ -plane (the old X-Z-plane!) would be randomly oriented, while those, e.g., parallel to the Y¶ Z¶ -plane (the old X-Y-plane!) were not, contradicting the previous result. Equivalent paradoxes are also encountered in the description of partial orientation (e.g. random orientation in a plane). One obvious solution to this dilemma is to ban the weighting factor sin Tfrom eqn. (A.10.2) and the idea behind as suggested by Fig.A.10.2. It should be safe to assume that an experimental investigation to solve the paradox is not possible by methods that are based on a continuous change of the reference frame or, alternatively, for which the reference frame is fixed inside the sample and the sample is

- 204 -

Appendix

rotated relative to an incoming wave or beam for reasons given below. A continuous change of the reference frame is, e.g., involved if pole figures are recorded. In case of X-ray methods like T2T -scans (Rietveld refinement), pole figures or rocking curves such a change is essential, since only those planes give signals, which fulfill the Bragg-condition. Therefore only a distribution of crystallites in the X-Y-plane contributes, but the planes perpendicular to the plane can only be investigated by either a rotation of the sample or a change of the position of the beam and the detector. From such changes of the reference frame certain difficulties arise, which require corrections, e.g., the volume of the sample that contributes to a signal is not known a priori and may change by a rotation of the sample. Besides, I RU ȥ ! 60° defocus of the beam poses problems in case of disk- like samples. Besides, ȥ 90° is always inaccessible since at this angle the X-ray beam misses the sample. In practice, the RY H U DO O L QW HQV L W \ EH F RPH V V PDO O H U D V ȥ J HW V O DU J H U $V D F RQ V H TX HQ F e, the accessible range of ȥ L V PX FK PRU H O L PL W H G I RU F RDU V H -crystalline samples (cf. Fig. A.9.4 with Fig. A.9.5) due to the smaller number of crystallites per volume of the sample. Therefore, even after a correction is F D U U L H G RX W SRO H I L JXU H V GR Q RW V K RZ W K H V DPH L QW HQV L W \ L QGH SHQ GHQW RI ĳ DQ G ȥ L Q F DV H RI random orientation. Further possible sources of error are flat detectors, which are scanning a ± necessarily curved ±solid angle. For an enhanced discussion and a comparison of different methods of X-ray texture-analysis see e.g. ref. [63]. All these problems do not occur in reflectance measurements: x The sample remains fixed relative to the beam; defocusing cannot occur. x The probed volume does not change; therefore a volume correction is superfluous. x Nearly all crystallites, regardless of orientation, contribute to the spectrum (those having their transition moment perpendicular to the polarization vector in case of Rs are covered by Rp if the angle of incidence D ). x Due to the mixed TO-LO nature of the modes for T the orientation distribution of a sample can be determined to a large extent (cf. section 8.3.2). This might be able to explain the absence of the factor sin Tin the modeling of optical properties whenever experimental and theoretical results show a good correspondence, e.g. in refs. [3,8,15]. Besides, usually a whole bunch of peaks occur in a spectrum, the relative intensities all of which indicate a possible non-random orientation and allow finding out the kind of the orientation distribution. Especially small deviations from random orientation can more easily be detected by optical methods.

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Acknowledgments

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Acknowledgments First of all I want to thank Prof. Dr. Helga Dunken for encouraging me to try to answer those questions, which could not be addressed in my Ph. D. work and also some of the multitude of questions that arose from the answers to the original questions. I am greatly indebted to Prof. Dr. Jürgen Popp, who took me under his wings after the retirement of Prof. Dunken and made it possible for me to finish this work. A special kind of thanks is dedicated to Mrs. Inge Weber and Mrs. Marion Ludwig. By carrying out the routine measurements that came up as part of my other duties, they made it easier for me to concentrate on this work. Besides, I want to mention that Mrs. Weber carried out some of the measurements the results of which are presented in section 8.3. Dipl. chem. Georg Peiter and PD Dr. Hartmut Hobert drew my attention to the 4×4 matrix formalisms. For that I want to express my sincere gratitude. Without knowledge of the existence of these formalisms most parts of this work would never have existed, especially because the equations, which I have used in my Ph. D. work instead, are excellent approximations for the reflectance from uniaxial crystals in particular at low angles of incidence. Without proper samples this work would have been a mere theoretical one. Since in my eyes W KH RU \ L V Q RW ³ U H DO ´ ZL W K RX W H[ SHU L PHQ W V W K DW SU RY HL W , gratefully acknowledge the preparation of the samples used in the course of this work and the work of those who characterized the samples by additional methods or helped me to characterize them: The Fresnoite single crystal was synthesized by Dr. Reinhard Uecker and Dr. Peter Reiche from the Institut für Kristallzüchtung im Forschungsverbund e.V. The polycrystalline samples were either prepared completely by PD Dr. Ralf Keding (Otto-Schott-Institut für Glaschemie der Universität Jena) or he contributed the powders, which were then compacted by Dr. Zhijian ³ DPH V ´Shen (Department of Inorganic Chemistry, Arrhenius Laboratory, Stockholm University, Sweden). PD Dr. Thomas Höche (Institut für Oberflächenmodifizierung Leipzig e.V.) performed the electron microscopy investigations and passed most of the samples on to Dr. Rudolf Hergt (Institut für Physikalische Hochtechnologie, Jena), who did all X-ray diffraction measurements (including the pole figures). Mr. Alfred Jacobi helped me to carry out the Scanning Force Microscopy measurements. I also want to thank Mrs. Gabriele Möller (Otto-Schott-Institut) who made all samples shine.

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Acknowledgments

Prof. Dr. Janice Musfeldt (Department of Chemistry, University of Tennessee, Knoxville, USA) gave me the opportunity to study cross-polarization effects in the far infrared. I enjoyed my stay at her laboratory very much (indeed a lab were nothing was broken!) and want to express to her my deepest thanks for this opportunity. I am also very grateful to my friends Bo Luttrell, Jason Haraldsen, Jinbo Cao and Roman Wesolowski, who worked at JDQ ¶ V O D E D W W K H time of my stay, for their comradeship, their help to organize my life in Knoxville and the fun we had together. I am looking forward to meeting you again! As a non-native speaker it was not a completely easy task for me to write this thesis in (QJO L V K , K RSH, GL GQ ¶ W I DL O W RRI W HQ 'U 9O D GL PL U , YDQ RY V NL ZK RZRQ D+XPERO GW scholarship and is currently working with me on several problems connected with the optical properties of low symmetry crystals, helped me to correct the manuscript (not only the language) for which I thank him very much. On this occasion, I also want to express my thanks to PD Dr. Michael Schmitt who corrected the language of some of my papers, which have been integrated into this thesis. If some of the sentences in this thesis have not become too complicated, then I have to thank again Jan, who convinced me to occasionally forget this typical-German liking. I also want to thank heartily Prof. Dr. David Tanner (Department of Physics, University of Florida, Gainesville, USA). We only met once, nevertheless he kindly declared himself willing to referee this habilitation thesis. Prof. Dr. =RU DQ 9 3RSRYL ü 8QL YH rsity of Belgrade) and Dr. Christopher C. Homes (Brookhaven National Laboratory) kindly provided their original data of polycrystalline CuO. For that I would like to express them my sincere thanks. I greatly appreciate the comradeship of all colleagues at the institute and want to thank PD Dr. Antje Kriltz and Dr. Beate Truckenbrodt deputizing for all other unnamed persons. Last but not least I want to thank my family who greatly supported me during the course of this work (strangely enough, quite often by distracting me from it).