modes of the corresponding coordination polyhedra that make up the crystal. That is, that the low energy modes correspond mainly to translational motion of the heavy Y atoms and small parts of rotational and translational motion of AlO4 tetrahedra, while the high energy modes contain mainly breathing (v1) andv4 AlO4 molecular modes. The calculated projected densities of state (figures 3.3, 3.4) and our subsequent analysis also strongly support this assumption.
Figure 4.2: Raman spectrum of YAG at ambient conditions reported by [3].
In many theoretical lattice dynamical studies, the most usual theoretical method to di- rectly determine the Raman (and IR) active modes is based on irreducible multiplier representations and on derivation of symmetry adapted vectors that are used to block diagonalize the dynamical matrix [25]. In this work, we have followed a more direct method. By implementing the Shell Model, where the existence of the O shells accounts for electronic polarizability, we can have a direct calculation of the Raman Tensor aij,s (Section 1). Then, the relative Raman intensity of a mode can be approximated by di- rectly evaluating the quantity
P
jaij,sej2
, according to equation 4.8. Therefore, we can determine the Raman active modes simply based on the values of the Raman Intensity Isc that is directly calculated by GULP for each mode:
Isc ∝ X
j
aij,sej
!2
(4.10) The intensity that is assigned to each mode is averaged among different scattering polar- izations.
In table 4.1 we present the calculated Raman active phonon modes as compared to the experimental values by [3] and [19]. It must be first emphasized that the model used for the calculations is by no means simple, as it contains a large number of parameters (19), for which no fitting procedure was considered, given the shear number of parameters.
Therefore, upon comparison, we find that the agreement is more than satisfactory, given the complexity of both the structure and the model used. In fact, it should be noted that our model exactly predicts an accidental degeneracy of an A1g and a T2g at ≈ 363cm−1. The agreement with experiment seems to be especially good for lower frequencies.
Table 4.1: Calculated Raman active phonon modes for YAG by the Shell Model. For compar- ison, experimentally measured mode frequencies reported by [[3] and [19] are also tabulated.
Mode Symmetry Wavenumber (calc.) Wavenumber (exper.) Raman Activity
(cm−1) (cm−1) (Arb. Units)
A1g 363 370 14.0775
A1g 528 559 121.8397
A1g 703 783 46.8545
Eg 176 163 0
Eg 298 310 0
Eg 360 340 0
Eg 401 402 0
Eg 512 523 0
Eg 556 536 0
Eg 659 712 0
Eg 738 754 0
T2g 153 145 0.006
T2g 224 220 1.8188
T2g 244 243 7.8442
T2g 273 261 7.4047
T2g 310 295 12.8115
T2g 364 372 2.6082
T2g 376 406 3.0102
T2g 467 438 0.0452
T2g 491 - 0.0262
T2g 549 545 22.2669
T2g 594 - 1.2891
T2g 672 691 185.9624
T2g 699 699 25.5233
T2g 834 834 71.4731
However, on first look, a very interesting observation is made. For all Eg modes, the calculated Raman Intensity is exactly zero. The reason for this result has not yet been exactly clear; we hypothesize that this likely has to do with the nature of our model. The assignment of these modes was made by direct methods; by noting that there are only 8Eg and 10Eu doubly degenerate modes, out of which only the Eg are Raman active, we were able to distinguish between the two by applying a simple inversion transformation of their corresponding eigenvectors with respect to an inversion center, such as an Aloct ion. The modes that were symmetric under this transformation were classified as Eg and therefore Raman active.
Furthermore, some discrepancies between experimentally measured and theoretically cal- culated frequencies exist, such as a major one at 783 cm−1 (while the calculated value was 703 cm−1), and a less significant one at 712 cm−1 (calculated value was 659 cm−1).
The reasons for these discrepancies will be made clear by the subsequent analysis of their corresponding eigenvectors that follows.
In the rest of this chapter, we will focus on the study of the calculated phonon polarization eigenvectors for some Raman active modes. In our subsequent discussion, we will follow the main argumentation line of [31], a similar lattice dynamical study of RE3Al5O12 compounds using the Rigid Ion Model. The results are comparable.
First of all, we must notice that all Raman active modes are gerade, that is, invariant under inversion. We remember from section 2.2 that the octahedrally coordinated Aloct ions are situated on inversion centers (site symmetryC3i). Therefore, since~eoct =i·~eoct =
−~eoct, we can deduce that
~eoct = 0 (4.11)
for all Raman active modes. This means that vibrations inside octahedra can be simply expressed in terms of bond bendings and stretchings.
Furthermore, given the complexity of the garnet structure, as was shown in figure 2.2, a full presentation of the eigenvectors for the entire unit cell would not be intuitively helpful in our understanding of the eigenvector relations. For this reason, only the eigenvectors belonging to the structural subunit of the unit cell that was shown in figure 2.3 are de- picted instead. We may remind ourselves that this subunit contains an Y-dodecahedron, an Al-octahedron and an Al-tetrahedron, all sharing an oxygen anion. As we shall dis- cover, although each oxygen belongs to four different polyhedra as was just mentioned, meaning that these polyhedra are intricately connected and closely packed, it is nonethe- less noteworthy of the fact that many modes exhibit strong molecular character, mainly of AlO4 and AlO8 molecular vibrations.
Hence, the eigenvectors for the 3 non-degenerate A1g modes, with frequencies ω1 =
363 cm−1, ω2 = 528 cm−1, ω3 = 703 cm−1 are presented in figure 4.3. The charac- teristic feature of all the A1g modes is the fact that only oxygen atoms vibrate, while all the metal cations rest immobile at their equilibrium positions. This means that an analysis in terms of bond bendings stretchings and molecular vibrations of the coordina- tion polyhedra is possible. The eigenvectors were visually depicted using the graphical user interface (GUI) computer program “DL Visualise” (DLV), developed at Daresbury Laboratory, UK. Also, color code was used to indicate magnitude, with blue indicating low and red indicating high vector magnitude.
Figure 4.3: Calculated eigenvectors for degenerate A1gmodes. With the exception of the high-frequency mode, they show an excellent agreement with previous studies [31] using the RIM model
We can see that the low-frequency mode at 363 cm−1 exhibits rotational character of the AlO6 octahedra, while two opposite bonds are symmetrically stretched, with no shear component, in agreement with [31]. The intermediate frequency mode at 528 cm−1 ex- hibits rotational components of the AlO6 octahedra and strongv2 quadrupolar character of the AlO4 tetrahedra, that is, the E mode of tetrahedral XA4 molecules belonging to the T2d point group, also in agreement with [papagelis]. On the other hand, the mode at 703 cm−1 exhibits mainly rotational components of the octahedra and tetrahedra in- stead of a v1 breathing A1 mode that was found in [31] to have an eigenfrequency that agreed well with experimental data. On the contrary, in our model the Al −O bonds in the octahedra and tetrahedra experience much less stretching. Therefore, we strongly consider this to be the main reason for the discrepancy between the calculated and ex- perimental value for the eigenfrequency. The vibrations of the corresponding octahedra and tetrahedra are depicted in figure 4.4.
Figure 4.4: Vibrations of the coordinationAlO6 octahedra andAlO4 tetra- hedra for the non-degenerate A1g modes
Also, the doubly degenerate Eg modes are depicted in figure 4.5. These modes, like the A1g modes also possess a characteristic behavior that is imposed by the crystal symmetry.
We have already mentioned that in all Raman active modes, all 8 Aloct do not vibrate, also due to symmetry. What is more, in the Eg modes, out of all the 12Altet and Y ions, only 8 Altet and 8 Y vibrate. Those ions vibrate in same-atom pairs, in antiphase with each other, not necessarily with the same amplitude and their corresponding eigenvectors are plane polarized on the bccrystallographic plane.
These modes can also be further classified into two categories. In the first one, a pair of Y −Altet atoms vibrate in phase with each other and both parallel to the +ˆb crystallo- graphic direction, and another pair vibrate in phase, parallel to the −bˆ crystallographic direction and in antiphase with the first pair. The same happens for four pairs ofY −Altet
atoms along the +ˆc and −ˆc directions. This group contains the modes with frequencies ω4 = 176 cm−1, ω5 = 512 cm−1, ω6 = 556 cm−1 and ω7 = 659 cm−1. For example, in the subunit depicted in figure 4.5 for ω3 = 176 cm−1, the upper dodecahedron and the tetrahedron vibrate in phase and parallel to each other towards the plane of the page (not clearly visible), while the lower dodecahedron is vibrating perpendicularly towards the lower end of the figure, along the −ˆc direction.
Figure 4.5: The calculated doubly degenerateEgmode eigen- vectors.
In the second group, consisting of the remaining Eg modes ω8 = 298 cm−1, ω9 = 360 cm−1, ω10 = 401 cm−1 and ω11 = 738 cm−1, the same pattern appears, with the only difference being that now the Y −Altet pair vibrate in antiphase with each other. It shall be noted that in most depicted eigenvectors except the one for the ω4 = 176 cm−1 mode, there seems to be no movement of the cations at all. This is mainly due to the fact that their polarization vectors have very small, but still non-zero lengths.
Figure 4.6: Vibrations of the coordinationAlO6 octahedra andAlO4 tetra- hedra for the first group of the doubly-degenerate Eg modes.
Figure 4.7: Vibrations of the coordinationAlO6 octahedra andAlO4 tetra- hedra for the second group of the doubly-degenerate Eg modes.
The vibrations of the tetrahedral and octahedral units for both groups can be clearly seen in figures 4.6 and 4.7. In the low-frequency ω4 = 176cm−1 mode, all ions except the Y cation have very small vector lengths; the mode vibration is almost entirely due to the heavy Y ion. The ω7 = 659 cm−1 and ω10 = 401 cm−1 modes exhibit Al −O bond stretching, and the ω5 = 512 cm−1 and ω6 = 556 cm−1 modes have a significant quadrupolar AlO4 character.
Figure 4.8: Calculated eigenvectors for the structural subunit. and the coordination AlO4 and AlO6 polyhedra for the lowest frequency (ω11) T2g mode.
Finally, the triply-degenerate T2g modes do not exhibit any notable pattern. For this reason, we shall only discuss some particular modes that are of special interest. These are the lowest and the highest-frequency T2g modes at ω12 = 153 cm−1 and ω13 = 834 cm−1 (which are also the lowest and the highest-frequency Raman active modes in general), as well as the two modes at ω14 = 491 cm−1 and ω15 = 594 cm−1 which are not observed experimentally, but which our model predicts nonetheless. These are depicted in figures 4.8-4.10
Figure 4.9: Calculated eigenvectors for the structural subunit. and the coordination AlO4 and AlO6 polyhedra for the highest frequency (ω12) T2g mode.
As is expected, the low frequency mode ω4 = 153cm−1 (figure 4.8) is mainly due to vibrations of the heavy Y ion, as well as to small translational and rotational motions of theAl−O polyhedra. On the other hand, the high frequency modeω13= 834cm−1 (figure 4.9) is composed mainly of av4(ofT2 symmetry) vibrational mode of theAlO4 tetrahedra and a v5 normal mode of the AlO6 octahedra (of T2g symmetry of the octahedral XY6 molecule belonging to the Oh point group). This further supports our assumption that the low frequency modes correspond to either vibrations of only the heavy Y ions or of entire rotational and translational motions of the coordination polyhedra, while the high frequency modes correspond roughly to internal molecular vibrations of the AlO4 tetrahedra and AlO6 octahedra.
Finally, in the predicted mode at 491 cm−1 that is not experimentally observed (figure 4.10), we can see that it corresponds mainly to pure shear bond bendings of the octahedral Al−O bonds that can be roughly classified as av5 vibrational mode of theXY6 molecule.
The other predicted mode at 594 cm−1features intense tetrahedralAl−Obond stretching and a strong octahedral v5 molecular character.
Figure 4.10: Calculated eigenvectors for the structural subunit and the coordination AlO4 and AlO6 polyhedra for the ω14 and ω15 modes that are not experimentally observed.
In this work we have conducted a thorough lattice dynamical study of the vibrational, thermodynamic and elastic properties of Y3Al5O12 based on the Shell Model and the Quasi-harmonic Approximation, using empirical potentials. Our study has shown that the material’s free energy is manly electrostatic, as is expected of an ionic compound.
Therefore, most thermodynamic properties are little affected by variations in the poten- tial parameters and our calculations closely reproduce experimental findings. Especially concerning the Density of States, our model correctly predicts the existence of a phonon frequency gap at around 650cm−1, in accordance with inelastic neutron scattering exper- iments. All our calculations are indicating that the material is hard and incompressible, as well as mechanically and chemically stable even at temperatures close to its melting point, with little variation along a temperature range from 0 K to 2000 K.
Our model also correctly predicts the existence of 17 (A1g andT2g) out of 25 total Raman active modes based on calculations of the Raman tensor, which directly provides the rel- ative intensities of the corresponding peaks. The remaining 8Eg modes were assigned by considering the symmetry of their corresponding eigenvectors. It also predicts the exis- tence of two additional Raman active modes ofT2g symmetry which are not experimentally observed, due to either accidental degeneracies or low scattering transition amplitudes.
An accidental degeneracy that is known to occur at around 370cm−1 is also accounted for. Finally, the Raman active mode eigenfrequencies and polarization eigenvectors were thoroughly examined; it was found that low frequency modes are mainly due to either vibrations of the heavy Y ions or translational and rotational movements of entire AlO4 tetrahedra and AlO6 octahedra, while the high-frequency modes exhibit mainly molec- ular AlO4 and AlO6 vibrational character, in accordance with previous similar lattice dynamical studies using the Rigid Ion Model.
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