84
3 Vibrational Up-Pumping: Predicting Impact Sensitivity of Some …
The optimised primitive unit cell obtained under the DFT-TS scheme had a volume
ca. 1.2% below the experimental volume (as compared to DFT-D2, which overestimated the volume by 1.1%). The frequencies that result from the DFT-TS scheme
show poorer agreement with the higher frequency lattice modes (226, 189 and
187 cm
−1 ), although it did lead to slight improvements of the lowest frequency lattice
modes (127 cm
−1 and 174 cm
−1 ). The DFT-D2 scheme was therefore selected for
further use. The internal vibrational modes are modelled less accurately. The bending
frequency (DFT-D2) is calculated to be ~ 606/610 cm
−1 , ca. 4.5% lower than the
measured INS frequency of 639 cm
−1 . The symmetric stretching mode is modelled
even more poorly at 1250 cm
−1 , ca. 8% lower than the INS value of 1358 cm
−1 .
However, the calculated δ R S frequency does agree well with previous simulations
and suggests an inherent inability of the PBE scheme to capture this mode [60].
Note that the band corresponding to δ R as was not observed in the INS spectrum
likely due to the low scattering cross section of [16] N and the low amplitude of the
asymmetric stretching mode. The calculated ν(δ R as ) can therefore be compared to
literature Raman spectra [88]. The frequency of δ R as is better reproduced by the
PBE-D2 than δ R s , simulated to occur at 2037 cm
−1 (1959.98 cm
−1 without LOTO correction) and the experimental Raman[88] frequency at 2043 cm
−1 , a 0.2%
underestimation). Overall, it therefore appears that the PBE-D2 based scheme leads
to a good correlation with experimental frequencies in the external mode region and
δ θ N N N . The latter is particularly important as it is the target frequency identified in
Sect. 3.5.1.1
Despite the agreement between zone-centre simulated low-frequency bands and
the INS spectrum, there are two striking differences:
• A well-defined band is observed at ca. 100 cm
−1 in the INS spectrum
• The experimental intensities are poorly reproduced by simulation.
Both effects can be explained by noting that the TOSCA spectrometer does
not probe the Brillouin zone centre, Chapter 2.2.2.2, but spans a broad range of
momentum transfer [89]. As scattering from both N and Na are dominated by coherent
scattering, vibration dispersion through the Brillouin zone becomes important.
Despite the high frequency associated with the top external bands near k = 0,
these frequencies represent only a small subset of the Brillouin zone, Fig. 3.9. By
comparison with the simulated INS spectra in Fig. 3.8, it can be inferred that these
frequencies are diluted (e.g. by powder averaging [90]) as they are not observed when
simulated scattering from the full Brillouin zone is considered. Instead, the highest
feature observed in the simulated INS band occurs at ca. 240 cm
−1 , consistent with
the average frequency of these external bands. This is only slightly higher than
the experimentally observed highest frequency (ca. 230 cm
−1 ). This explains why
the zone-centre calculation of α-NaN 3 without LO-TO correction offered a good
starting point in Fig. 3.8. In fact, inclusion of the LO-TO correction for the zone
centre calculation leads to gross overestimation of the INS frequencies of the external
modes.
3 Vibrational Up-Pumping: Predicting Impact Sensitivity of Some …
The optimised primitive unit cell obtained under the DFT-TS scheme had a volume
ca. 1.2% below the experimental volume (as compared to DFT-D2, which overestimated the volume by 1.1%). The frequencies that result from the DFT-TS scheme
show poorer agreement with the higher frequency lattice modes (226, 189 and
187 cm
−1 ), although it did lead to slight improvements of the lowest frequency lattice
modes (127 cm
−1 and 174 cm
−1 ). The DFT-D2 scheme was therefore selected for
further use. The internal vibrational modes are modelled less accurately. The bending
frequency (DFT-D2) is calculated to be ~ 606/610 cm
−1 , ca. 4.5% lower than the
measured INS frequency of 639 cm
−1 . The symmetric stretching mode is modelled
even more poorly at 1250 cm
−1 , ca. 8% lower than the INS value of 1358 cm
−1 .
However, the calculated δ R S frequency does agree well with previous simulations
and suggests an inherent inability of the PBE scheme to capture this mode [60].
Note that the band corresponding to δ R as was not observed in the INS spectrum
likely due to the low scattering cross section of [16] N and the low amplitude of the
asymmetric stretching mode. The calculated ν(δ R as ) can therefore be compared to
literature Raman spectra [88]. The frequency of δ R as is better reproduced by the
PBE-D2 than δ R s , simulated to occur at 2037 cm
−1 (1959.98 cm
−1 without LOTO correction) and the experimental Raman[88] frequency at 2043 cm
−1 , a 0.2%
underestimation). Overall, it therefore appears that the PBE-D2 based scheme leads
to a good correlation with experimental frequencies in the external mode region and
δ θ N N N . The latter is particularly important as it is the target frequency identified in
Sect. 3.5.1.1
Despite the agreement between zone-centre simulated low-frequency bands and
the INS spectrum, there are two striking differences:
• A well-defined band is observed at ca. 100 cm
−1 in the INS spectrum
• The experimental intensities are poorly reproduced by simulation.
Both effects can be explained by noting that the TOSCA spectrometer does
not probe the Brillouin zone centre, Chapter 2.2.2.2, but spans a broad range of
momentum transfer [89]. As scattering from both N and Na are dominated by coherent
scattering, vibration dispersion through the Brillouin zone becomes important.
Despite the high frequency associated with the top external bands near k = 0,
these frequencies represent only a small subset of the Brillouin zone, Fig. 3.9. By
comparison with the simulated INS spectra in Fig. 3.8, it can be inferred that these
frequencies are diluted (e.g. by powder averaging [90]) as they are not observed when
simulated scattering from the full Brillouin zone is considered. Instead, the highest
feature observed in the simulated INS band occurs at ca. 240 cm
−1 , consistent with
the average frequency of these external bands. This is only slightly higher than
the experimentally observed highest frequency (ca. 230 cm
−1 ). This explains why
the zone-centre calculation of α-NaN 3 without LO-TO correction offered a good
starting point in Fig. 3.8. In fact, inclusion of the LO-TO correction for the zone
centre calculation leads to gross overestimation of the INS frequencies of the external
modes.