4.5 Results and Discussion
133
follows quickly afterwards (1–2 ps). Hence, it is worth analysing the contributions
of combination pathways and their potential to rationalise impact sensitivity.
The combination pathways generated in the absence of temperature are simply
taken as the two-phonon density of state,
(2)
= δ(ω − ω 1 − ω 2 ), with ω 1 = ω 2 , in
line with Eq. 3.5. As discussed in Chap. 3, a restriction is placed on the generation of
these curves, such that ω 1 < 2 max and ω 2 < < max . Hence, the maximum allowed
target frequency is 3 max . This has the effect of ensuring that up-pumping occurs
by the addition of at least one mode from the phonon bath, which is initially excited
by the impact of the shock wave.
These
(2) curves are generated for the materials studied here, Fig. 4.11. As a
qualitative rule,
(2) appears to increase earlier and more rapidly for the sensitive
compounds of each structure type. For example, the onset of increase is roughly the
same between HNB (~260 cm
−1 ) and β-HMX (~250 cm
−1 ), although the former
rises much more rapidly. In contrast, α-FOX-7 has an onset frequency (~320 cm
−1 )
approximately 100 cm
−1 higher than in β-HMX. In general, each successive doorway
mode leads to an increase in
(2) that corresponds to the density of states about that
doorway mode. Hence, in line with Fermi’s Golden Rule, Eq. 3.6, a lower onset
frequency and more rapid increase in
(2) corresponds to a more rapid transfer of
energy into the internal modes. A notable exception to this generalization appears
to be NTO, which is based on max = 200 cm
−1 in Fig. 4.11. The rapid onset
of
(2) results from the low-lying vibrational band that sits just above max . If
NTO is instead recast based on max = 240 cm
−1 , the onset frequency is shifted to
~350 cm
−1 , Fig. 4.11. This further supports previous suggestions to include the −
NO 2 rocking motions within the phonon bath.
Raw integration of
(2) results in largely meaningless quantities, noting that uppumped energy again only contributes to the excitation of the internal modes if
an internal mode exists at a particular frequency. Hence,
(2) are again projected
onto the DOS curves, and P(
(2) ) are generated. An example is given for ABT in
Fig. 4.12. As is observed for ABT, and indeed holds across the energetic materials,
the large majority of
(2) sits between existing vibrational states and can therefore
be discarded.
With these preparations in mind, it is possible to analyse the combination mode
contributions to the energy transfer of the molecular EMs within the 0 K model.
In the first instance, this simply corresponds to an integration of the P(
(2) ) curves
generated above.
The integration of P(
(2) ), Fig. 4.13, does not reveal as promising a trend as
was observed for the azide materials in Chap. 3. There is no exponential decay
observed with increasing impact sensitivity, and the integrated values of α-FOX7 and β-HMX are very similar. In fact, α-FOX-7 is predicted to be slightly more
sensitive than β-HMX. This may point towards an error with the assignment of max
in the computational model. However, inspection of the INS spectra suggests that the
values employed in both cases are accurate, and that no peaks are added or omitted
to the region max → 3 max in the simulated spectra for either compound. These
therefore appear to be well representative of the systems under the current model.
133
follows quickly afterwards (1–2 ps). Hence, it is worth analysing the contributions
of combination pathways and their potential to rationalise impact sensitivity.
The combination pathways generated in the absence of temperature are simply
taken as the two-phonon density of state,
(2)
= δ(ω − ω 1 − ω 2 ), with ω 1 = ω 2 , in
line with Eq. 3.5. As discussed in Chap. 3, a restriction is placed on the generation of
these curves, such that ω 1 < 2 max and ω 2 < < max . Hence, the maximum allowed
target frequency is 3 max . This has the effect of ensuring that up-pumping occurs
by the addition of at least one mode from the phonon bath, which is initially excited
by the impact of the shock wave.
These
(2) curves are generated for the materials studied here, Fig. 4.11. As a
qualitative rule,
(2) appears to increase earlier and more rapidly for the sensitive
compounds of each structure type. For example, the onset of increase is roughly the
same between HNB (~260 cm
−1 ) and β-HMX (~250 cm
−1 ), although the former
rises much more rapidly. In contrast, α-FOX-7 has an onset frequency (~320 cm
−1 )
approximately 100 cm
−1 higher than in β-HMX. In general, each successive doorway
mode leads to an increase in
(2) that corresponds to the density of states about that
doorway mode. Hence, in line with Fermi’s Golden Rule, Eq. 3.6, a lower onset
frequency and more rapid increase in
(2) corresponds to a more rapid transfer of
energy into the internal modes. A notable exception to this generalization appears
to be NTO, which is based on max = 200 cm
−1 in Fig. 4.11. The rapid onset
of
(2) results from the low-lying vibrational band that sits just above max . If
NTO is instead recast based on max = 240 cm
−1 , the onset frequency is shifted to
~350 cm
−1 , Fig. 4.11. This further supports previous suggestions to include the −
NO 2 rocking motions within the phonon bath.
Raw integration of
(2) results in largely meaningless quantities, noting that uppumped energy again only contributes to the excitation of the internal modes if
an internal mode exists at a particular frequency. Hence,
(2) are again projected
onto the DOS curves, and P(
(2) ) are generated. An example is given for ABT in
Fig. 4.12. As is observed for ABT, and indeed holds across the energetic materials,
the large majority of
(2) sits between existing vibrational states and can therefore
be discarded.
With these preparations in mind, it is possible to analyse the combination mode
contributions to the energy transfer of the molecular EMs within the 0 K model.
In the first instance, this simply corresponds to an integration of the P(
(2) ) curves
generated above.
The integration of P(
(2) ), Fig. 4.13, does not reveal as promising a trend as
was observed for the azide materials in Chap. 3. There is no exponential decay
observed with increasing impact sensitivity, and the integrated values of α-FOX7 and β-HMX are very similar. In fact, α-FOX-7 is predicted to be slightly more
sensitive than β-HMX. This may point towards an error with the assignment of max
in the computational model. However, inspection of the INS spectra suggests that the
values employed in both cases are accurate, and that no peaks are added or omitted
to the region max → 3 max in the simulated spectra for either compound. These
therefore appear to be well representative of the systems under the current model.