3.5 Results and Discussion
99
3.5.3.2 Coupling Pathways and Impact Sensitivity
Within the first anharmonic approximation, only two phonons (ω j and ω j ) may
scatter to form a third (ω T ). This leads to two different scattering mechanisms:
1. ω j and ω j share the same branch index and frequency, i.e. ω j = ω j , imposing
the restriction that q
ω j
= −q
ω j
, and q(ω T ) = .
2. ω j = ω j such that q(ω T ) = q
ω j
+ q
ω j
These two scattering mechanisms are analogous to spectroscopic processes of (1)
overtone and (2) combination bands. This terminology will therefore be adopted for
ease of the following discussion.
While explicit solution of Eq. 3.5 requires the calculation of V
(3) , its calculation
is intractable for large, low symmetry systems [93]. Previous attempts at deriving
values of this term have suggested that structurally similar materials exhibit minimal
difference in the average value of V
(3) , and that its neglect is generally sufficient [21,
22, 94]. However, it follows from the definition of this term that vibrational eigenvectors that do not comprise the same atoms will not couple with any notable efficiency.
This is an important consideration for NH 4 N 3 and TAGZ, where large portions of the
vibrational structure contain no N
−
3 character. For these systems, coupling pathways
are therefore considered only for the azide-channel PDOS, Fig. 3.18.
Overtone Pathways
In accordance with Eq. 3.5, overtone pathways can be expected to occur more efficiently [19]. This assumption formed the base for previous work at understanding
impact sensitivity [16, 21]. However, the number of overtone pathways is far fewer
than the pathways available by the combination mechanism. Further, only the first
overtone pathway can be considered within the first anharmonic approximation.
Higher order overtones become increasingly improbable, making these pathways
less likely for materials in which ω T > 2 max . This restriction affects NaN 3 , TAGZ,
and BaN 3 . However, due to the high anharmonicity of phonon modes, it has been
suggested that quartic terms can occur with sufficient speed for further consideration.
[97] This extends the restriction to 3 max .
To describe the overtone pathways, the PDOS, g(ω) is scaled by N, the overtone
number,
g
N
(ω) =
g(ω)
N
; ω
N
= ωN
(3.7)
Within this model, the number of pathways through which energy can transfer
can then be taken as an integration of g
N
(ω) at each ω T , Fig. 3.20. To account for
the existence of resonant vibrational states, a sampling window of ω T ± 10 cm
−1
was used. This reflects the magnitude of Gaussian smearing that was applied during
generation of the PDOS.
The number of overtone pathways present in each material does appear to correlate loosely with the relative sensitivities, Fig. 3.20B. As expected, modes in which
99
3.5.3.2 Coupling Pathways and Impact Sensitivity
Within the first anharmonic approximation, only two phonons (ω j and ω j ) may
scatter to form a third (ω T ). This leads to two different scattering mechanisms:
1. ω j and ω j share the same branch index and frequency, i.e. ω j = ω j , imposing
the restriction that q
ω j
= −q
ω j
, and q(ω T ) = .
2. ω j = ω j such that q(ω T ) = q
ω j
+ q
ω j
These two scattering mechanisms are analogous to spectroscopic processes of (1)
overtone and (2) combination bands. This terminology will therefore be adopted for
ease of the following discussion.
While explicit solution of Eq. 3.5 requires the calculation of V
(3) , its calculation
is intractable for large, low symmetry systems [93]. Previous attempts at deriving
values of this term have suggested that structurally similar materials exhibit minimal
difference in the average value of V
(3) , and that its neglect is generally sufficient [21,
22, 94]. However, it follows from the definition of this term that vibrational eigenvectors that do not comprise the same atoms will not couple with any notable efficiency.
This is an important consideration for NH 4 N 3 and TAGZ, where large portions of the
vibrational structure contain no N
−
3 character. For these systems, coupling pathways
are therefore considered only for the azide-channel PDOS, Fig. 3.18.
Overtone Pathways
In accordance with Eq. 3.5, overtone pathways can be expected to occur more efficiently [19]. This assumption formed the base for previous work at understanding
impact sensitivity [16, 21]. However, the number of overtone pathways is far fewer
than the pathways available by the combination mechanism. Further, only the first
overtone pathway can be considered within the first anharmonic approximation.
Higher order overtones become increasingly improbable, making these pathways
less likely for materials in which ω T > 2 max . This restriction affects NaN 3 , TAGZ,
and BaN 3 . However, due to the high anharmonicity of phonon modes, it has been
suggested that quartic terms can occur with sufficient speed for further consideration.
[97] This extends the restriction to 3 max .
To describe the overtone pathways, the PDOS, g(ω) is scaled by N, the overtone
number,
g
N
(ω) =
g(ω)
N
; ω
N
= ωN
(3.7)
Within this model, the number of pathways through which energy can transfer
can then be taken as an integration of g
N
(ω) at each ω T , Fig. 3.20. To account for
the existence of resonant vibrational states, a sampling window of ω T ± 10 cm
−1
was used. This reflects the magnitude of Gaussian smearing that was applied during
generation of the PDOS.
The number of overtone pathways present in each material does appear to correlate loosely with the relative sensitivities, Fig. 3.20B. As expected, modes in which