94
3 Vibrational Up-Pumping: Predicting Impact Sensitivity of Some …
conserved by setting q = −q
− q
. Energy transfer to ω T is then largely dependent
on the number of pathways defined by
(2) .
By imposing the Einstein approximation (that ω is q-independent) for the internal
vibrational modes, it is possible to consider only the phonon density of states (PDOS),
rather than the full phonon dispersion curves. The latter are shown in Fig. 3.17. This
is based on the following:
1. To a good approximation, there is a continuum of vibrational states within the
phonon bath.
2. For phonon-phonon coupling involving two phonons with the same branch index,
j 1 = j 2 , momentum conservation and phase matching dictate that q 1 = −q 2 ,
such that ω j,q + ω j,−q = ω T,q = 0 . Thus, ω T is only accessible at the zone centre
and only the absolute frequency of the low frequency modes is important.
3. For phonon-phonon coupling involving two phonons with different branch index,
j 1 = j 2 , the q-independence of ω T imposes that for any combination of
ω q j , ω T
, there will be a ω q
j at the appropriate momentum and frequency
to satisfy Eq. (3.5). The same holds under the assumption that one of the phonon
modes is a doorway mode, regardless of its q-dependence.
4. In the absence of explicit consideration of V
(3) , coupling between all sets of
phonons can be taken to be approximately the same, provided they comprise
distortion of the same set of interacting atoms (i.e. are of the same molecule or
strongly bonding intermolecular atoms) [94, 95].
3.5.3.1 Partitioning of the Vibrational Structure
Following from the three terms contained in
(2) of Eq. 3.5, the PDOS can be
segmented into a series of physically meaningful regions, Fig. 3.18 [18]. The first
mode, ω q j , generally exhibits lattice character, and is held within the phonon bath,
which has an upper limit defined by max . While this value is not rigorously defined,
it can be qualitatively described as the highest lattice-based mode. Due to the high
anharmonicity of these lattice modes, and the high density of vibrational states, the
thermalisation of vibrational energy occurs very rapidly in this region. This imposes a
crude definition of max as being the first point in which the phonon density of states
drops to zero. The second frequency, ω q j , generally sits somewhere between max
and 2 max , and is termed the ‘doorway mode’. The upper limit of 2 max is significant
as it defines the highest frequency attainable by coupling of two phonon bath modes.
The third mode, ω T is the target vibrational frequency. It must fall within 3 max to
be accessed by coupling of a phonon mode with a doorway mode. The identity of ω T
was determined in Sect. 3.5.1 as being δθ N N N . Within the Einstein approximation, the
frequency of δθ N N N can be identified from the -point eigenvectors. This was done
by artificially extending the eigenvectors of each zone-centre normal coordinate and
led to easy identification of ω T , Fig. 3.19. Note that due to factor group splitting and
symmetry independent azido anions, multiple distinct δθ N N N frequencies can exist in
the same crystal. The covalent compounds exhibit a more extensive spread in δθ N N N
frequencies. While a cluster of δθ NNN exists around 600 cm
−1 in all of the azides (with
3 Vibrational Up-Pumping: Predicting Impact Sensitivity of Some …
conserved by setting q = −q
− q
. Energy transfer to ω T is then largely dependent
on the number of pathways defined by
(2) .
By imposing the Einstein approximation (that ω is q-independent) for the internal
vibrational modes, it is possible to consider only the phonon density of states (PDOS),
rather than the full phonon dispersion curves. The latter are shown in Fig. 3.17. This
is based on the following:
1. To a good approximation, there is a continuum of vibrational states within the
phonon bath.
2. For phonon-phonon coupling involving two phonons with the same branch index,
j 1 = j 2 , momentum conservation and phase matching dictate that q 1 = −q 2 ,
such that ω j,q + ω j,−q = ω T,q = 0 . Thus, ω T is only accessible at the zone centre
and only the absolute frequency of the low frequency modes is important.
3. For phonon-phonon coupling involving two phonons with different branch index,
j 1 = j 2 , the q-independence of ω T imposes that for any combination of
ω q j , ω T
, there will be a ω q
j at the appropriate momentum and frequency
to satisfy Eq. (3.5). The same holds under the assumption that one of the phonon
modes is a doorway mode, regardless of its q-dependence.
4. In the absence of explicit consideration of V
(3) , coupling between all sets of
phonons can be taken to be approximately the same, provided they comprise
distortion of the same set of interacting atoms (i.e. are of the same molecule or
strongly bonding intermolecular atoms) [94, 95].
3.5.3.1 Partitioning of the Vibrational Structure
Following from the three terms contained in
(2) of Eq. 3.5, the PDOS can be
segmented into a series of physically meaningful regions, Fig. 3.18 [18]. The first
mode, ω q j , generally exhibits lattice character, and is held within the phonon bath,
which has an upper limit defined by max . While this value is not rigorously defined,
it can be qualitatively described as the highest lattice-based mode. Due to the high
anharmonicity of these lattice modes, and the high density of vibrational states, the
thermalisation of vibrational energy occurs very rapidly in this region. This imposes a
crude definition of max as being the first point in which the phonon density of states
drops to zero. The second frequency, ω q j , generally sits somewhere between max
and 2 max , and is termed the ‘doorway mode’. The upper limit of 2 max is significant
as it defines the highest frequency attainable by coupling of two phonon bath modes.
The third mode, ω T is the target vibrational frequency. It must fall within 3 max to
be accessed by coupling of a phonon mode with a doorway mode. The identity of ω T
was determined in Sect. 3.5.1 as being δθ N N N . Within the Einstein approximation, the
frequency of δθ N N N can be identified from the -point eigenvectors. This was done
by artificially extending the eigenvectors of each zone-centre normal coordinate and
led to easy identification of ω T , Fig. 3.19. Note that due to factor group splitting and
symmetry independent azido anions, multiple distinct δθ N N N frequencies can exist in
the same crystal. The covalent compounds exhibit a more extensive spread in δθ N N N
frequencies. While a cluster of δθ NNN exists around 600 cm
−1 in all of the azides (with