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The emission at any instant from an excited state can
be described as

N(t) is the emission at any particular instant, N(0)
emission at t=0 and ? is the lifetime.

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The reciprocal of decay time, ? is determined by radiative
emission, nonradiative relaxation and cross relaxation between the adjacent
ions.  Therefore the total relaxation
rate can be expressed as

where ?A is the radiative (photon)
emission rate and ?WNR is the
nonradiative (multiphonon) emission rate. The term PCR is
the rate of cross relaxation between adjacent ions.

When comparing the decay time of BaSiF6 and BaF2 samples doped
with the same amount of rare earths, the contribution from PCR is
the same and hence ?A is the radiative (photon) emission rate and ?WNR
is the nonradiative (multiphonon) emission are the major factors contributing
to the decay time. The term corresponding to radiative emission is
dependent on the site symmetry and environment of the dopant ions whereas the
nonradiative (multiphonon) emission is determined by the lattice phonon energy.
Therefore, the lifetime can be change either by the local symmetry of the rare
earth ion in the host lattice or by the phonon energy of the host, unless the
level of dopants is changed. In the present case, the shortening of decay time
in the BaSiF6 lattice with respect to BaF2, can be the result change in the
local symmetry of the dopant ions in BaSiF6 and BaF2 matrices respectively and
the corresponding lattice phonon energy.

The non-radiative relaxation is in
turn related the number of phonons as,

where A and B are
constants, p is the number of phonons which can be calculated
using the equation,

where ?E=16949 cm?1 (corresponding
to the fluorescence at 594 nm, the energy gap between 5D0
? 7F1 levels)
and ?? is the energy of phonons, in wave
numbers, of the crystal lattice. The lattice phonon energy, ??, can be approximated to the lowest vibration energies in the
IR spectra of BaSiF6 and BaF2, and are 510 and 242 cm?1 a,
respectively. The corresponding number of phonons are 33.2 and 70.0 for BaSiF6
and BaF2, respectively. From equation (1), one can conclude that with an
increase in the number of phonons p, WNR decreases
and as a result the lifetime increases. Hence the difference in the life time
among BaSiF6 and BaF2 doped samples can be associated to the nonradiative
relaxation mechanisms with the phonon energy of host matrix, and low phonon
energy host is most favorable matrix for high luminescence lifetime. 

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