= Sequential excited-state population kinetics
\b[Sequential excited-state population kinetics] describes transfer from an initially prepared <excited state> into another <excited state> that subsequently decays. The measured rise and decay are consequences of both population flow and the detection model. A single fitted <exponential function> need not identify a unique molecular step.
Let upper-state loss rate be $a=k_{21}+k_{2x}$, lower-state loss rate $b$ and transfer rate $k_{21}$. Starting with $N_2(0)=N_0$, $N_1(0)=0$,
$$
\dot N_2=-aN_2,\qquad \dot N_1=k_{21}N_2-bN_1.
$$
For $a\ne b$,
$$
N_2=N_0e^{-at},\qquad N_1=\frac{N_0k_{21}}{a-b}(e^{-bt}-e^{-at}).
$$
The transfer branching fraction is $k_{21}/a$. If upper-state loss has competing channels, not every initial excitation reaches the lower state.
The equal-rate limit is finite, although the displayed difference quotient looks singular. Taking $a\to b$ gives
$$
N_1=N_0k_{21}t e^{-at}.
$$
For stipulated $a=b=k_{21}=1$ in consistent inverse-time units, the lower population peaks at $t=1$ with value $N_0/e$. The apparent denominator singularity is a numerical representation issue, not infinite population.
For unequal rates, the peak time is
$$
t_{\rm peak}=\frac{\log(a/b)}{a-b}.
$$
With $a=10^{12}\,\mathrm{s^{-1}}$ and $b=10^8\,\mathrm{s^{-1}}$, it is about 9.21 ps. This differs from the upper-state <time constant> of 1 ps: a population maximum and a single-step time constant are not interchangeable observations.
An <instrument-response limit in sequential population inference> enters when observed <fluorescence> is a <convolution> of the radiative population signal with the instrument response, plus background. Temporal broadening can obscure a very fast rise. Spectral overlap can also mix upper- and lower-state contributions. A proposed analysis fits the response-convolved model to multiple assigned signals rather than interpreting an unresolved rise as an exact molecular rate.
For same-multiplicity <electronic states>, <molecular internal conversion> supplies one physical transfer mechanism. IUPAC defines it as a <radiationless transition> with <vibrational excitation> possible in the lower electronic state.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book)>[internal conversion I03102]{ref} The equations above are original kinetic models; their parameters require experimental determination for the molecule and conditions studied.
\Image[/sequential-rate-coincidence.png]
{title=Exact sequential population solutions for coincident and separated loss rates. The equal-rate limit is finite and its lower-state population peaks after the upper-state population has begun to decay.}
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