= 2D IR response-factor ratio
{c}
The \b[2D IR response-factor ratio] $\kappa=\beta/\alpha$ compares the response per unit population of two specified spectral components. In the positive-component model used for <2D IR response-factor correction>,
$$
C=\beta Pq,\qquad D=\alpha P(1-q),\qquad
\frac CD=\kappa\frac q{1-q}.
$$
Here $q$ is the population fraction of interest and $P$ a shared survival factor. The <2D IR response-factor ratio> is a property of this measurement model and its conditions, rather than a universal property of every <2D IR cross peak>.
An independently characterized reference can identify $\kappa$ if the model applies. For reference fraction $0<q_{\rm ref}<1$,
$$
\kappa=\frac{C_{\rm ref}(1-q_{\rm ref})}{D_{\rm ref}q_{\rm ref}}.
$$
For an original example, $q_{\rm ref}=0.25$, $C_{\rm ref}=2$ and $D_{\rm ref}=3$ give $\kappa=2$. An unknown with $C=6$ and $D=4$ then has corrected fraction $q=6/(2\cdot4+6)=3/7$, whereas its raw <2D IR population normalization> ratio is $0.6$.
Using an unknown fraction to estimate $\kappa$ and then using that same $\kappa$ to recover the fraction would be circular. The reference needs independent information. Moreover, <2D IR response-factor transferability> must be tested when pulse spectrum, <polarization>, <2D IR waiting time>, detection or fitted response pathway changes. A common multiplicative gain cancels from $C/D$; a component-specific gain does not.
First-order <uncertainty propagation> is transparent on the logarithmic scale:
$$
d\ln\kappa=\frac{dC_{\rm ref}}{C_{\rm ref}}-\frac{dD_{\rm ref}}{D_{\rm ref}}-\frac{dq_{\rm ref}}{q_{\rm ref}(1-q_{\rm ref})}.
$$
At $q_{\rm ref}=0.25$, a <standard uncertainty> $0.01$ in the reference fraction alone contributes about $0.0533$ to the relative <standard uncertainty> of $\kappa$. Fractions near zero or one make that sensitivity larger. Shared inputs require <covariance> terms, rather than independent addition of every uncertainty contribution.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Sections 5.1–5.2, sensitivity coefficients and correlated inputs]{ref}
A phase-resolved experiment also needs a stated choice of real, imaginary or fitted component: the primary peptide experiment recorded phases and amplitudes of <vibrational photon echoes>.<Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes by M. C. Asplund and others>[Abstract, phase-controlled echoes and heterodyne readout]{ref} A ratio of signed or complex values does not automatically satisfy the positive population model.
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