## 3 Experimental Results
### Optical Cross-Section and Red Shift  
Given the absorbance and the molecular concentration \(\rho_{\mathrm{m}}\), we can estimate the absorption cross-section per molecule, \(\sigma_{\mathrm{m}}\), which is a fundamental physical measure of the strength of the absorption (or scattering) at the molecular (or atomic) scale. We apply the Beer-Lambert law for chemical solutions [25]:  
\[\mathrm{A_{S}}=[\mathrm{L}/\ln(10)]\cdot\sum_{i}^{\mathrm{N}}\rho_{\mathrm{m, i}}\sigma_{\mathrm{m,i}} \tag{2}\]  
where L is the physical path length through the absorbing medium, i is the molecule-type index, and N is the number of molecular types. For the si-RNA, the 17, 21, and 25-bp species all have nominally equal concentrations so that we can write As \(\approx\) [L/ln(10)] \(\rho_{\mathrm{m}}\)\(<\)\(\sigma_{\mathrm{m}}\)\(>\), where from Table 1 \(\rho_{\mathrm{m}}\) = 2.0 \(\times\) 10\({}^{14}\) cm\({}^{-3}\) is the total (50% diluted) concentration including all three species, \(<\)\(\sigma_{\mathrm{m}}\)\(>\) is the average cross-section, and L \(\approx\) h = 0.6 \(\mu\)m, where h is the height of the nanochannels. With this approximation, we obtain \(<\)\(\sigma_{\mathrm{m}}\)\(>\) = 1.9, 2.0, and 2.0 \(\times\) 10\({}^{4}\) nm\({}^{2}\) for the (1), (2), and (3) signatures, respectively. Although less applicable, we make the same approximation for the 1-kbp-DNA using \(\rho_{\mathrm{m}}\) = 6.2 \(\times\) 10\({}^{13}\) cm\({}^{-3}\) for the total (50% diluted) molecular concentration from Table 2. We find \(<\)\(\sigma_{\mathrm{m}}\)\(>\) = 9.8, 8.9, and 8.6 \(\times\) 10\({}^{3}\) nm\({}^{2}\) for the (1), (2), and (3), respectively.  
A subtle but important aspect of the Beer-Lambert expression of (2) is that the cross section can be represented spatially as a disk, similar to the concept of scattering cross-section in atomic physics. Hence, an interesting quantity is the cross-section diameter D = 2\(\cdot\)(\(\sigma_{\mathrm{m}}\)/\(\pi\))\({}^{1/2}\). This quantity is also entered in Tables 1 and 2, and it becomes more interesting when compared to the molecular length, L\({}_{\mathrm{m}}\). Fortunately, for our linear ds-RNA and -DNA samples, this length is very well known in aqueous solution. Accurate to about 1%, for linear dsDNA, L\({}_{\mathrm{m}}\) (nm) = 3.3 \(\times\) (N\({}_{\mathrm{bp}}\) - 1) and for ds-RNA, L\({}_{\mathrm{m}}\) (nm) = 2.7 \(\times\) (N\({}_{\mathrm{bp}}\) - 1), where N\({}_{\mathrm{bp}}\) is the number of base pairs in the double strand. Hence, L\({}_{\mathrm{m}}\) = 5.4 nm for the si-RNA, 168 nm for the 50-bp DNA, and 1305 nm for the 1-kbp DNA, all based on the average N\({}_{\mathrm{bp}}\) in each sample listed in Figure 2a. Taking the ratio, we get D/L\({}_{\mathrm{m}}\) = 29.3 for the si-RNA, = 0.25 for the 50-bp DNA, and =8.1 \(\times\) 10\({}^{-2}\) for the 1-kbp DNA, all calculated for signature (2). Qualitatively, the ratio D/L\({}_{\mathrm{m}}\) is a measure of the molecular oscillator strength, and in principle can be \(>>\) 1.0 for particularly strong absorbers. In the present context, it means that the small nucleic acid molecules are much more effective absorbers than the large ones.  
One other interesting aspect of our data is seen by plotting the center frequency for the (1), (2), and (3) signatures vs. sample for the si-RNA, 50-bp DNA, and 1-kbp DNA. As shown in Figure 7, there is a significant decrease in frequency for all three signatures. To the extent that these signatures are associated with the same physical mechanism (see the Discussion below), then this decrease is indicative of a spectroscopic red shift. This is a common effect in damped vibrational oscillators, and even occurs for the simple mass-spring harmonic oscillator of classical mechanics [26].