where E, A, V , and ρ are the Young's modulus, surface area, volume, and density of the fibril segment, respectively. L 0 = ⟨N⟩D is the average equilibrium length of a "continuous" β-sheet segment in the fibrils, with N being the average number of strands and D the average distance between the strands (∼4.5 Å). As discussed in more detail below, we define L 0 as the coherence length of the β-sheets in the fibrils. Eq. 3 predicts that the accordion mode frequency approximately scales as 1/N, suggesting that the frequency decreases with increasing β-sheet length (Figure ). If the features between ∼ 15 -35 cm -1 arise from this mode, eq. 3 estimates the average coherence length of Aβ 25-35 and amylin 20-29 fibrils to be about 5 strands, similar to globular proteins. Taken at face value, this suggests amyloid fibrils may have a relatively high defect density, limiting long-range ordering of the β-sheets. This disorder in the β-sheet lattice could potentially disrupt the vibrational coupling between βstrands. These defects could arise from the misincorporation (Figure ) or slight misalignment (Figure ) of strands in the β-sheet lattice. Evidence supporting this may come from the X-ray diffraction patterns of amyloid fibrils, which rarely show reflections beyond the first-or second-order, indicating limited long-range order in their β-sheets.