Fourier transform scaling dynamics

The dynamic structure factor is S(q, t) = (nq(r) q(0)), where nq(t) = Sam e q r is the Fourier transform of the total density of the polymer beads. The Zimm model predicts that this function should scale as S(q, t) = S(q, 0)J-(qat), where IF is a scaling function. The data in Fig. 12b confirm that this scaling form is satisfied. These results show that hydrodynamic effects for polymeric systems can be investigated using MPC dynamics. 

Figure 1.32 Fourier transforms of the two time domain waveforms shown in Figure 1.31. The signal is plotted on a logarithmic scale to show the dynamic range of the instrument. Pure rotation lines due to water vapor are seen on the background spectrum (above). The sample spectrum (below) is that of crystalline silicon. Reproduced with permission from Ref [79].

Inelastic neutron scattering is a technique that has been widely used both in the liquid and in the solid states to measure the stmcture and dynamics at small (that is, molecular) length scales. In an incoherent inelastic neutron-scattering experiment, the measured quantity is the self-dynamic structure factor Ss(Q, (o), which gives information, as in the liquid state, of the self-diSiision coefficient of the water molecules. Ss(Q, (o) is the Fourier transform of the intermediate self-scattering function Fg(Q, t), which is defined by... 

By definition, (t) represents the probabihty amphtude that, after a finite evolution time t, the system is stiU in the same state as at time t = 0. Referring back to Eq. (32), we see that (t) and the spectral intensity distribution P E) are related by Fourier transformation and thus contain the same information on the system dynamics. The square (t)P is also known as survival probability and provides a measure of noiuadiative decay processes occurring on picosecond or nanosecond time scales. ... 

What makes the neutral final state or complete screening condition approximate rather than exact is the time required for the system to respond to the core excitation (the various time scales are discussed by Gadzuk (1978)). In this work we are concerned only with the excitation energy, the threshold for absorption. Since the threshold corresponds to a long-time process (in the sense of a Fourier transform of a correlation function), the actual dynamics of complete screening will have negligible effect on the threshold energy but will affect the lineshape. 

Phase separation and spatial organization of membrane domains determine the state of water, fundamental interactions in the polymer/water/ion system, vibration modes of fixed sulfonate groups, and mobilities of water molecules and protons. Dynamic properties of the membrane can be probed at the microscopic scale with spectroscopic techniques, including Fourier transform infrared spectroscopy (FTIR) and nuclear magnetic resonance (NMR) (Mauritz and Moore, 2004). FTIR... 

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