Angular modulation frequency

The typical polymer or rubber sample would be classified as optically transparent or opaque and thermally thick except possibly for the strongest bands. In this case the signal intensity would be proportional to the product of the optical absorption coefficient (P) and the thermal diffusion length and show a - 3/2 dependence on the modulation frequency (to). The angular modulation frequency is a product of the interferometer mirror velocity and the wavenumber ... 

CO (1) Angular velocity (rad s 1) (2) Light modulation frequency DETR... 

In polarization modulated ENDOR spectroscopy (PM-ENDOR)45, discussed in Sect. 4.7, the linearly polarized rf field B2 rotates in the laboratory xy-plane at a frequency fr fm, where fm denotes the modulation frequency of the rf carrier. In a PM-ENDOR experiment the same type of cavity, with two rf fields perpendicular to each other, and the same rf level and phase control units used in CP-ENDOR can be utilized. To obtain a rotating, linearly polarized rf field with a constant magnitude B2 and a constant angular velocity Q = 2 fr (fr typically 30-100 Hz), double sideband modulation with a suppressed carrier is applied to both rf signals. With this kind of modulation the phase of the carrier in each channel is switched by 180° for sinQt = 0. In addition, the phases of the two low-frequency envelopes have to be shifted by 90° with respect to each other. The coding of the two rf signals is shown in Fig. 8. 

Here, p is the electron moment, y the magnetogyric ratio of the species of nucleus being observed ( H,13C, or possibly 19 F) cos is the angular precessional frequency of the electron in the field employed, and rc the characteristic average time of the modulation of the electron-nucleus interaction r is the distance from the metal to the observed nucleus i. When the ion is tightly bound to the protein, rc becomes the rotational correlation time of the protein, usually of the order of 10"8 s. 

As shown in previous sections of this chapter, when an external perturbation is applied to the polymer film (such as irradiation), the ATR guided modes shift their angular positions and the reflectivity is modulated (Fig. 31b). These angular shifts are very small in the case of electrooptic experiments they correspond to refractive index variations of the order of 10 . One has then to modulate the measuring electric field at a low frequency Q( = cos fit) and to detect the modulated signal with lock-in amplifiers. The lock-in signals detected at the modulation frequency and its second harmonic give, respectively, the linear (or Pockels) and the quadratic (or Kerr) electrooptic effects. The amplitude of the modulation of the thickness and the refractive indices is evaluated by a computer fit, and allows the determination of Pockels (r) and Kerr (s) coefficients (Eqs. 28) ... 

Here the modulating frequency is twice that in formula (2). The polarization modulator must be a fixed polarizer (o)2 = 0) and phaseshifter with a phase shift of 90° or the polarizer s angular rate must be twice the speed of angular rotation of the PSP (o)2=2a)i). Then formula (4) becomes... 

A much simpler expression for the peak height is obtained, however, if a digital modulation of the form shown in Figure 4.2B is applied in which Av(r) jumps between peak positive and negative values Av with an intermediate pause at zero. This is synchronised to waveform A and detected by waveform C. If the source is tuned to the peak frequency Vq its modulated frequency simply jumps back and forth between the line peak and a symmetrical offset on each side, generating a square wave of angular frequency 2(o and amplitude ... 

Pyroelectric currents are proportional to the time derivative of the temperature and, in case of infrared (IR) detectors, to the changes of IR radiation. In order to assess stationary IR sources as well, a chopper is used to modulate the IR radiation. Signals thus exhibit a well-defined angular chopping frequency a>. The smallest detectable temperature change is limited to the terr5>erature fluctuation of the detector system, the electronics or the intrinsic noise current jn of the p)Toelectric element, given here for a thin film by the expression... 

System (A8.2)-(A8.4) defines completely the time variation of orientation and angular velocity for every path X(t). One can easily see that (A8.2)-(A8.4) describe the system with parametrical modulation, as the X(t) variation is an input noise and does not depend on behaviour of the solution of (Q(t), co(r). In other words, the back reaction of the rotator to the collective motion of the closest neighbourhood is neglected. Since the spectrum of fluctuations X(t) does not possess a carrying frequency, in principle, for the rotator the conditions of parametrical resonance and excitation (unrestricted heating of rotational degrees of freedom) are always fulfilled. In reality the thermal equilibrium is provided by dissipation of rotational energy from the rotator to the environment and... 

For single exponential fluorescence decay, as is expected for a sample containing just one fluorophore, either the phase shift or the demodulation can be used to calculate the fluorescence lifetime t. When the excitation light is modulated at an angular frequency (o = 2itv, the phase angle f, by which the emission modulation is shifted from the excitation modulation, is related to the fluorescence lifetime by ... 

In a periodic dilatational experiment, the complex elasticity module is a function of the angular frequency ... 

Equation (1.8) represents a plane wave exp[i(A x — mt)] with wave number k, angular frequency m, and phase velocity m/A, but with its amplitude modulated by the function 2 cos[(AA x — Amt)/2]. The real part of the wave (1.8) at some fixed time to is shown in Figure 1.2(a). The solid curve is the plane wave with wavelength X = In jk and the dashed curve shows the profile of the amplitude of the plane wave. The profile is also a harmonic wave with wavelength... 

Thus, the wave packet P(jc, 0 represents a plane wave of wave number ko and angular frequency mo with its amplitude modulated by the factor B(x, i). This modulating function B x, i) depends on x and t through the relationship [x — (dm/dA )o/]. This situation is analogous to the case of two plane waves as expressed in equations (1.7) and (1.8). The modulating function B(x, t) moves in the positive x-direction with group velocity given by... 

As before, the wave packet is a plane wave of wave number ko and angular frequency mo with its amplitude modulated by a factor that moves in the positive x-direction with group velocity given by equation (1.16). Following... 

The cornerstone of the strong anharmonic coupling theory relies on the assumption of a modulation of the fast mode frequency by the intermonomer distance. This behavior is correlated by many experimental observations, and it is undoubtly one of the main mechanisms that take place in a hydrogen bond. Because the intermonomer distance is, in the quantum model, represented by the dimensionless position coordinate Q of the slow mode, the effective angular frequency of the fast mode may be written [52,53]... 

We restrict the study of the adiabatic approximation to the case where the modulation parameters (30,/0, and gD, defined in Eqs. (6) and (7), are neglected. The effective angular frequency of the fast mode reduces to ... 

Jonas et al. measured the proton rotating frame spin-lattice relaxation time (Tip) at pressures from 1 bar to 5000 bar and at temperatures of 50 to 70 °C for DPPC and at 5 to 35 °C for POPC. If intermolecular dipolar interactions modulated by translational motion contribute significantly to the proton relaxation, the rotating frame spin-lattice relaxation rate (1/Tip) is a function of the square root of the spin-locking field angular frequency... 

The phase-shift fluorimeter on the other hand provides measurements of the phase angle and degree of modulation m relative to those of the excitation signal modulated at an angular frequency decay constants of both emitting species by Ax and A2 which are readily computed from the relationships... 

---