Retardation superposed

The usefulness of the Mueller formulation becomes apparent when we realize that Mueller matrices give us a simple means of determining the state of polarization of a beam transmitted by an optical element for an arbitrarily polarized incident beam. Moreover, if a series of optical elements is interposed in a beam, the combined effect of all these elements may be determined by merely multif ying their associated Mueller matrices. As an example, let us consider how a circular polarizer can be constructed by superposing a linear polarizer and a hnear retarder. The beam is first incident on a linear polarizer with horizontal transmission axis ( = 0°), the Mueller matrix for which is obtained from (2.87) ... 

Fig. 9 et 17, observees toutes 40° C aprfes rechauffement. Ces courbes se superposent encore assez convenablement sur les isothermes experi-mentales, mais ici les ecarts sont plus importants que ceux observes avec les isothermes de contraction (Fig. 3). D autre part, la valeur de r40 obtenue ci partir de cette superposition augmente d une manure syste-matique avec l amplitude 6it et atteint 0,084 heure, qui est environ 6 fois superieure k la valeur moyenne (0,013 h) obtenue partir de l ensemble des courbes de contraction (Fig. 20). Ce disaccord grave semble indiquer que le spectre des temps de retard varie, et que sa variation est liee k celle de la configuration. Les experiences de cisaillement effectuees avec le meme echantillon dans le meme domaine de temperature montrent effectivement une telle anomalie (Kovacs, Stratton et Ferry, 1963).

Also in this case, several retardation phenomena with different retardation times may be superposed. 

Figure 9.7 Double logarithmic plots of the retardation spectra as a function of the retardation time for different fractions of polystyrene of narrow molecular weight distributions (0) 4.7 x 10 (0) 9.4 x 10 ( ) 1.9 x 10 (O) 6.0 x 10 and (3) 3.8 X 10. The spectra were shifted to superpose them in the glassy-like region. (Reference temperature 100°C molecular weight, 1.9 x 10 ). (From Ref. 8.)...

When studying a polymer on a large frequency/time scale, the response of a given material under a dynamic stimulus usually exhibits several relaxations. Moreover, the peaks are usually broad and sometimes and are associated with superposed processes. The relaxation rate, shape of the loss peak, and relaxation strength depend on the motion associated with a given relaxation process [41]. In general, the same relaxation/retardation processes are responsible for the mechanical and dielectric dispersion observed in polar materials [40]. In materials with low polarity, the dielectric relaxations are very weak and cannot be easily detected. The main relaxation processes detected in polymeric systems are analyzed next. 

Optical activity is, however, also shown in solution. It is due to molecules which themselves lack a plane of symmetry. In virtue of this lack they cannot be brought into coincidence with their own mirror images, any more than right-handed and left-handed helices can be superposed. They have in fact the same geometrical property as the helical media in causing a selective retardation of one of the two circular vibrations, each of the mirror image forms retarding a different component of the wave. 

A caution should be exercised in using Avrami exponent values for the copolymers. In contrast to homopolymer crystallization, the isotherms of copolymers do not superpose one with the other deviations from the Avrami relation occurs at low levels of crystallinity and retardation in crystallization rate is pronounced with the extent of transformation due to continuous change in both composition and sequence distribution during crystallization. See reference (28) for the detailed discussion. Avrami exponent values for selected mLLDPE are shown below. 

FIGURE 12.22. Comparison of retardation spectra for volum-inai Lo and shear deformation U. In this doubie logarithmic piot of the distribution functions of retardation times t the ordinate scaies have been adjusted to superpose the short-time resuits, r/ar is the reduced retardation time.Reproduced from Donaid J. Piazek and Craig A. Bero, Precise Glass Temperature, J. Phys. Condens. Matter. 15 (2003) 5789-5802 with permission from Institute of Physics Publishing. 

The basic scheme adapted for generating 2D IR spectra [3, 8] is shown in Figure 1-2. In a typical optical spectroscopy experiment, an electromagnetic probe (e.g., IR, X-ray, UV or visible light) is applied to the system of interest, and physical or chemical information about the system is obtained in the form of a spectrum representing a characteristic transformation (e.g., absorption, retardation, and scattering) of the electromagnetic probe by the system constituents. In a 2D IR experiment, an external physical perturbation is applied to the system [2, 3] with the incident IR beam used as a probe for spectroscopic observation. Such a perturbation often induces time-dependent fluctuations of the spectral intensity, toown as the dynamic spectrum, superposed onto the normal static IR spectrum. 

Fig. 2.27. Recoverable-compliance, Ji(t), data of PPMS 5000 at temperatures -32.2°C ( ), -35.0°C U),-38.6°C (t),-40.0°C ( ), -41.1 °C (x), -42.6 °C n, -44.5 °C ( ), -45.2 °C (V), -46.9 °C (A), and -50 °C (o). The data taken at different temperatures have been shifted horizontally along the log t axis by a temperature-dependent shift factor log ut in order to superpose the curves at the short-time end with the data for -35.0 °C. The inset shows the retardation spectrum, L, as a function of the reduced retardation time X with reference temperature To = -35.0 °C, which was obtained numerically from J lf) data.

The shapes of the curves in Fig. 11 -7 Cc i yield information about the retardation spectrum in bulk deformation, as discus ed in Chapter 18. From this analysis it follows, however, that their shapes should not be exactly the same, but that another somewhat more complicated function of volume when plotted against log t should be exactly superposable by horizontal shifts this is in fact observed. ... 

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