Debye scattering equation

Contributions to the line broadening from the sample is dealt with in detail in Chapter 13. One equation can be applied to powder diffraction data of any kind (with or without three-dimensional ordering liquids, amorphous or crystalline solids, and any intermediate stage) the Debye scattering equation ... 

This result was published by Debye in 1946. Since we shall also encounter a light-scattering equation associated with his name, we shall refer to Eq. (2.56) and its variations as the Debye viscosity equation. 

Equation (10.82) is a correct but unwieldy form of the Debye scattering theory. The result benefits considerably from some additional manipulation which converts it into a useful form. Toward this end we assume that the quantity srj, is not too large, in which case sin (srj, ) can be expanded as a power series. Retaining only the first two terms of the series, we obtain... 

Light scattering technique was used in determining the oil solubilization rate. Debye s equation ( ) was used in the interpretation. The basic principle involves the measurement of the surfactant aggregate size during the solubilization. As the oil goes into the surfactant micelle, the increased size will be reflected by the turbidity of the solution. 

Interpretation of Light Scattering. We used Debye s equation ( for micellar solution as a basis for the light scattering measurement ... 

Figure 5.3 The Debye function, Equation (5.31), for a random coil chain is plotted and compared with the independent scattering intensity function for a thin rod and a thin circular disk.

This is the three dimensional form of the Debye-Bueche equation which may be used to describe the scattering of oriented systems. For samples... 

Equation 4.130 is the usual form of the Debye-Bueche equation. This equation generally describes the scattering from a spherically symmetrical system provided a suitable expression can be found for the correlation function. For many systems, an exponential correlation... 

The Debye-Bueche equation is more general than just for light scattering and applies also to x-ray and neutron scattering, but in these cases, the p s and the K s will be different. 

For time-averaged light-scattering in the Rayleigh-Debye approximation, equation (18) reduces to ... 

The quantitative analysis of the scattering profile in the high q range can be made by using the approach of Debye et aJ as in equation (B 1.9.52). As we assume tiiat the correlation fiinction y(r) has a simple exponential fomi y(r) = exp(-r/a ), where is the correlation length), the scattered intensity can be expressed as... 

The Debye temperatures of stages two and one were determined by inelastic neutron scattering measurements [33], The total entropy variation using equation 8 is in the order of about 2 J/(mol.K). Although smaller in value, such variation accounts for 10-15% of the total entropy and should not be neglected. We are currently carrying on calculations of the vibrational entropy from the phonon density of states in LixC6 phases. 

Equation 3.27 forms the basis for determination of Molecular weight from light scattering data. Like Osmotic pressure measurements, it is essential to consider the non-ideality of solutions and the concentration dependence. Following Debye, eq. 3.27 gets modified to... 

Consider a mixture of acoustic-mode (rL) and ionized-impurity (r,) scattering. For tL t, we would expect r 0 = 1.18 and for r, tl, rn0 = 1.93. But for intermediate mixtures, r 0 goes through a minimum value, dropping to about 1.05 at 15% ionized-impurity scattering (Nam, 1980). For this special case (sL = i, s, = — f), the integrals can be evaluated in terms of tabulated functions (Bube, 1974). For optical-mode scattering the relaxation-time approach is not valid, at least below the Debye temperature, but rn may still be obtained by such theoretical methods as a variational calculation (Ehrenreich, 1960 Nag, 1980) or an iterative solution of the Boltzmann equation (Rode, 1970), and typically varies between 1.0 and 1.4 as a function of temperature (Stillman et al., 1970 Debney and Jay, 1980). 

Before turning to the applications of the Debye approximation, we should elaborate more fully on a point that was glossed over. This is the assumption —made at the outset, but explicated in going from Equation (58) to Equation (59) —that the scattering behavior of each scattering element is independent of what happens elsewhere in the particle. The approximation that the phase difference between scattered waves depends only on their location in the particle and is independent of any material property of the particle is valid as long as... 

Equation (67) shows clearly that i should be measured as a function of both concentration and angle of observation in order to take full advantage of the Debye theory. The light scattering photometer described in Section 5.4 is designed with this capability, so this requirement introduces no new experimental difficulties. The data collected then consist of an array of i/I0 values (i needs no subscript since it now applies to small and large particles) measured... 

In the method of trial the intensity I of scattered radiation at any angle d is given (for a solid containing two different types of atom) by the Debye equation... 

In contrast to osmotic pressure, light-scattering measurements become easier as the particle size increases. For spherical particles the upper limit of applicability of the Debye equation is a particle diameter of c. A/20 (i.e. 20-25 nm for A0 600 nm or Awater 450 nm or a relative molecular mass of the order of 10 ). For asymmetric particles this upper limit is lower. However, by modification of the theory, much larger particles can also be studied by light scattering methods. For polydispersed systems a mass-average relative molecular mass is given. 

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