Scattering integral

If the half angle of acceptance of our detector is 90 and we are concerned with 90 very small, that is, with forward scattering, integration with respect to 9 may be carried out analytically. We find23... 

The direct amplitudes involving are analogous to the distorted-wave Born approximation and are calculated by (10.31). The T-matrix element in the second amplitude of (10.51), which has the observed resonances, is calculated by solving the problem of electron scattering on He" ". The solution consists of half-on-shell T-matrix elements at the quadrature points for the scattering integral equations (6.87). The same points are used for the k integration of (10.51). 

For trajectory calculations the interaction potential is also needed. It enters into the scattering integral in the center of mass system via ... 

The scattering integral may also be readily evaluated for rod-shaped particles in which case, the intensity is dependent upon the angle of orientation of the rod... 

This implies a higher cutoff angle in the scattering integral. Approximating Q according to the beam attenuation... 

Thus V (r) can be computed and hence the socalled scattering integral... 

Now for the scattering-integral term. The total number of neutrons scattered into dr dv dli per unit time from all possible speeds v and directions of motion O is obtained by taking the number of neutrons in dr dv dO which experience scattering collisions, namely,... 

Expansion (7.117) may now be substituted into the scattering integral of (7.116) however, it is necessary first to express the functions P,(/io) in terms of the variables B and The addition theorem for... 

Collision densities S,(w ) f(r,u, 0 are slowly varying functions over the lethargy range of the scattering integral... 

Also, the substitution of (7.148) into the scattering integrals of Eqs. (7.142) through (7.144), along with the expression for i from (7.145), yields the results... 

As in the case of the scattering integrals which appear in the Boltzmann equation, we assume that S,(w )< (r,u ) is a slowly varying function over the interval of integration and therefore may be approximated by the first two terms of a Taylor series,... 

Identification of the second structural component is more complex. In the lower insert of Figure 5.3 the dependence of the reciprocal value of the scattering integral of this component intensity on the relative fraction of clusters is adduced. It is linear and passes through the origin of the coordinates. A similar dependence was obtained for amplitude intensity. This circumstance allows to connect the indicated scattering component to clusters. If to proceed from analogy with crystalline materials, then the reciprocal dependence (or /((p, )) presents itself as an unexpected one,... 

Figure 11 (a) SAXS from titania nanoparticles and unified fit. fp = 34.9nm, PDI = 14.4, / g = 44.2nm. (b) Particle size distribution for titania unified analysis of Porod regime, TEM particle counting, and a scattering integral method. Adapted from Beaucage, G. Kammler, H. K. Pratsinis, S. E. J. Appl. Cryst. 2004, 37,523-535, Figures 5(a) and 5(b). ... 

---