Differential scattering cross section defined

For more general interactions - i.e. not necessarily between two hard-spheres - we introduce a differential scattering cross section a v,8), defined by b db d(j> a u,8)dU. Boltzman s equation (equation 9.32) becomes... 

The most detailed possible photofragmentation cross section is the detailed final-state resolved differential photofragmentation cross section defined in Eq. (4.12), which measures the probability of the formation of a particular final state, vj,m.j scattered into a specified scattering direction, k = 0, 4>j. This cross section has been discussed in Ref. 80 in the context of time-independent... 

The differential scattering cross section dCsc /dQ, a familiar quantity in atomic physics, is defined as the energy scattered per unit time into a unit solid angle about a direction —which may be specified by two angles, the scattering angle 6 and the azimuthal angle (see Fig. 3.3)—for unit incident irradiance. It is expressed in terms of the scattered irradiance Is(0, ), the incident irradiance /, and the distance r to the detector as... 

We define a scattering (or deflection) angle, x, as in Fig. 2.2, an azimuth, solid angle, Q, so that d2Q = sin dx dq>. The differential scattering cross section is defined as the number AN of particles scattered every second into the solid angle A 2 around x,

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Hug and Surbeck 35) have proposed the use of Ada. = daL — daR, the difference of the Raman differential scattering cross sections in left and right circularly polarized incident light. This is the Raman equivalent of the circular dichroism Ae = el — eR, where s is the decadic molar extinction coefficient and, unlike the measured (but not the theoretical) IR — IL which depends on both sample and instrumental factors, is solely a molecular parameter. They introduced a chirality number q defined by... 

To calculate the differential scattering cross section we need to define a particular system of reference for the differential solid angle. A suitable reference system can be defined in the following manner [12]. Let and e / be unit vectors in the direction of g2i and g2i, so that g2i = gQg and g2i = Q g The element of the surface dfl includes the point in a perpendicular plane... 

But what is measured in fact in neutron scattering is the differential scattering cross-section, dafcIQ. (q), which is defined as the number of neutrons scattered per second towards a detector in a certain direction per incident beam flux and solid angle. In the case of a liquid or a glass sample for which the average structure is isotropic, only the vector norms (r = r and q= q ) are relevant. 

Energy exchange with the sample requires the differential scattering cross section to contain energy terms, d a/dQdE. In preparation for this we define, Eo and E as the energies of the ground state and first excited state of the vibrating system of atoms. 

We now define drr(0c). the differential scattering cross-section, to be given by... 

It has already been mentioned in Section 1.2.1 that the term intensity will be used to denote the differential scattering cross section dcr/dQ, i.e., the ratio J(q)IJo. The normalized amplitude defined here is therefore related to the intensity by I(q) = A(q) 2. 

The double differential scattering cross section, according to (8.15), is proportional to the space-time Fourier transform of G(r,f). This is analogous to the fact, discussed in Section 1.5.2, that in the static approximation the intensity I(q) (or the differential scattering cross section ds/dQ) is given by the spatial Fourier transform of the density-density autocorrelation function defined in (1.79). In the special case of t = 0, G(r,0) denotes the probability of finding a particle at r when there is already a particle at position 0. G(r,0) is therefore related to the pair distribution function g(r) discussed in Section 4.1.1, as in... 

Suppose now t t we have one scattering center and an incident flux of N particles per unit area per unit time, with the flux small enough for there to be no interference between particles in the beam. Then the number of particles scattered per unit time into a solid angle dCl is Na d, ) dCl. Here 0 and are defined with respect to the fixed center of mass, and is a proportionality factor known as the differential scattering cross-section. The total scattering cross-section is given as... 

We define the differential scattering cross-section per unit volume of the scatterer, per unit solid angle ( 2), per unit angular frequency change as... 

The scattered intensity is defined in terms of the normahzed differential scattering cross-section per unit volume, dX((2)/dQ (expressed in units of cm ), where dE((2) (neutrons s ) represents the number of neutrons scattered per second into a small solid angle, dfi. In general,... 

The differential scattering cross-section, or the Rayleigh ratio is defined as the flux of light scattered into a unit solid angle divided by the incident intensity and by the scattering volume, and may be written as... 

The differential scattering cross section in the RLM is defined as usual, the ratio of the spherically scattered flux into final state n originating from an incident plane wave in molecular state m,... 

After a short description of the experimental setup in the previous section, we now focus on the basic mathematical description of the intensity measured in a SAXS experiment. X-rays are electromagnetic waves. Assume an incident plane wave with flux 7o, which is scattered by the electrons in the sample. The scattered spherical waves interfere with each other, resulting in an angle-dependent flux J of scattered radiation. The flux of the incident plane wave corresponds to energy transmitted per unit area per unit time and the flux J of the scattered radiation to energy transmitted per unit solid angle per unit time. Now the differential scattering cross section or scattered intensity is defined as the ratio... 

S A N S can be employed to study both homogeneous as well as heterogenous blends and various information can be derived by analysis of the differential scattering cross-section, d2(Q)/dQ (cm ), where d2(Q) (neutron s ) is the number of neutrons scattered per second into a small solid angle, dQ. The measured intensity, I Q), is related to d 2(Q)/d Q and depends on various parameters such as the incident flux, resolution of the detector, detector efficiency, and so on. It is also a function of the contrast factor (Ag), which is defined as the square of the difference of the scattering length density of the scattering centers, for example, the particle and the matrix. This quantity is calculated from ... 

The total scattering cross section is another statistical quantity that can be derived from the differential cross section, defined as the integral of aA ] g) over all solid angle elements, hence ... 

The ET spectrum of trans-butadiene shows two well-defined resonances which we attributed (18) to occupation of the two empty it orbitals. The lower resonance lies below that of ethylene and exhibits sharper structure. Figure 2 shows these data on an expanded energy scale for both "high rejection" conditions In which the signal derives from the total scattering cross section, and "low rejection" which reflects the differential elastic scattering near 180° (IS). The symmetric C-C vibrations of the anion are the most pronounced, but there Is evidence for low frequency out-of-plane modes as well. The upper resonance lies above that of ethylene and Is featureless. 

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