Differential solid angle

The cross section for scattering into the differential solid angle dD centred in the direction (9,(l)), is proportional to the square of the scattering amplitude ... 

The differential solid angle dco subtended by a differential area dS on a sphere of radius r can be expressed as... 

Note that the area dS is normal to the direction of viewing since dS is viewed from the center of the sphere. In general, the differential solid angle ddifferential surface area dA when viewed from a point at a distance ) from dA is expressed as... 

FIGURE 12-18 The emission of radiation from a differential surface element into the surrounding hemispherical space through a differential solid angle. 

This is shown schematically in Fig. 3. A narrow beam of fast particles impinges on a thin uniform target that is wider than the beam. At a scattering angle 9 from the direction of incidence, an ideal detector is located that counts each particle scattered in the differential solid angle dO. In the simplest approximation, the scattering cross section is given by ... 

Basically, the solid angle is the three dimensional analog of an ordinary angle. A differential plane angle dx is defined as the ratio of the element of arc length dl on the circle to the radius r of the circle, hence d = dl/r. Analogous, the differential solid angle dQ subtended by an arbitrary differential surface dA is defined as the differential surface area dQ of a unit sphere covered by the arbitrary surface s projection onto the unit sphere [77] [42] [88] ... 

Fig. 2.5. An illustration of the differential solid angle, dfl angle has the units of the steradian (sr) or radian (rad ).

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... 

From sect 2.4.2 we recall that the differential solid angle df2 is defined by a region between the rays of a sphere and is measured as the ratio of the element of area dAs on the sphere to the square of the sphere s radius (see Figs 2.5 and 2.6 in which dA denotes any surface element) ... 

This is the flux through unit area of surface at the end of the ray OP that lies at an angle 9 from the z-axis. If dQ is the differential solid angle, then the area element s dA = dQ (Fig. 35.14), and the total radiation through dA is... 

Figure 2.15 defines the geometry of the excitation source, the specimen, and the detector. The excitation source may be an x-ray tube anode or a radioisotope. To simplify the discussion the source is considered to be a point source. The number of x-ray photons per second emitted by the excitation source in the energy interval Eo to Eo + dEo within the differential solid angle dJ i is defined as lo(Eo) dEo dJ2i. 

This photon rate is emitted isotropically in all directions, i.e., into a solid angle of 47t steradians. The photon rate emitted toward the detector into the differential solid angle d 2 at a takeoff angle yf/i is given by... 

Before we can derive a general relationship for the view factor between two finite bodies we must consider and discuss two quantities, a solid angle and the intensity of radiation. A solid angle cu is a dimensionless quantity which is a measure of an angle in solid geometry. In Fig. 4.11-3a the differential solid angle dm, is equal to the normal projection of d/lj divided by the square of the distance between the point P and area dAj. 

Thus, we will use the (< i-integrated) differential solid angle cross-section (Tr(0) given by cross-section is the most-used and the one normally referred to as differential solid-angle cross-section. For some types of study... 

The integral of this quantity over all r yields the contribution to the current through dA from all neutrons which made their last scattering collision somewhere in the differential solid angle d/i d (see Fig. 5.5). We define a second function... 

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