Cross-differentiation identity

From cross-differentiation identities one can derive some additional Maxwell relations for partial molar quantities ... 

Equation (2.9) is known as the reciprocity relation or cross-differentiation identity. It follows from the fact that the order of differentiation of our original function 2 == z x, y) with respect to x and y is immaterial. Mathematically this is written... 

Applying to Eq. (3) the cross-differentiation identities known as Schwarz relations yields the relations... 

Reciprocity relation. The reciprocity relation or the cross-differentiation identity is very useful in some thermodynamic derivations. Let us represent the partial derivatives in Eq. (1.77) by... 

Equations (4.30) and (4.31) have been developed and dehned within a time-dependent framework. These equations are identical to Eqs. (35) and (32), respectively, of Ref. 80. They differ only in that a different, more appropriate, normalization has been used here for the continuum wavefunction and that the transition dipole moment function has not been expanded in terms of a spherical harmonic basis of angular functions. All the analysis given in Ref. 80 continues to be valid. In particular, the details of the angular distributions of the various differential cross sections and the relationships between the various possible integral and differential cross sections have been described in that paper. 

Figure 4.20.A shows a more recent cell reported by Cobben et al. It consists of three Perspex blocks, of which two (A) are identical and the third (B) different. Part A is a Perspex block (1) furnished with two pairs of resilient hooks (3) for electrical contact. With the aid of a spring, the hooks press at the surface of the sensor contact pads (4), the back side of which rests on the Perspex siuface, so the sensor gate is positioned in the centre of the block, which is marked by an engraved cross as in the above-described wall-jet cell. Part B is a prismatic Perspex block (2) (85 x 24 x 10 mm ) into which a Z-shaped flow channel of 0.5 mm diameter is drilled. Each of the wedges of the Z reaches the outside of the block. The Z-shaped flow-cell thus built has a zero dead volume. As a result, the solution volume held between the two CHEMFETs is very small (3 pL). The cell is sealed by gently pushing block A to B with a lever. The inherent plasticity of the PVC membrane ensures water-tight closure of the cell. The closeness between the two electrodes enables differential measurements with no interference from the liquid junction potential. The differential signal provided by a potassium-selective and a sodium-selective CHEMFET exhibits a Nemstian behaviour and is selective towards potassium in the presence of a (fixed) excess concentration of sodium. The combined use of a highly lead-selective CHEMFET and a potassium-selective CHEMFET in this type of cell also provides excellent results.

Plots of each of these quantities as a function of particle size would look quite different and, therefore, would tell different stories. Except for a scale factor, each of them plotted as a function of wavelength for the same particle size would be identical. In our first example of extinction (Fig. 4.6) we displayed the efficiency Qext, as we shall often do in this chapter. In Chapter 12, however, our preference switches to the extinction cross section per unit particle volume. Unnormalized extinction cross sections (strictly speaking, the differential scattering cross section integrated over the acceptance angle of the detector) are more appropriate in Section 13.5 on particle sizing. 

The differential cross-section d/i/dfl is not invariant when we change the description from one coordinate system to another. Clearly, due to the relation in Eq. (4.55) a change in y will not lead to the same change in 0 and the space angle d 2c.m. = si n ydycif/) is not identical to the space angle dfl = sin GdGd in the laboratory frame. Thus,... 

However, it is important to realize that although the differential cross sections in the c.m. system at 0 and — 0 are identical, the same is not true of the corresponding lab quantities. The lab differential cross section, which is actually determined in a scattering experiment, is defined by an expression exactly analogous to that for /c.m (0), that is,... 

---