Self-Conjugate and Normal Response Modes

One can appreciate in (11.68) the intrinsic physical significance that the coexistence coordinate a possesses in an actual phase diagram, even though it involves simultaneous changes of coordinates T, P having different units (so that units of cr would be difficult to discuss). If T) and P) were measured in different units, the numerical value of the coefficient y(r would readjust to represent the same physical coordinate, so that no mathematical or physical inconsistency could result. As remarked previously, lengths in Ms depend (consistently) on the chosen units of measured response functions, whereas internal angles are dimensionless invariants of the system. 

Note also that (11.67) determines the coefficients in (11.68) only up to a multiplicative factor, which has been chosen so that V (rather than a multiple thereof) becomes the remaining conjugate variable in (11.70). 

Arbitrarily chosen intensive vectors Rai) are in general neither orthogonal nor normalized. It is therefore of interest to identify a particular choice of A for which these vectors become an orthonormal set. Such unit-like response vectors, denoted as E ), can be expressed as usual as linear transformations of standard reference intensities 

We note first that the matrix A leading to vectors with the property (11.75) is not uniquely defined, since if O is any orthogonal matrix satisfying [cf. (S9.1-13)] 

The orthonormality property (11.75) establishes, in conjunction with (11.15), that the normal intensities are self-conjugate, 

---