Beyond the WNA

It is possible now to consider the case of general deformation, and to go beyond the WNA. Given a general deformation we should project out (subtract) all the special components, leaving the normal component, and only then apply the WNA. In Fig. 4.8 we demonstrate this decomposition for the deformation (CO + W16) and the deformation SX. Here and elsewhere, all boundary deformation function integrals (of the form (4.5)) are evaluated using the technique of Appendix G.

The special deformations constitute a linear space
which is spanned by the basis functions: one dilation, translations,
and rotations.
(For they are listed in Table 3.2).
For a general cavity shape these basis functions are not orthogonal.
However, because they are linearly independent,
we can use standard linear algebra to build an orthonormal
basis
of special deformations.
The special () and the normal () components of any given
deformation
are therefore

where the coefficients are

The improved approximation for applies the WNA only to the normal component, giving

which we name the IFIF (Improved Fluctuations Intensity Formula). In the particular case of , substitution of this result into the microcanonical FD relation gives an `improved wall formula' consisting of the replacement of by in Eq. (4.1).