Roughening Transition of Solid Surfaces

For a solid surface, where the surface jumps in discrete steps, the continuum energy of Eq. (3.1) may not be accurate enough to account for all the thermodynamic properties. If one considers a fluctuating solid surface described by the height (in the z direction) of a column located in the xy plane, a more accurate accounting of the extra area is given by the expression 

The simplified model, which is a good approximation to Eq. (3.44) when the local jumps in the height between nearest neighbors are limited to either 

We shall investigate the system using the approximate variational approach described in Chapter 2. We consider a reference Hamiltonian Hq, which is tractable, and use the theorem that bounds the exact free energy Fg, by 

To perform the average of the term proportional to yo, we use the fact that 

We now minimize F with respect to Gdi) fitid the equation that defines G q) 

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