Isothermal compressibility

[sanctus]

The isothermal compressibility would usually be given by [math]\frac{-1}{V}\cdot \frac{\partial^2g}{\partial\lambda^2}[/math] with lambda the pressure and g the gibbs free energy.

 

Now in a two-dimensional lattice (double chain) it seems to be given by [math]\frac{-b}{2a}\frac{\partial a}{\partial \lambda}[/math] where [math]a=\frac{1}{2}\frac{\partial g}{\partial \lambda}[/math] is the average lattice spacing. and b is the (fixed) distance between the spins on each chain.

 

The derivation terms are OK, it's the same as in the usual 3-dim model, but where comes the term b/2a from? It should be more something like 1/(b*a) as volume becomes surface in 2-dim (the factor 2 comes just from definition of a, so it cancels with 1/2)?

[Qfwfq]

Offhand, I suspect it comes from chain derivation (function of function), I'd say the volume is replaced by the square of a, so I imagine the forms are coherent except that I don't get what is meant by "the distance between the spins".

[sanctus]

If you imagine the double chain horizontal,ie there is the first pair of spins in x=0 and one spin is in y=0 the other in y=b, then you have the next pair (in average) in x=a and again one in y=0 and one in y=b.

The model is so that you can compress the double chain horizontally but no vertically ie the "vertical distance" between the two spins of one pair is always b.

Hope that clears it up.