Widom Scaling - Derivation

Derivation

The scaling hypothesis is that near the critical point, the free energy, in dimensions, can be written as the sum of a slowly varying regular part and a singular part, with the singular part being a scaling function, i.e., a homogeneous function, so that

Then taking the partial derivative with respect to H and the form of M(t,H) gives

Setting and in the preceding equation yields

for

Comparing this with the definition of yields its value,

Similarly, putting and into the scaling relation for M yields

Hence

Applying the expression for the isothermal susceptibility in terms of M to the scaling relation yields

Setting H=0 and for (resp. for ) yields

Similarly for the expression for specific heat in terms of M to the scaling relation yields

Taking H=0 and for (or for yields

As a consequence of Widom scaling, not all critical exponents are independent but they can be parameterized by two numbers with the relations expressed as

The relations are experimentally well verified for magnetic systems and fluids.