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Widom scaling is a hypothesis in Statistical mechanics regarding the free energy of a magnetic system near its critical point which leads to the critical exponents becoming no longer independent so that they can be paramaterized in terms of two values.
Additional recommended knowledge
The critical exponents α,α',β,γ,γ' and δ are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows
The scaling hypothesis is that near the critical point, the free energy f(t,H) can be written as the sum of a slowly varying regular part fr and a singular part fs, with the singular part being a scaling function, ie, a homogeneous function, so that
Then taking the partial derivative with respect to H and the form of M(t,H) gives
Setting H = 0 and λ = ( − t) − 1 / p in the preceding equation yields
Comparing this with the definition of β yields its value,
Similarly, putting t = 0 and λ = H − 1 / q into the scaling relation for M yields
Applying the expression for the isothermal susceptibility χT in terms of M to the scaling relation yields
Setting H=0 and λ = (t) − 1 / p for (resp. λ = ( − t) − 1 / p for ) yields
Similarly for the expression for specific heat cH in terms of M to the scaling relation yields
Taking H=0 and λ = (t) − 1 / p for (or λ = ( − t) − 1 / p 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.
H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Widom_scaling". A list of authors is available in Wikipedia.|