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## Widom scaling
## Additional recommended knowledge## DefinitionsThe 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 - , for
- , for
where - measures the temperature relative to the critical point.
## DerivationThe scaling hypothesis is that near the critical point, the free energy *f*_{s}(λ^{p}*t*,λ^{q}*H*) = λ*f*_{s}(*t*,*H*)
Then taking the partial derivative with respect to - λ
^{q}*M*(λ^{p}*t*,λ^{q}*H*) = λ*M*(*t*,*H*)
Setting - for
Comparing this with the definition of β yields its value, Similarly, putting Applying the expression for the isothermal susceptibility χ - λ
^{2q}χ_{T}(λ^{p}*t*,λ^{q}*H*) = λχ_{T}(*t*,*H*)
Setting Similarly for the expression for specific heat - λ
^{2p}*c*_{H}(λ^{p}*t*,λ^{q}*H*) = λ*c*_{H}(*t*,*H*)
Taking 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 - γ = γ' = β(δ − 1)
The relations are experimentally well verified for magnetic systems and fluids. ## ReferencesH.E. Stanley, |

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. |