Editing Reference hyper-netted chain
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(Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1). | |||
Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality': | Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality': | ||
"...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential" | "...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential" | ||
The basis of the method is to solve the modified | The basis of the method is to solve the modified HNC equation | ||
(with inclusion of the one-parameter | (with inclusion of the one-parameter bridge functions | ||
appropriate to | appropriate to hard spheres), and determine the only free parameter <math>\eta</math> | ||
(related to the hard-sphere diameter) by requiring | (related to the hard-sphere diameter) by requiring thermodynamic consistency. | ||
Fred Lado (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3) | Fred Lado (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3) | ||
noticed that the | noticed that the OZ equation can always be written in the form | ||
:<math>\gamma_{12} = \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math> | :<math>\gamma_{12} = \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math> | ||
In view of this a hybrid solution between the [[HNC| hyper-netted chain]] | In view of this a hybrid solution between the [[HNC| hyper-netted chain]] | ||
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(See also Ref.s 6 and 7) | (See also Ref.s 6 and 7) | ||
The RHNC closure is given by (Eq. 17 Ref. 7) | The RHNC closure is given by (Eq. 17 Ref. 7) | ||
:<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \ | :<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \phi(r)}] + B_0(r) </math> | ||
along with the constraint (Eq. 18 Ref. 7) | along with the constraint (Eq. 18 Ref. 7) | ||
:<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math> | :<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math> | ||
Incorporating a reference potential <math>\phi_0(r)= \phi_0(r;\sigma,\epsilon)</math> | |||
Incorporating a reference potential <math>\ | |||
this equation becomes (Eqs. 19a and 19b in Ref. 7) | this equation becomes (Eqs. 19a and 19b in Ref. 7) | ||
:<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math> | :<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math> | ||
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These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math> | These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math> | ||
that minimise the free energy. | that minimise the free energy. | ||
The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and | The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 8, i.e. | ||
:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math> | :<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math> | ||
For highly asymmetric mixtures see Ref. 9. | For highly asymmetric mixtures see Ref. 9. | ||
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==References== | ==References== | ||
#[http://dx.doi.org/10.1103/PhysRevA.20.1208 Yaakov Rosenfeld and N. W. Ashcroft "Theory of simple classical fluids: Universality in the short-range structure", Physical Review A '''20''' pp. 1208 - 1235 (1979)] | #[http://dx.doi.org/10.1103/PhysRevA.20.1208 Yaakov Rosenfeld and N. W. Ashcroft "Theory of simple classical fluids: Universality in the short-range structure", Physical Review A '''20''' pp. 1208 - 1235 (1979)] | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | |||
#[http://dx.doi.org/ | |||
#[PRA_1973_08_002548] | |||
#[MP_1987_60_0663] | |||
#[MP_1989_67_0431] | |||
#[PLA_1982_89_0196] | |||
#[PRA_1983_28_002374] | |||
#[JPSJ_1957_12_00326] | |||
#[JCP_2005_123_174508] | |||
[[Category: Integral equations]] | [[Category: Integral equations]] | ||